PHYSICAL PROPERTIES AND TRANSFORMATIONS OF LOW-DIMENSIONAL

SYSTEMS

By

Dan Liu

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics – Doctor of Philosophy

2019

PHYSICAL PROPERTIES AND TRANSFORMATIONS OF LOW-DIMENSIONAL

ABSTRACT

SYSTEMS

By

Dan Liu

Evolving from the macroscopic scale to the nanometer scale, in particular by reducing the

dimensionality, fundamental properties (such as electronic and mechanical properties) of cer-

tain systems exhibit dramatic changes, which not only give rise to a wide range of emergent

phenomena, but also boost technology development including nanoelectronics, optoelectron-

ics and catalysis. In this thesis, I utilized combined techniques including density functional

theory (DFT), molecular dynamic simulations (MD), continuum elasticity approach, and

the tight-binding model to conduct a systematic study on low-dimensional nanostructures

regarding their electronic and mechanical properties as well as underlying microscopic trans-

formation mechanisms between different structural allotropes.

First, I briefly introduce the motivation and background of this thesis. Then, in Chapter

2, I describe the computational techniques, mainly the DFT approach, on which most of my

thesis is based.

In Chapters 3 and 4, I apply the continuum elasticity method to study the phonon

spectrum of two-dimensional (2D) and one-dimensional (1D) systems. My results highlight

advantages of the continuum elasticity approach especially for the flexural acoustic phonon

modes close to the Γ point, which are otherwise extremely hard to converge in atomistic

calculations that use very large supercell sizes.

From Chapter 5 to Chapter 7, I focus on allotropes of group III, V and VI elements and

study both their stability and microscopic transformation mechanisms from one allotrope

to another. First, I predicted a stable phosphorus coil structure, which may form by re-

construction of red phosphorous, and which was synthesized by filling a carbon nanotube

with sublimed red phosphorus. Second, I proposed two stable 2D allotropes of Se and Te.

I also suggested and evaluated a promising fabrication approach starting from natural 1D

structures of these elements. After considering low-dimensional charge neutral systems, I

changed my focus to study the effect of net charge on the equilibrium structure. Consid-

ering a heterostructure of alternating electron donor layers an monolayers of boron, I have

identified previously unknown stable 2D boron allotropes that may change their structure

under different levels of charge transfer.

From Chapter 8 to Chapter 10, I focus mainly on carbon-based nanomaterials and their

properties. In Chapter 8, I proposed a way to enhance the density of states at the Fermi

level in doped C60 crystals in order to increase their superconducting critical temperature

to room temperature. In Chapter 9, I have investigated a shear instability twisted bilayer

graphene using the tight binding model. This system is susceptible to very small structural

changes, since it becomes superconducting in a very narrow range of twist angles near the

’magic angle’. In Chapter 10, I introduced the cause of an unusual negative Poisson ratio

and a shape-memory behavior in porous graphene with an artificially designed pattern.

In Chapter 11, I finally present general conclusions of my PhD Thesis.

To all people whom I love and respect.

iv

ACKNOWLEDGMENTS

During the five and a half years of my Ph.D. study at Michigan State University, many

things happened, including not only happiness, but also embarrassment. Fortunately, I have

a family, friends, and colleagues to share the joy and to help me to get through tough times.

In retrospect, I consider my five years at MSU to be valuable. What I gained is not only

a more profound understanding of Physics, but also a more in-depth knowledge of what a

meaningful life should be, and what a valuable person should be like.

The thing I need to appreciate is the difficulties. Solving the challenges I met in Physics

and life help me to grow up, become humble, and let me know that nothing is easy in the life

of a mature person. I want to quote a sentence from the movie ’Angel has Fallen’: ’It’s our

moment of struggle that defines us’. It is defining who I am now and will determine who I

will be. The first person I should appreciate is my Ph.D. advisor Professor David Tomanek.

He is not just my advisor in Physics but also my friend in life. According to his advice, I

decided to stay in Physics even though it is not a field that can bring me wealth quickly.

And because of him, I know a scientist should value what he is doing and should not look

down at other people’s work. From him, I saw what integrity and selflessness is. For me, he

is very strict in Physics but very kind in life. He set a standard for me to be an instructor

in the future.

I need to thank all of my classmates and friends, including Kuan Zhu, Xingze Mao, Hao

Lin, Liangji Zhang, Faran Zhou, Shuyue Xue, Mengze Zhu, Didi Luo, we have a good time

together in MSU. Especially, I want to thank Professor Xianqinq Lin of Zhejiang University

of Technology, Professor Jie Guan of Southeast University and Jingwei Jiang of UC Berkeley,

we discussed lots of intriguing projects and collaborated on several publications. I also need

v

to appreciate the help from Professor Vincent Meunier of Rensselaer Polytechnic Institute

and Professor Arthur Every of the University of the Witwatersrand for understanding the

fundamental physics of nanomaterials and also for their recommendation letters that have

provided strong support for me when applying for a Postdoc position.

I acknowledge financial support from the National Science and Engineering Center for
High-Rate Nanomanufacturing (NSEC) program, grant number #EEC − 0832785, and the
NSF/AFOSR EFRI 2-DARE program, grant number #EF M A − 1433459. I also acknowl-

edge partial financial support by the Physics and Astronomy Department, and the Dis-

sertation Completion Fellowship from College of Natural Science, which has supported me

financially during the last semester of my Ph.D. study.

Last but not least, I want to apologize to my parents, my wife, and my daughter. As a

son, I have obtained much support from my parents, but I can not give them a hand now

when they become old and need my help. As a husband, I feel the respect and understanding

of my wife even though we are living apart on two continents. Still, I feel bad that I am not

available to her when she is sick. As a father, I have not seen my daughter for more then

one year (expect on we-chat in cyberland) and miss seeing her grow for close-up. For the

next two years, I still will unlikely stay with them and can not contribute my share to being

a warm-hearted family. I believe this is a temporary situation. In the near future, I will do

my best in being responsible as a son to my parents, a husband to my wife, and a father to

my daughter.

vi

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Chapter 1

and 1D nanomaterials

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Continuum approach of long-wavelength acoustic phonon modes of quasi-2D
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Microscopic mechanism of structural transitions in 2D and 1D systems
. . .
1.3 Mechanical and electronic properties of novel carbon-based nanomaterials . .
1.4 Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 From Hartree-Fock to Density Functional Theory . . . . . . . .
2.1 The Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . . . .
2.2 Hartree-Fock equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Exchange-correlation functionals . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Basis sets and pseudopotentials
. . . . . . . . . . . . . . . . . . . . . . . . .

1

1
2
3
5

7
7
8
10
15
16

uum limit

Chapter 3 Continuum approach for long-wavelength acoustic phonons in
quasi-2D structures . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18
18
3.1
3.2 Continuum approach for long-wavelength acoustic modes of an elastic membrane 21
3.3 Calculation of acoustic phonon modes of an elastic membrane in the contin-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Computational approach to determine the elastic constants . . . . . . . . . .
3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Phosphorene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.3 Vibration spectra of carbon fullerenes and nanotubes . . . . . . . . .
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Related Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.1 Lagrange function of an infinitely thin elastic plate under strain . . .
Stretching . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.1.1
Shearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.1.2
3.8.1.3 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.2 Derivation of Euler-Lagrange equations of motion for deformations of
an infinitely thin elastic plate using Hamilton’s principle . . . . . . .
3.8.2.1
Stretching . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.2.2
Shearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.2.3 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22
24
25
25
26
28
29
31
32
32
32
33
34

35
35
37
37

vii

3.8.3 Radial breathing mode of spherical fullerenes . . . . . . . . . . . . . .
3.8.4 Radial breathing mode of carbon nanotubes . . . . . . . . . . . . . .

39
41

Chapter 4

Long-wavelength deformations and vibrational modes in empty
and liquid-filled microtubules and nanotubes: A theoretical
study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Continuum elasticity approach . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Vibrational Modes of empty Nanotubes . . . . . . . . . . . . . . . . .
4.2.2 Vibrational Modes of Liquid-Filled Nanotubes . . . . . . . . . . . . .
4.3 Vibrational Modes of Nanotubes in a Surrounding Liquid . . . . . . . . . . .
4.4 Computational Approach to Determine the Elastic Response of Carbon Nan-
otubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Tubulin-Based Microtubules . . . . . . . . . . . . . . . . . . . . . . .
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Related Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.1 Lagrange Function of a Strained Nanotube . . . . . . . . . . . . . . .
4.8.1.1
Stretching . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.1.2 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.1.3 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.2 Derivation of Euler-Lagrange Equations of Motion for Deformations of
a Nanotube using Hamilton’s Principle . . . . . . . . . . . . . . . . .
4.8.2.1
Stretching . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.2.2 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.2.3 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.3 Coupling Between a Travelling Pressure Wave and the RBM in a
Liquid-Filled Carbon Nanotube . . . . . . . . . . . . . . . . . . . . .
4.8.4 Coupling Between the LA Mode and the RBM in Carbon Nanotubes

Chapter 5 Unusually stable helical coil allotrope of phosphorus . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2 Computational methods
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44
44
46
47
50
51

54
54
55
57
60
62
63
64
64
65
65

68
68
70
70

71
75

79
79
81
81
89

Chapter 6 Microscopic Mechanism of the Helix-to-Layer Transformation
90
in Elemental Group VI Solids . . . . . . . . . . . . . . . . . . . .
90
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
6.2 Computational techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

viii

Chapter 7 Effect of Net Charge on the Relative Stability of 2D Boron

Allotropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.2 Computational techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Chapter 8 Towards room-temperature superconductivity in low-dimensional

C60 nanoarrays: An ab-initio study . . . . . . . . . . . . . . . . . 129
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.2 Computational methods
8.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Chapter 9

Shear instability in twisted bilayer graphene . . . . . . . . . . . 140
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.1
9.2 Computational methods
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
9.3.1 Deformation Modes in Bilayer Graphene . . . . . . . . . . . . . . . . 142
9.3.2 Relaxations in Sheared and Twisted Bilayer Graphene
. . . . . . . . 143
9.3.3 Electronic Structure of Sheared, Twisted and Relaxed Bilayer Graphene147
9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.5 Related Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9.5.1 Mathematical formulation of the rigid shear-twist deformation in bi-

layer graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9.5.2 Mathematical background of the relaxation treatment in sheared and

twisted bilayer graphene . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.5.3 Local relaxation in sheared and twisted bilayer graphene . . . . . . . 157
9.5.4 Electronic structure changes in sheared and twisted bilayer graphene
157
9.5.5 Value of the magic angle in unsheared and sheared twisted bilayer

graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Chapter 10 Two-dimensional Mechanical Metamaterials with Unusual Pois-

son Ratio Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.2 Computational methods
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
10.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
10.5 Related Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
. . . . . . . 169
. . . . . 170

10.5.1 Deformation behavior in 2D isosceles triangle assemblies
10.5.2 Deformations in a 2D assembly of rigid equilateral triangles

Chapter 11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

ix

LIST OF TABLES

Table 4.1: Summary of expressions derived for the vibrational frequencies ω of 1D
tubules and 2D plates. ˜ω denotes the frequency of liquid-filled tubules.
Equation numbers refer to the present publication.
. . . . . . . . . . . . .

52

Table 4.2: Elastic behavior of a 2D graphene monolayer. c11, c22, c66 are 2D elastic
stiffness constants defined in Eq. (4.1). α is the in-plane Poisson ratio, D is
the flexural rigidity, and ρ2D is the areal mass density. These values can be
used directly to calculate long-wavelength acoustic frequencies ω(k) using
the expressions in Table 4.1 and the speed of sound vLA and vT A. Present
values are compared to published data.
. . . . . . . . . . . . . . . . . . .

Table 6.1: Cohesive energy Ecoh of various Se and Te allotropes in [eV/atom] units,
. . . . . . . . . . .

obtained using DFT-LDA and DFT-PBE calculations.

56

95

Table 7.1: Cohesive energy Ecoh of selected neutral borophene allotropes, obtained
using DFT-PBE calculations. n is the number of boron atoms per unit cell
and Z is the coordination number of individual atoms in the unit cell.

. . 115

x

LIST OF FIGURES

Figure 3.1:

(a) Schematic representation of possible distortions of an elastic mem-
brane. (b) Schematic dependence of the longitudinal acoustic (LA), in-
plane transverse acoustic (TA) and the flexural acoustic (ZA) mode vibra-
tion frequencies on the crystal momentum near the center of the Brillouin
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
zone.

Figure 3.2: Phonon spectra of (a) graphene, reproduced from Reference [1] and (b)
phosphorene, a monolayer of black phosphorus, reproduced from Refer-
ence [2], shown by solid lines. Superposed to the spectra are continuum
results for the three acoustic phonon modes in different high-symmetry
directions, evaluated near Γ, with the longitudinal acoustic (LA, dashed
lines), in-plane transverse acoustic (TA, dotted lines), and the flexural
acoustic (ZA, solid lines) modes. Ball-and-stick models of the structure,
including the primitive unit cells, are shown in the top panels. The Bril-
. . . . . . . . . .
louin zones are shown as insets in the phonon spectra.

Figure 3.3: Radial breathing mode (RBM) in (a) fullerenes and (b) carbon nan-
otubes as a function of their diameter d. The prediction is shown by
the solid line. Experimental and theoretical RBM frequencies for the
only observed fullerene with spherical symmetry, C60, are shown by the
data points in (a). The experimentally well-established relationship [3, 4]
ω = 248 cm−1×(1 nm/d) in nanotubes in (b) is represented by the dashed
. . . . . . . . . . . .
line and the calculation is shown by the solid line.

Figure 4.1: Schematic representation of important deformations of a tubular struc-
ture. (a) Longitudinal acoustic (LA, stretching), (b) torsional acoustic
(TA, torsion), (c) flexural acoustic (doubly degenerate ZA, bending), and
(d) the radial breathing mode (RBM). (e) Schematic dispersion relations
of the corresponding long-wavelength phonon modes. The tilde denotes
liquid-filled tubules.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.2:

(a) Frequency of vibrational modes depicted in Fig. 4.1(a) in empty and
water-filled carbon nanotubes. (b) Dependence of the flexural coefficient
cZA(R), defined in Eq. (4.9), on the radius R of empty and water-filled
carbon nanotubes. The tilde denotes liquid-filled nanotubes.
. . . . . .

Figure 4.3:

(a) Frequency of vibrational modes depicted in Fig. 4.1(a) in empty and
water-filled tubulin-based microtubules. (b) Dependence of the flexural
coefficient cZA(R), defined in Eq. (4.9), on the radius R of empty and
water-filled tubulin-based microtubules. The tilde denotes liquid-filled
tubules.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

27

28

46

55

57

xi

Figure 4.4: Nature and coupling of vibration modes ω(k) of a carbon nanotube that
may be filled with water. (a) Coupling between the dispersionless RBM of
a CNT, shown by the black line, and the pressure wave of water enclosed
in the CNT, shown by the green dotted line. (b) Coupling between the
RBM of a CNT, shown by the black line, and the longitudinal acoustic
mode of the CNT, shown by the green dotted line. Results are presented
for a CNT with a radius of 1 nm.
. . . . . . . . . . . . . . . . . . . . .

Figure 5.1:

(a) Optimum structure of an isolated, straight 1D P∞ chain with the P10
unit cell of length L. (b) Optimized P18, P28 and P38 segments of the iso-
lated chain, indicating the tendency to form rings with a radius R≈2.4 nm.
(c) The optimum structure of a single coil. (d) Optimum structure of a
2D assembly of P∞ chains separated by the distance d. (e) Possible sce-
nario for the formation of helical coils by connecting finite-length chain
segments inside a cylindrical cavity. The unit cells are indicated by dashed
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
lines in (a) and (d).

Figure 5.2:

(a) Strain energy ∆E per atom as a function of the radius R of an isolated
P coil. (b) Inter-chain interaction energy ∆E per atom in a 2D assembly of
phosphorus chains, depicted in Fig. 5.1(d), as a function of the inter-chain
distance d. (c) Interaction energy ∆E per phosphorus atom between a P
chain and a graphene monolayer as a function of the adsorption height h.
PBE results are shown by the solid red lines, LDA results by the dashed
blue lines.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 5.3: Electronic structure of (a) an isolated P chain and (b) a 2D layer of P
chains, shown in Figs. 5.1(a) and 5.1(d). Left panels depict the electronic
band structure based on PBE and the middle panels the corresponding
density of states. The Brillouin zone is shown as inset of the left panel
in (b). The right panels depict the charge distribution associated with
frontier states in the valence band region, indicated by the green hashed
region in (a) and (b), which extends from EF to 0.2 eV below the top
of the valence band. Charge density contours are superposed to struc-
tural models, with the unit cells indicated by the dashed lines. Due to
differences in the density of states between these systems, the contours
are presented at the electron density 0.008 e/˚A3 in (a) and 0.002 e/˚A3 in
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b).

Figure 6.1: Stable 1D and 2D structural allotropes of Se and Te. 1D structures of the
(a) a helix and (b) b chain and their 2D counterparts, the (c) δ and (d) η
allotrope. The δ allotrope is a covalently bonded 2D assembly of b chains.
The unit cells of the 2D structures are highlighted by the transparent
. . . . . . . . . . . . . . . . . . . . . . . . .
green areas in (c) and (d).

72

80

82

85

91

xii

Figure 6.2: Phonon spectra of (a) δ-Se and (b) η-Se calculated using the DFT-LDA
energy functional. The Brillouin zones and high-symmetry points are
shown schematically in the insets. Continuum elasticity results for long-
wavelength longitudinal acoustic modes are shown by the blue dash-dotted
lines, for transverse acoustic modes by the dotted green lines, and for
flexural modes by the dashed red parabolas.
. . . . . . . . . . . . . . .

92

Figure 6.3:

(a) Schematic growth mechanism of the energetically stable 2D δ structure
by zipper-like attachment of the b chain, which is being formed locally
locally at a defect in the native a helix and propagates by dislocation
motion. (b) Bond length d, bond angle θ and dihedral angle ψ used to
characterize chalcogen structures. (c) DFT-LDA based energy differences
∆E encountered during the stepwise conversion from 1D a-Se to b-Se as a
function of the reaction coordinate. The system is represented by a finite
Se9H2 chain, passivated by hydrogen at both ends, and the total energy is
given with respect to the final state. The dotted line is guide to the eye.
The energy of a 9-atom long segment of the defect-free infinite a-Se and
b-Se chains is indicated by asterisks. Ball-and-stick models show stable
Se9H2 geometries, labeled by A-D, and the transition states T . Location
of the unusually small dihedral angle in the transition states is indicated
by shaded triangles.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 6.4: Transformation of 1D to 2D structures of elemental Se. (a) Perspective
and end-on view of the initial 2D assembly of 1D a-chains. (b) End-on
view of the structure after artificial confinement to an infinitely thin slab.
(c) End-on view of the 2D δ-Se allotrope that formed spontaneously after
the structure in (b) was relaxed.
. . . . . . . . . . . . . . . . . . . . . .

normal to the lattice are shown by (cid:13) and atoms in front of these by(cid:74).

Figure 6.5: Structural transformation between the 2D δ and the η allotropes by a se-
ries of reflections in planes normal to the picture plane. (a)-(c) Schematic
side view of the initial and intermediate structures. Atoms in a plane
Atoms in one unit cell are labeled by the numbers 1− 6. Primed numbers
are used to label atoms in neighboring cells. Perspective views of (d) the
initial δ and (e) the final η structure.
. . . . . . . . . . . . . . . . . . .

94

97

98

Figure 6.6:

(a) Potential energy E as a function of the optimum reaction coordinate
during the A-D transformation. (b) Schematic transformation from the a-
helix to the b-chain by dislocation motion.Schematic geometry of hydrogen
terminated 9-atom segments of (c) the a-helix and (d) the b-chain. Atoms
are labeled for later use.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 102

xiii

Figure 6.7:

(a,c) Dihedral angle ψ and (b,d) potential energy E of the Se system dur-
ing (a,b) CG optimization and (c,d) MD calculations starting in transition
point TAB, identified energetically in Fig. 6.6(a). Atomic positions used
to define ψ6789 are identified in Fig. 6.6(c) and 6.6(d).

. . . . . . . . . . 103

Figure 6.8:

(a,c) Dihedral angle ψ and (b,d) potential energy E of the Se system dur-
ing (a,b) CG optimization and (c,d) MD calculations starting in transition
point TBC , identified energetically in Fig. 6.6(a). Atomic positions used
to define ψ2345 are identified in Fig. 6.6(c) and 6.6(d).

. . . . . . . . . . 104

Figure 6.9:

(a,c) Dihedral angle ψ and (b,d) potential energy E of the Se system dur-
ing (a,b) CG optimization and (c,d) MD calculations starting in transition
point TCD, identified energetically in Fig. 6.6(a). Atomic positions used
to define ψ789,10 are identified in Fig. 6.6(c) and 6.6(d).

. . . . . . . . . 105

Figure 6.10: Electronic band structure of an isolated (a) a-Se helix, (b) b-Se chain,
isolated (c) δ-Se and (d) η-Se monolayers. GW results, shown by solid red
lines, are compared to LDA results, shown by the black dashed lines.

. . 106

Figure 6.11: Electronic band structure of an isolated (a) a-Te helix, (b) b-Te chain,

isolated (c) δ-Te and (d) η-Te monolayers obtained using DFT-LDA.

. . 107

Figure 7.1: Previously unexplored neutral borophene allotropes formed by sponta-
neous conversion of an artificial honeycomb lattice: (a) 
-B and (b) ω-B.
The structures are shown in top and side view. The lattice vectors (cid:126)a1 and
(cid:126)a2, shown in red, delimit the highlighted primitive unit cells.

. . . . . . 114

Figure 7.2: Electron doped 2D borophene structures obtained by optimizing a dis-
torted boron honeycomb superlattice with 32 atoms per unit cell. The
average excess charge (cid:104)∆Q(cid:105) per boron atom, specified in the panels, in-
creases from (a) 
1-B to (f) 
6-B. The structures are shown in top and side
view. The lattice vectors (cid:126)a1 and (cid:126)a2, shown in red, delimit the highlighted
unit cells.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 7.3: Electron doped 2D borophene structures obtained by optimizing a dis-
torted boron honeycomb superlattice with 32 atoms per unit cell. The
average excess charge (cid:104)∆Q(cid:105) per boron atom, specified in the panels, in-
creases from (a) to (c). The structures are shown in top view (upper pan-
els) and side view (lower panels). The lattice vectors (cid:126)a1 and (cid:126)a2, shown in
red, delimit the highlighted unit cells.

. . . . . . . . . . . . . . . . . . . 117

xiv

Figure 7.4: Charge redistribution in 2D borophene layers induced by extra electrons
or by contact with monolayers of Ca2N electride. Charge density differ-
ence ∆ρ caused by placing an excess charge (a) (cid:104)∆Q(cid:105) = −0.406 e/atom
on 
3-B and (b) (cid:104)∆Q(cid:105) = −1.0 e/atom on 
6-B. Charge density redistri-

bution ∆ρ = ρ(B/Ca2N)−ρ(B)−(cid:80) ρ(Ca2N) in the bilayer structures (c)


3-B/Ca2N and (d) 
6-B/Ca2N as well as in the sandwich structures (e)
Ca2N/
3-B/Ca2N and (f) Ca2N/
6-B/Ca2N. ∆ρ is shown by isosurfaces
bounding regions of electron excess at +7×10−3 e/˚A3 (yellow) and elec-
tron deficiency at −2 × 10−3 e/˚A3 (blue). (cid:104)∆ρ(z)(cid:105) is averaged across the
x − y plane of the layers.

. . . . . . . . . . . . . . . . . . . . . . . . . . 118

Figure 7.5: Changes in the interlayer interaction energy ∆E and net average charge
(cid:104)∆Q(cid:105) on borophene in the Ca2N/B/Ca2N sandwich geometry as a func-
tion of the interlayer distance d. Results for 
3-B in (a) and (c) are com-
pared to those for 
6-B in (b) and (d).

. . . . . . . . . . . . . . . . . . . 119

Figure 7.6: Electron flow between borophene and bilayers of the Ca2N electride. Charge

density redistribution ∆ρ = ρ(B/[Ca2N]2)−ρ(B)−(cid:80) ρ(Ca2N) in (a) 
3-

B/[Ca2N]2 and (b) 
6-B/[Ca2N]2 triple-layer heterostructures. ∆ρ is
shown by isosurfaces bounding regions of electron excess at +7×10−3 e/˚A3
(yellow) and electron deficiency at −2× 10−3 e/˚A3 (blue). (cid:104)∆ρ(z)(cid:105) is av-
eraged across the x − y plane of the layers.

. . . . . . . . . . . . . . . . 120

Figure 7.7: Effect of net charge on the equilibrium geometry of 
-B. (a) Effect of
the net average charge (cid:104)∆Q(cid:105) on the orthogonal lattice constants ai with
i = 1, 2. Plotted are charge-induced relative changes ∆ai/a0
i , where a0
i
are the lattice constants in the neutral 
-B allotrope. (b) Strain energy
∆E as a function of in-layer strain σ at different levels of extra electrons
(cid:104)∆Q(cid:105).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Figure 7.8: Electronic band structure of (a) neutral 
-B, (b) 
1-B with (cid:104)∆Q(cid:105) = −0.156 e/B,

(c) 
3-B with (cid:104)∆Q(cid:105) = −0.406 e/B, (d) 
4-B with (cid:104)∆Q(cid:105) = −0.500 e/B, (e)

5-B with (cid:104)∆Q(cid:105) = −0.750 e/B, and (f) 
6-B with (cid:104)∆Q(cid:105) = −1.000 e/B,
calculated using the DFT-PBE functional. The indirect fundamental band
gap is indicated by the red arrow in (a).

. . . . . . . . . . . . . . . . . . 122

Figure 7.9: Electronic band structure of 
-B subject to in-layer strain of (a) -6%, (b)
-4%, (c) +4%, and (d) +6% along the (cid:126)a1 direction, and (e) -3%, (f) -1%,
(g) +1%, and (h) +3% along the (cid:126)a2 direction. The fundamental band gap
is indicated by the red arrows in semiconducting systems.

. . . . . . . . 123

xv

Figure 7.10: Charge density difference ∆ρ caused by placing an excess charge (cid:104)∆Q(cid:105) =
−0.0625 e/atom on selected borophene structures characterized in Fig-
ure 7.2 of the main manuscript and in Figure 7.3. (a) 
1-B of Figure 7.2(a),
(b) 
2-B of Figure 7.2(b), (c) 
3-B of Figure 7.2(c), (d) 
4-B of Fig-
ure 7.2(d), as well as B structures shown in (e) Figure 7.3(b) and (f)
Figure 7.3(c). ∆ρ is shown by isosurfaces bounding regions of electron
excess at +2.5×10−3 e/˚A3 (yellow).

. . . . . . . . . . . . . . . . . . . . 125

Figure 8.1: Schematic arrangement of C60 molecules in (a) a 3D fcc crystal, (b) a
2D triangular lattice, and (c) a 1D array inside a (10, 10) carbon nan-
otube peapod. (d) Width of the t1u-derived band in a 1D, 2D and 3D
arrangement of C60 molecules as a function of the C60-C60 center-to-
center separation dcc. The equilibrium value of dcc depends on the C60
orientation and is indicated by the gray strip in (d) for undoped struc-
tures. The ‘error bars’ reflect the effect of changing the C60 orientation
on the bandwidth. (e) Schematic arrangement of dopant atoms outside
the 1D peapod ((cid:78)), inside the nanotube but outside the fullerene ((cid:4)),
and inside the fullerene (+). Dark spheres in (a) represent clusters other
than C60 that separate quasi-1D percolating arrays of fullerenes from the
surrounding matrix. The planes in (b) are only a visual aid.

. . . . . . . 132

∆ρ = ρtot(C60@CNT+7K)−ρtot(C60@CNT) −(cid:80)

Figure 8.2: Density of states (DOS) and charge redistribution in doped C60 crys-
tals.
(a) DOS of the t1u LUMO-derived bands of C60 in 1D, 2D and
3D periodic C60 arrangements for the C60-C60 separation dcc = 9.8 ˚A.
(b) Total and projected DOS of a C60@(10, 10) CNT peapod doped ex-
(c) Charge density difference
ternally by 7 K donor atoms per C60.
(d)
DOS of the hu HOMO-derived bands of C60 in 1D, 2D and 3D peri-
odic C60 arrangements for the C60-C60 separation dcc = 9.8 ˚A. (e) Total
and projected DOS of a C60@(10, 10) CNT peapod doped externally by
8 F acceptor atoms per C60. (f) Counterpart of (c) for C60@CNT+8F.
In (c) and (f), blue contours for electron deficit are shown for ∆ρ =
−3.0×10−3 e/bohr3 and red contours for electron excess are shown for
∆ρ = +3.0×10−3 e/bohr3. All energies are with respect to EF .

at ρtot(K atom).

. . . . . 133

xvi

Figure 8.3: Properties of 1D arrays of C60 molecules found in CNT peapods.

(a)
Ball-and-stick models of different atomic arrangements at the C60-C60 in-
terface. Considered are double-bonds facing double-bonds (d), hexagons
facing hexagons (h), pentagons facing pentagons (p). α and β arrange-
ments are obtained by a 90◦ rotation of C60 molecules in p arrangement
about two different axes that are orthogonal to the chain axis and to each
other. (b) DOS of the t1u LUMO-derived bands of C60 for the C60 orien-
tations, defined in (a), at the C60-C60 separation dcc = 9.8 ˚A. All energies
are with respect to EF . (c) Maximum DOS value of t1u-derived bands in
1D, 2D and 3D crystals of C60. The ‘error bars’ reflect the effect of chang-
ing the C60 orientation. (d) Critical temperature for superconductivity
Tc based on the McMillan equation (8.1) and using N (EF ) from (c). The
lines in (c) and (d) are guides to the eye.

. . . . . . . . . . . . . . . . . 137

Figure 9.1: Shear and twist in bilayer graphene (BLG). (a) Definition of shear and
twist operations in BLG, initially a bilayer in AA stacking indicated by
the honeycomb lattice. A rectangular segment of the top layer, with one
side closing the angle β with respect to the highlighted armchair direction,
is indicated in yellow and surrounded by the dotted line. The segment
is first sheared by the angle α and subsequently rotated by the angle θ.
(b) Schematic top view of a uniformly twisted and sheared monolayer
graphene (MLG) on top of an undeformed MLG. The resulting relaxed
Moir´e pattern contains regions of AA stacking, highlighted by the white
circles, and those of AB or BA stacking. (c) Dark-field TEM image of
bilayer graphene reproduced from Ref. [ [5]]. (d) Schematic side view of a
relaxed sheared and twisted BLG with locally varying stacking.

. . . . . 143

Figure 9.2:

(a) Energy difference ∆E caused by shearing the top layer of BLG, which
had been twisted by θm = 1.08◦, by varying the shear angle α along the
β = 0◦ direction. Results for the BLG with unrelaxed atomic positions in
planar, sheared monolayers are shown by the red dotted line and for the
BLG with relaxed atomic positions by the black solid line. Contour plots
of the local shift vector length δ = |δ| in the BLG structure, which had
been twisted by θm = 1.08◦ and sheared by α = 0.08◦ along the β = 0◦
direction, (b) in absence and (c) in presence of lattice relaxation.

. . . . 144

Figure 9.3: Electronic density of states (DOS) in the BLG structure subject to the
magic twist angle θm = 1.08◦. Results for the unsheared structure are
shown in panels (a) and (c), and those for the top layer sheared by α =
0.08◦ along the β = 0◦ direction in (b) and (d). Results for the unrelaxed
structure in (a) and (b) are compared to those for the relaxed structure
in (c) and (d).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

xvii

Figure 9.4: Contour plots of the local shift vector length δ = |δ| in the BLG structure,
which had been twisted by θ = 0.4◦ and sheared by α = 0.08◦ along the
β = 0◦ direction, (a) in absence and (b) in presence of lattice relaxation.

155

Figure 9.5: Electronic density of states (DOS) in the BLG structure subject to the
magic angle twist θm = 1.08◦. Results for the unsheared structure are
shown in panels (a) and (c), and those for the top layer sheared by α =
0.08◦ along the β = 0◦ direction in (b) and (d). Results for the unrelaxed
structure in (a) and (b) are compared to those for the relaxed structure
in (c) and (d). The energy scale is extended in comparison to Fig. 9.3.

. 156

Figure 9.6: The width of the ’flat band’ in a relaxed twisted bilayer graphene structure
as a function of the twist angle θ. Data points for commensurate, relaxed
structures in the vicinity of the observed value θm = 1.08◦ are presented
by the black diamonds ((cid:7)) for the unsheared lattice and by the red circles
(•) for the structure with the top layer sheared by α≈0.08◦ along the
β = 0◦ direction. The dotted and dashed lines are guides to the eye.

. . 157

Figure 10.1: Deformations in a 2D assembly of rigid isosceles triangles. (a) Adjacent
triangles with opening angle α and mutual orientation defined by the
closing angle β, hinged tip-to-corner, forming the primitive unit cell. The
triangle height x0 and the length y0 of its base define the horizontal and
vertical length scales. (b) Snap shots of the α = 120◦ triangle assembly
for different values of β. The conventional rectangular unit cell is twice
the size of the primitive unit cell. (c) Contour plot of the Poisson ratio
νxy = −(dy/y)/(dx/x) as a function of α and β. The dotted red line
highlights behavior of the α = 120◦ triangle assembly. (d) Poisson ratio
νxy as a function of β in the α = 120◦ system. (e) Changes in the scaled
width x/x0 and height y/y0 of the conventional unit cell for α = 120◦
caused by changing the angle β.

. . . . . . . . . . . . . . . . . . . . . . 161

Figure 10.2: Deformations in porous graphene, a phenanthrene-based 2D mechanical
metamaterial. (a) Structure of the C14H10 phenanthrene molecule and its
relation to an isosceles α = 120◦ triangle of Fig. 10.1. (b) Equilibrium
structure of 2D porous graphene consisting of polymerized phenanthrene
molecules with β = 70◦. Saturating hydrogen atoms are shown by the
lighter and smaller spheres. Changes in the scaled width x/x0 (c) and
height y/y0 (d) of the conventional unit cell in the triangle assembly and
porous graphene as a function of the closing angle β. (e) Poisson ratio νxy
in porous graphene as a function of β. (f) Strain energy in the C56H28
conventional unit cell as a function of β. The dashed and dotted lines
connecting data points for porous graphene in (c)-(f) are guides to the eye. 162

xviii

Figure 10.3: Electronic structure of porous graphene, a phenanthrene-based 2D me-
chanical metamaterial, based on DFT-PBE calculations. (a) Band struc-
ture of the equilibrium structure with β = 70◦ obtained using the rectan-
gular C56H28 unit cell. High-symmetry points in the rectangular Brillouin
zone are shown in the inset. (b) Fundamental band gap Eg as a function
of the angle β.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Figure 10.4: Changes in the scaled width x/x0 and height y/y0 of the conventional unit
cell for different values of the opening angle α as a function of the closing
angle β. The relevant quantities are defined in Fig. 10.1.

. . . . . . . . . 170

Figure 10.5: Deformations in a 2D assembly of rigid equilateral triangles. (a) Adjacent
triangles with mutual orientation defined by the closing angle β, hinged
at the corners, forming the primitive unit cell. The triangle height x0 and
the length y0 of its base define the horizontal and vertical length scales.
(b) Snap shots of the triangle assembly for different values of β. The
conventional unit cells of width x and height y are indicated.

. . . . . . 171

xix

Chapter 1

Introduction

There are three main projects related to low-dimensional systems on the nanoscale in this

thesis. First of all, the continuum elasticity method was utilized to successfully solve a

very common problem, namely the occurrence of unphysical imaginary frequencies in long-

wavelength flexural acoustic phonon modes of 2D and 1D systems. The second topic mainly

addresses the stability and electronic properties, as well as a microscopic mechanism behind

structural transformations between different low-dimensional systems. The last topic are

electronic and mechanical properties of carbon-based materials. All of these topics have been

motivated by a desire to solve long-standing problems in the specific field and to explore the

bright future that may benefit from unique properties of 2D and 1D nanostructures.

1.1 Continuum approach of long-wavelength acoustic

phonon modes of quasi-2D and 1D nanomaterials

Atomistic calculations of phonon spectra based on ab initio density functional theory (DFT)

start with the calculation of the force-constant matrix, which is associated with finite dis-

placements of atoms in the unit cell. While addressing long-wavelength acoustic phonon

modes, this approach reaches it practical limitations in a computer calculation. For long-

wavelength vibration modes, especially as the wave number k is close to zero, a very large

1

supercell is typically required to obtain an accurate result. However, the computer limitations

lead to numerical inaccuracies in the force-constant matrix which usually cause imaginary

frequencies in the acoustic phonon modes near the Γ point.

In this case, imaginary frequencies are numerical artifacts that differ fundamentally from

the case, where imaginary frequencies indicate real structural instabilities. Imaginary fre-

quencies appearing as artifacts also affect related simulations that depend on a correct

phonon spectrum, such as thermal conductivity, charge transport, and others. Here, I pro-

pose a continuum elasticity approach to solve the problem, which does not suffer from the
large supercell sizes in 2D/1D systems. In this thesis, I introduced a 3×3 2D elastic stiff-

ness matrix, complemented by the flexural rigidity D, which describes the elastic behavior

of 2D/1D systems in the linear regime. Independent of thickness of the 2D/1D system, all

acoustic phonon modes can be determined using elements of the 2D elastic matrix. These

elements can be calculated using very few static ab initio density functional theory calcula-

tions or relaxed and distorted structures in a much more reliable and straight-forward way,

or can be obtained experimentally.

1.2 Microscopic mechanism of structural transitions in

2D and 1D systems

Low-dimensional nanostructures receive growing attention due to their different physical

properties in comparison with those of macroscopic 3D systems. These include high flexi-

bility, charge carrier mobility, tunable electronic structure, and others. Exploring 2D and

1D nanomaterials beyond those, which have been synthesised in experiment, plays an im-

portant role in developing electronic devices. Recent developments including data mining

2

and machine learning appear to have accelerated the process of predicting low dimensional

nanomaterials. However, many of such predicted nanomaterials are hard to synthesis and

some may only occur under extreme synthesis conditions. To quote one example, 2D al-

lotropes of elemental boron need a special substrate to grow. It is thus not only the stability

of a nanostructure in vacuum, but also the likely formation formation mechanism that is

essential to provide a guideline for future experiments.

In this thesis, I predicted previously unknown stable allotropes of elements in groups III,

V and VI. I found these allotropes, often with a peculiar structural symmetry, to be at least

as stable as existing materials. I also investigate the electronic properties of these allotropes

to explore their application prospect. More importantly, I studied the transformation mech-

anism between different allotropes, and proposed critical information about the synthesis

conditions under which these allotropes could be formed in the experiment. I found that

the most efficient transformation from a 1D helix of Se/Te to a 2D allotrope involves only

changes in the dihedral angle, while the bond lengths and bond angles are not affected. I also

found that when carrying one extra electron per atom, elemental boron will mimic carbon

in both its atomic and electronic structure. In reality, such a large electron charge can be

provided when in contact with a 2D electride such as Ca2N.

1.3 Mechanical and electronic properties of novel carbon-

based nanomaterials

Carbon is one of the most abundant elements on earth. Reducing dimensionality from 3D to

0D, we find many stable carbon allotropes, such as diamond in 3D, graphene/graphite in 2D,

nanotubes in 1D, and fullerenes in 0D. Different carbon allotropes also exhibit a rich spectrum

3

of physical properties. For example, twisted bilayer graphene will turn superconducting at a

special twist angle. Carbon nanotubes can be used for medication delivery. Functionalized

fullerenes are valuable candidate drugs. The high number os structurally stable 0D-3D

allotropes provides nanomaterials based on elemental carbon with a wide range of unexpected

properties and promises new fundamental physics insight.

I will discuss carbon materials from three aspects. First, the 3D alkali-metal-doped C60

crystal is a superconductor with Tc around 30 K, and Tc can be well explained based on

McMillan equation in theory. One critical factor, which determines the value of Tc, is the

density of state at the Fermi level.

Instead of changing the Fermi level by doping, my

calculation shows that the critical superconducting temperature of doped C60 crystals can

be elevated near to room temperature simply by reducing the dimensionality from 3D to

quasi-1D.

Pristine carbon has never been observed to superconduct. It is only in an alkali-metal-

doped C60 crystal that superconductivity has been observed and explained within the frame-

work of BCS theory. On the other hand, the Physics underlying the recently reported

emerging superconductivity in twisted bilayer graphene at the ’magic angle’ remains un-

clear. Conventional DFT calculations fail to reproduce the electronic structure for lack of

periodicity or in view of very large Moir´e supercells with a large number of atoms. Here,

by using the tight-binding model, I succeeded in exploring the importance of in-plane shear

and atomic relaxation in the bilayer graphene system. Both effects stabilize the system and

enhance the density of states at Fermi level.

Finally, I study an unexpected behavior of an important mechanical property, namely

the Poisson ratio ν, which can become negative and change sign in a designer carbon meta-

material. ν is positive in ordinary materials, which means that stretching the material in

4

one direction will shrink the material in an orthogonal direction.

In previously proposed

auxetic materials with a particular atomic structure, the Poisson ratio is negative, meaning

that stretching in one direction causes expansion in an orthogonal direction. Graphene is a

promising candidate from this perspective given the possibility to modify its structure, which

may be achieved by supramolecular assembly in the experiment. Here I propose that a struc-

turally tailored graphene, which could form by polymerization of phenanthrene molecules,

exhibits both a negative Poisson ration and a shape-memory effect, which changing reversibly

its fundamental band gap at will.

1.4 Outline of the dissertation

This thesis contains 11 Chapters in total. The introduction to all my Ph.D. research is

described in Chapter 1, and computational methods I used for most of my work are described

in Chapter 2.

In Chapters 3 and 4, I introduced the continuum elasticity approach to study acoustic

phonon modes of one-dimensional and two-dimensional systems. I have developed expres-

sions for the momentum-dependent frequencies of long-wavelength acoustic phonon modes

in terms of the 2D elastic constants of the system. My approach solves the problem of

numerical artifacts in atomistic calculations that result in imaginary frequencies near the Γ.

In Chapter 5 to Chapter 7, I predicted a variety of previously unknown stable low-

dimensional allotropes of elements in group III, V and VI. The stability of these allotropes

turns out to be comparable to the existing materials. In particular, a one-dimensional phos-

phorus coil, which can be formed inside carbon nanotubes, is as stable as black phosphorus.

As for Se/Te, I have predicted two previously unknown, stable allotropes. In Chapter 6,

5

I study the transformation of Se/Te allotropes from 1D to 2D structures and identify the

reaction coordinates for this transformation. My general findings should be of value to other,

related systems. Unlike other elements, boron is very sensitive to doping. Thus searching

for stable 2D electron-doped boron allotropes is of practical importance. Here I predict

different stable allotropes at different level of net charging. More importantly, I also discuss

the mechanism of forming such allotropes in detail.

In Chapter 8, I focused on the properties of carbon based materials, including the super-

conductivity of C60 crystals, the shear-induced instability of twisted bilayer graphene, and

the negative Poisson ratio of porous graphene. For the first topic, heavy alkali atoms have

been used to dope the C60 crystal and increase its lattice constant to increase its critical

temperature for superconductivity. I propose an alternative approach by reducing the coor-

dination number of C60 in 2D and quasi-1D systems, which promises an even more significant

in increasing Tc to close to room temperature. The second topic relates to twisted bilayer

graphene which is aperiodic in the most general case. By using the tight-binding model, I

found that during the twist, an in-plane shear provides further stabilization for the system.

This appears to be confirmed by dark-field transmission electron microscopy images. Finally,

I studied the peculiar negative Poisson ratio ν of porous graphene with specific structure.

I found that the designed metamaterial shows not only a negative value ν, but that ν may

change sign and even diverge for specific structural arrangements in the 2D metamaterial.

6

Chapter 2

From Hartree-Fock to Density

Functional Theory

In this Chapter, I will start with the Hatree-Fock approximation that begins with the

Schrodinger equation for a one-electron system and antisymmetrizes the many-electron wave-

function represented by a Slater determinant. Then I will introduce the Density Functional

Theory (DFT) where the density of electrons is the fundamental property that determines

the properties of ground state of the system.

In most cases, self-consistency in DFT is

obtained by solving a set of Kohn-Sham equations.

2.1 The Born-Oppenheimer approximation

For a material containing atoms and electrons, the Hamilton operator of the system is

following:

ˆH =

(cid:88)

I

− 2

2M

∇2
I +

(cid:88)

i

− 2

2m

∇2
i +

1
2

(cid:88)

I(cid:54)=J

ZI ZJ e2
|RI − RJ| +

1
2

(cid:88)

i(cid:54)=j

|ri − rj| −(cid:88)

e2

I,i

ZI e2
|ri − RI| (2.1)

This Hamiltonian contains five parts, namely the kinetic and interaction energy of elec-

trons, kinetic and interaction energy of ions, and the interaction between electrons and ions.

Uppercase indices refer to ions and lowercase indices to electrons. RI and RJ denote the

7

ion positions and ri and rj denote the electron positions. Since the mass of the nucleus

is thousands times higher than that of electron, the nucleus can be treated as static while

the electrons are moving. This is the Born-Oppenheimer approximation [6]. With this

approximation, the Hamiltonian can be written as:

(cid:88)

i

ˆH =

− 2

2m

∇2
i +

1
2

(cid:88)

i(cid:54)=j

|ri − rj| −(cid:88)

e2

I,i

ZI e2
|ri − RI| = ˆHi +

1
2

(cid:88)

i(cid:54)=j

e2

|ri − rj|.

(2.2)

Here, ˆHii is the Hamiltonian of a single electron.

2.2 Hartree-Fock equation

In order to get the ic wavefunction in a one-electron system, we need to solve the Schr¨odinger

equation

The total energy of the system is

ˆHΨ (r) = EΨ (r).

E = (cid:104)Ψ∗(r) | ˆH | Ψ (r)(cid:105).

(2.3)

(2.4)

In a many-electron system, if we do not consider the electron-electron interaction as described

as the second term in Eq. (2.2), the wavefunction of the whole system can be written as the

product of the one-electron wave functions for each electron. Since electrons are fermions,

exchange of two fermions will result in a sign change of the wavefunction. In that case, the

8

wave function of the whole system can be written as a Slater determinant,

Ψ (r) =

1√
N !

Φ1(r1) Φ1(r2)

Φ2(r1) Φ2(r2)

...

...

··· Φ1(rN)
··· Φ2(rN)

...

ΦN (r1) ΦN (r2)

··· ΦN (rN)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:90)

Here, Φi(rj) represents the orthonormal orbital of electron i at the position rj,

Φ∗
i (r)Φj(r) = δij.

(2.5)

The third term in Eq. (2.2) represents the crystal potential of the electron and can be written

as

(cid:88)

v(r − RI) = −(cid:88)

I

I

ZI e2
|r − RI| .

V (r) =

(2.6)

By substituting the Hamiltonian and the wave function into Eq. (2.6), we can get the average

energy

E =

(cid:90)
(cid:88)
(cid:88)

i

1
2

i(cid:54)=j

(cid:90)

(cid:88)

1
2

drΦ∗
(cid:90)

i (r) ˆHiΦi(r) +

drdr(cid:48)Φ∗

i (r)Φ∗

j (r(cid:48))

drdr(cid:48)|Φi(r)|2

i(cid:54)=j
e2
|r − r(cid:48)|Φj(r)Φi(r(cid:48)).

e2

|r − r(cid:48)||Φj(r(cid:48))|2 −

(2.7)

9

The energy of the ground state can be obtained by applying the variational principle on

Eq. (2.7). This leads to the Hartree-Fock equation:

(cid:90)

(cid:88)
dr(cid:48) e2Φ∗

j

[− 2

2m

∇2 + V (r) +

(cid:90)

(cid:88)

j

dr(cid:48)|Φj(r(cid:48))|2
j (r(cid:48))Φi(r(cid:48))
|r − r(cid:48)|

e2
|r − r(cid:48)|]Φi(r) −

Φj(r) = 
iΦi(r).

(2.8)

The Hartree-Fock equation is a wavefunction-based equation. Since the size of the Slater

determinant increases with the size of the system, this approach is not suited for infinite

solids.

2.3 Density Functional Theory

Two remarkable theorems form the exact basis of density functional theory (DFT). The first

theorem states that the total energy of the system in the ground state is a functional of only

the electron density ρ(r). The second theorem states that the ground-state charge density

is the one that minimizes the total energy.

While both theorems are exact, finding the energy functional E[ρ], which relates the total

energy to the charge density in the ground state, is a challenging problem. Minimization of

E[ρ] is achieved by solving

δ

E[ρ] − 


(cid:26)

(cid:90)

(cid:27)

ρ(r)dr

= 0,

(2.9)

where 
 is a Lagrange multiplier that keeps the total charge constant. This expression implies

that for the correct charge density,

δE[ρ]/δρ(r) = 
.

(2.10)

10

E[ρ] is usually separated into several terms:

E[ρ] = T [ρ] + Vs[ρ],

(2.11)

where T [ρ] is the kinetic energy functional and Vs[ρ] is a potential energy functional. The

potential energy functional is usually separated as

Vs[ρ] = Vc + VH + VXC ,

(2.12)

where Vc is the crystal potential energy describing the interaction of the electron with the

nuclei or external fields. VH [ρ] is the Hartree energy functional describing the Coulomb

repulsion between electrons. All other energy terms except for T [ρ], Vc and VH are combined

in the last potential energy term VXC , which describes all other contributions to E[ρ], mostly

related to electron exchange, correlation, relativistic effects, etc.

We may define an energy functional

leading to the expression

G[ρ] = T [ρ] + VXC ,

E[ρ] = Vc + VH + G[ρ].

(2.13)

(2.14)

All potential energy functionals may be represented by specific potentials. Vc[ρ] can be

written as

Vc[ρ] =

drvc(r)ρ(r).

(cid:90)

The space integral describes the interaction of the electron with the potential vc(r) that is

11

dominated by the Coulomb potential of the nuclei.

VH [ρ] is the Hartree energy that can be expressed as

(cid:90)

drρ(r)vH .

e2

|r − r(cid:48)| ρ(r(cid:48)) =

dr(cid:48)

e

|r − r(cid:48)| ρ(r(cid:48)).

VH [ρ] =

1
2

where

(cid:90)

drdr(cid:48)ρ(r)
(cid:90)

vH =

1
2

Then, the total electrostatic potential in the system is given by

v(r) = vc(r) + vH (r).

(2.15)

Similarly,

(cid:90)

VXC [ρ] =

drvXC (r)ρ(r),

where vXC [ρ] is the so-called exchange-correlation potential.

So, the total energy can be written as:

(cid:90)

E[ρ] =

drv(r) + G[ρ].

(2.16)

Inserting Eq. (2.16) to Eq. (2.10), we can get

v(r) + δG[ρ]/δρ(r) = 
.

(2.17)

It is also not easy to solve the above equation. To solve this problem, Kohn and Sham mapped

the real system of interacting electrons to a system of non-interacting quasielectrons with

12

the same charge density. For the quasielectron system, they derived a set of single-particle

Schr¨odinger equations. The self-consistent solution for the occupied states would provide

the same charge density as found in the interacting many-electron system.

First let us consider a system of N non-interacting quasi-electrons, which move in a static

external potential vext(r). The kinetic energy of this noninteracting system is

T [ρ] =< Ψ| − 2

2m

∇2|Ψ > .

(2.18)

For this noninteracting system, instead of Eq. (2.17), we will get:

vext(r) + δT [ρ]/δρ(r) = 
.

(2.19)

If we know the form of T [ρ], we can get the ρ(r) for solving Eq. (2.19), however, we can find

the charge density in an easier way.

Since it is a system of noninteracting fermions, the wave function will be the Slater

determinant of single-particle wave functions ψi of the Schr¨odinger equation

[− 2

2m

∇2 + vext(r)]ψi(r) = 
iψi(r).

The electron density is given by

(cid:88)

i

ρ(r) =

ψ∗
i (r)ψi(r).

(2.20)

(2.21)

Comparing with Eq. (2.19), it is much easier to solve Eq. (2.21) self-consistently.

Now, let us go back to the interacting electron system. Using Eq. (2.13), Eq. (2.17) can

13

be reformulated as

v(r) + δVXC [ρ]/δρ(r) + δT [ρ]/δρ(r) = v(r) + vXC (r) + δT [ρ]/δρ(r) = 
.

(2.22)

Next, we define the effective potential vef f (r),

vef f (r) = v(r) + vXC (r).

(2.23)

Then, Eq. (2.22) can be written as

vef f (r) + δT [ρ]/δρ(r) = 
,

(2.24)

which is similar to Eq. (2.19) by relating vext(r) to vef f (r). As a result, Eq. (2.20) becomes

the so-called Kohn-Sham equation

[− 2

2m

∇2 + vef f (r)]ψi(r) = 
iψi(r).

The charge density is then given by

ρ(r) =

(cid:88)

occ

ψ∗
i (r)ψi(r).

(2.25)

(2.26)

Equations (2.25) and (2.26) are solved self-consistently to obtain the correct charge density

ρ(r) in the ground state. Once ρ(r) is established, the total energy of the many-electron

system is obtained using Eq. (2.11) and the proper expressions for the potential energy

functional Vs[ρ].

14

2.4 Exchange-correlation functionals

As mentioned above, the properties of the system is the functional of electron density ρ(r),

also the exchange-correlation energy does. One frequently uses the Local Density Approxi-

mation (LDA) [7,8] which works especially very well for carbon systems. This approximation

depends only on the value of the electron density, not on the derivatives of the density. Within

this approximation, the exchange-correlation is:

V LDA
XC [ρ] =

drρ(r)vxc(ρ).

(2.27)

(cid:90)

(cid:19) 1

3

(cid:18) 3

π

where ρ is the electron density and vxc(ρ) = vx(ρ) + vc(ρ) is the exchange and correlation

potential per particle in a homogeneous electron gas.

For the homogenous electron gas, the exchange energy is:

vx(ρ) = −3
4

1
3 .

ρ(r)

(2.28)

The precise value of the correlation energy vc has been established by Quantum Monte Carlo

calculations [8] for a homogenous electron gas with different electron densities.

LDA is based on the homogenous electron gas, in which the electron density is the

same. In the more common case, the electron density is not constant. LDA works well for

systems, where the electron density does not change dramatically in space. It may not be

correct in systems with a strongly varying electron density. For solving this problem, other

approximations for Vxc have been introduced that consider charge density gradients. They

are usually called the Generalized Gradient Approximation (GGA). In GGA, the exchange-

15

correlation energy can be written as

V GGA
xc

[ρ] =

(cid:90)

drf (ρ(r),∇ρ(r)).

(2.29)

There are different functions f (ρ,∇ρ) for different systems. Currently, most commonly used

is the PBE [9] functional proposed by Perdew, Burke and Ernzerhof in 1996.

LDA and GGA can give the information of structure and energy, but none of them can

describe the long distance interaction, like van der Waals (VDW), very well.

In layered

systems, the interlayer interaction is important, which will be overestimated by LDA and

underestimated by GGA-PBE. Various corrections have been applied for layered systems,

like the DFT-D2 [10] and optB86b-vdW [11, 12] functionals.

2.5 Basis sets and pseudopotentials

In quantum mechanics, a basis set {χν(r)} is needed to solve the Kohn-Sham equations

Eqs. (2.25) and (2.26). The single quasi-electron wave function ψi(r) can be represented by

the basis functions:

(cid:88)

ν

ψi(r) =

cνiχν(r).

(2.30)

Here cνi are expansion coefficients.

Usually there are two methods used to construct the basis. One is using the linear

combination of atomic orbitals to build the molecular orbitals in quantum chemistry; this

method is so-called LCAO-MO approximation. Such basis of atomic orbitals is used in DFT

implemented in the SIESTA code [13]. The other basis, more proper for infinite systems, is

the plane wave basis which is used in DFT implemented in VASP code [14–17].

16

In DFT, there is also a approximation about the atomic potential, since most of the

properties of the system are determined by the valence electrons. The effect of the nucleus

and core electrons on the valence electrons are reproduced by a pseudopotential which re-

place the atomic all-electron potential. With pseudopotential, the wave function of valence

electrons have significantly fewer nodes, which result in the reducing of the computational

effort. In this thesis, norm-conserving pseudopotentials are used in the SIESTA code and

projector augmented wave (PAW) pseudopotentials [17, 18] are used in the VASP code.

17

Chapter 3

Continuum approach for

long-wavelength acoustic phonons in

quasi-2D structures

The following discussion is my original contribution to the related publication by Dan Liu,

Arthur G. Every, and David Tom´anek, Phys. Rev. B 94, 165432 (2016) [19].

3.1

Introduction

Since graphene has become one of the hottest research topic in recent years, interest in quasi-

2D materials has been rising steadily. Phonon spectra are one of the important characteris-

tics. State-of-the-art atomistic calculations of phonon frequencies based on ab initio Density

Functional Theory (DFT) start with the calculation of the Hessian (or force-constant) matrix

using energy differences associated with finite displacements of all atoms in the unit cell. The

eigenvalues of the dynamical matrix constructed from the Hessian matrix. This gives phonon

frequencies. This approach works very well for all phonons except for long-wavelength acous-

tic modes in quasi-2D systems. There, all practical implementations require the use of very

large supercells, dense k-meshes, and a highly converged basis to obtain a sufficiently ac-

18

Figure 3.1: (a) Schematic representation of possible distortions of an elastic membrane. (b)
Schematic dependence of the longitudinal acoustic (LA), in-plane transverse acoustic (TA)
and the flexural acoustic (ZA) mode vibration frequencies on the crystal momentum near
the center of the Brillouin zone.

curate dynamical matrix. Even small inaccuracies caused by computer resource limitations

commonly lead to imaginary frequencies in long-wavelength flexural ZA modes [20,21]. This

artifact does not mean the structure is instable, but relates to the way the dynamical ma-

trix is constructed and diagonalized. Even though this shortcoming does not affect other

modes much, its unphysical consequences will affect the subsequent calculation. So far, no

practicable, predictive and accurate alternative approach has been proposed to determine

the frequency of long-wavelength ZA modes, which – among others – play an important role

in thermal conductivity of graphene nanoribbons [22].

In order to address the long-standing problem with the computation of long-wavelength

flexural ZA modes in atomically thin layers, I propose a quantitatively predictive and phys-
ically intuitive approach based on continuum elasticity theory. I use a (3×3) elastic matrix

as well as the flexural rigidity D to describe elastic behavior of a layer, independent of its

19

Figure 1ωkLA (stretch)TA (shear)ZA (bend)(b)Γstretchbend(a)shearthickness, by a membrane. In the thesis, I present simple quantitative expressions for fre-

quencies of long-wavelength acoustic ZA and also for the longitudinal acoustic (LA) and

transverse acoustic (TA) modes, which I determine using 2D elastic constants calculated by

ab initio Density Functional Theory. The calculated spectra accurately reproduce observed

and calculated long-wavelength phonon spectra of graphene and monolayer of black phos-

phorus. This approach also describes the observed dependence of the radial breathing mode

frequency on the diameter of carbon fullerenes and nanotubes very well.

Using continuum elasticity theory to determine the frequencies of long-wavelength acous-

tic phonons is well established [23], including its extension to plates of finite thickness [24,25].

Elastic constants have also been related to specific phonon modes in graphene [26–29]. Inde-

pendent of thickness, any plate can be mapped onto a 2D elastic membrane. Here, I introduce
a (3×3) elastic stiffness matrix for 2D membranes which relate to the commonly used (6×6)

elastic matrix for 3D systems. I then derive simple expressions for long-wavelength acoustic

phonon frequencies in these structures based on the 2D elastic constants and flexural rigidity

D, which determined using static DFT calculations for the deformation energy of unit cells

with few atoms. The quadratic frequency dependence on the crystal momentum for ZA and

the linear dependence for LA and TA modes is quantitatively reproduced near the Brillouin

zone center using these 2D elastic constants and D. Clearly, this approach is limited to long-

wavelength acoustic modes. For the full phonon spectrum, these results can be combined

with those of atomistic DFT calculations, which do not have the convergence problem for

optical and short-wavelength acoustic modes.

20

3.2 Continuum approach for long-wavelength acoustic

modes of an elastic membrane

Independent of the thickness, a free-standing thin slab of any substance can be mapped

onto an elastic membrane that resists deformations, as illustrated in Fig. 3.1(a). Within
the harmonic region, a (3×3) 2D elastic stiffness matrix describe the elastic response of this
two-dimensional system,which lie in the x − y plane, the matrix is given in Voigt notation

by



c11

c12

c12

c22

0

0

0

0

c66

 .

(3.1)

c11 and c22 describe the longitudinal strain-stress relationship along the x− and y−direction,

respectively. c66 describes the elastic response to in-plane shear. In an elastic isotropic plate,
c11 = c22, c66 = (c11 − c12)/2, and the Poisson ratio α = c12/c11. In a 2D plate represented

by a membrane, the dimension of the elastic stiffness constants cij is (N/m).

The individual coefficients cij are obtained by subjecting the equilibrium 2D structure

to particular distortions in the harmonic regime, typically σ < 1%. For c11 as an example,

a unit square of the system would be stretched and compressed along the same direction.

The energies of the compressed and the stretched structure should be equal in the harmonic

regime and higher than that of the equilibrium structure. Three points of E(σ) define a

parabola and its curvature is the particular elastic constant, which is c11 in our example.

The elastic response of the corresponding 3D system consisting of weakly interacting
layers separated by interlayer spacing dil is described in Voigt notation by a (6×6) Cij

21

matrix of elastic stiffness constants with the dimension (N/m2). The relation between the
two elastic matrices is given by cij = dil·Cij.

The flexural response to out-of-plane stress is described by the flexural rigidity D of

the plate, which may be anisotropic. D can be calculated by considering the energy cost

associated with rolling up a rectangle of length L to a tube with diameter d. Assuming that

L and d are large enough to ignore edge effects, I obtain

D =

1
2


bd2 ,

(3.2)

where 
b is the bending strain energy divided by the surface area of the tube, which is close

to the area of the initial rectangle.

3.3 Calculation of acoustic phonon modes of an elastic

membrane in the continuum limit

Near the of the Brillouin zone center, it is well established that the frequency of LA and

TA show a linear dependence on the crystal momentum k, whereas the ZA mode frequency

increases as k2, as seen in Fig. 3.1(b). As shown in the Related Information section, the

frequencies of the three acoustic modes of an elastic membrane can be determined quanti-

tatively using only the elastic constants c11, c22, c66 and D. These elastic constants can be

either calculated or obtained experimentally. Also needed is the 2D mass density ρ2D, which

is easily determined using the atomic mass numbers and the area of the optimized unit cell.

22

For acoustic modes with linear dispersion, I get

(cid:114) c11

ρ2D

(cid:114) c22

ρ2D

(cid:114) c66

ρ2D

ωLA,1 =

k

(3.3)

for the longitudinal acoustic (LA) mode along the x−direction, with the value of the square
root giving the longitudinal speed of sound in the x−direction. Similarly, I get

ωLA,2 =

k

(3.4)

for the LA mode along the y−direction, with the value of the square root giving the longi-
tudinal speed of sound in the y−direction. Finally, I get

ωT A =

k

(3.5)

for the in-plane transverse acoustic (TA) modes in the x− and y− directions, with the value

of the square root giving the transverse speed of sound in those directions. In anisotropic

plates, acoustic modes in an arbitrary in-plane direction have mixed LA and TA character,

and their speed varies in a complicated way with the direction [24].

As shown in the Related Information section, the flexural ZA mode displays an unusual

quadratic frequency dependence on the crystal momentum. Its frequency is given by

(cid:115)

ωZA =

D
ρ2D

k2 ,

(3.6)

where the value of the flexural rigidity D depends on the bending direction in an anisotropic

material.

23

The schematic dependence of the acoustic mode frequencies ω of a zero-thickness plate

on the crystal momentum k, given by Eqs. (3.3)-(3.6), is shown in Fig. 3.1(b). I note that

these expressions, albeit not in the notation of the 2D elastic matrix, have been obtained

previously [29–31].

Expressions describing the deformation of an elastic membrane can also be used the

determine the frequency of the radial breathing mode (RBM). Previously derived theoretical

expressions [32] and experimental observations are being used commonly as an indirect way

to determine the diameter of carbon fullerenes and nanotubes [4].

As shown in the Related Information section, I obtain

(cid:115)

ωCn =

2
d

2c11
ρ2D

,

(3.7)

for the RBM frequency of a spherical Cn fullerene with diameter d. Similarly, the RBM

frequency of a carbon nanotube (CNT) of radius d is given by

(cid:114) c11

ρ2D

ωCN T =

2
d

.

(3.8)

3.4 Computational approach to determine the elastic

constants

I determine the elastic response of atomically thin graphene and phosphorene monolay-

ers using ab initio density functional theory (DFT) as implemented in the SIESTA [13]

code. I used the Perdew-Burke-Ernzerhof (PBE) [9] exchange-correlation functional, norm-

conserving Troullier-Martins pseudopotentials [33], and a double-ζ basis including polariza-

24

tion orbitals. To determine the energy cost associated with in-plane distortions, I sampled the
Brillouin zone of a 3D superlattice of non-interacting layers by a 20×20×1 k-point grid [34].

To determine the strain energy associated with flexural motion, I constructed and optimized
single-wall nanotubes and sampled their 3D superlattice by a 20×1×1 k-point grid. I used a

mesh cutoff energy of 180 Ry and an energy shift of 10 meV in the self-consistent total energy
calculations, which has provided us with a precision in the total energy of ≤2 meV/atom.

3.5 Results

3.5.1 Graphene

Graphene is known to display isotropic elastic behavior. The DFT calculations yield the
(3×3) 2D elastic stiffness matrix of Eq. (3.1)

 N/m .



 =



c11

c12

c12

c22

0

0

0

0

c66

352.6

59.6

59.6

352.6

0

0

0

0

146.5

(3.9)

These values are in very good agreement with experimental and other theoretical results [35],

including the value of the Poisson ratio α = c12/c11 = 0.17, which is very close to the

observed value of 0.19 based on HREELS [36].

The calculated 2D mass density of graphene is ρ2D = 0.743·10−6 kg/m2 and the calcu-
lated value of the flexural rigidity D = 1.40 eV= 0.224 GPa·nm3 lies close to the previously
estimated value [29] D≈1.0 eV. Using the numerical values listed in Eq. (3.9) and the above

value of D, I can determine the three acoustic branches of graphene near the Γ point using

25

Eqs. (3.3)-(3.6). The results are presented in Fig. 3.2(a), superposed to those of a more re-

cent ab initio calculation [1] that agrees very well with with the observed and fitted phonon

spectrum of graphene [37].

First of all, I notice an excellent agreement with the linear LA and TA modes, which

indicate that the calculated speed of sound agrees with the observation. Specifically, the cal-

culated speed of sound with in-plane longitudinal polarization, vLA,th = 22.1 km/s, agrees

very well with the observed value vLA,expt = 22.0 km/s obtained using HREELS [36]. Sim-

ilarly, the speed of sound with in-plane transverse polarization, vT A,th = 14.2 km/s, agrees

very well with the observed value [36] of vT A,expt = 14.0 km/s.

Agreement between calculated and observed ZA modes indicates that the calculated

flexural rigidity value correctly reproduces the elastic response of graphene to bending.

3.5.2 Phosphorene

Unlike graphene, phosphorene is strongly anisotropic. It is much softer under compression
along the x− (or a1−) direction than along the y− (or a2−) direction. The optimized

rectangular unit cell is defined by a1 = 4.63 ˚A and a2 = 3.35 ˚A according to the DFT studies.
With 4 atoms per unit cell, the 2D mass density of phosphorene is ρ2D = 1.34·10−6 kg/m2.

The numerical values c11 = 24.4 N/m and c22 = 94.6 N/m reflect the strong anisotropy in the

in-plane longitudinal elastic response. The calculated speed of sound with LA polarization

is vLAX ,th = 4.3 km/s along the soft Γ − X direction and vLAY ,th = 8.4 km/s along the
stiff Γ − Y direction.

The transverse acoustic phonon frequency depends on the in-plane shear and is described

by the calculated value c66 = 22.1 N/m. The corresponding speed of sound with TA polar-

ization is vT A,th = 4.1 km/s.

26

Figure 3.2: Phonon spectra of (a) graphene, reproduced from Reference [1] and (b) phos-
phorene, a monolayer of black phosphorus, reproduced from Reference [2], shown by solid
lines. Superposed to the spectra are continuum results for the three acoustic phonon modes
in different high-symmetry directions, evaluated near Γ, with the longitudinal acoustic (LA,
dashed lines), in-plane transverse acoustic (TA, dotted lines), and the flexural acoustic (ZA,
solid lines) modes. Ball-and-stick models of the structure, including the primitive unit cells,
are shown in the top panels. The Brillouin zones are shown as insets in the phonon spectra.

Finally, I find also the flexural rigidity to be highly anisotropic.

I find D(Γ − X) =
1.55 eV= 0.248 GPa·nm3 when bending phosphorene along the x− (or a1−) direction,
yielding a tube with its axis aligned along the y− (or a2−) direction. Bending along the y−
(or a2−) direction, I find D(Γ − Y ) = 7.36 eV= 1.179 GPa·nm3.

These data are sufficient to reproduce the three acoustic branches of phosphorene along
the Γ− X and Γ− Y direction near Γ using Eqs. (3.3)-(3.6) and are presented in Fig. 3.2(b).

In analogy to graphene, the results are superposed to phonon spectra , which – in absence of

experimental phonon spectra – are based on atomistic DFT calculations of phosphorene [2].
I notice a particularly good agreement between the two approaches along the stiff Γ − Y

direction. As I expand in the Discussion, the continuum results deviate from those of the

27

Figure 2(a)(b)Phosphorenezxya1a2160012008004000ω(cm-1)ΓK MΓM          K                      ΓLALATAZAZATAω(cm-1)ΓY          S     X          ΓZAZALALA6005004003002001000TATAY     S ΓX Grapheneyzxa2a1Figure 3.3: Radial breathing mode (RBM) in (a) fullerenes and (b) carbon nanotubes as a
function of their diameter d. The prediction is shown by the solid line. Experimental and
theoretical RBM frequencies for the only observed fullerene with spherical symmetry, C60,
are shown by the data points in (a). The experimentally well-established relationship [3, 4]
ω = 248 cm−1×(1 nm/d) in nanotubes in (b) is represented by the dashed line and the
calculation is shown by the solid line.

atomistic study along the soft Γ − X direction, which – at close inspection – predicts small

imaginary ZA frequencies near Γ.

3.5.3 Vibration spectra of carbon fullerenes and nanotubes

The radial breathing mode (RBM) frequency ωRBM of Cn fullerenes may be calculated

using Eq. (3.7) and that of carbon nanotubes using Eq. (3.8), in combination with the elastic

constants provided in the section on graphene. The expected dependence of ωRBM on the

diameter d is displayed in Fig. 3.3(a) for fullerenes and in Fig. 3.3(b) for carbon nanotubes

(CNTs). The RBM mode is Raman active and its frequency is known to depend primarily on

the diameter. Thus, the RBM frequency is commonly used to judge the diameter of carbon

nanostructures.

According to Eq. (3.7), the RBM frequency of spherical fullerenes should scale inversely

with their diameter. But only one spherical fullerene, namely C60 with d = 7.1 ˚A, forms a

28

(a)(b)ω(cm-1)d(nm)100080040020006000             0.5           1.0           1.5            2.0Figure 3ω(cm-1)d(nm)10008006004002000C600             0.5           1.0            1.5           2.0molecular solid. I find the predicted value ωRBM,th(C60) = 467 cm−1 to lie very close to
the observed RBM frequency ωRBM,expt(C60) = 497 cm−1.

The well-documented observed diameter dependence of the RBM in nanotubes [4], ωRBM,expt(CNT) =

248 cm−1×(1 nm/d), is reproduced by the dashed line in Fig. 3.3(b). Based on continuum
theory and Eq. (3.8), I find ωRBM,th(CNT) = 234 cm−1×(1 nm/d) in very good agreement

with the observed behavior.

3.6 Discussion

So far, the most common description of a layer by continuum elasticity theory has been that
of a finite-thickness plate consisting of a material characterized by a (6×6) elastic stiffness

matrix. As I show here, this approach is unnecessarily cumbersome, since every 2D system of

finite thickness may be mapped onto an elastic membrane with the same mass density. The

resistance of the realistic system, such as graphene or phosphorene, to stretching, shear and

bending becomes that of the elastic membrane. The advantage of this approach, which does

not suffer from ambiguities about the “real” thickness of an atomic layer, has been discussed

before [26, 27]. The unconventional units of the 2D elastic stiffness matrix in Eq. (3.1) are

well adapted to ultra-thin layers.

The approach appears particularly suitable when describing the ZA mode in soft, atom-

ically thin atomic layers such as phosphorene. As mentioned earlier during the discussion

of Fig. 3.2(b), the required precision of the dynamical matrix has not been reached in the
calculation of phonon modes near Gamma along the Γ − X direction [2]. This is a common

shortcoming of ab initio phonon calculations in layered solids. Among the eigenvalues of

the dynamical matrix, which are proportional to ω2, the one associated with the ZA mode

29

often turns out to be negative, yielding nominally an imaginary frequency, as a numerical

artifact. The small error in the ZA frequency eigenvalue is also reflected in other close-lying

eigenvalues, such as those of the LA and TA modes, at the same crystal momentum. This is

clearly reflected in Fig. 3.2(b). I believe that the present continuum approach is much better

adapted to describe long-wavelength acoustic modes and should be preferred to atomistic

phonon calculations near Γ.

As seen in Fig. 3.2(b), the LA and TA mode frequencies are very similar along the
soft Γ − X direction in phosphorene. This is unusual, but not unexpected in view of the

accordion-like structure depicted in the ball-and-stick model in the top panel of Fig. 3.2(b).

Making an analogy to a real accordion, it appears equally easy to produce a longitudinal and

a transversal motion while the instrument is being played. Since the TA speed of sound is
the same along the Γ− X and the Γ− Y , the TA mode is clearly distinguished in its softness
from the LA mode along the hard Γ − Y direction. In this rigid direction in space, small

imprecisions in the dynamical matrix play a much less important role than along the soft

direction. Therefore, the continuum results agree well with the phonon frequencies obtained

using the atomistic approach for all three acoustic branches.

In the Section on the radial breathing motion of carbon nanostructures, I have shown in

Eq. (3.61) that the RBM frequency of nanotubes does not show a pure 1/d behavior as a

function of the nanotube diameter. For single-wall carbon nanotubes with typical diameters
between 1 − 2 nm, the value of the correction 4D/(c11d2) is indeed negligibly small in

comparison to 1. This need not be the case in nanotubes of other substances with large

values of D and small values of c11. In postulated phosphorene nanotubes [38], c11 differs

significantly from c22, and D is strongly anisotropic.

Finally, I have shown in Eq. (3.2) how to estimate the value of D in a layered material

30

by calculating the strain energy in nanotubes with a very large diameter d. Optimizing

wide nanotubes using ab initio techniques is non-trivial. The values for D quoted in the

Section on phosphorene required DFT calculations containing more than 14 unit cells along
the perimeter of phosphorene nanotubes bent along the soft x− (or a1−) direction. For
nanotubes bent along the rigid y− (or a2−) direction, D(d→∞ was extrapolated using 10,

12, 14, 16 and 18 unit cells along the perimeter of the corresponding phosphorene nanotubes.

3.7 Summary and Conclusions

In conclusion, as a viable alternative to atomistic calculations of long-wavelength acoustic

modes of atomically thin layers, which are known to converge very slowly, I have proposed

a quantitatively predictive and physically intuitive approach based on continuum elasticity

theory. I describe a layer, independent of its thickness, by a membrane, and characterize its
elastic behavior by a (3×3) elastic matrix as well as the flexural rigidity D. I have derived

simple quantitative expressions for frequencies of long-wavelength acoustic ZA and – for the

sake of completeness – also for the longitudinal acoustic (LA) and transverse acoustic (TA)

modes. These frequencies are determined using 2D elastic constants obtained from ab initio

DFT calculations for the deformation energy of unit cells with few atoms. I found that the

calculated spectra accurately reproduce observed and calculated long-wavelength phonon

spectra of graphene and phosphorene, the monolayer of black phosphorus. The approach

also correctly describes the observed dependence of the radial breathing mode frequency on

the diameter of carbon fullerenes such as C60 and carbon nanotubes.

31

3.8 Related Information

The most fundamental way to determine vibration motion in any system starts with a

Lagrangian, from which the Euler-Lagrange equations of motion can be derived by applying

Hamilton’s principle of least action.

In the following, I determine the Lagrangians that

describe stretching, shearing and bending of an infinitely thin elastic plate. Using these

Lagrange functions, I derive the equations of stretching, shearing and bending motion of

the plate. Finally, I derive the Lagrangian describing the radial breathing motion of carbon

fullerenes and nanotubes and determine the corresponding equations of motion for the RBM

in these nanostructures.

3.8.1 Lagrange function of an infinitely thin elastic plate under

strain

3.8.1.1 Stretching

Let us consider a thin plate suspended in the xy−plane and its response to in-plane tensile
strain applied uniformly along the x−direction,

dux
dx

= const.

(3.10)

The resulting energy density caused by tensile strain along the x−direction is then given by

U =

1
2

c11

dx

,

(3.11)

(cid:18) dux

(cid:19)2

32

where c11 describes the elastic stiffness to tensile strain.

In the harmonic regime, I will

consider only small strain values. Releasing the strain will cause a vibration in the x-

direction with the velocity vx = dux/dt. Then, the kinetic energy density will be given

by

T =

1
2

ρ2D

(cid:18) dux

(cid:19)2

dt

where ρ2D is the 2D mass density. The Lagrangian density is then given by

,

(3.12)

(cid:18) dux

dt

(cid:19)2 − c11

(cid:18)dux

dx

(cid:19)2(cid:35)

.

(3.13)

,

dux
dt

, x, t

= T − U =

1
2

ρ2D

In an anisotropic plate, the x− and y−directions are not equivalent. To describe the
y−response to in-plane tensile strain applied uniformly along the y−direction, I have to

replace x by y and c11 by c22 in Eqs. (A1)-(A4), yielding the Lagrangian density

,

duy
dt

, y, t

= T − U =

1
2

ρ2D

(cid:18) duy

dt

(cid:19)2 − c22

(cid:18) duy

dy

(cid:19)2(cid:35)

.

(3.14)

3.8.1.2 Shearing

The derivation of the Euler-Lagrange equation for the shear motion is very similar to that

for the stretching motion. The main difference is that the displacement u is normal to the

propagation direction. To obtain the corresponding equations, I need to replace ux by uy

and c11 by c66 in Eqs. (A1)-(A4). The Lagrangian density is then given by

(cid:18) duy

dx

L

,

duy
dt

, x, t

= T − U =

1
2

ρ2D

(cid:18) duy

dt

(cid:19)2 − c66

(cid:18)duy

dx

(cid:19)2(cid:35)

.

(3.15)

(cid:18) dux

dx

L

(cid:18) duy

dy

L

(cid:19)

(cid:19)

(cid:19)

(cid:34)

(cid:34)

(cid:34)

33

3.8.1.3 Bending

Bending a thin plate suspended in the xy-plane plane in order to to achieve a radius of

curvature R requires displacements uz(x) along the normal z-direction that are described by

d2uz
dx2 =

1
R

.

(3.16)

The resulting bending energy density is then given by

(cid:18) d2uz

(cid:19)2

dx2

U =

1
2

D

,

(3.17)

where D is the flexural rigidity. In the harmonic regime, I will consider only small strain
values corresponding to R→∞. Releasing the strain will cause a vibration in the z-direction

with the velocity vz = duz/dt. Then, the kinetic energy density will be given by

(cid:18) duz

(cid:19)2

dt

T =

1
2

ρ2D

,

(3.18)

where ρ2D is the 2D mass density. This leads to the Lagrangian density

(cid:18) d2uz

L

dx2 ,

duz
dt

, x, t

(cid:19)

(cid:34)

ρ2D

(cid:18) duz

dt

(cid:19)2 − D

(cid:18) d2uz

dx2

(cid:19)2(cid:35)

.

(3.19)

= T − U =

1
2

34

3.8.2 Derivation of Euler-Lagrange equations of motion for defor-

mations of an infinitely thin elastic plate using Hamilton’s

principle

3.8.2.1 Stretching

The Lagrangian specified in Eq. (3.13) has the form L(dux/dx, dux/dt, x, t). In this case, I

have two independent variables, x and t, and can define the action S by

(cid:90) t2

(cid:90) x2

S =

dt

t1

x1

dxL(

dux
dx

,

dux
dt

, x, t) .

Hamilton’s principle of least action yields

(cid:90) t2

(cid:90) x2

δS = δ

dt

t1

x1

dxL(

dux
dx

,

dux
dt

, x, t) = 0 .

(3.20)

(3.21)

This can be modified to

(cid:90) t2

(cid:90) x2

dt

t1

x1

δS =

and

dx[L(

dux
dx

+ δ

dux
dx

,

dux
dt

+ δ

dux
dt

, x, t) − L(

dux
dx

,

dux
dt

, x, t)] .

(3.22)

(cid:34)

(cid:90) t2

(cid:90) x2

dt

t1

x1

dx

δS =

∂L
∂ dux
dx

δ

dux
dx

+

∂L
∂ dux
dt

δ

dux
dt

(cid:35)

.

(3.23)

This expression can be reformulated to

(cid:90) t2

(cid:90) x2

t1

x1

(cid:33)

(cid:34)

(cid:32)

d
dt

∂L
∂ dux
dt

(cid:32)

(cid:33)(cid:35)

+

d
dx

∂L
∂ dux
dx

δux .

(3.24)

δS =

dt

dx

35

For δS to vanish, the quantity in the square brackets in Eq. (3.24) must vanish. This leads

to the Euler-Lagrange equation

d
dt

(cid:33)

(cid:32)

∂L
∂ dux
dt

+

d
dx

(cid:33)

(cid:32)

∂L
∂ dux
dx

= 0 .

(3.25)

Inserting the Lagrangian of Eq. (3.13) in the Euler-Lagrange Eq. (3.25) yields the wave

equation for longitudinal in-plane vibrations

ρ2D

d2ux
dt2 − c11

d2ux
dx2 = 0 .

This wave equation can be solved using the ansatz

to yield

ux = ux,0ei(kx−ωt)

ρ2Dω2 = c11k2 .

This finally translates to the desired form

ω =

(cid:114) c11

ρ2D

k ,

which is identical to Eq. (3.3).

(3.26)

(3.27)

(3.28)

(3.29)

In an anisotropic plate, I need to use the Lagrangian of Eq. (3.14) to describe motion

(cid:114) c22

ρ2D

36

along the y−direction and obtain

ω =

k ,

(3.30)

which is identical to Eq. (3.4).

3.8.2.2 Shearing

The Lagrangian L(duy/dx, duy/dt, x, t) in Eq. (3.15), which describes the shearing motion,

has the same form as the Lagrangian in Eq. (3.13). To obtain the equations for shear motion

from those for stretching motion, I need to replace ux by uy and c11 by c66 in Eqs. (3.20)-

(3.29). Thus, the equation for shear motion becomes

(cid:114) c66

ρ2D

k ,

(3.31)

ω =

which is identical to Eq. (3.5).

3.8.2.3 Bending

The Lagrangian specified in Eq. (3.19) has the unconventional form L(d2uz/dx2, duz/dt, x, t).

In this case, I have two independent variables, namely x and t, and can define the action S

by

S =

(cid:90) t2

(cid:90) x2

dt

t1

x1

dx L(

d2uz
dx2 ,

duz
dt

, x, t) .

(3.32)

Hamilton’s principle of least action yields

(cid:90) t2

(cid:90) x2

t1

x1

δS = δ

dt

dx L(

d2uz
dx2 ,

duz
dt

, x, t) = 0 .

(3.33)

This can be modified to

(cid:90) t2

(cid:90) x2

δS =

dt

t1

x1

dx[L(

d2uz
dx2 + δ

d2uz
dx2 ,

duz
dt

+ δ

duz
dt

, x, t) − L(

d2uz
dx2 ,

duz
dt

, x, t)] .

(3.34)

37

and

(cid:90) t2

(cid:90) x2

dt

t1

x1

dx

δS =

Finally, I can rewrite this expression as

(cid:90) t2

(cid:90) x2

t1

x1

 d

dt

δS =

dt

dx

 ∂L
(cid:33)
(cid:32)

∂ d2uz
dx2

∂L
∂ duz
dt

δ

duz
dt

 .
 δuz .

δ

d2uz
dx2 +

∂L
∂ duz
dt

 ∂L

∂ d2uz
dx2

− d2
dx2

(3.35)

(3.36)

For δS to vanish, the quantity in the square brackets in Eq. (3.36) must vanish. This leads

to the Euler-Lagrange equation

(cid:32)

(cid:33)

∂L
∂ duz
dt

− d2
dx2

 = 0 .

 ∂L

∂ d2uz
dx2

(3.37)

d
dt

Inserting the Lagrangian of Eq. (3.19) for flexural motion in the Euler-Lagrange Eq. (3.37)

yields the wave equation for flexural vibrations

ρ2D

d2uz
dt2 + D

d4uz
dx4 = 0 .

This wave equation can be solved using the ansatz

to yield

uz = uz,0ei(kx−ωt)

ρ2Dω2 = Dk4 .

38

(3.38)

(3.39)

(3.40)

This finally translates to the desired form

ω =

(cid:115)

D
ρ2D

k2 ,

which is identical to Eq. (3.6).

(3.41)

3.8.3 Radial breathing mode of spherical fullerenes

Let us consider a spherical fullerene molecule with the equilibrium radius R, such as C20,

C60, ... Except for the presence of 12 pentagons, the surface is covered by hexagonal carbon

rings, so that the 2D mass density ρ2D can be taken as that of graphene. Similarly, R

can be estimated using the number of C atoms in the fullerene and the unit cell area in

graphene. According to continuum elasticity theory, the total bending strain energy of any

such spherical fullerene, independent of R, is given by [39]

Ub = 4πD(1 + α) ,

(3.42)

where D is the flexural rigidity and α is the Poisson ratio of graphene.

In other words,

changing the radius of the fullerene by δR will not affect the total bending energy.

Allowing the equilibrium radius R of the fullerene to change by δR results, on the other

hand, in a quadratic increase of the tensile strain energy

(cid:104)

Ut = 2×1
2

4πR2c11

(cid:105)(cid:18) δR

(cid:19)2

.

(3.43)

R

The (radial) kinetic energy of a radially expanding or contracting fullerene, shown schemat-

39

ically in the inset of Fig. 3.3(a), is given by

T =

1
2

and the Lagrangian by

(cid:104)

4πR2ρ2D

(cid:105)(cid:18) d

dt

(cid:19)2

δR

L = T − U = T − Ut

(cid:104)

=

1
2

(cid:34)

4πR2ρ2D

δR

(cid:105)(cid:18) d
(cid:18) d

dt

(cid:19)2 −(cid:104)
(cid:19)2 − c11

(cid:19)2

(cid:105)(cid:18) δR
(cid:19)2(cid:35)

R

.

4πR2c11

(cid:18)δR

R

= 4πR2

1
2

ρ2D

δR

dt

Hamilton’s principle of least action

∂L
∂(δR)

− d
dt

(cid:32)

∂L
∂( d
dt δR)

(cid:33)

= 0

leads, with the Lagangian of Eq. (3.45), to the Euler-Lagrange equation

With the ansatz

I obtain

which is identical to Eq. (3.7).

−c11

2δR
R2 = ρ2D

d2
dt2 δR .

δR = δR0 eiωt

(cid:115)

2c11
ρ2D

,

ω =

1
R

40

(3.44)

(3.45)

(3.46)

(3.47)

(3.48)

(3.49)

3.8.4 Radial breathing mode of carbon nanotubes

Let us now consider a carbon nanotube that has been rolled up from a rectangular graphene

nanoribbon of width 2πR0, length L, and the 2D mass density ρ2D. According to continuum

elasticity theory, the total bending strain energy of any such nanotube is given by [39]

Ub =

πDL
R0

,

(3.50)

where D is the flexural rigidity of graphene. Increasing the radius by the small amount δR,

while still δR/R0 << 1, changes the potential energy of the nanotube with respect to the

value at R0 by

∆Ub(δR) =

πDL
R0

−δR
R0

+

R0

(cid:34)

(cid:18) δR

(cid:19)2(cid:35)

,

(3.51)

where higher than quadratic terms have been neglected in the Taylor expansion.

I consider the initial nanotube of radius R0 to be free of tensile strain energy. In this

case, the change in tensile strain energy associated with the radius increase by δR is

∆Ut(δR) =

1
2

[2πR0c11] L

.

(3.52)

(cid:18) δR

(cid:19)2

R0

The equilibrium value of δR can be obtained by minimizing ∆Ub + ∆Ut,

This leads to

∂(∆Ub + ∆Ut)

∂(δR)

= 0 .

δR≈ D

2c11R0

41

(3.53)

(3.54)

and the equilibrium value of the nanotube radius becomes R = R0 + δR

(cid:32)

(cid:33)

D

2c11R2
0

R = R0

1 +

.

(3.55)

R is the radius, around which the RBM vibrations take place. I use the total potential energy

U of the nanotube at the equilibrium radius R as the reference energy. Then, changing the

equilibrium nanotube radius R by an arbitrarily small value δR increases the potential energy

of the nanotube by

U =

1
2

[2πRL]

(cid:19)

(cid:18) D

R4 +

c11
R2

(δR)2 .

(3.56)

The (radial) kinetic energy of a radially expanding or contracting nanotube, shown schemat-

ically in the inset of Fig. 3.3(b), is given by

T =

1
2

[2πRρ2DL]

(cid:19)2

δR

.

(cid:18) d

dt

(3.57)

The Lagrangian for this motion is

L = T − U = [πRρ2DL]

(cid:19)2 − [πRL]

δR

(cid:18) d

dt

(cid:18) D

R4 +

c11
R2

(cid:19)

(δR)2 .

(3.58)

Using Hamilton’s principle of Eq. (3.46), I get the Euler-Lagrange equation

(cid:19)

(cid:18) D

R4 +

c11
R2

δR .

(3.59)

ρ2D

d2(δR)
dt2 = −

Using the ansatz

δR = δR0 eiωt

(3.60)

42

and referring to the nanotube diameter d = 2R, I obtain

(cid:115)

(cid:18)

(cid:19)

.

(3.61)

ω =

2
d

c11
ρ2D

1 +

D

c11R2

For a nanotube with a typical diameter d = 1 nm, using the values of c11 and D for graphene,
I find that the bending correction 4D/(c11d2)≈1×10−3 is negligibly small. In this case, I

obtain

ω =

2
d

which is identical to Eq. (3.8).

(cid:114) c11

ρ2D

,

(3.62)

43

Chapter 4

Long-wavelength deformations and

vibrational modes in empty and

liquid-filled microtubules and

nanotubes: A theoretical study

The following discussion is my original contribution to the related publication by Dan Liu,

Arthur G. Every, and David Tom´anek, Phys. Rev. B 95, 165432 (2017) [40].

4.1

Introduction

There are plenty of tubular structures with diameters ranging from nanometers to meters in

nature to fill various functions. Low-frequency flexural acoustic (ZA) modes dominate the

elastic response of most tubular structures. Much attention has been paid on the nanometer-

wide carbon nanotubes (CNTs) [41], which are extremely stiff [42], and on their flexural

modes [42–45]. Correct description of soft flexural modes in stiff quasi-1D systems like

nanotubes and nanowires is essential for calibrating nanoelectromechanical systems used for

ultrasensitive mass detection and radio-frequency signal processing [43, 46]. In CNTs and

44

in related graphene nanoribbons, flexural ZA modes have also been shown to significantly

influence the unsurpassed lattice thermal conductivity [22]. Much softer microtubules formed
of tubulin proteins, with a diameter d≈20 nm, are key factors of the cytoskeleton and help

to maintain the shape of cells in organisms. Although they are important, there are little

experimental data available describing the elastic behavior of microtubules. The conventional

approach to calculate the frequency spectrum is based on an atomistic calculation of the

force-constant matrix. This approach often fails for long-wavelength acoustic modes, in

particular the soft flexural ZA modes, due to an excessive demand on supercell size and basis

convergence. Typical the failure of atomistic calculation will results in numerical artifacts

such as imaginary vibration frequencies [47].

Like Chapter 3, in this chapter, I also offer an alternative way based on continuum

elasticity theory [23] and its extension to planar [19] and tubular structures [48–50], to

predict the frequency of acoustic modes in quasi-1D structures such as empty and liquid-

filled tubes consisting of stiff graphitic carbon or soft tubulin proteins. While the approach

works very well for long-wavelength acoustic modes, the accuracy of vibration frequencies

calculated using the simple expressions I derive surpasses that of conventional atomistic ab

initio calculations. The approach covers longitudinal and torsional modes, flexural modes,

as well as the radial breathing mode. The longitudinal and flexural acoustic modes of

tubules are simply related to those of an elastic beam resembling a nanowire. Since the

native environment of tubulin nanotubes contains water, I specifically consider the effect of

a liquid on the vibrational modes of tubular structures. The numerical results for tubulin

microtubules and carbon nanotubes agree with available experimental data.

45

Figure 4.1: Schematic representation of important deformations of a tubular structure. (a)
Longitudinal acoustic (LA, stretching), (b) torsional acoustic (TA, torsion), (c) flexural
acoustic (doubly degenerate ZA, bending), and (d) the radial breathing mode (RBM). (e)
Schematic dispersion relations of the corresponding long-wavelength phonon modes. The
tilde denotes liquid-filled tubules.

4.2 Continuum elasticity approach

A 1D tubular structure of radius R can be thought of as a rectangular 2D plate of width 2πR

that is rolled up to a cylinder. Consequently, the elastic response of 1D tubules to strain,

illustrated in Figs. 4.1(a)-4.1(d), is related to that of the constituting 2D plate. To describe

this relationship in the linear regime and calculate the frequency of long-wavelength vibra-

tional modes in 1D tubular structures, I take advantage of a continuum elasticity formalism

that has been successfully adapted to 2D structures [19].

As shown earlier [19], elastic in-plane deformations of a plate of indefinite thickness may
be described by the (3×3) 2D elastic stiffness matrix, which is given in Voigt notation by

(4.1)

 .



c11

c12

c12

c22

0

0

0

0

c66

46

Figure 1(a)(b)(c)(d)(e)ωkRBMRRRLATAZA(2)ZA(2)~RResistance of a plate to bending is determined by the flexural rigidity D. For a plate
suspended in the x− y plane, c11 and c22 describe the longitudinal strain-stress relationship
along the x− and y−direction, respectively. c66 describes the elastic response to in-plane
shear. For an isotropic plate, which I consider here, c11 = c22, c66 = (c11 − c12)/2, and the

in-plane Poisson ratio α = c12/c11. Considering a 3D plate of finite thickness h, characterized
by the (6×6) elastic stiffness matrix Cij, the coefficients of the 2D elastic stiffness matrix
cij for the equivalent plate of indefinite thickness are related by cij = h·Cij(1 − α2⊥). This

expression takes account of the fact that stretching a finite-thickness slab of isotropic material

not only reduces its width by the in-plane Poisson ratio α, but also its thickness by the out-

of-plane Poisson ratio α⊥. For the weak inter-layer coupling layered compound such as
graphite, this consideration is not needed, so that cij = h·Cij. In near-isotropic materials
like tubulin, however, α⊥≈α and cij = h·Cij(1 − α2).

4.2.1 Vibrational Modes of empty Nanotubes

Now considering an infinitely thin 2D plate of finite width 2πR and an areal mass density
ρ2D rolled up to a nanotube of radius R that is aligned with the x−axis. The linear mass

density of the nanotube is related to that of the plate by

ρ1D = 2πRρ2D .

(4.2)

In the long-wavelength limit, represented by k = (2π/λ)→0, the longitudinal acoustic mode

of a tubular structure, depicted in Fig. 4.1(a), resembles the stretching mode of a 2D

plate [19]. As mentioned above, the equivalent plate I consider here is a strip of finite

width that is reduced during stretching due to the nonzero in-plane Poisson ratio α.

47

In the following, I illustrate the computational approach for a tubular structure by fo-

cussing on its longitudinal acoustic mode. The derivation, which is described in more detail

in Related Information sections 4.8.1 and 4.8.2, starts with the Lagrange function density

L

(cid:18) dux
(cid:34)
(cid:34)

1
2

dx

=

1
2

,

dux
dt

ρ2D

ρ1D

dt

, x, t

= T − U

(cid:19)
(cid:19)2 − c11
(cid:18) dux
1 − α2(cid:17)(cid:18) dux
(cid:16)
(cid:19)2(cid:35)
(cid:18) dux
(cid:19)2 − cLA
(cid:18) dux
1 − α2(cid:17)
(cid:16)

2πR

dx

dx

dt

,

cLA = c11

(cid:19)2(cid:35)

2πR

(4.3)

(4.4)

=

where

is the longitudinal force constant of a 1D nanowire equivalent to the tubule, and the relation

between ρ1D and ρ2D is defined in Eq. (4.2). The resulting Euler-Lagrange equation is

(cid:32)

(cid:33)

(cid:33)

(cid:32)

∂L
∂ dux
dt

+

d
dx

∂L
∂ dux
dx

= 0 .

(4.5)

d
dt

Using the ansatz

ux = ux,0ei(kx−ωt)

(4.6)

I obtain the vibration frequency of the longitudinal acoustic (LA) mode of the nanotube or

nanowire from

ωLA =

(cid:115)

c11(1 − α2)

ρ2D

k =

(cid:114) cLA

ρ1D

k .

(4.7)

The prefactor of the crystal momentum k is the longitudinal speed of sound. As already

noted in Ref. [ [51]], the frequency of the LA mode is independent of the nanotube radius.

48

The torsional mode, depicted in Fig. 4.1(b), is very similar to the shear mode of a plate.

Consequently, as shown in Related Information section 4.8.2, the vibration frequency of the

torsional acoustic (TA) mode of the nanotube and the transverse acoustic mode of the plate

should be the same. With c66 describing the resistance of the equivalent plate to shear, I

obtain

ωT A =

(cid:114) c66

ρ2D

k .

(4.8)

Again, prefactor of the crystal momentum k is the corresponding speed of sound. Similar to

the LA mode, the frequency of the TA mode is independent of the nanotube radius [51].

The doubly degenerate flexural acoustic (ZA) mode, depicted in Fig. 4.1(c), differs signif-

icantly from the corresponding bending mode of a plate [19] that involves the plate’s flexural

rigidity D. The continuum elasticity treatment of the bending deformation, described in

Related Information sections 4.8.1 and 4.8.2, leads to

(cid:115)

(cid:18)

(cid:19)

(cid:115)

ωZA =

πR3c11

ρ1D

1 +

D

c11R2

k2 =

Db
ρ1D

k2 = cZA(R) k2 .

(4.9)

Here, cZA is the effective bending force constant and Db is the effective beam rigidity of a

corresponding nanowire, defined in Eq. (4.36).

Finally, the radial breathing mode (RBM) of the nanotube, depicted Fig. 4.1(d), has a

nearly k−independent frequency given by [19]

ωRBM =

1
R

(cid:114) c11

ρ2D

.

(4.10)

The four vibration modes described above and their functional dependence on the momen-

tum k and radius R have been partially described before using an elastic cylindrical shell

49

model [49, 50]. The schematic dependence of the vibration frequencies of these modes on k

is shown in Fig. 4.1(e). The main expressions for the vibration frequencies of both 2D and

tubular 1D structures are summarized in Table 4.1.

4.2.2 Vibrational Modes of Liquid-Filled Nanotubes

Next, I consider the nanotubes completely filled with a compressible, but viscosity-free liquid

that can slide without resistance along the nanotube wall [52]. Since the nanotubes remain

straight and essentially maintain their radius during stretching and torsion, the frequency ˜ω

of the LA and TA modes is not affected by the liquid inside, which remains immobile during

the vibrations. I thus obtain

and

˜ωLA(k)≈ωLA(k)

˜ωT A(k) = ωT A(k) ,

where the tilde refers to filling by a liquid.

(4.11)

(4.12)

The only effect of filling by a liquid on the flexural modes is an increase in the linear

mass density to

˜ρ1D = ρ1D + πR2ρl ,

(4.13)

where ρl denotes the gravimetric density of the liquid. In comparison to an empty tube,

described by Eq. (4.9), I observe a softening of the flexural vibration frequency to

(cid:18)

1 +

50

(cid:115)

(cid:19)

(cid:115)

˜ωZA =

πR3c11

˜ρ1D

D

c11R2

k2 =

= ˜cZA(R) k2 .

Db
˜ρ1D

k2

(4.14)

Finally, as I expand in Related Information section 4.8.3, the effect of the contained liquid

on the RBM frequency will depend on the stiffness of the tubular container. For stiff carbon

nanotubes, the RBM mode is nearly unaffected, whereas the presence of an incompressible

liquid increases ˜ωRBM in soft tubules. Thus,

˜ωRBM≤ωRBM .

(4.15)

The schematic dependence of the four vibration modes on the momentum k in liquid-filled

nanotubes is shown in Fig. 4.1(e). The main expressions for the vibration frequencies of

liquid-filled tubular 1D structures are summarized in Table 4.1.

4.3 Vibrational Modes of Nanotubes in a Surrounding

Liquid

From among the four long-wavelength vibrational modes of nanotubes illustrated in the left

panels of Fig. 4.1, the stretching and the torsional modes are not affected by the presence

of a liquid surrounding the nanotube.

I expect the radial breathing mode in Fig. 4.1(d)

to couple weakly and be softened by a small amount in the immersing liquid. The most

important effect of the surrounding liquid is expected to occur for the flexural mode shown

in Fig. 4.1(c).

The following arguments and expressions have been developed primarily to accommodate

soft biological structures such as tubulin-based microtubules, which require an aqueous envi-

ronment for their function. I will describe the surrounding liquid by its gravimetric density ρl

and viscosity η. As suggested above, I will focus the concern on the flexural long-wavelength

51

Mode

LA

TA

Equation

2D Plates1

k

Eq. (4.7)

ωLA =

Eq. (4.7)

Eq. (4.8)

ωT A =

k

k

ρ2D

(cid:113) c11
(cid:113) c66
(cid:113) D
(cid:113) D

ρ2D

ρ2D

ρ2D

ωZA =

ωZA =

Table 4.1: Summary of expressions derived for the vibrational frequencies ω of 1D tubules
and 2D plates. ˜ω denotes the frequency of liquid-filled tubules. Equation numbers refer to
the present publication.

(cid:114)

1D Tubules
c11(1−α2)

ωLA =

k

k

ωLA =

ρ1D
˜ωLA = ωLA

ωT A =

ρ2D
˜ωT A = ωT A
πR3c11
ρ1D

ρ2D

(cid:113) cLA
(cid:113) c66
(cid:18)
(cid:18)
(cid:114)
(cid:18)
(cid:114)

Db
ρ1D

(cid:115)
(cid:115)
(cid:115)

(cid:19)
(cid:19)
(cid:19)

ωZA =

ZA

ωZA =

˜ωZA =

RBM

1 + D

c11R2

1 + D

c11R2

πR3c11
ρ1D

ωZA =

k2

πR3c11
˜ρ1D

1 + D

c11R2

Db
˜ρ1D

(cid:113) c11

k2

˜ωZA =
ωRBM = 1
ρ2D
R
˜ωRBM≥ωRBM

k2 Eq. (4.9)

k2 Eq. (4.9)

Eq. (4.9)

k2 Eq. (4.14)

Eq. (4.14)

Eq. (4.10)

Eq. (4.15)

vibrations of such structures.

I will show later on, the flexural modes of idealized, free-standing biological microtubules

are extremely soft.

In that case, the velocity of transverse vibrations will also be very

small and definitely lower than the speed of sound in the surrounding liquid. Under these

conditions, the motion of the rod-like tubular structure will only couple to the evanescent

sound waves in the surrounding liquid and there will be no radiation causing damping. The

main effect of the immersion in the liquid will be to increase the effective inertia of the rod.

I may assume that the linear mass density ρ1D of the tubule in vacuum may increase to

˜ρ1D = ρ1D + ∆ρ1D in the surrounding liquid. I can estimate ∆ρ1D = ∆Aρl, where ∆A

52

describes the increase in the effective cross-section area of the tubule due to the surrounding
liquid that is dragged along during vibrations. I expect ∆A(cid:46)πR2, where R is the radius of

the tubule. The softening of the flexural mode frequency ˜ωZA due to the increase in ρ1D is

described in Eq. (4.14).

Next I consider the effect of viscosity of the surrounding liquid on long-wavelength vibra-
tions of a tubular structure that will resemble a rigid rod for k→0. Since – due to Stoke’s

paradox – there is no closed expression for the drag force acting on a rod moving through

a viscous liquid, I will approximate the rod by a rigid chain of spheres of the same radius,

which are coupled to a rigid substrate by a spring. The motion for a rigid chain of spheres

is the same as of a single sphere, which is damped by the drag force F = 6πηRv according

to Stoke’s law, where v is the velocity.

The damped harmonic motion of a sphere of radius R and mass m is described by

m

d2u
dt2 = −mω2

0u − 6πηR

du
dt

.

With the ansatz u(t) = u0eiωt, I get

and thus

(cid:115)

ω = ±

0 −
ω2

(cid:18)3πηR

m

−mω2 = −mω2

0 − iω6πηR
(cid:19)2

+ i

3πηR

m

(4.16)

(4.17)

.

(4.18)

Assuming that the damping is small, I can estimate the energy loss described by the Q−factor

Q = ω0

m

3πηR

=

2
3

m
R

f0
η

,

(4.19)

53

where f0 = ω0/(2π) is the harmonic vibration frequency. In a rigid string of masses separated

by the distance 2R, the linear mass density is related to the individual masses by ρ1D =
m/(2R). Then, the estimated value of the Q−factor will be

Q =

4
3

ρ1D

f0
η

.

(4.20)

4.4 Computational Approach to Determine the Elastic

Response of Carbon Nanotubes

I use ab initio density functional theory (DFT) as implemented in the SIESTA [13] code to de-

termine the elastic response and elastic constants of an atomically thin graphene monolayer,

the constituent of CNTs. The exchange-correlation functional is Perdew-Burke-Ernzerhof

(PBE) [9] with norm-conserving Troullier-Martins pseudopotentials [33], and a double-ζ ba-

sis including polarization orbitals. To determine the energy cost associated with in-plane

distortions, I sampled the Brillouin zone of a 3D superlattice of non-interacting layers by a
20×20×1 k-point grid [34]. I used a mesh cutoff energy of 180 Ry and an energy shift of

10 meV in the self-consistent total energy calculations, which has provided us with a preci-
sion in the total energy of ≤2 meV/atom. The same static approach can be applied to other

layered materials that form tubular structures.

4.5 Results

I selected two extreme examples to show the usefulness of the approach for all tubular struc-

tures. One is nanometer-wide CNTs which have been characterized well as rigid structures

54

Figure 4.2: (a) Frequency of vibrational modes depicted in Fig. 4.1(a) in empty and water-
filled carbon nanotubes.
(b) Dependence of the flexural coefficient cZA(R), defined in
Eq. (4.9), on the radius R of empty and water-filled carbon nanotubes. The tilde denotes
liquid-filled nanotubes.

able to support themselves in vacuum. The other example is tubulin-based microtubules,

which are significantly wider and softer than carbon nanotubes. These biological structures

require an aqueous environment for their function.

4.5.1 Carbon Nanotubes

The elastic behavior of carbon nanotubes can be described using quantities previously ob-

tained using DFT calculations for graphene [19]. The calculated elements of the elastic

stiffness matrix (4.1) are c11 = c22 = 352.6 N/m, c12 = 59.6 N/m, and c66 = 146.5N/m,

all in very good agreement with experimental results [53]. The calculated in-plane Poisson

ratio α = c12/c11 = 0.17 is also close to the experimentally estimated value for graphene [53]
of αexpt = 0.19. The calculated flexural rigidity of a graphene plate is D = 0.22 GPa·nm3.
The calculated 2D mass density of graphene ρ2D = 0.743·10−6 kg/m2 translates to ρ1D =
0.743·10−6 kg/m2·2πR for nanotubes of radius R.

The phonon dispersion relations ω(k) depend primarily on the radius and not the specific

55

Figure 2cZA(cm-1nm2)R (nm)cZA(empty CNT)(a)(b)ZA(R=1nm)LATARBM (R=1nm)ω(cm-1)k (nm-1)ZA(R=1 nm)~cZA(water filled CNT)~Table 4.2: Elastic behavior of a 2D graphene monolayer. c11, c22, c66 are 2D elastic stiffness
constants defined in Eq. (4.1). α is the in-plane Poisson ratio, D is the flexural rigidity, and
ρ2D is the areal mass density. These values can be used directly to calculate long-wavelength
acoustic frequencies ω(k) using the expressions in Table 4.1 and the speed of sound vLA and
vT A. Present values are compared to published data.

R ωRBM =(cid:112)c11/ρ2D 116 cm−1nm

0.17

Quantity
c11 = c22
c66
α
D

vLA

vT A

352.6 N/m
146.5 N/m
0.22 GPa·nm3

Present result Literature values
342 N/m 2
144 N/m a
0.19 a
0.225 GPa·nm3 3
116 cm−1nm 4
108 cm−1nm 5
22 km/s a
≈21 km/s 6
14 km/s a,b

21.5 km/s

14.1 km/s

chiral index (n, m) of carbon nanotubes and are presented in Fig. 4.2(a) for the different

polarizations. The LA and TA mode frequencies are almost independent of the nanotube
radius for a given k. The corresponding group velocities at k→0 give the longitudinal speed

of sound of vLA = dωLA/dk = 21.5 km/s and the speed of sound with torsional polarization

of vT A = dωT A/dk = 14.1 km/s.

The flexural or bending ZA mode does depend on the nanotube radius through the

proportionality constant cZA(R), defined in Eq. (4.9), which is plotted as a function of R

in Fig. 4.2(b). The dispersion of the ZA mode in a CNT of radius R = 1 nm is shown

in Fig. 4.2(a). Also the RBM frequency depends on the nanotube radius according to

Eq. (4.10). I find the value(cid:112)c11/ρ2D = 116 cm−1nm of the prefactor of R−1 in Eq. (4.10)

to agree well with the published theoretical value [55] of 116 cm−1nm and with the value of
108 cm−1nm, obtained by fitting a set of observed Raman frequencies [56]. The calculated
value ωRBM = 116 cm−1 for CNTs with R = 1 nm is shown in Fig. 4.2(a).

Filling the CNT with a liquid of density ρl increases its linear density ρ1D according to

56

Figure 4.3: (a) Frequency of vibrational modes depicted in Fig. 4.1(a) in empty and water-
filled tubulin-based microtubules. (b) Dependence of the flexural coefficient cZA(R), defined
in Eq. (4.9), on the radius R of empty and water-filled tubulin-based microtubules. The
tilde denotes liquid-filled tubules.

Eq. (4.13). For a nanotube filled with water of density ρl = 1 g/cm3, the radius-dependent

quantity ˜cZA(R), defined in Eq. (4.14), is plotted as a function of R in Fig. 4.2(b). The

dispersion of the ˜ZA mode in a water-filled CNT of radius R = 1 nm is shown in Fig. 4.2(a).

Elastic constants calculated in this work, and results derived using the present continuum

elasticity approach are listed and compared to literature data in Table 4.2.

4.5.2 Tubulin-Based Microtubules

To describe phonon modes in tubulin-based microtubules, I used published experimental

data [57] for microtubules with an average radius R = 12.8 nm and a wall thickness h =

2.7 nm. The observed density of the tubule wall material ρ = 1.47 g/cm3 translates to ρ2D =
4.0·10−6 kg/m2. The estimated Young’s modulus of the wall material is E = 0.5 GPa and
the flexural beam rigidity of these microtubules with R = 12.8 nm is Db = 0.9·10−23 N·m2.

Using the relationship between Db and D, defined in Eq. (4.35), to map these values onto the
elastic 2D wall material and obtain c11 = E·h = 1.4 N/m and D = 2.71 GPa·nm3. In a rough

57

ω(cm-1)ZA (R=12.8nm)~LATARBM(R=12.8nm)k (nm-1)cZA(cm-1nm2)(a)(b)Figure 3cZA(empty MT)R (nm)h=2.7 nmcZA(water-filled MT)~RhZA(R=12.8nm)approximation, tubulin can be considered to be isotropic, with a Poisson ratio α = 0.25.
Further assuming that c11 = c22, I estimate c66 = (c11 − c12)/2 = c11(1 − α)/2 = 0.5 N/m.

There is one fundamental difference between tubulin-based microtubules and systems

such as CNTs is that the former necessitate an aqueous environment for their shape and func-

tion. So, the correct description of microtubule deformations and vibrations need to address

the complete microtubule-liquid system, which would exceed the scope of this study. I rather

resorted to the expressions derived in the Subsection on nanotubes in a surrounding liquid

to at least estimate their Q−factor in an aqueous environment. I used ρ1D = 7×10−13 kg/m
for a water-filled microtubule, η = 10−3 Pa·s and f0 = 109 Hz, which provided us with the
value Q≈1.2. In other words, the aqueous environment highly damped flexural vibrations in

microtubules, as a result, their frequency should be very hard to measure. Consequently, the

only available comparison between the calculations and experimental data should be made

for static measurements.

An appropriate way to experimentally determine the effective beam rigidity of individual

tubulin-based microtubules involves the measurement of buckling caused by applying an

axial buckling force using optical tweezers. Experimental results for the effective beam

rigidity have been obtained in this way for microtubules that are free of the paclitaxel

stabilizing agent and contain 14 tubulin protofilaments, which translates to the effective

radius R≈9.75 nm. The observed values range from Db = 3.7±0.8×10−24 Nm2 [58] to Db =
7.9±0.7×10−24 Nm2 [59], in good agreement with the calculated value Db = 4.2×10−24 Nm2.
The estimated value Db = 6.2×10−24 Nm2 for wider tubules with 16 protofilaments is 49%

larger than for the narrower tubules with 14 protofilaments. The corresponding increase by

49% has been confirmed in a corresponding experiment [59].

Next, I still use the oversimplifying assumption that tubulin microtubules which may

58

exist in the vacuum and could be described by the above-derived continuum values [50].

In this way, I compare the results to published theoretical results. The calculated phonon

dispersion relations ω(k) for most common microtubules with the radius R = 12.8 nm are

presented in Fig. 4.3(a). The LA and TA mode frequencies are independent of the tubule

radius. The longitudinal speed of sound vLA = dωLA/dk = 0.56 km/s and the speed of

sound with torsional polarization vT A = dωT A/dk = 0.36 km/s are got from the slope. In

order to compare, I extracted the vLA value based on the elastic cylindrical shells model with

E = 2.0 GPa from Ref. [50]. Extrapolating to the value E = 0.5 GPa in the set of parameters

using the relationship vLA∝√

E, I obtained vLA = 0.59 km/s, in excellent agreement with

the calculated value.

The flexural or bending ZA mode depends on the tubule radius through the proportion-

ality constant cZA(R), defined in Eq. (4.9), which is plotted as a function of R in Fig. 4.3(b).

The dispersion of the ZA mode in a microtubule of radius R = 12.8 nm is shown in Fig. 4.3(a).

Also the RBM frequency depends on the nanotube radius according to Eq. (4.10). For

R = 12.8 nm, I obtain ωRBM = 0.24 cm−1, as seen in Fig. 4.3(a).

For describing the increase in the linear density ρ1D of a microtubule filled with a liquid

of density ρl, I need to account for the finite thickness h of the wall and replace the radius R
by R − h/2 in Eq. (4.13). Considering water of density ρl = 1 g/cm3 as the filling medium,

I plot the radius-dependent quantity ˜cZA(R), defined in Eq. (4.14), as a function of R in

Fig. 4.3(b). The dispersion of the ˜ZA mode in a water-filled CNT of radius R = 12.8 nm is

shown in Fig. 4.3(a).

The results in Fig. 4.3(a) suggest soft vibration in the GHz range, in agreement with

other theoretical estimates [49, 50]. As mentioned above, all these vibrations will be hight
damped in an aqueous environment to the low Q−factor.

59

4.6 Discussion

Although the atomistic approach which is based on an atomistic calculation of the force-

constant matrix, is quite adequate to determine frequencies of the optical and of short-

wavelength acoustic modes. This approach for calculating the frequency spectrum, does

not provide accurate frequencies for long-wavelength soft acoustic modes in quasi-1D tubu-

lar structures, the excessive demand on supercell size and basis convergence often yields

imaginary vibration frequencies as an artifact of insufficient convergence.

An alternative to tedious atomistic calculations of the force-constant matrix of complex

tubular systems, I propose here a continuum elasticity approach to determine the frequency

of long-wavelength acoustic modes in tubular structures that does not require the thickness

of the wall as an input. The approach for quasi-1D structures is based on the successful

description of corresponding modes in 2D structures [19]. Using this approach, I obtain for

the first time quantitative results for systems ranging from stiff CNTs to much wider and

softer protein microtubules.

The elastic behavior of the wall material can be determined accurately by static calcula-

tions of 2D plate subjected to small deformation or by elastic measurements. This approach

works very well for long-wavelength vibrations and large-radius nanotubes, both of which

require extraordinary computational resources in atomistic calculations. In particular, the
flexural ZA modes with their ω∝k2 momentum dependence are known to be very hard to

reproduce by ab initio calculations near the Γ point [47].

For the sake of completeness, I have also derived the Euler-Lagrange equations of motion

required to describe all long-wavelength acoustic modes and present the detailed derivation

in the Related Information section.

60

Of course, the frequency of the ZA modes is expected to be much softer than that of the
LA mode in any nanotube or nanowire. Since ωZA∝k2 whereas ωLA∝k, expressions derived

here for the long-wavelength limit would lead to the unrealistic behavior ωZA > ωLA for
large values of k. This limits the k−range, for which the formalism is valid in the dispersion

relations presented in Figs. 4.2(a) and 4.3(a).

In a crystalline tubule, k is restricted to

typically an even smaller range given by the size of the 1D Brillouin zone.

The radial breathing mode (RBM) should be decoupled from the longitudinal acoustic or

stretching mode in systems with a vanishing Poisson ratio α. However, most systems have

a non-vanishing value of α as discussed in Related Information section 4.8.4. In that case,

these two modes mix and change their character beyond the wavevector k = 1/R, where
ωLA(k)≈ωRBM , as discussed previously [50]. For k with smaller values, coupling between
the LA mode and the RBM modifies the frequency of these modes by only ≈1% in CNTs.

The model allows a simple extension from empty to liquid-filled nanotubes. As shown

in Related Information section 4.8.4, I find that presence of a filling liquid does not affect

longitudinal acoustic and torsional acoustic modes to a significant extent, but softens the

flexural modes. I also expect the pressure wave of the liquid to couple to the RBM beyond
the wavevector k≈ωRBM /vp, where vp is the speed of the propagating pressure wave.

To demonstrate the validity of the approach, I also considered microtubules formed of

the proteins α- and β-tubulin. These are responsible for maintaining the shape and elasticity

of cells, but are too complex for an atomistic description to predict vibration spectra. The

necessity to include the aqueous environment in the description of tubulin-based microtubules

makes the problem becomes more complex for computation.

The basic finding that microtubule motion and vibrations are overdamped in the natural
aqueous environment, with a Q−factor of the order of unity, naturally explains the absence

61

of experimental data reporting observation of motion, dynamical shape changes and vibra-

tions in these protein-based systems. Among static measurements of the elastic behavior

of microtubules, optical tweezers appear to be the optimum way to handle and deform in-

dividual microtubules in order to determine their effective beam rigidity Db. In this static

scenario, I find the description of the beam rigidity precise enough to compare with experi-

mental data. The reported dependence of Db on the cube of the radius [59] is reflected in the

corresponding expression for Db in Eq. (4.35). In the case of tubulin-based microtubules, I

find the leading term in Db to be indeed proportional to R3 and to be much larger than the

second term, which is proportional to R.

4.7 Summary

For addressing the shortcoming of conventional atomistic calculations of long-wavelength

acoustic frequencies in tubular structures, I have developed an alternative computational

approach based on continuum elasticity theory to 2D and 1D structures. Since 1D tubular

structures can be viewed as 2D plates of finite width rolled up to a cylinder, I have taken

advantage of the correspondence between 1D and 2D structures to determine their elastic

response to strain. In the approach, computation of long-wavelength acoustic frequencies

does not require the determination and diagonalization of a large, momentum-dependent dy-

namical matrix. Instead, the simple expressions I have derived for the acoustic frequencies
ω(k) depends only four elements of a k−independent 2D elastic matrix, namely c11, c22, c12,

and c66, as well as the value of the flexural rigidity D of the 2D plate constituting the wall.

These five numerical values can easily be obtained using static calculations for a 2D plate.

The scope of the approach is limited to long-wavelength acoustic modes, but the accuracy

62

of the calculated vibration frequencies surpasses that of conventional atomistic ab initio cal-

culations. Starting with a Lagrange function describing longitudinal, torsional, flexural and

radial deformations of empty or liquid-filled tubular structures, I have derived correspond-

ing Euler-Lagrange equations to obtain simple expressions for the vibration frequencies of

the corresponding modes. I have furthermore shown that longitudinal and flexural acous-

tic modes of tubules are well described by those of an elastic beam resembling a nanowire.

Using the simple expressions, I were able to show that a pressure wave in the liquid con-

tained in a stiff carbon nanotube has little effect on its RBM frequency, whereas the effect

of a contained liquid on the RBM frequency in much softer tubulin tubules is significant. I

found that presence of water in the native environment of tubulin microtubules reduces the
Q−factor to such a degree that flexural vibrations can hardly be observed. I also showed

that the coupling between long-wavelength LA modes and the RBM can be neglected.

I

have found general agreement between the numerical results for biological microtubules and

carbon nanotubes and available experimental data.

4.8 Related Information

Material in the Related Information section provides detailed derivation of expressions used

in the main text and considers specific limiting cases. In Related Information section 4.8.1, I

derive the Lagrange function for stretching, torsional and bending modes of tubular structure.

In Related Information section 4.8.2, I derive analytical expressions for the frequencies of the

corresponding vibration modes using the Euler-Lagrange equations. The effect of a liquid

contained inside a CNT on its RBM frequency is investigated in Related Information section

4.8.3. The coupling between the longitudinal acoustic mode and the RBM in a CNT due to

63

its non-vanishing Poisson ratio is discussed in Related Information section 4.8.4.

4.8.1 Lagrange Function of a Strained Nanotube

4.8.1.1 Stretching

Let us consider a nanotube of radius R aligned with the x−axis and its response to tensile
strain dux/dx applied uniformly along the x−direction. The strain energy will be the same
as that of a 2D strip of width y = 2πR lying in the xy−plane that is subject to the same

condition.

Assuming that the width of the strip is constrained to be constant, the strain energy per

length is given by

Ux =

1
2

c11

(cid:18) dux

(cid:19)2

dx

2πR .

(4.21)

For a nonzero Poisson ratio α, stretching the strip by dux/dx will reduce its width by

duy/dy = αdux/dx and its radius R, as shown in Fig. 4.1(a). Releasing the constrained width
will release the energy Uy = −α2Ux. The total strain energy per length of a nanotubule or

an equivalent 1D nanowire is the sum U = Ux + Uy and is given by

(cid:16)

1 − α2(cid:17)(cid:18)dux

(cid:19)2

dx

U =

1
2

c11

(cid:18) dux

(cid:19)2

dx

2πR =

1
2

cLA

.

(4.22)

Here, cLA = 2πRc11(1−α2) is the the longitudinal force constant of a 1D nanowire equivalent

to the tubule, defined in Eq. (4.4).

In the harmonic regime, I will consider only small strain values. Releasing the strain will

cause a vibration in the x-direction with the velocity vx = dux/dt. Then, the kinetic energy

64

density of the strip will be given by

(cid:18) dux

dt

(cid:19)2

(cid:18) dux

(cid:19)2

dt

,

2πR =

1
2

ρ1D

(4.23)

T =

1
2

ρ2D

where ρ2D is the areal mass density of the equivalent strip that is related to ρ1D by Eq. (4.2).

The Lagrangian density is then given by

L

(cid:18) dux
(cid:34)
(cid:34)

1
2

dx

=

1
2

,

dux
dt

ρ2D

ρ1D

, x, t

= T − U

(cid:19)
1 − α2(cid:17)(cid:18) dux
(cid:18) dux
(cid:19)2 − c11
(cid:16)
(cid:19)2(cid:35)
(cid:19)2 − cLA
(cid:18) dux
(cid:18) dux

dx

dt

.

dx

dt

(cid:19)2(cid:35)

2πR ,

(4.24)

=

4.8.1.2 Torsion

The derivation of the Euler-Lagrange equation for the torsional motion is very similar to

that for the longitudinal motion. The main difference is that the displacement uφ is normal

to the propagation direction x. To obtain the corresponding equations, I need to replace ux
by uφ and c11(1 − α2) by c66 in Eqs. (A1)-(A4). The Lagrangian density is then given by

(cid:18) duφ

dx

L

(cid:19)

,

duφ
dt

, x, t

= T − U =

1
2

ρ2D

(cid:34)

(cid:18) duφ

dt

(cid:19)2 − c66

(cid:18) duφ

dx

(cid:19)2(cid:35)

2πR .

(4.25)

4.8.1.3 Bending

Bending a nanotube of radius R is equivalent to its transformation to a segment of a nan-

otorus of radius Rt. Initially, I will assume that c11 = 0 and D > 0 in the given nanotorus

segment, so the strain energy would contain only an out-of-plane component. I first con-

65

sider a straight nanotube of radius R formed by rolling up a plate of width 2πR. The

corresponding out-of-plane strain energy per nanotube segment length is

U =

1
2

D
R2 (2πR) =

πD
R

.

(4.26)

The corresponding expression for the total out-of-plane strain energy in a nanotorus is [60],

U = 2π2D

R(cid:112)(Rt + R)(Rt − R)

R2
t

.

(4.27)

Divided by the average perimeter length 2πRt, I obtain the out-of-plane energy of the torus

per nanotube segment length

R(cid:112)(Rt + R)(Rt − R)

Rt

.

U = πD

(4.28)

Assuming that the torus radius is much larger than the nanotube radius, Rt(cid:29)R, I can Taylor

expand U in Eq. (4.28) and neglect higher-order terms in (R/Rt), which leads to

.

(4.29)

(cid:32)

U =

πD
R

1 +

1
2

(cid:19)2(cid:33)

(cid:18) R

Rt

Comparing the out-of-plane strain energy of a bent nanotube in Eq. (4.29) and that of a

straight nanotube in Eq. (4.26), the change in out-of-plane strain energy per segment length

associated with bending turns out to be

U =

1
2

πDR

66

(cid:18) 1

(cid:19)2

Rt

.

(4.30)

During the flexural or bending vibration mode, the local curvature 1/Rt = d2uz/dx2 changes

along the tube, yielding the local in-plane strain energy per nanotube segment length of

(cid:18) d2uz

(cid:19)2

dx2

U =

1
2

πDR

.

(4.31)

Next, I consider the in-plane component of strain, obtained by assuming c11 > 0 and D = 0

in a given nanotorus segment. There is nonzero strain in a nanotube deformed to a very
wide torus with Rt(cid:29)R even if its cross-section and radius R were not to change in this

process. The reason is that the wall of the nanotube undergoes stretching along the outer

and compression along the inner torus perimeter in this process. This amounts to a total

in-plane strain energy [60]

U =

π2c11R3

Rt

(4.32)

for the entire torus with an average perimeter of 2πRt in relation to a straight nanotube of

length 2πRt. Thus, the in-plane strain energy within the torus per segment length is

.

(4.33)

Considering local changes in curvature 1/Rt = d2uz/dx2 during the bending vibrations of a

nanotube, the local in-plane strain energy per nanotube segment length becomes

(cid:18) 1

(cid:19)2

Rt

U =

1
2

πc11R3

(cid:18) d2uz

(cid:19)2

dx2

U =

1
2

πc11R3

67

.

(4.34)

The strain energy in the deformed nanotube per length is the sum of the in-plane strain

energy in Eq. (4.34) and the out-of-plane strain energy in Eq. (4.31), yielding

(cid:18)d2uz

(cid:19)2

dx2

=

1
2

Db

(cid:18)d2uz

(cid:19)2

dx2

,

(4.35)

U =

1
2

(πc11R3 + πDR)

where

Db = πc11R3 + πDR

(4.36)

is the effective beam rigidity of a corresponding nanowire. The kinetic energy of a bending

nanotube or nanowire segment is given by

T =

1
2

ρ1D

(cid:18) duz

(cid:19)2

dt

.

This leads to the Lagrangian density

L

(cid:18)d2uz
(cid:34)

dx2 ,
1
2

=

duz
dt

, x, t

(cid:18)duz

dt

ρ1D

(cid:19)
(cid:19)2− πc11R3

= T − U

(cid:18)

1 +

(cid:19)(cid:18)d2uz

(cid:19)2(cid:35)

D

c11R2

dx2

.

(4.37)

(4.38)

4.8.2 Derivation of Euler-Lagrange Equations of Motion for De-

formations of a Nanotube using Hamilton’s Principle

4.8.2.1 Stretching

The Euler-Lagrange equation for stretching a tube or a plate is [19]

(cid:33)

(cid:32)

d
dt

∂L
∂ dux
dt

(cid:33)

(cid:32)

∂L
∂ dux
dx

+

d
dx

68

= 0 .

(4.39)

Inserting the Lagrangian of Eq. (4.24) in the Euler-Lagrange Eq. (4.39) yields the wave

equation for longitudinal vibrations of the tubule or the equivalent nanowire

2πRρ2D

d2ux
dt2 − 2πRc11

dx2 = ρ1D

d2ux
dt2 − cLA

d2ux
dx2 = 0 .

(4.40)

The nanotube radius R drops out and I obtain

(cid:16)

1 − α2(cid:17) d2ux

(cid:16)

1 − α2(cid:17) d2ux

ρ2D

d2ux
dt2 − c11

dx2 = ρ1D

d2ux
dt2 − cLA

d2ux
dx2 = 0 .

(4.41)

This wave equation can be solved using the ansatz

to yield

for a tubular structure or

ux = ux,0ei(kx−ωt)

(cid:16)

1 − α2(cid:17)

k2

ρ2Dω2 = c11

ρ1Dω2 = cLAk2

for an equivalent 1D nanowire. This finally translates to the desired form

(cid:115)

(cid:114) cLA

ρ1D

c11(1 − α2)

ρ2D

k .

ωLA =

which is identical to Eq. (4.7).

k =

69

(4.42)

(4.43)

(4.44)

(4.45)

4.8.2.2 Torsion

The Lagrangian L(duφ/dx, duφ/dt, x, t) in Eq. (4.25), which describes the torsion of a tubule,

has a similar form as the Lagrangian in Eq. (4.24). To obtain the equations for torsional
motion from those for stretching motion, I need to replace ux by uφ and c11(1 − α2) by c66

in Eqs. (4.39)-(4.45). Thus, the frequency of the torsional acoustic mode becomes

(cid:114) c66

ρ2D

ω =

k ,

(4.46)

which is identical to Eq. (4.8). The torsional frequency is the same as frequency of the shear

motion in the equivalent thin plate [19].

4.8.2.3 Bending

The Euler-Lagrange equation for bending a tube or a plate is given by [19]

(cid:32)

(cid:33)

d
dt

∂L
∂ duz
dt

− d2
dx2

 = 0 .

 ∂L

∂ d2uz
dx2

(4.47)

Inserting the Lagrangian of Eq. (4.38) for flexural motion in the Euler-Lagrange Eq. (4.47)

yields the wave equation for flexual vibrations

(cid:18)

(cid:19) d4uz

ρ1D

d2uz
dt2 + πc11R3

1 +

D

c11R2

dx4 = ρ1D

d2uz
dt2 + Db

d4uz
dx4 = 0 .

(4.48)

This wave equation can be solved using the ansatz

uz = uz,0ei(kx−ωt)

(4.49)

70

to yield

(cid:18)

1 +

(cid:19)

D

c11R2

ρ1Dω2 = πc11R3

k4 = Dbk4 .

(4.50)

This finally translates to the desired form

ω =

πc11R3

ρ1D

1 +

D

c11R2

(cid:115)

(cid:19)

k2 =

Db
ρ1D

k2 ,

(4.51)

which is identical to Eq. (4.9).

For a liquid-filled nanotube, I only need to replace ρ1D by ˜ρ1D in Eq. (4.51) to get

(cid:115)

(cid:115)

(cid:18)

(cid:18)

(cid:19)

(cid:115)

ω =

πc11R3

˜ρ1D

1 +

D

c11R2

k2 =

Db
˜ρ1D

k2 ,

(4.52)

which is identical to Eq. (4.14).

4.8.3 Coupling Between a Travelling Pressure Wave and the RBM

in a Liquid-Filled Carbon Nanotube

Next I consider a long-wavelength displacement wave ux = ux,0 exp[i(kx − ωt)] of small

frequency ω and wave vector k travelling down the liquid column filling a carbon nanotube.

I assume the liquid to be compressible but viscosity-free. Thus, the travelling displacement
wave will result in a pressure wave p = p0 exp[i(kx − ωt)] that causes radial displacements
r = r0 exp[i(kx− ωt)] in the CNT wall. These radial displacements couple the pressure wave

in the liquid to the RBM, but not the longitudinal and torsional modes of the CNT.

At small frequencies ω, there will be little radial variation in the pressure. The local

71

Figure 4.4: Nature and coupling of vibration modes ω(k) of a carbon nanotube that may
be filled with water. (a) Coupling between the dispersionless RBM of a CNT, shown by the
black line, and the pressure wave of water enclosed in the CNT, shown by the green dotted
line. (b) Coupling between the RBM of a CNT, shown by the black line, and the longitudinal
acoustic mode of the CNT, shown by the green dotted line. Results are presented for a CNT
with a radius of 1 nm.

compressive strain in the liquid will thus be

−δV
V

= −

(cid:18) ∂ux

∂x

(cid:19)

+

2r
R

and the pressure becomes

(cid:18) ∂ux

∂x

(cid:19)

.

+

2r
R

p = −B

δV
V

= −B

(4.53)

(4.54)

Here, B is the bulk modulus of the filling liquid, which I assume is water with B = 2.2·109 Pa.

The local acceleration of water is given by

∂2ux
∂t2 = − ∂p

∂x

ρl

(4.55)

72

(a)(b)ω(cm-1)k (nm-1)ω(cm-1)k (nm-1)Figure 4and the radial acceleration of the CNT is given by

ρ2D

∂2r
∂t2 = p − c11
R2 r .

Inserting harmonic solutions for p, ux and r into Eqs. (4.54)-(4.56), I obtain



1
Bik
−ik ρlω2

2B/R

1

0 −(c11/R2 − ρ2Dω2)

0

p0

ux,0



 = 0
k2B − ρlω2(cid:17) − ρlω2 2B

r0

= 0 .

R

with the characteristic equation

R2 − ρ2Dω2(cid:17)(cid:16)
(cid:16)c11

This can be rewritten as

(cid:16)

0 − ω2(cid:17)(cid:16)

ω2

k2v2 − ω2(cid:17) − ω2γ2 = 0 ,

(4.56)

(4.57)

(4.58)

(4.59)

0 = c11/(ρ2DR2) and γ2 = 2B/(ρ2DR). For a CNT of radius R = 1 nm, I obtain

where ω2
0 = 474 ps−2 and γ2 = 5.92 ps−2.
ω2

Solving Eq. (4.59) leads to the dispersion relation ω(k), which is presented in Fig. 4.4(a).

In the following, I focus on the lowest lying branch of the dispersion relation describing a

long wavelength, low frequency pressure wave travelling down the liquid column.

In this

73

case, I can neglect ω2 in the first factor of Eq. (4.59) and obtain

vk(cid:114)

1 + γ2
ω2
0

(cid:115)

ω =

.

(4.60)

Considering the filling liquid to be water with the speed of sound v =(cid:112)B/ρl≈1483 m/s,

the velocity of the propagating pressure wave inside the CNT becomes

ω
k

= v

1(cid:114)

1 + γ2
ω2
0

= 1474 m/s .

(4.61)

This value is only slightly reduced from that of bulk water because of the relative rigidity of

the CNT.

In the corresponding low wavenumber range, the frequency of the RBM is changed to

˜ωRBM = ωRBM

1 +

γ2

ω2

RBM

.

(4.62)

For a CNT with radius R = 1 nm, (1 + γ2/ω2

RBM )1/2 = (1 + 5.92/474)1/2 = 1.006, yielding

only a 0.6% increase in frequency.

The situation is quite different for tubulin microtubules. Assuming a radius of R =

12.8 nm, I find ωRBM = 0.24 cm−1 corresponding to ω2
case, (1 + γ2/ω2

RBM )1/2 = (1 + 0.0859/0.00203)1/2 = 6.6. In other words, filling tubulin

RBM = 0.00203 ps−2.

In that

microtubules with water will increase their RBM frequency by a factor of 6.6.

As seen in the full solution of Eq. (4.59) in Fig. 4.4(a), at k≈15 nm−1, there is level

repulsion and interchange in character between the two dispersion curves. At very much

higher frequencies there will be radial modes in the water column that will couple to the

74

RBM of the CNT. These lie outside the scope of the present treatment.

4.8.4 Coupling Between the LA Mode and the RBM in Carbon

Nanotubes

Consider a longitudinal wave travelling along a CNT containing no liquid. The CNT of
radius R is aligned along the x−direction and can be thought of as a rolled up plate in
the xy−plane with a width of 2πR along the y−direction. Where the CNT is being locally

stretched, it will narrow down and where it is compressed, it will fatten due to the nonzero

value of c12 reflected in the Poisson ratio. For longitudinal displacement ux and radial

displacement r, the strains will be 
11 = ∂ux/∂x and 
22 = r/R.

The strain energy density is then

(cid:16)

(cid:17)

c11
2

11 + c11
2

22 + 2c12
22
11

U =

=

1
2
1
2

c11

(cid:32)(cid:18) ∂ux

(cid:19)2

∂x

(cid:33)

(cid:16) r

(cid:17)2

+

R

+ c12

∂ux
∂x

r
R

and the kinetic energy density is

T =

1
2

ρ2D

(cid:32)(cid:18) ∂ux

(cid:19)2

∂t

(cid:18)∂r

∂t

+

(cid:19)2(cid:33)

.

There are two Euler-Langrange equations for the radial and the axial motion,

(cid:32)

(cid:33)

d
dt

∂L
∂( dr
dt )

− ∂L

∂r

= 0

75

(4.63)

(4.64)

(4.65)

and

(cid:33)

(cid:32)

d
dt

∂L
∂( dux
dt )

+

∂
∂x

(cid:32)

∂L
∂( dux
dx )

(cid:33)

= 0 .

(4.66)

With the Lagrangian L = T − U given by Eqs. (4.63)-(4.64), the Euler-Lagrange equations

translate to partial differential equations

and

ρ2D

∂2r
∂t2 + c11

r
R2 +

c12
R

∂ux
∂x

= 0

ρ2D

∂2ux
∂t2 − c11

∂2ux
∂x2 − c12

R

∂r
∂x

= 0 .

(4.67)

(4.68)

Assuming harmonic solutions ux = ux,0 exp[i(kx − ωt)] and r = r0 exp[i(kx − ωt)], I get

 −ρ2Dω2 + c11/R2

−ikc12/R

ikc12/R

−ρ2Dω2 + c11k2)



 r0

ux,0

 = 0

with the characteristic equation

(cid:18) c11
ρ2DR2 − ω2

(cid:19)(cid:18) c11

ρ2D

(cid:19)

k2 − ω2

− c2
ρ2D

12k2
2R2 = 0 .

(4.69)

(4.70)

Solving Eq. (4.70) leads to the dispersion relations ω(k) that are shown in Fig. 4.4(b)

for a CNT with radius R = 1 nm, with the values c11/(ρ2DR2) = 474 ps−2, c11/(ρ2D) =
2DR2) = 6434 nm2ps−4. The results in Fig. 4.4(b) closely resemble
474 ps−2nm2 and c2

12/(ρ2

those of Ref. [50], obtained using a more complex formalism describing orthotropic elastic

cylindrical shells using somewhat different input parameters. Were c12 to be zero, then the

76

uncoupled solutions would be the dispersionless RBM of frequency

shown by the black solid line in Fig. 4.4(b), and the pure longitudinal mode of velocity

(cid:114) c11

ρ2D

ω =

1
R

(cid:114) c11

ρ2D

v =

ω
k

=

,

(4.71)

,

(4.72)

shown by the green dotted line in Fig. 4.4(b).

The coupling term induces level repulsion between the ω−(k) and ω+(k) branches, with
strong mode hybridization occurring near k≈1 nm−1. It is of interest to examine the limiting
forms of the two solutions for k→0 and k→∞.

For k→0, the larger solution ω+(k) approaches a constant value ω0

+. From Eq. (4.70) I

obtain

ω0
+ =

1
R

(cid:114) c11

ρ2D

+ O(cid:16)
k2(cid:17)

.

(4.73)

The lower solution ω−(k) approaches the value ω− = vk, where v is the velocity of longi-

tudinal mode, modified by its coupling to the RBM. Inserting this in Eq. (4.70) and taking
the limit k→0, I obtain

(cid:115)

(cid:0)1 − α2(cid:1)

ρ2D

v =

ω
k

=

c11

,

(4.74)

where α = c12/c11 is the Poisson ratio. The numerical value of the velocity obtained using

this expression, v = 21.45 nm/ps, is slightly smaller than the velocity of the longitudinal

mode

v =

ω
k

=

,

(4.75)

(cid:114) c11

ρ2D

77

which turns out to be v = 21.77 nm/ps. The 1% reduction by the factor of

√

1 − α2 is

caused by the coupling of the longitudinal mode to the RBM.

In the opposite limit k→∞, the lower solution ω−(k) approaches a value, which is a little

below the uncoupled RBM frequency [19] of a nanotube with R = 1 nm,

= 117 cm−1 .

(4.76)

(cid:114) c11

ρ2D

ω =

1
R

(cid:115)

(cid:0)1 − α2(cid:1)

ρ2D

ω =

1
R

c11

= 115 cm−1 .

I can obtain the coupled RBM frequency ω∞− from Eq. (4.70) by neglecting its value in
comparison with c11k2/ρ2D. This yields

(4.77)

√
1 − α2 is again

The 1% reduction of the RMB frequency in the k→∞ limit by the factor of

caused by the coupling of the longitudinal mode to the RBM.

For k→∞, the upper solution approaches the value ω+ = vk, where v is the velocity of

the uncoupled LA mode. Neglecting c11/(ρ2DR) in comparison with v2k2 and neglecting k2

terms in comparison with k4 terms in Eq. (4.70), I arrive at

v =(cid:112)c11/ρ2D

(4.78)

with no corrections due to the coupling to the RBM. This was to be expected, since for
kR(cid:29)1, this nanotube mode corresponds to the LA mode in a graphene sheet.

78

Chapter 5

Unusually stable helical coil allotrope

of phosphorus

The following discussion is my original contribution to the related publication by Dan Liu,

Jie Guan, Jingwei Jiang, and David Tom´anek, Nano Lett. 16, 7865-7869 (2016) [61].

5.1

Introduction

Elemental phosphorus has been known for its unusual properties since its isolation as the

white phosphorus allotrope [62], a P4-based molecular solid,

in 1669. Other observed

allotropes include violet phosphorus [63, 64], also known as Hittorf’s metallic phospho-

rus [65, 66], rather common fibrous red phosphorus [66, 67] with an amorphous structure,

and layered black phosphorus [68], known as the most stable crystalline allotrope. Other

bulk allotropes, including blue phosphorene, have been predicted [2] and subsequently syn-

thesized [69]. Other structures of elemental phosphorus, which have been studied, include Pn

clusters [70–72] and atomically thin P helices, which have been identified as constituents [73]

in the complex structure of SnIP. It thus appears quite possible that still more allotropes

may be synthesized in the future.

I report here theoretical results that identify an unusually stable helical coil allotrope of

phosphorus. The ab initio Density Functional Theory calculations indicate that the uncoiled,

79

Figure 5.1: (a) Optimum structure of an isolated, straight 1D P∞ chain with the P10 unit
cell of length L. (b) Optimized P18, P28 and P38 segments of the isolated chain, indicating
the tendency to form rings with a radius R≈2.4 nm. (c) The optimum structure of a single
coil. (d) Optimum structure of a 2D assembly of P∞ chains separated by the distance d. (e)
Possible scenario for the formation of helical coils by connecting finite-length chain segments
inside a cylindrical cavity. The unit cells are indicated by dashed lines in (a) and (d).

isolated straight 1D chain is equally stable as a monolayer of black phosphorus dubbed

phosphorene. The coiling tendency and the attraction between adjacent coil segments add
an extra stabilization energy of ≈12 meV/atom to the coil allotrope, similar in value to
the ≈16 meV/atom inter-layer attraction in bulk black phosphorus. Thus, the helical coil

structure is essentially as stable as black phosphorus, the most stable phosphorus allotrope

known to date. With an optimum radius of 2.4 nm, the helical coil of phosphorus may fit

well and even form inside wide carbon nanotubes.

80

Isolated P∞chainR(a)(b)(d)Figure 1P18P28P38●●●LCoiled Pnsegments(e)End-on view●●●Top viewdEnd-on viewPerspective side viewP∞helical coilRPossible helical coil formation scenarioLayer of P∞chains●●●●●●●●●●●●●●●●●●dABA(c)5.2 Computational methods

Global search for small structural fragments of phosphorus was performed using Adaptive

Genetic Algorithms based on a tight-binding Hamiltonian with universal parameters. [74]

Suitable structural candidates were then optimized using ab initio density functional the-

ory (DFT) as implemented in the SIESTA [13] code to obtain insight into the equilibrium

structure, stability and electronic properties of phosphorus structures reported in the main

manuscript. All isolated structures, including infinite 1D chains and bent chain segments,

have been represented using periodic boundary conditions and separated by a 15 ˚A thick

vacuum region in all directions. I have used the Perdew-Burke-Ernzerhof (PBE) [9] or al-

ternately the Local Density Approximation (LDA) [8, 75] forms of the exchange-correlation

functional, norm-conserving Troullier-Martins pseudopotentials [33], and a local numerical

double-ζ basis including polarization orbitals. The Brillouin zone of periodic structures has
been sampled by a fine grid [34] of 12×1×1 k-points for 1D structures and 12×8×1 k-points
for 2D structures. I found the basis, the k−point grid, and the mesh cutoff energy of 180 Ry

used in the Fourier representation of the self-consistent charge density to be fully converged,

providing us with a precision in total energy of 2 meV/atom. All geometries have been opti-

mized using the conjugate gradient method [76] until none of the residual Hellmann-Feynman
forces exceeded 10−2 eV/˚A.

5.3 Results and discussion

The unusually stable structure of a P10 cluster and its suitability to link up to an infinite

1D chain was discovered while developing and testing a Genetic Algorithm optimization

technique for phosphorus clusters based on a tight-binding formalism [74]. The optimum

81

Figure 5.2: (a) Strain energy ∆E per atom as a function of the radius R of an isolated P
coil. (b) Inter-chain interaction energy ∆E per atom in a 2D assembly of phosphorus chains,
depicted in Fig. 5.1(d), as a function of the inter-chain distance d. (c) Interaction energy
∆E per phosphorus atom between a P chain and a graphene monolayer as a function of
the adsorption height h. PBE results are shown by the solid red lines, LDA results by the
dashed blue lines.

structure of a P10 unit cell in a straight 1D chain, which resembles a narrow tube with a

pentagonal cross-section, is shown in Fig. 5.1(a). I notice a structural similarity with fibrous

red [66, 67] and violet phosphorus [63–66] structures, which also contain P10 subunits in

the interlinked chains. The postulated chain structure is also similar to P nanorods [77, 78]

and P tubes [79] observed in the AgP15 compound. The DFT-PBE calculations indicate

a binding energy Ecoh = 3.274 eV/atom for the postulated P10 structure with respect to

spin-polarized P atoms. This value is only negligibly larger than that of a monolayer of black

phosphorus, known as the most stable phosphorus allotrope, with Ecoh = 3.273 eV/atom.

Finite chain segments, shown in Fig. 5.1(b), display a tendency to form coils with an

average radius of 2.4 nm. I find this coiling, which had been identified earlier [70, 80], to be

associated with an energy gain of 6 meV/atom (PBE) and 9 meV/atom (LDA). Assuming

that an ordered P∞ system may form by connecting finite Pn segments end-to-end, the

resulting equilibrium structure will be a helical coil, depicted in Fig. 5.1(c). On a per-atom

basis, the elastic strain ∆E in the coil is shown in Fig. 5.2(a) as a function of radius R. The

82

h0.040.020.01-0.05-0.02-0.04-0.010.050.030.00-0.03h (Å)3.0       3.5      4.0       4.5       5.0      5.5       6.0       6.5       7.0Figure 2(b)(a)dd0.000.020.040.080.06-0.02-0.046.0   6.5    7.0    7.5     8.0     8.5    9.0    9.5   10.0  10.5  11.0d (Å)50.00              25.00              16.67              12.50   ∞R0.00                0.02                0.04                0.06                0.08R-1(Å-1)0.0000.0010.0020.0030.0040.0050.0060.0070.008-0.0010.0100.009R (Å)(c)data points are well represented by

∆E =

(cid:18) 1

R

1
2

k

− 1
Req

(cid:19)2

,

(5.1)

where Req is the equilibrium radius. This expression describes the local strain energy in a
finite-length 1D beam thought to be initially aligned with the x−direction and deformed to
a circular arc of radius R in the xz−plane. The local strain is σ = d2uz/dx2 = 1/R. Should

not a straight, but rather a bent beam of radius Req represent the equilibrium structure, then
the local strain would be σ = d2uz/dx2 − 1/Req = 1/R − 1/Req. Equation (5.1) describes

the corresponding local strain energy [81]. I find k = 7.2 eV˚A2 for the rigidity of the elastic

beam and Req = 24 ˚A for the optimum radius of curvature based on PBE. The LDA values

of k = 7.5 eV˚A2 and Req = 21 ˚A are in fair agreement with the PBE values.

Same as in the infinite chain, the stability of the helical coil is dominated by the covalent

interatomic bonds, which are described well by DFT calculations. The coil is further sta-

bilized by the weak attraction between neighboring strands that is similar in nature to the

inter-layer attraction in bulk black phosphorus. As shown in superior Quantum Monte Carlo

(QMC) calculations of the latter system [82], the fundamental nature of the inter-layer inter-

action is rather non-trivial, different from a van der Waals interaction, and not reproduced

well by DFT functionals with or without van der Waals corrections. When compared to the

more accurate QMC value of 81 meV/atom, the LDA value of 94 meV/atom overestimates

and the PBE value of 16 meV/atom underestimates the inter-layer interaction in bulk black
I also notice the large ratio of 5 − 6 between PBE and LDA values for the

phosphorus.

weak inter-layer interaction. Extrapolating what is known about the interlayer interaction

in black phosphorus to the inter-chain interaction in a 2D assembly of chains of Fig. 5.1(d)

83

or the related wall of the helical coil in Fig. 5.1(c), I expect that PBE will also underestimate

and LDA overestimate the value of this weak interaction.

In view of the fact that the optimum coil radius is much larger than the chain thickness,

the wall of the infinite helical coil in Fig. 5.1(c) is well represented by a 2D assembly of

chains of Fig. 5.1(d). I found that the most stable 2D arrangement is one with AB stacking

of chains. The inter-chain interaction energy ∆E is displayed as a function of the inter-chain

distance d in Fig. 5.2(b). As anticipated above, I expect a large difference between PBE and

LDA interaction energies. I obtain the optimum distance deq = 7.3 ˚A and the interaction

energy ∆E = 5.3 meV/atom based on PBE. LDA suggests a smaller separation deq = 6.5 ˚A

and a much larger interaction energy ∆E = 30.8 meV/atom. While still small, the LDA

interaction energy is roughly five times higher than the PBE value.

Postulating a new allotrope is of limited use without a plausible formation scenario. I note

that previously, the void inside carbon nanotubes has been successfully filled by sublimed

C60 fullerenes that eventually fused to an inner nanotube [83]. Similarly, functionalized

diamondoid molecules were observed to enter the nanotube void, where they converted to

carbon chains [84] or diamond nanowires [85]. Inspired by these results, I feel that the most

suitable scenario to form a helical coil phosphorus allotrope involves a cylindrical cavity,

shown in Fig. 5.1(e).

Suitable cavities with an optimum inner diameter of few nanometers may be found in

zeolites or in nanotubes of carbon, BN and other materials. The phosphorus feedstock could

be white, red or violet phosphorus that had been sublimed in vacuum, under exclusion of

air. The sublimed species would likely be finite-chain segments, shown in Fig. 5.1(b), which

may enter at the open end and benefit energetically from the interaction with the inner wall

of the cavity. As seen in Fig. 5.2(c), where I consider the related system of an isolated chain

84

Figure 5.3: Electronic structure of (a) an isolated P chain and (b) a 2D layer of P chains,
shown in Figs. 5.1(a) and 5.1(d). Left panels depict the electronic band structure based on
PBE and the middle panels the corresponding density of states. The Brillouin zone is shown
as inset of the left panel in (b). The right panels depict the charge distribution associated
with frontier states in the valence band region, indicated by the green hashed region in (a)
and (b), which extends from EF to 0.2 eV below the top of the valence band. Charge density
contours are superposed to structural models, with the unit cells indicated by the dashed
lines. Due to differences in the density of states between these systems, the contours are
presented at the electron density 0.008 e/˚A3 in (a) and 0.002 e/˚A3 in (b).

on graphene, this interaction is weak and similar in nature to the inter-chain interaction in

Fig. 5.2(b). The optimum arrangement is found by inspecting the adsorption energy ∆E

as a function of height h in Fig. 5.2(c). For the optimum geometry, I find ∆Eeq = 9 meV

per P atom at heq = 3.9 ˚A based on PBE and ∆Eeq = 44 meV per P atom at heq = 3.3 ˚A
based on LDA. I notice here again the adsorption energy ratio of ≈5 between LDA and PBE

results, consistent with the other results.

Once inside and near the wall of the cylindrical cavity, finite Pn chain fragments will

85

Top viewSide viewTop viewSide viewFigure 3Gap:Isolated=2.03 eVAB=1.82 eVLdos:Isolated:EF~TOV-0.2 eVIsosurface:0.008 e/A3AB:EF~TOV-0.2 eVIsosurface:0.002 e/A3(a)(b)benefit energetically from an end-to-end connection that eliminates open ends. The number

of atoms in the finite circular arc, preferentially oriented along the perimeter of the inner

cavity, will grow. At the elevated temperatures of subliming phosphorus, the growing Pn

ring fragments are very unlikely to interconnect with corresponding segments that contain

exactly the right number of atoms, which would complete a ring at the optimum distance

to the wall. Much more likely, the last segment to join before possible ring closure will be

too long and start the formation of a helical coil. Since transformation of the coil to one or

more adjacent rings would require bond breakage within the coil, it is unlikely to happen.

The electronic structure of the new allotrope, similar to that of phosphorene, is of utmost

interest [86]. The PBE results for the related phosphorus chains and their 2D assemblies are

shown in Fig. 5.3. As seen in Fig. 5.3(a), the P∞ chain has a direct fundamental band gap of

2.03 eV at X. Similarly, also the 2D chain assembly has a direct band gap of 1.82 eV at X as

seen in Fig. 5.3(b). Based on what is known theoretically and experimentally about few-layer

phosphorene [86], the PBE band gap values are strongly underestimated in comparison to

the experiment.

In view of the fact that phosphorus is typically p−doped, I are also interested in the

nature of the frontier states in the valence band region. I plotted the charge distribution of

these states, covering the energy range between EF and 0.2 eV below the top of the valence

band, in the right panels of Fig. 5.3. Similar to what is known about black phosphorus,

I observe lone-pair electron states in the isolated chain in Fig. 5.3(a) that contributes to

the electronic inter-chain coupling modifying the band structure, as seen in Fig. 5.3(b).

This demonstrated influence of the inter-chain coupling on the electronic structure is a clear

evidence that the interaction differs from a purely van der Waals interaction, similar to black

phosphorus [82].

86

In view of the relatively low beam rigidity of the coiling chain, I expect the coils to adjust

their radius freely for the optimum fit inside cylindrical cavities. The ability of the helical

coil strands to slide past each other allows the helix to adjust to a changing cavity diameter.

In view of the favorable inter-chain interaction of 16 meV/atom for an AB stacking in the

radial direction, with an optimum inter-chain distance of 0.6 nm based on PBE, I consider

it quite possible for a second helix forming inside the outer helix. Considering an outer

helical coil at its equilibrium radius Rout = 2.4 nm, the inner coil should have a radius of
Rin≈1.8 nm. In PBE, the strain energy in the inner structure of < 1 meV/atom according

to Fig. 5.2(a) is negligibly small when compared to the additional inter-chain interaction

energy of 16 meV/atom. I may even imagine additional helices forming inside the double-

helix structure.

In view of the low beam rigidity of the chain and the large inter-chain

interaction, a structure consisting of nested coaxial coils should be even more stable than

bulk black phosphorus, the most stable phosphorus allotrope known to date.

As suggested by the end-on view of a chain in Fig. 5.1(a), the cross-section of the helical

coil should appear as lines of pentagons near the walls and along the axis of the cylin-

drical cavity in Transmission Electron Microscopy (TEM) images. This is very similar to

recently observed TEM data [87]. I can imagine left- and right-handed helical coils forming

simultaneously and coexisting inside a suitable cylindrical cavity.

As a structural alternative, a black phosphorus monolayer may also roll up to a tube

inside a carbon nanotube with a 2.4 nm radius. Assuming an inter-wall distance of 0.5 nm,
the radius of the phosphorene nanotube should be R≈1.9 nm, and its strain energy should
be ≈8.6 meV/atom if bent along the soft direction to become an armchair P nanotube,

or 42.1 meV/atom if bent along the normal, harder direction [19]. Thus, energetically, a

black phosphorus nanotube is not favorable. Also a straight 1D phosphorus chain inside a

87

nanotube should be less stable by > 6 meV/atom than the coiled structure. Thus, I find the

coil to be more stable than competing phosphorene nanotube and straight chain structures.

Also, assuming that the new phase forms by sublimation of red phosphorus, it will more

likely resemble structural elements of red P than the completely different black P.

In layered black phosphorus, the observed bulk band gap value of 0.35 eV is known to

increase to 2 eV in the monolayer due to the change in the weak interlayer interaction. PBE

calculations underestimate the band gap significantly, suggesting a value of 0.04 eV for the

bulk and 0.9 eV for the monolayer. I thus expect also the calculated band gap in isolated

chains to lie below the experimental value and to decrease due to inter-chain interaction in

2D chain assemblies. In helical coils, a further reduction, which should furthermore depend

on the coil radius, is expected due to improved screening of the electron-hole interaction.

In view of this reasoning, it is not surprising that the calculated band gap values lie rather

close to the value of 1.95 eV that has been observed in fibrous red phosphorus [88].

Further electronic structure changes induced by coil deformation may open a wide range

of applications. Similar to bulk black phosphorus, where changes in the interlayer distance d

modify the band gap Eg significantly, I find that axial compression of the coil also modifies the

band gap. At the optimum value 0.73 nm for the inter-coil distance d, defined in Fig. 5.2(b),

the gap is direct and Eg = 1.8 eV according to Fig. 5.3(b). Reducing d to 0.6 nm turns the
gap indirect and reduces its value to ≈0.8 eV. Further reduction to d = 0.5 nm turns the

coil metallic. Increasing the inter-chain distance to d = 0.8 nm opens the gap to Eg 2.0 eV,

while maintaining its direct character. Of course, these changes in the fundamental band

gap may only be exploited inside semiconducting nanotubes, such as BN, which have even

larger band gaps.

88

5.4 Summary

In conclusion, I have identified computationally an unusually stable helical coil allotrope of

phosphorus. The ab initio Density Functional Theory calculations indicate that the uncoiled,

isolated straight 1D chain is equally stable as a monolayer of black phosphorus dubbed

phosphorene. The coiling tendency and the attraction between adjacent coil segments add
an extra stabilization energy of ≈12 meV/atom to the coil allotrope, similar in value to
the ≈16 meV/atom inter-layer attraction in bulk black phosphorus. Thus, the helical coil

structure is essentially as stable as black phosphorus.

In view of the low beam rigidity

of the chain and the large inter-chain interaction, a structure consisting of nested coaxial

coils should be even more stable than bulk black phosphorus, the most stable phosphorus

allotrope known to date. With an optimum radius of 2.4 nm, the helical coil of phosphorus

may fit well and even form inside wide carbon nanotubes. I find the coiled P structure to

be a semiconductor with a direct gap exceeding 1.8 eV. The size and character of the band

gap can further be modified by small structural changes in the coil.

89

Chapter 6

Microscopic Mechanism of the

Helix-to-Layer Transformation in

Elemental Group VI Solids

The following discussion is my original contribution to the related publication by Dan Liu,

Xianqing Lin and David Tom´anek, Nano Lett. 18, 4908-4913 (2018) [89].

6.1

Introduction

After much attention has been devoted to graphene, a 2D allotrope of group IV elemental

carbon, scientific interest turned to semiconducting 2D allotropes of group V elements P [86,

90] and As [91,92]. Recent observation of 2D allotropes of group VI elements Se and Te [93–

95] came as a surprise, since – unlike group IV and V elemental solids – the bulk structure

of Se and Te is not layered, but consists of helical chains of covalently bonded atoms packed

in a hexagonal array. For lack of well-defined layers, 2D Se and Te can not be obtained by

mechanical exfoliation used in group IV and V systems. Chalcogens are known for a large

number of stable allotropes and oxidation states [96]. The latter fact had been identified

as the key factor behind the stability of specific 2D allotropes of Se and Te [93]. Still, the

90

Figure 6.1: Stable 1D and 2D structural allotropes of Se and Te. 1D structures of the (a)
a helix and (b) b chain and their 2D counterparts, the (c) δ and (d) η allotrope. The δ
allotrope is a covalently bonded 2D assembly of b chains. The unit cells of the 2D structures
are highlighted by the transparent green areas in (c) and (d).

strong dissimilarity between the bulk structure containing weakly interacting, intertwined a

helices and covalently bonded, atomically thin layers raises the intriguing question about the

microscopic mechanism behind the transformation from quasi-1D to 2D structures, which

has not been addressed yet.

Here I present results of ab initio calculations, which help to identify the intermediate

steps of the observed transition from a helices in the native bulk structure to atomically thin

layers of elemental Se and Te [93]. The results unveil the energetics and the intermediate

steps encountered during this structural transition. I have discovered an intriguing mech-

anism that converts an a helix to a more stable, previously unknown b chain by moving a

point-dislocation connecting these two structures. In a zipper-like motion, the b chain may

reconnect to a previously unknown 2D δ structure, which is unusually stable, similar to the

91

(a)(c)(b)End-on viewSide viewb -chaina1a -helixa1Figure 1xzxa1a2δyb-chain(d)xzxya1a2ηFigure 6.2: Phonon spectra of (a) δ-Se and (b) η-Se calculated using the DFT-LDA energy
functional. The Brillouin zones and high-symmetry points are shown schematically in the
insets. Continuum elasticity results for long-wavelength longitudinal acoustic modes are
shown by the blue dash-dotted lines, for transverse acoustic modes by the dotted green lines,
and for flexural modes by the dashed red parabolas.

related η structure. The structural change from the a helix to the 2D δ allotrope is mildly
exothermic with −0.17 eV/atom for Se and −0.23 eV/atom for Te. The low number of

structural constraints allows the helical structure to exploit many degrees of freedom and
thus to lower the activation barriers along the reaction path to (cid:46)0.3 eV, indicating that the

transition may occur at moderate temperatures. The GW quasiparticle calculations of the

electronic structure indicate that all quasi-1D and 2D chalcogen allotropes are semiconduct-

ing.

6.2 Computational techniques

The calculations of the stability, equilibrium structure, the pathway and dynamics of struc-

tural transformations have been performed using density functional theory (DFT) as im-

92

Figure 2(b)(a)ω(cm-1)δ-SeГX                    S        Y                    ГΓXYSω(cm-1)η-SeГX     S         Y       ГYSΓXplemented in the SIESTA [13] and VASP [14, 17] codes. Periodic boundary conditions

have been used throughout the study, with monolayers represented by a periodic array of

slabs separated by a 30 ˚A thick vacuum region. I compared results using both the Local

Density Approximation (LDA) [8, 75] and the Perdew-Burke-Ernzerhof (PBE) [9] exchange-

correlation functionals, since LDA typically overbinds and PBE underbinds. I also checked

the importance of van der Waals corrections to the total energy by using the optB86b

exchange-correlation functional [97, 98] for selected structures. The SIESTA calculations

used norm-conserving Troullier-Martins pseudopotentials [33], a double-ζ basis including po-

larization orbitals, and a mesh cutoff energy of 180 Ry to determine the self-consistent charge
density, which provided us with a precision in total energy of (cid:46)2 meV/atom. The VASP

calculations were performed using the projector augmented wave (PAW) method [17] and
500 eV as energy cutoff. The reciprocal space has been sampled by a fine grid [34] of 10×10 k-

points in the 2D Brillouin zones (BZ) of the primitive unit cells of the δ and η structures

containing 6 atoms each, and 10 k-points in the BZ of 1D a and b chains with 3- and 6-atom

unit cells, respectively. Geometries have been optimized using the conjugate gradient (CG)
method [76], until none of the residual Hellmann-Feynman forces exceeded 10−2 eV/˚A. Mi-

crocanonical and canonical MD calculations were performed using 1 fs time steps. Electronic

structure has been calculated using the GW quasiparticle approach [99] as implemented in

the BerkeleyGW package [100] interfaced with QuantumEspresso [101]. In a periodic

arrangement, 1D and 2D structures were separated by 17 ˚A wide vacuum regions. The
Brillouin zone of quasi-1D structures was sampled by 36×1×1 k-points and that of quasi-2D
structures by 6×14×1 k-points. I used 10 Ry as energy cutoff for the plane wave expansion

of the dielectric matrix. The quasiparticle energies have been determined by considering the

lowest 220 unoccupied conduction bands and accounting for all higher-lying bands using the

93

Figure 6.3: (a) Schematic growth mechanism of the energetically stable 2D δ structure by
zipper-like attachment of the b chain, which is being formed locally locally at a defect in
the native a helix and propagates by dislocation motion. (b) Bond length d, bond angle θ
and dihedral angle ψ used to characterize chalcogen structures. (c) DFT-LDA based energy
differences ∆E encountered during the stepwise conversion from 1D a-Se to b-Se as a function
of the reaction coordinate. The system is represented by a finite Se9H2 chain, passivated by
hydrogen at both ends, and the total energy is given with respect to the final state. The
dotted line is guide to the eye. The energy of a 9-atom long segment of the defect-free
infinite a-Se and b-Se chains is indicated by asterisks. Ball-and-stick models show stable
Se9H2 geometries, labeled by A-D, and the transition states T . Location of the unusually
small dihedral angle in the transition states is indicated by shaded triangles.

modified static-remainder approximation [102].

6.3 Results and discussions

Understanding the observed 1D to 2D transformation is an unprecedented challenge due to

the large and constantly changing number of degrees of freedom that are actively involved

in lowering the activation barriers between intermediate states.

In this complex system,

the use of common techniques such as the nudged elastic band model becomes a futile

94

(c)(a)a-helixb-chain(b)Figure 2TABTCDTBCDa-helixb-chainReaction CoordinateABCθψd∆E (eV)0.250.00-0.25Table 6.1: Cohesive energy Ecoh of various Se and Te allotropes in [eV/atom] units, obtained
using DFT-LDA and DFT-PBE calculations.

a-helix b-chain
3.700
3.327
3.235
2.871

LDA 3.677
PBE
3.307
LDA 3.209
PBE
2.856

α

3.795
3.198
3.479
2.904

β

3.823
3.302
3.434
2.933

γ

3.569
2.953
3.323
2.716

δ

3.843
3.357
3.443
2.940

η

3.854
3.355
3.451
2.943

Se

Te

endeavor. Restricting the system’s freedom invariably increases the activation barriers, in-

correctly suggesting that the transformation should not occur under laboratory conditions.

I chose a different approach that will be discussed in the following.

Recently observed 2D Se and Te structures [93–95, 103] have been formed by initially

evaporating the bulk substances. It is likely that the vapor contained primarily short seg-

ments of a helices, shown in Fig. 6.1(a), which constitute the bulk structure. Consequently,

I will consider the a helix as the initial structure in the transformation process to 2D struc-

tures. I discovered a previously unknown, atomically thin and unusually stable 2D structure

of Se and Te, which I call the δ structure, by artificially compressing a 2D assembly of the

native a-Se helices in the direction normal to the 2D layer. The 2D δ allotrope, shown in

Fig. 6.1(c), emerged after the pressure was released. An impractical, yet computationally

feasible way to transform 1D to 2D structures is by artificial confinement in a 2D plane, as

shown in Fig. 6.4. The initial structure, depicted in Fig. 6.4(a), consists of a 2D assembly

of parallel Se helices separated by d = 3.83 ˚A. The atomic positions in the artificial flat

structure shown in Fig. 6.4(b) are obtained by projecting this structure onto a plane. By

re-optimizing this structure, I obtained 2D δ-Se, shown in Fig. 6.4(c).

Another previously unknown and stable allotrope, labeled η, is depicted in Fig. 6.1(d). It

is related to the δ structure by a series of reflections,while keeping the bond lengths and bond

angles constant throughout the structure. The space group of the δ structure is C4

2v in the

95

Sch¨onflies notation and its group number is #28. The space group of the η structure is D2
2

in the Sch¨onflies notation and its group number is #17. Both groups have only 4 symmetry

operations.The structural transformation from the δ to the η structure is depicted in Fig. 6.5.

The δ allotrope in Fig. 6.5(a) is transformed to the allotrope in Fig. 6.5(b) by first reflecting

the part of the structure on the right of sites 1, 2, 3 in a plane containing these atoms.

This is to be followed by an infinite series analogous reflections involving sites 4, 5, 6, then
1(cid:48), 2(cid:48), 3(cid:48), etc., resulting in a structure reminiscent of a staircase. The structure in Fig. 6.5(b)

is transformed to the final η allotrope in Fig. 6.5(c) by an analogous series of reflections.

This transformation starts by reflecting the part of the structure on the right of sites 3, 4

in a plane containing these atoms. This is to be followed by an infinite series analogous
reflections involving sites 6, 1(cid:48), then 3(cid:48), 4(cid:48), etc. A perspective view of the initial δ structure

is shown in Fig. 6.5(d) and that of the final η structure in Fig. 6.5(e). Numerical results for

the cohesive energies of all known Se and Te allotropes are summarized in Table 6.1.

I studied the stability of the new phases by determining their elastic response and their

phonon spectra. Since the 3D elastic modulus tensor is not defined for a truly 2D system,

which does not naturally form layered 3D compounds, I have determined the components

of the 2D elastic tensor defined earlier [19]. For the δ-phase of Se, I find c11 = 4.97 N/m,
c22 = 20.02 N/m, c66 = 5.92 N/m, D(Γ − X) = 0.33 eV, D(Γ − Y ) = 1.15 eV. For the
η-phase of Se, I find c11 = 11.25 N/m, c22 = 22.71 N/m, c66 = 7.09 N/m, D(Γ − X) =
0.39 eV, D(Γ − Y ) = 1.02 eV. Among others, these elastic constants allow a more accurate

representation of low-frequency acoustic modes in the vibrational band structure of the 2D

structures, which I present in Fig. 6.2. Due to the similarity of the phonon spectra, I expect

the zero-point motion to not to play an important role in the cohesive energy. I find a zero-

point energy of 23 meV/atom for β-Se and 24 meV/atom for δ-Se and η-Se, with energy

96

Figure 6.4: Transformation of 1D to 2D structures of elemental Se. (a) Perspective and end-
on view of the initial 2D assembly of 1D a-chains. (b) End-on view of the structure after
artificial confinement to an infinitely thin slab. (c) End-on view of the 2D δ-Se allotrope
that formed spontaneously after the structure in (b) was relaxed.

differences of (cid:46)0.1 meV/atom between the different phases.

In the following, I will first address structures of elemental Se and refer discussion of

Te structures for later. The DFT-LDA and DFT-PBE results suggest that δ-Se and η-

Se are energetically near-degenerate. The previously introduced [93, 94] β-Se allotrope is

less stable than δ-Se by 20 meV/atom (LDA) and 55 meV/atom (PBE) and thus the least

stable of the three. Still, in view of the relatively small energy differences and structural

similarities, I expect that all these structural allotropes, and possibly even others, may be

formed under synthesis conditions at elevated temperatures. In the following, I will focus on

the energetically stable δ structure and its microscopic formation mechanism starting from

the native a helix structure.

Inspecting the equilibrium structure of δ-Se in Fig. 6.1(c), I found that it can be viewed

97

Figure S5a -helicesda1(a)δ(b)(c)Figure 6.5: Structural transformation between the 2D δ and the η allotropes by a series of
reflections in planes normal to the picture plane. (a)-(c) Schematic side view of the initial
and intermediate structures. Atoms in a plane normal to the lattice are shown by (cid:13) and

atoms in front of these by (cid:74). Atoms in one unit cell are labeled by the numbers 1 − 6.

Primed numbers are used to label atoms in neighboring cells. Perspective views of (d) the
initial δ and (e) the final η structure.

as a 2D assembly of 1D chains, which I call b chains. I found the previously unknown b chain,

shown in Fig. 6.1(b), to be a stable allotrope of Se, even more stable than the 1D a helix

by 23 meV/atom (LDA) and 20 meV/atom (PBE). The b chain may be attached laterally

to a semi-infinite δ-Se layer in a zipper-like motion depicted in Fig. 6.3(a). Owing to the

multi-valent behavior of the chalcogens [93], this is an activation-free exothermic process

that releases 143 meV/atom (LDA) and 30 meV/atom (PBE).

Assuming that b-Se chains, which are entropically favored at high temperatures, are

indeed present during the formation of δ-Se, then the only task remaining to understand

the entire conversion path from a-Se to δ-Se is locating an energetically favorable pathway

98

Figure SF3ηδ(a)(b)(c)4 5     61’2’3’4’5’6’1   2     31   2     34  5    61’2’3’4’5’6’6  5    41’2’3’6’5’4’1   2     3(d)(e)δηfor the transformation from the a helix to the b chain. The most plausible transformation

begins by connecting the semi-infinite a helix and a b chain end-to-end by a covalent bond,

as seen in Fig. 6.3(a). The a-b connection is a dislocation defect or a 0D domain wall that

may propagate along the 1D chain, as indicated by the broken arrow in Fig. 6.3(a).For Se,

the step-wise a-to-b conversion is exothermic and requires only a finite activation energy to

be discussed below. Individual processes within the entire a-to-b-to-δ transformation may

occur concurrently, as shown in Fig. 6.3(a).

I noted that observed stable allotropes of Se and Te all share structural commonalities.
As defined in Fig. 6.3(b), these include the bond length d(Se)≈2.38 ˚A and d(Te)≈2.84 ˚A, the
bond angle θ≈100◦−130◦, and the dihedral angle ψ≈80◦−100◦ found experimentally in bulk

structures [104–108]. The step-wise dislocation motion corresponds to a series of a-Se to b-Se

structural changes at the dislocation, which were studied in finite chain segments containing

12 Se atoms, passivated by hydrogen at both ends. Relaxing the finite segment of a-Se yielded

structure A and relaxing the finite b-Se segment resulted in structure D. Interestingly, the

infinite a and b chains as well as their finite counterparts A and D displayed very similar

structural characteristics as the bulk structures. Later on, I found out that shorter, 9-

atom segments shown in Fig. 6.3(c), are sufficient to visualize and understand the step-wise

transformations in the 1D structure.

I discovered that the relative stability of the infinite a-Se and b-Se chains, as well as that

of the optimized finite segments, can be rationalized in terms of strain originating in the
deviation from the optimum bond length d = 2.36 ˚A, bond angle θ≈106◦ and dihedral angle
ψ≈83◦, defined in Fig. 6.3(b). Most robust of these parameters is the bond length, which is
close to its optimum value in all optimized structures. Whereas θ = 102◦ and ψ = 100◦ are
constant throughout the unit cell of a-Se, two thirds of the b-Se unit cell display θ = 105◦

99

and ψ = 83◦, and the rest is characterized by θ = 101◦ and ψ = 100◦. The closer proximity

of b-Se to the optimum angles θ and ψ is reflected in its higher stability by 23 meV/atom

(LDA) and 20 meV/atom (PBE) with respect to a-Se.

I found that the entire A to D transformation can be accomplished by a sequence of

bond rotations or reflection while maintaining the optimum bond lengths d and bond angles

θ throughout the structure. The transformation steps only involved changes in one dihedral

angle ψ at a time, as shown in Fig. 6.3(c), which required a typical activation energy of
(cid:46)300 meV. The DFT-LDA energies for the process described in Fig. 6.3(c) differ from
DFT-PBE results by (cid:46)30 meV and from van-der-Waals corrected DFT-optB86b results by
(cid:46)25 meV for the entire structure. The similarity in cohesive energies obtained using LDA

and PBE is also seen in Table 6.1. I identified two locally stable structures, labeled B and

C, along the A to D trajectory. The locally stable states A, B, C and D all displayed

near-optimum values of d, θ and ψ throughout the structure. The contiguous trajectory in

configurational space contains unstable transition states TAB between A and B, TBC between

B and C, and TCD between C and D. I traced back the lower stability of the transition states

to one of the dihedral angles being near zero, far from its optimum value. I verified that

all transition states T were unstable in the sense that perturbing the TN,N +1 structure in

whichever way and following up with microcanonical MD calculations or conjugate gradient

(CG) optimization always lead to optimum N or N + 1 geometries and to no other structure.

The relative energy and the structure of these states are depicted in Fig. 6.3(c).

The most favorable conversion from the infinite a-helix (related to state A) to the more

stable b-chain (related to state D) is a multi-state process that involves intermediate states

B and C, as indicated in Fig. 6.6(a). The energetically favored transformation path by

dislocation motion along the chain is illustrated schematically in Fig. 6.6(b).For the following

100

discussion, I label individual atoms by numbers in the finite segment of the a-helix structure

in Fig. 6.6(c) and in the finite segment of the b-chain structure in Fig. 6.6(d).

I expect the postulated transition process to be just one of many similar transformations

in the system that may occur with potentially even lower activation barriers.Results of

selected trajectories from the transition points T , identified in Fig. 6.6(c), are presented in

Figs. 6.7-6.9 in terms of the dihedral angle ψ and the potential energy E as a function of

the number of conjugate gradient (CG) steps or as a function of time in MD simulations. I

found it very useful to consider both CG and MD calculations. CG optimization reaches the

intermediates states only in case that there are no local minima between the transition states

T and the adjacent intermediate states. This limitation is eliminated in MD simulations at

the cost of large fluctuations in the potential energy E caused by corresponding fluctuations

in the kinetic energy, which increases in time. Of course, I found the total energy to be

conserved in the microcanonical MD simulations. Inspection of the results in Fig. 6.7 shows

that all trajectories starting in transition point TAB, whether obtained by CG or MD, lead

either to the geometry of the adjacent intermediates state A or B. Similarly, in Fig. 6.8, I

found all trajectories from TBC to lead either to state B or C. Finally, as seen in Fig. 6.9, I

found all trajectories from TCD to lead either to state C or the final state D. The potential

energy E has been decreasing along all trajectories starting from the frozen transition state

geometry.

Interestingly, I found the interatomic bond distances and bond angles to remain near

their optimum values during the entire A-D transformation. The third factor associated

with stability, the dihedral angle ψ, was found to change significantly during the process.

In all transition-state geometries, I could identify 4-atom segments with a very unfavorable
dihedral angle ψ near 0◦. Structural evolution of the system from the transition state to

101

Figure 6.6: (a) Potential energy E as a function of the optimum reaction coordinate during
the A-D transformation. (b) Schematic transformation from the a-helix to the b-chain by
dislocation motion.Schematic geometry of hydrogen terminated 9-atom segments of (c) the
a-helix and (d) the b-chain. Atoms are labeled for later use.

the adjacent intermediate state was in all cases accompanied by a change of the particular
dihedral angle ψ towards its optimum value ≈83◦. As a matter of fact, I could rationalize

the entire transformation from A to D by a sequence of bond rotations.

I should remember that the sequence and energetics of A to D transformations, iden-

tified in the finite chain segment, may differ in detail from the corresponding process at a

dislocation defect connecting infinite a and b chains, since the free-standing finite structure

has fewer constraints than the infinite structure. As seen in Fig. 6.3(c), the net energy gain

from the infinite a to the b structure is higher than the energy gain in the finite segment

changing from the A to the D structure. As a matter of fact, I should not place too much

emphasis on the relative stability of finite A, B, C and D structures, but rather realize

that the activation barriers for step-wise structural changes are similar in finite and infinite

structures. The energetics and structure of the free-standing finite chain segment will change

when connected to a semi-infinite a chain at the one and a semi-infinite b chain at the other

102

(cid:69)(cid:3)(cid:16)(cid:70)(cid:75)(cid:68)(cid:76)(cid:81)(cid:68)(cid:3)(cid:16)(cid:75)(cid:72)(cid:79)(cid:76)(cid:91)123456789101112HH410123567891112HH(c)(b)(a)(d)TABTCDTBCDABCEReaction coordinateFigure 6.7: (a,c) Dihedral angle ψ and (b,d) potential energy E of the Se system during
(a,b) CG optimization and (c,d) MD calculations starting in transition point TAB, identified
energetically in Fig. 6.6(a). Atomic positions used to define ψ6789 are identified in Fig. 6.6(c)
and 6.6(d).

end. Whatever differences in the relative stability of the intermediate states in the infinite

or finite segment connecting a and b chains, none will stop the attachment of the b chain to

δ-Se that occurs at a significant net energy gain of 143 eV/atom, thus driving the exothermic

reaction forward.

Since Te shares the same group VI with Se in the periodic table, I expect the chemical

behavior and bonding in the respective elemental solids to be very similar.

I found the

formation mechanism of the δ allotrope from native a helices via the b chains, discussed above

for Se, to be viable for Te as well, with small differences in reaction energies. According to

Table 6.1, the a-Te to b-Te conversion is exothermic, releasing 26 meV/atom (LDA) and

15 meV/atom(PBE). Attaching the b chain laterally to a semi-infinite δ-Te layer is also

exothermic, releasing 208 meV/atom (LDA) and 69 meV/atom (PBE). As mentioned earlier,

also the geometries of the initial, final and intermediate states encountered during the a-Te

103

Figure 6.8: (a,c) Dihedral angle ψ and (b,d) potential energy E of the Se system during
(a,b) CG optimization and (c,d) MD calculations starting in transition point TBC , identified
energetically in Fig. 6.6(a). Atomic positions used to define ψ2345 are identified in Fig. 6.6(c)
and 6.6(d).

to b-Te transformation are similar, the main difference being the Te-Te bond length, which
is larger than the value in Se allotropes. Most important, also activation energies of ∼0.3 eV

are similar in Te and Se.

The electronic band structure of the different 1D and 2D allotropes of Se is shown in

Fig. 6.10. The GW results, shown by the solid red lines, are considered a valid counterpart

to experimental observation. The LDA results, shown as a matter of reference by the dashed

black lines, underestimate the fundamental band gap Eg significantly. Among 1D structures,

a-Se in Fig. 6.10(a) has an Eg=5.3 eV wide direct gap at X, and the b-Se in Fig. 6.10(b) has

an Eg=5.1 eV wide direct gap at Γ. The GW band gap for a-Se compares favorably with a

recently reported Eg=5.46 eV value [109] and a smaller Eg=3.00 eV value [110] obtained

using a different approach. δ-Se in Fig. 6.10(c) has an Eg=3.1 eV wide indirect gap, and

η-Se in Fig. 6.10(d) has an Eg=2.4 eV wide indirect gap. In general, I see that the band

104

Figure 6.9: (a,c) Dihedral angle ψ and (b,d) potential energy E of the Se system during (a,b)
CG optimization and (c,d) MD calculations starting in transition point TCD, identified en-
ergetically in Fig. 6.6(a). Atomic positions used to define ψ789,10 are identified in Fig. 6.6(c)
and 6.6(d).

gaps in 1D structures are significantly larger than in the 2D layers.

The electronic band structure of different 1D and 2D allotropes of Te is displayed in

Fig. 6.11. I find the trends and main results for the different Te allotropes to be consistent

with those for Se, in particular the band gaps in 1D structures to be much larger than in 2D

structures. As seen in Fig. 6.11(a), the band gap of the isolated Te a helix is 1.4 eV wide and

indirect. The band gap of the isolated Te b chain, on the other hand, is direct at Γ and 1.5 eV

wide, as seen in Fig. 6.11(b). The band structure of 2D δ-Te, shown in Fig. 6.11(c), has an

indirect, 0.9 eV wide band gap. Results for the 2D allotrope η-Te, shown in Fig. 6.11(d),

indicate a direct, 0.3 eV wide band gap between Γ and Y . These numerical results indicate

that the LDA-based band gaps in Te are roughly one third of the LDA values found in Se. As

mentioned earlier, while DFT-LDA underestimates band gaps, it still provides useful insight

into trends in the electronic structure.

105

Figure 6.10: Electronic band structure of an isolated (a) a-Se helix, (b) b-Se chain, isolated
(c) δ-Se and (d) η-Se monolayers. GW results, shown by solid red lines, are compared to
LDA results, shown by the black dashed lines.

Reported experimental results for 2D chalcogen allotropes include a thin layer of a-Se

helices on silicon [95], a monolayer of a-Te helices on graphene [103], a substrate-free thin

layer of a-Te helices [111], and β-Te, a covalently bonded 2D assembly of a helices [93,94]. A

valid question to ask is, why the more stable 2D δ allotrope and the 1D b chain have not been

observed. Since interest in group VI elemental 2D structures took off only very recently, it

is quite possible that optimum conditions for the synthesis of the proposed allotropes have

not been found yet. Established information about 3D chalcogen allotropes consisting of

interacting a helices provides only a limited insight into how structures may grow on a 2D

substrate. There, the substrate-chalcogen interaction may play a significant role, such as

providing extra stabilization of the more reactive b chains over the a helices.

106

Figure 4(a)(b)(c)(d)4E (eV)321EF-3-2-1-4ΓXb-chainΓX4E (eV)321EF-3-2-1-4a-helixSeSeE (eV)EFη-SeΓX         S   Y          Γ-1-2-3-44321ΓX         S   Y          Γ4E (eV)321-3-2-1-4EFδ-SeFigure 6.11: Electronic band structure of an isolated (a) a-Te helix, (b) b-Te chain, isolated
(c) δ-Te and (d) η-Te monolayers obtained using DFT-LDA.

Without question, the substrate plays a significant role facilitating the 1D to 2D trans-

formation in chalcogen structures. Even a weak adsorbate-substrate interaction will confine

condensing chalcogen structures in the 2D space adjacent to the substrate, thus significantly

increasing the coalescence rate. To decouple the intrinsic chalcogen reaction energetics from

substrate-chalcogen interaction on a particular substrate, I performed all the calculations in

vacuum. Specific substrates can be selected that may change the relative stability order in

adsorbed chalcogen structures in comparison to such structures in vacuum.

As mentioned before, locating a transformation path in configurational space between
very dissimilar structures a and δ, with activation barriers not exceeding ∼0.3 eV, is a

nontrivial task. So far, state-of-the-art global structural searching techniques were unable to

locate such a path or other structures that were very different from the native a helices [93].

107

Figure 4-4a-helixTe4321EF-3-2-1-4(a)E (eV)b-chainTe4321EF-3-2-1(b)E (eV)4321-3-2-1-4EF(c)E (eV)-1-2-3-44321EF(d)E (eV)δ-Teη-TeΓXΓXΓX         S   Y          ΓΓX         S   Y          ΓI feel that for the time being, understanding the physical origin of strong bonds in terms

of d, θ, ψ and locating a pathway along which only the least energy sensitive structural

parameter is modified is a more promising approach to understanding the reaction energy.

I located such a path from A to D that involves only a sequence of changes in ψ within a

9-atom segment of the chain.

Maybe the most important lesson to learn was that – at least in the class of structures

discussed here – releasing structural constraints and increasing the number of degrees of

freedom may significantly lower the activation barriers for structural transformations. Even

though the initially considered artificial compression of a 2D assembly of a helices with 3

atoms per unit cell to a completely flat structure with 6 degrees of freedom per cell did

eventually yield the stable δ allotrope, the energy invested was unphysically high. Allow-

ing for concerted atomic motion in a 9-atom segment with 27 degrees of freedom lowered

the activation barriers significantly. In a related scenario of structural phase transitions in

monochalcogenindes, artificial spatial constraints [112] were also found to significantly af-

fect the energy barriers and thus the critical temperature [113]. The same behavior can be

expected for a wide range of systems undergoing structural phase changes.

6.4 Summary

In summary, based on DFT calculations, I have uncovered the microscopic mechanism of the

recently observed structural transition in elemental chalcogens Se and Te from their native

bulk structure consisting of a helices to atomically thin 2D layers. I found that the a helices

convert to more stable, previously unknown b chains in a multi-step process that involves a

point-dislocation motion along the helix. In a zipper-like motion, the b chain reconnects to

108

a related, previously unknown and unusually stable 2D δ structure of Se and Te. The 1D a
helix to 2D δ conversion is mildly exothermic with −0.17 eV/atom for Se and −0.23 eV/atom

for Te. The high structural flexibility allows the helix to exploit many degrees of freedom and
thus significantly lower the activation barriers along the complex reaction path to (cid:46)0.3 eV,

indicating that the conversion may occur at moderate temperatures. In view of the similar

stability of the structurally related β, δ and η structures, I expect that all these and maybe

even other allotropes should be formed at elevated temperatures. I found all 1D and 2D

chalcogen structures to be semiconducting.

109

Chapter 7

Effect of Net Charge on the Relative

Stability of 2D Boron Allotropes

The following discussion is my original contribution to the related publication by Dan Liu

and David Tom´anek, Nano Lett. 19, 1359-1365 (2019) [114].

7.1

Introduction

Identifying the most stable allotrope of a given compound is one of the key problems in

Physics and Chemistry. Whereas charge neutral systems have attracted most attention,

notable exceptions are low-dimensional systems that can be charged by extra electrons in

specific bulk geometries. Excess of electrons induced by Li intercalation does not affect

the honeycomb structure of two-dimensional (2D) graphene layers in graphite intercalation

compounds (GICs) [115], but does change the equilibrium structure of MoS2 monolayers from

the 2H to the 1T allotrope [116]. Even though the effect of excess charge on chemical bonding

and equilibrium geometry should be general, I expect the most drastic changes to occur in

structures of elemental boron that is notorious for its many stable allotropes [117,118]. What

appears as frustrated bonding in mostly metallic 2D structures of elemental boron, called

borophene, reflects the inability of the element to follow the octet rule of bonding due to

its electron deficiency. I speculate that both the electron deficiency and thus the chemical

110

bonding characteristic may be modified by placing a nonzero charge on B atoms. In that

case, the most stable borophene structure may differ from a puckered triangular lattice with

monatomic vacancies [119–123] or equally stable irregular structures [124–126] identified in
the neutral system. Indeed, placing a net charge of (cid:46)1 e per atom on borophene, provided

by an Al(111) substrate [127], or intercalating Mg ions in-between borophene layers in the

MgB2 compound [128], changes the most stable allotrope to a very different honeycomb

lattice. Structural changes induced by charging may significantly modify the electronic

structure, turning semiconducting 2H-MoS2 to metallic 1T’-MoS2 locally [129] and undoped

borophene to honeycomb lattices in diborides [130, 131] including MgB2, which displays

superconducting behavior [132].

In this study, I explore the effect of net charge on the bonding character and structural

stability of 2D allotropes of boron. The ab initio calculations for the neutral system reveal

a previously unknown stable 2D 
-B structure with a 0.2 eV wide fundamental band gap. I

find that the chemical bonding characteristic in this and other boron structures is strongly

affected by extra charge, including a 23% lattice constant change in 
-B, induced by changing

the net charge from 0.25 holes to 0.25 electrons per B atom. Beyond a critical degree of extra

electrons near 0.5 electrons/atom, the most stable allotrope changes from 
-B to a buckled

honeycomb structure. Additional extra electrons, mimicking a transformation of boron to

carbon, causes a gradual decrease in the degree of buckling of the honeycomb lattice that

can be interpreted as piezoelectric response. I propose that extra electrons can easily be

achieved by placing borophene in direct contact with layered electrides such as Ca2N. In this

system, I find that extra electrons accepted by borophene can be doubled by changing from

the B/Ca2N bilayer to the Ca2N/B/Ca2N sandwich geometry.

As mentioned above, the vast number of stable neutral 2D borophene allotropes, including

111

α−B, β−B, γ−B, δ−B and η−B structures [119,120], reflects a frustrated bonding character

of electron-deficient boron. This element may engage its three valence electrons in only three

covalent bonds, resulting in an electron sextet instead of the desirable octet noble-gas config-

uration. To partly compensate for lack of electrons in the octet configuration, boron atoms

often prefer increasing their coordination to six nearest neighbors in the triangular lattice.

This is essentially equivalent to adding electrons [133, 134] to the individual atoms that are

held together by pure three-center bonding. On the other hand, the stability of the trian-

gular lattice is often enhanced by removing atoms and forming hexagon-shaped monatomic

vacancies. This process somehow mimics subtracting electrons while the structure locally

converts to a honeycomb lattice, where atoms are held together by two-center bonding. The

competition between two-center and three-center bonding has been used to identify the op-

timum concentration of hexagonal vacancies in the neutral triangular lattice [133, 135, 136],

but likely controls also less common neutral structures with four- and five-fold coordinated

boron atoms [124, 125]. Biasing this competition by net charge has been shown to affect the

fraction of hexagonal vacancies in triangular borophene lattices [137]. Even though extra

electrons of B layers has been linked to unexpected superconducting behavior of MgB2 and

found useful to modulate CO2 capture [138], no systematic attention has been paid to the

possibility of deliberately changing the bonding and the equilibrium structure of boron by

excess charge.

A viable possibility to significantly dope 2D structures of elemental boron by electrons

is to place them in direct contact with electrides including Ca2N [139, 140] and Y2C [141].

In these highly electronegative systems with a layered structure, regions of large electron

density are found in-between the layers. The charge density in this electron layer amounts

to one electron per formula unit in Ca2N and two electrons per formula unit in Y2C. The

112

possibility of exfoliation down to a monolayer [139] has been demonstrated in Ca2N, so that

assembly of various vertical heterostructures is possible.

7.2 Computational techniques

The calculations of the stability, equilibrium structure and electronic structure have been

performed using density functional theory (DFT) as implemented in the SIESTA [13] code.

Periodic boundary conditions have been used throughout the study, with monolayers rep-

resented by a periodic array of slabs separated by a 30 ˚A thick vacuum region. I used the

Perdew-Burke-Ernzerhof (PBE) [9] exchange-correlation functional. The SIESTA calcula-

tions used norm-conserving Troullier-Martins pseudopotentials [33], a double-ζ basis includ-

ing polarization orbitals, and a mesh cutoff energy of 250 Ry to determine the self-consistent
charge density, which provided us with a precision in total energy of (cid:46)2 meV/atom. The
reciprocal space has been sampled by a fine grid [34] of 4×4 k-points in the 2D Brillouin

zones (BZ) of the primitive unit cells of neutral and doped borophene containing 32 atoms,
3×3 k-points in the BZ of supercell of heterostructure of Ca2N and 
6, and 3×1 k-points in

the BZ of supercell of heterostructure of Ca2N and 
3. Geometries have been optimized us-

ing the conjugate gradient (CG) method [76], until none of the residual Hellmann-Feynman
forces exceeded 10−2 eV/˚A.

7.3 Results and discussions

Inspired by the honeycomb structure of the negatively doped boron sublattice found in di-

borides including MgB2, I started the investigation of electron doped borophene structures

with the honeycomb lattice. To provide substantial configuration freedom for the lattice

113

Figure 7.1: Previously unexplored neutral borophene allotropes formed by spontaneous con-
version of an artificial honeycomb lattice: (a) 
-B and (b) ω-B. The structures are shown
in top and side view. The lattice vectors (cid:126)a1 and (cid:126)a2, shown in red, delimit the highlighted
primitive unit cells.

structure, I consider a superlattice with 32 atoms per unit cell and subject all atoms in

the supercell to random distortion. To study the effect of extra electrons on the equilib-

rium structure and the chemical nature of bonding, I changed the degree of extra electrons

gradually and optimized each system using the conjugate gradient optimization method.

Starting with no excess charge, I found the neutral honeycomb structure to be unstable

and to convert to rather stable allotropes that have not been reported previously. The first

allotrope, shown in Fig. 7.1(a) and called 
-B, is only 7 meV/atom less stable than the
most stable α(cid:48)-B phase [117] and displays a very uncommon morphology with triangles and

pentagons, quite distinct from the well-documented class of triangular lattices with vacancies.

The structure shown in Fig. 7.1(b), which I call ω-B, has a very similar morphology to known

allotropes containing triangles and hexagons only [119–122], but is 26 meV/atom less stable
than α(cid:48)-B. 
-B and ω-B add to the large number of known allotropes, and I expect many

more to follow.

Both 
-B and ω-B are buckled. Still, I could locate a locally stable, flat counterpart of

ω-B, which is 7 meV/atom less stable than ω-B. The cohesive energies of neutral borophene

114

(a)                                     (b)                     Figure 1a1xya1a2xzε-Byza1xxa1a2ω-BTable 7.1: Cohesive energy Ecoh of selected neutral borophene allotropes, obtained using
DFT-PBE calculations. n is the number of boron atoms per unit cell and Z is the coordi-
nation number of individual atoms in the unit cell.

Ecoh (eV/atom)

5.699a
5.680a
5.706a
5.762b
5.712b
6.23c
5.662b
5.723b
6.19c

n
8
32
8
8
5
5
4
4
4

Z
4

4, 5, 6

5, 6
5, 6

3, 5, 6
3, 5, 6

6

3, 5
3, 5

Allotrope

-B
ω-B
α(cid:48)-B

β12-B

δ6-B
χ3-B

a Present work.
b Reference [117].
c Reference [122].

allotropes are compared in Table 7.1.

Next, I added extra 5, 10, 13, 16, 24 and 32 electrons to the 32-atom unit cell with an ini-

tial honeycomb arrangements and optimized the geometry. I found the optimum geometries

not to depend on initial deformations imposed on the starting structure, which may not be
reachable in molecular dynamics trajectories. The optimum boron structures labelled 
1− 
6
and their average levels of extra electrons (cid:104)∆Q(cid:105) are displayed in Figs. 7.2(a)-7.2(f).Additional
results at the levels of extra electrons of −18 e, −20 e and −28 e in the 32-atom unit cell and
the corresponding average charge per boron (cid:104)∆Q(cid:105) are shown in Figure 7.3.I find all these

structures to differ significantly from the neutral 
-B and ω-B structures in Fig. 7.1, which

have been optimized in the same way.

The structure 
1-B in Fig. 7.2(a), which contains triangles, pentagons and hexagons, is

reminiscent of 
-B, but is completely flat. With increasing extra electrons, the density of

pentagons gradually diminishes and eventually vanishes in 
3-B in Fig. 7.2(c), representing

the buckled χ3 phase [117]. At the same time, the density of hexagons increases from 
1-B

115

Figure 7.2: Electron doped 2D borophene structures obtained by optimizing a distorted
boron honeycomb superlattice with 32 atoms per unit cell. The average excess charge (cid:104)∆Q(cid:105)
per boron atom, specified in the panels, increases from (a) 
1-B to (f) 
6-B. The structures
are shown in top and side view. The lattice vectors (cid:126)a1 and (cid:126)a2, shown in red, delimit the
highlighted unit cells.

to 
3-B until all other polygons are eliminated in 
4 − 
6, shown in Figs. 7.2(d)-7.2(f), as
the extra electrons exceeds |(cid:104)∆Q(cid:105)|(cid:38) 0.5 e/atom. The buckled honeycomb lattice of 
4-B

gradually flattens to the graphene-like structure of 
6-B with increasing extra electrons.

The interpretation of these structural changes is rather straight-forward. With one extra

electron per atom, boron behaves as sp2-bonded carbon with four valence electrons, with

atoms forming the 2D graphene honeycomb lattice. This structure is rather robust with

respect to extra electrons and holes, as evidenced in GIC structures. Similarly, also doped

boron should keep its optimum honeycomb lattice structure even if the net charge may

be smaller or larger than one extra electron per atom. As seen in Figs. 7.2(d)-7.2(f), the

honeycomb structure of borophene, with different degree of buckling, is preferred for the
excess charge (cid:104)∆Q(cid:105) ranging between −0.5 and −1.0 e/atom.

To understand the change in the electronic structure that caused a profound structural

116

Figure 2xzxya1a2a1ε2-B(b)<ΔQ> = −0.312 e/B-atom(d)xyxza1a2a1ε4-B<ΔQ> = −0.500 e/B-atom(f)xyxza1a2a1ε6-B<ΔQ> = −1.000 e/B-atomxyxza1a2a1ε5-B(e)<ΔQ> = −0.750 e/B-atomxyxza1a2a1ε3-B(c)<ΔQ> = −0.406 e/B-atom(a)xyxza1a2a1ε1-B<ΔQ> = −0.156 e/B-atomFigure 7.3: Electron doped 2D borophene structures obtained by optimizing a distorted
boron honeycomb superlattice with 32 atoms per unit cell. The average excess charge (cid:104)∆Q(cid:105)
per boron atom, specified in the panels, increases from (a) to (c). The structures are shown
in top view (upper panels) and side view (lower panels). The lattice vectors (cid:126)a1 and (cid:126)a2,
shown in red, delimit the highlighted unit cells.

change from neutral 
-B in Fig. 7.1(a) to negatively charged 
6-B− in Fig. 7.2(f), I first

probed the charge density changes ∆ρ associated with charging. The results for ∆ρ caused

by placing artificially extra electrons on free-standing borophene structures 
3-B and 
6-B

are shown in Figs. 7.4(a)-7.4(b). ∆ρ can be viewed as the crystal counterpart of the Fukui

function, and the results indicate that the extra electrons are accommodated in pz (or π)

states normal to the layer, very similar to graphene layers.

A realistic way to provide a high degree of extra electrons, I suggest, is to place borophene

layers in contact with the Ca2N electride. The optimum lattice constant of the triangular
lattice of Ca2N is a = 3.97 ˚A. This layered system has the nominal configuration [Ca2N]+·e−,

with layers of Ca2N separated by layers of excess electrons, and can be exfoliated down to

a monolayer chemically [139]. It is to be expected that borophene will be electron doped

when placed in the region of excess electrons outside a Ca2N monolayer. Since I focus on

general trends rather than minute details, I studied the charge redistribution using only

two prototype structures of doped borophene, namely 
3-B at low- and 
6-B at high-level

of extra electrons, in contact with Ca2N. To determine the degree of extra electrons in

borophene caused by a contact to Ca2N, I inspected the charge redistribution when placing

117

Figure S1xyxzxyxzxyxz(a)                                (b)                          (c)  <∆Q> = −0.563 e/B<∆Q> = −0.625 e/B<∆Q> = −0.875 e/Ba1a2a1a2a1a2a1a1a1Figure 7.4: Charge redistribution in 2D borophene layers induced by extra electrons or by
contact with monolayers of Ca2N electride. Charge density difference ∆ρ caused by placing
an excess charge (a) (cid:104)∆Q(cid:105) = −0.406 e/atom on 
3-B and (b) (cid:104)∆Q(cid:105) = −1.0 e/atom on 
6-B.

Charge density redistribution ∆ρ = ρ(B/Ca2N)−ρ(B)−(cid:80) ρ(Ca2N) in the bilayer structures

(c) 
3-B/Ca2N and (d) 
6-B/Ca2N as well as in the sandwich structures (e) Ca2N/
3-B/Ca2N
and (f) Ca2N/
6-B/Ca2N. ∆ρ is shown by isosurfaces bounding regions of electron excess
at +7×10−3 e/˚A3 (yellow) and electron deficiency at −2 × 10−3 e/˚A3 (blue). (cid:104)∆ρ(z)(cid:105) is
averaged across the x − y plane of the layers.


3-B and 
6-B on top of a Ca2N monolayer or, as in a sandwich geometry, in-between Ca2N

monolayers. The charge density differences caused by electron redistribution in the system

are shown in Figs. 7.4(c)-7.4(f).Electron flow from the Ca2N electride bilayer towards 
3-B is

shown in Figure 7.6(a) and towards 
6-B is shown in Figure 7.6(b). The net electron transfer

results are similar to those for Ca2N monolayers in Figure 3 of the main manuscript, namely
(cid:104)∆Q(cid:105) = −0.15 e/B in 
3-B and (cid:104)∆Q(cid:105) = −0.20 e/B in 
6-B. This indicates that the number

of Ca2N layers does not affect the charge transfer to borophene.

I should point out that incommensurate vertical heterostructures formed of doped borophene

and Ca2N layers can not be represented accurately in a periodic structure used in the com-

118

012246810-4     0       4     8z (Å)(a)(c)(b)(d)(e)(f)012246810141618-8     -4     0      4    8z (Å)012246810-4     0       4      8Δρ(z)(x10-3e/Å3)z (Å)012246810141618-8   -4   0     4   8   12z (Å)Figure 3Isosurfacee excess (-): 0.007 e/Å3Isosurfacee deficit (+): 0.002 e/Å3electron excesselectron deficitelectron excessB       Ca      NΔρ(z)(x10-3e/Å3)Δρ(z)(x10-3e/Å3)Δρ(z)(x10-3e/Å3)ε3-Bε3-Bε3-Bε6-Bε6-Bε3-Bε6-Bε6-BFigure 7.5: Changes in the interlayer interaction energy ∆E and net average charge (cid:104)∆Q(cid:105) on
borophene in the Ca2N/B/Ca2N sandwich geometry as a function of the interlayer distance
d. Results for 
3-B in (a) and (c) are compared to those for 
6-B in (b) and (d).

putational approach. In the calculation, I used the optimum interlayer distance d = 4.0 ˚A
in agreement with the results in Figs. 7.5(a) and 7.5(b). I furthermore matched 1×3 
3-B
supercells with 3×7 Ca2N supercells in the 
3-B/Ca2N superstructure and primitive unit
cells of 
6-B with 3×3 supercells of Ca2N in the 
6-B/Ca2N superstructure. The remaining
lattice mismatch of (cid:46)2% was accommodated by averaging the lattice constants of borophene

and Ca2N. Due to this minor lattice distortion, results presented in Figs. 7.4(c)-7.4(f) may

differ to a small degree from the charge redistribution in an incommensurate structure. Com-

paring results for the bilayer in Figs. 7.4(c)-7.4(d) with those in the sandwich structure in

Figs. 7.4(e)-7.4(f), I see clearly that borophene receives twice the number of electrons in the
sandwich in comparison to the bilayer structure. Specifically, (cid:104)∆Q(cid:105) in 
3-B almost doubles

119

(b)(a)(d)(c)Figure 4dd1.0-1.00.50.0-0.53.06.03.54.04.55.05.5d (Å)ΔE (eV/B atom)0.400.200.350.300.250.153.06.03.54.04.55.05.5<ΔQ> (e/B atom)d (Å)0.550.200.500.450.400.350.300.253.06.03.54.04.55.05.5<ΔQ> (e/B atom)d (Å)d1.51.00.5-0.5-1.0-1.53.06.03.54.04.55.05.5dΔE (eV/B atom)0.0d (Å)ε6-Bε3-Bε3-Bε6-Bdensity redistribution ∆ρ = ρ(B/[Ca2N]2)−ρ(B)−(cid:80) ρ(Ca2N) in (a) 
3-B/[Ca2N]2 and (b)

Figure 7.6: Electron flow between borophene and bilayers of the Ca2N electride. Charge


6-B/[Ca2N]2 triple-layer heterostructures. ∆ρ is shown by isosurfaces bounding regions of
electron excess at +7×10−3 e/˚A3 (yellow) and electron deficiency at −2× 10−3 e/˚A3 (blue).
(cid:104)∆ρ(z)(cid:105) is averaged across the x − y plane of the layers.

from −0.16 e/atom in Fig. 7.4(c) to −0.31 e/atom in Fig. 7.4(e). Similarly, (cid:104)∆Q(cid:105) in 
6-B
almost doubles from −0.21 e/atom in Fig. 7.4(d) to −0.41 e/atom in Fig. 7.4(f).

I have evaluated the dependence of the total energy on the interlayer distance d in the

Ca2N/B/Ca2N sandwich structure for the 
3-B and 
6-B allotropes and present the results
in Figs. 7.5(a) and 7.5(b). In both structures, the optimum interlayer distance d≈4.0 ˚A.

There is a significant electron accumulation with a maximum at d≈3 ˚A outside a free-

standing Ca2N electride layer [139], which is accommodated by an adjacent borophene layer.

In this case, the net charge on the borophene layer may be changed by changing the interlayer
distance d. The results for (cid:104)∆Q(cid:105) as a function of d are presented in Fig. 7.5(c) for 
3-B and

in Fig. 7.5(d) for 
6-B sandwiched in-between two Ca2N layers. I find it interesting that
(cid:104)∆Q(cid:105) decreases almost linearly with increasing interlayer distance.

At the equilibrium interlayer distance, I find (cid:104)∆Q(cid:105)≈− 0.31 e/B-atom in the system with

3-B and (cid:104)∆Q(cid:105)≈− 0.41 e/B-atom in the system with 
6-B. The higher value of (cid:104)∆Q(cid:105) in 
6-B

is associated with the better ability of this structure to accept electrons.

120

(a)(b)Figure S2-4     0       4      8Δρ(z)(x10-3e/Å3)z (Å)01224681014161820-4     0       4      8Δρ(z)(x10-3e/Å3)z (Å)01224681014161820electron excesselectron deficitB       Ca      Nε3-B/[Ca2N]2ε6-B/[Ca2N]2Figure 7.7: Effect of net charge on the equilibrium geometry of 
-B. (a) Effect of the net
average charge (cid:104)∆Q(cid:105) on the orthogonal lattice constants ai with i = 1, 2. Plotted are
charge-induced relative changes ∆ai/a0
i are the lattice constants in the neutral

-B allotrope. (b) Strain energy ∆E as a function of in-layer strain σ at different levels of
extra electrons (cid:104)∆Q(cid:105).

i , where a0

It is well known that in GICs, the net charge transferred from dopant atoms to the

graphene layers changes the lattice constant. This change is relatively small, amounting
to [142] ∆a/a≈0.8% in KC8 with (cid:104)∆Q(cid:105)≈ − 0.125 e/C-atom. The corresponding results for

the effect of extra electrons on the lattice constants in the stable 
-B allotrope are shown in

Fig. 7.7(a). The lattice changes are anisotropic and larger than found in graphene. I find

that extra electrons expands the lattice more along the softer (cid:126)a1 direction than along the
harder (cid:126)a2 direction. At the levels of extra electrons (cid:104)∆Q(cid:105)≈−0.3 to −0.5 e/B-atom discussed

above for the Ca2N/B/Ca2N heterostructures, the lattice expansion exceeds 10%.

The in-layer stiffness of 
-B with neutral, extra electrons and extra holes is addressed
I used the shorthand notation 0 for the neutral system, − for (cid:104)∆Q(cid:105) =
in Fig. 7.7(b).
−0.125 e/B and + for (cid:104)∆Q(cid:105) = +0.125 e/B. Irrespective of extra electron, the energy change

due to in-layer strain σ is much larger along the harder (cid:126)a2 direction than along the softer

(cid:126)a1 direction.
In terms of the 2D elastic constants [19], I find for the softer (cid:126)a1 direction
c11(−) = 67.99 N/m, c11(0) = 18.70 N/m, and c11(+) = 29.98 N/m. Along the harder (cid:126)a2

121

Figure 5(a)ε-B<∆Q> (e/B atom)hole dopedelectrondopedΔai/ai0Δa1/a10Δa2/a20-0.3    -0.2    -0.1    0.0     0.1     0.2     0.30.100.050.00-0.05-0.10-0.15(b)-6%-4%-2%2%4%6%000.10.20.00.10.20.00.10.2σalong a2along a1along a1along a2along a1along a2<ΔQ> = 0.0 (e/B)<ΔQ> = ‐0.125 (e/B)<ΔQ> = +0.125 (e/B)∆E (eV/B atom)Figure 7.8: Electronic band structure of (a) neutral 
-B, (b) 
1-B with (cid:104)∆Q(cid:105) = −0.156 e/B,
(c) 
3-B with (cid:104)∆Q(cid:105) = −0.406 e/B, (d) 
4-B with (cid:104)∆Q(cid:105) = −0.500 e/B, (e) 
5-B with
(cid:104)∆Q(cid:105) = −0.750 e/B, and (f) 
6-B with (cid:104)∆Q(cid:105) = −1.000 e/B, calculated using the DFT-PBE
functional. The indirect fundamental band gap is indicated by the red arrow in (a).

direction I find c22(−) = 446.79 N/m, c22(0) = 360.52 N/m, and c22(+) = 338.58 N/m.

The electronic band structure of selected boron 2D allotropes discussed in this study is

shown in Fig. 7.8. I should note that DFT calculations used in this study do not represent

the true quasi-particle band structure and typically underestimate band gaps. With this fact

in mind, I find that, according to the PBE results in Fig. 7.8(a), the neutral 
-B allotrope

of Fig. 7.1(a) is a semiconductor with a small indirect band gap of Eg = 0.2 eV. Whereas

stretching along the soft (cid:126)a1 direction by 4% turns 
-B into a direct-gap semiconductor,

compressing by 4% along (cid:126)a1 causes gap closure. Stretching 
-B by 1% along the hard (cid:126)a2

direction turns this allotrope metallic, whereas compression by 1% changes the its indirect

122

(a)(b)(c)Figure 6EF123‐1‐2‐3Energy (eV)ΓK1K2MΓK1        K2M      ΓEF123‐1‐2‐3Energy (eV)ΓK1K2MΓK1        K2M      ΓEF123‐1‐2‐3Energy (eV)ΓXYSΓYS        X     Γ(d)(e)EF123‐1‐2‐3Energy (eV)ΓK1K2MΓK1        K2M      ΓEF123‐1‐2‐3Energy (eV)ΓK1        K2M      ΓΓK1K2Mε1ε3ε4ε5(f)EF123‐1‐2‐3Energy (eV)ΓK1K2MΓK1        K2M      Γε6ε<∆Q> = 0 e/B<∆Q> = −0.156 e/B<∆Q> = −0.312 e/B<∆Q> = −0.500 e/B<∆Q> = −0.750 e/B<∆Q> = −1.000 e/BFigure 7.9: Electronic band structure of 
-B subject to in-layer strain of (a) -6%, (b) -4%,
(c) +4%, and (d) +6% along the (cid:126)a1 direction, and (e) -3%, (f) -1%, (g) +1%, and (h)
+3% along the (cid:126)a2 direction. The fundamental band gap is indicated by the red arrows in
semiconducting systems.

gap to a direct gap.The neutral 
-borophene monolayer is an indirect-gap semiconductor,

as seen in Figure 66(a) of the main manuscript. Its electronic structure changes, however,

significantly, when it is subject to in-layer strain, as shown in Figure 7.9. The effect of

strain along the softer (cid:126)a1 direction is shown in Figures 7.9(a)-7.9(d). The results indicate

that compression exceeding 4% closes the gap and turns 
-B metallic. Stretching opens the

gap and, when in excess of 4%, turns the gap direct. The effect of strain along the harder

(cid:126)a2 direction is shown in Figures 7.9(e)-7.9(h). The trend is opposite in this case, since it

is compression that opens the gap and turns it direct, and stretching beyond 3% turns the

system metallic. Ability to tune very narrow band gaps by moderate strain appears very

interesting for Terahertz applications.

I find all doped borophene allotropes to be metallic or semi-metallic, as shown in Figs. 7.8(b)-

123

EF123‐1‐2‐3EF123‐1‐2‐3Energy (eV)ΓY    S        X     ΓEF123‐1‐2‐3ΓY    S        X     Γ(a)(b)(c)(d)(f)(g)(h)Figure S4ΓXYS‐6% (a1)EF123‐1‐2‐3ΓY    S        X     ΓΓXYS‐4% (a1)ΓY    S        X     ΓΓXYS+4% (a1)EF123‐1‐2‐3ΓY   S        X     ΓΓXYS+6% (a1)EF123‐1‐2‐3ΓY    S        X     ΓΓXYS‐1% (a2)EF123‐1‐2‐3ΓY    S        X     ΓΓXYS+1% (a2)ΓXYS+3% (a2)(e)EF123‐1‐2‐3Energy (eV)ΓY    S        X     ΓΓXYS‐3% (a2)7.8(f). Inspection of the band structure reveals the formation of a Dirac cone at K1 and

K2 in the honeycomb structures in 
5-B in Fig. 7.8(e) and in 
6-B in Fig. 7.8(f). The Dirac
cone appears ≈1.4 eV above EF in 
5-B at the level of extra electrons (cid:104)∆Q(cid:105) = −0.750 e/B
and at EF in 
6-B at (cid:104)∆Q(cid:105) = −1.000 e/B, which mimics the structure and valence charge

of graphitic carbon.

The majority of the reported stable structures of neutral 2D borophene were triangular

lattices containing arrays of monatomic vacancies, reflecting the frustrated bonding character

of electron-deficient boron. When considering the effect of excess charge on the bonding

geometry, I observed a transition driven by increasing net negative charge from structures

containing triangles and higher polygons, depicted in Fig. 7.1(a) and Figs. 7.2(a)-7.2(c), for
|(cid:104)∆Q(cid:105)| < 0.5 e/B to all-hexagon structures, depicted in Figs. 7.2(c)-7.2(f), for |(cid:104)∆Q(cid:105)| >

0.5 e/B. The highest coordinations number of six that may be achieved in a network of

triangles reflects to some degree the vain attempt of neutral boron atoms to satisfy the octet
rule. Excess negative charge, with the maximum value |(cid:104)∆Q(cid:105)| = 1.0 e/B considered here,

offers the ability to satisfy this rule in the honeycomb network of 
6-B in Fig. 7.2(f).

To understand the chemical origin of these structural changes, I need to inspect the charge

redistribution caused by extra electrons, corresponding to the Fukui function for a crystal.In

parallel to Figure 63(a) and 63(b) of the main manuscript, where I showed the spatial

distribution of extra electrons added to two borophene structures, I present in Figure 7.10

corresponding results for the distribution of extra two electron charges added to the 32-
atom unit cell of borophene, equivalent to (cid:104)∆Q(cid:105) = −0.0625 e/atom. I considered negatively

charged structures 
1-B in Figure 7.10(a), 
2-B in Figure 7.10(b), 
3-B in Figure 7.10(c), and


4-B in Figure 7.10(d). In addition to these structures described in Figure 2 of the main

manuscript, I present results for the structure depicted in Figure 7.3(b) in Figure 7.10(e),

124

Figure 7.10: Charge density difference ∆ρ caused by placing an excess charge (cid:104)∆Q(cid:105) =
−0.0625 e/atom on selected borophene structures characterized in Figure 7.2 of the main
manuscript and in Figure 7.3. (a) 
1-B of Figure 7.2(a), (b) 
2-B of Figure 7.2(b), (c) 
3-B
of Figure 7.2(c), (d) 
4-B of Figure 7.2(d), as well as B structures shown in (e) Figure 7.3(b)
and (f) Figure 7.3(c). ∆ρ is shown by isosurfaces bounding regions of electron excess at
+2.5×10−3 e/˚A3 (yellow).

and for that of Figure 7.3(c) in Figure 7.10(f). The results indicate that the electrons

are initially occupying pz (or π) states, causing increasing buckling, which is evident when

comparing the side views in Figures 7.10(a)-7.10(c). After transition to an all-hexagon

lattice in Figure 7.10(d), electrons begin gradually occupying σ states, causing flattening

of the honeycomb lattice, until the excess charge of one electron per B atom is reached in

Figure 7.10(f). Occupation of the σ-states in the bond region even at levels of extra electrons

as low as half an electron per boron is facilitated by the fact that the σ band lies much closer

to the π band and to the Fermi level than in graphene.

Complementing the results presented in Fig. 7.4 for borophene interacting with Ca2N,

these results suggest that additional electrons are first accommodated in pz (or π) states

normal to the borophene layer, which cause buckling, and then in σ states, which reduce the

125

Figure S5(a) ε1: 5e/32B‐atom+2e (isosurface: 0.0025/A3)(b) ε2: 10e/32B‐atom+2e (isosurface: 0.0025/A3)(c) ε3: 13e/32B‐atom+2e (isosurface: 0.0025/A3)(d) ε4: 16e/32B‐atom+2e (isosurface: 0.0025/A3)(e) ε8: 20e/32B‐atom+2e (isosurface: 0.0025/A3)(f) ε9: 28e/32B‐atom+2e (isosurface: 0.0025/A3)(a)(c)(d)(e)(f)electron excessxxxxxxyzyzyzxxyzxxyzxxyz(b)<∆Q> = −0.156 e/B<∆Q> = −0.312 e/B<∆Q> = −0.406 e/B<∆Q> = −0.500 e/B<∆Q> = −0.625 e/B<∆Q> = −0.875 e/Bamount of buckling. Starting with the lightly electron doped 
1-B containing triangles, I ob-
serve an increasing degree of buckling with increasing extra electrons up to |(cid:104)∆Q(cid:105)|(cid:46)0.5 e/B.

At that point, the structure changes to the honeycomb structure mimicking hole-doped
graphene. For |(cid:104)∆Q(cid:105)| > 0.5 e/B, additional excess charge gets increasingly accommodated

in σ states, which are much closer to EF in borophene than in graphene, causing a reduction
of buckling down to zero for (cid:104)∆Q(cid:105) = −1.0 e/B.

Assuming that the Ca2N electride can transfer up to one electron per formula unit to a
borophene layer, I can expect the maximum average charge (cid:104)∆Q(cid:105) per boron atom in B/Ca2N
bilayers to range from −0.22 e in 
3-B/Ca2N to −0.28 e/atom in 
6-B/Ca2N. The transferred

charge may be up to twice as large in Ca2N/B/Ca2N sandwich structures. These values are

slightly larger, but close to those found in the actual heterostructures, reported in Fig. 7.4.

I compared the charge transfer between monolayers and multilayers of the Ca2N electride

in contact with borophene and found essentially no difference. Thus, the number of Ca2N
layers does not affect the maximum value of (cid:104)∆Q(cid:105).

It is not easy to achieve the level of extra electrons (cid:104)∆Q(cid:105) = −1.0 e/B by contacting an

electronegative material. According to the results for the Ca2N/B/Ca2N sandwich structure
in Fig. 7.5(d), the Ca2N electride can provide only up to ≈0.4 electrons per boron atom,

much less than the desired level of extra electrons of 1 e/B. An alternative to Ca2N is Y2C,

which can supply twice as many electrons as Ca2N, but is hard to exfoliate. Assuming

maximum charge transfer from Y2C to borophene, B could receive up to 0.8 electrons in

the Y2C/B/Y2C heterostructure. Such a large electron transfer should further augment

the Coulomb attraction between borophene and Y2C, thus further deducing the interlayer

distance, as seen in Figs. 7.4(a) and 7.4(b), and increase the amount of electron transfer,

possibly up to 1.0 e/B.

126

I have mentioned electronic structure parallels between the honeycomb structure of 
6-B

carrying one extra electron per atom and graphene. Even though the system of π electrons

near EF and the Dirac cone in the corner of the Brillouin zone occur in both systems, there

are notable differences between the systems. In graphene, the top of the σ-band lies more

than 3 eV below EF , whereas this energy difference is only 0.2 eV in 
6-B. Apparently,

the lower core charge of elemental boron is the main reason, why the σ and π bands are

energetically closer than in graphene. As mentioned before, this results in an increased role

of σ states in electron doped borophene structures.

According to the results in Fig. 7.7(a), changing the level of extra electrons (cid:104)∆Q(cid:105) from
−0.25 e to +0.25 e results in a 23% increase of the borophene lattice constant a1. Conversely,

I may speculate that changing the lattice constant should modulate the electron transfer

from the electride to the borophene layer, thus changing the dipole moment normal to the

interface. In that case, an in-plane vibration of the heterostructure will cause this dipole to

oscillate and to emit an electromagnetic signal of the same frequency.

Clearly, the most drastic effect of extra electrons is seen in the hexagonal borophene

structure stabilized in MgB2, which is the cause of superconductivity in this system. Extra

electrons may also be used as a way to change structures in a predictable way. As an

illustration, I find the χ3-B structure [122], which has been synthesized at 680 K on Ag(111),

to be very similar to the electron-doped, buckled 
3-B structure. In that case, a simpler way

to fabricate χ3-B may consist of bringing borophene in contact with an electride, allowing it

to relax to 
3-B while electron doped. The final flat structure of χ3-B may evolve after the

electride has been removed. These two examples illustrate new possibilities of using extra

electrons to change the structure of boron. Even though extra electrons may have a lesser

effect on the equilibrium structure of other systems, the findings about the interplay between

127

structure and excess charge have a general validity.

7.4 Summary

In summary, I have studied the effect of extra electrons on the bonding character and stability

of two-dimensional (2D) structures of elemental boron, called borophene, which is known

to form many stable allotropes. The ab initio calculations for the neutral system have

revealed previously unknown stable 2D 
-B and ω-B structures in addition to previously

reported triangular lattices with monatomic vacancies. I found that the chemical bonding

characteristic in this and other boron structures is strongly affected by extra charge, which

is first accommodated in the π and subsequently in the σ network of electrons. Beyond a

critical degree of extra electrons, the most stable allotrope was found to change from 
-B

containing triangles and higher polygons to a buckled honeycomb structure. With extra

electrons, mimicking a transformation of boron to carbon, causes a gradual decrease in the

degree of buckling of the honeycomb lattice that can be interpreted as piezoelectric response.

I also found that extra electrons can be achieved by placing borophene in direct contact with
layered electrides such as Ca2N. I found that extra electrons can be doubled to ≈0.4 e/B

atom by changing from the B/Ca2N bilayer to the Ca2N/B/Ca2N sandwich geometry.

128

Chapter 8

Towards room-temperature

superconductivity in low-dimensional

C60 nanoarrays: An ab-initio study

The following discussion is my original contribution to the related publication by Dogan Er-

bahar, Dan Liu, Savas Berber and David Tom´anek, Phys. Rev. B 97, 140505(R) (2018) [143].

8.1

Introduction

The quest for room-temperature superconductivity has lost nothing of its appeal during the

30-year long intense search following the observation of superconductivity in cuprate per-

ovskites, with the critical temperature Tc rising from the 30-K range in the La-Ba-Cu-O

system [144] to 77 K in Y-Ba-Cu-O [145]. Current record Tc values of 133 K in the doped

HgBa2Ca2Cu3O8 perovskite [146] and 203 K in sulfur hydride [147] have only been observed

under high pressure. Progress in raising Tc significantly further has lagged behind expecta-

tions. Whereas the microscopic origin of superconductivity is still being speculated about

in high-Tc compounds, the rather high Tc values observed in doped solid C60, possibly even

exceeding 60 K [148], results from strong electron-phonon coupling caused by the dynam-

129

ical Jahn-Teller effect on individual fullerene molecules [149, 150]. In alkali-doped M3C60

molecular solids, Tc could be quantitatively reproduced [149, 150] using the McMillan equa-

tion [151]. The key behind a substantial electron-phonon coupling constant is one of its

factors, namely a high density of states (DOS) at the Fermi level N (EF ), which depends on

the particular element M used to intercalate bulk C60.

Here I propose a way to further increase Tc for superconductivity by increasing N (EF )

and thus the electron-phonon coupling constant λ by reducing the C60 coordination number

Z in doped low-dimensional C60 nanoarrays.

I considered intercalation by both electron

donors and acceptors, as well as electron doping in a solid formed of La@C60 endohedral

complexes. I found that N (EF ) increases with decreasing bandwidth of the partly filled hu

HOMO and t1u LUMO derived frontier bands, which may be achieved by reducing the co-

ordination number of C60. Whereas N (EF ) increases significantly by changing from 3D C60

crystals to 2D arrays of doped fullerenes intercalated in-between graphene layers, N (EF )

reaches its maximum in doped quasi-1D arrays of C60 molecules inside (10, 10) carbon nan-

otubes (CNTs), forming C60@CNT peapods. Whereas partial filling of the t1u-derived band

of C60 may be achieved by adsorbing K atoms on the peapod surface, the desired depopu-

lation of the hu-derived band by adsorbed F is not possible. The results indicate that the

highest Tc value approaching room temperature may occur in electron-doped C60 peapod

arrays or in diluted 3D crystals, where quasi-1D arrangements of C60 form percolation paths.

8.2 Computational methods

I performed density functional theory (DFT) calculations to obtain insight into the effect of

geometrical arrangement of fullerenes on the electronic structure of C60 intercalation com-

130

pounds. I used the Perdew-Zunger [75] form of the spin-polarized exchange-correlation func-

tional in the local density approximation to DFT, as implemented in the SIESTA code [152],

which correctly reproduces the inter-layer spacing and interaction in graphitic structures.

The valence electrons were described by norm-conserving Troullier-Martins pseudopoten-

tials [33] with partial core corrections in the Kleinman-Bylander factorized form [153].

I

used a double-zeta polarized basis and limited the range of the localized orbitals in such a

way that the energy shift caused by their spatial confinement was no more than 10 meV [154].
The Brillouin zone of a 3D lattice of C60 molecules was sampled by 10×10×10 k-points, that
of a 2D lattice by 10×10 k-points, and that of decoupled 1D chains of C60 molecules inside
a nanotube by 10 k-points. The DOS was convoluted by 0.02 eV−1. In a periodic arrange-

ment, 1D structures were separated by 15 ˚A thick vacuum regions and 2D structures by

13 ˚A thick vacuum regions. The charge density and the potentials were determined on a

real-space grid with a mesh cutoff energy of 180 Ry, which was sufficient to achieve a total

energy convergence of better than 2 meV/atom.

8.3 Results and discussions

In alkali-doped M3C60 (M=K, Rb, Cs) fcc crystals, superconductivity with Tc(cid:46)40 K has been

observed [155] and explained by electron-phonon coupling that is modulated by the lattice

constant [149,150]. The same behavior is expected to occur in the isoelectronic La@C60 that

has been isolated from raw soot [156] and found to be stable [157]. When exohedrally doped

M3C60 crystals are exposed to ambient or harsh conditions, other than the dopant atoms M

may penetrate deep inside the lattice, react with the M atoms and destroy superconductivity.

This is much less likely to occur in endohedrally doped La@C60 crystals, since the dopant La

131

Figure 8.1: Schematic arrangement of C60 molecules in (a) a 3D fcc crystal, (b) a 2D
triangular lattice, and (c) a 1D array inside a (10, 10) carbon nanotube peapod. (d) Width
of the t1u-derived band in a 1D, 2D and 3D arrangement of C60 molecules as a function of
the C60-C60 center-to-center separation dcc. The equilibrium value of dcc depends on the
C60 orientation and is indicated by the gray strip in (d) for undoped structures. The ‘error
bars’ reflect the effect of changing the C60 orientation on the bandwidth. (e) Schematic
arrangement of dopant atoms outside the 1D peapod ((cid:78)), inside the nanotube but outside
the fullerene ((cid:4)), and inside the fullerene (+). Dark spheres in (a) represent clusters other
than C60 that separate quasi-1D percolating arrays of fullerenes from the surrounding matrix.
The planes in (b) are only a visual aid.

atoms are enclosed inside the protective C60 cage. As mentioned above, superconductivity in

3D M3C60 crystals is caused by strong electron-phonon coupling related to a dynamical Jahn-

Teller effect on individual C60 cages, made possible by retardation. The dominant role of the

intercalated alkali atoms is to partly fill the t1u LUMO of C60 that broadens to a narrow band

in the M3C60 molecular solid. Changes in Tc caused by pressure or changing the element

M can be traced back to changes in the electron-phonon coupling constant λ = V N (EF ) in

the McMillan equation [149–151]. Since the on-ball Bardeen-Pines interaction V does not

change, λ is proportional to the C60-projected DOS at the Fermi level N (EF ), which – for

electron doping – is roughly inversely proportional to the width of the t1u-derived band. In

hole-doped C60, EF is expected to be lowered into the hu-derived band with an even higher

132

(a)(b)(c)(e)Fig. 1(d)C60-C60separation (Å)t1ubandwidth (eV)ρtot(C60@CNT+7K)−ρtot(C60@CNT) −(cid:80)

Figure 8.2: Density of states (DOS) and charge redistribution in doped C60 crystals. (a)
DOS of the t1u LUMO-derived bands of C60 in 1D, 2D and 3D periodic C60 arrangements
for the C60-C60 separation dcc = 9.8 ˚A. (b) Total and projected DOS of a C60@(10, 10) CNT
peapod doped externally by 7 K donor atoms per C60. (c) Charge density difference ∆ρ =
(d) DOS of the hu HOMO-
derived bands of C60 in 1D, 2D and 3D periodic C60 arrangements for the C60-C60 separation
dcc = 9.8 ˚A. (e) Total and projected DOS of a C60@(10, 10) CNT peapod doped externally
by 8 F acceptor atoms per C60. (f) Counterpart of (c) for C60@CNT+8F. In (c) and (f),
blue contours for electron deficit are shown for ∆ρ = −3.0×10−3 e/bohr3 and red contours
for electron excess are shown for ∆ρ = +3.0×10−3 e/bohr3. All energies are with respect to
EF .

at ρtot(K atom).

N (EF ) value, which may be the cause of the high value Tc≥60 K that has been reported

earlier [148].

All experimental strategies used so far to raise Tc have been based on increasing the

C60-C60 separation dcc in a 3D fcc crystal, which would lower the width of the LUMO-

and HOMO-derived bands and thus increase N (EF ) in doped crystals. The approach is

quite different [158]: I consider increasing N (EF ) by reducing the number of C60 nearest

neighbors. As seen in Fig. 8.1(a), this may be achieved simply in a 3D crystal by mixing C60

133

(a)(b)(c)(d)(e)(f)Electron excessElectron deficitFig. 2C60@(10,10) CNT+ 7 KC60@(10,10) CNT+ 8 Ft1uof C60huof C60EFEFtotalC60CNTKC60@(10,10) CNT+ 7 KtotalC60CNTFC60@(10,10) CNT+ 8 F(a) and (d): FWHM=0.02 eV; (b) and (e): 100 kpts, FWHM=0.02 eVwith clusters of similar size that do not interact with C60, such as BN fullerenes. In this case,

the lowered C60 coordination number would decrease the width of the t1u and hu derived

bands and thus increase N (EF ) in doped crystals. Other C60 arrangements with a lower Z

include 2D arrays of C60 that could possibly be intercalated in graphite [159,160], as seen in

Fig. 8.1(b), or 1D arrays of C60 in C60@NT peapods [161–163] shown in in Fig. 8.1(c). As

seen in Fig. 8.1(d), the width of the t1u-derived band decreases both with increasing C60-C60

separation and with the reduction of dimensionality that translates to the reduction of the

coordination number, with 1D arrangements appearing optimal. Since superconductivity is

suppressed in truly 1D systems according to the Mermin-Wagner theorem [164], I consider

bundles of weakly interacting peapods instead of isolated 1D peapods. As I will show in the

following, the mail role of the nanotube in these systems is to provide a suitable enclosure

that aligns C60 molecules and protects them from the ambient. Due to their weak interaction,

bundles of nanotubes have a very similar DOS as isolated nanotubes. Since the same applies

to peapods, I will consider an isolated peapod a valid representative of a peapod bundle

from the viewpoint of electronic structure.

As indicated in Fig. 8.1(e), peapods with the narrowest bandwidth and potentially high-

est N (EF ) could be doped by donor or acceptor atoms that may be outside the nanotube,

inside the nanotube but outside C60, or inside the C60 molecule such as the La@C60 met-

allofullerene [157]. Within the peapod, the C60-C60 separation could be further increased

by enclosing dopant atoms along with the C60 molecules inside the nanotube. For most of

this study, I will focus on donor doping, causing partial filling of the t1u-derived band, and

will show later that acceptor doping may be hard to achieve.

The DOS shape of the t1u LUMO-derived band in quasi-1D, 2D triangular and 3D fcc

lattices of C60 is depicted in Fig. 8.2(a) and that of the hu HOMO-derived band in the same

134

lattices is shown in Fig. 8.2(d). Clearly, the DOS at EF reaches its maximum near half-

filling of these bands in quasi-1D structures. Since the t1u LUMO-derived band holds up to

6 electrons and the hu HOMO-derived band up to 10 electrons, half-filling of these bands

requires either 3 extra electrons for each C60 and EF inside the t1u-derived band, or depletion

of 5 electrons from each C60 and placing EF inside the hu-derived band. Comparing the

results in Figs. 8.2(a) and 8.2(d), I note that acceptor doping – if achievable – would result

in a significantly higher N (EF ) than donor doping.

The calculated DOS of a C60@(10, 10) peapod doped externally by 7 K atoms per C60 is

shown in Fig. 8.2(b) and the DOS of the corresponding peapod doped externally by 8 F atoms

per C60 is shown in Fig. 8.2(e). Comparing the partial densities of states in these two cases,

I conclude that C60-derived states are barely affected by those of the surrounding nanotube

due to a very small hybridization. In the case of donor doping by K depicted in Fig. 8.2(b), I

clearly observe partial filling of the t1u-derived band of C60 as well as the nearly-free electron

bands of the (10, 10) nanotube [165]. To get a better feel for the charge flow in the system, I
plotted the charge density difference defined by ∆ρ = ρtot(C60@CNT+7K)−ρtot(C60@CNT)

−(cid:80)

at ρtot(K atom) in Fig. 8.2(c). Obviously, there is a net electron flow from K atoms to

the C60@(10, 10) peapod, with the excess charge accommodated both on the C60 and the

nanotube. Integration of the C60-projected DOS in Fig. 8.2(b) up to EF indicates a partial

population of the t1u-derived band by 0.4 electrons.

The calculated DOS of an acceptor-doped peapod, shown in Fig. 8.2(e), presents a very

different picture. I selected F as a suitable electron acceptor due to its high electronegativity.

Unlike in previous studies of acceptor-doped C60, where covalently bonded halogen atoms

disrupted the π-electron network on the molecules [166], F atoms were bonded on the outside

of the nanotube surrounding C60 molecules. As expected, I found the hu-derived band of

135

C60 to be essentially unaffected by the presence of the surrounding nanotube and the 8 F

atoms per C60 outside the nanotube. Still, the C60 molecule remained charge neutral. The

hu-derived narrow band band remained completely filled, located about 1 eV below EF .

I found F atoms to be strongly bonded outside the nanotube, causing pyramidalization of

the CNT, disrupting its π-electron network and opening up a gap at the Fermi level, which

turned the system into a semiconductor. This can be clearly see when inspecting the charge

flow in this system in Fig. 8.2(f). I find that F atoms strongly hybridize with the C atoms of

the tube, redistributing the charge only within the F/CNT subsystem, with no effect on the

net charge of C60. Since hole doping of C60 appears very difficult, I will focus on electron

doping of the t1u-derived band of C60 chains in the following.

As mentioned earlier, the electronic band structure of C60 arrays should depend to a

nontrivial degree on the orientation of the C60 molecules that will affect the interaction

between adjacent molecules [167]. I studied 5 different orientations, identified in Fig. 8.3(a),

which result in a different degree of inter-ball hybridization. Since the effect of orientation on
the total energy of C60 chains amounts to ≤2 meV/atom, I expect many C60 orientations to

coexist within a quasi-1D C60 array inside a peapod. The DOS for a chain of C60 molecules

at different orientations and the C60-C60 separation dcc = 9.8 ˚A is shown in Fig. 8.3(b).

I note that the maximum DOS value changes significantly with orientation. Therefore, in

Fig. 8.3(c), I plotted the range of achievable maxima of N (EF ) as ‘error bars’ for different

C60-C60 orientations. I find that very high values of N (EF ) may be achieved in 1D chains

of C60 molecules, since doping further increases C60-C60 separations beyond the value found

in pristine peapods.

To estimate the critical temperature for superconductivity Tc, I used McMillan’s equa-

tion [149–151]

136

Figure 8.3: Properties of 1D arrays of C60 molecules found in CNT peapods.
(a) Ball-
and-stick models of different atomic arrangements at the C60-C60 interface. Considered
are double-bonds facing double-bonds (d), hexagons facing hexagons (h), pentagons facing
pentagons (p). α and β arrangements are obtained by a 90◦ rotation of C60 molecules in p
arrangement about two different axes that are orthogonal to the chain axis and to each other.
(b) DOS of the t1u LUMO-derived bands of C60 for the C60 orientations, defined in (a), at
the C60-C60 separation dcc = 9.8 ˚A. All energies are with respect to EF . (c) Maximum
DOS value of t1u-derived bands in 1D, 2D and 3D crystals of C60. The ‘error bars’ reflect
the effect of changing the C60 orientation. (d) Critical temperature for superconductivity Tc
based on the McMillan equation (8.1) and using N (EF ) from (c). The lines in (c) and (d)
are guides to the eye.

(cid:20) −1.04(1 + λ)

λ − µ∗ − 0.62λµ∗

(cid:21)

.

Tc =

ωlog
1.2kB

exp

(8.1)

This equation describes the solution of the Eliashberg equations in superconductors with a
strong electron-phonon coupling (λ≤2) in a semi-empirical way that is physically appealing.

It also has been found to correctly reproduce the observed Tc values in M3C60 solids as

a function of the C60-C60 separation [150], shown by the data points for 3D systems in
Fig. 8.3(d). I used the parameters of Ref. [ [150]], namely ωlog/kB = 2800 K, µ∗ = 0.2 for

the effective mass and V = 52 meV for the Bardeen-Pines interaction, which are not affected

137

(a)(b)(c)(d)Fig. 3dβhpαC60C60C60C60C60-C60separation (Å)C60-C60separation (Å)αβDOS (e/eV/spin)by the local arrangement of C60 molecules. Using λ = V N (EF ) for the electron-phonon

coupling constant, I were able to convert N (EF ) values for 3D, 2D and quasi-1D systems

with different C60-C60 distances to potentially achievable Tc values and present the results

in Fig. 8.3(d). The estimates indicate that, in the best imaginable scenario, Tc near room

temperature may be achievable using bundles of donor-doped peapods.

Clearly, there are limits to the range of C60-C60 separations dcc compatible with super-

conductivity. Increasing dcc decreases the inter-ball hopping integral t, while not affecting

the on-ball Coulomb integral U . At large C60-C60 separations, the U/t ratio should increase

beyond a critical value that would change doped C60 from a metal to a Mott-Hubbard

insulator [168, 169].

8.4 Summary

In summary, I have proposed a viable way to further increase Tc for superconductivity by

increasing the C60-projected density of states (DOS) at the Fermi level N (EF ) and thus the

electron-phonon coupling constant in doped low-dimensional C60 nanoarrays. I considered

intercalation by both electron donors and acceptors, as well as electron doping in a solid

formed of La@C60 endohedral complexes.

I found that N (EF ) increases with decreasing

bandwidth of the partly filled hu HOMO- and t1u LUMO-derived frontier bands, which may

be achieved by reducing the coordination number of C60. N (EF ) increases significantly

by changing from 3D C60 crystals to 2D arrays of doped fullerenes intercalated in-between

graphene layers and reaches its maximum in doped quasi-1D arrays of C60 molecules inside

a (10, 10) carbon nanotube, forming a C60@CNT peapod. Whereas partial filling of the

t1u-derived band may be achieved by adsorbing alkali atoms outside the 1D peapod, the

138

desired depopulation of the hu-derived band could not be achieved by F atoms adsorbed

on the nanotube surrounding the C60 molecules. The results indicate that the highest Tc

values may occur in electron-doped C60 peapods or in dilute 3D crystals, where quasi-1D

arrangements of C60 form percolation paths. Only experimental evidence will show if low-

dimensional arrays of doped C60 will become superconducting with Tc approaching room

temperature, or rather turn to a Mott-Hubbard insulator.

139

Chapter 9

Shear instability in twisted bilayer

graphene

The following discussion is my original contribution to the related publication by Xianqing

Lin, Dan Liu and David Tom´anek, Phys. Rev. B 98, 195432 (2018) [170].

9.1

Introduction

Theoretically postulated [171–174] drastic changes in the electronic structure at the Fermi

level EF of bilayer graphene (BLG) near the ‘magic’ twist angle θm(cid:39)1.08◦ have been recently

confirmed experimentally [175, 176] and ignited a feverish research effort in twisted BLG

(TBLG). Superconductivity [176] and strongly correlated electronic behavior [175], observed

at θm, are associated with a meV-wide flat band around the Dirac point [171] at EF with a

vanishing density of states (DOS). This flat band, which is formed only within an extremely
narrow range ∆θ < 0.1◦ around θm, is separated by band gaps above and below from the rest

of the electronic spectrum [171, 175–177]. From the viewpoint of atomic structure, nonzero

twist causes a Moir´e pattern with domains of AB, BA and AA stacking to change rapidly in

size especially at small values of the twist angle θ. In view of the unusual sensitivity of the

electronic structure to twist angle θ alone, I study here the effect of two other soft deformation

modes, namely global shear and atomic relaxation in TBLG. Published data suggest that

140

shear does affect the electronic structure of untwisted and unrelaxed BLG [178], but do not

report associated energy changes. Many calculations have explored atomic relaxations in

untwisted [179] and twisted BLG and their effect on the electronic structure [180–188], but

ignored inhomogeneities in the stacking structure observed by high-resolution transmission

electron microscopy (TEM) [5, 189].

Here I combine continuum elasticity theory with a tight-binding description of the elec-

tronic structure [190] to study the behavior of BLG under combined twist and shear.
I
focus on geometries with a twist angle θ near the observed magic angle θm≈1.08◦, where the

electronic structure, including the appearance and disappearance of a flat band near EF as

well as band gaps above and below, shows extreme sensitivity to structural deformations. I

find TBLG near θm to be energetically unstable with respect to global shear by the angle
α≈0.08◦. Also, I find that the effect of shear on the electronic structure is as important as
that of atomic relaxation. Under optimum global shear, calculated θm is reduced by 0.04◦

and agrees with the observed value.

9.2 Computational methods

I use precise experimental data for monolayer and bilayer graphene as input for the calcu-

lations. For unstable geometries, where such data are not available, I use ab initio density

functional theory (DFT) as implemented in the VASP code [14, 16, 191]. I used projector-

augmented-wave (PAW) pseudopotentials [17,18] and the SCAN+rVV10 exchange-correlation

functional [192], which provides a proper description of the van der Waals interaction [193].

I used 800 eV as the electronic kinetic energy cutoff for the plane-wave basis and a total
energy difference between subsequent self-consistency iterations below 10−5 eV/atom as the

141

criterion for self-consistency. I reached full convergence, since the calculations were limited

to very small unit cells of AA- and AB-stacked BLG.

9.3 Results and discussions

9.3.1 Deformation Modes in Bilayer Graphene

The softest deformation modes of BLG are relative translation and rotation of the constituent

graphene layers. BLG subject to uniform twist creates a Moir´e lattice of equilateral triangles,

and rigid displacement only translates this lattice with no effect on the atomic and electronic

structure. Atomic relaxations including local bending in TBLG, as well as global shear

and stretch of the monolayers, require more energy. Global stretch, which involves bond

stretching, is energetically harder than global shear, which involves only bond bending.

Global shear creates a stripe pattern of domains [178]. The combination of twist and shear

has not been explored so far.

The deformation of BLG under shear and twist is defined in Fig. 9.1(a). In the initially

AA-stacked BLG, I define the shear direction within the top layer by the angle β with

respect to the armchair direction. After first being sheared by the angle α, the to player

is subsequently rotated by the angle θ. The mathematical formulation of the shear-twist

deformation in BLG is detailed in the Related Information section. The general Moir´e

pattern of a sheared and twisted BLG is a lattice asymmetric triangles with AA-stacked

regions forming the vertices, as shown schematically in Fig. 9.1(b). The triangular regions

of AB and BA stacking between the vertices have no symmetry in general, but become
equilateral in the case of pure twist in the BLG associated with α = 0◦. Figure 9.1(c)

contains a reproduction of a dark-field TEM image of the BLG reported in Ref. [5]. The

142

Figure 9.1: Shear and twist in bilayer graphene (BLG). (a) Definition of shear and twist
operations in BLG, initially a bilayer in AA stacking indicated by the honeycomb lattice.
A rectangular segment of the top layer, with one side closing the angle β with respect to
the highlighted armchair direction, is indicated in yellow and surrounded by the dotted line.
The segment is first sheared by the angle α and subsequently rotated by the angle θ. (b)
Schematic top view of a uniformly twisted and sheared monolayer graphene (MLG) on top of
an undeformed MLG. The resulting relaxed Moir´e pattern contains regions of AA stacking,
highlighted by the white circles, and those of AB or BA stacking. (c) Dark-field TEM image
of bilayer graphene reproduced from Ref. [ [5]]. (d) Schematic side view of a relaxed sheared
and twisted BLG with locally varying stacking.

triangular lattice in this figure is strongly distorted due to inhomogeneous twist and strain.

Quantities associated with atomic relaxation are defined in the schematic side view of sheared

and twisted BLG in Fig. 9.1(d). The local inter-layer distance d0 depends on the 2D position

vector R within the BLG plane, which distinguishes regions with different local stacking

sequences.

9.3.2 Relaxations in Sheared and Twisted Bilayer Graphene

To quantify the effect of relaxation in the BLG, I consider both in-plane and out-of-plane

deformations and calculate changes in the total energy Etot = Eel + Eint in terms of the

elastic deformation energy Eel of the individual layers and the interlayer interaction energy

Eint. The displacement of the atom i at rn,i, defined in Fig. 9.1(d) for layers n = 1, 2, is

143

Figure 1αβθBAAB(d)rr1,id0r2,j12RAAAB(a)                                     (b)                     (c)Figure 9.2: (a) Energy difference ∆E caused by shearing the top layer of BLG, which had
been twisted by θm = 1.08◦, by varying the shear angle α along the β = 0◦ direction. Results
for the BLG with unrelaxed atomic positions in planar, sheared monolayers are shown by
the red dotted line and for the BLG with relaxed atomic positions by the black solid line.
Contour plots of the local shift vector length δ = |δ| in the BLG structure, which had been
twisted by θm = 1.08◦ and sheared by α = 0.08◦ along the β = 0◦ direction, (b) in absence
and (c) in presence of lattice relaxation.

described by a continuous displacement field with an in-plane component u(n)(R) and an

out-of-plane component h(n)(R). The elastic deformation energy is given by [194]

(cid:33)2

Eel =

dR

∂2h(n)
∂x2 +

∂2h(n)

∂y2

(cid:90)
(cid:32)

n=1

2(cid:88)
(cid:32)

2

(cid:32)

(cid:26) κ

2

+

(cid:33)2
(cid:32)
(cid:33)2

+

+

+

λ + µ

∂u

(n)
x
∂x

∂u

(n)
y
∂y

µ
2

∂u

(n)
x
∂x

(n)
− ∂u
y
∂y

(cid:33)2(cid:27)

,

∂u

(n)
y
∂x

∂u

(n)
x
∂y

+

(9.1)

where the integral extends over the Moir´e pattern supercell. I use κ = 1.4 eV for the flexural

rigidity [19], and λ = 4.23 eV/˚A2 and µ = 9.04 eV/˚A2 for the 2D elastic Lam´e factors [188]

144

Figure 2-0.05     0.00     0.05     0.10     0.15α(degrees)ΔE (meV/nm2)1.00.50.0unrelaxedrelaxed(c)(a)                                      (b)0             10            200            10         200          10            201.51.00.50.0δ(Å)x (nm)y (nm)y (nm)AAAAAAABBABABAAAAAAAABBABABAof graphene.

To calculate the interlayer interaction energy Eint, I first characterize the local stacking

at position R = r1,i of atom i in layer 1, shown in Fig. 9.1(d), by calculating the connection
vectors r2,j − r1,i to all atoms j of the same sublattice in layer 2 and their projections onto
the x − y plane of the lattice. The shortest among these vectors is then called the shift
√
vector [179] δ(R). With this definition, |δ| = 0 in AA-stacked and |δ| = a/
3 in AB-

stacked BLG, where a = 2.46 ˚A is the lattice constant of graphene. The observed interlayer

distance in the stable AB-stacked BLG is dAB

0 = 3.35 ˚A. The corresponding calculated value

for AA-stacked BLG, which is less stable by ∆E = 0.38 eV/nm2, is dAA

0 = 3.60 ˚A.

The interlayer interaction energy depends on the local interlayer separation d0(R) and

the local shift vector δ(R). Since the Moir´e supercells are much larger than the interatomic

distance in the twist angle range of interest, both quantities can be represented well by a

continuous field. Then, the interlayer interaction energy is given by the integral

(cid:90)

Eint =

V (R)dR ,

(9.2)

which extends over the Moir´e supercell.

I represent the interlayer interaction potential

V (R) = V [δ(R), d0(R)], as well as the quantities d0 and δ, by a Fourier expansion over

the BLG lattice [181], which is detailed in the Related Information section. The expansion

requires only few of the shortest reciprocal superlattice vectors of the BLG and is trivial
for an untwisted BLG with AA and AB stacking, where V = const. and V AA − V AB =

0.38 eV/nm2.

The optimum geometry of the relaxed BLG that had been subject to global shear and

twist is obtained by globally minimizing the total energy, which has been determined using

145

a Fourier expansion detailed in the Related Information section.

Whereas lattice relaxation is driven by energy gain, shear distortion of a monolayer

requires energy investment. For a given twist angle θ, it is conceivable that the energy

invested in shear may be outweighed by an additional energy gain associated with relaxation

to a more favorable structure. This situation is illustrated in Fig. 9.2(a) that shows energy
changes in TBLG with θm = 1.08◦. I found the energy values, which are shown for shear
along the β = 0◦ direction, to be nearly identical for shear along β = 30◦, an conclude that

the shear energy does not depend on β. When lattice relaxation is suppressed, as shown
by the dotted line, the most stable geometry is unsheared with α = 0◦. When allowing for
lattice relaxation, the optimum geometry has acquired a global shear angle α = 0.08◦.

The absolute value of the local shift vector δ = |δ| as a function of R = (x, y) is shown in
Figs. 9.2(b) and 9.2(c) for BLG structures that have been twisted by θm = 1.08◦ and sheared
by the small angle α = 0.08◦ along the β = 0◦ direction. The corners of the Moir´e supercell,

shown in white, are the unshifted AA regions with δ = 0.

In the unrelaxed structure of

Fig. 9.2(b), the energetically favorable regions of AB or BA stacking are rather small. Upon

relaxation, these favorable stacking regions increase in size, as seen in Fig. 9.2(c). The effect

of relaxation becomes much more visible at smaller twist angles θ associated with very large

Moir´e domains. As seen in Fig. 9.4, the AB and BA domains then acquire a distinctly

triangular shape upon relaxation, which has been observed in TEM images [5,189] including

Fig. 9.1(c).

146

9.3.3 Electronic Structure of Sheared, Twisted and Relaxed Bi-

layer Graphene

To study the combined effect of shear, twist and atomic relaxation on the electronic struc-

ture of BLG, I use an extension of the minimum Hamiltonian [190] that had successfully

reproduced the electronic structure of twisted BLG. Due to the high in-plane stiffness and

flexural rigidity of graphene, the atomic relaxation is rather small and smooth across the

BLG lattice, so that the intra-layer nearest-neighbor Hamiltonian with V 0

ppπ = 3.09 eV is

not affected. Since the interlayer separation changes between the value dAA

0

and dAB

0 within

each Moir´e domain, as seen in Fig. 9.1(d), I modify the expressions for the interlayer hopping

in Ref. [ [190]] to

where

t(r, R) = V 0

ppσ(d0)e

(cid:113)

−(

r2+d2

0−d0)/λ

d2
0

r2 + d2
0

,

V 0
ppσ(d0) = V 0

ppσ(dAB

0

−(d0−dAB

0

)e

)/λ(cid:48)

.

(9.3)

(9.4)

In these expressions, the only quantity that depends on the position R within the layer is the

local interlayer separation d0 = d0(R), defined in Fig. 9.1(d). For the unrelaxed geometry

0 = const., the values V 0

with d0 = dAB
0 = 3.35 ˚A, λ = 0.27 ˚A have
been established in Ref. [ [190]]. I furthermore use the parameter λ(cid:48) = 0.58 ˚A to adequately

) = 0.39 eV, dAB

ppσ(dAB

0

describe the dependence of V 0

ppσ on the interlayer separation to match the DFT calculations.

In TBLG, where atomic relaxation has not been considered explicitly in a related previous

study [190], I determined the electronic structure at the interlayer distance dAB

0

using the

same continuum method for the description of eigenstates. In unrelaxed TBLG subject to

147

shear, the interlayer Hamilton matrix elements of Ref. [ [190]] are modified as

(cid:88)

G,G(cid:48)

Ω

(cid:104)ψ1,η(k1)|H|ψ2,ξ(k2)(cid:105) =

˜t(|k1 + G|)

i(G·τη−G(cid:48)·τ(cid:48)
ξ)

e

δk2−k1,G−G(cid:48) .

(9.5)

Here I use η and ξ to represent the two sublattices with the basis vectors τ in the unsheared
bottom layer 1 and τ(cid:48) in the top layer 2 that has been sheared and twisted. kn is the

momentum vector in layer n. ˜t(k) is the 2D Fourier transform of t(r) that is independent of

the position within the unrelaxed bilayer, which is kept at the constant optimum interlayer
. G is the reciprocal lattice vector of the bottom layer 1 and G(cid:48) is the

separation dAB

0

reciprocal lattice vector of the sheared and twisted top layer 2. For non-specific values of

α, β and θ, the BLG lattice is generally incommensurate, but can be approximated by a

commensurate Moir´e superlattice with large unit cells.

For small twist angles, the reciprocal lattice vectors of the undeformed bottom layer 1 in

Eq. (9.5) can be approximated by G = n1b1 + n2b2 and those of the deformed top layer 2
by G(cid:48) = n1b(cid:48)
the two vectors spanning the reciprocal lattice of the bottom layer and b(cid:48)

2, with n1 and n2 being small integers. In these expressions, b1/2 are

1 + n2b(cid:48)

1/2 those spanning

the reciprocal lattice of the deformed top layer. In relaxed commensurate BLG structures, I

do not use Eq. (9.5), but rather diagonalize the tight-binding Hamiltonian directly to obtain

the electronic band structure.

The effect of shear and atomic relaxation on the DOS of TBLG subject to twist by the

magic angle θm = 1.08◦ is discussed in Fig. 9.3 for energies close to EF and in Fig. 9.5

for a wider energy range. The DOS characteristics at θm are a narrow ‘flat band’ around

the Dirac point with vanishing DOS at EF and band gaps above and below. The DOS of

unsheared and unrelaxed TBLG in Fig. 9.3(a) is the same as in a previous report [190] for

148

Figure 9.3: Electronic density of states (DOS) in the BLG structure subject to the magic
twist angle θm = 1.08◦. Results for the unsheared structure are shown in panels (a) and
(c), and those for the top layer sheared by α = 0.08◦ along the β = 0◦ direction in (b) and
(d). Results for the unrelaxed structure in (a) and (b) are compared to those for the relaxed
structure in (c) and (d).

the same structure, where shear and atomic relaxation have not been considered explicitly.
Subjecting BLG to global shear by α = 0.08◦, while suppressing any atomic relaxation,

reduces the width of the flat significantly, as seen in Fig. 9.3(b), to almost half its value in

unsheared TBLG. This result alone proves that even minor shear plays a significant role in

the electronic structure of TBLG. As seen in Fig. 9.3(c), atomic relaxation alone increases

the width of the band near EF significantly with respect to the structure in Fig. 9.3(a) to

the degree that the designation ‘flat band’ is no longer appropriate. Subjecting this relaxed
structure to minor shear of α = 0.08◦, however, narrows down this band to resemble that of

Fig. 9.3(a) for TBLG irrespective of shear and relaxation. Thus, taking shear into account

is essential to properly identify the θm value in relaxed TBLG. I compared the width of

the narrow band around EF for unsheared and sheared TBLG with atomic relaxation. The

149

Figure 3(a)                                           (b)               DOS (eV-1nm-2)40200DOS (eV-1nm-2)40200α=0.00ºunrelaxedα≈0.08ºunrelaxed(c)                                            (d)              DOS (eV-1nm-2)40200Energy (meV)DOS (eV-1nm-2)40200Energy (meV)-10      -5         EF5       10α=0.00ºrelaxedα≈0.08ºrelaxed-10      -5         EF5       10results, reproduced in Fig. 9.6, suggest that the minimum bandwidth is found at the magic
angle θm = 1.12◦ in relaxed TBLG with suppressed shear. This value is larger than the
magic angle value θm = 1.08◦ that is observed in sheared and relaxed TBLG.

As reported in the discussion of shear in Fig. 9.2(a), the deformation energy ∆E is

nearly independent of the shear direction β. Nevertheless, β modifies strongly the shape of

the triangles in the Moir´e superlattice. Local changes in the shear angle α and direction β

are clearly visible in the TEM images of BLG reported in Fig. 9.1(c) and Refs. [ [5]] and

[ [189]]. I also note that the high energy cost of in-layer deformations including shear limits

the degree of atomic relaxation. Distributing shear from one to both layers of BLG should

cut this energy cost in half in the harmonic regime, thus reducing the limits imposed on

atomic relaxation and providing extra energy gain for the system.

9.4 Summary

In conclusion, I studied the effect of combined shear and twist on the energy as well as

the atomic and electronic structure of BLG. I found that the observed drastic changes in

the electronic structure near the Fermi level, caused by minute changes of the twist angle
away from the magic value θm≈1.08◦, are strongly affected by lattice relaxation and global

shear. Using precise experimental data for monolayer and bilayer graphene as input in a

simplified formalism for the electronic structure and elastic energy, I found TBLG near θm
to be unstable with respect to global shear by the angle α≈0.08◦. I also found that shear

and atomic relaxation modified the electronic structure of TBLG to a similar degree. At

optimum global shear, the calculated value of the magic angle θm in relaxed TBLG is reduced
by 0.04◦ to agree with the observed value of 1.08◦.

150

9.5 Related Information

9.5.1 Mathematical formulation of the rigid shear-twist deforma-

tion in bilayer graphene

Before describing the shear-twist deformation in bilayer graphene (BLG), I need to recall

that the honeycomb lattice of graphene consists of two triangular sub-lattices A and B. The
3/2,−1/2) and

√
Bravais lattice of the bottom layer 1 is spanned by the vectors a1 = a(

√
3/2, 1/2) in Cartesian coordinates, where a = 2.46 ˚A is the lattice constant of

a2 = a(

graphene. The basis vectors of the sublattices are τ A = 0 and τ B = (a1 + a2)/3. Initially,

the top layer 2 of the BLG is on top of layer 1 in AA stacking.

The sheer-twist transformation of the BLG top layer with respect to the bottom layer

is shown schematically in Fig. 9.1(a). Pure shear in the graphene top layer by the angle α

along the shear direction angle β with respect to the armchair direction is described by the

transformation matrix

Sα =

1 − sin β cos β tan α

−(sin β)2 tan α

 .

(cos β)2 tan α

1 + sin β cos β tan α

(9.6)

Pure counter-clockwise twist of the top layer by the angle θ with respect to the bottom layer

is described by the unitary transformation matrix

cos θ − sin θ

 .

(9.7)

Tθ =

sin θ

cos θ

The shear-twist operation transforms the two Bravais lattice vectors aj of the top layer to

151

a(cid:48)
j = TθSαaj with j = 1, 2. Except for specific values of θ, α and β, the shear-twisted BLG

(STBLG) lattice is incommensurate. For small values of θ and α, such an incommensurate

structure can be approximated by a commensurate superlattice with with large Moir´e su-

percells. The electronic structure of this system can be obtained to a good accuracy using

the continuum method described in Ref. [ [190]].

The reciprocal Moir´e superlattice is spanned by the two vectors b

(s)

1 = b2 − b(cid:48)

2 and

(s)

1 + b(cid:48)

2 = (b(cid:48)
b
spanning the reciprocal lattice of the bottom layer and the primed vectors b(cid:48)

2) − (b1 + b2).

In the notation, bj with j = 1, 2 are the two vectors

j span the

(s)
2 define the
(s)
(s)
1 and a
2

reciprocal lattice of the top layer. The two reciprocal lattice vectors b

(s)
1 and b

lattice vectors a

(s)
1 and a

(s)
2 of the direct Bravais STBLG superlattice. When a

both belong to the Bravais lattice of the individual layers, STBLG becomes commensurate.

This condition is fulfilled when a

(s)
1 = M1a1 + N1a2 and a

(s)
2 = M2a1 + N2a2 with integer

values of M1, N1, M2, and N2.

The supercell area of the sheared top layer remains the same as that of the unsheared
bottom layer if M2 + N2 = N1 − 1. The quantities M1, N1, M2, and N2 define uniquely

the twist angle θ, the shear angle α and the shear direction angle β of the top layer. One

geometry near the first magic angle, characterized by (M1, N1) = (30, 31) and (M2, N2) =
(−31, 61), describes an unsheared twisted BLG layer with θ = 1.0845◦ and α = 0◦. Another
similar geometry, characterized by (M1, N1) = (31, 32) and (M2, N2) = (−36, 67), describes
a sheared twisted BLG layer with θ = 1.0845◦, α = 0.0872◦, and β = −0.5423◦. I determine

the lattice relaxation and its effect on the electronic structure of the sheared and twisted

BLG using the commensurate structures.

152

9.5.2 Mathematical background of the relaxation treatment in

sheared and twisted bilayer graphene

BLG subject to rigid shear and twist, which is characterized by θ, α and β, is further stabi-

lized by atomic relaxation. In the bottom layer 1 with θ = α = 0, the in-plane displacement

u(1)(R) of an atom with respect to its initial position R can be represented by a Fourier

expansion as

(cid:88)

j

u(1)(R) =

˜u(1)(G

(s)
j ) sin(Gj

(s)·R) .

(9.8)

The sum extends over all vectors G

(s)
j

(s)
is spanned by b
1

and b

(s)
2 .

of the reciprocal Moir´e superlattice of the BLG, which

In the twisted and sheared top layer 2, the in-plane displacement u(2)(R) of an atom with

respect to its position R in the unrelaxed, unsheared, but twisted layer can be expressed by

(cid:88)

u(2)(R) =

˜u(2)(G

(s)
(s)
j ) sin(G
j

·SαR) + SαR − R .

(9.9)

j

Here, the direction angle β of the shear transformation, defined as Sα in Eq. (9.6), has been

rotated by the twist angle θ. I note that atomic displacements described by Eqs. (9.8) and

(9.9) maintain the shape and size of the Moir´e supercells. In BLG subject to twist angles
near θm = 1.08◦, the summation in Eqs. (9.8) and (9.9) requires only three shortest vectors

(s)
of the reciprocal Moir´e superlattice G
2 .
In BLG subject to smaller twist angles θ around 0.4◦, I include three additional vectors

(s)
2 , and G

(s)
1 = b

(s)
2 = b

(s)
1 , G

(s)

3 = −b

(s)

1 − b

G

(s)
4 = 2b

(s)
1 , G

(s)
5 = 2b

(s)
2 , and G

(s)

6 = −2b

(s)

1 − 2b

(s)
2

for convergence.

Expressions for the in-plane atomic displacements u(n)(R) in layers n = 1, 2 allow us to

153

determine the shift vector δ(R) at the position R by

(cid:104)
(cid:105)
u(2)(TθR) − u(1)(R)

δ(R) =

+ [TθR − R] .

(9.10)

The periodicity of u(1)(R) and u(2)(R) of the Moir´e superlattice, described by the Fourier

sum in Eqs. (9.8) and (9.9), describes the same periodic behavior of δ(R).

Similarly, the periodicity in the interlayer separation d0(R) can be evaluated by the

Fourier sum

d0(R) =< d0 > +∆d0

3(cid:88)

j=1

cos(G

(s)
j

·R) .

(9.11)

Here, < d0 > is the average interlayer distance and ∆d0 describes the modulation of d0.

Since d0 varies smoothly across the Moir´e supercell, an adequate description of d0(R) can be

(s)
obtained using only three shortest vectors G
of the reciprocal Moir´e superlattice defined
j
above for j = 1 − 3. Having specified the position dependence of the interlayer distance

d0(R), the out-of-plane displacement h(n) of each individual layer n, used to evaluate Eel

using Eq. (9.1), is given by

h(1)(R) =

h(2)(R) =

(−d0(R) + 3∆d0+ < d0 >) and
(+d0(R) − 3∆d0− < d0 >) .

1
2
1
2

(9.12)

(9.13)

With the Fourier expansions of the in-layer displacement u(n)(R) and the interlayer

distance d0(R), I can determine the elastic energy Eel of STBLG using Eq. (9.1).

To determine the interlayer interaction energy Eint specified in Eq. (9.2), I have to locate

a proper expression for the interlayer interaction potential V (R). Since V (R) is periodic in

154

Figure 9.4: Contour plots of the local shift vector length δ = |δ| in the BLG structure, which
had been twisted by θ = 0.4◦ and sheared by α = 0.08◦ along the β = 0◦ direction, (a) in
absence and (b) in presence of lattice relaxation.

STBLG, I can also express it as a Fourier expansion [181]

(cid:88)

V (R) =

˜V (G

(s)
j ) cos(Gj

(s)·R) .

(9.14)

j

(s)
The sum extends over all vectors G
j

of the reciprocal Moir´e superlattice of the BLG.

In reality, V (R) = V (δ(R), d0(R)) depends not on the global position R, but rather the

quantities δ and d0, which show the same periodicity, as expressed in Eqs. (9.10) and (9.11).

Using straight-forward algebra, I arrive at the expression

V (δ, d0) = VAA(d0) − 6∆V (d0) + 2∆V (d0)

3(cid:88)

j=1

cos(Gj·δ) ,

(9.15)

where the sum is limited to the smallest three vectors G1 = b1, G2 = b2, and G3 =
−b1 − b2, since V varies smoothly with δ.

I have used the expression ∆V (d0) = (1/9)[V AA(d0) − V AB(d0)] for the periodic mod-
ulation of V in Eq. (9.15), where V AA(d0) and V AB(d0) are the total energies of AA- and

AB-stacked BLG per area, which both depend on the interlayer distance d0. Furthermore, I

155

Figure S10               30             60δ(Å)AAAAAAABBABABAAAAAAAABBABABA0            30            600            30             60y (nm)y (nm)x (nm)1.51.00.50.0(a)                                 (b)x (nm)0               30             60Figure 9.5: Electronic density of states (DOS) in the BLG structure subject to the magic
angle twist θm = 1.08◦. Results for the unsheared structure are shown in panels (a) and
(c), and those for the top layer sheared by α = 0.08◦ along the β = 0◦ direction in (b) and
(d). Results for the unrelaxed structure in (a) and (b) are compared to those for the relaxed
structure in (c) and (d). The energy scale is extended in comparison to Fig. 9.3.

have set V AA(dAA

0

) = 0 as a reference value. Using the DFT results, I have fitted VAA(d0)

and VAB(d0) by third-order polynomials. The expressions in units of eV/˚A2 are V AA(d0) =
[0.113(d0 − dAA
)2 −
0.224(d0 − dAB

)2 − 0.340(d0 − dAA
)3]/Ω0 when using d0 in ˚A units and the value Ω0 = 5.24 ˚A2 for the area of

)3]/Ω0 and VAB(d0) = [−0.020 + 0.174(d0 − dAB

0

0

0

0

the graphene unit cell.

With expressions for Eel and Eint in place, I can evaluate the total energy Etot =

Eel + Eint for any BLG geometry. For given global shear characterized by α and β and given

global twist given by θ, I can determine the atomic relaxations by globally minimizing the

total energy with respect to ˜u(n)(G(s)), ∆d0 and < d0 >.

156

Figure S2(a)                                         (b)                 (c)                                        (d)                  DOS (eV-1nm-2)50α=0.00ºunrelaxedDOS (eV-1nm-2)50Energy (meV)-80     -40        EF40       80α=0.00ºrelaxedDOS (eV-1nm-2)50Energy (meV)α≈0.08ºrelaxedDOS (eV-1nm-2)α≈0.08ºunrelaxed50-80     -40        EF40       80Figure 9.6: The width of the ’flat band’ in a relaxed twisted bilayer graphene structure as
a function of the twist angle θ. Data points for commensurate, relaxed structures in the
vicinity of the observed value θm = 1.08◦ are presented by the black diamonds ((cid:7)) for the
unsheared lattice and by the red circles (•) for the structure with the top layer sheared by
α≈0.08◦ along the β = 0◦ direction. The dotted and dashed lines are guides to the eye.

9.5.3 Local relaxation in sheared and twisted bilayer graphene

The pattern of the local shift vectors δ depends primarily on the twist angle θ, which de-

termines the domain size. The pattern in the bilayer graphene (BLG) lattice twisted by a
very small angle θ = 0.4◦ is presented in Fig. 9.4. The domains of the Moir´e superlattice
are significantly larger than in BLG with the magic twist angle θm = 1.08◦ discussed earlier.

The δ pattern in the larger domains is changed substantially by atomic relaxation. In par-

ticular, the domains of AB or BA stacking acquire a distinct triangular character that has

been observed in dark-field TEM images [5, 189].

9.5.4 Electronic structure changes in sheared and twisted bilayer

graphene

The electronic density of states (DOS) in sheared and twisted BLG is shown in Fig. 9.5

in a larger energy window around EF than in Fig. 9.3.

I note that both atomic relax-

157

Figure S3θ(degrees)Bandwidth (meV)ation and global shear modify the spectrum of the unsheared and unrelaxed lattice, shown

in Fig. 9.5(a), significantly. Band gaps above and below the ‘flat band’ region are signifi-

cant only in the relaxed structure. Only the DOS of the sheared and relaxed structure in

Fig. 9.5(d) displays a combination of a narrow flat band and band gaps above and below.

9.5.5 Value of the magic angle in unsheared and sheared twisted

bilayer graphene

Since shear plays an important role in the relaxed twisted bilayer graphene, it is expected to

affect the magic angle θm as well. In Fig. 9.6 I show the width of the narrow band around EF

as a function of the twist angle θ for both an unsheared BLG and for BLG with its top layer
sheared by α≈0.08◦ along the β = 0◦ direction. Only few data points are shown, since only

few commensurate structures with reasonably small unit cells exist in the narrow twist angle

region shown. I notice that the (first) magic angle, associated with the narrowest bandwidth,
occurs at 1.12◦ in the unsheared and at 1.08◦ in the globally sheared BLG. The latter value
in the sheared lattice agrees well with the observed value [175, 176] θm(expt)≈1.08◦.

158

Chapter 10

Two-dimensional Mechanical

Metamaterials with Unusual Poisson

Ratio Behavior

The following discussion is my original contribution to the related publication by Zhibin

Gao, Dan Liu and David Tom´anek, Phys. Rev. Appl. 10, 064039 (2018) [195].

10.1

Introduction

There is growing interest in mechanical metamaterials, man-made structures with counter-

intuitive mechanical properties [196]. Unlike in ordinary uniform materials, deformations in

such metamaterials derive from the geometry of the assembly rather than the elastic prop-

erties of the components. This behavior is scale independent, covering structures from the

macro- to the nanoscale. Most attention in this respect seems to be drawn by the Poisson

ratio ν [197], the negative ratio of lateral to applied strain. Ordinary materials with typical

values 0 < ν < 0.5 contract laterally when stretched, with unusually large values reported

for cellular materials [198]. Auxetic metamaterials with ν < 0, on the other hand, expand in

both directions when stretched [198–202] leading to advanced functionalities [203,204]. Aux-

159

etic systems with macroscopic components have been utilized for shock absorption in auto-

mobiles [205], in high-performance clothing [206–208], in bioprostheses [209] and stents [210]

in medicine, and for strain amplification [211]. Auxetic 2D mechanical metamaterials with

nanostructured components, some of which have been described previously [212–215], may

find their use when precise micromanipulation of 2D structures including bilayer graphene

is required [176].

Here I report the design of 2D mechanical metamaterials that may be deformed substan-

tially at little or no energy cost. Unlike origami- and kirigami-inspired metamaterials, which

derive their functionality from folding a 2D material into the third dimension [216–219], the

structures I describe are confined to a plane during deformation. Such confinement may be

achieved by a strong attraction to a planar substrate or in a sandwich geometry. Specifi-

cally, I consider infinite assemblies of rigid isosceles triangles hinged in their corners on the

macro-scale [220] and polymerized phenanthrene molecules forming ‘porous graphene’ on

the nano-scale. In these and in a large class of related structures, consisting of connected

and near-rigid isosceles triangles, the Poisson ratio ν diverges at particular strain values.

ν also changes its magnitude and sign, and displays a ‘shape memory’ effect in a specific

range of deformations, meaning that this quantity depends on previously applied strain. The

corresponding results are scale invariant.

10.2 Computational methods

I have studied the electronic and structural properties as well as the deformation energy

of polyphenanthrene dubbed ‘porous graphene’ using ab initio density functional theory

(DFT) as implemented in the VASP code [14, 16, 191]. I represented this 2D structure by

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Figure 10.1: Deformations in a 2D assembly of rigid isosceles triangles. (a) Adjacent triangles
with opening angle α and mutual orientation defined by the closing angle β, hinged tip-to-
corner, forming the primitive unit cell. The triangle height x0 and the length y0 of its base
define the horizontal and vertical length scales. (b) Snap shots of the α = 120◦ triangle
assembly for different values of β. The conventional rectangular unit cell is twice the size
of the primitive unit cell. (c) Contour plot of the Poisson ratio νxy = −(dy/y)/(dx/x) as
a function of α and β. The dotted red line highlights behavior of the α = 120◦ triangle
assembly. (d) Poisson ratio νxy as a function of β in the α = 120◦ system. (e) Changes in
the scaled width x/x0 and height y/y0 of the conventional unit cell for α = 120◦ caused by
changing the angle β.

imposing periodic boundary conditions in all directions and separating individual layers by

a vacuum region of 20 ˚A. I used projector-augmented-wave (PAW) pseudopotentials [17, 18]

and the Perdew-Burke-Ernzerhof (PBE) [9] exchange-correlation functional. The Brillouin
zone of the conventional unit cell of the 2D structure has been sampled by an 5×3×1 k-

point grid [34]. I used 500 eV as the electronic kinetic energy cutoff for the plane-wave basis
and a total energy difference between subsequent self-consistency iterations below 10−4 eV

as the criterion for reaching self-consistency. All geometries have been optimized using

the conjugate-gradient method [76], until none of the residual Hellmann-Feynman forces

161

Figure 1(b)(a)xy(e)(d)(c)βy0x0αβ=00yxβ=600xyα= 1200triangle assemblyβ=1000xy0.00.0-1.0-2.0-3.0-5.0-8.0+8.0+1.0+2.0+5.0-1.0-2.0-3.0-5.00β(degrees)α(degrees)1001502005020406080100120140160+8.0+6.0+4.0+2.00.0-2.0-4.0-6.0-8.00νxyνxy10β(degrees)α=120°50-5-10050100150200νxy>0νxy<0νxy<0y/y012340123x/x0α=120°β=0°β=60°β=158.2°β=210°νxy&νyx>0νxy&νyx<0Figure 10.2: Deformations in porous graphene, a phenanthrene-based 2D mechanical meta-
material. (a) Structure of the C14H10 phenanthrene molecule and its relation to an isosceles
α = 120◦ triangle of Fig. 10.1. (b) Equilibrium structure of 2D porous graphene consisting
of polymerized phenanthrene molecules with β = 70◦. Saturating hydrogen atoms are shown
by the lighter and smaller spheres. Changes in the scaled width x/x0 (c) and height y/y0
(d) of the conventional unit cell in the triangle assembly and porous graphene as a function
of the closing angle β. (e) Poisson ratio νxy in porous graphene as a function of β. (f) Strain
energy in the C56H28 conventional unit cell as a function of β. The dashed and dotted lines
connecting data points for porous graphene in (c)-(f) are guides to the eye.

exceeded 10−2 eV/˚A.

10.3 Results and discussions

Figure 10.1 depicts the macro-scale 2D mechanical metamaterial I consider, namely an infi-

nite assembly of rigid isosceles triangles hinged in the corners and described using periodic

boundary conditions. There are two identical triangles with different orientation in the prim-

itive unit cell of the lattice, as seen in Fig. 10.1(a). The conventional unit cell, shown in

Fig. 10.1(b), is rectangular and twice the size of the primitive unit cell. The deformation be-

havior of such constrained lattices of polygons including rectangles [221] and connected bars,

some of which display a Poisson ratio that changes sign, value, and even diverges, has been

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Figure 2(a)(c)(d)(f)β(degrees)νxyPorous grapheney/y0β(degrees)α= 1200triangle assemblyPorous grapheneβ(degrees)ΔE (eV/cell)(e)β(degrees)α= 1200triangle assemblyPorous graphenex/x0xyPorous grapheneβ(b)y0x0yxdescribed and classified earlier [220, 222]. In the system, structural changes are regulated by
the only independent variable, the angle β. The full range of β is 0 ≤ β ≤ α+180◦ for α≤60◦
and 0 ≤ β ≤ 270◦ − α/2 for α≥60◦. Since there is no energy involved when changing β,

the structure maintains its geometry after deformation. Snap shots of the triangle assembly

and the conventional unit cell at different values of β, shown in Fig. 10.1(b), illustrate the

unusual flexibility of the system.

For a system of triangles aligned with the Cartesian coordinate system as shown in

Fig. 10.1(a), I can determine the strain in the y-direction in response to strain applied along

the x-direction. The negative ratio of these strains is the Poisson ratio νxy, which is given

by

νxy =− dy/y
dx/x

=

cos( α
cos( α

2 ) sin( β
2 ) cos( β

2 ) − 3 sin( α
2 ) + 3 sin( α

2 ) cos( β
2 )
2 ) sin( β
2 )

(cid:18)α+β

(cid:19)

.

(10.1)

tan

2

Dependence of νxy on α and β is presented as a contour plot in Fig. 10.1(c). Several aspects of

this result are noteworthy when inspecting the behavior of νxy(β) for a constant value of the
opening angle α. With the exception of α = 60◦ describing equilateral triangles [223,224], νxy

changes magnitude and sign with changing β. Presence of the tangent function in Eq. (10.1)

causes νxy to diverge to ±∞ for βcrit(νxy) = 180◦ − α, with βcrit(νxy) = 60◦ for α = 120◦.
For α > 60◦, νxy changes sign twice across the full range of β values, as shown in Fig. 10.1(d)
for α = 120◦. The condition for the divergence of νyx = 1/νxy, describing strain in the x-

direction in response to strain applied in the y-direction, is tan(βcrit(νyx)/2) = 3 tan(α/2).
For α = 120◦, νyx will diverge at βcrit(νyx) = 158.2◦.

Maybe the most unexpected aspect of the result is the ‘shape memory’ effect displayed

by both νxy and νyx if the angle β becomes a hidden variable in the system. To explain

163

what I mean, I first inspect the (x(β)/x0, y(β)/y0) trajectory given by

(cid:20)

(cid:16) α
(cid:17)
(cid:19)
(cid:18) β

2

2

(cid:18) β
(cid:19)
(cid:17)
(cid:16)α

2

2

(cid:18) β
(cid:19)(cid:21)
(cid:19)
(cid:18) β

2

.

2

x
x0
y
y0

= 2

tan

cos

+ sin

= 3 sin

+ cot

cos

,

(10.2)

(10.3)

The (x(β)/x0, y(β)/y0) trajectory, describing the changing shape of the unit cell, is shown
for α = 120◦ in Fig. 10.1(e), and for other values of α in Fig. 10.4 in the Related Information

section. The sign of the slope of the trajectory, opposite to the sign of νxy and νyx, changes

twice as the structure unfolds with increasing β. Regions of positive and negative νxy and

νyx, delimited by the above-mentioned critical values βcrit(νxy) for νxy and βcrit(νyx) for

νyx, are distinguished graphically in Fig. 10.1(e). For any x in the range 3.46 < x/x0 < 4.00,

there are two different values of y associated with different values of β and different signs of

νxy. Similarly, for any y in the range 2.75 < y/y0 < 3.06, there are two different solutions

for x associated with different values of β and different signs of νyx.

Let us now consider a macroscopic piece of ‘material’ consisting of hinged triangles,

which are so small that their mutual orientation cannot be made out. With no information

about the deformation history, the material may exhibit either a positive or a negative

Poisson ratio. The only way to change the material so that it would exhibit a definite

positive or negative sign of the Poisson ratio is to subject it to a sequence of deformations.

Assume that this material is first stretched to its maximum along a given direction such

as x. Subsequent stretching along a direction normal to the first will result in a positive,

subsequent compression in a negative Poisson ratio. I may say that the system retains a

memory of previous deformations.

What happens microscopically can be clearly followed in Fig. 10.1(e). Even though the

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value of β is hidden, I know that it becomes 60◦ for maximum stretch along x and 158.2◦

for maximum stretch along y. Subsequent deformation normal to the first direction then

dictates the sign of ν. This behavior derives from the nonlinearity in the system and, in

some aspect, parallels the behavior of shape memory alloys.

Whereas macroscopic triangular assemblies with various values of α will find their use

in particular applications, I turn the interest to 2D nanostructures that can be formed

by coordination chemistry and macromolecular assembly. Microstructures including col-

loidal Kagom´e lattices [225–227] and graphitic nanostructures [228,229] including polypheny-

lene [230], sometimes dubbed nanoporous graphene, have been synthesized, but do not dis-

play a negative Poisson ratio. In the following, I focus on polyphenanthrene, a 2D structure

of phenanthrene molecules shown in Fig. 10.2(a). There is a strong similarity between this
molecule and α = 120◦ triangles depicted in Fig. 10.1. In particular, 2D assemblies of struc-

tures in Figs. 10.1(a) and 10.2(a) display strong similarities in their Poisson ratio behavior

discussed below.

The calculated equilibrium structure of 2D porous graphene formed of polymerized
phenanthrene molecules with the optimum angle β = 70◦, shown in Fig. 10.2(b), illus-
trates the relationship between this structure and the α = 120◦ triangle assembly. The

unusual flexibility of polyphenanthrene is owed to the connection of phenanthrene molecules

by strong C-C σ bonds, which are also responsible for the strength and flexibility of polyethy-

lene. The DFT calculations indicate only small structural distortions of the phenanthrene

molecules, which nevertheless break their initial mirror symmetry.

In Fig. 10.2(c) I compare changes in the scaled width x/x0 of the conventional unit cell

as a function of the closing angle β for the assembly of triangles and for porous graphene.

The corresponding changes in the scaled height y/y0 are shown in Fig. 10.2(d) in the same

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Figure 10.3: Electronic structure of porous graphene, a phenanthrene-based 2D mechanical
metamaterial, based on DFT-PBE calculations. (a) Band structure of the equilibrium struc-
ture with β = 70◦ obtained using the rectangular C56H28 unit cell. High-symmetry points
in the rectangular Brillouin zone are shown in the inset. (b) Fundamental band gap Eg as
a function of the angle β.

range of β values. Interestingly, x(β)/x0 reaches its maximum at βcrit(νxy) for both systems,

whereas y(β)/y0 increases monotonically with increasing β. According to the definition of
the Poisson ratio νxy = −(dy/y)/(dx/x), νxy diverges at βcrit(νxy) = 60◦ in the triangular
assembly, as seen in Fig. 10.1(d). Similarly, νxy diverges at βcrit(νxy) = 70◦ in porous

graphene, as shown in Fig. 10.2(e). The slope of x(β)/x0 changes sign at βcrit, resulting in

νxy < 0 for β < βcrit(νxy) and νxy > 0 for β > βcrit(νxy) in both systems.

The energy investment ∆E associated with deforming the polyphenanthrene structure is

shown in Fig. 10.2(f). The results were obtained by optimizing the structure for selected val-

ues of the angle β that defines the relative orientation of the two inequivalent phenanthrene
molecules in the unit cell. With β≈70◦ representing the structural optimum, I found that
changing β by ±10◦ required ∆E < 3 eV per unit cell, corresponding to an energy investment
of only ≈50 meV per C atom, about 1% of the bond breaking energy. Thus, the polyphenan-

threne structure is rather soft and represents a valid counterpart to the isoenergetic model

166

Figure 3(b)(a)β(degrees)Eg(eV)ΓX   S       Y   ΓDOSEnergy (eV)β=70°EFΓSXYsystem of Fig. 10.1.

Phenanthrene is a tricyclic organic molecule with a 3.36 eV wide DFT-PBE gap between

the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital

(HOMO). When polymerized to the 2D polyphenanthrene structure depicted in Fig. 10.2(b),

the HOMO broadens to the valence and the LUMO to the conduction band. This is seen

in Fig. 10.3(a), which depicts the band structure and the density of states of the optimum
geometry of polyphenanthrene with β = 70◦, with the Brillouin zone shown in the inset. The

DFT-PBE results indicate that the fundamental band gap Eg is reduced from the molecular

value to 1.75 eV in the equilibrium structure of the layer, but still does not vanish for
55◦ < β < 80◦. The gap is near-direct due to the flatness of bands, and decreases from
1.9 eV at β = 55◦ to 1.7 eV at β = 80◦. I should remember that Kohn-Sham eigenvalues in

all DFT calculations including ours do not correctly represent the electronic structure and

typically underestimate the band gaps.

The decrease of Eg and its dependence on β upon polymerization is caused by the pres-

ence of covalent C-C bonds that connect individual phenanthrene molecules elastically and

electronically. Unfolding of the polyphenanthrene structure with increasing angle β rotates

individual phenanthrene molecules and modifies the bonding at the connection between

adjacent monomers, causing the the electronic structure to depend on β. The range of de-

formations in polyphenanthrene is smaller than in triangular assemblies due to the steric

hindrance caused by hydrogen termination. In absence of planar confinement, phenanthrene

molecules rotate out-of-plane at large tensile strain values not considered here.

Elastic response of materials is commonly described by elastic constants constituting the

elastic matrix, which describe stress-strain relationships and thus contain energy in their

dimension. The Poisson ratio is fundamentally different. It is a dimensionless quantity that

167

describes deformations induced by strain, independent of the energy cost. According to its

definition in Eq. (10.1), it depends on the choice of the coordinate system. The trace of

the strain matrix, however, which describes the fractional change of the area induced by

the mechanism, is independent of the choice of coordinates and could couple naturally to

external fields such as pressure.

I believe that changes in pore size caused by the deformation of the 2D unit cell may

find their use in tunable sieving in a layered system [231, 232], including application in

desalination membranes. 2D mechanical metamaterials may also find unusual applications

in micro-manipulation. In particular, a 2D layer in partial contact with an in-plane junction

of 2D metamaterials with different values of ν, including ν > 0 and ν < 0, may experience a

torque normal to the plane when in-plane strain is applied at the junction of the 2D systems.

Also the observation of strain-related electronic structure changes in polyphenanthrene opens

new possibilities. Since polyphenanthrene and a wide range of porous graphene structures

can be viewed as a system of covalently connected quantum dots, in-layer strain may be used

to tune the coupling between such quantum dots and thus change the electronic structure

of the system.

10.4 Summary

In summary, I have designed 2D mechanical metamaterials that may be deformed substan-

tially at little or no energy cost. Unlike origami- and kirigami-based mechanical metamate-

rials that derive their functionality from folding a 2D material to the third dimension, the

structures I design are confined to a plane during deformation. In reality, such confinement

may be achieved by a strong attraction to a planar substrate or in a sandwich geometry. On

168

the macro-scale, the structures I describe are assemblies of rigid isosceles triangles hinged

in their corners. Their nanoscale counterpart are molecules such as phenanthrene that may

be polymerized using coordination chemistry or macromolecular assembly to form specific

geometries with a porous graphene structure. In these and in a large class of related struc-

tures, consisting of connected and near-rigid isosceles triangles confined to a plane, the

Poisson ratio ν diverges for particular strain values. ν also changes its magnitude and sign,

depending on the applied uniaxial strain, and displays a shape memory effect with respect

to the deformation history.

10.5 Related Information

10.5.1 Deformation behavior in 2D isosceles triangle assemblies

As discussed earlier, for a given value y of the unit cell height in a 2D assembly of isosceles
triangles with α > 60◦, I can find two different values x of the unit cell width, with the

two structures displaying opposite signs of ν. Similarly, I can find two different values y

for a given value of x, with the two structures displaying opposite signs of ν. This unusual

behavior results from the presence of a hidden variable, the relative triangle orientation β,

and causes ν to depend not only on the overall sample shape, but also the history of the
system. The unfolding of an assembly of triangles with α = 120◦ and its history dependence
has been characterized by the x − y trajectory in Fig. 10.1(e) in the range of accessible β

angles.

x − y trajectories for several values of α are shown in Fig. 10.4. The particular shape of
these x − y trajectories indicates that also for opening angles other than α = 120◦ discussed

above, the value and sign of ν may depend on sample history. Only in the specific case of

169

Figure 10.4: Changes in the scaled width x/x0 and height y/y0 of the conventional unit cell
for different values of the opening angle α as a function of the closing angle β. The relevant
quantities are defined in Fig. 10.1.

equilateral triangles with α = 60◦, discussed in the following, the y−x trajectory in Fig. 10.4

is linear and ν is history independent.

10.5.2 Deformations in a 2D assembly of rigid equilateral triangles

I mentioned above that the behavior of α = 60◦ triangle systems, depicted in Fig. 10.5,

is unique among the 2D assemblies of corner-sharing isosceles triangles. As discussed in

the main manuscript and above, the Poisson ratio changes drastically for triangle systems
with opening angle α other than 60◦. While hinged equilateral triangles gradually unfold

when β increases, the width x of the unit cell remains proportional to its height y, resulting
in a constant, β-independent Poisson ratio νxy = −1, as noted earlier [223, 224]. For the
particular angle β = 120◦, the structure of the assembly resembles the Kagom´e lattice.

170

Figure S1y/y0x/x01.0              2.0              3.0              4.0              5.0              6.0          0.00.51.01.52.02.53.03.5α=60°α=80°α=100°α=120°α=140°(a) Adjacent
Figure 10.5: Deformations in a 2D assembly of rigid equilateral triangles.
triangles with mutual orientation defined by the closing angle β, hinged at the corners,
forming the primitive unit cell. The triangle height x0 and the length y0 of its base define
the horizontal and vertical length scales. (b) Snap shots of the triangle assembly for different
values of β. The conventional unit cells of width x and height y are indicated.

171

β=00y0βx0α(b)(a)Figure S1α=600xyβ=300xyβ=600xyβ=900xyβ=1200xyChapter 11

Conclusions

Low-dimensional materials have been attracting unusual attention due to their exotic prop-

erties and their potential application in miniaturized devices. The vision of what is desirable

and possible and what is not requires physical understanding first, provided by theory and

quantified by numerical calculation. It is important to recognize that each approach has its

limitations. As an example, atomistic calculations are limited in their prediction of long-

wavelength acoustic phonon modes due to the required large supercell sizes and often produce

numerical artifacts. Applying the continuum elasticity approach to low-dimensional systems

appears as a promising alternative to address this problem. This approach does not depend

on supercells, is universal and efficient. It is interesting that results for graphene and black

phosphorene obtained using this approach agree with available observed spectra and cover

a substantial part of the Brillouin zone centered at the Γ-point.

To achieve progress in nanoscience and nanotechnology, searching for low-dimensional

materials with promising properties is a key prerequisite. Since the explosion of interest in

graphene, there has been significant progress in developing computational approaches for low-

dimensional nanomaterials. Besides black phosphorene, which is the most stable 2D allotrope

of phosphorus, I have identified in this thesis an equally stable 1D structure that prefers to

bend. Long segments, formed from sublimed red phosphorus inside a carbon nanotube, like

to form coils. Since the coil stability is almost independent of the radius, nested coils should

172

be formed, as confirmed by high-resolution transmission electron microscopy.

Despite extensive experimental efforts to synthesize 2D allotropes of other elements ex-

cept for few studies of group VI elements. What we know about Se or Te, both bulk materials

are composed by 1D helices. I started from the 1D helices and proposed a way to transform

the helices to a stable 2D allotrope by searching the high-dimensional configuration space

for the optimum microscopic reaction path. Following this path, the activation barriers have

been reduced to below 0.3 eV, which is sufficiently low to allow a synthesis.

What has been underestimated in many studies is the potential effect of net charge on

the relative stability of structural allotropes. I found that elemental boron is very sensitive

to net charge and that stable 2D allotropes of charged boron are very different from neutral

boron. Gradually increasing the electron concentration in a 2D geometry causes boron to

behave more and more like carbon. With one excess electron per boron atom, the opti-

mum structure becomes a honeycomb lattice and the electronic structure resembles that of

graphene, including the occurrence of a Dirac cone at a corner of the Brillouin zone. In this

study, and in the general situation necessitating excess electrons, a 2D electride appears to

be the most promising option to supply the extra charge.

A key objective in the field of superconductors is to increase the critical temperature

Tc to hopefully near the room temperature. In an alkali-doped C60 crystal, reducing the

dimension from the 3D fcc structure in the bulk to a quasi-1D structure inside a nanotube

causes a significant increase in the density of states at the Fermi level, which may increase

Tc to possibly even room temperature.

Availability of high-quality graphene monolayers has ignited broad interest in graphene-

inspired 2D nanomaterials. A graphene bilayer has one additional degree of freedom that had

been overlooked in the past, namely relative twist. I have found that twisted bilayer graphene

173

is unstable with respect to in-plane shear and out-of-plane atomic relaxation. The emergence

of an ultra-narrow band near the Fermi level at the ’magic angle’ is likely responsible for the

observed superconductivity in this system. The honeycomb structure of graphitic carbon is

also found in many organic molecules that may be interconnected to a contiguous 2D layer

that may be viewed as a metamaterial that can be deformed. Tailoring the deformation

affects not only the electronic structure, but also the mechanical behavior. I have studied

a particular 2D porous graphene system, namely a phenanthrene-based 2D metamaterial,

which exhibits an unusual behavior of the Poisson ratio ν. Applying in-layer strain, I have

found that ν may change its sign and even diverge at specific in-layer deformations.

174

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