STABILITYANDELECTRONICPROPERTIESOFLOW-DIMENSIONALNANOSTRUCTURESbyJieGuanADISSERTATIONSubmittedtoMichiganStateUniversityinpartialentoftherequirementsforthedegreeofPhysics{DoctorofPhilosophy2017ABSTRACT STABILITY AND ELECTRONIC PROPERTIES OF LOW-DIMENSIONAL NANOSTRUCTURES By Jie Guan As the devices used in daily life become smaller and more concentrated, traditional three-dimensional (3D) bulk materials have reached their limit in size. Low-dimensional nanomaterials have been attracting more attention in research and getting widely applied in many industrial fields because of their atomic-level size, unique advanced properties, and varied nanostructures. In this thesis, I have studied the stability and mechanical and electronic properties of zero-dimensional (0D) structures including carbon fullerenes, nanotori, metallofullerenes and phosphorus fullerenes, one-dimensional (1D) structures including carbon nanotubes and phosphorus nanotubes, as well as two-dimensional (2D) structures including layered transition metal dichalcogenides (TMDs), phosphorene and phosphorus carbide (PC). I first briefly introduce the scientific background and the motivation of all the work in this thesis. Then the computational techniques, mainly density functional theory (DFT), are reviewed in Chapter 2. In Chapter 3, I investigate the stability and electronic structure of endohedral rare-earth metallofullerene La@C60 and the trifluoromethylized La@C60(CF3)n with n  5. Odd n is preferred due to the closed-shell electronic configuration or large HOMO-LUMO gap, which is also meaningful for the separation of C60-based metallofullerenes. Mechanical and electronic properties of layered materials including TMDs and black phosphorus are studied in Chapter 4 and 5. In Chapter 4, a metallic NbSe2/semiconducting WSe2 bilayer is investigated and besides a rigid band shift associated with charge transfer, the presence of NbSe2 does not modify the electronic structure of WSe2. Structural similarity and small lattice mismatch results in the heterojunction being capable of efficiently transferring charge acrossthe interface. In Chapter 5, I investigate the dependence of stability and electronic band structure on the in-layer strain in bulk black phosphorus.  In Chapters 6, 7 and 8, novel 2D structures are predicted theoretically. In Chapter 6, I propose two new stable structural phases of layered phosphorus besides the layered -P (black) and -P (blue) phosphorus allotropes. A metal-insulator transition caused by inlayer strain or changing the number of layers is found in the new -P phase. An unforeseen benefit is the possibility to connect different structural phases at no energy cost, which further leads to a paradigm of constructing very stable, faceted phosphorus nanotube and fullerene structures by laterally joining nanoribbons or patches of different planar phosphorene phases, which is discussed in Chapter 7. In Chapter 8, I propose previously unknown allotropes of PC in the stable shape of an atomically thin layer. Different stable geometries, which result from the competition between sp2 bonding found in graphitic C and sp3 bonding found in black P, display different electronic properties including metallic, semi-metallic with an anisotropic Dirac cone, and direct-gap semiconductors with their gap tunable by in-layer strain.  In Chapter 9, I propose a fast method to determine the local curvature in 2D systems with arbitrary shape. The curvature information, combined with elastic constants obtained for a planar system, provides an accurate estimate of the local stability in the framework of continuum elasticity theory. This approach can be applied to all 2D structures. Finally, I present general conclusions from the PhD Thesis work in Chapter 10. iv To my parents.  v ACKNOWLEDGMENTS  During my five year Ph.D study, I learnt a lot from my advisor, my colleagues, my family and my friends. I got so much help from so many people when I was suffering confuse, loss and helplessness during these five years. I want deeply thank all of them and without them I can never finish my Ph.D degree. First of all, I am grateful to my advisor Prof. David Tomanek. He is a great advisor knowing how to guide a fresh student who just start his academic career. From him I learnt physics, research skills, as well as how to efficiently work and communicate. I deeply respect his attitude was encouraged a lot as a early bird in the theoretical research field. During these years, I learnt many skills including understanding references, writing codes, doing calculations, summarizing and presenting results, all of which are very important for an independent researcher. Besides, I also learnt a lot from the discussion with Prof. Tomanek about the society and life. This is so important for me in my future life. I want to thank all my guidance committee members, including Prof. Phillip M. Duxbury, Prof. John A. McGuire, Prof. Chong-Yu Ruan and Prof. Matthew Comstock, for their precious time for my committee meetings and helpful advices about my research. I appreciate the recommendation letters on my behalf when I was applying for a job from Prof. Gottard Seifert of TU Dresden and Prof. Li Yang of Washington University in St. Louis. I want to thank my other collaborators, including Prof. nori Shinohara of Nagoya University, Prof. Zhixian Zhou of Wayne State University, Dr. Luke Shulenburger of Sandia National Labs, Prof.  vi Salvador Barra-Lopez of University of Arkansas and Prof. Teng Yang of Institute of Metal Research, Chinese Academy of Sciences. Dr. Zacharias Fthenakis gave me a lot of help at my first year in the group when I knew little about the computational skills. Special thanks to Dr. Zhen Zhu who brought me to MSU and to Prof. Tofriends at MSU. I learnt a lot from him at my early time and we had a lot of discussions in research. I also got a lot of help besides research from him during my MSU life. I also want to thank all the visitors to our group, including Dr. Ceren Tayran, Prof. Changhong Yao, Prof. Pavel B. Sorokin and Anja Foerster. I specially thank Dan Liu who discussed a lot with me in research and gave me a lot of help in my life. I also want to thank my collaborators from chemistry, Zhongqi Jin and Pengchao Hao, who brought me different sight from another field. I want to thank Baojuan Dong and Jizhang Wang who helped me a lot when I was at IMR in Shenyang, Liguo Ma and Yijun Yu who helped me a lot when I was at Fudan University in Shanghai. I want to thank all my claseemates in physics including Mengze Zhu, Xukun Xiang, Faran Zhou, Bing Hwang, Nan Du, Yanlian Xin, Bo Xiao, Chaoyue Liu. I wish all of them enjoying their current life. I also want to thank Yaxing Zhang, Jun Hong, Yanhao Tang, Zhensheng Tao, Xu Lu, Xueying Huyan, Fengguo Xu, Didi Luo, Xingze Mao, Kuan Zhu, Liangji Zhang and Hao Lin.  I want to thank my friends who came to MSU together with me, including Xiaoran Zhang, Ruiqiong Guo, Shuang Liang, Yuan Gao, Chuanpeng Jiang, Yijing Dai, Wei Shen, Jun Zhang from Chemistry, Li Ke from Education, Jinyao Yan from Engineering, Jun Du from Mathematics, Xiyao Jia and all those who have already left MSU. The first year here was my best year and I  vii still miss the time licing in Spartan Village. I also want to thank my friends Keyin Wen, Yiqiu Zhang, Feifei Wu, Yahui Tang, Zhiyi Su, Weichan Tang, Xiaoyun Zhen, Mi Ding, Ge Shi, Lanxing Li, Jiankun Liu, Weijing Liu, Ke Ma, Yongle Pang, Chenpeng Chen, Li Zhen, Xiaojing Shen, Yujue Wang, Zhihui Liu, Yaohui Ding, Peng Gao and others. I had an enjoyable life during my Ph.D study because of you. I also want to thank Wanqi Li and all other friends from MSU Chinese Students and Scholars Association.  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 #EFMA-1433459. I also acknowledge partial financial support by the Physics and Astronomy Department, and the Dissertation Completion Fellowship from College of Natural Science, which has supported me during the last semester of my Ph.D study. Finally, I wish to thank the China Scholarship Council for selecting me as winner of 2015 Chinese Government Award for Outstanding Students Abroad and providing future travel support.  Last but not least, I want to thank my family and especially my parents. They support me all the time and give me most useful advices when I meet troubles. All my achievement and honor belong to them. I owe them too much these years and I will do my best to repay them.     viii TABLE OF CONTENTS  LIST OF TABLES.........................................................................................................................x LIST OF FIGURES......................................................................................................................xi Chapter 1  Introduction................................................................................................................1 1.1  Stable and extractable metallofullerenes..................................................................................1 1.2  Electronic properties and applicationas of layered materials...................................................2 1.3  Novel 2D structures..................................................................................................................4 1.4  Curvature and stability for 2D systems.....................................................................................5 1.5  Outline of the dissertation.........................................................................................................5 Chapter 2  Introduction to Density Functional Theory.............................................................8 2.1  Hartree-Fock approximation.....................................................................................................8 2.2  Hohenberg-Kohn theorem......................................................................................................11 2.3  Kohn-Sham equations.............................................................................................................13 2.4  Exchange-correlation functionals...........................................................................................16 2.5  Basis sets and pseudopotentials..............................................................................................18 Chapter 3  Stability and electronic structure of La@C60 functionalized by CF3                   radicals.........................................................................................................................20 3.1  Introduction.............................................................................................................................20 3.2  Computational methods..........................................................................................................21 3.3  Results and discussion............................................................................................................22 3.4  Summary.................................................................................................................................28 Chapter 4  Electronic structure and charge injection across transition metal                   dichalcogenide heterojunctions....................................................................................29 4.1  Introduction.............................................................................................................................29 4.2  Computational methods..........................................................................................................31 4.3  Results and discussions...........................................................................................................31 4.4  Summary.................................................................................................................................35 Chapter 5  Structure and electronic properties of strained bulk black phosphorus.......................36 5.1  Introduction.............................................................................................................................36 5.2  Computational methods..........................................................................................................37 5.3  Results and discussion............................................................................................................39 5.4  ...........................................................................................................................44 Chapter 6  Various phases of few-layer phosphorene.............................................................45 6.1  Introduction.............................................................................................................................45 6.2  Computational techniques.......................................................................................................46 6.3  Results and discussions...........................................................................................................47  ix 6.4  ...........................................................................................................................56 Chapter 7  Faceted nanotubes and fullerenes of multi-phase phosphorene...................................57 7.1  Introduction.............................................................................................................................57 7.2  Computational techniques.......................................................................................................58 7.3  Results and discussions...........................................................................................................59 7.4  Summary.................................................................................................................................70 Chapter 8  Combination of sp2 and sp3 bonding: two-dimensional phosphorus                   carbide..........................................................................................................................71 8.1  Introduction.............................................................................................................................71 8.2  Computational methods..........................................................................................................73 8.3  Results and discussions...........................................................................................................74 8.4  Summary.................................................................................................................................87 Chapter 9  Curvature and stability for two-dimensional systems..................................................88 9.1  Introduction.............................................................................................................................88 9.2  Continuum elastic method......................................................................................................90 9.3  Validation of the continuum elasticity approach....................................................................94 9.4  Discussion.............................................................................................................................102 9.5  Summary...............................................................................................................................104 Chapter 10  Conclusions..............................................................................................................106 APPENDIX..................................................................................................................................109 BIBLIOGRAPHY........................................................................................................................115         x LIST OF TABLES  Table 6.1: Observed and calculated properties of the four layered bulk phosphorus  allotropes.  and  are the in-plane lattice constants defined inFig. 6.1. d is the inter layer separation and Eil is the inter-layer separation energy per atom. Ecoh is the cohesive energy with respect to isolated atoms.  is the relative stability of the layered allotropes with respect to the                     most stableblack phosphorene (or -P) phase.  ... Table 6.2: The fundamental band gap Eg in monolayers of -P, -P, -P and -P, based on                   DFT_PBE calculations.  . Table 6.3: Energy cost per edge length Ec/l and connection angle , defined in Fig.6.6, associated with connecting two semi-infinite phosphorene                     monolayers.  .55 Table 8.1: Calculated properties of different 2D-PC allotropes. <Ecoh> is the cohesive energy  describes the relative stability of  a system with respect to the most stable structure.  and  are the in-plane lattice constants defined in Fig. 8.1. dP-P, dP-C and dC-C are the equilibrium bond lengths between the respective species. In 0-PC, the P1-C bonds differ from the P2-C bonds in                     length.  .                       xi LIST OF FIGURES  Figure 3.1: Ball-and-stick model of the optimized structure of the most stable isomers of                    La@C60(CF3)n for (a) n = 0, (b) n = 1, (c) n = 2, (d) n = 3, (e) n = 4, (f)                     n = 5.  ..22 Figure 3.2: Energy change Et based on DFT-PBE for attaching one trifluoromethyl of different La@C60(CF3)n isomers. Here Et is defined as  is the total energy for the most stable isomer of La@C60(CF3)n, which are indicated by the larger symbols in                     green.  .. Figure 3.3: Hydrogenated La@C60(CF3)n for (a) n = 0, n = 1 with hydrogen and  triflu-oromethyl at (b) para-position and (c) ortho-position in a hexagon. The first row is the ball-and-stick model of the structures without La atom inside the cage and the second row is the structures with La atom inside the cage. The last row is the corresponding Schlegel diagram of C60 with hydrogen site indicated by blue dot and trifluoromethyl site by red. EH here is defined as , or for that without La atom                     Inside.  . Figure 3.4: Electronic eigenstates for molecular orbitals based on DFT-PBE of (a) C60 and La@C60(CF3)n with (b) n = 0, (c) n = 1, (d) n = 2, (e) n = 3, (f) n = 4, (g) n = 5. The LUMO and HOMO of C60 and the corresponding states in La@C60 are                     indicated in red and orange squares, respectively, in (a) and (b).  . Figure 4.1: Ball-and-stick model of the AB-stacked NbSe2/WSe2 bilayer in (a) top view and (b) side view. The unit cell is indicated by yellow shading in the top view. a1 and a2 are the lattice constants spanning the layers and d is the inter-layer                     distance.  . Figure 4.2: Electronic structure of an NbSe2/WSe2 bilayer and its decomposition. The band structure of the bilayer in (a) is compared in (b) to a superposition of individual monolayers of NbSe2 (solid red lines) and WSe2 (dashed green lines). The band structure in (c) is obtained from (b) by shifting the bands of semiconducting WSe2 rigidly up by 0.293 eV. The resulting band structure is superimposed on                     that of (a) representing the bilayer.  . Figure 4.3: Charge redistribution (r) and spatial variation of the electron potential associated with assembling the NbSe2/WSe2 bilayer from isolated monolayers. (a) Charge density difference .  is shown by isosurfaces bounding regions of electron excess (yellow) at +2.5×10-4 e/Å3 and electron deficiency (blue) at -2.5×10-4 e/Å3. (b) <(z/c)>  xii averaged across the x-y plane of the layers. The raw data, shown by the red solid line, have been convoluted by a Gaussian with a full-width at half-maximum (z/c) = 0.17, shown by the blue dashed line. (c) Electrostatic potential <V(z/c)> averaged across the x-y plane of the layers. z/c indicates the                     relative position of the plane within the unit cell.  . Figure 5.1: (a) Ball-and-stick model of the structure of bulk black phosphorus in top and side views. (b) Fractional change of the interlayer distance a3 as a function of the in-layer strain  along the a1 and a2 directions. (c) Dependence of the strain energy E per unit cell on the in-layer strain along the a1 and a2                               directions.  . Figure 5.2: Electronic band structure (left panels) and density of states (right panels) of bulk black phosphorus (a) without strain, (b) when stretched by 1% along the a1 direction, and (c) when stretched by 2% along the a1 direction. GW results are shown by the solid black lines. DFT-PBE results, which underestimate the band                     gap, are shown by the dashed red lines.  . Figure 5.3: (a) Correlation between quasiparticle (GW) energies Eqp and Kohn-Sham energy values EPBE, obtained using the DFT-PBE functional, for states along high symmetry lines in the Brillouin zone. The straight lines in the valence and conduction band regions are drawn to guide the eye. The results represent bulk black phosphorus subject to strain values (a1) = 1% and (a2) = 0. The Fermi level is at zero energy. (b) Dependence of the quasiparticle (GW) electronic band gap Eg on the in-layer strain applied along the a1 and a2                     directions.  . Figure 6.1: Equilibrium structure of (a) an -P (black), (b) -P (blue), (c) -P and (d) -P monolayer in both top and side views. Atoms at the top and bottom of the non-planar layers are distinguished by color and shading and the Wigner-Seitz cells                     are shown by the shaded regions.  . Figure 6.2: Vibrational band structure (k) of a monolayer of (a) -P and (b) -P. High-                     symmetry lines are shown in the insets.  . Figure 6.3: Snap shots of canonical molecular dynamics simulations depicting structural changes in -P at (a) T = 300 K and (b) T = 1,000 K. Corresponding snap shots of -P are shown in (c) for T = 300 K and (d) for T = 1,000 K. For better comparison, both top and side views are presented for all structures. The unit                     cell of the lattice contains 96 atoms in -P and 128 atoms in -P.  . Figure 6.4: Electronic band structure and density of states (DOS) of (a) -P and (b) -P monolayers. Results for bilayer and bulk systems are shown for comparison in the DOS plots only. Top and side views of the electron density vc near the top of the valence and the bottom of the conduction bands of (c) -P and (d) -P. Only states in the energy range EF-0.4 eV < E < EF+0.4 eV are considered, as  xiii indicated by the green shaded region in (a) and (b). vc is represented at the isosurface value vc = 1.110-3 e/Å3 for -P and -P and superposed with a                     ball-and-stick model of the structure.  . Figure 6.5: (a) Dependence of the fundamental band gap Eg on the slab thickness in N-layer slabs of -P (black), -P (blue), -P and -P. Dependence of the fundamental band gap on in-layer strain  is presented in (b) for -P and in (c) for -P. The strain direction is defined in Fig. 6.1. The shaded regions in (a) and (b) highlight conditions, under which -P become metallic. Dashed vertical lines in                     (b) and (c) indicate a direct-to-indirect band gap transition.  . Figure 6.6: Energetically favorable in-layer connections between (a) -P and -P, (b) -P and -P, and (c) -P and -P, shown in perspective and side view. l represents the edge length at the interface and  is the connection angle. The color scheme                    for the different allotropes is the same as in Fig. 6.1.  . Figure 7.1: (a) Atomic structure of -, -, - and -P in top view and the side view of zigzag and armchair edges. The orthogonal lattice vectors a1 and a2 define the unit cells or supercells used in this study. Schematic and atomic structure of (b) an armchair and (d) a zigzag PNT, with the different structural phases distinguished by color and shading. The cross-sections of (c) an armchair and (e) a zigzag nanotube illustrate the symmetry and the distribution of phases along                     the perimeter.  . Figure 7.2: Geometric cross-section of selected (a) a-PNTs and (b) z-PNTs. Different                      structural phases are distinguished by color.  . Figure 7.3: Source of ambiguity in the nomenclature of faceted nanotubes. In two identical junctions of phases  and , atoms along the connection line, indicated by the                     dotted ellipse, may be assigned to the phase on either side.  . Figure 7.4: Phosphorene-based fullerene structures with (a-c) octahedral and (d-f) icosahedral symmetry. The structural models in (a) and (d) indicate how triangular facets of -P are connected by -P along the edges. The stick models of P72 in (b), P200 in (c), P80 in (e) and P180 in (f) depict the relaxed atomic                     structures of octahedral and icosahedral fullerenes.  .63 Figure 7.5: (a) Average strain energy per atom  in PNTs of different radius R with respect to a planar -P monolayer. The shaded region indicates the range of stabilities of different planar phases. I also present data points for pure-phase PNTs obtained by rolling up - and -P to a tube. (b) Strain energy per atom in octahedral (o) and icosahedral (i) fullerenes of radius R. The dashed lines in (a) and (b) represent the 1/R2 behavior based on continuum elasticity theory for pure-phase nanostructures. (c) Band gaps Eg values in pure planar phosphorene monolayers. (d) HOMO-LUMO gaps in o- and i-fullerenes. The red and blue                     shaded regions indicate the range of values for a-PNTs and z-PNTs.  .  xiv Figure 7.6: Snap shots of canonical molecular dynamics simulations at T = 300 K and T = 1000 K depicting structural changes in phosphorus nanotubes and fullerenes. Nanotubes are represented by (a) the armchair a-PNT(3,3,0,0) and (b) the zigzag z-PNT(5,0,5,0). Fullerenes are represented by (c) P72 and (d) P80. The                     nanotubes are shown in end-on view.  . Figure 7.7: Electronic band structure of (a) the z-PNT(3,0,3,0) and (c) the z-PNT(5,0,5,0) nanotube. The region associated with frontier orbitals is highlighted by the green shading in the energy range EF-0.1 eV < E < EF+0.1 eV in (a) and EF-0.2 eV < E < EF +0.2 eV in (c). Electron density f associated with frontier states in (b) the z-PNT(3,0,3,0) and (d) the z-PNT(5,0,5,0) nanotube, superposed with ball-and-stick models of the structures. f is represented by the isosurface value                     810-4 e/Å3.  . Figure 7.8: (a) Cross-section and (b) DOS of the double-wall z-PNT (5,0,5,0)@(9,0,9,0). The solid line in (b) shows the total DOS and the dashed line depicts the superposition of the densities of states of the isolated nanotube components. The Fermi level EF is set at 0. (c) Perspective view of the a-PNT(3,0,9,0) and (d)                     dependence of the gap energy Eg on axial strain.  . Figure 8.1: Possible stable structures of an atomically thin PC monolayer, represented by (a-c) a tiling pattern and (d-i) by ball-and-stick models in both top and side view. The number of like nearest neighbors defines the structural category N. There are two stable allotropes,  and , for each N. The primitive unit cells are highlighted and the lattice vectors are shown by red arrows. Two inequivalent P                     sites are distinguished by a subscript in (d).  . Figure 8.2: Bonding configuration in (a) N = 1, (b) N = 2, and (c-e) N = 0 category 2D-PC allotropes. Green-shaded regions indicate sites that satisfy the octet rule discussed in the text. Bonding in 0-PC is characterized by panel (d) and                     bonding in 0-PC is bed by panel (e).  . Figure 8.3: Spontaneous structural transformation of 0-PC from its initial armchair profile in (a) to the final structure with a zigzag profile in (d). Snap shots of the intermediate structures, shown in side view in the upper panels, are accompanied by the corresponding band structure in the lower panels. Noteworthy is the transition from a metallic structure in (a) to a semiconductor in (d). The significance of the inequivalent P1 and P2 sites is discussed in the                     main text.  . Figure 8.4: Phonon spectra of (a) 0-PC and (b) 1-PC monolayers.   Figure 8.5: Electronic band structure, density of states (DOS), and charge density vc associated with valence frontier states of N and N allotropes, where N is the structural category. The energy range associated with vc is indicated by the green shaded region in the band structure and DOS panels and extends from  xv EF-0.45 eV < E < EF in (a), EF-0.40 eV < E < EF in (c) and (d), EF-0.10 eV < E < EF in (b), (e) and (f). Isosurface plots of vc are displayed in the right-side panels and superposed with ball-and-stick models of the structure in top and side view. The isosurface values of vc are 1.010-3 e/Å3 in (a), 2.010-3 e/Å3 in (b), 0.510-3 e/Å3 in (c) and (d), and 0.510-4 e/Å3 in (e) and                     (f).  ... Figure 8.6: Band structure of semi-metallic (a) 2-PC and (b) 2-PC near the Dirac point. (c) Definition of the high-symmetry points in the Brillouin zone of the structures in                     (a) and (b).  . Figure 8.7: Electronic band structure of (a) 0-PC, (b) 1-PC, (c) 2-PC, (d) 0-PC, (e) 1-PC, and (f) 2-PC monolayer obtained using DFT calculations with PBE and HSE06                     exchange-correlation functionals.  . Figure 8.8: Effect of uniaxial in-layer strain on (a) the relative binding energy tot and (b) the fundamental band gap in different PC allotropes. Results for 0-PC, 0-PC, 1-PC, 1-PC, 2-PC and 2-PC, are distinguished in Fig. 8.1, are shown by solid lines and for strain in the y-direction by dashed                     lines.  . Figure 9.1: Principal radii of curvature R1, R2 and the Gaussian curvature G on the surface of (a) a sphere and (b) a cylinder and (c) at a saddle point. (d) Determination of the local curvature at point P using the atomic lattice (grey dashed lines) and the dual lattice (blue solid lines). P is the center atom and Fi (i = 1~3) are three first                     neighbors. Vi (i = 1~3) are vertices of due lattice.  . Figure 9.2: Local Gaussian curvature G (upper panels) and local curvature energy C/A (lower panels) across the surface of (a) the least stable C38 isomer, (b) the most stable C38 isomer, (c) a (10,10) carbon nanotube, and (d) a schwarzite structure with 152 atoms per unit cell. The values of G and C/A have been interpolated                    across the surface.  .. Figure 9.3: Strain energy  in carbon nanostructures with respect to the graphene reference system. (a) DFT-based total strain energy totDFT for selected fullerenes, with the most stable isomers indicated by the larger symbols. (b) Strain energy  in different C38 isomers. Total energy differences totDFT based on DFT, totTersoff based on the Tersoff potential and totKeating based on the Keating potential are compared to curvature energies CKeating based on Keating-optimized geometries. To further compare curvature energy based on different geometries, totDFT are compared to curvature energies CDFT based on continuum elasticity theory for structures optimized by DFT, and CDFT in                     (c).  . Figure 9.4: (Color online) Strain energy  in carbon nanostructures with respect to the graphene reference system. (a) Comparison between DFT-based total energies  xvi totDFT and the curvature energy CKeating based on Keating-optimized geometries for all fullerene isomers considered in Fig. 9.3(a). (b) Comparison between DFT-based strain energies totDFT/n and curvature energies per atom CKeating/n for Keating-optimized geometries of fullerenes, nanotubes and schwarzites. Dashed lines represent agreement between DFT and continuum                     elasticity results.  . Figure 9.5: (a)Atomic structure of a polygonal nanotorus containing non-hexagonal rings. (b) Energy differences per atom totDFT based on DFT and totKeating based on the Keating potential are presented next to curvature energies CKeating for                     Keating-optimized geometries.  . Figure 9.6: Local Gaussian curvature G (left panels) and local curvature energy C/A (right panels) across the surface of selected nanotori. Representative examples shown are (a) the flattened C320 nanotorus with 320 atoms and (b) the elongated C280 nanotorus with 280 atoms. G and C/A are interpolated from their values at the                     atomic sites.  . Figure 9.7: (a) Perspective view of the planar structure of a blue phosphorene monolayer (top), which has been rolled up to a nanotube with radius R (bottom). (b) Comparison between the strain energy per atom C/n based on continuum elasticity theory and totDFT/n based on DFT in blue phosphorene nanotubes. The dashed line represents agreement between DFT and continuum elasticity                      results.  . Figure 9.8: Average curvature energy <C> per area of a nanocapsule tessellated by a honeycomb lattice with different numbers of vertices. The vertical dash-dotted red line indicates that the capsule represents the C120 structure. The inset shows, how the capsule surface can be tessellated by a honeycomb lattice with 120 vertices or atoms, shown by the white lines, and also with 480 vertices, shown by the red lines. The horizontal dashed black line represents an extrapolation to                     an infinitely dense tessellation.  . Figure A.1: Ten different isomers of La@C60(CF3)n with n = 2. In each subfigure ball-and-stick model of the optimized structure is shown at the top, the relative DFT-PBE energy with respect to the most stable isomer is listed in the middle and the corresponding Schlegel diagram of C60 with trifluoromethyl sites indicated                     by red dots is shown at the bottom.  . Figure A.2: Six different isomers of La@C60(CF3)n with n = 3. In each subfigure ball-and-stick model of the optimized structure is shown at the top, the relative DFT-PBE energy with respect to the most stable isomer is listed in the middle and the corresponding Schlegel diagram of C60 with trifluoromethyl sites indicated                     by red dots is shown at the bottom.  .  xvii Figure A.3: Ten different isomers of La@C60(CF3)n with n = 4. In each subfigure ball-and-stick model of the optimized structure is shown at the top, the relative DFT-PBE energy with respect to the most stable isomer is listed in the middle and the corresponding Schlegel diagram of C60 with trifluoromethyl sites indicated                     by red dots is shown at the bottom.  . Figure A.4: Ten different isomers of La@C60(CF3)n with n = 5. In each subfigure ball-and stick model of the optimized structure is shown at the top, the relative DFT-PBE energy with respect to the most stable isomer is listed in the middle and the corresponding Schlegel diagram of C60 with trifluoromethyl sites indicated                     by red dots is shown at the bottom.  . Chapter1IntroductionInthisthesis,Ihavestudiedfourttopicsrelatedtolow-dimensionalnanostructuresandtheirassociatedwithstabilityandelectronicproperties:stableandextractablemetallo-fullerenes,electronicpropertiesandapplicationsoflayeredmaterials,noveltwo-dimensional(2D)structures,aswellascurvatureandstabilityfor2Dsystems.Allthesetopicsarestud-iedtoexploretheadvancedpropertiesofexistingnanomaterialsandtodiscoverunknownpromisingnanomaterials,andtoleadtoapplicationsint1.1StableandextractablemetallofullerenesThediscoveryofC60in1985[1]startedthenanotechnologyrevolution.Inthemeantime,metallofullereneLaC60wasalsoobservedinamassspectrumoftheproductsfromlaservaporizationofaLaCl3impregnatedgraphiterod[2].Asmoreendohedralmetallofullerensweresuccessfullysynthesized[3{6],therehaveemergedwideapplicationsintsuchassolarcells[7,8]andbiomedicine[9,10].However,unlikehollowfullerenes,itisnoteasytoseparatethemetallofullerenesexceptM@C82fromtherawsootcontainingbothhollowfullerenesandmetallofullereneswitherentsizes[3,11{14].TheextractionofsmallmetallofullerenesespeciallyformostabundantM@C60,hasbecomeabigchallengeduetotheirinsolubilityinnormalfullerenesolventssuchastolueneandCS2[11,15,16].Thesemetallofullerenesarechemicallyreactiveduetotheiropen-shellelectronic1tionsorsmallHOMO-LUMOgapsandconsequenttendencytoforminsolublepolymerizedsolids[17].Recently,Shinohara'sgroupatNagoyaUniversityreportedanewstrategytoopentheHOMO-LUMOgapsofsmallmetallofullerenesincludingM@C60andM@C70byylation[18,19],andanumberofylatedmetallofullereneshavebeenInthisthesisIinvestigatedthestabilityandelectronicstructureoftheendohedralrare-earthmetallofullereneLa@C60andylizedLa@C60(CF3)nwithn5.UnliketheylderivativesofC60,La@C60(CF3)nprefersoddnumbersofn,duetothe3extravalenceelectronsfromtheencapsuledmetalatomLa.CalculatedelectroniceigenstatesofmolecularorbitalsfurtherthatwithanoddnumbernLa@C60(CF3)nismorestablebyformingaclosed-shellelectronicionorlargeHOMO-LUMOgap,whichisalsotfortheseparationofC60-basedmetallofullerenes.1.2Electronicpropertiesandapplicationsoflayeredmaterials2Dmaterialshavebeenattractingtheelectronicscommunity'sresearchinterestsincethesuccessfulexfoliationofsingle-layergraphenefrombulklayeredgraphite[20].Applicationsin2Delectronicsarelimitedforgrapheneduetoitszerobandgap.Layeredtransition-metaldichalcogenides(TMDs)havebeenconsideredaspromisingcandidatesfor2Delectronicsinthepost-grapheneera,sincetheisolationofsingle-layermolybdenum(MoS2)anditsapplicationintransistors[21,22].Oneofthebiggestchallengesforthefabricationofhigh-performanceelectronicdevicesisthepoorcontactbetweenmetalandtheTMDsurface[23{26].2Inthisthesis,insearchofanimprovedstrategytoformlow-resistancecontactstosemi-conductingTMDs,IproposetousemetallicTMDas2D/2Ddrain/sourcecontacts.AmetallicNbSe2/semiconductingWSe2bilayerisinvestigated.Verytfromtraditionalcontactswithrelativelythickdepletionlayer,IfoundthattheelectronicstructureofWSe2isnotmobytheNbSe2layerbesidesarigidbandshiftassociatedwithchargetransfer.SincethetwoTMDsarestructurallysimilaranddisplayonlyasmalllatticemismatch,theyformaheterojunctioncapableoftlytransferringchargeacrosstheinterface,thusimprovingcurrentinjectionintoWSe2.Layeredblackphosphorushasbeenanotherpromising2Dmaterialsincethesuccessfulapplicationsoffew-layerphosphoreneinelectronicdevices[27,28].Bulkblackphosphorushasbeenknownforacentury[29]anddisplaysaninterestingchangeinitselectronicandtopologicalpropertiesundercompression[30{32].Similartobulkblackphosphorus,itsbandgapdisplaysastrongandanisotropicresponsetoin-layerstraininphosphorenemonolayersandfew-layersystems[27,28,33{38],whichisnowwellestablished.However,nodependabledataareavailableforthecorrespondingresponseinthebulksystem.Inthisthesis,Iinvestigatethedependenceofstabilityandelectronicbandstructureonthein-layerstraininbulkblackphosphorusandfoundthatthestrainenergyandinterlayerspacingdisplayastronganisotropywithrespecttotheuniaxialstraindirection.Thebandgapdependssensitivelyonthein-layerstrainandevenvanishesatcompressivestrainvaluesexceedingˇ2%,thussuggestingapossibleapplicationofblackPinstrain-controlledinfrareddevices.31.3Novel2DstructuresEventhoughinterestin2Delectronicsandthefamilyof2Dsemiconductorsisgrowingfast,aconsensushasnotbeenreachedregardingtheoptimumcandidateforchannelmaterials.Semi-metallicgraphene,withanexcellentcarriermobility,hasreceivedthemostattentionsofar,butallattemptstoopenupasizeable,robust,andreproduciblebandgaphavefailedduetothenegativesideofthetmo[39{42].TheTMDfamily,suchasMoS2[22,43]hassizeablefundamentalbandgaps,butlowercarriermobility.Recentlyisolatedfew-layerofblackphosphorus,includingphosphorenemonolayers,combinehighcarriermobilitywithasizeableandtunablefundamentalbandgap[27,28],buthavelimitedstabilityinair[44].Novellow-dimensionalstructuresandmaterialswithadvancedpropertiesarestilldesirable.Inthisthesis,Iproposetwonewstablestructuralphasesoflayeredphosphorusbesidesthelayered-P(black)and-P(blue)phosphorus[33]allotropes.Thepossibilitytoconnectthesetstructuralphasesatnoenergycostfurtherleadstotheparadigmofconstruct-ingverystable,facetedphosphorusnanotubeandfullerenestructuresbylaterallyjoiningnanoribbonsorpatchesoftplanarphosphorenephases.Ialsoproposepreviouslyun-knownallotropesofphosphoruscarbide(PC)inthestableshapeofanatomicallythinlayer.tstablegeometries,whichresultfromthecompetitionbetweensp2bondingfoundingraphiticCandsp3bondingfoundinblackP,displaytelectronicpropertiesinclud-ingmetallic,semi-metallicwithananisotropicDiraccone,anddirect-gapsemiconductorswiththeirgaptunablebyin-layerstrain.41.4Curvatureandstabilityfor2DsystemsOneofthemostimportantadvantagesof2Dsystemsismechanicaly,whichallowstuningofthemorphologyofthe2Dsurface.Themostprominentexample,graphiticcarbon,isthestructuralbasisnotonlyofgraphene[45],butalsofullerenes,nanotubes,toriandschwarzites[46{50].Eventhoughthestructuralmotifinallofthesesystemsmaybethesame,theirmechanicalandelectronicpropertiesdependsensitivelyonthelocalmorphology[51{53].Ontheotherhand,abinitiocalculationsareimpracticaltoestimatetheglobalorlocalstabilityforverylargestructures.EmpiricalrulesorparameterizedforceincludingtheTpotentialandmolecularmechanics[54{57],haveoftenbeenusedtoestimatestabilitybutaresometimesunsatisfactory.Applicationofcontinuumelasticitytheory,whichcandescribestabilitychangesduetodeviationfromplanarity,hasbeensuccessful,butitislimitedtosystemswithawconstantcurvature[58,59].Sincestrainenergyisdominatedbylocalgeometryandindependentoftheglobalmor-phology,hereIproposeafastmethodtodeterminethelocalcurvaturein2Dsystemswitharbitraryshape.Thecurvatureinformation,combinedwithelasticconstantsobtainedforaplanarsystem,providesanaccurateestimateofthelocalstabilityintheframeworkofcontinuumelasticitytheory.Thisapproachcanbeappliedtoall2Dstructures.1.5OutlineofthedissertationThisPhDthesiscontains9chapters,includingChapter1asanintroductorychapterandChapter2todescribethecomputationalmethodsusedthroughoutthisthesis.InChapter3,Iinvestigatedstabilityandelectronicstructureofendohedralrare-earthmetallofullereneLa@C60andtheylizedLa@C60(CF3)nwithn5.Preferred5oddnumberofnarefoundbfromtheclose-shellelectronicorlargeHOMO-LUMOgap,whichisalsomeaningfulfortheseparationofC60-basedmetallo-fullerenes.MechanicalandelectronicpropertiesoflayeredmaterialsincludingTMDsandblackphosphorusarestudiedinChapter4and5.InChapter4,ametallicNbSe2/semiconductingWSe2bilayerisinvestigatedandbesidesarigidbandshiftassociatedwithchargetransfer,thepresenceofNbSe2willnotmodifytheelectronicstructureofWSe2.Structurallysimilarandsmalllatticemismatchresultintheheterojunctionbeingcapableoftlytransferringchargeacrosstheinterface.InChapter5,Iinvestigatethedependenceofstabilityandelectronicbandstructureonthein-layerstraininbulkblackphosphorus.InChapters6,7and8,novel2Dstructuresarepredictedtheoretically.InChaper6,Iproposetwonewstablestructuralphasesoflayeredphosphorusbesidesthelayered-P(black)and-P(blue)phosphorusallotropes.Metal-insulatortransitioncausedbyin-layerstrainorchangingthenumberoflayersarefoundinthenew-Pphase.Anunforeseenbisthepossibilitytoconnecttstructuralphasesatnoenergycost,whichfurtherleadstoanewparadigminconstructingverystable,facetedphosphorusnanotubeandfullerenestructuresbylaterallyjoiningnanoribbonsorpatchesoferentplanarphosphorenephases,whichisdiscussedinChapter7.InChapter8,IproposepreviouslyunknownallotropesofPCinthestableshapeofanatomicallythinlayer.tstablegeometries,whichresultfromthecompetitionbetweensp2bondingfoundingraphiticCandsp3bondingfoundinblackP,displaytelectronicpropertiesincludingmetallic,semi-metallicwithananisotropicDiracconeanddirect-gapsemiconductorswiththeirgaptunablebyin-layerstrain.InChapter9,Iproposeafastmethodtodeterminethelocalcurvaturein2Dsystems6witharbitraryshape.Thecurvatureinformation,combinedwithelasticconstantsobtainedforaplanarsystem,providesanaccurateestimateofthelocalstabilityintheframeworkofcontinuumelasticitytheory.Thisapproachcanbeappliedtoall2Dstructures.FinallyinChapter10,IprovidemyconclusionsforthewholeworkofmyPhDstudy.7Chapter2IntroductiontoDensityFunctionalTheoryInthischapter,IwillstartwiththeHartree-Fock(HF)approximationandshowthedisad-vantageofthissingle-particleapproximation.ThenIwillintroducedensityfunctionaltheory(DFT)whichisbettermethodformanymaterials.IwilltalkabouttheHohenberg-Kohn(HK)theorem,Kohn-Sham(KS)equations,aswellastheapproximationsandmethodsusedintheapplicationofDFT.2.1Hartree-FockapproximationInacomplexnanostructureincludingmanyatomsandelectrons,theHamiltonianoperatorofthewholesystemcanbewrittenas:^H=XI~22Mr2I+Xi~22mr2i+12XI6=JZIZJe2jRIRJj+12Xi6=je2jrirjjXI;iZIe2jriRIj(2.1)Here,theuppercaseindicesdescribetheionsandthelowercaseindicesdescribeelectrons.Thetwotermsrepresentthekineticenergyoftheionsandtheelectrons.Thelastthreetermsare,respectively,ion-ion,electron-electronandtheelectron-ioninteractions.RIandRJdenotetheionpositionsandriandrjdenotetheelectronpositions.Asthenucleusis8thousandsoftimesheavierthantheelectron,theionmotionandtheelectronmotioncanbeseparatedbytheBorn-Oppenheimerapproximation[60].Basedonthisapproximation,theionscanbeseenaswhenconsideringtheelectronsystem.Inthiscase,theHamiltonianofelectronscanbewrittenas^H=Xi~22mr2i+12Xi6=je2jrirjjXI;iZIe2jriRIj:(2.2)TogetthestatesofelectronsweneedtosolvetheScodinger'sequation^H(r)=E(r):(2.3)InasystemincludingNelectrons,theHFapproximationassumesthatthewavefunctionofthewholesystemcanbewrittenasaSlaterdeterminant(r)=1pN!1(r1)1(r2)1(rN)2(r1)2(r2)2(rN).........N(r1)N(r2)N(rN):Herethei(rj)representstheelectronorbitaliatpositionrj,andtheyareorthonormalZi(r)j(r)=ij:(2.4)9ThelastterminEq.(2.2)representstheperiodicpotentialinacrystalandcanbewrittenasV(r)=XIv(rRI)=XIZIe2jrRIj:(2.5)ThetotalenergyofthesystemisE=Z(r)^H(r)dr:(2.6)BysubstitutingtheHamiltonianandwavefunctionintoEq.(2.6),wecangettheaverageenergyE=XiZdri(r)[~22mr2+V(r)]i(r)+12Xi6=jZdrdr0ji(r)j2e2jrr0jjj(r0)j212Xi6=jZdrdr0i(r)j(r0)e2jrr0jj(r)i(r0):(2.7)Wecanseethatiftheelectronsinthesystemdonotinteractwitheachother,thentheenergywillonlyincludetheterm.Obviouslythisisnotthetruth.ThesecondtermistheCoulombinteractionbetweentelectrons.Thethirdtermdescribestheexchangeinteractionbetweenelectronswiththesamespin.Tothegroundstateofthesystemweneedtominimizethetotalenergy.Wecanusethevariationalprincipletodoitand10thati(r)shouldsatisfythefollowingequation:[~22mr2+V(r)+XjZdr0jj(r0)j2e2jrr0j]i(r)XjZdr0e2j(r0)i(r0)jrr0jj(r)=ii(r):(2.8)HereweobtainedthefamousHFequation.Noticethatinthesecondtermofthelefthandsideoftheequationtheoperatorisrelatedtothewavefunctionfortheelectronweconsider,sotheequationcanonlybesolvedinaselfconsistentway.TheHFmethodisonlyanapproximationandsometimesdoesnotgiveasatisfactoryresult.ThisisbecauseinHFweassumethattheelectronwavefunctioncanbewrittenasasingleSlaterdeterminant,whichiswrong.Inthiswayweassumethateachelectroninteractswithanaveragechargedistributionduetotheotherelectrons.Thisintroducesanerrorinthewavefunctionandtheenergyoftheelectrons.TheenergyerrorhereignoredinHFiscalledtheelectroncorrelationenergy.Theerrorsometimescanberatherbigandgivewrongresults.2.2Hohenberg-KohntheoremAlthoughtheHFmethodhasbeenappliedforalongtimeinsolidstatephysics,itisnotastricttheoreticalbasisforthesingle-particleapproximationofthemany-bodysystem.DensityfunctionaltheorybasedontheHohenberg-Kohntheorem[61]andthesubsequentKohn-Shamequations[62]givesstrictevidencethatthegroundstateproblemofamany-bodysystemcanbetransformedtoagroundstateproblemofasinglequasi-particleprobleminaneexternalpotential.11TheessentialideaoftheHohenberg-Kohntheoremisthatinamany-bodysystem,theelectrondensityˆ(r)isthebasicvariablethatdeterminesthephysicalquantities.In1964,HohenbergandKohnprovedabasiclemmawhichstatesthatoneexternalpotentialV(r)onamany-electronsystemonlycorrespondstoonegroundstateelectronicdensityˆ(r).WithaknownV(r),boththegroundstatewavefunctionandthetotalenergyofthesystemaredecided.Inthisway,wereachtheHohenberg-Kohntheorem,whichstatesthattheground-statepropertiesofamany-electronsystemdependonlyontheelectronicdensityˆ(r).Thus,inthegroundstate,itturnsoutthatthetotalelectronicenergyofthesystemcanbeexpressedbyafunctionaloftheelectronicdensityˆ(r),includingthekineticenergyofelectronsT[ˆ],theinteractionbetweenelectronsVee[ˆ],andtheinteractionwiththeexternalpotentialV(r)asE[ˆ]=T[ˆ]+Vee[ˆ]+ZdrV(r)ˆ(r):(2.9)ThesecondHohenberg-KohntheoremstatesthatthecorrectgroundstatedensityforasystemistheonethatminimizesthetotalenergythroughthefunctionalE[ˆ].OncetheexpressionforthekineticenergyT[ˆ]andelectron-electroninteractionenergyVeeareknown,thegroundstateelectrondensityˆandtotalenergyE[ˆ]canbecalculatedbasedonthevariationalprinciple,whichminimizesthetotalenergywhilekeepingthetotalnumberofelectronsNconstant.122.3Kohn-ShamequationsTheHohenberg-Kohntheoremsjustifythattheground-statepropertiesofthesystemaredeterminedbythefunctionalofthegroundstateelectrondensityˆ(r).However,asexactexpressionsforthekineticenergyT[ˆ]andelectron-electroninteractionVee[ˆ]areunknown,itisstillunclearhowtoobtainE[ˆ].KohnandShamintroducedtheideathatthegroundstateelectrondensityofamany-electronsystemcanbeapproximatedusingcontributedbyNindependentorbitals:ˆ(r)=Xi i(r) i(r):(2.10)Heref i(r)g(i=1;2;;N)isasupposedlynoninteractingquasi-electronsystem,whichjusthappenstohavethesamegroundstateelectrondensityˆ(r)astheinteractingsys-temweareconsideringhere.SincetheHohenberg-Kohntheoremtellsusthatthegroundstatepropertiesonlydependonˆ(r),wecanobtainthegroundstatepropertiesfromthisequivalentnoninteractingquasi-electronsystem.Usingthenoninteractingquasi-electronwavefunctions i(r),wecanwritedowntheCoulombenergy(Hartreeenergy)asVH[ˆ]=12Zdrdr0ˆ(r)e2jrr0jˆ(r0)=12Xi;j< i jje2r12j i j>;(2.11)13andthekineticenergyforthenoninteractingquasi-electronsystemasT0[ˆ]=XiZdr i(r)(~22mr2) i(r)=Xi< ij~22mr2j i>:(2.12)Then,thegroundstateenergyasafunctionalofˆ(r)forthesystemcanberewrittenasEKS[ˆ;V]=T0[ˆ]+VH[ˆ]+Exc[ˆ]+ZdrV(r)ˆ(r):(2.13)HereExc[ˆ]isthetermforelectronexchangeandcorrelation,whichisasExc[ˆ]=(T[ˆ]T0[ˆ])+(Vee[ˆ]VH[ˆ]):(2.14)Itdescribestheenergycausedbytheexchangeinteractionandelectroncorrelationintheelectron-electroninteractionenergyVee[ˆ]aftersubtractingtheHartreeenergy;andtheinthekineticenergybetweentheinteractingelectronsandthenon-interactingquasi-electrons.WecanusevariationalmethodtominimizeEKS[ˆ;V]bykeepingthetotalnumberofelectronsNunchangedastheconstraintcondition:N=Zdrˆ(r)=Xi< ij i>:(2.15)Thequasi-electronwavefunctions< ij j>areorthonormal.ThegroundstateenergyandelectrondensityaredecidedbythevariationalextremumofEKS[ˆ;V]Pi(i< ij i>1),whereiaretheLagrangemultipliers.SubstitutingEq.(2.13)intothevariationaltermand14doingthevariationwithrespectto< ij,weget< iT0[ˆ]ˆ+VH[ˆ]ˆ+Exc[ˆ]ˆ+V(r)i i>=0:(2.16)Theequationisforarbitrary< ij,sothecotof< ijshouldbezero.Thenweget[~22mr2+Veff(r)] i(r)=i i(r):(2.17)Itisverysimilartothesingle-electronequationexceptVeffistheepotentialandhastheformVeff(r)=VH[ˆ]ˆ+Excˆ+V(r)=VC(r)+Vxc(r)+V(r);(2.18)VC(r)=Zdr0ˆ(r0)e2jrr0j:(2.19)Eq.(2.17)describesthemotionofaquasi-electronundertheepotentialVeff(r).However,thisepotentialdependsontheground-stateelectronicdensityˆ(r)=Xij i(r)j2:(2.20)Eqs.(2.17)and(2.20)arecalledKohn-Shamequationsandtheyneedtobesolvedinaself-consistentway.ThisisanimportanttheoreticalresultbecauseitdemonstratesthatthegroundstateproblemcanbesolvedbyKohn-Shamself-consistentequations,whichdescribethemotionofsinglequasi-electrons.However,theelectronexchangeandcorrelationpartExc[ˆ]andcorrespondingVxc[ˆ]areunknown,andsolvingtheKohn-Shamequationsisstillachallenge.15AnotherthingthatmustbenoticedisthattheLagrangemultipliersiarenotthesingle-electronenergiesinthemany-bodysystem.2.4Exchange-correlationfunctionalsTheapplicationofDFTdependsonthechoiceoftheexchange-correlationfunctionalExc[ˆ].Toconstructsuitableformsoftheexchange-correlationfunctionals,approximationsatdif-ferentlevelshavebeenmade.Thelocaldensityapproximation(LDA)[62,63],whichisappliedformostofthecarbonsystemsinthisthesis,isthemoststraightforwardapproximationoftheexchange-correlationfunctionalwheretheexchange-correlationenergyisafunctionofthelocalelectrondensityˆ.Theexchange-correlationenergycanbewrittenasELDAxc[ˆ]=Zdrˆ(r)xc(ˆ):(2.21)Here,xc(ˆ)istheexchangeandcorrelationenergydensityofahomogenouselectrongaswithdensityˆ.xc(ˆ)canbefurtherseparatedintoanexchangetermandacorrelationtermxc(ˆ)=x(ˆ)+c(ˆ):(2.22)Inauniformelectrongas,theexchangepartisknown:x(ˆ)=343ˇ13ˆ(r)13:(2.23)Theremainingpartofxc,thecorrelationenergyc,hasbeendeterminedbyCeperleyand16Alder[63]usingquantumMonteCarlocalculationsofauniformelectrongaswithtdensities.LDAisexactfortheuniformelectrongas.However,theelectrondensityisnothomoge-nousinrealsystems.ApplicationofLDAisbasedontheassumptionthattheexchange-correlationenergyofthenonuniformsystemcanbeobtainedapproximatelybyseparatingthesystemintoitelymanysmallportionswithconstantelectrondensityandbytreatingtheseasalocallyuniformelectrongasofthesamedensity.LDAprovidesagoodapproximationtotheexchange-correlationenergywhenthechargedensitydoesnotchangedramatically.However,ignoringthevariationoftheelectrondensitywillleadtoerrorsespeciallyforsystemswithsignitvariationsintheelectrondensity.Tocorrectthiserror,Exccanbeconsideredasafunctionalofnotonlythelocalelectrondensitybutalsotheelectrondensitygradient,whichistheessentialideaoftheso-calledgeneralizedgradientapproximation(GGA).InGGA,theexchange-correlationenergycanbewrittenasEGGAxc[ˆ]=Zdrf(ˆ(r);rˆ(r)):(2.24)Herethefunctionf(ˆ;rˆ)canhavetchoicesandtherearemanyerentGGAsdependingontf(ˆ;rˆ).TheGGAfunctionalusedinthisthesisappliedtomostlayeredstructuresisthePBE[64]functionalproposedbyPerdew,BurkeandErnzerhofin1996,whichiscurrentlythemostpopularGGAfunctional.AlthoughGGAsaresometimescalled\nonlocal"functionals,thedensityanditsgradientstillsupplyonlylocalinformation.NeitherLDAandGGAcangiveagooddescriptionforlong-rangedispersion(orvanderWaals(vdW)interaction),whichisimportantlayeredstructures.VariousfunctionalsincludingvdWinteractioncorrectionshavebeendeveloped17recently.Inthisthesis,theoptB86b-vdW[65,66]functionalhasbeenusedtodescribetheinter-layerinteractionofthelayeredstructuresstudied.However,theresultsusingvdW-correctedfunctionalshavebeenshowntodependsensitivelyonthechoiceoffunctionalsandcannotcomparedtoquantumMonteCarlocalculations[67]intheirabilitytocorrectlydescribethephysicsoftheinter-layerinteraction.Asdiscussedabove,intheKohn-ShamequationsEq.(2.17),iareLagrangemultipliersandhavenoparticularphysicalmeanings.However,itturnsoutthatwhilethisisstrictlyapproximate,theenergyspectrumofioftenresemblestheeigenvaluesofelectronicstates.Inaddition,thereisusuallyanunderestimationoftheelectronicbandgapofsemiconductorsinbothLDAandGGA.AlthoughDFTisnotdesignedtoreproducethebandgapinsolids,havebeenmadetocorrecttheunderestimationofthebandgapinLDAandGGA.Hybridfunctionalmethods,whichcombinepartialHartree-Fockexactexchangewiththepartialexchange-correlationenergyofDFT,havebeenappliedtocorrectlyreproducethebandgapinsolids.ThemixtureofHF,whichtypicallyoverestimatesthebandgap,withDFT,whichtypicallyunderestimatesthebandgap,usuallypredictsareasonableelectronicbandgapvalue.Inthisthesis,theHSE06hybridfunctionalapproximationintroducedbyHeyd,Scuseria,andErnzerhof[68]areusedtopredictthebandgapvaluesofcertainsystems.2.5BasissetsandpseudopotentialsInordertosolvetheKohn-ShamequationsEq.(2.17),apreselectedbasissetf˜(r)gneedstobechosenandthenthesinglequasi-electronwavefunction i(r)canbeprojectedtothebasisfunctions: i(r)=Xci˜(r):(2.25)18Hereciarethecots.f(r)=~22mr2+Veff(r);(2.26)Eq.(2.17)canberewrittenasf(r)Xci˜(r)=iXci˜(r):(2.27)Asthebasisfunctionsf˜(r)gareknown,whatweneednowarejustthecotsci.Inthisway,theproblemcanbetransformedtoamatrixproblem,whichismucheasiertodealwithbyacomputer.Inquantumchemistry,atomicorbitalsareusuallyusedasthebasisforthemolecularorbitals,basedontheLCAO-MOapproximation,whichassumesthatthemolecularorbitals(MO)arealinearcombinationofatomicorbitals(LCAO).Inphysics,planewavebasissetsarealsooftenusedinsolids.Inthisthesis,atomic-orbitalbasissetsareusedinDFTasimplementedintheSIESTAcode[69].PlanewavebasissetsareusedinDFTasimplementedintheVASPcode[70{73].Sincethechemicalpropertiesaredecidedbythevalenceelectronsofeachatom,thecoreelectronshavelittleonthepropertiesofsolids.Therefore,theironthevalenceelectronscanberepresentedbyapseudopotential,whichthecomplicatedrealall-electronpotentialandreducesthecomputationalsubstantially.Therearemanyentformulastoconstructpseudopotentials.Inthisthesis,norm-conservingpseudopotentialsareusedintheSIESTAcodeandprojectoraugmentedwave(PAW)pseudopotentials[73,74]areusedintheVASPcode.19Chapter3StabilityandelectronicstructureofLa@C60functionalizedbyCF3radicalsThisstudyisacollaborationwithProf.HisanoriShinohara'sgroupatNagoyaUniversity,Japan.3.1IntroductionAtthesametimeasthefamousfullereneC60wasobservedexperimentallyin1985[1],thecomplexesLaC60andotherLaC2nwerealsofoundbylaservaporizationofaLaCl3impregnatedgraphiterod[2].Lateron,moremetalandnonmetalatomsencapsuledbyfullerenesweresynthesisedwithtmethods[3{6,75{78].Unlikepreviousexpectations,onlyM@C82-typemetallofullereneswereseparatedfromtherawsootcontaininghollowfullerenesandothermetallofullereneswithtsizes[3,11{14].IthasbeentoextractthemostabundantM@C60andM@C70metallofullerenesduetotheirinsolubilityinnormalfullerenesolventssuchastolueneandCS2[11,15,16].AlthoughseveralM@C60metallofullereneshavebeenextractedbysolventssuchaspyridineandaniline[16,79{81],andisolationoftheM@C60metallofullerenesisstillabigchallengesincepyridineandanilinearenotsuitablesolventsforHPLC(high-performanceliquidchromatography)separationmethods[15].RightnowitisstillnotclearwhytheM@C60metallofullerenes20behavesotlyfromM@C82,whichiseasytoextractandisolate.Recently,Wangetal.reportedanewstrategytoisolatesmallmetallofullerenesincludingM@C60andM@C70byylation[18,19].Thekeypointhereisthatthesmallmetallofullereneswithopen-shellelectronictionsarechemicallyunstable[17]andthiscanbesolvedbyylation.InthischapterIinvestigatedstabilityandelectronicstructureofendohedralrare-earthmetallofullereneLa@C60andtheylizedLa@C60(CF3)n.IsomerswithromethylsattsitesontheC60arecalculatedbyabinitioDensityFunctionalTheoryandthemoststableisomersfor2n5areidenUnliketheylderivativesofC60,La@C60(CF3)nprefersoddnumbersofn,duetothe3extravalenceelectronsfromtheencapsuledmetalLa.Calculatedelectroniceigenstatesofmolecularorbitalsfurthercon-thatwithanoddnumbernLa@C60(CF3)nismorestablebyformingaclose-shellelectronicorlargegapbetweenhighestoccupiedmolecularorbital(HOMO)andlowestunoccupiedmolecularorbital(LUMO),whichisalsorelevantfortheseparationofC60-basedmetallofullerenes.3.2ComputationalmethodsIutilizeabinitiodensityfunctionaltheory(DFT)asimplementedintheVASPcode[70{72]toobtaintheoptimizedstructureandenergyoftheendohedralmetallofullerene,aswellastheelectronicstructure.Iusedprojector-augmented-wave(PAW)pseudopotentials[82]andthePerdew-Burke-Ernzerhof(PBE)[64]exchange-correlationfunctionals.Allisolatedstructureshavebeenrepresentedusingperiodicboundaryconditionsandseparatedbya12Athickvacuumregion.Iused500eVastheelectronickineticenergyfortheplane-wavebasis21Figure3.1:Ball-and-stickmodeloftheoptimizedstructureofthemoststableisomersofLa@C60(CF3)nfor(a)n=0,(b)n=1,(c)n=2,(d)n=3,(e)n=4,(f)n=5.andatotalenergybetweensubsequentiterationsbelow105eVasthecriterionforreachingself-consistency.Allgeometrieshavebeenoptimizedusingtheconjugate-gradientmethod[83]untilnoneoftheresidualHellmann-Feynmanforcesexceeded102eV/A.3.3ResultsanddiscussionTheequilibriumgeometryofLa@C60andLa@C60(CF3)nforn5asdeterminedbyDFT-PBEcalculationareshowninFig.3.1.Forn2,theylscanhavemorethanoneontheC60surface.Manytisomersfornfrom2to5arecalculatedandthedetailedresultsarediscussedinAppendixA.HereIshowthemoststableisomersfornfrom2to5inFig.3.1(c)-(f).Icanseethatforn=2andn=3,theylsprefertobeclosetoeachotherwithaseparationdistanceofpara-positioninonehexagon(thirdnearestneighbor)onthesurfaceoffullerene.Forn=4theylsareseparatedintotwopara-(CF3)2pairs,andforn=5itprefersoneCF3separatedfromthe22Figure3.2:EnergychangeEtbasedonDFT-PBEforattachingoneyloftLa@C60(CF3)nisomers.HereEtisasEt(n)=Etot(n)Emintot(n1)1=2Etot((CF3)2).Emintot(n)isthetotalenergyforthemoststableisomerofLa@C60(CF3)n,whichareindicatedbythelargersymbolsingreen.otherpara-(CF3)4group.Ialsonoticedthatforn=3andn=4,themoststableisomershaveC2vsymmetry,asshowninFig.3.1(d)and(e).TheBaderchargeanalysis[84{87]isperformedandabout1.8electronsarefoundtobetransferredfromtheLaatomtothecageforallthemoleculesshowninFig.3.1.Anaverageterdistanceofabout1.2AfortheLaatomisfound,andanaverageenergyofabout3.3eVisgainedbymovingtheLaatomfromthecentertotheoptimizedposition.Thiscanbesimplyinterpretedastheelectrostaticpolarizationenergyofthemodelwithapointchargeinsideathinmetallicsphericalshell,whichhasbeensuccessfullyappliedformetallofullerenesbefore[88,89].Usingthe3.5AradiusofC60astheradiusoftheshell,Iobtainedanenergygainof1.7eV,whichisthesameorderasthevalueobtainedbyDFTcalculation.TooutthepreferentialnumberofylsattachedontheLa@C60,Icalcu-latedtheenergychangeEtforattachingoneylofLa@C60(CF3)nbyDFT-23Figure3.3:HydrogenatedLa@C60(CF3)nfor(a)n=0,n=1withhydrogenandoromethylat(b)para-positionand(c)ortho-positioninahexagon.Therowistheball-and-stickmodelofthestructureswithoutLaatominsidethecageandthesecondrowisthestructureswithLaatominsidethecage.ThelastrowisthecorrespondingSchlegeldia-gramofC60withhydrogensiteindicatedbybluedotandylsitebyred.EHhereisasEH=Etot(La@C60(CF3)n-H)Etot(La@C60(CF3)n)1=2Etot(H2),orEH=Etot(C60(CF3)n-H)Etot(C60(CF3)n)1=2Etot(H2)forthatwithoutLaatomin-side.PBE.ThespitionforEtasafunctionofnisEt(n)=Etot(n)Emintot(n1)1=2Etot((CF3)2).HereEmintot(n)isthetotalenergyforthemoststableisomerofLa@C60(CF3)n.ThenegativevalueofEtindicatesanexothermicreactionforadsorb-ingoneCF3radicalfromaethanemolecule.ThelowerEtis,themorestablethestructurewouldbe.AccordingtotheresultsplottedinFig.3.2,Ithattherecanbeanenergyofasmuchas3eVfortisomerswiththesamen.Comparingthemoststableisomers,Iseethat,ingeneral,La@C60moleculesprefertohaveanoddnumber24ofCF3s.ThreeistheindicatedbestnumberofattachedCF3sonLa@C60,whichhasthelowestEof-0.44eV.n=4hasthehighestEof-0.11.eV,0.33eVhigherthann=3.ThisresultisconsistentwithpreviousworkonM@C70molecules[18,19].However,thisiscontrarytothehollowfullerenes,whichpreferevennumberofattachedCF3s[90,91].Itiseasytounderstandthatonahollowfullerene,whenaddingCF3s,theC-CdoublebondsofthefullereneneedtobebrokenandtheCatomwhereCF3isattachedwillchangetoasp3bondingfromtheoriginalsp2bonding.Inthatcase,anoddnumberofCF3radicalsattachedmustproduceoneCradicalonthefullerenesurface,whichisunstable.Thisdif-ferenceiscausedbytheencapsuledmetalatomLa,whichhasthreevalenceelectrons.Theoddnumberofextrafreeelectronswilltrytotransfertothecageandmodifytheelectronicenvironmentofthesurface,whichconsequentlyresultsinapreferenceforanoddnumberofattachedCF3s.TofurtherinvestigatetheoftheencapsulatedLaatominLa@C60(CF3)nmoleculesonthecagesurface,IcalculatedthehydrogenationprocessofLa@C60(CF3)nmoleculeswithandwithoutLainsideforn=0andn=1.TheenergychangeEHforadsorbingonehydrogenatomfromahydrogenmoleculeonLa@C60(CF3)nwascalcu-lated.TheresultsareshowninFig.3.3.ThespforEHhereisEH=Etot(La@C60(CF3)n-H)Etot(La@C60(CF3)n)1=2Etot(H2)formoleculeswithLaen-caplsuledorEH=Etot(C60(CF3)n-H)Etot(C60(CF3)n)1=2Etot(H2)forthosewithoutLaatom.FromtherowofFig.3.3IcanseethatforthehollowfullereneC60-H,Iob-tainedapositivevalueof0.205eVforEH,whichindicatesaendothermicprocess.Similaraspreviouslydiscussedforyls,theattachedHatommustbreakonedoublebondonC60andchangetheoriginalsp2CatomwhereHisattachedtosp3.ThentheCatomontheotherendoftheoriginaldoublebondwillhaveanunpairedelectronleftand25causeinstability.Thisisconsistentwithpreviousworkonhydrogenationofcarbonnan-otubesandfullerenes[92,93].However,fortheC60(CF3)-Hmolecule,theattachedHatomandCF3radicalcanthetwounstablesiteswithunpairedelectronscausedbybreakingonedoublebond.IgotnegativevaluesforboththeHandCF3inaparaarrangement(-0.807eV)andanorthoarrangement(-1.083eV)ontheC60surface.AlowervalueofEHisobtainedfortheorthoarrangementwhereHandCF3areattachedtothetwoendsofadoublebondonC60,whichcaneasilybeseenintheSchlegeldiagramshowninthelastrowofFig.3.3.ThesituationwillbetotallyntwhenaLaatomisencapsulated.AsshowninthesecondrowofFig.3.3,IalwaysgetnegativeEHregardlessofwhetherorwheretheCF3isattached.TheextrathreefreeelectronscomingfromthevalenceelectronsofLachangetheelectronicenvironmentonthesurfaceofthewholemoleculeandmakeitalwaysfavorabletoadsorbHatoms.TheofEHforLa@C60(CF3)-HandLa@C60-Hisabout0.1eV(from-0.485eVto-0.592eVand-0.332eV),whichismuchsmallerthanthe1:0eVchange(from0.205eVto-0.807eVand-1.083eV)intheabsenceofencapsulatedLa.Moreover,forLa@C60(CF3)-H,theHandCF3nolongerliketheorthoarrangementontheendsofabrokendoublebondbutprefertheparaarrangement,whichistheoptimizedarrangementforLa@C60(CF3)2asshowninFig.3.1(c).TheresultofthecomparisonofhydrogenationforthemetallofullereneswithLaencapsuledandhollowfullerenesthattheLaatomisthekeyfactorforwhyLa@C60(CF3)nprefersanoddnumbernratherthantheevenn,whichispreferredforhollowC60.ThethreevalenceelectronsofLaalsoagreewellwiththreebeingthebestnumbern.Sincetheelectronicstrucureswilltlythechemicalstabilityofthemet-allofullerenemolecules,IcalculatedtheelectroniceigenstatesofC60andLa@C60(CF3)n26Figure3.4:ElectroniceigenstatesformolecularorbitalsbasedonDFT-PBEof(a)C60andLa@C60(CF3)nwith(b)n=0,(c)n=1,(d)n=2,(e)n=3,(f)n=4,(g)n=5.TheLUMOandHOMOofC60andthecorrespondingstatesinLa@C60areindicatedinredandorangesquares,respectively,in(a)and(b).moleculeswithnfrom0to5attheDFT-PBElevelandshowtheresultsinFig.3.4.InFig.3.4(a),IshowthemolecularstatesofC60.Ithasawidegapof1.6eV.TheHOMO(inorangedashedsquare)ise-folddegenerate,andtheLUMO(inreddashedsquare)isthree-folddegenerateduetothehighsymmetryofC60.Incomparison,inFig.3.4(b)IseethatforLa@C60,sincethesymmetryofC60isbrokenbytheencapsulatedLaatom,thee-folddegenerateHOMOinC60splitsintotwodoublydegeneratestatesandoneseparatestate,whichareveryclosetoeachother(showninorangesquare).Thethree-folddegenerateLUMOinC60splitsintoadoublydegeneratestateandaseparatestateabout0.2eVhigherinenergy(shownintheredsquare).La@C60displaysmetalliccharacterwiththeFermilevellyingonthedoublydegeneratestatewhichusedtobetheLUMOinC60.ThisisduetothethreevalenceelectronsfromLapartlyoccupyingthedoublydegeneratestate.ForLa@C60(CF3)nwhenn1,theeigenstateschangealotduetotheattachedCF3sandIcannotdistinguishtheoriginalHOMOandLUMOinC60anymore.ThemostimportantthingIisthatforevennIalwaysgetametalliccharacterwiththeFermilevelonapartlystate,asshowninFig.3.4(b),(d),and(f).Ontheotherhand,foroddnIalways27getanon-zerogap,asshowninFig.3.4(c),(e),and(g).Thisalsoagreeswithpreviousdiscussionsthatoddnispreferred,sincestructureswithanon-zerogaparechemicallymorestablethanthosewithnogap.IalsofoundthatLa@C60(CF3)nmoleculeswithlargernwillhavealargergap.Thereforeforoddnumberofn,n=5withagapof1.0eVshouldbethemostchemicallystable,followedbyn=3withagapof0.7eV.n=1willberelativelytheleaststablewithanarrowgapof0.1eV.IexpectthatLa@C60(CF3)nmoleculeswithn=3and5willbeeasytoseparateinexperimentsincetheyhaverelativelylargegaps.3.4SummaryInsummary,Iinvestigatedthestabilityandelectronicstructureofendohedralrare-earthmetallofullereneLa@C60andtheylizedLa@C60(CF3)n.IsomerswithoromethylsatrentsitesontheC60arecalculatedusingabinitioDensityFunctionalTheory,andthemoststableisomersfor2n5areidenUnliketheylderivativesofC60,La@C60(CF3)nprefersoddn,duetothe3extravalenceelectronsfromtheencapsuledmetalLa.Calculatedelectroniceigenstatesofmolecularorbitalsfurthercon-thatwithanoddnLa@C60(CF3)nismorestableduetoformationofaclosed-shellelectronicorlargeHOMO-LUMOgap,whichisalsorelevantfortheseparationofC60-basedmetallofullerenes.28Chapter4ElectronicstructureandchargeinjectionacrosstransitionmetaldichalcogenideheterojunctionsThisstudyisacollaborationwithProf.ZhixianZhou'sgroupatWayneStateUniversity.4.1IntroductionSincethesuccessfulisolationofsingle-layermolybdenum(MoS2)anditsapplicationintransistors[21,22],layeredtransition-metaldichalcogenides(TMDs)havebeenconsideredaspromisingcandidatesfortwo-dimensional(2D)electronicsandoptoelectronicsapplications[94{112].Eventhoughmuchhasbeendevotedtothefabricationofhigh-performanceelectronicdevices,thedeviceperformancetypicallyersfrominadequateelectricalcontactscausedbytSchottkybarriers(SB)betweenthecontactmetalandtheTMD[23{26].Localdopingnearthemetal-semiconductorinterface,whichhasbeensuccessfullyusedintraditionalthree-dimensional(3D)silicon-baseddevices,cannotbeeasilyrealizedin2Dcases[113{117].MetalelectrodeswithaproperworkfunctionandFermilevelclosetotheconductionbandminimum(CBM)orvalencebandmaximum(VBM)areexpectedtohavealowerSchottkybarrierheight(SBH)andtodecreasethe29contactresistance[118{121].However,thecomplexFermilevelpinningattheelectrode-semiconductorinterfacemakesOhmiccontactsrathertoachieve[122{124].Themetal-TMDinterfaceandthustheextentofFermilevelpinningishighlysensitivetotheprocessingenvironmentandthecrystallineorderinTMDs[125,126].Severalotherstrategiestoreducethecontactresistancehavebeenproposedandused,suchasin-layerjunctionsobtainedbyphaseengineering[127,128]and2D/2Dinter-layerjunctionswithweakFermi-levelpinning[129],includinghexagonalboronnitride(hBN)[130],graphene[131{134]anddopedTMDs[135].Insearchofanimprovedstrategytoformlow-resistancecontactstosemiconductingTMDs,IproposetousemetallicTMDsasdrain/sourcecontacts.ItheSchottky/tunnelingbarrierbetweenmetallicandsemiconductingTMDsisfundamentallytfromacon-ventionalmetal-semiconductorSchottkybarrierforthefollowingimportantreasons.First,theformationofinterfacestatesissuppressedinTMDheterojunctionsduetothelackofdanglingbondsonTMDsurfacessimilartographene/TMDjunctions.Second,anabruptpotentialdrop,ratherthanarelativelythickdepletionlayer,formsacrossthevanderWaalsgapofTMDheterojunctions.Asaresultofthesences,FermilevelpinningisexpectedtobetlyreducedatmetallicandsemiconductingTMDinterfaces.Tovalidatethisbehavior,IperformabinitiodensityfunctionalcalculationsofaNbSe2/WSe2bilayer,con-sistingofsemiconductingWSe2andmetallicNbSe2monolayers.Besidesarigidbandshiftassociatedwithchargetransfer,IthatpresenceofNbSe2doesnotmodifytheelectronicstructureofWSe2.Sincethetwotransitionmetaldichalcogenidesarestructurallysimilaranddisplayonlyasmalllatticemismatch,theyformaheterojunctioncapableoftlytransferringchargeacrosstheinterface,thusimprovingcurrentinjectionintoWSe2.304.2ComputationalMethodsSameDFTmethodastheformerchapterisutilizedtoobtaintheoptimizedstructureandelectronicpropertiesofWSe2andNbSe2monolayers,theNbSe2/WSe2bilayer,andthecorrespondingbulkstructureswithABstackingoflayers.Thetwo-dimensional(2D)struc-turesarerepresentedbyaperiodicarrayofslabsseparatedbyavacuumregioninexcessof15A.BesidesthePerdew-Burke-Ernzerhof(PBE)[64]exchange-correlationfunctionals,theoptB86b-vdW[65,66]functionalisusedtoaddresstheweakinterlayerinteractions.TheBrillouinzoneoftheprimitiveunitcellofthe2Dstructureswassampledby881k-pointsandthatofbulkstructuresby882[136].Atotalenergybetweensubsequentself-consistencyiterationsbelow106eVasthecriterionforreachingself-consistency.Theotherparametersarethesameasusedbefore.4.3ResultsanddiscussionsIoptimizedthebulkstructuresofWSe2andNbSe2inthestable2Hphase.ResultsofmyDFT-optB86bcalculationsshowthattheABlayerstackingisenergeticallyfavorablecomparedtotheAAstackinginbothsystems.ForbulkNbSe2,Ifoundthatmycalculatedin-planelatticeconstantatheor=3:46Aliesclosetotheobservedvalue[137]aexpt=3:44Aandtheout-of-planelatticeconstant,coveringtwointerlayerdistances,isctheor=12:74A,closetotheobservedvalue[137]cexpt=12:48A.ForbulkWSe2,Ialsoobtainedverygoodagreementwithexperiment[138]:atheor=3:30Aandaexpt=3:28Aforthein-layerlatticeconstantandctheor=13:10Aandcexpt=12:96Afortheout-of-planelatticeconstant.Inoticethatthein-layerlatticeconstantofNbSe2isonly5%largerthanthatofWSe2inthebulk,suggestingthelikelihoodofepitaxialstackingespeciallyinfew-layersystems.31Figure4.1:Ball-and-stickmodeloftheAB-stackedNbSe2/WSe2bilayerin(a)topviewand(b)sideview.Theunitcellisindicatedbyyellowshadinginthetopview.a1anda2arethelatticeconstantsspanningthelayersanddistheinter-layerdistance.IalsofoundapreferentialABstackingforthe2H-NbSe2/2H-WSe2bilayer,depictedinFig.4.1.Thebilayerformsahoneycomblatticewith6atomsperunitcell,3ofwhichareintheNbSe2andtheother3intheWSe2layer.Theoptimumin-layerlatticeconstantatheor=3:37A,whichliesin-betweenthevaluesfortheindividualbulkcomponents.Alsothevaluedtheor=6:41Afortheoptimuminter-layerdistanceliesin-betweenthecorrespondingvaluesinbulkNbSe2andWSe2.Thebindingenergyofthetwolayersis0.19eVperunitcellbasedonDFT-optB86b,indicatingaweakinterlayercoupling.ElectronicbandstructureresultsfortheNbSe2/WSe2bilayeranditsmonolayercom-ponentsareshowninFig.4.2.AsseeninFig.4.2(a),theNbSe2/WSe2bilayerismetallic.Tointerpretthebandstructureofthebilayer,IcalculatedseparatelythebandstructureofisolatedNbSe2andWSe2monolayersanddisplayeditinFig.4.2(b).TheseresultsindicatethattheNbSe2monolayerismetallic,whereastheWSe2monolayerissemiconductingwithadirectgapofabout1:35eVattheKpoint.Eventhoughtheabsolutevalueoftheband32Figure4.2:ElectronicstructureofanNbSe2/WSe2bilayeranditsdecomposition.Thebandstructureofthebilayerin(a)iscomparedin(b)toasuperpositionofindividualmonolayersofNbSe2(solidredlines)andWSe2(dashedgreenlines).Thebandstructurein(c)isobtainedfrom(b)byshiftingthebandsofsemiconductingWSe2rigidlyupby0:293eV.Theresultingbandstructureissuperimposedonthatof(a)representingthebilayer.gapistypicallyunderestimatedinDFTcalculations,thedispersionofindividualbandsistypicallyingoodagreementwithmoreadequateself-energycalculations.Ifoundthatin-dividualbandsinisolatedNbSe2andWSe2monolayersinFig.4.2(b)canbeideninthebandstructureofthebilayer.Toprovethispoint,Isuperposedthebandstructureofthetwoconstituentparts,shiftingrigidlythebandsofWSe2upby0:293eV.Theresultingbandstructure,presentedinFig.4.2(c),issuperposedwiththatofthebilayerofFig.4.2(a).Theagreementbetweenthebandstructureofthebilayerandthesuperpositionofmonolayersisnear-perfect,suggestingtheapplicabilityofarigid-bandmodelandasmallchargetransferbetweenotherwiseectedNbSe2andWSe2monolayers.Tobetterunderstandtheinter-layerinteractionintheNbSe2/WSe2bilayer,Icalculatedthechargedensityˆ=ˆtot(bilayer)Pˆtot(monolayers)associatedwiththeassemblyofthebilayerfromisolatedmonolayers.ThischargedensityisvisualizedbycontourplotsinFig.4.3(a).Ithatthechargeredistributionisverysmall,with33Figure4.3:Chargeredistributionˆ(r)andspatialvariationoftheelectronpotentialas-sociatedwithassemblingtheNbSe2/WSe2bilayerfromisolatedmonolayers.(a)Chargedensityˆ=ˆ(NbSe2=WSe2)ˆ(NbSe2)ˆ(WSe2ˆisshownbyisosurfacesboundingregionsofelectronexcess(yellow)at+2:5104e/A3andelectron(blue)at2:5104e/A3.(b)hˆ(z=c)iaveragedacrossthexyplaneofthelayers.Therawdata,shownbytheredsolidline,havebeenconvolutedbyaGaussianwithafull-widthathalf-maximumz=c)=0:17,shownbythebluedashedline.(c)ElectrostaticpotentialhV(z=c)iaveragedacrossthexyplaneofthelayers.z=cindicatestherelativepositionoftheplanewithintheunitcell.electronstransferringmainlyfromtheWSe2layertotheinter-layerregionandtotheNbSe2layer.Tofurtherelucidatethechargeredistribution,IaveragedthechargedensityerenceinplaneswithconstantzthatareparalleltotheNbSe2andWSe2layers.Theaveragedquantityhˆ(z=c)i,shownbythesolidlineinFig.4.3(b),displaysmanyoscillationseveninindividualTMDlayers.Tobetterunderstandthenetchargew,IconvolutedtherawdatabyaGaussianwithafull-widthathalf-maximumof0.17.Theresultingfunction,displayedbythebluedashedlineinFig.4.3(b),indicatesaverysmallnetwofelectronsfromtheWSe2layertotheNbSe2layer.ThisresultisconsistentwiththeupwardshiftofWSe2bandsdiscussedinrelationtoFig.4.2(c).WhatIfoundhereagreedwithpreviousworkontheVS2/MoS2bilayersystem[129].Asamorequantitativemeasureofthecharge34redistribution,IperformedaBaderchargeanalysis[84{87]intheNbSe2/WSe2bilayerandfoundthat0:017electronsperunitcellaretransferredfromtheWSe2totheNbSe2layer.Sincechargeinjectionacrosstheinterfaceismodulatedbypotentialbarriers,whichappearastunnelorSchottkybarriers,IalsoinvestigatedthelocalelectrostaticpotentialVinthewholeNbSe2/WSe2bilayerregion.Foreasyinterpretation,IaveragedVinconstantzplanes,similartoˆ(z=c)inFig.4.3(b),andpresentresultsforhV(z=c)iinFig.4.3(c).ThebetweentheelectrostaticpotentialinthevacuumregionandtheFermilevel,observedatV(vac:)EFˇ5:6eV,correspondstotheworkfunction.IalsoobserveanarrowSchottkybarriercharacterizedbyz)ˇ3:4AandVˇ14eVwhenmeasuredfromthebottomoftheclosestpotentialwell.ForelectronsattheFermilevel,theebarriertotunnelthroughreducestoonethirdofVandthebarrierthicknessreducesz)bymorethanhalf,thustlyincreasingtheprobabilityoftunnelingacrosstheinterface[119].4.4SummaryInsummary,IperformedabinitiodensityfunctionalcalculationsofaNbSe2/WSe2bilayer,consistingofsemiconductingWSe2andmetallicNbSe2monolayers.Thissystemcanbeconsideredaheterojunctionbetweentworelatedmaterialsandmaybeviewedasanewparadigmforhigh-transparencymetalcontactstotransitionmetaldichalcogenides.BesidesarigidbandshiftassociatedwithasmallelectrontransferfromNbSe2toWSe2,IthatthepresenceofNbSe2doesnotmodifytheelectronicstructureofthesemiconductingchannel.Sincethetwotransitionmetaldichalcogenidesarestructurallysimilaranddisplayonlyasmalllatticemismatch,theirheterojunctioniscapableoftlytransferringchargeacrosstheinterface,thusimprovingoncurrentmetalcontactstoWSe2.35Chapter5StructureandelectronicpropertiesofstrainedbulkblackphosphorusThefollowingdiscussionismyoriginalcontributiontotherelatedpublicationbyJieGuan,WenshenSong,LiYangandDavidTanek,Phys.Rev.B94,045414(2016)[139].ThisstudyisacollaborationwithProf.LiYang'sgroupatWashingtonUniversityinSt.Louis.5.1IntroductionLayeredbulkblackphosphorus(BP),discoveredonlyonecenturyago[29,140],isadirect-gapsemiconductorwithanobservedfundamentalbandgap[141{144]of0:310:36eV.TheelectronicresponsedistinguishesBPasfavorablefromotherwell-studiedlayeredsystemsincludingsemimetallicgraphiteandtransitionmetaldichalcogenides(TMDs)suchasMoS2,whichareindirect-gapsemiconductors.Undercompression,bulkBPdisplaysaninterestingchangeinitselectronicandtopologicalproperties[30{32].SimilartobulkBPbutamuchwiderdirectfundamentalbandgapisfoundinphosphorenemonolayersandfew-layersys-tems,suggestingpromisingapplicationsin2Dsemiconductorelectronics[27,28,33].Whereasitisnowwellestablishedthatthegapdisplaysastrongandanisotropicresponsetoin-layerstraininphosphorenemonolayersandfew-layersystems[27,28,33{38],nodependabledataareavailableforthecorrespondingresponseinthebulksystem.36Tointhismissinginformation,Istudytheoreticallythestructuralandelectronicresponseoflayeredbulkblackphosphorustoin-layerstrain.Myabinitiodensityfunctionaltheory(DFT)calculationsrevealthatthestrainenergyandinterlayerspacingdisplayastronganisotropywithrespecttotheuniaxialstraindirection.Tocorrectlydescribethedependenceofthefundamentalbandgaponstrain,IcombinedmyDFTresultswiththecomputationallymoreinvolvedGWquasiparticleapproach,performedbyLiYang'sgroup,whichisfreeofparametersandsuperiortoDFTstudies,whichareknowntounderestimategapenergies.Ithatthebandgapdependssensitivelyonthein-layerstrainandevenvanishesatcompressivestrainvaluesexceedingˇ2%,thussuggestingapossibleapplicationofblackPinstrain-controlledinfrareddevices.5.2ComputationalmethodsIutilizedabinitiodensityfunctionaltheory(DFT)asimplementedintheSIESTA[69]codetooptimizethestructureandtodeterminethestructuralresponsetoin-planestrain.IusedthePerdew-Burke-Ernzerhof(PBE)[64]exchange-correlationfunctional,norm-conservingTroullier-Martinspseudopotentials[145],andadouble-basisincludingpolarizationorbitals.Reciprocalspacewassampledbyagrid[136]of884k-pointsintheBrillouinzoneoftheprimitiveunitcellcontaining8atoms.Iusedameshenergyof180Rytodeterminetheself-consistentchargedensity,whichprovideduswithaprecisionintotalenergyof2meV/atom.Thesameconjugategradientmethod[83]asformerchaptersisusedinasameaccuracy.DFTcalculationsarenotdesignedtoreproducetheelectronicbandstructurecorrectly.EventhoughDFTbandstructureusuallyresemblesobservedresults,thefundamentalband37Figure5.1:(a)Ball-and-stickmodelofthestructureofbulkblackphosphorusintopandsideviews.(b)Fractionalchangeoftheinterlayerdistancea3asafunctionofthein-layerstrainalongthea1anda2directions.(c)DependenceofthestrainenergyEperunitcellonthein-layerstrainalongthea1anda2directions.gapisusuallyunderestimated.Theproperwaytocalculatethebandstructurewithoutad-justableparametersinvolvessolvingtheself-energyequation.MycollaboratorsfromWash-ingtonUniversityinSt.LouisperformsuchcalculationsusingtheGWapproximation[146]asimplementedintheBerkeleyGWpackage[147],wherethedynamicalelectronicscreeningiscapturedbythegeneralplasmonpolemodel.[146]Wechosethisstate-of-the-artap-proachtocomputationallylessinvolvedhybridDFTfunctionalssuchasHSE06[68],whichmixHartree-FockandDFT-PBEexchange-correlationenergiesusinganadjustableparam-eter.Theyusesingle-shotG0W0calculationswitha14104k-pointgrid,whichprovidesconvergedresultsfortheself-energiesandquasiparticleenergygapsinstrainedbulkblackphosphorus.385.3ResultsanddiscussionTheoptimumstructureofbulkblackphosphorus,asobtainedbyDFT-PBEcalculations,isshowninFig.5.1(a).Asseeninthebottompanel,individuallayersinthelayeredstructurearenotduetothenon-planarsp3hybridizationofthePatoms.TheABstackingiscausedbydisplacingeveryotherlayeralongthe~a2-direction,yieldinganorthorhombiclatticespannedbytheorthogonallatticevectors~a1,~a2and~a3,with~a3extendingovertwointer-layerdistances.Thecovalentin-planebondingisadequatelydescribedbyDFT-PBE,assuggestedbytheagreementbetweenthecalculatedlatticeconstants,a1(PBE)=4:53Aanda2(PBE)=3:36A,andtheexperimentalvalues[148]a1(expt)=4:38Aanda2(expt)=3:31A.AsindicatedbyrecentQuantumMonteCarlostudies[67],thenatureoftheweakinter-layerinteractioninbulkblackphosphorusinanon-trivialmannerfromavanderWaalsinteraction.Inviewofthisfact,thecalculatedvalueoftheout-of-planelatticeconstanta3(PBE)=11:15Aagreesratherwellwiththeobservedvalue[148]a3(expt)=10:50A.Asaresultoftheweakinter-layerinteractioncontrastingthestrongin-layercovalentbonding,Idonotexpecttheinterlayerdistancetochangemuchwhenthelatticeissubjectedtoin-layerstrain.Iconsideredbothtensileandcompressivein-layerstrainupto2%.Resultsforthefractionalchangea3=a3fortstraincombinations(a1);(a2)arepresentedinFig.5.1(b).Thecontinuouscontourplotisbasedonacubicsplineinterpolationofa55gridofdatapointsfortstrainvaluecombinations.Theseresults,aswellasmysuggestthattheinterlayerspacingchangesmuchlessthan1%forthestrainrangeconsideredhere.Suchsmallchangesintheinterlayerdistanceareunlikelytobebythechoiceofthetotalenergyfunctional,whichplaysonlyaminorroleinthe39interlayerseparation[67]andareconsistentwithveryweaklycoupledlayers.MyresultsinFig.5.1(b)indicateatrendthattheinter-layerdistanceincreasesbystretch-inganddecreasesbycompressingthecrystalalongthesoft,accordion-likea1direction.Incontrast,bothstretchingandcompressionalongthea2directioncauseareductionoftheinter-layerspacing.Eventhoughthesearesmall,theyclearlytheanisotropyofthesystem.TheytranslatetoaverysmallnegativePoissonratiobetweenthesoft~a1in-layerdirectionandthe~a3directionnormaltothelayers.ThePoissonratiobetweenthehard~a2in-layerdirectionandthe~a3directionisalsoverysmallinmagnitude,butchangessignneara2=0.ThisitionofthePoissonratiointhebulkfromthe\Poissonratio"inaphosphorenemonolayer,whichrelatesthemonolayerthicknesstothein-layerstrainandanegativevalueforthatquantity[149].Withtheoptimumvalueoftheinter-layerspacinga3;opt(a1;a2)forthetstraincombinationsathand,IhavecalculatedthestrainenergyEasafunctionof(a1)and(a2)andpresenttheresultsinFig.5.1(c).Theprominentlyellipticalshapeoftheisoenergeticcontoursisanothermanifestationoftheelasticanisotropyinthesystem.TheobservedtiltoftheellipticalaxesfromthehorizontalandverticaldirectionindicatesapositivePoissonratio21=(a2)(a1)=0:19withinthephosphoreneplane,indicatingthatstretchinginone(in-plane)directionresultsinalatticecontractioninthenormal(in-plane)direction.Ithelatticetoberathersoftwithrespecttoin-planecompression,sincestretchingby2%evenalongthea2directionrequiresanenergyinvestmentofonlyˇ0:06eVperunitcell.Resultsofmycalculationsfortheelectronicbandstructureanddensityofstates(DOS)ofunstrainedandstrainedbulkblackphosphorusareshowninFig.5.2.TheDFT-PBEresults,representedbythedashedredlinesinFig.5.2(a),predictanextremelysmalldirect40Figure5.2:Electronicbandstructure(leftpanels)anddensityofstates(rightpanels)ofbulkblackphosphorus(a)withoutstrain,(b)whenstretchedby1%alongthea1direction,and(c)whenstretchedby2%alongthea1direction.GWresultsareshownbythesolidblacklines.DFT-PBEresults,whichunderestimatethebandgap,areshownbythedashedredlines.fundamentalbandgapvalueEg(PBE)ˇ0:05eVforunstrainedbulkblackphosphorus.Thebandgapbecomeslargerasthestructuresarestretchedalongthea1direction.Asmen-tionedbefore,theDFTresultsareknowntosubstantiallyunderestimatethebandgapinsemiconductors.BandstructureresultsobtainedusingthemoreproperGWapproachbymycollaboratorsarerepresentedbythesolidblacklinesinFig.5.2(a).ThesedatasuggestalargerquasiparticlebandgapofEg(qp)ˇ0:35eVinunstrainedbulkblackphosphorus,veryclosetothepublishedvaluebasedonGWcalculations[150,151]andtotheobservedvalue[141]ofˇ0:33eV.betweenquasi-particlespectra,Eqp,andDFTbandstructureresults,EPBE,aresummarizedinFig.5.3(a).Asshownpreviously[146,152],theself-energy(orGW)cor-rectionisroughlyrepresentedbya\scissoroperator",whichwouldshiftDFT-basedvalencestatesrigidlydownandconductionstatesrigidlyupby.0:2eVinbulkblackphosphorus,thusopeningupthefundamentalbandgap.AmoreprecisecomparisonbetweenthequasiparticlespectraandDFTeigenvaluesre-vealsthattheEqpEPBEdoesdependontheenergy,butisindependentofthe41Figure5.3:(a)Correlationbetweenquasiparticle(GW)energiesEqpandKohn-ShamenergyvaluesEPBE,obtainedusingtheDFT-PBEfunctional,forstatesalonghighsymmetrylinesintheBrillouinzone.Thestraightlinesinthevalenceandconductionbandregionsaredrawntoguidetheeye.Theresultsrepresentbulkblackphosphorussubjecttostrainvalues(a1)=1%and(a2)=0.TheFermilevelisatzeroenergy.(b)Dependenceofthequasiparticle(GW)electronicbandgapEgonthein-layerstrainappliedalongthea1anda2directions.crystalmomentumk.Iconsideredblackphosphorusstretchedby1%alongthesofta1di-rection,whichhasanonzerobandgapinDFT-PBE,anddisplayedthecorrelationbetweenquasiparticleenergiesEqpandcorrespondingDFTeigenvaluesEPBEatselectedhighsym-metrypointsintheBrillouinzoneinFig.5.3(a).BesidesthediscontinuityattheFermilevel,IfoundthatthequasiparticleenergiesdisplayalinearrelationshipwithDFTeigenvalues,givenbyEqp(CB)=1:10EPBE(CB)+0:11eV(5.1)Eqp(VB)=1:07EPBE(VB)0:18eV:(5.2)AssumingthattheFermilevelzeroenergy,thelinearrelationshipbetweenEqpandEPBEisslightlytintheconductionband(CB)region,idenbyEqp>0,andthevalencebandregion,idenbyEqp<0.SinceGWenergieshavebeencalculated42onlyatafewk-points,Ihaveusedexpressions(5.1)and(5.2)togeneratethecontinuousGWbandstructureshowninFig.5.2.ComparingDFTandGWvaluesattstrains,IfoundthatthemodulationofthebandgapEg(PBE)ˇg(qp)isthesameupto.0:01eVinthestrainrangestudiedhere.Withthequasiparticlebandgapofunstrainedblackphosphorusathand,Ithuscandeducethequasiparticlebandgapinphosphorussubjecttotstrainvalues(a1)and(a2)bycombiningtheDFTbandgapvaluesEg(PBE)withEqs.(5.1)and(5.2).Myresults,basedonacubicsplineinterpolationofa55gridofdatapoints,areshowninFig.5.3(b).Myresultsindicatethat,withintherangejj.2%ofstrainappliedalongthea1anda2direction,thebandgapEgofbulkblackphosphorusvariessmoothlybetween0:05eVand0:70eV.Independentofthestraindirection,Egincreasesuponstretchinganddecreasesuponcompression.NoticingthenearlyequidistantspacingbetweenthecontourlinesinFig.5.3(b),Icanextrapolatetolargerstrainsandexpectthebandgaptocloseatcompressivestrainsalongbotha1anda2directionsexceeding2%.Sofar,muchlessattentionhasbeenpaidtothemoderate0.3eVbandgapofbulkBPthantothe2eVwidedirectbandgapofaphosphorenemonolayer[28].Theinter-layercouplinginbulkBP,whichisresponsibleforthelargeinthebandgap,displaysanontrivialcharacter[67].Therefore,ithasnotbeenclearaprioriifpreviouslyreachedconclusionsforthebandgapdependenceonuniaxialin-layerstraininaphosphorenemonolayerwillalsobeapplicableforthebulksystem.Thereisanindependent,morepracticalconcernaboutbandgapmodulationinBPsystemsbystrain.Pristinephosphorenemonolayersandfew-layersystemsareveryunstableunderambientconditions[153{155]andmustbecapped,typicallybysandwichingin-betweeninerth-BNlayerstoremainuseful[156].Itisunclearifapplyingin-layerstrainwillnotdestroytherigidcappinglayerbeforereachingdesirablestrainvalues43intheenclosedphosphorene.Thislimitationappliestoamuchlesserdegreetobulkblackphosphorus,whichischemicallymuchmorestableandthuscanbehandledmoreeasilyintheexperiment.Thesensitivityofthebandgaptothein-layerstrainsuggeststheuseofbulkormulti-layerblackphosphorusininfrareddevicestunablebystrain.Opticalmeasurementsshouldbeabletorevealthebandgapvaluediscussedhere,sinceobservedopticalspectrashouldnotbemodibyexcitonicstatesduetothenegligiblysmallexcitonbindingenergyinbulkblackphosphorus[150].5.4SummaryInsummary,Ihavestudiedtheoreticallythestructuralandelectronicresponseoflayeredbulkblackphosphorustoin-layerstrain.Abinitiodensityfunctionaltheory(DFT)calcu-lationsrevealthatthestrainenergyandinterlayerspacingdisplayastronganisotropywithrespecttotheuniaxialstraindirection.Tocorrectlydescribethedependenceofthefunda-mentalbandgaponstrain,IcombinedthecomputationallymoreinvolvedGWquasiparticleapproachperformedbymycollaboratorsfromWashingtonUniversityinSt.LouisthatisfreeofparametersandsuperiortoDFTstudies,whichareknowntounderestimategapenergies.IfoundthatthemainbetweenGWquasiparticleenergiesEqpandDFTeigenval-uesEPBEisadiscontinuityattheFermilevelandhaveidentherelationshipbetweenEqpandEPBEinthevalenceandconductionbandregions.Similartoaphosphorenemono-layer,Ifoundthatthebandgapdependssensitivelyonthein-layerstrainandevenvanishesatcompressivestrainvaluesexceedingˇ2%,thussuggestingapossibleapplicationofblackPinstrain-controlledinfrareddevices.44Chapter6Variousphasesoffew-layerphosphoreneThefollowingdiscussionismyoriginalcontributiontotherelatedpublicationbyJieGuan,ZhenZhuandDavidTanek,Phys.Rev.Lett.113,046804(2014)[157].6.1IntroductionLayeredblackphosphorusisconsideredasaviablecontenderinthecompetitiveoftwo-dimensional(2D)semiconductors[158,159].Unlikethepopularsemi-metallicgraphene,itdisplaysatbandgapwhilestillmaintainingahighcarriermobility[27,28,160].Thebandgapinfew-layerphosphorus,dubbedphosphorene,isbelievedtodependsensitivelyonthenumberoflayersandin-layerstrain[28,33,116,161].Layeredbluephosphorus,previouslydescribedastheA7phase[162,163],hasbeenpredictedtobeequallystableasblackphosphorusbuttohaveatelectronicstructure[33].Itisintriguingtoout,whethertherearemorethanthesetwostablelayeredphosphorusallotropes,andtowhatdegreetheirdielectricresponsemaybemofromsemiconductingtometallic.HereIintroduce-Pand-Pastwoadditionalstablestructuralphasesoflayeredphos-phorusbesidesthelayered-P(black)and-P(blue)phosphorusallotropes.Basedonmyabinitiodensityfunctionalcalculations,Ithesenewstructures,showninFig.6.1,to45Figure6.1:Equilibriumstructureof(a)an-P(black),(b)-P(blue),(c)-Pand(d)-Pmonolayerinbothtopandsideviews.Atomsatthetopandbottomofthenon-planarlayersaredistinguishedbycolorandshadingandtheWigner-Seitzcellsareshownbytheshadedregions.bealmostasstableastheotherlayeredallotropes.Monolayersofsomeoftheseallotropeshaveawidebandgap,whereasothers,including-P,showametal-semiconductortransitioncausedbyin-layerstrainorchangingthenumberoflayers.Anunforeseenbisthepossibilitytoconnecttstructuralphasesatnoenergycost.Thisbecomesparticu-larlyvaluableinassemblingheterostructureswithwmetallicandsemiconductingregionsinonecontiguouslayer.6.2ComputationaltechniquesTheDFTmethodusedtoobtaininsightintotheequilibriumstructure,stabilityandelec-tronicpropertiesof-Pand-Pisthesameaslastchapter.Reciprocalspacewassampledbyagrid[136]of881k-pointsintheBrillouinzoneoftheprimitiveunitcelloritsequivalentinsupercells.EquilibriumstructuresandenergiesbasedonSIESTAwerecheckedagainstvaluesbasedontheVASP[70]code.TheoptB86b-vdWfunctional[65,66],sameasusedinChapter4,isusedtoestimatethetofvanderWaalsinteractionson46theinter-layerdistancesandinteractionsinthelayeredsystems.Forselectedsystems,IperformedGWself-energycalculationsusingVASP.6.3ResultsanddiscussionsReferringtothewell-establishedblackphosphorusstructureas-Pandtobluephospho-rus[33]as-P,Ipresenttheoptimizedstructureofthesetwoandtwoadditionallayeredphosphorusallotropes,called-Pand-P,inFig.6.1.Allsharethethreefoldcoordinationofallatomsandanonzerointrinsicthicknessofthelayers,causedbythepreferenceofphos-phorusforatetrahedralarrangementofitsnearestneighbors.Infact,theamongthesestructuresarisefromthetwaystoconnecttetrahedrallycoordinatedPatomsina2Dlattice.Thereare4atomsintherectangularWigner-Seitzcellof-Pand8atomsinthatof-P.Theridgestructureofthesephasesisanalogoustothatoftheanisotropic-P,butsfromtheisotropic-PwithahexagonalWigner-Seitzcellcontainingonlytwoatoms.TheoptimumstructuralparametersaresummarizedinTable6.1.ResultsofmytotalenergycalculationsinTable6.1indicatethatalllayeredstructuresarenearlyequallystable,withcohesiveenergybelow0.1eV.Thiscomesasnosurprise,sincethelocalenvironmentoftheatomsisverysimilar,resultinginallbondlengthsbeingcloseto2.29A.Duetothewell-knownoverbindingindensityfunctionalcalculations,myDFT-PBEcohesiveenergiesarelargerthantheexperimentalvalue.Independentofitscohesiveenergy,astructuremayonlybeconsideredstableifitdoesnotspontaneouslychange.Atendencyforspontaneousstructuralchangeisindicatedbythepresenceofsoftmodes(orimaginaryfrequencyvalues)inthevibrationalspectrum.Todemonstratethestabilityandstructuralrigidityof-and-P,Ipresentthevibrational47Table6.1:Observedandcalculatedpropertiesofthefourlayeredbulkphosphorusallotropes.j~a1jandj~a2jarethein-planelatticeconstantsinFig.6.1.distheinter-layersepara-tionandEilistheinter-layerseparationenergyperatom.Ecohisthecohesiveenergywithrespecttoisolatedatoms.Ecoh=EcohEcoh(-P)istherelativestabilityofthelayeredallotropeswithrespecttothemoststableblackphosphorene(or-P)phase.Phase-P-P-P-P-P(expt.)(calc.)(calc.)(calc.)(calc.)j~a1j(A)4.3814.5323.33b3.41b5.56bj~a2j(A)3.31a3.36b3.33b5.34b5.46bd(A)5.25a5.55b5.63b4.24b5.78b{5.3034.20c4.21c5.47cEil(eV/atom){0.02b0.01b0.03b0.02b{0.12c0.10c0.13c0.11cEcoh(eV/atom)3.4343.30b3:29b3:22b3:23bEcoh(eV/atom){0.00b0:01b0:08b0:07b{0.00c0:04c0:09c0:08cspectraofthesephasesinFigs.6.2(a)and6.2(b).Absenceofsoftmodesinthesespectraindicatesthat-and-Parestable.Icomparethevibrationalspectrawiththoseof-Pand-Pmonolayersinapreviousstudy[33]andthemtoberathersimilar,similarbondingcharacterinallfourphosphoreneallotropes.Thehardestopticalmodefrequencyin-Pnear500cm1atliesveryclosetothatof-Pduetothecloserelationshipofthetwostructures.Similarto-P,thevibrationspectrumof-Pisnearlyisotropic.Thiscanbeexplainedbyacloseinspectionofthestructuralwhichindicatethatthelevelofbucklingin-Pismuchsmallerthanin-Pand-P.Thevibrationalspectrumof-Pbestmatchesthatof-Pintermsofthehardestmodefrequencyandtheanisotropyofthelongitudinalacousticmodes.Theinthevibrationalspectraapromisingroutetoidentifythepresenceof-Pand-PusingRamanorinfraredspectroscopy.StructuralstabilityatT=0,evidencedinthevibrationalspectra,sayslittleaboutstabilityathightemperatures,wherethermallyactivatedstructuralchangesmaytakeplace.-P,-P,-Pand-Parestructurallyrelatedbytwaystoconnecttetrahedrally48coordinatedPatomsina2Dlattice.Localstructuralchangesmaybeachievedbybondsinabi-stableatanenergycostof0:5eV/atom,identinthe-P!-Ptransition[33].Therelativelyhighactivationenergyvaluesuggeststhatstructuralchangeswillonlyrarelyoccurbeforethesystemmelts.Amoredirectwaytoprobethestabilityof-Pand-Pathightemperaturesisbyperformingmoleculardynamics(MD)simulations.ResultsofmycanonicalMDsimulationsarepresentedasstructuralsnapshotsof-Pand-PmonolayersatT=300KandT=1;000KinFig.6.3.Toavoidartifactsassociatedwithconstraintsimposedbyunitcells,Iusedverylargesupercellscontaining96atomsfor-Pand128atomsfor-P.Duetothelargeunitcellsize,Ineededtolimitthesimulationtimeto1.6psfor-Pand0.6psfor-Pwhenusing1fstimesteps.Thissimulationtimestillcoversabout10vibrationalperiodsoftheopticalmodesandthusshouldindicatethepropensitytowardsstructuralchanges.Atroomtemperature,Ithestructuralchangesinthelayerstobeminimal.MoretchangesareexpectedatT=1;000K,whichlieswellaboveTM=863K,themeltingpointofredphosphorus[165].Eventhoughthechangesarelargerinthiscase,theyresemblemoretheonsetofameltingprocessthanaconcertedstructuralchange.Ialsoexpectthatfew-layerstructuresmayconverttoanamorphousstructureresemblingredphosphorusabovethemeltingpoint.Thehigh-temperatureMDsimulationssuggestthat-Pand-Pshoulddisplaysimilarthermalstabilityaspreviouslyfound[33]for-Pand-P.Inviewofcohesiveenergydencesbetweenthestructuresnotexceeding0.1eVperatom,Iexpectthefourphosphoreneallotropestocoexistsunderexperimentalconditions.Myresultsfortheoptimuminter-layerseparationd(d=j~a3jfortheAAlayerstacking)49Figure6.2:Vibrationalbandstructure!(~k)ofamonolayerof(a)-Pand(b)-P.High-symmetrylinesareshownintheinsets.andinter-layerinteractionenergyEilinthefourphasesaresummarizedinTable6.1.BynottakingproperaccountofthevanderWaalsinteractions[166,167],DFT-PBEcalculationstendtounderestimateEilandoverestimated[28,33].Probablythebest,albeitcomputa-tionallyextremelydemandingwaytocorrectthisencyistheQuantumMonteCarlo(QMC)approach[168].QMCresultsfor-PindicateEilˇ40meV/atom[169],twicethe20meV/atomvaluebasedonDFT-PBE,ascitedinTable6.1.Ialsolistthevalueobtainedfor-PusingvanderWaals-correctedoptB86b-vdWfunctional,Eilˇ120meV/atom,whichistlylargerthanthemorereliableQMCvalue.Verysimilarcorrectionstotheinter-layerinteractionof0:1eVandareductionoftheinter-layerdistancearealsoobtainedfortheotherlayeredallotropes.InspiteoftheseminorIdtheinter-layerinterac-tionsanddistancestoberathersimilarinalltheseallotropesandinreasonableagreementwithobserveddataintheonlypreviouslyknownallotropeblackphosphorus(-P).ThecalculatedenergybetweentheAA,ABandABCstackingoflayersoffewmeV/atomrepresentonlyafractionoftheinter-layerinteractionEil.Sincetheinter-layerdistancesarelargeandinter-layerinteractionsaresmallinallfourphases,theoptimized50Figure6.3:Snapshotsofcanonicalmoleculardynamicssimulationsdepictingstructuralchangesin-Pat(a)T=300Kand(b)T=1;000K.Correspondingsnapshotsof-Pareshownin(c)forT=300Kand(d)forT=1;000K.Forbettercomparison,bothtopandsideviewsarepresentedforallstructures.Theunitcellofthelatticecontains96atomsin-Pand128atomsin-P.layerstructuresofthebulksystemandthemonolayerarenearlyindistinguishable.Thefactthattheinter-layerinteractionissimilarlysmallinallphasesindicatesthepossibilityoflayer-by-layerexfoliationnotonlyofblackphosphorus[28],butalsotheotherlayeredallotropes.IpresentresultsofDFT-PBEelectronicbandstructurecalculationsfor-Pand-PmonolayersinFig.6.4.AscanalsobeinferredfromthenumericalresultsintherelatedTable6.2:ThefundamentalbandgapEginmonolayersof-P,-P,-Pand-P,basedonDFT-PBEcalculations.Phase-P-P-P-PEg(eV)0.901.980.500.4551Figure6.4:Electronicbandstructureanddensityofstates(DOS)of(a)-Pand(b)-Pmonolayers.ResultsforbilayerandbulksystemsareshownforcomparisonintheDOSplotsonly.Topandsideviewsoftheelectrondensityˆvcnearthetopofthevalenceandthebottomoftheconductionbandsof(c)-Pand(d)-P.OnlystatesintheenergyrangeEF0:4eV<E<EF+0:4eVareconsidered,asindicatedbythegreenshadedregionin(a)and(b).ˆvcisrepresentedattheisosurfacevalueˆvc=1:1103e/A3for-Pand-Pandsuperposedwithaball-and-stickmodelofthestructure.Table6.2,thefundamentalbandgapsin-Pand-Paresomewhatsmallerthanthoseof-Pand-Pmonolayers,butstillt.SincemyGWself-energycalculationsindicatethattheseDFT-PBEbandgapvaluesareunderestimatedbyˇ1eV,asexpectedforDFTcalculations,allfourphasesshoulddisplayafundamentalbandgapinexcessof1eVinthemonolayer.Whereas-Phasanindirectbandgap,-Pisadirectbandgapsemiconductor.Besidestheelectronicbandstructureofthemonolayers,IpresenttheassociateddensityofstatesofamonolayerandofthebulksysteminFigs.6.4(a)and6.4(b).Asalreadynoticedforthe-Pand-Pstructuresinpreviousworks[28,33],theelectronicstructurenearEFincludingthebandgapdependssensitivelyonthenumberoflayersinallphosphoreneallotropes,including-Pand-P.Themostnotableinthedensityofstatesof-P52Figure6.5:(a)DependenceofthefundamentalbandgapEgontheslabthicknessinN-layerslabsof-P(black),-P(blue),-P,and-P.Dependenceofthefundamentalbandgaponin-layerstrain˙ispresentedin(b)for-Pandin(c)for-P.ThestraindirectionisinFig.6.1.Theshadedregionsin(a)and(b)highlightconditions,underwhich-Pbecomesmetallic.Dashedverticallinesin(b)and(c)indicateadirect-to-indirectbandgaptransition.inFig.6.4(a)isbetweenasemiconductorforN=1andametalforN2.WhereasDFTcalculationstypicallyunderestimatethefundamentalbandgap,theyarebelievedtocorrectlyrepresenttheelectronicstructureinthevalenceandtheconductionbandregion.Togetabetterimpressionaboutthenatureoftheconductingstatesindoped-Pand-P,IdisplaythechargedistributionassociatedwithstatesneartheFermilevelinFigs.6.4(c)and6.4(d),superposedwiththeatomicstructure.Thesestatesandtheirhybridswithelectronicstatesofthecontactelectrodeswillplayacrucialroleinthecarrierinjectionandquantumtransport.Itheseconductionstatestohavethecharacterofp-statesnormaltothelayers,similartographene.Inmulti-layersystems,thesestateshybridizebetweenadjacentlayers,causingabanddispersionnormaltotheslab.Thiscausesachangeinthedensityofstatesinthegapregionbetweenamonolayerandthebulkstructure.Tojudgehowthefundamentalbandgapdependsontheslabthickness,IpresentmyDFT-PBEbandgapresultsfor-P,-P,-Pand-PasafunctionofthenumberoflayersNinFig.6.5(a).MymostimportantisthatthebandgapvanishesforN2in-P,53Figure6.6:Energeticallyfavorablein-layerconnectionsbetween(a)-Pand-P,(b)-Pand-P,and(c)-Pand-P,showninperspectiveandsideview.lrepresentstheedgelengthattheinterfaceand'istheconnectionangle.ThecolorschemeforthetallotropesisthesameasinFig.6.1.turningbilayersandthickerslabsmetallic.Asimilarlyintriguingpictureemergeswhenstudyingthedependenceofthefundamentalbandgaponthein-layerstrain.ResultsforstrainappliedintwoorthogonaldirectionsareshowninFig.6.5(b)for-PandinFig.6.5(c)for-P.Again,mymosttisthatstretchingbeyond4%shouldturna-Pmonolayermetallic.Asalreadyreportedfor-Pand-P[28,33,116,161],applyingevenrelativelylowlevelsofin-layerstraincausesdrasticchangesinthebandgap,andmayevenchangeitscharacterfromdirecttoindirect.Thelatterfactresultsfromthepresenceofseveralvalleysintheconductionband,whichmaychangetheirrelativedepthduetolatticedistortions.Strainofuptoafewpercentmaybeaccomplishedwhenphosphoreneisgrownepitaxiallyonaparticularsubstrate.Wemayevenconsiderthepossibilityofin-layerbandgapengineeringbysubstratepatterning.Evenricherpossibilitiesforbandstructureengineeringshouldarisebyin-layerconnec-tionsbetweenthetphases.In-layerconnections,whichhavebeenobservedinhybridsystemsofgrapheneandhexagonalBN[170],fromlargeinterfaceenergypenalties54Table6.3:EnergycostperedgelengthEc=landconnectionangle',inFig.6.6,associatedwithconnectingtwophosphorenemonolayers.Phaseconnection---Ec=l<1meV/A<17meV/A<6meV/AAngle'142160145duetothelackofcommensurability.Thesituationisverytinphosphorene,sincethefourlayeredallotropessharethesamestructuralmotifofthreefoldcoordinatedPatomssurroundedbynearestneighborsinatetrahedralarrangement.Ithatthistetrahedralarrangementcanbemaintainedevenwithinspcin-layerconnectionsofthetstructures,resultinginanextremelylowenergypenalty.Ihaveoptimizedthestructureofin-layerconnectionsbetween-Pand-P,between-Pand-P,andbetween-Pand-P.Myresults,depictedinFig.6.6,indicatethatanoptimumconnectioninvolvestorientationsofthejoinedplanes.Theoptimizationcalculations,performedinasupercellgeometrywithvaryingcellsizes,allowedustodeterminetheenergycostperedgelengthEc=ltoconnecttwostructuralphases.Toobtainthisquantityforaconnectionbetweenphases1and2,IconsideredN1atomsofphase1andN2atomsofphase2perunitcellandvariedtheN1=N2ratiowhilekeepingthesamelengthoftheinterfaceboundary.Forareliableestimateoftheenergypenaltyassociatedwithforminganinterfacebetweenthetwophases,Icomparedtotalenergiesofoptimizedstructureswithcoexistingphasestothoseofpure,defect-freephases.MyresultsforEc=lfortheconnectionsshowninFig.6.6arelistedinTable6.3,alongwiththeoptimumvaluesoftheconnectionangle'.TheenergyresultsinTable6.3indicatethattheenergycosttoconnectstable,butt,structuralphasesisnegligibleincomparisontothecohesiveenergy.Theimplicationthatcoexistenceofseveralphaseswithinonelayerisnotenergeticallypenalizedisextremelyuncommoninnature.Icanenvisagethepossibilityofformingsuchmulti-phasestructures55bydepositingphosphorenemonolayersonasubstratewithaspecistepstructure,suchasavicinalsurface,usingChemicalVaporDeposition.Thedomainwallboundariesbetweentphasesmayalsomovetooptimizeadhesiontoaninhomogeneousornon-planarsubstrate.Theelectronicpropertiesofaheterostructurewithinonelayerwilldependnotonlyontheelectronicstructureofthepurephases,butalsotheirwidthorsizeandthedefectbandsassociatedwiththeinterfaces.Inprinciple,itshouldbepossibletoformacomplexdevicestructurebyajudiciousarrangementoftstructuralphaseswithinonephosphorenemonolayer.Monolayerscontainingthefourlayeredphosphorenephasesareexpectedtobenotonlystable,butalsoConsequently,thenon-planarityofmulti-phasestructuresdoesnotposearealproblem.Itmayevenprovideanadvantageinformationofcomplexfoamstructures,similartographiticcarbonfoams,withunusualelectronicproperties[171,172].6.4SummaryInconclusion,basedonabinitiodensityfunctionalcalculations,Ihaveproposed-Pand-Pastwoadditionalstablestructuralphasesoflayeredphosphorusbesidesthelayered-P(black)and-P(blue)phosphorusallotropes.Monolayersofsomeoftheseallotropeshaveawidebandgap,whereasothers,including-P,showametal-insulatortransitioncausedbyin-layerstrainorchangingthenumberoflayers.Anunforeseenbisthepossibilitytoconnecttstructuralphasesatnoenergycost.Thisbecomesparticularlyvaluableinassemblingheterostructureswithwmetallicandsemiconductingregionsinonecontiguouslayer.56Chapter7Facetednanotubesandfullerenesofmulti-phasephosphoreneThefollowingdiscussionismyoriginalcontributiontotherelatedpublicationbyJieGuan,ZhenZhuandDavidTanek,Phys.Rev.Lett.113,226801(2014)[173].7.1IntroductionOnereasonfortheunprecedentedinterestingraphiticcarbonisitsabilitytoformnotonlyself-supportinggraphenelayers[174,175],butalsosingle-andmulti-wallnanotubes[47]andfullerenes[46].Similartographite,whichistheparentcompoundofthesecarbonallotropes,thestableblackphosphorusallotropeisalayeredcompoundthatcanbeexfoliatedtophos-phorenemonolayers[27,28].Phosphorusnanotubes[176,177]andfullerenes[178{180]havebeenpostulatedtoforminanalogytotheircarboncounterpartsbydeformingaphosphorenemonolayer,typicallyattenergycost.Incontrasttotheuniquestructureofplanargraphene,atleastfourequallystablephaseswithtproperties,-P,-P,-Pand-P,canbedistinguishedinthepuckeredstructureofaphosphorenemonolayer[33,163,181].Theabilityofthedtphasestoformnon-planarin-layerconnectionsatessentiallyzeroenergycostsuggeststhepossibilitytoformfacetednanotubeandfullerenestructuresthatareasstableasplanarphosphorene.Thepossibilitytomixtphaseswithineach57wallofsphericalandcylindricalsingle-andmulti-wallstructureswouldunprecedentedrichnessnotonlyofformbutalsotheassociatedelectronicproperties.Bulkquantitiesofcarbonnanotubesandersarecurrentlyusedasaperformance-enhancingadditivetographiteinLi-ionbatteries(LIBs)[182].SinceblackphosphorusisconsideredsuperiortographiteforLIBapplications[183,184],asimilarbcouldbederivedfromthepresenceofphosphorenenanotubesandrelatedstructures.HereIpresentanewparadigminconstructingverystable,facetednanotubeandfullerenestructuresbylaterallyjoiningnanoribbonsorpatchesoferentplanarphosphorenephases.Myabinitiodensityfunctionalcalculationsindicatethatthesephasesmayconnectlaterallyatanangle.Unlikefullerenesandnanotubesobtainedbydeformingasingle-phaseplanarmonolayeratsubstantialenergypenalty,Ifacetedfullerenesandnanotubestobenearlyasstableasplanarsingle-phasemonolayers.Theresultingrichvarietyofpolymorphsallowsonetotunetheelectronicpropertiesofphosphorenenanotubesandfullerenesnotonlybythechiralindexbutalsobythecombinationoftphosphorenephases.InselectedPNTs,ametal-insulatortransitionmaybeinducedbystrainorbychangingthenumberofwalls.7.2ComputationaltechniquesThesamemethodaslastchapterisusedtoobtaininsightsintotheequilibriumstructure,stabilityandelectronicpropertiesofnanotubesandfullerenesbasedondtlayeredphosphorusallotropes.Iuseperiodicboundaryconditionsthroughoutthestudy,withnan-otubesandfullerenesseparatedbyavacuumregionexceeding15A.Isamplereciprocalspacebyaegrid[136]of8k-pointsforthe1DBrillouinzoneofnanotubesandonly1k-point58Figure7.1:(a)Atomicstructureof-,-,-and-Pintopviewandthesideviewofzigzagandarmchairedges.Theorthogonallatticevectors~a1and~a2theunitcellsorsupercellsusedinthisstudy.Schematicandatomicstructureof(b)anarmchairand(d)azigzagPNT,withthetstructuralphasesdistinguishedbycolorandshading.Thecross-sectionsof(c)anarmchairand(e)azigzagnanotubeillustratethesymmetryandthedistributionofphasesalongtheperimeter.forthesmallBrillouinzoneofisolatedfullerenes.7.3ResultsanddiscussionsThenanotubeandfullerenestructurespresentedinthisstudyareformedbylaterallycon-nectingthetstableallotropesoflayeredphosphorus,namely-,-,-and-P,whichareshowninFig.7.1(a).Whereas-and-ParethemoststableallotropeswithEcoh=3:28eV/atominthemonolayer,thestabilityof-and-Pareloweronlyby<0:1eV/atom[181].Allthesestructuressharetheunderlyinghoneycomblatticewithgraphenebut{incontrasttographene{arenotInanalogytographene,IthearmchairandzigzagedgesoftherentphosphorenephasesinFig.7.1(a).Thevectors~a1and~a2,whichspantheselattices,mayalsobeusedtoidentifytheedgesofphosphorene59Figure7.2:Geometriccross-sectionofselected(a)a-PNTsand(b)z-PNTs.tstruc-turalphasesaredistinguishedbycolor.Figure7.3:Sourceofambiguityinthenomenclatureoffacetednanotubes.Intwoidenticaljunctionsofphasesand,atomsalongtheconnectionline,indicatedbythedottedellipse,maybeassignedtothephaseoneitherside.nanoribbons(PNRs).Consideringtheequilibriumnon-planarconnectionsbetween-,-,-Palongzigzagedgesand-,-Palongarmchairedges[181],Icandesigntwotypesoffacetednanotubes.Theexactmorphologyofthemorecommoncarbonnanotubes(CNTs)isbythechiralindex(n1;n2),whichisassociatedwiththechiralvector~Ch=n1~a1+n2~a2onagraphenemonolayer.Thisvectorthewrappingintoananotubeandidenitsedge.Thereisacommondistinctionbetweenarmchairnanotubes(a-NTs)withanarmchairedgeandzigzagnanotubes(z-NTs)withazigzagedge.Asimilarconventioncanbeusedwhenbendingmonolayersof-,-,-and-Pintocorrespondingnanotubes.60ThenanotubesIconsiderhereareveryt,astheyareformedbyconnectingnarrowplanarnanoribbonsoftphosphoreneallotropes.Armchairnanotubes(a-PNTs),showninFig.7.1(b)and7.1(c),formbyconnectinglaterally-PNRswith-and-PNRsalongtheirzigzagedges.VirtuallynodeformationisrequiredtoformananotubewithC3symmetryandapolygonalcross-sectionlikethatshowninFig.7.1(c).Thethreeidentical120segmentsinthecross-sectionofthisa-PNTcontain,inthissequence,an-PNRconnectedtoa-PNR,-PNR,and-PNR.ThewidthofeachindividualPNRmaybezeroornonzero,givingrisetomanytmorphologies.Sincethetwo-PNRsinthissegmentmayalsohavetwidths,Idistinguishthembyasubscript.Next,IimaginejoininglaterallyallnanoribbonsofagivenphasetoawiderribbonofwidthW=nj~a1j.Obtaininginthiswaythevaluesn,n1,nandn2,Imaycharacterizeanarmchairnanotubeasa-PNT(n,n1,n,n2)andidentifythenanotubeinFig.7.1(c)asa-PNT(6,3,3,3).Inanalogytoa-PNTs,zigzagnanotubes(z-PNTs),asshowninFig.7.1(d)and7.1(e),formbyconnectinglaterally-and-PNRsalongtheirarmchairedges.Virtuallynode-formationisrequiredtoformananotubewithC2symmetryandapolygonalcross-section,showninFig.7.1(e).Thetwoidentical180segmentsinthecross-sectionofthisz-PNTcontain,inthissequence,a-PNRconnectedtoa-PNR,-PNR,and-PNR.ThewidthofeachindividualPNRmaybezeroornonzero,givingrisetomanytmorphologies.Sincethetwo-andthetwo-PNRsinthissegmentmayalsohavetwidths,Idistinguishthembyasubscript.Next,IimaginejoininglaterallyallnanoribbonsofthesamephasetoawiderribbonofwidthW=nj~a2j.Obtaininginthiswaythevaluesn1,n1,n2andn2,Imaycharacterizeazigzagnanotubeasz-PNT(n1,n1,n2,n2)andidentifythenanotubeinFig.7.1(e)asz-PNT(5,2,5,4).Idonotdiscussherethenar-61rowestz-PNT(1,0,1,0)withaP4squareinthecross-section,whichisinrealityananowire.AdditionalexamplesofnarrowarmchairnanotubesareshowninFig.7.2(a)andofzigzagnanotubesinFig.7.2(b).Eventhoughthesenanotubeshaveasmallaverageradius,theyareunusuallystableasIwilldiscusslater.Ialsowishtopointoutthatthefacetednanotubesdescribedherearenotcompletelyfreeofstrain.Assumingthattheoptimumconnectionanglesbetweennarrowstripsarethesameasbetweensemi-iplanesprovidedinReference[181],Idothatnarrowstripsofthetphases,connectedattheoptimumangle,wouldnotformaperfecttube.Sp,constituentsofanarmchairnanotubewouldideallyconnectat348insteadof360,requiringasmallamountofadditionalbendingtoformaarmchairnanotube.Similarly,constituentsofazigzagnanotubewouldconnectat280insteadof360,requiringalargeramountofadditionalbendingtoformazigzagnanotube.Whilethisresidualstrainisverysmall,itisnotnegligible.Asstatedbefore,thestabilityofaparticularnanotubeismainlydeterminedbythetstabilitiesoftheconstituentphasesand,toamuchsmallerdegree,bythisresidualstrain.Whereasthedesignationa-PNT(n,n1,n,n2)thewaytoconstructauniquearmchairnanotubefromPNRs,agivennanotubemaybecharacterizedbytsetsofchiralindices.Ambiguityarisesfromassigningatomsalongtheconnectionlinetoeitherthephaseontheoneorontheothersideoftheconnectionline.AsshowninFig.7.3,thesamestructurecouldbedescribedby(n;n)and(n+1;n1).Thiswouldleadtotvaluesofninthesamea-PNTandtvaluesofn1andn2inthesamez-PNT.Thisambiguitycanbeavoidedbyselectingn=max.Asimilarambiguityinthenomenclatureofz-PNTscanbeavoidedbyselectingn1tobemaximumandn2tobemaximum.Similartotheconstructionofnanotubesbyconnectingnanoribbonsoftphases,62Figure7.4:Phosphorene-basedfullerenestructureswith(a-c)octahedraland(d-f)icosahe-dralsymmetry.Thestructuralmodelsin(a)and(d)indicatehowtriangularfacetsof-Pareconnectedby-Palongtheedges.ThestickmodelsofP72in(b),P200in(c),P80in(e)andP180in(f)depicttherelaxedatomicstructuresofoctahedralandicosahedralfullerenes.fullerenes,too,maybeconstructedbyconnectingplanartriangularsegmentsof-Pmono-layersbynarrow-Pstripsattheedges,asshowninFig.7.4.Ihaveconsideredoctahe-dralfullerenes,illustratedschematicallyinFig.7.4(a),andicosahedralfullerenes,illustratedschematicallyinFig.7.4(d).IdealPnoctahedralfullerenescontainn=8m2atomsandicosahedralfullerenescontainn=20m2atoms,wheremisaninteger.Twoexamplesofoc-tahedralfullerenesarepresentedinFigs.7.4(b)and7.4(c),andtwoexamplesoficosahedralfullerenesinFigs.7.4(e)and7.4(f).Sincethesestructuresdonotrequiretdefor-mationoftheplanarmonolayerstructure,butratherresultfromanoptimumconnectionbetween-Pand-P,theyalsoareexpectedtobenearlyasstableastheplanarsingle-phaseallotropes.MyresultsfortherelativestabilityofphosphorenenanotubesarepresentedinFig.7.5(a)63Figure7.5:(a)AveragestrainenergyperatomE=ninPNTsoftradiusRwithrespecttoaplanar-Pmonolayer.Theshadedregionindicatestherangeofstabilitiesoftplanarphases.Ialsopresentdatapointsforpure-phasePNTsobtainedbyrollingup-and-Ptoatube.(b)Strainenergyperatominoctahedral(o)andicosahedral(i)fullerenesofradiusR.Thedashedlinesin(a)and(b)representthe1=R2behaviorbasedoncontinuumelasticitytheoryforpure-phasenanostructures.(c)BandgapsEginfaceteda-PNTsandz-PNTs.ThehorizontallinesdepictEgvaluesinpureplanarphosphorenemonolayers.(d)HOMO-LUMOgapsino-andi-fullerenes.Theredandblueshadedregionsindicatetherangeofvaluesfora-PNTsandz-PNTs.andthoseforfullerenesinFig.7.5(b).Inbothgures,thedashedlinesdisplaytheex-pected1=R2behaviorofthestrainenergyperatomE=nontheradiusRthatenergeticallypenalizesstructureswithsmallradii.AsseeninFig.7.5(a),thisprojectedbehavior,basedoncontinuumelasticitytheory[58],agreescloselywithmyresultsforpure-and-Pnanotubesandpreviouslypublishedresultsfor-Pnanotubes,calculatedbydensityfunctionalbasedtight-binding(DFTB)method[177].Asanticipatedoriginally,thefacetedmulti-componentnanotubesaremuch64Figure7.6:SnapshotsofcanonicalmoleculardynamicssimulationsatT=300KandT=1000Kdepictingstructuralchangesinphosphorusnanotubesandfullerenes.Nanotubesarerepresentedby(a)thearmchaira-PNT(3,3,0,0)and(b)thezigzagz-PNT(5,0,5,0).Fullerenesarerepresentedby(c)P72and(d)P80.Thenanotubesareshowninend-onview.morestablethanthese.Idthat(i)theirstrainenergiesarenearlyindependentoftheradiusand(ii)theirrelativestabilitieslieintherangedelimitedbythestabilitiesofthepureplanarcomponents,indicatedbytheshadedregion.Sincez-PNTscontaintheleaststableandphases,theyarealsotheleaststableamongthefacetednanotubes.Thepresenceofthemorestableandphases,ontheotherhand,makesa-PNTsconsistentlymorestablethanz-PNTs.StabilityenhancementcausedbythecoexistenceofmultiplephasescanalsobeobservedinmyresultsforfullerenesinFig.7.5(b).Asinthenanotubes,Imoststrainenergieswithinthevaluerangedelimitedbythepureplanar-and-Pphases.Thestabilityenhancementismostvisibleinverysmallfullerenes.Interestingly,IthesmallfullerenestructuresmorestablethanP4,thebuildingblockofthe(mostreactive)bulkphosphorusallotrope.StructuralstabilityatT=0sayslittleaboutstabilityathightemperatures,where65Figure7.7:Electronicbandstructureof(a)thez-PNT(3,0,3,0)and(c)thez-PNT(5,0,5,0)nanotube.TheregionassociatedwithfrontierorbitalsishighlightedbythegreenshadingintheenergyrangeEF0:1eV<E<EF+0:1eVin(a)andEF0:2eV<E<EF+0:2eVin(c).Electrondensityˆfassociatedwithfrontierstatesin(b)thez-PNT(3,0,3,0)and(d)thez-PNT(5,0,5,0)nanotube,superposedwithball-and-stickmodelsofthestructures.ˆfisrepresentedbytheisosurfacevalue8104e/A3.thermallyactivatedstructuralchangesmaytakeplace.Adirectwaytoprobethestabilityoffacetedphosphorusnanotubesandfullerenesathightemperaturesisbyperformingmoleculardynamics(MD)simulations.ResultsofmycanonicalMDsimulationsarepresentedinFig.7.6asstructuralsnapshotsofthea-PNT(3,3,0,0)andthez-PNT(5,0,5,0)nanotubeandtheP72aswellastheP80fullereneatT=300KandT=1;000K.Toavoidartifactsassociatedwithconstraintsimposedonnanotubesbyunitcells,Iusedsupercellscontaining3primitiveunitcells.Iused1fstimestepstocover0.8-3ps,dependingonsystemandtemperature.Ifoundstructuralchangestobeminimalatroomtemperature.IobservedlargeratomicmotionatT=1;000K,whichliesslightlyaboveTM=863K,themeltingpointofredphosphorus[165],butnoindicationofconcertedstructuralchangestoadtallotrope.Ithusconcludethatfacetednanotubesandfullerenesdisplayasimilarthermalstabilityasmonolayersof-,-,-and-phosphorenediscussedpreviously[33,181]andalsoinearlierchapters.Sincethecohesiveenergiesoffacetednanotubesandfullerenesareatmost0:2eV/atomlowerthanbulkblackphosphorus,Iexpectthesenanostructures66Figure7.8:(a)Cross-sectionand(b)DOSofthedouble-wallz-PNT(5,0,5,0)@(9,0,9,0).Thesolidlinein(b)showsthetotalDOSandthedashedlinedepictsthesuperpositionofthedensitiesofstatesoftheisolatednanotubecomponents.TheFermilevelEFissetat0.(c)Perspectiveviewofthea-PNT(3,0,9,0)and(d)dependenceofthegapenergyEgonaxialstrain.tocoexistwiththebulkstructureunderexperimentalconditions.Incarbonnanotubesandfullerenes,theoccurrenceofafundamentalbandgapisasig-natureofquantumtintheunderlyingsemi-metallicgraphenestructure.Theadvantageofphosphoreneovergrapheneisthepresenceofafundamentalbandgapinalllayeredallotropesdiscussedhere.Ithusexpectthefundamentalbandgaps,Eg,ofnan-otubesandfullerenestoapproximatelyspanthevaluerangeofthepurecomponentsofplanarstrucutures,indicatedbytheshadedregionsinFigs.7.5(c)and7.5(d).Eventhoughadditionalcorrectionsareexpectedduetoquantumcontandstructuralrelaxation,suchcorrectionsareapparentlynotasimportant,sincemostofmydatapointslieinthe67rangedelimitedbythepurecomponents.Atthispoint,Iwishtopointoutthatmyelec-tronicstructureresultsinFigs.7.5(c)and7.5(d),obtainedbyDFT-PBE,areexpectedtounderestimatethefundamentalbandgaps[28,33].AsseeninFig.7.5(c),IlargerbandgapvaluesinarmchairPNTscontaining-,-and-P,sinceeachofthepureplanarcomponentshasabandgapinexcessof0.5eVinthemonolayer.Since-and-Phavethesmallestbandgapsamongthephosphoreneallotropes,Ialsoseethesmallestbandgapsinz-PNTs,whichcontainthesetwopases.Onlythenarrowestz-PNT(3,0,3,0)ismetallic,withtwobandscrossingtheFermilevel,asseenfromitsbandstructureinFig.7.7(a).Widerz-PNTsareallnarrow-gapsemiconductors,asseenintherepresentativebandstructureofthez-PNT(5,0,5,0)showninFig.7.7(c).Tothereasonfortheanomalouselectronicstructureofthez-PNT(3,0,3,0),IexaminedthechargedistributioninthefrontierorbitalsanddisplayitinFig.7.7(b).Forthesakeofcomparison,Idisplaytheanalogouschargedistributioninthewiderz-PNT(5,0,5,0)inFig.7.7(d).IusedasomewhatarbitraryoffrontierorbitalsascorrespondingtotheenergyrangeEF0:1eV<E<EF+0:1eVinthemetallicnanotubeinFigs.7.7(a)and7.7(b).Sincenostatesarefoundinthisenergyrangeinthesemiconductingz-PNT(5,0,5,0),IexpandedtheenergyrangeassociatedwiththefrontierorbitalstoEF0:2eV<E<EF+0:2eVinFigs.7.7(c)and7.7(d).Ithefrontierorbitalstodisplayadominantpcharacterwithasmallsadmixture,causingthemtopointradiallyinorout.Whereasthereisnooverlapbetweensuchorbitalsonneighboringsitesinthewiderz-PNT(5,0,5,0)inFig.7.7(d),Iobserveatoverlapbetweenthird-neighborsitesalongtheinnerperimeter,whichcomeclosertoeachotherinthenarrowz-PNT(3,0,3,0),showninFig.7.7(b).Theoriginofconductioninthenarrowz-PNT(3,0,3,0)istheconductionchannelsformedbytheserehybridizedstates.Oneboftherehybridizationacrossthediameterisan68improvedrigidityofthenanotubes.Infullerenes,Egrepresentsthegapbetweenthehighestoccupiedmolecu-larorbital(HOMO)andthelowestunoccupiedmolecularorbital(LUMO).MyresultsinFig.7.5(d)suggestthattheHOMO-LUMOgapsinicosahedralfullerenesarelargerthaninoctahedralfullerenes.EventhoughthevaluesaresimilartothoseofnanotubesinFig.7.5(c),Icannoteasilyrationalizethevaluerangeforo-andi-fullerenes,sincebothstructuresconsistofthesame-Pand-Pallotropes.Asreportedpreviously[28,33,116,161,181],thefundamentalbandgapinphosphorenedependssensitivelyonthenumberoflayersandonin-layerstrain.MyresultsinFig.7.8indicatethatthesamebehavioroccursalsoinPNTs.Imulti-wallPNTstobesta-bilizedbyaninter-wallinteractionof50meV/atom,roughlythesameasinthelayeredcompounds[168,169].Duetoalargefractionof-Pinthewall,whichhasbeenshowntoundergoametal-semiconductortransition,Iinvestigatedthe(5,0,5,0)@(9,0,9,0)double-wallz-PNT,showninFig.7.8(a).Thedensityofstates(DOS)ofthisdouble-wallPNT,showninFig.7.8(b),indicatesthattheinter-wallinteractionmayturntwosemiconductingnanotubesmetallicuponbeingcombinedtoformadouble-wallnanotube.AsseeninFigs.7.8(c)and7.8(d)forthesingle-walla-PNT(3,0,9,0),evenamodest5%stretchmayturnasemicon-ductingnanotubecontainingatfractionof-Pmetallic.Thislowlevelofstrainmaybeappliedexternallyorinducedbyepitaxy,includingstructuralchangesinducedinmulti-wallnanotubes.Themostimportantimplicationofmyclaimthatfacetednanotubesandfullerenesareasstableasplanarphosphoreneisthattheyshouldexistinnatureandwillbeobservedeventually,aswasthecasewithboronnanostructures[185,186].Phosphorusnanotubesandfullerenesmayformduringballmillingofblackphosphorus[184]underinert,oxygen-free69atmosphere.Thisprocessmayalsoproducestructureswithalargeaccessiblesurfaceareaforphosphorus-basedLIBapplications[183,184].7.4SummaryInconclusion,Ihavepresentedanewparadigminconstructingverystable,facetednanotubeandfullerenestructuresbylaterallyjoiningnanoribbonsorpatchesoferentplanarphos-phorenephases.Myabinitiodensityfunctionalcalculationsindicatethatthesephasesmayformverystable,non-planarjoints.Unlikefullerenesandnanotubesobtainedbydeformingasingle-phaseplanarmonolayeratsubstantialenergypenalty,Ifacetedfullerenesandnanotubestobenearlyasstableastheplanarsingle-phasemonolayers.Theresultingrichvarietyofpolymorphsallowsonetotunetheelectronicpropertiesofphosphorenenanotubesandfullerenesnotonlybytheirchiralindexbutalsobythecombinationoftphos-phorenephases.InselectedPNTs,ametal-insulatortransitionmaybeinducedbystrainorchangingthenumberofwalls.70Chapter8Combinationofsp2andsp3bonding:two-dimensionalphosphoruscarbideThefollowingdiscussionismyoriginalcontributiontotherelatedpublicationbyJieGuan,DanLiu,ZhenZhuandDavidTanek,NanoLett.16,3247-3252(2016)[187].8.1IntroductionThereisgrowinginterestin2Dsemiconductors,bothforfundamentalreasonsandaspoten-tialcomponentsinlow-powerelectroniccircuitry.Alargenumberofsubstanceswithuniqueadvantagesandlimitationshasbeenstudiedinthisrespect,butconsensushasnotbeenreachedregardingtheoptimumcandidate.Semi-metallicgraphenewithanexcellentcarriermobilityhasreceivedthemostattentionsofar,butallattemptstoopenupasizeable,robust,andreproduciblebandgaphavefailedduetothenegativesideoftheentmo[39{42].Transitionmetaldichalchogenides(TMDs)suchasMoS2[22,43]orTcS2[188]dohaveasizeablefundamentalbandgap,buttheyalsohavelowercarriermobilities.Recentlyisolatedfew-layerofblackphosphorus,includingphosphorenemonolayers,combinehighcarriermobilitywithasizeableandtunablefundamentalbandgap[27,28],buttheyhavelimitedstabilityinair[44].Sincebothelementalcarbonandphosphorusformstable2Dmonolayers,whichhavebeen71Figure8.1:PossiblestablestructuresofanatomicallythinPCmonolayer,representedby(a-c)atilingpatternand(d-i)byball-and-stickmodelsinbothtopandsideview.ThenumberoflikenearestneighborsthestructuralcategoryN.Therearetwostableallotropes,and,foreachN.Theprimitiveunitcellsarehighlightedandthelatticevectorsareshownbyredarrows.TwoinequivalentPsitesaredistinguishedbyasubscriptin(d).studiedextensively,itisintriguingtooutwhetherthecompoundphosphoruscarbide(PC),alsocalledcarbonphosphide,mayalsobestableasamonolayeranddisplaypropertiesthatmayevenbesuperiortobothconstituents.Theplausibilityofa2DstructureofPCderivesfromthesamethree-foldcoordinationfoundbothingrapheneandphosphorene.Ontheotherhand,the2Dstructurewilllikelyfromacompetitionbetweentheplanarsp2bondingcharacteristicofgrapheneandthetlytnon-planarsp3bondingfoundinphosphorene.Thepostulated2DstructureofPCwith1:1stoichiometryisfunda-mentallytfromtheamorphousstructureobservedindepositedthinsolid[189],thepostulatedfoam-like3Dstructure[190],orthepostulatedGaSe-likemulti-layerstruc-72turesofPCcontainingCandPwiththesamesp3hybridization[191,192].Ontheotherhand,2DallotropesofPCarerelatedtopostulatedandobservedfullerene-likestructuresofCPx[193,194]andCNx[195{198],andtog-C3N4,calledgraphiticcarbonnitride.[199]InthisLetter,basedonabinitiodensityfunctionalcalculations,Iproposepreviouslyunknownallotropesofphosphoruscarbideinthestableshapeofanatomicallythinlayer.Ithattstablegeometries,whichresultfromthecompetitionbetweensp2bondingfoundingraphiticCandsp3bondingfoundinblackP,maybemappedonto2Dtilingpatternsthatsimplifycategorizingofthestructures.IintroducethestructuralcategoryN,bythenumberoflikenearestneighbors,andthatNcorrelateswiththestabilityandtheelectronicstructurecharacteristic.Dependingonthecategory,Iidentify2D-PCstructuresthatcanbemetallic,semi-metallicwithananisotropicDiraccone,ordirect-gapsemiconductorswiththeirgaptunablebyin-layerstrain.8.2ComputationalmethodsInsightintotheequilibriumstructure,stabilityandelectronicpropertiesof2D-PCallotropesreportedinthischapterisobtainedbythesamemethodasusedinlastchapter.Reciprocalspaceissampledbyagrid[136]of8121k-pointsintheBrillouinzoneoftheprimitiveunitcellof4atomsoritsequivalentinsupercells.SincethefundamentalbandgapisusuallyunderestimatedinDFT-PBEcalculations,IhaveresortedtotheHSE06[68,200]hybridexchange-correlationfunctional,asimplementedintheVASP[70{73]code,togetat(possiblysuperior)descriptionofthebandstructure.Iuse500eVastheenergyandthedefaultmixingparametervalue=0:25inthesestudies.DFT-PBEandDFT-HSE06bandstructureresultsarecompared.73Figure8.2:Bondingin(a)N=1,(b)N=2,and(c-e)N=0category2D-PCallotropes.Green-shadedregionsindicatesitesthatsatisfytheoctetrulediscussedinthetext.Bondingin0-PCischaracterizedbypanel(d)andbondingin0-PCisdescribedbypanel(e).8.3ResultsanddiscussionsAsmentionedabove,allatomsinthe2D-PCallotropesarethreefoldcoordinated,similartotheplanarhoneycomblatticeofgraphene.Thus,thestructurecanbetopologicallymappedontoa2DlatticewithsitesoccupiedeitherbyPorCatoms.Bisectingallnearest-neighborbondsbylinesyieldsa2Dtilingpattern,whereeachtriangulartilewithacharacteristiccolorrepresentseitheraPoraCatom.Next,IastructuralcategoryNforeachallotrope,withNgivenbythenumberoflikenearestneighbors.ForN=0,noneoftheatomsareconnectedtoanylikeneighbors.EachCorPatomhasonlyonelike(CorP)neighborforN=1,andtwolikeneighborsforN=2.ThereisnoN=3structure,whichwouldimplyapurecarbonorphosphoruslattice.Thetilingpatternsfort2D-PCallotropesareshowninFig.8.1(a)-8.1(c).Asimilarcategorizationschemehasbeenusedpreviouslytodistinguishbetweentallotropesof2Dphosphorene[201],whereNwasthenumberof\like"neighborseitherintheupperorlowerpositionwithinthelattice.Whereasthetilingpatternisusefulforsimplecategorization,itdoesnotprovideinfor-74mationaboutthenontrivialoptimumstructureshowninFig.8.1(d)-8.1(i),whichresultsfromacompetitionbetweenthefavoredplanarsp2hybridizationofCandnon-planarsp3hybridizationofP.ThesideviewofstructuresdisplayedinFig.8.1bestillustratesthatal-lotropeswiththesamevalueofNmaybestructurallyt.Inanalogytothetpostulatedphosphoreneallotropes[33,157],IdistinguishN,whichdisplayablack-P-likearmchairstructureinsideview,fromNphasesofPC,whichdisplayablue-P-like(orgrey-As-like)zigzagstructureinsideview,andusetheindexNtoidentifythestructuralcategory.IstartmydiscussionwiththeN=1allotropes1-PCand1-PC,showninthemiddlecolumninFig.8.1.AccordingtotheofN,eachatomhasoneneighborofthesamespeciesandtwooftheotherspecies,formingisolatedP-PandC-Cdimers,asseeninthetilingpatternandtheatomicstructures.AsseeninFig.8.2(a),thechemicaloctetrule[202]isbothonCsitesinthegraphiticsp2andonPsites,containingaloneelectronpair,insp3indicatingstability.Bothallotropeshaverectangularunitcellsconsistingofdistortedhexagons.Theunitcellof1-PCwith8atomsislargerthanthatof1-PCwithfouratoms.IntheN=2allotropes2-PCand2-PC,shownintherightcolumnofFig.8.1,eachatomhastwolikeneighborsandoneunlikeneighbor.Inthesideview,theseallotropeslookverysimilartothoseoftheN=1category.Themainbecomesapparentinthetopview.WhereasN=1structurescontainethylene-likeC2unitsthatareinterconnectedbyP2dimers,N=2systemscontaincontiguoustrans-polyacetylene-likeall-carbonchainsthatareseparatedbyP-chains.Duetothebetweenthelocallyplanarsp2bondingofCatomsandlocallynon-planarsp3bondingofPatoms,andduetothebetweenequilibriumC-CandP-Pbondlengths,thehexagonsfoundinN=1structureschangeto75Table8.1:Calculatedpropertiesoft2D-PCallotropes.<Ecoh>isthecohesiveenergyper\average"atomwithrespecttoisolatedatoms.<E>=<Ecoh><Ecoh;max>describestherelativestabilityofasystemwithrespecttothemoststablestructure.j~a1jandj~a2jarethein-planelatticeconstantsinFig.8.1.dPP,dPCanddCCaretheequilibriumbondlengthsbetweentherespectivespecies.In0-PC,theP1-CbondsfromtheP2-Cbondsinlength.Structure0-PC0-PC1-PC1-PC2-PC2-PC<Ecoh>(eV/atom)4.804.755.055.065.205.20<E>(eV/atom)0:400:450:150:140.000.00j~a1j(A)8.415.128.734.769.8410.59j~a2j(A)2.942.952.952.955.115.11dPP(A){{2.362.362.292.29dPC(A)1.86(P1)1.781.841.841.851.851.71(P2)dCC(A){{1.381.381.441.44pentagon-heptagonpairsintheoptimumN=2structureresemblingthepentheptiteorhaeckelitestructuresofgraphiticcarbon.AsseeninFig.8.2(b),similartoN=1structures,thechemicaloctetruleisonbothCandPsites.Thelatticeof2-PCand2-PCallotropescontainsrectangularunitcellswithsixteenatoms.In2DPCcompoundsofcategoryN=0,shownintheleftcolumnofFig.8.1,eachatomissurroundedbythreeunlikeneighbors.Thereisnobondingrationthatwouldsatisfytheoctetrulesonallsites.ThebondingdepictedinFig.8.2(c)theoctetruleonlyattheCsites,whereastheconinFig.8.2(d)favorsonlythePsites.ThebondingcondepictedinFig.8.2(e)containsalternatingP-CchainscontainingPsiteswithloneelectronpairsandCatomsinsp2whichsatisfytheoctetrule,andP-Cchainsthatdonotsatisfyit.Inwhateverbondingarrangement,thebondingationinN=0structuresisfrustrated.Asaconsequence,the0-PCstructureconvertsspontaneouslyfromaninitialarmchairtion,similarto1-PCand2-PC,tothezigzagstructuredepictedinFig.8.1(d).Thereconstructionprocessof0-PCcanbeseeninthesnapshotsoftheoptimizing76Figure8.3:Spontaneousstructuraltransformationof0-PCfromitsinitialarmchairin(a)tothestructurewithazigzagin(d).Snapshotsoftheintermediatestructures,showninsideviewintheupperpanels,areaccompaniedbythecorrespondingbandstructureinthelowerpanels.Noteworthyisthetransitionfromametallicstructurein(a)toasemiconductorin(d).TheoftheinequivalentP1andP2sitesisdiscussedinthemaintext.processshownintheupperpanelsofFig.8.3(a)-8.3(d).Thechangingbandstructure,calcu-latedusingtheDFT-PBEfunctional,ispresentedbelowthestructuralsnapshotsinFig.8.3.Theseresultsallowustofollowthegradualtransitionfromtheinitialmetallicstructurein(a)toasemiconductorwitha0:7eVbandgapin(d).Thebandgapopeningcomesalongwithasymmetryreduction,bestseeninthetransformationofthelocalenvironmentattheP1andP2sites,whichareequivalentin(a)andbecomesignitlytin(d).The0-PCstructurewithinequivalentP1andP2siteststhebondinginFig.8.2(e)containingP1siteswithloneelectronpairsandP2siteswithloneelectrons.The0-PCstructure,depictedinFig.8.1(g),remainslocallystableintheelectronicshowninFig.8.2(d).Tomakesurethatthereconstructionprocessof0-PCisuniqueandthatallstructuresdiscussedinthischapterrepresentthetrueequilibriumratherthananartifactofunitcelllimitations,Idoubledtheunitcellsofallallotropesandfoundnofurtherchanges.77Figure8.4:Phononspectraof(a)0-PCand(b)1-PCmonolayers.Ialsofoundallstructuresresistantunderarbitrarydeformations,whichisvalidatedbyshowingthatthephononspectraofunsupportedmonolayersarefreeofimaginaryfrequenciesthatwouldindicatespontaneouscollapseofthestructureinawaydictatedbytheparticularsoftmodecharacter.Computationofphononspectraof2Dstructureswithlowrigidityisverydemanding,sincesuchcalculationsrequireverylargesupercellstoyieldconvergedresultsinparticularfortheZmodewithan!(k)/k2dispersionnearthepoint.Forthesakeofillustration,Ipresentthephononspectraof0-PCand1-PCmonolayersinFig.8.4.Thesoftout-of-planeacousticZmodescanbeclearlydistinguishedfromthein-planetransverseacousticandthein-planelongitudinalacousticmodes,whicharethehardestofthethree.Clearly,furtherstructuralstabilizationthatsuppressestheZmodeswilloccurwhenPCisdepositedonasubstrate.Structuralcharacteristicsandthebindingenergyofthetallotropesaresumma-rizedinTable8.1.MyenergyresultsareobtainedusingtheDFT-PBEfunctional(includingspinpolarizationwhererequired),whichisknowntooverbindtosomedegree.Ithecohesiveenergyperatom,<Ecoh>,bydividingthetotalatomizationenergybythetotal78numberofatoms,irrespectiveofspecies.TheenergyvaluesintherowsindicatethatforgivenN,theandphasesarealmostequallystable,thatcategorizingstruc-turesbythenumberoflikeneighborsatanysitemakessenseintermsofstability.Clearly,N=2systemsaremoststable,followedbyN=1andN=0allotropes.Inparticular,thecohesiveenergyofN=2monolayersexceedthe5:14eV/atomvalueofthepostulatedGaSe-likePCmulti-layerstructures[191,192]by0:06eV/atom.ThelowerstabilityofN=0systemshasbeenanticipatedabove,sincetheoctetrulecannotbeatallsites.Ialsonotethatthe0phaseisslightlymorestablethanthe0phaseofPC.Thestabilityadvantageof0-PCderivesfromthelargervariationalfreedomwithintheunitcell,whichallowsthedistinctionoftwotPsites(P1andP2),asshowninFig.8.1(d)andFig.8.2(e).The0-PCstructureconsistsofP1(sp3)-C(sp2)chains,whichobeytheoctetruleandformstableridges,alternatingwithP2-Cchains,whichdonotobeytheoctetruleandformterraces.AdditionalsupportfortheplausibilityofthebondingdepictedinFig.8.2comesfromtheequilibriumbondlengths,whicharelistedinTable8.1.WiththeexceptionofN=0structures,thebondlengthsdependprimarilyonNandareratherinsensitivetothephase(or).ForN=1andN=2structures,theC-Cbondlengthslieclosetothe1:42Avalueinsp2bondedgraphite(orgraphene)andtheP-Pbondlengthsareclosetothe2:262:29Arangefoundinlayeredblackphosphorus(orphosphorene).AsseeninFig.8.2(a)and8.2(b),PandCatomsareconnectedbyasingle-bondwithdPCˇ1:85AinN=1andN=2categorystructures.Assuggestedabove,thebondingisfrustratedatleastinpartsofN=0structures.Inthetlyreconstructed0-PCsystem,depictedinFig.8.1(d),IcandistinguishP1sitesatridgesfromP2sitesatterraces.ThelengthsofthethreeP-CbondsareverysimilarateachofthePsitesofonetype,but79Figure8.5:Electronicbandstructure,densityofstates(DOS),andchargedensityˆvcassociatedwithvalencefrontierstatesofNandNallotropes,whereNisthestructuralcategory.TheenergyrangeassociatedwithˆvcisindicatedbythegreenshadedregioninthebandstructureandDOSpanelsandextendsfromEF0:45eV<E<EFin(a),EF0:40eV<E<EFin(c)and(d),EF0:10eV<E<EFin(b),(e)and(f).Isosurfaceplotsofˆvcaredisplayedintheright-sidepanelsandsuperposedwithball-and-stickmodelsofthestructureintopandsideview.Theisosurfacevaluesofˆvcare1:0103e/A3in(a),2:0103e/A3in(b),0:5103e/A3in(c)and(d),and0:5104e/A3in(e)and(f).theytlybetweenP1andP2.AtP1sitesthatsatisfytheoctetrule,asseeninFig.8.2(e),theP1-Cbondlengthof1:86AisverysimilartoN=1andN=2structures.AtP2sites,whichdonotsatisfytheoctetrule,thefrustratedbondsaremuchshorterwithdPC=1:71A.AsseeninFig.8.1(g),thereisnoreconstructioninthe0-PCstructure.AsseeninthecorrespondingFig.8.2(c)or8.2(d),theoctetruleisonlyateitherthePortheCsites.TheP-Cbondsarefrustratedandtheirlengthof1:78Aliesin-betweentheP1-CandP2-Cbondlengthsin0-PC.80ResultsofmyDFT-PBEelectronicbandstructurecalculationsformonolayersofthesixproposedPCallotropesarepresentedinFig.8.5.Theelectronicbandstructureandassociateddensityofstates(DOS)ofN=0systemsisshowninFig.8.5(a)and8.5(b).MyresultsinFig.8.5(a)suggestthat0-PCisanindirect-gapsemiconductorwithabandgapofˇ0:7eV.Instarkcontrast,thestructurallysimilar0-PCallotropeismetallicaccordingtoFig.8.5(b).Assuggestedearlier,allbondsandelectronicarefrustratedin0-PC,withallCsitesengagingonlythreevalenceelectronsinsp2-likebonds,leavingoneloneelectronbehind,andtheangleatthePridgebeingtoolargefortypicalsp3bonding.Thisding,inparticularthepresenceofanon-bondingelectronintheC2p?orbital,isseeninthefrontierstatesof0-PCthataredepictedintherightpanelofFig.8.2(b).0-PCisquiteerentfrom0-PC,asitcontainstwoinequivalentPandCsites.TheP1siteattheridgedisplaysthefavoredsp3bondingcharacteristicanditslonepairorbitalispresentinthefrontierstatedisplayedintheright-handpanelofFig.8.5(a).Incontrast,thebondingisverytattheP2site,wherethelonepairorbitaldoesnotcontributetothefrontierstate.Thebondinggeometrynearthissiteisreminiscentofsp2bondingattheCsites.Theaddedilityprovidedbyalargerunitcellallowsforadditionalstabilizationof0-PCduetotheopeningofabandgap,inroughanalogytothePeierlsinstability.AsseeninFig.8.5(c)and8.5(d),both1-PCand1-PChaveadirectbandgap,whichIattributetothepresenceofisolatedethylene-likeunitsmentionedabove.Thetwoallotropesdisplayaverysimilarchargedistributionintheirvalencefrontierstates,whichcontainlonepairorbitalsonPsitesandsp2bondingbetweenCsites.Themainbetweenthetwostructuresisthatthe0.4eVwidegapin1-PCisatthepoint,whereasthe0.3eVgapin1-PCisattheXpoint.Inbothstructures,thebanddispersionisratheranisotropic81nearthetopofthevalenceandbottomoftheconductionband,whichcausesananisotropyintheemass.Itheemassofbothelectronsandholestobemuchsmalleralongx-directionthanalongthey-direction,whichisreminiscentofthesituationinblackphosphorene[27,28].Theemassanisotropyatadvantageintransport,sinceitcombineshighmobilityofcarrierswithalargeDOSnearthebandedges.AccordingtoFig.8.5(e)and8.5(f),thetwoN=2allotropes,2-PCand2-PC,shareverysimilarbandstructure,DOS,andfrontierorbitalsduetostructuralsimilarities.Theelectronicstructureofthesesystemsisneverthelessverytfromtheothertwocate-gories,cyduetothepresenceoftrans-polyacetylene-likechainsmentionedabove.Both2-PCand2-PCdisplayaDiracconeattheFermilevel,atacrystalmomentumbetweenandY.Asmentionedbefore,thedistinguishingfeatureofN=2structuresisthealternationbetweenchainsconsistingofpurePorpureCatoms.Fig.8.2(b)indicatesthatallPsiteshaveoccupiedlonepairorbitals,whicharealsointhefrontierstates.ThePchainsformridgeswithinthestructure,withbondanglescharacteristicofthesp3bondingfoundinblackphosphorus.Thestructureofthecarbonchains,alsoillustratedinFig.8.2(b),resem-blesthatofconjugatedtrans-polyacetyleneorgraphenewithsp2bonding,andthepresenceofC2p?orbitalsinthefrontierstatesisclearlyseenintheright-sidepanelsofFig.8.5(e)and8.5(f).betweenequilibriumbondlengthandbondanglesofthePandCchainsareaccommodatedbyintroducingpentagon-heptagonpairs.TheconjugationwithinCchainsandtheirsuppresseddimerizationcausedbytheirbondingtoadjacentPchainsliesbehindtheformationoftheDiraccone.Duetothestronganisotropyinthesystem,causedbythedirectionofthetrans-polyacetylene-likechains,theDiracconeisanisotropicintheplaneofthelayer.82Figure8.6:Bandstructureofsemi-metallic(a)2-PCand(b)2-PCneartheDiracpoint.(c)ofthehigh-symmetrypointsintheBrillouinzoneofthestructuresin(a)and(b).Moreinformationaboutthebandstructureofsemi-metallic2and2allotropesofPCisdisplayedinFig.8.6(a)and8.6(b).EachstructurehasaDiracpointatpointAlocatedin-betweenandYintheBrillouinzoneshowninFig.8.6(c).TheunusuallocationoftheDiracpointsinsidetheBrillouinzonefromitslocationinBrillouinzonecornersingraphene.Duetothestrongstructuralanisotropyinthesystems,theDiracconeisanisotropicandthusnotatruecone.Theopeningangleofthecone,givenbytheE(k)dispersionnearA,islargestalongtheABdirectionandthesmallestalongtheAorAYdirection.IhavefoundthatuniaxialstrainmaybeusedtoeliminatetheanisotropyoftheDiraccone.Atthesametime,however,alsothelocationoftheDiracpointalongYlineintheBrillouinzonechanges.EventhoughDFT-PBEcalculationsnotoriouslyunderestimatethefundamentalbandgapbetweenoccupiedandunoccupiedstates,thecalculateddispersionE(k)ofindividualbandsisbelievedtocloselyresembleexperimentalvalues.Forthesakeofcomparison,IhavealsoperformedDFT-HSE06[68,200]calculationswithahybridexchange-correlationfunctionalforthesamestructures.AsseeninFig.8.7,myHSE06studiesusethedefault83Figure8.7:Electronicbandstructureof(a)0-PC,(b)1-PC,(c)2-PC,(d)0-PC,(e)1-PC,and(f)2-PCmonolayerobtainedusingDFTcalculationswithPBEandHSE06exchange-correlationfunctionals.valueof0.25forthemixingparameterand,forthesakeoffaircomparison,theidenticalgeometrythathadbeenoptimizedbyDFT-PBE.BandstructureresultsinFig.8.7indicatethatforthesemiconducting0-PC,1-PCand1-PCstructures,HSE06shiftsoccupiedstatesrigidlydownandunoccupiedstatesupwithrespecttotheFermilevel,thusincreasingthebandgapvalue.Asaresult,HSE06givesbandgapvaluesof1.5eVfor0-PC,1.1eVfor1-PC,and0.8eVfor1-PC.Asaconsequenceoftheincreasedbandgap,thecompressivein-layerstrainrequiredtoclosethisgapin1-PCand1-PCwillbelargerthanwhatwasexpectedbasedonDFT-PBEresults.HSE06doesnotthemetalliccharacterof0-PCandsemi-metalliccharacterof2-PCand2-PCpredictedbyDFT-PBE,butincreasesthebanddispersionandthusthewidthofthevalenceandconductionbands.84Figure8.8:ofuniaxialin-layerstrainon(a)therelativebindingenergyEtotand(b)thefundamentalbandgapintPCallotropes.Resultsfor0-PC,0-PC,1-PC,1-PC,2-PCand2-PCaredistinguishedbycolorandsymbols.Resultsforstraininthex-direction,inFig.8.1,areshownbysolidlinesandforstraininthey-directionbydashedlines.Similartoothernon-planar2Dsystemslikephosphorene,PCissusceptibletoevenminutein-planestress,whichcancausemajordistortionsinthegeometry,atheelectronicstructureandbonding.ToquantifythisIhavedeterminedtheoftensileandcompressivestrainonthestabilityandthefundamentalbandgapinthetPCallotropesandpresenttheresultsinFig.8.8.Ihaveconsidereduniaxialstrainalongthex-andthey-direction,inFig.8.1.Sinceallallotropesdiscussedherearenon-planar,applyingin-layerstrainchangesthevethicknessofthelayersandviceversa.Asexpected,layerthicknessisreducedundertensilestrainandincreasedundercompressivestrain.Forstrainvaluesbelow5%,Ihaveobservedchangesinlayerthicknessofupto10%.Thedistinctstructuralanisotropy,bestseeninthesideviews,translatesintoadistinctanisotropyofthestrainenergywithrespecttothestraindirection,showninFig.8.8(a).Similartoblackphosphorene,thesystemappearssoftwhenstrainedalongthex-directionnormaltotheridgesandvalleys,whereasitismuchstwhendistortedalongthey-direction.Ithephasetobeparticularlysoftinthex-direction,withcompressiveor85tensilestrainrequiringE.5meV/atominstrainenergy.Thedependenceofthefundamentalbandgaponthein-layerstrain,asobtainedbymyDFT-PBEcalculations,isshowninFig.8.8(b).Ithatcompressionalongthesoftx-directiondoesnotthebandgapmuch,quiteunlikewhatisexpectedtooccurinblackphosphorene[28].Thisisquitetfrommyresultsforstrainalongthey-direction.There,Iobservethefundamentalbandgaptodisappearatcompressivestrainexceeding4%for1-PCand3%for1-PC.Ialsothatthemetalliccharacterof0-PCandsemi-metalliccharacterof2-PCand2-PCarenotbytensileorcompressivestrainsupto5%appliedalongthex-orthey-direction.Sinceverticalstraincausinga10%reductionofthelayerthicknessisequivalenttoanetensilein-layerstrainbelow5%,IcanjudgeitsontheelectronicstructurebasedontheaboveEventhoughthecohesiveenergyofthe2DstructurespresentedhereexceedsthatofpreviouslydiscussedPCsystems,thecalculatedcohesiveenergyperformulaunitstillfalls0:54eVshortofthesumofthecohesiveenergiesofpureblackphosphorene,3:27eV,andpuregraphene,7:67eVaccordingtoDFTcalculation.EventhoughthePCallotropesdiscussedhereareallstable,theslightenergeticpreferenceforpurecomponentsoverthePCcompoundshouldchallengesinthesynthesis.Recentadvancesinsupramolecularassemblymaysolvethisproblem.Similartomyrequirements,preciselydesignedstructuresincludinggraphdiyne[203,204],graphenenanoribbons[205]andcarbonnanotubes[206]havebeenassembledusingwetchemicalprocessesfromspmolecularprecursors.Inthesameway,Iexpectthatthepostulated2D-PCstructuresmaybeformedofpropermolecularprecursorsthatcontainsp2bondedcarbonandsp3bondedphosphorus.868.4SummaryInconclusion,Ihaveperformedabinitiodensityfunctionalcalculationsandidenpre-viouslyunknownallotropesofphosphoruscarbide(PC)inthestableshapeofanatomicallythinlayer.Ifoundthattnon-planarstablegeometries,whichresultfromthecom-petitionbetweensp2bondingfoundingraphiticCandsp3bondingfoundinblackP,maybemappedonto2Dtilingpatternsthatsimplifycategorizingofthestructures.Ihavein-troducedthestructuralcategoryN,bythenumberoflikenearestneighborsrangingfrom0to2,andfoundthatNcorrelateswiththestabilityandtheelectronicstructurechar-acteristic.IfoundstructuresoftheN=0categoryeithertobemetallicortoreconstructspontaneouslytoamorestablestructurewithalargerunitcellandasizeablefundamentalgap.SystemsoftheN=1categoryaremorestablethanN=0systemsanddisplayat,directbandgapandatanisotropyoftheemassofcarriers.N=2systemsarethemoststableofall,aresemi-metallic,anddisplayananisotropicDiracconeattheFermilevel.Duetotheirnon-planarcharacter,allsystemscansustainin-layerstrainatlittleenergycost.ThefundamentalbandgapisnotverysensitivetostraininmostsystemswiththeexceptionofN=1allotropes,whereitclosesuponapplyingcompressivestrainof.5%alongtheridgesandvalleys.87Chapter9Curvatureandstabilityfortwo-dimensionalsystemsThefollowingdiscussionismyoriginalcontributiontotherelatedpublicationsbyJieGuan,ZhongqiJin,ZhenZhu,ChernChuang,Bih-YawJinandDavidTanek,Phys.Rev.B90,245403(2014)[207],andbyChernChuang,JieGuan,DavidWitalka,ZhenZhu,Bih-YawJinandDavidTanek,Phys.Rev.B91,165433(2015)[208].9.1IntroductionLayeredstructuresincludinggraphite,hexagonalboronnitride,blackphosphorus,transitionmetaldichalcogenidessuchasMoS2,andoxidesincludingV2O5areverycommoninNature.Thepossibilitytoformstabletwo-dimensional(2D)structuresbymechanicalexfoliationofthesestructuresappearsveryattractiveforavarietyofapplications.[45,209]Themostprominentexampleofsuch2Dsystems,graphiticcarbon,isthestructuralbasisnotonlyofgraphene,[45]butalsofullerenes,nanotubes,tori,andschwarzites.[46{50]Eventhoughthestructuralmotifinallofthesesystemsmaybethesame,theirmechanicalandelectronicpropertiesdependsensitivelyonthelocalmorphology.[51{53]Notonlydoesthenaturalabundanceofstructuralallotropesandisomerstheirnetenergeticstability,butalsotherelativechemicalreactivityofspsitesinagivenstructurecorrelateswellwiththe88Figure9.1:PrincipalradiiofcurvatureR1;R2andtheGaussiancurvatureGonthesurfaceof(a)asphereand(b)acylinderand(c)atasaddlepoint.(d)DeterminationofthelocalcurvatureatpointPusingtheatomiclattice(greydashedlines)andtheduallattice(bluesolidlines).PisthecenteratomandFi(i=13)arethreeneighbors.Vi(i=13)areverticesofduelattice.localcurvatureandlocalstability.[51{53]ThisrelationshiphasbeenwellestablishedforthereactivesitesintheC50fullerene,[51]usedtoinducestructuralcollapseleadingtochemicalunzippingofcarbonnanotubes,[93,210,211]andusedtodestroycollapsedcarbonnanotubes.[53]Forverylargestructures,estimatingtheglobalorlocalstabilityusingabinitiocalcu-lationshasprovenimpracticable.Instead,insuchstructuresthestabilityhasoftenbeenestimatedusingempiricalrulesorparameterizedforceincludingtheTersopoten-tialandmolecularmechanics,[54{57]withsometimesunsatisfactoryresults.Applicationofcontinuumelasticitytheory,whichcandescribestabilitychangesduetodeviationfrom89planarity,hasbeensuccessful,butitislimitedtosystemswithawconstantcur-vature.[58,59]Sincestrainenergyisdominatedbylocalgeometryandindependentoftheglobalmorphology,itisintriguingtoexplore,whetherthelocaldeformationenergymaybeaccuratelydeterminedfromlocalmorphologyestimatesusingtheatomicgeometry.Ifso,thenthelocalstabilityinevenarbitrarilyshapedstructurescouldbeestimatedaccurately.HereIproposeafastmethodtodeterminethelocalcurvaturein2Dsystemswithacom-plexmorphologyusingthelocalatomicgeometry.Curvatureinformationalone,combinedwithelasticconstantsobtainedforaplanarsystem,providesaccuratestabilityestimatesintheframeworkofcontinuumelasticitytheory.Ithatrelativestabilitiesofgraphiticstructuresincludingfullerenes,nanotori,nanotubes,andschwarzites,aswellasphosphorenenanotubes,calculatedusingthisapproach,agreecloselywithabinitiodensityfunctionalcalculations.Thecontinuumelasticityapproachcanbeappliedtoall2Dstructuresandisparticularlyattractiveincomplexsystemswithknownstructure,wherethequalityofparameterizedforcehasnotbeenestablished.9.2ContinuumelasticmethodThelocalcurvatureataparticularlocationonasurfaceisgivenbythetwoprincipalradiiofcurvatureR1andR2,asshowninFig.9.1.Onasphericalsurface,R1=R2.Onacylindricalsurface,R1isthecylinderradiusandR2!1.Finally,asaddlepointonasurfaceischaracterizedbyoppositesignsofR1andR2.Knowingtheprincipalradiiofcurvatureeverywhere,Imayusecontinuumelasticitytheorytodeterminethecurvature90energyECwithrespecttoaplanarlayerusing[212]EC=12DZsurfacedA 1R21+1R22+2R1R2!:(9.1)Here,theintegralextendsacrosstheentiresurface,DistherigidityandisthePoissonratio.SimpleexpressionsforECcanbeobtainedforsimplemorphologiessuchasasphereoracylinder,whereR1andR2areconstanteverywhere.[58]Thisis,however,notthecaseingeneral.Iitconvenienttointroducethelocalmeancurvaturek=121R1+1R2(9.2)andthelocalGaussiancurvatureG=1R1R2:(9.3)Usingthesequantities,IcanrewriteEq.(9.1)asEC=DZsurfacedAh2k2(1)Gi:(9.4)Inthefollowing,Iwillconsidertheequilibriumarrangementofatomsinaplanar2Dstructureasthereferencestructureandwilldeterminethelocalcurvaturefromchangesinthelocalmorphology.ThediscretecounterpartofEq.(9.4)forthecurvatureenergyECisasumoveratomicsitesi,ECˇDAXih2k2i(1)Gii;(9.5)91whereAistheareaperatom.TouseEq.(9.5)forcurvatureenergyestimates,IneedtoknowthelocalcurvatureskandGatallatomicsites.MyapproachtoestimatethesevaluesatagivensitePisillustratedinFig.9.1(d).AccordingtoEq.(9.2),thelocalmeancurvaturekshouldbeclosetotheaverageinverseradiusofcurvatureatthatpoint,kˇ˝1R˛:(9.6)SincetheatomicsitePanditsnearestthreeneighborsF1,F2andF3thesurfaceofasphereofradiusR,Itakek=1=R.ThepositionsoffouratomsdonotallowonetodistinguishwhetherPisonaplane,asphereoracylinderoratasaddlepoint.Imayobtainthisadditionalinformationusingtheconceptofanangulardefect.OnanysurfacethatcanbetriangulatedasinFig.9.1(d),theangulardefectatarepresentativevertexV1isbyV1)=2ˇPi'iinunitsofradians.ThelocalGaussiancurvatureatV1isthengivenby[213]G(V1)=V1)=At= 2ˇXi'i!=At;(9.7)whereAtisthetotalareaofthetriangulatedsurfacedividedbythenumberofvertices.Fortrivalentmoleculargraphscontaining5-,6-and7-memberedringsfoundinfullerenes,carbonnanotubes,andschwarzites,auniquetriangulationmaybeobtainedbyconnectingthecentersofadjacentpolygons.Thismethodisreferredtoasthedualgraphingraphtheory[214]anditsuseisillustratedinFig.9.1(d).SincePisnotavertexinthedualgraph,butratherthecenterofthetriangle4V1V2V3,ImustinferthelocalGaussiancurvatureat92PfromtheangulardefectsatV1,V2andV3.IfvertexVjissurroundedbynjtriangles,ImayassigntopointPtheangulardefectP)=V1)=n1+V2)=n2+V3)=n3.Then,IcanestimatethelocalGaussiancurvatureatPasG(P)=P)=A;(9.8)whereAistheaverageareaperatom.IuseA=2:62A2,thevaluefoundinthehoneycomblatticeofgraphene,forallgraphiticstructures.TheaboveofthelocalGaussiancurvatureexactlytheequalityAXatomsG(Pj)=AtXverticesG(Vj)=2ˇ˜:(9.9)Here,˜istheEulercharacteristicofthesurface,givenby˜=22g,wheregisthegenus,meaning`numberofholes'.Ofinteresthereisthefactthat˜=2forsphericalobjectslikefullerenesand˜=0forcylindricalobjectssuchasnanotubes.Equation(9.9)isthediscretizedversionoftheGauss-Bonnettheorem[215]regardingtheintegraloftheGaussiancurvatureoveranentireclosedsurface,calledthesumofthedefect,whichisusuallyformulatedasRsurfaceGdA=2ˇ˜.ThevariationofthelocalGaussiancurvatureGandthelocalcurvatureenergyEC=Aacrossthesurfaceofcarbonpolymorphs,includingtwofullereneisomers,ananotube,andaschwarzitestructure,isdisplayedinFig.9.2.Thelocalcurvatureenergyinthesesp2-bondedstructureshasbeenevaluatedusingtheelasticconstantsofgraphene[58]:D=1:41eVand=0:165.ThehigherstabilityoftheC38(17)isomerinFig.9.2(b)isinaratheruniformcurvatureenergyandGaussiancurvaturedistribution.Thelowstabilityofthe93Figure9.2:LocalGaussiancurvatureG(upperpanels)andlocalcurvatureenergyEC=A(lowerpanels)acrossthesurfaceof(a)theleaststableC38isomer,(b)themoststableC38isomer,(c)a(10,10)carbonnanotube,and(d)aschwarzitestructurewith152atomsperunitcell.ThevaluesofGandEC=Ahavebeeninterpolatedacrossthesurface.C38(2)isomerinFig.9.2(a)isinalargevariationofcurvatureenergyandGaussiancurvature,clearlyindicatingthemostreactivesites.Cylindricalcarbonnanotubes,suchasthe(10,10)nanotubedisplayedinFig.9.2(c),havezeroGaussiancurvatureandaconstantcurvatureenergycausedbythemeancurvature.SchwarzitessuchastheC152structure,displayedinFig.9.2(d),haveonlynegativeGaussiancurvaturethatmayvaryacrossthesurface,causingvariationsinthelocalcurvatureenergy.9.3ValidationofthecontinuumelasticityapproachIwillnexttesttheaccuracyofthecontinuumelasticityapproachbycalculatingtherel-ativestabilityofnon-planarstructuresbasedongraphiticcarbon.Annumberofmorphologiesincludingnanotubes,fullerenesandschwarzitesmaybeproducedbydeform-ingasegmentofagraphenelayerandreconnectingitsedgessothatallcarbonatomsare94Figure9.3:StrainenergyEincarbonnanostructureswithrespecttothegraphenereferencesystem.(a)DFT-basedtotalstrainenergyEDFTtotforselectedfullerenes,withthemoststableisomersindicatedbythelargersymbols.(b)StrainenergyEintC38isomers.TotalenergyEDFTtotbasedonDFT,ETersofftotbasedontheTpotentialandEKeatingtotbasedontheKeatingpotentialarecomparedtocurvatureenergiesEKeatingCbasedonKeating-optimizedgeometries.Tofurthercomparecurvatureenergybasedonntgeometries,EDFTtotarecomparedtocurvatureenergiesEDFTCbasedoncontinuumelasticitytheoryforstructuresoptimizedbyDFT,andEKeatingCin(c).threefoldcoordinated.Inmanycases,thenon-planarstructurescontaincarbonpentagonsandheptagonsinthegraphitichoneycombarrangementofatomsasrequiredbyEuler'stheorem.[215]Tovalidatethecontinuumelasticitytheoryresults,Icalculatedthetotalenergyofagraphenemonolayerandselectedgraphiticstructuresusingabinitiodensityfunctionalthe-ory(DFT)asimplementedintheSIESTAcode.[69]IusedtheLocalDensityApproximation(LDA)[216,217]andPerdew-Burke-Ernzerhof(PBE)[64]exchange-correlationfunctionals,norm-conservingTroullier-Martinspseudopotentials[145],andadouble-basisincludingpolarizationorbitals.The1DBrillouinzoneofnanotubeswassampledby16k-pointsandthe2DBrillouinzoneofgrapheneby1616k-points.[136]ThesmallBrillouinzonesofschwarziteswithseveralhundredCatomsperunitcellweresampledbyonly1k-point.Alltheotherparametersusedherearethesameaspreviouschapters.MyDFT-LDAresultsfortherelativeenergyEDFTtotofoptimizedCnfullerenes[218]95withrespecttographeneareshowninFig.9.3(a).Thevariousdatapointsforonesizecorrespondtotstructuralisomers,whichincreasefastinnumberwithincreasingn.Ifallfullereneswereperfectspheres,Eq.(9.4)wouldsimplifyto[58]EC=4ˇD(1+).Usingtheproperelasticconstantsforgraphene[58]D=1:41eVand=0:165,IwouldestimateEC=20:6eVforallfullerenesindependentofsize.ThenumericalvaluesforthetoptimizedfullereneisomersinFig.9.3(a)arealllarger,indicatingthatvariationsinthelocalcurvatureandbondlengthscauseatenergypenalty.AsIshowinthefollowing,consideringonlylocalcurvaturevariationsacrossthesurface(andignoringpreciseatomicpositions)allowscontinuumelasticitytheorytoquantitativelypredictthestrainenergywithaprecisioncompetingwithabinitiocalculations.Toillustratethispoint,IpresentinFig.9.3(b)thetotalstrainenergyEinseventeenisomersofC38obtainedusingvariousapproaches.ThestrainenergyEDFTtotbasedonDFT,whichisexpectedtorepresentcloselytheexperimentalresults,isnotonlycantlylowerthanthepredictedvalues,ETersofftot,basedontheTpotential[54]butalsofromthispopularbond-orderpotentialinthepredictionofrelativestabilities.NextIdemonstratethataccurateenergyestimatesmaybeobtainedevenforgeometriesoptimizedusingsimplepotentialswithonlybondstretchingandbondbendingtermssuchastheKeatingpotential[219,220]EK=12KX<i;j>i<j(r2ijR2)2R2+12KX<i;j;k>j<k(rijrik+12R2)2R2:(9.10)Thetermsumsovernearestneighborpairsandthesecondtermovernearestneighbortriplets,wherejandksharethesameneighbori.DFTcalculationsforgrapheneyieldR=1:42Aasbondlength,120asbondangle,K=11:28eV/A2andK=4:14eV/A2.96StrainenergiesforKeatingoptimizedfullerenesareshowninFig.9.3(b).WhereastheKeatingoptimizedgeometryisclosetotheDFToptimizedgeometry,theKeatingstrainenergyEKeatingtotclearlyunderestimatestheDFTvaluesanddoesnotcorrectlyrepresenttherelativestabilitiesofthetisomers.Asanalternative,IusedtheKeatingop-timizedgeometrytoobtainthecurvaturestrainenergyEKeatingCusingthecontinuumapproach.IfoundthatthisapproachrepresentstherelativestabilitiesofisomersadequatelyandcompareswelltoEDFTtot.ThecurvaturestrainenergyvaluesaresomewhatlowerthantheDFTvalues,sinceenergypenaltiesassociatedwithbondstretchingandbendingdonotappearinthecontinuumapproach.Thesmallvalueofsuchcorrectionsthefactthatinequilibratedstructures,bondlengthsandanglesareneartheiroptimum.Thelargesterrorsareexpectedinfrustratedstructures,wherenotallbondlengthsandanglescanbeoptimizedsimultaneously.Oneofthekeyofthisstudyisthatcontinuumelasticitytheoryprovidesnotonlyafast,butalsoarelativelyrobustwaytodeterminerelativestabilitiesthatare,tosomedegree,insensitivetotheprecisegeometry.IillustratethispointinFig.9.3(c),whereIcomparetwaystodeterminethetotalstrainenergyEinallC38isomersdiscussedinFig.9.3(b).EDFTtot,shownbythesolidline,isthebetweenthetotalenergyinDFTofDFT-optimizedC38isomersand38carbonatomsinthegraphenestructure.EDFTC,givenbythedashedline,isthecurvatureenergybasedontheDFT-optimizedgeometry.EKeatingC,givenbythedash-dottedline,isthecurvatureenergybasedontheKeating-optimizedgeometry.Inotethatallexpressionsprovideanaccuraterepresentationofrelativestabilities.Asmentionedabove,thefactthatECisabout10%lowerthanEtotiscausedbymyneglectingthestretchingandbendingofdiscreteatomicbondsinthecontinuumapproach.97Figure9.4:(Coloronline)StrainenergyEincarbonnanostructureswithrespecttothegraphenereferencesystem.(a)ComparisonbetweenDFT-basedtotalenergiesEDFTtotandthecurvatureenergyEKeatingCbasedonKeating-optimizedgeometriesforallfullereneisomersconsideredinFig.9.3(a).(b)ComparisonbetweenDFT-basedstrainenergiesEDFTtot=nandcurvatureenergiesperatomEKeatingC=nforKeating-optimizedgeome-triesoffullerenes,nanotubesandschwarzites.DashedlinesrepresentagreementbetweenDFTandcontinuumelasticityresults.EncouragedbythelevelofagreementforC38,IpresentinFig.9.4(a)thecorrelationbetweenthecurvatureenergyEKeatingCandEDFTtotbasedonDFTforallfullerenesdiscussedinFig.9.3(a).ThenarrowspreadofthedatapointsclosetotheEKeatingC=EDFTtotlinecothatthecontinuumelasticityapproachiscompetitiveinaccuracywithcomputationallymuchmoreinvolvedabinitiocalculations.Todemonstratethegeneralityofmyapproach,Iextenditfromnear-sphericalfullerenestonanotubeswithcylindricalsymmetryandschwarziteswithlocalnegativeGaussiancurva-ture.Sincenanotubesandschwarzitesarelarge,Icomparestabilitiesonaper-atombasisinthesestructures.BesidesresultsforthefullerenesdiscussedinFigs.9.3and9.4(a),Fig.9.4(b)displaysresultsfornanotubeswithradiirangingbetween2:59:0Aandforschwarzitestructureswith152,192and200carbonatomsperunitcell.TheseresultsagainindicateexcellentagreementbetweencurvatureenergiesinKeating-optimizedstructuresand98Figure9.5:(a)Atomicstructureofapolygonalnanotoruscontainingnon-hexagonalrings.(b)EnergydperatomEDFTtotbasedonDFTandEKeatingtotbasedontheKeat-ingpotentialarepresentednexttocurvatureenergiesEKeatingcforKeating-optimizedgeometries.DFT-basedstrainenergies.Thisagreementisparticularlyimpressive,sincethespreadofatomicbindingenergiesextendsovermorethan1eV.Mycontinuumelasticmethodwasalsoappliedtopolygonalcarbonnanotori.Unlikethepolyhexnanotorirolledupfromcarbonnanutubes,thepolygonalnanotorialsocontainnon-hexagonalringstoreleasethelargein-planestrain,asanexampleshowninFig.9.5(a).Theresultsofmymethodologytoasubsetof22polygonalnanotoriareshowninFig.9.5(b).HerethestrainenergycalculatedthroughEq.(9.5)withgeometryoptimizedbyKeatingpo-tentialisrepresentedbythegreensquares,energycalculatedwiththeaccurateDFTmethodbyblackdots,andtheKeatingpotentialenergyfortheKeating-optimizedoptimizedgeom-etrybyredrhombi.MyresultsshowclearlythatforKeating-optimizedgeometries,strainenergiesbasedonEq.(9.5)reproducemyabinitioresultsratherwell.Ontheotherside,strainenergiesestimatedusingKeatingpotentialalonenotonlytlyunderestimatesthestrain,butalsodonotfollowthecorrectgeneraltrend.Thisestablishestheapplicabilityofthecontinuummethodologytopolygonalnanotoriunderinvestigation.99Figure9.6:LocalGaussiancurvatureG(leftpanels)andlocalcurvatureenergyEc=A(rightpanels)acrossthesurfaceofselectednanotori.Representativeexamplesshownare(a)theC320nanotoruswith320atomsand(b)theelongatedC280nanotoruswith280atoms.GandEc=Aareinterpolatedfromtheirvaluesattheatomicsites.ThedistributionofthelocalGaussiancurvatureGiandthelocalcurvatureenergyperarea,E(i)c=A0=D[2k2i(1)Gi],acrossthesurfaceoftworepresentativenanotoriisshowninFig.9.6.Inbothcasesthepositivelycurvedsegments,showninredandyellowintheleftpanels,areconcentratednearthelociofpentagonsalongtheouterperimeter.Thenegativelycurvedsegments,showninblue,areconcentratednearthelociofheptagonsalongtheinnerperimeter.Sp,theheptagonsalongtheinnerperimeteroftheC320nan-otorusinFig.9.6(a)arewellseparatedfromthepentagonsalongtheouterperimeter.ThisistfromtheaxiallyelongatedC280torusofFig.9.6(b),wherepentagon-heptagonpairsintheupperandlowerplanesformanazulene-likepattern.Asaconsequence,theGaussiancurvatureismoreevenlydistributedinthelatter.Iemphasizeagainthatforclosednanotori,thesummationofthelocalGaussiancurvaturesisstrictlyzeroasdictated100bytheGauss-Bonnettheorem.Thedistributionofthelocalcurvatureenergyisevenmoreintriguing.IobservedsomedegreeofcorrelationbetweentheabsolutevalueofthelocalGaussiancurvatureandthecurvatureenergy.Thisshouldimply,atleastforthenanotoriinvestigated,thatthetwolocalcurvatureskiandGiarenotentirelyindependent.Yetthevalueofthiscorrelationhasitslimitations,asshowninFig.9.6.WhereasintheC320nanotorusinFig.9.6(a)thecurvatureenergyisratherevenlydistributedacrossthestructure,thestrainisclearlylargestneartheupperandlowerendsoftheC280nanotorusinFig.9.6(b).ThiscurvatureenergydistributionintherightpanelsobviouslyfromtheGaussiancurvaturedistributionintheleftpanels.Thereasonforthisisthatintheseextremestructures,IcannottrulydecouplekiandGi.InC280,Giˇ0andkiisconstantinthecentral`tubularsegments'.Onlyattheupperandlowerends,alargemeancurvaturekiisrequiredtoconnecttheinnerandtheoutertube.TheC320nanotoruslacks`tubularsegments'withGiˇ0.Therefore,theGaussiancurvatureandcurvatureenergyarebettercorrelatedandmoreevenlydistributedinthisisomer.Assuggestedattheoutset,myapproachtoestimaterelativestabilitiesisparticularlyvaluableforunexploredsystemssuchasmonolayersofbluephosphorus,[33]wheremodelpo-tentialshavenotyetbeenproposed.MyDFT-PBEresultsforabluephosphorenemonolayerindicateA=4:78A2astheprojectedareaperatom,D=0:84eVand=0:10.Themono-layerstructure,showninthetoppanelofFig.9.7(a),hasanethicknessof1.27A.ThisstructurecanberolledupintophosphorenenanotubeswithtradiiRusingtheapproachusedintheconstructionofcarbonnanotubes.[50]AsseeninFig.9.7(b),thestrainenergyforthisgeometry,obtainedusingcontinuumelasticitytheory,agreesverywelldowntoverysmallradiiwithresultsobtainedusingmuchmoreinvolvedDFTcalculations[173].101Figure9.7:(a)Perspectiveviewoftheplanarstructureofabluephosphorenemonolayer(top),whichhasbeenrolleduptoananotubewithradiusR(bottom).(b)Compari-sonbetweenthestrainenergyperatomEC=nbasedoncontinuumelasticitytheoryandEDFTtot=nbasedonDFTinbluephosphorenenanotubes.Thedashedlinerepresentsagree-mentbetweenDFTandcontinuumelasticityresults.9.4DiscussionGivenasetofpointsinspace,suchasatomicpositions,itispossibletoconstructasmoothsurfacethatcontainsallthesepointsinordertocharacterizeitsshapeeverywhere,andtoeventuallydeterminethedeformationenergyusingthecontinuumelasticityapproach.Iillustratethispointbytessellatingthesmoothsurfaceofagraphiticnanocapsule,con-sistingofacylindercappedbyhemispheresatbothendsandrepresentingC120,intways.MyresultsinFig.9.8showthattheaveragecurvatureenergy<EC>isratherin-sensitivetothetessellationdensity.Thehorizontaldashedlineat<EC>=0:099eV/A2,representinganextrapolationtoadensetessellation,isˇ5%higherthantheexactcontinuumelasticityvalueof0:093eV/A2,obtainedforanidealcapsulewithcylinderandhemisphereradiusR=3:55A.Thesmallarisesfrommyapproximatewaytoestimatethemeancurvaturekonthecylindersurfaceandattheinterfacebetweenthecylinderandthehemisphere.Theextrapolatedvalueisalsoclosetothevalueof<EDFTtot>=0:100eV/A2102Figure9.8:Averagecurvatureenergy<EC>perareaofananocapsuletessellatedbyahoneycomblatticewithtnumbersofvertices.Theverticaldash-dottedredlineindicatesthatthecapsulerepresentstheC120structure.Theinsetshows,howthecapsulesurfacecanbetessellatedbyahoneycomblatticewith120verticesoratoms,shownbythewhitelines,andalsowith480vertices,shownbytheredlines.Thehorizontaldashedblacklinerepresentsanextrapolationtoandensetessellation.basedontheDFT-optimizedC120capsule.Thereverseprocesstodetermineatomicpositionsfromtheshapealoneisnotunique.Aninformativeexampleisthestructureofacarbonnanotube.Whereasthepreciseatomicstructurewithineachnanotubeisbythechiralindex,manynanotubeswithdtchiralindicesshareessentiallythesamediameterandthesamelocalcurvature.Thus,givenonlythediameterofa(wide)hollowcylinderrepresentingananotube,itisimpossibletouniquelyidentifythechiralindexandthustheatomicposition.Asamatteroffact,iden-tifyingthepreciseatomicpositionsisnotnecessary,sinceaccordingtocontinuumelasticitytheory,supportedbyexperimentalevidence,thestabilityofnanotubesdependsonlyonthe103tubediameter.[50]Fromitsconstruction,thecontinuumelasticitydescriptionoflocalandglobalstabilityisbestsuitedforverylargestructureswithsmalllocalcurvatures.Therefore,thehighlevelofagreementbetweenitspredictionsandabinitioresultsinstructureswithlargelocalcurvaturesisratherimpressive.Amongthetallotropes,Idthecontinuumelasticitydescriptiontobemostaccurateforcarbonnanotubes,whereallbondlengthsareattheirequilibriumvalue.Infullerenesandschwarzites,thepresenceofnon-hexagonalrings,includingpentagonsandheptagons,preventsaglobaloptimizationofbondlengthsandbondangles,reducingtheagreementwithDFTresults.Mystabilityresultsareconsistentwiththepentagonadjacencyrulethatprovidesanenergypenaltyof0:70:9eVforeachpairofadjacentpentagons,[221{223]whichcausesanincreaseofthelocalcurvature.Whilethisruleissurelyuseful,itcannotcomparethestabilityofisomerswithisolatedpentagonsorstructuresoftsize.WhatIconsiderthemosttbofmyapproachtodeterminelocalstrainistoidentifytheleaststablesitesinastructure.Localcurvatureandin-planestrainplaythekeyroleinbothlocalstabilityandlocalelectronicstructure,[52]whichalsocontrolsthechemicalreactivity.[51,53]Thus,myapproachcanidentifythemostreactiveandtheleaststablesites,whichcontrolthestabilityoftheentiresystem.9.5SummaryInconclusion,Ihaveintroducedafastmethodtodeterminethelocalcurvaturein2Dsystemswitharbitraryshape.Thecurvatureinformation,combinedwithelasticconstantsobtainedforaplanarsystem,providesanaccurateestimateofthelocalstabilityintheframeworkof104continuumelasticitytheory.Relativestabilitiesofgraphiticstructuresincludingfullerenes,nanotori,nanotubesandschwarzites,aswellasphosphorenenanotubescalculatedusingthisapproach,agreecloselywithabinitiodensityfunctionalcalculations.Thecontinuumelasticityapproachcanbeappliedtoall2Dstructuresandisparticularlyattractiveincomplexsystemswithknownstructure,wherethequalityofparameterizedforcehasnotbeenestablished.105Chapter10Conclusions2Dmaterialshaveattractedalotofattentionsincethesuccessfulexfoliationofgraphene.Lackofafundamentalbandgapingrapheneturnedoutintoanunsurmountableobstacleforelectronicsapplications.TMDs,whichhavenonzerobandgapandarechemicalstable,havebecomethemostpromising2Dcandidatesforthesemiconductingchannelmaterials.However,theperformanceofTMD-baseddevicesisstillnotsatisfactory,mainlyduetotheirlowcarriermobility.Additionalchallengescomefromthesmallsizeofsynthesizedsamples,defects,contacts,andthelimitationsinthetuningofbandgapsinelectronicdevices.Few-layerblackphosphorushasemergedasanew,promisingmaterialin2014.Thisgrowsfastduetothenonzerobandgapandhighcarriermobilityinblackphosphorus.Anisotropictransport,tunablebandgap,andvariousstablephasesmakephosphorene,themonolayerofblackphosphorusaunique2Dmaterial.Thebiggestchallengefortheapplica-tionofphosphoreneisitscientchemicalstabilityinair.Inallphosphorenestructures,eachphosphorusatomissp3hybridized,hasthreecovalentbondsandoneloneelectronpair.ThelonepairwithenergyneartheFermileveleasilyreactswithoxygenatomsintheair.Thiswillformphosphorusoxidesorphosphoricacids,resultinginthedegradationofthephosphorenestructure.Themosttwaytoavoidthisistoprotecttheposphorenechannelsbycappingthemonbothsides.Thecappinglayerscanbethechemicallystableandinsulatinghexagonalboronnitride(h-BN),aluminumoxide,orpolymerovercoating.Sincephosphorushasseveralphaseswithnearlyequivalentstability,itmaybeto106synthesizehigh-qualitypure-phasephosphoreneonalargescale.Multi-layerphosphorenehasamuchbetterstabilitythanamono-layerandisalsoeasiertoobtainintheexperiment.Thuscurrentphosphorene-baseddevicesaremostlyusingmulti-layerphosphorene,andtheirperformancecanbeoptimizedbycontrollingthethicknessofphophorenesamples.Asstatedbefore,allmaterialsinthecurrent2D-semiconductormarkethavetheirpartic-ularadvantagesanddisadvantages.Itisthusstillanimportantissuetosearchanddesignnovel2Dsemiconductorstoaddressparticulardisadvantagesanddisplayadvancedmechan-icalandelectricalproperties.Besidestraditionallayeredblackphosphorusandtherecentlydiscoveredbluephosphorus,morestructuralphasesforlayeredphosphorushavebeenfound,includingsemiconductorsawithtunablebandgapandananisotropiccarriermobility.Sim-ilartophosphorus,allthegroupVelements,suchasarsenicandantimony,havelayeredallotropesduetotheirsimilarchemicalproperties.Even2Dnitrogenhasbeenpredictedtobestableatlowtemperatures.Inadditiontotheelementalsemiconductors,alsocom-poundsofgroupVelements,suchasAsxP1xandSbxAs1x,arepromisingcandidatesfor2Dsemiconductors,withpropertiestunablebychangingthestoichiometry.Ontheotherhand,similartoh-BNwhichisisoelectronictographeneandsharesitshoneycombstructure,groupIVandgroupVIcompounds,suchasSiS,SnSe,areisoelectronictophosphporusandmaysharesomeofitsproperties.ThesecompoundshavestablelayeredstructuressimilartogroupVelements,butcanhaveverytproperties.Itisverypossibletosolvetheproblemthatmostofthe2DgroupVstructuresarechemicallyunstablebyexploringthebehaviorofotherisoelectronicstructures.EventhoughnotisoelectronictogroupVsystems,combinationofsp2carbonandsp3phosphorusinasingle2Dstructurefurtherwidensthescopeofthe2Dstructuralfamily.Identifyingtheoptimummethodforsynthesisofallthese2Dstructureswillremainthemainchallengefortheupcoming107generationofexperimentalscientists.108 109 APPENDIX  110 APPENDIX  As mentioned in Chapter 3, there is more than one isomer for La@C60(CF3)n with n  2. To find the most stable isomers of La@C60(CF3)n, I calculated different geometries with n trifluomethyls attached on C60 in different arrangements. The results for n = 2, 3, 4 and 5 are shown in Fig. A.1, A.2, A.3 and A.4, respectively. In general, coupled with more than one CF3 prefer to be separated in a para arrangement, which can be best seen in the case of n = 2 as shown in Fig. A.1. The most stable isomer has the two CF3s attached at the para-position in one hexagon shown in Fig. A.1(a). The two CF3s are too crowded and less stable when they are closer as shown in Fig. A.1 (d), (e), (j). It is also less stable when they are two far away as shown in Fig. A.1 (b), (c), (f), (g), (h), (i). For the case of n = 3, the most stable isomer is the one with all three CF3s in para arrangement, which has a C2v symmetry as shown in Fig. A.2 (a). The second most stable one shown in Fig. A.2 (f) has a C3 symmetry, which is only 0.108 eV less stable. For n = 4 and n = 5 they do not like all the CF3s para arranged any more. As shown in Fig. A.3 (a), when there are four CF3s, they prefer to be separated to two para-arranged pairs of CF3s. The one with all four CF3s in para arrangement is slightly less stable with 0.051 eV in energy, as shown in Fig. A.3 (e). Both these most stable two isomers for n = 4 have a C2v symmetry. When n = 5, four of the five CF3s like to be in para arrangement and with the left one a little far away, as shown in Fig. A.4 (a). The one with all five CF3s is para arrangement in 0.857 eV less stable, as shown in Fig. A.4 (b).  FigureA.1:TentisomersofLa@C60(CF3)nwithn=2.Ineachball-and-stickmodeloftheoptimizedstructureisshownatthetop,therelativeDFT-PBEenergywithrespecttothemoststableisomerislistedinthemiddleandthecorrespondingSchlegeldiagramofC60withylsitesindicatedbyreddotsisshownatthebottom.111FigureA.2:SixerentisomersofLa@C60(CF3)nwithn=3.Ineachgureball-and-stickmodeloftheoptimizedstructureisshownatthetop,therelativeDFT-PBEenergywithrespecttothemoststableisomerislistedinthemiddleandthecorrespondingSchlegeldiagramofC60withylsitesindicatedbyreddotsisshownatthebottom.112FigureA.3:TentisomersofLa@C60(CF3)nwithn=4.Ineachball-and-stickmodeloftheoptimizedstructureisshownatthetop,therelativeDFT-PBEenergywithrespecttothemoststableisomerislistedinthemiddleandthecorrespondingSchlegeldiagramofC60withylsitesindicatedbyreddotsisshownatthebottom.113FigureA.4:TentisomersofLa@C60(CF3)nwithn=5.Ineachball-and-stickmodeloftheoptimizedstructureisshownatthetop,therelativeDFT-PBEenergywithrespecttothemoststableisomerislistedinthemiddleandthecorrespondingSchlegeldiagramofC60withylsitesindicatedbyreddotsisshownatthebottom.114 115 BIBLIOGRAPHY 116 BIBLIOGRAPHY   [1] H. 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