NON-EQUILIBRIUM PHASE TRANSFORMATIONS IN

CHARGE-DENSITY WAVE AND STRONGLY CORRELATED

SYSTEMS STUDIED BY COHERENT FEMTOSECOND ELECTRON

DIFFRACTION

By

Faran Zhou

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics – Doctor of Philosophy

2019

ABSTRACT

NON-EQUILIBRIUM PHASE TRANSFORMATIONS IN CHARGE-DENSITY WAVE

AND STRONGLY CORRELATED SYSTEMS STUDIED BY COHERENT

FEMTOSECOND ELECTRON DIFFRACTION

By

Faran Zhou

By photo-exciting electrons in a material causing subsequent electron-electron and electron-

phonon interactions, an energy landscape is created that is very different from that in thermal

equilibrium. This distinction sets the baseline that photo-induced phase transitions (PIPT)

could go through very different pathways compared with the thermally-induced phase tran-

sitions in equilibrium. This opens up a new dimension for complex materials research in

non-equilibrium with ultrafast tools. In recent years, with the discovery of photo-induced

superconductivity (SC) and other hidden quantum states, the study of these metastable and

hidden phases in quantum materials has drawn intense interest in the ultrafast community.

With femtosecond electron diffraction, we have observed intriguing phenomena in a few

charge-density wave (CDW) materials (CeTe3, ErTe3 and 1T-TaS2) and iron chalcogenide

systems (FeTe, FeSe0.44Te0.56 and FeSe). We explore the metastable hidden phases and

observe universal dynamics in these materials far from equilibrium.

CeTe3 exhibits uni-directional stripe CDW order in thermal equilibrium. Bi-directional

CDW is thermodynamically forbidden. After femtosecond laser pulse excitation, the system

is driven to a bi-directional order as it crosses a nonthermal fixed point. The new state

is formed through associated symmetry changes that cause softening/hardening of CDW-

related phonons. The CDW wavevector change proves that Fermi surface nesting (FSN)-

enhanced electron-phonon coupling plays a central role in driving the CDW instabilities.

Based on these results, we propose a nonthermal phase transition pathway in the non-

equilibrium phase diagram.

The work in ErTe3 is one step further based on the CeTe3 results. ErTe3 is on the opposite

side of the rare-earth tritelluride (RETe3) series to host two orthogonal CDW orders at low

temperature. Together with CeTe3 data, the ultrafast results at various temperatures in

ErTe3 indicate that the system becomes more symmetric after laser excitation. Given the

robustness of the data, the conclusion here may be extended to similar systems as well.

The generic features of CDW dynamics in 1T-TaS2 are very similar to those in the quench

dynamics of isolated quantum systems (e.g. cold atoms). After laser excitation, the system

goes to the prethermalization plateau region before the thermalization stage. We find that

both regimes follow universal scalings due to the existence of two nonthermal fixed points.

Microscopically, we propose a chiral-symmetry-breaking mechanism that mediates the phase

transformation. With a 2500 nm excitation laser, we emphasize the photo-doping, instead

of the photo-thermal effect, in driving the phase transition. Due to the lack of thermal

energy, the phase transition induced by the 2500 nm laser is more first order-like with faster

switching speeds than 800 nm excitation. This high-speed switching with little thermal

energy deposition holds promise for better future optoelectronic applications.

In FeTe, we directly observed the ultrafast structural transition by ultrafast electron

diffraction (UED) for the first time. In the studies of optimally-doped compound FeSe0.44Te0.56,

we observed the large-amplitude acoustic phonon excitations at right above the transition

temperature Tc. While experiments with better spatial-temporal resolutions are needed, the

acoustic phonon observed here might be important for SC in iron chalcogenide. In FeSe, we

observed the laser-enhanced local stress that is known to be coupled to the nematic phase

and superconductivity. Future UED experiments targeted for understanding the local stress

would be very helpful.

To my family

iv

ACKNOWLEDGMENTS

First of all, I would like to thank my advisor Prof. Chong-Yu Ruan for his guidance and

support. He taught me how to do research and how to manage a project. I don’t have much

research experience before joining the group and learned significantly from Prof. Ruan over

the years. I am grateful to him for all the guidance and encouragement.

My Ph.D. work can not be done without the collaboration and help of my lab colleagues.

Tzong-Ru Han taught me most of the things in the lab when I first joined the group and

he is also a great mentor to me in life. I also learned a lot from Zhensheng Tao and Kiseok

Chang. Great thanks to my colleague Joseph Williams who I worked with for a few years

and helped me a lot in the lab. Also thank other labmates that I worked with: Dr. Tianyin

Sun, Dr. Yingjie Li, Dr. Shuaishuai Sun, Xiaoyi Sun, Alan McLean, Anna Turnbull, Daniel

Bartles, Elliot Wozniak, and Nicholas Breslin, etc.

To my committee members who give me great advice on improving myself and finishing

the Ph.D. program: Prof. Phillip Duxbury, Prof. Norman Birge, Prof. Pengpeng Zhang,

Dr. Frederique Pellemoine. Also, I would like to thank my collaborators who contributed to

my Ph.D. work in different ways. Thanks to Prof. Mercouri G. Kanatzidis and Dr. Christos

D. Malliakas for providing us high quality samples for the experiments reported in chapters

3 to 5. Thank Prof. Zhiqiang Mao for providing the iron chalcogenides samples. Thank

Prof. Nelson Sepulveda and David Torres for providing VO2 thin film samples. Thank Prof.

Alexander F. Kemper from North Carolina State University for the collaboration in CeTe3

project. Thank Prof. Mohammad Maghrebi for the help in 1T-TaS2 project. I would also

like to thank Dr. Kai Sun from University of Michigan for his help in CeTe3 measurements.

Dr. Reza Lolloee and Dr. Baokang Bi are of great help in teaching me how to use

v

various instruments and gives very helpful advice when I come into a problem in the lab. To

the machine shop staff Tom Palazzolo, Tom Hudosn, Jim Muns and Rob Bennett for their

friendliness and patience when I seek them for help. Also, thank Dr. Xudong Fan in the

electron microscopy center for teaching me how to use TEM and giving me helpful advice.

Barry Tigner from the electronics shop is the guy that I would first think of and turn to

whenever I have anything related with electronics and he is always very patient to help. I

am very grateful to Art Camire from Spectra Physics who taught me quite a lot in laser

technology. I would also like to thank our secretaries for their support: Cathy Cords, Jessica

Cords, Kim Crosslan.

To a lot of friends who supported me along the way and made my life more interesting:

Xukun Xiang, Nan Du, Mengze Zhu, Jie Guan, Didi Luo, Xueying Huyan, Yanhao Tang,

Dan Liu, Zheng Gu, Zixi Chen, Li Ke, Jun Zhang, Jie Yang, and many others.

On my personal side, I would thank my parents, uncle, grandma, and sisters for their

unconditional love and support. I would not be me today without their consistent support

and encouragement. I also thank my love Shanshan Wang who has been always on my side

encouraging me and always has faith in me. I always owe them for not being able to be get

together with them. Thank them all for their support and understanding.

Studying at MSU is a life-changing experience for me. I am so grateful to get to know so

many colleagues and friends who have helped me along the way. Finally, sincere thank you

to all of them.

vi

TABLE OF CONTENTS

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

ix

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

x

KEY TO ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Complex quantum materials . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Landau’s theory of phase transitions
. . . . . . . . . . . . . . . . . .
1.2.2 From mean-field theory to renormalization group . . . . . . . . . . .
1.2.3

1
2
4
8
14
Systems driven far from equilibrium and photo-induced phase transitions 20
25
27
30
35
38

1.3 Charge-density wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
1.3.1 One-dimensional electron gas
1.3.2 Coupled one-dimensional electron-phonon linear chain . . . . . . . . .
1.3.3 Fluctuation effect versus mean-field model
. . . . . . . . . . . . . . .
1.3.4 Origin of charge-density wave: Fermi surface nesting and beyond . . .

Chapter 2 Diffraction theory and ultrafast electron diffraction . . . . . . .
2.1 Basic formulation of diffraction . . . . . . . . . . . . . . . . . . . . . . . . .
Scattering from a point target: atomic form factor . . . . . . . . . . .
2.1.1
2.1.2
Scattering from a crystal: Bragg’s law and Laue condition . . . . . .
2.1.3 Reciprocal space and Ewald’s sphere . . . . . . . . . . . . . . . . . .
Structure factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4
2.1.5 Diffraction from imperfect crystals
. . . . . . . . . . . . . . . . . . .
2.2 Ultrafast electron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Pump-probe technique . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Data interpretation in ultrafast electron diffraction . . . . . . . . . .

Chapter 3 Observation of an emergent bi-directional order in CeTe3 . . .
. . . . . . . . . . . . . . . . . . . . . .
3.1 Background and properties of CeTe3
3.2 Ultrafast dynamics of CeTe3 . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Mechanism for the non-equilibrium phase transition . . . . . . . . . . . . . .
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4

Symmetry breaking and phase transitions in ErTe3 . . . . . . .
4.1 Background about ErTe3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
. . . . . . . . . . . . . . . . . . . . . . .
4.3 Phase transitions at intermediate temperature . . . . . . . . . . . . . . . . .
4.4 Non-equilibrium phase diagram and summary . . . . . . . . . . . . . . . . .

Investigations at low temperatures

40
41
41
43
46
48
52
55
56
60

66
67
69
76
84

86
87
89
92
94

vii

97
Chapter 5 Dynamical scaling and phase transitions observed in 1T-TaS2
5.1 General properties of 1T-TaS2 . . . . . . . . . . . . . . . . . . . . . . . . . .
99
5.2 Phase transition mechanism of 1T-TaS2 . . . . . . . . . . . . . . . . . . . . . 104
5.2.1 Reconstruction of real-space structures . . . . . . . . . . . . . . . . . 104
5.2.2 CDW peak dynamics and domain evolution . . . . . . . . . . . . . . 107
5.2.3 General behaviors of dynamical phase transition . . . . . . . . . . . . 108
5.2.4 The Friedel oscillation and chiral symmetry breaking . . . . . . . . . 109
5.2.5 Fluence-dependent studies . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3 Universal behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3.1 Universal scaling in the prethermalization plateau region . . . . . . . 114
5.3.2 Universal scaling in the thermalization stage . . . . . . . . . . . . . . 116
5.4 Phase transitions induced by 2500 nm laser . . . . . . . . . . . . . . . . . . . 120
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Chapter 6

Study of iron chalcogenides by ultrafast electron diffraction . 131
6.1 Background and properties of iron chalcogenide compounds . . . . . . . . . . 133
. . . . . . . . . . . . . . . . . . . . . . . 136
6.2 Ultrafast phase transitions in FeTe
6.3 Ultrafast structural dynamics in FeSe0.44Te0.56
. . . . . . . . . . . . . . . . 140
6.4 Ultrafast dynamics in FeSe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Chapter 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Appendix A 2D sample preparation procedure . . . . . . . . . . . . . . . . . . . . 154
Appendix B Transfer matrix method for optical absorption calculations . . . . . . 156
Appendix C Photo diode box-car integrator (PD-BCI) system for pump power

measurement, monitor, and calibration . . . . . . . . . . . . . . . . . . . . . 164

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

viii

LIST OF TABLES

Table 1.1: Critical exponents for different models at different dimensions. References
are included in the last column for different models. For cell values that are
integer or fraction, they come from exact solutions. For cells with decimal
values, they usually come from RG based field theory methods.
. . . . . .

Table 1.2: Classification of non-equilibrium dynamical systems based on alphabetical
In the first row title line, the “No.” refers the designation of
order [1].
alphabet for each model. “D” is the dimension of the system. “N.C. fields”
represents the non-conserved fields. “Cons. fields” represents “conserved
fields”.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 1.3: Comparison of the CDW transition temperature T3D and the temperature
predicted by mean-field theory TM F in several materials. The TM F is
estimated based on CDW gap in Eq. 1.52. For CeTe3, the CDW gap is 400
meV. For 1T-TaS2 at zero temperature, there are both the CDW gap and
Mott-Hubbard gap. The CDW gap based on different measurements are
estimated to be 100-400 meV. . . . . . . . . . . . . . . . . . . . . . . . . .

Table 2.1: Comparison of several typical UED/UEM techniques and some typical pa-
rameters. The Group column just list one or two groups as examples and
definitely not a complete list. All the numbers are typical numbers and
may not represent a specific system. Ne is electron number per pulse. Ek
is the electron kinetic energy. τ is electron pulse duration. . . . . . . . . .

19

21

38

59

Table 5.1: 1T-TaS2 phases across different temperatures. Φ refers to the CDW orien-

tation angle away from the lattice plane. . . . . . . . . . . . . . . . . . . . 101

Table 6.1: Summary of the properties and experimental results for all three iron
chalcogenide compounds. “DWF” represents the relative intensity at + 40
ps of the Bragg peaks analyzed. The “Estd. ∆T (K)” means the estimated
sample temperature rise based on applied fluences and the DWF analyses. 148

Table B.1: Comparison of the optical coefficients for 50 nm-thick 1T-TaS2 by using
the regular method and transfer matrix method for 800 nm and 2500 nm
lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

ix

LIST OF FIGURES

Figure 1.1: The schematic phase diagram near the quantum critical point. Adapted
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

from Ref. [2].

Figure 1.2: A generic phase diagram for high-Tc superconductors. . . . . . . . . . . .

Figure 1.3: The profiles of free energy as a function of the order parameter at different
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a2 values.

Figure 1.4: The order parameter Ψ evolution as a function of temperature T , at both
below and above Tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.5: The magnetic susceptibility χ evolution as a function of temperature near
the critical point. On both sides of Tc, χ is inversely proportional to the
distance from Tc and diverges at Tc.
. . . . . . . . . . . . . . . . . . . .

Figure 1.6: The order parameter evolution as a function of temperature for first-order
phase transition near Tc. Cooling from high temperature, the system
follows the red curve until T = Tc where it becomes unstable and phase
transition will happen. The system jumps to one of the blue curves. If the
system is heated from low temperature, then the system would remain on
the blue curve until T = T1, after which it jumps to the red curve and
phase transition occur. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.7: The evolution of the exciton density observed in photo-excited carbon
nanotubes [3]. In the log-log plot, the system switches from a - 1 power-law
decay to a - 1/2 exponent, indicating a transition from the reaction-limited
regime (RLR) to diffusion-limited regime (DLR).
. . . . . . . . . . . . .

4

5

10

11

12

14

25

Figure 1.8: The electron susceptibility for 1D, 2D, and 3D systems at zero temperature. 28

Figure 1.9: Examples of Fermi surface nesting for different Fermi surface topologies.
a, 1D electron gas, perfect FSN occurs. b, 2D free electron gas. There is
no “parallel” parts in the circular FS, so no FSN. c, FS of a 2D square
lattice. The diamond shape of FS results in relatively good FSN.
. . . .

Figure 1.10: The phonon softening effect when the temperature is close to T M F

CDW . It
. . . . . . . . . . . . . . . . .

becomes frozen phonon when T = T M F

CDW .

x

29

32

Figure 1.11: An example of CDW formation in a half-filled 1D linear chain. a, Metallic
state. Charge-density is uniform and atomic ions are undistorted. Elec-
trons fill up to half of the band and it is a metal. b, CDW state. The
charge-density re-modulate and atomic ions distort every pair. The band
opens up a gap at ± kF .
. . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Figure 1.12: The temperature-dependent correlation length of the parallel and perpen-
dicular to the chains as well as single 1D chain correlation length. T3D is
the real transition temperature in materials, below which is CDW state.
TM F is the mean-field transition temperature based on mean-field models. 37

Figure 2.1: Diagram of the momentum transfer in a single atom scattering event. ki
and kf are the incident and scattered wave-vector, respectively, defined
as ki = 2π/λi and kf = 2π/λf , where λi and λf are the incident and
scattered wavelengths of the probe particles. s is the momentum transfer.
θ is the scattering angle, θ/2 is equal to the so-called Bragg angle θb in
the Bragg’s law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.2: The atomic form factor fel(s) evolutions for electron scattering at different
s and for different atoms [4]. The atomic form factor decreases with the
increasing scattering angle and increases with the increasing target atomic
number.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.3: The schematic drawing of scattering diagram for the Laue condition (a)
. . . . . . . . . . . . . . . . . . . . . . . . . .

and the Bragg’s Law (b).

Figure 2.4: The Ewald’s sphere intersecting with reciprocal lattice points. a, Ewald’s
sphere for a square 2D lattice. b, A more realistic scale for the Ewald’s
sphere and reciprocal spots.
. . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.5: The intensities of Bragg peak and satellite peak for a dimerized 1D atomic
chain. a, Dimerization of a linear atomic chain. b, Bragg peak and satel-
lite peak intensity evolutions as a function of relative distortion amplitude
u/a.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.6: Timescales of different electron and lattice processes after laser excitation
[5]. Each green bar represents the typical time scales and may vary for
different materials and different excitation conditions. . . . . . . . . . . .

42

44

45

47

51

56

xi

Figure 2.7: The schematic drawing of ultrafast electron diffraction experimental setup.
Near-infrared 800 nm laser pulse is split into two paths: one goes through
an optical delay stage to serve as a pump pulse, the other goes through
third-harmonic generation (THG) to generate ultraviolet (UV) laser pulse
for electron pulse generation. Of course, this is just a way simplified
schematic drawing. Electron gun, sample, and CCD camera have to be in
ultra-high vacuum (UHV) chamber. The electron optics, such as magnetic
lens and radio-frequency (RF) cavity, are also omitted out in here. . . . .

Figure 2.8: The laser-induced lattice constant change in graphite captured by ultrafast
electron diffraction [6]. a, Experimental geometry. b, Lattice constant c
change and the excitation of the breathing mode phonon. . . . . . . . . .

Figure 2.9: MeV ultrafast electron diffraction experiment on Weyl semimetal WTe2
[7]. a, Lattice structure of layered WTe2. b, Experimental setup. THz
pump, MeV electron probe is employed in the experiment. c, Responses of
different Bragg peaks. d, Schematic drawing of the energy potential with
changes of the shear distortion. e, The time-dependent shear distortion
. . . . . . . . . . . . . . . . . . . . . . .
at two different pump fluences.

Figure 2.10: The Debye-Waller effect and electron-phonon coupling analyses in poly-
crystalline copper [8]. a, MeV UED pump-probe experimental setup. b,
The Debye-Waller analysis of Bragg peaks at different q-values and differ-
ent time delays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.11: The correlation length analysis in a phase transition in CDW material
1T-TaS2 [9]. a, The ultrafast low-energy electron diffraction setup. b,
The growth of correlation length follows a 1/2 power-law scaling growth.

Figure 2.12: The diffusive scattering background phonon analyses in graphite [10]. a,
Original diffraction pattern of graphite with electron beam along the c
axis. (b-f ), The difference of diffraction patterns at different time delays
showing diffusive scattering background evolution. g, The dynamics of
Bragg peak (2 1 0). h, The dynamics of diffusive scattering background
at different points of the Brillouin zone, corresponding to different phonon
branches in graphite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.1: Structure, FS, and CDW formation in the RETe3 series. a, Ball-stick
model of RETe3 structure. The solid cuboid is the unit cell with a, b, c
labeled. b, Top view of the square net Te layer in which CDW is formed.
c, The electron orbitals in the Te layer are 5px and 5pz orbitals from Te
atoms. d, The schematic FS of the square Te net. Dashed line is the
BZ boundary. e, The transition temperature of different RE elements. f,
Freestanding thin sample flakes on 1000-mesh gold TEM grid.
. . . . . .

58

61

62

63

64

65

68

xii

Figure 3.2: The diffraction patterns of CeTe3. a, Ground state full diffraction pattern
of CeTe3 at RT. b, Zoom-in view of the cyan square box in a. The Bragg
peaks are labeled and Bragg and CDW wave-vectors are also indicated. c,
Zoom-in view of the black dashed square box in b at negative time delay.
d, The same area as in c, but at + 1 ps after laser excitation. c and d are
in the same scale as indicated by the scale bar on the left. New emergent
CDW satellite peaks are clearly observed along the a∗ axis (a-CDW). . .

Figure 3.3: The integrated intensity changes of c- and a-CDW peaks. Both c- and a-
CDW peak intensities are normalized to the c-CDW intensity at negative
time delay. Inset is the difference diffraction pattern: diffraction pattern
. . . . . . . . . . . . . . . . . . . . . . . .
at + 1 ps minus that of - 1 ps.

Figure 3.4: The pseudo-Voigt function peak fitting for extracting the correlation length,
static, and fluctuating components of a peak. a, c-CDW peak at nega-
tive time where the fluctuation is small, so a single Voigt function fits it
well. Neither a Gaussian nor a Lorentzian would fit. b, Voigt function
fitting of several CDW peak profiles. The sharpest black curve is a Bragg
peak with Gaussian fitting and others are by Voigt function fitting. c,
c-CDW peak at + 1 ps, at which a single Voigt function (black curve)
does not fit. We have to use two Voigt functions to do the fitting and
the two components are in dashed lines corresponding to the static and
fluctuating/critical components. . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.5: Analyses of the fluctuating order and real space reconstruction. a, Schematic

demonstration of the static and fluctuating orders in correlation length
and population near the thermal critical point. b, Dynamics of intensity
evolution. Top panel is the intensity evolution of peak shoulder (δq = 0.07)
and peak center. Inset are images before and after to show the suppres-
sion of c-CDW and appearance of a-CDW. The bottom panel dynamics
shows the integrated intensity of c-CDW static portion and a-CDW. In-
set shows the linear relation of correlation length and CDW intensity. c,
Peak profiles and 2-Voigt function fitting. Pie chart shows the population
change of each component: blue is c-CDW static part, pink is c-CDW
fluctuating order, green is a-CDW portion. d, Real space reconstruction
based on diffraction patterns, check text for details.
. . . . . . . . . . .

70

71

73

74

Figure 3.6: Comparison of the experimental pattern with simulated pattern. . . . . .

76

xiii

Figure 3.7: The fluence-dependent data and evolution of topological defects. a, c-
CDW intensity dynamics at different pump fluences. Inset: the charac-
teristic time of the recovery time for c-CDW order at different fluences.
b, The extracted c-CDW and a-CDW amplitude evolution from fluence
scan. The corresponding c- and a-CDW thresholds are indicated. c, Dy-
(a)
namics at two fluences above F
. Inset: evolution of topological defects
c
at F = 1.85 mJ/cm2.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Figure 3.8: Simulations of the quench profile and the Kibble-Zurek freeze-out time. .

78

Figure 3.9: The q-dependent Debye-Waller analysis and FSN wave-vector evolution.
a, Three representative Bragg peaks dynamics at the first 10ps.
Inset:
the schematic diffraction pattern of CeTe3. Bragg peaks in red-circle in
similar s value are used for Debye-Waller analysis in b. b, the q-dependent
Debye-Waller analysis at different time delays. c, a- and c-CDWs FS
nesting vector changes as a function of time. d, Schematic FS structure
of the square net Te layer (solid black line). Square gray dashed line
is the BZ boundary, dashed orange lines are projected FS change after
laser excitation, drawn based on the CDW nesting vector changes in c. e,
Theoretical calculations of the susceptibility (χ) and nesting wave-vector.
f, Peak position of the electronic susceptibility along the (11) direction .

Figure 3.10: Static TEM measurements of CeTe3, a, Full TEM diffraction patterns at
RT. b, Cropped diffraction pattern at several temperatures at fixed scale.
c, c-CDW peak intensity evolution normalized by Bragg peak intensity.
d, The CDW wave-vector evolution towards transition temperature. e,
Line cut along c-CDW peak at three characteristic temperatures. f, The
same line cut along the potential a-CDW peak position.
. . . . . . . . .

80

83

Figure 3.11: The non-equilibrium phase diagram and phase transition pathways of CeTe3. 85

Figure 4.1: The diffraction patterns of ErTe3. a&b, Full diffraction patterns at 40 K
and 180 K. c&d, Zoom-in view in a and b. The red circles are c-CDW
peaks and green circles are a-CDW peaks. At 40 K, there are both a-
and c-CDW orders while at 180 K there is only CDW order along the c∗
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
direction.

Figure 4.2: The CDW (a) and Bragg (b) peak dynamics at base temperature of 50 K
and pump fluence of 2.9 mJ/cm2. The a-CDW dynamics data in the first
5 ps are 2-binned for better s/n. . . . . . . . . . . . . . . . . . . . . . . .

88

89

xiv

Figure 4.3: The fluence-dependence of the a- and c-CDW peaks relative intensities
at 25 K. The a-CDW data are averaged every two points (binning of 2)
for better s/n. The data points for 50 K (2.9 mJ/cm2) and 100 K (2.5
mJ/cm2) are extracted from the delay scan dynamics.
. . . . . . . . . .

Figure 4.4: The a- and c-CDW peaks dynamics at 100 K. Data points for a-CDW
are binned by 2 for better s/n. . . . . . . . . . . . . . . . . . . . . . . . .

90

91

Figure 4.5: The dynamics of CDW (a) and Bragg (b) peaks at 165 K.

. . . . . . . .

92

Figure 4.6: The a- and c-CDWs satellite peaks dynamics at 200 K. . . . . . . . . . .

93

Figure 4.7: The schematic phase diagram of ErTe3 under non-equilibrium conditions.
The phase diagram is based on dynamics and fluence-dependent data at
various temperatures. The thin gray line shaded-region represents the
.
uncertainties of determining the a-CDW suppression phase boundaries.

Figure 5.1: The structure of 1T-TaS2. a, 3D view of the structure. Blue balls are
Tantalum atoms and yellow represents Sulfur atom as labeled. The red
box represents the unit cell. b, Side view. c, Top view. d, Ta layer
structure when forming commensurate CDW. Every 13 atoms form Star-
of-David supercell with surrounding 12 atoms moving inwards. The CDW
superlattice is oriented 13.9◦ relative to the original lattice. . . . . . . . .

95

99

Figure 5.2: The optical images of the sample and electron diffraction patterns. a,
Freestanding sample flakes on 1000 mesh gold TEM grid. b, Zoom-in view
of a thin sample piece used for experiment. c, The diffraction pattern of
1T-TaS2 by 30 keV electron beam at 150 K C-CDW phase. d, Zoom-in
view of the diffraction pattern in c. The relative angle between the CDW
q -vector and the Bragg plane is 13.9◦.

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

Figure 5.3: The diffraction patterns of 1T-TaS2 in the NC-CDW phase. a, The orig-
inal diffraction pattern in NC-CDW state. b, The diffraction pattern in
IC-CDW state. c, A schematic diffraction pattern of the NC-CDW phase.
Pattern is in the same orientation as that in a. The 1st, 2nd, and 3rd CDW
peaks are labeled as q(1), q(2), q(3), respectively, with each have two peaks
differentiated in the subscript by “1” and “2”. d, The difference diffrac-
tion pattern of after laser excitation (IC-CDW phase) minus the pattern
before laser excitation (NC-CDW phase). . . . . . . . . . . . . . . . . . . 103

xv

Figure 5.4: The dynamics of different order parameters after laser excitation. a, The
dynamics of the (first-order) NC and IC peak intensities at fluence of
1.73 mJ/cm2. Shaded area represents the correlation length of NC and
IC order, fitted in the transverse (T) and longitudinal (L) direction. b,
The dynamics of the 1st (red), 2nd (green), and 3rd (blue)-order CDW
Inset shows the ratio changes of the 2nd− and 3rd-order peak
peaks.
normalized by the 1st-order peak intensity. c, The reconstructed CDW
intensity map at several critical time delays. d, The 1st-order CDW peak
radius changes normalized by Bragg peak distance. e, The dynamics of
1st-order CDW peak angle relative to Bragg plane.
f, The NC-CDW
domain size-dependence on the NC-CDW angle changes.

. . . . . . . . . 107

Figure 5.5: An example of the Friedel oscillation in vanadium-doped blue bronze.

Adapted from Ref. [11, 12].

. . . . . . . . . . . . . . . . . . . . . . . . . 110

Figure 5.6: The chiral symmetry breaking and Friedel oscillations. a, The diffraction
patterns at 3 critical time delays. The pattern at + 1.5 ps corresponds to
the chiral symmetry breaking and FO region (X state). b, Phase modula-
tion imposed on the reconstruction process due to FO. On the right shows
the fit of the reconstructed pattern to experimental data. k = q − Q1
N C .
c, The reconstructed atomic displacement map. The dark spots repre-
sent the loss of atomic density (atoms moving away) and the white spots
represent an increase in atomic density (atoms moving towards that spot). 111

Figure 5.7: The dynamics of NC and IC peaks at various fluences. a, The NC peak
intensity dynamics. The gray line represents the end of prethermaliza-
tion and the beginning of thermalization stage. b, The IC peak intensity
dynamics. c, The NC peak intensity changes at several critical time de-
lays. d, The normalized IC change at several time delays. Thresholds of
Fc,1, Fc,2 are as marked in the figure. . . . . . . . . . . . . . . . . . . . . 113

Figure 5.8: The universal scaling dynamics in the prethermalization regime (before
tpre) for NC and IC. The horizontal time axis is scaled based on power-
law with exponent of κ while the vertical axis is scaled with power-law
exponent of η. The r and rREF refer to the distance from the critical
threshold r = F − Fc and the distance from the critical threshold for the
reference curve, respectively. The reference curves are chosen as the first
curve above the thresholds Fc,1 and Fc,2. The curves between Fc,1 and
Fc,2 are scaled to the reference curve of F = 1.05 mJ/cm2; the curves
above Fc,2 are scaled to the reference curve at fluence of F = 1.59 mJ/cm2.115

xvi

Figure 5.9: The dynamical scaling of IC peak dynamics and IC phase evolution. a,
The IC peak profile evolution at different time delays. The profile line
scan is from Bragg peak (O point in the lower left) to IC peak (QIC ) in
the radial direction outwards. Intensity at each time frame is normalized
by IC peak to highlight the sharpening of IC peak. b, Schematic IC peak
profile at two different time delays to illustrate the scaling in peak width
Wq and peak height Aq. c, The power scaling of IC peak width and peak
height scaling across several orders of time evolution (gray shaded area).
Pump fluence is 1.73 mJ/cm2. d, The fluence-dependence of the scaling
exponents α and β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Figure 5.10: The universal scaling of IC peak dynamics across all fluences. a, Un-
scaled IC dynamics after prethermalization. Horizontal time is delay after
prethermalization, t − tpre. Vertical scale is intensity above prethermal-
ization, I(t)− I(tpre). b, The fluence-dependence of diffusion coefficients.
Dξ is from correlation length fitting and Dm is from IC universal scaling
in the thermalization stage. c, IC universal dynamics scaled by diffu-
sion coefficient D ξ. In the horizontal axis notation, t2 = t − tpre. d, IC
universal dynamics scaled by diffusion coefficient Dm.

. . . . . . . . . . 119

Figure 5.11: The schematic drawing of phase transitions by 2500 nm and 800 nm lasers.
a, Phase switching between NC- and IC-CDWs by means of either photo-
doping or photo-thermal effect. b, The schematic drawing of density of
states (DOS) near the FS for 1T-TaS2 at RT, adapted from Ref. [13]. c,
The schematic drawing of the transition from NC-CDW to IC-CDW via
chiral symmetry breaking.

. . . . . . . . . . . . . . . . . . . . . . . . . . 121

Figure 5.12: The fluence-dependent data of 2500 nm dynamics and thresholds compar-
ison. a, The NC-CDW peak intensity evolution at various pump fluences
by 2500 nm laser. b, IC-CDW peak growth at various pump fluence, the
same color code as a. c, Comparison of the IC formation time for 800 nm
and 2500 nm and determination of photo-doping, photo-thermal thresholds.122

Figure 5.13: The CDW and Bragg peak dynamics at ρhν ≈ 0.75 nm−3 for both 800 nm

and 2500 nm pump excitation. a, NC and IC peak intensity evolution.
b, The Bragg peak dynamics comparison and corresponding fit guidelines
for the fit model.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Figure 5.14: The CDW peak dynamics at excitation density of ρhν = 0.9 and 1.0 nm−3

for 800 nm and 2500 nm lasers, respectively.

. . . . . . . . . . . . . . . . 128

Figure 5.15: The comparison of IC formation for the two excitation lasers. a, IC rela-
tive intensity at 10 ps and 1 ns by 2500 nm laser excitation. b, Comparison
of IC peak relative intensity at 10 ps with 800 nm and 2500 nm excitation. 129

xvii

Figure 6.1: The structures of FeTe/FeSe in the high temperature tetragonal phase. a,
3D view of the ball-stick model structures. Cuboid with solid line repre-
sents the unit cell. b&c, Side and top view of FeTe/FeSe structure, respec-
tively. d, Lattice constant changes across the tetragonal-orthorhombic
structural transition for FeSe0.92, reprinted from Ref. [14].

. . . . . . . . 134

Figure 6.2: The phase diagrams of iron chalcogenides. a, The phase diagram of
FeTe1−xSex, figure from Ref. [15]. b, Neutron scattering measurement
of FeTe to demonstrate simultaneous emergence of AFM order and struc-
tural transition, figure from Ref. [16]. c, The pressure-dependence of Se
height and superconducting temperature Tc in FeSe. d, The anion height
dependence of Tc of typical Fe-SC materials. Figure is reprinted from
Ref. [15]. The overall trend indicates there exists an optimal anion height
for the highest Tc.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Figure 6.3: The structure and diffraction patterns of FeTe in the RT tetragonal and
low-temperature monoclinic phase. a, Top view of the RT tetragonal
structure at 23 K. Lattice constants a and b expand by a factor of ≈√
structure. Lattice constants a = b = 3.822 ˚A. b, Top view of monoclinic
2,
with a = 5.418 ˚A, b = 5.350 ˚A due to unit cell doubling. c, FeTe lattice
constant changes across the structural transition at ≈ 72 K [16]. d&e,
Full diffraction pattern of FeTe at RT and 23 K, respectively. Electron
diffraction pattern of FeTe at RT. 30 keV electron beam is along the [0 0 1]
zone axis. The labeling is based on a∗ = 2π/a and b∗ = 2π/b, where a and
b are the tetragonal lattice constants. f, The difference diffraction pattern
of RT minus that of 23 K. Both patterns are acquired at the same beam
conditions and exposure times. The peak shift and sharpening/broadening
can be clearly visible from the difference image.

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

Figure 6.4: The time evolution of FeTe lattice constants and peak width at base tem-
perature of 23 K (sample a, 2500 nm pump laser, 2.7 mJ/cm2). a, The
lattice constant changes along a and b directions, extracted from (4 0 0)
and (0 4 0) peak movement relative to the center (0 0 0) peak. b, c&d,
Peak sharpening after laser excitation with Gaussian profile fit for peak
(-4 0 0), (0 4 0), and (-2 2 0), respectively. The width σ here refers to σ
of a Gaussian fitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

xviii

Figure 6.5: FeTe peak intensity evolution at low and high temperatures. a, Four
groups of peaks evolution in sample a (λ = 2500 nm, F = 2.7 mJ/cm2).
b, Peak notation of 4 different groups in an artificial diffraction pattern.
c, The oscillations observed in another two samples at low temperature,
with a period of 60 ps instead of 40 ps seen in sample a. Experimental
for sample b: λ = 2500 nm, F = 2.3 mJ/cm2; for sample c:
conditions:
λ = 606 nm, F = 1.4 mJ/cm2. d, The peak intensity evolution of sample
b at 225 K, λ = 2500 nm, F = 1.8 mJ/cm2. No oscillations are observed
at high temperature.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Figure 6.6: Diffraction patterns of FeSe0.44Te0.56. a, Original diffraction pattern of
FeSe0.44Te0.56 at 23 K. b, Zoom-in view of pattern in a. c, Difference
diffraction pattern of positive time (+ 40 ps) minus negative time (- 20 ps)
under 2500 nm laser pumping at F = 1.1 mJ/cm2. . . . . . . . . . . . . . 142

Figure 6.7: The peak dynamics of FeSe0.44Te0.56 at low and high temperatures (λ = 2500

nm, F = 1.1 mJ/cm2). a, The lattice constant changes after laser exci-
tation at base temperature of 23 K. b, The peak width changes at 23
K. c, The intensity oscillations observed at 23 K. d, The peak intensity
dynamics at base temperature of 225 K. No oscillations are observed.

. . 143

Figure 6.8: The diffraction patterns and dynamics of FeSe at 23 K (2500 nm pump
laser, F = 1.1 mJ/cm2). a, The diffraction pattern of FeSe at 23 K. b,
Lattice constant change of FeSe at 23 K, a and b axes are similar: no
clear lattice constant changes are observed. c, The peak broadening after
laser excitation observed at 23 K. d, The peak intensity change at 23 K.

145

Figure 6.9: The dynamics of FeSe at 225 K (λ = 2500 nm, F = 1.1 mJ/cm2). a, Peak
intensity evolution at base temperature of 225 K. b, Peak shift along a∗
and b∗ direction. c&d, Snapshots of different images at - 10 ps and 0 ps
(both minus the diffraction pattern at - 100 ps). All peaks exhibit similar
drifts in amplitude and phase, mainly along b∗ direction. e, The sample
grid and zoom-in view of the sample used for the experiments. . . . . . . 146

Figure A.1: Typical sample images during sample preparation process. a, 1T-TaS2
sample on scotch tape. Light is reflected from sample surface so we see
the gold metallic color. b, Typical 1T-TaS2 samples on scotch tape after
a few times of exfoliation. Light is transmitting through the sample to
show the transparency. c, Sample flakes on silicon wafer. d, Typical
optical image of the samples transferred to the TEM grid.

. . . . . . . . 155

Figure B.1: The schematic drawing of N -layer sample. The electric field amplitude is

labeled at each interface for both forward and backward waves.

. . . . . 158

xix

Figure B.2: The schematic drawing of the wave propagation in (a) 1T-TaS2 and (b)

VO2 film deposited on 9 nm amorphous silicon film. . . . . . . . . . . . . 160

Figure B.3: 1T-TaS2 optical reflection, transmission, and absorption coefficients as
a function of sample thickness for 800 nm (a) and 2500 nm (b) lasers.
The pump laser is 45◦ incident on the sample and p-polarized for the
calculation, the same as experimental conditions.

. . . . . . . . . . . . . 162

Figure B.4: The optical reflection, transmission, and absorption coefficients as a func-
tion of VO2 film thickness for 800 nm (a) and 2500 nm (b) lasers. Inset of
b are the same data but plotted up to 4 um thick. VO2 film is deposited
on 9 nm amorphous silicon film. Laser pulse is 45◦ incident on the sample
and p-polarized for the calculation, the same as experimental conditions.

163

Figure C.1: The photo diode-box car integrator system schematic drawing and signal
wave forms for input/output signals. a, The schematic drawing for the
box-car integrator system. The pump laser is split to feed into a photo
diode. The four input/output signals and three main adjusting knobs are
drawn. b, The wave forms of input/output signals . . . . . . . . . . . . . 166

Figure C.2: Applications of the PD-BCI system in UED experiments. a, Calibration
of BCI output voltage relative to pump power. b, Delay scan of 1T-
TaS2 IC peak dynamics with PD-BCI system calibration with 2500 nm
pump. The arrows indicate strong correlation between the data noise and
pump power fluctuations. c, The fluence-dependent data of 1T-TaS2 NC-
CDW peak changes at several critical delays before calibration. Horizontal
labeled power is measured by power meter. The few black circles are
“noisy/jumping” data points before calibration. d, Fluence scan data the
same as that in c but after power calibration with the PD-BCI system.

. 168

xx

KEY TO ABBREVIATIONS

Here are some abbreviations and acronyms that appear in this dissertation. The full name is

shown in the first appearance of each abbreviation. For a better reading experience and ease

of reference, the list of abbreviations used in this work are summarized below in alphabetical

order.

AFM . . . . . . . . Anti-ferromagnetic

ARPES. . . . . . Angle-resolved photoemission

BCS . . . . . . . . . Bardeen-Cooper-Schrieffer

BS . . . . . . . . . . Beam splitter

BZ . . . . . . . . . . Brillouin zone

CDW. .. . . . . . Charge-density wave

DWF . . . . . . . . Debye-Waller factor

EELS . . . . . . . Electron energy loss spectroscopy

FEL . . . . . . . . . Free electron laser

FO . . . . . . . . . . Friedel oscillation

fs . . . . . . . . . . . . Femtosecond

FS . . . . . . . . . . Fermi surface

FT . . . . . . . . . . Fourier transform

FWHM. . . . . . Full width at half maximum

IC . . . . . . . . . . . Incommensurate

IMT . . . . . . . . . Insulator to metal transition

KZM . . . . . . . . Kibble-Zurek mechanism

MFT . . . . . . . . Mean field theory

xxi

NC . . . . . . . . Nearly-commensurate

PIPT. . . . . . Photo-induced phase transition

PD . . . . . . . . Photo diode

PT . . . . . . . . Phase transition

QCP . . . . . . Quantum critical point

RF . . . . . . . . Radio-frequency

rms . . . . . . . Root mean square

RG . . . . . . . . Renormalization group

RT . . . . . . . . Room temperature

SC . . . . . . . . Superconductivity

SSB . . . . . . . Spontaneous symmetry breaking

Tc . . . . . . . . Critical temperature

TEM . . . . . . Transmission electron microscopy

THG . . . . . . Third harmonic generation

TMD. . . . . . Transition metal dichalcogenides

UED . . . . . . Ultrafast electron diffraction

UEM. . . . . . Ultrafast electron microscopy

UHV . . . . . . Ultra-high vacuum

WRT. . . . . . With respect to

xxii

Chapter 1

Introduction

In modern physics, nearly all physical processes and phenomena can be explained under

the framework of quantum mechanics. Although in many cases quantum effects can be

approximated to classical descriptions on the macroscopic level, in recent years, more and

more materials and exotic phenomena are discovered to exhibit effects that might only be

explained quantum mechanically, such as unconventional superconductors, topological insu-

lators, quantum spin liquids, etc. In many complex quantum materials, often referred to as

strongly correlated electron systems, the interplays between charge, spin, orbital, and lat-

tice degrees of freedom result in multiple competing ground states such as anti-ferromagnet,

superconductor, and density waves.

By tuning the external parameters, phase transitions between different quantum phases

can be realized. Instead of just tuning the system’s temperature that usually causes ther-

mal fluctuations, one can also manipulate the system in a nonthermal fashion by applying

external pressure, magnetic field, or changing the electron density distribution by doping.

Ultrafast excitation of the electron systems by intense femtosecond (fs) laser pulses could

be another way of driving the system nonthermally under non-equilibrium conditions. In

the past 3 decades, fs lasers have been developed to cover almost all the energy scales com-

parable to physical processes in condensed matter physics, varying from milli-electron-volt

(meV) teraherz wave generation to tens of keV hard X-ray in the free-electron laser facili-

1

ties. The ultrafast laser holds a special position in studying quantum materials in that it can

selectively excite certain modes or subsystems and also track, in real time, the subsequent

order parameter evolution through the associated laser-based probes.

The work presented here focuses on understanding phase transitions between different

symmetry-broken ground states in quantum materials and exploring the phase diagrams

under non-equilibrium conditions.

In these studies, we excite the systems with fs laser

pulses and track the structural order parameter changes and associated instabilities by the

ultrafast electron diffraction method.

In this chapter, the background and an overview of the complex quantum materials are

given in Sec. 1.1. In Sec. 1.2, phase transition is first introduced based on simple Landau’s

theory. Then the development of renormalization group theory to address the limitations

of the mean-field theory is discussed. We will also describe phase transitions under non-

equilibrium, especially on photo induced phase transitions. In Sec. 1.3, a general theoretical

description of charge-density wave (CDW) is formulated based on mean-field theory in the

weak coupling limit.

1.1 Complex quantum materials

In condensed matter physics, complex quantum materials are a class of materials in which

quantum effects are non-negligible due to strong electron-electron, electron-phonon, or spin-

orbital interactions [17]. They include, but are not limited to superconductivity, ferro- and

antiferromagnetism, charge- and spin-density waves, topological insulators, Weyl semimetal,

Mott insulators, etc.

Due to the intricate interactions between different degrees of freedom in quantum mate-

2

rials, perturbations can suppress or enhance certain interactions or order. Above a certain

threshold, phase transitions would occur between different quantum states. These occur

via tuning of the control parameter and the phase transition could be either thermal (i.e.

via temperature change) or nonthermal (e.g. through externally applied pressure, magnetic

field, or doping).

For a thermally-induced phase transition, it is driven by the competitions between the

interaction energies of the system and the entropy of its thermal fluctuations. By contrast,

a nonthermal phase transition is achieved by varying other external parameters instead of

temperature. More interestingly, there is a type of transition called quantum phase transition

that happens at absolute zero temperature. Under this condition, thermal fluctuations are

completely absent and the transition is purely driven by quantum fluctuations, which follow

explicitly the Heisenberg uncertainty principle [2,17]. In the quantum phase transition, there

is usually a interaction-mediated threshold at which the phase transition happens and this

threshold at zero temperature is called the quantum critical point (QCP). Of course, the

QCP is not accessible experimentally because it is at absolute zero temperature. But its

influence extends way above zero temperature to a region above it usually referred to as the

quantum critical region.

Fig. 1.1 provides an example schematic drawing of the phase diagram near a QCP [2]. The

horizontal axis r is the nonthermal control parameter and the vertical axis is temperature.

The solid line crosses the vertical axis at the transition temperature in a classical phase

transition. With increasing r, the transition temperature is suppressed until all the way

down to zero temperature at the critical threshold rc at QCP. Above the QCP, the shaded

area is the quantum critical region whose character is governed by both thermal and quantum

fluctuations. It is widely believed that the quantum critical region and QCP hold the key

3

Figure 1.1: The schematic phase diagram near the quantum critical point. Adapted from
Ref. [2].

to understanding these complex quantum materials.

Fig. 1.2 is another example of a general phase diagram of quantum materials, hosting

high-temperature (high-Tc) superconductivity, adapted from Ref. [18]. In a phase diagram,

for high-Tc cuprate superconductors, the unknown phase X usually exhibits the pseudogap

character that competes with superconductivity.

1.2 Phase transitions

A phase transition is a transformation of a thermodynamic system from one state to another.

The discontinuous jump in the thermodynamic behavior is indicative of a phase transition.

Classification of phase transitions is the following: when there is a discontinuity in the first

order derivative of the Gibbs free energy (i.e. a jump in entropy), it is called a first order

4

Figure 1.2: A generic phase diagram for high-Tc superconductors.

phase transition; when there is a discontinuity in the second order derivative of the Gibbs

free energy (i.e. entropy is continuous but a jump in specific heat), it is called a second order

phase transition, or a continuous phase transition. The position in the phase diagram where

a continuous phase transition occurs is called the critical point.

Condensed matter physicists are dealing with systems that have atom numbers on the

order of 1023. Such complicated many-body systems consist of many small individuals

interacting with each other. The many-body problem is generally very difficult to solve

precisely except in some special cases like the one-dimensional (1D) Ising model. With

mean-field theory, or mean-field approximations, the interactions from all other individuals

on a specific individual are approximated by an averaged effect, thus reducing the many-body

problem to a single-body problem. In condensed matter physics, mean-field theory (MFT)

5

is one of the most successful ideas that has applications in many contexts, for example the

Ising model, the Landau’s theory of phase transitions, and the Bardeen–Cooper–Schrieffer

(BCS) theory.

Phase transitions are fascinating in that they involve changes in the atomic arrange-

ments and also changes of the electronic parameters such as electronic density or magnetic

moment. Ever since the development of thermodynamics, researchers have been trying to

theoretically formulate, describe, or even predict phase transitions.

In the late 19th and

early 20th centuries, pioneering work on phase transitions was done by Johannes van der

Waals, Pierre Curie, James Clerk Maxwell, Paul Ehrenfest, Ernst Ising, etc., all under the

framework of MFT. Their work was typically related to transitions such as the gas-liquid

transition, or the magnetic transition. In 1937 [19,20], Landau realized many kinds of phase

transitions are just manifestations of broken symmetries. He used the concept of order pa-

rameter to describe the extent of symmetry breaking: the order parameter is non-zero in

the symmetry-broken phase and zero otherwise.

As we will show later on, Landau’s theory is essentially a mean-field model, meaning

that it treats the system as spatially uniform and no fluctuations exist. It was designed to

predict behaviors near the critical point based on a free energy landscape, but ironically,

near the critical point, fluctuations diverge and dominate the behavior of phase transitions.

So Landau’s theory of phase transitions can provide a qualitative but numerically inaccurate

description near the critical point.

It fails to explain the lack of phase transition in a

finite system, or an infinite but 1D system. For example, independent of the dimensions

of the systems or the interaction range, MFT always predicts a 1/2 critical exponent in

order parameter change, which also contradicts many experimental observations. On the

other hand, Landau’s theory is mathematically simple and explains the general scaling and

6

universal behaviors of the seemingly very different types of transitions. It is typically the

starting point in most textbooks that explain phase transitions.

Landau’s theory does not specify which dimension it should be applied to. But as we

know, the dimension of a system affects the number of the nearest neighbors. With increasing

dimension, each particle has more nearest neighbors, thus more interaction constraints, so

it is easier for them to line up. Effectively speaking, there is less freedom for fluctuations

to play a role. Ginzburg’s calculations show that if the dimension is below an upper critical

dimension dc, phase transition near the critical point is dominated by fluctuations and mean-

field theory fails to predict correct critical exponents, which is a power-law evolution during

phase transition. But when the dimension is greater than dc, mean-field theory is self-

consistent and predicts the precise scaling behavior. This is called the Ginzburg criterion.

See Table 1.1, for the Ising model, O(n) model, and directed percolation model, the upper

critical dimension dc is 4 while for standard percolation d c is 6.

To better understand and predict the scaling and universal behaviors near a phase tran-

sition, different approaches were tried and breakthrough happened in 1971 [21, 22], when

Kenneth Wilson came up with an approach called renormalization group, in which he in-

cluded many parameters and coupling terms and performed repeated renormalization trans-

formations until a fixed point was reached. This fixed point is where the phase transition

occurs. Wilson’s renormalization group (RG) theory successfully predicted phase transitions

and explained the critical phenomena that mean-field theory failed to do. But it is mathe-

matically very complicated. Given that mean-field theory is mathematically simple and yet

predicts qualitatively most of the critical behaviors of phase transition, in the Sec. 1.2.1, we

first introduce Landau’s theory.

7

1.2.1 Landau’s theory of phase transitions

In Landau’s mean-field theory of phase transitions, the order parameter is built on the

macroscopic quantity that is zero on the high temperature side, but non-zero on the lower

temperature side of the phase transition. For example, for the ferromagnetic to paramagnetic

transition, the order parameter is magnetization; for the gas-liquid-solid transition, the order

parameter is usually the density difference; for the orthorhombic-to-tetragonal structural

transition, the order parameter is the difference between two lattice constants.

When it is near a phase transition, the order parameter is usually small so one can

perform power expansion on the free energy with respect to (WRT) the order parameter Ψ.

The free energy f has the following expression

f (Ψ, T ) = f0(T ) + a1(T )Ψ +

1
2

a2Ψ2 +

1
3

a3Ψ3 +

1
4

a4Ψ4 +

1
5

a5Ψ5 +

1
6

a6Ψ6 + ...

(1.1)

where f0(T) is free energy for the high-temperature state; a0, a1, etc., are the coefficients.

The values of both coefficients an and order parameter Ψ depend on temperature (T ).

To have a stable state, free energy requires minimum(s). Mathematically, this means the

following two conditions have to be met:

(1), The first derivative of f WRT to Ψ is zero:

∂f (Ψ, T )

∂Ψ

= a1 + a2Ψ + a3Ψ2 + a4Ψ2 + a5Ψ4 + a6Ψ5 + ... = 0.

(1.2)

(2), The second derivative of f WRT to Ψ is larger than or equal to zero:

∂2f (Ψ, T )

∂Ψ2

= a2 + 2a3Ψ + 3a4Ψ2 + 4a5Ψ3 + 5a6Ψ4 + ... ≥ 0.

(1.3)

8

At a high temperature where Ψ = 0, the system is a stable state, so ∂f (Ψ=0,T )

∂Ψ

= a1 = 0.

This means the linear term cannot exist in the expansion series. Another obvious conclusion

to make is the highest order of the expansion must be an even number.
If it is an odd
number, f(Ψ,T) would go to −∞ as Ψ goes to −∞ or +∞. This condition yields no stable

state globally. For the same reason, the coefficient for the highest even order should be

positive and the highest order in the expansion should be the 4th order or higher. Next we

will consider two classical forms of the expansion to the 4th and 6th orders, respectively:

f (Ψ, T ) = f0(T ) +

1
2

a2Ψ2 +

1
4

a4Ψ4,

and

f (Ψ, T ) = f0(T ) +

1
2

a2Ψ2 +

1
4

a4Ψ4 +

1
6

a6Ψ6.

(1.4)

(1.5)

In Eq. 1.4, where a4 > 0, the free energy has the forms as illustrated in Fig. 1.3.

When a2 > 0, the system is convex and there is only one minimum at Ψ = 0. But when

a2 becomes negative, the original minimum at Ψ = 0 is no longer the minimum of the free

energy, so the system would spontaneously move to new minimum positions. This process

is also called spontaneous symmetry breaking (SSB).

To a first order approximation, a2 can be written as a2 = a0

2(T − Tc), where a0

2 is a

positive constant; Tc refers to the critical temperature. So

∂f (Ψ, T )

∂Ψ

= a2Ψ + a4Ψ3 = Ψ[a0

2(T − Tc) + a4Ψ2] = 0.

(1.6)

9

Figure 1.3: The profiles of free energy as a function of the order parameter at different a2
values.

The solution for Eq. 1.6 is

(cid:114)



±

0,

Ψ =

2(Tc−T )
a0

a4

, T < Tc
T ≥ Tc.

(1.7)

The evolution trend for Ψ is shown in Fig. 1.4. Below Tc, Ψ is continuously approaching

the critical point with the exponent of 1/2; above Tc, there is no jump in the order parameter

so the expression in Eq. 1.4 corresponds to a second order transition.

Plugging the solution in Eq. 1.7 to the free energy terms in Eq. 1.4, we can get free

energy expression wrt temperature only. With this free energy form, other parameters, such

as entropy, latent heat, specific heat, etc., can be readily obtained. It is easy to show that

for a second order phase transition, entropy is continuous across the phase transition so the

latent heat is zero.

10

-101-0.4-0.20.00.20.4f - f0Ψa2>0a2<0a2=0Figure 1.4: The order parameter Ψ evolution as a function of temperature T , at both below
and above Tc

We have shown the 1/2-scaling behavior of the order parameter near a phase transition.

Next, we will consider applying a magnetic field B during, for example, a paramagnetic-

to-ferromagnetic transition. Here, the free energy has an additional magnetic field term

compared to Eq. 1.4

f (Ψ, T ) = f0(T ) +

2(T − Tc)Ψ2 +
a0

1
2

a4Ψ4 − ΨB.

1
4

Similarly, to minimize the free energy, its first derivative wrt to Ψ is zero

∂f (Ψ, T )

∂Ψ

= a0

2(T − Tc)Ψ + a4Ψ3 − B = 0.

11

(1.8)

(1.9)

-1.0-0.50.00.51.0ΨTTcNo orderOrdered phaseLow temperature:High temperature:Since Ψ is small near the critical point, we can ignore the Ψ3 term. Then the order

parameter becomes

Ψ =

B

2(T − Tc)
a0

.

(1.10)

As mentioned, in a paramagnetic-to-ferromagnetic transition, Ψ usually corresponds to

the magnetic moment. So the magnetic susceptibility would have the following form

χ =

Ψ
B

=

1

2|T − Tc|.
a0

(1.11)

The magnetic susceptibility χ diverges at T = Tc and on either side of phase transition,

its decay follows power of - 1, as shown in Fig. 1.5.

Figure 1.5: The magnetic susceptibility χ evolution as a function of temperature near the
critical point. On both sides of Tc, χ is inversely proportional to the distance from Tc and
diverges at Tc.

12

01234χTTcFor a second order phase transition, the critical scaling behavior is demonstrated from

order parameter and the divergence in susceptibility. But for a first order transition, which,

as we will show later, takes the form of Eq. 1.5, where there is no such scaling behavior due

to the jump in the order parameter. We take first derivative of Eq. 1.5 wrt Ψ to minimize

the free energy

∂f (Ψ, T )

∂Ψ

= a0

2(T − Tc)Ψ + a4Ψ3 + a6Ψ5 = 0.

(1.12)

(cid:113)

−a4+

The solution for this is

Ψ2 =



0,

0,

4−4a6a0
a2
(cid:113)
2a6
4−4a6a0
a2
2a6

2(T−Tc)

,
2(T−Tc)

−a4+

T < Tc

, Tc < T < T1

(1.13)

T > T1,

where T1 =

a2
4
4a0
2a6

+ Tc, a0

2 > 0, a4 < 0, a6 > 0.

The temperature dependence of the order parameter Ψ is plotted in Fig. 1.6.

As we can see from the plot, there is a jump in the order parameter at the phase transition.

It is easy to show that there is also a jump in the entropy curve, so the latent heat is nonzero

for a first order phase transition. Due to the existence of latent heat, first order transitions

are harder to study/formulate than second order. This is because Landau’s theory is based

on power-law expansion. It is thus a question of whether the expansion method would work

well for a first order transition, when the order parameter is not necessarily small enough

for power expansion near Tc. So in the following sections we will focus on second order

transitions.

13

Figure 1.6: The order parameter evolution as a function of temperature for first-order phase
transition near Tc. Cooling from high temperature, the system follows the red curve until
T = Tc where it becomes unstable and phase transition will happen. The system jumps to
one of the blue curves. If the system is heated from low temperature, then the system would
remain on the blue curve until T = T1, after which it jumps to the red curve and phase
transition occur.

1.2.2 From mean-field theory to renormalization group

In the previous section, we discussed how Landau used a simple formulation to explain the

critical phenomena and the universality of different phase transitions. His theory is general,

simple, and elegant in many ways. But as we mentioned, it can only give a qualitative but

not quantitative description of phase transitions, especially for the critical scaling.

Landau’s theory predicts a universal critical exponent of 1/2, independent of the inter-

actions or dimension of the system. But this is not always true experimentally and also

14

-1.0-0.50.00.51.0ΨTTcT1conflicts with other theories. For example, in 1945, E. A. Guggenheim summarized different

types of gas-liquid transitions [23] and fitted the universal critical exponent to be 1/3 instead

of 1/2.

When studying magnetic transitions, one usually refers to the Ising model. By solving

the 2D Ising model [24, 25], the critical exponent is determined to be 1/8. For 3D, it is

determined to be about 0.33. The critical exponents for both 2D and 3D Ising models

disagree with the MFT.

Another aspect is that MFT predicts a discontinuous but finite jump in specific heat.

But experimentally, for example in reference [26], the authors studied the normal-superfluid

transition in He4 and liquid-vapor transition of He3 and He4, where the specific heat is

observed to have singularities and divergent behaviors near the critical points.

The validity of MFT is constrained by the Ginzburg criterion, which states that mean-

field theory is self-consistent and valid only for dimensions greater than dc. So for experiments

in system whose dimension is below dc, MFT can not predict the precise scaling behaviors.

Critical scaling means that the system appears the same at different scales (the so-called

self-similarity or scale invariance). Consider a 2D spin system, each spin interacts with its

nearest neighbors with certain coupling J at temperature T . The physics of this system is

described by Hamiltonian H(T, J). Then we divide the 2D lattice into 2x2 blocks and try to

describe the system in terms of the blocks instead of individual spin. We assume the block
variables are described by Hamiltonian with different temperature and coupling: H(T(cid:48), J(cid:48)).

Originally we may have too many problems and degrees of freedom, now with the “block

spin” method [27], in the renormalized problem we have only 1/4 of them. By doing another

iteration, we will reduce the problem down to 1/16 of the original value. With each RG step,

we are increasing the observable scale and decreasing the degrees of freedom. This iteration

15

process is equivalent of finding the long range behavior of RG transformation from (T, J) to
(T(cid:48), J(cid:48)), (T(cid:48), J(cid:48)) to (T(cid:48)(cid:48), J(cid:48)(cid:48)), etc. Often, when iterated enough times, the RG transformation

will lead to a certain number of fixed points. For a magnetic system, the coupling term J

has a tendency of aligning neighboring spins while the temperature term T tends to disorder

the spins. For many models of this kind, there are three fixed points:

1. T = 0 and J → ∞, meaning that disorder disappear and the spin system is aligned

in the large scale. This is the ferromagnetic phase.

2. T → ∞ and J = 0, meaning that the high temperature and disorder dominate. This

is the so-called paramagnetic phase.

3. T = Tc and J = Jc. At this point, changing scales does not change the properties of

the system and the system is in a self-similar or fractal state. It corresponds to the critical

point at a Curie ferromagnetic to paramagnetic phase transition.

This block spin idea was brought up by Leo Kadanoff in 1966 [27] whereas the fixed

point concept was Kenneth Wilson’s genius idea to bridge RG with phase transitions [21,22].

Wilson considered many couplings in a system in repeated RG transformations. Each trans-

formation increases the size of length scale and the phase transition would occur when the

transformation is brought to a fixed point. That is, after enough transformations, the cou-

plings will not further modify the system and the system becomes scale-invariant. Further-

more, the correlation length remains unchanged under transformations at the fixed point.

Wilson pointed out two ways of stabilizing the correlation length: for a continuous phase

transition, correlation length is infinite at the critical point; for a first order phase transition,

correlation length is finite, reflecting that the local interactions drive the phase transition.

To generalize the RG theory to phase transitions, in 1972, K. Wilson worked with M.
Fisher [28] to apply the RG theory on a system that has dimension of d = 4 − 
 to calculate

16

the critical exponents. This method, later known as epsilon-expansion, was generalized not

just to small 
 but also to three dimensions and beyond.

Regarding critical scaling near the critical point, we have mentioned the scaling of the

order parameter, susceptibility and specific heat based on Landau’s theory of phase transi-

tions. Next we will summarize the key scaling orders and scaling exponents in 2D and 3D

Ising models. Following conventions in this field, we define parameters as follows:

Ψ: order parameter.

It can be density for liquid-gas transition, or magnetization for

paramagnetic to ferromagnetic transition

τ : reduced temperature, τ = T−Tc
Tc

f : free energy
C: specific heat, C = −T ∂2f
∂T 2

, distance away from the critical point

J: source field, it could be magnetic field, reduced chemical potential, or pressure, etc.

χ: susceptibility, compressibility, etc., χ = ∂Ψ
∂J

ξ: correlation length

d: spatial dimension of a system

(cid:104)ψ(−→x )ψ(−→y )(cid:105): the correlation function

r: spatial distance

Then, there will be the following scaling relations:

Ψ ∝ |τ|β, for τ < 0 only; Ψ = 0 when τ ≥ 0,

ξ ∝ |τ|−ν,

χ ∝ |τ|−γ,

17

(1.14)

(1.15)

(1.16)

C ∝ |τ|−α,

J ∝ Ψδ,

(cid:104)ψ(−→x )ψ(−→y )(cid:105) ∝ r−d+2−η.

(1.17)

(1.18)

(1.19)

From Eq. 1.14 to Eq. 1.19, there are 6 critical exponents shown in the scaling powers:

α, β, γ, δ, ν, η. Due to the confinements and definition of the physical parameters, the

critical exponents have the following scaling relations (Note: the relations apply for d < dc,

where dc is the upper critical dimension; above dc, some of the exponents are no longer

spatial dimension-dependent and some of the relations will no longer hold):

νd = 2 − α = 2β + γ = β(δ + 1) = γ

δ + 1
δ − 1

=

γd
2 − η

.

(1.20)

Therefore, for a d−dimension system, of the 6 critical exponents (α, β, γ, δ, ν, η), only two

of them are independent. These exponents are universal in a sense that they only depend on

the spatial dimension d and the type of model used. Systems very different from each other

could share a set of critical exponents and such systems together are said to belong to the

same universality class.

Table 1.1 is a summary of critical exponents for some well-known models at different

dimensions. In the table, the O(n) model is also called n-vector model. It is a generalization

of the Ising model, XY model, and Heisenberg model. The majority of these models are

calculated based on RG theory which laid the foundation for quantum field theory in particle

physics and the theory of continuous phase transition in condensed matter physics.

18

Model/Class

d

α

Mean-field

all d

2D Ising

4D Ising

Self-avoiding walk/ O(0)

3D Ising / O(1)

XY model / O(2)

Heisenberg / O(3)

O(4)

Standard percolation

Standard percolation

Standard percolation

Standard percolation

Standard percolation

Standard percolation

Directed percolation

Directed percolation

Directed percolation

Directed percolation

2

4

3

3

3

3

3

1

2

3

4

5
≥ 6

1

2

3
≥ 4

β

1/2

γ

1

1/8

7/4

1/2

1

δ

3

15

3

ν

1/2

η

0

Ref.

[19,29]

1

1/4

[24,25]

1/2

0

[30]

0

0

0

0.2350

0.3024 1.1596 4.8347 0.5882

0.0284

[31]

0.1090

0.3258 1.2396 4.8048 0.6304

0.0335

[31]

- 0.011

0.3470 1.3169 4.7951 0.6703

0.0354

[31]

- 0.122

0.3662 1.3895 4.7944 0.7073

0.0355

[31]

- 0.223

0.3830

1.456

1

0

1

4.8016 0.7410
∞

1

0.0350

[31]

1

[32]

- 2/3

5/36

43/18

91/5

4/3

5/24

[33]

- 0.616

0.4050 1.8050 5.4593 0.8720

- 0.070

[34]

- 0.712

0.6390 1.4350 3.2441 0.6780

- 0.120

[34]

- 0.855

0.8350 1.1850 2.4192 0.5710

- 0.075

[34]

- 1

1

1

2

1/2

0

[32]

0.1595

0.2765 2.2777 0.1595 1.0969

0.3137

[35]

0.4505

0.5834 1.5950 0.4505 0.7330

0.2295

[36]

0.7320

0.8130 0.7320 1.2370 0.5840

0.1200

[37]

- 1

1

1

2

1/2

0

[38]

Table 1.1: Critical exponents for different models at different dimensions. References are
included in the last column for different models. For cell values that are integer or fraction,
they come from exact solutions. For cells with decimal values, they usually come from RG
based field theory methods.

19

1.2.3 Systems driven far from equilibrium and photo-induced phase

transitions

Thus far the phase transitions are discussed under equilibrium conditions. But everything

in nature is moving, interacting, or evolving even in equilibrium state. Much less is under-

stood about phase transitions for systems driven out of equilibrium, both theoretically and

experimentally.

Generally, dynamical systems can be classified into several categories based on whether

the order parameter is conserved or not, the dimensionality, and symmetry of the system.

Table 1.2 shows such classification of currently known dynamical models treated by the RG

method (from Ref. [1]).

It is generally believed that many concepts and techniques used in describing the equi-

librium phase transitions, such as scaling and universality, can still be applied to the non-

equilibrium dynamical systems. Systems of the the same universality class share the same

scaling exponents and sometimes even the same scaling function. However, non-equilibrium

systems have an additional degree of freedom: time. Not only does this gives more freedom

for expressing universality, it also allows one to actively manipulate and predict the behav-

ior of the systems following universal dynamics. In terms of the universality and scaling,

besides the aforementioned 6 static critical exponents (2 independent exponents), there are

additional critical exponents for the non-equilibrium systems defined as below.

The divergences of the relaxation time τ and correlation length ξ are related by the

dynamical exponent z

τ ∝ ξz.

(1.21)

The dynamical exponent z can also be defined as the ratio of the temporal (ν||) and spatial

20

Model

No.

System

D

N.C.

Cons.

Non-vanishing

Relaxational

A

Kinetic Ising

Relaxational

B

Kinetic Ising uniaxial

anisotropic magnets

Relaxational

C

Anisotropic magnets

ferromagnet

structural transition

Gas—liquid binary

fluid

Easy-plane magnet,

H

E

hz = 0

Fluid

Symmetric

planar

magnet

Asymmetric

F

planar

magnet

Easy-plane magnet,
hz (cid:54)= 0, superfluid

helium

Isotropic an-

G

Heisenberg

tiferromagnet

antiferromagnet

Isotropic

J

Heisenberg

ferromagnet

ferromagnet

n

n

n

1

2

2

3

3

fields

fields

Poisson bracket

Ψ

None

None

None

Ψ

None

Ψ

m

None

None

Ψ, j

Ψ

m

{Ψ, j}

{Ψ, m}

Ψ

m

{Ψ, m}

Ψ

m

None

Ψ

{Ψ, m}

{Ψ, Ψ}

Table 1.2: Classification of non-equilibrium dynamical systems based on alphabetical order
[1]. In the first row title line, the “No.” refers the designation of alphabet for each model.
“D” is the dimension of the system. “N.C. fields” represents the non-conserved fields. “Cons.
fields” represents “conserved fields”.

(ν⊥) correlation length exponents: z = ν||/ν⊥.

To describe the two-time correlations in a spin system {si} of size L relaxing to the

21

critical state from a disordered initial condition, another dynamical exponent λ is needed

(cid:104)(cid:88)

i

A(t, 0) =

1
Ld

si(0)si(t)(cid:105) ∝ t−λ/z.

(1.22)

Persistence exponent θ is introduced by Derrida et al in Ref. [39] to describe a quench

to the critical point. The probability p(t) that the local or global order parameter does not
change sign at time t after the quench follows a power-law: p(t) ∝ t−θ.

Of the several dynamical critical exponents, z is the most common and widely discussed.

The same universality classes share the same critical exponents. For example, in the Ising

model of model A dynamics, dynamic exponent z is around 2 for all dimensions [40]. With

dynamic exponent z, the relaxation time τ diverges as the system approaches the critical

point. This is usually called critical slowing down. Critical slowing down enforces a local

choice of the broken symmetry and leads to the formation of topological defects [41].

One classical example of non-equilibrium dynamics is the relaxational dynamics after

quenching the system from a homogeneous phase to a broken-symmetry phase. The phase

ordering kinetics can be understood in terms of domain coarsening processes, in which the
domain size grows with time following the relation: L(t) ∝ t1/z. Such a dynamical process

involving spontaneous symmetry breaking (SSB) was studied in cosmology in the 1970s and

also in condensed matter physics. According to Tom W. B. Kibble [42, 43], right after the

“big bang”, the universe is in a high-temperature “normal” phase with high symmetry. Then

the universe expands and cools down and at a certain point when the temperature drops

below the transition temperature Tc, the symmetry is spontaneously broken and domains are

expected to form. Under certain conditions, the universe is expected to form domain walls,

cosmic strings, or “monopoles”, all of which are called topological defects. The evolution of

22

the universe can be understood in such a way that it is similar to phase transitions in con-

densed matter physics. For example, in a paramagnetic-to-ferromagnetic phase transition,

it also involves SSB and formation of domains.

To quantitatively investigate the effects of topological defects on structural evolution,

Wojciech H. Zurek studied the domain coarsening process in the context of dynamical scaling
[44, 45]. The defined criticality is by a reduced temperature 
: 
 = Tc−T
Tc

, where Tc is the

critical temperature and T is system temperature. As discussed earlier, near critical point,

the equilibrium correlation length and relaxation time τ diverge: ξ =

ξ0

ν , τ =

τ0

zν . Now

consider a linear (in time) quench that drives the system across the critical point. When the

system is far away from the critical point, the relaxation time is small and system can catch

up with the quench, so the dynamics is adiabatic. On the other hand, when the system is

close to the critical point, dynamics become almost frozen due to divergence of the relaxation

time in critical slowing down, the adjustment of the system is unable to catch up with the

quench. Such a non-adiabatic time is called Kibble-Zurek freeze-out time.

Near the critical point, the reduced temperature 
(t) is written linear with respect to

time


(t) =

t
τQ

,

(1.23)

where τQ is the quench period. The process starts at negative time. At time-zero, 
(0) = 0

and it is at critical point. To estimate the Zurek freeze-out time, the relaxation time is equal

to the time elapsed (t)

τ (
(t)) =

τ0

(t)zν = τ0(

t
τQ

)−zν = t.

(1.24)

23

With this, we can estimate the freeze-out time (tKZ ) to be

tKZ = (τ0τ zν
Q )

1

1+zν .

Correspondingly, the order parameter at this non-adiabatic-to-adiabatic transition is


KZ = 
(tKZ ) = (

1

1+zν .

τ0
τQ

)

The corresponding correlation length at 
KZ is

ξKZ = ξ(
KZ ) = ξ0(

ν

1+zν .

τQ
τ0

)

With this correlation length, the density of topological defects is estimated to be

n ∼ ξd
KZ
ξD
KZ

=

1
ξD−d

0

(

τ0
τQ

(D−d)
)

ν

1+zν ,

(1.25)

(1.26)

(1.27)

(1.28)

where D and d are dimensions of the system and defects, respectively. The results above

from Eq. 1.25 to Eq. 1.28 are the main predictions by theory, which is frequently referred

to as the so-called Kibble-Zurek mechanism (KZM). KZM can be experimentally tested by

examining whether the defect density follows the scaling law described in Eq. 1.28.

Photo-induced phase transitions (PIPT), by definition, are phase transitions initiated by

laser pulses and typically involve non-equilibrium, non-thermal processes. After laser excita-

tions, the system is initially driven far from equilibrium then relaxes. The relaxation process

could exhibit scaling behaviors that can be described with different universality classes. Fig.

1.7 is an example of PIPT from Ref. [3], in which the authors performed transient fs optical

24

Figure 1.7: The evolution of the exciton density observed in photo-excited carbon nanotubes
[3]. In the log-log plot, the system switches from a - 1 power-law decay to a - 1/2 exponent,
indicating a transition from the reaction-limited regime (RLR) to diffusion-limited regime
(DLR).

transmission measurements on carbon nanotubes and determined the power-law decay of the

exciton density. It is found that the exciton-exciton recombination process switches from the

reaction-limited to a diffusion-limited process, belonging to different universality classes.

1.3 Charge-density wave

Charge-density wave (CDW) is a symmetry-broken ground state of metals and its formation

has been usually described by Fermi surface nesting that enhances electron-phonon coupling.

25

At the wave-vector of the CDW, the translational symmetry is broken and the charge-density

system is characterized by periodic modulation, which couple to the lattice in terms of

periodic lattice distortion.

The idea of CDW was first proposed by Rudolf Peierls [46] in 1955 via a mechanism

later know as Peierls transition. When Peierls was studying an ideal 1D atomic chain, he

found that a one-dimensional metallic chain is intrinsically unstable at low temperatures: the

electrons prefer to open up a gap at the Fermi surface (FS) and atomic ions will periodically

distort to form a new symmetry-broken ground state. Such a state is called the charge-

density wave state and the transition is called the Peierls transition.

Due to the formation of CDW, the phonon spectrum is strongly renormalized. At the

wave-vector of the CDW, the phonon is frozen into a static atomic distortion. In 1959, Walter

Kohn [47] proposed that in the phonon dispersion relation for a metal, the frequency of the

phonon at specific wave-vector is lowered when there is a discontinuity in the derivative of

phonon dispersion relations. Such a phenomenon is called Kohn anomaly. The CDW state

is an extreme case of a Kohn anomaly with phonons softened to zero frequency.

Experimentally, CDWs were first found in a 1D conductor compound K2Pt(CN)4Br0.30xH2O

[48] by X-ray diffraction in 1973. In the following several years, CDWs were found in the

1D charge-transfer organic salt tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ)

[49], blue bronze K0.3MoO3 [50], transitional metal tri-chalcogenide NbSe3 [51], and also in

two-dimensional (2D) layered compounds [52, 53].

According to Peierls’ theory, CDW formation usually happens in low dimensional materi-

als or materials with very high anisotropy. In the following part of this section, the simplest

and most classical theoretical description of CDW will be formulated based on a 1D metallic

atomic chain.

26

1.3.1 One-dimensional electron gas

One-dimensional (1D) electron gas refers to electrons arranged in a 1D array. Consider such

a system with length of L, the Fermi energy is given by


F =

¯h2k2
F
2me

,

(1.29)

where me is electron mass and kF is the Fermi wave-vector, kF =

N0π
2L .

The electron density ρ changes in the presence of an external potential φ, such that

ρ((cid:126)q) = χ((cid:126)q)φ((cid:126)q),

(1.30)

where χ((cid:126)q) is the electronic susceptibility defined by Lindhard response function

(cid:90)

χ((cid:126)q) =

1

(2π)d

fk − fk+q

k − 
k+q

d(cid:126)k,

(1.31)

where fk = f (
k) is the Fermi function, d is dimension of the system [54].

For 1D electron gas, the dispersion relation near the FS can be approximated to the

following relation:


k − 
F = ¯hvF (k − kF ),

(1.32)

where vF is Fermi velocity.

From Eq. 1.32, susceptibility in Eq. 1.31 is calculated to be

χ(q, T = 0) =

−e2
π¯hvF

ln| q + 2kF
q − 2kF

| = −e2n(
F )ln| q + 2kF
q − 2kF

|,

(1.33)

27

where n(
F ) is the density of states at the Fermi level. Here we consider T = 0 for χ, but

generally χ should also depend on temperature.

Immediately we can see that susceptibility diverges at q = 2kF , meaning that at q = 2kF

the electron density can reorganize even without any external field. So the 1D electron gas is

intrinsically unstable at zero temperature. The 1D susceptibility is plotted in Fig. 1.8 along

with the 2D and 3D cases, where the curves are generated based on expressions in Ref. [55].

Divergence of the susceptibility for 1D case clearly stands out.

Figure 1.8: The electron susceptibility for 1D, 2D, and 3D systems at zero temperature.

When a large segment of the FS can be connected to another large segment of FS via

a reciprocal lattice vector, we call this phenomenon Fermi surface nesting (FSN) and the

wave-vector is called FSN vector qχ. If there is strong FSN in a large portion of FS, then by

examining the Lindhard response function in Eq. 1.31, one usually sees a peak in electron

28

0.00.51.01.52.02.53D2Dχ(q)/χ(q = 0)qT = 0 K2kF01Ddispersion or divergence of χ((cid:126)q) at q = qχ. So a strong FSN always results in divergence in

susceptibility and increased instabilities in the electron system.

For 1D linear chain, the FS is just two points, so one can completely connect the entire

FS with a single wave-vector q = 2kF . This is the case of perfect FSN. For 2D or 3D cases,

the FS topology is more complicated and exhibits much less symmetry, so only partial FSN

occurs. Fig. 1.9 shows examples of FSN for different FS topologies.

Figure 1.9: Examples of Fermi surface nesting for different Fermi surface topologies. a, 1D
electron gas, perfect FSN occurs. b, 2D free electron gas. There is no “parallel” parts in the
circular FS, so no FSN. c, FS of a 2D square lattice. The diamond shape of FS results in
relatively good FSN.

At a finite temperature T , the susceptibility can be calculated based on the finite-

temperature Fermi function for density of state n(
F ) [54]

χ(q = 2kF , T ) = −e2n(
F )

tanh(x)

x

dx,

(1.34)

(cid:90) 
0/2kB T

0

where 
0 is the cutoff energy of the integral, which is usually taken to be equal to the Fermi

energy 
F .

29

-kFkFqqqabcThe integral can be calculated to be

χ(q = 2kF , T ) = −e2n(
F ) ln

1.14
0
kBT

.

(1.35)

This temperature-dependent susceptibility is a finite value at finite temperatures and

diverges only when T → 0.

1.3.2 Coupled one-dimensional electron-phonon linear chain

Next, we will consider a 1D free electron gas coupled to the underlying chain of ions through

electron-phonon coupling. The Hamiltonian for the electron gas can be written as

ˆHel =

†
kak,


ka

(1.36)

k

where a

†
k and ak are the creation and annihilation operators for the electron state with

energy 
k = ¯h2k2/2m.

The Hamiltonian for the lattice can be written as

(cid:88)

(cid:88)

q

(cid:88)

k,q

30

ˆHph =

†
qbq +

¯hωq(b

1
2

),

(1.37)

where b

†
q and bq are the creation and annihilation operators for the phonons at frequency ωq.

For the electron-phonon interaction term, the Hamiltonian can be written as

ˆHph =

†
−q + bq)a

†
k+qak,

gq(b

(1.38)

where gq is the electron-phonon coupling constant defined as

gq = i(

¯h

2M ωq

)1/2|q|Vq,

(1.39)

where M is the ionic mass, Vq is the ionic potential. For detailed derivation check Ref. [54].

So summing up Eqs. 1.36, 1.37, 1.38, we get the total Hamiltonian of the system:

†
qbq +

¯hωq(b

1
2

) +

†
−q + bq)a

†
k+qak.

gq(b

(1.40)

(cid:88)

k

ˆH =

†
kak +


ka

(cid:88)

q

(cid:88)

k,q

Based on Heisenberg equation of motion, assuming Qq is the atomic position, for small

displacements, the equation of motion is

¯h2 ¨Qq = −[[Qq, ˆH], ˆH].

(1.41)

Then based on commutation relations of [Qq, Pq(cid:48)] = i¯hδq,q(cid:48), Eq. 1.41 becomes

¯h2 ¨Qq = −ω2

q Qq − g(

2ωq
M ¯h

)1/2ρq,

where ρq is the electron density determined by susceptibility: ρq = χ(q, T )g(

so the equation of motion becomes

(1.42)

2M ωq

¯h

)1/2Qq,

¨Qq = −[ω2

q +

2g2ωq

¯h

χ(q, T )]Qq.

(1.43)

This gives a normalized phonon frequency of ω2

ren,q = ω2

q +

2g2ωq

¯h χ(q, T ). At FS nesting

vector of q = 2kF , plugging in the susceptibility expression in Eq. 1.35, the phonon frequency

31

becomes

ω2
ren,2kF

= ω2

2kF

− 2g2n(
F )ω2kF

¯h

(cid:18)1.14
0

(cid:19)

.

(1.44)

ln

kBT

From Eq. 1.44, we can see with decreasing temperature, the renormalized phonon fre-

quency goes down to zero to form “frozen” phonons. This defines the transition temperature

kBT M F

CDW = 1.14
0e−1/λ,

where λ is a dimensionless electron-phonon coupling constant defined as λ =

(1.45)

g2n(
F )
¯hω2kF

.

This Kohn anomaly effect can be visualized in Fig. 1.10, which is adapted from Ref. [54].

Figure 1.10: The phonon softening effect when the temperature is close to T M F
frozen phonon when T = T M F

CDW . It becomes

CDW .

32

ωqq2kFT = TCDWMFT (cid:843)(cid:7716)TCDWMFT (cid:843)(cid:843)(cid:7716)TCDWMFWe can define an order parameter (∆ is actually the CDW gap) as

|∆|eiφ = g((cid:104)b2kF

(cid:105) + (cid:104)b

†
−2kF

(cid:105)).

(1.46)

Replace the electronic part Hamiltonian in Eq. 1.36 with this order parameter and with

some algebra transformations (Ref. [54]), in the weak coupling limit, the energy loss by the

electron system due to CDW formation is

Eel = n(
F )[−∆2

2

− ∆2 log

(cid:18)2
F

∆

(cid:19)

].

On the other hand, the energy gain due to lattice distortion is

Elatt =

N
2

M ω2

2kF

(cid:104)u(x)(cid:105)2 =

∆2

¯hω2kF
2g2

=

∆2n(
F )

2λ

,

(1.47)

(1.48)

where the lattice distortion amplitude u is actually directly connected to the CDW gap
∆ as (cid:104)u(x)(cid:105) = (

g cos(2kF x + φ). That is why in Eq. 1.48 there is that

)1/2 2|∆|

¯h

2N M ω2kF

replacement.

Consider the total energy change by summing up Eq. 1.47 and Eq. 1.48

Etot = Eel + Elatt = n(
F )[−∆2

2

− ∆2 log

Minimizing this energy, we get the CDW gap

+

∆2
2λ

].

(1.49)

(cid:18)2
F

(cid:19)

∆

∆ = 2
F e−1/λ

(1.50)

33

and the condensation energy

Econd = Enormal − ECDW =

n(
F )

2

∆2.

(1.51)

This positive condensation energy indicates that the formation of CDW reduces the total

energy and the CDW is now a stable ground state.

Figure 1.11: An example of CDW formation in a half-filled 1D linear chain. a, Metallic
state. Charge-density is uniform and atomic ions are undistorted. Electrons fill up to half
of the band and it is a metal. b, CDW state. The charge-density re-modulate and atomic
ions distort every pair. The band opens up a gap at ± kF .

Comparing Eq. 1.50 and Eq. 1.45, we can get the well-known relation between the zero-

temperature gap and the transition temperature

2∆(T = 0) = 3.52kBT M F

CDW .

(1.52)

The temperature-dependent gap can be deduced by integrating the Fermi function that

defines T M F

CDW in Eq. 1.45, yielding the BCS-like order parameter temperature evolution

|∆(T )|

|∆(T = 0)| = 1.74

1 − T
T M F
CDW

, for T < T M F

CDW .

(1.53)

(cid:115)

34

-π/aπ/a-kFkF0E(k)aρ(r)atomEF-π/aπ/a-kFkF0E(k)ρ(r)atomEF2akkgapab1.3.3 Fluctuation effect versus mean-field model

As we have discussed in Sec. 1.2, mean-field theory ignores fluctuations. On the other hand,

RG theory predicts that the fluctuation will diverge at the fixed point. At low dimensions

where atomic motions are less constrained by fewer nearest neighbors, the fluctuation effect

is expected to be more dominant. The divergence of correlation length in the fluctuating

order parameter near the critical point is discussed here.

For a 1D chain described by Hamiltonian in Eq. 1.40, the order parameter is complex. So

both the phase and amplitude should be considered when calculating the correlation length.

As drawn in Fig. 1.3, when temperature is above T M F , the potential has one minimum in the

middle, and the system moves inside like a pendulum, the amplitude fluctuation dominates

over phase; when T is below T M F , the system’s orbit is more likely along the minimum

circle, so phase fluctuation dominates.

The phase-phase correlation length for T < T M F can be written as the following (for

detailed derivation, check Ref. [54])

ξ1D =

¯hvF
πkBT

.

(1.54)

On the other hand, at high temperature, the correlation length due to amplitude fluctu-

ation takes the form of

ξ1D =

[7ζ(3)]1/2vF

4πkBT

,

(1.55)

where ζ(3) is the third-order zeta function and this equation is derived from Ref. [56].

In real systems, a 1D chain does not exist and it is always multiple chains coupled

together with some kind of interactions. Due to this interaction, CDW fluctuations are

35

better stabilized and this leads to a transition to a ground state with 3D long range order.

Assuming a square lattice chain with lattice constant d⊥, the inter chain interaction can be

written as

Eint = n(
F )

c⊥

2
d2⊥

(cid:88)

i,j

dxRe[∆∗

i (x), ∆j(x)],

(1.56)

where n(
F ) is density of states at the Fermi level, c⊥ is the inter chain interaction, ∆ is

complex order parameter, i, j are the chain index.

In the mean-field approximation, the susceptibility can be generally written as

χ(q(cid:107), q⊥) =

χ1D(q(cid:107))

1 − Zc⊥(0)χ1D(q(cid:107))T

,

(1.57)

where Z is the number of the nearest neighbors, χ1D(q(cid:107)) is the common susceptibility in a

1D chain discussed in Sec. 1.2. The divergence of χ(q(cid:107), q⊥) defines a transition temperature

T3D

1 − Zc⊥(0)χ1D(q(cid:107))T3D = 0.

(1.58)

According to Ref. [54], the correlation length parallel and perpendicular to the chains

can be defined as

ξ(cid:107) = (

ξ⊥ = (

c

a(cid:48) )1/2(T − T3D)−1/2
c⊥
a(cid:48) )1/2(T − T3D)−1/2,

(1.59)

(1.60)

where c, c⊥ are the interaction terms, a is a parameter related to temperature and order

parameter, detailed definitions can be seen in chapter 5 of Ref. [54].

From Eq. 1.59 and Eq. 1.60, in directions both parallel and perpendicular to the chain,

the correlation length diverges at T = T3D. At near T3D, both correlation length is large

36

Figure 1.12: The temperature-dependent correlation length of the parallel and perpendicular
to the chains as well as single 1D chain correlation length. T3D is the real transition temper-
ature in materials, below which is CDW state. TM F is the mean-field transition temperature
based on mean-field models.

and ξ⊥ is larger than d⊥, which is the inter chain distance. So in this region, the neighboring

chains are strongly coupled and it is more 3D like. At high temperature where ξ⊥ is smaller

than d⊥, each chain is decoupled and it is more 1D character. From 3D to 1D, the crossover

happens when

d⊥ = ξ⊥ = (

c⊥
a(cid:48) )1/2(T∗ − T3D)−1/2.

Thus the crossover temperature T∗ is

T∗ = T3D +

d2⊥a(cid:48)
c⊥ .

(1.61)

(1.62)

The correlation length of 3D ξ(cid:107), ξ⊥ and 1D ξ1D is plotted in Fig. 1.12 for comparison.

Crossover temperature T∗ is also labeled in the plot.

In Table 1.2, the CDW transition temperature T3D and MFT predicted temperature TM F

are compared for several different CDW materials. From the table, we can see all the T3D

37

0T3Dd⊥T*TMFξ1Dξ⊥Correlation lengthξ||Ordered stateis smaller than the mean-field value, thus confirming the point of inter-chain interactions.

Materials

T3D (K)

TM F (K)

Ref.

K2Pt(CN)4Br0.3· 3H2O

TTF-TCNQ(2kF )

TTF-TCNQ(4kF )

NbSe3

BaVS3

K0.3MoO3

CeTe3

1T-TaS2

100

54

49

145

70

180

650

150

600

700

170

500

[57]

[58]

[59]

[60]

[61]

[62]

> 500

2700

[63, 64]

543

700-2700

[65]

Table 1.3: Comparison of the CDW transition temperature T3D and the temperature pre-
dicted by mean-field theory TM F in several materials. The TM F is estimated based on CDW
gap in Eq. 1.52. For CeTe3, the CDW gap is 400 meV. For 1T-TaS2 at zero temperature,
there are both the CDW gap and Mott-Hubbard gap. The CDW gap based on different
measurements are estimated to be 100-400 meV.

1.3.4 Origin of charge-density wave: Fermi surface nesting and

beyond

Based on Peierls’ theory of CDW, CDW formation is initially driven by electronic instabilities

(Fermi surface nesting and thus divergence of electron susceptibility) then stabilized through

electron-phonon coupling.

In this picture, the lattice distortion is just a secondary “side

effect” of CDW formation but not the driving force. The CDW Q-vector matches well with

the FSN vector, especially in 1D chains where there is a perfect FSN.

But in many 2D or higher dimensional CDW materials, researchers have shown [55,66,67]

38

that FSN could be weakened by carrier scattering and Fermi function broadening and FSN

alone is not sufficient to drive the CDW. Instead, strong momentum-dependent electron-

phonon interactions are attributed strongly to drive CDW formation. Examples of this type

include NbSe2, TaSe2, etc.

In some strongly correlated materials, for example VO2, the metal-insulator transition

involves a large gap opening and lattice atomic distortion (e.g. dimerization). Sometimes

this is also called CDW formation. But this type of transitions is completely different from

the Peierls mechanism, more to do with electron-electron interactions and Mott physics.

The CDW is also found to form in some high-temperature superconductors in cuperates

[68, 69]. But it is still debated whether FSN or strong electron phonon interactions is the

driving force of CDW [55].

39

Chapter 2

Diffraction theory and ultrafast

electron diffraction

Scattering is one of the most powerful experimental tools to study physics throughout history.

In scattering experiments, probe particles are shot onto the target material and interact

with the target, then the probe particles are collected for detection. By monitoring the

probe particle’s momentum or energy change, information about the target material can be

revealed. Such a probe could be photons (from THz to hard X-ray), electrons, neutrons,

alpha particles or other charged particles, etc.

Different probe particles interact differently with the materials being tested. For example,

X-ray photons mainly interact with electrons in a material through their electromagnetic

(EM) field. Electrons in the specimen oscillate with the EM field in an X-ray beam then emit

their own EM field that is identical in wavelength and phase to the incident X-ray. Hence,

X-ray diffraction can be used to map the electron density of the material. Electrons are light,

negatively charge particles. They interact with materials by passing through the Coulomb

potential formed by both electrons and core ions in the materials. So electron diffraction is

a measure of the Coulomb potential of the material. On the other hand, neutrons are heavy,

charge-neutral but non-zero spin particles. Therefore, neutrons interact with materials either

through strong nuclear force with nucleus or through magnetic interactions with outer shell

40

electrons.

In this chapter, the basic formulations of (electron/X-ray) diffraction are discussed in

Sec. 2.1, from atomic scattering to crystal scattering. Factors that affect scattering will also

be discussed, especially the structural factor and the Debye-Waller factor (DWF). Following

this, the pump-probe technique and ultrafast electron diffraction will be introduced in Sec.

2.2.

2.1 Basic formulation of diffraction

Diffraction is a process of wave interference from multiple scattered waves. Crystalline

materials consist of atoms arranged in a certain periodic way. First scattering from single

atom will be discussed, then we will extend it to crystalline materials.

2.1.1 Scattering from a point target: atomic form factor

When a probe particle, either X-ray photon or electron, is scattered by an isolated atom, the

scattering amplitude depends on the type of interaction, properties of the probe and target

atom, scattering angle, etc. Such a scattering amplitude is called atomic form factor.

For X-rays, the atomic form factor involves a Fourier transform of the spatial density

distribution of the scattering target from real space to reciprocal space. For a target atom

with spatial density distribution of ρ(r), the form factor f (s) is defined as

(cid:90)

f (s) =

ρ(r)eis·rd3r,

(2.1)

where s is called momentum transfer defined below.

41

Figure 2.1: Diagram of the momentum transfer in a single atom scattering event. ki and
kf are the incident and scattered wave-vector, respectively, defined as ki = 2π/λi and
kf = 2π/λf , where λi and λf are the incident and scattered wavelengths of the probe
particles. s is the momentum transfer. θ is the scattering angle, θ/2 is equal to the so-called
Bragg angle θb in the Bragg’s law.

As shown in Fig. 2.1, for elastic scattering where there is no energy exchange, the incident

and scattered wave-vectors are ki and kf , respectively. ki = kf = 2π/λ. Using trigonometric

relation, it is easy to prove that

s = kf − ki =

4π
λ

(cid:18) θ

2

sin

(cid:19)

ˆs,

(2.2)

with ˆs being the unit length vector along s direction.

The scattering intensity I is proportional to atomic form factor squared: f 2.

The electron charge density about nucleus ρ(r) in Eq. 2.1 can be assumed to be a spherical

distribution. For X-ray scattering, the atomic form factor is usually written as sum of

42

kfkisθGaussians [70]

(cid:16)−bi(

ai exp

(cid:17)

)

s
4π

+ c,

4(cid:88)

i=1

fx(s) =

(2.3)

where ai, bi, and c are parameters from different atomic species. Their values can be found

in Ref. [71].

For electron scattering, the atomic form factor is different but is related to that of X-rays

by the Mott–Bethe formula [72]

fel(s) =

me2

32π3¯h2
0

1

s2 (Z − fx(s)),

(2.4)

where m and e are electron mass and charge, respectively, 
0 is vacuum permittivity, Z is

atomic number of the target atom.

The electron atomic form factor in Eq. 2.4 can be plotted in Fig. 2.2, which is adapted

from Ref. [4].

2.1.2 Scattering from a crystal: Bragg’s law and Laue condition

Consider two atoms separated by distance a as shown in Fig. 2.3a, an incident plane wave

with wavelength λ and incidence angle θi (relative to the atomic line) scattered by the two

atoms. The scattered angle is θf .

In order to have constructive interference, the path

difference of the two rays should be equal to

a(cos θf − cos θi) = hλ,

(2.5)

where h is an integer number.

For a 3D lattice with lattice constants a, b, c forming constructive interferences along all

43

Figure 2.2: The atomic form factor fel(s) evolutions for electron scattering at different s
and for different atoms [4]. The atomic form factor decreases with the increasing scattering
angle and increases with the increasing target atomic number.

3 crystal periodicities, it requires

a(cos θf − cos θi) = hλ
b(cos θf − cos θi) = kλ
c(cos θf − cos θi) = lλ,

(2.6)

where h, k, l are all integer numbers. These equations are called Laue equations. Sometimes

Laue equations are written in a vector form. We can rewrite Eq. 2.6 in the wave-vector

format

a(

2π
λ

) cos θf − a(

2π
λ

) cos θi = a(kf − ki) = a · s = 2πh
b · s = 2πk
c · s = 2πl.

44

(2.7)

This is equivalent to Eq. 2.6.

Figure 2.3: The schematic drawing of scattering diagram for the Laue condition (a) and the
Bragg’s Law (b).

The Laue condition is a general form of identifying diffraction spots. By contrast, Bragg’s

law is built on reflection from the lattice plane. Consider the schematic drawing of Fig. 2.3(b),

the incident beam hits on a lattice plane, then gets “reflected” from the surface. Both the

incident angle and scattering angle are the same θb, the so-called Bragg angle.

It equals

half the scattering angle θ that we mentioned in Fig. 2.1. Similarly, to have a constructive

interference, the path difference should be equal to integer numbers of the probe wavelength

2d sin(θb) = nλ,

(2.8)

where n is an integer number. It is easy to prove that Bragg’s Law is a special case of the

Laue equations. Bragg’s law reduces Laue’s three equations into only one and has been

widely used.

45

θiθθfθbθbdθbaIncident beamScattered beamab2.1.3 Reciprocal space and Ewald’s sphere

When discussing diffraction, we are talking about momentum transfers and wave interfer-

ences. It is actually easier to treat it in a new imaginary space called momentum space or

reciprocal space.

Consider a 3D lattice that has basis vectors of a, b, c. Any atomic positions in the lattice

can be written as r = la + mb + nc, where l, m, n are integers. Then we can define a new
set of basis vectors a∗, b∗, c∗ such that they represent the reciprocal space:

a∗ = 2π

b∗ = 2π

c∗ = 2π

b × c

a · (b × c)

c × a

b · (c × a)
a × b

c · (a × b)

.

(2.9)

In such a definition, a∗ is perpendicular to the (b, c)-plane; b∗ is perpendicular to the

(c, a)-plane; c∗ is perpendicular to the (a, b)-plane.

We can define a reciprocal lattice vector G to be

G = ha∗ + kb∗ + lc∗.

(2.10)

By comparing G with Eq. 2.7, we find that G = s: the reciprocal lattice vector coincides

with the momentum transfers in diffraction. Based on this idea, we use a 2D square reciprocal

lattice as shown in Fig. 2.4a as an example. If we draw a circle of radius R = 2π

λ = ki in such

a way that the incident wave-vector ki starts from circle center and points to a reciprocal

lattice point, then any other points that fall onto the circle would define a kf that can be

shown to produce constructive diffraction spot. This circle (or sphere in 3D) is called Ewald’s

46

sphere. This Ewald’s sphere construction is very useful in identifying diffraction spots once

we know the lattice orientation, beam energy, and beam direction.

Fig. 2.4b shows a more realistic example of the Ewald’s sphere cutting reciprocal lattice

spots for high-energy beams. The wavelength of 100 keV electron beam is 0.037 ˚A. Thus, the
Ewald’s sphere radius will be 2π/λ = 167 ˚A−1. For a real material, for example CeTe3, the
lattice constant is a = 4.4 ˚A. The corresponding reciprocal lattice spot distance is 2π/a = 1.4
˚A−1. So the Ewald’s sphere is about 100 times larger than the reciprocal lattice unit. Given

the Fourier relationship between real space and reciprocal space, if we have a finite object,

the diffraction spots would not be infinitesimal but instead be finite-sized. For a very thin

specimen, the diffraction spots will be elliptical as shown in Fig. 2.4b. If the probe beam

is tilted or has a finite angle (e.g. convergent beam electron diffraction), then we need to

tilt the Ewald’s sphere to see how the cutting is changed to derive the diffraction intensity

based on the overlap. On the other hand, if sample is tilted, we just need to correspondingly

rotate the reciprocal lattice.

Figure 2.4: The Ewald’s sphere intersecting with reciprocal lattice points. a, Ewald’s sphere
for a square 2D lattice. b, A more realistic scale for the Ewald’s sphere and reciprocal spots.

47

kfkisab2.1.4 Structure factor

With Laue equations, Bragg’s Law, and Ewald’s sphere reconstruction, we can quantitatively

determine where and how strong a diffraction spot would show up. Similarly, if one measures

the position, intensity, and peak width of the diffraction spot, then periodicity, orientation

and size of the sample may be determined.

Consider a system with N atoms and the jth atom is at position rj. fj is the atomic

form factor of the jth atom. Then the scattered wave equals to the sum from all atoms

N(cid:88)

Ψ(s) =

−is·rj ,

fje

where s is the momentum transfer defined earlier in Eq. 2.2.

j=1

Then the scattering intensity is obtained by the following expression:

N(cid:88)

−is·rj × N(cid:88)

fje

fjeis·rk

I(s) = Ψ(s) · Ψ∗(s) =

N(cid:88)

N(cid:88)

=

j=1

k=1

−is·(rj−rj ).

fjfke

(2.11)

(2.12)

The structure factor for this system is defined as intensity normalized by 1/(cid:80)N

k=1

j=1

j=1 f 2
j

1(cid:80)N

j=1 f 2
j

N(cid:88)

N(cid:88)

j=1

k=1

S(s) =

−is·(rj−rk).

fjfke

(2.13)

If the object is randomly oriented, define rjk = |rj − rk|. After averaging over all

48

orientations, Eq. 2.13 becomes what is known as the Debye scattering equation

S(s) =

1(cid:80)N

j=1 f 2
j

N(cid:88)

N(cid:88)

j=1

k=1

sin(cid:0)srjk

(cid:1)

srjk

fjfk

.

(2.14)

For scattering from crystalline sample, we consider the lattice unit cell has n atoms. The

jth atom has an atomic coordinate of (xj, yj, zj) and G = ha∗ + kb∗ + lc∗ for the Bragg

reflection. Then the structure factor can be written as

n(cid:88)

Fhkl =

−2πi(hxj +kyj +lzj ).

fje

j=1

The intensity of diffraction spot (h k l) is square of the structural factor

Ihkl = F 2

hkl.

(2.15)

(2.16)

So, for a crystal with specific symmetry, the structural factor for a certain Bragg peak (h

k l ) could be zero. Such peaks are referred to as symmetry forbidden or symmetry absent

peaks. For example, for a body-centered cubic (BCC) Bravais lattice, there are two atoms

per unit cell. Coordinates for the two atoms are xj, yj, zj = (0, 0, 0) and (1/2, 1/2, 1/2). The

structural factor becomes

So,

Fhkl = f [1 + e−iπ(h+k+l)] = f [1 + (−1)h+k+l].

Fhkl = 2f, Ihkl = 4f 2

Fhkl = 0, Ihkl = 0

49

, when (h+k +l ) is even,

, when (h+k +l ) is odd.

(2.17)

(2.18)

Now, we consider another example of the structure factor for a dimerized 1D atomic

chain. As shown in Fig. 2.5a, assume a 1D lattice with N atoms, where each atom is

separated by length a. Every other atom is distorted to form dimerization with a period of

2a. The atomic distortion is u. Each unit cell contains two atoms. The structure factor can

be written as

Consider two conditions:

2(cid:88)

j=1

F =

fje−iq·r.

1, at wave-vector q = 2π

a h, the hth Bragg peak (h is an integer)

FBragg = f [e−2πih u

a + e−2πih(1− u

a )] = 2f cos

(cid:16)

(cid:17)

.

2πh

u
a

So the Bragg peak intensity is

IBragg = 4f 2 cos2(cid:16)

(cid:17)

2πh

u
a

(cid:16)

(cid:17)

u
a

(2.19)

(2.20)

(2.21)

where in the last step, for small u

= 2f 2[1 + cos

4πh

] ≈ 4f 2[1 − 4π2h2(

)2],

u
a

a , we use approximation cos(cid:0)4πh u

a

(cid:1) = 1 − 1

2(4πh2 u

a )2.

From this result, the Bragg peak intensity is decreasing as distortion u increases and for the

same distortion u Bragg peak intensity change also increases as the order of the Bragg peak

increases (h2).

2, at wave-vector of q = 2π

a (h + 1

2) (satellite peak)

FCDW = f [e

−2πi(h+ 1

2 ) u

a + e

−2πi(h+ 1

2 )(1− u

a )] = −i2f sin[2π(h +

1
2

)

u
a

].

(2.22)

50

Figure 2.5: The intensities of Bragg peak and satellite peak for a dimerized 1D atomic
chain. a, Dimerization of a linear atomic chain. b, Bragg peak and satellite peak intensity
evolutions as a function of relative distortion amplitude u/a.

So the satellite peak is

ICDW = 4f 2 sin2

(cid:18)

2π(h +

(cid:19)

(cid:18)

= 2f 2[1 − cos

(cid:19)

]

4π(h +

1
2

)

u
a

(2.23)

≈ 4f 2[4π2(h +

1
2

u
a

)

1
2
)2 u2
a2 ],

51

0.000.020.040.060.080.100.000.050.100.150.200.25Bragg peak at a*Peak intensity (arb. u.)u/aSatellite peak at 0.5a*0.00.20.40.60.81.0Peak intensity (arb. u.)UniformDistortedauablatticelatticewhere an approximation is made for small distortions: 1−cos

(cid:16)

(cid:17)

4π(h + 1

2) u
a

= 1

2[4π(h+ 1

2) u

a ]2.

From this result, the satellite peak intensity is proportional to distortion squared. For the

same distortion u, the satellite peak intensity also changes more with increasing scattering

angle (h + 1

2)2.

The intensity evolution of the Bragg peak and satellite peak is plotted in Fig. 2.5 as a

function of the relative distortion amplitude u/a.

2.1.5 Diffraction from imperfect crystals

In previous discussions, diffraction is on perfect crystals: the sample has no atomic vibrations;

lateral size is infinitely large; the sample is thin enough to justify that there is only single

elastic scattering events; there is no defects or disorders in the sample.

Here, we will discuss three aspects of lattice imperfections:

1, temperature effect, lattice vibrations, or in other words, Debye-Waller effect;

2, finite sample size effect;

3, effect of strains, disorder, or defects.

Debye-Waller Effect:

Consider the structure factor in Eq. 2.11, the atomic coordinate now is no longer a

constant but instead time-dependent: rj(t) = rj + uj(t), where uj(t) is the jth atomic

distortion away from equilibrium position rj. Since they are random atomic vibrations and

each atom is independent, the time average of the structure factor can be written as

Ψ(s) =

−is·rj(cid:104)e−is·u(cid:105).

fje

(2.24)

N(cid:88)

j=1

52

The scattering intensity has an additional factor compared with Eq. 2.12

N(cid:88)

N(cid:88)

IDW F (s) = (cid:104)e−is·u(cid:105)2

−is·(rj−rj ) = (cid:104)e−is·u(cid:105)2I0(s).

fjfke

(2.25)

j=1

k=1

Mathematically, if s · u is small or u follows a Gaussian distribution, then (cid:104)e−is·u(cid:105) =
− 1
2(cid:104)(s·u)2(cid:105)
. So the term (cid:104)e−is·u(cid:105)2 in Eq. 2.25 becomes: e−(cid:104)(s·u)2(cid:105). This term is called the
e

Debye-Waller factor (DWF)

DW F = e−(cid:104)(s·u)2(cid:105).

(2.26)

If the system is isotropic, then the average of (cid:104)(s · u)2(cid:105) = s2(cid:104)u2(cid:105)| cos θ|2 over spherical
3 s2(cid:104)u2(cid:105). The DWF is reduced to a simpler form

4π is 1

DW F = e

− 1
3 s2(cid:104)u2(cid:105)

.

(2.27)

From the DWF expression, we can see it is s- or q-dependent. At large s, the DWF is
smaller so Bragg peak intensity is attenuated more. In the − ln(I(s)) vs. s plot, the curve

is quadratic.

Another important thing to note is that thermal vibrations or DWF only attenuates the

Bragg peak intensity. It does not broaden the peak.

Finite sample size effect:

Consider a lattice of finite size consisting of N parallel planes with each neighboring plane

separated by distance a. Then calculated structure factor is the same as shown in Eq. 2.14.

With some transformation, it can be written as

S(s) =

1
N

sin2(N sa/2)
sin2(sa/2)

,

53

(2.28)

which gives a series of Bragg peaks at sP K = 0, 2π/a, 4π/a, ..., and peak height N .

To determine the peak width (FWHM), assume at s = sP K + ∆s, the peak is at half

height of N/2. The solution for this would be: N a∆s = 5.56.

The peak’s FWHM β (after deconvoluting the instrumental width) in radians is correlated

with ∆s by: β = 2∆s/

ds
dθ

= 2∆ 4π

λ cos(θb), where θb is the Bragg angle. Comparing this

with N a∆s = 5.56, we get the following relation

particle size: p =

0.88λ

β cos(θb)

.

(2.29)

This equation is called the Scherrer equation. This can be generalized to equation

particle size: p = K

λ

β cos(θb)

,

(2.30)

where K is dimensionless shape factor. The typical value of K is 0.9 but varies with the

actual shape of crystalline and crystal symmetries.

With finite particle size, from Eq. 2.28 and Eq. 2.30, we can conclude that with decreasing

crystal size, peak amplitude is decreasing while peak width is increasing.

Effect of strains, disorder, or defects:

In materials with defects or disorder, there will be strains around the defects. Such strain

will cause atoms to move away from perfect positions, thus broadening the Bragg peak.

Quantitatively, there have been a lot of derivations on the strain effect, refer to Refs.

[73–75]. The effect can be summarized by the following relation

e =

β

2 tan θb

,

54

(2.31)

where e is a measure of the micro-strain distribution, β is the FWHM of the Bragg peak in

radians after deconvoluting instrumental width, θb is the Bragg angle.

With a perfect crystal and ideal scattering event, the Bragg peak is an infinitely sharp

Delta function. But with these crystal imperfections, the peak is broadened to finite width

with some kind of profile. A lot of times, assuming a Lorentzian profile of the finite size and

strain effect gives a relatively good fit. By combining the instrumental broadening, which

is usually a Gaussian profile, the real peak profile is usually a combination of Gaussian and

Lorentzian, i.e. a Voigt function. So in the following chapters when fitting peak width to

extract correlation length, etc, a Voigt function is used for the fitting.

2.2 Ultrafast electron diffraction

One of the ultimate dreams in the scientific community is to watch the motion of atoms

or electrons. To fulfill such a dream, scientists have been designing experimental tools to

improve the time resolution of various measurements.

After the construction of the first laser in 1960, the field of lasers grew exponentially

from the 1960s to 1980s. The idea of using ultrafast electron diffraction (UED) to study

chemical reactions was first proposed by Prof. Ahmed Zewail in 1991 [76], then experi-

mentally demonstrated in the following several years [77–80]. Over the past 25 years, UED

has blossomed into a multi-disciplinary field that has been applied to chemistry, biophysics,

condensed matter physics, accelerator physics, and material science, etc.

Ultrafast electron diffraction is a pump-probe technique that uses a laser pulse to excite

the sample, which defines the time-zero, then uses a short electron bunch to probe the

subsequent structural changes induced by the laser in real time.

55

2.2.1 Pump-probe technique

Particles (electrons, atomic ions or phonons, etc.) in materials can get excited, interact with

each other, or propagate through the sample medium to transport heat or electricity. Such

excitation or interaction processes take place from fs to ns scales. To better understand

these ultrafast processes at microscopic scales, there is a need for a tool that is fast enough

to capture these processes.

Figure 2.6: Timescales of different electron and lattice processes after laser excitation [5].
Each green bar represents the typical time scales and may vary for different materials and
different excitation conditions.

A famous example in history similar to this is the “horse galloping” problem in 1878.

People at the time were trying to solve the argument of whether a race horse’s feet were

actually all off the ground at any single time during a gallop. This problem was solved by

the development of instantaneous photography technology, with which they could actually

capture the whole course of the horse’s motion in real time and solve the debate definitely.

56

Similarly, to capture the microscopic interaction processes in real time, we need to have a

“camera” that is fast enough to resolve the material responses in fs scales. Fig. 2.6 (adapted

from Ref. [5]) summarizes the characteristic timescales of microscopic interaction processes

after fs laser pulse excitation.

Of course, there is not a mechanical camera capable of such a fast response. To achieve

fs resolution, scientists have come up with the idea of a “pump-probe” scheme, in which one

uses a fs laser pulse to first perturb the system, which defines a time-zero, then uses another

short pulse to probe the subsequent parameter changes. The probe pulse could be a fs laser

pulse (THz pulse, visible, or X-ray, etc.), or a fs electron pulse. The overall time resolution

of this pump-probe technique is the convolution of pump and probe pulse lengths. After

deconvoluting the instrumental resolution width from the experimental data, we can get the

intrinsic response of the material.

The typical interaction between probe pulse and material depends on the probe type

and the interpretation could be different. For example, THz pulses mainly interact with

conduction electrons, thus giving direct information on a material’s conductivity. UV or soft

X-ray pulses can photo-emit electrons from the material surface, the energy vs. momentum

relation of the photo-electrons is key to resolve electronic behavior near the Fermi surface.

On the other hand, hard X-rays or high energy electron pulses directly map out the lattice

vibrations and atomic movements.

Fig. 2.7 is a simplified schematic drawing of the UED setup.

In conventional transmission electron microscopy (TEM), the system has been optimized

in the source, optics, and detector such that it can routinely obtain an image spatial resolu-

tion of 0.4 ˚A (Ref. [81]) and an energy resolution of sub-20 meV (Ref. [82]). But the time

resolution in conventional TEMs is typically limited to the camera’s acquisition speed.

57

Figure 2.7: The schematic drawing of ultrafast electron diffraction experimental setup. Near-
infrared 800 nm laser pulse is split into two paths: one goes through an optical delay stage to
serve as a pump pulse, the other goes through third-harmonic generation (THG) to generate
ultraviolet (UV) laser pulse for electron pulse generation. Of course, this is just a way
simplified schematic drawing. Electron gun, sample, and CCD camera have to be in ultra-
high vacuum (UHV) chamber. The electron optics, such as magnetic lens and radio-frequency
(RF) cavity, are also omitted out in here.

In a UED setup, to achieve a good time, spatial, energy resolution, the key lies in im-

proving the probe electron pulse characteristics. Several different approaches are adopted

to improve on this. For example, to have a good time resolution, one can employ an RF

cavity to longetudinally compress the electron pulse to overcome the space charge effect, or

accelerate electrons to relativistic energies. To have a high coherence length or high bright-

ness, we can either use a tip source or optimize the photo-emitted laser pulse and electron

optics. Table 2.1 lists a few approaches in optimizing the UED and UEM (ultrafast electron

microscopy) systems.

A compact UED setup has the advantage of a relatively simple instrumental design, and

a moderate bunch electron number. But the time resolution is usually limited by the space

charge effect when pushing for a high intensity.

The few-electron UEM system takes advantage of the existing TEM electron optics and

58

THGfs laser sourceCCD cameraDelaystagee- gune- pulseSample 266 nm800 nm800 nm∆tMethod

Groups

Rep.

Rate(kHz)

Ne

Ek(keV)

τ (ps)

Compact UED

MSU/Toronto-Max

1

Few e− UEM

Planck Inst.

Caltech

RF UED(M)

MSU/McGill

Low-energy UED

G¨ottingen

MeV UED

DTEM

FEL (X-ray)

SLAC

LLNL

SLAC

104

1

10

0.12
10−3

0.12

103

5

106

100

105

> 109
∼ 1013

50

100

100

0.4

3000

200

5

≤ 0.5

0.8
≤ 0.1

2

≤ 0.1

104

0.05

Table 2.1: Comparison of several typical UED/UEM techniques and some typical parameters.
The Group column just list one or two groups as examples and definitely not a complete
list. All the numbers are typical numbers and may not represent a specific system. Ne is
electron number per pulse. Ek is the electron kinetic energy. τ is electron pulse duration.

has a relatively good time resolution. The disadvantage lies in the low bunch electron number

and the required high repetition rate that may not be suitable for some experiments.

The RF-enabled UED or UEM utilizes a large number of electrons per bunch and can also

achieve a very good time resolution. It is one of the actively developed forefront techniques

in the ultrafast community.

For a low-energy UED system with a sharp electron source, it has a relatively high

transverse brightness to afford a high coherence length beam without demanding electron

optics. The disadvantage lies in relatively poor time resolution, a low electron number, and

the electron-laser velocity mismatch problem.

The relativistic MeV UED system is proven to be capable of delivering high-intensity

59

beams with a very good time resolution. It is also one of the cutting edge techniques at the

current stage. The disadvantages may lie in the high-cost instrumentation, the high demand

on electron optics, and a relatively low repetition rate compared with the RF-enabled UEM.

The dynamical-TEM (DTEM) is a different branch compared with the above described

methods. DTEM pushes for a high bunch electron number, a high spatial resolution, and

capabilities for the single-shot imaging or diffraction experiments by operating in the ns

timescales.

In Table 2.1, the free electron laser (FEL) is also listed mainly for comparison. The FEL

is the state-of-the-art facility-based technique that generates extremely high-brightness and

ultrashort pulsed X-rays. The FEL X-ray source can be used for both spectroscopy and

diffraction.

2.2.2 Data interpretation in ultrafast electron diffraction

Diffraction patterns are Fourier transform of the real space lattice distribution. So in theory,

any change induced by a pump laser will be reflected in the diffraction pattern. Up to now,

we have discussed a few key aspects related to diffraction. These are the guiding principles

for analyzing diffraction data. For example, for single crystal diffraction patterns, peak shift

usually corresponds to lattice constant change; Bragg peak intensity change usually comes

from either the structure factor change or the DWF. Peak width would usually correspond

to the formation of defects and correlation length change. Diffusive background in-between

Bragg peaks is from inelastic scattering and has valuable information on lattice phonons.

Next I will give a few UED experiments from other groups to demonstrate each case.

Peak position shift:

Laser excitation may result in sample symmetry change. For example from orthorhombic

60

to tetragonal lattices, such a transition involves a lattice constant change and results in

Bragg peak position shift. Sometimes, optical excitation can also cause lattice expansion or

contraction. As shown in Fig. 2.8 (Ref. [6]), the authors performed experiments on graphite.

The incident electron beam has a certain angle relative to the graphite c-axis, so the peak
with (hkl) indexes of l (cid:54)= 0 can be monitored to check the inter-layer spacing change. In

this case, they observed inter-layer spacing, c ,increases, accompanied by the lowest order

longitudinal acoustic phonon mode (sometimes called the breathing mode).

Figure 2.8: The laser-induced lattice constant change in graphite captured by ultrafast
electron diffraction [6]. a, Experimental geometry. b, Lattice constant c change and the
excitation of the breathing mode phonon.

Peak intensity structure factor analysis:

Atoms are confined in the lattice with specific arrangements by Coulomb interactions.

After absorbing photons with electronic excitation, the overall energy landscape will change,

so as the atomic positions. Such atomic movement can be tracked in diffraction patterns

and simulated based on structure factor calculations.

In Fig. 2.9 (Ref. [7]), the authors

examined the intensity change of many Bragg peaks and came up with the idea of inter-layer

61

abshear phonon modes. By writing down the structure factor considering shear phonon modes,

the simulated peak intensity changes agree with the experimental data quite well. So the

shear displacement can be extracted and further used to explore the intricate physics in this

topological quantum material.

Figure 2.9: MeV ultrafast electron diffraction experiment on Weyl semimetal WTe2 [7]. a,
Lattice structure of layered WTe2. b, Experimental setup. THz pump, MeV electron probe
is employed in the experiment. c, Responses of different Bragg peaks. d, Schematic drawing
of the energy potential with changes of the shear distortion. e, The time-dependent shear
distortion at two different pump fluences.

Peak intensity Debye-Waller analysis:

As discussed earlier, in photo-induced phase transitions (PIPT) in almost any system,

electron phonon coupling will show up as an electronic energy relaxational channel. Such

a coupling results in coherent or incoherent phonon excitations and thermal heating in the

62

acdebsystem. The thermal effect is manifested as Debye-Waller effect in the diffraction patterns. In

Fig. 2.10 (Ref. [8]), the authors study a simple system, polycrystalline copper. By examining

the Bragg peak intensity changes at different q values and different time delays, they are able

to track the sample temperature evolution at different times and fluences with a simple two-

temperature model. Such a study allows them to determine the electron phonon coupling

strength and thermal transport rate for copper.

Figure 2.10: The Debye-Waller effect and electron-phonon coupling analyses in polycrys-
talline copper [8]. a, MeV UED pump-probe experimental setup. b, The Debye-Waller
analysis of Bragg peaks at different q-values and different time delays.

Peak width correlation length analysis:

Crystal defects induce strains in the sample and reduce the correlation length of the

system. Such an effect manifests as peak broadening in diffraction. Correlation length is one

of the key parameters in tracking phase transitions and it can be deduced from the peak width

analysis. In Fig. 2.11 (Ref. [9]), the authors used an ultrafast low-energy electron diffraction

method to study phase transitions in a CDW material 1T-TaS2, in which they monitored

the peak width evolution and extracted the corresponding correlation length of the new

phase. It is found that a 1/2 power-law increase is independent of the pump fluence. Such

63

aba correlation growth is attributed to decay of CDW dislocations based on a time-dependent

Landau-Ginzburg theoretical modeling.

Figure 2.11: The correlation length analysis in a phase transition in CDW material 1T-
TaS2 [9]. a, The ultrafast low-energy electron diffraction setup. b, The growth of correlation
length follows a 1/2 power-law scaling growth.

Diffusive scattering background phonon analysis:

Phonons are lattice vibrations with certain energy and momentum. They interact and

transfer energy/momentum with probing particles like X-ray photons, electrons or neutrons.

For example, one of the most noticeable phonons in graphite is the in-plane E2g phonon at Γ
point. It shows up in the Raman spectrum and has a frequency of 1582 cm−1, corresponding

to an energy of 190 meV. This is among the highest energy phonons. Other phonons could

have energy of less than 1 meV. For phonons with such energies, the transferred energy or

momentum to the high-energy electron probe particles is very small and hard to detect. So

normally, such optical phonons are be detected by Raman spectrum. Meanwhile, the phonon

spectrum can be mapped out by low-energy (meV) neutron scattering or high precision

inelastic X-ray scattering. An alternative approach to probe phonons is through diffusive

scattering. With ultrafast electron diffraction, it is now possible to map out the momentum-

dependent phonon responses in the Brillouin zone (BZ) through the differential difference

method.

64

abFigure 2.12: The diffusive scattering background phonon analyses in graphite [10]. a, Orig-
inal diffraction pattern of graphite with electron beam along the c axis. (b-f ), The differ-
ence of diffraction patterns at different time delays showing diffusive scattering background
evolution. g, The dynamics of Bragg peak (2 1 0). h, The dynamics of diffusive scatter-
ing background at different points of the Brillouin zone, corresponding to different phonon
branches in graphite.

In Ref. [10], the authors demonstrated the ability of UED for mapping the momentum-

dependent phonon relaxations across the entire BZ by following the diffusive scattering

patterns. As shown in Fig. 2.12, the phonon momentum transfer at different BZ positions

are identified along with their time evolution. Such a time-dependent study may reveal the

energy flow in graphite.

65

ghbcfedaChapter 3

Observation of an emergent

bi-directional order in CeTe3

Spontaneous symmetry breaking (SSB) is one of the most common phenomena in physics.

For example, during metal-to-superconductor, paramagnetic-to-ferromagnetic, metal-to-charge

or spin-density wave transitions, they all involve SSB by breaking a certain kind of symmetry.

By lowering the system temperature across Tc, the system goes to a lower symmetry and

lower energy state. On the other hand, with laser excitations, the system absorbs energy and

typically goes to a higher symmetry state. For instance, with fs laser excitation, researchers

studied ultrafast melting of the charge- or spin-density waves, or melting of certain magnetic

orders. However, it is rare to see laser-induced superconductivity, laser-induced charge or

spin-density waves, etc. Here in this chapter, we will show such an example of laser-induced

new symmetry-breaking route and exploration of the hidden states in a CDW system.

In this chapter, I will first give a little background on the materials in Sec. 3.1. The

ultrafast results will be presented in Sec. 3.2. More discussions will follow in Sec. 3.3. Then

a quick summary is given in Sec. 3.4.

66

3.1 Background and properties of CeTe3

CeTe3 belongs to a class of materials called rare-earth tritellutrides (RETe3), with rare-earth

elements in the lanthanide series. The RETe3 series share the same structure, as shown in

Fig. 3.1a. It consists of Te square nets sandwiching the corrugated RETe plane and CDW

emerges within the Te planes (Fig. 3.1b). Bands near the Fermi surface (FS) in the Te

layer are partially filled by Te 5p electrons. Due to the perpendicular hopping (t⊥ in Fig.

3.1c), the FS (Fig. 3.1d) is in a warped diamond shape rather than perfect square. From

photoemission studies [83, 84], the FS structure has parallel portions and the FS nesting

vector matches well with the CDW wave-vector, suggesting that the CDW is most likely

driven by FS nesting. The nesting vector is drawn in the figure.

The crystal structure of RETe3 is slightly orthorhombic such that a and c axes are not

equivalent. For light rare-earth elements like La, Ce, Sm, and Gd, the atomic radius is large

and only one CDW is observed along c axis. For heavier elements with smaller atomic radius,

the CDW is observed to be shown in both directions at different temperatures (see Fig. 3.1e).

There is a well defined trend of transition temperature under chemical pressure [85]. For

CeTe3 studied here, as seen in Fig. 3.1e, there is only one CDW formed along the c-axis

(stripe phase).

In the CeTe3 experiment, we perform experiments on the freestanding thin flakes of

single-crystal (Fig. 3.1f), prepared by the scotch tape exfoliation method (details about

sample preparation can be found in Appendix A). The sample is exfoliated and transferred

to the 1000-mesh gold TEM grid. The sample thickness is characterized by TEM zero-loss
EELS to be 30± 5 nm. The electron beam is along the crystal b axis [0 1 0], perpendicular

to the a-c plane. The high quality sample yields very high signal-noise-ratio (s/n) diffraction

67

Figure 3.1: Structure, FS, and CDW formation in the RETe3 series. a, Ball-stick model of
RETe3 structure. The solid cuboid is the unit cell with a, b, c labeled. b, Top view of the
square net Te layer in which CDW is formed. c, The electron orbitals in the Te layer are 5px
and 5pz orbitals from Te atoms. d, The schematic FS of the square Te net. Dashed line is
the BZ boundary. e, The transition temperature of different RE elements. f, Freestanding
thin sample flakes on 1000-mesh gold TEM grid.

68

abcTeCeacbxyzabcTeRETe layer top viewact//t//tƦtƦpxpzxzc*-qca*-qaacdefb4.264.304.344.380100200300400500TbDyHoEr c-CDW a-CDWTemperature (K)Lattice constant a (Å)CeSmGdTbDyHoErTmUniaxial CDWBi-directionalCDWNormal phaseTc1Tc225 µmpatterns, as shown in Fig. 3.2a.

The diffraction pattern of CeTe3 consists of alternating weak and strong lattice Bragg

peaks due to its unit cell structure factor:

for peak (h 0 l ), if h + l is even, then it is a

strong Bragg peak; otherwise the Bragg peak is weak, as seen from Fig. 3.2b. In addition,
there is a pair of satellite peaks along the weak Bragg peak in the c∗ direction, indicating

CDW formation along the c-axis. This CDW wave-vector agrees with what was reported

earlier [86]: q c ≈ 0.28c∗.

Figs. 3.2c&d show the scaled-up view of CDW satellites on the same color scale. Before
laser excitation (- 1 ps), satellite peaks only exist at along the c∗ axis. After laser excitation

(+ 1 ps), the original CDW peaks become weaker. But at the same time, a pair of new
satellite peaks are observed along the a∗ axis. Such new satellite peak formation is a direct
signature of new CDW order along the a∗ axis, thus forming a bi-directional CDW that is

thermodynamically forbidden.

From the Bragg peak distance analysis, the lattice constant changes in a and c are

both within the instrument resolution of 0.05% — much less than the required 1.4% lattice

constant change to form bi-directional CDW order (from CeTe3 to TbTe3 in Fig.3.1e). So

there is no way that the new CDW formation is due to laser-induced stress or lattice constant

change.

3.2 Ultrafast dynamics of CeTe3

The observation of this new CDW order is surprising because such bi-directional CDW order

in CeTe3 has been ruled out in the steady-state phase diagram as shown in Fig. 3.1e. We

examined the full dynamical evolution of the order parameters. From the result in Fig. 3.3,

69

Figure 3.2: The diffraction patterns of CeTe3. a, Ground state full diffraction pattern of
CeTe3 at RT. b, Zoom-in view of the cyan square box in a. The Bragg peaks are labeled
and Bragg and CDW wave-vectors are also indicated. c, Zoom-in view of the black dashed
square box in b at negative time delay. d, The same area as in c, but at + 1 ps after laser
excitation. c and d are in the same scale as indicated by the scale bar on the left. New
emergent CDW satellite peaks are clearly observed along the a∗ axis (a-CDW).

the c-CDW peaks get suppressed to ≈ 15% in less than 500 fs then partially recovers to 70%

in a time scale of 4 ps. Note, the full recovery to the same state as negative time takes

beyond ns range but within 1 ms before the next pump-probe cycle starts. For a-CDW,
with a delay of ∼ 300 fs after time-zero, it starts to form and the maximum intensity at this

70

a*c*qcabcd(4 0 2)(5 0 1)(3 0 1)(4 0 0)(4 0 1)025050075010001250(4 0 1)(4 0 1)qaqc-1 ps+1 psfluence is ≈ 10% of c-CDW intensity at the negative time. As the c-CDW peak partially

recovers, the a-CDW intensity also decays, suggesting that the two symmetry-broken ground

states are competing.

Figure 3.3: The integrated intensity changes of c- and a-CDW peaks. Both c- and a-CDW
peak intensities are normalized to the c-CDW intensity at negative time delay. Inset is the
difference diffraction pattern: diffraction pattern at + 1 ps minus that of - 1 ps.

Scattering experiments with X-ray, neutrons or high coherence length electrons could

measure both the static and fluctuating orders. To better understand this, the peak width,

which is a direct representation of the corresponding order’s correlation length, is examined.

As discussed in Sec. 2.1.5, due to crystal defects, the diffraction peak will be broadened

instead of infinitely sharp. In theory, the broadened profile due to imperfections could be

71

-202468100.00.20.40.60.81.0 c-CDW a-CDW, x 2Relative intensityDelay, t (ps)F =1.85 mJ/cm2Gaussian, Lorentzian, or more complicated functions, depending on the type of imperfections

and assumptions made. But most of the time, broadening with a Lorentzian profile gives a

good fit for the presence of inhomogeneities [74]. In this case, while the instrument resolution

is typically described by a Gaussian, the CDW peak can be described by a Voigt function,

which is a Lorentzian function convoluted with a Gaussian. The CDW peak in Fig. 3.4a

would neither be fitted with a Gaussian nor Lorentzian. Only a Voigt function can fit the

profile well.

In Fig. 3.4b, the black curve represents a lattice Bragg peak. Due to the infinite corre-

lation length of Bragg peak, the broadened width (Lorentzian width) is close to zero and a

Gaussian function can fit the entire peak relatively well. This Gaussian function is set as the

instrumental width for deconvoluting other correlation lengths. The other several colored

curves in Fig. 3.4b are examples of several peak profiles with different correlation lengths.

In addition, we find that after laser excitation, one single Voigt function does not fit the

c-CDW peak well, as demonstrated in Fig. 3.4c. The black curve is single-Voigt function

and does not fit the data. Instead, we have to use two-Voigt functions to do the fitting. Red

curve is the sum of two Voigt functions and the two individual Voigt functions are in dashed

lines. Such two-components fitting is appropriate for describing coexistence of static and

fluctuating orders, which occurs when a system undergoes a continuous phase transition [87],

where additional broadened peak emerges due to critical fluctuations [87]. However here, as

the system is spontaneously quenched across the critical point, strong fluctuations are created

due to instability under a new energy landscape. The unstable CDW fields represented by

these fluctuations will coarsen over time to form the new broken-symmetry order.

To further look at the fluctuations, we examined the peak intensity at the shoulder region

(Fig. 3.5b). It is found that while the peak center (red curve) is decreasing in intensity, the

72

Figure 3.4: The pseudo-Voigt function peak fitting for extracting the correlation length,
static, and fluctuating components of a peak. a, c-CDW peak at negative time where the
fluctuation is small, so a single Voigt function fits it well. Neither a Gaussian nor a Lorentzian
would fit. b, Voigt function fitting of several CDW peak profiles. The sharpest black curve is
a Bragg peak with Gaussian fitting and others are by Voigt function fitting. c, c-CDW peak
at + 1 ps, at which a single Voigt function (black curve) does not fit. We have to use two
Voigt functions to do the fitting and the two components are in dashed lines corresponding
to the static and fluctuating/critical components.

shoulder part (purple curve) normalized to the peak center is actually gaining intensity. Such

a phenomenon is indicative of a coarsening process. As seen from the pie chart population

change in Fig. 3.5c, the original static order in the blue color is losing population in the first
1 ps, then partially recovers; at the same time, a new c-CDW (hereafter referred as c†) in

the pink color and a-CDW in green gain population.

Based on KZM, for a initially disordered state after quench, the system goes through the

critical point to form an ordered state. Naturally, topological defects would be formed so as

to reconcile the phases of the independently grown ordered states in disconnected regions.

Such formation of defects could be long-living if the system is rapidly quenched.

73

-0.10.00.1100020003000400050006000 Bragg peak, ξ→∞  ξ=33 nm  ξ=2 nm  ξ=10 nmIntensity (arb. u.)q-qc (r.l.u.)-0.050.000.05100020003000400050006000 c-CDW at -1 ps Pseudo-Voigt fit Gaussian Fit Lorentzian FitIntensity (arb. u.)q-qc (r.l.u.)abc-0.2-0.10.00.10.2 c-CDW at  +1 ps 2-Voigt fit Static, c portion Critical, c* portion 1-Voigt fitIntensity (arb. u.)q-qc (r.l.u)100020003000400050006000Figure 3.5: Analyses of the fluctuating order and real space reconstruction. a, Schematic
demonstration of the static and fluctuating orders in correlation length and population near
the thermal critical point. b, Dynamics of intensity evolution. Top panel is the intensity
evolution of peak shoulder (δq = 0.07) and peak center. Inset are images before and after
to show the suppression of c-CDW and appearance of a-CDW. The bottom panel dynamics
shows the integrated intensity of c-CDW static portion and a-CDW. Inset shows the linear
relation of correlation length and CDW intensity. c, Peak profiles and 2-Voigt function
fitting. Pie chart shows the population change of each component: blue is c-CDW static
part, pink is c-CDW fluctuating order, green is a-CDW portion. d, Real space reconstruction
based on diffraction patterns, check text for details.

74

025050075010001250-1 ps(4 0 1)0.3 ps  qcqa0.10.20.30.40.5q (r.l.u)1 ps  3 ps  10 ps  δq=0.07c*-2qc-1 ps  log(I)M(r, t)+1 psqcqa(4 0 1)δq=0.07-12305001000150020002500 Δq=0 Δq=0.07x 50x -1Amplitude I(q) (arb. u)02468100.00.51.01.5 c-CDW a-CDW    Integrated intensity IintDelay, t (ps) ξ15 nmx 2ξcξaChart Title1234x 2.2x 6.7x 3.5x 1.60.00.20.40.60.81.0010203040ξc (nm)Integrated intensity Iint0.3ps12343ps123410ps123427 nmTCorder parameterLog I(q)qqqqTotal  Static Critical correlation lengthT10-110-210-31c-CDW a-CDW ScatteringCritical Static Real spaceDiffraction line profileabcdThe real-space CDW maps are reconstructed based on the experimental data. First,
CDW peak intensities I (a-, c†-, and c-CDWs), Q-values, and correlation lengths ξ are

extracted from data. Then they are used as input parameters for the CDW order parameters

(cid:88)

(cid:113)
Iint,j(Qj(t))ei(Qj (t)·r+φj )e

Ψe(r, t) =

− r−r0
2ξj (t) .

(3.1)

j

For no other good reasons, we simply assign φj = 0 and set the reference position r0 to

be at the origin. The real-space CDW map (Fig. 3.5d) is calculated to be

M (r, t) = |Ψe(r, t)|2.

(3.2)

Performing Fourier transform on M (r, t), we can obtain the simulated diffraction pattern

(see Fig. 3.6). To refine the reconstruction parameters, simulated diffraction curve is com-

pared with the experimental one (See Fig. 3.5c) by doing the same line scan over the CDW

peaks to retrieve the peak profiles.

Besides the change at a single pump fluence, the fluence-dependent responses are also

studied and shown in Fig. 3.7. It is found that the a- and c-CDWs have different thresholds

with F

(a)
c = 0.6 mJ/cm2, F

(c)
c = 1.9 mJ/cm2, as indicated in Fig. 3.7b. The a-CDW intensity

quickly rises at F

(a)
c

, then plateaus while by contrast, c-CDW intensity drops linearly with

fluence in both short and long times.

From the simulation results in Fig. 3.7c, from time zero to 1 ps, the bi-directional orders

become more and more prominent. After 1 ps, the bi-directional order starts to “shrink” and

decay. Also, topological defects are visible at the bi-directional phase domain boundaries.

75

Figure 3.6: Comparison of the experimental pattern with simulated pattern.

3.3 Mechanism for the non-equilibrium phase transi-

tion

From previous results, there is clear evidence that a bi-directional phase forms under a

nonthermal scenario. But we have not gone into details about why this happens, how we

may take advantage of it, and what general implication this experiment might have on other

systems.

According to KZM discussed in Sec. 1.2.3, if the system is quenched from a disordered

phase to an ordered state at a finite rate, the relaxation time near the critical point would

diverge (critical slowing down). As a result, the order parameter might not follow the quench,

so there is a lag in the system response, usually behaving as a delay in the growth of the

order parameter. Such a delay or lag period is called Kibble-Zurek (KZ) freeze-out time

τKZ . Experimentally, it is not easy to apply a finite rate quench while tracking the order

parameter change. The conventional protocol is to track the post-quench topological defect

76

qaqcG(0 0 0)QcQaExperimentSimulated patternqaqcFigure 3.7: The fluence-dependent data and evolution of topological defects. a, c-CDW
intensity dynamics at different pump fluences. Inset: the characteristic time of the recovery
time for c-CDW order at different fluences. b, The extracted c-CDW and a-CDW amplitude
evolution from fluence scan. The corresponding c- and a-CDW thresholds are indicated. c,
Dynamics at two fluences above F
. Inset: evolution of topological defects at F = 1.85
mJ/cm2.

(a)
c

density, which is related to the observed correlation length.

In this experiment, we have

enough time resolution to follow the early non-adiabatic evolution of the order parameter,

therefore can shed light on the Kibble-Zurek freeze-out phenomenon.

77

acb0.00.51.01.52.00.00.20.40.60.81.0 c-CDW at 1 ps c-CDW at 1 ns a-CDW at 1 psRelative intensityF (mJ/cm2)Fc(c)Fc(a)-10121010010000.00.20.40.60.81.01.21.42  0.99 mJ/cm2  1.85 mJ/cm22Relative IntensityDelay, t (ps)-1 ps0.3 ps1 ps3 ps10 ps1 ns20 nm-2024681010010000.00.20.40.60.81.01.2Fluence, F (mJ/cm2) 0.28 0.57 0.77 0.99 1.19 1.42 1.62 1.85Relative intensityDelay, t (ps)0.00.51.01.52.02345Recovery time   τ  (ps) F (mJ/cm2)Figure 3.8: Simulations of the quench profile and the Kibble-Zurek freeze-out time.

From Fig. 3.3, after fs quench of the c-CDW state (time-zero), a-CDW order is not imme-

diately formed and there is about 300 fs time delay in the onset of a-CDW ordering. Such a

delay, as we will try to explain later, might fit into the KZM freeze-out time hypothesis. Fig.

3.8 is a schematic demonstration of the quench profile and freeze-out time. The top panel

is the quench profile and induced carrier density change, which is calculated based on the

sample optical constants [88, 89] and the thin film transfer matrix method [90]. The further

decay at longer than 1 ps is based on correlation length partial recovery from 1 ps to about

10 ps. In the bottom panel, the orange curve is the relaxation time, calculated based on
non-equilibrium relaxation time: τ = |
|zν, where 
 is reduced temperature: 
 = Tc−T
Tc

. The

78

 0.01 0.1 1 10 102 103−150−100−50050 100 150 200 250 Delay, t (fs)Time (fs)τKZRelaxation time τ1ps6ps11psQuench time τQKZMQuench profileF =1.85 mJ/cm2Critical level, nCDensity (nm-3)Laser pulseQuenchFreeze-outPhase orderingRecovery0.11.0green curve corresponds to the quench time, which is inverse of the quench rate at t. When
the relaxation rate (∼ 1/τ ) catches up with the quench rate, the system becomes adiabatic

again.

However, within the freeze-out time, local defects are formed and annihilated, driven

by short-range fluctuations. Meanwhile, the coarsening is mediated by the long-wavelength
fluctuation to grow to the domain size. At ∼ 1 ps, the static a-CDW is established. However,

due to the thermalization in the system at longer times, the c-CDW starts to recover and

competes with a-CDW. So the a-CDW becomes unstable. But topological defects remain

until more than 1 ns as shown in Fig. 3.7c.

Aside from the CDW peaks, we also analyzed Bragg peaks and found interesting dynamics

as shown in Fig. 3.9a. From Fig. 2.5, we can see in a CDW system, the Bragg peak and CDW

peak intensities are anti-correlated. So in theory, when c-CDW get suppressed, the Bragg

peaks along the c-axis should gain intensity, and vice versa for those along the a direction.

But in the experiment, what we see is contrary to that: Bragg peak along the a-axis, e.g.

peak (6 0 0), gains intensity first then drops; Bragg peak along the c-axis, e.g. peak (0 0

6), has no rise and directly goes down. Such an orthogonal correlation clearly demonstrates

that the softened modes associated with both the a- and c-CDWs formations are transverse

phonons. This is consistent with a recent density functional perturbation theory study of

RETe3, which predicts the soft modes consisting of transverse optical modes [91]. To better

explore the CDW formation, we examined all the Bragg peaks of similar s values with peak

(6 0 0), as shown in Fig. 3.9a inset.

Bragg peak intensity changes can be treated through the Debye-Waller analysis. Based

79

Figure 3.9: The q-dependent Debye-Waller analysis and FSN wave-vector evolution. a, Three
representative Bragg peaks dynamics at the first 10ps.
Inset: the schematic diffraction
pattern of CeTe3. Bragg peaks in red-circle in similar s value are used for Debye-Waller
analysis in b. b, the q-dependent Debye-Waller analysis at different time delays. c, a-
and c-CDWs FS nesting vector changes as a function of time. d, Schematic FS structure
of the square net Te layer (solid black line). Square gray dashed line is the BZ boundary,
dashed orange lines are projected FS change after laser excitation, drawn based on the CDW
nesting vector changes in c. e, Theoretical calculations of the susceptibility (χ) and nesting
wave-vector. f, Peak position of the electronic susceptibility along the (11) direction

80

cbadefa*c*(0 0 0)(0 0 2)(0 0 4)(0 0 6)(2 0 0)(4 0 0)(6 0 0)02468100.70.80.91.01.1 (6 0 0) (4 0 4) (0 0 6)Delay, t (ps)I(G,t)/I002468100.2800.2850.2900.2950.300  q (r.l.u.)Delay, t (ps)qcqa0.1Åa*-axisc*-axisτKZt = -1 ps0.2 ps0.6 ps1.5 ps3.0 ps10 psc*-qca*-qa∆q/qeqVπ (eV)0.60.70.80.91.00.50.000-0.005-0.030-0.025-0.020-0.015-0.010PK1PK2q/qeqVπ (eV)1.00.8750.750.6250.50.60.71.10.80.91.01.20.600.300.400.500.550.350.45χ(q,ω=0)  (arb. unit.) PK1PK2on Eq. 2.26, we get

DW F =

I(t)

I(t < 0)

= e−(cid:104)(s·u(t))2−(s·u(t<0))2(cid:105).

(3.3)

Then the atomic vibration along that specific Bragg peak direction can be calculated as

(cid:104)u(t)2(cid:105) = (cid:104)u(t < 0)2(cid:105) − 1

s2 ln

I(t < 0)

(cid:18) I(t)

(cid:19)

.

(3.4)

Initial atomic vibration (cid:104)u(t < 0)2(cid:105) is estimated to be about 0.008 ˚A2 from X-ray mea-

surements [92]. So with Bragg peak intensity change known, we can extract the true root

mean square (rms) atomic vibration (u) at different directions and different time delays

(Fig. 3.9b). Here, we make an assumption that the initial thermal phonon rms amplitude

is isotropic. The black circle with radius of 0.1 ˚A represents negative time delay; at 0.2 ps

after pump excitation, the rms vibration along the a axis becomes smaller and that along

the c axis increase, as shown by the purple curve; at 1 ps, the anisotropy becomes maximal;

after 1 ps, the system starts to recovery back to an isotropic state; at about 10 ps, the rms

vibrations along a- and c- axes are already very close.

On the other hand, since CDW formation in RETe3 series is shown to be strongly associ-

ated with FS nesting from the photo-emission measurements, it is necessary to explore how

the CDW nesting vector changes associated with a laser pump excitation. By tracking the

CDW wave-vectors qa and qc, it is found that the CDW FS nesting vectors are changing,

as shown in Fig. 3.9c. The a-CDW wave-vector is relatively noisy due to weak signal, so

only a few averaged data points are shown. But for c-CDW wave-vector, it is clear that the

wave-vector is increased by 2.5% at + 1 ps. With a delay of 300 fs that matches with the

81

KZ freeze-out time, qc peaks at + 1.5 ps, then partially decays back. Fig. 3.9d shows how

the FS change after laser excitation (orange dashed line).

In photoemission measurement [93], it has been shown that there is a reduction in trans-

verse coupling t⊥. With theoretical calculations, we show that this reduction leads to the

observed change in CDW wave-vector qc. We evaluate the pairing vector by calculating the

static charge fluctuation susceptibility χ(qc, ω = 0) based on a model band structure that

incorporates the reduction in t⊥. Here, we focus on the underlying susceptibility and its
behavior as the Vπ (∝ t⊥) is reduced. Fig. 3.9e shows the line cuts of the χ along (11)

direction (where the electron-phonon coupling vertex is peaked) for several values of Vπ.

For both the local maxima labeled as PK1 and PK2, as Vπ is reduced, they shift toward

smaller nesting wave-vectors. Fig. 3.9f plots the extracted two peak positions at several Vπ

values to show the trend.

By comparing the Bragg peak and CDW wave-vector dynamics, and associating them

with the intensity dynamics, it is clear that the shift in the wave-vector goes hand in hand

with the intensity change: both have a freeze-out period, both peaks at about 1 to 1.5 ps.

In contrast, the Bragg peak dynamics has no delay and peaks much early.

To prove that the changes we observed here are induced mainly by a nonthermal effect,

we performed static TEM measurements to study the thermal phase transition. The TEM

measurement is done using a thermal stage in the JEOL 2010F TEM equipped with Gatan

OneView 4k x 4k camera (at University of Michigan microscopy center with the help of

Dr. Kai Sun). Fig. 3.10a is the full diffraction pattern of CeTe3 at RT while Fig. 3.10b

are cropped CDW peaks at 4 different temperatures. The CDW wave-vector is also seen

to gradually increase at the beginning then increase significantly faster after 400 K. At the
estimated sample temperature (≈ 370 K based on optical absorption, see next paragraph

82

Figure 3.10: Static TEM measurements of CeTe3, a, Full TEM diffraction patterns at RT. b,
Cropped diffraction pattern at several temperatures at fixed scale. c, c-CDW peak intensity
evolution normalized by Bragg peak intensity. d, The CDW wave-vector evolution towards
transition temperature. e, Line cut along c-CDW peak at three characteristic temperatures.
f, The same line cut along the potential a-CDW peak position.

for details), the CDW wave-vector is much smaller than the photo-induced change. So the

observed q c change in the UED data is not due to thermal heating. The plot in Fig. 3.10c

shows the square root of ICDW /IBragg. The overall trend defines a transition temperature

83

cdabef373 K543 K493 Kqc(4 0 1)293 K3003504004505000.2790.2800.2810.2820.2830.2840.285 qc qc (r.l.u)Temperature (K)Estimated Tsample at F=1.85 mJ/cm20.0%0.5%1.0%1.5%2.0%∆qc/qc3004005006000.00.10.20.30.4[ICDW/IBragg]1/2Temperature (K)Tc =540±15 K[ICDW/IBragg]1/20.200.240.280.320.36103104105 293 K 373 K 493 KIntensity (arb. u.)q (r.l.u.)qc0.200.240.280.320.36103104105Intensity (arb. u.)q (r.l.u.)Expected qa 293 K 373 K 493 K030002500200010001500500[010](0 0 0)of 540± 15 K. For c-CDW and a-CDWs intensity profiles as shown in Figs. 3.10e&f, it is

clear that the thermal effect can not induce CDW order in the a direction.

Based on optical measurements of CeTe3 [88, 89], we extract the refractive index n and

extinction coefficient κ to be 1.04 and 3.45, respectively. With n and κ as the inputs and

transfer matrix calculation (refer to Appendix B), the optical absorption is calculated to
be 0.32 (45◦ incidence, p-polarized, 30 nm sample thickness). At F = 1.85 mJ/cm2, the

absorbed energy density is calculated to be 1.22 eV/nm3. According to thermodynamic

measurements on CeTe3 and similar compounds [94, 95], the specific heat for CeTe3 at RT
is estimated to be 120 J/(mol·K). Then we can estimate the sample temperature rise to be
≈ 75 K.

Finally, based on the previous discussions, we can establish the following phase diagram in

Fig. 3.11, in which vertical axis is the equilibrium state and the horizontal axis is the photo-

induced non-equilibrium direction. A new hidden bi-directional phase emerges solely due to

the interaction-mediated symmetry-recovery in the electronic structure and the decoupling

between the microscopic relaxation and CDW collective modes may provide a key window

for the hidden state to grow into a stable, meta-stable phase.

3.4 Summary

In conclusion, we have observed a bi-directional CDW state induced by laser pulse illumina-

tion, which is thermodynamically forbidden at RT. Such a state is formed over the transiently
suppressed c-CDW state, with a threshold of ≈ 0.6 mJ/cm2 — much smaller than what is
needed to suppress c-CDW, which is ≈ 1.9 mJ/cm2. This threshold defines a nonthermal

critical point. The analysis of Bragg peak dynamics and anisotropic atomic vibrational re-

84

Figure 3.11: The non-equilibrium phase diagram and phase transition pathways of CeTe3.

sponses to the laser excitations prove that the associated symmetry changes involve soft

modes, in particular a transverse mode that drives the softening/hardening of phonon dis-

persions leading to formation/suppression of CDWs. The CDW wave-vector change clearly

demonstrates that CDW order parameter evolution follows the FS curvature change resolved

by angle-resolved photoemission spectroscopy (ARPES) [93]. This shows that FS nesting

enhanced electron-phonon coupling plays a central role in driving the CDW instabilities.

Finally, we propose a nonthermal pathway to explain the introduction and stabilization of

the hidden interaction-mediated CDW order.

85

TCFC(a)FC(c)t=0PhaseτKZt = KZ regimeInteraction-driven, non-equilibriumThermal, equilibriuma-CDWc-CDWBi-directional CDWNonthermalThermalizationInteractionorderingquenchsteady stateHidden stateChapter 4

Symmetry breaking and phase

transitions in ErTe3

In the previous chapter, the spontaneous symmetry breaking and hidden states are explored

in CeTe3. CeTe3 is optically driven into a hidden bi-directional state via a SSB process

involving a nonthermal fixed point. Following this thread, we studied a compound of the

same rare-earth tritellutride group: ErTe3, which is on the opposite side of RETe3 series to

CeTe3.

ErTe3 has two CDW transition temperatures: above 267 K, the material is in a metallic

phase and exhibits no CDW order. Below 267 K, CDW order along the c direction is formed

(uni-directional stripe CDW). When temperature is further reduced to 155 K, another CDW

order along the a direction is formed (bi-directional CDW). Similar to CeTe3, the CDW

is also believed to be formed due to FS nesting: the FS nesting wave-vector matches well

with the CDW wave-vector [84]. However, ErTe3 exhibits a richer phase diagram. There

is a metallic state at RT. Between 155 K and 267 K, ErTe3 exhibits a stripe CDW state

resembling what we see in CeTe3 at RT. Below 155 K, it is likely to have more interesting

phenomena due to the interactions between the two CDW orders in a more symmetric FS.

So ErTe3 is naturally a further step to give a comprehensive picture for the equilibrium and

non-equilibrium CDW for materials in the RETe3 family.

86

In this chapter, the background on ErTe3 will be introduced first in Sec. 4.1. The

dynamics and fluence-dependence data will be discussed in Sec. 4.2. The non-equilibrium

melting thresholds for a- and c-CDWs will be investigated at various temperatures. Then

at an intermediate temperature between the two transition temperatures, dynamics are

discussed in Sec. 4.3.

In Sec. 4.4, a new phase diagram is made and the chapter will be

summarized.

4.1 Background about ErTe3

According to previous reports [84, 85], ErTe3 has a normal-to-stripe CDW transition at 267

K and stripe-to-bidirectional CDW transition at 155 K. This transition is directly reflected

in the diffraction patterns. As shown in Fig. 4.1, at 40 K, the CDW satellite peaks show up
in both the a∗ (green circles) and c∗ (red circles) directions. At an intermediate temperature
of 180 K, only CDW satellite peaks along the c∗ direction show up.

According to the diffraction and photoemission measurements [84, 85], the CDW along

the c-axis has the wave-vector of q c = 0.30c∗ with a band gap of 175 meV. The CDW along
the a-axis has the wave-vector of q a = 0.315a∗ and a CDW gap of 50 meV. For comparison,
CeTe3 has the transition temperature of ≈ 540 K, q c = 0.285c∗, and the CDW gap of 400

meV. These large differences result in very different s/n of the data.

In our electron diffraction pattern of CeTe3 at RT, the peak intensity of CDW : Bragg
≈ 1 : 15. By contrast, for ErTe3 at 25 K, the intensity ratio of c-CDW : Bragg≈ 1 : 50 and

the a-CDW intensity is another factor of 5 to 10 weaker than c-CDW’s. This makes ErTe3

experiments extremely difficult. It requires a factor of 10 or more probe electrons than CeTe3

in order to get the same level of s/n for studying the CDW dynamics. Another factor that

87

Figure 4.1: The diffraction patterns of ErTe3. a&b, Full diffraction patterns at 40 K and
180 K. c&d, Zoom-in view in a and b. The red circles are c-CDW peaks and green circles
are a-CDW peaks. At 40 K, there are both a- and c-CDW orders while at 180 K there is
only CDW order along the c∗ direction.

makes ErTe3 experiments harder is in the sample preparation step. ErTe3 is very easily

oxidized in air, and the surface will be oxidized and sample will become unusable in a few

hours. So during sample preparation, it has to be done in a nitrogen-filled glove box. Also,

the inter-layer bonding strength in the ErTe3 crystal is much larger than CeTe3 or TaS2,

88

40 Kabcd180 Kc*a*making it very hard to exfoliate to very thin flakes. For our collaborators in Northwestern

University, they also spent quite some time before resolving the twining problem (mixture of

the a and c axes) in ErTe3. After trials and errors, we managed to run a series of experiments

to unveil the interesting physics in ErTe3, though the s/n is slightly less than the CeTe3 data

reported in the previous chapter.

4.2

Investigations at low temperatures

We first examine the phase transitions at 50 K. With 800 nm pump at the fluence of 2.9

mJ/cm2, both a- and c-CDW peaks are suppressed to almost zero within the first 1 ps then
partially recover to ∼ 40% of the original level (Fig. 4.2a).

Figure 4.2: The CDW (a) and Bragg (b) peak dynamics at base temperature of 50 K and
pump fluence of 2.9 mJ/cm2. The a-CDW dynamics data in the first 5 ps are 2-binned for
better s/n.

According to phase transitions in the steady state, if one continuously warms the sample

up from 50 K across the CDW transition temperatures, the a-CDW will be melted first at 155

K before the c-CDW is melted at 267 K. Naturally, one expects that with laser excitation,

89

01020300.750.800.850.900.951.001.05 6 0 0 4 0 4 0 0 6Relative intensityDelay, t (ps)50 K, Bragg peaksab01020300.00.20.40.60.81.0 a-CDW c-CDWRelative intensityDelay, t (ps)50 K, CDW peaksone would drive the melting of a-CDW more efficiently than c-CDW if the thermal heating

plays a key role. But experimentally, at about 2 ps, c-CDW is fully melted while a-CDW
still has ∼ 20% remaining (Fig. 4.2a).

Figure 4.3: The fluence-dependence of the a- and c-CDW peaks relative intensities at 25
K. The a-CDW data are averaged every two points (binning of 2) for better s/n. The data
points for 50 K (2.9 mJ/cm2) and 100 K (2.5 mJ/cm2) are extracted from the delay scan
dynamics.

The anomaly observed in time-dependent dynamics is consistent with what we see in the

fluence-dependent data in Fig. 4.3, where we monitor the peak intensity changes at + 30 ps

relative to the negative time value. The c-CDW data in red curve have a relatively good s/n
and the trend is quite linear with about ± 5% uncertainty bar. The extracted threshold for

c-CDW is 6.6 mJ/cm2. On the other hand, a-CDW in green color is much noisier, limited
by s/n. The error bar could be as large as ± 15%. In the plot, we average every two data

90

012345670.00.20.40.60.81.0         a-CDW   c-CDW25 K:       50 K:       100 K:     Relative changeFluence, F (mJ/cm2)Changes at +30 pspoints (binning of 2) for better s/n. However, following the trend line, it is evident that the

melting threshold determined at this time scale for a-CDW is not much different from that

of c-CDW, or even slightly higher.

Figure 4.4: The a- and c-CDW peaks dynamics at 100 K. Data points for a-CDW are binned
by 2 for better s/n.

This phenomenon can be explained from previous CeTe3 experiments. After photo excita-

tion, there is a reduction in transverse hopping integral. This change leads to a modification

in FS curvatures, resulting in a FS nesting that favors a-CDW. This nonthermal effect makes

a-CDW state as competitive as c-CDW. Therefore laser-induced quench does not obviously

favor one particular stripe order in our presentation.

In terms of anisotropic lattice phonons that we deduce based on the Bragg peak dynamics,

ErTe3 shows a smaller anisotropy compared with CeTe3. As shown in Fig. 4.2b, since the

91

0102030601000.00.20.40.60.81.0 c-CDW a-CDW, 2-binnedRelative intensityDelay, t (ps)changes along [1 0 0] & [0 0 1] are influenced by soft modes associated with a- and c-CDW

states, the smaller anisotropy here reflects a more symmetric energy landscape in ErTe3 at

low temperatures than CeTe3 at RT.

We further raise the sample temperature to 100 K and examine the a- and c-CDW

peaks responses at pump fluence of 2.5 mJ/cm2, as shown in Fig. 4.4. While the c-CDW
is suppressed to 20% and shows over-damped dynamics before recovering to ≈ 50%. The a-
CDW intensity is suppressed to ∼ 40% without a clear recovery. We note that judging from

the changes at + 30 ps, both 50 K and 100 K ErTe3 system follow similar fluence-dependence

as 25 K (Fig. 4.3). Their early time dynamics reflect disordering of the initial state.

4.3 Phase transitions at intermediate temperature

At temperature of 165 K, which is in between the two transition temperatures, only the

c-CDW satellite peak is present. At the pump fluence of 2.9 mJ/cm2, the c-CDW peaks are
suppressed to ∼ 10% (Fig. 4.5a).

Figure 4.5: The dynamics of CDW (a) and Bragg (b) peaks at 165 K.

92

-50510152025-0.20.00.20.40.60.81.01.2 a-CDW c-CDWRelative intensityDelay, t (ps)165 K, CDW peaks-505101520250.800.850.900.951.001.05 6 0 0 4 0 4 0 0 6165 K, Bragg peaksRelative intensityDelay, t (ps)abAt the location where there is supposed to have a-CDW peak, we gather scattering weight

above the background to extract changes induced by laser excitations. As shown in Fig. 4.5,

the data are very noisy but do show evidence of an enhancement during the first 5 ps. In the

plot, the a-CDW intensity is normalized based on c-CDW intensity (also shown) at negative

time.

The state of ErTe3 at 165 K should be very similar to that of CeTe3 at RT. Both have

CDW orders only along the c direction. In CeTe3 the laser quench of c-CDW results in the

formation of a-CDW. In ErTe3, as we mentioned, the c-CDW order is already a factor of 5

to 10 weaker than that in CeTe3, so the s/n is relatively low and the formation of a-CDW

is a little ambiguous. But given the fact that a-CDW order is induced in CeTe3, we expect

the signs of a-CDW formation in ErTe3 may be real.

Figure 4.6: The a- and c-CDWs satellite peaks dynamics at 200 K.

93

-505101520253035-0.20.00.20.40.60.81.01.2 a-CDW c-CDWRelative intensityDelay, t (ps)For understanding the soft modes dynamics, symmetry breaking/recovery associated

Bragg peak intensities are examined. While the weak anisotropic Bragg peak intensity

remains to be less suppressed (Fig. 4.5b), but a lower (0 0 6) intensity, instead of (6 0 0), is

observed in the low temperature state, which may reflect the stronger soft mode instabilities

here associated with a-CDW, which is no longer static at this temperature.

At base temperature of 200 K and fluence of 2.7 mJ/cm2, c-CDW is suppressed to

less than 10% of negative time. We are not able to resolve any presence of a-CDW fluc-

tuation at this temperature. Limited by the s/n in the current UED system, the early

non-adiabatic evolution for the two CDW states are not resolved. Further experiments with

higher s/n possible with the high-brightness UEM system may resolve these subtle differences

in temperature-dependent investigations.

4.4 Non-equilibrium phase diagram and summary

In summary, due to the limited time resolution and s/n, we do not resolve the early symmetry

breaking dynamics for ErTe3. But the results presented here do support the key conclusions

from CeTe3 that fs laser excitation promotes a-CDW formation. At low temperatures, the

energy landscape, compared to that of CeTe3, is already more symmetric. The soft mode

features are more isotropic as well.

Based on the Bragg and CDW peak dynamics at various temperatures, we propose the

following phase diagram in Fig. 4.7. At low temperatures, the material already resides in

the bi-directional CDW state. After laser excitation, both CDW orders are quenched but

the a-CDW order has a similar (or even larger) nonthermal melting threshold as the c-

CDW. At intermediate temperature of 165 K, which is right above the a-CDW transition

94

Figure 4.7: The schematic phase diagram of ErTe3 under non-equilibrium conditions. The
phase diagram is based on dynamics and fluence-dependent data at various temperatures.
The thin gray line shaded-region represents the uncertainties of determining the a-CDW
suppression phase boundaries.

temperature (155 K), it is in a uni-directional stripe CDW phase. But with laser excitation,

there is signature that a-CDW order is induced after laser excitation, similar to the CeTe3

case at RT. But due to the more than a factor of 10 weakness in CDW intensity compared

with CeTe3, the a-CDW is much noisier than the CeTe3 data presented in previous chapter.

When the base temperature is further above the 155 K transition temperature (at 200 K),

the a-CDW formation is too weak to be observed and only a quench in c-CDW order is

95

TFC(a)FC(c)Photo doping, phase transition directiona-CDWc-CDW267 K155 KBi-directional CDWobserved.

The non-equivalent axes (a and c) become more symmetric and degenerate after laser

excitation. In the non-equilibrium regime, the CDW order is transiently enhanced after laser

excitation in the otherwise non-exist a-axis, demonstrating the completely different energy

landscape compared with thermodynamic states. Such a non-equilibrium window opens

up a new channel for exploring the metastable and hidden phases for possible functional

materials. Given the consistency of the results in CeTe3 and ErTe3, we believe the results

and conclusions can be extended to other rare-earth tri-telluride compounds as well and

other similar symmetry-broken systems with non-equivalent crystal axes.

96

Chapter 5

Dynamical scaling and phase

transitions observed in 1T-TaS2

Research on quantum materials is a central topic in contemporary condensed matter physics.

Of particular interest are 2D materials due to the presence of strong electronic instabilities

which are the results of competing interactions at low dimension, frequently leading to a rich

phase diagram. 1T-TaS2 represents such an example of the transition metal dichalcogenides

compound where electron-electron interaction leads to novel Mott insulating ground state

and even superconductivity at low temperatures, which are entwined with density-wave

formations.

In the early years from 1970s to 1990s, by means of X-ray and electron diffraction, and

STM measurements, researchers have spent a great deal of effort on understanding how

the CDW is formed in 1T-TaS2 [52, 96], including a series of drastically different CDW

formations. Among them, the most intriguing is the periodic domain texture formed in

the NC-CDW state [97–100]. In recent years, the low-temperature Mott insulator-to-metal

transition (IMT) has attracted a lot of attention with the advances in the time-resolved tools

through which one may seek to separate the electron correlation effect from the electron-

phonon coupling in the CDW phase transitions [101–107]. It is generally believed that the

strong electron-electron correlations are directly responsible for IMT in the low-temperature

97

C-CDW state. This Mott gap is located in the Γ point of BZ which is decoupled from the

CDW gap. With chemical doping [13] or under a high pressure [108, 109], the materials

could exhibit superconductivity, which may emerge due to tuning of the electronic structure.

An important recent development that draws intense interest is the work by Stojchevska et

al [110], where they showed that with a single fs laser pulse, 1T-TaS2 goes from an insulator

to a hidden metallic state that lasts very long. This metallic state can be erased by another

strong laser pulse. Apart from the nature of this switching, such insulating and metallic

transitions are highly promising for optoelectronic devices. Ensuing work [111–116] shows

that the hidden states in 1T-TaS2 may also occur under electrostatic chemical injection,

gating with potentials in practical applications.

Even with all the previous work, there are still fundamental questions left unanswered.

For example, how exactly the phase transition may happen that to drive the rearrangement

of more than billions of atoms in a relatively short time; how an ultrafast laser quench may

lead to a hidden state; and what limits the speed of phase switching. In this chapter, we

will try to address some of these questions.

In this chapter Sec. 5.1, the background information on 1T-TaS2 is given and general

current problems related to this system is summarized. Then our study with ultrafast

electron diffraction is presented in Sec. 5.2, including time-dependent dynamics and the

microscopic physical picture of the transition.

In Sec. 5.3, we will discuss the universal

scaling behavior under non-equilibrium. Then in Sec. 5.4, the ultrafast photo-doping and

photo-thermal effects for driving the phase transition will be discussed.

98

5.1 General properties of 1T-TaS2

1T-TaS2 and other transition metal dichalcogenides were studied as early as the 1970s by J.

Wilson [52] et al. As shown in Fig. 5.1, the layered structure of 1T-TaS2 with weak inter-layer

van der Waals binding makes it easily exfoliatable by the scotch tape for preparing the thin

TEM specimen. Within one layer, it has C3 symmetry with two sulfur layers sandwiching

the tantalum layer.

Figure 5.1: The structure of 1T-TaS2. a, 3D view of the structure. Blue balls are Tantalum
atoms and yellow represents Sulfur atom as labeled. The red box represents the unit cell. b,
Side view. c, Top view. d, Ta layer structure when forming commensurate CDW. Every 13
atoms form Star-of-David supercell with surrounding 12 atoms moving inwards. The CDW
superlattice is oriented 13.9◦ relative to the original lattice.

99

Top viewTa layer:3D viewSide viewTaSTaSTaSabcd13.9 ODespite the simple structure, it exhibits a very rich phase diagram. Below 180 K, the

material is in commensurate CDW (C-CDW) state entwined with the Mott insulator state

[108]. The C-CDW state unit cell consists of 13 atoms in the so-called Star-of-David (SD)

structure formation (Fig. 5.1d). Within one SD supercell, there is in total one electron in
the center. The C-CDW is oriented 13.9◦ from the lattice plane. Upon warming at ≈ 223

K, the system’s long range correlation is broken by the formation of stripe order [99, 117].
The CDW orientation angle Φ rotates from 13.9◦ to about 13◦. Further heating to above
283 K, the hexagonal domains with a size of ≈ 7.2 nm are formed and the system is often
referred to as the nearly-commensurate CDW (NC-CDW) state with Φ ≈ 12◦ [97–100, 118].

At T = 353 K, it undergoes yet another transition to the incommensurate CDW (IC-CDW)
state, in which CDW is aligned with the lattice plane (Φ = 0◦) [98, 119]. At T = 543 K, the

IC-CDW is melted and the system becomes metallic.

Details about 1T-TaS2 phases can be found in Table 5.1.

Based on McMillan’s free energy formulation of CDW in transition metal dichalcogenides

[120], Nakanishi and Shiba et al proposed that domains in the NC-CDW phase are formed

by a series of CDW harmonics [96, 121]. Within the domain, these harmonic states are

phase-locked and commensurate with the SD building blocks. Domains are separated by

discommensurations characterized by phase jumps. This prediction was later experimentally

confirmed in the STM studies by Wu et al [97] and Burk et al [118], where they observed the

7.2 nm hexagonal domain structures. From a Fourier transform (FT), the high-order CDW

wave-vector components obtained from STM images are in agreement with the diffraction

results.

The sample studied here was grown by the chemical vapor transport technique by Prof.

Kanatzidis group in Northwestern University. Single-crystalline bulk sample is exfoliated

100

by the Scotch tape method down to ≈ 30 nm thickness with a lateral size of 20 µm. The

thickness is then confirmed by zero-loss electron energy loss spectroscopy (EELS). Details

about sample preparation can be found in Appendix A.

Phase

Temperature (K)

Φ(◦)

Comments

Normal

T > 543

NA

Metallic phase, no CDW

IC-CDW

353 < T < 543

0

283 < T < 353(warming)

11-13

NC-CDW

183 < T < 353(cooling)

Incommensurate CDW
≈ 7 nm hexagonal domain

Triclinic

223 < T < 283 (warming)

12-13

Striped domain

C-CDW

T < 223 (warming)

13.9

Lock-in commensurate

T < 183 (cooling)

CDW, no domain

Table 5.1: 1T-TaS2 phases across different temperatures. Φ refers to the CDW orientation
angle away from the lattice plane.

Figs. 5.2a&b are typical optical images of the freestanding samples supported on the

TEM grid used in the experiments. The high quality single-crystal sample is reflected in the

diffraction patterns as shown in Figs. 5.2c&d. As shown in Fig. 5.2d, in the C-CDW state,
the relative angle Φ between the CDW and the lattice is 13.9◦. For comparison, Φ is about
12◦ in the NC-CDW phase (Fig. 5.3a) and 0◦ in the IC-CDW phase (Fig. 5.3b).

Optical and spectroscopic approaches such as transient optical reflectivity or angle-

resolved-photoemission spectroscopy (ARPES) can track the electronics state evolution and

closing of the CDW gap. They do not directly confirm the structural transition, which is

believed to strongly couple to the evolution in the electronic subsystem. There have been

101

Figure 5.2: The optical images of the sample and electron diffraction patterns. a, Freestand-
ing sample flakes on 1000 mesh gold TEM grid. b, Zoom-in view of a thin sample piece used
for experiment. c, The diffraction pattern of 1T-TaS2 by 30 keV electron beam at 150 K
C-CDW phase. d, Zoom-in view of the diffraction pattern in c. The relative angle between
the CDW q -vector and the Bragg plane is 13.9◦.

several studies of 1T-TaS2 structural phase transitions by UED and X-ray diffractions. The

pioneering UED study of 1T-TaS2 by Eichberger et al [122] discussed the cooperative in-

teraction between the electron and lattice subsystems. But they did not explicitly discuss

the mechanism for driving the phase transitions. Han et al [106] explored the metastable

and hidden phases, transition pathways and photo-doping effect during phase transitions. In

102

021221028262422a*b*13.9 ObacdFigure 5.3: The diffraction patterns of 1T-TaS2 in the NC-CDW phase. a, The original
diffraction pattern in NC-CDW state. b, The diffraction pattern in IC-CDW state. c, A
schematic diffraction pattern of the NC-CDW phase. Pattern is in the same orientation
as that in a. The 1st, 2nd, and 3rd CDW peaks are labeled as q(1), q(2), q(3), respectively,
with each have two peaks differentiated in the subscript by “1” and “2”. d, The difference
diffraction pattern of after laser excitation (IC-CDW phase) minus the pattern before laser
excitation (NC-CDW phase).

103

b*a*Bragg1st order CDW2nd order CDW3rd order CDWb*q2(1)q1(1)q1(3)q2(3)q2(2)q1(2)a*b*a*DecreaseIncrease0b*a*bdcaanother UED paper, Haupt et al [123] discussed the NC-to-IC CDW phase transformation

and suggested IC domain growth after excitation. The work based on X-ray diffraction by

Laulhe et al [124] tracked the IC phase coarsening process following a power-law with ex-

ponent of 1/2. The work by Vogelgesan et al [9] discussed the phase ordering kinetics and

the power-law growth, but further implicated the involvements of topological defects. From

these earlier works, it is clear that the NC to IC transition occurs in the ultrafast time scales

and the formation of IC tracked by correlation length follows a power-law. But it is still not

entirely clear how the microscopic process can lead to effective switching in the macroscopic

systems involving a great number of particles in such relatively short time scales. There

are also questions to be reconciled from different approaches with respect to the roles of

photo-doping and photo-thermal effects for driving the phase transition.

5.2 Phase transition mechanism of 1T-TaS2

The key to understand the transformation pathways involves the annihilation of the topo-

logical defects (discommensurations) to evolve from NC to IC states. We first discuss how

with coherent electron pulses one can resolve the domain structures via UED.

5.2.1 Reconstruction of real-space structures

Diffraction pattern encodes coordinates in the momentum space. To go back and retrieve

the real-space structures, a simple Fourier transform (FT) is insufficient because the phase

information is missing in the diffraction pattern, which is the so-called phase problem. In

a typical microscopy setup with lenses to form an image, the phase information is restored.

One method of using coherent diffraction pattern to reconstruct the real-space structure,

104

generally referred to as coherent diffractive imaging, relies on algorithms to retrieve the

phase information. The reconstruction process usually applies a random phase first, then an

iterative algorithm involving FT to be performed under preset constraints allows the phase

to be retrieved in a self-consistent way. Here in our reconstruction protocol, we do not follow

exactly the same protocols of coherent diffractive imaging described above, but instead we

rely on the phase information credible to enable successful prediction of the structures at

ground state as the constrains for reconstructing the CDW states in real space.

The reconstruction process starts with the effective CDW order parameter

(cid:88)

(cid:113)

ψe(r, t) =

Iint,j(t)e

−i(q j (t)·r+φj )e

−|r−r0|
2ξj (t) ,

(5.1)

j

where q j, φj, ξj are the wave-vector, phase, and correlation length of the CDW branch

(indexed j ) centered on a reference position r0.

In the NC-CDW phase, considering three-fold rotational symmetry, it has one 1st-order,

two 2nd-order, and two 3rd-order NC peaks. The diffraction pattern in Fig. 5.3c mimics that

in Fig. 5.3a and has labels for all the CDW peaks. The expression for CDW peaks are as

follows. First-order CDW peaks:

1 = 0.249a∗ + 0.063b∗;
q (1)

2 = −0.312a∗ + 0.249b∗.
q (1)

(5.2)

Second-order CDW peaks:

q (2)
1 = q (1)

1 − q (1)
2 ;

2 = a∗ − 2q (1)
q (2)
1 .

(5.3)

105

Third-order CDW peaks:

1 = a∗ − 2q (1)
q (3)

1 + q (1)
2 ;

2 = a∗ − q
q (3)

(1)

1 + 2q (1)
2 .

(5.4)

With three-fold rotational symmetry, all the NC-CDW peaks of different orders as well

as the IC CDW peak, are included for reconstruction in Eq. 5.1. The key to form the com-

mensurate domain is the requirement of “locked-in”, namely the phases φj from individual

harmonic state of NC-CDW are the same at the center of domain defined as position r0.

Here we simply set all φj = 0, then the CDW density map can be calculated as

M (r, t) = |ψe(r, t)|2.

(5.5)

amplitude of δri = (cid:80)

With CDW present, the lattice is distorted from high symmetry positions with distortion

j δcejRe[ψe(ri, t)], where ej is the distortion polarization, δc is the

static distortion of ≈ 0.25 ˚A for 1T-TaS2. Then the distorted lattice can be written as

(cid:88)

i,j

I(r, t) =

− δr2
i
u2
i e

−|r−r0|
ξj (t) .

Z2
i e

(5.6)

Then we can get the reconstructed diffraction pattern by doing a FT of the distorted

lattice image

Itheo(q , t) = FFT[I(r, t)] ⊗ G(q ),

(5.7)

where G(q ) is the Gaussian convolution function representing instrumental resolution.

We can compare the reconstructed theoretical diffraction pattern with experimental pat-

terns to refine the parameters. Fig. 5.4c are examples of CDW density map.

106

Figure 5.4: The dynamics of different order parameters after laser excitation. a, The dy-
namics of the (first-order) NC and IC peak intensities at fluence of 1.73 mJ/cm2. Shaded
area represents the correlation length of NC and IC order, fitted in the transverse (T) and
longitudinal (L) direction. b, The dynamics of the 1st (red), 2nd (green), and 3rd (blue)-
order CDW peaks. Inset shows the ratio changes of the 2nd− and 3rd-order peak normalized
by the 1st-order peak intensity. c, The reconstructed CDW intensity map at several critical
time delays. d, The 1st-order CDW peak radius changes normalized by Bragg peak distance.
e, The dynamics of 1st-order CDW peak angle relative to Bragg plane. f, The NC-CDW
domain size-dependence on the NC-CDW angle changes.

5.2.2 CDW peak dynamics and domain evolution

In our experiment, we directly track the Fourier components of the order parameters with

electron diffraction and use those to reconstruct the real-space CDW structures, as discussed

107

0.00.20.40.60.81.01.21.4 ξNCT  Normalized Peak intensity= 25 nmξ   ξNCLξICTξICL I(Q1NC) I(Q1IC)0.2800.2810.2820.2830.284Bragg-NC distance (r.l.u.)| Qnc | / | G1 |-202468100100010.010.511.011.5NC CDW angle NC angle (deg)Delay, t (ps)-10123451010010000.00.20.40.60.81.01.2Peak relative changeDelay, t (ps) I(Q1NC) I(Q2NC) I(Q3NC)-202468100.040.080.120.160.20 Ratio, I(Q2NC)/I(Q1NC)Ratio, I(Q3NC)/I(Q1NC)Intensity ratiosDelay, t (ps)10.010.511.011.512.04.55.05.56.06.57.07.5 Domain sizeReconstructed domain size (nm)NC-CDW angle (deg)bcfeda-1 ps3 ps25 ps750 ps135 ps0.3 ps7 nmin Sec. 5.2.1. Fig. 5.4 shows the evolution of NC-CDW peak of various orders (1st-, 2nd-,

and 3rd-orders) and IC-CDW peaks probed by UED. Key parameters, including the CDW

amplitude, correlation length, and the wave-vector are extracted from the diffraction anal-

ysis. Fig. 5.4c shows the reconstructed CDW density maps. At negative time, the 7 nm

NC domains are clearly visible. After laser excitation, the high-order NC-CDW peaks are

suppressed more than the first-order peaks (Fig. 5.4b). This is reflected in the less sharp

domain structures in the reconstructed images. In the longer time scale, IC phase grows and

the visible range expands.

Since the domains resulted from different modulations of the CDW waves, the domain

size is found to be almost linearly correlated with NC-CDW angle. In the steady-state mea-

surements, as the temperature increase, the NC-CDW angle decreases [98] and the domain

size also shrinks [125]. These measurements are consistent with our reconstruction in Fig.

5.4f, indirectly proving the validity of our method.

5.2.3 General behaviors of dynamical phase transition

There are some generic features in the CDW phase evolution. First off, the NC peak shows

well-defined plateau after the initial quench for all the excitations, with different fluences

show different plateau duration periods. After the plateau region, the NC peak further drops

and IC peak further develops. So this plateau can be seen as a precursor for thermalization

and has been observed widely in the post-quench relaxation process in isolated quantum

systems [126, 127] — a phenomenon called prethermalization.

The term prethermalization comes from the field of cold atoms physics out of equilibrium,

where the isolated quantum many-body systems (i.e. cold atoms) reach a quasi-steady state

after initial quench but prior to the long-time thermalization. Prethermalization is a fast

108

loss of memory of the initial conditions due to dephasing after the system is driven out of

equilibrium [128–134], typically in nearly-integrable systems [127]. It has been extensively

studied in non-equilibrium dynamics in isolated quantum many-body systems theoretically

[128–134] and experimentally [126, 135, 136]. This prethermalization plateau is a generic

feature of post-quench relaxational dynamics in isolated many-body systems, which may

occur due to multiple conserved quantities in the system [40, 137].

5.2.4 The Friedel oscillation and chiral symmetry breaking

There has been extensive literature discussing impacts of chemical doping in a CDW system

[11,13,138]. In the presence of charge impurities, there will be local perturbations effect to the

CDW system. Such an effect is called Friedel oscillations (FO). It is a quantum mechanical

analog to the electric charge screening of charged particle in a pool of ions. It depicts the

characteristic decay behavior in the fermionic density near the perturbation followed by a

damped sine wave. Fig. 5.5 is an example of FO, where the authors use X-ray diffraction

to probe vanadium-doped blue bronze. The doped vanadium atoms serve as the defects and

modulate the charge-density amplitude and phase nearby, as drawn in the figure. As a result

of this modulation, the diffraction peak shows up as an asymmetric tail as shown in the left.

This is one of the key phenomena of showing existence of FO and charge impurities.

Here in our data, at ≈ 1 ps, the NC-CDW peak also shows asymmetric tails, as seen in

Fig. 5.6a and intensity profile in Fig. 5.6c. To best reconstruct and reproduce experimental

data, we go through an iterative process in the reconstruction and find that to explain the

emergence of the asymmetric diffraction tail and peak broadening in the plateau region, we
only need to apply a phase adjustment that changes from q N C to q IC (Q F O ∝ q IC−q N C ).

This phase-only modulation protocol applied here is only similar to the approach applied to

109

Figure 5.5: An example of the Friedel oscillation in vanadium-doped blue bronze. Adapted
from Ref. [11, 12].

model the FO phenomenon in 1D CDW system. The adjustment made here reflect a defect-

induced field created by the photo-doping. Unlike the static chemical doping, the mobile

charge impurities can self-organize by adjusting to the new energy landscape that drives the

NC-CDW to IC-CDW transition. This empirical phase adjustments results in a spiral CDW

order emanating from the domain center, manifesting a chiral symmetry breaking within

the commensurate domain. FT of this spiral order creates an asymmetric CDW peak that

reproduce the experimental feature. Further adjustments in the phase create a tight fit to

the data intensity profile as shown in Fig. 5.6c.

The correlation length of FO is about 7 nm with best fit to data and the intensity is the

NC intensity after initial quench. Here the chiral symmetry breaking as constructed through

phase modulations around the domain center can be seen as a state that exhibits intertwined

NC and IC components. This spiral state further develops after prethermalization plateau

to shift NC (FO) population into IC state, evidenced from the peak profile evolution where

110

Figure 5.6: The chiral symmetry breaking and Friedel oscillations. a, The diffraction patterns
at 3 critical time delays. The pattern at + 1.5 ps corresponds to the chiral symmetry breaking
and FO region (X state). b, Phase modulation imposed on the reconstruction process due
to FO. On the right shows the fit of the reconstructed pattern to experimental data. k =
q− Q1
N C . c, The reconstructed atomic displacement map. The dark spots represent the loss
of atomic density (atoms moving away) and the white spots represent an increase in atomic
density (atoms moving towards that spot).

IC peaks grows from the asymmetric tail.

111

ab-1 ps+1.5 ps+1 ns+∆φXXIC  QFOPhaseFourier -1.0-0.50.00.51.01.51010010000.01.0      Delay, t (ps)NCtransformadjustc3.5 nm0.00-0.040.040.080.12Q1ICData  Fit -1 ps, x 0.6 +0.5 ps +3 ps +8 ps +25 ps +1 nsIntensity (arb. u.)k (r.l.u)Q1NCQ1ICData  Fit -1 ps, x 0.6 +0.5 ps +3 ps +8 ps +25 ps +1 nsIntensity (arb. u.)k (r.l.u)Q15.2.5 Fluence-dependent studies

Fig. 5.7 reports the NC and IC peak dynamics at various pump excitation fluences, where the

symbol m represents normalized intensity of the CDW peak. For NC, the peak intensity is

normalized to that at the negative time IN C (t < 0) and for IC, peak intensity is normalized

by dividing 0.45IN C (t < 0) — a normalization protocol based on the steady states where

the intensity of IC is 0.45 that of NC.

Another key feature is that the changes at short and long times show two thresholds, Fc,1

and Fc,2, respectively. Since the change in long time scales corresponds to thermalization, we

define Fc,1 as the thermal threshold. By contrast, Fc,2 is defined as the nonthermal (doping)

threshold.

To better understand the thermal heating effect, we also estimate the temperature rise

due to laser heating. Based on 1T-TaS2 optical measurements on the bulk sample [139],

reflectivity R = 0.405 is derived for 800 nm laser at normal incidence. According to the thin

film transfer matrix method [90] and Appendix B, the total reflectivity and transmission are

R = 0.35, T = 0.12, respectively (note: experimental condition: sample thickness d = 50 nm,
45◦ incidence, p-polarized). So the net photon absorption coefficient is: 1− R− T = 0.43. At
fluence F, the absorbed energy density on average is calculated to be: ρE = F (1−R−T )
critical threshold of Fc,2 = 1.52 mJ/cm2, the absorbed energy density ρE = 1.0 eV/nm3. By

. At

d

integrating the specific heat [119] of 1T-TaS2 from RT (296 K) to the transition temperature

of 353 K, the required thermal density is estimated to be 0.85 eV/nm3 (including latent

heat). At threshold of Fc,2, assuming all the absorbed photon is converted into heat, then

the sample temperature increase would be 71 K (from RT to 367 K). This result is consistent

with other measurements in 1T-TaS2 [122, 123]. As to the doping and nonthermal effect,

112

Figure 5.7: The dynamics of NC and IC peaks at various fluences. a, The NC peak intensity
dynamics. The gray line represents the end of prethermalization and the beginning of ther-
malization stage. b, The IC peak intensity dynamics. c, The NC peak intensity changes at
several critical time delays. d, The normalized IC change at several time delays. Thresholds
of Fc,1, Fc,2 are as marked in the figure.

more will be discussed in Sec. 5.4.

5.3 Universal behaviors

As discussed in Chapter 1, phase transitions exhibit critical behavior near the critical point

[140]. Even in the non-equilibrium regime, it is predicted to also have universal scaling

near the nonthermal fixed point. Recently it has been experimentally observed that the

post-quench relaxation process of 1D Bose gas follows a critical scaling [141, 142]. In these

experiments, the Bose gas is confined in a 1D tube and perturbed by external field that

113

0.00.20.40.60.81.0  024681010010000.00.20.40.60.81.0  Delay, t (ps)1.051.161.311.451.591.731.791.882.02F (mJ/cm2)mNC (t)NC state decayIC state formationmIC (t)Fluence, F (mJ/cm2)0.00.20.40.60.81.0Fc,1Fc,21.20.80.41.62.02.4mIC (t)IC0.00.20.40.60.81.0mNC (t)NC2 ps100 ps400 ps1 nsbadcmimics a strong cooling quench. Then the post-quench state evolution is monitored with

distribution function in real space probe. A FT of the real space distribution yields the

momentum space distribution, or correlation function. It is found that within the universal

scaling time window, the distribution function can scale horizontally and vertically to collapse

into one curve. This critical scaling behavior is a proof of nonthermal fixed point and

existence of certain universality class far from equilibrium.

In the case of 1T-TaS2, the fluence-dependence data unambiguously prove two thresholds

in both NC and IC data. In this section, we will discuss the universal scaling behavior at

both the prethermalization plateau region and also at the thermalization stage.

5.3.1 Universal scaling in the prethermalization plateau region

The fluence-dependent data reported earlier already indicate that the prethermalization

plateau is dependent on fluence, so as the IC switching speed. The life span of prethermal-

ization plateau tpre exhibits slowing down near the thermal threshold of Fc,1. Similarly, the

IC formation time also follows the diverging trend as the pump fluence is tuned near Fc,1.

The divergence of two characteristic times indicates that the dynamics themselves may

be collapsed into a single scaling function (or fixed point). Indeed, with proper assignment of

scaling functions and scaling exponents, we found that both the NC and IC curves collapse

— into groups of above Fc,1 and above Fc,2. In the time domain, we separate the scaling

into two parts with different physical origins: prethermalization plateau region and long-time

thermalization region. This collapse is performed for both horizontal (t)and vertical (m)

axis. We base our scaling using the first data curve after each threshold as the reference
curve. The scaling exponents are applied to the control parameter, r = F − Fc, where Fc
could be Fc,1 or Fc,2. Now with scaling, the horizontal axis is changed to t1·(r/rRef )κ, where

114

Figure 5.8: The universal scaling dynamics in the prethermalization regime (before tpre)
for NC and IC. The horizontal time axis is scaled based on power-law with exponent of κ
while the vertical axis is scaled with power-law exponent of η. The r and rREF refer to the
distance from the critical threshold r = F − Fc and the distance from the critical threshold
for the reference curve, respectively. The reference curves are chosen as the first curve above
the thresholds Fc,1 and Fc,2. The curves between Fc,1 and Fc,2 are scaled to the reference
curve of F = 1.05 mJ/cm2; the curves above Fc,2 are scaled to the reference curve at fluence
of F = 1.59 mJ/cm2.

t1 = t− 0.5 ps to ignore the initial quench period. κ refers to the dynamical scaling exponent

and here we set κ = 0.5. In the vertical axis, in the original dynamics plot, it is integrated

intensity m N C or m IC . In the scaling plot, it is changed to m 0 +∆m j(t)/(r/rRef )η, where
m 0 is the offset value for the reference curves, ∆m j(t) = m j(t) − 1 for NC so the scaling

part is the absolute change while for IC peaks, ∆m j(t) = m j(t) itself. The scaling exponent

η is set at 0.5. By applying both scaling, the data collapse well into one curve.

The prethermalization region is by definition region before the system thermalizes. There

has been some reports of universal scaling in the thermalization coarsening process [9, 124],

115

Universal collapse before tpre-1011010010000.00.20.40.60.81.0m0+∆mj(t)/(r/rRef)ηt1⋅ (r/rRef)κtpreXNCPrethermalization(ps)but not in the prethermalization region. This is the first measurement indicating that uni-

versal scaling may be observed before thermalization.

5.3.2 Universal scaling in the thermalization stage

After the system passes chiral-symmetry-breaking prethermalization region, the emergent

IC-CDW state starts to gain both intensity and correlation length, as shown in Fig. 5.4a.

The increase in correlation length is reflected in the 2D map of IC peak evolution shown

in Fig. 5.9a, where at each horizontal line scan intensity profile, it is normalized to the IC

peak center. So the IC peak center is always at the same color scale. This normalization

is to highlight the sharpening in IC peak. A more vivid schematic drawing of the IC peak

evolution is drawn in Fig. 5.9b, where the peak profile is a Lorentzian function. Inverse of

Lorentzian width scales as tβ and the amplitude scales as tα. Fig. 5.9c is an example of the

(inverse) peak width and height evolution. Note, the peak is fitted with a Voigt function

where Gaussian part represents the instrumental resolution and Lorentzian part represents

the overall distribution function. All the Lorentzian width discussed in this section refers

to the full width at half maximum of the Lorentzian function. In the plot of Fig. 5.9c, the

data in purple are the correlation length, the inverse of the Lorentzian HWHM (half width

at half maximum); the data in green are the Lorentzian peak height, which based on a 2D

Lorentzian distribution, is the integrated intensity divided by HWHM squared, then divided

by the integrated intensity to represent the true profile height.

The scaling region (gray shaded area) starts from tpre and ends until IC is fully formed

to a saturated level. The universal scalings in the width and height of the structure factor

presented in Fig. 5.9c is just for one sample fluence.

In fact, the data at all excitation

fluences show such scaling behavior and the scaling exponents α and β are found to be

116

Figure 5.9: The dynamical scaling of IC peak dynamics and IC phase evolution. a, The
IC peak profile evolution at different time delays. The profile line scan is from Bragg peak
(O point in the lower left) to IC peak (QIC ) in the radial direction outwards.
Intensity
at each time frame is normalized by IC peak to highlight the sharpening of IC peak. b,
Schematic IC peak profile at two different time delays to illustrate the scaling in peak width
Wq and peak height Aq. c, The power scaling of IC peak width and peak height scaling
across several orders of time evolution (gray shaded area). Pump fluence is 1.73 mJ/cm2.
d, The fluence-dependence of the scaling exponents α and β.

fluence-independent, with β = 0.5 ± 0.05 and α = 1.0 ± 0.1. Both scaling exponents has

physical meanings.

β is for scaling in the correlation length during coarsening, which is the characteristic

size of IC domain size. β = 0.5 means that when the time is 4 times longer, the IC domain

117

acdb0.010.1103104|k| (r.l.u.) 3 ps, x 3.2 30 ps0tαtβ0.010.1I(k)k=q-QICωkIIC(k,t)OkQICDelay, t (ps)t<00.11101001030.00.20.40.60.81.0024101001000-1.00.01.010100103I(k) (t)Delay, t (ps)Ak scaling-1.00.01.010100ξ(t) (nm)(Wk)-1 scalingα=1.0β=0.51.01.21.41.61.82.00.00.20.40.60.81.01.2βαScaling exponentsFluence, F (mJ/cm2)BZ boundaryokwould grow twice larger. Also, β = 0.5 is universal for phase ordering kinetics for systems

with non-conserved order parameter [1].

α is for scaling in the local order parameter of IC-CDW. In our experiments, we show

that α = dβ, where d = 2 is the dimensionality — a well-accepted result for coarsening under

a quench [143].

The scaling exponents are independent of pump fluence. Clearly, the dynamics do show

strong fluence-dependence. The difference in dynamics between different excitation fluences
lies in the level of scaling: all the correlation length follows 1/2 power-law scaling: ξ ∝ t0.5,
2D · t, where D

but the coefficients for each fluence are different. Here we can define ξ =

√

is the diffusion coefficient and it has unit of nm2/ps [143]. Fig. 5.10b shows the diffusion

coefficient at different laser fluences. It increases with fluence and changes by an order of

magnitude in the applied fluence range.

From the raw dynamics curves at various fluences, the curves are very different. Fig.

5.10a is a plot of the raw data with offset of the prethermalization turning point: delay
t is offset by tpre, intensity m IC (t) is offset by m IC (t) − m IC (tpre). By simply scaling

the horizontal time axis with a coefficient for each fluence, the cures would collapse into a

single curve — see Figs. 5.10c&d. The coefficient Dm for each fluence is very similar to the

diffusion constant extracted from correlation length fittings Dξ.

Recent high-impact papers on isolated Bose gas [141, 142] measurements are two of the

best examples to demonstrate universal dynamics in the post-quench relaxation process and

existence of a nonthermal fixed point, even though theory has been established decades ago.

In these experiments, they work on isolated cold atoms, the measurements rely on Fourier

transform to extract the distribution and scaling functions. In comparison, our data measure

condensed matter systems in the real materials. Our probe is based on scattering and by

118

Figure 5.10: The universal scaling of IC peak dynamics across all fluences. a, Unscaled IC
dynamics after prethermalization. Horizontal time is delay after prethermalization, t − tpre.
Vertical scale is intensity above prethermalization, I(t)− I(tpre). b, The fluence-dependence
of diffusion coefficients. Dξ is from correlation length fitting and Dm is from IC universal
scaling in the thermalization stage. c, IC universal dynamics scaled by diffusion coefficient
In the horizontal axis notation, t2 = t − tpre. d, IC universal dynamics scaled by
D ξ.
diffusion coefficient Dm.

nature it is a direct representation of correlation functions (scaling functions). The high co-

herence and time resolution of our electron probe allow us to directly monitor the correlation

function (i.e. peak profile) evolution. This level of resolution and sensitivity represents a

significant step forward in studying non-equilibrium dynamics in complex materials.

119

0200400600800100012000.00.20.40.60.81.0Fluence(mJ/cm2):1.051.161.31  1.511.681.731.791.881.96mIC(t)-mIC(tpre)Delay, t2 (ps)Unscaled dynamics-2024681010010000.00.20.40.60.81.0Scaled dynamicsbased on DmmIC(t)-mIC(tpre) 2Dm*t2 (nm2)1.01.21.41.61.82.00.000.250.500.751.00 Dξ from ξIC fitting Dm from mIC scaling D (nm2/ps)Fluence, F (mJ/cm2)acbd-2024681010010000.00.20.40.60.81.0 2Dξ*t2 (nm2)Scaled dynamicsbased on DξmIC(t)-mIC(tpre)5.4 Phase transitions induced by 2500 nm laser

In recent years, there has been a lot of interests in developing novel optoelectronic devices

based on graphene, transition metal dichalcogenides (TMD), and heterostructures of layered

materials [144–149]. Due to the high flexibility in controlling photodoping density, fast

write/erase cycles and vast variety of materials to choose from, laser-based optoelectronic

devices are promising to find vast applications in the future.

For a material such as 1T-TaS2 in optoelectronic applications, there are many concerns

one like to get answers for. For example, what is the intrinsic switching speed and can we

improve it? How do we deal with heat dissipation problem? In this section, based on our

results, we will provide a possible solution for improving the switching speed and at the same

time depositing much less energy to avoid heating problem in applications.

In a rigid lattice, for example in silicon or gold [150–152], there is no lattice symme-

try change by laser excitation. So the post-excitation behavior is mainly governed by the

carrier relaxation and subsequent lattice heating effect. By contrast, in symmetry-broken

complex materials for example VO2 [153–155] or 1T-TaS2 [106, 122], where there are both

strong electron-electron and electron-phonon couplings, disruption to the electronic system

instantaneously modifies the lattice potential energy surface and ions will spontaneously

proceed to a high symmetry state. Such a process does not necessarily require lattice tem-

perature rise, although lattice heating is inevitable through electron-phonon coupling in the

long timescale. To clarify whether the phase transition is primarily thermally-driven or via

photo-doping effect, we designed experiments by using 2500 nm and 800 nm lasers to induce

phase transitions in 1T-TaS2 and compare the results.

The photon energy between 2500 nm and 800 nm differs by a factor of 3, as illustrated

120

Figure 5.11: The schematic drawing of phase transitions by 2500 nm and 800 nm lasers.
a, Phase switching between NC- and IC-CDWs by means of either photo-doping or photo-
thermal effect. b, The schematic drawing of density of states (DOS) near the FS for 1T-TaS2
at RT, adapted from Ref. [13]. c, The schematic drawing of the transition from NC-CDW
to IC-CDW via chiral symmetry breaking.

in Fig. 5.11b, while for both of them, each absorbed photon creates one pair of electron

and hole, ignoring nonlinear effects. This gives us a unique opportunity to do comparative

experiments to clarify whether phase transition is induced by photo-thermal or photo-doping

effect. Photo-thermal refers to the thermal and lattice heating effect by laser excitation.

Photo-doping is similar to chemical doping, which is generally related to changes in the

electronic states independent of lattice heating.

Fig. 5.12 shows the fluence-dependent dynamics induced by 2500 nm laser. As we can

see, there is also distinct prethermalization plateaus as seen in 800 nm. From the peak profile

analysis, the 2500 nm excitation also has asymmetric tails in the NC peaks — signatures of

FO and chiral symmetry breaking (see Fig. 5.11c).

From the fluence-dependent data, we can extract the characteristic time of IC state

formation (Fig. 5.12c). Here, we use the time it takes to reach 1/2 of the IC intensity and

define this time as τ1/2. In both cases, the photo-doping enables a new dynamical pathway

driving phase transitions. Accordingly, we determine the excitation density to be 0.68 and

121

qNCqICSymm. Chiral symm.recoverySymm. breakingbreakingNC-CDWIC-CDWPhoto dopingPhoto thermalhν?EFE800 nm2500 nmDOSabc0.75 nm−3 for 800 nm and 2500 nm, respectively. The difference is within 10%. But in

contrast, the corresponding absorbed energies are 1.05 and 0.37 eV/nm3, respectively — a

factor of 3 difference.

Figure 5.12: The fluence-dependent data of 2500 nm dynamics and thresholds comparison.
a, The NC-CDW peak intensity evolution at various pump fluences by 2500 nm laser. b,
IC-CDW peak growth at various pump fluence, the same color code as a. c, Comparison of
the IC formation time for 800 nm and 2500 nm and determination of photo-doping, photo-
thermal thresholds.

Note: the absorbed photon and energy density are calculated based on the following:

122

abc04010010000.00.20.40.60.81.0Normalized NC intensityDelay, t (ps)  0.63 0.75 0.83 0.92 1.00 1.08 1.17 1.38F (mJ/cm2):NC suppression04010010000.00.20.40.60.81.0Normalized IC IntensityDelay, t (ps) IC formation0.40.60.81.01.31.51.7101001000IC τ1/2 2500 nmExpected 2500 nm  800 nmIC formation time (ps)Photo-doping density (nm-3)0.20.30.40.50.650.750.852500 nm photon absorbed energy (eV/nm3)800 nm photon absorbed energy (eV/nm3)800 nm photo- 1.01.21.40.80.62.02.32.6thermal thresholdPhoto-dopingSpecific/latent heatthresholds:2500 nm800 nmphoto-thermal thresholdbased on 1T-TaS2 optical measurements on the bulk sample [139], for 2500 nm laser, reflec-

tivity is R = 0.534 at normal incidence. According to thin film transfer matrix method [90]

and Appendix B, the total reflectivity and transmission are R = 0.42, T = 0.22, respectively
(experimental conditions: sample thickness d = 59 nm, 45◦ incidence, p-polarized). So the
photon absorption coefficient is: 1−R−T = 0.36. The absorbed energy density is calculated
to be: ρE = F (1−R−T )
density ρE = 0.37 eV/nm3, corresponding to absorbed photon density of 0.75 photon/nm3.

. At critical threshold of Fc = 1.04 mJ/cm2, the absorbed energy

d

The coincidence in photo-doping density at the phase transition threshold suggests that

the phase transition is more coupled to the photo-doping effect (excited carrier density)

than photo-thermal effect (thermal heating). The absorbed photon density at the threshold
level of ∼ 0.7 photon/nm3 corresponds to 0.52 absorbed photon per Star-of-David, which

coincides with the iron-doped 1T-TaS2 [13]. Comparing with the specific heat measurement

[119], 800 nm at the critical level Fc,2 is above the thermal requirement but the 2500 nm

threshold is way below that. This observation is consistent with a previous report by Han

and coworkers [106].

More evidences on doping and thermal effects are shown in the time-resolved dynamics

in Fig. 5.13, where both 800 nm and 2500 nm are at a similar absorbed photon density
level of ≈ 0.75 nm−3. In Fig. 5.13a, we examine the satellite intensity integral. The overall

changes in NC satellite intensities are very similar whereas clear differences are shown in IC

intensities. IC peak change with 2500 nm excitation is seen to rise very fast to 20% in the

first 2 ps while 800 nm excitation does not lead to an obvious initial rise. Another difference

is 2500 nm excitation only reaches to 0.8 while 800 nm data rise to 1. This difference is also

revealed in the previous fluence scan plot. Without enough thermal energy deposition, the

2500 nm data can only reach 80% of the level reached in 800 nm data in the thermalization

123

process.

Figure 5.13: The CDW and Bragg peak dynamics at ρhν ≈ 0.75 nm−3 for both 800 nm

and 2500 nm pump excitation. a, NC and IC peak intensity evolution. b, The Bragg peak
dynamics comparison and corresponding fit guidelines for the fit model.

The Bragg peak dynamics plotted in Fig. 5.13b reveals more information. To simulate

124

0.00.20.40.60.81.0  800 nm:NC  IC2500 nm:NC  ICNormalized NC & ICρhν= 0.75 nm-3-2024681010010000.80.91.01.11.21.3  800 nm:Bragg      Fit2500 nm:Bragg      FitNormalized BraggDelay, t (ps)Three portions for the fit:ICNCDWFaband understand the Bragg peak data, we propose the following simple model. Bragg peak

intensity change has three contributions: 1, the intensity increase due to NC state suppression

(symmetry recovery); 2, the intensity drop due to IC phase formation (symmetry breaking);

3, the intensity drop due to thermal heating (Debye-Waller effect). For the Debye-Waller

(DW) factor, it is a convention to assume an exponential decay (two-temperature model)

with the characteristic electron-phonon coupling time. For considering the static lattice

distortion due to the NC and IC states, it is necessary to refer to the structure factors of

CDW.

According to Giuliani and Overhauser [156], the structure factor of a CDW system can

be written in the following way

(cid:88)

+∞(cid:88)

n=−∞

G

S(q) =

δ[q − (G + nQ)]J 2

n(q · A)F φ

n F A

n F DW (q ),

(5.8)

where q is the scattering wave-vector, G is the reciprocal lattice vector, Q and A are

the CDW wave-vector and lattice distortion vector, respectively. Jn is Bessel function of

the first kind of order n, with n = 0 corresponds to Bragg peak, n = 1 (2) corresponds to

first- (second-) order CDW peaks. F φ

F DW (q , u) = exp(cid:0)−(cid:104)(q · u)2(cid:105)(cid:1) is DW factor with atomic vibration amplitude u. With

n are the phase and amplitude factors and

n and F A

small amplitude approximations at an initial CDW distortion amplitude A0, CDW intensity

can be written as: ICDW (q , A0) = J 2

1 (q · A0) ≈ 1

4(q · A0)2F φ

1 F A

1 . Bragg peak intensity:

IBragg(q , A0) = J 2

0 (q · A0)F DW (q , u) ≈ [1 − 1

2(q · A0)2]F DW (q , u).

After laser excitation, assuming that the lattice distortion changes from A0 to A1, the

125

normalized CDW peak intensity can be written as

Inorm
CDW =

ICDW (q , A1)
ICDW (q , A0)

=

A2
1
A2
0

.

Normalized Bragg peak intensity can be written as

Inorm
Bragg =

IBragg(q , A1)
IBragg(q , A0)

=

1 − 0.5(q · A0)2Inorm

CDW

1 − 0.5(q · A0)2

exp

(cid:104)−q2(u2

1 − u2
0)

(5.9)

(5.10)

(cid:105)

.

Assuming NC- and IC-CDWs are independent, the normalized 1T-TaS2 Bragg peak

intensity can be written as follows

Inorm
Bragg = [

1 − 0.5(q · A0)2 · Inorm

N C

1 − 0.5(q · A0)2

] · [1 − 0.5(q · A0)2 · 0.45Inorm

IC ] · [a · exp[−t/τDW ] + 1 − a],

(5.11)

where the first part is the contribution from NC (dashed line in Fig. 5.13b), the second part

is related to IC (dotted line in Fig. 5.13b), the third part is the DWF, where we assume

an exponential decay with a time constant τDW and amplitude of a. In this simple model,

Inorm
N C and Inorm

IC

are independently measured (Fig. 5.13a). τDW is estimated to be 1 ps,

a typical value for electron-phonon coupling time constant and it fits our data well. So the
only fitting parameters left are (q · A0) and DWF amplitude a.

To best fit the 2500 nm data, |q · A0| = 0.68 and a = 0.06 are obtained. The Bragg
peaks in the plot has a q-value of 5.6 ˚A−1. Considering the maximum lattice distortion for
1T-TaS2 is ∼ 0.2 ˚A [157] at RT, |q · A0| = 0.68 is a reasonable estimate. Using the same
|q · A0|, τDW , and DW amplitude a to fit the 800 nm data, the data and fit agree well

in the first several ps. But there is another decay process that is much longer than 1 ps

in the thermalization region. To capture the longer time scale, we have to impose another

126

DW exponential decay with a time constant of ∼ 300 ps and amplitude of 0.09. With this

addition, it fits the data quite well. The total effective DW decay amplitude for 800 nm is

0.15, a factor of 2.5 that of 2500 nm data. Based on absorbed energy density calculations,

the temperature rises from 2500 nm and 800 nm pump are 23 K and 71 K, respectively. This

a factor of 3 in energy deposition in general agreement with the factor of 2.5 in DW decay

amplitudes.

In the fit model of Bragg peak dynamics, we first use the CDW intensities (both NC and

IC) to extract the static distortions. Then the static distortions are applied in the Bragg

peak intensity formula to fit the Bragg peak dynamics.

In this process, both CDW and

Bragg peak intensity formulas use small-amplitude approximations in the Bessel functions.

To verify the validity of this approximation, we estimate the error bars based on NC-CDW

intensity (IC -CDW has weaker intensity thus smaller error bars). When NC-CDW intensity

is 1 at negative time, the calculations with both Bessel function and approximations reveal

the same result: the error bar is zero. When NC-CDW intensity drops to 50%, the difference

between Bessel function and small-amplitude approximation is 1.2% in Bragg peak intensity

calculation. When NC-CDW intensity drops to 10%, the difference is 2.2%. When NC-CDW

intensity drops to 0, the difference is zero. So through out the NC-CDW peak evolution, the

error bar is less than 2.5%. This small difference between Bessel function and approximation

verifies that the small-amplitude approximation is good enough for the calculations.

With a higher excitation fluences, at similar doping level of 0.9 and 1.0 nm−3 for 800

nm and 2500 nm, respectively, the NC and IC peak evolutions are plotted for comparison

in Fig. 5.14. From the results, while NC peaks also show similar levels of change initially,

IC peak dynamics are drastically different. It takes 2 ps and 15 ps to reach 40% and 80%,

respectively, by 2500 nm laser excitation. In comparison, to reach the same level of changes,

127

Figure 5.14: The CDW peak dynamics at excitation density of ρhν = 0.9 and 1.0 nm−3 for

800 nm and 2500 nm lasers, respectively.

it takes 14 ps and 80 ps, respectively, for 800 nm laser — more than a factor of 5 slower than

2500 nm excitation. A faster switching speed from 2500 nm excitation is obvious.

From previous data, one of the main differences between the two excitation lasers is the

IC formation. Fig. 5.15a shows the IC relative intensity at various fluences at 10 ps ant 1

ns with 2500 nm laser excitation. In the short time scale (10 ps), there is a sharp jump that

defines the photo-doping threshold. In the long time scale (1 ns), the sub-threshold doping

produces partial IC growth in activated processes than thermalization. In comparison with

800 nm data as shown in Fig. 5.15b, 800 nm data do not have such a sharp jump at the

threshold.

In summary, the NC to IC phase transition in 1T-TaS2 is closely coupled to the photo-

doping density rather than the energy density. With 2500 nm laser excitation, it deposits

128

-2-10123451010010000.00.20.40.60.81.0  800 nm (0.9 nm-3): NC IC2500 nm (1.0 nm-3): NC ICNormalized changeDelay, t (ps)Figure 5.15: The comparison of IC formation for the two excitation lasers. a, IC relative
intensity at 10 ps and 1 ns by 2500 nm laser excitation. b, Comparison of IC peak relative
intensity at 10 ps with 800 nm and 2500 nm excitation.

less energy while showing a faster initial switching speed. In the long time thermalization

regime, both 800 nm and 2500 nm data show scaling in the IC coarsening process. But 2500

nm has a much slower coarsening rate than 800 nm due to the lack of thermal energy. The

less energy deposition, faster initial switching speed and distinct threshold behavior of 2500

nm excitation all point toward a better choice than 800 nm in optoelectronic applications.

129

ab0.00.20.40.60.81.0 10 ps 1 nsRelative intensity2500 nm:ActivatedIC preferredNCICNCIC0.60.81.01.21.40.00.20.40.6 800 nm 2500 nmRelative intensityF/FcResponse at 10 ps:5.5 Summary

To summarize, in this chapter, we explored phase transitions in 1T-TaS2. The dynamics

separates well in the time domain — initial order parameter quench in the first 0.5 ps,

followed by prethermalization plateau (chiral-symmetry-breaking period), then by thermal-

ization and coarsening in the longer timescale. The dynamics have well-defined universal

scaling behaviors in each regime. Microscopically, a chiral-symmetry-breaking mechanism

is proposed and proved by reconstruction and simulations. With 2500 nm laser excitation

for comparison, the phase transition is proved to be connected to the photo-doping effect.

Phase transition thresholds for such a novel transition pathway are primarily determined by

photo-doping densities by comparing the data from 800 nm and 2500 nm laser excitations.

Lack of excess thermal energy in the 2500 nm data proves to be more first order-like with

faster switching speed.

130

Chapter 6

Study of iron chalcogenides by

ultrafast electron diffraction

Superconductivity (SC) is an old but fascinating topic ever since its first discovery in 1911 by

H. K. Onnes [158]. Five Nobel Prizes in physics have been given related to superconductivity

(in the years of 1913, 1972, 1973, 1987, 2003) across almost a century. Realization and

understanding of SC is the dream of many condensed matter physicists. A major step

forward in this field was the discovery of SC in copper oxide in 1986 [159]. In the following

20 years, most records in Tc were related to the copper-based SC. This made people believe

that copper oxide plays a magic role in SC. But in 2008, an iron pnictide compound was

found to have SC with Tc of 26 K [160], which promoted a new field of iron-based SC (Fe-

SC). In the same year, another iron pnictide compound was reported to have a Tc of 55

K [161] and iron chalcogenide FeSe was also shown to have SC with Tc of 8 K [162]. In 2012,

it was demonstrated that single-layer FeSe on the SrTiO3 substrate exhibits Tc of 77 K —

ten times higher than the bulk FeSe. Another work reported the Tc could reach 109 K in

single-layer FeSe on SrTiO3 substrate [163]. Iron chalcogenide is the simplest form of Fe-SC

materials and will be the main discussion in this chapter, particularly from structural point

of view.

Superconductivity in Fe-SC arises from the proximity of anti-ferromagnetic (AFM) order.

131

The electron pairing mechanism is not BCS-type so Fe-SC is unconventional SC [164]. It is

believed that the AFM correlations are important for understanding SC in Fe-SC. Moreover,

the occurrence of AFM is always accompanied by a structural transition. On the high

temperature side, it is paramagnetic with tetragonal structure. On the low temperature side,

it is AFM or spin-density wave order with a lower structural symmetry, usually orthorhombic

or monoclinic order. The low temperature state is often referred to as the nematic order

because the 4-fold rotational symmetry is reduced to 2-fold. This nematic phase is a precursor

of SC. So it is crucial to understand how the structural phase transition happens and the

formation mechanism of the nematic phase in order to better understand and manipulate

the SC in Fe-SC.

FeTe is the most stable iron chalcogenide, but it is non-superconducting. On the other

hand, FeSe is superconducting with Tc of 8 K (bulk). With substitution of Te by Se, the

compound with a formula of FeSe0.5Te0.5 is reported to have the highest Tc of 15 K [165]. In

this chapter, we will show the UED studies of all three compounds: FeTe, FeSe0.44Te0.56, and

FeSe. In Sec. 6.1, the background information on iron chalcogenide systems will be provided.

Then ultrafast studies of FeTe at both low and high temperatures will be presented in Sec.

6.2. Structural transition from low temperature monoclinic to high temperature tetragonal

phase was observed and will be discussed. In Sec. 6.3, we will show the ultrafast responses

in an optimally-doped sample of FeSe0.44Te0.56. We report ultrafast dynamics of FeSe in

Sec. 6.4. Finally, a summary is presented in Sec. 6.5.

132

6.1 Background and properties of iron chalcogenide

compounds

Iron chalcogenide is the simplest form of Fe-SC class. It is composed of layered structure

as shown in Fig. 6.1.

In one unit cell, there are three layers with two chalcogen layers

sandwiching the Fe layer, which is responsible for SC, as well as magnetic orders. A zoom-in

view of the structural order is also plotted. One Se/Te atom forms bonds with 4 Fe atoms

in a pyramid shape. Around one Fe atom, 4 Se/Te atoms form a tetrahedron structure. At
above ≈ 75 K, both FeSe and FeTe have tetragonal structure with lattice constant a = b.

Below that temperature, it transforms into the orthorhombic (FeSe) or monoclinic (FeTe)
phase with a change of the unit cell and lattice constant a (cid:54)= b (Fig. 6.1d).

The binary phase diagram between FeTe and FeSe is presented in Fig. 6.2a (from Ref.

[15]). The green curve represents the Tc.
It goes from 8 K in FeSe to about 15 K at
x ≈ 0.5. With Te concentration higher than 0.5, the Tc drops and it is non-superconducting

in FeTe. The structural phase transition temperature Ts is anti-correlated with Tc. At
the optimal-doping of x ≈ 0.5, there is no longer a structural transition all the way down

to zero temperature.
In FeTe, the structural transition is accompanied by a AFM/SDW
orders along a∗ direction. The structural transition is observed by peak broadening or peak

splitting while SDW order is seen through emergence of new peak — see Fig. 6.2b.

Superconductivity in FeSe is found to be enhanced under external pressure. The Tc

increases to above 20 K at 4 GPa pressure (Fig. 6.2c). At the same time, the height of Se

above Fe layer is decreasing with pressure (Fig. 6.2d). A more direct way of demonstrating

how the structural order affects SC is shown in Fig. 6.2d, where all the Fe-SC materials

are summarized. There exists an optimal anion height that generates the highest Tc. The

133

Figure 6.1: The structures of FeTe/FeSe in the high temperature tetragonal phase. a, 3D
view of the ball-stick model structures. Cuboid with solid line represents the unit cell. b&c,
Side and top view of FeTe/FeSe structure, respectively. d, Lattice constant changes across
the tetragonal-orthorhombic structural transition for FeSe0.92, reprinted from Ref. [14].

FeSe under pressure is plotted in open red circle symbols and it follows the trend relatively

well. For our measurements using the electron probe, we are interested in exploring how

the structure transforms after laser excitation to provide evidences and potential answers for

134

abccaFeSeabFeSe3D viewSide viewTop viewa, bacdbFigure 6.2: The phase diagrams of iron chalcogenides. a, The phase diagram of FeTe1−xSex,
figure from Ref. [15]. b, Neutron scattering measurement of FeTe to demonstrate simul-
taneous emergence of AFM order and structural transition, figure from Ref. [16]. c, The
pressure-dependence of Se height and superconducting temperature Tc in FeSe. d, The an-
ion height dependence of Tc of typical Fe-SC materials. Figure is reprinted from Ref. [15].
The overall trend indicates there exists an optimal anion height for the highest Tc.

understanding SC in Fe-SC.

135

½ 0 ½0 0 12 0 0222381828384852θ (deg)52.562.572.582.5Temperature (K)FeTeacbd6.2 Ultrafast phase transitions in FeTe

At room temperature, FeTe is in tetragonal phase with space group of P4/nmm. The unit

cell contains 2 Te and 2 Fe atoms (lattice constants: a = b = 3.822 ˚A, c = 6.285 ˚A). At 72.5

K, it transforms into monoclinic phase with space group of P2/1 m. Each unit cell contains

4 Te and 4 Fe atoms (lattice constants: a = 5.331 ˚A, b = 5.308 ˚A, c = 5.485 ˚A, Ref. [16]). For

the diffraction pattern taken at RT, it is labeled in the following way: Bragg peak (1 0 0) to
(0 0 0) has length of a∗ = 2π/a and Bragg peak (0 1 0) to (0 0 0) has length of b∗ = 2π/b,

where a and b are lattice constants in tetragonal phase. In the monoclinic phase, there is a

change in unit cell thus in the labeling. The (2 2 0) peak in tetragonal phase splits into (4 0

0) and (0 0 4) peak in monoclinic state (Ref. [14]). In fact all the peaks with component in

the a-b plane exhibit splitting or broadening, as shown by X-ray diffraction measurements in

a previous paper [16]. This is also what we observed for the first time with ultrafast electron

diffraction.

Diffraction patterns in Figs. 6.3d&e are taken at similar electron beam conditions (30
keV electron energy, ≈ 5000 e−/pulse) but at different sample temperatures. The differences

are quite visible: all the peaks are very sharp at RT while the peaks at 23 K are blurry

(broadened), especially at higher s values. The difference image between RT and 23 K

shown in Fig. 6.3f indicates the movement and broadening of the peaks.

The time-dependent measurements in Fig. 6.4 track the peak position and width induced

by laser excitation at the following experimental conditions: the sample is cooled to 23 K

and pumped by 2500 nm laser at F = 2.7 mJ/cm2 and 1 kHz repetition rate. The anisotropy

in the lattice along a and b axes can be distinguished from the static diffraction pattern,

where the distances between (6 0 0) to (-6 0 0) and (0 6 0) to (0 -6 0) differs by about 2

136

Figure 6.3: The structure and diffraction patterns of FeTe in the RT tetragonal and low-
temperature monoclinic phase. a, Top view of the RT tetragonal structure. Lattice constants
expand by a factor of ≈√
a = b = 3.822 ˚A. b, Top view of monoclinic structure at 23 K. Lattice constants a and b
2, with a = 5.418 ˚A, b = 5.350 ˚A due to unit cell doubling. c, FeTe
lattice constant changes across the structural transition at ≈ 72 K [16]. d&e, Full diffraction
pattern of FeTe at RT and 23 K, respectively. Electron diffraction pattern of FeTe at RT.
30 keV electron beam is along the [0 0 1] zone axis. The labeling is based on a∗ = 2π/a and
b∗ = 2π/b, where a and b are the tetragonal lattice constants. f, The difference diffraction
pattern of RT minus that of 23 K. Both patterns are acquired at the same beam conditions
and exposure times. The peak shift and sharpening/broadening can be clearly visible from
the difference image.

pixels. After laser excitation, the peaks move toward more symmetric positions. The lattice

constant changes are plotted in Fig. 6.4a. The lattice constant a decreases by 0.1% while b

increases by 0.2%. By comparing with thermodynamic X-ray measurements [16], the lattice

constants change at the first-order phase transition are - 0.42% and + 0.76% for a and b,

respectively. Our measurements are overall a factor of 4 smaller than the reported values. It

137

abcRTRT, tetragonal23 K, monoclinic23 KRT - 23 Kdabcef(0 1 0)(0 0 0)b*a*(1 0 0)ababFeTe01002003003.783.803.793.823.833.816.256.276.266.28Lattice constant (Å)Lattice constant (Å)T (K)is likely that either our laser fluence is not high enough or only 25% of the sample is driven

to the tetragonal phase while the rest 75% is still in the monoclinic state.

To validate this, we estimate the temperature increase induced by laser. Based on optical

measurements of Fe1.05Te [166], the optical reflectivity R = 0.65 and the real part of optical
conductivity σ1 = 2.7 ∗ 105 Ω−1m−1 at 2500 nm at 23 K. We assume the same values

apply to FeTe. With this, the imaginary part of electrical permittivity 
2 is calculated to

be 4.14 and optical refractive index n = 0.834. The extinction coefficient κ = 2.48, which

derives a penetration depth of 80 nm for 2500 nm photon. With transfer matrix method

described in Appendix B, the optical absorption is calculated to be 0.34. The absorbed energy
density with fluence of 2.7 mJ/cm2 is calculated to be 0.72 eV/nm3 (sample thickness ≈ 80

nm). Based on the specific heat of FeTe [167], the absorbed energy would raise the sample

temperature from 23 K base temperature to 130 K, assuming all the absorbed energy is

converted into heat and no heat loss.

This estimated final temperature is way above the structural transition temperature of

72.5 K. This discrepancy might because of several reasons: 1, the temperature estimate is

not considering heat dissipation from sample to TEM grid; 2, the absorbed energy density is

overestimated due to the inaccurate values of optical constant input, which are extrapolated

from Fe1.05Te measurements instead of FeTe; 3, in the specific heat integration, the latent

heat is not mentioned in the paper and it is underestimated in the calculation. The latent

heat in a first-order phase transition could be very large compared to the specific heat

integration. For example, in VO2, from 0 K to 340 K, the integration of specific heat

(not including latent heat) is only twice that of latent heat in the monoclinic to tetragonal

transition at 340 K, which is the same type of structural transition as in FeTe. The specific

heat is generally increasing from 0 at 0 K and gradually plateaus at some value around 300

138

K. For FeTe, the transition temperature (72.5 K) is much lower than that of VO2 (340 K).

If the latent heat in FeTe is similar to that of VO2, then it would be easily several times

higher than the total specific heat integration from 23 K to 72.5 K, reducing the estimated

temperature rise to below 72.5 K.

Figure 6.4: The time evolution of FeTe lattice constants and peak width at base temperature
of 23 K (sample a, 2500 nm pump laser, 2.7 mJ/cm2). a, The lattice constant changes along
a and b directions, extracted from (4 0 0) and (0 4 0) peak movement relative to the center
(0 0 0) peak. b, c&d, Peak sharpening after laser excitation with Gaussian profile fit for
peak (-4 0 0), (0 4 0), and (-2 2 0), respectively. The width σ here refers to σ of a Gaussian
fitting.

Aside from lattice constant change to demonstrate the structural transition, there is also

peak sharpening (merging). Figs. 6.4b-d shows three typical peaks become sharper after

139

-100102030200400600-0.2%-0.1%0.0%0.1%0.2%0.3%0.4%aRelative lattice constant changeDelay, t (ps)Lattice constant changeb0204060801000.900.951.001.05 σ along a* direction σ along b* direction GuidelineRelative peak widthDelay, t (ps)Peak (-4 0 0) width change0204060801000.700.750.800.850.900.951.001.051.10Peak (0 4 0) width change σ along a* direction σ along b* direction GuidelineRelative peak widthDelay, t (ps)0204060801000.850.900.951.001.05Peak (-2 2 0) width change Radial (a*+b*) direction Tangential (a*-b*) directionDelay, t (ps)Relative peak widthabcdlaser excitation. Peaks are fitted with Gaussian fitting and the presented peak width is
normalized to sigma at the negative time. While peaks along a∗ and b∗ direction sharpens

isotropically, the peaks with index of (h h 0) shows anisotropic change with sharpening

mainly in the tangential direction (relative to center peak (0 0 0)) and almost no change in

the radial direction.

In terms of peak intensity, it is found that the peak intensity has oscillations with pe-
riod of ≈ 40 ps — see Fig. 6.5a and different peak oscillations have different phase about
90◦ apart. This oscillation corresponds to phonon frequency of 0.025 THz, which is most

likely an acoustic phonon. Fig. 6.5b is a schematic diffraction pattern that shows the peaks

with oscillations. About 1/3 of the peaks have very clear intensity oscillations. This peak

oscillation is also reproducible in other samples. Fig. 6.5c shows oscillations in another two

samples. But interestingly, instead of 40 ps as seen in sample a, the other two samples

(b&c) have an oscillation period of 60 ps. The oscillations last for about two periods be-

fore damping away. These very large amplitude oscillations without consistent pattern and

the periodicity seem to suggest that the oscillations are sample-dependent.

It’s likely to

originate from sample warping after laser pulse excitation. The change of sample flatness

affects the Ewald’s sphere cutting of the reciprocal rods thus the intensities. However, at

high temperatures, the oscillations are not clearly visible, as shown in Fig. 6.5d.

6.3 Ultrafast structural dynamics in FeSe0.44Te0.56

FeSe0.44Te0.56 is reported to have the highest Tc among the FeSexTe1−x series. At the same

time, it seems the structural and magnetic transitions are competing with SC. So with the

highest Tc, the structural transition is quench down to unmeasurable — no thermodynamical

140

Figure 6.5: FeTe peak intensity evolution at low and high temperatures. a, Four groups
of peaks evolution in sample a (λ = 2500 nm, F = 2.7 mJ/cm2). b, Peak notation of 4
different groups in an artificial diffraction pattern. c, The oscillations observed in another
two samples at low temperature, with a period of 60 ps instead of 40 ps seen in sample a.
Experimental conditions: for sample b: λ = 2500 nm, F = 2.3 mJ/cm2; for sample c: λ = 606
nm, F = 1.4 mJ/cm2. d, The peak intensity evolution of sample b at 225 K, λ = 2500 nm,
F = 1.8 mJ/cm2. No oscillations are observed at high temperature.

structural transition has been observed and the system remains in tetragonal phase down

to 0 K. The lack of structural transformation is confirmed with the ultrafast measurements,

where even at a low temperature (23 K), the diffraction pattern peaks remain sharp and no

peak broadening is seen (Figs. 6.6a&b) after applying a laser fluence of 1.1 mJ/cm2 (2500

141

0204060801001201400.60.81.01.21.4 Group 1 Group 2 Group 3 Group 4Relative intensityDelay, t (ps)23 KSample aBeam stop1 0 02 0 00 1 00 2 0Group 1Group 2Group 3Group 4acdb0501001502002500.70.80.91.01.11.2 Sample b Sample c, group 1 Sample c, group 2Relative intensityDelay, t (ps)23 K30 ps0501001502002500.70.80.91.0Relative intensityDelay, t (ps)Sample b, 225 Knm laser). Also, from the difference image, the lack of peak movement and sharpening is

evident (Fig. 6.6c).

Figure 6.6: Diffraction patterns of FeSe0.44Te0.56.
a, Original diffraction pattern of
FeSe0.44Te0.56 at 23 K. b, Zoom-in view of pattern in a. c, Difference diffraction pat-
tern of positive time (+ 40 ps) minus negative time (- 20 ps) under 2500 nm laser pumping
at F = 1.1 mJ/cm2.

While the lack of lattice structure modifications is observed within our observable window
(up to ≈ 200 ps), as shown in Figs. 6.7a&b, the peak intensity at different Bragg reflection

142

(1 0 0)(0 1 0)23 Kacb-20000-200-400-800-1000-1200-1400-1600-1800Figure 6.7: The peak dynamics of FeSe0.44Te0.56 at low and high temperatures (λ = 2500 nm,
F = 1.1 mJ/cm2). a, The lattice constant changes after laser excitation at base temperature
of 23 K. b, The peak width changes at 23 K. c, The intensity oscillations observed at 23 K.
d, The peak intensity dynamics at base temperature of 225 K. No oscillations are observed.

exhibit oscillations with a period of ≈ 40 ps, similar to the case of FeTe, as shown in Fig. 6.7c.

But these oscillations are only seen at low temperatures. Such oscillation occur over a single-

step decay in the intensity integral, indicating that optical energy is indeed transferred to

the lattice system. Interestingly, raising the sample temperature to 225 K, the peak intensity

does not show oscillatory behaviors but only a one-step drop.

143

04080120160200-0.4%-0.2%0.0%0.2%0.4%Relative lattice constant changeDelay, t (ps)Lattice constant change, 23 K040801201602000.60.81.01.21.4Relative peak widthDelay, t (ps)Peak width relative change, 23 K040801201602000.60.70.80.91.01.1Group 1Group 2Relative intensityDelay, t (ps)Peak intensity change, 23 K040801201602000.800.850.900.951.00Peak intensity change, 225 KRelative intensityDelay, t (ps)acdb6.4 Ultrafast dynamics in FeSe

In FeSe, we expect to see a structural transition similar to that of FeTe in terms of ab-plane

lattice constant change (see Fig. 6.1d and Fig. 6.3d) when pumping at low temperatures.

However, the data we acquired do not show any signs of such phase transition: the lattice

constant is not observed to change within the measurement resolution; the peak is not seen

to sharpen as expected, but rather an overall of 2.5% broadening is seen, as shown in Figs.

6.8b&c. While we believe, given what is observed in FeTe at F = 2.7 mJ/cm2, the applied

fluence here (1.1 mJ/cm2) may be too low to reach the thermal threshold for the phase

transition. The peak broadening phenomenon points to a different type of dynamics from

phase transformation. The peak intensity is also not seen to have any oscillations, even at

low temperature of 23 K.

Another interesting phenomenon observed in the FeSe dynamics is oscillations at absolute

peak positions at 225 K. As shown in Fig. 6.9a, the peak intensity evolution also has no
oscillations. But the peak position is seen to vibrate quite obviously, mainly along the b∗

direction. All peaks show oscillations with the same amplitude and phase along the same

direction. Figs. 6.9c&d are snap-shots at two time delays. It is clear from these images

that all the peaks move together. Such an oscillation starts even at negative time and the

oscillation frequency damps down over time.

This collective shifts in absolute positions of the Bragg peaks in the diffraction pattern

could be due to either the electron beam drifting or the sample tilts/oscillates. It is unlikely

to have periodic modulation to the electron beam, so it is more likely that the sample is

vibrating. From the sample geometry as shown in Fig. 6.9e, one side of the sample is in

good contact with the TEM grid while the other side is freestanding. From the oscillation

144

Figure 6.8: The diffraction patterns and dynamics of FeSe at 23 K (2500 nm pump laser,
F = 1.1 mJ/cm2). a, The diffraction pattern of FeSe at 23 K. b, Lattice constant change of
FeSe at 23 K, a and b axes are similar: no clear lattice constant changes are observed. c,
The peak broadening after laser excitation observed at 23 K. d, The peak intensity change
at 23 K.

direction shown, if the sample vibrates around the contact side, then this matches well with

the observed oscillation direction.

It is very likely that the various oscillation phenomena here are nuanced, resulting from

a large pump area that generates acoustic wave across the interface between the supporting

metal grid and the free-standing specimen. The observation that the onset of the oscillation

(in FeSe) even begins at the negative time, indicating the source for those oscillations does

145

050100150200-0.4%-0.2%0.0%0.2%0.4%Lattice constant changeDelay, t (ps)Lattice constant change, 23 K0501001502000.991.001.011.021.031.04Relative peak widthDelay, t (ps)Relative peak width change, 23 K0501001502000.80.91.0Relative intensityDelay, t (ps)Peak intensity change, 23 KabcdFigure 6.9: The dynamics of FeSe at 225 K (λ = 2500 nm, F = 1.1 mJ/cm2). a, Peak
intensity evolution at base temperature of 225 K. b, Peak shift along a∗ and b∗ direction.
c&d, Snapshots of different images at - 10 ps and 0 ps (both minus the diffraction pattern at
- 100 ps). All peaks exhibit similar drifts in amplitude and phase, mainly along b∗ direction.
e, The sample grid and zoom-in view of the sample used for the experiments.

146

-4004080120160200-0.2%0.0%0.2%0.4%0.6%0.8% Shift along a* Shift along b*Absolute position shift (r.l.u.)Delay, t (ps)0DecreaseIncrease-10 ps0 psabceda*b*a*b*e- beam: ab20 um040801200.850.900.951.00Relative intensityDelay, t (ps)Peak intensity change, 225 Knot start at the probed sample region. We have also observed such oscillations in other

specimen with wide irradiation footprint. But reducing the pump size by an aperture to

limit the exposure of the laser pulse only to the specimen has suppressed such nuanced

oscillation.

6.5 Summary

To summarize, the iron chalcogenide compounds studied here show very interesting results

in the Bragg peak dynamics. The properties and main experimental results are summarized

in Table 6.1.

In FeTe at below the transition temperature, the peaks are observed to sharpen and

the changes in lattice constants agree with the structural phase transition. The observation

of ultrafast phase transitions in FeTe is never reported before with time-resolved X-ray or

electron diffraction tools. This is the first report of FeTe phase transitions in ultrafast

time scale. Based on the estimated temperature rise, it is likely that the sample is indeed

heated above the transition temperature. The Bragg peak intensities show the sophisticated

large-amplitude oscillations, likely due to laser-induced acoustic phonons in the system. We

do not yet know exactly the sources that drive this interesting phenomenon, but given

that it occurs only in the monoclinic state, such acoustic wave generation may be tied to

the hydrodynamics modes in the system which involves collective orders (AFM). At high

temperature, as expected, there is no phase transition observed.

In the low-temperature state of FeSe0.44Te0.56 (23 K) right above Tc (15 K), there is no

phase transition observed as expected. But we observed very strong Bragg peak intensity

oscillations. The period of this oscillation (40 ps) coincides with that observed in FeTe. Given

147

Compound

FeTe

FeSe0.44Te0.56

FeSe

SC

AFM order

Structural PT

F (mJ/cm2)

DWF

Estd. ∆T (K)

Peak width

a, b change

Intensity osc.

Position osc.

Phase transition

F (mJ/cm2)

DWF

Estd. ∆T (K)

Peak width

a, b change

Intensity osc.

Position osc.

Phase transition

No

Yes

Yes

Yes

No

No

Experiments at 23 K (λ = 2500 nm):

2.7

0.69

90

1.1

0.78

40

Yes

No

Yes

1.1

0.80

40

Sharpen

No change

Broaden

Yes

Yes

No

Yes

No

Yes

No

No

Experiments at 225 K (λ = 2500 nm):

1.8

0.76

60

1.1

0.83

35

No

No

No

No

1.1

0.86

35

No change

No change

Broaden

No

No

No

No

No

No

No

No

No

No

Yes

No

Table 6.1: Summary of the properties and experimental results for all three iron chalcogenide
compounds. “DWF” represents the relative intensity at + 40 ps of the Bragg peaks analyzed.
The “Estd. ∆T (K)” means the estimated sample temperature rise based on applied fluences
and the DWF analyses.

that the oscillations have a very high s/n and last for more than 5 periods with little changes

in frequency, the observed acoustic phonons here might also be intrinsic to FeSe0.44Te0.56.

148

In the low-temperature state of FeSe (23 K), the system is expected to exhibit laser-

induced structural transition but our data do not show signs of any lattice constant change

or peak sharpening. This is likely due to the low excitation fluence. Based on both the

specific heat estimate and the DWF analysis, the final sample temperature is indeed below

the transition temperature. The observation of Bragg peak broadening is contradictory to

the expected peak sharpening effect. According to a recent work on FeSe [168], there are

nanoscale local nematicities that persist in a large range of temperature even above the

structural transition temperature. From the point of local nematicity and local stress, the

peak broadening phenomenon observed here can be explained based on laser-enhanced local

stress. This laser-induced instability in the stress-field exhibits peak broadening, which is

seen at both low and high temperatures. (peak broadening at high temperature is similar

to that observed at low temperature but not plotted in the figures.) At high temperature,

the peak positions are observed to have very strong oscillations. Given the facts that the

oscillation starts at negative time, oscillation frequency is damping down, and that the

oscillation polarization matches well with sample orientations, this phenomenon is very likely

due to the sample vibrations, not related to the intrinsic structural responses of these systems.

The phase transition observed in FeTe has never been observed by UED or other ultrafast

measurements. The very strong acoustic phonon oscillations observed in FeSe0.44Te0.56 are

also new and highly interesting. The enhanced local stress seen in FeSe is surprising and

may give information to help understand the nematic and superconductivity phase.

In

the meantime, these results call for experiments with better spatial/temporal resolutions to

further clarify the questions such as the nature of these excited acoustic phonons and role

of local stress/nematicity to SC formation in FeSe.

149

Chapter 7

Conclusion

The results presented here demonstrate the great potential of using ultrashort laser pulses

to explore phase transitions in complex materials in the non-equilibrium regime. With all

the results presented in this dissertation, it is evident that photo-induced phase transitions

(PIPT) follow different paths from that of steady state. The PIPT process, while frequently

proceeding at a high temperature, acts very much like an interaction-mediated phase transi-

tion, which shares key characteristics with the quantum phase transitions. This thus opens

up a new dimension in studying novel states in the material phase diagrams. Moreover, in

a quantum phase transition, there exists a quantum critical point due to competing phase

orders. The same is true for PIPT processes.

In a PIPT, the competition between dif-

ferent states also leads to the existence of a nonthermal fixed point. Near the nonthermal

fixed point, the system order parameters follow universal scaling laws and exhibit critical

divergence behaviors.

In CeTe3, the system is shown to be driven to a hidden bi-directional order that cannot

exist at thermal equilibrium.

In the first 200 to 300 fs after laser excitation, the order

parameter field is highly unstable, exhibiting characteristics of the so-called Kibble-Zurek

freeze-out time. After the KZ freeze-out time, the a-CDW order starts to form, together
with the c†-CDW order. The coincidence in time scales and amplitude between a-CDW and
c†-CDW orders indicates that the hidden state may emerge in a more symmetric electronic

150

energy landscape that prefers a checkerboard order over a stripe order. By examining the

CDW wave-vector change, it is found the nesting vector change is in agreement with the

photoemission measurements [93]. The photo-excited electrons cause a reduction in the

transverse coupling constant, and thus the FS nesting vector. The discovery here proves

another solid example of photo-induced metastable and hidden states in complex materials.

The study in ErTe3 is one step further based on the CeTe3 results. ErTe3 exhibits an

even richer phase diagram due to the existence of two CDW orders. The dynamics at

various temperatures show a-CDW may have a larger non-thermal melting threshold like in

the case of CeTe3. This unexpected result, together with those in CeTe3, provide a more

comprehensive picture of PIPT in the RETe3 series: the system follows transition pathways

different from that in thermal equilibrium and the ac-plane becomes more symmetric (non-

degenerate to degenerate). The same may be true for other “asymmetric” systems with

nonequivalent crystal axes.

In 1T-TaS2, the system is shown to observe universal scaling at both non-thermal and

thermalization regimes. Such universal dynamics in the nonthermal regime is the first exam-

ple of such universality observed in the prethermalization stage. The phase ordering kinetics

are also highly interesting in the thermalization stage. Although such results are predicted

theoretically and validated recently in cold quantum gas experiments, our experiments ex-

hibit subtleties that warrant further theoretical investigation. By comparison of the 2500 nm

and 800 nm results, we emphasize the photo-doping effect in driving the non-thermal phase

transition in complex materials. The phase transition by 2500 nm shows much less energy

deposition and faster switching speed, giving promises for better optoelectronic applications

with long-wavelength laser.

In the study of iron chalcogenides, we observed the ultrafast phase transition and exci-

151

tation of acoustic phonons. SC in iron-based superconductor arises near the AFM order.

The pairing mechanism is believed to be correlated with the spin order. At the same time,

the spin order is directly coupled with the structural order. The ultrafast structural phase

transition in non-superconducting compound FeTe presented here is first observed with UED

measurements. In the superconducting compound of FeSe0.44Te0.56, at a temperature close

to Tc, the observed strong acoustic phonon oscillations might be intrinsic to the system. But

further high-resolution experiments are needed to clarify the roles of the acoustic phonons

and how they are related to SC. The observed laser-induced local stress in FeSe is also highly

interesting. It is still unclear what the role of nematic order is in the formation of SC. The

observation of enhanced stress field after laser excitation may promote understanding of SC

in FeSe. However, given the limited s/n and available data set, experiments with better

temporal and higher coherence length are needed.

These results presented here show promising directions of using fs laser to explore metastable

and hidden phases in complex materials, as well as helping to understand the physics in the

equilibrium states. One of the ultimate goals in condensed matter physics is to realize photo-

induced superconductivity at high temperature or even room temperature. Even though it is

a long way ahead, the results presented here, along with other similar experiments in recent

years with different approaches, show promises towards that goal.

152

APPENDICES

153

Appendix A

2D sample preparation procedure

For the two-dimensional (2D) materials discussed in this dissertation, the popular “scotch

tape” method [169] is used for sample preparation.

The 2D bulk material is first cleaved to have a flat and clean surface, which may require

a few times of exfoliation. Then press a clean scotch tape onto the clean surface to peel off

a big and uniform sample piece, which could be more or less 1 µm thick (Fig. A.1a). Use

another clean tape to exfoliate with the one that has sample on for several times. As the

exfoliation times increase, the sample pieces would be thinner and smaller. Typically after

exfoliating 6 to 8 times, the sample pieces should be good for the next step (Fig. A.1b). The

sample quality can be checked by examining the transparency under optical microscope.

Fig. A.1 is an example. After measuring thickness of some freestanding sample on TEM

grid with TEM zero-loss energy spectroscopy thickness map, the sample transparency can

be determined to correspond to thickness. This thickness-transparency correlation can be

used to quickly determine the sample thickness on tape

Once the sample is properly exfoliated on the tape (usually make sure there is a significant

percentage of samples below 50 nm), we can drop cast two drops of acetone onto 1 cm x 1

cm silicon wafer and press the tape evenly onto the wafer with a glass piece. Press and hold

the glass piece for 2 minutes until acetone all evaporates. Then gently separate the tape

from Si wafer. Now most of the samples are on the Si wafer (Fig. A.1c). Place the wafer in

154

Figure A.1: Typical sample images during sample preparation process. a, 1T-TaS2 sample
on scotch tape. Light is reflected from sample surface so we see the gold metallic color. b,
Typical 1T-TaS2 samples on scotch tape after a few times of exfoliation. Light is transmitting
through the sample to show the transparency. c, Sample flakes on silicon wafer. d, Typical
optical image of the samples transferred to the TEM grid.

a clean beaker and put several ml of acetone. Put the beaker in a ultrasonicator for a few

seconds to make sure samples come off from the wafer into the acetone.

Drop cast two drops of acetone sample solution onto TEM grid. Wait for about 1 minute

to let acetone evaporate. Then the samples would be left on the grid for experiment use.

The density of sample is kept more or less as the one shown in Fig. A.1d to avoid sample

overlap (too dense) as well as enough samples for experiments.

155

100 um100 um400 um400 umabcdAppendix B

Transfer matrix method for optical

absorption calculations

In UED or other optical experiments, to estimate the absorbed optical density or energy

density, it is necessary to know the optical constants to calculate the absorption coefficient.

Since we are dealing with thin film samples of tens of nm, the interference effect at sample

interfaces should be considered in order to calculate the true absorption coefficient. This is

where the transfer matrix method is involved for the calculation.

In the following part, I will first list a few common optical parameters that are frequently

seen in literature then list their relationships, then introduce the transfer matrix method.

Reflectivity: R

Electrical permittivity: 


Refractive index: n

Extinction coefficient: κ

Attenuation coefficient (absorption coefficient): α

Optical conductivity: σ

The relative electrical permittivity 
r is related with refractive index by the relation:


r = n2 (for non-magnetic materials). Considering that both of them are complex numbers,
0 − κ2 = 
1 and 2n0κ = 
2.


r = 
1 + i
2, n = n0 + iκ, we can get the following relation: n2

156

Optical conductivity σ is also complex: σ = σ1 + iσ2 and by definition it is related with

electrical permittivity by: 
r = 1 + i


0ω σ1, where 
0 is
vacuum permittivity and ω is the photon angular frequency. The attenuation coefficient α is


0ω σ2 and 
2 = 1


0ω σ. Thus, 
1 = 1 − 1

related with extinction coefficient κ by: α = 4πκ

λ , where λ is optical wavelength. The optical

penetration depth δ is usually defined as inverse of the attenuation coefficient: δ = 1

α = λ

4πκ.

From optical measurements in literature, the most direct measurement is reflectivity

spectrum normal to bulk sample surface. Through the Kramers-Kronig analysis (for example

refer to the paper [139]), the real and imaginary part of optical conductivity (σ1 and σ2)

or permittivity (
1 and 
2) can be derived, with which we can get the refractive index (n),

extinction coefficient (κ), etc. n0 and κ are our input parameters for the following transfer

matrix calculations.

Consider a N -layer sample structure as shown in Fig. B.1. We assume the mth layer

has a thickness of δm and refractive index of nm. The incident optical wave has an electric

field of amplitude E+

0 . The forward field is labeled as + sign and backward field has - sign.

At the mth interface, the electric field at before and after the interface has the following





E+

E−

m

m

(B.1)

relationship:

 =

E+

m−1
E−
m−1



1

tm−1,m

1

rm−1,m

rm−1,m

1

where rm−1,m, tm−1,m are the reflectance and transmittance coefficients, respectively, at

the mth interface. They can be calculated based on the Fresnel equations.

At the mth medium, the electric field has the following relation:

E

E

m

(cid:48)+
(cid:48)−

m

 =

e−iδm

0





E+

E−

m

m

0

eiδm

(B.2)

157

Figure B.1: The schematic drawing of N -layer sample. The electric field amplitude is labeled
at each interface for both forward and backward waves.

At this point, the answer is straight forward for the N -layer problem: simply multiply all

the transfer matrices from the left to right for each medium layer and medium interface:

 =

E+

E−

0

0



e−iδ1

 1


r0,1

1
t0,1

1

r0,1

1

1

...

tN,N +1

rN,N +1

rN,N +1

1



e−iδ2

0

r1,2

1



0

eiδ2

0

 1
E

(cid:48)+



t1,2

 1


0

eiδ1

r1,2

= [T0,1T1,2 ... TN−1,N ] ∗

1

tN,N +1

where the Tm−1,m =

1

tm−1,m



1

rm−1,m

rm−1,m

1

N +1

1

0


e−iδm


rN,N +1

0



E

(cid:48)+

N +1

0

(B.3)



rN,N +1

1

.

0

eiδm

For Eq. B.3, with each side divided by E+
0 , the total reflectance and transmittance (in
(cid:48)+
E-field amplitude) are r =
N +1, respectively. The total reflectivity and
transmission (in laser intensity) would be R = (r∗)∗ (r), T = (t∗)∗ (t). Correspondingly, the

and t = E

E−
0
E+
0

158

Layer 1Layer m-1Layer mLayer NVacuumVacuumE0+E0-E’  -m-1E’  +m-1E’  -mE’  +mE -m-1E +m-1E -mE +mE’  -NE’  +NE’  +N+1E -N-1E +N-1E -NE +NE’1-E’1+n0n1nm-1nmnNnN+1Interface: 1m-1mm+1NN+1absorption coefficient A can be extracted from the reflectivity and transmission: A = 1 −
R − T . With absorption coefficient estimated, for one-layer sample, assuming homogeneous

excitation for thin samples, the absorbed energy density is

F (1 − R − T )

δ

ρE =

where F is the laser fluence, δ is the sample thickness.

The absorbed photon density is the energy density divided by photon energy hν

ρhν =

ρE
hν

(B.4)

(B.5)

For two dimensional layered materials under study discussed in this dissertation, for example

1T-TaS2, CeTe3, ErTe3, etc., the system is a one-layer problem, as shown by the schematic
drawing in Fig. B.2a. In our real experiment, the pump laser incident angle is 45◦ as shown

in the figure. For 1T-TaS2, from reference [139], with 800 nm laser, n0 = 3.00, κ = 2.09;

with 2500 nm laser, n0 = 5.60, κ = 1.79. With n0 and κ as the inputs, we can calculate the

optical reflection, transmission, and absorption coefficients by the above discussed transfer

matrix method in Mathematica. These parameters are plotted as a function of sample

thickness in Fig. B.3. From the results, 800 nm data get stable after 100 nm while 2500 nm

data become stable after 400 nm. The interference effect is observable from the oscillations

as a function of sample thickness variation.

For more direct comparisons of the regular

calculation (ignore interference effect) and transfer matrix calculations, Table. B.1 is made

for a 50 nm thick sample for 800 nm and 2500 nm laser.

In this table, for the regular

method without considering interference effect, the optical reflectivity is taken from literature

159

Figure B.2: The schematic drawing of the wave propagation in (a) 1T-TaS2 and (b) VO2
film deposited on 9 nm amorphous silicon film.

[139]; transmission is calculated based on formula: T = (1 − R)e−a/δ and the absorption is
calculated based on: A = (1 − R)(1 − e−a/δ), where a is the sample thickness and δ is the

optical penetration depth. From the result, the absorption coefficient is very similar for 800

nm with two methods; but the absorption for 2500 nm pump differs by a factor of two.

For the thin film VO2 deposited on 9 nm amorphous silicon film, the sample is a two-

layer system. From optical measurements in reference [155, 170], for 800 nm laser, n0 =

2.88, κ = 0.42; for 2000 nm laser, n0 = 3.25, κ = 0.13. With this input, the optical reflection,

transmission, and absorption coefficients are calculated as a function of VO2 film thickness,

plotted in Fig. B.4. For this two-layer system, the interference effect is more pronounced from

the oscillation amplitude. Note, for the absorption calculations in VO2, the absorption by

amorphous silicon film is negligible. So the absorption by VO2 film takes the total absorption

of VO2 and silicon film combined.

160

1T-TaS2vacuumvacuumE0+E0-E1’+E1’-E2’+E1+E1-VO2a-SivacuumvacuumE0+E0-E1’+E1’-E2’+E2’-E3’+E1+E1-E2+E2-abCondition

Reflection,

Transmission,

Absorption, A

R

0.405

T

0.113

0.482

800 nm, w/o

interference

800 nm, transfer

0.352

0.122

0.526

matrix calc.

2500 nm, w/o

0.534

0.291

0.175

interference

2500 nm, transfer

0.423

0.222

0.355

matrix calc.

Table B.1: Comparison of the optical coefficients for 50 nm-thick 1T-TaS2 by using the
regular method and transfer matrix method for 800 nm and 2500 nm lasers.

161

Figure B.3: 1T-TaS2 optical reflection, transmission, and absorption coefficients as a function
of sample thickness for 800 nm (a) and 2500 nm (b) lasers. The pump laser is 45◦ incident
on the sample and p-polarized for the calculation, the same as experimental conditions.

162

0.000.20.40.60.81.0Sample thickness (nm)AbsorptionRe(cid:31)ectionTranmission1T-0.001002003004005006000.20.40.60.81.0Sample thickness (nm)AbsorptionRe(cid:31)ectionTranmission1T-abFigure B.4: The optical reflection, transmission, and absorption coefficients as a function of
VO2 film thickness for 800 nm (a) and 2500 nm (b) lasers. Inset of b are the same data but
plotted up to 4 um thick. VO2 film is deposited on 9 nm amorphous silicon film. Laser pulse
is 45◦ incident on the sample and p-polarized for the calculation, the same as experimental
conditions.

163

0.00.20.40.60.81.00100200300400500Sample thickness (nm)AbsorptionRe(cid:31)ectionTranmission0.00.20.40.60.81.00100200300400500Sample thickness (nm)AbsorptionRe(cid:31)ectionTranmission0.00.20.40.60.81.001000200030004000AbsorptionRe(cid:31)ectionTranmissionLonger rangeabAppendix C

Photo diode box-car integrator

(PD-BCI) system for pump power

measurement, monitor, and

calibration

In a UED pump-probe experiment, it is crucial to have a good pump-probe overlap and

monitor the pump conditions. Due to the temperature or humidity change in the lab, the

pump power might change or pump position could drift (which eventually also results in

pump power change at the sample). In the long time delay experiment over 2 or 3 ns, the

delay stage might not be perfectly aligned so the pump beam would drift at different time
delays. Another issue is that the regular power meter can only measure up to ± 0.1 mW

precision. Such a resolution might not be fine enough in a fluence scan, especially near the

threshold where 5% change in pump power could result in 100% change in diffraction signals.

To solve these issues, we designed a photo diode-box car integrator (PD-BCI) system to

measure and monitor the pump fluence in real time for each diffraction image at each time

delay. This system is computer controlled while acquiring diffraction images. The pump

fluence for each diffraction pattern is automatically stored in the image info. Ideally, with

164

this system and computer-controlled variable NDQ pump laser filter, students do not even

need to be in the lab to measure or adjust the pump power when running delay scan or

fluence scan experiments — minimal perturbation to the experiment conditions.

The schematic for this setting is drawn in Fig. C.1a, where the pump laser is split into

two paths: one goes into the chamber as excitation pump, the other is directed to a photo-

diode. A 50 µm pin hole is placed right in front of the photo diode (PD) and the distance

from beam splitter (BS) to sample and BS to pin hole is the same. Also, the 50 µm pin

hole mimics the sample size to allow the same amount of photon goes into the PD as the

amount of photon illuminating on the sample. The PD output is fed into a device called

box-car integrator (BCI), which performs the function of integrating and averaging the PD

signal and outputs a voltage signal that is proportional to laser power. This voltage signal

is automatically recorded in the diffraction images.

The BCI (Stanford Research Systems, model: SR250) is a gated device that needs proper

input signals to trigger and reset. Fig. C.1b shows the signal diagram into and out of BCI.

The PD input and trigger signals are both at 1 kHz repetition rate. The trigger signal is

chosen to use the camera image intensifier (I.I.) gate signal instead of direct trigger from laser

Timing Delay Generator (TDG) box because the I.I. gate signal is “1” only in the period

of camera acquisition while the laser TDG box trigger signal is always on independent of

camera acquisition. This avoids the noise accumulation when camera is not acquiring.

With the trigger signal, PD input, and proper settings in the delay and integration width,

the BCI would integrate only tens of ns duration when the input signal is present and avoids

accumulating the noise. BCI calculates the exponential moving average of the signal inputs.

For example, if we set the number of average to be 1000, then BCI would reach 66.7% of its

true value in one second (1000 pulses). To reach 99% of its true value, it usually requires

165

Figure C.1: The photo diode-box car integrator system schematic drawing and signal wave
forms for input/output signals. a, The schematic drawing for the box-car integrator system.
The pump laser is split to feed into a photo diode. The four input/output signals and three
main adjusting knobs are drawn. b, The wave forms of input/output signals

166

UHV ChamberPDCCDSamplee- pulsePump laserUV laser50um pin holePhoto-diodeInput signalTrigger signalReset signalOutput signalDelayWidthAverageGPIBMultimeterPCBox-car integratorBSNot ScanTime point when PC (SanGabriel) reads GPIB(Delay/exteneded Not scan signal by PDG)(Image Intensi(cid:31)er gate signal)Trigger signalOutput signalReset signalInput signalaba few seconds. To record the pump power reading after diffraction image acquisition, the

computer would takes one to two seconds to read the GPIB voltage. So the output has to

hold for about two seconds for the computer to read the GPIB output, but at the same time

it gets reset before the next image acquisition. To fulfill such requirements, we use the NOT

SCAN signal from camera and extend it for three seconds using a pulse delay generator.

This three seconds period is when the computer reads GPIB output. After the three seconds

period, the BCI output gets reset to get ready for the next image acquisition.

To use such a system to run experiments, for example on a fluence scan, we first find

pump-probe overlap on the sample by scanning pump beam on the sample grid at certain

positive time delay. The maximum response from sample would be overlap position. At the

same time delay, we optimize the PD and pin hole positions to have a maximum reading.

With proper settings on the BCI, we calibrate the output vs. power meter reading as shown

in Fig. C.2a. Then without people in the lab, the computer can control the variable laser

attenuation filter then measure the pump power with this system. At proper pump fluence,

we can start the delay scan or fluence scan. Once the scan finishes, one uses the computer

to adjust pump fluence remotely and moves to the next fluence point and restarts the scan.

In this way, very consistent and convenient way of running experiments can be achieved at

high precision.

Fig. C.2b is an example of 1T-TaS2 phase transition delay scan with 2500 nm laser

excitation. 2500 nm laser is generated by an optical parametric amplifier through nonlinear

processes. So if there is some fluctuation in source 800 nm laser, the 2500 nm laser power

could be unstable. Fig. C.2b is such an example of unstable 2500 nm laser power as well

as bad alignment in the delay stage. As we can see, the pump power stays relatively stable

in the first 500 ps. But after 500 ps, it starts to increase due to bad alignment and has

167

Figure C.2: Applications of the PD-BCI system in UED experiments. a, Calibration of BCI
output voltage relative to pump power. b, Delay scan of 1T-TaS2 IC peak dynamics with
PD-BCI system calibration with 2500 nm pump. The arrows indicate strong correlation
between the data noise and pump power fluctuations. c, The fluence-dependent data of 1T-
TaS2 NC-CDW peak changes at several critical delays before calibration. Horizontal labeled
power is measured by power meter. The few black circles are “noisy/jumping” data points
before calibration. d, Fluence scan data the same as that in c but after power calibration
with the PD-BCI system.

≈ 40% rise in power at + 2.2 ns. The black curve is the typical raw dynamics we would see

after analyzing the data. Typically, these noisy, jumping points marked by black arrows are

just treated as noise from somewhere we do not know. But with this PD-BCI system, it is

strikingly clear that the jumping data points are directly correlated with jumps in the pump

168

123450.080.120.160.200.24 BCI output calibration data Linear fitBCI output (V)Pump power (mW)05010015050010001500200025000.00.20.40.6IC normalized intensityDelay, t (ps) IC intensity pump power percentage change01234560.00.20.40.60.81.0NC relative changePump power (mW) NC, 20ps NC, 400 ps NC, 1 ns NC, 2.3 nsFluence scan with power meter0246810120.00.20.40.60.81.0NC relative changeCalibrated pump power (mW)Fluence scan after BCI calibrationabcdpower.

Figs. C.2c&d are another examples of fluence scan done with PD-BCI system calibration.

Fig. C.2c is the typical fluence scan we would get in 1T-TaS2 NC-CDW peak response. As

we can see, there are some kinks and jumping points, making it hard to judge which kinks

are real which kinks are noise. With this PD-BCI system installed, after calibration, the

curves become very smooth and noisy points are gone, demonstrating the high dependence

of data noise on pump power.

With these examples and demonstrations, we believe this is a very reliable system that

can greatly improve the s/n of UED experiments, especially when the laser power is unstable

or the delay stage alignment is not perfect. Moreover, it allow a person to do experiments

remotely without being in the lab to perturb the experimental conditions.

169

BIBLIOGRAPHY

170

BIBLIOGRAPHY

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Mod. Phys., vol. 49, pp. 435–479, 1977.

[2] M. Vojta, “Quantum phase transitions,” Reports on Progress in Physics, vol. 66, no. 12,

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