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This is to certify that the

dissertation entitled

ANGLE RESOLVED PHO‘IOEMISSION STUDIES
OF TRABEITION METAL DICHALCOGENIDES

presented by

WAYNE C . TONJES

has been accepted towards fulfillment
of the requirements for

PH . D . degree in PHYSICS

 

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R. Liu

Date 06/12/01

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6/01 cJCIRC/DateDue.p65-p.15

ANGLE RESOLVED PHOTOEMISSION STUDIES OF
TRANSITION METAL DICHALCOGENIDES
By

Wayne C. Tonjes

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

2001

ABSTRACT

Angle Resolved Photoemission Studies of Transition Metal Dichalcogenides

By

Wayne C. Tonjes

High resolution angle resolved photoelectron spectroscopy was used to study the
electronic structures of several transition metal dichalcogenides. These materials are of
interest as quasi-two-dimensional layered materials which undergo charge density wave
(CDW) phase transitions. Five specific transition metal dichalcogenides were studied:
2H-TaSe2, 2H-TaSz, 2H-NbSe2, 1T-TaSz, and lT-TiSez. The band dispersions and Fermi
surfaces of each material were investigated. Results were compared to those of band
structure calculations. Comparing spectra taken in the high temperature states and the
lower temperature CDW state, information about the CDW energy gaps and the CDW
mechanisms were obtained.

For the 2H polytype materials, the normal state electronic structures have the
same basic characteristics. Their Fermi surface consists of a pocket around I‘ and another
one around K. They all exhibit a saddle band near E1: along F-K. In the CDW phases, a
finite energy gap is observed in the saddle band region and on the Fermi surface around
K in 2H-TaSz and 2H-TaSe2. Gapping on the Fermi surface around 1" is not observed.
Thus, the charge density waves in this family of materials originate from a combination

of the partial nesting of the Fermi surface around K and the saddle band.

For lT-TaSZ, a Fermi surface was observed in the quasi-commensurate CDW
phase. In addition to the pockets around the M points as predicted by earlier non-
relativistic non-self-consistent band structure calculations for the normal state, a small
electron pocket around 1" was observed. The latter pocket is consistent with results of
recent self-consistent band structure calculations including spin-orbit interaction. In the
commensurate CDW phase, a k-independent energy gap of approximately 120 meV
magnitude opens at the Fermi level as a result of a CDW-induced Mott transition.

lT-TiSez shows significant three-dimensional electronic structure despite its
layered morphology. It also shows interesting photoelectron spectroscopy effects, such
as emission from thermally populated states above the Fermi level. The CDW formation
seems to result from coupling of hole and electron states in an excitonic insulator

transition.

Copyright by
Wayne C. Tonjes

2001

ACKNOWLEDGEMENTS

Thanks are given to Dr. C. G. Olson and Dr. S. D. Mahanti for helpful
discussions. This work was partially supported by the Center for Fundamental Materials
Research, Michigan State University, and by the NSF under Award No. DMR-9801776.
This work is based upon research conducted at the Synchrotron Radiation Center,
University of Wisconsin-Madison, which is supported by the NSF under Award No.
DMR-95-31009. Ames Laboratory was operated for the U. S. DOE by Iowa State
University under Contract No. W-7405-ENG-82. Support of the Natural Sciences and

Engineering Research Council of Canada is also acknowledged.

Table of Contents

List of Tables ................................................................................................................. ix
List of Figures ................................................................................................................. x
Abbreviations ................................................................................................................ xv
Chapter 1 Introduction ............................................................................................. 1
Chapter 2 Angle Resolved Photoelectron Spectroscopy ............................................ 3
2.1 Photoemission Process .......................................................................................... 3
2.1.1 Photoexcitation ............................................................................................... 4
2.1.2 Transport to the Surface .................................................................................. 5
2.1.3 Escape From the Crystal ................................................................................. 6

2.2 Energy Distribution Curves ................................................................................... 7
2.3 Angle Resolved Photoemission ............................................................................ 10
2.3.1 Momentum Relations .................................................................................... 10
2.3.2 Band Mapping .............................................................................................. 13
2.3.3 Fermi Surface Mapping ................................................................................ 13
2.3.4 Measurement of the Energy Gap ................................................................... 14
Chapter 3 Charge Density Waves ........................................................................... 15
3.1 Electronic Susceptibility and Fermi Surface Nesting ............................................ 15
3.2 The Energy Gap and the Kohn Anomaly ............................................................. 17
3.3 Other Physical Properties Related to CDW .......................................................... 20
3.3.1 Resistivity ..................................................................................................... 21
3.3.2 Magnetic Susceptibility ................................................................................. 21
3.3.3 Hall Coefficient ............................................................................................ 21
3.3.4 Seebeck Coefficient ...................................................................................... 22
Chapter 4 Transition-Metal Dichalcogenides .......................................................... 23

vi

4.1 Structure .............................................................................................................. 23

4.2 Electronic Structures ........................................................................................... 24
4.2.1 2H-TaSe2, 2H-TaSz, and 2H-NbSez .............................................................. 25
4.2.2 lT-TaSZ ........................................................................................................ 27
4.2.3 lT-TiSez ....................................................................................................... 29

4.3 Charge Density Wave Transitions ........................................................................ 30
4.3.] 2H-TaSe2, 2H-TaSz, and 2H-NbSe2 .............................................................. 31

4.3.1.1 Resistivity .............................................................................................. 33
4.3.1.2 Magnetic Susceptibility .......................................................................... 35
4.3.1.3 Hall and Seebeck Coefficients ................................................................ 36
4.3.2 lT-TaSz ........................................................................................................ 38
4.3.2.1 Resistivity .............................................................................................. 39
4.3.2.2 Magnetic Susceptibility .......................................................................... 40
4.3.2.3 Hall and Seebeck Coefficients ................................................................ 41
4.3.3 lT-TiSez ....................................................................................................... 42
4.3.3.1 Resistivity .............................................................................................. 43
4.3.3.2 Magnetic Susceptibility .......................................................................... 44
4.3.3.3 Hall and Seebeck Coefficients ................................................................ 44
Chapter 5 Experimental Details .............................................................................. 46

5.1 Sample Preparation .............................................................................................. 46

5.2 Photon Source ..................................................................................................... 48

5.3 Electron Energy Analyzer .................................................................................... 50

Chapter 6 2H-TaSez, 2H-TaSz, and 2H-NbSe2 ........................................................ 52

6.1 Band Dispersions and the Saddle Band ................................................................ 53
6.1.1 2H-TaSe2 ...................................................................................................... 53
6.1.2 2H-TaSz ........................................................................................................ 56
6.1.3 2H-NbSe2 ..................................................................................................... 56

6.2 Fermi Surfaces .................................................................................................... 58

vii

6.2.1 2H-TaSe2 ...................................................................................................... 58

6.2.2 2H-TaSz ........................................................................................................ 61
6.2.3 2H-NbSe2 ..................................................................................................... 62

6.3 Energy Gap in the CDW State ............................................................................. 63
6.3.1 2H-TaSe2 ...................................................................................................... 63
6.3.2 2H-TaSz ........................................................................................................ 68
6.3.3 2H-NbSe2 ..................................................................................................... 70

6.4 Summary and Discussion ..................................................................................... 70
Chapter 7 lT-TaSz ................................................................................................. 74
7.1 Band Dispersions ................................................................................................. 74
7.2 Fermi Surfaces .................................................................................................... 79
7.3 Energy Gap in the Commensurate CDW State ..................................................... 81
7.4 Summary and Discussion ..................................................................................... 83
Chapter 8 lT-TiSez ................................................................................................ 86
8.1 Band Dispersions ................................................................................................. 86
8.2 Fermi Surfaces .................................................................................................... 89
8.3 Energy Gap in the CDW State ............................................................................. 93
8.4 Summary ............................................................................................................. 96
Chapter 9 Summary ............................................................................................... 97
References .................................................................................................................. 1 00

viii

List of Tables

Table 4-1 ....................................................................................................................... 31

ix

List of Figures

Figure 2-1: In the photoexcitation process an electron in some initial state i below the
Fermi level is excited to some final state f above it. In the process, energy and
momentum are conserved. Ex is the measurable kinetic energy of the excited electron
afier it escapes through the work function e¢ of the material into vacuum. ...................... 5

Figure 2-2: The universal curve for the scattering length of electrons as a function of
kinetic energy. The data are taken from numerous materials. At kinetic energies of
interest to a photoemission experiment, the escape depth is of the order of 10 A. ............ 6

Figure 2-3: A sample EDC from an angle resolved experiment ........................................ 7

Figure 2-4: The parallel component of the electron wavevector [C] is conserved as it passes
from the material into vacuum with momentum K. By measuring 9, the parallel

components of K and kf can be determined. ................................................................... 10

Figure 2-5: The energy levels of a free electron material. The inner potential V0 is simply
the distance from the minimum of the free electron parabola to the vacuum level. ......... 12

Figure 3-1: The electronic band structure opens a gap, A, when the material goes from a)

the normal state to b) the CDW state. The solid and dashed vertical lines define the
normal state and PLD Brillouin zone respectively. ........................................................ 15

Figure 3-2: Examples of Fermi nesting for a) one-dimensional free electrons, b) arbitrary
two-dimensional material and c) closed, two-dimensional cylinder. In a) and b), the
nesting is perfect, but c) shows very imperfect nesting. ................................................. 17

Figure 3-3: Electronic susceptibility for a free electron gas normalized to the undisturbed
system as a function of disturbance wavevector q. Depending on the dimensionality of
the material, a disturbance of Q=2kp can create a singularity, a sharp step or a weak
shoulder. ....................................................................................................................... 18

Figure 3-4: The temperature dependence of the CDW gap in the weak coupling limit. .. 19

Figure 3-5: Longitudinal acoustic phonon dispersion relation of a one-dimensional metal
at various temperatures above the mean field transition temperature T09. ....................... 20

Figure 4-1: Basic TMD sandwich structures of a) octahedral and b) trigonal prismatic
coordination. The octahedral has the two chalcogen layers offset by l/2ao, while the
trigonal prismatic has them aligned. .............................................................................. 23

Figure 4-2: The unit cells of polytypes a) 2H and b) IT. The stacking arrangement of the
chalcogen-metal-chalcogen sandwiches and relative orientation are shown. .................. 24

Figure 4-3: The hexagonal Brillouin zone in a) two- and b) three-dimensions ................ 25

Figure 44: Band calculations for a) 2H-NbSe2, b) 2H-TaSz, and c) 2H-TaSe2 from
Wexler and Woolley”. .................................................................................................. 26

Figure 4-5: Fermi surfaces of a) 2H-TaSe2, b) 2H-TaS2, and c) 2H-NbSe2. Curves A and
B are at equivalent height in the conventional zone. The middle, unlabeled curve is at the
top of zone. ................................................................................................................... 26

Figure 4-6: Fermi surface of lT-TaS2 extrapolated from calculations without relativistic
spin orbit corrections. .................................................................................................... 27

Figure 4-7: lT-TaSz band calculations a) with and b) without relativistic spin orbit
corrections along the FM and FK symmetry lines. ........................................................ 28

Figure 4-8: lT-TiSez band calculation by Benesh et. al.23 with valence bands and the top
most core levels ............................................................................................................. 29

Figure 4-9: Schematic Fermi surface of lT-TiSez based on carrier concentrations and the
measured CDW wavevector by Wilson24. ...................................................................... 30

Figure 4-10: The 3 x 3 superlattice effect on a) the real space cell and b) the reciprocal
space Brillouin zone. The lattice change generates three inequivalent atomic sites a, b,
and c as labeled in ratios of 1::6::2 ................................................................................ 32

Figure 4-11: Resistivity data parallel to the layers versus temperature for 2H polytypes a)
2H-TaSe2 and 2H-TaS2 from Di Salvo and Rice10 (log plot) and b) 2H-NbSe2 from
Edwards and Frindt (log-log plot) .................................................................................. 33

Figure 4-12: The schematic Fermi surface Rice and Scott28 used. The real system shows
a Fermi surface similar to the thick solid curves shown in the lower right K region. The
saddle point is the crossing of the dashed lines. ............................................................. 34

Figure 4-13: Magnetic susceptibility of 2H-TaSe2 (squares), 2H-TaS2 (triangles), and 2H-
NbSez (circles) normalized at T= 300K by Ayache et. al.29 ............................................ 35

Figure 4-14: The Hall coefficient versus temperature for a) 2H-TaSe2 by Lee et. al.30 and
b) 2H-TaSz by Thompson et. al.3‘. ................................................................................. 36

Figure 4-15: The Hall coefficient versus temperature for 2H-NbSe2 at 28.6 kG (empty
squares) and 150 kG (solid squares) from Whitney et. a1.32 ............................................ 37

Figure 4-16: Seebeck coefficient plotted against the log of temperature for two samples
of 2H-NbSe2 from Huntley and Frindt. .......................................................................... 38

Figure 4-17: The V13x*/13 superlattice effects on a) the real space cell and b) the
reciprocal space Brillouin zone of the CDW in lT-TaSz. The reconstruction generates
three inequivalent atomic sites a, b, and c in the ratios of 1::6::6 .................................... 39

xi

Figure 4-18: Parallel resistivity of lT-TaS2 from Manzke et. al. .................................... 40

Figure 4-19: Temperature dependence of the magnetic susceptibility for lT-TaSZ by
Wilson et. a1. ................................................................................................................. 40

Figure 4-20: Hall coefficient versus temperature for lT-TaSZ by Mutka et. al. 36 The
vertical scale above zero is 1000 times larger than the scale below zero to bring out

details ............................................................................................................................ 41
Figure 4-21: The reciprocal space reconstruction caused by a 2x2x2 superlattice ........... 42
Figure 4-22: Resistivity data for lT-TiSez by R. V. Coleman et. a1. ............................... 43

Figure 4-23: Magnetic susceptibility of lT-TiSe2 versus temperature, taken by Salvo et.
a1. .................................................................................................................................. 44

Figure 4-24: Hall and Seebeck coefficient by Salvo et al.39. They both become negative
as the CDW develops with temperature decrease. The Hall scale is on the left axis; the
Seebeck coefficient on the right ..................................................................................... 45

Figure 5-1: Synchrotron Radiation Center electron storage ring Aladdin. Focussed light
is emitted past the bending magnets and employed along beam lines. The beamlines are
labeled by the type of monochromator used upon them. ................................................ 49

Figure 5-2: ERG-Seya combined monochromators. The metal flat is a mirror inserted
into the photon beam to select the Seya (top) over the ERG (bottom). ........................... 50

Figure 5-3: Hemispherical electron energy analyzer with focussing lenses. The inner and
outer potential shells are set to specific voltages to allow only electrons with a desired
energy to reach the electron multiplier to be pulse counted. The entrance aperture limits
the accepted electrons to a small angle for angular resolution. ....................................... 51

Figure 6-1: a) Normal state EDCs (solid lines) for the F-K line of 2H-TaSe2 with
modeled fits (dashed curve). The tick marks indicate the peak positions. b) The extracted
band dispersions from a). .............................................................................................. 54

Figure 6-2: Energy distribution curves (EDCS) in the normal state (T=125 K), for k-
points along two lines parallel to F-K at a) Icy = -0.07 and b) -0.15 A". The inset of the

Brillouin zone’s irreducible wedge (top) shows the amount of Isa-dispersion of the Fermi
surfaces predicted by band calculations 4 (A: k, = 0; B: k2 = 21t/c; C: 19 = file.) and the
lines of a) and b) ............................................................................................................ 55

Figure 6-3: EDCs for 2H-NbSe2 for k-points a) along l"-K and b) along a line
perpendicular to F-K intersecting it at approximately the midpoint, as shown in the insets.

c) and (1) Band dispersions (symbols) determined by modeling the spectra in a) and b),
respectively. .................................................................................................................. 57

xii

Figure 6-4: The Fermi surface cylinders (shaded and open circles) of 2H-TaSez
determined from the EDCs. The open triangle is the second wall seen along F-K. Points

a, b, and c are shown in Figure 6-10. Point (1 (diamond) was used for the temperature
dependence of the gap shown in Figure 6-11. ................................................................ 59

Figure 6-5: Fermi level intensity map of 2H-TaSe2. Bright regions indicate greater
intensity. The data between the dashed wedge were actually measured and symmetrized
to give the rest of the Brillouin zone. ............................................................................. 60

Figure 6-6: Fermi level intensity map of 2H-TaS2. The data were measured on one sixth
of the Brillouin zone, rotated to give a comparable orientation to the other materials, and
then symmetrized. ......................................................................................................... 61

Figure 6-7: Fermi level intensity map of 2H-NbSe2. The arrow indicates a nesting vector
of the same direction and magnitude as the CDW wavevector. ...................................... 63

Figure 6-8: EDCs in the commensurate CDW state (T=20 K, solid curves) of 2H-TaSe2,
for k-points along I‘-K. The EDCs in the normal state are superimposed as dashed curves

for comparison. ............................................................................................................. 64

Figure 6-9: The energy shift vs. kx along F-K of 2H-TaSe2. The energy shift is taken as

the binding energy difference of the leading edge midpoints in the normal and CDW state
spectra. .......................................................................................................................... 65

Figure 6-10: EDCs for three FS crossing points of 2H-TaSe2, labeled as a, b, and c in
Figure 6-4. The solid and dashed lines are spectra measured in the CDW (T=20 K) and
normal (T=125 K) states respectively. ........................................................................... 66

Figure 6-11: a) EDCs for point d in Figure 6-4 in the vicinity of the extended saddle band
of 2H-TaSe2, measured at various temperatures. The normal state spectrum is
superimposed on each curve as dashed lines for comparison. b) The energy shifts of the

leading edge midpoint (solid squares) as a function of temperature. The solid line is the
BCS relation .................................................................................................................. 68

Figure 6-12: a) EDCs for 2H-TaSz measured in the normal (T=8O K, dashed lines) and
CDW (T=20 K, solid lines) states, along a line at approximately 36° with respect to F—K,

as shown (inset). Enlarged spectra at point A and B are reproduced in c) and b)
respectively. .................................................................................................................. 69

Figure 6-13: EDCs for 2H-NbSe2 along F—K measured in the normal (T=40 K, dashed

lines) and CDW (T=13 K, solid lines) states. The bottom curve is for a k-point on the
Fermi surface around F while the other spectra are in the saddle band region. ............... 71

Figure 7-1: EDCs of the QC-CDW (T= 250 K) state for the a) F—M and b) l"-K lines of
lT-TaSZ. The arrows indicate Fermi crossings .............................................................. 75

xiii

Figure 7-2: Second derivative of the spectral intensity plotted versus energy and
wavevector along a) F—M and b) F-K of 1T-TaSz. Darker regions indicate peak positions.

...................................................................................................................................... 76

Figure 7-3: Band dispersions in the normal state from new band structure calculation, a)

without, and b) and c) both with spin-orbit corrections included. Second derivatives of
EDCs along 1"-M and F-K in the QC-CDW phase (see text for details) are also shown in

c) ................................................................................................................................... 78

Figure 7-4: Fermi level intensity maps of lT-TaSz. The data between the dashed wedge
were actually measured and symmetrized to give the rest of the Brillouin zone. Bright
regions correspond to high intensity. ............................................................................. 80

Figure 7-5: EDCs in the C-CDW state (T=20 K, solid curves), for the same set of k-
points along F-M as in Figure 7-1 a). The EDCs in the QC-CDW state are superimposed

as the dashed curves for comparison. ............................................................................. 82

Figure 8-1: Normal state EDCs for the projections along a) F—M and b) F-K lines of IT-
TiSez. ............................................................................................................................ 87

Figure 8-2: EDCs along the normal emission directions of a) F-A and b) M-L of IT-
TiSez. The EDCs of a) are labeled with the k, (A") distance from F. The EDCs of b) are
labeled with their (kx, k,) A'1 values. .............................................................................. 88

Figure 8-3: Fermi level intensity maps of lT-TiSez taken with 21eV photons. The data
between the dashed lines were actually measured and rotated to give the rest of the
Brillouin zone. .............................................................................................................. 90

Figure 8-4: EDCs taken near apparent Fermi crossing in the normal state of lT-TiSez
near a) the L point and b) midway between A and I". The (kx, k2) points are in Al. ...... 92

Figure 8-5: The temperature dependence in lT-TiSez of bands close in energy to the

Fermi level near a) the L-M line and b) the A point. The normal state EDCs are shown in
dashed lines and the CDW state in solid lines. ............................................................... 95

xiv

ARPES
BCS

BZ
CDW
C-CDW
DOS
EDC
EELS
ERG
FLAPW
F S

GGA
I-CDW
LDA
LEED
PES
PLD
QC-CDW
SRC

STM

Abbreviations

Angle Resolved Photoelectron Spectroscopy
Bardeen-Cooper-Shriffer

Brillouin zone

Charge density wave

Commensurate charge density wave

Density of states

Energy distribution curve

Electron energy loss spectroscopy

Extended range grasshopper monochromator
Full-potential linearized augmented plane-wave
Fermi surface

Generalized gradient approximation
Incommensurate charge density wave

Local density approximation

Low energy electron diffraction
Photoelectron spectroscopy

Periodic lattice distortion
Quasi-commensurate charge density wave
Synchrotron Radiation Center

Scanning tunneling microscopy

XV

T-CDW Triclinic charge density wave

TMD Transition metal dichalcogenide

xvi

Chapter 1 Introduction

Quasi-low—dimensional materials have anisotropic structures that inhibit transport
in one or two directions. They exhibit a rich variety of interesting physical phenomena.
Many of them adopt new phases with lowered temperature such as superconductivity,
periodic lattice distortions (PLDs), Peierls transitions that drive a metal to an insulating
phase, and charge density waves (CDWs). One group of quasi-two-dimensional
materials that has attracted tremendous attention in recent years is the high temperature
superconducting cuprates, such as YBazCu3O7-5l, BIZSI2CaCU203+52, and L82-
x(Sr,Ba),,CuO43. Despite intense studies for more than ten years, high temperature
superconductivity, though, is still not thoroughly understood. Another interesting group
of quasi-two-dimensional materials is the transition metal dichalcogenides (TMDs), such
as TaSez and TaSz. A variety of CDWs have been observed in their various polytypes.
Many aspects of CDWs are similar to those of superconductivity and they are not well
understood either. The electronic structure near the Fermi energy plays an important role
in the CDW formation. Therefore, understanding the electronic structure in these
materials is important for understanding their CDW mechanisms.

Angle resolved photoelectron spectroscopy (ARPES) is a powerful tool for
studying the electronic structure of solids. This technique is especially straightforward
when applied to quasi-two-dimensional materials. Layered materials are also ideal for
ARPES studies because they can be cleaved easily in situ. The cleavages occur between
weakly bonded layers usually resulting in minimal surface disruptions.

This thesis describes high resolution ARPES studies of five TMDs. They are:

2H-TaSez, 2H-TaSz, 2H-NbSe2, lT-TaSz, and lT-TiSez. The three 2H—polytype

compounds have very similar CDW properties. The transitions are weak and are of
second order. lT-TaSZ has a very complicated CDW phase diagram. It undergoes a
series of strong first order transitions when cooled. lT-TiSez exhibits yet another type of
CDW with a single broad phase transition. In the ARPES experiments, band dispersions
near the Fermi energy and the Fermi surfaces were investigated. The results are
compared with band structure calculations. The CDW energy gaps were studied by
comparing spectra taken in high and low temperature states. Important insights about the
CDW mechanisms in these materials are obtained.

The thesis is organized as follows. Chapter 2 describes the ARPES technique in
detail. Chapter 3 outlines the theory and properties of CDWs. Chapter 4 reviews the
crystal structure, electronic structure and CDW properties of the five materials. Chapter
5 describes the specifics of the experimental system used in this work. Chapters 6, 7, and
8 present the ARPES results and discussions on the five materials. Chapter 9 concludes

with a general summary.

Chapter 2 Angle Resolved Photoelectron Spectroscopy

The photoelectric effect was first considered as an experimental tool by Einstein
in 1905 in one of the three papers4 that he published that revolutionized physics. In this
paper, Einstein proposed the photon model of light that could be proven by quantitative
measurements of the photoelectric effect. In 1914, R. A. Millikan performed such
experiments and determined that the photoelectron effect had a frequency dependence,
fully confirming Einstein’s proposed photon model of light.

In the 19503, it was found that the photoelectric effect could be used as a powerful
tool to probe the electronic structure of matter through the simple relations of momentum
and energy conservation. During the 1960s, this technique of photoelectron spectroscopy
(PBS) was used as a probe of the density of states (DOS) of materials. During the 19703,
the development of angular resolved electron energy analyzers allowed this technique to
be used to map the energy bands in materials. With the refinement of these instruments
and the development of synchrotron light sources, angle resolved photoelectron
spectroscopy (ARPES) has become a key tool to understanding the electronic properties

of solids.
2.1 Photoemission Process

There are several explanations of photoemission, one being the one step model.
It explains the photoemission in terms of an inverse low energy electron diffraction
(LEED) process. Here we follow the three step model suggested by Berglund and
Spicer’. Although this approach lacks rigor, it provides an intuitive phenomenological

model. The three step model designates the three steps of the photoemission process as:

1) Excitation: An electron in some initial state absorbs a photon and is excited
to a final state.
2) Transport: The photoexcited electron migrates to the surface of the material.
3) Escape: The photoexcited electron overcomes the work function of the
material to escape.
Further details of the three steps are considered below.
2.1.1 Photoexcitation
In the photoexcitation process, both momentum and energy have to be conserved
as shown in Figure 2-1. For a bound electron, the conservation of momentum is met by
the crystal lattice, which can absorb or provide momentum to the electron in units of the
reciprocal lattice vectors. The initial and final states are constrained to wavevectors

related by:
hkf = h(k,- + G) (2.1)

where lgand k,- are the final and initial state wavevectors and G is a reciprocal lattice
vector. The photon momentum is negligible in comparison to the electron momenta. For
a reduced Brillouin zone scheme, this means that only direct, vertical transitions of the
electrons are of consideration.

The conservation of energy requires:

Ef = E, + 7160 (2.2)
where E, and Ef are the initial and final state energies respectively and he) is the photon

energy. The initial states must be occupied and the final states must be empty for

photoemission to occur.

 

 

    
 

 

 

 

 

k

Figure 2-1: In the photoexcitation process an electron in some initial state i
below the Fermi level is excited to some final state f above it. In the process,
energy and momentum are conserved. BK is the measurable kinetic energy of
the excited electron after it escapes through the work function e¢ of the

material into vacuum.

2.1.2 Transport to the Surface

As the excited electron migrates to the surface, it can be inelastically scattered by
other electrons, phonons and plasmons. The mean free path between scattering events is
nearly material independent as shown in Figure 2-2. For the kinetic energy range
normally encountered in a photoemission experiment, the mean free path is in the range
of 10-15A. This makes the photoemission technique inherently surface sensitive. The
inelastic scattering also introduces a smooth background of secondary electrons that adds

to the primary electron signal.

 

 

 

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I l

 

 

I IO 100 1000
ELECTRON KINETIC ENERGY (8V)

Figure 2-2: The universal curve for the scattering length of electrons as a
function of kinetic energy‘. The data are taken from numerous materials. At

kinetic energies of interest to a photoemission experiment, the escape depth is
of the order of 10 A.

2.1.3 Escape From the Crystal
The excited electrons that reach the surface of the material have to overcome the

work function to escape and be detected. Therefore, the measured kinetic energy of the
photoelectron, Ex, is:

EK = hoe—EB —e¢ (2.3)
where E B is the binding energy of the initial state and 64) is the work function of the

material.

As the electron escapes the sample, its momentum changes. There is an effective

refraction effect at the surface due to the work function. The normal component of the

electron’s momentum is not conserved, whereas the parallel component is conserved. As

a result, the electron’s direction of travel is bent as it exits the material.

2.2 Energy Distribution Curves

In a photoemission experiment, the photoelectrons are collected and energy

analyzed by an electron energy analyzer. By measuring the photoelectron yield as a
function of Ex at fixed photon energy, one gets an energy distribution curve (EDC).

Figure 2-3 shows an example of an EDC. Usually, the kinetic energy is converted to

binding energy, using (2.3).

 

I I l l

Normal State
he): 21 eV

(9, ¢)=(22,0)
k,,=(0.77,0.00)

   
  
  

 

Intensity (Arb. units)

  
   

 

 

 

l l

0.4 03 072 0.1 0.0 -0.1 02
Binding Energy (eV)

Figure 2-3: A sample EDC from an angle resolved experiment.

 

The incident photon and the electron interaction is governed by the Hamiltonian:
h
H=—(Aok+k-A) (2.4)
2mc

where A is the vector potential of the photon and k is the momentum operator:

k = —i V (2.5)
In the Coulomb gauge, the momentum operator acting upon the vector potential is zero.
Since A is discontinuous at the surface of the material, the second term of (2.4) leads to a
specific surface photoelectric effect. It is usually ignored, as it does not contribute any
sharp features to the photoemission spectrum. The transition probability of some electron

being excited by this interaction is governed by the matrix element:
Mfi = <f, kfIHll, ki>
. (2.6)
—zh .
: <f, k f l, ki>

— A - V
2mc
where Ii, k.) and 1f: kf) are the initial and final state wavefunctions of the electron

 

 

respectively. The photocurrent at a specific photon energy 710) is given by:

E‘Mfi '2 5(Ef ‘ Ei ‘ hw)5(ki + G - kf) (2.7)
,l

where the 5 fimctions ensure energy and momentum conservation.

The energy distribution curve that results from this photocurrent is given by:

N(EK,ha))oc 2|Mfi|25(Ef —E,- —hw)6(k,- +G—kf)
f,i

Xd(Ef,kf)6(kf_L) (2.8)
X T(Ef,K//)6(EK — Ef +e¢)5(K,, — kf/l)
where dis the transport coefficient and T is the transmission function of the surface. The

terms are organized by row according to the process of the three step model which

generates them with the energy and momentum constraints explicitly included as 8-

functions and the 9 step-function. The transport coefficient d can be written as:

a). ldEf
dE ,k =—zaA=ar —— 2.9
(f f) 1+a2 hdkf ( )

where 0t is the optical absorption coefficient of light for the material, A. is the electron
inelastic mean free path, I is the isotropic mean period between scattering, and the last
term in parenthesis is the electron’s average group velocity. The approximation is made
because A is on the order of 10-20A and 0t is on the order of 10'3 A’1 for typical
photoemission energies. The transmission function through the material surface can be
written as:

h2

T(Ef,K//)=9 Ef —EV-%

K//2 6(ka. +G'J.) (2.10)

where E V is the vacuum energy.

Electrons that scatter inelastically at least once as they migrate to the surface are
called secondary electrons. They have lower kinetic energy than the primary electrons
and contribute a smooth background intensity to the measured intensity. There are
different ways to estimate this background. Electron energy loss spectroscopy (EELS)
can be used to estimate the background directly. The Tougaard method uses a
generalized form of the scattering function extracted from EELS7 to fit the background
instead. Finally, the proportional “Shirley” procedures, used here, assumes that the
background at any given kinetic energy is proportional to the primary intensity at higher

kinetic energies.

2.3 Angle Resolved Photoemission

In an angle resolved photoemission (ARPES) experiment, an analyzer with a
small entrance aperture is used to select a specific emission direction of electrons of

N(EK, hm). Thus, both the kinetic energy and the momentum of the photoelectrons are

determined. Because of the energy and momentum conservation relations, ARPES can
be used to study the initial state of the electron before it was excited.
2.3.1 Momentum Relations

The magnitude of the wavevector of the photoelectron is given by:

K = ’2’:ng (2.11).

The parallel and perpendicular components of this wavevector can be written as:

 

 

 

K” = 2:5 81116
(2.12)
K I — 2’1ng cosO

where O is the polar angle in the geometry shown in Figure 2-4. Since the parallel

component is conserved in the escape process, this allows the determination of the
parallel components of the lg. Since the momentum is also conserved for the excitation

process, this gives:

kil/ = K” = gig-E-ESIHQ (2.13).

The perpendicular component of the momentum, however, is not conserved due to the

surface potential. This implies that the wavevector component k), is indeterminate.

10

Analyzer

Z
7 Electron Energy
ha) I

 

 

 

 

 

 

Figure 24: The parallel component of the electron wavevector kf is
conserved as it passes from the material into vacuum with momentum K. By
measuring 9, the parallel components of K and kf can be determined.

For a two-dimensional material, the band dispersions are completely determined
by the parallel component of the momentum. Thus, the indeterminacy of k), is not a
problem. Most of the work in this thesis deals with quasi-two-dimensional materials.
The data interpretation is therefore particularly simple.

For three-dimensional materials, there are ways to overcome this problem. One
of them is to take spectra at normal emission. In this geometry, the parallel component is
zero. One can probe the band dispersion as a function of k,-i by changing the photon
energy. The effective value of k), can be determined from the symmetries of the bands
seen in the spectra.

For off normal emission, k), can also be determined if one makes the free electron
approximation for the final state:

_ .4;

E
K 2m

V0 (2.14)

11

where V0 is the inner potential. For a real material, V0 is the potential an electron has to
overcome to reach the vacuum level and may have spatial or angular dependence. In the
free electron approximation, the determination of this potential is simply the difference
between the bottom of the free electron parabola and the vacuum level as shown in
Figure 2-5. With this approximation, the normal component of the initial state’s

momentum is given by:

 

2
ki_L =\/(7i;(EK+VO)_ki//2) (2.15).

V0 is material specific and can be estimated from the normal emission data as mentioned

previously. The free electron approximation is valid for photon energies above 100 eV.
However, it has been shown that it can also be applied as a guide for much lower

energies.

 

Energy

=Ef0 I III-I'lIlIl-IIOIOCICUII III-IIIIIDOI IO.

   

inner potential
(Va = Ev - Ea)

 

wave vector in crystal

Figure 2-5: The energy levels of a free electron material. The inner potential
V0 is simply the distance from the minimum of the free electron parabola to
the vacuum level.

12

2.3.2 Band Mapping

Band mapping is done by taking EDCs at various K-points. The binding energy
of an electronic state is determined from the peak position in the spectrum. For a quasi-
two-dimensional material, K” is varied by changing the analyzer angle. The
corresponding ky/ values can be calculated using (2.13). For a three-dimensional
material, K can be varied by changing both the analyzer angle and the photon energy.
2.3.3 Fermi Surface Mapping

Since the Fermi surface is crucial to most physical properties of a material, its
determination is important. High resolution ARPES can be used to determine the Fermi
surface by observing band dispersions near the Fermi level. The Fermi level is usually
determined by measuring the EDC of a good metal, such as a Pt foil which is in electrical
contact with the sample. Fermi wavevectors can be determined by observing locations
where bands disperse across the Fermi level. Due to the Fermi function, the true binding
energy of a peak near EF is often not its apparent peak position. Therefore, spectra
modeling is sometimes required to determine the Fermi crossings. The F errni wavevector
can be further confirmed by checking the slope of the spectral leading edge. At a Fermi
crossing, the slope should have a local maximum.

Another technique for determining the Fermi surface is to make an intensity map
at the Fermi level. In this technique, intensity is measured as a function of wavevector
while the kinetic energy is fixed at the Fermi level. The contour of high intensity should
correspond to the two-dimensional Fermi surface. The Fermi intensity map gives an

overall picture of the intensity distribution in k-space. This technique is complimentary

to the EDC method.

13

2.3.4 Measurement of the Energy Gap

As a material undergoes certain phase transitions, such as a superconducting or
charge density wave transition, an energy gap often opens at the Fermi level. The
property of the gap is important for understanding the phase transition. High resolution
ARPES is a very powerful tool for studying the gap properties.

A gap opening can be observed by measuring the spectra at a k-point where there
is a Fermi crossing in the normal state at temperatures above and below the phase
transition. The gap suppresses the spectral intensity at the Fermi level. As a result, the
leading edge of the spectrum would appear shifted toward higher binding energy. The

shift of the leading edge mid-point gives a good approximation to the size of a gap.

14

Chapter 3 Charge Density Waves

Some metals, when cooled, undergo phase transitions to a new type of state. One
of the possible states is a charge density wave (CDW). A CDW is a modulation of the
conduction electron density and an associated modulation of the lattice atom positions
into a periodic lattice distortion (PLD). A one—dimensional case is shown in Figure 3-1.

In the CDW state, the charge density p(x) in the crystal is given by:

p(x) = p0[1+ p cos(Q . x)] (3.1)
where p0 is the mean electron density, p is the amplitude of the charge modulation, and Q
is the modulation wavevector.

The modulation of charge causes each ion to experience a different potential. The
resulting forces cause each ion to be displaced to a new equilibrium position. Electron,
x-ray, or neutron diffraction can detect this PLD effect. New diffraction satellites appear
in addition to the Bragg spots from the underlying lattice. The charge modulation can be
detected by x-ray spectroscopy, nuclear magnetic resonance, and scanning tunneling
microscopy (STM).

The CDW wavevector Q may be incommensurate with the spacing of the
underlying lattice. This alters the material so that it is no longer truly periodic, having
two unrelated periods. For some materials, the wavevector Q may become
commensurate with the lattice at lower temperatures. When this occurs, the modulation

of the charge and atomic lattice positions simply gives the crystal a larger unit cell.
3.1 Electronic Susceptibility and Fermi Surface Nesting

The onset of a CDW is caused by the instability of the metallic Fermi surface

resulting from the electron-phonon interactions. The response of an electron gas to an

15

a)

I O O O
C C O I

 

 

 

 

 

 

 

:(k)

"z, :5
' I

I p I

Figure 3-1: The electronic band structure opens a gap, A, when the material

goes from a) the normal state to b) the CDW state. The solid and dashed
vertical lines define the normal state and PLD Brillouin zone respectively.

 

 

 

 

external perturbation is described by the electronic susceptibility:

fk — fk
1.1 = 2 +q (3.2)
k €k+q - 8k

 

where fl, is the Fermi-Dirac function, 8., is the energy and k indicates the wavevector of
the state. The system becomes unstable when x., diverges. From equation (3.2), the most
significant contributions to )0. come from states whose wavevectors differ by Q=2kp, the
distance between two points of the Fermi surface where the states have the same energy.

For a one dimensional electron gas, the Fermi surface consists of two parallel planes, as

16

 

 

 

 

 

 

 

 

Figure 3-2: Examples of Fermi nesting for a) one-dimensional free electrons,

b) arbitrary two-dimensional material and c) closed, two-dimensional

cylinder. In a) and b), the nesting is perfect, but c) shows very imperfect

nesting.
shown in Figure 3-2 a). All states on one sheet of the Fermi surface are related to the
states on the other sheet by the same wavevector Q=2kp. Such Fermi surface topology is
called perfect nesting. In this case, 1., diverges at q = Q as shown in Figure 3-3. In
higher dimensions, the number of nesting states is significantly reduced. This leads
directly to a weakening of the singularity as indicated.

In a real material, the shape of the Fermi surface is distorted from that of a free
electron gas by the periodic potential. It is possible to have a large number of states
paired by the same connecting wavevector Q=2k|7. Figure 3-2 b) shows a hypothetical
example for a two-dimensional case. In such cases, xq still develops a singularity at
Q=2kp. In most systems, there is only imperfect, partial nesting of the Fermi surface as

shown in Figure 3-2 c). Some systems can still be driven into new states by even this

moderate nesting.
3.2 The Energy Gap and the Kohn Anomaly

The microscopic theory of CDW9 formation can be approached by using the

17

 

xl‘II/xI‘FOI

 

 

211, ‘I

Figure 3-3: Electronic susceptibility for a free electron gas normalized to the
undisturbed system as a function of disturbance wavevector q'o. Depending
on the dimensionality of the material, a disturbance of Q=2kp can create a
singularity, a sharp step or a weak shoulder.

mean-field approximation. The formalism is very similar to that of the Bardeen-Cooper-
Shriffer (BCS) theory of superconductivity. The simplest model is that of Peierlsll and
Frtihlich12 who considered noninteracting electrons in a one-dimensional metal.
Electrons moving among a chain of equally spaced atoms with wavevecter k have an
energy 8(k) as shown in Figure 3-1 a). At zero temperature, the states are filled up to the
Fermi energy 81: and the states above 8]: are empty. Applying a periodic potential with

wavevector Q=2kp creates an energy gap at the Fermi surface as shown in Figure 3-1 b).

18

The energies of the occupied states below 8]: are lowered and the empty states above it
are raised. The electronic energy of the system, which only depends on the occupied
states, is thereby reduced. If this electronic energy reduction is larger than the energy
cost of the lattice distortion, a CDW will spontaneously develop. For a true one-
dimensional system at zero temperature, this must always occur. As the temperature
increases, thermally excited electrons will reduce the gap at the Fermi level until at some
transition temperature To it closes entirely. The dependence of the gap on temperature is

the same as the BCS relation for superconductivity shown in Figure 3-4. In the weak

1.
NT)

 

Energy (Arb. Units)

 

 

0 T, T.

Figure 3-4: The temperature dependence of the CDW gap in the weak
coupling limit.

coupling limit, the gap can be related to the onset temperature by the BCS relation:
2Amax = 3.5k,,TO (3.3)

where To is the onset temperature of the CDW. In a strong coupling limit, this

relationship may not hold. Therefore, the gap size can be used in some degree to probe

the coupling strengths in a CDW material.

19

It can be shown that the electron-phonon interaction leads to a renormalized
phonon frequency spectrum. Due to the divergence of the susceptibility at Q = 2k}: at
low temperature, the phonon frequency is reduced or softened at this wavevector as
shown in Figure 3-5. This phenomenon is called the Kohn anomaly. With decreasing
temperature, the renormalized phonon frequency goes to zero and this defines a transition

A

0),,

 

 

' Ir-
2k, ‘1

 

Figure 3-5: Longitudinal acoustic phonon dispersion relation of a one-
dimensional metal at various temperatures above the mean field transition
temperature T010.

temperature, where a frozen-in distortion occurs. In many systems, the incommensurate

CDW sets in before the phonon energy goes entirely to zero and the commensurate CDW

phase results when the phonon energy reaches zero.

3.3 Other Physical Properties Related to CDW

The development of a CDW affects other physical properties of the material such

as the resistivity, magnetic susceptibility, Hall coefficients, and Seebeck coefficient.

20

3.3.1 Resistivity

The resistivity usually shows an anomaly at the CDW transition temperature.
Since the CDW opens an energy gap on the Fermi surface, normally the resistivity
increases at the onset of the CDW. However, there are exceptions, such as the resistivity
decrease in some 2H-polytypes of the transition metal dichalcogenides. These will be
discussed in the next chapter.
3.3.2 Magnetic Susceptibility

The magnetic susceptibility indicates how the system responds to an external
magnetic field. As such, it is sensitive to the rearrangement of the electron density by the
CDW and the underlying ionic lattice by the PLD. Thus, it shows anomalies at CDW
transitions, either as a discontinuity or a change in slope.
3.3.3 Hall Coefficient

The Hall coefficient depends on the difference of hole and electron carrier
concentration and their respective mobilities. Thus, it is sensitive to the removal of the
Fermi surface in limited regions. Only a rough qualitative approximation of the Hall
coefficient, R”, is necessary to highlight the likely effects of the change in relative
concentrations of electron and hole carriers. For this purpose, the simplified two-carrier

model equation from Wilsonl3 :

H = p—n(u,,/up)2
[‘0 +"(fln/f‘p)]2

where p indicates the concentration of the positive hole carriers, n the concentation of the

 

(3.4)

negative electron carriers, and up and 11,. their respective mobilities. The opening of an

21

energy gap in the CDW phase causes a change in R”. If the carrier concentrations or
mobilities are altered by the CDW significantly enough, RH can even change sign.
3.3.4 Seebeck Coefficient

Probing the therrnopower, or the Seebeck coefficient, gives further information on
the carrier concentrations of a material. The Seebeck coefficient, S, also depends on the
hole and electron carrier concentrations. A qualitative two carrier model from Wilson13
is given by:

S = plsp|_nISn|
p+n

 

(3.5)

where Sp and S, are the Seebeck coefficients of the p- and n-type carriers respectively.

22

Chapter 4 Transition-Metal Dichalcogenides

The transition-metal dichalcogenides (TMDs) have the chemical formula MXz,
where M is a transition metal such as V, Ti, Ta, Nb, Hf or Mo, and X is a chalcogen atom
of S, Se, or Te. This group of materials is interesting because they are simple compounds
but show a broad variety of properties. For example, many of them show

superconducting and CDW transitions at low temperatures.
4.] Structure

The basic crystal structure of the TMDs consists of a sheet of hexagonally packed
metal atoms sandwiched between two sheets of hexagonally packed chalcogen atoms.
The atoms within a sandwich are covalently bonded. These covalently bonded
sandwiches can have one of two types of coordination as shown in Figure 4-1 depending
on the relative orientation of the chalcogen layers. In the octahedral coordination shown

in Figure 4-1 a), the two chalcogen layers are oriented 180° with respect to each other. In

 

O Chalcogen Atom . Metal Atom

Figure 4-1: Basic TMD sandwich structures of a) octahedral and b) trigonal
prismatic coordination. The octahedral has the two chalcogen layers offset
by 1/2ao, while the trigonal prismatic has them aligned.

 

23

the trigonal prismatic coordination shown in Figure 4-1 b), they are oriented in the same
orientation. The sandwiches are stacked along the c-axis via van der Waals forces.
Various stacking sequences and registries of the sandwich layers give rise to a large
number of polytypes.

The 2H and IT polytypes, shown in the (110) plane in Figure 4-2 a) and b)
respectively, are the simplest of the stacking orders. They have trigonal prismatic and
octahedral coordination respectively. The 2H polytype has two trigonal prismatic
coordinated sandwiches per unit cell. There are three variants of the 2H stacking order,
but only the 2H. polytype was under study here. All references to the 2H polytype are
therefore of the 2H, variant. The IT polytype has a single octahedral coordinated

sandwich per unit cell.

 

Orepresents two lined up chacogen (El

atoms
0 represents a transition metal atom

Figure 4-2: The unit cells of polytypes a) 2H and b) IT. The stacking
arrangement of the chalcogen-metal-chalcogen sandwiches and relative
orientation are shown.

4.2 Electronic Structures

For both 2H and IT polytypes, the Brillouin zone (B2) is hexagonal as shown in

24

Figure 4-3 with the major symmetry points indicated. All polytypes have a hexagonal
Brillouin zone, although they have slightly different symmetries depending on their
coordination. The flat projection of the zone can be used in quasi-two-dimensional cases

where the Ila-dispersion is of no concern.

a) -<: kzll M
K 1"—
n 3Z>
' L
:O L!

b) L' /

 

 

 

 

 

 

H A

L a: L. L
K o..- a---..-....o.l-‘

..04.. 0

M

 

 

 

H """::==**:A /
L' :

Figure 4-3: The hexagonal Brillouin zone in a) two- and b) three-dimensions.

4.2.1 2H-TaSe2, 2H-TaSz, and 2H-NbSe2

Figure 4-4 shows the band dispersions near E: for a) 2H-NbSe2, b) 2H-TaSz and
c) 2H-TaSe2 calculated by Wexler and Woolleyl4 using the layer method. They are all
very similar, with two bands crossing E}: along the FM line and two bands briefly
dispersing below and then above the Fermi level along FK. The two bands are not
degenerate due to the interlayer coupling. The location where the bands have maximum
binding energies along 1"K is identified as a saddle point.

Figure 4-5 shows the Fermi surfaces of a) 2H-TaSe2, b) 2H-TaSz, and c) 2H-
NbSez from the same calculations. Because of their layered structures, the Fermi
surfaces are highly two-dimensional in that they are cylinders slightly corrugated along

the k, -direction. The three contours, A, B, and unlabelled, in Figure 4-5 correspond to

25

 

 

 

Elli-319V (Film)

 

 

 

l" M K FA L H A
Figure 4-4: Band calculations for a) 2H-NbSe2, b) 2H- TaSz, and c) 2H-TaSe2
from Wexler and Woolley”.

 

a) \‘w

AMA
'2th

 

 

 

 

c) g“ ‘1
A) (TA/3“,.
'1 \ga’ (\V

um. /M’

Figure 4-5: Fermi surfaces of a) 2H-TaSe2, b) 2H-TaSz, and c) 2H-NbSe2.

Curves A and B are at equivalent height in the conventional zone. The
middle, unlabeled curve is at the top of zone.

26

Fermi surfaces at k2 = 0, 21t/c and WC respectively. The separations between these
contours indicate the amount of kz-dispersion. As can be seen, the Fermi surfaces for
these three materials all consist of a cylinder around FA and another cylinder around KB.
The cylinders are double walled except at the top of the BZ because there are two
sandwiches per unit cell. It can be shown that the double walled cylinder can be unfolded
into a single walled cylinder in the double-zone scheme along kl”.
4.2.2 lT-TaSz

Figure 4-6 shows the Fermi surface of 1T-TaSz in the normal state extracted from
the non-self consistent non—relativistic band structure calculation by Mattheiss's. It is
comparable to other band calculationsw'n’ls. The FS is also highly two-dimensional,
consisting of elliptical cylinders around ML.

Recently, Larson and Mahanti19 performed self-consistent band structure
calculations using the full-potential linearized augmented plane-wave (FLAPW)
method20 within the density functional theory”. The generalized gradient approximation

(GGA) of Perdew et. 211.22 was used for the exchange and correlation potential. The band

 

 

 

 

 

 

 

Figure 4-6: Fermi surface of 1T-TaSz extrapolated from calculations without
relativistic spin orbit corrections.

27

 

 

—L
'01

  

 

 

EF

A f7<

\EF
F;

E M K A I.

Figure 4-7: lT-TaSZ band calculations a) with and b) without relativistic spin
orbit corrections along the FM and FK symmetry lines.

.C'J
U'I D
1

Energy (eV)

 

I
—II.
:-l-
9/
I

 

 

 

 

_._'.
-i.n'tn

;
5'
v

 

P
D UT
.1

/

 

 

Energy (eV)

I
—L

 

 

 

\

 

 

~15 I‘

dispersions near Ep both with and without spin orbit corrections are shown in Figure 4-7
a) and b) respectively. As can be seen, the spin orbit corrections have an important affect
on the Ta 5d-band near Ep. It lifts the degeneracy and brings one band below the Fermi
level near F as demonstrated in Figure 4-7 b). This creates a small electron pocket
around F. This new calculation is in better agreement with ARPES results as will be

shown later.

28

4.2.3 lT-TiSez

A band calculation by Benesh et. al.23 of lT-TiSez is shown in Figure 4-8. As can
be seen, bands briefly cross the Fermi level near the F and L points. These would lead to
small hole and electron Fermi surface pockets respectively. Also, there is significant
dispersion along the FA line, the direction perpendicular to the layers indicating three-

dirnensional electronic structure.

4
TV;

“v.4

 

m

   
         

M

 

 

D—l

(

I
A

r

 

 

4

J

a;
4“ f
"k
i

p.

Energy (eV)
tn 1'“

 

 

 

 

 

 

-4
-5
-6
-12 ' :5-
43’
-14
1" M K L H A

Figure 4-8: lT-TiSez band calculation Aby Benesh et. a].23 with valence
bands and the top most core levels.

Wilson24 proposed a schematic Fermi surface as shown in Figure 4-9, with the
relative size of the Fermi pockets based upon carrier concentrations. The dotted cylinder
about the LM-line is the case for a Fermi surface consisting solely of electrons. The solid
ellipsoids around F and L are the Fermi surface expected for hole and electron carriers
respectively. Later band structure calculations performed after Wilson’s proposal predict

the hole and electron Fermi surface pockets.

29

 

 

 

Figure 4-9: Schematic F errni surface of lT-TiSez based on carrier
concentrations and the measured CDW wavevector by Wilson“.

4.3 Charge Density Wave Transitions

A variety of CDW phase transitions have been observed in the various polytypes
of the TMDs. The structural distortions accompanied by the CDW transitions have been
extensively studied by electron diffraction, x-ray diffraction, neutron scattering and
scanning tunneling microscopy. In general, the 2H compounds have weak second-order
CDW transitions and the IT compounds have stronger first-order CDW transitions. The
1T materials have greater material specific variations, as seen in lT-TaSz which has a
very complicated CDW phase diagram and lT-TiSez which has a single transition as will
be discussed below in detail.

Listed in Table 4-1 are the CDW transition temperatures, superlattice structure of

the commensurate CDW and crystal lattice dimensions of 2H-TaSe2, 2H-TaSz, 2H-

30

NbSez, lT-TaSZ, and lT-TiSez. The following sections will review the CDW properties

and other relevant physical properties such as resistivity, magnetic susceptibility, the Hall

coefficient and the Seebeck coefficient of each material.

 

 

 

 

 

 

 

Table 4-1

Material To (K) T; (K) Superlattice Lattice Constants (A)
2H-TaSe2 122.3 ~90 3a., x 3a. a0 = 3.436, co = 12.696
2H-TaS2 75.3 ~14l 3a., x 3ao a0 = 3.315, co = 12.104
2H-NbSe2 33 <52 3a., x 3a., a0 = 3.444, co = 12.552

lT-TaSZ 5523 180 ‘113 aox #13 a0 a0 = 3.346, co = 5.859

or 2404
lT-TiSez -- 315 2ao x 2ao x 2co a0 =3.542, co =6.015

 

 

 

 

 

 

To is the onset temperature of the incommensurate CDW phase. T1 is the
commensurate CDW lock-in temperature.

1. The commensuration below this transition may be only local, broken by regions
of discommensuration forming a quasi-commensurate phase.

2. The incommensurate phase clearly exists until this temperature. Below this
temperature, the incommensuration is too small to be detected within scanning
tunneling microscopy resolution.

3. Between the incommensurate transition and the commensurate lock-in
temperature, lT-TaSZ exhibits a quasi-commensurate phase transition at 360K and
a triclinic phase on warming up above the commensurate phase to 280K.

4. The first temperature is for cooling; the second is on the warming cycle.

4.3.1 2H-TaSez, 2H-TaSz, and 2H-NbSez
For the 2H materials considered here, the CDW phase transitions occur well

below room temperature. In the case of 2H-TaSe2, the weak second-order transition to
the I-CDW phase occurs at 122.3 K followed by the C-CDW transition at 90 K. 2H-TaSz

undergoes a weak second-order transition to the I-CDW phase at 75.3 K. Evidence for a

31

locally commensurate phase below 14 K is suggested by nuclear magnetic resonance
(NMR) studies” taken at 4.2 K. 2H-NbSe2 has the lowest I-CDW transition temperature
of 33 K. The CDW remains incommensurate down to at least 5 K, where the
incommensuration is nearly undetectable by scanning tunneling microscopy (STM)
measurements. The incommensuration does decrease with temperature, so there may be
a C-CDW phase below 5 K that is just undetected presently. All three compounds have a
near 3 x 3 superlattice in the CDW phases. The lattice reconstruction in the transition
metal layer is shown in Figure 4-10. The enlargement of the real space unit cell causes
an inversely proportional reduction of the reciprocal space Brillouin zone.

2H-TaSe2, 2H-TaSz, and 2H-NbSe2 also become superconducting at 0.14 K, 0.7
K, and 7.4 K respectively“. The apparent correlation between higher superconducting
transition temperature and lower CDW transition temperature suggests competition

between superconductivity and CDW in these materials.

 

Figure 4-10: The 3 x 3 superlattice effect on a) the real space cell and b) the
reciprocal space Brillouin zone. The lattice change generates three
inequivalent atomic sites a, b, and c as labeled in ratios of 1::6::2.

32

43.1.1 Resistivity

Figure 4-11 shows the in-plane resistivity as a function of temperature for 2H-

TaSez and 2H-TaS2 and 2H-NbSez. As can be seen, all three materials exhibit a

resistivity decrease at the onset of the CDW transition. This is rather unusual because

normally a CDW should cause a resistivity increase due to gapping at the Fermi level.

Because of this unusual behavior, the CDW mechanism for this group of materials has

been a subject of debate for decades. Several models were proposed to explain the

resistivity decrease.

)

Resistivity (9 cm)

 

 

 

 

 

 

 

 

 

10-3 W 10-3 - .
2H-Tas, " "
(E: _ ZH-NbSq
1:: 104 - .
10-5 2,- E 2
IE
3% — .
10-5 — —
I: : :
. , | 1 n l l lllll n 1 1
107 200 400 10 so 100 300
Temperature (K) Temperature (K)

Figure 4-11: Resistivity data parallel to the layers versus tern erature for 2H
polytypes a) 2H-TaSe2 and 2H-TaSz from Di Salvo and Rice (log plot) and
b) 2H-NbSe2 from Edwards and F rindt27 (log-log plot).

Rice and Scott28 proposed a saddle band model. It was based on a schematic

FS shown in Figure 4-12 derived from band calculations by Mattheis's. On this FS, there

are a set of six saddle points at approximately the midpoints of FK symmetry lines of the

33

Brillouin zone. Using susceptibility arguments, they concluded that if the saddle point
was within ~k3To = 0.01 eV of the Fermi level, then a CDW would result. They further
argued that the saddle points act as scattering sinks in the normal state. Therefore, their
removal in the CDW state would lead to the resistivity decrease observed experimentally.
However, later band calculations were made with improved techniques that
indicated the saddle point was not close enough in energy to the Fermi level.
Furthermore, the newer band calculations suggested the saddle point was not at the

midpoint of the FK symmetry line. These predictions removed support for the Rice-Scott

 

 

Figure 4-12: The schematic Fermi surface Rice and Scott28 used. The real
system shows a Fermi surface similar to the thick solid curves shown in the
lower right K region. The saddle point is the crossing of the dashed lines.

model. An alternative model was suggested by Wilson". He suggested that the CDW is

caused by nesting of region of the Fermi surface with heavy effective canier masses.

34

Gapping on those parts of the Fermi surface can also lead to a resistivity decrease by
removing carriers that worsen the conduction of the 2H polytypes. As will be shown in
Chapter 6, direct measurements of the band dispersions and the CDW energy gap using
high resolution ARPES clarify if either of these theories is applicable.
4.3.1.2 Magnetic Susceptibility

The magnetic susceptibility gives information on the density of state near the
Fermi level. From the relative strength of the raw data, which is not shown here, these
materials have moderately high concentrations of carriers near the Fermi level, which
agrees with the predictions of band structure calculations. In Figure 4-13, magnetic
susceptibility data, normalized at 300K for comparison, of 2H-TaSez, 2H-TaSz, and 2H-
NbSez by Ayache et. a1.29 is shown. Each material shows a distinct downturn with

decreasing temperature near the CDW transition temperatures.

 

1.3
“0......

 

 

 

o 100 200 300 400
Temp erature (K)

Figure 4-13: Magnetic susceptibility of 2H-TaSe2 (squares), 2H-TaSz
(triangles), and 2H-NbSe2 (circles) normalized at T= 300K by Ayache et.
al.

35

4.3.1.3 Hall and Seebeck Coefficients

The Hall coefficient is plotted versus temperature for the three materials

Figure 4-14, and Figure 4-15. As can be seen in all three cases, the Hall coefficient

undergoes a sharp decrease and a change in sign near the CDW onset transitions. The

11,, (10" c BIC)

RH (10" c 3/C)

2.0
1 .5

1 .0
0.5
0
-0.5
-1 .0

-1.5

30,31,32 1

 

I I 1 I I I I I I I r I I

_—'I
a) I'd-0% - 3 : —._ -

_i"

 

 

 

 

0 40 an 120 160 200 240 280

Temperature (K)

 

4. b)

 

2-
1..
0

 

 

 

T I I I I I U f I I ‘ I
0 20 4o 50 an 100120 140 160 180 200 220 240
Temperature (K)

Figure 4-14: The Hall coefficient versus temperature for a) 2H-TaSe2 by Lee
et. a1.30 and b) 2H-TaSz by Thompson et. . .

36

n

 

lo I l I I I I I

 

 

 

 

 

 

0
.5 - d
1 l
100 10 20 30 40

Temperature (K)

Figure 4-15: The Hall coefficient versus temperature for 2H-NbSe2 at 28.6
kG (empty squares) and 150 kG (solid squares) from Whitney et. a1.32

change in sign fi'om positive to negative indicates that there is an improvement in the
electron carrier term in comparison to the hole carrier term in the CDW state.

The Seebeck coefficient is negative for all three materials at room temperature,
approximately -10 uV/K for 2H-TaSe2, -7 uV/K for 2H-TaSz, and -10 uV/K for 2H-
NbSez. The Seebeck coefficient also undergoes a sign change as the CDW forms as
demonstrated in example Figure 4-16 for 2H-NbSe2 from Huntley and Frindt”. For a
one-band system, the Seebeck and Hall coefficients are expected to have the same Sign.
Since their Hall coefficients have the sign opposite of the Seebeck coefficients in both
states, this indicates that there is an overlapping two-band model. In the normal state,
there are more electrons than holes, but a greater mobility for the holes than the electrons.
The opposite sign of the Seebeck coefficients at room temperature indicates that there is
more than just a change in relative carrier concentrations since it still has the sign

opposite to the Hall coefficient in the CDW state.

37

 

 

 

 

+2-
A 0
A‘ -2.
Z.
V _4.
CD
-5.
-8-
-10..

l l l l n I n n

4 10 20 40 100
Temperature (K)

 

rq—r-I . ..T1.
+4)- 1
q
C
A
200 400

Figure 4-16: Seebeck coefficient plotted against the log of temperature for
two samples of 2H-NbSe2 from Huntley and Frindt”.

4.3.2 lT-TaSz

lT-TaSz can be obtained by quenching it from at least 1023 K to room
temperature. It remains in the normal state down to 552 K, but it tends to destabilize into
the 2H polytype much above this transition. Below 552 K, the material forms an I-CDW.
At 360 K, the material enters a quasi-commensurate phase (QC-CDW) with
commensurate domains. Another phase, the triclinic CDW (T-CDW) phase, is seen
between 280 K and 360 K on warming. The transition from the QC-CDW to the C-CDW
shows a hysteresis, shifting at 180 K on cooling and at 240 K on warming. The QC-
CDW to C-CDW phase transition occurs at 180 K on cooling and at 240 K on warming.
In the C-CDW phase, it forms a 413x413 superlattice rotated 13.54° to the original lattice
which generates three inequivalent sites of a, b, and c. The reconstructed structure is

shown in Figure 4-17. The ring of six b atoms around each a shift straight towards a.

38

 

 

Figure 4-17: The ‘/l3X\/13 superlattice effects on a) the real space cell and b)
the reciprocal space Brillouin zone of the CDW in lT-TaSz. The
reconstruction generates three inequivalent atomic sites a, b, and c in the
ratios of 1::6::6.

The next ring of six c atoms around a are both drawn inwards and rotated slightly around
a. This produces the pattern of a Star of David.
4.3.2.1 Resistivity

Figure 4-18 shows the in-plane resistivity as a function of temperature for IT-
TaSz. As can be seen, there is a discontinuous resistivity increase upon cooling through
each CDW transition, indicating that the transitions are of first order. In particular, the
resistivity increases by nearly a factor of five at the onset of the C-CDW phase,
exceeding the metallic limit. The formation of the C-CDW is not enough to explain this
dramatic change. It has been suggested a combination of a Mott localization and an
Anderson localization concurrently occurs in the C-CDW phase. The Mott localization
could drive the system into an insulator state at the QC-C transition. The Anderson
localization is more important at 60 K (not shown) where the resistivity starts to increase

somewhat sharply with temperature.

39

 

9
on
I

.0
.b
I

p
03
fi

C-CDW QC-CDW

Resistance (52)
p
M
F

 

 

.0
I

 

 

 

0.0|:. LI.|_|_.I_|_|_|_I_.I_L|_|_|_.I_I_I_|_I_I._I_I_I_I_I_I_I_'J
100 200 300

Temperature (K)
Figure 4-18: Parallel resistivity of lT-TaS2 from Manzke et. a1.34
4.3.2.2 Magnetic Susceptibility
The magnetic susceptibility of 1T-TaSz, as shown in Figure 4-19, shows

discontinuous changes and hysteresis at the I-CDW to QC-CDW and QC-CDW to C-

 

 

50 *
transform to

3 ZHBR
E o
B
E fj”)
f9, -50 ..
>5

-100 - l . 1 . 1 . l - l - 1

 

 

0 1 00 200 300 400 500 600
Temperature (K)

Figure 4-19: Temperature dependence of the magnetic susceptibility for 1T-
TaSz by Wilson et. al.”

40

CDW transitions. The T-CDW phase is not resolved in this set of measurements as the
transition is rather subtle. The data are also cut off at high temperature because the
material tends to change polytype above 552 K.
4.3.2.3 Hall and Seebeck Coefficients

The Hall coefficient for lT-TaSz also shows a change of sign as the CDW
develops. Figure 4-20 shows the Hall coefficient versus temperature as measured by
Mutka et. a1.36 At 200 K, there is a sign reversal and a very sharp change in scale. The
sign reversal indicates a change from electron like carriers to hole carriers. The change in
scale indicates a decrease of carrier concentration on the order of two magnitudes. The

subsequent low temperature behavior of the Hall coefficient is similar to that of an

I 0 . - .
'0 B: 5.4 KG

 

I11

RH (cm3/ C)
.1.

 

 

 

 

 

. A I e
. l e
g.
.- C
n I l I n I l
'5" 1040 100 200 300 400

Temperature (K)

Figure 4-20: Hall coefficient versus temperature for lT-TaSZ by Mutka et. a1.
36 The vertical scale above zero is 1000 times larger than the scale below
zero to bring out details.

41

extrinsic semiconductor with impurity doping. This supports the view that more than a
simple CDW is responsible for the low temperature ground state.
4.3.3 lT-TiSez

The physical properties of lT-TiSez are very different from those of its Ta
counterpart because Ti has one less d—electron. This has significant effects on the
electronic structure. There has been much debate on whether lT-Ti862 is a semi-metal or
a small gap semiconductor and subsequent questions on the nature of the CDW
mechanism”. lT-TiSez has only one commensurate CDW transition around 202 K. The
CDW is of particular interest as it has a three dimensional characteristic of forming a
2x2x2 superlattice. The effect upon the reciprocal space Brillouin zone is shown in
Figure 4-21. Under this reconstruction, the Brillouin zone is reduced to one eighth its

former size.

 

 

 

 

 

 

Figure 4-21: The reciprocal space reconstruction caused by a 2x2x2
superlattice.

42

4.3.3.1 Resistivity

Figure 4-22 shows the resistivity of lT-TiSez as a function of temperature.
Unlike lT-TaSZ, the transition is not a sharp, first order transition. This suggests a more

complicated CDW mechanism than the Fermi surface nesting mechanism. An excitonic
insulator model involving electron-hole interactions has been proposed for this material
by Wilson and Mahajan”. This mechanism can cause a metallic material to become
insulating under a CDW transition. Instead of the nesting of Fermi surfaces, this
mechanism arises from a coupling of hole states to electron states to form excitons. This

can occur as a specialized case of the Fermi surface nesting mechanism, if the nested

 

 

 

 

surfaces are of opposite sort.

'5 t I' I r I I I I I I I I I I I I I
u . I
13 - .h. -
l2 . .' '. .

O
I] - . 1 .
e
u) .. . I. .
C
A 9 u .. . u
I
7 - f 'e. -
s n I. . O . . I . u
5 - I .
4 I .. I
Q
3 . . . .0 u
z I
I I
20 60 m I40 180 229 260 300 340

Temperature (K)
Figure 4-22: Resistivity data for lT-TiSez by R. V. Coleman et. a1. 38

43

4.3.3.2 Magnetic Susceptibility
The magnetic susceptibility for lT-TiSez, as shown in Figure 4—23, indicates the

of the CDW transition clearly by a sharp change in slope. The maximum in this slope

change is at 203K, at the detected transition temperature of lT-TiSez.

 

 

0.05 —

10‘5 emu/G)

”g; -o.05-

 

 

'0.‘ I" l J I l
0 200 400 600 800
Temperature (K)

Figure 4-23: Magnetic susceptibility of 1T-TiSe2 versus temperature, taken
by Salvo et. al.39

 

4.3.3.3 Hall and Seebeck Coefficients

Both the Hall and Seebeck coefficients show a sign reversal near the onset of the
CDW as seen in Figure 4-24. However, unlike the 2H materials, the signs are the same
for both coefficients. Both coefficients change from positive to negative as the CDW

transition is approached. This indicates a shift from hole like to electron like carriers.

44

 

S (IN/K)

 

 

 

 

l I
O 1 00 200 300

Temperature (K)
Figure 4-24: Hall and Seebeck coefficient by Salvo et a1.3 9. They both

become negative as the CDW develops with temperature decrease. The Hall
scale is on the left axis; the Seebeck coefficient on the right.

45

Chapter 5 Experimental Details

5.1 Sample Preparation

The 2H-TaSe2 samples were provided by Professor F rindt from Simon Fraser
University. The other samples were provided by Dr. Molinié at Nantes cedex in France.
All samples were grown using the iodine vapor transport method. In general, the samples
have a thin flake like morphology.

Single crystals of the 2H-TaSez, 2H-TaSz, and 2H-NbSe2 were grown from
powder material directly synthesized from elements (99.95% pure Nb and Ta, 99.9995%
pure Se, and 99.999% pure S). The samples were allowed to cool normally from the
growth temperature of 1100 K to room temperature to solidify. Electrical resistivity
measurements indicate incommensurate CDW transition temperatures of 122 K, 75.3 K,
and 33.5 K40 respectively. 2H-TaSe2 undergoes a second transition into a commensurate
CDW phase at 90 K, as expected from good quality samples.

Single crystals of lT-TaSZ were grown at 1100 K and quenched to room
temperature to preserve the IT polytype“. The starting powder material was directly
synthesized from elements (99.95% pure Ta and 99.9999% pure S). Electrical resistivity
measurements indicate an incommensurate CDW (I-CDW) to quasi-commensurate CDW
(QC-CDW) transition temperatures of 352K. There is a hysteresis at the QC-CDW to
commensurate CDW (C-CDW) transition. On cooling the transition is at 191 K and on
warming it is at 209 K. Measurements of the Hall effect show the same transitions. The
measured transition temperatures match the expectations for stoichiometric samples.

Single crystals of the material were grown using iodine vapor transport around

800 K. lT-TiSez does not need to be quenched to room temperature to preserve the IT

46

polytype, since it only forms in the one polytype. The starting powder material was
directly synthesized from elements with an excess of Se to provide a pure stoichiometry.
Any impurities of off-stoichiometric samples are usually additional Ti atoms that occupy
interstitial sites between layers. Electrical resistivity measurements indicate a broad
commensurate CDW transition around 200 K.

X-ray Laue diffraction pictures were taken from the mounted samples prior to
being entered into the chamber. The Laue diffraction pattern indicates the sample
orientation, which can be used to mount the samples inside the chamber with a desired
orientation. For hexagonal lattices, the Laue diffraction patterns give preferential
emission from the FM line, allowing for direct determination of the sample orientation.
The quality of the Laue diffraction patterns gives some information on sample quality.

Clean surfaces were obtained by cleaving the samples in a vacuum better than 3 x
10''1 Torr just before measurements. The samples were cut and mounted to sample
holders using high vacuum epoxy, with a pin epoxied to the opposite face. Cleaving was
achieved by knocking off the pin epoxied. Since the TMDs are layered materials, this
usually results in a smooth surface.

The samples were attached to a copper arm, which was connected to a closed
cycle helium refrigerator. This allowed the samples to be cooled to 20 K as needed. A
resistor was used to heat the samples to as high as 250 K to attain the high temperature
phases for some samples. The thermal expansion of the copper arm had to be accounted
for when raising the temperature from 20 K. For the lT-TaSZ and lT-TiSez samples, the
temperatures had to be raised to between 250 K and room temperature producing a

thermal expansion of approximately 0.7 mm. For the 2H materials, the highest

47

temperature required was 130 K for 2H-TaSe2, producing only a 0.2 mm thermal
expansion. Comparably, the photon spot was about 0.6 mm in diameter. Measurements
were made both with and without the sample height adjusted to compensate the thermal
expansion. The spectra were qualitatively similar but did show some intensity difference,
part of which may be due to sample inhomogeneity and part of which may be due to

slightly different surface alignment.
5.2 Photon Source

These ARPES measurements were carried out at the Ames-Montana ERG-Seya
beam line at the Synchrotron Radiation Center (SRC), in Stoughton, Wisconsin. A
schematic of the Aladdin storage ring and the beamline layout is shown in Figure 5-1.
Electrons orbit around the storage ring at 800 MeV energy travelling at nearly the speed
of light. Each time that the electrons’ trajectory is bent by bending magnets, they emit
light along the tangential direction of the trajectory. The emitted light contains a
continuous spectrum of wavelengths. Monochromators are built at ports along the
tangential emission directions to allow selection of photon energies for experiments. The
ability to select photon energy in a continuous range is the main advantage of using a
synchrotron light source versus using other lab sources such as a He lamp. In particular,
probing the band dispersion along kl direction in the normal emission geometry is only
possible if a synchrotron light source is used. The light from a synchrotron source also

has an advantage of being 60-70% linearly polarized.

43

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/ 4m t-
NIM
, 6m TGM
_ Mark v
Grasshopper
Mulli-
Layer
ERG-
SEYA
9 \wmte
10m Light
TGM
l; 0 108 MeV +
‘ Mlcmtron N Main RF k
Cavrty T GM
cxn. @ SGM
41h Harmonic Q Fluorescence
C XrL Cavrty
E ERG-
Mark n An. 3W"
' J1 u fims- Seya
Opper Double White GMS‘ 4m NIM
l'ag NIM CW8] Light “WW pGM
I‘! I 1 I l l\ ’
l [ l fi

 

F991 Q—' ‘ 'Set' I o tiral
6—1‘0 up Sgt up
Figure 5-1: Synchrotron Radiation Center electron storage ring Aladdin.

Focussed light is emitted past the bending magnets and employed along beam

lines. The beamlines are labeled by the type of monochromator used upon
them.

The beamline primarily used in these experiments is the Ames/Montana’s ERG-
Seya, designed by C. G. Olson“. As shown in Figure 5-2, it combines an Extended
Range Grasshopper (ERG) monochromator for high energy photons of 40-1 500 CV with
a Seya monochromator for 5-40 eV photons. Both monochromators use spherical

gratings, with a design based on the focussing property of a Rowland’s circle. The light

49

from the Seya monochromator is nearly linearly polarized in the horizontal plane, which
is the plane of incidence. Most of the spectra present in this thesis were measured using
the Seya.

13 beymdtluga
c. an
124.6cmabove tloor Seya Grahg

Seya Cyindiml Glass
3' Grazing

    
 

Entmca
Seya Flat Mirror Slit

Slit + I _if :3

   

 

  

 

Aladdin
Flat Intermpt Mina “f.” 'W'“ Mam Fla Sauna
13' Gazing 1 Grazing f Grazing
2‘? “'0' M0 c and Mirror
Sm grains ( orizontal)

Figure 5-2: ERG-Seya combined monochromators. The metal flat is a mirror
inserted into the photon beam to select the Seya (top) over the ERG (bottom).

5.3 Electron Energy Analyzer

A movable 50-mm mean radius hemispherical electron energy analyzer was used
to collect and analyze the photoelectrons. Figure 5-3 shows its schematic construction.
The analyzer is mounted on a goniometer that allows precise rotation of the analyzer
about its focal point horizontally ~340° and vertically «40°. The electrons first pass
through a few focussing lenses at the entrance slit. They are then filtered by the
electrostatic potential formed between the inner and outer spheres. The electrons that
pass through the exit slit are collected by an electron multiplier. The electron and photon
combined energy resolution of the spectrometer was 30 meV at 1.5 eV pass energy and

21 eV photon energy. The angular resolution of the electron analyzer was i1°, which

50

at 21 eV photon energy.

corresponds to a momentum resolution of 0.07 A" for measurements near the Fermi level

 

 

 

 

Entrance
SW p le Aperture
Lenses

Inner Potental
Shell

Electron
Multiplier

Outer Pobntial

Figure 5-3: Hemispherical electron energy analyzer with focussing lenses.
The inner and outer potential shells are set to specific voltages to allow only
electrons with a desired energy to reach the electron multiplier to be pulse
counted. The entrance aperture limits the accepted electrons to a small angle

for angular resolution.

51

Chapter 6 2H-TaSe2, 2H-TaSz, and 2H-NbSe;

As has been discussed, ARPES is a very useful tool for studying the electronic
phase transitions in solids. In particular, high resolution ARPES studies of the high Tc
superconducting cuprates have been very fruitful”. In contrast, the CDW materials have
not been studied as thoroughly and need to be explored. Although there has been some
ARPES study of 2H-TaSe2 by Smith et a1.43 and Dardel et a1.44 with moderate energy and
moderate angular resolution respectively, there were no high resolution studies of this
group of materials prior to this work.

As discussed in Chapter 4, the CDW mechanism in 2H-TaSe2, 2H-NbSe2 and 2H-
TaSz is still not well understood45’46. Two different viewpoints have been put forward to
explain the CDW. One of them is the Fermi surface (FS) nesting mechanism, either
involving the F S around F" or the FS around K13. However, the FS nesting around F
cannot explain the resistivity decrease in the CDW phase. In contrast, Rice and Scott28
proposed a CDW mechanism which invokes saddle points on the F S. They showed that
the susceptibility X(Qo), where Q. is a wavevector connecting two saddle points, diverges
logarithmically leading to a CDW instability. These saddle points act as scattering sinks
in the normal state and their removal can reduce the resistivity.

In these high resolution ARPES experiments48’49'50, the band dispersions and
Fermi surfaces were studied of 2H-TaSe2, 2H-NbSe2 and 2H-TaSz in the normal and
CDW states. The following results provide important insights on the CDW mechanism

in this group of materials.

52

6.1 Band Dispersions and the Saddle Band

6.1.1 2H-TaSez

Figure 6-1 a) shows the EDCs (solid lines) for k-points along F-K in the normal
state at a temperature of 125 K for 2H-TaSe2. Note that, as k, is increased from 0.34 to
0.42 Al, the spectral weight at E increases significantly. This indicates that an energy
band has just dispersed from above By to below By. Therefore, kJr =0.42 A'1 is a FS
crossing point. As kx is further increased, this band splits into two bands. One disperses
below By; the other stays very close to E1: for an extended region as illustrated by the
vertical tick marks in Figure 6-1 a). As k, is increased from 0.87 to 0.90 Al, the spectral
weight at E; decreases, suggesting that It, =0.87 A'1 is another FS crossing point.
However, this FS crossing is not so obvious and it is also obscured by the deeper binding
energy band approaching Ep. To determine the FS crossing points more accurately, the
spectra were modeled by Lorentzian peaks multiplied by the Fermi-Dirac function and
convolved with a Guassian instrument resolution term. The best fits are shown as dashed
lines in Figure 6-1 a). As can be seen, the modeled spectra reproduce the measured ones
very well. The corresponding band dispersions are shown as the solid circles in Figure
6-1 b). In order to obtain a good fit at the higher binding energy region, an additional
weak feature, whose binding energies are indicated by the crosses in Figure 6-1 h), is
needed. The modeling shows that the two bands cross E: at kJr = 0.87 A‘1 and 1.08 A'1
respectively. Because of the splitting of the two bands, the FS of 2H-TaSe2 is double
walled. As pointed out by Wexler and Woolley”, due to time-reversal symmetry, the
double walled FS can be folded out into a single continuous F8 in a double Brillouin zone

scheme along the c-axis. Note that the low binding energy band is extremely close to Ep.

53

 

a) Normal State
F—K

 

Intensity (Arb. units)

_ v i 0.42
53“" I; l I I 0.34 F

0.4 0.3 0.2 0.1 0.0 -O.1

 

 

 

 

 

 

 

 

 

Binding Energy (eV)
A 0.0 b) t . relic 0"." .10.-
E 0.1- , :0
LL? 0'2- + l++.....+++++++
O 3-1+ + +l++ r r
0.0 0.2 0.4 0.6 0.8 10 12
k (A1)

Figure 6-1: a) Normal state EDCs (solid lines) for the PK line of 2H-TaSe2

with modeled fits (dashed curve). The tick marks indicate the peak positions.

b) The extracted band dispersions from a).

Figure 6-2 shows the EDCs of 2H-TaSe2 in the normal state for k-points along

two additional lines parallel to but slightly below F-K as shown in the inset. The k,- and

ky-values marked along each curve in A", are selected to be the momentum components

parallel and perpendicular to F-K, respectively. The band dispersions of these lines have

54

a general trend similar to those along F-K. Well defined Fermi crossings can be observed

at (k,, Icy) = (0.36, -0.07) and (0.33, -0.15) in Figure 6-2 a) and b), respectively. These

Fermi crossings, along with the one at (k,, Icy) = (0.44, 0), form a Fermi surface locus

disperses significantly below E1: in the mid-range, as indicated by the vertical dashes. As

Intensity (Arb. units)

 

 

 

2H-TaSe2 r‘x A13} it“ A
hOJ=21 CV A
T = 125K

 

 

 

 

 

.9
\l

0.65

 

 

‘A

 

 

 

v ‘F 1' ‘Iflo‘3

 

0.4 0.3 0.2 0.1 0.0 .0.1
Binding energy (eV)

0.4 0.3 0.2 0.1 0.0 0.1
Binding energy (eV)

Figure 6-2: Energy distribution curves (EDCS) in the normal state (T=125 K),
for k-points along two lines parallel to l"-K at a) ky = -0.07 and b) -0.15 A'l.
The inset of the Brillouin zone’s irreducible wedge (top) shows the amount of
k,-dispersion of the Fermi surfaces predicted by band calculations14 (A: It:z =

0;B:k¢=

21t/c; C: k2 = 1t/c.) and the lines of a) and b).

55

the band disperses back toward Ep, it appears to cross E1: at (kx, ky) = (0.85, -0.07). In
Figure 6-2 b) where k-points are along a line further away from F-K, the band disperses
even further below E1: in the mid-range, and appears to cross 13;: at (kx, ky) = (0.85, -0.15).
The second Fermi crossings in Figure 6-2 a) and b) are more obvious than the one in
Figure 6-1 because the band has a finite dispersion. The fact that this band stays close to
E; for an extended region along l"-K and disperses below Ep in the direction
perpendicular to F—K in the mid-range suggests that it is an extended saddle band. As
will be discussed, this extended saddle band places an important role in the CDW
formation of this material.
6.1.2 2H-TaSz

The 2H-TaSz samples produced ring or are Laue diffraction patterns, indicating
some distribution of crystal orientations. As a result, the samples were mounted with
unknown orientations. The ARPES spectra taken from those samples show distinct
angular dependence and clear band dispersions, but the directions were not clear enough

to probe the F—K line for saddle band behavior. The clear band dispersions suggest that

there is a stacking disorder, which gives misaligned layers, but that the depth probed by
the ARPES has a unique orientation. Further experiments using in-situ low energy
electron diffraction (LEED) to determine the surface crystal orientation were attempted to
rectify this situation, but the resulting signal was too low to help.
6.1.3 2H-NbSe2

Shown in Figure 6-3 are the EDCs for 2H-NbSe2 taken in the normal state (T=40
K) for k-points a) along F-K and b) along a line perpendicular to F-K, on a line

intersecting it at approximately the midpoint, as shown in the insets. Along F—K, two

56

bands are observed dispersing below ER They both reach maximum binding energies at

approximately the midpoint of F—K. As can be seen in Figure 6—3 h), both bands disperse

 

 

l l
a) 2H-NbSe2
hv=21 eV

 

 

 

 

£19; 0.6:;
skv=

7:
ES
7:

  
  

 

 

 

Intensity (Arb. units)
Intensity (Arb. units)

 

 

0.5:
, r

N “A I
.4

 

 

I I l

I I l
0.4 0.3 0.2 0.1 0.0 -0.10.4 0.3 0.2 0.1 0.0 -O.l
Binding Energy (eV) Binding Energy (eV)

 

 

A
.

 

 

-0.1 . ,
0 0 Id) Perpendicular to l"—K

c O
0.0-—)—OQ-G.i—;°O—'o-
O
0.1 - 0
oz- ’0‘

 

 

0.1 O

02 '
O . .
03

0.4lllllI-O.4,IIIIL
0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.00 0.05 0.10 0.15 0.20 0.25

1“ k, (13.4) K k, (K1)

Figure 6-3: EDCs for 2H-NbSe2 for k-points a) along l"-K and b) along a line
perpendicular to I‘-K intersecting it at approximately the midpoint, as shown
in the insets. c) and d) Band dispersions (symbols) determined by modeling
the spectra in a) and b), respectively.

 

 

0.3 -

 

Brndrng Energy (eV)
0
0

Binding Energy (eV)
0

57

toward higher binding energies in the orthogonal direction away from the F—K line,
indicating their saddle band nature.

To determine the energy versus momentum relations for 2H-NbSe2, modeling was
used. The results are shown as symbols in Figure 6-3 c) and d). Along F—K, the lower
and higher binding energy bands have maximum binding energies of 0.055 eV and 0.187
eV respectively. Two things are of note. While the dispersion trend is similar to that in
2H-TaSe2, the lower binding energy band has a greater maximum binding energy than
that of the corresponding band in 2H-TaSe2. Similarly, the higher binding energy band
has a smaller maximum binding energy than that of the corresponding band in 2H-TaSe2

(0.22 eV maximum binding energy).
6.2 Fermi Surfaces

6.2.1 2H-TaSez

EDCs were measured for k-points in the entire Brillouin zone for 2H-TaSez. By
observing where bands cross Ep, the Fermi surface is determined, as summarized in
Figure 6—4 as shaded and open circles. The open triangle along I"-K indicates the second
Fermi crossing determined from the modeling as shown in Figure 6-1 b) which
corresponds to the inner wall of the F S around the K point. Because the two walls of the
double walled F8 are very close to each other, it is difficult to determine both walls
accurately by just visually inspecting the EDCs. Only one set of the doubled walled FS
was determined and shown for the other parts of the Brillouin zone. The size of the
circles corresponds approximately to the experimental k-resolution. The data in the
wedge bounded by the two dashed lines are original; the rest are obtained by a six-fold

rotational symmetry operation. Note that the data join smoothly at the borders between

58

 

 

 

 

 

Figure 6-4: The Fermi surface cylinders (shaded and open circles) of 2H-
TaSez determined from the EDCs. The open triangle is the second wall seen
along l"-K. Points a, b, and c are shown in Figure 6-10. Point d (diamond)

was used for the temperature dependence of the gap shown in Figure 6-11.

the rotated segments, which reassures the validity of the symmetry operation. As can be
seen, the shaded circles form a near hexagon centered at the F point; the open circles
form a pocket centered at the K point. The PS topology was also investigated by
measuring the intensity map at Ep, which can be considered as a two-dimensional cut of
the F8. The resulting E1: intensity map for the normal state of 2H-TaSe2 is shown in
Figure 6-5, presented in a linear gray scale (bright for high intensity). The intensities
were measured with 21eV photons and were normalized to the photon flux. One sixth of
the map was measured in each case as shown by the dashed wedge in Figure 6-5; the rest
were obtained by a mirror operation about F-K, then a three-fold rotational symmetry
Operation. Note that there is a lack of symmetry between the F—K line and F-K’ line at
60° with respect to it. This is partially caused by the unequal matrix elements of the two

lines resulting from the horizontal photon polarization and plane of incidence which

59

 

1-

0e—

g“ 02—
5; -
xr-02_
-oe—

-1_

-i4

 

 

 

 

-1.4

3 '-de'-62' d2' 65' 1 i4
'1
kxiA I

Figure 6-5: Fermi level intensity map of 2H-TaSe2. Bright regions indicate
greater intensity. The data between the dashed wedge were actually
measured and symmetrized to give the rest of the Brillouin zone.

process5 1.

contains l"-K, and partially caused by the finite escaping depth of the photoemission

The 2H-TaSe2 intensity map shows a bright ring around the F point, indicating a

cylinder.

FS locus. There is also a somewhat faint bright arc surrounding the K point, indicating
another FS locus. The extended bright regions along l"-K are caused by the extended
saddle band near Ep as discussed previously. The F S locus around the F point has a
rounded, three fold symmetry which is not aligned with the Brillouin zone. This is in
contrast to the near hexagon shape in the FS determined fi'om the EDCs. This difference

may be due the overlap of the two walls of the FS distorting the apparent shape of the

60

6.2.2 2H-TaSz

The BF intensity map of 2H-TaS2 shown in Figure 6-6 has the same qualitative
features as 2H-TaSe2. Here, the contour around K appears to have the double-wall
structure, and the contour around 1" does have an apparent, but distorted hexagon shape.
In this case, the raw data were rotated to correct for the sample misalignment prior to the
symmetry orientation. This generated the wide swath of intensity along the F—K line for
better comparison with the other materials. The breadth of the saddle band and the
distortion of the hexagonal shape of the FS cylinder around 1" may well be from a

misadjustment of the data set’s orientation.

1.4

 

 

 

 

 

' l I l | | I I I T I
-1.4 -1 -U.5 -U.2 0.2 0.5 1 1.4
-1
kal 1

Figure 6-6: Fermi level intensity map of 2H-TaS2. The data were measured
on one sixth of the Brillouin zone, rotated to give a comparable orientation to
the other materials, and then symmetrized.

6.2.3 2H-NbSe2

For 2H-NbSe2, the EF intensity map in Figure 6-7 shows a bright ring around I"

61

and another one around K. This FS pattern is qualitatively consistent with the band
structure calculation results”. The double walls of the FS were not resolved here,
although Straub et 2.11.52 used a two-dimensional gradient to distinguish them. The bright
strip along F—K between the two FS pockets arises from the saddle band. The F8 pocket
around F is nearly hexagonal with a three-fold degenerate self-nesting wavevector along
the F—M direction indicated by the arrow. The magnitude of the nesting wavevector is
estimated from the middle of the bright contour on one side of the hexagon to that of the
opposite side. An estimation of error is based upon the blurred appearance of the
contour. The direction and the magnitude (0.68:1:027 A!) of the nesting wavevector are

consistent with those of the observed CDW wavevector”.

 

 

14
it;
oej
4* 02:
g _
xi-u2_
-ee:
-10:
-t4- .

 

 

 

r l r I I l l 1
-r4 -io -d5 -dz d2 dB to 14
kxorh

Figure 6-7: Fermi level intensity map of 2H-NbSe2. The arrow indicates a
nesting vector of the same direction and magnitude as the CDW wavevector.

62

6.3 Energy Gap in the CDW State

6.3.1 2H-TaSe2

The formation of a CDW usually opens an energy gap at the Fermi energy and
truncates parts of the FS that are connected by the CDW modulation’s wavevector. In
both 2H-TaSe2 and 2H-NbSe2, there appears to be a nesting vector Q0 that spans the FS

cylinder around F. The nesting vectors for each material are shown in Figure 6-4 and

Figure 6-7 respectively. Straub et al. suggest that the nesting of this FS is the cause of the
CDW formation in 2H-NbSe252. However, this mechanism should result in a very strong
resistivity increase instead of the decrease measured in experimentsg’27. To make a
thorough investigation of the CDW mechanism in these materials, the energy gap
dependence on k-point was studied by measuring EDCs in the normal and CDW states at
specific k-points.

Figure 6-8 shows the EDCs along F-K for 2H-TaSe2 in the commensurate CDW
state at 20 K with solid lines and the normal state at 250 K with dashed lines. Both sets
of EDCs were measured in terms of the unreconstructed 1x1 Brillouin zone. It can be
seen that the CDW-induced spectral changes are highly k-dependent. At (k,, k,.) = (0.44,
O) A", which is a Fermi surface crossing point, the change near E1: is relatively small.
The leading edge is slightly sharper at low temperature, which can be accounted for by
the reduced thermal broadening. The Fermi level remains above the mid-point of the
leading edge, indicating that the CDW-induced energy gap is very small or near zero.
For (k,, k,) beyond (0.44, 0), the spectral changes are much more dramatic. Note that the
leading edges shift significantly toward higher binding energies at low temperature,

indicating the removal of electronic states from the Fermi level. At some k-points, such

63

 

 

Intensity (Arb. units)

 

 

 

 

~——+~“ i~ ~--"i"-"_"‘—' 036i

0.4 0.3 0.2 0.1 0.0 -0.1
Brndrng Energy (eV)

Figure 6-8: EDCs in the commensurate CDW state (T=20 K, solid curves) of

2H-TaSe2, for k-points along F—K. The EDCs in the normal state are

superimposed as dashed curves for comparison.
as (k,, k,) = (0.85, 0), (0.92, 0), (0.98, 0), and (1.05, 0) A", weak additional spectral
features appear at higher binding energies, indicating band splitting. Doran and
Woolley54 performed band calculations on the 3x3 CDW superlattice and predicted

similar band splitting. During this experiment, measurements were repeated several

times as the sample went through temperature cycles between 20 K and 125 K. The

spectra were reproducible and no surface degradation was detected. The spectra were

64

qualitatively similar but did show some small intensity difference, part of which may be
due to sample inhomogeneity and part of which may be due to slightly different surface
alignment.

To quantify the momentum dependence of the energy gap, the energy shift
between the leading edge midpoints in the normal and CDW state was measured.
Different methods of determining the gap might change the numbers slightly, but will not
change the curve shape. The energy shift as a function of kJr along l"-K is plotted in
Figure 6-9. Strictly speaking, a gap can be defined only on the Fermi surface. Here, the
term “gap” is also used for the extended saddle band region, since there are states near Ep
at those k-points which are affected by the CDW formation. As can be seen, the energy
shift is near zero at k, = 0.44 A", which is a Fermi crossing point. It increases to finite
along F—K, the energy shift remains large comparable to the gap at kJr = 1.05 Al, but the

spectral intensity at EF diminishes. A finite energy gap of up to 25 meV wide is observed

 

30
a)

25

{it _
‘° hit“ :

Energy Shift (meV)

(II

o
r—O—q

I l J I I I

04 06 08 10 12
k(A") K
Figure 6-9: The energy shift vs. k, along F-K of 2H-TaSe2. The energy shift

is taken as the binding energy difference of the leading edge midpoints in the
normal and CDW state spectra.

 

pm
.1 C)

O

N

65

in the extended saddle band region and on the FS around K.
To confirm the effects of the CDW on the FS around the 1" point, this comparison
of the normal and CDW state EDCs was made for several points on the cylinder. Figure

6-10 shows the EDCs for three F S crossing points, labeled as a, b, and c in Figure 6-4.

 

I T l

I I
2H-TaSe2 hm =216V

a) k=(0.44,0.00)A'1
20K
----------- 125K

  
   
 
 

 

 
   
 
 

   

Intensity (Arb. Units)

 

ibuoooconunnnnno

 

O
O
I
O
p—s

l l

0.4 0.3 0.2 0. 1
l 1—_T'

b) It=(o.36,-o.07)i'i'1

  

 

 

 

Intensity (Arb. Units)

 

—.-..-.-....-.-

   

l l
0.4 0.3 0.2 0.1 . 0.0 -0. 1

 

 

c) It=(o.33,-o.15)ifii1 ..-‘

 

 

Intensity (Arb. Units)

 

 

l l I V
0.4 0.3 0.2 0.1 0.0 01
Binding energy (eV)

Figure 6-10: EDCs for three FS crossing points of 2H-TaSe2, labeled as a, b,
and c in Figure 6-4. The solid and dashed lines are spectra measured in the
CDW (T=20 K) and normal (T=125 K) states respectively.

66

Point c is on F—M, the direction of the CDW wavevector. The solid and dashed lines are
spectra measured in the CDW (T=20 K) and normal (T=125 K) states respectively. As
can be seen, the spectral changes are very small at these k-points. The leading edges are
slightly sharper at low temperature due to reduced thermal broadening. The Fermi
level,indicated by the vertical dashed lines, remains at the upper portion of the leading
edges, indicating a sharply peaked density of states at E1: and hence nearly zero energy
gaps. Thus, the FS centered at the T‘ point is not gapped, despite the apparent nesting
vector Q0.

The temperature dependence of the gap in the extended saddle band region along
F—K is of some interest as it can give information on the nature of the CDW mechanism.
Figure 6-11 a) shows the spectral dependence on temperature for (k, , ky)=(1.05, 0.00)A'l
labeled as d in Figure 6-4. At this k-point, the shallow band has just crossed BF and the
deeper band is about to cross 131:. The normal state spectrum taken at T=125 K is
superimposed on each curve as the dashed lines for comparison. As can be seen, there is
significant spectral weight at E1: in the normal state. As the temperature is lowered below
the incommensurate CDW transition temperature (To=122 K), the spectral weight at E]: is
suppressed and the leading edge midpoint shifts toward higher binding energies,
indicating the opening of an energy gap. The temperature dependence clearly shows that
the spectral changes are related to the CDW transition.

Figure 6-11 b) shows the midpoint energy shifts as a function of temperature.
The energy shift decreases for increasing temperature and approaches zero near To=122
K. The decreasing rate appears to be faster than the Bardeen-Cooper-Schrieffer (BCS)

relation (solid line) of the weak coupling limit of the superconducting gap to temperature

67

is. There is no apparent anomaly near the commensurate transition at Tt=90 K. Further
theoretical work would help clarify what basis there is for this deviation from the weak

coupling limit.

 

 

 

Intensity (Arb.Units)

 

 

 

    

 

 

 

«W i 100K
. . 110K
I l I l I 125K
0.4 0.3 0.2 0.1 0.0 01
Binding Energy (eV)
2? 25 -. b)
Z: 20%
E 15-
g 10-
S 5 -
t5 0 .
0 40 80 120
Temperature (K)

Figure 6-11: a) EDCs for point d in Figure 6-4 in the vicinity of the extended
saddle band of 2H-TaSe2, measured at various temperatures. The normal
state spectrum is superimposed on each curve as dashed lines for comparison.
b) The energy shifts of the leading edge midpoint (solid squares) as a
function of temperature. The solid line is the BCS relation.

68

6.3.2 ZH-Tasz
Shown in Figure 6-12 a) are the EDCs measured on a 2H-TaSz sample in the
normal (T=80 K, dashed lines) and CDW (T=20 K, solid lines) states. The spectra were

normalized to photon flux. The corresponding k-points are along a line at approximately.

 

a) 2H-TaSz ' '

_ ------ 80 K
.133be — 20 K A K

 

 

 

I I I

  
   
 
    

 

 

 

 

 

  

 

 

 

 

b) PointB
35.7 .
'E
3
3
E: .
a 0.4 0.3 0.2 0.1 0.0 or
c: . -.
£3 53.". Binding Energy (eV)
.5 _ .. ‘ a. -o.7 I I ,
.0" '.
"O . 0) Point A
0-6 —-20K
, ‘ ----- 80K _ '
0.5
0.50 A L
I l I
l l I l

 

 

l -
0.4 0.3 0-2 0-1 0-0 '0'] 0.4 0.3 0.2 0.1 0.0 or

Binding Energy (6V) Binding Energy (eV)

Figure 6-12: a) EDCs for 2H-TaSz measured in the normal (T=80 K, dashed
lines) and CDW (T=20 K, solid lines) states, along a line at approximately
36° with respect to F—K, as shown (inset). Enlarged spectra at point A and B

are reproduced in c) and b) respectively.

69

36° with respect to F-K, as shown in the inset. At point A, the enlarged spectra Figure
6-12 c), which is a point on the FS around I", there is no apparent energy shift between

the leading edges at the two temperatures, indicating a nearly zero energy gap. For k-
points beyond A such as point B in Figure 6-12 b), the midpoint of the leading edge shifts
toward higher binding energies (up to 20 meV) at the lower temperature, indicating a
finite energy gap. Note that, in addition to the leading edge shift, the spectral intensity
also decreases in the energy range between zero and 0.3 eV binding energies at the lower
temperature. This might be due to the electronic structure change in addition to the
opening of a gap.
6.3.3 2H-NbSe2

Figure 6-13 shows the EDCs for 2H-NbSe2 measured at temperatures above
(dashed lines) and below (solid lines) the CDW transition for some k—points along F—K.
These spectra were taken at the Undulator-NIM beam line using a Scienta SES200
hemispherical analyzer for better resolution. The energy resolution was 24 meV; and the
angular resolution was 0.7° corresponding to a momentum resolution of 0.026 A'1 for 21-
eV photons. There is no apparent indication of a gap at any k-point, including the Fermi
crossing around F and the saddle band region. Since the CDW transition temperature is
relatively low in this material, the energy gap is expected to be small. ARPES studies

with higher energy resolution are needed in order to study the gap properties.

6.4 Summary and Discussion

In summary, the normal and CDW state electronic structures of 2H-TaSe2, 2H-
NbSez and 2H-TaSz were studied using high resolution ARPES. The essential electronic

structure properties are similar within this family of materials. The Fermi surfaces

70

 

 
    
 
        

l l I l
zit-Niasd2
hv=21 6V
F—K
K
"3
'53
55..
3“
3
.5".
.FS crossing near I‘ ”‘4' r
l l l

 

 

 

l
0.4 0.3 0.2 0.1 0.0 -0.1
Binding Energy (eV)

Figure 6-13: EDCs for 2H-NbSe2 along l"-K measured in the normal (T=40
K, dashed lines) and CDW(T=13 K, solid lines) states. The bottom curve is
for a k-point on the Fermi surface around F while the other spectra are in the

saddle band region.

71

consist of a pocket around I" and another one around K which are in excellent agreement
with the band calculation results by Wexler and Woolley”. In 2H-TaSe2, the Fermi
surface pocket around F has an apparent hexagon shape. For 2H-NbSe2 and 2H-TaSe2,
the F S pocket around I‘ is nearly hexagonal with a three-fold degenerate nesting
wavevector along the F-M direction whose magnitude corresponds well with that of the
observed CDW wavevectors. The nesting conditions are not as apparent in 2H-TaSz, due
to the orientation problem of the samples available.

A saddle band was clearly observed in 2H-TaSe2 and 2H-NbSe2 near EF along I‘-
K with a saddle point at approximately the midpoint of F-K. This band is closer to E;
than predicted by band structure calculation”, indicating possible band renormalization
by electron-electron or electron-phonon interactions. While the saddle band in 2H-NbSe2
has a finite dispersion, the saddle band in 2H-TaSe2 stays very close to By. It shows
minimal dispersion for an extended region of k-points along F-K, forming an extended
saddle band. In the CDW phase of 2H-TaSe2 and 2H-TaSz, the energy gap is nearly zero
on the F S around F. In 2H-TaSe2, the energy gap is finite in the saddle band region and
on the F S around K. No spectral indication of a gap was observed in 2H-NbSe2 at any k
points within our energy resolution (AE=24 meV). Since the CDW transition
temperature in this material is relatively low (33 K), its gap is very small (e. g., ~2-3
meV). ARPES studies with higher energy resolution are needed in order to study the gap
properties.

The favorable nesting condition of the F S pocket around F observed in 2H-TaSe2
and 2H-NbSe2 might lead to the conclusion that the CDWs in the 2H-polytypes are

driven by PS nesting of this cylinder”. However, this nesting mechanism is not

72

consistent with the absence of a finite energy gap on that F S in 2H-TaSe2 and 2H-TaSz.
In addition, a CDW driven by such a nesting should be fairly strong, since ~1/3 of the FS
would be affected. Instead, only a 10-20% decrease in the susceptibility was observed in
this family of materials at the CDW onset”. Furthermore, this nesting mechanism cannot
explain the observed resistivity decrease at the CDW transition. Alternatively, the
resistivity decrease can be explained by the “saddle point” mechanism proposed by Rice
and Scott”. This model is also consistent with the observed saddle band near E: and the
finite energy gap in the saddle band region. However, the location of the saddle point in
k-space (~1/2 _) is not consistent with the observed CDW wavevector.

The CDWs in these materials are caused by a combination of the partial nesting of
the F S pocket around K and the saddle band. As illustrated by Wilson”, parts of the F S
around K can be brought into a nesting relationship by a displacement of roughly 1/3 FM,
giving rise to a broad maximum in the electronic susceptibility, x(q)47. Significant
contributions to x(q) are associated with points close to 2/3 17?, where the band is just
coming out of the saddle point region and is still fairly flat. The removal of the high
density of states at E1: associated with these k points in the CDW phase can enhance the
conductivity. Our observation of a finite gap at both the saddle band region and the FS
around K in 2H-TaSe2 and 2H-TaSz is consistent with this scenario. The failure to
observe the gap in 2H-NbSe2 is most likely a simple function of its low transition

temperature and may be resolved with further improvement of energy resolution.

73

Chapter 7 1T-TaS,

Among the transition metal dichalcogenides, the IT polytype of TaSz has been of
particular interest due to its complex CDW phase diagram”. Angle resolved

photoemission (ARPES) has been extensively used to study the CDW transitions in IT-

Tasz43,55,56,57,58,59,60,6l 55-59, a

, especially the QC to C phase transition. In most experiments
metallic Fermi edge was observed in the QC phase. As the material undergoes the QC to
C phase transition, the spectral weight at the Fermi level (Ep) abruptly decreases and an
intense peak appears at approximately 180 meV binding energysuo. This spectral
behavior was interpreted as the opening of a pseudogap in the C phase59 due to a Mott
transition concrurent with the CDW transition. As emphasized by Fazekas and Tosattiss,
it is not the Mott transition which drives the reconstruction; rather, it is the reconstruction
which brings about favorable conditions for the Mott transition to take place. Despite
numerous ARPES studies, the electronic structure of 1T-TaSz in the QC and C phases is
still a matter of controversy“. Contributing to this long standing issue, the band
dispersions and Fermi surfaces of 1T-TaSz in both the QC and C phases were studied

with high resolution ARPES.

7.1 Band Dispersions

Figure 7-1 shows the EDCs for k-points along a) F—M and a) F-K in the QC-CDW
state at a temperature of 250 K for 1T-TaSz. The k—points, labeled in Al, are the
momentum components kx for F-K and lcy for F-M which are parallel to the respective
symmetry lines. In these units, the F, M, and K points are at (kx, k,) = (O, 0), (0,1.08) and
(1.25, 0) Al, respectively. Note that the thermal broadening is on the order of 100 meV

in this phase, which is larger than the 30 meV instrument resolution. As can be seen, at

74

 

 

I I I I l _,
3:731 - .. F-M ikyrfds)

 

Intensity (Arb. units)
Intensity (Arb. units)

 

 

 

 

 

 

I‘ l l I r l P
0.5 0.4 0.3 0.2 0.1 0.0 -0.1 0.5 0.4 0.3 0.2 0.1 0.0 -0.1

Binding Energy (eV) Binding Energy (eV)

Figure 7-1: EDCs of the QC-CDW (T= 250 K) state for the a) F-M and b) F-
K lines of lT-TaSz. The arrows indicate Fermi crossings.

 

 

the F point, there is an intense spectral feature just below By. Along the M and K lines,
this feature crosses the Fermi level within ky=0.07 A" and kx=0.15 A" respectively as
indicated by the upward arrows. This creates a very small electron pocket around the I‘

point. As It, is scanned toward M, the feature above Ep disperses back below it at

75

ky=0.36, as indicated by the downward arrow. The crossings are estimated at the k-values
where the leading edge midpoint is closest to the Fermi level. These estimates are also
confirmed by checking the slope of the spectral leading edge. At Fermi crossings, this
slope has a local maximum. This feature continues to disperse well below Ep along the
F—M line. There is another broad spectral feature whose centroid is at about 300 meV
binding energy at the P point. This feature shows very little dispersion along F—K,
whereas it disperses toward higher binding energies along F-M.

To determine the band dispersions accurately, the second derivative of the EDC
intensity was determined as a function of energy. At peaks, the second derivative goes to
a negative extremum. The resulting band dispersions are shown in Figure 7-2 as the gray
scale map where bright indicates positive values and dark negative values. As can be

seen, there are dark regions just below By at the I‘ point along both lines. The regions

 

0.5 0.5
o a b

    
 
  

Energy (eV)

  

 

 

-4.5 -4.5
F M F K

Figure 7-2: Second derivative of the spectral intensity plotted versus energy
and wavevector along a) I"-M and b) F-K of lT-TaSz. Darker regions
indicate peak positions.

76

disperse away from E1: along both lines. Along F-M, the band disperses below E; in a
continuous are down to 1 eV binding energy. Along F—K, the band disperses above E]:
where it no longer contributes to the spectral intensity. There are some other darker
regions from a band starting between 1 eV and 1.5 eV at F that disperse to deeper binding
energies along both lines. These are more prominent along the F-M line. Overall, these
dispersions are consistent with the qualitative descriptions from the EDCs.

Larson and Mahantil9 recently performed new band structure calculations using
the self-consistent F LAPW method20 within density functional theory”. The calculations
were made both without and with spin-orbit corrections. The generalized gradient
approximation (GGA) of Perdew, Burke, and Emzerhof22 was used for the exchange and
correlation potential. The calculations were performed using the WIEN97 packagefiz.

The atomic radii were chosen identically at 2.29 a.u. for each atom in such a manner so as
to minimize the regions between the atomic spheres. Adjustments of these radii within a
reasonable range showed little dependence of the final band structure on this variation.
The lattice parameters used in the calculation were taken from experiment”.
Convergence of the self-consistent iterations was performed with 20 k-points in the
reduced BZ with a cutoff between valence and core states of -6.0 Ry. Scalar relativistic
effects such as mass-velocity and Darwin corrections63 were included in the calculation.
The spin-orbit interactions were included using a second variational procedure of the self-
consistent potential“. The calculated band dispersions are shown in Figure 7-3 a) with
and b) without spin orbit corrections respectively. In Figure 7-3 a), the band which
crosses EF is primarily a Ta 5d-band. Without spin-orbit correction, this band is slightly

above Be at the I‘ point. Spin-orbit correction splits the bands and places one of the Ta

77

 

 

Energy (eV)

 

 

 

Energy (eV)

I
_r.
I

 

I
_L —L
-

 

._A
l

.0
(II
I

 

 

 

:5
01

Energy (eV)

I
—A

 

 

 

 

-1.5

 

l" 2 M K A 1" A A

Figure 7-3: Band dispersions in the normal state fiom new band structure
calculation, a) without, and b) and c) both with spin-orbit corrections
included. Second derivatives of EDCs along F-M and F-K in the QC-CDW
phase (see text for details) are also shown in c).

5d-bands slightly below E; at the F point. As a result, the FS has a small electron pocket

78

around the F point in addition to the large electron pocket around the M point. Since the
energy bands around the F point are sensitive to the interstitial potential”, the
convergence was carefully checked. Three methods were used: 1) increasing the number
of k-points from 20 to 100, 2) increasing the cutoff energy from 1.5 to 5.5 Ry, and 3)
changing the relative sizes of the atomic sphere radii of Ta and S. The convergence was
better than 1% indicating that the results give the correct band structures within LDA and
GGA. Also, the S 4p-band from the calculation is much closer to the Ta 5d-band than
previous calculations show's'”, resulting in a much better agreement with ARPES data“.
As can be seen in Figure 7-3 c), the agreement between the observed and
calculated dispersions is quite good. The features of the measurements match the
dispersions of the Ta 5d-band nearest the Fermi level. The calculated gap between the
valence band and the top most core level state is also well matched by the observed

features.
7.2 Fermi Surfaces

The PS topology was investigated by measuring the photoelectron intensity
distribution at Ep as a function of the momentum component parallel to the surface. The
resulting Ep emission intensity maps for the QC-CDW state of lT-TaSz is shown in
Figure 7-4. The intensities were normalized to the 21eV photon flux and are presented in
a linear gray scale with high intensities corresponding to white. The solid lines mark the
unreconstructed Brillouin zone (BZ). One sixth of the data were measured in each map
as shown by the dashed wedge in Figure 7-4; the rest were obtained by a mirror operation
about F-K, then a three-fold rotational symmetry operation. Note that there is a lack of

symmetry between the F—K line and F-K’ line at 60° with respect to it. This is partially

79

 

Figure 7-4: Fermi level intensity maps of lT-Tasz. The data between the

dashed wedge were actually measured and symmetrized to give the rest of the

Brillouin zone. Bright regions correspond to high intensity.
caused by the unequal matrix elements for the two lines since the photon polarization and
plane of incidence were both in the horizontal plane which contains F-K, and partially
caused by the finite escaping depth of the photoemission process“.

As can be seen, there is an extended high intensity region around the I‘ point,
which forms a clear border with the low intensity region in the rest of the B2. The
strength of intensity throughout the 1" area suggests there is some sort of band structure
near the Fermi level. The contour of bright region also hints at the shape of an elliptical

FS around the M points. This is consistent with predictions by the earlier band structure

calculations for the normal statels'”. However, the intensity does not extend to the outer

80

edge of the Brillouin zone as one would expect for a FS around M. Pillo et. al. 6‘ cited a
complete ‘remnant’ F S around the M point. In order to determine this structure, they
divided the intensities on each azimuthal scan, i.e., on concentric circles around I‘, by its
mean intensity to remove the overall polar intensity variation, which they considered an
artifact of ARPES. This procedure is rather unphysical, since there are intense features
seen at deep binding energies near the M point. The overall polar intensity variation is
real and is not an artifact, therefore should not be removed in this fashion. Alternatively,
the expected FS intensity could be removed by the QC-CDW or that state specific matrix
elements could be damping the intensity within a few eV of the Fermi level around M. A
detailed calculation of the element effects of the QC-CDW or the photoexcitation matrix

would clarify this discrepancy.
7.3 Energy Gap in the Commensurate CDW State

Figure 7-5 shows the EDCs measured along I‘-M in the unreconstructed BZ for
the C phase (T=20 K, solid lines). For comparison, the corresponding EDCs for the QC
phase (T=250 K) are shown as dashed lines. During the experiment, measurements were
repeated several times as the sample went through temperature cycles between 20 K and
250 K. The spectra were reproducible and no surface degradation was detected.
Measurements with small temperature increments between 20 K and 250 K showed that
the spectral change is correlated with the QC-CDW to C-CDW transition.

Two major spectral changes occur when the material undergoes the QC to C
phase transition. First, the spectral intensity at E; is nearly completely suppressed at all
k-points. This occurs in the entire BZ, as evidenced from EDC measurements taken on a

fine grid of points over the zone. This indicates that the entire F S is truncated and a k-

81

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Intensity (Arb. units)

 

 

 

 

 

 

 

 

 

 

 

0.5 0.4 0.3 0.2 0.1 0.0 or
Binding Energy (eV)

Figure 7-5: EDCs in the C-CDW state (T=20 K, solid curves), for the same
set of k-points along l"-M as in Figure 7-1 a). The EDCs in the QC-CDW
state are superimposed as the dashed curves for comparison.

82

independent energy gap (or pseudogap) opens at Ep in the C phase. To quantify the size
of the energy gap, the energy shift between the leading edge midpoints in the normal and
CDW state is used. From this, it is clear that the gap is on the magnitude of 120 meV.
Different methods of determining the gap might change the numbers slightly, but have
little other effect. Second, an intense peak develops at about 180 meV binding energy.
However, this band is flatter than the one seen in the QC-CDW phase. It does not
disperse in energy towards the Fermi level but simply loses intensity until it is
indistinguishable from background. This is very similar to the loss of intensity in the FS

seen around the M point in the QC-CDW phase.
7.4 Summary and Discussion

Previous ARPES experiments found some disagreement between the QC-CDW
phase and the calculated normal state band structures. For example, recently Pillo et. al.61
reported that there is a pseudogapped, remnant FS around the M point. Instead of
resolving clear Fermi crossings, they found peaks which only had thermal tails that just
reached Ep. In contrast, the data in Figure 7-1, Figure 7-2, and Figure 7-4 show that there
exists a F8 in the QC phase. There are two clear crossings distinguished along the F-M
line and another crossing along the F-K line. The second crossing along the F-M line is
part of the normal state FS predicted by previous, non-relativistic band structure
calculations”"7 that still exists in the QC-phase. This Fermi surface is harder to
distinguish at higher k-values, as seen in Figure 7-3, possibly due to some effects of the
QC-CDW or band dependent matrix element effects.

The PS determined from these measurements is slightly different from the

previous band calculations, since there is an apparent small electron pocket near the 1"

83

point. This result is consistent with those of many other ARPES studies43’5560, though.

Several groups‘""56'57

previously saw in the QC-CDW state a set of bands just below the
Fermi level near F. To understand this feature, new band calculations were performed by
Larson and Mahantil9 including relativistic and spin orbit corrections. The inclusion of
the spin orbit corrections makes one small, but crucial change as can be seen by
comparing Figure 7-3 a) and b). The spin orbit corrections pull a Ta 5d-band just below
Ep in a region near the 1" point. This matches the additional crossing seen in these
measurements and implied by the other ARPES experiments.

In the C phase, the entire F S is truncated and the top most band is pushed down
to approximately 180 meV below By. This suppresses the intensity within 120 meV of
the Fermi level and it occurs across the entire Brillouin zone, indicating a k-independent
energy gap. These spectral changes have been seen previously at normal emission”60
and it was shown that they occur within a few degrees at the transitions 9. The dramatic
spectral changes occurring at the QC to C phase transition are the results of a CDW-
induced Mott transition as suggested by Fazekas and Tosatti”. As they pointed out, the
resistivity in the C phase is higher than the two-dimensional metallic limit, indicating that
the material has become a Mott insulator. There are two arguments that this gap in the C-
CDW phase indicates a Mott insulator. First, the gap is entirely k-independent. A purely
CDW nesting gap should only affect states near the Fermi surface regions that are nested.
Second, the size of the gap is 120 meV, which is larger than the expected gap size of 75
meV estimated using the BCS formula.

In summary, using high energy and high angular resolution ARPES, a F S in the

QC-CDW phase of 1T-TaSz was observed. Its shape was quite comparable with that of

84

15-17 and is

the normal state F S predicted by previous band structure calculations
consistent with previous ARPES measurements. A small electron pocket was found
around the F point, though, that was not previously determined. The observed Fermi
crossing near the F point is consistent with new band structure calculations that include

spin orbit corrections. Under the QC to C-CDW transition, a large, k-independent gap

resulting from a Mott insulator transition is resolved.

85

Chapter 8 lT-TiSez

The electrical and CDW properties of lT-TiSez have been subjects of interest for
years. There has been an outstanding question of whether lT-TiSez is a semimetal or
semiconductor. ARPES experiments39’65"56'67’68’69’m'7l'72'73 have been performed to
address this issue but the results are inconsistent. The CDW properties have also been
studied by ARPES. However, all but one recent experiment74 had rather poor energy and
momentum resolutions. In this study, high resolution ARPES studies of lT-TiSez were

performed in both the normal and CDW states.
8.1 Band Dispersions

As shown in Chapter 4, unlike the other TMDs studied here, lT-TiSez has a rather
three-dimensional like electronic structure in spite of its quasi-two-dimensional layered
morphology. Therefore, a three-dimensional band mapping is necessary. In these
experiments, the free-electron approximation for the states was used to estimate the k;-
components of the initial electron state. The three-dimensional band dispersions were
mapped by varying the photon energy and the analyzer angle simultaneously, as
described in Chapter 2.

Figure 8-1 shows the EDCs for k-points with parallel components along the
directions of a) 1"-M and b) F—K at 21 eV photon energy. They were taken in the normal
state for lT-TiSez at room temperature. Spectra along both lines show a double peak
feature at the Fermi level right next to the F point. These bands disperse quickly away
from Ep to deeper binding energies. This indicates that the Fermi surface around T‘ is a
hole pocket, since it surrounds a region of unoccupied bands. Along the F-M direction,

there is a second feature that appears at Ep near the M point. Since no band disperses to

86

 

 

l

a) 1 F-M ' lku=0.00 (IA-:1 b) F—K I lky=0.00 (A-
NormalState k,=1.11(A') “in???” _ lee-1.14m"

&’__‘.’——

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fl ": its _ ,8)
4 r , i
0.9 - _ 075'
A :-
£1 __- A“ . a T 002]
D 1"" - 0.7 D ~. - - -:

' ‘ - A '07 - "w :Zl
"E .. WE . "
V __ _-‘ I034 V \~ 047'
.g l0.51| 4:” ’ 040'

- - e- m .

e .. a i
H r 8 0.33
3 x1, I _
a 0.26
. . 0.18
1% V‘om

I L I m l '0'04

 

 

 

, 7 L I
2.0 1.5 1.0 0.5 0.0 2.0 1.5 1.0 0.5 0.0
Binding Energy (eV) Binding Energy (eV)

Figure 8-1: Normal state EDCs for the projections along a) F-M and b)
F-K lines of lT-TiSez.

deeper binding energy from this feature, it must disperse down from the unoccupied
states instead. This makes the Fermi surface pocket out near the M point an electron

pocket, since it encloses occupied electron bands.

87

EDCs taken along the normal emission direction which correspond to k-points

along F-A are shown in Figure 8-2 a). For the normal emission spectra, the electrons

have no parallel k—component so that by varying the photon energy, one directly probes

 

 

 

Intensity (Arb. units)

 

 

 

 

 

 

 

 

   

 

  

l l 1.00 F. W l l l
0.92 “I" (104073)
0.83 “~—
11 (0990.65)
0.78 '
(1020.55)
0.74 N
”a?
0.69 ,4: (100,051)
. a
e \
:5 L
a (104,044)
0.55 \ v A\
00.5 3‘
'71
Q4 5 (1020.40)
a \
0'39 (1.05033)
0.34 A
(102,029)
0.29
0.23
101,018

 

 

 

 

 

 

I l l I l I l l
2.0 . 1.5 1.0 0.5 0.0 2.0 . 1.5 1.0 0.5 0.0M
Bmdrng Energy (eV) Bmdrng Energy (eV)

Figure 8-2: EDCs along the normal emission directions of a) 1"-A and b) M-L
of lT-TiSez. The EDCs of a) are labeled with the k. (A") distance from r.
The EDCs of b) are labeled with their (k,, k.) A" values.

88

 

 

 

 

 

the krdispersion. As can be seen, there are three bands below the Fermi level within 2
eV. The first band disperses from above Ep to below E: at k2 = 0.18 A". It disperses to
about 2eV below E1: and then disperses back to By at higher kfpoints. The large amount
of dispersion demonstrates that the electronic structure of lT-TiSe; is quite three-
dimensional. The second hand appears around IQ = 0.23 A" and disperses down to a
binding energy of ~0.5 eV. The third feature appears as a shoulder between the second
band and the Fermi level. It becomes evident near k2 = 0.29-0.34 A" and remains within
0.1 eV of BF at all times. By careful consideration of the symmetry of the dispersion

along the F-A line, an estimation of the inner potential can be made. The kz-values

labeled in Figure 8-2 were determined from this estimation. The A point is estimated at
Ie = 0.50 A".

Figure 8-2 b) shows EDCs approximately along M-L. The k-values were selected
by varying the analyzer angle and the photon energy simultaneously. There is a feature at
deep binding energy. It disperses from well below 2 eV binding energy to a minimum
around 1.8 eV binding energy at the L point and then disperses to deeper binding energies
again. This further demonstrates the three dimensional nature of the band dispersion.
There is a second feature that appears as a small shoulder on this first band that seems to
move with it. Finally, there is a feature near the Fermi level that only shows small
dispersions. This feature is a very intense and sharp at (1.02, 0.29) and several other
crossings as well without any bands dispersing much below the Fermi level. This

suggests that this is a small electron pocket from some band above the Fermi level.

8.2 Fermi Surfaces

The PS topology was investigated by measuring the photoelectron intensity

89

distribution at E}: at a constant photon energy of 21 eV. The resulting Ep emission
intensity map for the normal state of lT-TiSez is shown in Figure 8-3, labeled with the
parallel k-components of the momentum. Because of the three-dimensional characteristic
of lT-TiSez, the change in the kz-component must also be considered. The intensity map
is effectively over a parabolic surface with a foci at (O, 0) upon the A point and the edges
of the hexagon roughly halfway between L and M.

-1.3

 

-1.1—
-0.9—
-0.7—
-0.5—
-O.3—
>-0.1—
x
0.1—

 

~H

1-3 l | I 1 1 T l l 1 1 I l

-1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3
Kx

Figure 8-3: Fermi level intensity maps of 1T-TiSe2 taken with 21eV photons.

The data between the dashed lines were actually measured and rotated to give
the rest of the Brillouin zone.

 

 

 

One third of the data were measured as shown by the dashed wedge in Figure 8-3;
the rest were obtained by a three-fold rotational symmetry operation. Note that there is a

lack of symmetry between the A-L line and A-L’ line at 60° with respect to it. There are

90

two causes of this distinction. First, there are unequal matrix elements for the two lines
since the photon polarization and plane of incidence were both in the horizontal plane of
AHKI‘. Second, the finite escaping depth of the photoemission processs ' may not
include the effective sources of the two lines equally. This intensity map shows that there

is some spectral intensity near the zone center around the F—A and near the zone edge

around M-L within the energy window of the analyzer. According to band calculations,
the F S of lT-TiSez consists of two small ellipsoids, one around F and one around L.
Although the intensity map is qualitatively consistent with this picture, the Ig-dependence
of the FS needs to be determined more closely by EDCs.

To determine the Fermi crossings along the k,-direction, EDCs were taken on a
grid of points around the F-A and L-M lines in the normal state at room temperature.
Figure 8-4 a) shows some EDCs near the L-M line. As can be seen, there are actually
peaks above the Fermi level. It is not clear that the band actually disperses below the
Fermi level at all, because the leading edge is significantly above E: in all cases. In
contrast, the bands near the F—A line, shown in Figure 84 b), do not actually seem to
reach the Fermi level, because the Fermi level only intersects the spectra at the thermal
tails of the peaks. It appears as if there is a pseudogap in the F—A region, with the
spectral intensity leading edge midpoint at a binding energy of 80 meV. The results were
repeatedly found on different samples. Unfortunately, the locations of Fermi crossings
are not apparent in these EDCs. Therefore, the exact FS shape along the k,-direction is
still not resolved.

In principle, ARPES is not able to probe the unoccupied states above the Fermi

level. In this material, there is a combination of unique factors that allow the unoccupied

91

 

1 r 1 1
a) ho): 23eV

(kx : k Z )=
118,038)

 

(1.11,0.4l) F a

(1.04.0.44)

Intensity (Arb. Units)

 

  

 

 

 

 

l l l I
0.3 0.2 0.1 0.0 -O.1 -0.2

 

 

 

 

 

 

 

 

 

Binding Energy (eV)
b)l ' ' hw='26ev
'E (kx. kz)=
D (0.17,0.78)
£3 - -
i
2? (008,078)
.g _ --
O.)
E
(000,078)
I I I h:—-l------—-m
0.3 0.2 0.1 0.0 -0.1 -0.2
Binding Energy (eV)

Figure 8-4: EDCs taken near apparent Fermi crossing in the normal state of
lT-TiSez near a) the L point and b) midway between A and 1'”. The (kx, kz)

points are in A".

states to be probed. The spectral leading edge along the L-M line has a midpoint 65 meV

92

above Ep. This location indicates that the band is above the Fermi level and is detected
only because of thermally excited electrons. These features never cross the Fermi level,
but simply come within a minimal distance of the Fermi level near the L point. The
actual peak position may not be directly indicative of the band location, since the spectral
shape is distorted by the Fermi function.

In order for ARPES to yield this sort of information, there are three requirements
which must be met. First, the peak that is above the Fermi level has to be narrow and
strong. The peak has to be intrinsically less than one half the width of its nearest
approach. Otherwise, the observed peak will have enough intensity on the occupied side
of the Fermi level that it will dominate and the peak position will appear below the Fermi
level as normally seen. Titanium 3d-bands fulfill these requirements. Second, there has
to be limited momentum resolution. If the peak disperses much and there is poor
momentum resolution, the apparent position will be smeared out by the peak’s own
dispersion. Third, the Fermi function has to be rather broad. This condition is easily met
by high temperatures, since the system temperature broadens the Fermi function. At
room temperature, this thermal broadening gives an effective 4.2kBT width of 110 meV.
The observation of photoemission intensity above Ep in this material was also reported by
Pillo et. al.74 at the time these data were taken and has been seen in other Ti compounds

as well”.

8.3 Energy Gap in the CDW State

The formation of a CDW usually opens an energy gap at the Fermi energy and
truncates parts of the FS that are connected by the CDW modulation’s wavevector. In

1 T—TiSez, there is only limited evidence of there being a Fermi surface, though. Instead,

93

 

 

there are a set of Se bands just below El: around I‘ and a set of Ti bands above By

around the L points. There is a vector Q0 which is consistent with the observed CDW

wavevector that would connect these respective electron and hole states into an exciton.
This suggests that lT-TiSez has an excitonic insulator transition driving the CDW
formation, as suggested by Wilson and Mahaj an76‘24.

The property of the energy gap was investigated by measuring spectra near the L
and I“ points above and below the CDW transition temperature. During theexperiment,
measurements were repeated on several different samples and through temperature cycles
between 20 K and 240 K. The spectra were reproducible and no surface degradation was
detected. The EDCs are plotted in Figure 8-5. Strictly speaking, a gap can be defined
only on a Fermi surface. Here, we also use the term “gap” for the pseudogap region
around the F point and the near crossings of the L point. This is justified since there are
states near E}: at those k-points and they are affected by the CDW formation.

The spectra in Figure 8-5 a) show a dramatic change as a function of temperature.
The single broad peak in the normal state is split into two peaks in the CDW state. At
20K, the spectral leading edge is pushed toward higher binding energy by 55 meV by the
formation of the CDW. However, there is still significant spectral intensity at the Fermi
level. It appears as if the band is lowered and now crosses Be. In addition, another
intense peak appears at approximately a binding energy of 180 meV. This dramatic
restructuring of bands might be due to the superlattice of the CDW state, which can cause
band folding.

As can be seen in Figure 8-5 b), the spectral change near the F-point is also

significant, although not as dramatic. There, the spectral intensity near EF is suppressed

94

 

 

 

Intensity (Arb. Units)

‘‘‘‘‘

 
   

 

Q
1".

 

hv = 23e‘}
-— CDW State
.;,---- Normal State

I on.“ (kx , kZ )=

211,180.38)

|
s
\
\
a
\
Q
a
T.
s“
. .....

“a

(111,041)

0
‘0
‘0
s
C
'\
.
§
‘‘‘‘‘

' (104,044)

 

.
‘.
¥ ‘~
Q
L 24--A.

 

I l
0.3 0.2 0.1 0.0 -O.1 -O.2

Binding Energy (eV)

 

hv = 26e\'/
— CDW State
----- Normal State

(kx. k z )=
(0.17.0.78)

Uh-_-— _ A

 

(0.08078)

0
-------- A ‘

 

 

Intensity (Arb. Units)

 
   

0

 

(0.00,0.78)

 

A

 

1 I
0.3 0.2 0.1

___________ 1
0.0 —O.1 -O.2

Binding Energy (eV)

Figure 8-5: The temperature dependence in lT-TiSez of bands close in energy
to the Fermi level near a) the L-M line and b) the A point. The normal state

EDCs are shown in dashed lines and the CDW state in solid lines.

95

 

 

and the leading edge is shifted toward higher binding energy by 85 meV. It appears that
the pseudogap is significantly widened by the CDW. There is no peak splitting evident in

these bands.
8.4 Summary

In summary, the normal state band dispersions of lT-TiSez was investigated. It
was seen that the electronic structure of this material is quite three-dimensional, showing
significant dispersions in the direction perpendicular to the layers. Both the EDCs and
the EF intensity map show emission intensities near E1: in the regions around the F—A and
M-L, qualitatively consistent with the FS predicted by the band structure calculations.
However, the high resolution EDC spectra show that, the band is actually located
approximately 65 meV above E; at the L point. At the F point, the band is located
approximately 80 meV below Ep. Therefore, lT-TiSe2 is actually a semiconductor with
an indirect gap of ~145 meV. In the CDW state, bands in both regions are dramatically
affected. There is still intensity at E: as the band near the L point is pulled down to cross
Ep. This provides carriers for the CDW state to remain in a semimetallic phase with a
resistivity comparable to its semiconducting normal phase.

The normal state band structure and the CDW’s dramatic effects upon it support
the excitonic-insulator modelm’24 of the CDW mechanism for this material. The band
near 1" is the top of the valence band and provides electron carriers. In contrast, the band
near L is the bottom of the conduction band and consists of hole carriers. The two

regions also directly connect with the CDW wavevector.

96

 

 

Chapter 9 Summary

In summary, high resolution ARPES was used to study the electronic structure of
the transition metal dichalcogenides 2H-TaSe2, 2H-TaSz, 2H-NbSe2, 1T-TaS2, and 1T-
TiSez. Their normal state band dispersions and Fermi surfaces, and the CDW energy gap
properties were investigated in detail. A number of new spectral properties in these
materials were discovered.

Although 2H-TaSe2 is one of the first materials to be studied by ARPES, a
detailed study of the Fermi surface and CDW gap property was lacking until the present
study. Early studies focussed on band dispersions below EFS 1. Some of the more recent
studies addressed spectral changes in the CDW state43’44. However, they suffered either
from insufficient energy resolution or from insufficient momentum resolution. The new
findings in this study include: 1) An extended saddle band very close to the Fermi level
was observed along FK. 2) The Fermi surface was determined for the first time. It
consists of a pocket around 1" and another pocket around K, qualitatively consistent with
band structure calculations. 3) It was found that there is no gapping on the Fermi
cylinder around I‘ in the CDW state, indicating that this Fermi cylinder does not
contribute to the CDW mechanism. A gap of 18 meV was observed in the saddle band
region and on the Fermi surface around K. These results suggest that the CDW in the
material originates from both the partial nesting of the Fermi surfaces around K and the
saddle band along F-K as proposed by Rice and Scott”. 4) The temperature dependence
of the CDW gap indicates a deviation from the BCS relation, suggesting that the weak
coupling theory is not sufficient to describe the CDW in this material. Two other closely

related compounds of 2H-TaSz and 2H-NbSe2 were also studied. Although the results are

97

 

not as complete as those for 2H-TaSe2 due to experimental or material limitations, they
are consistent with the conclusions made for the 2H-TaSe2 data.

For lT-TaSz, its electronic properties near the Fermi energy in both the QC-CDW
and C-CDW phases have been studied by ARPES very extensively43'55'61. It was
observed that the spectral shape near Ep undergoes dramatic changes at the QC- to C-
CDW transition. The emission intensity at Ep is nearly completely suppressed and a
strong peak appears at approximately 180 meV binding energy in the C-CDW phase.
These changes occur within a few degrees of the transition”. These measurements were
made at normal emission with poor angular resolution of i5°. The k-dependence of the
spectral changes was not investigated. It was also observed that the bands collapse into
three subbands separated by gaps in the C-CDW43. The new findings in the present study
include: 1) In the QC-CDW phase, the EDCs and Fermi intensity map indicate significant
emission intensity near the F point and there is a Fermi pocket near I" in addition to the
large one around M. This result is inconsistent with the early non-self-consistent non-

relativistic band structure calculations”17

which predict only the Fermi surface pocket
around M. Although this discrepancy was noted in previous ARPES experimentss 1, it
was not investigated in detail. From recent self-consistent band structure calculations by
Larson and Mahanti”, it was determined that the emission intensity and the Fermi
crossing near G was caused by spin-orbit interactions which were not included in
previous calculations. 2) By measuring EDCs in both the QC-CDW and C-CDW phases

over a large portion of the Brillouin zone, it was determined that a k-independent gap of

120 meV opened. This gap opens at the QC- to C-CDW transition throughout the

98

 

 

 

Brillouin zone. This supports the argument for a Mott localization occurring at the QC-
CDW to C-CDW transition”.

For lT-TiSez, previous ARPES studies with moderate energy and moderate
angular resolutions found that the band structure near Ep near the L point is significantly
affected by the CDW transition”. There also existed a controversy on whether this

material is a semimetal or small gap semiconductor‘s‘s’73

. The new findings in the present
study include: 1) Significant band dispersions in the direction perpendicular to the layers
were observed, indicating a strong three-dimensional characteristic despite its layered
structure. 2) It was observed that in the normal state a band stays slightly below Ep near
the F point and another band stays slightly above Ep near the L point. The latter was
observed only because of thermal population of the band. This suggests that lT-TiSez is
a narrow gap semiconductor in the normal phase. 3) In the CDW state, the band near F is
pushed to deeper binding energy and the band near L is lowered below E; with an
additional band appearing at higher binding energy. The lowering of the band near L to
the Fermi level indicates that the material becomes a semimetal in the CDW state. The
fact that both bands are affected by the CDW supports the excitonic insulator CDW
mechanism”. Similar results were reported in a paper by Pillo et. al.74 at the time these
measurements were made.

In conclusion, the present high resolution ARPES studies of these materials have
provided more details about the electronic structures relevant to the CDW formation.

Some spectral phenomena are still not well understood and require fiu'ther theoretical and

experimental investigation.

99

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