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First-principles Study of Charge Density Waves andElectron-phonon Coupling in Transition Metal Dichalcogenides, and
Magnetism of Surface Adatoms
A Dissertationsubmitted to the Faculty of the
Graduate School of Arts and Sciencesof Georgetown University
in partial fulfillment of the requirements for thedegree of
Doctor of Philosophyin Physics
By
Oliver Ruben Albertini, B.A.
Washington, DCAugust 11, 2017
Copyright c© 2017 by Oliver Ruben AlbertiniAll Rights Reserved
ii
First-principles Study of Charge Density Waves andElectron-phonon Coupling in Transition Metal
Dichalcogenides, and Magnetism of Surface Adatoms
Oliver Ruben Albertini, B.A.
Dissertation Advisor: Amy Y. Liu, Ph.D.
Abstract
Recently, low-dimensional materials have attracted great attention, not only
because of novel physics, but also because of potential applications in electronic
devices, where performance drives innovation. This thesis presents theoretical and
computational studies of magnetic adatoms on surfaces and of charge density wave
(CDW) phases in two-dimensional materials.
We investigate the structural, electronic, and magnetic properties of Ni adatoms
on an MgO/Ag(100) surface. Using density functional theory, we find that strong
bonding is detrimental to the magnetic moment of an atom adsorbed on a surface.
Previously, it was shown that Co retains its full gas-phase magnetic moment on the
MgO/Ag(100) surface. For Ni we find the total value of spin depends strongly on
the binding site, and there are two sites close in energy.
Next, we study the 1T polymorph of the transition metal dichalcogenide TaS2.
Bulk 1T -TaS2 is metallic at high temperatures, but adopts an insulating
commensurate CDW below ∼ 180 K. The nature of the transition is under debate,
and it is unclear whether the transition persists down to a monolayer. Transport
and Raman measurements suggest that the commensurate CDW is absent in
samples thinner than ∼ 10 layers. Our theoretical and experimental study of the
vibrational properties of 1T -TaS2 demonstrates that the commensurate CDW is
robust upon thinning—even down to a single layer.
iii
Bulk 2H-TaS2 also has a CDW instability at low temperatures. A recent study
of a single layer of 1H-TaS2 grown on Au(111) revealed that the CDW phase is
suppressed down to temperatures well below the bulk transition temperature (75
K), possibly due to electron doping from the substrate. We present a first-principles
study of the effects of electron doping on the CDW instability in monolayer
1H-TaS2, finding that electron doping: 1) shifts the electronic bands by ∼ 0.1 eV; 2)
impacts the strength and wave vector dependence of electron-phonon coupling in
the system, suppressing the CDW instability. Furthermore, our work identifies
electron-phonon coupling—not peaks in the real or imaginary parts of the electronic
susceptibility—as the main cause of CDW instabilities in real 2D systems.
Index words: density functional theory, charge density wave,electron-phonon coupling, transition metal dichalcogenides,electron doping, magnetism, TaS2, theses (academic)
iv
Acknowledgments
I thank my advisors, Amy Liu and Jim Freericks for their consistent support and
dedication. Luckily, my advisors are also two of my favorite teachers (one of them
since I was an undergraduate) and this made the research process much more
enjoyable.
Amy, in particular, spent a lot of time helping me develop as a physicist and
researcher, and for her Herculean efforts, I am so grateful. Thanks to her, I have a
new appreciation for the scientific process and how to collaborate with others in a
meaningful way. And because of her thoughtful guidance, I have a thesis and body
of work to really be proud of, and a future to really look forward to.
I also thank the rest of my dissertation committee: Miklos Kertesz, Makarand
Paranjape, and Barbara Jones for their insightful questions and comments. I thank
Barbara for hosting me at IBM in 2013; it was a wonderful experience I will always
remember fondly.
This is a short list, in no particular order, of people I had the pleasure of
meeting and working with as a student at Georgetown: Chen Zhao, Andre De
Vincenz, Jesús Cruz-Rojas, Woonki Chung, Shruba Gangopadhyay, Jasper Nijdam,
Wes Mathews, Bryce Yoshimura, Lydia Chiao-Yap, Alex Jacobi, Matteo Calandra,
Paola Barbara, Pasha Tabatabai, Tingting Li, Jennifer Liang, David Egolf, Leon
Der, Ed Van Keuren, Xiaowan Zheng, Joe Serene, James Lavine, Amy Hicks, Rui
Zhao, Peize Han, Josh Robinson, Baoming Tang, Mary Rashid, Max
Lefcochilos-Fogelquist, Janet Gibson, Sona Najafi, and Mark Esrick.
v
Finally, I thank my family–LeRoy Jr., Sonia, LeRoy III, and Graciela
Albertini–and girlfriend, Sheryl Lun, who supported me throughout these last few
years. This was a challenge for all of us, and they share my joy in completing this
great chapter of my life.
vi
Table of Contents
Chapter1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Magnetic Adatoms on Surfaces . . . . . . . . . . . . . . . . . . . 21.2 Two Dimensional Materials . . . . . . . . . . . . . . . . . . . . . 4
2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1 Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . 212.2 Hohenberg-Kohn Theorems . . . . . . . . . . . . . . . . . . . . . 222.3 Kohn-Sham Formulation . . . . . . . . . . . . . . . . . . . . . . 232.4 Exchange and Correlation . . . . . . . . . . . . . . . . . . . . . 262.5 Implementation: Basis Sets . . . . . . . . . . . . . . . . . . . . . 272.6 Lattice Dynamics in DFT . . . . . . . . . . . . . . . . . . . . . 31
3 Site-dependent Magnetism of Ni Adatoms on MgO/Ag(001) . . . . . . 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Zone-center Phonons of Bulk, Few-layer, and Monolayer 1T -TaS2: Detec-tion of the Commensurate Charge Density Wave Phase Through RamanScattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Description of Crystal Structures . . . . . . . . . . . . . . . . . 564.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 594.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5 Effect of Electron Doping on Lattice Instabilities of Single-layer 1H-TaS2 735.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.3 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . . 775.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
vii
List of Figures
1.1 Periodic table showing the ingredients for transition metal dichalco-
genides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Two types of coordination geometry for transition metal dichalcogenides. 7
1.3 Impact of coordination geometry on electronic bandstructure. . . . . 8
1.4 Charge density wave phases of 1T -TaS2. . . . . . . . . . . . . . . . . 12
1.5 1T -TaS2 charge density wave viewed from above. . . . . . . . . . . . 14
1.6 An atomic displacement pattern for 1H-TaS2. . . . . . . . . . . . . . 17
1.7 Another atomic displacement pattern for 1H-TaS2. . . . . . . . . . . 18
2.1 Simplified DFT work flow. . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Diagram depicting the augmented plane wave scheme. . . . . . . . . . 29
3.1 Top view of the three binding sites for Ni on MgO/Ag(100). . . . . . 41
3.2 Total energy per Ni adatom on MgO/Ag, calculated for structures
relaxed within GGA and GGA+U (U = 4 eV) from PAW calculations. 42
3.3 Densities of states from inside the muffin-tins of all-electron LAPW
calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Spin density isosurfaces of Ni adsorbed on MgO/Ag, from GGA+U
all-electron LAPW calculations. . . . . . . . . . . . . . . . . . . . . . 49
4.1 An illustration of the Ta plane and Brillouin zone reconstruction under
the lattice distortion of the C phase. . . . . . . . . . . . . . . . . . . 58
4.2 Optical modes of 1T bulk and monolayer structures. . . . . . . . . . . 60
4.3 Calculated phonon spectra of undistorted 1T -TaS2. . . . . . . . . . . 61
viii
4.4 Simplified illustration of the low-frequency phonon modes of a 1T bilayer. 62
4.5 Raman spectra of bulk (> 300 nm) and thin (14.6 nm, 13.1 nm, 5.6
nm, and 0.6 nm) samples, measured at 250 K (NC phase) and 80 K (C
phase). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6 Comparison of experimental and calculated Raman frequencies for the
bulk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.7 Comparison of experimental and calculated Raman frequencies for a
single layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.8 Measured polarized Raman spectra of the C phase for (top) a mono-
layer and (bottom) a 120 nm sample measured at 100 K. . . . . . . . 69
5.1 Electronic bands of single-layer 1H-TaS2. . . . . . . . . . . . . . . . . 79
5.2 Phonon dispersions of single-layer 1H-TaS2 with and without spin-
orbit coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Phonon dispersions of single-layer 1H-TaS2 with and without doping
and lattice constant relaxation. . . . . . . . . . . . . . . . . . . . . . 82
5.4 Brillouin-zone maps of the phonon linewidth [(a-c)] and the geometric
Fermi-surface nesting function [(d-f)]. . . . . . . . . . . . . . . . . . . 84
5.5 Momentum dependence of the square of the self-consistent (ω2q) and
bare (Ω2q) phonon energies for the branch with instabilities, self-energy
correction (2ωqΠq) for that branch, and real part of the bare suscepti-
bility (χ′0(q)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
ix
List of Tables
3.1 Relaxed distance between adatom and nearest-neighbor Mg and O
atoms, and height of these atoms above the MgO ML. . . . . . . . . . 43
3.2 Total magnetization per adatom (in µB) for different sites, calculated
within GGA and GGA+U , from all-electron LAPW calculations. . . 45
3.3 Net charge of Ni atom and its nearest-neighbor O and Mg atoms
according to Bader analysis. . . . . . . . . . . . . . . . . . . . . . . . 46
x
Chapter 1
Introduction
The physics of low-dimensional materials generates great curiosity because the
behavior of electrons in confined environments is fascinating and beautiful. And the
prospect of using these materials in real-world applications is becoming ever more
desirable and achievable.
The miniaturization of computer components, whether microchip transistors or
magnetic sectors of hard drives has led to tremendous advances. But recently, the
performance of transistors (power consumption and speed) has not kept up with
miniaturization. The search for materials that perform better has led us to
two-dimensional (2D) and even one-dimensional (1D) systems.
Materials like transition metal dichalcogenides are quasi 2D in their bulk form,
and often take on new and interesting properties in the few-layer limit. Even the
bandstructure of graphene opens up a gap when it is reduced to thin strips, or
nanoribbons. If we can harness these new properties, it may lead to flexible,
transparent, faster, and more efficient computing devices.
Taking a bottom-up approach, scientists have also developed techniques to
arrange individual atoms on a surface, working in the ultimate limit of
miniaturization and low-dimensionality. It has been shown, for example, that just a
single magnetic Ho atom can hold information for hours [1]. Besides exotic
information storage, research like this brings us closer to spin-based logic gates [2]
and even the realm of quantum computing.
1
1.1 Magnetic Adatoms on Surfaces
As part of the degree requirements of the Industrial Leadership in Physics (ILP)
track of the Ph.D. program of the Georgetown University physics department, I did
a year-long internship at the IBM Almaden Research Center in San Jose, CA. The
idea behind the ILP program is to expose students not just to academia, but also
industry, where there are many promising careers for scientists.
From January 2013 to January 2014, I worked at IBM under the guidance of
Barbara Jones and her postdoctoral fellow Shruba Gangopadhyay. We collaborated
with the scanning tunneling microscope (STM) team there, led by Andreas
Heinrich. Doing density functional theory (DFT) calculations, we provided the STM
team with theoretical understanding about the properties of transition metal atoms
adsorbed to a substrate.
The story begins in 1989, when Don Eigler made an STM under vacuum and
ultra-cold temperatures (≤ 4 K) at Almaden. He used this STM to manipulate the
positions of atoms and molecules on a substrate. This began a long tradition of
famous experiments: in 1993, the creation of a quantum corral of Fe atoms which
beautifully reflected electronic states of the Cu substrate [3]; in 2012, the formation
of bistable ‘bits’ of 12 Fe atoms, arranged antiferromagnetically, which held their
spin arrangements for hours [4]; and even the world’s first atomic-scale animation in
2013 [5].
Part of the Almaden STM team’s motivation is to approach magnetic storage
from the bottom up; the smallest possible magnet is the atom. Mastery of the
physics of magnetic atoms on a surface may launch us past the next hurdles in
magnetic storage and other areas, since stable quantum magnets have potential
computing applications. Furthermore, the basic science of magnetism in reduced
2
dimensionalities (e.g. surface magnetism) mixes elements of quantum and classical
magnetism, helping to bridge our scientific understanding.
In recent years, the team has focused on magnetic adatoms adsorbed onto a few
atomic layers of an insulating material such as Cu2N or MgO. They studied
adatoms from almost the entire 3d transition metal row, including Mn [6, 7], Cu [8],
Ti [9], Fe [4], Co [10], and finally Ni [11].
For a magnetic atom on a surface, magnetism must compete with the tendency
for bonding. For an adatom to exhibit some level of magnetism, the bonding
orbitals must not become too diffuse or overpower the energy benefit of a magnetic
configuration. While at Almaden, I worked with the STM team to understand the
case of Co adsorbed onto MgO/Ag(001). Remarkably, the adatom preserves almost
all of its spin and orbital angular momentum from the gas phase (Lz = 3 and
Sz = 3/2). We showed that Co bonds to an O site of the MgO surface, where the
ligand field is approximately axial (C∞), preserving the energetic ordering of the Co
3d orbitals. Both the bonding and charge transfer are weak, which explains why the
orbital and spin moments are preserved. The experiments also reported a strong
magnetic anisotropy for Co on MgO/Ag, which is a critical property for nanoscale
magnets [10].
In contrast, through DFT studies, we find that Ni, which sits just one column
over from Co in the periodic table, forms a strong bond at the O site and loses its
gas phase spin moment (Sz = 1) altogether [11]. This study is presented in Chapter
3 of this thesis.
3
1.2 Two Dimensional Materials
While every student of quantum mechanics can recall simple low dimensional
versions of the simple harmonic oscillator and quantum well, our understanding of
the behavior of real electrons mostly centered on three-dimensional (3D) systems. In
condensed matter, electrons generally occupy an environment where they are
surrounded by matter in each direction. Sometimes theoretical physicists would
study a two dimensional version of a material, as P. R. Wallace did in 1947 when he
calculated the bandstructure of single honeycomb sheets of carbon atoms to better
understand the properties of graphite [12].
Presumably, single layers of graphite have been made by students with pencils
for centuries. The first time that students intentionally made a sheet of graphene
was by mechanically exfoliating graphite with Scotch tape [13]. Graphene has many
interesting and useful properties including inertness, transparency, extreme tensile
strength, and high electron mobility. The exfoliation and experimental
characterization of graphene led not only to a Nobel Prize in 2010, but also an
explosion in research on 2D materials.
The exfoliation and, later, growth techniques which allowed the production of
graphene sheets have also been used to synthesize other 2D materials, including
transition metal dichalcogenides, hexagonal boron nitride, black phosphorous, and
more. Like graphite, many of these were already well-known as layered bulk
materials with weak van der Waals interlayer interactions.
We now know that there are new properties, advantages, and disadvantages in
2D. For Wallace, one consequence was that the pz orbitals of carbon did not interact
with neighbors outside the plane. These half-occupied pz orbitals give rise to a Dirac
4
cone at the Fermi level, which is the origin of many of the exotic conduction
properties of graphene.
More generally, thinned-down layered materials have reduced screening
compared to the bulk, due to the absence of neighboring layers. This leads to
stronger Coulomb interactions, and strong exciton binding [14]. Many materials also
have a fundamentally different symmetry in their single- and few-layer forms, and in
particular the lack of inversion symmetry gives rise to new properties. Finally, a thin
flake of material is more sensitive to its environment than its bulk counterpart. This
can prove detrimental to experiment since in ambient conditions, samples can
quickly degrade, altering the properties of interest. On the other hand, an upside to
this environmental sensitivity is the possibility of using these materials for chemical
sensing applications. Similarly, these materials are sensitive to their substrate,
making it possible to enhance desirable properties or suppress undesirable ones by
choosing the appropriate substrate.
1.2.1 Transition Metal Dichalcogenides
Because of their weak interlayer interaction, transition metal dichalcogenides
(TMDs) like MoS2 have traditionally been used as dry lubricants. Now, since these
materials can be exfoliated and synthesized in monolayer and few-layer form, they
are sought for their more interesting and diverse electronic and chemical properties,
as well as their potential for small feature height (typical monolayer thickness is
0.6-0.7 nm).
They have the general chemical formula MX2, where M is a transition metal
(TM) and X is a chalcogen (S, Se, Te). The structure of a TMD consists of a
triangular lattice of M atoms, sandwiched between two layers of X atoms (here we
will refer to this trilayer as a monolayer). In the bulk, these layers are stacked.
5
1
HHydrogen
3
LiLithium
11
NaSodium
19
KPotassium
37
RbRubidium
55
CsCaesium
87
FrFrancium
4
BeBeryllium
12
MgMagnesium
20
CaCalcium
38
SrStrontium
56
BaBarium
88
RaRadium
21
ScScandium
39
YYttrium
57-71
La-LuLanthanide
89-103
Ac-LrActinide
22
TiTitanium
40
ZrZirconium
72
HfHafnium
104
RfRutherfordium
23
VVanadium
41
NbNiobium
73
TaTantalum
105
DbDubnium
24
CrChromium
42
MoMolybdenum
74
WTungsten
106
SgSeaborgium
25
MnManganese
43
TcTechnetium
75
ReRhenium
107
BhBohrium
26
FeIron
44
RuRuthenium
76
OsOsmium
108
HsHassium
27
CoCobalt
45
RhRhodium
77
IrIridium
109
MtMeitnerium
28
NiNickel
46
PdPalladium
78
PtPlatinum
110
DsDarmstadtium
29
CuCopper
47
AgSilver
79
AuGold
111
RgRoentgenium
30
ZnZinc
48
CdCadmium
80
HgMercury
112
UubUnunbium
31
GaGallium
13
AlAluminium
5
BBoron
49
InIndium
81
TlThallium
113
UutUnuntrium
6
CCarbon
14
SiSilicon
32
GeGermanium
50
SnTin
82
PbLead
114
UuqUnunquadium
7
NNitrogen
15
PPhosphorus
33
AsArsenic
51
SbAntimony
83
BiBismuth
115
UupUnunpentium
8
OOxygen
16
SSulphur
34
SeSelenium
52
TeTellurium
84
PoPolonium
116
UuhUnunhexium
9
FFlourine
17
ClChlorine
35
BrBromine
53
IIodine
85
AtAstatine
117
UusUnunseptium
10
NeNeon
2
HeHelium
18
ArArgon
36
KrKrypton
54
XeXenon
86
RnRadon
118
UuoUnunoctium
3 4 5 6 7 8 9 10 11 12
Figure 1.1: Periodic table showing the ingredients for transition metaldichalcogenides. Transition metal dichalcogenides have the composition MX2,where M (X) can be one of the transition metal (chalcogen) atoms highlighted above.
TMDs are generally considered quasi-2D materials because the coupling between
monolayers is mostly due to the van der Waals interaction.
As shown in Figure 1.1, the groups of the periodic table that most commonly
combine with chalcogens to form TMDs are 4, 5, 6, 7, 9 & 10. The coordination of a
M atom in a TMD material is either trigonal prismatic or approximately octahedral
(trigonal antiprismatic), as shown in Figure 1.2. The M coordination strongly
influences d level splittings, and thus the electronic properties of the material. Each
M atom gives 4 valence electrons to bonding orbitals with two X atoms, and the
remaining valence electrons occupy d levels. Depending on the number of remaining
electrons, and the splitting of the d levels, TMDs can be insulators (HfS2),
semiconductors (MoS2,WS2), semimetals (WTe2,WSe2), and metals (NbS2,VSe2)
[15], as shown schematically in Fig. 1.3.
6
(a) trigonal prismatic (side and top views)
M: X:
(b) octahedral (side and top views)
Figure 1.2: Two types of coordination geometry for transition metaldichalcogenides. TMD coordination can be trigonal prismatic (D3h) or approxi-mately octahedral (trigonal antiprismatic, D3d). For the former (a), the two chalcogen(X) layers align vertically, while for the latter (b), they are rotated 60 from each other.The top views show this vertical alignment. For monolayer systems, these are the onlytwo polymorphs, and are referred to as 1H (trigonal prismatic) and 1T (octahedral).
7
σ∗
σ
t2g
eg
4
567
910
DOS
E
(a) D3d
σ∗
σ
dz2
4
56
7
910
DOS
E
(b) D3h
Figure 1.3: Impact of coordination geometry on electronic bandstructure.Transition metal dichalcogenides can coordinate by D3d or D3h. This can play a rolein what types of electronic properties they have, ranging from insulating to metallic.Density of states are drawn schematically, indicating possible scenarios for band filling.The blue/red peaks represent bonding/antibonding TM-chalcogen bands, and thegreen nonbonding d orbitals of TM. Numbers indicate the group number of the TMcoordination center. Four M valence electrons are assumed to be in bonding bandsformed with X atoms. The other M valence electrons occupy nonbonding d orbitals(green). For (a), perfect octahedral coordination is assumed.
8
For thicknesses greater than a monolayer, interlayer stacking can also play a role
in the material properties. Octahedral coordination materials tend to stack directly
(AA stacking), and are referred to as 1T . Trigonal prismatic materials tend to stack
with a c-axis periodicity of two or three monolayer units, referred to as 2H (ABAB
stacking) and 3R (ABCABC stacking). For these polymorphs, the M atoms do not
always align vertically.
Rich phase diagrams exist for many of these materials, indicating multiple
ordering tendencies like charge density waves (CDW), magnetism and
superconductivity. An important area of study involves probing the behavior of
these materials as they are thinned from the bulk down to a single monolayer. A
noteworthy example of this is the semiconductor MoS2, which in the bulk exhibits
an indirect band gap, while the monolayer has a direct band gap. This sort of
material tunability shows great application promise, but requires investigation on a
fundamental science level.
1.2.2 Charge Density Waves
A charge density wave (CDW) is a modulation of charge density in a periodic solid
which lowers the total energy of the system of ions and electrons. For a simple
model of a one-dimensional metal Rudolf Peierls showed that the energy lowering
from an appropriate charge density modulation is always greater than the energy
cost of the accompanying lattice distortion [16], meaning that the system is
fundamentally unstable against CDW formation.
In Peierls’ model, the origin of this energy lowering is purely electronic and is
rooted in a familiar concept from solid state physics—the opening of a band gap at
the Brillouin zone (BZ) edge. Peierls considered a one-dimensional chain of atoms
9
with lattice constant a and one electron per atom. This leads to a half-filled band
with a Fermi “surface” consisting of two points, kF = ± π2a.
Peierls noted that a charge modulation with twice the lattice periodicity would
halve the BZ and open a gap at the boundary of the new BZ (i.e. at ± π2a), lowering
the energy of occupied electronic states near the BZ edge. Such a charge density
modulation would be accompanied by a dimerization of the lattice since the
electrons and lattice are coupled. For a one-dimensional metal at low temperature,
Peierls showed that the elastic energy cost is less than the electronic energy
reduction, so the CDW is favorable at low temperature. At high temperature, the
excitation of electrons to states above the gap increases the electronic energy,
destabilizing the CDW phase. Thus a Peierls transition occurs when a 1D metal is
cooled to the transition temperature, where it undergoes a spontaneous doubling of
the unit cell (dimerization), as well as a metal to insulator transition.
At its core, the Peierls transition is driven by a divergence in the real part of the
bare electronic susceptibility:
Re[χω=0(q)] =2
Nk
∑kmn
f(εk+qn)− f(εkm)
εk+qn − εkm, (1.1)
where εkn is the energy and f(εkn) the Fermi-Dirac occupation of the eigenstate
with wavevector k and band index n. In 1D, the real part of χω=0 diverges at
q = 2kF . This is the ordering vector that describes the doubled unit cell in the
CDW phase. This is also the geometric nesting vector that connects the two points
of the 1D Fermi surface. For this reason, the Peierls transition is often associated
with Fermi surface nesting in the literature. In general, however, geometric nesting
is related to the imaginary part of χω=0 rather than the real part:
Im[χω=0(q)] =2
Nk
∑kmn
δ(εkm − εF )δ(εk+qn − εF ). (1.2)
10
The imaginary part of χω=0 peaks at wavevectors q that connect large parallel
sections of the Fermi surface.
The quasi-one-dimensional crystals NbSe3 [17], NbTe4, and TaTe4 [18] are
real-world examples of the Peierls transition. Below some transition temperature,
they form CDWs, in which, in some cases, no ionic displacement is detected, at least
within the uncertainty of the measurements [17]. Indeed the lattice distortion is
considered secondary in the Peierls transition.
In quasi-two-dimensional materials, in particular the transition metal
dichalcogenides, there are various materials that form CDWs, including TaS2, TaSe2,
and NbSe2. It is becoming increasingly clear, however, that CDW formation in these
materials deviates from the Peierls picture. In many cases, the CDW ordering vector
coincides with neither Fermi surface nesting nor a divergence of the real part of the
electronic susceptibility [19, 20, 21, 22, 23]. The CDW phases do involve structural
transitions, but these structural transitions are not always accompanied by a metal
to insulator transition.
The wavevector dependence of the electron-phonon coupling, instead, seems to
dictate the parts of the BZ where a CDW ordering vector is likely to appear. In
these cases, one or more phonon modes soften due to strong screening by electrons.
The extreme softening of a lattice vibration, in which the frequency approaches
zero, is like a softened spring that never returns to its unstretched position. Thus in
2D and quasi-2D materials, it has been argued that CDWs form when strong
electron-phonon coupling softens selected phonon modes to the point of instability.
Note that in the Peierls transition, a phonon mode also softens (so-called Kohn
anomaly), but its wavevector is dictated by the q dependence of Re[χ], rather than
that of the electron-phonon matrix elements.
11
0 K 180 K 350 K 540 K T
C-CDW NC-CDW I-CDW undistorted lattice
Figure 1.4: Charge density wave phases of 1T -TaS2. 1T -TaS2 forms a chargedensity wave lattice distortion that is fully commensurate with the lattice below 180K. Between 180 - 350 K, the CDW is nearly commensurate with the lattice, andincommensurate between 350 - 540 K. Finally, above 540 K, the lattice is undistortedby any CDW formation.
It may serve the scientific community to clearly distinguish these two types of
charge modulation phenomena, as they are often conflated in the literature. Zhu, et
al. have proposed that charge modulations that follow the Peierls mechanism based
on the divergence of the real part of the electronic susceptibility be dubbed Type I
CDWs. Type II CDWs would then be the much more common CDWs linked to
electron-phonon interaction that do not necessarily display a metal-insulator
transition [24].
1.2.3 TaS2
This thesis centers mostly on TaS2, a TMD that can adopt either the 2H or 1T
polymorph. The 2H polymorph is favored at low temperatures. To realize the
metastable 1T phase at e.g. room temperature, the material must be heated above
∼ 1050 K and quenched back down [25, 26]. Annealing (heating and slowly cooling)
will return the material to the more stable 2H phase [27]. Both polymorphs are
metallic at high temperatures and undergo CDW phase transitions at lower
temperatures, but with very different effects on electronic transport [25, 28].
12
1T -TaS2
As shown in Fig. 1.4, above 540 K, 1T -TaS2 has no lattice distortion (undistorted
phase) and is metallic. When cooled to 540 K, the material develops CDW order
that is not commensurate with the lattice (incommensurate CDW, I-CDW). At 350
K, the lattice enters a more ordered phase, known as the nearly-commensurate
CDW (NC-CDW) phase, which is accompanied by an abrupt increase in resistivity.
Finally, at 180 K, the CDW becomes fully commensurate with the lattice, and an
even larger jump in resistivity occurs. At 180 K and below, the material is
considered to be insulating [25, 29]. This material has drawn considerable attention
because a material with large changes in resistivity can be useful in device
applications such as transistors, especially if there is a way to externally induce the
metal to insulator transition (MIT).
As seen in Figure 1.3, the group 5 TMDs have one valence electron per formula
unit (f.u.) available to occupy d bands. When the C-CDW occurs, two concentric
rings of 6 Ta atoms contract in towards a central Ta atom, forming 13-atom clusters
referred to as the star of David (Fig. 1.5). Hybridization of the d orbitals in the
cluster leads to 6 bonding and 6 antibonding bands and one nonbonding band. The
bonding bands become occupied, leaving one half-filled nonbonding band, mainly of
dz2 character, around the central Ta. Many questions about this transition remain
unanswered, including the driving mechanism.
It has long been thought that the electronic transition that accompanies the
formation of the star of David clusters is driven by electronic correlations (Mott
physics), where a singly-occupied orbital localizes on each cluster. DFT calculations
for the bulk indeed find a half-filled band at the Fermi level, with very little in-plane
dispersion. However, the bandwidth is not negligible in the out-of-plane direction.
13
Figure 1.5: 1T -TaS2 charge density wave viewed from above. Only Ta atomsare shown for simplicity. Two concentric rings of atoms (blue and black dots) contracttoward central Ta atoms (red dots) to form a star of David configuration. The
√13×√
13 supercell is outlined in green dashed lines.
14
Hence it has been suggested that the observed insulating behavior of the bulk
C-CDW phase may arise from stacking disorder rather than Mott physics [30]. In
this scenario, the single layer C-CDW would be a Mott insulator. In another recent
publication [31], it was shown through the combined use of DFT calculations and
angle-resolved photoemission spectroscopy (ARPES) that because the stacking for
the C-CDW and NC-CDW are quite different, it is possible to explain the MIT
without using Mott physics, but rather just considering changes in out-of-plane
stacking1.
Previously, Yizhi Ge studied the effects of two types of stacking on the electronic
bandstructure of 1T -TaS2 and 1T -TaSe2: 1) hexagonal, where star of David centers
line up vertically, and 2) triclinic, where star of David centers in adjacent layers are
separated by e.g. 2a + c2. Similar to what was reported in Ref. [31], he found that
for hexagonal stacking, the half-filled band at the Fermi level has an out-of-plane
bandwidth of ∼ 0.5 eV. In the plane, the band is very flat. This contrasts with the
triclinic stacking, where the in-plane dispersion is much stronger and out-of-plane
somewhat weaker (both are about 0.3-0.4 eV). The actual stacking in this material
seems to be neither the triclinic or hexagonal stacking sequences, but perhaps some
mixture of these, as shown in diffraction experiments and subsequent theoretical
analysis [32, 33].
There has been a lot of interest in whether the C-CDW transitions (both
structural and electronic) persist as 1T -TaS2 is thinned down to a monolayer, where
stacking is no longer a concern. Early experiments suggested that the material may1Note that CDW stacking differs from the stacking of atomic layers in the 2H or 3R
polymorphs, for example. In this context, stacking refers to the relative positions of theCDWs in the vertical direction.
2This results in a 13 layer sequence, where the 14th layer aligns with the first.
15
not undergo this transition if less than 14 or so layers thick, and that the transition
temperature decreases upon thinning [34, 35].
Theoretical calculations, however, indicate that the C-CDW exists even for the
monolayer [30, 36]. A transmission electron experiment showed that formation of
surface oxide prevents any CDW order in thin samples [37]. In Chapter 4 of this
thesis, I present a theoretical and experimental Raman study that shows the
presence of the C phase vibrational signature in a monolayer of 1T -TaS2 [38]. More
recently, the C phase was observed using scanning transmission electron microscopy
(STEM) in a freestanding trilayer flake at room temperature, suggesting that not
only is the structural transition present in few-layer samples, but that the transition
temperature may actually increase [39], corroborating other work that suggests a
very different transition temperature for the bulk and surface commensurate CDWs
[40].
Although evidence for the commensurate phase CDW in ultrathin 1T -TaS2
continues to appear, the question of electronic transport in these thin samples
remains. Tsen, et al. saw the electronic transition temperature decrease upon
thinning (as did Yu, et al. [35]), and no evidence for the commensurate phase
structural transition in a h-BN encapsulated trilayer [37, 40]. Whether
environmental issues like oxidation in Yu’s study or substrate effects in Tsen’s
experiment impacted the transition remains to be seen.
2H-TaS2
2H-TaS2 is the trigonal prismatic polymorph of TaS2. The 2H polymorph of TaS2
differs from that of some other 2H materials (e.g. MoS2) in that the metal atoms lie
directly atop one another. The S atoms alternate positions between trilayers (TaS2
“monolayers”). Bulk 2H-TaS2 undergoes a low-temperature CDW phase transition
16
Figure 1.6: An atomic displacement pattern for 1H-TaS2. This displacementpattern is similar to that found by Moncton, et al. for 2H-TaSe2 [41]. The displace-ments are exaggerated for visibility.
at 75 K, with only a small increase in the electrical resistivity. Lowering the
temperature further, the resistivity decreases steadily, as for a metal [28].
The CDW superstructure for bulk 2H is generally believed, based on electron
diffraction, to be a 3× 3 distortion (qCDW = a/3) [28]. However, others have
proposed, based on Raman spectroscopy, that the distortion is incommensurate with
the primitive lattice [42]. Like other materials including NbSe2 and TaSe2, the
precise atomic displacement pattern in the 2H CDW phase is not known for TaS2.
For 2H-TaSe2, a displacement pattern similar to that shown in Fig. 1.6 was
proposed by Moncton based on neutron diffraction [41]. Later, Bird et al. used
electron diffraction to characterize the same material and found a different type of
displacement pattern, like the one shown in Fig. 1.7 [43]. Unlike the neutron beam
in Ref. [41], the spot size of the electron beam in Ref. [43] was smaller than the
17
Figure 1.7: Another atomic displacement pattern for 1H-TaS2. This dis-placement pattern is similar to that found by Bird, et al. for 2H-TaSe2 [43]. Thedisplacements are exaggerated for visibility.
average CDW domain size in 2H-TaSe2, which accounts for the difference in
symmetry of the proposed structures.
DFT calculations indicate that TaS2 has many competing 3× 3 structures close
in energy. Our calculations indicate that these 3× 3 distortion patterns result in a
1-2 meV / TaS2 lowering of the energy of the lattice, with maximum Ta
displacements of δRTa ≈ 0.05 Å. For example, structures similar to that described
by Moncton for 2H-TaSe2 (see Fig. 1.6) are lower in energy than the undistorted
structure by about 1 meV / TaS2 [41]. The structure we found to have the lowest
energy (energy lowering of about 2 meV / TaS2) is similar to the structure described
by Bird, et al. also for 2H-TaSe2 (see Fig. 1.7) [43].
Recently, a paper by Sanders, et al. indicated that when grown epitaxially on Au
(111), single-layer TaS2 adopts the 1H polymorph, and cooling well below the 75 K
bulk CDW transition temperature reveals no lattice distortion. The authors suggest
18
that electron doping from the substrate may explain the absence of a CDW in their
monolayer [44]. That raises the question, however, about whether the observed
absence of a CDW phase is intrinsic to the monolayer or induced by the substrate.
This question is explored in Chapter 5 of this thesis.
With all the materials that exhibit a CDW, the question of what causes the
distortion to occur is fundamental. In the case of 1H-TaS2, the CDW phase is
metallic, and so the Peierls mechanism does not appear to be responsible for the
distortion. Instead, strong electron-phonon coupling at the CDW wavevector is
likely responsible for the lattice distortion. This is explored in detail in Chapter 5.
1.2.4 Overview of Thesis
The rest of this dissertation is organized as follows. Chapter 2 gives a description of
the theoretical background of density functional theory (DFT) and density
functional perturbation theory (DFPT), indispensable tools for modern theoretical
physicists and chemists.
Chapter 3 discusses work that I did in conjunction with IBM Almaden Research
Center on the adsorption of Ni on a MgO/Ag(001). Using DFT and DFT+U
approaches, we find that the electronic and magnetic properties of the adatom
depend strongly on the binding site. In our calculations, the two lowest energy
binding sites are close in energy, and an on-site Coulomb interaction U changes
their energetic ordering. We find that bonding and magnetism compete to give the
most energetically-favorable configuration of charge transfer, orbital hybridization,
and spin channel occupation.
In Chapter 4 we present first-principles (DFPT) calculations of the Raman mode
frequencies of 1T -TaS2 in the high-temperature (undistorted) phase and the
low-temperature commensurate charge density wave phase. We also present
19
measurements of the Raman spectra for bulk and few-layer samples at low
temperature. Our measurements show that the low temperature commensurate
charge density wave remains stable even for a monolayer. We also calculate the
vibrational spectra for the strained material, and propose polarized Raman
spectroscopy as a way to identify the c-axis orbital texture (stacking) in the low
temperature phase. The orbital texture has been predicted to strongly impact the
electronic transport in this material.
Chapter 5 delves into the effects of electron doping on the lattice instability of
single-layer 1H-TaS2. Recent ARPES measurements suggest that when grown on
Au(111), strong electron doping from the substrate occurs. Furthermore, STM/STS
measurements on this system show suppression of the charge density wave
instability seen in bulk 2H-TaS2. Our ab initio DFT and DFPT calculations of a
free-standing monolayer of 1H-TaS2 in the harmonic approximation show a lattice
instability along the ΓM line, consistent with the bulk 3× 3 CDW ordering vector.
Electron doping removes the CDW instability, in agreement with the experimental
findings. The doping and momentum dependence of both the electron-phonon
coupling and of the bare phonon energy (unscreened by metallic electrons)
determine the stability of lattice vibrations. Electron doping also causes an
expansion of the lattice, so strain is a secondary but also relevant effect.
Chapter 6 concludes my thesis with a summary of the main results and potential
directions for future work, some of which I am currently pursuing.
20
Chapter 2
Theoretical Background
Density Functional Theory (DFT) has become ubiquitous in the study of condensed
matter due to its ability to give accurate representations of the electronic structure
of solids and molecules. This technique relies on a number of approximations and
simplifications that allow the mapping of the quantum many-body problem to a
procedure of solving for the eigenstates and eigenvalues of a single-particle
Hamiltonian self-consistently. The basic ideas of DFT are summarized here.
2.1 Born-Oppenheimer Approximation
We start with the full many-body Hamiltonian of a system of interacting electrons
and nuclei (ions):
H = Telectron + Tion + Velectron−electron + Velectron−ion + Vion−ion (2.1)
Because nuclei are much more massive than electrons (mp/me ≈ 1.8× 103), they
move much more slowly by comparison. To a good approximation, the nuclei appear
stationary on the time scale of electronic motion. Thus, our first step is to remove
the kinetic energy of the nuclei and replace the interaction of the nuclei with each
other with a constant. This is known as the Born-Oppenheimer Approximation [45].
HBO = Telectron + Velectron−electron + Velectron−ion + Eion−ion (2.2)
21
Grouping the last two terms together into Vext, we arrive at our simplified
Hamiltonian for electrons:
HBO = Telectron + Velectron−electron + Vext, (2.3)
which is still too complicated to solve directly except for the simplest of systems.
2.2 Hohenberg-Kohn Theorems
It is clear that a system with a nondegenerate ground-state, under external
potential Vext, cannot have two ground-state densities n,1(r), n,2(r). The first
theorem of Hohenberg-Kohn proves the inverse: density n(r) cannot be the
ground-state density of two different systems [46].
Let there be two external potentials: Vext,1, Vext,2. The ground-states of H1 and
H2 are Ψ1 and Ψ2, respectively: E1 = 〈Ψ1|H1 |Ψ1〉, E2 = 〈Ψ2|H2 |Ψ2〉. Assuming
that the ground states are non-degenerate,
Eo1 < 〈Ψ2|H1 |Ψ2〉 = 〈Ψ2|H2 |Ψ2〉+ 〈Ψ2|H1 −H2 |Ψ2〉
and
Eo2 < 〈Ψ1|H2 |Ψ1〉 = 〈Ψ1|H1 |Ψ1〉+ 〈Ψ1|H2 −H1 |Ψ1〉
If Ψ1 and Ψ2 give the same density, n(r), then this leads to an absurd result:
E1 < E2 +∫n(r)(Vext,1 − Vext,2)dr
and
E2 < E1 +∫n(r)(Vext,2 − Vext,1)dr
⇒ E1 + E2 < E1 + E2 .
The density n(r) appears after integration over N − 1 spatial coordinates, e.g.:∫drdr2 . . . drNΨ∗(r, r2 . . . rN)V (r)Ψ(r, r2 . . . rN) =
∫drn(r)V (r).
22
So, each Hamiltonian is uniquely linked to its ground-state electron density, and
this density determines E, the ground-state energy. The energy, E, depends on the
function n(r), and thus it is a functional of the density (denoted as E[n(r)]).
The second theorem of Hohenberg-Kohn simply states that the ground-state
density is that which gives the lowest energy, and thus the ground-state density may
be found according to the variational principle. The variational principle essentially
states that for a trial wavefunction |Ψ〉,
E ≤〈Ψ|H |Ψ〉〈Ψ|Ψ〉 , (2.4)
meaning that the energy of any trial wavefunction energy is an upper bound for the
ground-state energy, and by altering the trial wavefunction in a way that lowers the
energy, the trial wavefunction becomes a better approximation for the ground state.
Note that the only state discussed thus far has been the ground state, and so it can
be said that DFT is a ground-state theory, and essentially models systems at 0 K.
2.3 Kohn-Sham Formulation
The next theorem of Walter Kohn, known as the Kohn-Sham (KS) formulation,
transforms the above theorems into a method for actually finding the ground state
of an electronic system [47]. In essence, this formulation maps the many-body
Hamiltonian onto a single-particle Hamiltonian, in which the many-body effects,
namely exchange and correlation, are represented in an effective potential term. The
many-body wavefunction Ψ is replaced by N single-particle wavefunctions ψi, which
together form the density n(r) =∑
i=1 fi|ψi(r)|2 where fi is the occupation of state
i. Kohn-Sham makes the ansatz that the true ground-state density of the system
can be represented this way.
23
Kohn-Sham starts by taking the kinetic energy of a non-interacting N-electron
gas, TS[n(r)] =∑N
i=1−〈ψi| ~22m∇2 |ψi〉 and adds to it the classical Hartree energy
(Coulombic interaction only) EH [n(r)] = e2
2
∫ ∫ n(r)n(r′)|r−r′| drdr
′. This, along with the
interaction energy of the electrons with the external potential,
Eext[n(r)] =∫Vext(r)n(r)dr, would characterize a non-interacting system’s energy.
The rest of the interactions, that of the electrons with each other, can now be
expressed in an unknown energy term, which like the others, is a functional of the
density. We call this the exchange-correlation energy, Exc[n(r)]. We would like to
minimize the energy of all these terms, which we do by implementing a Lagrangian
constraint on the system electron number:
0 =∂
∂ψ∗i
(TS[n] + EH [n] + Eext[n] + Exc[n]−
∑i
λi
∫ψ∗i (r)ψi(r)dr− 1
),
(2.5)
which yields the following Kohn-Sham Schrödinger equation:(−1
2∇2 + VH + Vext + Vxc
)ψi(r) = εiψi(r), (2.6)
where εi = λi, the energies of the single-particle states. We can now write down the
Kohn-Sham Hamiltonian operator:
HKS = TS + VH + Vext + Vxc, (2.7)
where VH = δEH [n]δn
and Vxc = δExc[n]δn
.
Guessing a trial electron density gives us a trial Hamiltonian to solve, which
yields a new electron density. After some iterations of this procedure, the density
converges on what can only be the ground-state density of the system. This
self-consistent method of solving for the ground-state density is summarized in
Fig. 2.1.
24
guess n(r)
HKS = TS + VH [n(r)] + Vext + Vxc[n(r)]
solve (HKS − ǫi)ψi(r) = 0for ψi(r), ǫi
nnew(r) =∑i
fi |ψi(r)|2
Is n(r) ≈nnew(r),i.e. self-
consistent?
nnew(r) = n(r).
nnew(r) ⇒ n(r).
yes
no
Figure 2.1: Simplified DFT work flow.
25
Some practical considerations for the KS formulation are important to keep in
mind. The single-particle KS states do not represent multi-electron configurations;
they are fictitious quasi-particles, and the only link between the single-particle
states and the real system is through the ground-state electron density n. Also, the
theorems of DFT are exact, but in practice, the approximation of Exc is a source of
error.
2.4 Exchange and Correlation
A simple but useful approximation to the exchange and correlation energy term is
known as the Local Density Approximation (LDA). It is based on numerical
solutions for the correlation energy and the exact exchange energy for a
homogeneous electron gas, i.e. jellium. In the LDA, Exc is calculated using the
uniform electron gas results for the density at each point, r:
ELDAxc [n(r)] =
∫εxc (n(r))n(r)dr, where εxc is a function of n(r), not a functional.
One would expect this to be accurate when n(r) is slowly varying, but LDA works
surprisingly well in many situations.
The next approximation is called the Generalized Gradient Approximation
(GGA) which takes into account, using gradients, the local density, and how that
density changes with position: EGGAxc [n(r)] =
∫εxc(n(r), |∇n(r)|)n(r)dr [48]. GGA
often improves the tendency of LDA to overestimate cohesive energies.
Other more sophisticated methods exist to correct inaccuracies in the DFT
representation of exchange-correlation. Hybrid functionals mix some amount of
Hartree-Fock, or exact exchange into Vxc, using
EHFx [n(r)] = e2
2
∑i,j
∫ ∫ψ∗i (r)ψ
∗j (r2) 1
r12ψj(r)ψi(r2)drdr2 [49]. This mixing of EHF
x
into Exc is somewhat empirical, but nonetheless, can improve accuracy in certain
26
cases. Meta-GGA is yet another approach, which incorporates the kinetic energy
density, τ(r) = ~22m
∑i |∇ψi(r)|
2 into Exc; we have Exc[n(r),∇n(r), τ(r)] [50]. It may
be considered as a refinement to GGA, since it incorporates the Laplacian, ∇2. This
additional degree of freedom allows the functional to capture not just slowly varying
densities, but also non-local self-energy corrections.
A different approach is to include a Coulombic energy penalty U , when an
orbital becomes doubly occupied [51]. This can be helpful when working with
transition metal elements or rare earths whose d or f electron correlations are
underrepresented by Exc. This is computationally inexpensive; however, it is not
always clear what value of U is appropriate.
2.5 Implementation: Basis Sets
Next, we consider the accuracy and computational cost of different basis sets. The
ideal basis set gives an efficient path towards the desired solution without being
biased. For particles in a periodic potential, such as electrons in a crystalline lattice,
Bloch’s Theorem states that eigenfunctions can be written as
ψnk = unk(r)eik·r, (2.8)
where k is any point in the first Brillouin zone, n is the band index, and unk(r) has
the periodicity of the potential. The most natural choice is to express our
single-particle wave-functions in terms of plane waves, so that after expanding unk(r)
in terms of a plane-wave basis set, we have
ψnk(r) =Kmax∑K
ck,nK ei(k+K)·r (2.9)
where K is a reciprocal lattice vector that serves as the index for a particular basis
function, e.g. φkK(r) = ei(k+K)·r. This means that by diagonalizing the Hamiltonian
27
matrix (which is simplified by the fact that plane waves are orthogonal), one can
arrive at the eigenvalues and eigenvectors for all the included bands (determined by
Kmax) at that particular k-point.
In principle, plane waves are the perfect choice for periodic potentials. They are
unbiased, and the basis can be systematically improved by increasing Kmax.
However, near the atomic cores, wave functions oscillate with very short
wavelengths, requiring a quite large Kmax for an accurate description. This results
in large matrix diagonalizations and increased computational cost. To remedy this
hindrance, there are various solutions.
Pseudopotentials that represent the potential due to the nuclei and the core
electrons can be used to replace the bare nuclear potential. In essence, the
wavefunctions of the core electrons are assumed to be unaffected by the presence of
other atoms, which is reasonable since these particles do not participate in chemical
bonding. Since the wavefunctions of core electrons oscillate with very short
wavelengths, eliminating them from the calculation allows for a smaller plane wave
basis set. Furthermore, the pseudopotential can also soften the oscillations of the
valence wavefunctions very close to the core of the atom, while allowing the
wavefunction to be realistic beyond a cut-off radius. A pseudopotential’s ability to
reduce the number of plane waves required for accurate calculations is known as its
softness. Making pseudopotentials with transferability, i.e. that work well in many
environments, is also important, especially since charge transfer can depend greatly
on the bonding situation.
Another solution is the all-electron augmented plane wave (APW) method as
implemented, for example, in the DFT code Wien2k [53]. Like the pseudopotential
method, the APW method defines a sphere around the atoms, called the muffin-tin
sphere. Inside, solutions to the spherical Schrödinger equation serve as the basis set.
28
Figure 2.2: Diagram depicting the augmented plane wave scheme. Thisdiagram shows the muffin-tin regions and the interstitial regions, where APW andplane waves, respectfully, reside. This image taken from Ref. [52].
29
Outside, in the interstitial region, plane waves are used. This is advantageous
because the core electrons remain in the calculation with reduced computational
cost, since radial functions are more efficient than plane waves in the muffin tin.
Thus an APW is defined as:
φkK(r, E) =
1√Vei(k+K)·r interstitial∑l,mA
(k+K)lm Rl(|r− rsphere|, E)Y l
m(θsphere, φsphere) spheres
(2.10)
where the coefficients Alm are solved by matching the wavefunctions at the
muffin-tin surface, rsphere is the coordinate of a muffin-tin center, θsphere, φsphere use
rsphere as their origin, and Y lm are spherical harmonics [54, 55]. Since the radial
wavefunctions Rl only live in the muffin tin, E above is a parameter that must be
solved for each basis function (boundary conditions). This, along with the fact that
APWs are not orthogonal, makes the APW method inefficient, but some
improvements have sped it up. These include the linearization of the muffin-tin
portion of φkK(r, E), which fixes E as a set of El (this method is called linearized
APW, or LAPW). Also, adding local orbitals makes the basis set larger, yet more
complete in calculating the band structure. The main advantage of this method is
that, unlike the pseudopotential method, the core electrons are accurately
considered. With APW, it is possible to calculate hyperfine parameters, electric field
gradients and core level excitations.
The projector augmented wave (PAW) method [56], which is used in plane wave
codes such as VASP [57], essentially bridges the pseudopotential and APW
methods. Pseudo-wavefunctions are represented in a ‘soft’ basis of plane waves.
Projector functions, defined for each atomic species inside augmentation spheres,
allow transformation of the wavefunctions to ‘all-electron’ ones, in a basis similar to
that of the muffin tins of APW. Although core electrons are ‘frozen’ (not
30
determined self-consistently), the behavior of valence electrons near the ionic cores
is somewhat better described than in the pseudopotential method, and with a
smaller workload than the APW method.
2.6 Lattice Dynamics in DFT
In order to calculate phonon frequencies and eigenvectors, one must build and
diagonalize the dynamical matrix. Consider first a simple one-dimensional
mass-spring system. For an ideal spring, the potential is harmonic, V = 12kx2, and
the force on the mass is proportional to the derivative with respect to position,
F = −dVdx
= −kx. The equation of motion is md2xdt2
= −kx which has a sinusoidal
solution oscillating with frequency ω =√
km.
For a solid, the system of masses and springs consists of ions with electronic
density between. The total energy of the lattice can be Taylor-expanded about the
equilibrium atomic positions R0I as
E (RI) = E(
R0I
)+∑Iα
∂E
∂RIα
∣∣∣∣RIα=R0
Iα
(RIα − R0Iα)
+1
2
∑Iα,Jβ
∂2E
∂RIα∂RJβ
∣∣∣∣RIα=R0
Iα,RJβ=R0Jβ
(RIα − R0Iα)(RJβ − R0
Jβ) + . . . (2.11)
where α, β indicate Cartesian axes and I, J label the atoms of the crystal. Since we
are expanding about the equilibrium positions, the linear term vanishes. In the
harmonic approximation, we neglect cubic and higher terms in the expansion. The
real-space force constant matrix
ΦαβIJ =
∂2E
∂RIα∂RJβ
∣∣∣∣RIα=R0
Iα,RJβ=R0Jβ
(2.12)
plays a role analogous to that of the spring constant in the mass-spring system,
relating the force felt by atom I to the displacement of atom J (and vice versa).
31
The equation of motion for atom I in direction α is
MId2RIα
dt2= −
∑Jβ
ΦαβIJ
RJβ − R0
Jβ
. (2.13)
For a periodic system, we assume the solutions to the equation of motion consist of
traveling plane waves with wave vector q. The equation of motion can then be
expressed in reciprocal space as
∑κ′β
Dαβκκ′(q)ηβκ′ν = ω2
ν(q)ηακν (2.14)
where the dynamical matrix is given by:
Dαβκκ′(q) =
1√MκMκ′
∑l
Φαβlκ0κ′e
−iq·Rl (2.15)
where ν is the phonon branch index. Here, because the lattice is periodic, atoms are
labeled by lκ instead of I, where l specifies the cell, κ specifies the atom within the
basis and Rl is the position of cell l. The eigenvalues of the dynamical matrix are
the squared frequencies (ω2ν) and the eigenvectors are related to the polarizations of
the phonon normal modes ε (ηκν =√Mκεκν).
Within DFT, there are two main approaches for lattice dynamics of periodic
systems. In finite displacement methods, supercells with appropriate atomic
displacements are built and the forces on the atoms are used to populate the
force-constant matrix. This has the advantage of allowing for the inclusion of
anharmonic effects by increasing the size of the atomic displacements. However, the
method can become computationally expensive due to the size of the supercells.
Another approach is to use linear response to calculate the first-order change in the
charge density, from which the second-order (and even third-order1) change in total
energy can be obtained.1This is the DFT extension of the 2n + 1 theorem of perturbation theory, which was
demonstrated by Gonze, et al. [58].
32
2.6.1 Linear Response
First-order perturbation theory applied to the Kohn-Sham system leads to the
relation
(HKS − εi) |δψi〉 = −(δVSCF (r)− δεi) |ψi〉 , (2.16)
where HKS, εi, and |ψi〉 are the unperturbed Kohn-Sham Hamiltonian, eigenvalues,
and eigenstates, respectively, and δVSCF , δεi and |δψi〉 are the first-order corrections
to the self-consistent potential, eigenvalues, and eigenstates, respectively. The linear
variation of the SCF potential (VSCF = VH + Vext + Vxc) in DFT is given by
δVSCF (r) = δVext(r) +
∫dr1
e2
|r− r1|+
δ2Excδn(r)δn(r1)
δn(r1), (2.17)
where the two terms in the integral represent the Hartree and exchange-correlation
contributions, respectively. The first-order correction to the eigenfunctions is given
by
δψi(r) =∑j 6=i
〈ψj| δVSCF |ψi〉εi − εj
ψj(r). (2.18)
Thus to first order, the variation in density is
δn(r) =∑i
fi (ψ∗i (r)δψi(r) + c.c.)
=∑i
∑j 6=i
fi − fjεi − εj
ψ∗i (r)ψj(r) 〈ψj| δVSCF |ψi〉 , (2.19)
with fi being the Fermi-Dirac occupation of state i. Equations 2.16-2.19 can be
solved self-consistently with a scheme similar to that described in Sec. 2.3 for the
Kohn-Sham equations. With δψ and δn, we calculate δVSCF ; solving the linear
system in Eq. 2.16 leads to a new set of δψ and δn, and so on. This is called Density
Functional Perturbation Theory (DFPT) [59, 60, 61].
33
According to the Hellmann-Feynman Theorem, the first derivative of the energy
with respect to a parameter λ is
∂E
∂λ=
∂
∂λ〈Ψ|H |Ψ〉
= 〈Ψ| ∂H∂λ|Ψ〉
=
∫n(r)
∂V
∂λdr. (2.20)
The second derivative is then
∂2E
∂µ∂λ=
∫∂n(r)
∂µ
∂V
∂λdr +
∫n(r)
∂2V
∂µ∂λdr. (2.21)
So knowing δn, one can calculate the second-order change in the total energy.
Application to Lattice Dynamics
In applying DFPT to lattice dynamics, the atomic positions RI are parameters in
the electronic Hamiltonian in the Born Oppenheimer approximation which can be
varied as in the Hellmann-Feynman Theorem:
HBO = Telectron + Velectron−electron + Velectron−ion (RI) + Eion−ion (RI) , (2.22)
where the last term is constant for each set of RI. The real-space force constants
between atoms I and J are then given by
ΦIJ =∂2EBO∂RI∂RJ
=
∫∂n(r)
∂RI
∂Velectron−ion∂RJ
dr +
∫n(r)
∂2Velectron−ion∂RI∂RJ
dr +∂2Eion−ion∂RI∂RJ
. (2.23)
For a periodic system with perturbations in atomic positions of the form
RI = R0lκ + uκ(l) = R0
lκ +[uκ(q)eiq·Rl + c.c.
], (2.24)
34
the derivative of the system energy with respect to atomic displacements can be
Fourier transformed to momentum space:
∂E
∂uακ(q)=∑l
∂E
∂uακ(l)e−iq·Rl . (2.25)
Due to translational invariance in the crystal, the second derivative is Fourier
transformed as so:
∂2E
∂uα∗κ (q)∂uβκ(q)=∑l,m
∂2E
∂uακ(l)∂uβκ(m)e−iq·(Rl−Rm) = NC
∑l
∂2E
∂uακ(l)∂uβκ(0)e−iq·Rl ,
(2.26)
with NC the number of unit cells in the crystal. Using this, we can write the
dynamical matrix for wavevector q as
Dαβκκ′(q) =
1
NC
√MκMκ′
∂2E
∂uα∗κ (q)∂uβκ(q)
=1
NC
√MκMκ′
[∫∂n
∂uα∗κ
∂Vion
∂uβκ′dr +
∫n
∂2Vion
∂uα∗κ ∂uβκ′
dr +∂2Eion−ion
∂uα∗κ ∂uβκ′
], (2.27)
with Velectron−ion shortened to Vion, and the derivatives evaluated about uκ(q) = 0.
The dynamical matrix can be computed on a uniform grid of q, then Fourier
transformed to obtain the real-space force constants. The range of interatomic forces
captured will depend on the density of the q point grid. Conveniently, the
dynamical matrix can then be transformed back to momentum space at arbitrary q.
This procedure is sometimes called Fourier interpolation.
Electron-phonon Coupling
In the above procedure, we also obtain all the ingredients for calculating the
electron-phonon matrix elements:
gqνk+qν′,kν =
(~
2ωqν
) 12
〈ψk+q,m|∆V νSCF (q) |ψk,n〉 (2.28)
35
with
∆V νSCF (q) =
∑κα
1√Mκ
∂VSCF∂uακ(q)
ηακν(q), (2.29)
where ηκν(q) is the eigenvector of the dynamical matrix for phonon branch ν at
wavevector q. The electron-phonon matrix elements tell us about the interaction
between electrons and phonons, which contributes to electrical resistivity in metals,
electron-phonon mediated superconductivity, phonon linewidths, electron mass
renormalization in metals, phonon renormalization due to electron screening, and
many other properties in solids.
36
Chapter 3
Site-dependent Magnetism of Ni Adatoms on MgO/Ag(001)
3.1 Introduction
The study of magnetic adatoms on surfaces has drawn recent attention due to
possible applications in the realm of magnetic storage and quantum computation.
The density of magnetic storage has enjoyed exponential growth for several decades,
but this growth will eventually slow as particle sizes approach the
superparamagnetic regime [62]. An alternative, bottom-up approach is to start from
the atomic limit. With the scanning tunneling microscope (STM), single atoms can
be moved around on a surface to construct desired nanostructures, and the STM
can also be used to probe the electronic and magnetic properties of those
nanostructures. Such experiments have found, for example, that a single Fe atom on
a Cu2N monolayer (ML) on Cu has a large magnetic anisotropy energy [7], and that
a magnetic bit consisting of an array of 12 Fe atoms on the same surface has a
stable moment that can stay in the ‘on’ or ‘off’ state for hours at cryogenic
temperatures (∼ 1 K) [4].
A recent STM study of a single Co atom on a ML of MgO on the Ag(001) surface
found that the Co atom maintains its gas phase spin of S = 3/2, and, because of the
This chapter is reprinted from O. R. Albertini, A. Y. Liu and B. A. Jones, Phys. Rev. B91, 214423 (2015). Copyright (2015) by the American Physical Society.
37
axial properties of the ligand field, it also maintains its orbital moment on the
surface (L = 3). The resulting magnetic anisotropy is the largest possible for a 3d
transition metal, set by the spin-orbit splitting and orbital angular momentum. The
measured spin relaxation time of 200 µs is three orders of magnitude larger than
typical for a transition-metal atom on an insulating substrate [10].
Here we investigate the structural, electronic, and magnetic properties of Ni
adatoms on the same substrate, MgO/Ag(001), using density functional methods.
Previous authors [63, 64, 65, 66, 67, 68, 69, 70] have studied the adsorption of
transition metal atoms on an MgO substrate using ab-initio techniques. These
studies, which employed various surface models and approximations for the
exchange-correlation functional, are in general agreement that the preferred binding
site of a Ni adatom on the MgO(001) surface is on top of an O atom, and that
structural distortion of the MgO surface upon Ni adsorption is minimal. The
situation is less clear when it comes to the spin state of the adatom, since the s-d
transition energy is so small in Ni. Calculations that employ the generalized
gradient approximation for the exchange-correlation functional generally predict a
full quenching of the Ni moment on MgO, while partial inclusion of Fock exchange
as in the B3LYP hybrid functional predicts that the triplet spin state is slightly
more favorable [66, 67, 65].
The system studied in the current work differs from previous studies of Ni
adsorption on MgO because the substrate consists of a single layer of MgO atop the
Ag(001) surface. This aligns more closely with STM experiments that use a thin
insulating layer to decouple the magnetic adatom from the conducting substrate
that is necessary for electrically probing the system. We show that, in contrast to
the surface layer of an MgO substrate, the MgO ML on Ag can deform significantly
due to interactions with the Ni, resulting in a very different potential energy surface
38
for adsorption. Further, we investigate how the interaction of Ni with the MgO/Ag
substrate is affected by on-site Coulomb interactions and find that the preferred
binding site depends on the interaction strength U . Unlike the case of Co adatoms
on the MgO/Ag substrate, the Ni spin moment is always lower than that of the
isolated atom. The degree to which the moment is reduced depends strongly on the
binding site. These results can be understood by considering the Ni 3d-4s hybrid
orbitals that participate in bonding at different sites. We conclude with a discussion
about experiments that could corroborate our findings.
3.2 Method
3.2.1 Computational Methods
Two sets of density-functional-theory (DFT) calculations were carried out, one using
the linearized augmented plane wave (LAPW) method as implemented in WIEN2K
[53], and the other using the projector augmented wave (PAW) method [56] in the
VASP package [57]. Both started with the Perdew, Burke and Ernzerhof formulation
of the generalized gradient approximation (GGA) for the exchange-correlation
functional [71]. In the LAPW calculations, structures for each binding site were
relaxed within the GGA. These structures were then used to calculate Hubbard U
values for each site using the constrained DFT method of Madsen and Novák [72].
GGA+U calculations were then carried out using those site-specific values of U to
examine electronic and magnetic properties, without further structural relaxation.
On the other hand, PAW calculations were used to optimize structures within both
GGA and GGA+U . However, since the total energy in DFT+U methods depends
on the value of the on-site Coulomb interaction strength, the same value of U was
used for each adsorption site to allow comparison of binding energies. In all
39
GGA+U calculations, a rotationally invariant method in which only the difference
U − J is meaningful was employed [73].
Within GGA, the two sets of calculations yielded very similar results for
adsorption geometries, binding energies, and electronic and magnetic structure, as
expected. Although the calculations were used for different purposes in assessing the
effect of on-site Coulomb repulsion, the GGA+U results from the two methods were
generally consistent and displayed the same trends.1
3.2.2 Supercell Geometry and Binding Sites
STM experiments require a conducting substrate, yet for magnetic nanostructures,
it is desirable to suppress interaction between the adatoms and a metallic substrate.
Hence the adatom is often placed on a thin insulating layer at the surface of the
substrate. It has been shown experimentally that ultrathin ionic insulating layers
can shield the adatom from interaction with the surface [74]. In a recent
experimental study of Co adatoms, a single atomic layer of MgO was used as the
insulating layer above an Ag substrate [75, 10]. The lattice constants of MgO and
Ag are well matched (4.19 Å and 4.09 Å, respectively), and DFT calculations of a
single layer of MgO on the Ag(100) surface have found that is it energetically
favorable for the O atoms in the MgO layer to sit above the Ag atoms [10].
Here we used this alignment of MgO on Ag(001) in inversion symmetric (001)
slabs of at least five Ag layers sandwiched between MLs of MgO. We found only1All-electron LAPW calculations used RmtKmax = 7, where Rmt is the smallest muffin-
tin radius in the unit cell, for all calculations. Muffin-tin radii in atomic units (ao) forAg/Mg/O/Ni at O, hollow and Mg sites were 2.50/1.89/1.55/1.81, 2.50/1.89/1.66/1.92,2.50/1.91/1.91/2.50, respectively. Structural relaxations were carried out with k-pointmeshes of 6 × 6 × 1 and finite temperature smearing of 0.001 Ry. Denser k-point gridsof up to 19× 19× 1 were used for more accurate magnetic moments and densities of states.VASP calculations were carried out using a plane-wave cutoff of 500 eV, k-point samplingof 8× 8× 1 grids, and a Gaussian smearing width of 0.02 eV.
40
(a) O site (b) Mg site (c) hollow site
Figure 3.1: Top view of the three binding sites for Ni on MgO/Ag(100).Cyan spheres represent Mg, small red spheres O, and the yellow spheres Ni.
minor differences in results for slabs containing five versus seven Ag layers. The
optimized in-plane lattice constants of the Ag slabs with and without MgO MLs
were found to be within ∼ 1% of the experimental bulk Ag value (4.09 Å), and so
we used the experimental value in this work. An isolated Ni adatom on the MgO/Ag
surface was modeled using a 3/√
2× 3/√
2 supercell of the slab with one Ni atom on
each surface, corresponding to a lateral separation of 8.68 Å between adatoms. In
the out-of-plane direction, the supercells contained 7 to 8 layers of vacuum. The
supercell lattice parameters were held fixed and all atoms were allowed to fully relax.
Three high-symmetry binding sites on the MgO surface were considered, as
shown in Fig. 3.1: Ni on top of an O atom, Ni on top of an Mg atom, and Ni above
the center of the square formed by nearest-neighbor Mg and O sites. These will be
referred to as the O site, the Mg site, and the hollow site, respectively.
41
hollow O Mg
Binding site
-0.5
0.0
0.5
1.0
1.5
2.0
∆E
tot (
eV /
ad
ato
m) Ni on MgO/Ag, GGA
Ni on MgO/Ag, GGA+U
Ni on MgO, GGA
Figure 3.2: Total energy per Ni adatom on MgO/Ag, calculated for struc-tures relaxed within GGA and GGA+U (U = 4 eV) from PAW calculations.For comparison, energies calculated for Ni adatoms on MgO are also shown. All ener-gies are plotted relative to the O site energy.
3.3 Results & Discussion
3.3.1 Binding Energetics and Geometries
Since GGA and GGA+U do not fully describe important correlation effects in the
isolated Ni atom, the calculated binding energies are not as reliable as differences in
binding energy between different adsorption sites. Figure 3.2 shows the total energy
relative to that of the O site. Within GGA, the O site is slightly favored over the
hollow site, while the Mg site is about 1 eV higher in energy. For comparison, we
also plot our results for Ni on an MgO substrate. Similar to previous reports [70],
we find that on the MgO substrate, Ni clearly favors the O site, with the hollow and
Mg sites lying about 1 and 1.7 eV higher in energy, respectively.
42
Table 3.1: Relaxed distance between adatom and nearest-neighbor Mg andO atoms, and height of these atoms above the MgO ML. All distances are inÅ. Results were obtained within the GGA using the all-electron LAPW method.
Adatom Binding site dad−O dad−Mg ∆zO ∆zMg
Ni O site 1.79 2.87 +0.15 +0.06Ni hollow site 1.90 2.53 +0.48 −0.18Ni Mg site 3.63 2.73 −0.15 +0.20Co O site 1.85 2.95 −0.06 −0.20
The difference between the potential energy surfaces for Ni adsorption on
MgO/Ag versus on MgO can be attributed to the greater freedom that the MgO
ML has to deform to accommodate the adatom. The pure MgO substrate remains
very flat upon Ni adsorption. When Ni is on the O or Mg site, the displacement of
surface atoms is negligible, and when it is on the hollow site, the neighboring O
atoms displace out of the plane by about 0.09 Å. The situation is very different for
Ni on MgO/Ag. Table 3.1 lists the distance between the adatom and its nearest Mg
and O neighbors, as well as the vertical displacement ∆z, of those Mg and O atoms.
For each adsorption site, at least one of the neighboring atoms is displaced vertically
by more than 0.1 Å. The deformation of the MgO ML is most striking when Ni is on
the hollow site, with neighboring O atoms being pulled up out of the MgO ML by
nearly 0.5 Å. This allows the hollow site to be competitive in energy with the O site
on MgO/Ag, in contrast to what happens on the MgO surface.
To examine the influence of local correlations on the binding energetics of Ni on
MgO/Ag, we present GGA+U results. Since total energies can only be compared for
calculations that use the same value of U , structures were relaxed assuming U = 4
43
eV for all sites. GGA+U predicts that the O and hollow sites reverse order, as
shown in Fig. 3.2. The site above an Mg atom still remains much less favorable than
either the O or the hollow site. Of course the Coulomb interaction U depends on the
local environment of the Ni atom, so it is an approximation to assume a fixed value
for all adsorption sites. Nevertheless, these results demonstrate that on-site
correlations can change the preferred adsorption geometry. A similar effect has been
reported for the adsorption of 3d transition metals on graphene [76].
For a Ni adatom above an O atom, the Ni-O bond is weakened by on-site
correlations, as evidenced by an increase of about 8% in the Ni-O bond length in
going from U = 0 to U = 4 eV. On the hollow and Mg sites, differences between
geometries relaxed within GGA and GGA+U are much smaller.
Given the limitations of the DFT+U method, we are not able to definitively
predict whether the O site or hollow site is preferred. However, it is likely that the
Mg site is not competitive, and not experimentally feasible. Hence the remainder of
the paper deals primarily with the hollow and O binding sites, comparing their
electronic and magnetic properties.
3.3.2 Electronic and Magnetic Properties
Based on the geometries relaxed within GGA, our constrained DFT calculations
yield Coulomb parameters U = 4.6, 6.0, and 5.0 eV for Ni 3d electrons on the O,
hollow, and Mg sites, respectively. The GGA+U results presented in this section use
these site-specific values of U , but assume the GGA-relaxed geometries. As
mentioned, the O-site geometry is somewhat sensitive to the inclusion of on-site
Coulomb repulsion, but the Mg-site and hollow site geometries are not.
Atomic Ni has a ground state configuration of 3d84s2 (3F4) with the next highest
state 3d94s1, belonging to a different multiplet (3D), only 0.025 eV away [77]. The
44
Table 3.2: Total magnetization per adatom (in µB) for different sites, cal-culated within GGA and GGA+U , from all-electron LAPW calculations.Site-specific values of U are listed in eV.
Adatom Binding site mGGA mGGA+U U
Ni O site 0.00 0.16 4.6Ni hollow site 1.04 1.06 6.0Ni Mg site 1.40 1.66 5.0Co O site 2.68 2.78 6.9
average energy of the 3D multiplet, however, is 0.030 eV lower in energy than that
of the 3F4 multiplet [77]. The proximity of these atomic states sets the stage for a
competition between chemical bonding and magnetism [66].
Our GGA calculations give a 3d94s1 ground-state configuration for atomic Ni,
which under-represents the s occupation compared to experiment, while inclusion of
a local Coulomb interaction (U = 6.0 eV ) yields a 3d84s2 configuration. In both
cases, the Ni moment is calculated to be 2.0 µB.
The calculated magnetic moments for Ni on MgO/Ag are listed in Table 3.2.
The Ni spin moment is reduced from the atomic value on all sites, though by
varying amounts. When Ni is on the O site, the magnetic moment is calculated to
be zero within GGA, but it increases slightly upon inclusion of U . (When the
structure is relaxed within GGA+U and the Ni-O bond length increases by 8%, the
moment increases to about 0.3 µB.) On the hollow site, the magnetic moment of
about 1 µB is insensitive to U , while the larger moment of about 1.5 µB on the Mg
site increases with U .
45
Table 3.3: Net charge of Ni atom and its nearest-neighbor O and Mgatoms according to Bader analysis. Ag/MgO is the bare slab without adatoms.These results were obtained from all-electron LAPW calculations.
GGA GGA+U
Ni Onn Mgnn Ni Onn Mgnn
O site −0.17 −1.50 +1.72 −0.13 −1.52 +1.71hollow site +0.38 −1.51 +1.71 +0.43 −1.54 +1.71Mg site −0.26 −1.62 +1.68 −0.40 −1.61 +1.69MgO/Ag −1.64 +1.71
In Table 3.3, we present the results of a Bader charge analysis [78] (space filling)
for the adatom and its nearest neighbor Mg and O atoms, as well as for a bare slab
of MgO/Ag. Because this analysis includes the interstitial electrons, these results
can provide insight on the nature of the bonding between the adatom and the
surface. On the O site, the Ni adatom gains electrons from its O neighbor (which is
less negatively charged than it would be on bare MgO/Ag). In contrast, on the
hollow site, Ni loses electrons to the underlying Ag substrate. These charge transfer
results are relatively insensitive to local Coulomb interactions.
To understand the site dependence of the Ni moment, we examine the bonding
and electronic density of states, starting with Ni on the O site within GGA. While
the transfer of electrons from the substrate to a 3d94s1 configuration can only reduce
the spin moment, the gain of ∼ 0.2 is not enough to account for the full quenching
of the moment. The fact that most of the electrons are transferred from the
neighboring O atom, and the fact that Ni adsorption causes the O atom to displace
upwards out of the MgO ML (Table 3.1), suggest the formation of a covalent bond.
46
-6 -4 -2 0 2E-E
F (eV)
-6 -4 -2 0 2E-E
F (eV)
dz
2
dxy
dx
2- y
2
s
dxz
+ dyz
dz
2
dxy
dx
2- y
2
s
dxz
+ dyz
dz
2
dxy
dx
2- y
2
dxz
dyz
s
dz
2
dxy
dx
2- y
2
dxz
dyz
s
Mgnn
total
Onn
total
total
Mgnn
total
Onn
total
total
Mgnn
total
Onn
total
total
Mgnn
total
Onn
total
total
(a) (b)
(c) (d)
Ni
Ni
Ni
Ni
DO
SD
OS
O s
ite
ho
llo
w s
ite
GGA GGA+U
Figure 3.3: Densities of states from inside the muffin-tins of all-electronLAPW calculations. (a) GGA results for Ni on O site. (b) GGA+U results for Nion O site. (c) GGA results for Ni on hollow site. (d) GGA+U results for Ni on hollowsite. In all graphs, majority spin is plotted on the positive ordinate and minority spinon the negative ordinate. Total DOS has been scaled down significantly, and includescontributions from the interstitial region. The relative scales of nearest-neighbor Mgand O, as well as Ni s and d DOS have been adjusted to enhance viewability.
47
Indeed, previous studies of Ni on MgO discuss a bonding mechanism involving Ni
hybrid orbitals of 4s and dz2 character [63, 64, 66, 65, 69]. One of the two orthogonal
orbitals, which we denote as 4s+ dz2 , is oriented perpendicular to the surface and
interacts strongly with O pz orbitals, forming a bonding combination (mostly O pz)
about 6 eV below the Fermi level (EF ) and an antibonding combination (mostly Ni
4s+ dz2) about 1 eV above the Fermi level. This can be seen in the electronic
density of states plotted in Fig. 3.3(a). The other hybrid orbital, denoted 4s− dz2 , is
concentrated in the plane parallel to the surface, and hence interacts much more
weakly with the surface. This planar hybrid orbital lies just below the other Ni d
states, about 1 eV below the Fermi level. The covalent interaction between Ni and O
pushes the antibonding orbital high enough to be unoccupied in both spin channels,
so the other d and s-d hybrid orbitals are fully occupied, yielding a zero net spin.
When the orbital-dependent Coulomb interaction U is included within GGA+U ,
the density of state changes significantly since the d and s states are affected
differently. The hybridization between Ni dz2 and s orbitals weakens, and the
interaction between these orbitals and O p states is affected. The partial density of
states plots in Fig. 3.3(b) show the majority and minority spin dz2 orbitals split by
about 4 eV (∼ U). The antibonding Ni - O orbital, now mostly composed of Ni s
states, has a small spin splitting of the opposite polarity: the spin-down states lie
just below EF and the spin-up ones lie just above EF . Hence even though U induces
a moment in the d shell within the Ni muffin tin, this moment is screened by the
oppositely polarized antibonding orbital formed from Ni 4s and O 2pz orbitals. This
is illustrated in Fig. 3.4(a), which shows a majority-spin (red) isosurface with dz2
character on the Ni atom, surrounded by the antibonding Ni s - O pz minority-spin
(blue) isosurface (corresponding to minority spin states near ∼ −0.5 eV in
Fig. 3.3(b)). The total magnetization of the system remains close to zero.
48
(a) O site
(b) hollow site
Figure 3.4: Spin density isosurfaces of Ni adsorbed on MgO/Ag, fromGGA+U all-electron LAPW calculations. Red and blue surfaces correspond tomajority and minority spin densities of 0.00175 e/a.u.3, respectively. (a) O site: themajority spin dz2 is screened by the antibonding Ni s - O p orbital. (b) hollow site:the antibonding Ni dyz - O p orbital is spin polarized. The y axis is parallel to theline joining the two O atoms that neighbor the Ni. These two O atoms are verticallydisplaced out of the MgO plane. O atoms are represented by small red spheres.
49
The situation differs for the hollow site. The presence of the Ni at the hollow site
disrupts the MgO bond network, and the two O atoms that neighbor the Ni atom
are displaced out of the surface by nearly 0.5 Å to form bonds with the Ni, as is
evident in Fig. 3.4(b). The resulting polar covalent Ni-O bond length is only about
6% larger than the corresponding O-site bond (Table 3.1). The hollow site has C2v
symmetry, meaning that the dxz, dyz degeneracy is broken. The lobes of the dyz
orbital point towards the O neighbors and interact with O py orbitals. However,
since dyz transforms differently than 4s, symmetry considerations prevent the dyz
orbital from forming diffuse hybrids with 4s analogous to the 4s± dz2 orbitals on
the O site that enhance the orbital overlap between the adatom and surface atoms.
While symmetry permits hybridization between Ni s and dx2−y2 orbitals, the hybrid
that interacts strongly with O p states has mostly Ni s character, while the
orthogonal hybrid orbital is mostly Ni dx2−y2 . Hence the Ni d orbitals all retain a
higher degree of localization than they do on the O site. Within GGA, it becomes
favorable for the antibonding Ni dyz - O p orbitals to become spin polarized and this
largely accounts for the total moment of about 1 µB. This can be seen in Fig. 3.3(c).
Because the effect of U in the hollow site case is to increase the splitting between
the dyz majority and minority spin states, the occupation of s and d states and the
total moment are not significantly impacted, as can be seen in Fig. 3.3(d). Figure
3.4(b) shows the spin density for this case. The majority-spin isosurface in red has
antibonding Ni dyz - O py character. While it is hard to see from the angle shown,
the minority-spin isosurface in blue has the character of a 4s− dx2−y2 hybrid.
We return now to the question of why the local Coulomb interaction stabilizes
the hollow site relative to the O site. When Ni adsorbs to the O site it forms a
strong covalent bond with O through diffuse sd hybrids that overlap significantly
with O p orbitals. Since the orbital-dependent U disrupts the sd hybridization, it
50
weakens the binding of the Ni to the O. On the hollow site, geometric and
symmetry considerations prevent the formation of diffuse sd hybrids that overlap
strongly with O, so the impact of U , both in terms of binding energy and spin
moment, is smaller. As a result, increasing the strength of the local Coulomb
interaction decreases the stability of the O site relative to the hollow site.
The presence of magnetization in adatoms depends largely on the relative
strengths of the covalent interaction with surface atoms and the intra-atomic
exchange coupling for the adatom. In a previous study [10], our calculations found
that Co adsorbs to the O site of the MgO/Ag surface. We also found that, in
contrast to Ni, it maintains a magnetization close to the atomic value (Table 3.2).
This can be explained by differences in the Ni-O and Co-O interactions. Unlike Ni,
Co remains charge neutral when adsorbed, according to a Bader analysis, and the
neighboring O and Mg atoms displace downward rather than being attracted to the
adatom (Table 3.1). Since the s-d transition energy is larger in Co than Ni, the
hybridization between s and d orbitals is reduced, leading to less interatomic
overlap and weaker covalent interactions. The d orbitals remain localized enough to
support magnetism.
3.4 Conclusions
Using first-principles calculations, we have found a strong site dependence of the
electronic and magnetic properties of Ni on MgO/Ag. In particular, the magnetic
moment varies from S = 0 to nearly 1 as Ni is moved away from surface O. This is
in stark contrast with the situation of Co on MgO/Ag, in which the O site
facilitates the preservation of Co spin and orbital moments on the surface.
51
Furthermore, we see that the energy surface for adsorption of Ni on MgO/Ag is
very different from that for Ni on MgO. On MgO/Ag, the O site and hollow site are
competitive in energy, and it is not clear from theory alone which site will be
realized in experiments. However, the properties of the Ni adatom on the two sites
differ significantly, not just in the magnetic moment, but also in the charge transfer
and bonding.
The energetic proximity of the hollow and O sites raises the interesting
possibility of being able to select one site or the other, and tune the magnetic
moment by modification of the substrate through application of strain, introduction
of defects, etc. For example, our tests show that while a ∼2% increase in the
in-plane lattice constant does not change the preferred binding site and has
negligible effect on magnetic moments, it does alter the relative energies of the O
and hollow sites. It would also be interesting to explore how the properties of the
system change with the thickness of the MgO layer, both experimentally and
theoretically. In going from mono- to bi- to few-layer MgO, we expect the O site to
become increasingly favored as the MgO layer becomes more rigid.
52
Chapter 4
Zone-center Phonons of Bulk, Few-layer, and Monolayer 1T -TaS2:
Detection of the Commensurate Charge Density Wave Phase
Through Raman Scattering
4.1 Introduction
The 1T polymorph of bulk TaS2, a layered transition-metal dichalcogenide (TMD),
has a rich phase diagram that includes multiple charge density wave (CDW) phases
that differ significantly in their electronic properties. At temperatures above 550 K
the material is metallic and undistorted. Upon cooling, it adopts a series of CDW
phases. An incommensurate CDW structure (I) becomes stable at 550 K, a nearly
commensurate phase (NC) appears at 350 K, and a fully commensurate CDW (C)
becomes favored at about 180 K. At each transition, the lattice structure distorts
and the electrical resistivity increases abruptly. In the C phase, 1T -TaS2 is an
insulator [25, 79, 29]. Upon heating, an additional triclinic (T) nearly commensurate
CDW phase occurs roughly between 220-280 K [80]. While the 1T -TaS2 phase
diagram is well studied [80, 25, 81, 79, 82, 83, 84, 85, 29, 35], the exact causes of the
electronic and lattice instabilities remain unclear. Electron-electron interactions,
This chapter is reprinted from O. R. Albertini, R. Zhao, R. L. McCann, S. Feng, M. Ter-rones, J. K. Freericks, J. A. Robinson, and A. Y. Liu, Phys. Rev. B 93, 214109 (2016).Copyright (2016) by the American Physical Society.
53
electron-phonon coupling, interlayer interactions, and disorder may all play a role. It
has long been assumed, for example, that electron correlations open up a
Mott-Hubbard gap in the C phase [86]. Recent theoretical work, however, suggests
that the insulating nature of the C phase may come from disorder in orbital
stacking instead [30, 31].
Advances in 2D material exfoliation and growth have enabled studies of how
dimensionality and interlayer interactions affect the phase diagram of 1T -TaS2.
Some groups have reported that samples thinner than about 10 nm lack the jump in
resistivity that signals the transition to the C CDW phase [35]. However, this may
not be an intrinsic material property, as oxidation may suppress the transition.
Indeed, samples as thin as 4 nm—when environmentally protected by encapsulation
between hexagonal BN layers—retain the NC to C transition [37]. Whether the C
phase is the ground state for even thinner samples, and whether this leads to an
insulating state, remain open questions.
Materials that exhibit abrupt and reversible changes in resistivity are attractive
for device applications, particularly when the transition can be controlled
electrically. Ultrathin samples may exhibit easier switching than bulk materials due
to reduced screening and higher electric-field penetration. Hollander et al. have
demonstrated fast and reversible electrical switching between insulating and
metallic states in 1T -TaS2 flakes of thickness 10 nm or more [34]. Tsen and
coworkers have electrically controlled the NC-C transition in samples as thin as 4
nm and identified current flow as a mechanism that drives transitions between
metastable and thermodynamically stable states [37]. Focusing on the C phase
rather than the NC-C transition, theoretical work indicates that in few-layer
1T -TaS2 crystals, the existence or absence of a band gap depends on the c-axis
54
orbital texture, suggesting an alternate approach to tuning or switching the
electronic properties of this material [31].
Raman spectroscopy has been a powerful tool for characterizing 2D TMDs
[87, 88, 89, 90, 91, 92, 93, 94, 95]. Because the phonon frequencies evolve with the
number of layers, Raman spectroscopy offers a rapid way to determine the thickness
of few-layer crystals [96]. Raman spectra can also reveal information about
interlayer coupling [93, 94, 97, 98], stacking configurations [99], and environmental
effects like strain and doping [92]. For these materials, the interpretation of
measured Raman spectra has benefited greatly from theoretical analyses utilizing
symmetry principles and materials-specific first-principles calculations. Thus far,
theoretical studies of TMD vibrational properties have focused on the 2H structure
favored by group-VI dichalcogenides like MoS2 and WSe2 [92].
We present a symmetry analysis and computational study of the Raman
spectrum of 1T -TaS2 in the undistorted normal 1T phase as well as in the
low-temperature C phase, alongside Raman measurements of the latter. The
symmetry of the 1T structure differs from that of the 2H structure, and different
modes become Raman active in the normal phase. In the C phase, a Brillouin zone
reconstruction yields two groups of folded-back modes, acoustic and optical,
separated by a substantial gap. The low-frequency folded-back acoustic modes serve
as a convenient signature of the C phase. In our measured Raman spectra, this
signature is evident even in a monolayer. This is in contrast to some previous
studies[35, 37, 100, 101] which suggest a suppression or change in nature of the C
phase transition in thin flakes. We also investigate the effects of c-axis orbital
texture on the Raman spectra and use polarized Raman measurements to narrow
down the possible textures.
55
4.2 Description of Crystal Structures
A single ‘layer’ of TaS2 consists of three atomic layers: a plane of Ta atoms
sandwiched between two S planes. In each plane, the atoms sit on a triangular
lattice, and in the 1T polytype, the alignment of the two S planes results in Ta sites
that are nearly octahedrally coordinated by S atoms; the structure is inversion
symmetric.1 The bulk material has an in-plane lattice constant of a = 3.365 Å, and
an out-of-plane lattice constant of c = 5.897 Å[85, 81].
In the C CDW phase, two concentric rings, each consisting of six Ta atoms,
contract slightly inwards towards a central Ta site (∼ 6% and 3% for the
first-neighbor and second-neighbor rings, respectively), forming 13-atom star-shaped
clusters in the Ta planes (Fig. 4.1). The in-plane structure is described by three
CDW wave vectors oriented 120 to each other, resulting in a√
13a×√
13a
supercell that is rotated φCDW = 13.9 from the in-plane lattice vectors of the
primitive cell. The S planes buckle outwards, particularly near the center of the
clusters, inducing a slight c-axis swelling of the material [102].
The vertical alignment of these star-of-David clusters determines the so-called
stacking of the C phase. Because the electronic states near the Fermi level are well
described by a single Wannier orbital centered on each cluster, the stacking
arrangement of the clusters is also referred to as c-axis orbital texture. In hexagonal
stacking, stars in neighboring layers align directly atop each other. In triclinic
stacking, stars in adjacent layers are horizontally shifted by a lattice vector of the
primitive cell, resulting in a 13-layer stacking sequence. Some experiments suggest
that the clusters in the C phase align in a 13-layer triclinic arrangement
[103, 104, 105], while others have suggested mixed[33] or even disordered[81, 32]
1Please see Fig. 1.2(b).
56
stacking. Interestingly, the precise stacking sequence strongly affects the in-plane
transport, due to the overlap between laterally shifted Wannier orbitals in adjacent
layers [31, 22].
4.3 Methods
4.3.1 Computational Methods
Density functional theory (DFT) calculations were performed with the Quantum
Espresso[106] package. The electron-ion interaction was described by ultrasoft
pseudopotentials [107], and the exchange-correlation interaction was treated with
the local density approximation (LDA) [108]. Electronic wave functions were
represented using a plane wave basis set with a kinetic-energy cut-off of 35 Ry. For
2D materials (monolayer, bilayer, and few-layer), we employed supercells with at
least 16 Å of vacuum between slabs. For the undistorted 1T structure, we sampled
the Brillouin zone using grids of 32× 32× 16 and 32× 32× 1 k-points for bulk and
few-layer systems, respectively. For the C phase, 6× 6× 12 and 6× 6× 1 meshes
were used for bulk and 2D systems, respectively. An occupational smearing width of
0.002 Ry was used throughout.
Bulk structures were relaxed fully, including cell parameters as well as the
atomic positions. For few-layer structures, the atomic positions and the in-plane
lattice constants were optimized. The frequencies and eigenvectors of zone-center
phonon modes were calculated for the optimized structures using density functional
perturbation theory, as implemented in Quantum Espresso [106].
To investigate the effect of c-axis orbital texture in the bulk C phase, we
considered two different supercells, referred to as the hexagonal and triclinic cells.
Both cells have in-plane lattice vectors A = 4a + b and B = −a + 3b, where a and
57
a
b
2a + 2b
A
B
Mundistorted BZ
C CDW BZ
qcdw
Γ
Figure 4.1: An illustration of the Ta plane and Brillouin zone reconstruc-tion under the lattice distortion of the C phase. The blue cells represent thereal space unit cell of the undistorted lattice (defined by the lattice vectors a and b)and the corresponding first Brillouin zone. The green cells represent the super cell ofthe C CDW lattice (defined by the lattice vectors A and B) and the correspondingfirst Brillouin zone. The dotted lines indicate the lateral shift of a supercell in anadjacent Ta plane under triclinic stacking.
b are the in-plane lattice vectors of the undistorted cell, as shown in Fig. 4.1. The
hexagonal cell has C = c, where c is perpendicular to the plane, while the triclinic
cell is described by C = 2a + 2b + c (Fig. 4.1). In the triclinic cell, the central Ta
atom in a cluster sits above a Ta in the outer ring of a cluster in the layer below.
4.3.2 Experimental Methods
Multilayer 1T -TaS2 was obtained by mechanical exfoliation onto Si/SiO2 (300 nm)
from bulk 1T -TaS2 crystals synthesized via chemical vapor transport (CVT) [109].
58
Optical microscopy was used to identify various thicknesses of exfoliated flakes, and
atomic force microscopy (AFM) was used to measure flake thickness. The
Supplemental Material[38] contains the results and corresponding images for these
measurements. Raman spectra were collected via a Horiba LabRAM HR Evolution
Raman spectrometer with 488 nm Ar/Kr wavelength laser and 1800/mm grating.
The laser power at the sample surface and the integration time were sufficiently low
to prevent damage to the sample. Back-scattered measurement geometry was
adopted and the laser was focused on the sample through a 50× objective, with a 1
µm spot size. The sample was kept in a sealed temperature stage, with pressure
being less than 1×10−2 torr to minimize oxidation during the measurement. The
measurement temperature was controlled from 80 K to 600 K in a cold-hot cell with
a temperature stability estimated to be ±1 K. Polarized Raman spectra were
collected with the relative angle of polarization between the incident and measured
light (θ) varied between 0 and 90.
4.4 Results and Discussion
4.4.1 Undistorted 1T -TaS2
The undistorted phase of bulk 1T -TaS2 is stable above 550 K. We did not measure
Raman spectra in this temperature range because our samples showed signs of
damage. Nevertheless, a theoretical study of the vibrational modes is a useful
starting point for understanding the more complicated C phase.
The point group of the undistorted 1T structure is D3d for both bulk and 2D
crystals. The zone-center normal modes belong to the representations
Γ = n(A1g + Eg + 2A2u + 2Eu), where n represents the number of layers except for
the bulk, where n = 1. One of the A2u and one of the Eu representations correspond
59
(a) (b)
Eg Eu
A1gA2u
Raman-active IR-active
inversion symm. inversion symm.preserving breaking
Figure 4.2: Optical modes of 1T bulk and monolayer structures. (a) Eg andA1g modes (b) Eu and A2u modes. Gray spheres represent Ta; yellow spheres representS.
to acoustic modes. The two-fold degenerate Eg and Eu optical modes involve purely
in-plane atomic displacements, while the A1g and A2u optical modes involve only
out-of-plane displacements. The displacement patterns of the optical modes for bulk
and monolayer (1L) crystals are shown in Fig. 4.2. Since the point group contains
inversion, the A1g and Eg modes, which preserve inversion symmetry, are Raman
active, whereas the A2u and Eu modes are IR active.
The zone-center phonon frequencies calculated for monolayer and few-layer
1T -TaS2 are plotted in Fig. 4.3(a), and those for the bulk are shown in Fig. 4.3(b).
The modes separate into three frequency ranges: acoustic and interlayer modes
below about 50 cm−1, in-plane Eg and Eu optical modes between about 230 and 270
cm−1, and high-frequency out-of-plane A1g and A2u optical modes between about
370 and 400 cm−1. The low-frequency interlayer modes are present when n > 1 and
60
0 100 200 300 400
wave number (cm-1
)
2optical modes q
cdwtri.
hex.6
(c) acoustic modes qcdw
12
(b) bulk
12
6L
12
num
ber
of
modes
5L
12
4L
12
3L
12
2Linterlayer modes
12
(a)IR-activeRaman-active 1L
Figure 4.3: Calculated phonon spectra of undistorted 1T -TaS2. (a) Γ phononsin few-layer crystals, (b) Γ phonons in the bulk crystal, and (c) bulk phonons at qcdw
for hexagonal and triclinic stacking. In (a) and (b), the doubly and singly degeneratemodes have E and A symmetry, respectively. In (c), the degeneracy corresponds to thenumber of equivalent qcdw points in the Brillouin zone. Unstable modes (imaginaryfrequencies) at qcdw are omitted.
correspond to vibrations in which each layer displaces approximately rigidly, as
illustrated in Fig. 4.4 for the bilayer (2L).
The 1T structure retains inversion symmetry regardless of the number of layers
n. This is in contrast to the 2H structure, in which the symmetry depends on
whether n is odd or even. However, since the center of inversion for an n-layer 1T
crystal alternates between being on a Ta site for odd n and in-between TaS2 layers
for even n (Fig. 4.4), the displacement patterns of Raman-active modes differ
depending on whether n is even or odd. We see this most clearly in the
ultra-low-frequency regime. The highest mode in this regime is always an interlayer
61
breathing mode (Fig. 4.4(b)) in which the direction of displacement alternates from
layer to layer. For odd n, this mode is IR active, while for even n it is Raman active.
The same effect is evident in the high-frequency out-of-plane modes.
(a) (b)
shear breathing
Figure 4.4: Simplified illustration of the low-frequency phonon modes of a1T bilayer. (a) interlayer shear and (b) interlayer breathing. Gray spheres representTa; yellow spheres represent S. The dots represent inversion centers.
To facilitate interpretation of the calculated and measured Raman spectra of the
C CDW materials in Sec. 4.4.2, we show in Fig. 4.3(c) the phonon frequencies of
bulk undistorted 1T -TaS2 at the CDW wave vectors, qcdw, for hexagonal and
triclinic stacking2. Modes from the qcdw and equivalent points fold back to the Γ
point in the Brillouin zone of the C supercell (Fig. 4.1) [110]. For both types of
stacking, we find unstable modes (not shown), indicating that the structure is
dynamically unstable against a periodic distortion characterized by qcdw. Based on
Fig. 4.3(c), we expect that upon C CDW distortion, the folded-back acoustic modes
will form a group near 100 cm−1 (shown in orange), separated from the folded-back
2For the hexagonal stacking, two inequivalent reciprocal lattice vectors of the CDW cell liewithin the Brillouin zone of the undistorted cell: g1 and g1 + g2. Each is 6-fold degenerate.For triclinic stacking, there are 6 such vectors: g1,g2,g1 − g2, 2g1 − g2,g1 − 2g2, andg1 + g2 + g3. Each is 2-fold degenerate.
62
optical modes (shown in blue) by a gap of ∼ 100 cm−1. These are in addition to the
original zone center modes.
4.4.2 Commensurate CDW Phase of 1T -TaS2
As in previous calculations [31, 30, 36], we find the C phase energetically favorable
compared with the undistorted structure for all thicknesses. In the bulk, the system
lowers in energy by ∼ 9 meV/TaS2 and ∼ 13 meV/TaS2 for hexagonal and triclinic
stacking, respectively. For a monolayer, the energy lowers ∼ 23 meV/TaS2.
In the C CDW phase, the point group symmetry of a monolayer reduces from
D3d to C3i; inversion symmetry is preserved. The normal modes at Γ are classified
as 19Eg + 19Ag + 20Eu + 20Au, with 57 Raman and 60 IR modes. For bulk or
few-layer C CDW with hexagonal stacking, the point group remains C3i. With
triclinic stacking (e.g. C = 2a + 2b + c), the symmetry further reduces to Ci; only
inversion symmetry remains, and the normal mode classification at Γ becomes
n(57Ag + 60Au).
The black curves in Fig. 4.5 show the measured Raman spectrum of a bulk
sample of 1T -TaS2 on an SiO2 substrate. At 250 K (top panel), the sample is in the
NC phase. At 80 K (bottom panel), the sample is well below the NC–C transition
temperature (180 K). In going from NC to C, new peaks appear in the 60–130 cm−1
and 230–400 cm−1 ranges. These spectra are consistent with previous Raman
studies of the bulk material [111, 112, 100, 113].
We calculated the phonon modes at Γ for the bulk C structure with hexagonal
and triclinic stacking. Fig. 4.6 shows the Raman-active modes alongside the
frequencies extracted from bulk Raman measurements of Fig. 4.5. Our calculations
and experiment are in general agreement in that both find two groups of
Raman-active modes, one ranging from about 50-130 cm−1 and the other from
63
250 K
50 100 150 200 250 300 350 400Raman shift (cm
-1)
Ram
an i
nte
nsi
ty (
arb
. u
nit
s)
80 K
0.6 nm (~1 layer)
5.6 nm (~9 layers)
13.1 nm (~21 layers)
14.6 nm (~23 layers)
bulk
Figure 4.5: Raman spectra of bulk (> 300 nm) and thin (14.6 nm, 13.1 nm,5.6 nm, and 0.6 nm) samples, measured at 250 K (NC phase) and 80 K(C phase). All the spectra are normalized to their strongest Raman peak, indicatedfor the 0.6 nm curve by the arrow.
about 230-400 cm−1. These correspond to the folded-back acoustic and optic modes,
as anticipated in Fig. 4.3. (A more detailed comparison of the theoretical and
experimental Raman lines is presented later in this section.) In 1T -TaSe2, unlike
what we observe for 1T -TaS2, the folded-back acoustic and optical groups overlap
[110]. The smaller mass of S compared to Se raises the frequencies of the optical
group in TaS2. The presence of folded-back acoustic modes and thus the C phase is
easier to identify in this material due to the frequency gap.
64
0 100 200 300 400
wave number (cm-1
)
1
2
nu
mb
er o
f m
od
es
triclinicstacking
(c)
1
2hexagonalstacking
(b)
bulk Ramanon SiO
2
80 K(a)
Figure 4.6: Comparison of experimental and calculated Raman frequenciesfor the bulk. (a) Raman lines measured for a bulk flake, at 80 K. Calculated C phaseRaman-active modes for bulk: (b) hexagonal stacking (c) triclinic stacking. Eg modesare doubly degenerate. Triclinic stacking has Ag modes only.
The red, blue, green and orange curves in Fig. 4.5 are the measured Raman
spectra for thin samples of 1T -TaS2: 14.6 nm (∼23 layers), 13.1 nm (∼21 layers),
5.6 nm (∼9 layers), 0.6 nm (1 layer). As far as we are aware, this is the first report
in the literature of the Raman spectrum of monolayer 1T -TaS2.
We focus on the data taken at 80 K, where the bulk material is in the C CDW
phase. As the thickness is reduced from bulk to monolayer, the Raman spectrum
remains largely unchanged, strongly suggesting that the C CDW structural phase
persists as the samples are thinned, even down to a monolayer. The monolayer
spectrum is remarkably similar to that of the bulk (as well as to that of the 14.6 nm
and 13.1 nm samples). On the other hand, the spectrum of the 5.6 nm sample, while
similar, shows some deviations. In particular, some of the features are broadened,
and there appears to be a higher background in the 80 - 130 cm−1 range, which
could be due to the presence of additional, closely spaced peaks. These changes do
not necessarily indicate a change in the C CDW lattice distortion. In general, in
65
going from a monolayer to a few-layer crystal, we expect weak interlayer interactions
to shift and split each monolayer phonon line, as can be seen, for example, in our
calculated results for the high-temperature undistorted phase of few-layer 1T -TaS2
in Fig. 4.3. The differences in the 5.6 nm (∼9 layers) spectrum compared to those of
the bulk-like and monolayer flakes (Fig. 4.5) could arise from such few-layer effects.
The thickness dependence of the NC to C phase transition has previously been
investigated experimentally using transport [35, 37], electron diffraction [37], and
Raman measurements [101]. Most of these studies have reported difficulty in
obtaining the C phase in thin samples, though, to our knowledge, none of them
probed monolayer samples. One previous Raman study suggested that the C phase
is suppressed below thicknesses of about 10 nm at 93 K [100]. It is not clear though,
whether those samples were protected from surface oxidation, which can affect the
transition. Studies of thin 1T -TaS2 samples that are environmentally protected by a
h-BN overlayer indicate that the C phase persists down to 4 nm (∼ 7 layers)
[37, 101]. In addition, the diffraction and transport measurements suggest the
presence of large C phase domains in a 2 nm sample (∼ 3 layers) at 100 K, although
a conducting percolating path between the domains still exists. The Raman spectra
of this 2 nm sample at 10 K loses many of the well-defined peaks [101]. Since our
Raman measurements for the monolayer strongly indicate the C phase at 100 K, it
may be that the barrier for the NC-C transition is smaller for the monolayer than
for few-layer samples. This would be consistent with the suggestion that there are
distinct bulk and surface transitions, with the bulk having a larger activation barrier
[101]. Alternatively, it is possible that the encapsulating layer affects the transition
in ultra thin samples.
The measured Raman spectra for the bulk and monolayer are remarkably
similar, both in peak locations and relative peak heights. Most of the discernible
66
peaks shift by less than 1 cm−1 in going from bulk to monolayer. Our calculations,
on the other hand, show a stronger dependence on dimensionality. Fig. 4.7 shows the
calculated 1L spectrum (c), alongside the hexagonally-stacked bulk spectrum (e). To
highlight mode shifts, we show the Ag modes in red and the Eg modes in black. As in
the experiments, the 1L and bulk spectra are similar, except the spectrum generally
hardens upon thinning, particularly in the folded-back acoustic group, where the Ag
modes harden by ∼ 1 cm−1 and the Eg modes harden by up to ∼ 20 cm−1.
This discrepancy between theory and experiment may be caused in part by
approximations made in our DFT calculations. For similar materials, LDA
frequencies are typically within 5-10 cm−1 of experiment [114]. In addition, it is
likely that substrate-induced effects, such as strain and screening, which are not
taken into account in our calculations, are important.
In general, positive strain and enhanced screening tend to soften vibrational
frequencies. In the absence of a substrate, the reduced screening in a monolayer
tends to harden the phonon modes, as seen in our calculations. The presence of a
dielectric substrate in the experiments mitigates this effect. To explore the effect of
strain, we calculated the Raman modes (Fig 4.7(b)) for a 1L under 3% strain as an
example. Straining the monolayer in our calculations generally softens the modes,
bringing the folded-back acoustic group into better agreement with the
measurement, while over-softening the folded-back optical group. Since we did not
include substrate screening in our simulations, the relative importance of strain and
screening remains unclear.
As with the undistorted case in Sec. 4.4.1, the bilayer (Fig. 4.7(d)) has ultra-low
frequency modes, associated with interlayer shear and breathing. The breathing
mode (Ag) is calculated to be ∼ 30 cm−1 lower than in the undistorted case,
indicating that the coupling between layers is weaker in the C phase; indeed, the
67
0 100 200 300 400
wave number (cm-1
)
0
1
2 bulk(e)
0
1
2
num
ber
of
modes
2L(d)
0
1
2 1L(c)
0
1
2 1L(b) +3% strain
(a) 1L Ramanon SiO
2
80 K
Figure 4.7: Comparison of experimental and calculated Raman frequenciesfor a single layer. (a) Raman lines measured for a flake 0.6 nm thick, at 80 K.Calculated C phase Raman-active modes: (b) monolayer, 3% strain (c) monolayer,no strain (d) bilayer, hexagonal stacking (e) bulk, hexagonal stacking. For clarity,red lines represent Ag modes and black lines represent Eg modes. The SupplementalMaterial[38] contains the calculated C phase Raman-active modes for a bilayer withtriclinic stacking.
structure swells by 0.1 Å in the distortion. In comparing Fig. 4.7(c) and (d), two 2L
Raman modes stand out above and below the 1L optical group range: an Eg mode
around 210 cm−1 and a Ag mode around 425 cm−1. These modes evolve from
IR-active 1L modes near 215 cm−1 and 415 cm−1 respectively (not shown). As
discussed in Sec. 4.4.1 for the undistorted case, the modes can switch from IR to
Raman depending on whether n is even or odd.
Polarized Raman measurements differentiate modes of different symmetries. For
the low-temperature phase of 1T -TaS2, this allows for a more detailed comparison of
theory and experiment while potentially distinguishing between different c-axis
orbital textures. In the C phase monolayer and hexagonally-stacked bulk, the Ag
68
60 70 80 90 100 110 120 130 140 150
wavenumber (cm-1
)
Ram
an i
nte
nsi
ty (
arb
. u
nit
s)
bulk
θ = 0°
↓↓
θ = 90°
↓↓
↓
↓
↓
↓
↓↓
↓
↓ ↓↓
↓
↓
↓
↓
↓
100 K
100 K
50 100 150 200 250
1
2hex. stacking
polarized expt.
Ram
an i
nte
nsi
ty (
arb
. u
nit
s)
monolayer
θ = 0°
↓
θ = 90°
↓↓
↓
↓
↓
↓ ↓↓
↓
↓ ↓↓
↓↓↓↓
↓
50 100 150 200 250
1
23% strain
polarized expt.
Figure 4.8: Measured polarized Raman spectra of the C phase for (top) amonolayer and (bottom) a 120 nm sample measured at 100 K. We show thefrequency range which gave the strongest signal. The spectrum changes significantlybetween θ = 0 (parallel polarization) and θ = 90 (cross polarization). Red and blackarrows indicate peaks with and without polarization dependence, respectively. Theinsets show comparisons of the measured polarized Raman lines and the calculatedspectra.
and Eg modes depend differently on the relative angle θ between the polarization of
the incident and scattered light. In back-scattering geometry with parallel
polarization (θ = 0), both Ag and Eg modes are detectable. If incident and scattered
light are cross polarized (θ = 90), the intensity of the Ag modes become zero,
leaving only the Eg modes. In contrast, with triclinic stacking of the bulk C phase,
69
all Raman modes have the same symmetry, meaning they should all have the same
dependence on θ.
In Fig. 4.8, we plot the measured polarized Raman spectra for bulk and
monolayer samples at 100 K comparing parallel and cross-polarized configurations.
These plots focus on the frequency range of the folded-back acoustic modes since
these modes provide the clearest signature of the C phase. The results are similar for
the bulk and monolayer, and agree well with an early polarized Raman study [113].
For both the bulk and monolayer, we measure two types of modes with different
θ dependence, consistent with Ag and Eg modes. A line by line comparison of the
measured bulk frequencies shows strong agreement with the Ag and Eg modes
calculated for hexagonal stacking. Most modes are a few cm−1 higher than the
measured frequencies, as is typical within LDA, which tends to over-bind the lattice
[114]. For the monolayer, neither the strained nor unstrained calculations match the
measured spectrum as closely. We conclude that the primary reason is a lack of
substrate effects in our calculations.
The good agreement with the calculated Ag and Eg modes for
hexagonally-stacked bulk is surprising since diffraction data do not support pure
hexagonal stacking in bulk samples [103, 104, 105]. It is likely that the c-axis orbital
texture is more complex, involving both hexagonal and triclinic alignment between
adjacent layers, as suggested by Ref. 33. Our data are inconsistent with a random or
triclinic stacking unless the individual layers are so weakly coupled in the bulk that
they vibrate like nearly independent monolayers [113].
70
4.5 Conclusions
We have calculated the phonon modes of both the undistorted and C phases of
1T -TaS2 in the bulk and ultrathin limits. We also measured the C phase Raman
spectrum in bulk and thin samples. Through our calculations, we identified the
low-frequency folded-back acoustic Raman modes as a signature of the C phase;
these modes conveniently lie in a low-frequency range that does not overlap with
optical modes, and in which there are no zone-center modes in the undistorted
phase and very few in the NC phase. In our measured Raman spectra, these modes
persist down to 1L, which is below the critical thickness previously suggested for the
NC-C transition [35]. The NC phase consists of C phase domains separated by
metallic interdomain regions. As the temperature is lowered, the domains grow as
the transition is approached. Tsen, et al. observed commensuration domains ∼ 500
Å in length at 100 K in a 2 nm-thick sample (∼ 3 layers) [37]. Our measurements
show that at 100K, the monolayer has transformed into the C phase, or at least has
C domains larger than our laser spot size of 1 µm. This suggests that the barrier for
the NC-C transition is lower in the monolayer than in few-layer samples, though
further investigation is needed, particularly regarding the impact of substrates and
overlayers on the transition.
Recently, there has been interest in the effect of c-axis orbital texture on
conduction, with the hope of controlling it [31]. It would be useful to have a way to
quickly identify the c-axis order in these materials. Since different stackings have
different symmetries, comparison of cross-polarized and parallel-polarized Raman
experiments could provide information about the orbital texture of the bulk or
few-layer crystals. For example, Ritschel, et al. have proposed a 2L 1T -TaS2 device
operating in the C phase. The ‘on’ state would be conducting with triclinic stacking,
71
the ‘off’ state would be insulating with hexagonal stacking, and the switching
mechanism could perhaps be an external field [31]. In testing the proposed switching
mechanism, polarized Raman could provide an easy way to determine a change in
the orbital texture of the sample.
More generally, our analysis of the 1T vibrational modes complements the
existing literature that focuses on the 2H phase of TMDs. Although the 2H
structure is more common, synthesis of metastable 1T phases has been investigated
[115, 116, 117]. Besides aiding interpretation of the Raman modes of C phase
1T -TaS2, our analysis of the undistorted phase is more broadly applicable to other
1T materials.
72
Chapter 5
Effect of Electron Doping on Lattice Instabilities of Single-layer
1H-TaS2
5.1 Introduction
A charge-density wave (CDW) is a common collective phenomenon in solids
consisting of a periodic modulation of the electron density accompanied by a
distortion of the crystal lattice. For a one-dimensional system of non-interacting
electrons, the phenomenon is described by the Peierls instability, where a divergence
in the real part of the electronic susceptibility at twice the Fermi wave vector drives
a metal-to-insulator transition. In extensions of this idea to real materials with
anisotropic band structures and quasi-1D Fermi surfaces, Fermi-surface nesting is
often cited to explain charge-ordering tendencies. However, geometric Fermi-surface
nesting (i.e., existence of parallel regions of the Fermi surface separated by a single
wave vector q) is related to the imaginary part of the noninteracting susceptibility
rather than the real part, so it is not directly connected to the Peierls mechanism
[20]. Indeed, calculations for a number of CDW materials, including NbSe2, TaSe2,
TaS2, and CeTe3, have found little or no correlation between the CDW ordering
vector qCDW, and peaks in the geometric nesting function (imaginary part of χ0)
This chapter is reprinted from O. R. Albertini, A. Y. Liu and M. Calandra, Phys. Rev. B95, 235121 (2017). Copyright (2017) by the American Physical Society.
73
[20, 19, 21, 22, 23]. Nor does qCDW coincide with sharp features in the real part of
χ0 in most of these cases. Instead, these studies pointed to the importance of the
momentum-dependent electron-phonon coupling, which softens selected phonon
modes to the point of instability.
Advances in the synthesis of low-dimensional materials provide new
opportunities to test these ideas. In going from quasi-2D bulk NbSe2 to the 2D
monolayer, for example, the number of bands crossing the Fermi level decreases
from three to one, which could reveal information about the role played by different
bands in driving the CDW instability. The stability and structure of the CDW
phase in monolayer NbSe2, however, remains under debate. Density functional
calculations predict that both the monolayer and bilayer have CDW instabilities,
but with a shifted qCDW and a larger energy gain compared to the bulk [21]. A
recent experimental study of bulk and 2D NbSe2 on SiO2 used the Raman signature
of the CDW amplitude mode to estimate the CDW transition temperature and
reported an increase of TCDW from about 33 K in the bulk to 145 K in the
monolayer [118]. This result, attributed in Ref. [118] to an enhanced
electron-phonon coupling due to reduced screening in two dimensions, is
qualitatively consistent with the DFT predictions, but does not address the question
of the CDW superstructure. On the other hand, another study of single-layer NbSe2,
this time on bilayer graphene, reported scanning tunneling microscopy/spectroscopy
(STM/STS) measurements showing a CDW transition at a lower temperature than
the bulk but with the same qCDW ordering vector [119]. Discrepancies between the
two experimental studies and between experiment and theory could be due in part
to substrate effects, as 2D materials tend to be highly sensitive to the environment
and to the substrate. Initial studies of 1T -TaS2, for example, suggested that it
completely loses the low-temperature commensurate CDW phase upon thinning to
74
about 10-15 layers [35], but later it was shown that oxidation can suppress the
formation of the commensurate phase [37]. Raman signatures of the commensurate
CDW were later found in single-layer 1T -TaS2 samples with limited exposure to air
[38]. In using 2D materials to probe the CDW transition and the effect of
dimensionality, it is therefore crucial to differentiate behavior intrinsic to each
material from effects due to the environment.
In bulk form, TaS2 is a layered transition-metal dichalcogenide for which the 1T
and 2H polymorphs are competitive in energy, and both can be synthesized. Both
the 1T and 2H polymorphs undergo CDW transitions as the temperature is
lowered. Recently, it was reported that single-layer TaS2 grown epitaxially on a Au
(111) substrate adopts the 1H structure rather than 1T [44]. Low-energy electron
diffraction (LEED) and STM data indicate that on Au (111), single-layer 1H-TaS2
does not undergo a CDW transition, at least down to T = 4.7 K. This is in contrast
to bulk 2H-TaS2, which develops a 3× 3 CDW periodicity below about 75 K. In
addition, a comparison of angle-resolved photoemission spectroscopy (ARPES) data
to the band structure calculated for a free-standing monolayer suggests that the
substrate causes the material to become n-doped with a carrier concentration of
approximately 0.3 electrons per unit cell [44].
In this paper, we use DFT calculations to investigate whether the observed
suppression of the CDW in monolayer 1H-TaS2 is intrinsic or a consequence of its
interaction with the metallic substrate. In the harmonic approximation, we find that
a free-standing single layer of 1H-TaS2 has lattice instabilities that are very similar
to its bulk counterpart. When the monolayer is electron-doped, however, the 1H
structure becomes dynamically stable, implying that substrate-induced doping is
the likely reason for the suppression of the CDW observed in Ref. [44]. The strong
effect of doping and a secondary effect of lattice strain on the CDW transition in
75
this material give insight on what drives the transition. We examine the role of
electron-phonon coupling by calculating the phonon self-energy. We find that the
contribution to the real part of the phonon self-energy from the single band that
crosses the Fermi level is largely responsible for the momentum dependence of the
soft mode, but the bare phonon frequency (unscreened from electrons at the Fermi
level) also plays a role.
5.2 Methods
We performed density-functional theory calculations using the
QUANTUM-ESPRESSO [106] package. The exchange-correlation interaction was
treated with the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation
[71]. To describe the interaction between electrons and ionic cores, we used ultrasoft
[120] Ta pseudopotentials and norm-conserving [121] S pseudopotentials. Electronic
wave functions were expanded in a plane-wave basis with kinetic energy cutoffs of
47 (53) Ry for scalar (fully) relativistic pseudopotentials. Integrations over the
Brillouin zone (BZ) were performed using a uniform grid of 36× 36× 1 k points,
with an occupational smearing width of σ = 0.005 Ry.
To simulate electronic doping, electrons were added to the unit cell along with a
compensating uniform positive background. A single layer of 1H-TaS2 was modeled
using a supercell with the out-of-plane lattice parameter fixed at c = 12 Å,
corresponding to about 9 Å of vacuum between layers. This ensured that for the
range of electron doping explored, spurious vacuum states that originate from the
periodic boundary conditions stay well above the Fermi level. The in-plane lattice
constant and the atomic positions were fully relaxed at each doping level.
76
Phonon spectra and electron-phonon matrix elements were calculated with
density functional perturbation theory [122] on a q-point grid of 12× 12× 1 phonon
wave vectors and Fourier interpolated to denser grids. To test the effect of spin-orbit
coupling on the phonon frequencies, we used a less dense grid of 8× 8× 1 q-points.
While this does not capture all the fine structure in the phonon dispersions, it is
sufficient to test the effect of spin-orbit coupling. In fact, even Fourier interpolation
on the 12× 12× 1 q-grid does not fully capture the sharp features along certain
directions in the Brillouin zone where direct calculations of the phonon frequency
are needed. The real and imaginary parts of electronic susceptibilities and phonon
self energies were calculated using dense k-point grids of at least 72× 72× 1 points.
5.3 Results & Discussion
The 1H-TaS2 lattice has point group D3h, reduced from D6h in the bulk. Sulfur
atoms adopt a trigonal prismatic coordination about each Ta site, and the crystal
lacks inversion symmetry. Our calculations for the undoped (x = 0) monolayer give
a relaxed in-plane lattice parameter of ax=0 = 3.33 Å and a separation between Ta
and S planes of zS = 1.56 Å. These values are similar to what we calculate for the
bulk. For comparison, the experimental in-plane lattice parameter for single-layer
1H-TaS2 on Au [44] and bulk 2H-TaS2 [123] are a = 3.3(1) Å and 3.316 Å,
respectively. With the addition of 0.3 electrons per cell (x = −0.3), the lattice
expands roughly 2.5% to ax=−0.3 = 3.41 Å, while the separation between Ta and S
planes decreases slightly to zS = 1.53 Å. In analyzing the impact of doping on the
CDW instability, we examine the effect of adding charge carriers as well as the effect
of the lattice expansion.
77
In single-layer 1H-TaS2, in the absence of spin-orbit coupling (SOC), a single
isolated band (two-fold spin degenerate) crosses the Fermi level, as shown in
Fig. 5.1(a). This half-filled band, which is about 1.4 eV wide, has strong Ta d
character and is separated by about 0.6 eV from the manifold of S p bands below.
As a consequence, metallic screening is due to a single band isolated from all the
others. The Fermi surface consists of a roughly hexagonal hole sheet of primarily Ta
dz2 character around Γ and a roughly triangular sheet of primarily in-plane Ta d
character (dx2+y2 and dxy) around K. The spin-orbit interaction splits the half-filled
d band by as much as ∼ 0.3 eV, except along the ΓM line where symmetry dictates
that the band remains degenerate. The number of Fermi sheets around the Γ and K
points doubles with SOC.
Electron doping of x = −0.3 causes a nearly rigid downward shift of the partially
occupied Ta d band by about 0.1 eV, and a somewhat larger downward shift of the
occupied S p bands. This is shown in Fig. 5.1(c) for the scalar relativistic bands.
The fully relativistic bands behave similarly with doping, as can be seen in
Figs. 5.1(a,b). The hole pocket around Γ thus gets slightly smaller while the one
around K shrinks more significantly. The doping-induced lattice expansion, on the
other hand, has a negligible effect on the Fermi surface, as shown in Fig. 5.1(d).
Despite the non-negligible effect that SOC has on the electronic structure, we
find little difference between the phonon spectra calculated with and without SOC.
This is shown in Fig. 5.2 where the phonon dispersion curves were obtained from a
Fourier interpolation on a 8× 8× 1 phonon q-grid. Hence, for the remainder of this
paper, we focus on the phonon properties calculated in the absence of SOC.
Figure 5.3(a) shows the phonon dispersion curves calculated for the undoped
monolayer (x = 0) from a Fourier interpolation on a 12× 12× 1 phonon q-grid. An
acoustic phonon branch is found to be unstable over a large area of the BZ
78
-1
0
1
2
-1
0
1
2
-1
0
1
2
-1
0
1
2
x = 0
x = 0
x = −0.3
x = −0.3
x = −0.3
relaxed
unrelaxed
ΓΓ KM
(a)
(b)
(c)
(d)
no SOC
no SOC
no SOC
no SOC
with SOC
with SOCenergy(eV)
Figure 5.1: Electronic bands of single-layer 1H-TaS2. Effect of: (a) SOC forthe undoped case, (b) SOC for the n-doped (x = −0.3) case, (c) n-doping (x = −0.3vs. x = 0), (d) lattice constant relaxation (ax=0 vs. ax=−0.3) on the doped (x = −0.3)system. Internal atomic coordinates were optimized in all cases.
79
-10
0
10
20
30
40
50
-10
0
10
20
30
40
50
x = 0
x = −0.3(relaxed)
phononen
ergy(m
eV)
ΓΓ KM
(a)
(b)
no SOC
with SOC
Figure 5.2: Phonon dispersions of single-layer 1H-TaS2 with and withoutspin-orbit coupling. Effect of SOC for: (a) the undoped case, (b) the n-doped(x = −0.3) case. In (b), the curves with and without SOC lie almost directly ontop of each other. Dispersion curves were obtained from Fourier interpolation from aq-grid of 8× 8× 1 points. Imaginary frequencies are shown as negative.
80
surrounding the M point and even has a dip at K. (Imaginary frequencies are
plotted as negative.) This branch involves in-plane Ta vibrations. This region of
instability arises primarily from softening of the phonon branch near the bulk CDW
wavevector (2/3 along the Γ to M line), but there is also a secondary point of
instability along the M to K line (close to M). 1 We will refer to these points as
qCDW and qMK , respectively.
When electrons are added while keeping the lattice constant fixed at ax=0 but
allowing zS to relax, the instabilities at qCDW and qMK are progressively
suppressed. With doping of x = −0.3, a weak instability at the M point remains, as
shown in Fig. 5.3(b). Once the lattice constant is allowed to expand to its optimal
value at x = −0.3, the lattice becomes dynamically stable, as seen in Fig. 5.3(c),
though an anomalous dip in the phonon dispersion remains near qMK . For both
doped cases, there is no longer a dip at K.
At intermediate values of electron doping, we find that in going from x = 0 to
x = −0.3, with the lattice constant optimized at each level of doping, the point of
strongest instability shifts from qCDW (2/3 ΓM) to the M point as the mode
gradually hardens. In these harmonic calculations, the mode becomes stable
between x = −0.25 and x = −0.3. With positive (hole) doping, the instability at
qCDW shifts in the other direction (close to 1/2 ΓM for x = 0.3), and various other
strong instabilities appear throughout the BZ. See Supplemental Material for more
information on intermediate and positive doping values.[124]
Since the unstable branch involves in-plane displacements of Ta ions, it is likely
that these phonons couple strongly to in-plane Ta d states near the Fermi level. To1While the instability at qMK is not captured by Fourier interpolation of the 8×8×1 q-
grid in Fig. 5.2(a), the Fourier interpolation based on a 12×12×1 q-grid in Fig. 5.3(a) clearlyshows the instability. Even so, direct calculation at qMK yields an even more imaginaryphonon frequency [Fig. 5.5(a)].
81
-10
0
10
20
30
40
50
-10
0
10
20
30
40
50
-10
0
10
20
30
40
50
x = 0
x = −0.3
x = −0.3
(relaxed)
(unrelaxed)phononen
ergy(m
eV)
qCDW
ΓΓ KM
(a)
(b)
(c)
Figure 5.3: Phonon dispersions of single-layer 1H-TaS2 with and withoutdoping and lattice constant relaxation. (a) undoped (x = 0), (b) n-doped withunrelaxed lattice constant (x = −0.3, ax=0), (c) n-doped with relaxed lattice constant(x = −0.3, ax=−0.3). Dispersion curves were obtained from Fourier interpolation froma q-grid of 12× 12× 1 points. Imaginary frequencies are shown as negative.
82
investigate the role of the electron-phonon interaction in the CDW transition in this
material, we consider the phonon self-energy due to electron-phonon coupling. In
the static limit, the real part of the phonon self-energy for phonon wave vector q
and branch ν is given by
Πqν =2
Nk
∑kjj′
|gνkj,k+qj′|2f(εk+qj′)− f(εkj)
εk+qj′ − εkj, (5.1)
while the phonon linewidth, which is twice the imaginary part of the phonon
self-energy, can be expressed as
γqν =4πωqν
Nk
∑kjj′
|gνkj,k+qj′|2δ(εkj − εF )δ(εk+qj′ − εF ). (5.2)
Here Nk is the number of k points in the Brillouin zone, j and j′ are electronic band
indices, f(ε) is the Fermi-Dirac function, εF is the Fermi energy, and ωqν is the
phonon frequency. The electron-phonon matrix element is
gνkj,k+qj′ =
√~
2ωqν
〈kj| δVSCF/δuqν |k + qj′〉 , (5.3)
where uqν is the amplitude of the phonon displacement and VSCF is the Kohn-Sham
potential. Both the linewidth and the product ωqνΠqν are independent of ωqν , so
they remain well defined even when the frequency is imaginary.
In Fig. 5.4, we show Brillouin zone maps of the phonon linewidth (summed over
the two acoustic modes with in-plane Ta displacements). In the undoped material
[Fig. 5.4(a)], the linewidth is sharply peaked at qCDW. For comparison, the
geometric nesting function, which is defined as the Fermi surface sum in Eq. 5.2
with constant matrix elements, does not have sharp structure near qCDW
[Fig. 5.4(d)], and instead has three sharp peaks surrounding the K point. This
means that the sharp peak in the linewidth at qCDW must be due to large
electron-phonon matrix elements between states on the Fermi surface, rather than
83
(a) (b) (c)
(d) (e) (f)
K
Γ Γ Γ
Γ Γ Γ
M M M
M M M
qCDW
qCDW
Figure 5.4: Brillouin-zone maps of the phonon linewidth [(a-c)] and thegeometric Fermi-surface nesting function [(d-f)]. (a,d) undoped (x = 0), (b,e)n-doped with unrelaxed lattice constant (x = −0.3, ax=0), (c,f) n-doped with relaxedlattice constant (x = −0.3, ax=−0.3). White (black) represents high (low) values, withone scale used for all the linewidth plots (a-c) and another used for all the nestingfunction plots (d-f). Dotted lines show high-symmetry lines from Γ to K and fromK to M . The plots in (a-c) represent the sum of the phonon linewidths for the twoacoustic phonon branches with in-plane Ta displacements.
84
-100
-50
0
50
100
150
200
250
300
-3
-2
-200
-150
-100
-50
0
x = 0 x = −0.3x = −0.3(relaxed)(unrelaxed)
ω2,Ω2(m
eV2) Ω2
q
ω2q
χ′ 0(eV
−1)
2ωΠ
(meV
2)
Γ ΓΓK KKM MM
(a) (b) (c)
Figure 5.5: Momentum dependence of the square of the self-consistent (ω2q)
and bare (Ω2q) phonon energies for the branch with instabilities, self-energy
correction (2ωqΠq) for that branch, and real part of the bare susceptibility(χ′0(q)). (a) undoped (x = 0), (b) n-doped with unrelaxed lattice constant (x =−0.3, ax=0), (c) n-doped with relaxed lattice constant (x = −0.3, ax=−0.3). The red,dashed lines in the top panels represent the square of the bare phonon energies (Ω2
q),calculated using Eq. 5.4. Only the response of the band crossing the Fermi level isincluded in 2ωqΠq and χ′0(q).
to the geometry of the Fermi surface itself. With electron doping (x = −0.3), the
maximum values of the linewidth are much smaller and occur elsewhere in the
Brillouin zone, whether the lattice constant is allowed to relax [Fig. 5.4(c)] or not
[Fig. 5.4(b)]. This suggests that the momentum dependence of the electron-phonon
coupling of states at the Fermi level plays an important role in picking out the
primary instability at qCDW. However, focusing on states at the Fermi level is not
sufficient to understand the broader region of instability.
85
The real part of the phonon self-energy Πqν is directly related to the
renormalization of phonon frequencies due to electron screening. From perturbation
theory,
ω2qν = Ω2
qν + 2ωqνΠqν , (5.4)
where Ωqν is the bare phonon frequency. While it is conceptually appealing to define
the bare frequencies as the completely unscreened ionic frequencies, such a starting
point leads to results that lie outside the range of validity of the perturbative
expression in Eq. 5.4. Instead, we define the bare frequencies to be the frequencies
obtained neglecting the metallic screening due to electrons in the isolated Ta d band
crossing the Fermi level. It has been shown that this non self-consistent definition of
the self-energy is equivalent to a fully self-consistent solution of the linear response
equations including both the real and imaginary parts of the self-energy [125]. In
this case, we limit the sum in the phonon self-energy (Eq. 5.1) to intraband
transitions within that band and denote it as Πqν . The corresponding bare (in the
sense of unscreened by metallic electrons) frequencies are denoted Ωqν . While the
bare frequencies are not directly accessible, we can estimate them from the fully
screened frequencies ωqν (as obtained from density functional perturbation theory)
and the perturbative correction 2ωqνΠqν by inverting Eq. 5.4.
For the branch with instabilities, Fig. 5.5 shows the square of the phonon
frequencies and the self-energy correction, 2ωqνΠqν , along high-symmetry directions
in the Brillouin zone. For comparison, χ′0(q), the real part of the bare electronic
susceptibility, given by Eq. 5.1 with constant matrix elements and limited to the
band at the Fermi level, is also plotted. For the undoped material, χ′0 has a
minimum at qCDW, but the momentum dependence between qCDW and M does not
match that of the calculated frequencies. Similarly, the momentum dependence of χ′0
does not correlate with the phonon softening when the material is doped to
86
x = −0.3, with or without relaxation of the lattice constant. From Fig. 5.5, we see,
however, that the momentum dependence of the phonon self-energy roughly follows
that of the phonon softening, indicating the importance of the electron-phonon
matrix elements in Πqν . The self-energy includes contributions from states
throughout the band crossing the Fermi level, in contrast to the linewidth, which
provides information about the electron-phonon coupling at the Fermi level.
The square of the bare frequencies (Ω2qν) estimated using Eq. 5.4 are plotted
with dashed lines in the top panels of Fig. 5.5. Note that in the regions of
instabilities, the self-energy correction is comparable to or larger than the square of
the bare frequency, possibly raising into question the degree to which the
perturbative expression is valid. Nevertheless, if we use the expression to estimate
the bare frequencies, we find some surprising results. For the undoped material, the
dispersion of the bare frequency has sharp dips at both qCDW and qMK . Thus, while
phonon softening due to screening of the band at the Fermi level accounts for most
of the momentum dependence of the unstable modes, structure in the bare
frequencies contributes as well. Upon doping to x = −0.3, but holding the lattice
constant fixed at ax=0, the local minima at qCDW and qMK disappear in both the
self-energy and the bare frequency, stabilizing those modes. At M , however, neither
the self-energy nor the bare frequency change significantly with doping, so a residual
instability remains at M . When the lattice constant of the doped system is relaxed,
the self-energy undergoes relatively minor changes, while the bare frequency at M
hardens significantly. Thus it is the lattice-constant dependence of the bare
frequency that stabilizes the M point phonon at a doping of x = −0.3. The
dramatic hardening of this mode upon a 2.5% lattice expansion is surprising, as
phonons usually soften with lattice expansion. Indeed most of the other phonon
modes in the doped material soften slightly with the lattice expansion.
87
To summarize, we find that the primary CDW ordering vector coincides with
very strong electron-phonon coupling at the Fermi surface, but the full momentum
dependence of the phonons in the wide region of instability is described more
completely by the momentum dependence of ωΠ, the phonon self-energy due to
screening from the band crossing the Fermi level. The momentum dependence of the
corresponding bare phonon frequencies, Ω, however, also plays a role. Regarding the
behavior of the instabilities with doping and lattice strain, the bare frequencies
provide the primary contributions, but the changes in the phonon self-energy,
especially with doping, are essential. While recent studies of CDW materials have
focused on the role of electron-phonon coupling as the driving mechanism, we find
that, for monolayer 1H-TaS2, taking into account screening arising from
electron-phonon coupling in the band crossing the Fermi level by itself is not
enough, and one must consider how the bare frequencies depend on momentum and
doping as well. Unfortunately it is not possible to determine if the momentum,
doping, and strain dependencies deduced for the bare frequencies come from the
response of other bands away from the Fermi level or appear as features in the
completely unscreened ionic frequencies. Nevertheless, our finding that screening by
states near the Fermi surface alone are not responsible for the CDW instability
means that the instability is generated by not only the long-range part of the force
constants but also the short-range part, which is included in Ω.
5.4 Conclusions
Our calculations show that in the harmonic approximation the free-standing
monolayer of 1H-TaS2 is unstable to CDW distortions with the same ordering
vector as the bulk. We also find that electron doping stabilizes the lattice. These
88
results indicate that the CDW suppression found in experiments of 1H-TaS2 on Au
is not intrinsic but rather induced by the substrate [44]. According to our harmonic
calculations, in addition to increasing the number of charge carriers, electron doping
induces a lattice expansion. While the addition of charge carriers itself stabilizes
most of the soft phonon modes, at the harmonic level a residual instability remains
if the lattice constant is not allowed to relax. In the experiments on the Au (111)
substrate, the uncertainty in the measured lattice constant was about 3% [44],
which is slightly larger than the lattice expansion we predict for electron doping of
x = −0.3. Hence in the experimental system, it remains an open question as to how
much the substrate affects the lattice constant, and how that in turn influences the
lattice instabilities.
We believe that the suppression of CDW by electron doping is robust also
against anharmonic effects. Indeed in metals, anharmonic effects tend to enhance
phonon frequencies and suppress CDW instabilities [126, 127]. These effects, not
considered in this work, could reduce the tendency towards CDW at x = 0, but will
not change the qualitative result that electron doping removes the CDW in this
system. It would be interesting to explore the interplay between anharmonicity and
doping in future studies.
Recently, it was pointed out that for metallic 2D materials on substrates, the
shift in the bands measured in ARPES experiments may not provide a reliable
estimate of charge transfer across the interface [128]. In particular, if there is
significant hybridization between the substrate and 2D material, the actual charge
transfer will be reduced. The impact of substrate hybridization on the CDW
instability in monolayer 1H-TaS2 warrants future investigation.
This system, like other transition-metal dichalcogenides, offers an opportunity to
better understand the intrinsic origins of CDWs. Going beyond previous studies
89
that emphasized the importance of the momentum dependence of the
electron-phonon interaction in driving the CDW, we find that the momentum,
doping, and strain dependencies of the bare phonon frequencies also play a role.
This system also underscores the importance of disentangling environmental and
intrinsic effects in 2D materials, and provides an example of using the substrate to
tune the properties of the material.
90
Chapter 6
Conclusion
At IBM Almaden, I studied single atoms adsorbed on a surface in order to
understand what is necessary for a 3d transition-metal atom to preserve its
magnetic moment on a substrate. For Co adatoms on MgO/Ag, a weak interaction
with the surface atop an O site results in a weak chemical bond and full
preservation of the Co gas-phase magnetic moment. On the other hand, Ni has two
binding sites close in energy. At the hollow site, it loses only some of its spin
moment, and at the O site it loses its spin moment entirely. On both binding sites,
the strength of the chemical bond is greater than for Co.
The main limitation to the survival of magnetism when a chemical bond forms
between an adatom and a substrate is the bond itself. In particular, unpaired
electrons can become paired when there is a significant charge transfer. Furthermore,
the formation of diffuse hybrid orbitals with the substrate atoms reduces the overall
correlation of the unpaired electrons. And finally, a strong crystal field shifts the
energy levels of the magnetic atom, and can significantly lower its magnetic moment.
Magnetism in the single atom, or zero-dimensional limit, is quite different from
the macroscopic, collective phenomenon in solids. Approaching the problem of
magnetism from this atomic limit is not just an exotic approach to magnetic
storage, but also a novel way to bridge the quantum and classical worlds. More work
is needed in this field, not just to continue the search for single atom magnets with
large anisotropy, but also to explore the viability of reading and writing these ‘bits’.
91
Of course, we must also understand the thermal limitations, since these experiments
are mostly around 4 K.
In our work with the transition-metal dichalcogenide 1T -TaS2, we identified the
vibrational signature of the low-temperature commensurate CDW phase, which
features two sets of vibrational modes separated by a 100 cm−1 gap. Our
experimental collaborators measured the low-temperature Raman spectra of these
materials, and uncovered that the commensurate phase is robust to thinning down
to even one layer. This result is interesting because earlier work did not detect the
electronic or structural transition for few-layer flakes.
The question of whether the well-known metal-insulator transition still
accompanies the structural transition in the 2D limit remains open. Transport
measurements and scanning tunneling spectroscopy, for example, could reveal if a
clean single-layer sample undergoes an electronic transition to an insulating state.
Also, the exfoliation or growth of flakes of varying few-layer thicknesses could reveal
whether the structural and/or electronic transition is always present upon thinning,
or only reentrant. Finally, the bulk electronic transition may be best explained by
c-axis order in the commensurate CDW phase. A theoretical investigation into the
energetics and electronic structure of different stackings would be an important
work.
Our calculations for a freestanding flake of single-layer 1H-TaS2 showed that it
is susceptible to a charge density wave similar to that found in the bulk. We also
showed that electron doping tends to suppress this charge density wave. This is
important for separating the intrinsic properties of the material from the impact of
its immediate environment (which becomes increasingly important with thinning).
When 1H-TaS2 interfaces with a metal substrate such as Au(111), this becomes an
important consideration. Looking more deeply at the origin of the charge density
92
wave and its suppression, we studied quantities associated with the electron-phonon
interaction. Surprisingly, the electronic screening from the lone metallic band was
not enough to account for all the phonon instabilities that we found in our
calculations, indicating that the ‘bare’ frequencies have strong momentum, doping,
and strain dependencies as well.
Of course, charge carrier doping, as considered in our DFT calculations, is not
the only way the substrate could affect the charge density wave of 1H-TaS2. For
example, hybridization between substrate and TaS2 electronic states could also be
important. One could actually study the system of 1H-TaS2 on Au(111) (and other
substrates, like graphene, hexagonal BN, etc.), with supercells of different sizes and
orientations between the two materials. This would provide additional insight on the
observed absence of a charge density wave in single-layer 1H-TaS2 on Au(111),
taking into account both charge transfer and hybridization effects.
93
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