First-principles study of charge density waves, electron-phonon coupling, and superconductivity in transition-metal dichalcogenides
A Dissertation submitted to the Faculty of the Graduate School of Arts and Sciences of Georgetown University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics
By
Yizhi Ge, B.S.
Washington, DC July 22, 2013 Copyright c 2013 by Yizhi Ge All Rights Reserved
ii First-principles study of charge density waves, electron-phonon coupling, and superconductivity in transition-metal dichalcogenides
Yizhi Ge, B.S.
Dissertation Advisor: Amy Y. Liu, Ph.D.
Abstract
In this thesis we investigate the electronic and vibrational properties of sev eral transition metal dichalcogenide materials through first principles calculations.
First, the charge density wave (CDW) instability in 1T TaSe2 is studied as a func tion of pressure. Density functional calculations accurately capture the instability at ambient pressures and predict the suppression of the CDW distortion under pressure. The instability is shown to be driven by softening of selected phonon modes due to enhanced electron phonon matrix elements, rather than by nesting of the Fermi sur face or other electronic mechanisms. We also discuss the possibility of electron phonon superconductivity in compressed 1T TaSe2.
Another polymorph of TaSe2 is then investigated. We focus on the origin of the
CDW instability in bulk and single layer 2H TaSe2. The role of interlayer interactions and the effect of spin orbit coupling are examined. The results show that the CDW instability has weak dependence on interlayer interactions and spin orbit coupling, which is in contrast to the closely related 2H NbSe2 material, where the CDW ordering vector is predicted to depend on dimensionality.
The electron phonon interaction in electron doped single layer MoS2 is also studied. The calculation predicts a weak coupling at low doping levels. It then grows rapidly to a maximum of λ 1.7, after which it begins to decrease with additional ≈
iii doping. The superconducting transition temperature is expected to follow the same trends. This behaviour is explained by the appearance, disappearance, growth, and shrinkage of Fermi sheets with different orbital character. These results, which are similar to the experimentally observed superconducting dome in gate tuned thin
flakes of MoS2, reveal the importance of having the right mix of states at the Fermi level to enhance the electron phonon interaction in this material. Finally, we present a new implementation of an iterative algorithm for the cal culation of lattice thermal conductivity. By applying it to a simple two dimensional system with interatomic interactions described by pair potentials, we show that the algorithm works, but further work is needed to improve the computational efficiency of the method.
Index words: density functional theory, charge density wave, superconductivity, transition metal dichalcogenides, TaSe2, MoS2
iv ACKNOWLEDGEMENTS
First of all, I would like to thank my thesis advisor, Dr. Amy Liu, for her patience and encouragement during the past years. Amy is a great advisor. She has been involved in every aspect of my research experience, gave me lessons from the fun damental physics to conducting research work of complex topics. Since I joined her group, the way she does research has shaped my attitude towards it. And the impact would be far reaching into the future. It has been an unforgettable experience to work with Dr. Liu. I would like thank Prof. Freericks, Prof. Barbara, and Prof. Kertesz for their guidance and support in my research work and in final defense. One of the rewarding aspects of my graduate life has been working with a strong group. I would like to thank Wen Shen and Jesus Cruz for helpful discussions in the oretical physics and computational techniques whenever I faced a problem. Working together with them has been constructive and fun. Finally, I would like to thank my wife, my parents, and the whole family for their never ending support and love.
v Table of Contents
Chapter 1 Introduction...... 1 1.1 Transition Metal Dichalcogenide Materials ...... 1 1.1.1 Bonding and Electronic Structure ...... 2 1.1.2 Charge Density Waves in TMDs ...... 5 1.1.3 Superconductivity in TMDs ...... 8 1.1.4 Interlayer Interactions and Dimensionality ...... 9 1.2 Electronic Structure Calculations ...... 10 1.3 OutlineofThesis ...... 13 Bibliography...... 15 2 TheoreticalBackground ...... 19 2.1 Density Functional Theory ...... 19 2.1.1 Thomas Fermi Theory ...... 21 2.1.2 Hohenberg Kohn Theorems ...... 21 2.1.3 The Kohn Sham Ansatz ...... 23 2.1.4 LDA and GGA approximations ...... 24 2.1.5 Solving the KS equations ...... 26 2.1.6 Plane Waves and Pseudopotentials ...... 26 2.2 Density Functional Perturbation Theory ...... 27 2.2.1 LatticeDynamics...... 28 2.2.2 Electron Phonon Coupling ...... 31 2.3 WannierFunction...... 32 Bibliography...... 34 3 First principles investigation of the charge density wave instability in 1T TaSe2 ...... 36 3.1 Introduction...... 36 3.2 ComputationalMethod...... 38 3.3 DescriptionofStructures ...... 39 3.4 ResultsandDiscussion ...... 40 3.4.1 Structural Instability ...... 40 3.4.2 ElectronicStructure ...... 43 3.4.3 Origin of the instability ...... 48 3.4.4 Superconductivity under pressure ...... 52
vi 3.5 Conclusions ...... 54 Bibliography...... 55 4 Effect of dimensionality and spin orbit coupling on charge density wave transition in 2H TaSe2 ...... 57 4.1 Introduction...... 57 4.2 ComputationalMethod...... 60 4.3 DescriptionofStructures ...... 60 4.4 ResultsandDiscussion ...... 62 4.4.1 Structural instability ...... 62 4.4.2 Electronicstructure...... 65 4.4.3 Origin of the instability and effect of interlayer interactions 69 4.5 Conclusions ...... 75 Bibliography...... 78
5 Phonon mediated superconductivity in electron doped single layer MoS2: Afirst principlesprediction ...... 81 5.1 Introduction...... 81 5.2 Method ...... 83 5.3 ResultsandDiscussion ...... 84 Bibliography...... 93 Appendix A LatticeThermalConductivity ...... 96 A.1 Introduction...... 96 A.2 Method ...... 97 A.2.1 LatticeDynamics ...... 97 A.2.2 Boltzmann Transport Equation ...... 99 A.2.3 Iterative Approach to Solve Linearized BTE ...... 102 A.3 Algorithm and Implementation ...... 102 A.4 Test Case: Two Dimensional Argon Solid ...... 105 Bibliography...... 108 B PublicationListofYizhiGe ...... 109
vii List of Figures
1.1 Schematics of the two common structural polytypes. (a) Hexagonal(H) symmetry with trigonal prismatic coordination around metal sites. (b) Trigonal(T) symmetry with approximately octahedral coordina tion around metal sites. Yellow dots represents chalcogens; blue dots representmetalatoms...... 3 1.2 Schematic illustration showing that the electronic structure of TMDs depends on the number of valence electrons and the coordination envi ronment. (a) corresponds to a group 5 metallic TMD with octahedral coordination, such as 1T TaSe2. The lower d subband is partially filled, making it a metal. (b) corresponds to a group 6 TMD with trigonal prismatic coordination, such as 2H MoS2. The lowest d band is filled, making it a semiconductor. The dashed line represents the Fermi level of a TMD with group 5 metal atom, such as 2H TaSe2. In this case, the lowest dz2 band is half filled, making it a metal...... 4 1.3 (a) √13 √13 CDW super lattice of 1T TaSe2. The Ta sites form a 13 atom cluster× in a Star of David shape. (b)3 3 supercell of the CDW × phase in 2H TaSe2. Different colors distinguish symmetry inequivalent metalsitesinthesupercell...... 6 1.4 The Peierls picture of charge density wave instability in a 1D lattice. Black dots represent lattice sites and the red line represents electron charge density. A gap is opened at the new zone boundary when the periodicity of the lattice is doubled. Credit: Ref. [27] ...... 7 1.5 (a) Temperature pressure phase diagram of the CDW and supercon ducting transitions in 2H NbSe2. Inset shows the dependence of super conducting Tc on pressure. Credit: Ref. [34, 35] (b) Schematic plot of the temperature pressure phase diagram of 1T TaS2. NCCDW denotes the nearly commensurate CDW phase. The low temperature commen surate CDW (CCDW) phase is also a Mott state. Pressure suppresses the CDW and superconductivity develops within the NCCDW state. Credit:Ref.[9]...... 9
viii 3.1 1T TaSe2 crystal structure and qz = 0 plane of the Brillouin zone. The large spheres (gray) represent Ta atoms and the small spheres (yellow) represent Se atoms. The in plane components of the ordering vectors for the ICDW and CCDW phases are shown. In both cases, the structure is characterized by a triplet of ordering wave vectors. Since the structure has trigonal symmetry, the M and M′ points are labeled separately...... 40 3.2 (a) Phonon spectrum of 1T TaSe2 plotted along high symmetry direc tions in the Brillouin zone. Results are shown for pressures of P = 0 and 45 GPa. (b) Dependence of P = 0 unstable acoustic modes on qz. Results are plotted for wave vectors with in plane projections of q = b1/2, corresponding to the arrow in (a). In both (a) and (b), imaginary frequencies are plotted as negative. Data points are con nectedbylinetoguideeyes...... 44 3.3 Density of states calculated for TaSe2 in the undistorted 1T structure. The Ta d and Se p projected density of states are also plotted. The Fermi level is set to zero. For the total density of states, the tetrahedron method was used to integrate over the Brillouin zone, while for the site and orbital projected densities of states, Gaussian broadening was used...... 45 3.4 Calculated bands near the Fermi level in the CCDW structure of 1T TaSe2 with (a) triclinic and (b) hexagonal stacking. The Fermi level is set to zero. To facilitate comparison, the bands are plotted along high symmetry directions of the Brillouin zone of the undistorted 1T structure. In both panels, the Wannier interpolation of the band that crosses the Fermi level is plotted as a solid (red) curve...... 47 3.5 Maximally localized Wannier function constructed for the band crossing the Fermi level in the triclinic CCDW structure of TaSe2. The dark and light (blue and yellow) coloring of surfaces distinguishes between positive and negative isosurfaces...... 48 3.6 Calculated Fermi surface nesting function for 1T TaSe2 at pressures of (a) 0 GPa and (b) 45 GPa. From left to right, the panels correspond to qz =0,π/2c, and π/c, respectively. The linear gray scale ranges from 0 (black) to 3.205 (white). The in plane component of the ICDW wave vectorisindicatedwitharrows...... 51 3.7 Calculated electron phonon coupling parameter λq for 1T TaSe2 at pressures of (a) 45 GPa and (b) 60 GPa. From left to right, the panels correspond to qz =0,π/2c, and π/c, respectively. The linear gray scale ranges from 0 (black) to 3.351 (white). The in plane component of the ICDW wave vector is indicated with arrows...... 52
ix 4.1 Crystal structure and Brillouin zone of 2H TaSe2. The large spheres (gray) correspond to Ta atoms and the small spheres (yellow) represent Se atoms. The bulk 2H structure contains two trilayers per unit cell, while the single layer structure corresponds to an isolated trilayer unit. High symmetry points in the qz = 0 plane of the hexagonal Brillouin zone are labeled, as well as a triplet of ordering vectors Qcdw. .... 61 4.2 Phonon dispersion of 2H TaSe2. (a) Bulk results are shown for three cases: scalar relativistic (SR) dispersion curves at two electronic tem peratures and fully relativistic (FR) results along Γ M for the unstable acoustic branch. (b) Single layer dispersion curves are plotted for both scalar relativistic and fully relativistic calculations. Negative values indicate imaginary frequencies. The inset in (a) shows the qz disper sion of the soft branch along the line that projects onto the CDW orderingvector...... 63 4.3 Calculated electronic band structure of (a) bulk and (b) single layer 2H TaSe2. The Fermi level is set to zero. Solid lines show the spin orbit split fully relativistic Ta d bands. The fully relativistic bulk bands were calculated at the experimental value of the c lattice constant. . . . . 66 4.4 Calculated Fermi surfaces arising from Ta d bands in 2H TaSe2. (a) Bulk Fermi surface in the scalar relativistic approximation. (b) Bulk Fermi surface calculated fully relativistically, showing spin orbit split ting of bands in the kz = π/2 plane. (c) Single layer Fermi surface in the scalar relativistic approximation. (d) Single layer Fermi surface calculated fully relativistically, with new sheets appearing as a result of the spin orbit interaction. The Γ point lies at the corners of the zone shown. The K centered sheets have strong dxy/dx2 y2 character (red), particularly near the Γ K lines, while the Γ centered− sheets have weak dxy/dx2 y2 character(blue)...... 68 4.5 Wave vector− dependence of electron phonon coupling and electronic susceptibility in bulk and single layer 2H TaSe2. Panels (a), (c), and (e) show λqν and for the soft acoustic branch in the SR bulk (σ =0.02 Ry), the SR single layer (σ =0.03 Ry), and the FR single layer (σ =0.2 Ry), respectively. Black denotes λqν = 0, while white represents the maximum value, which is different in each case. Panels (b), (d), and (f)
show the real part of the susceptibility χq′ per formula unit calculated for the SR bulk, the SR single layer, and the FR bulk, respectively, using a common gray scale. Arrows indicate Qcdw...... 71 5.1 (Color online) (a) Top view of the atomic structure of single layer MoS2. Large spheres represent Mo atoms on a triangular lattice. Small spheres represent S atoms, which lie in planes above and below the Mo plane. (b) High symmetry points in the Brillouin zone...... 84
x 5.2 (Color online) Conduction band and Fermi surface of single layer MoS2. Panels (a) and (b) correspond to a doping concentration of x = 0.03 electrons/f.u., panels (c) and (d) show results for x = 0.10 electrons/f.u., and panels (e) and (f) are for x = 0.22 electrons/f.u. Some representative phonon wave vectors that connect Fermi sheets areshown...... 85 5.3 Phonon dispersion curves calculated for single layer MoS2. Solid curves are for the undoped material, while circles show results for x = 0.06 electrons/f.u...... 87 5.4 (Color online) Electron phonon coupling parameter calculated as a function of electron doping for single layer MoS2. Contributions from different phonon wave vectors are indicated by the shading. Arrows on the horizontal axis show where Fermi sheets appear or disappear. . . 88 A.1 The lattice thermal conductivity of two dimensional argon solid cal culated using iterative method (black dots) and molecular dynamics simulationmethod[15](redsquare)...... 107
xi List of Tables
3.1 Comparison of calculated and measured structural parameters of TaSe2 in the undistorted 1T phase and the commensurate CDW phase. The lattice parameters a and c are given in Å. In the CCDW phase, δd repre sents a fractional change in the Ta Ta distance compared to the undis torted structure, with d1 (or d2) referring to the distance between a Ta site at the center of a cluster and a nearest neighbor (or next nearest neighbor) Ta site. The energy difference between the CCDW structure and the undistorted 1T structure, ∆E, is given in mRy/formula unit. Two stacking sequences were considered for the CCDW phase, triclinic (tri) and hexagonal (hex), as described in the text...... 41 3.2 Calculated electronic, vibrational, and superconducting properties of 1T TaSe2. The pressure P is in GPa, the electronic density of states at the Fermi level N(0) is in states/Ry/spin, the characteristic phonon energies are in meV, and the superconducting Tc isinK...... 53 4.1 Comparison of bulk and single layer (1L) CDW parameters. Scalar rel ativistic and fully relativistic results are presented for the total energy (per formula unit) of the 3 3 CDW structure relative to that of the undistorted structure, ∆E, and× for the fractional change in the distance from the center Ta site to its nearest neighbors in the CDW supercell, ∆R/R...... 65
xii Chapter 1
Introduction
1.1 Transition-Metal Dichalcogenide Materials
Transition metal dichalcogenides (TMDs) are materials that can be denoted as
MX2, where M is a transition metal within group 4 10, and X represents a chalcogen element. The physical and electronic properties of TMDs are dominated by anisotropy effects that are caused by the layered crystal structure and the chemical bonding these
materials adopt. Bulk TMDs can be insulators such as HfS2, semiconductors such as
MoS2, semimetals such as TiSe2, or metals such as TaS2 and TaSe2. These materials have been used as hydrodesulfurization catalysts [1], photoactive materials [2], solid lubricants [3], and energy storage media [4]. A lot of the TMDs have rich phase diagrams when external parameters such as temperature or pressure are changed. The phase diagrams show many types of insta bilities, including charge density wave (CDW) formation with commensurate, incom mensurate, and chiral order [5, 6], superconductivity [7], exciton condensation [8], and Mott insulator transitions [9]. Particularly, there is a long standing debate about the origin of the CDW instability and about the interplay between CDW formation and superconductivity in some metallic TMDs. Recently developed experimental methods have helped researchers probe these questions from new perspectives. For example, high resolution angle resolved photoemission spectroscopy (ARPES) allows a detailed mapping of the Fermi surface to look for possible gap formation in the CDW phase
1 [10]. In addition, time resolved spectroscopy can be used to decouple electronic and lattice degrees of freedom [11]. When an ultrafast laser pulse is used to excite the CDW material, on short time scales, the charge order is observed to weaken while the lattice remains unchanged. On another front, exfoliation techniques originally devel oped to make graphene [12] have been adapted to TMDs to produce few layer and single layer structures [13]. This allows investigation of how interlayer interactions and dimensionality affect the various instabilities. In this thesis, we study several TMD materials through first principles calcu lations. The effects of pressure, dimensionality, and doping on the electronic and vibrational properties are investigated to probe the origin of CDW instabilities, to study the interplay between superconductivity and CDW formation, and to explore field induced superconductivity in semiconducting TMDs.
1.1.1 Bonding and Electronic Structure
There exists a large number of different polymorphs of MX2, but the basic crystal structure consists of a sheet of metal atoms M, sandwiched by two sheets of chalco
gens X. The coordination of the metal can be either trigonal prismatic (D3h sym metry, H) or approximately octahedral (D3d symmetry, T), as illustrated in Fig. 1.1. In either case, there is fairly strong intralayer covalent bonding between the metal and the chalcogens, and weak interlayer van der Waals bonding [14]. The quasi two dimensional structure of MX2 gives rise to anisotropy in physical properties. The weak interlayer bonding permits intercalation of metal atoms, ions, or even organic molecules [15, 16], which can introduce dramatic changes in the physical properties of the host. It also allows bulk TMDs to be easily cleaved along the surface to make few layer or single layer samples [17].
2 Figure 1.1: Schematics of the two common structural polytypes. (a) Hexagonal(H) symmetry with trigonal prismatic coordination around metal sites. (b) Trigonal(T) symmetry with approximately octahedral coordination around metal sites. Yellow dots represents chalcogens; blue dots represent metal atoms.
The electronic structure of TMDs strongly depends on the number of valence electrons in the metal and its coordination environment [4]. Based on atomic energy levels, it is typical that the valence p orbitals from the chalogen X are lower than d orbitals in the metal M. In the crystal, there are 6 p bands (because there are two X atoms) per unit cell which are occupied by 8 electrons from X and another 4 electrons from M, leaving n 4 (where n is the group number of M) electrons to occupy the d bands. Depending on the coordination environment, the d bands are split by the crystal
field. When the M atoms are octahedrally coordinated, the d bands split into dz2,x2 y2 −
and dyz,xz,xy subbands of Eg and T2g symmetry respectively. With trigonal prismatic
′ ′ coordination, the d bands split into groups of dz2 , dx2 y2,xy, and dxz,yz of A1, E , and − ′′ E symmetries, respectively.
3 For example, based on these arguments, 1T TaSe2 (n=5) is expected to have one electron occupying the d bands and be metallic, as shown in Fig 1.2 (a). On the other hand, in 2H MoS2 (n=6), there should be one filled d band and a gap to the next set of d bands, as shown in Fig. 1.2 (b). Of course, Fig. 1.2 just illustrates a simple ionic model of the electronic structure. In a real compound, because of hybridization between different atomic orbitals, there are no pure p or d bands. Moreover, some of the bands could overlap in energy or have further splittings, making the picture more complicated. Nevertheless, the simple ionic model provides a good starting point for understanding trends in the electronic structure within the MX2 family.
Figure 1.2: Schematic illustration showing that the electronic structure of TMDs depends on the number of valence electrons and the coordination environment. (a) corresponds to a group 5 metallic TMD with octahedral coordination, such as 1T TaSe2. The lower d subband is partially filled, making it a metal. (b) corresponds to a group 6 TMD with trigonal prismatic coordination, such as 2H MoS2. The lowest d band is filled, making it a semiconductor. The dashed line represents the Fermi level of a TMD with group 5 metal atom, such as 2H TaSe2. In this case, the lowest dz2 band is half filled, making it a metal.
4 1.1.2 Charge Density Waves in TMDs
Several TMD materials have charge density wave (CDW) phases under certain conditions. A CDW is a periodic distortion in the density of charge carriers with spatial period characterized by the CDW wave vector [18]. Typically, TMD materials with group 5 metals (V, Nb, Ta) are rich in CDW phases [19, 20]. Below some onset temperature, CDW phases emerge, characterized by a new periodicity in both the electron charge density and the lattice. Several experimental techniques can be used to characterize the CDW. Electron [20] and neutron diffraction [21] experiments allow determination of the crystal lattice structure, and the scanning tunnelling microscope (STM) which image the charge density modulation in the CDW phase [22] can be used to characterize the real space orientation and phase of the CDW relative to underlying lattice. Furthermore, measurements of specific heat and resistivity [23] show the phase transition between the normal state and the CDW state. It is interesting to observe that the effect of the CDW transition on electrical resistivity in different polytypes of TMDs are strikingly
different. For example, in 1T TaS2, which undergoes a series of CDW phase transitions as temperature is lowered, the resistivity jumps to a higher value at each transition, and within each CDW phase, the resistivity has a semiconductor like temperature dependence. On the other hand, in 2H TaS2, there is a subtle change in the slope of the resistivity versus temperature curve at the CDW transitions, and the resistivity has a metallic temperature dependence in the CDW phases [24].
In the 1T family of TaSe2 and TaS2, the first CDW phase to appear upon cooling is incommensurate with the original lattice [20]. Upon further cooling (T < 473 K for 1T TaSe2 and T < 180 K for 1T TaS2), the CDW ordering vector changes from nearly commensurate to commensurate with the underlying lattice. Relative to the
5 hexagonal arrangement of M atoms in the undistorted phase, the lattice now develops a √13 √13 supercell structure as seen in Fig. 1.3 (a), where the M atoms displace × to create thirteen atom clusters. On the other hand, the 2H polymorph of NbSe2 [21],
TaSe2, and TaS2 [20] all develop CDW phases (T < 39 K for NbSe2, T < 120 K for TaSe , T < 78 K for TaS ) with a 3 3 supercell structure, where seven atom clusters 2 2 × form as shown in Fig. 1.3 (b). The driving mechanism behind CDW formation has long been debated. Explanations of the CDW instability include Fermi surface nesting [20], van Hove singularities [25], and electron phonon coupling [26].
Figure 1.3: (a) √13 √13 CDW super lattice of 1T TaSe . The Ta sites form a 13 atom × 2 cluster in a Star of David shape. (b)3 3 supercell of the CDW phase in 2H TaSe2. Different colors distinguish symmetry inequivalent× metal sites in the supercell.
Peierls [27] first presented the idea that in a one dimensional (1D) metal, CDW formation is related to perfect nesting of the Fermi surface. Consider a chain of atoms in 1D, with one electron per atom. There is a half filled band. If the periodicity is doubled, the new Brillouin zone (BZ) is half of the original zone. The original half filled band folds into a completely filled band and an empty band, with a gap at the BZ boundary. The system gains energy through lowering of the electronic states while elastic energy is paid for the accompanying ionic displacements. Since the electronic
6 density of states diverges at the band edge in 1D, this system is always unstable to formation of a CDW with a wave vector that nests the Fermi surface ( QCDW =2kF ). Because of the Peierls picture, it has been widely assumed that CDW transitions in real materials are driven by Fermi surface nesting.
Figure 1.4: The Peierls picture of charge density wave instability in a 1D lattice. Black dots represent lattice sites and the red line represents electron charge density. A gap is opened at the new zone boundary when the periodicity of the lattice is doubled. Credit: Ref. [27]
However, angle resolved photoemission studies (ARPES) on 2H NbSe2 [28] don’t find the expected gap opening on the Fermi surface during the CDW transition. In addition, the nesting function for the measured Fermi surface is not the strongest at the CDW wave vector. Furthermore, if Fermi surface nesting and the associated diver gence of the electronic susceptibility is driving the instability, all phonon branches should soften at the ordering wave vector, but this is not what is observed. These experimental findings are all in contrast with the Peierls picture. Some theoretical papers [26] have argued that a wave vector dependent electron phonon interaction plays an indispensable role in CDW transitions. While Fermi surface nesting, which is a purely electronic effect, may contribute to the CDW tran
7 sition, it is not the only consideration. In fact, based on calculated Fermi surfaces, a quantitative search for strong nesting at CDW wave vectors often fails [29].
1.1.3 Superconductivity in TMDs
TaSe2 [30], TaS2 [9], and NbSe2 [31] are examples of metallic TMDs with group
5 metal atoms where superconductivity has been found. 2H NbSe2 is the best char acterized superconducting TMD. It has the highest T value ( 7.4 K) and a sharp c ∼ superconducting transition [31]. Superconductivity and CDW order both result from instabilities that lead to cooperative electronic behavior and they coexist in this mate rial. Pressure and doping studies have been used to probe the superconducting and CDW transitions in some TMD materials [32, 33]. Often, pressure or doping reduces
the CDW transition temperature and increases the Tc of the superconducting state.
As shown in Fig.1.5 (a), in 2H NbSe2, there appears to be competition between CDW and superconductivity transitions as pressure varies [34, 35].
1T TaS2 provides a very different example for how CDW order and superconduc
tivity can coexist [9]. At ambient pressure and low temperature, 1T TaS2 is in the commensurate CDW (CCDW) phase described by the √13 √13 supercell in Fig. × 1.3 (a). According to band theory, the CCDW phase should be metallic, with the Fermi level in the middle of a narrow d band. Yet, resistivity measurements find it to
be insulating. Thus 1T TaS2 is believed to simultaneously undergo CCDW and Mott transitions. By applying pressure, the material adopts a nearly commensurate CDW phase (NCCDW) and superconductivity develops within the NCCDW state. Super conductivity persists at high pressure even after the CDW has already disappeared.
Fig. 1.5 (b) shows the temperature pressure phase diagram of 1T TaS2. Because the
superconducting Tc appears insensitive to the CDW order, it is believed that the NCCDW phase is textured, consisting of insulating CCDW domains separated by
8 metallic interdomain regions. Thus in this material the CDW and superconducting phases are separated in real space.
Figure 1.5: (a) Temperature pressure phase diagram of the CDW and superconducting transitions in 2H NbSe2. Inset shows the dependence of superconducting Tc on pres sure. Credit: Ref. [34, 35] (b) Schematic plot of the temperature pressure phase dia gram of 1T TaS2. NCCDW denotes the nearly commensurate CDW phase. The low temperature commensurate CDW (CCDW) phase is also a Mott state. Pressure sup presses the CDW and superconductivity develops within the NCCDW state. Credit: Ref. [9].
1.1.4 Interlayer Interactions and Dimensionality
Advances in nanoscale material fabrication techniques have opened up new oppor tunities for layered TMD materials in nanoelectronics and optoelectronics. Removal of interlayer interactions, introduction of quantum confinement, and changes in sym metry can lead to dramatic changes in the electronic structure of single layer or few layer TMDs compared to their bulk. For example, bulk MoS2 is a semiconductor with an indirect gap of 1.2 eV [36]. When it is exfoliated into an atomically thin single layer, the original indirect band gap is increased by quantum confinement, and the smallest gap is now a direct gap of 1.9 eV. The large direct gap makes single layer
9 MoS2 very attractive for optoelectronic as well as other nano devices, possibly on flexible substrates. Researchers are also interested in how dimensionality affects CDW instabilities.
A density functional theory investigation of single layer NbSe2 [37] predicts that, like the bulk, it should also undergo a CDW instability, but that in single layers, the CDW wave vector differs from that of the bulk. Moreover, an order of magnitude larger reduction of total energy during the CDW transition has been predicted for the single layer as compared to the bulk. This indicates that a much higher CDW
transition temperature is expected in atomically thin layers of NbSe2. On the other
hand, experiments have found that an exfoliated few layer patch of 2H TaS2 shows no CDW phase at low temperature [38]. This is in contrast with its bulk and indicates that reduced interlayer interactions can affect whether a CDW instability even exists. In general, interlayer interactions, dimensionality, and pressure act as good control parameters for tuning the structural, electronic, and vibrational properties of the TMDs. This in turn can provide insight about the various electronic orders exhibited by these fascinating materials.
1.2 Electronic Structure Calculations
Density functional theory (DFT), which is based on the Hohenberg Kohn theorem [39], states that all ground state properties of a many body system can be completely determined through the ground state charge density. The central concept of DFT is to describe a system of interacting quantum particles via its charge density rather than its many body wave function. For a system of N electrons in a solid, this means that the key quantity depends on three spatial coordinates rather than 3N degrees of freedom. Moving from many body wave functions to charge density significantly
10 reduces the computational complexity and gives the possibility of developing algo rithms that allow very large systems to be treated. The promise of DFT based electronic structure calculations was limited by com putational power from the 1960s to 1980s. It was the rapid advancement in computer technology in the 1990s that made DFT more and more popular in condensed matter physics and materials science. Now, it is the most widely used approach for studies of ground state properties in metals, semiconductors, and insulators, and it is increas ingly important for quantitative studies of molecules and other finite systems. It usually gives satisfactory results for total energies, forces acting on atoms, structural parameters, and charge densities. For example, in molecular structures, the calculated bond lengths can be accurate to within 1 2 % [40]. A wide variety of physical properties of materials depend on their lattice dynamics. Examples include specific heat, thermal expansion, and lattice thermal conductivity. Other phenomena like electrical resistivity and superconductivity arise from the inter action between electrons and phonons. So being able to calculate vibrational proper ties of materials is important. Since vibrational frequencies are related to differences in total energy for different atomic positions, they are accessible via DFT. Density Functional Perturbation Theory (DFPT) [41] offers an efficient way to calculate vibra tional spectra within DFT. DFPT calculates the interatomic force constants (IFC) with the knowledge of the ground state charge density n(r), as well as of its linear response to a distortion of the ionic positions. The IFC defines the dynamical matrix whose eigenvalues determine the normal mode frequencies. The power of DFPT is that it avoids large super cell calculations required by the "frozen phonon" method in which total energy differences are calculated directly. Also DFPT can treat any generic phonon wave vector.
11 The "2n+1 theorem" [42] states that knowledge of the wave function to all orders 0 through n in a perturbation determines the energy to order 2n+1. This mean our knowledge of the first order derivative of the wave function through DFPT calcula tions allows determination of the third order derivative of energy. This opens up the door for us to study anharmonic force constants arising from phonon phonon scat tering processes. It is the anharmonic phonon scattering process that gives rise to the intrinsic lattice thermal conductivity in materials. With the phonon spectrum as well as the third order IFC one can solve the Boltzmann transport equation to determine lattice thermal conductivity. In spite of the success of DFT, there exists a large class of materials that DFT fails to adequately describe. It is the approximate nature of the exchange correlation func tional that is responsible for the success and failure of DFT. The first type of these functionals is the local density approximation (LDA), in which the exchange and cor relation energy densities are approximated by those for a homogeneous electron gas with density equal to the local density in the homogeneous system. The Generalized Gradient Approximation (GGA) is the first step beyond conventional LDA, and it takes into account the local gradient of charge density. The next major advance is the development of hybrid functionals [43] which combine the orbital dependent Hartree Fock exchange energy with a LDA or GGA functional. The van der Waals functional [44] is another example of an effort to improve the functional, in this case focusing on the non bonded interaction in layered materials. New methodologies, such as LDA+U [45], many body perturbation theory (e.g. GW method [46]), and TD DFT [47] have been developed out of the DFT framework to tackle problems of strong electron cor relations and time dependent phenomena. In addition, a combined density functional theory dynamical mean field theory (DFT DMFT) [48] approach has shown promise for describing real materials that contain both itinerant electrons, which are well
12 described by band theory, and local orbitals, which have correlation effects that are well described by DMFT.
1.3 Outline of Thesis
The remainder of this thesis is organized as follows. Theoretical background about density functional theory is presented in Ch. 2. In Ch. 3 we apply DFT methods to study the electronic and vibrational properties of 1T TaSe2 as well as the origin of the CDW transition. 1T TaSe2 is similar to 1T TaS2 in crystal structure but offers a cleaner phase diagram, i.e. without the Mott transition and the textured NCCDW
phase. Comparison of the electronic structure of 1T TaSe2 and 1T TaS2 reveals that there is stronger p d hybridization in the selenide. Using pressure as a probe, we find −
that the CDW instability in 1T TaSe2 is driven by phonon softening due to strong electron phonon coupling rather than by an electronic mechanism such as Fermi surface nesting. Next, we study how dimensionality affects the CDW instability in
2H TaSe2 in Ch. 4. We predict that the CDW manifests at the same (or nearly
the same) wave vector in single layer and bulk 2H TaSe2. This is in contrast to 2H
NbSe2, where previous calculations [49] indicated that the ordering vector depends on dimensionality. The instability is found to be determined by the momentum space structure of both the electron phonon coupling and the dielectric response function.
This is also different from our findings for 1T TaSe2 (Ch. 3), where the ordering wave vector is completely determined by the momentum dependence of the electron phonon coupling constant. In Ch. 5, we study the electron phonon interaction in electron
doped single layer MoS2 and show that it depends sensitively on which valleys of the conduction band are occupied. The predicted trends in the electron phonon coupling
constant and superconducting Tc include abrupt changes at doping levels where Fermi
13 surface sheets appear or disappear. The results are similar to recent experimental
findings of field induced superconductivity in thin flakes of MoS2. Finally, in the appendix, we discuss an algorithm for calculating lattice thermal conductivity. Using a Lennard Jones solid as an example, we show that the algorithm works, but further work is needed to make it a practical method for more complex materials.
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18 Chapter 2
Theoretical Background
2.1 Density Functional Theory
Density Functional Theory (DFT) is a highly successful theory for describing structural and electronic properties of atoms, molecules, and crystals [1]. It emphasizes the charge density as the relevant quantity for calculating ground state properties of the system. It has become a common tool for doing calculations for real materials in a way that does not require empirical input. The Hamiltonian of a system of interacting electrons and ions is
~2 ~2 2 2 2 1 ZI ZJ e Htot = RI ri + + − 2MI ∇ − 2me ∇ 2 RI RJ I i I=J | − | (2.1) 1 e2 Z e2 I . 2 ri rj − RI ri i=j I,i | − | | − | The indices I, J run through all the ions, whereas i, j are electron indices. RI and ri are
the ion and electron positions, respectively. ZI is the atomic number of ion I. The first term on the right side of the equation is the kinetic energy of the ions, and the second term is the electron kinetic energy. The third one describes the Coulomb interaction between ions. The fourth item is the Coulomb interaction between electrons. The final term is the potential energy between ions and electrons. The time independent Schrodinger equation for the system is
HΦ(RI , RJ ,...ri, rj,...)= EΦ(RI , RJ ,...ri, rj,...), (2.2)
19 where Φ(RI , RJ ,...ri, rj,...) is the many body wave function of the system. However, for systems containing more than a few particles, it is impossible to solve this equation in practice due to its complexity. The first step in tackling this problem is to apply the Born and Oppenheimer [2] approximation (BO). Because ionic masses are usually much larger than the mass of an electron, the ions move much more slowly than electrons. Then the movement of ions and electrons can be treated separately. It is reasonable to assume the ions are fixed in position when we deal with movement of electrons. So the total wave function becomes
Φ( R , r )=Ψ ( R )Ψ ( r ; R ), (2.3) { I } { i} ion { I } ele { i} { I }
where Ψ ( R ) is the wave function for the ions and Ψ ( r ; R ) is the wave ion { I } ele { i} { I } function of electrons given a particular ionic configuration. R and r are the { I } { i} collective set of ion and electron position coordinates, respectively. Given the BO approximation, Eq. 2.1 can now be divided into two separate equa tions: H Ψ ( r ; R )= V ( R )Ψ ( r ; R ), (2.4) ele ele { i} { I } { I } ele { i} { I } and ~2 2 R R R [ RI + V ( I )]Ψion( I )= EΨion( I ), (2.5) − 2MI ∇ { } { } { } I where
~2 2 2 2 2 1 ZI ZJ e 1 e ZI e Hele = ri + + . (2.6) − 2me ∇ 2 RI RJ 2 ri rj − RI ri i I=J i=j I,i | − | | − | | − | Eq. 2.4 considers the electronic wave function with a fixed ionic configuration. Eq. 2.5 can be solved with the knowledge of V ( R ) from Eq. 2.4. Thus we now can { I } focus on the problem of electrons moving in a static external potential produced by the fixed ions.
20 2.1.1 Thomas-Fermi Theory
In 1927, Thomas and Fermi [3] proposed a density based theory to deal with many electron systems. In the Thomas Fermi (TF) model, the kinetic energy density of a many electron system is approximated by that of a homogeneous independent electron system [4] with charge density equal to the local density n(r):
5/3 TT F [n]= A n(r) dr, (2.7) where n(r) is the charge density of the homogeneous electron gas and A is a constant. The total energy is determined by adding electrostatic energy and interaction with external potential Vext(r) to the kinetic energy, so that
5/3 1 n(r)n(r′) E [n]= A n(r) dr + n(r)V (r)dr + drdr′. (2.8) T F ext 2 r r | − ′| The second term is the electrostatic energy of the ion electron interaction, and the final term is the classical Hartree energy. The original Thomas Fermi method overes timates the total energy because the exchange and correlation effects, which always lower the energy, are not taken into account. The ground state density and energy can be obtained by minimizing Eq. 2.8 with the constraint of constant electron number N. Because of the crude approximations used, the theory misses important physics of electrons in real materials. However, it provides a simple example of using the electron density as a fundamental variable, which was later developed by DFT theory.
2.1.2 Hohenberg-Kohn Theorems
At first glance, from Eq. 2.8, the ground state energy is a functional of both the density and external potential. If a mapping between the two physical quantities
21 can be established, then the total energy would only be a functional of one of these quantities. This is exactly what Hohenberg and Kohn did in 1964 [5]. They proved that DFT can be applied to a system of interacting particles in an external potential
Vext(r). There are two theorems. First theorem: For any system of interacting particles in an external potential
Vext(r), the potential Vext(r) is determined uniquely, except for a constant, by the ground state particle density n(r). This means ground state particle density deter mines all properties of the system. Second theorem: There exists a universal functional F[n(r)] of the particle den sity, independent of the external potential, such that the global minimum value of the energy functional E[n(r)]= n(r)Vext(r)dr+F[n(r)] is the exact ground state energy of the system and the exact ground state density n0(r) minimizes this functional. So the exact ground state energy and density are fully determined by the functional E[n(r)]. The Hohenberg Kohn theorems formulate the quantum many body problem in terms of the particle density n(r) as the basic variable. But practically, it is still impos sible to calculate any property of a system because the universal functional F[n(r)] is unknown. This leads to the Kohn Sham [6] approach, which assumes the ground state density of the interacting system equals that of a fictitious non interacting system with all the difficult many body terms incorporated into the exchange correlation functional of the charge density. While the exact exchange correlation functional is still unknown, various approximation can be made, allowing for a practical imple mentation of DFT within the Kohn Sham framework.
22 2.1.3 The Kohn-Sham Ansatz
Utilizing the Hohenberg Kohn theorem, Kohn and Sham introduced a method to minimize the energy functional E[n(r)] with respect to charge density to find the ground state energy under the constraint that the total number of electrons is constant: δ [E[n(r)] n(r)dr]=0 (2.9) δn(r) − where is a Lagrange multiplier. The corresponding Euler equation is:
δE[n(r)] = . (2.10) δn(r)
Kohn and Sham separated F[n(r)] into three parts, so that E[n(r)] becomes
1 n(r)n(r′) E[n(r)] = T [n(r)] + drdr′ + E [n(r)] + n(r)V (r)dr, (2.11) 2 r r XC ext | − ′| where T[n(r)] is defined as the kinetic energy of a non interacting electron gas with
charge density n(r). Using Hartree atomic units with ~ = me = 4π/ǫ0 =1,
N 1 2 T [n(r)] = ψ ∗(r) ψ (r)dr. (2.12) −2 i ▽ i i=1
We can now rewrite Eq. 2.10 in terms of an effective KS potential, VKS(r), as follows δT [n(r)] + V (r)= , (2.13) δn(r) KS where δE [n(r)] δE [n(r)] V (r)= V (r)+ H + XC KS ext r r δn( ) δn( ) (2.14)
= Vext(r)+ VH (r)+ VXC (r),
and
n(r′) V (r)= dr′. (2.15) H r r | − ′| 23 Now, if we consider a system that is truly of non interacting electrons with an
external potential equal to VKS(r), then the same analysis would lead to exactly the same Eq. 2.13. To find the ground state energy and density of such a system, one has to solve the one electron equation
1 ( 2 + V (r))ψ (r)= ǫ ψ (r) (2.16) −2 ▽ KS i i i where ψi(r) is the one electron wavefunction. The charge density is then determined by N n(r)= ψ (r) 2. (2.17) | i | i=1 Equations 2.16, 2.17 have to be solved self consistently because VKS(r) depends on the density through the exchange correlation potential VXC (r) and Hartree potential
VH (r), and the density depends on the wavefunctions. So far the Kohn Sham approach has mapped the original interacting system to a non interacting system with an effec tive Kohn Sham single particle potential VKS(r), and the two systems have the same ground state density. In practice, the Kohn Sham approach is an approximation to the original many body problem because the exchange correlation functional EXC [n(r)] is unknown. In order to obtain a satisfactory description of a real system, it is important to have an accurate functional EXC [n(r)]. Two of the most widely used approximations for the exchange correlation potential are the local density approximation (LDA) [6] and the generalized gradient approximation (GGA) [7].
2.1.4 LDA and GGA approximations
In the LDA, the exchange correlation energy per electron at a point r is taken to be the same as that for a homogeneous electron gas with density equal to that at
24 point r. Kohn and Sham already pointed out this idea even in their original paper
[6]. The exchange correlation functional EXC [n(r)] can then be expressed as:
LDA hom EXC [n(r)] = n(r)ǫXC (n(r))dr hom hom = n(r)[ǫX (n(r)) + ǫC (n(r))]dr (2.18) LDA LDA = EX [n(r)] + EC [n(r)]
LDA The exchange energy functional EX [n(r)] can be written analytically using the result for the homogeneous electron gas [8]:
LDA hom EX [n(r)] = n(r)ǫX (n(r))dr (2.19) 3 3 1/3 = ( ) n(r)4/3dr −4 π where 3 3 1/3 ǫhom = ( ) n(r)1/3 (2.20) X −4 π is the exchange energy density. Analytical expression of the correlation energy func
LDA tional EC [n(r)] in the homogeneous electron gas is unknown except in the high
and low density limits. Most LDA formulas use analytic forms for ǫC , fit to accurate quantum simulation results of the correlation energy at intermediate density, and designed to reproduce the exact result at the high and low density limits. A few examples of the most popular LDA formulas include Perdew Zunger (PZ81) [9] and Perdew Wang (PW92) [10]. In principle, the LDA should work best in systems where the density varies slowly. In fact, it turns out to work surprisingly well even in some systems where the electron density is rapidly varying. In general, the LDA tends to underestimate bond length by 1 2% [11], overestimate binding energies by 10 50% [12, 13], and underestimate band gaps by 50% [14]. ∼ 25 The XC energy of an inhomogeneous charge density could be significantly different from that of a homogeneous electron gas. The GGA addresses this problem by allowing the functional to depend on not just the local density, but the local gradient of the density as well. It generally works better than LDA in predicting bond lengths, crystal lattice constants, binding energies, and so on, but often overcorrects [15]. However, it should be noted that for materials where the electrons tend to be very localized and strongly correlated, both GGA and LDA can give qualitatively wrong results.
2.1.5 Solving the KS equations
The KS equations must be solved self consistently because the KS potential VKS and the electron density n(r) are related. Usually these equations are solved by starting with an initial guess for the electron density. For example, the initial guess could be a superposition of atomic electron densities. Then the KS potential VKS is calculated and Eq. 2.16 is solved to obtain the one electron eigenvalues and wave functions. Given the wave functions, a new electron density is calculated. For stability, the calculated electron density is often mixed with the electron density from the pre vious iteration to get a new density for the next iteration. The KS equations with the new VKS are solved again to obtain the next round of charge density. The itera tion continues until some self consistent conditions have been met. For example, the change of total energy or electron density between two iterations is smaller than some convergence criterion. After that, various quantities of interest can be calculated.
2.1.6 Plane Waves and Pseudopotentials
One approach for solving the KS equations is to expand the wave functions in a basis. Plane waves provide a simple unbiased choice of basis [16]. According to the Bloch theorem [17], the eigenfunctions of the KS equations with wave vector k can
26 be written as:
ik r ψk(r)= e uk(r), (2.21) where uk(r) is a periodic function with crystal periodicity and can be expanded in a Fourier series:
1 iG r u (r)= c (G)e , (2.22) k √ k Ω G where G is a reciprocal lattice vector, Ω is the crystal unit cell volume, ck(G) is the
iG r expansion coefficient, and e is a plane wave with wave vector G. So the eigenfunc tion ψk can be expanded in plane wave basis as:
1 i(k+G) r ψ (r)= c (G)e . (2.23) k √ k Ω G Since it is mainly valence electrons that are important for chemical interactions, it is convenient to treat only valence electrons, and to use pseudopotentials [18] to represent the interaction between valence electrons and ion cores (i.e. the nucleus plus frozen core electrons). In addition to reducing the number of electrons in the problem, pseudopotentials have another important advantage, particularly when used in con junction with a plane wave basis. Pseudopotential wave functions are constructed to match the true atomic valence wave function beyond a given radius, but within the core region, they can be much smoother than atomic wave functions. This means that the number of plane waves needed to expand the wave function can be significantly reduced. With a plane wave basis, the accuracy of the calculation can be improved in a systematic and unbiased way simply by increasing the size of the basis set.
2.2 Density Functional Perturbation Theory
Many physical properties of a solid depend upon its response to some perturba tions such as ionic displacements, lattice strain, and electric field. Density Functional
27 Perturbation Theory (DFPT) is a powerful technique that allows calculation of the corresponding properties (e.g., phonon spectra, elastic tensors, dielectric polariza tion) within the DFT framework. Detailed reviews about DFPT can be found in Ref.[19, 20].
2.2.1 Lattice Dynamics
The adiabatic Born Oppenheimer approximation allows one to decouple the ionic degrees of freedom of a solid state system from its electronic degrees of freedom. In the Born Oppenheimer approximation, it is assumed that electrons respond instanta neously to changes in ionic positions, so the electrons are always in the ground state of the current ionic configuration. The total energy of the system E( R ) can be { I } calculated from DFT. Near the equilibrium configuration of R , E( R ) can be { I } { I } expanded in powers of small deviations. The equilibrium of the system is given by
∂E( RI ) FI = { } =0, (2.24) − ∂RI
th where RI is the position of the I ion, and FI is the force acting on the ion. In the harmonic approximation, the vibrational frequencies are related to the second derivative of the total energy:
1 ∂2E( R ) det { I } ω2 =0. (2.25) √M M ∂R ∂R − I J I J The direct approach to calculating vibrational frequencie s is called the "frozen phonon" method, in which the total energy or forces are computed as a function of nuclear positions. This method is widely used since it is conceptually simple to carry out using standard DFT codes. However, it may involve using large supercells that
28 are computationally expensive, and it is applicable to only phonon wave vectors that are commensurate with the lattice. Another common procedure uses linear response within the density functional framework. It is called density functional perturbation theory (DFPT). In this method, the second order perturbation of the DFT total energy δ2E is obtained by expanding the DFT total energy with respect to the changes in electron wave functions to the first order and external potentials up to the second order. Within DFT, δ2E is calculated using the Hellmann Feynman theorem [21]:
∂E ∂H λ = Ψ λ Ψ , (2.26) ∂λ λ| ∂λ | λ
where Ψλ is the eigenfunction of the Hamiltonian Hλ corresponding to the Eλ eigen value, and λ is a parameter on which the Hamiltonian depends. The force acting on ion I in the electronic ground state is:
∂E( RI ) ∂Hele( RI ) FI = { } = Ψ( RI ) { } Ψ( RI ) , (2.27) − ∂RI − { } | ∂RI | { }
where Ψ( R ) is the ground state wave function. The Hamiltonian H ( R ) { I } ele { I } depends on the ionic positions R via the electron ion interaction which couples { I } the electronic degree of freedom through electron charge density. In such a case
∂V RI (r) ∂EION ( RI ) F = n(r) { } dr { } , (2.28) I − ∂R − ∂R I I where V RI (r) is the electron ion interaction term { } 2 ZI e V RI (r)= . (2.29) { } − RI ri I,i | − | and E ( R ) is the ion ion interaction term ION { I } 2 e ZI ZJ EION ( RI )= . (2.30) { } 2 RI RJ I=J | − | 29 The second order derivative of the total energy appearing in Eq. 2.25 is obtained by differentiating the Hellmann Feynman forces with respect to the ionic coordinates,
2 2 2 ∂ E( RI ) ∂FI ∂n(r) ∂V RI (r) ∂ V RI (r) ∂ EION ( RI ) { } = = { } dr+ n(r) { } dr+ { } . ∂R ∂R −∂R ∂R ∂R ∂R ∂R ∂R ∂R I J J J I I J I J (2.31) Eq. 2.31 reveals that in order to calculate the second order derivative of total energy with respect to ionic positions, it is necessary to know the ground state charge density n(r) as well as its linear response to the distortion of ionic positions ∂n(r) . ∂RJ In the Kohn Sham formulation, the ground state charge density is given as a sum
over the density of the occupied Kohn Sham orbitals ψi(r). Therefore its derivative with respect to the ionic position is:
N/2 ∂n(r) ∂ψn(r) =4Re ψ∗(r) . (2.32) ∂R n ∂R I n=1 I The derivative of Kohn Sham orbitals, ∂ψn(r) is ∂RI
∂ψn(r) ∂VSCF (r) ∂ǫn (HKS ǫn) = ( )ψn, (2.33) − ∂RI − ∂RI − ∂RI
where HKS and ψn are the unperturbed Kohn Sham Hamiltonian and orbitals.
′ ′ ′ ′ ∂VSCF (r) ∂V RI (r) 2 1 ∂n(r ) dVXC (n) ∂n(r ) = { } + e ′ dr + dr (2.34) ∂R ∂R r r ∂R dn ∂R I I | − | I I ∂ǫn ∂VSCF is the first order derivative of the self consistent potential, and = ψn ψn ∂RI | ∂RI | is the first order derivative of the Kohn Sham eigenvalue ǫn. Equations 2.32, 2.33, 2.34 should be solved self consistently in the same way as the Kohn Sham equations in the unperturbed case. The solutions give the first order change in the electron density n(r). Then the second derivative matrix of the total energy can be calculated through Eq. 2.31.
30 It can be shown that within the DFPT framework, the charge density response to perturbations with different wave vectors can be determined independently [19]. So we can do a DFPT calculation for any phonon wave vector q starting from ground state results in a primitive cell without using large supercells.
2.2.2 Electron-Phonon Coupling
The concept of Cooper pairs is the basis of the BCS theory for phonon mediated superconductivity [22]. The physical picture in this theory is that an electron attracts its nearby ions (forming phonons) which polarizes its surrounding environment with a positive domain. This in turn attracts another electron, resulting in an attractive retarded interaction between electrons. If the interaction is strong enough, it could exceed the repulsive Coulomb interaction and bind the two electrons to form a Cooper pair. So the interaction between electrons and phonons plays a crucial role in this theory. The electron phonon matrix element for scattering an electron from state nk (n
is a band index) to state n′k+q by absorption of a phonon with wave vector q and branch index ν is
~ g(nk,n′k + q, qν)= <ψnk ǫˆqν ▽ VSCF ψn′k+q >, (2.35) 2Mωqν | |
where ǫˆqν is the phonon polarization vector, and ▽ VSCF is the gradient of the self con sistent potential with respect to atomic displacement, which can be easily calculated with DFPT. The linewidth of the phonon qν from this scattering process is:
2 γq =2πωq δ(E k E )δ(E ′k q E ) g(nk,n′k + q, qν) . (2.36) ν ν n − F n + − F | | ′kk q nn +
31 The δ functions specify that the electron phonon scattering process must conserve both momentum and energy. Here the phonon energy ~ω is assumed to be much
smaller than the electron energy Enk and the Fermi energy EF .
The wave vector and mode dependent electron phonon coupling parameter λqν is related to a Fermi surface average of the matrix element as
2 qν 2 ′ λqν = gnk,n′k+q δ(Enk EF )δ(En k+q EF ) ~ωq N(E ) | | − − ν F ′kk q nn + (2.37) γ = qν . ~ 2 π ωqνN(EF )
Adding up all the λqν we have
λ = λqν, (2.38) qν where λ is a dimensionless coupling constant that characterizes the strength of electron phonon coupling.
The superconducting critical temperature Tc can be estimated using the Allen Dynes equation [23] in the strong coupling limit where λ>1 as
1.04(1+λ) ωln ∗ T = e− λ−µ (1+0.62λ) . (2.39) c 1.2
Here, ∗ is the screened Coulomb potential, and ωln is the average phonon frequency defined as
1 ω = exp( λ ln(ω )). (2.40) ln λ qν qν qν 2.3 Wannier Function
Wannier functions [24] are a set of orthonormal wave functions that are con structed from Bloch functions ψk(r), and are usually localized on atomic sites. The Wannier function centered at lattice site R is defined as:
32 1 ik r Φ (r)= e− ψ (r). (2.41) R √ k N k where N is the number of primitive cells in the crystal, and sum over k extends over the Brillouin zone. The Bloch functions can be recovered from the Wannier functions by
1 ik R ψ (r)= e Φ (r). (2.42) k √ R N R Each Bloch function is determined only within an overall phase factor, so Wannier functions can be written as:
1 ik r+iφ(k) Φ (r)= e− ψ (r), (2.43) R √ k N k where φ(k) is an arbitrary real function. One can choose the φ(k) to try to make the Wannier functions decay as fast as possible when r moves away from R. Maximally Localized Wannier Function (MLWFs) [25] can provide an insightful real space picture of chemical bonding that is missing from the Bloch picture.
33 Bibliography
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34 [17] N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Harcourt, 1976)
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35 Chapter 3
First-principles investigation of the charge-density-wave instability
in 1T-TaSe2
This chapter is reprinted from Y. Ge and A. Y. Liu, Phys. Rev. B 82, 155133 (2010). Copyright (2010) by the American Physical Society.
3.1 Introduction
Recently, there has been renewed interest in materials that exhibit charge density wave (CDW) transitions. Investigations on the driving mechanism for the transition have variously emphasized the importance of Fermi surface nesting [1], van Hove singularities [2], electronic states away from the Fermi level [3], and electron phonon coupling [3, 4]. While theoretical arguments suggest that the Peierl’s mechanism, a purely electronic effect commonly taken to be synonymous with CDW formation, is not at play in real materials [3], the Fermi surface topology continues to receive significant attention in the literature on CDW materials. Interest in CDW materials has also been driven by the fact that many of these materials exhibit multiple types of electronic order that may compete or cooperate. In the 1T family of dichalcogenides, for example, recent studies have discovered a variety of ways in which superconductivity and CDW order can co exist. In compressed 1T
TaS2, it has been suggested that this occurs via a real space separation of insulating
CDW domains by metallic and superconducting interdomain regions [5]. In 1T TiSe2,
36 both pressure and doping can be used to melt the CDW order [6, 7]. In either case, a superconducting dome appears in the phase diagram near the point where the CDW melts. The nature of the superconductivity may be different in the two domes (electron phonon vs. excitonic), raising the possibility of multiple ways for CDW order and superconductivity to coexist, even starting from the same parent compound [7].
A density functional study of compressed 1T TaS2 was able to correctly describe the suppression of the CDW instability with pressure, as well as the observed super conductivity in the pure undistorted phase at high pressures [8]. However, because the
low temperature commensurate CDW (CCDW) transition in 1T TaS2 is accompanied by a metal insulator transition believed to be driven by strong electron correlations, the DFT study was not able to fully address the driving forces for the CDW transi tion. While conventional electron phonon superconductivity was consistent with the observed superconducting Tc at high pressures in the undistorted 1T structure, the assumption that the structure of the metallic interdomain region in the textured phase is well approximated by the undistorted 1T structure led to a stronger pres sure dependence in Tc near the CDW transition than what has been observed in experiments.
1T TaSe2 has a low temperature commensurate CDW structure that is very sim
ilar to that of TaS2 [9]. However, unlike in TaS2, the structural transition in TaSe2 is not accompanied by an electronic transition, and the low temperature CCDW phase
remains metallic. Thus 1T TaSe2 presents a simpler test case for examining the role of Fermi surface topology and electron phonon coupling in driving the CDW transition. In a previous first principles study of 1T Ta dichalcogenides, Fermi surface spanning vectors close to the CDW ordering vector were identified, and it was shown that the total energy of the system could be reduced by relaxing atomic positions in supercells corresponding to a single spanning wave vector [10]. It was argued that the CDW
37 order is driven by an electronic instability associated with the Fermi surface nesting, but also requires electron phonon coupling, which causes the ions to displace as the electronic charge rearranges. In this paper, we investigate the structural, electronic, and vibrational properties
of 1T TaSe2 using first principles methods. We show that density functional theory correctly describes the sequence of transitions at zero pressure, starting with the undistorted to incommensurate transition as temperature is lowered, followed by the
incommensurate to commensurate transformation. The electronic structure of TaSe2
and TaS2 are compared to understand why one undergoes a metal insulator transi tion and the other does not. Calculations show that pressure stabilizes the 1T struc ture above about 30 GPa. By examining how different factors vary with pressure, we deduce that strong momentum dependent electron phonon coupling, rather than purely electronic effects, is key to driving the CDW transition. We also predict that pressure destabilizes the CDW phase and that the undistorted high pressure phase should be superconducting with a Tc in the range of a few K.
3.2 Computational Method
Calculations were carried out within density functional theory (DFT) using the PWSCF code [11]. The interaction between electrons and ionic cores was described by ultrasoft pseudopotentials [12]. Nonlinear core corrections were used to treat the overlap between core and valence charge densities in Se. The exchange correlation interaction was treated with the local density approximation (LDA) using the Perdew Zunger parameterization of the correlation energy [13]. The generalized gradient approximation was found to yield similar results, with the primary difference being weaker interlayer binding. The Kohn Sham orbitals were expanded in a plane wave
38 basis set with a kinetic energy cutoff of 35 Ry. For the undistorted 1T structure, a 16 16 8 uniform mesh of k points was used to sample the Brillouin zone, while × × for the CCDW structure, a 4 4 6 mesh of k points was used. Unless otherwise × × specified, the Vanderbilt Marzari smearing method with a width of σ =0.02 Ry was used to accelerate convergence. Vibrational spectra and electron phonon coupling constants were calculated using density functional perturbation theory [14]. For the undistorted 1T structure, a grid of 8 8 4 phonon wave vectors q was sampled. The double Fermi surface integrals × × for the nesting function and the average of the electron phonon matrix element were calculated using the tetrahedron method with a 32 32 16 sampling of k points. × ×
3.3 Description of Structures
At high temperature, 1T TaSe2 has a trigonal structure consisting of Se Ta Se trilayer units in which atoms in each layer are arranged on a triangular lattice (Fig. 3.1) [15]. The spacing between trilayer units is large compared to the interlayer spacing
within a trilayer unit. At about 600 K, TaSe2 transforms into an incommensurate
ICDW CDW (ICDW) phase with ordering wave vector Q 0.278b1 + b3/3, where ≈ b1 and b3 are respectively the primitive in plane and out of plane reciprocal lattice vectors of the undistorted 1T structure. In fact, the structure is characterized by a triplet of equivalent CDW wave vectors with in plane components oriented at 120◦ with respect to each other, as marked in Fig. 3.1. Upon further lowering of the temperature, the ICDW phase transforms to a com mensurate phase around 473 K. At this transition, the in plane projection of the ordering wave vectors rotates by about 13.5◦, yielding a √13 √13 supercell within × the basal plane [9]. Within each supercell, the six nearest and six next nearest Ta
39 neighbors of the central Ta atom are displaced inwards, forming a 13 atom star of David cluster. The neighboring Se planes buckle to help relieve the stress. The alignment of clusters between adjacent Ta planes depends on the stacking sequence
for the trilayer units in the commensurate CDW (CCDW) phase. TaSe2 adopts a triclinic bravais lattice, which can be alternatively viewed as a hexagonal lattice with a period 13 stacking sequence.
Figure 3.1: 1T TaSe2 crystal structure and qz = 0 plane of the Brillouin zone. The large spheres (gray) represent Ta atoms and the small spheres (yellow) represent Se atoms. The in plane components of the ordering vectors for the ICDW and CCDW phases are shown. In both cases, the structure is characterized by a triplet of ordering wave vectors. Since the structure has trigonal symmetry, the M and M′ points are labeled separately.
3.4 Results and Discussion
3.4.1 Structural Instability
The calculated lattice parameters for TaSe2 in the undistorted 1T structure at zero pressure are compared to measured values [15] in Table 3.1. The largest discrepancy
40 is in the distance between Se planes in adjacent trilayer units. This is not surprising since the LDA does not do a good job of describing the weak interlayer interactions. Since the CDW transitions primarily involve in plane distortions, the overbinding between trilayer units is not expected to affect our ability to describe the transition.
1T a c zSe calc 3.41 6.08 0.271 expt[15] 3.48 6.27 0.25
CCDW a c δd1 δd2 ∆E calc (tri) 12.33 6.13 6.5% 3.8% 1.8 calc (hex) 12.33 6.16 5.7% 3.6% 1.2 expt [9] (tri) 12.54 6.26 5.6% 5.3% N/A
Table 3.1: Comparison of calculated and measured structural parameters of TaSe2 in the undistorted 1T phase and the commensurate CDW phase. The lattice parameters a and c are given in Å. In the CCDW phase, δd represents a fractional change in the Ta Ta distance compared to the undistorted structure, with d1 (or d2) referring to the distance between a Ta site at the center of a cluster and a nearest neighbor (or next nearest neighbor) Ta site. The energy difference between the CCDW structure and the undistorted 1T structure, ∆E, is given in mRy/formula unit. Two stacking sequences were considered for the CCDW phase, triclinic (tri) and hexagonal (hex), as described in the text.
The calculated phonon dispersion curves for TaSe2 in the undistorted 1T struc ture at zero pressure are shown in Fig. 3.2a. Imaginary frequencies, corresponding to unstable modes, are plotted as negative. One of the acoustic branches involving mainly in plane displacements of Ta atoms is unstable in a region along the Γ M line, in the vicinity of the in plane projection of the ordering vectors QICDW and
CCDW Q . This instability persists at all values of qz, as shown in Fig. 3.2b. While the existence of imaginary frequencies indicates the dynamical instability of the 1T structure at low temperatures, it does not reveal what the stable structure is. We have carried out total energy calculations using the in plane √13 √13 supercell × observed in the CCDW phase, trying two different stacking sequences, as discussed below. With both types of stacking, the structure was relaxed after Ta atoms were
41 slightly displaced from their high symmetry positions. In the resulting structures, the 13 Ta atoms in each supercell condensed into a star of David cluster, with atomic positions close to the x ray diffraction results [9], as shown in Table 3.1. Both stacking configurations resulted in structures with lower total energies than the undistorted structure. The stacking sequences considered were a period one hexagonal structure in which centers of Ta clusters in one tri layer unit line up exactly with those in adjacent tri layer units, and a triclinic lattice in which the center of Ta clusters in one layer align with the (undistorted) positions of Ta sites on the edge of clusters in adjacent layers. The calculations indicate that the triclinic stacking is preferred at low temperatures (Table 3.1), consistent with experimental determinations of the structure.
Previous DFT calculations for CCDW 1T TaS2 found stabilization energies of roughly half the values for TaSe2 [8]. This is consistent with the lower onset tem perature of the CCDW transition in the sulfide (180 K vs. 470 K). That calculation also found the difference between triclinic and hexagonal stacking configurations to be within the error bars, which is consistent with the observed disordered stacking of trilayer units in TaS2. Because of the difference in atomic size between S and Se, the outward buckling of the chalcogen layers near the center of the Ta clusters is calculated to be more pronounced in the selenide than in the sulfide. The preference for the triclinic structure in TaSe2 may be because the staggered alignment of clusters better accommodates the buckling of the chalcogen layers. By varying the width over which electronic states near the Fermi level are smeared, we can examine the effect of electronic temperature on stability. Only the soft modes display significant dependence on the electronic temperature. At large electronic tem peratures (σ > 0.05 Ry ) the undistorted 1T structure is calculated to be dynamically stable. As the electronic temperature is lowered, the instability first manifests at a
42 wave vector close to the location of the minimum in the curve in Fig. 3.2b. Of the q points sampled, this is the one closest to the ICDW ordering vector QICDW . Thus our calculation not only describes the CCDW phase correctly, but also captures the initial instability with respect to an incommensurate structure. In many CDW materials, the application of pressure causes the CDW order to
melt. The present calculations likewise predict that the unstable modes in 1T TaSe2 harden with pressure, and that the undistorted structure becomes stable above about
30 GPa. In 1T TaS2, experiments [5] and calculations [8] find that the CCDW phase melts around 5 GPa. The difference in transition pressures between the two materials reflects the larger stabilization energy of the CCDW phase in the selenide. Total
energy calculations for the TaSe2 CCDW supercell show that the CDW amplitude decreases with compression and goes to zero around 30 GPa. Even above the transition pressure, soft (but stable) modes persist at some wave vectors, as shown in Fig. 3.2a.
3.4.2 Electronic Structure
The electronic density of states calculated for TaSe2 in the undistorted 1T struc ture is shown in Fig. 3.3. The Se s bands lie about 15 eV below the Fermi level and are not shown. In the range of 7 to 1 eV, there are six bands primarily of Se p character, though there is significant hybridization with Ta d states. Between 1 and 3 eV, there are three bands that are derived primarily from Ta d states, but with some
Se p contributions. States within about 1 eV of the Fermi level have significant d 2 ± z
character. The band structure is very similar to that of 1T TaS2 [16, 17]. However, since the Se 4p states lie higher in energy than the S 3p states, there is stronger p d − hybridization in the selenide. Upon formation of the clusters of 13 formula units in the CCDW phase, the original dz2 band near the Fermi level is expected to fold into 13 bands: six filled bonding
43 50 -4 ( a ) ( b ) 40 -5 30 -6 20 (meV) (meV) ω ω _ h _ 10 -7 h
0 P = 45 GPa P = 0 GPa -8 -10 Γ MK Γ A 0 0.5 π qz (2 /c)
Figure 3.2: (a) Phonon spectrum of 1T TaSe2 plotted along high symmetry directions in the Brillouin zone. Results are shown for pressures of P = 0 and 45 GPa. (b) Dependence of P = 0 unstable acoustic modes on qz. Results are plotted for wave vectors with in plane projections of q = b1/2, corresponding to the arrow in (a). In both (a) and (b), imaginary frequencies are plotted as negative. Data points are connected by line to guide eyes.
bands, one half filled nonbonding band, and six empty antibonding bands. The bands near the Fermi level in the triclinic and hexagonal CCDW structures are plotted in Fig. 3.4. In both cases, as in the undistorted structure, there is significant overlap and hybridization between Se p and Ta d states. As a result, the nonbonding band at the Fermi level does not split off from the other occupied bands, as it does in TaS2 [17, 18, 19], though there is still only a single band that crosses the Fermi level. The maximally localized Wannier function [20] constructed from the half filled band at the Fermi level in the triclinic CCDW structure is shown in Fig. 3.5, and the corresponding Wannier interpolated band is plotted as a solid curve in Fig. 3.4a. For both stackings, the Wannier functions are similar, with dz2 like symmetry on the central Ta site.
44 5 Total DOS Ta d Se p 4
3
2
1 DOS (states/eV/formula unit)
0 -8 -6 -4 -2 0 2 4 E (eV)
Figure 3.3: Density of states calculated for TaSe2 in the undistorted 1T structure. The Ta d and Se p projected density of states are also plotted. The Fermi level is set to zero. For the total density of states, the tetrahedron method was used to integrate over the Brillouin zone, while for the site and orbital projected densities of states, Gaussian broadening was used.
and spreads (as defined in Ref. [20]) between 55 and 60 Å2, indicating that they are reasonably well localized on a single cluster of 13 formula units. For comparison, while
the maximally localized Wannier functions for TaS2 look qualitatively similar, they have spreads of about 20 Å2, indicating a much greater degree of localization. Thus although the condensation into clusters gives rise to narrow bands at the Fermi level in the CCDW phases of both the selenide and sulfide, the greater p d overlap and − hybridization results in more diffuse Wannier orbitals in the selenide, making it less vulnerable to the Mott instability.
45 Figure 3.4 shows that the band crossing the Fermi level in the CCDW structures exhibits a striking dependence on the stacking sequence. In the triclinic case, the band is 3D like, with roughly the same amount of dispersion parallel and perpendicular to the plane, while in the hexagonal case, the band is very one dimensional, with almost no dispersion in the plane. This seems a little surprising given the quasi 2D nature of the crystal structure. The same effect is observed in calculations for TaS2 with triclinic [17] and hexagonal [18, 19] stacking, where the contrast is even more pronounced. The dependence of the dimensionality of the band on stacking can be understood by picturing the superposition of Wannier orbitals on the two bravais lattices. In the hexagonal structure, the center of clusters in one plane line up with those in adjacent planes, so there is some overlap between the Wannier orbitals, which leads to dispersion in the out of plane direction. In the in plane directions, there is very little overlap between Wannier orbitals centered on neighboring clusters, so the band is very flat in those directions. If the centers of clusters in adjacent planes are horizontally offset, as happens with triclinic stacking, the overlap between Wannier orbitals in the out of plane direction is slightly diminished, but there is an enhancement of the effective hopping in in plane directions because an electron can hop to a neighboring cluster in the same plane via a two step process involving a cluster in an adjacent plane.
46 (a) 1
0 Energy (eV)
-1 Γ MK Γ A (b) 1
0 Energy (eV)
-1 Γ MK Γ A
Figure 3.4: Calculated bands near the Fermi level in the CCDW structure of 1T TaSe2 with (a) triclinic and (b) hexagonal stacking. The Fermi level is set to zero. To facilitate comparison, the bands are plotted along high symmetry directions of the Brillouin zone of the undistorted 1T structure. In both panels, the Wannier interpo lation of the band that crosses the Fermi level is plotted as a solid (red) curve.
47 Figure 3.5: Maximally localized Wannier function constructed for the band crossing the Fermi level in the triclinic CCDW structure of TaSe2. The dark and light (blue and yellow) coloring of surfaces distinguishes between positive and negative isosurfaces.
3.4.3 Origin of the instability
To elucidate the role of Fermi surface topology in the CDW instability, we have examined how the Fermi surface of the undistorted phase of 1T TaSe2 changes with pressure. Figure 3.6 shows the Fermi surface nesting factor calculated for undistorted
1T TaSe2 at two pressures: 0 GPa, where the undistorted structure is unstable, and 45 GPa, where the undistorted structure is stable. In these linear gray scale plots, black indicates a small value and white indicates a large value. If Fermi surface nesting is a key driving mechanism for the instability, the nesting factor should have a pronounced maximum at the CDW ordering vector below the transition pressure but not above it. This is clearly not what is seen in Fig. 3.6, where the in plane projection of the ICDW ordering vector is indicated with arrows.
48 We have also investigated the electron phonon coupling as a function of pres sure in the undistorted phase to examine its relationship to the CDW instability. Starting from high pressures, where the structure is stable, the coupling increases as the CDW instability is approached. The coupling is highly wave vector dependent, and wave vectors with modes that become unstable below the transition pressure are the strongest contributors to the coupling parameter λ, as shown in Fig. 5.4. These wave vectors are all in the vicinity of the CDW ordering vectors. For each phonon wave vector q and branch ν, the electron phonon coupling param eter is γ λ = qν , (3.1) qν ~ 2 πN(0)ωqν where N(0) is the density of states per spin and γqν is the phonon linewidth due to scattering with electrons. Since the linewidth is proportional to a doubly constrained Fermi surface average of the square of the matrix element of the self consistent change
2 in potential due to ionic displacements, ∆Vq , the coupling parameter can be | ν| expressed as 2 1 ∆Vqν λq = N(0) 1 | | . (3.2) ν ω2 1 qν Hence a large coupling parameter can be due to a large density of states at the Fermi level, which would affect all wave vectors and branches; a large nesting factor 1 , which is a function of q; a small phonon frequency at a particular q and ν; and/or a large average matrix element, as represented by the term in the square brackets in
Eq. (2). In TaSe2 at both P = 45 and 60 GPa, the largest λqν are associated with acoustic modes that either soften or become unstable at lower pressures. The nesting factor is not particularly large at most of these wave vectors. What these modes have in common is relatively large average matrix elements as well as low frequencies. They don’t necessarily have the largest matrix elements or the lowest frequencies among
49 all the modes, but they have a combination that yields a large λqν. Of course the occurrence of large electron phonon matrix elements and low phonon frequencies is not completely independent, as the matrix element determines the degree to which metallic electrons screen the bare phonons in the material [21]. These results indicate that the CDW transition is driven by a lattice instability arising from strong screening of selected phonon modes, rather than by an instability of the electronic system itself.
Recent DFT based studies of other transition metal chalcogenides such as NbSe2
[4], CeTe3 [3], and TaS2 [3, 8] have similarly cast doubt on the relevancy of Fermi surface nesting to the CDW transition. Of course, questions can always be raised about the accuracy of the DFT Fermi surface. In 2H TaSe2, for example, details of the topology of the calculated Fermi surface depend on whether the experimental or calculated lattice parameters are used, and whether the spin orbit interaction is included. Even when the spin orbit interaction is taken into account, a small shift in the Fermi energy is needed to obtain good agreement with ARPES results [3].
The fact that the present calculations capture the CDW instability in 1T TaSe2 so accurately, even down to the first wave vector to display an instability upon cooling, suggests that the calculated Fermi surface is good enough and that the instability does not depend on details of the Fermi surface topology. Instead, the instability is caused by phonon softening due to strong electron phonon interactions.
50 Figure 3.6: Calculated Fermi surface nesting function for 1T TaSe2 at pressures of (a) 0 GPa and (b) 45 GPa. From left to right, the panels correspond to qz =0,π/2c, and π/c, respectively. The linear gray scale ranges from 0 (black) to 3.205 (white). The in plane component of the ICDW wave vector is indicated with arrows.
51 Figure 3.7: Calculated electron phonon coupling parameter λq for 1T TaSe2 at pres sures of (a) 45 GPa and (b) 60 GPa. From left to right, the panels correspond to qz = 0,π/2c, and π/c, respectively. The linear gray scale ranges from 0 (black) to 3.351 (white). The in plane component of the ICDW wave vector is indicated with arrows.
3.4.4 Superconductivity under pressure
The electron phonon coupling parameters calculated for compressed 1T TaSe2 are listed in Table 3.2. Using the Allen Dynes approximate formula for Tc [22], we predict that compressed 1T TaSe2 should be superconducting with a transition temperature
52 P N(0) ~ωlog ~ωave λ Tc 45 8.1 14.1 26.5 0.69 3.8 60 7.6 17.2 28.0 0.57 2.3 Table 3.2: Calculated electronic, vibrational, and superconducting properties of 1T TaSe2. The pressure P is in GPa, the electronic density of states at the Fermi level N(0) is in states/Ry/spin, the characteristic phonon energies are in meV, and the superconducting Tc is in K.
of a few Kelvin. (This estimate assumes a Coulomb repulsion parameter of ∗ =0.14, which is typical for transition metal compounds.) The calculations show that the superconducting transition temperature should grow with decreasing pressure as the phonon modes soften and the structural instability is approached. In this sense, the predicted superconductivity and CDW instability are related in that both are driven by strong electron phonon coupling of selected modes. This is a similar picture to what was found in a DFT study of compressed 1T TaS2 [8]. However, the existence of an intermediate textured phase consisting of insulating CCDW domains and metallic interdomain regions in TaS2 complicated the comparison of the DFT results with the measured pressure dependence of superconducting properties. Since the textured phase is connected to the nearly commensurate CDW phase at zero pressure, and
TaSe2 does not have a nearly commensurate phase, the present predictions for super conductivity in compressed TaSe2 offer a cleaner test case for comparison with exper iments. As yet, there have not been any experimental studies of superconductivity in
1T TaSe2, either at ambient pressure or under compression. In 2H TaSe2, supercon ductivity has been observed in the CDW phase at very low temperatures (Tc between 100 200 mK) [23, 24, 25], also the behavior under pressure has been explored [26].
53 3.5 Conclusions
We have studied the CDW transitions in 1T TaSe2 and shown that density func tional theory does a remarkably good job of describing the transition, including the initial instability to the incommensurate phase upon cooling. A comparison of the electronic structure of TaSe and TaS reveals that there is significantly stronger p d 2 2 − hybridization in the selenide. Consequently, when atoms condense into clusters in the CCDW structure, the maximally localized cluster centered Wannier orbitals corre sponding to the band that crosses the Fermi level are more diffuse in the selenide. This explains, at least in part, why the structural transition in the selenide is not accompanied by the same metal insulator transition observed in the sulfide. Using pressure as a probe, the present work shows that the CDW instability is driven by phonon softening due to strong electron phonon coupling rather than by an electronic mechanism such as Fermi surface nesting. In the undistorted structure at high pressures, the electron phonon coupling is calculated to be strong enough to expect a superconducting Tc of a few K. Though beyond the scope of this work, it would also be very interesting to study the possibility of co existence of superconduc tivity and CDW order in 1T TaSe2 at ambient pressure.
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56 Chapter 4
Effect of dimensionality and spin-orbit coupling on
charge-density-wave transition in 2H-TaSe2
This chapter is reprinted from Y. Ge and A. Y. Liu, Phys. Rev. B 86, 104101 (2012). Copyright (2012) by the American Physical Society.
4.1 Introduction
The driving mechanism behind charge density wave (CDW) formation has been a long standing question in solid state physics. Explanations of the CDW instability have variously emphasized Fermi surface nesting [1], van Hove singularities [2], and electron phonon coupling [3, 4], among others. A criterion [5] for a stable CDW phase characterized by a wave vector q is
2 4 gq 1 | ν| + (2U V ), (4.1) ~ bare q q ωqν ≥ χq −
2 where ν is a phonon branch index, gq is the average of the squared electron | ν| bare phonon matrix element, ωqν is the unrenormalized phonon energy, χq is the dielectric
susceptibility of the electrons, and Uq and Vq are the average Coulomb and exchange interaction of electrons. This condition expresses a balance between the lowering of the electronic energy and the increase in the lattice energy upon CDW formation. In the Peierls’ model for noninteracting electrons in 1D, the CDW transition results from the
divergence of the electronic susceptibility χq due to perfect nesting when q = 2kF ,
57 where kF is the Fermi wave vector. In this case, the instability is fundamentally electronic in origin and directly related to nesting. However, in the case of layered quasi 2D CDW materials, the role of Fermi surface nesting and purely electronic origins for the CDW instability have increasingly been called into question [3, 4, 6, 7, 8]. In Ref. [4], for example, Johannes and Mazin pointed out that the degree of Fermi surface nesting is measured by the imaginary part of the electronic response function, while it is the real part that appears in Eq. (1). Indeed, first principles investigations of layered CDW materials such as 1T TaS2 [6] ,1T TaSe2 [7], and 2H NbSe2[3, 8] have found no correlation between strong nesting vectors and the CDW ordering vector. In the layered transition metal dichalcogenide CDW systems, the origin of the CDW instability can be probed by varying the interlayer separation, as different mechanisms are expected to respond differently. Consider, for example, the 2H poly types of NbSe2, TaSe2, and TaS2, which have bulk CDW phases with similar ordering wave vectors of Q b1/3, where b1 is a primitive in plane reciprocal lattice vector cdw ≈ of the normal or undistorted structure. In NbSe2, pressure decreases the CDW transi tion temperature, eventually suppressing the transition completely [9]. In TaSe2, the normal to incommensurate CDW transition temperature has a weak positive pres sure coefficient, while the incommensurate to commensurate transition temperature decreases with pressure [10]. The response to intercalation has also been mixed. In some cases (e.g., silver intercalated NbSe2 [11]) the CDW transition temperature increases, while in others (e.g., sodium doped TaS2 [12]) it decreases. In addition, in some systems the CDW structure at the surface differs from that in the bulk.
The room temperature surface phonon spectrum of 2H TaSe2 [13], for example, has an anomaly at q = b1/2 rather than at Qcdw, where the anomaly is located in the bulk spectrum [14]. Furthermore, scanning tunneling microscopy studies show that
58 the CDW structure at the surface of Ag intercalated 2H NbSe2 is described by a √13 √13 supercell instead of the 3 3 bulk CDW structure [11]. × × Recently, mechanical and chemical exfoliation techniques similar to those used to make graphene have been applied to layered dichalcogenides to produce few and single layer samples [15, 16]. This offers a clean way to assess the role of dimensionality and interlayer interactions on the CDW transition in these materials. Experiments on thin multi layer 2H TaS2 patches, for example, find no evidence for a CDW transition
[17]. On the other hand, a density functional investigation of 2H NbSe2 predicts that compared to the bulk, the CDW phase in monolayers should be more stable and be
characterized by a different Qcdw than the bulk [18]. The change in Qcdw is attributed to the momentum dependence of the electron phonon matrix elements. In this paper, we investigate the structural, electronic, and vibrational properties of bulk and single layer 2H TaSe2 using first principles methods. The origin of the CDW instability and the effect of dimensionality are explored. We find that the CDW instability remains robust upon removal of the interlayer interactions in 2H
TaSe2, with little or no change in the ordering wave vector, even though the wave vector dependences of the electron phonon interaction and of the electronic response function both vary with the strength of interlayer interactions. We also find that the Fermi surfaces of both the bulk and single layer materials are very sensitive to spin orbit coupling, yet the phonon spectra, including the location of unstable modes, are not significantly impacted. The robustness of the ordering wave vector in 2H TaSe2 is explained through the interplay between the two sides of Eq. (1). These results also provide a basis for a more general discussion about the variety of behaviors exhibited by the 2H dichalcogenides as interlayer interactions are varied.
59 4.2 Computational Method
The calculations were performed within density functional theory (DFT) using the Quantum ESPRESSO suite of codes [19]. The exchange correlation interaction was treated with the local density approximation (LDA) using the Perdew Zunger parameterization of the correlation energy [20]. For most of the results presented, the interaction between electrons and ionic cores was described by scalar relativistic ultrasoft pseudopotentials [21]. The energy cut off for the plane wave basis set was 35 Ry. A 18 18 6 uniform mesh of k points was used to sample the Brillouin × × zone, and the Vanderbilt Marzari Fermi smearing method was used to accelerate con vergence [22]. Unless otherwise noted, a smearing parameter of σ = 0.02 Ry was used. Phonon dispersion curves and electron phonon coupling parameters were cal culated using density functional perturbation theory [23]. The double Fermi surface averages of electron phonon matrix elements were calculated using the tetrahedron method on grids of up to 90 90 12 k points. To investigate the effect of spin orbit × × coupling, some fully relativistic calculations were carried out with norm conserving pseudopotentials [24] requiring an energy cut off of 72 Ry.
4.3 Description of Structures
The high temperature bulk structure of 2H TaSe2 consists of stacked Se Ta Se trilayer units [25]. Each atomic sheet within a trilayer is close packed, and the sheets are aligned so that each Ta site is at the center of a trigonal prism formed by Se sites. The distance between trilayer units is large compared to the spacing between sheets in a trilayer. The primitive cell contains two trilayers with their metal sites aligned vertically and their trigonal prisms rotated 60◦ with respect to each other, as shown in Fig. 4.1. The single layer structure is defined as one trilayer unit.
60 Figure 4.1: Crystal structure and Brillouin zone of 2H TaSe2. The large spheres (gray) correspond to Ta atoms and the small spheres (yellow) represent Se atoms. The bulk 2H structure contains two trilayers per unit cell, while the single layer structure corresponds to an isolated trilayer unit. High symmetry points in the qz =0 plane of the hexagonal Brillouin zone are labeled, as well as a triplet of ordering vectors Qcdw.
In the bulk material, an incommensurate charge density wave forms at about 122
(1 δ) K, with an ordering wave vector of − b1, where δ 0.02. In a second transition, 3 ≈ near 90 K, δ becomes zero, leading to a commensurate structure. In fact, the structure
is characterized by a triplet of equivalent wave vectors that are oriented 120◦ with respect to each other in the Brillouin zone, resulting in a 3 3 supercell in real space. × In each supercell, six Ta atoms surrounding a central Ta site displace roughly radially inward, forming a seven atom cluster [25, 26].
61 4.4 Results and Discussion
4.4.1 Structural instability
The structure of bulk 2H TaSe2 was optimized, resulting in lattice parameters
a = 3.39 Å, c = 12.23 Å, and zSe = 0.135. With the exception of the separation between trilayer units, the underestimation of which is expected within the LDA, the structure is in good agreement with experimental findings [25]. For the single layer calculations, the position of Se atoms was optimized at the bulk value for the in plane lattice constant and with c = 15 Å, ensuring at least 11 Å of vacuum between adjacent trilayers.
The phonon dispersion curves calculated for bulk 2H TaSe2 are shown in Fig. 4.2(a). The width over which electronic states near the Fermi level are smeared was varied in order to examine the effect of electronic temperature on the stability of the structure. At the electronic temperature of σ = 0.02 Ry, two low lying branches display anomalous dips along the Γ M line. Along this line, both of these branches involve longitudinal displacements of Ta atoms. As the electronic temperature is low ered, the softening increases and at σ = 0.01 Ry, the longitudinal acoustic branch is found to be unstable over an extended region of the Brillouin zone. The instability first occurs close to the experimentally observed CDW wave vector, which is at two thirds along the Γ M line. The inset in Fig. 4.2(a) displays the out of plane dispersion of the soft acoustic branch along the line with an in plane projection of q = b1/3, showing that the instability is strongest in the qz =0 plane.
62 Figure 4.2: Phonon dispersion of 2H TaSe2. (a) Bulk results are shown for three cases: scalar relativistic (SR) dispersion curves at two electronic temperatures and fully relativistic (FR) results along Γ M for the unstable acoustic branch. (b) Single layer dispersion curves are plotted for both scalar relativistic and fully relativistic calculations. Negative values indicate imaginary frequencies. The inset in (a) shows the qz dispersion of the soft branch along the line that projects onto the CDW ordering vector.
63 Figure 4.2(b) shows the calculated phonon dispersion curves for single layer 2H
TaSe2. As in the bulk, acoustic modes that involve in plane motion of Ta atoms soften and become unstable as the electronic temperature is lowered. Results are shown only for an electronic temperature where the structure is dynamically unstable. The instability occurs at approximately the same wave vector as in the bulk, but the anomaly has a narrower, more cusp like shape, though softening still occurs over an extended region of the Brillouin zone. Results of fully relativistic (FR) calculations are plotted as triangles in Fig. 4.2. For the single layer, phonon wave vectors were sampled on the same grid used in the scalar relativistic (SR) calculations. Only the soft acoustic branch shows significant dependence on spin orbit coupling. The location of the instability remains essentially unchanged upon inclusion of the spin orbit interaction, but the region of the instability broadens and is more similar to the bulk result in Fig. 4.2(a). In light of these results for the single layer, we limited our fully relativistic calculations for the bulk to selected wave vectors, primarily along the Γ M direction. As shown in Fig. 4.2(a), the low lying branch along Γ M in the bulk is very similar whether or not spin orbit coupling is included. Since the existence of imaginary phonon frequencies indicates dynamical instabil ities but does not reveal what the stable structure is, we have carried out total energy calculations using the 3 3 supercell for the commensurate CDW phase. Holding the × lattice constants fixed, the atomic positions in the bulk and in the single layer were relaxed after Ta atoms were slightly displaced from their high symmetry positions. In both systems, seven atom Ta clusters formed, as observed in neutron scattering [25]. The results of the supercell calculations, both scalar relativistic and fully relativistic, are summarized in Table 4.1. The amplitude of the CDW distortion is predicted to be larger in the single layer than in the bulk, but the energy gained from the distortion
64 is similar for the two structures. This is in contrast to the case of 2H NbSe2, where calculations predict a much larger energy gain for the single layer than for the bulk [18]. These results are qualitatively consistent with the observation that near ambient pressure, the pressure derivative of the normal to incommensurate transition temper
ature has been measured to be very weakly positive for TaSe2[10] and negative for
NbSe2 [9].
Bulk 1L Bulk 1L Bulk SR SR FR FR Expt[25] ∆E (mRy/f.u.) 0.08 0.22 0.12 0.10 N/A ∆R/R 1.1% 1.9% 1.1% 1.6% 1.3%
Table 4.1: Comparison of bulk and single layer (1L) CDW parameters. Scalar rela tivistic and fully relativistic results are presented for the total energy (per formula unit) of the 3 3 CDW structure relative to that of the undistorted structure, ∆E, and for the fractional× change in the distance from the center Ta site to its nearest neighbors in the CDW supercell, ∆R/R.
4.4.2 Electronic structure
As discussed in Ref. [4], spin orbit coupling affects the bulk 2H TaSe2 bands near the Fermi level in a nontrivial way, and it is necessary to include spin orbit coupling to bring the calculated Fermi surface in alignment with results from angle resolved photoemission spectroscopy measurements (ARPES) [27]. Since the single layer struc ture is noncentrosymmetric, one might anticipate spin orbit coupling to have an even more significant effect on the band splittings and the Fermi surface in the single layer.
The band structures of bulk and single layer 2H TaSe2 are shown in Fig. 4.3, with and without spin orbit coupling. In the single layer scalar relativistic calculation, a single band (doubly degenerate due to spin), primarily of Ta d character, crosses
the Fermi level. The corresponding Wannier function localized at each Ta site has dz2
symmetry near the Ta atom, but tails of dxy or dx2 y2 character extending out towards − 65 neighboring Ta sites [26]. This band gives rise to two roughly cylindrical Fermi sheets, with one centered around Γ and one centered around K, as shown in Fig. 4.4(c). The
states on the Γ centered hole sheet project strongly onto Ta dz2 orbitals, while those
on the K centered hole sheet have large contributions from dxy/dx2 y2 orbitals. On −
both types of sheets, the in plane dxy/dx2 y2 character is strongest near the Γ K lines. −
Figure 4.3: Calculated electronic band structure of (a) bulk and (b) single layer 2H TaSe2. The Fermi level is set to zero. Solid lines show the spin orbit split fully rela tivistic Ta d bands. The fully relativistic bulk bands were calculated at the experi mental value of the c lattice constant.
When the inter trilayer distance is decreased and the trilayers are rotated rel ative to each other to form the bulk, the Ta d band at the Fermi level broadens and splits into a bonding and antibonding pair. The splitting is largest at the
Γ point, while the two bands remain degenerate on the kz = π/c plane since
66 the intracell bonding/intercell antibonding orbitals are equivalent to the intracell antibonding/intercell bonding ones [3]. In additional, the Se pz band, which lies below the Fermi level in the single layer, also broadens and splits into a bonding and antibonding pair in the bulk. The antibonding pz band is pushed up in energy and crosses the Fermi level. Increasing the c lattice constant from the LDA value to the experimental value reduces the width of the Se pz band so that it remains below the Fermi level, as found in photoemission experiments [27]. Figure 4.4(a) shows the four Fermi sheets associated with the Ta d bands in the scalar relativistic approximation: two centered around the Γ A line, and two around the K H line. All four Ta d hole sheets show considerable warping along the kz axis due to interlayer interactions, with concentric sheets having maximum separation in the kz = 0 plane and coinciding in the kz = π/c plane. Nevertheless, the projections of in plane d orbitals on the Fermi sheets have weak kz dependence and are similar overall to the single layer case.
67 Figure 4.4: Calculated Fermi surfaces arising from Ta d bands in 2H TaSe2. (a) Bulk Fermi surface in the scalar relativistic approximation. (b) Bulk Fermi surface calcu lated fully relativistically, showing spin orbit splitting of bands in the kz = π/2 plane. (c) Single layer Fermi surface in the scalar relativistic approximation. (d) Single layer Fermi surface calculated fully relativistically, with new sheets appearing as a result of the spin orbit interaction. The Γ point lies at the corners of the zone shown. The K centered sheets have strong dxy/dx2 y2 character (red), particularly near the Γ K − lines, while the Γ centered sheets have weak dxy/dx2 y2 character (blue). −
68 Since the bulk 2H structure is inversion symmetric, the fully relativistic bands remain two fold degenerate, and the number of Ta d bands crossing the Fermi level does not change. Four Fermi sheets arise from the Ta d bands, but as shown in Fig.
4.4(b), the sheets around the K H line show less dispersion along kz because the spin orbit interaction splits the degeneracy in the kz = π/c plane. While the Fermi surface in Fig. 4.4(b) does not have the M centered dog bone shaped electron pockets observed in ARPES studies [27], a small downward shift of the calculated Fermi level by about 20 meV eliminates the Fermi level crossing of one of the d bands along the Γ K direction, bringing the theoretical and experimental Fermi surfaces into better agreement. As discussed in Ref. [4], although the Fermi surface looks very different when this slight shift in the Fermi level is applied, properties such as the nesting function (discussed below) do not change appreciably.
Unlike the bulk, a single layer of 2H TaSe2 lacks a center of inversion. Thus with spin orbit coupling, the doubly degenerate Ta d band at the Fermi level splits into two nondegenerate bands (except in the vertical mirror plane containing the Γ M line.) As a result, new electron sheets, which have a dog bone shape, appear in the fully relativistic Fermi surface, as shown in Fig. 4.4(d).
4.4.3 Origin of the instability and effect of interlayer interactions
So far we have shown that the CDW instability occurs at nearly the same wave
vector for bulk and single layer 2H TaSe2, with and without spin orbit coupling, even though the Fermi surface topologies differ significantly in these different cases. This strongly suggests that the instability is not closely tied to the topology of the Fermi surface. The Fermi surface nesting function is related to the imaginary part of the
69 noninteracting susceptibility in the static limit,
′ lim χq′′ (ω)/ω = δ(ǫkj EF )δ(ǫk+qj EF ). (4.2) ω 0 − − → k ′ jj
It was shown in Ref. [4] that the nesting factor for bulk 2H TaSe2 (including spin orbit effects) is sharply peaked at the K point in the Brillouin zone, and is otherwise relatively flat (expect near Γ, where it diverges). Though the detailed structure of the nesting function varies in the four cases in Fig. 4.4, the general features are similar, and in no case is the nesting function peaked near Qcdw. Geometric nesting of the Fermi surface can therefore be ruled out as the driving mechanism for the CDW instability in either the bulk or the single layer material. Figure 4.5(a) shows the wave vector dependence of the electron phonon coupling parameter in bulk 2H TaSe2, calculated at σ =0.02 Ry, where the structure is dynam ically stable, and limited to the acoustic branch that becomes unstable as the elec tronic temperature is lowered. The coupling of this phonon branch to electrons is moderately large over an extended region of the Brillouin zone, and is particularly large near Qcdw, at about 1/3 along the M K line (which we will call Q′), and along the line connecting neighboring Qcdw points that map onto each other under a 60◦ rotation.
70 Figure 4.5: Wave vector dependence of electron phonon coupling and electronic sus ceptibility in bulk and single layer 2H TaSe2. Panels (a), (c), and (e) show λqν and for the soft acoustic branch in the SR bulk (σ =0.02 Ry), the SR single layer (σ =0.03 Ry), and the FR single layer (σ =0.2 Ry), respectively. Black denotes λqν =0, while white represents the maximum value, which is different in each case. Panels (b), (d), and (f) show the real part of the susceptibility χq′ per formula unit calculated for the SR bulk, the SR single layer, and the FR bulk, respectively, using a common gray scale. Arrows indicate Qcdw.
71 The structure of λqν in momentum space is qualitatively different from what we
previously found for the 1T polymorphs of TaSe2 and TaS2. In the 1T materials, λqν was calculated to be sharply peaked at the experimentally observed CDW vector, so it was clear that the momentum dependence of the electron phonon coupling was
determining the ordering vector. In 2H TaSe2, the electron phonon coupling is large
at many wave vectors, so from the plot of λqν alone, it would be difficult to pre dict the q vector at which the lattice first becomes unstable. This suggests that we need to consider the right hand side of Eq. (1). Figure 4.5(b) shows the wave vector dependence of the real part of the electronic susceptibility,
f(ǫk+qj′ ) f(ǫkj) χq′ = − , (4.3) ǫk ǫk q ′ k ′ j + j jj − where f(ǫ) is the Fermi Dirac function. The susceptibility has a broad maximum near
Qcdw and lines of minima along M K and Γ K. Hence while the CDW instability is driven by strong interactions between electrons and phonons that give rise to enhanced electronic screening of certain phonon modes, we find that the wave vector at which the strongest softening occurs depends not just on λqν, but rather on the interplay between the q dependence of the electron phonon interaction and that of the electronic response function.
Figure 4.5(c) and (d) show λqν and χq′ for a single layer of 2H TaSe2. The electron phonon coupling parameter is dominated by sharp peaks near Qcdw and Q′ (1/3 the way from M to K), with λqν at these points being more than twice as large as in the bulk. One the other hand, the wave vector dependence of χq′ in the single layer is similar to that in the bulk, though there is an overall enhancement in the values due to the narrowing of the d band as the interaction between trilayers weakens. The global maximum in λqν lies on the M K line, but this line lies in a shallow trench in the susceptibility function. At Qcdw, both λqν and χq′ are close to their maximum values,
72 resulting in the softening and instability seen at this point in Fig. 4.2(b). Thus even though the electron phonon coupling as a function of wave vector is rather different in the single layer as compared to the bulk, the interplay with the electronic response function selects out the same ordering wave vector in both cases.
Figure 4.5(e) and (f) show λqν and χq′ for a single layer with spin orbit coupling.
The electron phonon coupling is large at Qcdw and Q′ as in the scalar relativistic case, but the region over which the function is large is extended. On the other hand the response function develops fine structure, with a narrower peak near Qcdw. Both the coupling and the response function are maximal at Qcdw, again selecting this wave vector for the instability. We consider now the factors that lead to a strong interaction for a particular phonon mode and how they depend on the strength of interlayer interactions. The branch and wave vector dependent λqν is proportional to an average of the square of the electron phonon matrix element,
~ g(kj; k + qj′; qν)= kj ǫˆqν δVSCF k + qj′ , (4.4) 2Mωqν | | where ǫˆq δV is the self consistent change in the potential due to displacements ν SCF 2 following the phonon eigenvector ǫˆq . The average of g is taken as in Eq. 4.2, where ν | | k and k + q are constrained to the Fermi surface. In both the bulk and single layer materials, the low lying phonon branch that becomes unstable involves primarily in plane motion of the Ta atoms (and a combination of in plane and out of plane motion of the Se atoms). These motions couple strongly to electronic states that have charge concentrated in the Ta planes. Thus we expect that substantial matrix elements for this phonon branch will arise at phonon wave vectors q that span points on the Fermi surface dominated by in plane Ta d electronic states. On the Fermi surfaces in Fig. 4.4, each K centered sheet has three regions with large in plane d projection. These “hot
73 zones” lie near the Γ K lines on the K centered sheets [and on the dog bone sheets in panel Fig. 4.4(d)]. Although the Fermi surfaces differ, in each case, the phonon wave vectors with strong electron phonon coupling in Fig. 4.5 connect “hot zones” on the corresponding Fermi surfaces. However, not all vectors that connect in plane states show up as strong peaks in the plots of the coupling strength since the matrix element depends on how strongly a particular phonon displacement pattern impacts those electronic states. For the single layer in the scalar relativistic approximation, there are three inequiv alent ways to connect pairs of “hot zones”, and for each pair, a range of wave vectors spans points in these regions. The intrasheet spanning vectors are centered around
Qcdw, while one set of intersheet connections is centered around Q′. Introducing inter
actions between trilayers warps the Fermi surface along kz, so the phonon wave vectors that connect in plane states on the Fermi surface now depend on kz. This explains how the sharp peaks in the single layer λqν at just two inequivalent wave vectors [Fig.
4.5(c)] evolve into weaker peaks and an extended region of relatively large λqν in the bulk [Fig. 4.5(a)]. In the single layer, spin orbit coupling doubles the K and Γ centered sheets, and the outer sheets around these points touch to form the M centered dog bones. Con sidering only phonon wave vectors parallel to b1, there are now three inequivalent ways to connect “hot zones” on the K centered and dog bone sheets, with overlap ping ranges. The vector Qcdw lies near the center of this range, and connects in plane states on the K sheet to those on the dog bone sheet. This accounts for the broader region of phonon softening that is still centered around the same point, as seen in Fig. 4.2(b), when the spin orbit effect is taken into account. In the case of the bulk, spin orbit coupling reduces the warping of the K sheets, which tends to narrow the region of softening, but this is counteracted by a wider range of phonon wave vectors
74 that connect in plane states due to the expansion of one sheet and the shrinking of the other. The result is that the region of softening in the bulk is not as strongly affected by spin orbit coupling as in the single layer. As noted above, just because a phonon wave vector connects “hot spots" on the Fermi surface doesn’t necessarily mean it has a large electron phonon coupling param eter. To examine this, it is useful to adopt a real space picture. Consider for example a 3 1 supercell for the bulk, which has a periodicity consistent with the ordering × vector Q . The Ta d states originally at k = Q (about 0.3 eV below the Fermi cdw ± cdw level) fold back to the Γ point in the supercell. The displacement of ions according to the eigenvectors of the longitudinal acoustic mode compresses (or expands) regions in which the charge associated with one (or the other) of these states is concentrated, leading to an upward (or downward) shift in the band and a significant coupling matrix element g. For other wave vectors, the displacement pattern has less impact on the volume in which charge is concentrated, resulting in weaker coupling. This real space description is conceptually similar to the local bonding picture developed by Whangbo and Canadell in Ref. [28]. To explain the tendency for 2H d1 transition metal dichalcogenides to undergo a 3 3 modulation, they used an extended Hückel × model to show that the experimentally determined distortion pattern changes the net overlap population of metal metal bonds, leading to a lowering of the total energy.
4.5 Conclusions
This work shows that density functional theory provides an accurate description of the CDW instability in bulk 2H TaSe2 and predicts the instability to persist in single layer 2H TaSe2, at the same or nearly the same wave vector. The instability is found to be due to strong coupling of in plane Ta d states near the Fermi level
75 with phonons that involve in plane Ta displacements. However, since the electron phonon coupling is large at multiple phonon vectors, the momentum dependence of the electron phonon coupling alone is not sufficient for determining the wave vector at which the instability manifests. Instead, the momentum space structure of the dielectric response function must also be considered. This is in contrast to previous
findings for 1T TaSe2, where λqν was sharply peaked only near Qcdw. Although geometric nesting of the Fermi surface is found to be irrelevant to the instability, the shape of the Fermi surface nevertheless plays a role in that it affects the wave vector dependence of the electron phonon coupling. A necessary (but not sufficient) condition for large coupling is that the phonon momentum vector connects “hot zones” on the Fermi surface, where the in plane character of the electronic states
is large. Since the interlayer interaction affects the kz dispersion of the Fermi surface, and the spin orbit interaction can change the Fermi surface topology and even the number of Fermi sheets, the momentum dependence of λqν changes as the strength of these interactions is varied. Thus it is somewhat surprising that the phonon instability is calculated to occur at or about the same wave vector in the bulk and in the single
layer, with or without spin orbit coupling. By considering the structure of χq′ together with that of λqν, we are able to understand this result. Our prediction that the CDW manifests at the same (or nearly the same) wave vector in single layer and bulk 2H TaSe2 is in contrast to 2H NbSe2, where calculations indicate that the ordering vector depends on dimensionality [18]. Due to the smaller mass of the metal atom, spin orbit effects are not expected to be as important in
NbSe2 as in TaSe2. Comparing the scalar relativistic Fermi surfaces for the single layer structures, we notice a difference in the shape of the K centered sheet. This sheet has a more triangular cross section in single layer NbSe2, which decreases the length of the vector that spans regions of in plane states on this sheets. This could
76 explain, at least in part, the shift of the instability to a smaller wave vector in single
layer NbSe2. Our results are also in contrast to observations of a shifted surface phonon anomaly in 2H TaSe2 [13]. Given that the location and degree of phonon softening is deter mined by multiple factors, each of which reacts differently to changes in the strength of interlayer interactions and spin orbit coupling, it is plausible that the reduced sym metry at the surface affects the ordering vector. Further work is needed, however, to fully understand the difference between the single layer and the surface. More gener ally, with the interplay of different factors, we expect that differences in the electronic structure of isostructural, isoelectronic materials, doping effects in intercalated mate rials, and changes in symmetry could impact the ordering vector or even the stability of a CDW phase. So while this work has focused on 2H TaSe2, it helps to eluci date the variety of behaviors observed in the CDW dichalcogenides when interlayer interactions are tuned.
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80 Chapter 5
Phonon-mediated superconductivity in electron-doped single-layer
MoS2: A first-principles prediction
This chapter is reprinted from Y. Ge and A. Y. Liu, Phys. Rev. B 87, 241408(R) (2013). Copyright (2013) by the American Physical Society.
5.1 Introduction
Molybdenum disulfide (MoS2) is a transition metal dichalcogenide that consists of S Mo S trilayers that are weakly bound to each other [1]. A semiconductor with an indirect gap of about 1.2 eV [2], bulk MoS2 has been of interest for its tribo logical [3], catalytic [4, 5], and photovoltaic[6, 7] properties. In addition, bulk MoS2 is easily metallized through insertion of alkali or alkaline earth metals into the Van der Waals gap between trilayers, and many of these intercalated compounds become superconducting when cooled [8]. KxMoS2, for example, has a maximum supercon ducting transition temperature of about 7 K when x 0.4 [9]. This is similar to the ≈
transition temperature in NbSe2, which has the highest superconducting Tc among metallic transition metal dichalcogenides.
Recently, it has been demonstrated that single layer MoS2 can be produced by mechanical and chemical exfoliation techniques[10] and by direct growth methods
[11]. Unlike the bulk material, single layer MoS2 (consisting of one S Mo S trilayer) has a direct band gap of 1.9 eV [12, 13, 14]. With its reduced dimensionality and large
81 direct band gap, single layer MoS2 has quickly attracted attention for use in nanoscale electronic and photonic devices. Ultrathin MoS2 has been successfully fabricated into field effect transistors (FET) with high on off ratios [15], and it is being investigated for its chemical sensing[16] and photovoltaic properties [17].
Raman spectroscopy of field effect doped single layer MoS2 has found that the
Raman active A1g mode softens and broadens significantly with electron doping [18]. This is attributed to strong coupling of the mode to electronic states at the bottom of the conduction band. This indication of a strong electron phonon interaction in the monolayer, combined with the existence of superconducting phases in intercalated bulk compounds, raises the question of whether phonon mediated superconductivity can be induced in single layer MoS2. If so, the material could be useful for novel nanoscale devices that integrate electronic, optical, and superconducting components.
Recent experiments on liquid gated thin flake MoS2 transistors have discovered a dome shaped superconducting region in the temperature versus doping plane of the phase diagram [19]. While the samples in these experiments are not atomically thin, it is argued that the carriers are concentrated in the top layer. For carrier densities below
14 2 n =6 10 cm− , corresponding to a monolayer doping concentration of x 0.05 2d × ≈ electrons per formula unit, no superconducting transition is observed (down to 2 K).
Above this doping concentration, the superconducting Tc rises sharply, reaching a maximum of about 11 K near x =0.1 electrons/f.u. Then the superconducting phase line turns over. While it was not possible in these experiments to achieve higher doping levels comparable to the intercalated bulk compounds, it was suggested that the data for the alkali intercalated bulk superconductors might connect smoothly to the superconducting dome mapped out for the liquid gated thin flakes [19]. In this work we use density functional theory (DFT) calculations to investigate the electronic and vibrational properties of electron doped single layer MoS2 to
82 explore the possibility of phonon mediated superconductivity in a true monolayer. The electron phonon coupling strength is studied as a function of doping. We find the onset of superconductivity at a low but non zero doping concentration, as well
as a subsequent turnaround in Tc, both of which can be attributed to changes in the number and size of Fermi sheets as the carrier concentration is varied. These results
suggest that further enhancements in Tc might be possible via appropriate tuning of the band structure.
5.2 Method
Calculations were carried out using the Quantum Espresso package [20]. Ultra soft pseudopotentials were used to describe the interaction between electrons and ionic cores [21], and the local density approximation was adopted for the exchange correlation potential [22]. A plane wave basis with a 35 Ry energy cut off was used to represent electronic wave functions. The Brillouin zone was sampled with a 24 24 1 mesh of k points for calculation of the total energy and electron charge × × density. Phonon spectra and electron phonon coupling constants were calculated using density functional perturbation theory [23]. The double Fermi surface average of electron phonon matrix elements was done using grids of 288 288 1 k points with × × a smearing parameter of σ = 0.002 Ry. Doping was simulated by adding electrons to the system, together with a compensating uniform positive background.
Figure 5.1 (a) shows the top view of the crystal structure of single layer MoS2. Each atomic sheet consists of a hexagonal close packed lattice, and the sheets are stacked so that each Mo atom is coordinated by a trigonal prism of S sites. To model the single layer, we used a supercell with at least 14.5 Å of vacuum to avoid interaction between layers. The in plane lattice constant was held fixed at the optimized value for
83 the undoped material (3.122 Å), while the distance between Mo and S atomic planes
3 was relaxed at each doping concentration until forces were less than 10− Ry/au.
Previous DFT studies of MoS2 have found that the choice of exchange correlation functional affects the optimized geometry, which in turn affects how vibrational fre quencies evolve in going from bulk to monolayer [24]. The present work considers only the monolayer and focuses on the effects of doping. While details of the elec tronic structure and vibrational spectrum depend on geometry and functional, the qualitative effects of doping are relatively insensitive to these choices.
Figure 5.1: (Color online) (a) Top view of the atomic structure of single layer MoS2. Large spheres represent Mo atoms on a triangular lattice. Small spheres represent S atoms, which lie in planes above and below the Mo plane. (b) High symmetry points in the Brillouin zone.
5.3 Results and Discussion
Undoped single layer MoS2 is calculated to have a conduction band minimum at the K point in the Brillouin zone, and a second conduction band minimum located about halfway between Γ and K (along the Λ line), about 110 meV higher in energy.
The conduction band minimum at K has strong Mo dz2 character, while states near
84 the second minimum are derived from a combination of in plane d orbitals on the Mo sites and p orbitals on the S sites. These results are consistent with previous DFT results [25, 26, 27]. Both spin orbit interactions[28] and many body effects beyond DFT [27, 29, 30] have been shown to have significant impact on the band structure. However, the conduction band, which is the focus of the present work, retains the same qualitative features and trends.
Figure 5.2: (Color online) Conduction band and Fermi surface of single layer MoS2. Panels (a) and (b) correspond to a doping concentration of x = 0.03 electrons/f.u., panels (c) and (d) show results for x = 0.10 electrons/f.u., and panels (e) and (f) are for x =0.22 electrons/f.u. Some representative phonon wave vectors that connect Fermi sheets are shown.
85 Figure 5.2 shows the conduction band along the Λ line (Γ to K) as a function of doping, along with the corresponding Fermi surfaces. At low doping concentrations, such as x = 0.03 electrons/f.u., electrons occupy states near the conduction band minimum at K, as seen in Fig. 5.2(a). Figure 5.2(b) shows the corresponding Fermi sheets around the two K points in the Brillouin zone. With more doping, the second valley, centered at Λmin, becomes occupied as well, and new Fermi sheets appear, as in Figs. 5.2(c) and (d), which show results for x = 0.10 electrons/f.u. Note that as the concentration of charge carriers increases, the energy difference between the Λmin and K valleys decreases, and eventually becomes negative. At x =0.22 electrons/f.u., the Λmin valley lies below the K valley and there is barely a Fermi surface around the K points [Figs. 5.2(e) and (f)]. The relative shift in energy of the conduction band valleys with doping can be explained by considering the character of the electronic states involved. Dominated by Mo dz2 character, states near the original conduction band minimum at K are more localized than the Λmin states, which project strongly onto both Mo dxy/x2 y2 − and S p orbitals. As charge carriers are added, Coulomb repulsion pushes the more localized states near K up in energy, and leads to the eventual reversal in order of the two valleys.
The phonon spectra calculated for undoped and electron doped single layer MoS2 are presented in Fig. 5.3. The Raman active modes at the zone center are in reasonable agreement with experiments and prior calculations.[18, 24, 31, 32, 33] As noted in Ref.
1 [18], the A1g mode, which is slightly above 400 cm− , softens considerably when doped,
1 1 while the two fold degenerate E2g mode below 400 cm− is relatively insensitive to doping. However, the most significant softening occurs not in the optical branches near the zone center, but rather in the highest acoustic branch, particularly near the M and K points at the zone boundary. The optical A1g mode involves out of
86 plane displacement of the S atoms, which interact strongly with the Mo dz2 states at the conduction band minimum. When these electronic states become occupied, the phonon frequency is renormalized due to increased screening. On the other hand, the highest acoustic phonon branch involves longitudinal in plane displacement of Mo atoms (the direction of S displacements depends on the phonon wave vector). The softening of the longitudinal acoustic branch arises from strong coupling of this mode
to electrons in the Λmin valley of the conduction band, as discussed below.
Figure 5.3: Phonon dispersion curves calculated for single layer MoS2. Solid curves are for the undoped material, while circles show results for x =0.06 electrons/f.u.
The Fermi surface average of the square of the electron phonon matrix element is
2 ′ gk k q ′ q δ(ǫk )δ(ǫk q ′ ) 2 kjj | j; + j ; ν| j + j gqν = 2 , (5.1) | | [N(0)] where the Fermi level is set to zero, N(0) is the density of states at the Fermi level, and gkj;k+qj′;qν is the matrix element for scattering an electron from state kj (j is
a band index) to state k+qj′ by a phonon with wave vector q and branch index ν. Since the Fermi surface consists of multiple small sheets, a limited set of phonon wave vectors connect electronic states at the Fermi level. For example, q = K (and
87 nearby wave vectors) allows transitions between the two K centered Fermi sheets, as shown in Fig. 5.2(b). Electron phonon matrix elements were calculated for phonon
1 wave vectors q = Γ, K, M, and 2 ΓK. For each of these wave vectors, we also sam pled a few nearby points to estimate the phase space weight wq, corresponding to a fraction of the area of the Brillouin zone, to assign each of the representative wave vectors. The dimensionless electron phonon coupling parameter is then given
2 by λ = 2N(0) gq wq/~ωq . qν | ν| ν
Figure 5.4: (Color online) Electron phonon coupling parameter calculated as a func tion of electron doping for single layer MoS2. Contributions from different phonon wave vectors are indicated by the shading. Arrows on the horizontal axis show where Fermi sheets appear or disappear.
In Fig. 5.4, the total λ and the contributions from each representative phonon wave vector are plotted as a function of carrier concentration. For x 0.03 electrons/f.u., the total coupling parameter λ is very small and only includes contributions from q
= Γ (intrasheet) and K (intersheet). The A1g mode at the zone center has a large average electron phonon matrix element, as previously noted, but the Brillouin zone weight of this contribution to λ is limited by the small size of the Fermi sheets. For
88 intersheet coupling, the average matrix element is found to be small, as is the weight.
So even though the A1g frequency is noticeably softened by electron phonon coupling, the overall coupling constant, λ, is small.
Once the Fermi level reaches the Λmin valley of the conduction band (or within a phonon energy of it), the coupling parameter acquires new contributions involving the Mo dxy/x2 y2 like electronic states near Λmin. The contributions to λ from phonon − 1 wave vectors q = K and 2 ΓK are dominated by transitions from electronic states
on one Λmin Fermi sheet to another through absorption or emission of longitudinal acoustic phonons. In contrast, the longitudinal acoustic mode at q = M strongly couples electronic states near Λmin with those near K. At the doping level of x = 0.06 electrons/f.u., taking into account the Brillouin zone weights, each of the three phonon wave vectors that give rise to intersheet coupling have similar contributions to the total λ. As the carrier concentration is further increased, the coupling constant grows due to both matrix element and phase space factors. Above about x = 0.18 electrons/f.u., however, λ starts to decrease. This is mainly due to the shrinking of the Fermi sheets near K, which are important for processes involving q = M phonons. To estimate the superconducting transition temperature, we used the Allen Dynes formula[34] with ∗=0.13. At low doping, before the Λmin valley is occupied, Tc is cal culated to be close to zero. Following the behavior of λ, Tc increases rapidly when the Λmin Fermi sheets appear and grow, peaking near x = 0.18 electrons/f.u., with T 27 K. It then decreases as the K centered Fermi sheets shrink. Although the c ≈ maximum λ of about 1.7 is larger than the value calculated for hole doped graphane [35], which has been predicted to be superconducting at liquid nitrogen temperatures, the Tc estimates for electron doped single layer MoS2 are significantly lower because, unlike in graphane, where high frequency C C stretching modes dominate the cou pling, it is the acoustic modes that contribute most to the electron phonon coupling
89 in single layer MoS2. At x =0.18 electrons/f.u., for example, the logarithmic average
1 phonon frequency that sets the scale for Tc is about 160 cm− .
Because the K and Λmin centered Fermi sheets are very different in electronic
character, interband anisotropy could enhance Tc in the doping regime when both valleys are occupied [36]. Using a two band model [37], we estimate that the effective coupling constant for superconductivity at x = 0.18 electrons/f.u. would increase to about 2.3. On the other hand, a BCS like mean field treatment neglects phase
fluctuations, which can renormalize Tc in 2D systems [38]. Since there are considerable uncertainties in the calculated transition temperatures, we focus instead on the trends, which should be robust.
The trends predicted for doping induced superconductivity in single layer MoS2 are remarkably similar to the superconducting dome reported for electrostatically doped thin flakes of MoS2 [19]. However, it is not obvious that our monolayer results should match the experimental data for thin flakes. Even if the doping caused by gate tuning is limited to the first layer, the order of valleys in the conduction band is sensitive to interlayer interactions. In addition, our calculation does not take into account the effect of external electric fields on the electronic structure. Thus the order in which various Fermi sheets appear and disappear may be different in the thin flake FET devices as compared to the single layer system we investigated. Nevertheless, it seems likely that the features observed in the experiments, such as the onset of superconductivity at a critical carrier concentration, and the subsequent decrease in
Tc at larger x, are associated with changes in the number and type of Fermi sheets as the gate is tuned.
To optimize electron phonon coupling in electron doped single layer MoS2, our work suggests that it is important to have carriers in both the K valley and Λmin valley of the conduction band. This can be achieved over a very limited range of electrostatic
90 doping. Since intercalated bulk MoS2 exhibits superconductivity, chemical doping of
single layer MoS2 may be an alternative route to superconductivity. As an example, we have investigated single layer KxMoS2, where potassium adatoms are dispersed on one side of the MoS2 layer. Supercells were used to examine three doping levels: x = 0.11, 0.25, and 0.33. At low doping, the potassium 4s band lies well above the bottom of the lowest MoS2 conduction band, so the dopant simply transfers electrons into the MoS2 conduction band. However, as the doping increases, the formation of a dipole layer at the interface causes the substrate bands to shift up in energy relative to the potassium s band. For x =0.33, the bottom of the potassium s band lies lower
than the MoS2 conduction band edge, though the two bands overlap in energy. For K MoS , we find λ 0.6, indicating that the potassium 4s band couples much more 0.33 2 ∼
weakly to phonons than the MoS2 conduction band. This highlights the importance
of optimizing the character of electronic states at the Fermi level to enhance Tc, and demonstrates that different methods of doping (chemical vs. electrostatic) may lead to very different results. In neither case is a rigid band model adequate.
In summary, we find that the electron phonon interaction in electron doped MoS2 depends sensitively on which valleys of the conduction band are occupied, since the orbital character of electronic states differ substantially in the different valleys. When
both the K and Λmin valleys are occupied, the electron phonon coupling parameter
is significant and suggests a superconducting Tc on the order of 20 K. The predicted
trends in Tc, such as rapid changes when sheets of the Fermi surface appear and grow (or shrink and disappear), are similar to what has been observed in field induced
superconducting thin flakes of MoS2 [19], but further work is needed to bring the experimental and theoretical systems into closer contact. Given the difference in char acter between low lying states in the conduction band and the importance of having the right mix of states at the Fermi level, mechanical or chemical tuning of the elec
91 tronic structure could be promising routes for further enhancing the electron phonon interaction in single layer MoS2 and related materials.
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95 Appendix A
Lattice Thermal Conductivity
A.1 Introduction
The understanding of heat conduction in solids is a long standing problem. The Fourier law q = κ▽T, (A.1) − describes the conductive heat transfer in a solid. The heat flux vector q is propor tional to the spatial gradient of temperature and the proportionality κ is the thermal conductivity tensor. The energy carriers in solids are electrons and phonons. Electrons conduct the majority of heat flux in metals, while phonons (lattice vibration) domi nate the thermal conductivity in semiconductors and insulators. Peierls first carefully studied the lattice thermal conductivity [1]. He introduced the concept of "umk lapp" process in which the phonon momentum in the scattering process is changed by a reciprocal lattice vector. It is this umklapp phonon phonon scattering process that produces finite lattice thermal conductivity in semiconductors and insulators. To calculate κ requires knowledge of heat capacities, velocities, and scattering rate of phonons.
96 A.2 Method
A.2.1 Lattice Dynamics
Suppose a crystal has multiple atoms per unit cell. Let u(lk) represent the dis placement of the kth atom in the lth unit cell. Assuming the total potential energy V of the crystal is a function of the instantaneous positions of all atoms, V can be expanded in powers of the atomic displacement u(lk) [2]:
∂V V =V0 + uα(lk) ∂uα(lk) lkα 0 ′ ′ ′ ′ 1 + φαβ (lk,l k )uα(lk)uβ(l k ) (A.2) 2 ′ ′ lk,l k αβ 1 ′ ′ ′′ ′′ ′ ′ ′′ ′′ + Φαβγ(lk,l k ,l k )uα(lk)uβ(l k )uγ(l k )+ ... 3! ′ ′ ′′ ′′ lk,l k ,l k αβγ
where V0 is the equilibrium value,
2 ′ ′ ∂ V φαβ(lk,l k )= ′ ′ (A.3) ∂uα(lk)∂uβ(l k ) 0 is the second order interatomic force constant (IFC), and
3 ′ ′ ′′ ′′ ∂ V Φαβγ(lk,l k ,l k )= ′ ′ ′′ ′′ (A.4) ∂uα(lk)∂uβ(l k )∂uγ(l k ) 0 is the third order IFC. αβγ are Cartesian coordinates. The first term in Eq. A.2 is not important and can be set to zero. The second term vanishes for equilibrium ionic configurations. In the harmonic approximation, the above expansion can be kept at the second order to determine the phonon frequencies and eigenvectors, where for each phonon wave vector q, the dynamical matrix is
′ ′ 1 iq Rl Dαβ(q)= φαβ(lk,l k )e . (A.5) ′ MkMk l 97 th Rl is the lattice vector of the l unit cell, k specifies an atom in this cell, Mk is the
′ ′ atomic mass, and the sum is over all unit cells. φαβ(lk,l k ) is the second order force constant as in Eq. A.3. When all atoms are equally displaced, there is no force on any atom, which leads to the sum rule
′ ′ φαβ(lk,lk)= φαβ(lk,l k ). (A.6) − ′ ′ l k =lk If there is only one type of atom in the unit cell. We can simplify Eq. A.5 as:
′ ′ 1 iq Rl Dαβ(q)= φαβ(0k,l k )e + φαβ(0k, 0k) M ′ l =0 (A.7) ′ ′ 1 iq Rl = (e 1)φαβ(0k,l k ) . M ′ − l =0 Here, we have used the relation that
′ ′ ′ ′ φ (lk,l k )= φ 0k, (l l)k , (A.8) αβ αβ − which comes from the translation invariance of the crystal. The harmonic approximation gives the picture of non interacting normal modes. However, in reality the crystal has anharmonic forces. Since the anharmonic terms are usually a small fraction of the total Hamiltonian, they can be treated as a perturbation on the non interacting phonon states. The effect of anharmonicity is to introduce interaction among independent phonon states. For example, the cubic term causes the interaction between three phonons in first order perturbation. There are two types of three phonon scattering events. which we will call class 1
′ and class 2. In a class 1 event, a phonon λ interacts with another phonon λ , they
′′ + both get annihilated, and a third phonon λ is generated. W ′ ′′ is the scattering λλ λ probability for this process. Here λ is a shorthand notation for phonon wave vector q
′ and branch index ν. In a class 2 event, one phonon λ splits into two phonons λ and
′′ λ . W − ′ ′′ is its scattering probability. λλ λ 98 The scattering probabilities W ± ′ ′′ are determined from Fermi’s golden rule [3]: λλ λ
~ ′ ′′ π (n0λ + 1)(n0λ +1/2 1/2)n0λ ′ ′′ 2 W ± ′ ′′ = ± V (λ,λ ,λ ) δ(ωλ ω ′ ω ′′ ) λλ λ ′ ′′ λ λ 4N ωλωλ ωλ | ± | × ± − (A.9)
′ ′′ where N is the number of unit cells, and V (λ,λ ,λ ) is the three phonon scattering ± matrix element given by:
′ ′′ ′′ ′′ ′ ′ ′′ ′′ V (ν, q; ν , q ; ν , q )= Φαβγ(0k,l k ,l k ) − ∓ ′ ′ ′′ ′′ k l k l k αβγ ′′ ′′ ν ν ′ ν ′′ (A.10) ′ ′′ e (q)e ′ (q )e ′′ (q ) iq R ′ iq R ′′ αk βk γk e l e l . × ′ MkMk M ′′ k
′ ′ ′′ ′′ They depend on third order anharmonic interatomic force constants Φαβγ(0k,l k ,l k ).
ν Here eαk(q) is the phonon eigenvector. In first principles calculations, the phonon eigenvalues, eigenvectors, and three phonon scattering probability can all be deter mined through density perturbation theory (DFPT) method.
A.2.2 Boltzmann Transport Equation
The Boltzmann transport equation (BTE) can be used to analyze transport phe nomena within systems that have a gradient in carrier density and temperature. In the lattice thermal conductivity problem, the BTE for a specific phonon mode is:
∂nλ ∂nλ vλ ▽T + =0, (A.11) − ∂T ∂t scatt. where nλ and vλ are the phonon distribution function and group velocity, respectively. The equilibrium phonon distribution at temperature T follows Planck’s law:
1 n = , (A.12) 0λ exp(~ω/k T ) 1 B −
where kB is the Boltzmann constant. The first term on the right hand side of Eq. A.11 describes a diffusive system in the presence of a temperature gradient ▽T . The
99 second term accounts for all possible phonon scattering processes. The total rate of
change of nλ due to the two processes must vanish in the steady state of heat flow. Solution of Eq. A.11 is required for calculating κ. However, this form of the BTE is quite complicated because it requires knowledge of the distribution function nλ for
′ all possible phonon states λ and the transition rate from λ to λ. If the temperature gradient ▽T is small, the deviation of the phonon distribution function from equilibrium is linear in ▽T , so nλ = n0λ + n1λ, where n1λ is linear in ▽T . Keeping terms linear in ▽T , the first term in Eq. A.11 can be written as
∂n ∂n v ▽T λ v ▽T 0λ (A.13) λ ∂T ≃ λ ∂T The second term in Eq. A.11 describes the scattering process. Following the con
vention of Ziman [4], we introduce a new function Ψλ defined by 1 n = λ exp(~ω /k T Ψ ) 1 λ B − λ − ∂n0λ n0λ Ψλ (A.14) ≃ − ∂~ωλ
= n0λ +Ψλn0λ(n0λ + 1).
Obviously Ψλ measures the deviation from equilibrium of the distribution of phonon mode λ. Now, let’s consider a scattering process where three phonons are involved
′ ′′ λ + λ λ . (A.15) ↔ Then the second term in Eq. A.11 can be written as
∂n ∂n ′′ ′′′ ∂n ′ ′′ λ = λ (λ + λ λ) λ (λ + λ λ ) ∂t ∂t → − ∂t → scatt. + = W ′ ′′ W − ′ ′′ λλ λ λλ λ (A.16) ′ ′′ − λ λ + = n ′′ n ′ (n + 1) L ′ ′′ (n ′′ + 1)(n ′ + 1)n L− ′ ′′ λ λ λ λλ λ λ λ λ λλ λ ′ ′′ { } − { } λ λ 100 ′ ′′ where L± ′ ′′ are the intrinsic transition probabilities between states λλ λ , and λλ λ + + W ′ ′′ = n ′′ n ′ (n + 1)L ′ ′′ . The first and second bracketed term correspond λλ λ λ λ λ λλ λ
to class 1 and class 2 scattering events, respectively. If we substitute nλ = n0λ +
Ψλn0λ(n0λ +1) to the right hand side of Eq. A.16 and retain only terms of first order
in Ψλ, then the above equation can be expressed as
∂nλ + 1 = [W ′ ′′ (Ψ ′′ Ψ ′ Ψ )+ W − ′ ′′ (Ψ ′′ +Ψ ′ Ψ )] (A.17) λλ λ λ λ λ λλ λ λ λ λ ∂t ′ ′′ − − 2 − scatt. λ λ 1 The factor of 2 is to avoid double counting. In deriving this result, we have used the relation
′′ ′ ′′ ′ nλ (nλ + 1)(nλ +1)=(nλ + 1)nλ nλ. (A.18)
Now, we can rearrange Eq. A.11 using the linearized terms as
∂n0λ + 1 v ▽T = [W ′ ′′ (Ψ ′′ Ψ ′ Ψ )+ W − ′ ′′ (Ψ ′′ +Ψ ′ Ψ )]. (A.19) λ λλ λ λ λ λ λλ λ λ λ λ ∂T ′ ′′ − − 2 − λ λ The above linearized BTE only includes three phonon scattering, but in general higher order phonon scattering terms could be included. In either type of scattering event, the processes must conserve both momentum and energy, so
′ ′′ ′ ′′ ω(λ) ω(λ )= ω(λ ), q q = q + G, (A.20) ± ± where G is a reciprocal lattice vector that is zero for normal processes and nonzero for umklapp processes. In a normal process, the total momentum and energy do not change, so the heat current is independent of temperature gradient, and the thermal conductivity is infinite [5]. It is the umklapp process that gives rise to finite thermal conductivity.
101 A.2.3 Iterative Approach to Solve Linearized BTE
As first proposed by Omini et. al. [6], we define Ψλ = Fλα∂T/∂xα, put it into α Eq. A.19, and rearrange it as:
0 1 + 1 F = F + [W ′ ′′ (F ′′ F ′ )+ W − ′ ′′ (F ′′ + F ′ )], (A.21) λα λα λλ λ λ α λ α λλ λ λ α λ α Qλ ′ ′′ − 2 λ λ + 1 0 ~ where Qλ = [W ′ ′′ + W − ′ ′′ ] and Fλα = ωλn0(n0 +1)νλα/(TQλ). This is a set ′ ′′ λλ λ 2 λλ λ λ λ of equations that couples different phonon modes. To initiate the iterative procedure,
0 the second term in the right hand side of Eq. A.21 is set to zero. Then Fλα is the zeroth order solution which would be plugged into the right hand side for all the modes to obtain the first order solution. The process continues until F n+1 F n . λα ∼ λα The phonon thermal conductivity tensor is:
καβ = Cλvλαvλβτλβ. (A.22) λ A temperature gradient in the β direction and heat current in the α direction are related. vλα is the velocity of phonon mode λ in the α direction. Cλ = [ω(λ)]2/k T 2 n (n +1) is a phonon mode’s contribution to the specific heat, and { B } 0λ 0λ
τλβ = TFλβ~ωλvλβ is the scattering time. The necessary inputs to solve the linearized and transformed Eq. A.21 are the harmonic and anharmonic interatomic force constants (IFCs). Phonon frequencies and eigenvectors can be determined with harmonic IFCs and the three phonon scattering matrix element requires anharmonic IFCs.
A.3 Algorithm and Implementation
There are different ways of implementing this iterative method. One method sam ples the possible three phonon triplets in a quadrature grid [7]. In this way, for a given
102 ′ ′′ phonon mode qν, the branch index for the other two phonon modes ν , ν are fixed.
′ ′′ Then, there is a six dimensional space of q and q that can complete two possible scattering events. The requirement of momentum and energy conservation give four
′ constraints, so there is a two dimensional surface in reciprocal space for possible q .
′ ′ ′ In this algorithm, qx and qy are chosen from a quadrature grid, then qz is determined ′′ by energy conservation equation, and q is determined by momentum conservation
′ ′ ′′ ′′ equation. The resulting triplet qν, q ν , q ν is included in the evaluation of W ± ′ ′′ λλ λ
and Qλ. In this way, the lattice thermal conductivity has been calculated for bulk sil icon and silicon/germanium superlattices using IFCs from empirical potentials [8] or from DFPT results [9]. The main drawback of this method is the huge computational effort required for the sampling grid. We have tested an algorithm that is concep tually more simple. We choose Gaussian functions to approximate the δ function in Eq. A.9, so we can sample the Brillouin zone with a linear grid. Then the accuracy of calculation can be systematically improved by increasing the size of sampling grid and making the Gaussians narrower.
Our program consists of both parallel and serial parts. The value of each W ± ′ ′′ is λλ λ
needed in the iterative process as described in Eq. A.21, and Qλ which describes the
total scattering rate associated with phonon mode λ is needed to normalize W ± ′ ′′ λλ λ ′′ values. Since the wave vector of the third phonon mode λ can always be determined through momentum conservation, and wave vectors of all three modes are on the
′′ same linear grid, we can omit the index λ in W ± ′ ′′ . We set up a very large matrix λλ λ ′ to store individual W ± ′ , with rows correspond to λ and columns correspond to λ , λλ respectively. The matrix has to be initialized before the iteration process starts. We distribute the initialization process to several computer nodes. The size of the matrix is M M, where M = N d, N is the number of wave vectors in the Brillouin × × zone grid to sample phonon wave vectors, and d is the number of phonon branches
103 at each wave vector. Choice of N depends on the convergence criterion. By dividing the total matrix along the rows into several matrices, each small matrix is assigned to a different node, and is independent of the others. Each row i or column j in these divided matrices represents a particular phonon mode λ. The value at ij represents
+ 1 the summed three phonon scattering rates W ′ + W − ′ , which includes all scattering λλ 2 λλ ′ ′′ events that has phonon λ(i) and λ (j) associated. The third phonon λ is determined by momentum conservation requirement on the grid. We have used the MPI library for managing work on different computer nodes. The communication between nodes has been set to minimal and workloads are equally divided. After all the divided matrices have been filled, they are restored into a single large matrix. At that time, a single node is assigned to do the iterative step. Finally, κ is calculated according to Eq. A.22. We should note that some permutation and symmetry operations can be applied when evaluating the matrix element in Eq. A.10. Let’s define P as a permutation operation, where Pij switches the i and j element in the triplet of A.10. It is obvious to see that: ′ ′′ ′ ′′ (q, q , q ) P (q, q , q ) V ′ ′′ = V ′ ′′ (A.23) (ν,ν ,ν ) P (ν,ν ,ν ) ′ ′′ 1 ′ ′′ (q, q , q ) P − (q, q , q ) V ′ ′′ = V ′ ′′ (A.24) P (ν,ν ,ν ) (ν,ν ,ν ) Using the invariance of the potential energy under the space group of the crystal and the law of transformation for the eigenvectors [10], one finds that
′ ′′ ′ ′′ (q, q , q ) (q, q , q ) ′ ” V ′ ′′ = V ′ ′′ , q + q + q = G (A.25) R(ν,ν ,ν ) (ν,ν ,ν ) where R belongs to the space group of the crystal lattice, and G is a reciprocal lat tice vector. In principle, Eq. A.23 through Eq. A.25 can be used to define a set of
′ ′′ irreducible triples of wave vectors qq q , which can be used to generate any triplet 104 [11]. However, use of this kind of symmetry information has not been implemented in our code because of the conflict between the concept of finding the irreducible set
and the current design of the matrix that stores W ± ′ ′′ . The current MPI scheme λλ λ lets each node calculate the matrix element for arbitrary triplets without incurring communication between nodes that would be needed to make use of symmetry oper ations. In a complex system, making use of symmetries could potentially reduce the total workload.
A.4 Test Case: Two-Dimensional Argon Solid
Argon is an insulating crystal with face centered cubic bravais lattice up to 84 Kelvins. Here we consider a two dimensional analog, with the argon atoms sitting in a triangular lattice with lattice constant 3.87 Å. Like other condensed rare gas systems, the atomic potential of solid argon can be accurately described by the simple Lennard Jones (LJ) pair potential:
σ 12 σ 6 φ(rij)=4ǫ[( ) ) ], (A.26) rij − rij where rij is the distance between atoms i and j. The value of the parameters ǫ and
σ that we use is ǫ/kB = 119.8 K and σ = 3.405 Å, which best reproduces its ther modynamics [12]. Because of the simplicity of this pair potential, argon has become a benchmark system to test various methodologies of calculating lattice thermal con ductivity [13, 14]. In this example, the harmonic force constants in Eq. A.3 are computed directly as the second derivative of the L J potential between ions and are summed up to third nearest neighbours for the dynamical matrix in Eq. A.7. The diagonalization process is completed by a LAPACK subroutine. The anharmonic force constants in Eq. A.10 are computed similarly as the harmonic force constants. Because anharmonic force
105 constants are independent of phonon wave vector or branches, we calculate them once and store them for use in arbitrary scattering matrix calculations. Both types of IFCs have analytical forms. This is one reason that we use a model potential to test our algorithms. The phonon velocity of a particular mode is defined as
v(qν)= ▽qω(q) (A.27) and it is calculated using the midpoint method once the spectrum of phonon frequen cies has been obtained on a fine grid. We sample the first Brillouin zone with a grid of 48 48 1 q points. We use − 2 × × ( (x µ) ) 1 − (σ)2 Gaussian functions σ√π e to approximate δ functions in Eq. A.9 , with the smearing factor σ chosen as 5% of the maximum frequency. The iteration speed of Eq. A.21 is usually quite fast. Within less than 1000 iterations, the result converges reasonably well. This step takes only a few percent of the total calculation time. The
major effort of this program is in calculating the scattering rates W ± ′ ′′ . Results for λλ λ the lattice thermal conductivity of a 2D argon lattice as a function of temperature are shown in Fig. A.1. Our results are in good agreement with prior calculations based on molecular dynamics [15]. Using a 2D argon solid as an example, we have shown that the algorithm works, but further work is needed to make it a practical method for more complex materials. The main obstacle is that the design of the program is conceptually simple, but it requires a lot of memory. A typical three dimensional, one atom per unit cell calculation would require the order of 100 Giga bytes computer memory. And this could grow very fast as the number of phonon branches increases. The majority of time has been spent on filling the large scattering rate matrix. There are possible ways to improve the efficiency of the code, for example, focusing on phonon branches that
106 Figure A.1: The lattice thermal conductivity of two dimensional argon solid calculated using iterative method (black dots) and molecular dynamics simulation method [15] (red square).
could contribute to the scattering process, utilizing symmetry operations, calculating the value of variables on the fly without storing them, etc.
107 Bibliography
[1] R. Peierls, Ann. Phys. Leipzig 3, 1055 (1929)
[2] G. P. Srivastava, The Physics of Phonons, (Taylor & Francis, 1990)
[3] E. Fermi, Nuclear Physics, (University of Chicago Press, 1950)
[4] J. M. Ziman, Electrons and Phonons, (Oxford, 1960)
[5] C. Kittel, Introduction of Solid State Physics, (Wiley, 2004)
[6] M. Omini and A. Sparavigna, Il Nuovo Cimento 19, 1537 (1997)
[7] D. A. Broido, A. Ward, and N. Mingo, Phys. Rev. B 72, 014308 (2005)
[8] L. Lindsay and D. A. Broido, Phys. Rev. B 80, 125407 (2009)
[9] A. Ward, D. A. Broido, D. A. Stewart, and G. Deinzer, Phys. Rev. B 80, 125203 (2009)
[10] A. A. Maradudin and S. H. Vosko, Rev. Mod. Phys. 40, 1 (1968)
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[15] R. D. Mountain, R. A. MacDonald, Phys. Rev. B 28, 3022 (1983)
108 Appendix B
Publication List of Yizhi Ge
1. J. Sun, Y. Ge and S. Li, Evolving network with different edges, Phys. Rev. E 76 046108 (2007)
2. J. K. Freericks, H. R. Krishnamurthy, Y. Ge, A. Y. Liu, Th. Pruschke, The-
oretical description of time-resolved pump/probe photoemission in TaS2: a single-band DFT+DMFT(NRG) study within the quasiequilibrium approxima- tion, Physica Status solidi (b) 246, 948 (2009)
3. Y. Ge and A. Y. Liu, First-principles investigation of the charge-density-waves
instability in 1T-TaSe2, Phys. Rev. B 82, 155133 (2010)
4. P. Wang, R. Sullivan and Y. Ge, Risk-Based Dynamic Asset Allocation with Extreme Tails and Correlations, Journal of Portfolio Management, 38, 26 (2012)
5. Y. Ge and A. Y. Liu, Effect of dimensionality and spin-orbit coupling on charge-
density wave transition in 2H-TaSe2, Phys. Rev. B 86, 104101 (2012)
6. Y. Ge and A. Y. Liu, Phonon-mediated superconductivity in electron-doped
single-layer MoS2: A first-principles prediction, Phys. Rev. B (R) 87, 241408 (2013)
7. W. Shen, Y. Ge, A. Y. Liu, H. R. Krishnamurthy, T. P. Devereaux, and J. K. Freericks, Nonequilibrium "melting" of a charge density wave insulator via an ultrafast laser pulse, submitted (2013)
109