First-Principles Study of Charge Density Waves and Electron-Phonon Coupling in Transition Metal Dichalcogenides, and Magnetism of Surface Adatoms

First-principles Study of Charge Density Waves and Electron-phonon Coupling in Transition Metal Dichalcogenides, and Magnetism of Surface Adatoms

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

Oliver Ruben Albertini, B.A.

Washington, DC August 11, 2017 Copyright c 2017 by Oliver Ruben Albertini

ii First-principles Study of Charge Density Waves and Electron-phonon Coupling in Transition Metal Dichalcogenides, and Magnetism of Surface Adatoms

Oliver Ruben Albertini, B.A.

Dissertation Advisor: Amy Y. Liu, Ph.D.

Abstract

Recently, low-dimensional materials have attracted great attention, not only because of novel physics, but also because of potential applications in electronic devices, where performance drives innovation. This thesis presents theoretical and computational studies of magnetic adatoms on surfaces and of charge density wave (CDW) phases in two-dimensional materials. We investigate the structural, electronic, and magnetic properties of Ni adatoms on an MgO/Ag(100) surface. Using density functional theory, we ﬁnd that strong bonding is detrimental to the magnetic moment of an atom adsorbed on a surface. Previously, it was shown that Co retains its full gas-phase magnetic moment on the MgO/Ag(100) surface. For Ni we ﬁnd the total value of spin depends strongly on the binding site, and there are two sites close in energy.

Next, we study the 1T polymorph of the transition metal dichalcogenide TaS2.

Bulk 1T -TaS2 is metallic at high temperatures, but adopts an insulating commensurate CDW below 180 K. The nature of the transition is under debate, ∼ and it is unclear whether the transition persists down to a monolayer. Transport and Raman measurements suggest that the commensurate CDW is absent in samples thinner than 10 layers. Our theoretical and experimental study of the ∼

vibrational properties of 1T -TaS2 demonstrates that the commensurate CDW is robust upon thinning—even down to a single layer.

iii Bulk 2H-TaS2 also has a CDW instability at low temperatures. A recent study of a single layer of 1H-TaS2 grown on Au(111) revealed that the CDW phase is suppressed down to temperatures well below the bulk transition temperature (75 K), possibly due to electron doping from the substrate. We present a first-principles study of the effects of electron doping on the CDW instability in monolayer 1H-TaS , finding that electron doping: 1) shifts the electronic bands by 0.1 eV; 2) 2 ∼ impacts the strength and wave vector dependence of electron-phonon coupling in the system, suppressing the CDW instability. Furthermore, our work identifies electron-phonon coupling—not peaks in the real or imaginary parts of the electronic susceptibility—as the main cause of CDW instabilities in real 2D systems.

Index words: density functional theory, charge density wave, electron-phonon coupling, transition metal dichalcogenides, electron doping, magnetism, TaS2, theses (academic)

iv Acknowledgments

I thank my advisors, Amy Liu and Jim Freericks for their consistent support and dedication. Luckily, my advisors are also two of my favorite teachers (one of them since I was an undergraduate) and this made the research process much more enjoyable. Amy, in particular, spent a lot of time helping me develop as a physicist and researcher, and for her Herculean efforts, I am so grateful. Thanks to her, I have a new appreciation for the scientific process and how to collaborate with others in a meaningful way. And because of her thoughtful guidance, I have a thesis and body of work to really be proud of, and a future to really look forward to. I also thank the rest of my dissertation committee: Miklos Kertesz, Makarand Paranjape, and Barbara Jones for their insightful questions and comments. I thank Barbara for hosting me at IBM in 2013; it was a wonderful experience I will always remember fondly. This is a short list, in no particular order, of people I had the pleasure of meeting and working with as a student at Georgetown: Chen Zhao, Andre De Vincenz, Jesús Cruz-Rojas, Woonki Chung, Shruba Gangopadhyay, Jasper Nijdam, Wes Mathews, Bryce Yoshimura, Lydia Chiao-Yap, Alex Jacobi, Matteo Calandra, Paola Barbara, Pasha Tabatabai, Tingting Li, Jennifer Liang, David Egolf, Leon Der, Ed Van Keuren, Xiaowan Zheng, Joe Serene, James Lavine, Amy Hicks, Rui Zhao, Peize Han, Josh Robinson, Baoming Tang, Mary Rashid, Max Lefcochilos-Fogelquist, Janet Gibson, Sona Najafi, and Mark Esrick.

v Finally, I thank my family–LeRoy Jr., Sonia, LeRoy III, and Graciela Albertini–and girlfriend, Sheryl Lun, who supported me throughout these last few years. This was a challenge for all of us, and they share my joy in completing this great chapter of my life.

vi Table of Contents

Chapter 1 Introduction...... 1 1.1 Magnetic Adatoms on Surfaces...... 2 1.2 Two Dimensional Materials...... 4 2 Theoretical Background...... 21 2.1 Born-Oppenheimer Approximation...... 21 2.2 Hohenberg-Kohn Theorems...... 22 2.3 Kohn-Sham Formulation...... 23 2.4 Exchange and Correlation...... 26 2.5 Implementation: Basis Sets...... 27 2.6 Lattice Dynamics in DFT...... 31 3 Site-dependent Magnetism of Ni Adatoms on MgO/Ag(001)...... 37 3.1 Introduction...... 37 3.2 Method...... 39 3.3 Results & Discussion...... 42 3.4 Conclusions...... 51

4 Zone-center Phonons of Bulk, Few-layer, and Monolayer 1T -TaS2: Detec- tion of the Commensurate Charge Density Wave Phase Through Raman Scattering...... 53 4.1 Introduction...... 53 4.2 Description of Crystal Structures...... 56 4.3 Methods...... 57 4.4 Results and Discussion...... 59 4.5 Conclusions...... 71

5 Eﬀect of Electron Doping on Lattice Instabilities of Single-layer 1H-TaS2 73 5.1 Introduction...... 73 5.2 Methods...... 76 5.3 Results & Discussion...... 77 5.4 Conclusions...... 88 6 Conclusion...... 91 Bibliography...... 94

vii List of Figures

1.1 Periodic table showing the ingredients for transition metal dichalcogenides...... 6 1.2 Two types of coordination geometry for transition metal dichalcogenides.7 1.3 Impact of coordination geometry on electronic bandstructure.....8

1.4 Charge density wave phases of 1T -TaS2...... 12

1.5 1T -TaS2 charge density wave viewed from above...... 14

1.6 An atomic displacement pattern for 1H-TaS2...... 17

1.7 Another atomic displacement pattern for 1H-TaS2...... 18 2.1 Simplified DFT work flow...... 25 2.2 Diagram depicting the augmented plane wave scheme...... 29 3.1 Top view of the three binding sites for Ni on MgO/Ag(100)...... 41 3.2 Total energy per Ni adatom on MgO/Ag, calculated for structures relaxed within GGA and GGA+U (U = 4 eV) from PAW calculations. 42 3.3 Densities of states from inside the muffin-tins of all-electron LAPW calculations...... 47 3.4 Spin density isosurfaces of Ni adsorbed on MgO/Ag, from GGA+U all-electron LAPW calculations...... 49 4.1 An illustration of the Ta plane and Brillouin zone reconstruction under the lattice distortion of the C phase...... 58 4.2 Optical modes of 1T bulk and monolayer structures...... 60

4.3 Calculated phonon spectra of undistorted 1T -TaS2...... 61

viii 4.4 Simpliﬁed illustration of the low-frequency phonon modes of a 1T bilayer. 62 4.5 Raman spectra of bulk (> 300 nm) and thin (14.6 nm, 13.1 nm, 5.6 nm, and 0.6 nm) samples, measured at 250 K (NC phase) and 80 K (C phase)...... 64 4.6 Comparison of experimental and calculated Raman frequencies for the bulk...... 65 4.7 Comparison of experimental and calculated Raman frequencies for a single layer...... 68 4.8 Measured polarized Raman spectra of the C phase for (top) a monolayer and (bottom) a 120 nm sample measured at 100 K...... 69

5.1 Electronic bands of single-layer 1H-TaS2...... 79

5.2 Phonon dispersions of single-layer 1H-TaS2 with and without spin- orbit coupling...... 80

5.3 Phonon dispersions of single-layer 1H-TaS2 with and without doping and lattice constant relaxation...... 82 5.4 Brillouin-zone maps of the phonon linewidth [(a-c)] and the geometric Fermi-surface nesting function [(d-f)]...... 84

2 5.5 Momentum dependence of the square of the self-consistent (ωq) and ˜ 2 bare (Ωq) phonon energies for the branch with instabilities, self-energy ˜ correction (2ωqΠq) for that branch, and real part of the bare suscepti-

bility (χ0(q))...... 85

ix List of Tables

3.1 Relaxed distance between adatom and nearest-neighbor Mg and O atoms, and height of these atoms above the MgO ML...... 43

3.2 Total magnetization per adatom (in µB) for diﬀerent sites, calculated within GGA and GGA+U, from all-electron LAPW calculations... 45 3.3 Net charge of Ni atom and its nearest-neighbor O and Mg atoms according to Bader analysis...... 46

x Chapter 1

Introduction

The physics of low-dimensional materials generates great curiosity because the behavior of electrons in confined environments is fascinating and beautiful. And the prospect of using these materials in real-world applications is becoming ever more desirable and achievable. The miniaturization of computer components, whether microchip transistors or magnetic sectors of hard drives has led to tremendous advances. But recently, the performance of transistors (power consumption and speed) has not kept up with miniaturization. The search for materials that perform better has led us to two-dimensional (2D) and even one-dimensional (1D) systems. Materials like transition metal dichalcogenides are quasi 2D in their bulk form, and often take on new and interesting properties in the few-layer limit. Even the bandstructure of graphene opens up a gap when it is reduced to thin strips, or nanoribbons. If we can harness these new properties, it may lead to flexible, transparent, faster, and more efficient computing devices. Taking a bottom-up approach, scientists have also developed techniques to arrange individual atoms on a surface, working in the ultimate limit of miniaturization and low-dimensionality. It has been shown, for example, that just a single magnetic Ho atom can hold information for hours [1]. Besides exotic information storage, research like this brings us closer to spin-based logic gates [2] and even the realm of quantum computing.

1 1.1 Magnetic Adatoms on Surfaces

As part of the degree requirements of the Industrial Leadership in Physics (ILP) track of the Ph.D. program of the Georgetown University physics department, I did a year-long internship at the IBM Almaden Research Center in San Jose, CA. The idea behind the ILP program is to expose students not just to academia, but also industry, where there are many promising careers for scientists. From January 2013 to January 2014, I worked at IBM under the guidance of Barbara Jones and her postdoctoral fellow Shruba Gangopadhyay. We collaborated with the scanning tunneling microscope (STM) team there, led by Andreas Heinrich. Doing density functional theory (DFT) calculations, we provided the STM team with theoretical understanding about the properties of transition metal atoms adsorbed to a substrate. The story begins in 1989, when Don Eigler made an STM under vacuum and ultra-cold temperatures ( 4 K) at Almaden. He used this STM to manipulate the ≤ positions of atoms and molecules on a substrate. This began a long tradition of famous experiments: in 1993, the creation of a quantum corral of Fe atoms which beautifully reﬂected electronic states of the Cu substrate [3]; in 2012, the formation of bistable ‘bits’ of 12 Fe atoms, arranged antiferromagnetically, which held their spin arrangements for hours [4]; and even the world’s ﬁrst atomic-scale animation in 2013 [5]. Part of the Almaden STM team’s motivation is to approach magnetic storage from the bottom up; the smallest possible magnet is the atom. Mastery of the physics of magnetic atoms on a surface may launch us past the next hurdles in magnetic storage and other areas, since stable quantum magnets have potential computing applications. Furthermore, the basic science of magnetism in reduced

2 dimensionalities (e.g. surface magnetism) mixes elements of quantum and classical magnetism, helping to bridge our scientiﬁc understanding. In recent years, the team has focused on magnetic adatoms adsorbed onto a few

atomic layers of an insulating material such as Cu2N or MgO. They studied adatoms from almost the entire 3d transition metal row, including Mn [6,7], Cu [8], Ti [9], Fe [4], Co [10], and finally Ni [11]. For a magnetic atom on a surface, magnetism must compete with the tendency for bonding. For an adatom to exhibit some level of magnetism, the bonding orbitals must not become too diffuse or overpower the energy benefit of a magnetic configuration. While at Almaden, I worked with the STM team to understand the case of Co adsorbed onto MgO/Ag(001). Remarkably, the adatom preserves almost

all of its spin and orbital angular momentum from the gas phase (Lz = 3 and

Sz = 3/2). We showed that Co bonds to an O site of the MgO surface, where the ligand ﬁeld is approximately axial (C ), preserving the energetic ordering of the Co ∞ 3d orbitals. Both the bonding and charge transfer are weak, which explains why the orbital and spin moments are preserved. The experiments also reported a strong magnetic anisotropy for Co on MgO/Ag, which is a critical property for nanoscale magnets [10]. In contrast, through DFT studies, we ﬁnd that Ni, which sits just one column over from Co in the periodic table, forms a strong bond at the O site and loses its gas phase spin moment (Sz = 1) altogether [11]. This study is presented in Chapter 3 of this thesis.

3 1.2 Two Dimensional Materials

While every student of quantum mechanics can recall simple low dimensional versions of the simple harmonic oscillator and quantum well, our understanding of the behavior of real electrons mostly centered on three-dimensional (3D) systems. In condensed matter, electrons generally occupy an environment where they are surrounded by matter in each direction. Sometimes theoretical physicists would study a two dimensional version of a material, as P. R. Wallace did in 1947 when he calculated the bandstructure of single honeycomb sheets of carbon atoms to better understand the properties of graphite [12]. Presumably, single layers of graphite have been made by students with pencils for centuries. The ﬁrst time that students intentionally made a sheet of graphene was by mechanically exfoliating graphite with Scotch tape [13]. Graphene has many interesting and useful properties including inertness, transparency, extreme tensile strength, and high electron mobility. The exfoliation and experimental characterization of graphene led not only to a Nobel Prize in 2010, but also an explosion in research on 2D materials. The exfoliation and, later, growth techniques which allowed the production of graphene sheets have also been used to synthesize other 2D materials, including transition metal dichalcogenides, hexagonal boron nitride, black phosphorous, and more. Like graphite, many of these were already well-known as layered bulk materials with weak van der Waals interlayer interactions. We now know that there are new properties, advantages, and disadvantages in

2D. For Wallace, one consequence was that the pz orbitals of carbon did not interact

with neighbors outside the plane. These half-occupied pz orbitals give rise to a Dirac

4 cone at the Fermi level, which is the origin of many of the exotic conduction properties of graphene. More generally, thinned-down layered materials have reduced screening compared to the bulk, due to the absence of neighboring layers. This leads to stronger Coulomb interactions, and strong exciton binding [14]. Many materials also have a fundamentally diﬀerent symmetry in their single- and few-layer forms, and in particular the lack of inversion symmetry gives rise to new properties. Finally, a thin ﬂake of material is more sensitive to its environment than its bulk counterpart. This can prove detrimental to experiment since in ambient conditions, samples can quickly degrade, altering the properties of interest. On the other hand, an upside to this environmental sensitivity is the possibility of using these materials for chemical sensing applications. Similarly, these materials are sensitive to their substrate, making it possible to enhance desirable properties or suppress undesirable ones by choosing the appropriate substrate.

1.2.1 Transition Metal Dichalcogenides

Because of their weak interlayer interaction, transition metal dichalcogenides

(TMDs) like MoS2 have traditionally been used as dry lubricants. Now, since these materials can be exfoliated and synthesized in monolayer and few-layer form, they are sought for their more interesting and diverse electronic and chemical properties, as well as their potential for small feature height (typical monolayer thickness is 0.6-0.7 nm).

They have the general chemical formula MX2, where M is a transition metal (TM) and X is a chalcogen (S, Se, Te). The structure of a TMD consists of a triangular lattice of M atoms, sandwiched between two layers of X atoms (here we will refer to this trilayer as a monolayer). In the bulk, these layers are stacked.

5 1 2 H He Hydrogen Helium 3 4 5 6 7 8 9 10 Li Be B C N O F Ne Lithium Beryllium Boron Carbon Nitrogen Oxygen Flourine Neon 11 12 13 14 15 16 17 18 Na Mg Al Si P S Cl Ar Sodium Magnesium 3 4 5 6 7 8 9 10 11 12 Aluminium Silicon Phosphorus Sulphur Chlorine Argon 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Potassium Calcium Scandium Titanium Vanadium Chromium Manganese Iron Cobalt Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine Krypton 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Rubidium Strontium Yttrium Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium Palladium Silver Cadmium Indium Tin Antimony Tellurium Iodine Xenon 55 56 57-71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 Cs Ba La-Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Caesium Barium Lanthanide Hafnium Tantalum Tungsten Rhenium Osmium Iridium Platinum Gold Mercury Thallium Lead Bismuth Polonium Astatine Radon 87 88 89-103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 Fr Ra Ac-Lr Rf Db Sg Bh Hs Mt Ds Rg Uub Uut Uuq Uup Uuh Uus Uuo Francium Radium Actinide Rutherfordium Dubnium Seaborgium Bohrium Hassium Meitnerium Darmstadtium Roentgenium Ununbium Ununtrium Ununquadium Ununpentium Ununhexium Ununseptium Ununoctium

Figure 1.1: Periodic table showing the ingredients for transition metal dichalcogenides. Transition metal dichalcogenides have the composition MX2, where M (X) can be one of the transition metal (chalcogen) atoms highlighted above.

TMDs are generally considered quasi-2D materials because the coupling between monolayers is mostly due to the van der Waals interaction. As shown in Figure 1.1, the groups of the periodic table that most commonly combine with chalcogens to form TMDs are 4, 5, 6, 7, 9 & 10. The coordination of a M atom in a TMD material is either trigonal prismatic or approximately octahedral (trigonal antiprismatic), as shown in Figure 1.2. The M coordination strongly inﬂuences d level splittings, and thus the electronic properties of the material. Each M atom gives 4 valence electrons to bonding orbitals with two X atoms, and the remaining valence electrons occupy d levels. Depending on the number of remaining electrons, and the splitting of the d levels, TMDs can be insulators (HfS2), semiconductors (MoS2,WS2), semimetals (WTe2,WSe2), and metals (NbS2,VSe2) [15], as shown schematically in Fig. 1.3.

6 (a) trigonal prismatic (side and top views)

M: X:

(b) octahedral (side and top views)

Figure 1.2: Two types of coordination geometry for transition metal dichalcogenides. TMD coordination can be trigonal prismatic (D3h) or approximately octahedral (trigonal antiprismatic, D3d). For the former (a), the two chalcogen (X) layers align vertically, while for the latter (b), they are rotated 60◦ from each other. The top views show this vertical alignment. For monolayer systems, these are the only two polymorphs, and are referred to as 1H (trigonal prismatic) and 1T (octahedral).

7 E E

σ∗ σ∗

10 9 10 9 7 t2g 7 6 5 6 dz2 5 4 4

σ σ

DOS DOS

(a) D3d (b) D3h

Figure 1.3: Impact of coordination geometry on electronic bandstructure. Transition metal dichalcogenides can coordinate by D3d or D3h. This can play a role in what types of electronic properties they have, ranging from insulating to metallic. Density of states are drawn schematically, indicating possible scenarios for band ﬁlling. The blue/red peaks represent bonding/antibonding TM-chalcogen bands, and the green nonbonding d orbitals of TM. Numbers indicate the group number of the TM coordination center. Four M valence electrons are assumed to be in bonding bands formed with X atoms. The other M valence electrons occupy nonbonding d orbitals (green). For (a), perfect octahedral coordination is assumed.

8 For thicknesses greater than a monolayer, interlayer stacking can also play a role in the material properties. Octahedral coordination materials tend to stack directly (AA stacking), and are referred to as 1T . Trigonal prismatic materials tend to stack with a c-axis periodicity of two or three monolayer units, referred to as 2H (ABAB stacking) and 3R (ABCABC stacking). For these polymorphs, the M atoms do not always align vertically. Rich phase diagrams exist for many of these materials, indicating multiple ordering tendencies like charge density waves (CDW), magnetism and superconductivity. An important area of study involves probing the behavior of these materials as they are thinned from the bulk down to a single monolayer. A noteworthy example of this is the semiconductor MoS2, which in the bulk exhibits an indirect band gap, while the monolayer has a direct band gap. This sort of material tunability shows great application promise, but requires investigation on a fundamental science level.

1.2.2 Charge Density Waves

A charge density wave (CDW) is a modulation of charge density in a periodic solid which lowers the total energy of the system of ions and electrons. For a simple model of a one-dimensional metal Rudolf Peierls showed that the energy lowering from an appropriate charge density modulation is always greater than the energy cost of the accompanying lattice distortion [16], meaning that the system is fundamentally unstable against CDW formation. In Peierls’ model, the origin of this energy lowering is purely electronic and is rooted in a familiar concept from solid state physics—the opening of a band gap at the Brillouin zone (BZ) edge. Peierls considered a one-dimensional chain of atoms

9 with lattice constant a and one electron per atom. This leads to a half-ﬁlled band with a Fermi “surface” consisting of two points, k = π . F ± 2a Peierls noted that a charge modulation with twice the lattice periodicity would halve the BZ and open a gap at the boundary of the new BZ (i.e. at π ), lowering ± 2a the energy of occupied electronic states near the BZ edge. Such a charge density modulation would be accompanied by a dimerization of the lattice since the electrons and lattice are coupled. For a one-dimensional metal at low temperature, Peierls showed that the elastic energy cost is less than the electronic energy reduction, so the CDW is favorable at low temperature. At high temperature, the excitation of electrons to states above the gap increases the electronic energy, destabilizing the CDW phase. Thus a Peierls transition occurs when a 1D metal is cooled to the transition temperature, where it undergoes a spontaneous doubling of the unit cell (dimerization), as well as a metal to insulator transition. At its core, the Peierls transition is driven by a divergence in the real part of the bare electronic susceptibility:

2 X f(k+qn) f(km) Re[χω=0(q)] = − , (1.1) Nk k+qn km kmn −

where kn is the energy and f(kn) the Fermi-Dirac occupation of the eigenstate with wavevector k and band index n. In 1D, the real part of χω=0 diverges at q = 2kF . This is the ordering vector that describes the doubled unit cell in the CDW phase. This is also the geometric nesting vector that connects the two points of the 1D Fermi surface. For this reason, the Peierls transition is often associated with Fermi surface nesting in the literature. In general, however, geometric nesting is related to the imaginary part of χω=0 rather than the real part:

2 X Im[χω=0(q)] = δ(km F )δ(k+qn F ). (1.2) Nk − − kmn

10 The imaginary part of χω=0 peaks at wavevectors q that connect large parallel sections of the Fermi surface.

The quasi-one-dimensional crystals NbSe3 [17], NbTe4, and TaTe4 [18] are real-world examples of the Peierls transition. Below some transition temperature, they form CDWs, in which, in some cases, no ionic displacement is detected, at least within the uncertainty of the measurements [17]. Indeed the lattice distortion is considered secondary in the Peierls transition. In quasi-two-dimensional materials, in particular the transition metal dichalcogenides, there are various materials that form CDWs, including TaS2, TaSe2, and NbSe2. It is becoming increasingly clear, however, that CDW formation in these materials deviates from the Peierls picture. In many cases, the CDW ordering vector coincides with neither Fermi surface nesting nor a divergence of the real part of the electronic susceptibility [19, 20, 21, 22, 23]. The CDW phases do involve structural transitions, but these structural transitions are not always accompanied by a metal to insulator transition. The wavevector dependence of the electron-phonon coupling, instead, seems to dictate the parts of the BZ where a CDW ordering vector is likely to appear. In these cases, one or more phonon modes soften due to strong screening by electrons. The extreme softening of a lattice vibration, in which the frequency approaches zero, is like a softened spring that never returns to its unstretched position. Thus in 2D and quasi-2D materials, it has been argued that CDWs form when strong electron-phonon coupling softens selected phonon modes to the point of instability. Note that in the Peierls transition, a phonon mode also softens (so-called Kohn anomaly), but its wavevector is dictated by the q dependence of Re[χ], rather than that of the electron-phonon matrix elements.

11 C-CDW NC-CDW I-CDW undistorted lattice

T 0 K 180 K 350 K 540 K

Figure 1.4: Charge density wave phases of 1T -TaS2. 1T -TaS2 forms a charge density wave lattice distortion that is fully commensurate with the lattice below 180 K. Between 180 - 350 K, the CDW is nearly commensurate with the lattice, and incommensurate between 350 - 540 K. Finally, above 540 K, the lattice is undistorted by any CDW formation.

It may serve the scientiﬁc community to clearly distinguish these two types of charge modulation phenomena, as they are often conﬂated in the literature. Zhu, et al. have proposed that charge modulations that follow the Peierls mechanism based on the divergence of the real part of the electronic susceptibility be dubbed Type I CDWs. Type II CDWs would then be the much more common CDWs linked to electron-phonon interaction that do not necessarily display a metal-insulator transition [24].

1.2.3 TaS2

This thesis centers mostly on TaS2, a TMD that can adopt either the 2H or 1T polymorph. The 2H polymorph is favored at low temperatures. To realize the metastable 1T phase at e.g. room temperature, the material must be heated above 1050 K and quenched back down [25, 26]. Annealing (heating and slowly cooling) ∼ will return the material to the more stable 2H phase [27]. Both polymorphs are metallic at high temperatures and undergo CDW phase transitions at lower temperatures, but with very diﬀerent eﬀects on electronic transport [25, 28].

12 1T -TaS2

As shown in Fig. 1.4, above 540 K, 1T -TaS2 has no lattice distortion (undistorted phase) and is metallic. When cooled to 540 K, the material develops CDW order that is not commensurate with the lattice (incommensurate CDW, I-CDW). At 350 K, the lattice enters a more ordered phase, known as the nearly-commensurate CDW (NC-CDW) phase, which is accompanied by an abrupt increase in resistivity. Finally, at 180 K, the CDW becomes fully commensurate with the lattice, and an even larger jump in resistivity occurs. At 180 K and below, the material is considered to be insulating [25, 29]. This material has drawn considerable attention because a material with large changes in resistivity can be useful in device applications such as transistors, especially if there is a way to externally induce the metal to insulator transition (MIT). As seen in Figure 1.3, the group 5 TMDs have one valence electron per formula unit (f.u.) available to occupy d bands. When the C-CDW occurs, two concentric rings of 6 Ta atoms contract in towards a central Ta atom, forming 13-atom clusters referred to as the star of David (Fig. 1.5). Hybridization of the d orbitals in the cluster leads to 6 bonding and 6 antibonding bands and one nonbonding band. The bonding bands become occupied, leaving one half-ﬁlled nonbonding band, mainly of

dz2 character, around the central Ta. Many questions about this transition remain unanswered, including the driving mechanism. It has long been thought that the electronic transition that accompanies the formation of the star of David clusters is driven by electronic correlations (Mott physics), where a singly-occupied orbital localizes on each cluster. DFT calculations for the bulk indeed ﬁnd a half-ﬁlled band at the Fermi level, with very little in-plane dispersion. However, the bandwidth is not negligible in the out-of-plane direction.

13 Figure 1.5: 1T -TaS2 charge density wave viewed from above. Only Ta atoms are shown for simplicity. Two concentric rings of atoms (blue and black dots) contract toward central Ta atoms (red dots) to form a star of David conﬁguration. The √13 √13 supercell is outlined in green dashed lines. ×

14 Hence it has been suggested that the observed insulating behavior of the bulk C-CDW phase may arise from stacking disorder rather than Mott physics [30]. In this scenario, the single layer C-CDW would be a Mott insulator. In another recent publication [31], it was shown through the combined use of DFT calculations and angle-resolved photoemission spectroscopy (ARPES) that because the stacking for the C-CDW and NC-CDW are quite different, it is possible to explain the MIT without using Mott physics, but rather just considering changes in out-of-plane stacking1. Previously, Yizhi Ge studied the effects of two types of stacking on the electronic bandstructure of 1T -TaS2 and 1T -TaSe2: 1) hexagonal, where star of David centers line up vertically, and 2) triclinic, where star of David centers in adjacent layers are separated by e.g. 2a + c2. Similar to what was reported in Ref. [31], he found that for hexagonal stacking, the half-filled band at the Fermi level has an out-of-plane bandwidth of 0.5 eV. In the plane, the band is very flat. This contrasts with the ∼ triclinic stacking, where the in-plane dispersion is much stronger and out-of-plane somewhat weaker (both are about 0.3-0.4 eV). The actual stacking in this material seems to be neither the triclinic or hexagonal stacking sequences, but perhaps some mixture of these, as shown in diffraction experiments and subsequent theoretical analysis [32, 33]. There has been a lot of interest in whether the C-CDW transitions (both

structural and electronic) persist as 1T -TaS2 is thinned down to a monolayer, where stacking is no longer a concern. Early experiments suggested that the material may

1Note that CDW stacking diﬀers from the stacking of atomic layers in the 2H or 3R polymorphs, for example. In this context, stacking refers to the relative positions of the CDWs in the vertical direction. 2This results in a 13 layer sequence, where the 14th layer aligns with the ﬁrst.

15 not undergo this transition if less than 14 or so layers thick, and that the transition temperature decreases upon thinning [34, 35]. Theoretical calculations, however, indicate that the C-CDW exists even for the monolayer [30, 36]. A transmission electron experiment showed that formation of surface oxide prevents any CDW order in thin samples [37]. In Chapter 4 of this thesis, I present a theoretical and experimental Raman study that shows the

presence of the C phase vibrational signature in a monolayer of 1T -TaS2 [38]. More recently, the C phase was observed using scanning transmission electron microscopy (STEM) in a freestanding trilayer ﬂake at room temperature, suggesting that not only is the structural transition present in few-layer samples, but that the transition temperature may actually increase [39], corroborating other work that suggests a very diﬀerent transition temperature for the bulk and surface commensurate CDWs [40].

Although evidence for the commensurate phase CDW in ultrathin 1T -TaS2 continues to appear, the question of electronic transport in these thin samples remains. Tsen, et al. saw the electronic transition temperature decrease upon thinning (as did Yu, et al. [35]), and no evidence for the commensurate phase structural transition in a h-BN encapsulated trilayer [37, 40]. Whether environmental issues like oxidation in Yu’s study or substrate eﬀects in Tsen’s experiment impacted the transition remains to be seen.

2H-TaS2

2H-TaS2 is the trigonal prismatic polymorph of TaS2. The 2H polymorph of TaS2

diﬀers from that of some other 2H materials (e.g. MoS2) in that the metal atoms lie

directly atop one another. The S atoms alternate positions between trilayers (TaS2

“monolayers”). Bulk 2H-TaS2 undergoes a low-temperature CDW phase transition

16 Figure 1.6: An atomic displacement pattern for 1H-TaS2. This displacement pattern is similar to that found by Moncton, et al. for 2H-TaSe2 [41]. The displacements are exaggerated for visibility.

at 75 K, with only a small increase in the electrical resistivity. Lowering the temperature further, the resistivity decreases steadily, as for a metal [28]. The CDW superstructure for bulk 2H is generally believed, based on electron diﬀraction, to be a 3 3 distortion (q = a/3)[28]. However, others have × CDW proposed, based on Raman spectroscopy, that the distortion is incommensurate with the primitive lattice [42]. Like other materials including NbSe2 and TaSe2, the precise atomic displacement pattern in the 2H CDW phase is not known for TaS2.

For 2H-TaSe2, a displacement pattern similar to that shown in Fig. 1.6 was proposed by Moncton based on neutron diffraction [41]. Later, Bird et al. used electron diffraction to characterize the same material and found a different type of displacement pattern, like the one shown in Fig. 1.7[43]. Unlike the neutron beam in Ref. [41], the spot size of the electron beam in Ref. [43] was smaller than the

17 Figure 1.7: Another atomic displacement pattern for 1H-TaS2. This displacement pattern is similar to that found by Bird, et al. for 2H-TaSe2 [43]. The displacements are exaggerated for visibility.

average CDW domain size in 2H-TaSe2, which accounts for the diﬀerence in symmetry of the proposed structures. DFT calculations indicate that TaS has many competing 3 3 structures close 2 × in energy. Our calculations indicate that these 3 3 distortion patterns result in a ×

1-2 meV / TaS2 lowering of the energy of the lattice, with maximum Ta displacements of δR 0.05 Å. For example, structures similar to that described Ta ≈ by Moncton for 2H-TaSe2 (see Fig. 1.6) are lower in energy than the undistorted structure by about 1 meV / TaS2 [41]. The structure we found to have the lowest energy (energy lowering of about 2 meV / TaS2) is similar to the structure described by Bird, et al. also for 2H-TaSe2 (see Fig. 1.7)[43]. Recently, a paper by Sanders, et al. indicated that when grown epitaxially on Au

(111), single-layer TaS2 adopts the 1H polymorph, and cooling well below the 75 K bulk CDW transition temperature reveals no lattice distortion. The authors suggest

18 that electron doping from the substrate may explain the absence of a CDW in their monolayer [44]. That raises the question, however, about whether the observed absence of a CDW phase is intrinsic to the monolayer or induced by the substrate. This question is explored in Chapter 5 of this thesis. With all the materials that exhibit a CDW, the question of what causes the

distortion to occur is fundamental. In the case of 1H-TaS2, the CDW phase is metallic, and so the Peierls mechanism does not appear to be responsible for the distortion. Instead, strong electron-phonon coupling at the CDW wavevector is likely responsible for the lattice distortion. This is explored in detail in Chapter 5.

1.2.4 Overview of Thesis

The rest of this dissertation is organized as follows. Chapter 2 gives a description of the theoretical background of density functional theory (DFT) and density functional perturbation theory (DFPT), indispensable tools for modern theoretical physicists and chemists. Chapter 3 discusses work that I did in conjunction with IBM Almaden Research Center on the adsorption of Ni on a MgO/Ag(001). Using DFT and DFT+U approaches, we find that the electronic and magnetic properties of the adatom depend strongly on the binding site. In our calculations, the two lowest energy binding sites are close in energy, and an on-site Coulomb interaction U changes their energetic ordering. We find that bonding and magnetism compete to give the most energetically-favorable configuration of charge transfer, orbital hybridization, and spin channel occupation. In Chapter 4 we present first-principles (DFPT) calculations of the Raman mode

frequencies of 1T -TaS2 in the high-temperature (undistorted) phase and the low-temperature commensurate charge density wave phase. We also present

19 measurements of the Raman spectra for bulk and few-layer samples at low temperature. Our measurements show that the low temperature commensurate charge density wave remains stable even for a monolayer. We also calculate the vibrational spectra for the strained material, and propose polarized Raman spectroscopy as a way to identify the c-axis orbital texture (stacking) in the low temperature phase. The orbital texture has been predicted to strongly impact the electronic transport in this material. Chapter 5 delves into the eﬀects of electron doping on the lattice instability of

single-layer 1H-TaS2. Recent ARPES measurements suggest that when grown on Au(111), strong electron doping from the substrate occurs. Furthermore, STM/STS measurements on this system show suppression of the charge density wave instability seen in bulk 2H-TaS2. Our ab initio DFT and DFPT calculations of a free-standing monolayer of 1H-TaS2 in the harmonic approximation show a lattice instability along the ΓM line, consistent with the bulk 3 3 CDW ordering vector. × Electron doping removes the CDW instability, in agreement with the experimental ﬁndings. The doping and momentum dependence of both the electron-phonon coupling and of the bare phonon energy (unscreened by metallic electrons) determine the stability of lattice vibrations. Electron doping also causes an expansion of the lattice, so strain is a secondary but also relevant eﬀect. Chapter 6 concludes my thesis with a summary of the main results and potential directions for future work, some of which I am currently pursuing.

20 Chapter 2

Theoretical Background

Density Functional Theory (DFT) has become ubiquitous in the study of condensed matter due to its ability to give accurate representations of the electronic structure of solids and molecules. This technique relies on a number of approximations and simpliﬁcations that allow the mapping of the quantum many-body problem to a procedure of solving for the eigenstates and eigenvalues of a single-particle Hamiltonian self-consistently. The basic ideas of DFT are summarized here.

2.1 Born-Oppenheimer Approximation

We start with the full many-body Hamiltonian of a system of interacting electrons and nuclei (ions):

H = Telectron + Tion + Velectron electron + Velectron ion + Vion ion (2.1) − − −

Because nuclei are much more massive than electrons (m /m 1.8 103), they p e ≈ × move much more slowly by comparison. To a good approximation, the nuclei appear stationary on the time scale of electronic motion. Thus, our ﬁrst step is to remove the kinetic energy of the nuclei and replace the interaction of the nuclei with each other with a constant. This is known as the Born-Oppenheimer Approximation [45].

HBO = Telectron + Velectron electron + Velectron ion + Eion ion (2.2) − − −

21 Grouping the last two terms together into Vext, we arrive at our simpliﬁed Hamiltonian for electrons:

HBO = Telectron + Velectron electron + Vext, (2.3) − which is still too complicated to solve directly except for the simplest of systems.

2.2 Hohenberg-Kohn Theorems

It is clear that a system with a nondegenerate ground-state, under external potential Vext, cannot have two ground-state densities n ,1(r), n ,2(r). The ﬁrst ◦ ◦ theorem of Hohenberg-Kohn proves the inverse: density n(r) cannot be the ground-state density of two diﬀerent systems [46].

Let there be two external potentials: Vext,1,Vext,2. The ground-states of H1 and

H are Ψ and Ψ , respectively: E◦ = Ψ H Ψ , E◦ = Ψ H Ψ . Assuming 2 1 2 1 h 1| 1 | 1i 2 h 2| 2 | 2i that the ground states are non-degenerate,

Eo < Ψ H Ψ = Ψ H Ψ + Ψ H H Ψ 1 h 2| 1 | 2i h 2| 2 | 2i h 2| 1 − 2 | 2i and Eo < Ψ H Ψ = Ψ H Ψ + Ψ H H Ψ 2 h 1| 2 | 1i h 1| 1 | 1i h 1| 2 − 1 | 1i

If Ψ1 and Ψ2 give the same density, n (r), then this leads to an absurd result: ◦ R E1◦ < E2◦ + n (r)(Vext,1 Vext,2)dr ◦ − and R E2◦ < E1◦ + n (r)(Vext,2 Vext,1)dr ◦ − E◦ + E◦ < E◦ + E◦. ⇒ 1 2 1 2

The density n (r) appears after integration over N 1 spatial coordinates, e.g.: ◦ − R R drdr2 . . . drN Ψ∗(r, r2 ... rN ) V (r)Ψ(r, r2 ... rN ) = dr n(r)V (r).

22 So, each Hamiltonian is uniquely linked to its ground-state electron density, and this density determines E , the ground-state energy. The energy, E, depends on the ◦ function n(r), and thus it is a functional of the density (denoted as E[n(r)]). The second theorem of Hohenberg-Kohn simply states that the ground-state density is that which gives the lowest energy, and thus the ground-state density may be found according to the variational principle. The variational principle essentially states that for a trial wavefunction Ψ , | i Ψ H Ψ E h | | i, (2.4) ◦ ≤ Ψ Ψ h | i meaning that the energy of any trial wavefunction energy is an upper bound for the ground-state energy, and by altering the trial wavefunction in a way that lowers the energy, the trial wavefunction becomes a better approximation for the ground state. Note that the only state discussed thus far has been the ground state, and so it can be said that DFT is a ground-state theory, and essentially models systems at 0 K.

2.3 Kohn-Sham Formulation

The next theorem of Walter Kohn, known as the Kohn-Sham (KS) formulation, transforms the above theorems into a method for actually finding the ground state of an electronic system [47]. In essence, this formulation maps the many-body Hamiltonian onto a single-particle Hamiltonian, in which the many-body effects, namely exchange and correlation, are represented in an effective potential term. The

many-body wavefunction Ψ is replaced by N single-particle wavefunctions ψi, which together form the density n(r) = P f ψ (r) 2 where f is the occupation of state i=1 i| i | i i. Kohn-Sham makes the ansatz that the true ground-state density of the system can be represented this way.

23 Kohn-Sham starts by taking the kinetic energy of a non-interacting N-electron

gas, T [n(r)] = PN ψ ~2 2 ψ and adds to it the classical Hartree energy S i=1 − h i| 2m ∇ | ii e2 RR n(r)n(r0) (Coulombic interaction only) EH [n(r)] = 2 r r0 drdr0. This, along with the | − | interaction energy of the electrons with the external potential, R Eext[n(r)] = Vext(r)n(r)dr, would characterize a non-interacting system’s energy. The rest of the interactions, that of the electrons with each other, can now be expressed in an unknown energy term, which like the others, is a functional of the density. We call this the exchange-correlation energy, Exc[n(r)]. We would like to minimize the energy of all these terms, which we do by implementing a Lagrangian constraint on the system electron number: ! ∂ X Z 0 = TS[n] + EH [n] + Eext[n] + Exc[n] λi ψi∗(r)ψi(r)dr 1 , ∂ψ∗ − − i i (2.5) which yields the following Kohn-Sham Schrödinger equation:

1 2 + V + V + V ψ (r) = ψ (r), (2.6) −2∇ H ext xc i i i

where i = λi, the energies of the single-particle states. We can now write down the Kohn-Sham Hamiltonian operator:

HKS = TS + VH + Vext + Vxc, (2.7)

δEH [n] δExc[n] where VH = δn and Vxc = δn . Guessing a trial electron density gives us a trial Hamiltonian to solve, which yields a new electron density. After some iterations of this procedure, the density converges on what can only be the ground-state density of the system. This self-consistent method of solving for the ground-state density is summarized in Fig. 2.1.

24 guess n(r)

HKS = TS + VH [n(r)] + Vext + Vxc[n(r)]

solve (H ǫ ) ψ (r)=0 KS − i i for ψ (r) , ǫ { i } { i}

nnew(r) = 2 fi ψi(r) i | | P

Is n(r) ≈ nnew(r), nnew(r) n(r). ⇒ no i.e. self- consistent?

yes

nnew(r)= n (r). ◦

Figure 2.1: Simpliﬁed DFT work ﬂow.

25 Some practical considerations for the KS formulation are important to keep in mind. The single-particle KS states do not represent multi-electron conﬁgurations; they are ﬁctitious quasi-particles, and the only link between the single-particle states and the real system is through the ground-state electron density n . Also, the ◦ theorems of DFT are exact, but in practice, the approximation of Exc is a source of error.

2.4 Exchange and Correlation

A simple but useful approximation to the exchange and correlation energy term is known as the Local Density Approximation (LDA). It is based on numerical solutions for the correlation energy and the exact exchange energy for a homogeneous electron gas, i.e. jellium. In the LDA, Exc is calculated using the uniform electron gas results for the density at each point, r:

LDA R Exc [n(r)] = xc (n(r)) n(r)dr, where xc is a function of n(r), not a functional. One would expect this to be accurate when n(r) is slowly varying, but LDA works surprisingly well in many situations. The next approximation is called the Generalized Gradient Approximation (GGA) which takes into account, using gradients, the local density, and how that density changes with position: EGGA[n(r)] = R (n(r), n(r) )n(r)dr [48]. GGA xc xc |∇ | often improves the tendency of LDA to overestimate cohesive energies. Other more sophisticated methods exist to correct inaccuracies in the DFT representation of exchange-correlation. Hybrid functionals mix some amount of

Hartree-Fock, or exact exchange into Vxc, using

HF e2 P RR 1 HF E [n(r)] = ψ∗(r)ψ∗(r ) ψ (r)ψ (r )drdr [49]. This mixing of E x 2 i,j i j 2 r12 j i 2 2 x into Exc is somewhat empirical, but nonetheless, can improve accuracy in certain

26 cases. Meta-GGA is yet another approach, which incorporates the kinetic energy density, τ(r) = ~2 P ψ (r) 2 into E ; we have E [n(r), n(r), τ(r)] [50]. It may 2m i |∇ i | xc xc ∇ be considered as a reﬁnement to GGA, since it incorporates the Laplacian, 2. This ∇ additional degree of freedom allows the functional to capture not just slowly varying densities, but also non-local self-energy corrections. A diﬀerent approach is to include a Coulombic energy penalty U, when an orbital becomes doubly occupied [51]. This can be helpful when working with transition metal elements or rare earths whose d or f electron correlations are underrepresented by Exc. This is computationally inexpensive; however, it is not always clear what value of U is appropriate.

2.5 Implementation: Basis Sets

Next, we consider the accuracy and computational cost of diﬀerent basis sets. The ideal basis set gives an eﬃcient path towards the desired solution without being biased. For particles in a periodic potential, such as electrons in a crystalline lattice, Bloch’s Theorem states that eigenfunctions can be written as

n n ik r ψk = uk(r)e · , (2.8)

n where k is any point in the ﬁrst Brillouin zone, n is the band index, and uk(r) has the periodicity of the potential. The most natural choice is to express our

n single-particle wave-functions in terms of plane waves, so that after expanding uk(r) in terms of a plane-wave basis set, we have

Kmax n X k,n i(k+K) r ψk(r) = cK e · (2.9) K where K is a reciprocal lattice vector that serves as the index for a particular basis

k i(k+K) r function, e.g. φK(r) = e · . This means that by diagonalizing the Hamiltonian

27 matrix (which is simpliﬁed by the fact that plane waves are orthogonal), one can arrive at the eigenvalues and eigenvectors for all the included bands (determined by

Kmax) at that particular k-point. In principle, plane waves are the perfect choice for periodic potentials. They are unbiased, and the basis can be systematically improved by increasing Kmax. However, near the atomic cores, wave functions oscillate with very short wavelengths, requiring a quite large Kmax for an accurate description. This results in large matrix diagonalizations and increased computational cost. To remedy this hindrance, there are various solutions. Pseudopotentials that represent the potential due to the nuclei and the core electrons can be used to replace the bare nuclear potential. In essence, the wavefunctions of the core electrons are assumed to be unaffected by the presence of other atoms, which is reasonable since these particles do not participate in chemical bonding. Since the wavefunctions of core electrons oscillate with very short wavelengths, eliminating them from the calculation allows for a smaller plane wave basis set. Furthermore, the pseudopotential can also soften the oscillations of the valence wavefunctions very close to the core of the atom, while allowing the wavefunction to be realistic beyond a cut-off radius. A pseudopotential’s ability to reduce the number of plane waves required for accurate calculations is known as its softness. Making pseudopotentials with transferability, i.e. that work well in many environments, is also important, especially since charge transfer can depend greatly on the bonding situation. Another solution is the all-electron augmented plane wave (APW) method as implemented, for example, in the DFT code Wien2k [53]. Like the pseudopotential method, the APW method defines a sphere around the atoms, called the muffin-tin sphere. Inside, solutions to the spherical Schrödinger equation serve as the basis set.

28 Figure 2.2: Diagram depicting the augmented plane wave scheme. This diagram shows the muﬃn-tin regions and the interstitial regions, where APW and plane waves, respectfully, reside. This image taken from Ref. [52].

29 Outside, in the interstitial region, plane waves are used. This is advantageous because the core electrons remain in the calculation with reduced computational cost, since radial functions are more efficient than plane waves in the muffin tin. Thus an APW is defined as:  1 ei(k+K) r interstitial k  √V · φK(r,E) =  P A(k+K)R ( r r ,E)Y l (θ , φ ) spheres l,m lm l | − sphere| m sphere sphere (2.10) where the coefficients Alm are solved by matching the wavefunctions at the muffin-tin surface, rsphere is the coordinate of a muffin-tin center, θsphere, φsphere use

l rsphere as their origin, and Ym are spherical harmonics [54, 55]. Since the radial wavefunctions Rl only live in the muffin tin, E above is a parameter that must be solved for each basis function (boundary conditions). This, along with the fact that APWs are not orthogonal, makes the APW method inefficient, but some improvements have sped it up. These include the linearization of the muffin-tin

k portion of φK(r,E), which fixes E as a set of El (this method is called linearized APW, or LAPW). Also, adding local orbitals makes the basis set larger, yet more complete in calculating the band structure. The main advantage of this method is that, unlike the pseudopotential method, the core electrons are accurately considered. With APW, it is possible to calculate hyperfine parameters, electric field gradients and core level excitations. The projector augmented wave (PAW) method [56], which is used in plane wave codes such as VASP [57], essentially bridges the pseudopotential and APW methods. Pseudo-wavefunctions are represented in a ‘soft’ basis of plane waves. Projector functions, defined for each atomic species inside augmentation spheres, allow transformation of the wavefunctions to ‘all-electron’ ones, in a basis similar to that of the muffin tins of APW. Although core electrons are ‘frozen’ (not

30 determined self-consistently), the behavior of valence electrons near the ionic cores is somewhat better described than in the pseudopotential method, and with a smaller workload than the APW method.

2.6 Lattice Dynamics in DFT

In order to calculate phonon frequencies and eigenvectors, one must build and diagonalize the dynamical matrix. Consider ﬁrst a simple one-dimensional

1 2 mass-spring system. For an ideal spring, the potential is harmonic, V = 2 kx , and the force on the mass is proportional to the derivative with respect to position,

dV d2x F = dx = kx. The equation of motion is m dt2 = kx which has a sinusoidal − − q − k solution oscillating with frequency ω = m . For a solid, the system of masses and springs consists of ions with electronic density between. The total energy of the lattice can be Taylor-expanded about the equilibrium atomic positions R0 as { I }

0 X ∂E 0 E ( RI ) = E RI + (RIα RIα) { } ∂RIα 0 − Iα RIα=RIα 2 1 X ∂ E 0 0 + (RIα RIα)(RJβ RJβ) + ... (2.11) 2 ∂RIα∂RJβ 0 0 − − Iα,Jβ RIα=RIα,RJβ =RJβ where α, β indicate Cartesian axes and I,J label the atoms of the crystal. Since we are expanding about the equilibrium positions, the linear term vanishes. In the harmonic approximation, we neglect cubic and higher terms in the expansion. The real-space force constant matrix

2 αβ ∂ E ΦIJ = (2.12) ∂RIα∂RJβ 0 0 RIα=RIα,RJβ =RJβ plays a role analogous to that of the spring constant in the mass-spring system, relating the force felt by atom I to the displacement of atom J (and vice versa).

31 The equation of motion for atom I in direction α is

2 d RIα X M = Φαβ R R0 . (2.13) I dt2 − IJ Jβ − Jβ Jβ For a periodic system, we assume the solutions to the equation of motion consist of traveling plane waves with wave vector q. The equation of motion can then be expressed in reciprocal space as

X αβ β 2 α Dκκ0 (q)ηκ0ν = ων(q)ηκν (2.14) κ0β where the dynamical matrix is given by:

1 αβ X αβ iq Rl Dκκ0 (q) = Φlκ0κ0 e− · (2.15) √M M 0 κ κ l where ν is the phonon branch index. Here, because the lattice is periodic, atoms are labeled by lκ instead of I, where l speciﬁes the cell, κ speciﬁes the atom within the

basis and Rl is the position of cell l. The eigenvalues of the dynamical matrix are

2 the squared frequencies (ων) and the eigenvectors are related to the polarizations of

the phonon normal modes (ηκν = √Mκκν). Within DFT, there are two main approaches for lattice dynamics of periodic systems. In finite displacement methods, supercells with appropriate atomic displacements are built and the forces on the atoms are used to populate the force-constant matrix. This has the advantage of allowing for the inclusion of anharmonic effects by increasing the size of the atomic displacements. However, the method can become computationally expensive due to the size of the supercells. Another approach is to use linear response to calculate the first-order change in the charge density, from which the second-order (and even third-order1) change in total energy can be obtained.

1This is the DFT extension of the 2n + 1 theorem of perturbation theory, which was demonstrated by Gonze, et al. [58].

32 2.6.1 Linear Response

First-order perturbation theory applied to the Kohn-Sham system leads to the relation (H ) δψ = (δV (r) δ ) ψ , (2.16) KS − i | ii − SCF − i | ii where H , , and ψ are the unperturbed Kohn-Sham Hamiltonian, eigenvalues, KS i | ii and eigenstates, respectively, and δV , δ and δψ are the ﬁrst-order corrections SCF i | ii to the self-consistent potential, eigenvalues, and eigenstates, respectively. The linear variation of the SCF potential (VSCF = VH + Vext + Vxc) in DFT is given by

Z e2 δ2E δV (r) = δV (r) + dr + xc δn(r ), (2.17) SCF ext 1 r r δn(r)δn(r ) 1 | − 1| 1 where the two terms in the integral represent the Hartree and exchange-correlation contributions, respectively. The ﬁrst-order correction to the eigenfunctions is given by

X ψj δVSCF ψi δψi(r) = h | | iψj(r). (2.18) i j j=i 6 − Thus to ﬁrst order, the variation in density is

X δn(r) = fi (ψi∗(r)δψi(r) + c.c.) i X X fi fj = − ψi∗(r)ψj(r) ψj δVSCF ψi , (2.19) i j h | | i i j=i 6 − with fi being the Fermi-Dirac occupation of state i. Equations 2.16-2.19 can be solved self-consistently with a scheme similar to that described in Sec. 2.3 for the

Kohn-Sham equations. With δψ and δn, we calculate δVSCF ; solving the linear system in Eq. 2.16 leads to a new set of δψ and δn, and so on. This is called Density Functional Perturbation Theory (DFPT) [59, 60, 61].

33 According to the Hellmann-Feynman Theorem, the ﬁrst derivative of the energy with respect to a parameter λ is

∂E ∂ = Ψ H Ψ ∂λ ∂λ h | | i ∂H = Ψ Ψ h | ∂λ | i Z ∂V = n(r) dr. (2.20) ∂λ

The second derivative is then

∂2E Z ∂n(r) ∂V Z ∂2V = dr + n(r) dr. (2.21) ∂µ∂λ ∂µ ∂λ ∂µ∂λ

So knowing δn, one can calculate the second-order change in the total energy.

Application to Lattice Dynamics

In applying DFPT to lattice dynamics, the atomic positions R are parameters in { I } the electronic Hamiltonian in the Born Oppenheimer approximation which can be varied as in the Hellmann-Feynman Theorem:

HBO = Telectron + Velectron electron + Velectron ion ( RI ) + Eion ion ( RI ) , (2.22) − − { } − { }

where the last term is constant for each set of R . The real-space force constants { I } between atoms I and J are then given by

2 ∂ EBO ΦIJ = ∂RI ∂RJ Z Z 2 2 ∂n(r) ∂Velectron ion ∂ Velectron ion ∂ Eion ion = − dr + n(r) − dr + − . (2.23) ∂RI ∂RJ ∂RI ∂RJ ∂RI ∂RJ

For a periodic system with perturbations in atomic positions of the form

0 0 iq Rl RI = Rlκ + uκ(l) = Rlκ + uκ(q)e · + c.c. , (2.24)

34 the derivative of the system energy with respect to atomic displacements can be Fourier transformed to momentum space:

∂E ∂E X iq Rl = e− · . (2.25) ∂uα(q) ∂uα(l) κ l κ Due to translational invariance in the crystal, the second derivative is Fourier transformed as so:

∂2E ∂2E ∂2E X iq (Rl Rm) X iq Rl = e− · − = NC e− · , α β α β α β ∂uκ∗(q)∂uκ(q) l,m ∂uκ (l)∂uκ(m) l ∂uκ (l)∂uκ(0) (2.26) with NC the number of unit cells in the crystal. Using this, we can write the dynamical matrix for wavevector q as

2 αβ 1 ∂ E Dκκ0 (q) = β 0 α NC √MκMκ ∂uκ∗(q)∂uκ(q) "Z Z 2 2 # 1 ∂n ∂Vion ∂ Vion ∂ Eion ion = dr + n dr + − , (2.27) α β β β N √M M 0 ∂u α α C κ κ κ∗ ∂uκ0 ∂uκ∗∂uκ0 ∂uκ∗∂uκ0 with Velectron ion shortened to Vion, and the derivatives evaluated about uκ(q) = 0. − The dynamical matrix can be computed on a uniform grid of q, then Fourier transformed to obtain the real-space force constants. The range of interatomic forces captured will depend on the density of the q point grid. Conveniently, the dynamical matrix can then be transformed back to momentum space at arbitrary q. This procedure is sometimes called Fourier interpolation.

Electron-phonon Coupling

In the above procedure, we also obtain all the ingredients for calculating the electron-phonon matrix elements:

1 2 qν ~ ν gk+qν0,kν = ψk+q,m ∆VSCF (q) ψk,n (2.28) 2ωqν h | | i

35 with

X 1 ∂VSCF ∆V ν (q) = ηα (q), (2.29) SCF √M ∂uα(q) κν κα κ κ where ηκν(q) is the eigenvector of the dynamical matrix for phonon branch ν at wavevector q. The electron-phonon matrix elements tell us about the interaction between electrons and phonons, which contributes to electrical resistivity in metals, electron-phonon mediated superconductivity, phonon linewidths, electron mass renormalization in metals, phonon renormalization due to electron screening, and many other properties in solids.

36 Chapter 3

Site-dependent Magnetism of Ni Adatoms on MgO/Ag(001)

3.1 Introduction

The study of magnetic adatoms on surfaces has drawn recent attention due to possible applications in the realm of magnetic storage and quantum computation. The density of magnetic storage has enjoyed exponential growth for several decades, but this growth will eventually slow as particle sizes approach the superparamagnetic regime [62]. An alternative, bottom-up approach is to start from the atomic limit. With the scanning tunneling microscope (STM), single atoms can be moved around on a surface to construct desired nanostructures, and the STM can also be used to probe the electronic and magnetic properties of those nanostructures. Such experiments have found, for example, that a single Fe atom on a Cu2N monolayer (ML) on Cu has a large magnetic anisotropy energy [7], and that a magnetic bit consisting of an array of 12 Fe atoms on the same surface has a stable moment that can stay in the ‘on’ or ‘oﬀ’ state for hours at cryogenic temperatures ( 1 K) [4]. ∼ A recent STM study of a single Co atom on a ML of MgO on the Ag(001) surface found that the Co atom maintains its gas phase spin of S = 3/2, and, because of the This chapter is reprinted from O. R. Albertini, A. Y. Liu and B. A. Jones, Phys. Rev. B 91, 214423 (2015). Copyright (2015) by the American Physical Society.

37 axial properties of the ligand field, it also maintains its orbital moment on the surface (L = 3). The resulting magnetic anisotropy is the largest possible for a 3d transition metal, set by the spin-orbit splitting and orbital angular momentum. The measured spin relaxation time of 200 µs is three orders of magnitude larger than typical for a transition-metal atom on an insulating substrate [10]. Here we investigate the structural, electronic, and magnetic properties of Ni adatoms on the same substrate, MgO/Ag(001), using density functional methods. Previous authors [63, 64, 65, 66, 67, 68, 69, 70] have studied the adsorption of transition metal atoms on an MgO substrate using ab-initio techniques. These studies, which employed various surface models and approximations for the exchange-correlation functional, are in general agreement that the preferred binding site of a Ni adatom on the MgO(001) surface is on top of an O atom, and that structural distortion of the MgO surface upon Ni adsorption is minimal. The situation is less clear when it comes to the spin state of the adatom, since the s-d transition energy is so small in Ni. Calculations that employ the generalized gradient approximation for the exchange-correlation functional generally predict a full quenching of the Ni moment on MgO, while partial inclusion of Fock exchange as in the B3LYP hybrid functional predicts that the triplet spin state is slightly more favorable [66, 67, 65]. The system studied in the current work differs from previous studies of Ni adsorption on MgO because the substrate consists of a single layer of MgO atop the Ag(001) surface. This aligns more closely with STM experiments that use a thin insulating layer to decouple the magnetic adatom from the conducting substrate that is necessary for electrically probing the system. We show that, in contrast to the surface layer of an MgO substrate, the MgO ML on Ag can deform significantly due to interactions with the Ni, resulting in a very different potential energy surface

38 for adsorption. Further, we investigate how the interaction of Ni with the MgO/Ag substrate is affected by on-site Coulomb interactions and find that the preferred binding site depends on the interaction strength U. Unlike the case of Co adatoms on the MgO/Ag substrate, the Ni spin moment is always lower than that of the isolated atom. The degree to which the moment is reduced depends strongly on the binding site. These results can be understood by considering the Ni 3d-4s hybrid orbitals that participate in bonding at different sites. We conclude with a discussion about experiments that could corroborate our findings.

3.2 Method

3.2.1 Computational Methods

Two sets of density-functional-theory (DFT) calculations were carried out, one using the linearized augmented plane wave (LAPW) method as implemented in WIEN2K [53], and the other using the projector augmented wave (PAW) method [56] in the VASP package [57]. Both started with the Perdew, Burke and Ernzerhof formulation of the generalized gradient approximation (GGA) for the exchange-correlation functional [71]. In the LAPW calculations, structures for each binding site were relaxed within the GGA. These structures were then used to calculate Hubbard U values for each site using the constrained DFT method of Madsen and Novák [72]. GGA+U calculations were then carried out using those site-speciﬁc values of U to examine electronic and magnetic properties, without further structural relaxation. On the other hand, PAW calculations were used to optimize structures within both GGA and GGA+U. However, since the total energy in DFT+U methods depends on the value of the on-site Coulomb interaction strength, the same value of U was used for each adsorption site to allow comparison of binding energies. In all

39 GGA+U calculations, a rotationally invariant method in which only the difference U J is meaningful was employed [73]. − Within GGA, the two sets of calculations yielded very similar results for adsorption geometries, binding energies, and electronic and magnetic structure, as expected. Although the calculations were used for different purposes in assessing the effect of on-site Coulomb repulsion, the GGA+U results from the two methods were generally consistent and displayed the same trends.1

3.2.2 Supercell Geometry and Binding Sites

STM experiments require a conducting substrate, yet for magnetic nanostructures, it is desirable to suppress interaction between the adatoms and a metallic substrate. Hence the adatom is often placed on a thin insulating layer at the surface of the substrate. It has been shown experimentally that ultrathin ionic insulating layers can shield the adatom from interaction with the surface [74]. In a recent experimental study of Co adatoms, a single atomic layer of MgO was used as the insulating layer above an Ag substrate [75, 10]. The lattice constants of MgO and Ag are well matched (4.19 Å and 4.09 Å, respectively), and DFT calculations of a single layer of MgO on the Ag(100) surface have found that is it energetically favorable for the O atoms in the MgO layer to sit above the Ag atoms [10]. Here we used this alignment of MgO on Ag(001) in inversion symmetric (001) slabs of at least ﬁve Ag layers sandwiched between MLs of MgO. We found only

1 All-electron LAPW calculations used RmtKmax = 7, where Rmt is the smallest muffin- tin radius in the unit cell, for all calculations. Muffin-tin radii in atomic units (ao) for Ag/Mg/O/Ni at O, hollow and Mg sites were 2.50/1.89/1.55/1.81, 2.50/1.89/1.66/1.92, 2.50/1.91/1.91/2.50, respectively. Structural relaxations were carried out with k-point meshes of 6 6 1 and finite temperature smearing of 0.001 Ry. Denser k-point grids × × of up to 19 19 1 were used for more accurate magnetic moments and densities of states. VASP calculations× × were carried out using a plane-wave cutoff of 500 eV, k-point sampling of 8 8 1 grids, and a Gaussian smearing width of 0.02 eV. × ×

40 (a) O site (b) Mg site (c) hollow site

Figure 3.1: Top view of the three binding sites for Ni on MgO/Ag(100). Cyan spheres represent Mg, small red spheres O, and the yellow spheres Ni.

minor differences in results for slabs containing five versus seven Ag layers. The optimized in-plane lattice constants of the Ag slabs with and without MgO MLs were found to be within 1% of the experimental bulk Ag value (4.09 Å), and so ∼ we used the experimental value in this work. An isolated Ni adatom on the MgO/Ag surface was modeled using a 3/√2 3/√2 supercell of the slab with one Ni atom on × each surface, corresponding to a lateral separation of 8.68 Å between adatoms. In the out-of-plane direction, the supercells contained 7 to 8 layers of vacuum. The supercell lattice parameters were held fixed and all atoms were allowed to fully relax. Three high-symmetry binding sites on the MgO surface were considered, as shown in Fig. 3.1: Ni on top of an O atom, Ni on top of an Mg atom, and Ni above the center of the square formed by nearest-neighbor Mg and O sites. These will be referred to as the O site, the Mg site, and the hollow site, respectively.

41 2.0

Ni on MgO/Ag, GGA 1.5 Ni on MgO/Ag, GGA+U Ni on MgO, GGA 1.0

0.5 (eV / adatom) tot E

∆ 0.0

-0.5 hollow O Mg Binding site

Figure 3.2: Total energy per Ni adatom on MgO/Ag, calculated for structures relaxed within GGA and GGA+U (U = 4 eV) from PAW calculations. For comparison, energies calculated for Ni adatoms on MgO are also shown. All energies are plotted relative to the O site energy.

3.3 Results & Discussion

3.3.1 Binding Energetics and Geometries

Since GGA and GGA+U do not fully describe important correlation effects in the isolated Ni atom, the calculated binding energies are not as reliable as differences in binding energy between different adsorption sites. Figure 3.2 shows the total energy relative to that of the O site. Within GGA, the O site is slightly favored over the hollow site, while the Mg site is about 1 eV higher in energy. For comparison, we also plot our results for Ni on an MgO substrate. Similar to previous reports [70], we find that on the MgO substrate, Ni clearly favors the O site, with the hollow and Mg sites lying about 1 and 1.7 eV higher in energy, respectively.

42 Table 3.1: Relaxed distance between adatom and nearest-neighbor Mg and O atoms, and height of these atoms above the MgO ML. All distances are in Å. Results were obtained within the GGA using the all-electron LAPW method.

Adatom Binding site dad O dad Mg ∆zO ∆zMg − − Ni O site 1.79 2.87 +0.15 +0.06 Ni hollow site 1.90 2.53 +0.48 0.18 Ni Mg site 3.63 2.73 0.15− +0.20 Co O site 1.85 2.95 −0.06 0.20 − −

The difference between the potential energy surfaces for Ni adsorption on MgO/Ag versus on MgO can be attributed to the greater freedom that the MgO ML has to deform to accommodate the adatom. The pure MgO substrate remains very flat upon Ni adsorption. When Ni is on the O or Mg site, the displacement of surface atoms is negligible, and when it is on the hollow site, the neighboring O atoms displace out of the plane by about 0.09 Å. The situation is very different for Ni on MgO/Ag. Table 3.1 lists the distance between the adatom and its nearest Mg and O neighbors, as well as the vertical displacement ∆z, of those Mg and O atoms. For each adsorption site, at least one of the neighboring atoms is displaced vertically by more than 0.1 Å. The deformation of the MgO ML is most striking when Ni is on the hollow site, with neighboring O atoms being pulled up out of the MgO ML by nearly 0.5 Å. This allows the hollow site to be competitive in energy with the O site on MgO/Ag, in contrast to what happens on the MgO surface. To examine the influence of local correlations on the binding energetics of Ni on MgO/Ag, we present GGA+U results. Since total energies can only be compared for calculations that use the same value of U, structures were relaxed assuming U = 4

43 eV for all sites. GGA+U predicts that the O and hollow sites reverse order, as shown in Fig. 3.2. The site above an Mg atom still remains much less favorable than either the O or the hollow site. Of course the Coulomb interaction U depends on the local environment of the Ni atom, so it is an approximation to assume a fixed value for all adsorption sites. Nevertheless, these results demonstrate that on-site correlations can change the preferred adsorption geometry. A similar effect has been reported for the adsorption of 3d transition metals on graphene [76]. For a Ni adatom above an O atom, the Ni-O bond is weakened by on-site correlations, as evidenced by an increase of about 8% in the Ni-O bond length in going from U = 0 to U = 4 eV. On the hollow and Mg sites, differences between geometries relaxed within GGA and GGA+U are much smaller. Given the limitations of the DFT+U method, we are not able to definitively predict whether the O site or hollow site is preferred. However, it is likely that the Mg site is not competitive, and not experimentally feasible. Hence the remainder of the paper deals primarily with the hollow and O binding sites, comparing their electronic and magnetic properties.

3.3.2 Electronic and Magnetic Properties

Based on the geometries relaxed within GGA, our constrained DFT calculations yield Coulomb parameters U = 4.6, 6.0, and 5.0 eV for Ni 3d electrons on the O, hollow, and Mg sites, respectively. The GGA+U results presented in this section use these site-speciﬁc values of U, but assume the GGA-relaxed geometries. As mentioned, the O-site geometry is somewhat sensitive to the inclusion of on-site Coulomb repulsion, but the Mg-site and hollow site geometries are not.

8 2 3 Atomic Ni has a ground state conﬁguration of 3d 4s ( F4) with the next highest state 3d94s1, belonging to a diﬀerent multiplet (3D), only 0.025 eV away [77]. The

44 Table 3.2: Total magnetization per adatom (in µB) for diﬀerent sites, calculated within GGA and GGA+U, from all-electron LAPW calculations. Site-speciﬁc values of U are listed in eV.

Adatom Binding site mGGA mGGA+U U Ni O site 0.00 0.16 4.6 Ni hollow site 1.04 1.06 6.0 Ni Mg site 1.40 1.66 5.0 Co O site 2.68 2.78 6.9

average energy of the 3D multiplet, however, is 0.030 eV lower in energy than that

3 of the F4 multiplet [77]. The proximity of these atomic states sets the stage for a competition between chemical bonding and magnetism [66]. Our GGA calculations give a 3d94s1 ground-state conﬁguration for atomic Ni, which under-represents the s occupation compared to experiment, while inclusion of a local Coulomb interaction (U = 6.0 eV ) yields a 3d84s2 conﬁguration. In both

cases, the Ni moment is calculated to be 2.0 µB. The calculated magnetic moments for Ni on MgO/Ag are listed in Table 3.2. The Ni spin moment is reduced from the atomic value on all sites, though by varying amounts. When Ni is on the O site, the magnetic moment is calculated to be zero within GGA, but it increases slightly upon inclusion of U. (When the structure is relaxed within GGA+U and the Ni-O bond length increases by 8%, the

moment increases to about 0.3 µB.) On the hollow site, the magnetic moment of

about 1 µB is insensitive to U, while the larger moment of about 1.5 µB on the Mg site increases with U.

45 Table 3.3: Net charge of Ni atom and its nearest-neighbor O and Mg atoms according to Bader analysis. Ag/MgO is the bare slab without adatoms. These results were obtained from all-electron LAPW calculations.

GGA GGA+U

Ni Onn Mgnn Ni Onn Mgnn O site 0.17 1.50 +1.72 0.13 1.52 +1.71 hollow site −+0.38 −1.51 +1.71− +0.43 −1.54 +1.71 Mg site 0.26 −1.62 +1.68 0.40 −1.61 +1.69 MgO/Ag − −1.64 +1.71 − − −

In Table 3.3, we present the results of a Bader charge analysis [78] (space ﬁlling) for the adatom and its nearest neighbor Mg and O atoms, as well as for a bare slab of MgO/Ag. Because this analysis includes the interstitial electrons, these results can provide insight on the nature of the bonding between the adatom and the surface. On the O site, the Ni adatom gains electrons from its O neighbor (which is less negatively charged than it would be on bare MgO/Ag). In contrast, on the hollow site, Ni loses electrons to the underlying Ag substrate. These charge transfer results are relatively insensitive to local Coulomb interactions. To understand the site dependence of the Ni moment, we examine the bonding and electronic density of states, starting with Ni on the O site within GGA. While the transfer of electrons from the substrate to a 3d94s1 conﬁguration can only reduce the spin moment, the gain of 0.2 is not enough to account for the full quenching ∼ of the moment. The fact that most of the electrons are transferred from the neighboring O atom, and the fact that Ni adsorption causes the O atom to displace upwards out of the MgO ML (Table 3.1), suggest the formation of a covalent bond.

46 GGA GGA+U d 2 d 2 z Ni z Ni (a)dxy (b) dxy 2 2 d 2 2 dx - y x - y

dxz+ dyz dxz+ dyz s s DOS O site Mgnn total Mgnn total Onn total Onn total

total total

d 2 d 2 z Ni z Ni (c)dxy (d) dxy 2 2 2 2 dx - y dx - y

dxz dxz dyz dyz s s DOS Mgnn total Mgnn total hollow site Onn total Onn total

total total

-6 -4 -2 0 2 -6 -4 -2 0 2

E-EF (eV) E-EF (eV)

Figure 3.3: Densities of states from inside the muﬃn-tins of all-electron LAPW calculations. (a) GGA results for Ni on O site. (b) GGA+U results for Ni on O site. (c) GGA results for Ni on hollow site. (d) GGA+U results for Ni on hollow site. In all graphs, majority spin is plotted on the positive ordinate and minority spin on the negative ordinate. Total DOS has been scaled down signiﬁcantly, and includes contributions from the interstitial region. The relative scales of nearest-neighbor Mg and O, as well as Ni s and d DOS have been adjusted to enhance viewability.

47 Indeed, previous studies of Ni on MgO discuss a bonding mechanism involving Ni hybrid orbitals of 4s and dz2 character [63, 64, 66, 65, 69]. One of the two orthogonal orbitals, which we denote as 4s + dz2 , is oriented perpendicular to the surface and interacts strongly with O pz orbitals, forming a bonding combination (mostly O pz) about 6 eV below the Fermi level (EF ) and an antibonding combination (mostly Ni

4s + dz2 ) about 1 eV above the Fermi level. This can be seen in the electronic density of states plotted in Fig. 3.3(a). The other hybrid orbital, denoted 4s d 2 , is − z concentrated in the plane parallel to the surface, and hence interacts much more weakly with the surface. This planar hybrid orbital lies just below the other Ni d states, about 1 eV below the Fermi level. The covalent interaction between Ni and O pushes the antibonding orbital high enough to be unoccupied in both spin channels, so the other d and s-d hybrid orbitals are fully occupied, yielding a zero net spin. When the orbital-dependent Coulomb interaction U is included within GGA+U, the density of state changes significantly since the d and s states are affected differently. The hybridization between Ni dz2 and s orbitals weakens, and the interaction between these orbitals and O p states is affected. The partial density of states plots in Fig. 3.3(b) show the majority and minority spin dz2 orbitals split by about 4 eV ( U). The antibonding Ni - O orbital, now mostly composed of Ni s ∼ states, has a small spin splitting of the opposite polarity: the spin-down states lie just below EF and the spin-up ones lie just above EF . Hence even though U induces a moment in the d shell within the Ni muffin tin, this moment is screened by the oppositely polarized antibonding orbital formed from Ni 4s and O 2pz orbitals. This is illustrated in Fig. 3.4(a), which shows a majority-spin (red) isosurface with dz2 character on the Ni atom, surrounded by the antibonding Ni s -O pz minority-spin (blue) isosurface (corresponding to minority spin states near 0.5 eV in ∼ − Fig. 3.3(b)). The total magnetization of the system remains close to zero.

48 (a) O site

(b) hollow site

Figure 3.4: Spin density isosurfaces of Ni adsorbed on MgO/Ag, from GGA+U all-electron LAPW calculations. Red and blue surfaces correspond to majority and minority spin densities of 0.00175 e/a.u.3, respectively. (a) O site: the majority spin dz2 is screened by the antibonding Ni s -O p orbital. (b) hollow site: the antibonding Ni dyz -O p orbital is spin polarized. The y axis is parallel to the line joining the two O atoms that neighbor the Ni. These two O atoms are vertically displaced out of the MgO plane. O atoms are represented by small red spheres.

49 The situation diﬀers for the hollow site. The presence of the Ni at the hollow site disrupts the MgO bond network, and the two O atoms that neighbor the Ni atom are displaced out of the surface by nearly 0.5 Å to form bonds with the Ni, as is evident in Fig. 3.4(b). The resulting polar covalent Ni-O bond length is only about

6% larger than the corresponding O-site bond (Table 3.1). The hollow site has C2v symmetry, meaning that the dxz, dyz degeneracy is broken. The lobes of the dyz orbital point towards the O neighbors and interact with O py orbitals. However, since dyz transforms diﬀerently than 4s, symmetry considerations prevent the dyz orbital from forming diﬀuse hybrids with 4s analogous to the 4s d 2 orbitals on ± z the O site that enhance the orbital overlap between the adatom and surface atoms.

While symmetry permits hybridization between Ni s and dx2 y2 orbitals, the hybrid − that interacts strongly with O p states has mostly Ni s character, while the orthogonal hybrid orbital is mostly Ni dx2 y2 . Hence the Ni d orbitals all retain a − higher degree of localization than they do on the O site. Within GGA, it becomes favorable for the antibonding Ni dyz -O p orbitals to become spin polarized and this largely accounts for the total moment of about 1 µB. This can be seen in Fig. 3.3(c). Because the effect of U in the hollow site case is to increase the splitting between the dyz majority and minority spin states, the occupation of s and d states and the total moment are not significantly impacted, as can be seen in Fig. 3.3(d). Figure 3.4(b) shows the spin density for this case. The majority-spin isosurface in red has antibonding Ni dyz -O py character. While it is hard to see from the angle shown, the minority-spin isosurface in blue has the character of a 4s dx2 y2 hybrid. − − We return now to the question of why the local Coulomb interaction stabilizes the hollow site relative to the O site. When Ni adsorbs to the O site it forms a strong covalent bond with O through diffuse sd hybrids that overlap significantly with O p orbitals. Since the orbital-dependent U disrupts the sd hybridization, it

50 weakens the binding of the Ni to the O. On the hollow site, geometric and symmetry considerations prevent the formation of diﬀuse sd hybrids that overlap strongly with O, so the impact of U, both in terms of binding energy and spin moment, is smaller. As a result, increasing the strength of the local Coulomb interaction decreases the stability of the O site relative to the hollow site. The presence of magnetization in adatoms depends largely on the relative strengths of the covalent interaction with surface atoms and the intra-atomic exchange coupling for the adatom. In a previous study [10], our calculations found that Co adsorbs to the O site of the MgO/Ag surface. We also found that, in contrast to Ni, it maintains a magnetization close to the atomic value (Table 3.2). This can be explained by diﬀerences in the Ni-O and Co-O interactions. Unlike Ni, Co remains charge neutral when adsorbed, according to a Bader analysis, and the neighboring O and Mg atoms displace downward rather than being attracted to the adatom (Table 3.1). Since the s-d transition energy is larger in Co than Ni, the hybridization between s and d orbitals is reduced, leading to less interatomic overlap and weaker covalent interactions. The d orbitals remain localized enough to support magnetism.

3.4 Conclusions

Using ﬁrst-principles calculations, we have found a strong site dependence of the electronic and magnetic properties of Ni on MgO/Ag. In particular, the magnetic moment varies from S = 0 to nearly 1 as Ni is moved away from surface O. This is in stark contrast with the situation of Co on MgO/Ag, in which the O site facilitates the preservation of Co spin and orbital moments on the surface.

51 Furthermore, we see that the energy surface for adsorption of Ni on MgO/Ag is very different from that for Ni on MgO. On MgO/Ag, the O site and hollow site are competitive in energy, and it is not clear from theory alone which site will be realized in experiments. However, the properties of the Ni adatom on the two sites differ significantly, not just in the magnetic moment, but also in the charge transfer and bonding. The energetic proximity of the hollow and O sites raises the interesting possibility of being able to select one site or the other, and tune the magnetic moment by modification of the substrate through application of strain, introduction of defects, etc. For example, our tests show that while a 2% increase in the ∼ in-plane lattice constant does not change the preferred binding site and has negligible effect on magnetic moments, it does alter the relative energies of the O and hollow sites. It would also be interesting to explore how the properties of the system change with the thickness of the MgO layer, both experimentally and theoretically. In going from mono- to bi- to few-layer MgO, we expect the O site to become increasingly favored as the MgO layer becomes more rigid.

52 Chapter 4

Zone-center Phonons of Bulk, Few-layer, and Monolayer 1T -TaS2: Detection of the Commensurate Charge Density Wave Phase Through Raman Scattering

4.1 Introduction

The 1T polymorph of bulk TaS2, a layered transition-metal dichalcogenide (TMD), has a rich phase diagram that includes multiple charge density wave (CDW) phases that diﬀer signiﬁcantly in their electronic properties. At temperatures above 550 K the material is metallic and undistorted. Upon cooling, it adopts a series of CDW phases. An incommensurate CDW structure (I) becomes stable at 550 K, a nearly commensurate phase (NC) appears at 350 K, and a fully commensurate CDW (C) becomes favored at about 180 K. At each transition, the lattice structure distorts and the electrical resistivity increases abruptly. In the C phase, 1T -TaS2 is an insulator [25, 79, 29]. Upon heating, an additional triclinic (T) nearly commensurate

CDW phase occurs roughly between 220-280 K [80]. While the 1T -TaS2 phase diagram is well studied [80, 25, 81, 79, 82, 83, 84, 85, 29, 35], the exact causes of the electronic and lattice instabilities remain unclear. Electron-electron interactions, This chapter is reprinted from O. R. Albertini, R. Zhao, R. L. McCann, S. Feng, M. Ter- rones, J. K. Freericks, J. A. Robinson, and A. Y. Liu, Phys. Rev. B 93, 214109 (2016). Copyright (2016) by the American Physical Society.

53 electron-phonon coupling, interlayer interactions, and disorder may all play a role. It has long been assumed, for example, that electron correlations open up a Mott-Hubbard gap in the C phase [86]. Recent theoretical work, however, suggests that the insulating nature of the C phase may come from disorder in orbital stacking instead [30, 31]. Advances in 2D material exfoliation and growth have enabled studies of how dimensionality and interlayer interactions affect the phase diagram of 1T -TaS2. Some groups have reported that samples thinner than about 10 nm lack the jump in resistivity that signals the transition to the C CDW phase [35]. However, this may not be an intrinsic material property, as oxidation may suppress the transition. Indeed, samples as thin as 4 nm—when environmentally protected by encapsulation between hexagonal BN layers—retain the NC to C transition [37]. Whether the C phase is the ground state for even thinner samples, and whether this leads to an insulating state, remain open questions. Materials that exhibit abrupt and reversible changes in resistivity are attractive for device applications, particularly when the transition can be controlled electrically. Ultrathin samples may exhibit easier switching than bulk materials due to reduced screening and higher electric-field penetration. Hollander et al. have demonstrated fast and reversible electrical switching between insulating and metallic states in 1T -TaS2 flakes of thickness 10 nm or more [34]. Tsen and coworkers have electrically controlled the NC-C transition in samples as thin as 4 nm and identified current flow as a mechanism that drives transitions between metastable and thermodynamically stable states [37]. Focusing on the C phase rather than the NC-C transition, theoretical work indicates that in few-layer

1T -TaS2 crystals, the existence or absence of a band gap depends on the c-axis

54 orbital texture, suggesting an alternate approach to tuning or switching the electronic properties of this material [31]. Raman spectroscopy has been a powerful tool for characterizing 2D TMDs [87, 88, 89, 90, 91, 92, 93, 94, 95]. Because the phonon frequencies evolve with the number of layers, Raman spectroscopy offers a rapid way to determine the thickness of few-layer crystals [96]. Raman spectra can also reveal information about interlayer coupling [93, 94, 97, 98], stacking configurations [99], and environmental effects like strain and doping [92]. For these materials, the interpretation of measured Raman spectra has benefited greatly from theoretical analyses utilizing symmetry principles and materials-specific first-principles calculations. Thus far, theoretical studies of TMD vibrational properties have focused on the 2H structure favored by group-VI dichalcogenides like MoS2 and WSe2 [92]. We present a symmetry analysis and computational study of the Raman spectrum of 1T -TaS2 in the undistorted normal 1T phase as well as in the low-temperature C phase, alongside Raman measurements of the latter. The symmetry of the 1T structure differs from that of the 2H structure, and different modes become Raman active in the normal phase. In the C phase, a Brillouin zone reconstruction yields two groups of folded-back modes, acoustic and optical, separated by a substantial gap. The low-frequency folded-back acoustic modes serve as a convenient signature of the C phase. In our measured Raman spectra, this signature is evident even in a monolayer. This is in contrast to some previous studies[35, 37, 100, 101] which suggest a suppression or change in nature of the C phase transition in thin flakes. We also investigate the effects of c-axis orbital texture on the Raman spectra and use polarized Raman measurements to narrow down the possible textures.

55 4.2 Description of Crystal Structures

A single ‘layer’ of TaS2 consists of three atomic layers: a plane of Ta atoms sandwiched between two S planes. In each plane, the atoms sit on a triangular lattice, and in the 1T polytype, the alignment of the two S planes results in Ta sites that are nearly octahedrally coordinated by S atoms; the structure is inversion symmetric.1 The bulk material has an in-plane lattice constant of a = 3.365 Å, and an out-of-plane lattice constant of c = 5.897 Å[85, 81]. In the C CDW phase, two concentric rings, each consisting of six Ta atoms, contract slightly inwards towards a central Ta site ( 6% and 3% for the ∼ ﬁrst-neighbor and second-neighbor rings, respectively), forming 13-atom star-shaped clusters in the Ta planes (Fig. 4.1). The in-plane structure is described by three

CDW wave vectors oriented 120◦ to each other, resulting in a √13a √13a × supercell that is rotated φCDW = 13.9◦ from the in-plane lattice vectors of the primitive cell. The S planes buckle outwards, particularly near the center of the clusters, inducing a slight c-axis swelling of the material [102]. The vertical alignment of these star-of-David clusters determines the so-called stacking of the C phase. Because the electronic states near the Fermi level are well described by a single Wannier orbital centered on each cluster, the stacking arrangement of the clusters is also referred to as c-axis orbital texture. In hexagonal stacking, stars in neighboring layers align directly atop each other. In triclinic stacking, stars in adjacent layers are horizontally shifted by a lattice vector of the primitive cell, resulting in a 13-layer stacking sequence. Some experiments suggest that the clusters in the C phase align in a 13-layer triclinic arrangement [103, 104, 105], while others have suggested mixed[33] or even disordered[81, 32]

1Please see Fig. 1.2(b).

56 stacking. Interestingly, the precise stacking sequence strongly aﬀects the in-plane transport, due to the overlap between laterally shifted Wannier orbitals in adjacent layers [31, 22].

4.3 Methods

4.3.1 Computational Methods

Density functional theory (DFT) calculations were performed with the Quantum Espresso[106] package. The electron-ion interaction was described by ultrasoft pseudopotentials [107], and the exchange-correlation interaction was treated with the local density approximation (LDA) [108]. Electronic wave functions were represented using a plane wave basis set with a kinetic-energy cut-off of 35 Ry. For 2D materials (monolayer, bilayer, and few-layer), we employed supercells with at least 16 Å of vacuum between slabs. For the undistorted 1T structure, we sampled the Brillouin zone using grids of 32 32 16 and 32 32 1 k-points for bulk and × × × × few-layer systems, respectively. For the C phase, 6 6 12 and 6 6 1 meshes × × × × were used for bulk and 2D systems, respectively. An occupational smearing width of 0.002 Ry was used throughout. Bulk structures were relaxed fully, including cell parameters as well as the atomic positions. For few-layer structures, the atomic positions and the in-plane lattice constants were optimized. The frequencies and eigenvectors of zone-center phonon modes were calculated for the optimized structures using density functional perturbation theory, as implemented in Quantum Espresso [106]. To investigate the effect of c-axis orbital texture in the bulk C phase, we considered two different supercells, referred to as the hexagonal and triclinic cells. Both cells have in-plane lattice vectors A = 4a + b and B = a + 3b, where a and −

57 undistorted BZ M qcdw

Γ 2a + 2b B C CDW BZ

Figure 4.1: An illustration of the Ta plane and Brillouin zone reconstruction under the lattice distortion of the C phase. The blue cells represent the real space unit cell of the undistorted lattice (defined by the lattice vectors a and b) and the corresponding first Brillouin zone. The green cells represent the super cell of the C CDW lattice (defined by the lattice vectors A and B) and the corresponding first Brillouin zone. The dotted lines indicate the lateral shift of a supercell in an adjacent Ta plane under triclinic stacking.

b are the in-plane lattice vectors of the undistorted cell, as shown in Fig. 4.1. The hexagonal cell has C = c, where c is perpendicular to the plane, while the triclinic cell is described by C = 2a + 2b + c (Fig. 4.1). In the triclinic cell, the central Ta atom in a cluster sits above a Ta in the outer ring of a cluster in the layer below.

4.3.2 Experimental Methods

Multilayer 1T -TaS2 was obtained by mechanical exfoliation onto Si/SiO2 (300 nm) from bulk 1T -TaS2 crystals synthesized via chemical vapor transport (CVT) [109].

58 Optical microscopy was used to identify various thicknesses of exfoliated flakes, and atomic force microscopy (AFM) was used to measure flake thickness. The Supplemental Material[38] contains the results and corresponding images for these measurements. Raman spectra were collected via a Horiba LabRAM HR Evolution Raman spectrometer with 488 nm Ar/Kr wavelength laser and 1800/mm grating. The laser power at the sample surface and the integration time were sufficiently low to prevent damage to the sample. Back-scattered measurement geometry was adopted and the laser was focused on the sample through a 50 objective, with a 1 × µm spot size. The sample was kept in a sealed temperature stage, with pressure

2 being less than 1 10− torr to minimize oxidation during the measurement. The × measurement temperature was controlled from 80 K to 600 K in a cold-hot cell with a temperature stability estimated to be 1 K. Polarized Raman spectra were ± collected with the relative angle of polarization between the incident and measured light (θ) varied between 0 and 90◦.

4.4 Results and Discussion

4.4.1 Undistorted 1T -TaS2

The undistorted phase of bulk 1T -TaS2 is stable above 550 K. We did not measure Raman spectra in this temperature range because our samples showed signs of damage. Nevertheless, a theoretical study of the vibrational modes is a useful starting point for understanding the more complicated C phase.

The point group of the undistorted 1T structure is D3d for both bulk and 2D crystals. The zone-center normal modes belong to the representations

Γ = n(A1g + Eg + 2A2u + 2Eu), where n represents the number of layers except for the bulk, where n = 1. One of the A2u and one of the Eu representations correspond

59 (a) (b) Raman-active IR-active

Eg Eu

A2u A1g

inversionsymm. inversionsymm. preserving breaking

Figure 4.2: Optical modes of 1T bulk and monolayer structures. (a) Eg and A1g modes (b) Eu and A2u modes. Gray spheres represent Ta; yellow spheres represent S.

to acoustic modes. The two-fold degenerate Eg and Eu optical modes involve purely

in-plane atomic displacements, while the A1g and A2u optical modes involve only out-of-plane displacements. The displacement patterns of the optical modes for bulk and monolayer (1L) crystals are shown in Fig. 4.2. Since the point group contains inversion, the A1g and Eg modes, which preserve inversion symmetry, are Raman active, whereas the A2u and Eu modes are IR active. The zone-center phonon frequencies calculated for monolayer and few-layer

1T -TaS2 are plotted in Fig. 4.3(a), and those for the bulk are shown in Fig. 4.3(b). The modes separate into three frequency ranges: acoustic and interlayer modes

1 below about 50 cm− , in-plane Eg and Eu optical modes between about 230 and 270

1 cm− , and high-frequency out-of-plane A1g and A2u optical modes between about

1 370 and 400 cm− . The low-frequency interlayer modes are present when n > 1 and

60 2 (a) Raman-active 1L 1 IR-active 2 2L 1 interlayer modes 2 3L 1 2 4L 1 2 5L 1 2 6L 1

number of modes 2 1 (b) bulk

6 acoustic modes (c) hex. qcdw 2 optical modes tri. qcdw 0 100 200 300 400 -1 wave number (cm )

Figure 4.3: Calculated phonon spectra of undistorted 1T -TaS2. (a) Γ phonons in few-layer crystals, (b) Γ phonons in the bulk crystal, and (c) bulk phonons at qcdw for hexagonal and triclinic stacking. In (a) and (b), the doubly and singly degenerate modes have E and A symmetry, respectively. In (c), the degeneracy corresponds to the number of equivalent qcdw points in the Brillouin zone. Unstable modes (imaginary frequencies) at qcdw are omitted.

correspond to vibrations in which each layer displaces approximately rigidly, as illustrated in Fig. 4.4 for the bilayer (2L). The 1T structure retains inversion symmetry regardless of the number of layers n. This is in contrast to the 2H structure, in which the symmetry depends on whether n is odd or even. However, since the center of inversion for an n-layer 1T crystal alternates between being on a Ta site for odd n and in-between TaS2 layers for even n (Fig. 4.4), the displacement patterns of Raman-active modes diﬀer depending on whether n is even or odd. We see this most clearly in the ultra-low-frequency regime. The highest mode in this regime is always an interlayer

61 breathing mode (Fig. 4.4(b)) in which the direction of displacement alternates from layer to layer. For odd n, this mode is IR active, while for even n it is Raman active. The same eﬀect is evident in the high-frequency out-of-plane modes.

(a) (b)

shear breathing

Figure 4.4: Simpliﬁed illustration of the low-frequency phonon modes of a 1T bilayer. (a) interlayer shear and (b) interlayer breathing. Gray spheres represent Ta; yellow spheres represent S. The dots represent inversion centers.

To facilitate interpretation of the calculated and measured Raman spectra of the C CDW materials in Sec. 4.4.2, we show in Fig. 4.3(c) the phonon frequencies of

bulk undistorted 1T -TaS2 at the CDW wave vectors, qcdw, for hexagonal and

2 triclinic stacking . Modes from the qcdw and equivalent points fold back to the Γ point in the Brillouin zone of the C supercell (Fig. 4.1)[110]. For both types of stacking, we ﬁnd unstable modes (not shown), indicating that the structure is

dynamically unstable against a periodic distortion characterized by qcdw. Based on Fig. 4.3(c), we expect that upon C CDW distortion, the folded-back acoustic modes

1 will form a group near 100 cm− (shown in orange), separated from the folded-back 2For the hexagonal stacking, two inequivalent reciprocal lattice vectors of the CDW cell lie within the Brillouin zone of the undistorted cell: g1 and g1 + g2. Each is 6-fold degenerate. For triclinic stacking, there are 6 such vectors: g , g , g g , 2g g , g 2g , and 1 2 1 − 2 1 − 2 1 − 2 g1 + g2 + g3. Each is 2-fold degenerate.

62 1 optical modes (shown in blue) by a gap of 100 cm− . These are in addition to the ∼ original zone center modes.

4.4.2 Commensurate CDW Phase of 1T -TaS2

As in previous calculations [31, 30, 36], we ﬁnd the C phase energetically favorable compared with the undistorted structure for all thicknesses. In the bulk, the system lowers in energy by 9 meV/TaS and 13 meV/TaS for hexagonal and triclinic ∼ 2 ∼ 2 stacking, respectively. For a monolayer, the energy lowers 23 meV/TaS . ∼ 2 In the C CDW phase, the point group symmetry of a monolayer reduces from

D3d to C3i; inversion symmetry is preserved. The normal modes at Γ are classiﬁed as 19Eg + 19Ag + 20Eu + 20Au, with 57 Raman and 60 IR modes. For bulk or few-layer C CDW with hexagonal stacking, the point group remains C3i. With triclinic stacking (e.g. C = 2a + 2b + c), the symmetry further reduces to Ci; only inversion symmetry remains, and the normal mode classiﬁcation at Γ becomes n(57Ag + 60Au). The black curves in Fig. 4.5 show the measured Raman spectrum of a bulk sample of 1T -TaS2 on an SiO2 substrate. At 250 K (top panel), the sample is in the NC phase. At 80 K (bottom panel), the sample is well below the NC–C transition

1 temperature (180 K). In going from NC to C, new peaks appear in the 60–130 cm−

1 and 230–400 cm− ranges. These spectra are consistent with previous Raman studies of the bulk material [111, 112, 100, 113]. We calculated the phonon modes at Γ for the bulk C structure with hexagonal and triclinic stacking. Fig. 4.6 shows the Raman-active modes alongside the frequencies extracted from bulk Raman measurements of Fig. 4.5. Our calculations and experiment are in general agreement in that both ﬁnd two groups of

1 Raman-active modes, one ranging from about 50-130 cm− and the other from

63 250 K

80 K

0.6 nm (~1 layer)

5.6 nm (~9 layers)

Raman intensity (arb. units) 13.1 nm (~21 layers)

14.6 nm (~23 layers)

bulk 50 100 150 200 250 300 350 400 -1 Raman shift (cm ) Figure 4.5: Raman spectra of bulk (> 300 nm) and thin (14.6 nm, 13.1 nm, 5.6 nm, and 0.6 nm) samples, measured at 250 K (NC phase) and 80 K (C phase). All the spectra are normalized to their strongest Raman peak, indicated for the 0.6 nm curve by the arrow.

1 about 230-400 cm− . These correspond to the folded-back acoustic and optic modes, as anticipated in Fig. 4.3. (A more detailed comparison of the theoretical and

experimental Raman lines is presented later in this section.) In 1T -TaSe2, unlike

what we observe for 1T -TaS2, the folded-back acoustic and optical groups overlap [110]. The smaller mass of S compared to Se raises the frequencies of the optical

group in TaS2. The presence of folded-back acoustic modes and thus the C phase is easier to identify in this material due to the frequency gap.

64 (a) bulk Raman 80 K on SiO2

2 (b) hexagonal 1 stacking

2 (c) triclinic 1 stacking number of modes 0 100 200 300 400 -1 wave number (cm ) Figure 4.6: Comparison of experimental and calculated Raman frequencies for the bulk. (a) Raman lines measured for a bulk ﬂake, at 80 K. Calculated C phase Raman-active modes for bulk: (b) hexagonal stacking (c) triclinic stacking. Eg modes are doubly degenerate. Triclinic stacking has Ag modes only.

The red, blue, green and orange curves in Fig. 4.5 are the measured Raman spectra for thin samples of 1T -TaS : 14.6 nm ( 23 layers), 13.1 nm ( 21 layers), 2 ∼ ∼ 5.6 nm ( 9 layers), 0.6 nm (1 layer). As far as we are aware, this is the ﬁrst report ∼ in the literature of the Raman spectrum of monolayer 1T -TaS2. We focus on the data taken at 80 K, where the bulk material is in the C CDW phase. As the thickness is reduced from bulk to monolayer, the Raman spectrum remains largely unchanged, strongly suggesting that the C CDW structural phase persists as the samples are thinned, even down to a monolayer. The monolayer spectrum is remarkably similar to that of the bulk (as well as to that of the 14.6 nm and 13.1 nm samples). On the other hand, the spectrum of the 5.6 nm sample, while similar, shows some deviations. In particular, some of the features are broadened,

1 and there appears to be a higher background in the 80 - 130 cm− range, which could be due to the presence of additional, closely spaced peaks. These changes do not necessarily indicate a change in the C CDW lattice distortion. In general, in

65 going from a monolayer to a few-layer crystal, we expect weak interlayer interactions to shift and split each monolayer phonon line, as can be seen, for example, in our

calculated results for the high-temperature undistorted phase of few-layer 1T -TaS2 in Fig. 4.3. The differences in the 5.6 nm ( 9 layers) spectrum compared to those of ∼ the bulk-like and monolayer flakes (Fig. 4.5) could arise from such few-layer effects. The thickness dependence of the NC to C phase transition has previously been investigated experimentally using transport [35, 37], electron diffraction [37], and Raman measurements [101]. Most of these studies have reported difficulty in obtaining the C phase in thin samples, though, to our knowledge, none of them probed monolayer samples. One previous Raman study suggested that the C phase is suppressed below thicknesses of about 10 nm at 93 K [100]. It is not clear though, whether those samples were protected from surface oxidation, which can affect the transition. Studies of thin 1T -TaS2 samples that are environmentally protected by a h-BN overlayer indicate that the C phase persists down to 4 nm ( 7 layers) ∼ [37, 101]. In addition, the diffraction and transport measurements suggest the presence of large C phase domains in a 2 nm sample ( 3 layers) at 100 K, although ∼ a conducting percolating path between the domains still exists. The Raman spectra of this 2 nm sample at 10 K loses many of the well-defined peaks [101]. Since our Raman measurements for the monolayer strongly indicate the C phase at 100 K, it may be that the barrier for the NC-C transition is smaller for the monolayer than for few-layer samples. This would be consistent with the suggestion that there are distinct bulk and surface transitions, with the bulk having a larger activation barrier [101]. Alternatively, it is possible that the encapsulating layer affects the transition in ultra thin samples. The measured Raman spectra for the bulk and monolayer are remarkably similar, both in peak locations and relative peak heights. Most of the discernible

66 1 peaks shift by less than 1 cm− in going from bulk to monolayer. Our calculations, on the other hand, show a stronger dependence on dimensionality. Fig. 4.7 shows the calculated 1L spectrum (c), alongside the hexagonally-stacked bulk spectrum (e). To

highlight mode shifts, we show the Ag modes in red and the Eg modes in black. As in the experiments, the 1L and bulk spectra are similar, except the spectrum generally

hardens upon thinning, particularly in the folded-back acoustic group, where the Ag

1 1 modes harden by 1 cm− and the E modes harden by up to 20 cm− . ∼ g ∼ This discrepancy between theory and experiment may be caused in part by approximations made in our DFT calculations. For similar materials, LDA

1 frequencies are typically within 5-10 cm− of experiment [114]. In addition, it is likely that substrate-induced effects, such as strain and screening, which are not taken into account in our calculations, are important. In general, positive strain and enhanced screening tend to soften vibrational frequencies. In the absence of a substrate, the reduced screening in a monolayer tends to harden the phonon modes, as seen in our calculations. The presence of a dielectric substrate in the experiments mitigates this effect. To explore the effect of strain, we calculated the Raman modes (Fig 4.7(b)) for a 1L under 3% strain as an example. Straining the monolayer in our calculations generally softens the modes, bringing the folded-back acoustic group into better agreement with the measurement, while over-softening the folded-back optical group. Since we did not include substrate screening in our simulations, the relative importance of strain and screening remains unclear. As with the undistorted case in Sec. 4.4.1, the bilayer (Fig. 4.7(d)) has ultra-low frequency modes, associated with interlayer shear and breathing. The breathing

1 mode (A ) is calculated to be 30 cm− lower than in the undistorted case, g ∼ indicating that the coupling between layers is weaker in the C phase; indeed, the

67 (a) 1L Raman 80 K on SiO2 2 (b) +3% strain 1L 1 0 2 (c) 1L 1 0 2 (d) 2L 1 0 number of modes 2 (e) bulk 1 0 0 100 200 300 400 -1 wave number (cm ) Figure 4.7: Comparison of experimental and calculated Raman frequencies for a single layer. (a) Raman lines measured for a ﬂake 0.6 nm thick, at 80 K. Calculated C phase Raman-active modes: (b) monolayer, 3% strain (c) monolayer, no strain (d) bilayer, hexagonal stacking (e) bulk, hexagonal stacking. For clarity, red lines represent Ag modes and black lines represent Eg modes. The Supplemental Material[38] contains the calculated C phase Raman-active modes for a bilayer with triclinic stacking.

structure swells by 0.1 Å in the distortion. In comparing Fig. 4.7(c) and (d), two 2L

Raman modes stand out above and below the 1L optical group range: an Eg mode

1 1 around 210 cm− and a Ag mode around 425 cm− . These modes evolve from

1 1 IR-active 1L modes near 215 cm− and 415 cm− respectively (not shown). As discussed in Sec. 4.4.1 for the undistorted case, the modes can switch from IR to Raman depending on whether n is even or odd. Polarized Raman measurements differentiate modes of different symmetries. For the low-temperature phase of 1T -TaS2, this allows for a more detailed comparison of theory and experiment while potentially distinguishing between different c-axis orbital textures. In the C phase monolayer and hexagonally-stacked bulk, the Ag

68 monolayer↓ 100 K polarized expt. ↓ 2 ↓ 3% strain ↓↓ 1 50 100 150 200 250 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ θ = 0° ↓ ↓ ↓ ↓ ↓

Raman intensity (arb. units) θ = 90°

↓ bulk polarized expt. ↓ 100 K 2 1 hex. stacking ↓ ↓↓ 50 100 150 200 250 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ θ = 0° ↓ ↓ ↓ ↓ ↓

Raman intensity (arb. units) θ = 90°

60 70 80 90 100 110 120 130 140 150 -1 wavenumber (cm ) Figure 4.8: Measured polarized Raman spectra of the C phase for (top) a monolayer and (bottom) a 120 nm sample measured at 100 K. We show the frequency range which gave the strongest signal. The spectrum changes signiﬁcantly between θ = 0◦ (parallel polarization) and θ = 90◦ (cross polarization). Red and black arrows indicate peaks with and without polarization dependence, respectively. The insets show comparisons of the measured polarized Raman lines and the calculated spectra.

and Eg modes depend diﬀerently on the relative angle θ between the polarization of the incident and scattered light. In back-scattering geometry with parallel polarization (θ = 0), both Ag and Eg modes are detectable. If incident and scattered light are cross polarized (θ = 90◦), the intensity of the Ag modes become zero,

leaving only the Eg modes. In contrast, with triclinic stacking of the bulk C phase,

69 all Raman modes have the same symmetry, meaning they should all have the same dependence on θ. In Fig. 4.8, we plot the measured polarized Raman spectra for bulk and monolayer samples at 100 K comparing parallel and cross-polarized conﬁgurations. These plots focus on the frequency range of the folded-back acoustic modes since these modes provide the clearest signature of the C phase. The results are similar for the bulk and monolayer, and agree well with an early polarized Raman study [113]. For both the bulk and monolayer, we measure two types of modes with diﬀerent

θ dependence, consistent with Ag and Eg modes. A line by line comparison of the measured bulk frequencies shows strong agreement with the Ag and Eg modes

1 calculated for hexagonal stacking. Most modes are a few cm− higher than the measured frequencies, as is typical within LDA, which tends to over-bind the lattice [114]. For the monolayer, neither the strained nor unstrained calculations match the measured spectrum as closely. We conclude that the primary reason is a lack of substrate eﬀects in our calculations.

The good agreement with the calculated Ag and Eg modes for hexagonally-stacked bulk is surprising since diﬀraction data do not support pure hexagonal stacking in bulk samples [103, 104, 105]. It is likely that the c-axis orbital texture is more complex, involving both hexagonal and triclinic alignment between adjacent layers, as suggested by Ref. 33. Our data are inconsistent with a random or triclinic stacking unless the individual layers are so weakly coupled in the bulk that they vibrate like nearly independent monolayers [113].

70 4.5 Conclusions

We have calculated the phonon modes of both the undistorted and C phases of

1T -TaS2 in the bulk and ultrathin limits. We also measured the C phase Raman spectrum in bulk and thin samples. Through our calculations, we identified the low-frequency folded-back acoustic Raman modes as a signature of the C phase; these modes conveniently lie in a low-frequency range that does not overlap with optical modes, and in which there are no zone-center modes in the undistorted phase and very few in the NC phase. In our measured Raman spectra, these modes persist down to 1L, which is below the critical thickness previously suggested for the NC-C transition [35]. The NC phase consists of C phase domains separated by metallic interdomain regions. As the temperature is lowered, the domains grow as the transition is approached. Tsen, et al. observed commensuration domains 500 ∼ Å in length at 100 K in a 2 nm-thick sample ( 3 layers) [37]. Our measurements ∼ show that at 100K, the monolayer has transformed into the C phase, or at least has C domains larger than our laser spot size of 1 µm. This suggests that the barrier for the NC-C transition is lower in the monolayer than in few-layer samples, though further investigation is needed, particularly regarding the impact of substrates and overlayers on the transition. Recently, there has been interest in the effect of c-axis orbital texture on conduction, with the hope of controlling it [31]. It would be useful to have a way to quickly identify the c-axis order in these materials. Since different stackings have different symmetries, comparison of cross-polarized and parallel-polarized Raman experiments could provide information about the orbital texture of the bulk or few-layer crystals. For example, Ritschel, et al. have proposed a 2L 1T -TaS2 device operating in the C phase. The ‘on’ state would be conducting with triclinic stacking,

71 the ‘oﬀ’ state would be insulating with hexagonal stacking, and the switching mechanism could perhaps be an external ﬁeld [31]. In testing the proposed switching mechanism, polarized Raman could provide an easy way to determine a change in the orbital texture of the sample. More generally, our analysis of the 1T vibrational modes complements the existing literature that focuses on the 2H phase of TMDs. Although the 2H structure is more common, synthesis of metastable 1T phases has been investigated [115, 116, 117]. Besides aiding interpretation of the Raman modes of C phase

1T -TaS2, our analysis of the undistorted phase is more broadly applicable to other 1T materials.

72 Chapter 5

Effect of Electron Doping on Lattice Instabilities of Single-layer

1H-TaS2

5.1 Introduction

A charge-density wave (CDW) is a common collective phenomenon in solids consisting of a periodic modulation of the electron density accompanied by a distortion of the crystal lattice. For a one-dimensional system of non-interacting electrons, the phenomenon is described by the Peierls instability, where a divergence in the real part of the electronic susceptibility at twice the Fermi wave vector drives a metal-to-insulator transition. In extensions of this idea to real materials with anisotropic band structures and quasi-1D Fermi surfaces, Fermi-surface nesting is often cited to explain charge-ordering tendencies. However, geometric Fermi-surface nesting (i.e., existence of parallel regions of the Fermi surface separated by a single wave vector q) is related to the imaginary part of the noninteracting susceptibility rather than the real part, so it is not directly connected to the Peierls mechanism

[20]. Indeed, calculations for a number of CDW materials, including NbSe2, TaSe2,

TaS2, and CeTe3, have found little or no correlation between the CDW ordering vector qCDW, and peaks in the geometric nesting function (imaginary part of χ0) This chapter is reprinted from O. R. Albertini, A. Y. Liu and M. Calandra, Phys. Rev. B 95, 235121 (2017). Copyright (2017) by the American Physical Society.

73 [20, 19, 21, 22, 23]. Nor does qCDW coincide with sharp features in the real part of

χ0 in most of these cases. Instead, these studies pointed to the importance of the momentum-dependent electron-phonon coupling, which softens selected phonon modes to the point of instability. Advances in the synthesis of low-dimensional materials provide new

opportunities to test these ideas. In going from quasi-2D bulk NbSe2 to the 2D monolayer, for example, the number of bands crossing the Fermi level decreases from three to one, which could reveal information about the role played by diﬀerent bands in driving the CDW instability. The stability and structure of the CDW

phase in monolayer NbSe2, however, remains under debate. Density functional calculations predict that both the monolayer and bilayer have CDW instabilities,

but with a shifted qCDW and a larger energy gain compared to the bulk [21]. A

recent experimental study of bulk and 2D NbSe2 on SiO2 used the Raman signature of the CDW amplitude mode to estimate the CDW transition temperature and

reported an increase of TCDW from about 33 K in the bulk to 145 K in the monolayer [118]. This result, attributed in Ref. [118] to an enhanced electron-phonon coupling due to reduced screening in two dimensions, is qualitatively consistent with the DFT predictions, but does not address the question of the CDW superstructure. On the other hand, another study of single-layer NbSe2, this time on bilayer graphene, reported scanning tunneling microscopy/spectroscopy (STM/STS) measurements showing a CDW transition at a lower temperature than the bulk but with the same qCDW ordering vector [119]. Discrepancies between the two experimental studies and between experiment and theory could be due in part to substrate eﬀects, as 2D materials tend to be highly sensitive to the environment and to the substrate. Initial studies of 1T -TaS2, for example, suggested that it completely loses the low-temperature commensurate CDW phase upon thinning to

74 about 10-15 layers [35], but later it was shown that oxidation can suppress the formation of the commensurate phase [37]. Raman signatures of the commensurate

CDW were later found in single-layer 1T -TaS2 samples with limited exposure to air [38]. In using 2D materials to probe the CDW transition and the effect of dimensionality, it is therefore crucial to differentiate behavior intrinsic to each material from effects due to the environment.

In bulk form, TaS2 is a layered transition-metal dichalcogenide for which the 1T and 2H polymorphs are competitive in energy, and both can be synthesized. Both the 1T and 2H polymorphs undergo CDW transitions as the temperature is

lowered. Recently, it was reported that single-layer TaS2 grown epitaxially on a Au (111) substrate adopts the 1H structure rather than 1T [44]. Low-energy electron

diﬀraction (LEED) and STM data indicate that on Au (111), single-layer 1H-TaS2 does not undergo a CDW transition, at least down to T = 4.7 K. This is in contrast to bulk 2H-TaS , which develops a 3 3 CDW periodicity below about 75 K. In 2 × addition, a comparison of angle-resolved photoemission spectroscopy (ARPES) data to the band structure calculated for a free-standing monolayer suggests that the substrate causes the material to become n-doped with a carrier concentration of approximately 0.3 electrons per unit cell [44]. In this paper, we use DFT calculations to investigate whether the observed

suppression of the CDW in monolayer 1H-TaS2 is intrinsic or a consequence of its interaction with the metallic substrate. In the harmonic approximation, we ﬁnd that

a free-standing single layer of 1H-TaS2 has lattice instabilities that are very similar to its bulk counterpart. When the monolayer is electron-doped, however, the 1H structure becomes dynamically stable, implying that substrate-induced doping is the likely reason for the suppression of the CDW observed in Ref. [44]. The strong eﬀect of doping and a secondary eﬀect of lattice strain on the CDW transition in

75 this material give insight on what drives the transition. We examine the role of electron-phonon coupling by calculating the phonon self-energy. We ﬁnd that the contribution to the real part of the phonon self-energy from the single band that crosses the Fermi level is largely responsible for the momentum dependence of the soft mode, but the bare phonon frequency (unscreened from electrons at the Fermi level) also plays a role.

5.2 Methods

We performed density-functional theory calculations using the QUANTUM-ESPRESSO [106] package. The exchange-correlation interaction was treated with the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation [71]. To describe the interaction between electrons and ionic cores, we used ultrasoft [120] Ta pseudopotentials and norm-conserving [121] S pseudopotentials. Electronic wave functions were expanded in a plane-wave basis with kinetic energy cutoﬀs of 47 (53) Ry for scalar (fully) relativistic pseudopotentials. Integrations over the Brillouin zone (BZ) were performed using a uniform grid of 36 36 1 k points, × × with an occupational smearing width of σ = 0.005 Ry. To simulate electronic doping, electrons were added to the unit cell along with a compensating uniform positive background. A single layer of 1H-TaS2 was modeled using a supercell with the out-of-plane lattice parameter ﬁxed at c = 12 Å, corresponding to about 9 Å of vacuum between layers. This ensured that for the range of electron doping explored, spurious vacuum states that originate from the periodic boundary conditions stay well above the Fermi level. The in-plane lattice constant and the atomic positions were fully relaxed at each doping level.

76 Phonon spectra and electron-phonon matrix elements were calculated with density functional perturbation theory [122] on a q-point grid of 12 12 1 phonon × × wave vectors and Fourier interpolated to denser grids. To test the effect of spin-orbit coupling on the phonon frequencies, we used a less dense grid of 8 8 1 q-points. × × While this does not capture all the fine structure in the phonon dispersions, it is sufficient to test the effect of spin-orbit coupling. In fact, even Fourier interpolation on the 12 12 1 q-grid does not fully capture the sharp features along certain × × directions in the Brillouin zone where direct calculations of the phonon frequency are needed. The real and imaginary parts of electronic susceptibilities and phonon self energies were calculated using dense k-point grids of at least 72 72 1 points. × ×

5.3 Results & Discussion

The 1H-TaS2 lattice has point group D3h, reduced from D6h in the bulk. Sulfur atoms adopt a trigonal prismatic coordination about each Ta site, and the crystal lacks inversion symmetry. Our calculations for the undoped (x = 0) monolayer give a relaxed in-plane lattice parameter of ax=0 = 3.33 Å and a separation between Ta and S planes of zS = 1.56 Å. These values are similar to what we calculate for the bulk. For comparison, the experimental in-plane lattice parameter for single-layer

1H-TaS2 on Au [44] and bulk 2H-TaS2 [123] are a = 3.3(1) Å and 3.316 Å, respectively. With the addition of 0.3 electrons per cell (x = 0.3), the lattice − expands roughly 2.5% to ax= 0.3 = 3.41 Å, while the separation between Ta and S −

planes decreases slightly to zS = 1.53 Å. In analyzing the impact of doping on the CDW instability, we examine the eﬀect of adding charge carriers as well as the eﬀect of the lattice expansion.

77 In single-layer 1H-TaS2, in the absence of spin-orbit coupling (SOC), a single isolated band (two-fold spin degenerate) crosses the Fermi level, as shown in Fig. 5.1(a). This half-filled band, which is about 1.4 eV wide, has strong Ta d character and is separated by about 0.6 eV from the manifold of S p bands below. As a consequence, metallic screening is due to a single band isolated from all the others. The Fermi surface consists of a roughly hexagonal hole sheet of primarily Ta dz2 character around Γ and a roughly triangular sheet of primarily in-plane Ta d character (dx2+y2 and dxy) around K. The spin-orbit interaction splits the half-filled d band by as much as 0.3 eV, except along the ΓM line where symmetry dictates ∼ that the band remains degenerate. The number of Fermi sheets around the Γ and K points doubles with SOC. Electron doping of x = 0.3 causes a nearly rigid downward shift of the partially − occupied Ta d band by about 0.1 eV, and a somewhat larger downward shift of the occupied S p bands. This is shown in Fig. 5.1(c) for the scalar relativistic bands. The fully relativistic bands behave similarly with doping, as can be seen in Figs. 5.1(a,b). The hole pocket around Γ thus gets slightly smaller while the one around K shrinks more significantly. The doping-induced lattice expansion, on the other hand, has a negligible effect on the Fermi surface, as shown in Fig. 5.1(d). Despite the non-negligible effect that SOC has on the electronic structure, we find little difference between the phonon spectra calculated with and without SOC. This is shown in Fig. 5.2 where the phonon dispersion curves were obtained from a Fourier interpolation on a 8 8 1 phonon q-grid. Hence, for the remainder of this × × paper, we focus on the phonon properties calculated in the absence of SOC. Figure 5.3(a) shows the phonon dispersion curves calculated for the undoped monolayer (x = 0) from a Fourier interpolation on a 12 12 1 phonon q-grid. An × × acoustic phonon branch is found to be unstable over a large area of the BZ

78 2 (a) x = 0 1 no SOC

0 with SOC -1

2 (b) x = 0.3 − 1 no SOC

-1 with SOC

2 (c) no SOC 1 energy (eV) x = 0 0 x = 0.3 -1 −

2 (d) x = 0.3 − 1 relaxed no SOC

-1 unrelaxed

Γ M K Γ

Figure 5.1: Electronic bands of single-layer 1H-TaS2. Eﬀect of: (a) SOC for the undoped case, (b) SOC for the n-doped (x = 0.3) case, (c) n-doping (x = 0.3 − − vs. x = 0), (d) lattice constant relaxation (ax=0 vs. ax= 0.3) on the doped (x = 0.3) system. Internal atomic coordinates were optimized in− all cases. −

79 50

30 no SOC 20 with SOC

0 x = 0 -10 (a) 50

30 phonon energy (meV) 20

0 x = 0.3 -10 (b) (relaxed)− Γ M K Γ

Figure 5.2: Phonon dispersions of single-layer 1H-TaS2 with and without spin-orbit coupling. Eﬀect of SOC for: (a) the undoped case, (b) the n-doped (x = 0.3) case. In (b), the curves with and without SOC lie almost directly on top of− each other. Dispersion curves were obtained from Fourier interpolation from a q-grid of 8 8 1 points. Imaginary frequencies are shown as negative. × ×

80 surrounding the M point and even has a dip at K. (Imaginary frequencies are plotted as negative.) This branch involves in-plane Ta vibrations. This region of instability arises primarily from softening of the phonon branch near the bulk CDW wavevector (2/3 along the Γ to M line), but there is also a secondary point of instability along the M to K line (close to M). 1 We will refer to these points as qCDW and qMK , respectively.

When electrons are added while keeping the lattice constant ﬁxed at ax=0 but allowing zS to relax, the instabilities at qCDW and qMK are progressively suppressed. With doping of x = 0.3, a weak instability at the M point remains, as − shown in Fig. 5.3(b). Once the lattice constant is allowed to expand to its optimal value at x = 0.3, the lattice becomes dynamically stable, as seen in Fig. 5.3(c), − though an anomalous dip in the phonon dispersion remains near qMK . For both doped cases, there is no longer a dip at K. At intermediate values of electron doping, we ﬁnd that in going from x = 0 to x = 0.3, with the lattice constant optimized at each level of doping, the point of −

strongest instability shifts from qCDW (2/3 ΓM) to the M point as the mode gradually hardens. In these harmonic calculations, the mode becomes stable between x = 0.25 and x = 0.3. With positive (hole) doping, the instability at − −

qCDW shifts in the other direction (close to 1/2 ΓM for x = 0.3), and various other strong instabilities appear throughout the BZ. See Supplemental Material for more information on intermediate and positive doping values.[124] Since the unstable branch involves in-plane displacements of Ta ions, it is likely that these phonons couple strongly to in-plane Ta d states near the Fermi level. To

1While the instability at q is not captured by Fourier interpolation of the 8 8 1 q- MK × × grid in Fig. 5.2(a), the Fourier interpolation based on a 12 12 1 q-grid in Fig. 5.3(a) clearly × × shows the instability. Even so, direct calculation at qMK yields an even more imaginary phonon frequency [Fig. 5.5(a)].

81 50

0 x = 0 -10 (a) qCDW 50

0 x = 0.3 phonon energy-10 (meV) (b) (unrelaxed)− 50

0 x = 0.3 -10 (c) (relaxed)− Γ M K Γ

Figure 5.3: Phonon dispersions of single-layer 1H-TaS2 with and without doping and lattice constant relaxation. (a) undoped (x = 0), (b) n-doped with unrelaxed lattice constant (x = 0.3, a ), (c) n-doped with relaxed lattice constant − x=0 (x = 0.3, ax= 0.3). Dispersion curves were obtained from Fourier interpolation from a q-grid− of 12 − 12 1 points. Imaginary frequencies are shown as negative. × ×

82 investigate the role of the electron-phonon interaction in the CDW transition in this material, we consider the phonon self-energy due to electron-phonon coupling. In the static limit, the real part of the phonon self-energy for phonon wave vector q and branch ν is given by

2 X ν 2 f(k+qj0 ) f(kj) Πqν = gkj,k+qj0 − , (5.1) Nk | | k+qj0 kj kjj0 − while the phonon linewidth, which is twice the imaginary part of the phonon self-energy, can be expressed as

4πωqν X ν 2 γqν = gkj,k+qj0 δ(kj F )δ(k+qj0 F ). (5.2) Nk | | − − kjj0

Here Nk is the number of k points in the Brillouin zone, j and j0 are electronic band indices, f() is the Fermi-Dirac function, F is the Fermi energy, and ωqν is the phonon frequency. The electron-phonon matrix element is s ν ~ gkj,k+qj0 = kj δVSCF/δuqν k + qj0 , (5.3) 2ωqν h | | i where uqν is the amplitude of the phonon displacement and VSCF is the Kohn-Sham potential. Both the linewidth and the product ωqνΠqν are independent of ωqν, so they remain well deﬁned even when the frequency is imaginary. In Fig. 5.4, we show Brillouin zone maps of the phonon linewidth (summed over the two acoustic modes with in-plane Ta displacements). In the undoped material

[Fig. 5.4(a)], the linewidth is sharply peaked at qCDW. For comparison, the geometric nesting function, which is deﬁned as the Fermi surface sum in Eq. 5.2 with constant matrix elements, does not have sharp structure near qCDW [Fig. 5.4(d)], and instead has three sharp peaks surrounding the K point. This means that the sharp peak in the linewidth at qCDW must be due to large electron-phonon matrix elements between states on the Fermi surface, rather than

83 (a) (b) (c)

M M M qCDW Γ Γ Γ

(d) (e) (f)

M M M qCDW Γ Γ Γ Figure 5.4: Brillouin-zone maps of the phonon linewidth [(a-c)] and the geometric Fermi-surface nesting function [(d-f)]. (a,d) undoped (x = 0), (b,e) n-doped with unrelaxed lattice constant (x = 0.3, a ), (c,f) n-doped with relaxed − x=0 lattice constant (x = 0.3, ax= 0.3). White (black) represents high (low) values, with one scale used for all− the linewidth− plots (a-c) and another used for all the nesting function plots (d-f). Dotted lines show high-symmetry lines from Γ to K and from K to M. The plots in (a-c) represent the sum of the phonon linewidths for the two acoustic phonon branches with in-plane Ta displacements.

84 300 250 (a) (b) (c) 200 Ω˜ 2 ) q 2 150 100 (meV

2 50

˜ 2 Ω ωq

, 0 2

ω -50 x = 0 x = 0.3 x = 0.3 -100 (unrelaxed)− (relaxed)− 0 ) 2 -50 -100

˜ -150 Π (meV ω

2 -200 )

1 -2 −

(eV -3 ′ 0 χ

Γ M K Γ M KΓ M K

2 Figure 5.5: Momentum dependence of the square of the self-consistent (ωq) ˜ 2 and bare (Ωq) phonon energies for the branch with instabilities, self-energy ˜ correction (2ωqΠq) for that branch, and real part of the bare susceptibility (χ0(q)). (a) undoped (x = 0), (b) n-doped with unrelaxed lattice constant (x = 0.3, ax=0), (c) n-doped with relaxed lattice constant (x = 0.3, ax= 0.3). The red, − − − ˜ 2 dashed lines in the top panels represent the square of the bare phonon energies (Ωq), calculated using Eq. 5.4. Only the response of the band crossing the Fermi level is ˜ included in 2ωqΠq and χ0(q).

to the geometry of the Fermi surface itself. With electron doping (x = 0.3), the − maximum values of the linewidth are much smaller and occur elsewhere in the Brillouin zone, whether the lattice constant is allowed to relax [Fig. 5.4(c)] or not [Fig. 5.4(b)]. This suggests that the momentum dependence of the electron-phonon coupling of states at the Fermi level plays an important role in picking out the primary instability at qCDW. However, focusing on states at the Fermi level is not suﬃcient to understand the broader region of instability.

85 The real part of the phonon self-energy Πqν is directly related to the renormalization of phonon frequencies due to electron screening. From perturbation theory,

2 2 ωqν = Ωqν + 2ωqνΠqν, (5.4) where Ωqν is the bare phonon frequency. While it is conceptually appealing to define the bare frequencies as the completely unscreened ionic frequencies, such a starting point leads to results that lie outside the range of validity of the perturbative expression in Eq. 5.4. Instead, we define the bare frequencies to be the frequencies obtained neglecting the metallic screening due to electrons in the isolated Ta d band crossing the Fermi level. It has been shown that this non self-consistent definition of the self-energy is equivalent to a fully self-consistent solution of the linear response equations including both the real and imaginary parts of the self-energy [125]. In this case, we limit the sum in the phonon self-energy (Eq. 5.1) to intraband ˜ transitions within that band and denote it as Πqν. The corresponding bare (in the ˜ sense of unscreened by metallic electrons) frequencies are denoted Ωqν. While the bare frequencies are not directly accessible, we can estimate them from the fully screened frequencies ωqν (as obtained from density functional perturbation theory) ˜ and the perturbative correction 2ωqνΠqν by inverting Eq. 5.4. For the branch with instabilities, Fig. 5.5 shows the square of the phonon ˜ frequencies and the self-energy correction, 2ωqνΠqν, along high-symmetry directions in the Brillouin zone. For comparison, χ0(q), the real part of the bare electronic susceptibility, given by Eq. 5.1 with constant matrix elements and limited to the band at the Fermi level, is also plotted. For the undoped material, χ0 has a minimum at qCDW, but the momentum dependence between qCDW and M does not match that of the calculated frequencies. Similarly, the momentum dependence of χ0 does not correlate with the phonon softening when the material is doped to

86 x = 0.3, with or without relaxation of the lattice constant. From Fig. 5.5, we see, − however, that the momentum dependence of the phonon self-energy roughly follows that of the phonon softening, indicating the importance of the electron-phonon ˜ matrix elements in Πqν. The self-energy includes contributions from states throughout the band crossing the Fermi level, in contrast to the linewidth, which provides information about the electron-phonon coupling at the Fermi level.

˜ 2 The square of the bare frequencies (Ωqν) estimated using Eq. 5.4 are plotted with dashed lines in the top panels of Fig. 5.5. Note that in the regions of instabilities, the self-energy correction is comparable to or larger than the square of the bare frequency, possibly raising into question the degree to which the perturbative expression is valid. Nevertheless, if we use the expression to estimate the bare frequencies, we ﬁnd some surprising results. For the undoped material, the

dispersion of the bare frequency has sharp dips at both qCDW and qMK . Thus, while phonon softening due to screening of the band at the Fermi level accounts for most of the momentum dependence of the unstable modes, structure in the bare frequencies contributes as well. Upon doping to x = 0.3, but holding the lattice −

constant fixed at ax=0, the local minima at qCDW and qMK disappear in both the self-energy and the bare frequency, stabilizing those modes. At M, however, neither the self-energy nor the bare frequency change significantly with doping, so a residual instability remains at M. When the lattice constant of the doped system is relaxed, the self-energy undergoes relatively minor changes, while the bare frequency at M hardens significantly. Thus it is the lattice-constant dependence of the bare frequency that stabilizes the M point phonon at a doping of x = 0.3. The − dramatic hardening of this mode upon a 2.5% lattice expansion is surprising, as phonons usually soften with lattice expansion. Indeed most of the other phonon modes in the doped material soften slightly with the lattice expansion.

87 To summarize, we find that the primary CDW ordering vector coincides with very strong electron-phonon coupling at the Fermi surface, but the full momentum dependence of the phonons in the wide region of instability is described more completely by the momentum dependence of ωΠ˜, the phonon self-energy due to screening from the band crossing the Fermi level. The momentum dependence of the corresponding bare phonon frequencies, Ω˜, however, also plays a role. Regarding the behavior of the instabilities with doping and lattice strain, the bare frequencies provide the primary contributions, but the changes in the phonon self-energy, especially with doping, are essential. While recent studies of CDW materials have focused on the role of electron-phonon coupling as the driving mechanism, we find that, for monolayer 1H-TaS2, taking into account screening arising from electron-phonon coupling in the band crossing the Fermi level by itself is not enough, and one must consider how the bare frequencies depend on momentum and doping as well. Unfortunately it is not possible to determine if the momentum, doping, and strain dependencies deduced for the bare frequencies come from the response of other bands away from the Fermi level or appear as features in the completely unscreened ionic frequencies. Nevertheless, our finding that screening by states near the Fermi surface alone are not responsible for the CDW instability means that the instability is generated by not only the long-range part of the force constants but also the short-range part, which is included in Ω˜.

5.4 Conclusions

Our calculations show that in the harmonic approximation the free-standing monolayer of 1H-TaS2 is unstable to CDW distortions with the same ordering vector as the bulk. We also ﬁnd that electron doping stabilizes the lattice. These

88 results indicate that the CDW suppression found in experiments of 1H-TaS2 on Au is not intrinsic but rather induced by the substrate [44]. According to our harmonic calculations, in addition to increasing the number of charge carriers, electron doping induces a lattice expansion. While the addition of charge carriers itself stabilizes most of the soft phonon modes, at the harmonic level a residual instability remains if the lattice constant is not allowed to relax. In the experiments on the Au (111) substrate, the uncertainty in the measured lattice constant was about 3% [44], which is slightly larger than the lattice expansion we predict for electron doping of x = 0.3. Hence in the experimental system, it remains an open question as to how − much the substrate affects the lattice constant, and how that in turn influences the lattice instabilities. We believe that the suppression of CDW by electron doping is robust also against anharmonic effects. Indeed in metals, anharmonic effects tend to enhance phonon frequencies and suppress CDW instabilities [126, 127]. These effects, not considered in this work, could reduce the tendency towards CDW at x = 0, but will not change the qualitative result that electron doping removes the CDW in this system. It would be interesting to explore the interplay between anharmonicity and doping in future studies. Recently, it was pointed out that for metallic 2D materials on substrates, the shift in the bands measured in ARPES experiments may not provide a reliable estimate of charge transfer across the interface [128]. In particular, if there is significant hybridization between the substrate and 2D material, the actual charge transfer will be reduced. The impact of substrate hybridization on the CDW instability in monolayer 1H-TaS2 warrants future investigation. This system, like other transition-metal dichalcogenides, offers an opportunity to better understand the intrinsic origins of CDWs. Going beyond previous studies

89 that emphasized the importance of the momentum dependence of the electron-phonon interaction in driving the CDW, we ﬁnd that the momentum, doping, and strain dependencies of the bare phonon frequencies also play a role. This system also underscores the importance of disentangling environmental and intrinsic eﬀects in 2D materials, and provides an example of using the substrate to tune the properties of the material.

90 Chapter 6

Conclusion

At IBM Almaden, I studied single atoms adsorbed on a surface in order to understand what is necessary for a 3d transition-metal atom to preserve its magnetic moment on a substrate. For Co adatoms on MgO/Ag, a weak interaction with the surface atop an O site results in a weak chemical bond and full preservation of the Co gas-phase magnetic moment. On the other hand, Ni has two binding sites close in energy. At the hollow site, it loses only some of its spin moment, and at the O site it loses its spin moment entirely. On both binding sites, the strength of the chemical bond is greater than for Co. The main limitation to the survival of magnetism when a chemical bond forms between an adatom and a substrate is the bond itself. In particular, unpaired electrons can become paired when there is a significant charge transfer. Furthermore, the formation of diffuse hybrid orbitals with the substrate atoms reduces the overall correlation of the unpaired electrons. And finally, a strong crystal field shifts the energy levels of the magnetic atom, and can significantly lower its magnetic moment. Magnetism in the single atom, or zero-dimensional limit, is quite different from the macroscopic, collective phenomenon in solids. Approaching the problem of magnetism from this atomic limit is not just an exotic approach to magnetic storage, but also a novel way to bridge the quantum and classical worlds. More work is needed in this field, not just to continue the search for single atom magnets with large anisotropy, but also to explore the viability of reading and writing these ‘bits’.

91 Of course, we must also understand the thermal limitations, since these experiments are mostly around 4 K.

In our work with the transition-metal dichalcogenide 1T -TaS2, we identiﬁed the vibrational signature of the low-temperature commensurate CDW phase, which

1 features two sets of vibrational modes separated by a 100 cm− gap. Our experimental collaborators measured the low-temperature Raman spectra of these materials, and uncovered that the commensurate phase is robust to thinning down to even one layer. This result is interesting because earlier work did not detect the electronic or structural transition for few-layer flakes. The question of whether the well-known metal-insulator transition still accompanies the structural transition in the 2D limit remains open. Transport measurements and scanning tunneling spectroscopy, for example, could reveal if a clean single-layer sample undergoes an electronic transition to an insulating state. Also, the exfoliation or growth of flakes of varying few-layer thicknesses could reveal whether the structural and/or electronic transition is always present upon thinning, or only reentrant. Finally, the bulk electronic transition may be best explained by c-axis order in the commensurate CDW phase. A theoretical investigation into the energetics and electronic structure of different stackings would be an important work.

Our calculations for a freestanding ﬂake of single-layer 1H-TaS2 showed that it is susceptible to a charge density wave similar to that found in the bulk. We also showed that electron doping tends to suppress this charge density wave. This is important for separating the intrinsic properties of the material from the impact of its immediate environment (which becomes increasingly important with thinning).

When 1H-TaS2 interfaces with a metal substrate such as Au(111), this becomes an important consideration. Looking more deeply at the origin of the charge density

92 wave and its suppression, we studied quantities associated with the electron-phonon interaction. Surprisingly, the electronic screening from the lone metallic band was not enough to account for all the phonon instabilities that we found in our calculations, indicating that the ‘bare’ frequencies have strong momentum, doping, and strain dependencies as well. Of course, charge carrier doping, as considered in our DFT calculations, is not

the only way the substrate could aﬀect the charge density wave of 1H-TaS2. For

example, hybridization between substrate and TaS2 electronic states could also be

important. One could actually study the system of 1H-TaS2 on Au(111) (and other substrates, like graphene, hexagonal BN, etc.), with supercells of diﬀerent sizes and orientations between the two materials. This would provide additional insight on the observed absence of a charge density wave in single-layer 1H-TaS2 on Au(111), taking into account both charge transfer and hybridization eﬀects.

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