Computational Methods for Charge Density Waves in 2D Materials

Abstract

Two-dimensional (2D) materials that exhibit charge density waves (CDWs)-spontaneous reorganization of their electrons into a periodic modulation-have generated many research endeavors in the hopes of employing their exotic properties for various quantum-based technologies. Early investigations surrounding CDWs were mostly focused on bulk materials. However, applications for quantum devices require few-layer materials to fully utilize the emergent phenomena. The CDW field has greatly expanded over the decades, warranting a focus on the computational efforts surrounding them specifically in 2D materials. In this review, we cover ground in the following relevant theory-driven subtopics for TaS₂ and TaSe₂: summary of general computational techniques and methods, resulting atomic structures, the effect of electron-phonon interaction of the Raman scattering modes, the effects of confinement and dimensionality on the CDW, and we end with a future outlook. Through understanding how the computational methods have enabled incredible advancements in quantum materials, one may anticipate the ever-expanding directions available for continued pursuit as the field brings us through the 21st century.

Keywords: charge density waves; density functional theory; transition metal dichalcogenides.

Figure 1

(a) Real part of the bare static electronic susceptibility (in arbitrary units) with constant matrix elements. This was calculated by using the band structure of monolayer NbSe₂. (b) The corresponding Fermi surface is shown. (c) Angle-resolved photoemission spectroscopy (ARPES) intensity maps of 2HTaSe₂ and (d) 2HNbSe₂ are shown at 125 and 65 K. The darker grayscale indicates higher photoemission intensity. The small-dotted hexagons are the Brillouin zone scheme for the super-lattice. (e) The crystal structure and Brillouin zone are shown for 2HTaSe₂. The large spheres (gray) correspond to Ta atoms, and the small spheres (yellow) represent Se atoms. (a,b) Reprinted figure with permissions from Reference [46]. Copyright 2009 by the American Physical Society. (c,d) Reprinted figure with permissions from Reference [41]. Copyright 2005 by the American Physical Society. (e,f) Reprinted figure with permissions from Reference [47]. Copyright 2012 by the American Physical Society.

Figure 2

(a) Fermi surfaces are calculated for monolayer 2HTaSe₂, (b) monolayer 1T-TaSe₂, and (c,d) their respective bulk counterparts with spin–orbit coupling (SOC) explicitly included [82]. The crystal structures of monolayer TaSe₂ are shown for the (e) 2H phase and the (f) 1T phase. Reference [82] is an open-access article distributed under the terms of the Creative Commons CC BY license, which permits unrestricted use, distribution, and reproduction in any medium.

Figure 3

Comparison between (a) experimental [97] and (b) DFT [108] lattice expansion for TaSe₂ when the temperature dependence is introduced in the DFT modeling solely through changes in the electronic temperature. Reprinted figure with permissions from Reference [97]. Copyright 1978 by the American Physical Society. Reprinted figure with permission from Reference [108]. Copyright 2019 by the American Physical Society.

Figure 4

Thermal expansion of the c lattice constant for TaS₂. (a) The experimental data are found in Reference [98]. (b) Relevant computational result may be found in Reference [109]. The linear trend seen in the experimental findings is well reproduced in the DFT modeling by using only the electronic temperature. Reprinted figure with permission from Reference [109]. Copyright 2019 by the American Physical Society.

Figure 5

Temperature dependence of the experimentally observed CDW mode at 79 cm⁻¹. Including incommensurability through a small compression in the simulation cell (triangular data) leads to an excellent reproduction of the experimental data.

Figure 6

Decomposition of the total energy as a function of the separation of the S-Ta-S layers for the 1T and 2H polytypes is shown and plotted as such: (a) the total energy, (b) the RPA correlation energy, (c) the difference in the polytypes’ exact-exchange (EXX), and RPA energies (d) EXX. (e) The starlike arrangement of the atoms in the C-CDW phase from the in-plane view (left) and from the side view (right). Reprinted figure with permission from Reference [114]. Copyright 2015 by the American Physical Society.

Figure 7

Electronic structure of monolayer TaS₂ measured with ARPES. (a) Photoemission intensity at the Fermi energy. (b) Photoemission intensity along high-symmetry directions of the 2D Brillouin zone. (c) Data in (b) with the calculated 1H-TaS₂ band structure superimposed in orange. The calculated bands were shifted by 0.12 eV to higher binding energy. The Au surface state and projected bulk bands of Au(111) are indicated with blue, as guides to the eye. Reprinted figure with permission from Reference [118]. Copyright 2016 by the American Physical Society.

Figure 8

Atomic and electronic structure of TaS₂ nanoribbons as shown by Cain et al. [119]. (a) The planar atomic structure of a 2.99 nm–wide nanoribbon is shown without the zigzag defect, whose CDW distortions are too small to be seen. (b) The electronic band structure is shown. (c) The partial density of states (PDOS, DOS) of the nanoribbon is plotted. (d) A planar view of the atomic structure of a 3.08 nm–wide nanoribbon is shown, with zigzag boundaries of S vacancies represented by black dashed lines denoted as L1 and L2. (e) The electronic band structure and (f) the PDOS of the nanoribbon are calculated. Localized edge states are denoted as ψ₁ and ψ₂. Structures in (g,h) show real-space wave functions of those localized edges states, with isosurfaces for the positive and negative wave function values shaded in cyan and orange, respectively. Ta and S atoms are represented by red and yellow spheres, respectively. Reprinted with permission from Reference [119]. Copyright 2021 by the American Chemical Society.

Figure 9

Normal (red) and commensurate reconstructed (green) Brillouin zone of monolayer 2H-TaSe₂. The equivalent Γ points in the first extended Brillouin zone and second extended Brillouin zone are connected to Γ by red and blue vectors, respectively. Reprinted with permission from Reference [127]. Copyright 2015 by the American Chemical Society.

Figure 10

Electron-band structure and Fermi surface of TaSe₂. (a) ARPES and (b) calculated Fermi surface map in the normal state (150 K). (c) ARPES and (d) calculated Fermi surface for the CDW state (15 K). Solid lines in panels (a,c) mark the 2D Brillouin zone of TaSe₂. Insets at the right bottom corner in (c,d) are the region marked by yellow dotted square around Γ with different color scales. Calculated band structure and Fermi surface of normal-state TaSe₂ in its (e,f) bulk form with SOC, (g,h) monolayer form without SOC, and (i,j) monolayer form with SOC. Reprinted with permission from Reference [130]. Copyright 2018 by the American Chemical Society.

Figure 11

(a) Electronic-band structure and (b) the DOS for monolayer 2HTaSe₂. (c,d) Results for monolayer 1T-TaSe₂. Band structures and the DOS of bulk are shown as red dashed lines. The Fermi level was shifted to zero. Reference [82] is an open-access article distributed under the terms of the Creative Commons CC BY (https://creativecommons.org/licenses/ (accessed on 14 December 2021).) license, which permits unrestricted use, distribution, and reproduction in any medium.

Figure 12

(a) Mode diagrams are shown and correspond to those seen in the normal and CDW phases. (b) Room-temperature Raman spectra of bulk TaS₂ reveals several modes. The two-phonon mode is shaded brown. Purple and golden shades represent modes with E and A symmetry, respectively. Gray areas represent fits to the laser Rayleigh line and other features not predicted to be part of the material’s Raman spectrum. (c) Excitation-dependent Raman measurements performed at 4 K with 458, 476, 514, and 633 nm laser wavelengths. Low-frequency peaks correspond to CDW amplitude modes and a zone-folded CDW mode. Reprinted figure with permission from Reference [109]. Copyright 2019 by the American Physical Society.

Figure 13

(a) Phonon dispersions and lattice instabilities of monolayer TaS₂ under different electronic conditions at 0 K (calculated). Acoustic phonon dispersions for different levels of hybridization with the substrate Γ (half width at half maximum of the electronic broadening) and charge doping x (electrons per Ta atom); x < 0 refers to electron addition. The character of the phonon modes, i.e., longitudinal (LA), transverse (TA), or out-of-plane (ZA), is marked in color. Imaginary phonon mode energies indicate that the lattice is unstable toward corresponding periodic lattice distortions. (b) Phase diagram of lattice instabilities in monolayer TaS₂. The CDW region is defined by the presence of an imaginary phonon energy. Regions with instabilities are shaded in color. Some experimentally realized situations are in the phase diagram. Reprinted with permission from Reference [134]. Copyright 2018 by the American Chemical Society.

Figure 14

(a) Low-frequency Raman spectra show four modes emerge at low temperatures: Amp 1, Amp 2, P1, and P2. Amp 1 and Amp 2 are the CDW amplitude modes, whereas P1 and P2 are the phase modes. The transition temperature for the IC- (C-) phase is indicated in the legend by the blue (red) star. (b) Temperature dependence of the frequencies and full width at half maximum of Amp 1, Amp 2, P1, and P2. The DFT-calculated frequencies are plotted as a function of electronic temperature, using the same temperature axis. Solid lines guide the eye. The transition temperature for the IC-CDW (C-CDW) is indicated by a dark (light) gray dashed line. (c) DFT-calculated vibrations for the phase (side view) and amplitude (top view) modes. Reprinted figure with permission from Reference [108]. Copyright 2019 by the American Physical Society.

Figure 15

Raman spectroscopic investigation of the CDW transition in 6.3 nm– and 3 nm–thick 1T-TaSe₂. Raman spectra are shown for (a) 6.3 nm and (b) 3 nm 1T-TaSe₂ at selected temperatures for the heating cycle. The temperature dependence of the phonon frequencies is shown for the (c) 6.3 nm and (d) 3 nm 1T-TaSe₂. Soft modes are indicated with red dots. Reprinted figure with permission from Reference [149]. Copyright 2020 by John Wiley and Sons, Inc.

Figure 16

(a) Plot of ARPES intensity at the Fermi energy is shown for pristine 1T-TaS₂ as a function of in-plane wave vector. (b) ARPES intensity as a function of binding energy. (c,d) Show the same content as (a,b) but for 8BL Bi(111) on 1T-TaS₂. (e,f) Shows ARPES data and their second derivative intensities, respectively, for 8BL Bi(111)/1T-TaS₂ as a function of binding energy and wave vector. Solid curves in (f) are calculated band dispersions for free-standing 8BL Bi(111). The ARPES data were recorded at 30 K. Reprinted with permission from Reference [152]. Copyright 2018 by the American Chemical Society.

Figure 17

DFT calculations of the electronic structure and spectral signatures of 1T-TaS₂ are shown for two C-CDW interlayer stacking arrangements. (a) The 13 a × 13 a Star-of-David reconstruction of the Ta layers is illustrated. All atoms of each star are displaced (depicted with arrows) toward the center atom. Bottom left: CDW stacking of the Ta planes with the Star-of-David centers on top of each other (T_s = c). Bottom right: CDW stacking with a stacking vector having an in-plane component (T_s = 2a + c). (b) DFT calculations of the PDOS of the polarized S 3p states with on-site S 2s core hole for T_s = c (left) and T_s = 2a + c (right). Purple shows the out-of-plane PDOS, and green shows the in-plane PDOS. The Fermi level is located at 0 eV. (c) Simulations of the sulfur scattering planes based on the PDOS from (b). Intensity increases from blue to brown. Reference [157] is an open-access article distributed under the terms of the Creative Commons CC BY (available online at https://creativecommons.org/licenses/ (accessed on 14 December 2021)) license, which permits unrestricted use, distribution, and reproduction in any medium.

Figure 18

(a) Illustration of superstructure and stacking arrangement in 1T-TaSe₂. The periodic lattice displacements form a 13-Ta-atom Star-of-David cluster (solid bonds). Blue dashed triangles indicate the threefold symmetric stacking displacements. Se atoms are color coded according to their c-axis coordinate (in fractions of c), as listed in the lower part of (a). The Se atoms at the center of a Star-of-David cluster are at highest (most protruding, red-shaded areas) positions, whereas other Se atoms at the lowest positions are shown as blue-shaded areas. (b) Stacking arrangement along the c₀ direction for two adjacent layers of the C-CDW phase. Projections of the C-CDW phase at (c) [001] and (f) [102] zone axis are shown with the corresponding experimental diffraction patterns in (d) and (g), respectively. The simulated diffraction patterns are shown along [001] (e) and [102] (h). Reference [166] is an open-access article distributed under the terms of the Creative Commons CC BY (https://creativecommons.org/licenses/ (accessed on 14 December 2021)) license, which permits unrestricted use, distribution, and reproduction in any medium.

Figure 19

(a) Thickness dependence of the CDW transition temperature in 2HMX₂. Error bars are standard deviations obtained from the least-squares fits to the temperature-dependent amplitude mode intensity. (b) The calculated values of electron–phonon coupling constant (λ) for monolayer 1H-MX₂. (c) Schematic illustration of the possible phase diagram describing the CDW response in a layered material in terms of ionic charge transfer (ΔQ_I), electron–phonon coupling constant (λ), and the spatial extension of electronic wave functions (1/<r²>). Reference [168] is an open-access article distributed under the terms of the Creative Commons CC BY license (Available online: https://creativecommons.org/licenses/ (accessed on 14 December 2021)), which permits unrestricted use, distribution, and reproduction in any medium.