Stable charge density wave phase in a $1T--{\mathrm{TiSe}}_{2}$ monolayer

Stable charge density wave phase in a $1 T - {TiSe}_{2}$ monolayer

Bahadur Singh, Chuang-Han Hsu, Wei-Feng Tsai, Vitor M. Pereira, and Hsin Lin

Phys. Rev. B 95, 245136 – Published 28 June 2017

Abstract

Charge density wave (CDW) phases are symmetry-reduced states of matter in which a periodic modulation of the electronic charge frequently leads to drastic changes of the electronic spectrum, including the emergence of energy gaps. We analyze the CDW state in a $1 T - {TiSe}_{2}$ monolayer within a density functional theory framework and show that, similarly to its bulk counterpart, the monolayer is unstable towards a commensurate $2 \times 2$ periodic lattice distortion (PLD) and CDW at low temperatures. Analysis of the electron and phonon spectrum establishes the PLD as the stable $T = 0$ K configuration with a narrow band gap, whereas the undistorted and semimetallic state is stable only above a threshold temperature. The lattice distortions as well as the unfolded and reconstructed band structure in the CDW phase agree well with experimental results. We also address evidence in our results for the role of electron-electron interactions in the CDW instability of $1 T - {TiSe}_{2}$ monolayers.

Received 9 February 2017
Revised 22 April 2017

DOI:https://doi.org/10.1103/PhysRevB.95.245136

Physics Subject Headings (PhySH)

Charge density waves

Dichalcogenides Transition metal dichalcogenides

Density functional theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Bahadur Singh^1,2, Chuang-Han Hsu^1,2, Wei-Feng Tsai^1,2,3, Vitor M. Pereira^1,2,*, and Hsin Lin^1,2

¹Centre for Advanced 2D Materials and Graphene Research Centre, National University of Singapore, 117546 Singapore
²Department of Physics, National University of Singapore, 117542 Singapore
³School of Physics, Sun Yat-sen University, Guangzhou, 510275 China

^*Corresponding author: vpereira@nus.edu.sg

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Vol. 95, Iss. 24 — 15 June 2017

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Figure 1
(a) Top and (b) side views of a ${TiSe}_{2}$ monolayer structure. Blue and green balls denote Ti and Se atoms, respectively. $Se 1$ (light green) and $Se 2$ (dark green) represent Se atoms in the top and bottom sublayers, respectively. (c) Local octahedral structure of ${TiSe}_{2}$ monolayer. Ti atoms lie at the center of the octahedral coordination unit. (d) First Brillouin zones (BZs) of the monolayer in the normal phase (black solid lines) and the CDW phase (blue dashed lines) with three high-symmetry points $Γ, M$ , and $K$ . The high-symmetry points of the reduced BZ are marked with a star ( $*$ ). (e) Band structure of monolayer ${TiSe}_{2}$ in the $1 \times 1$ normal phase. Ti $d$ and Se $p$ orbital characters are shown by red and blue colors, respectively. (f) The corresponding constant energy contours at $E_{F}$ . Blue and red color represent hole and electron pockets, respectively.
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Figure 2
(a)–(c) Phonon dispersion of the $1 \times 1 {TiSe}_{2}$ monolayer with different electronic smearing parameter $σ$ . The frequency of the soft phonon mode at $M$ point changes from imaginary (represented with a negative value) to a positive value with increasing $σ$ . (d) Evolution of the lowest phonon mode frequency at the $M$ point as a function of $σ$ . The strong dependence of this phonon's frequency on $σ$ and its quenching signals a CDW phase transition. (e) Total and partial phonon density of states for $σ = 0.1$ eV. The imaginary (negative) phonon frequencies arise mainly from the Ti atoms. (f) Charge density distribution in the Ti-Se plane containing Ti-Se bonds and (g) in the Ti plane of an undistorted $2 \times 2$ superstructure of ${TiSe}_{2}$ monolayer.
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Figure 3
(a) Illustration of the atomic displacements in the CDW phase. The length of each arrow is proportional to the magnitude of the atomic displacement. (b) The two types of local octahedral structure in the CDW state. Red arrows in the first depict the movement of Se atoms, while the inward and outward black arrows in the second illustrate the contraction and expansion, respectively, of the Ti-Se bonds. (c) Evolution of the total energy of the $2 \times 2$ superstructure computed using LDA and GGA as a function of CDW distortion $δ_{Ti}$ . Energies are given relative to the normal phase energy ( $δ_{Ti} = 0$ ). (d) Energy per chemical unit of the normal phase (red) and distorted phase (black) as a function of smearing parameter $σ$ . Energies are given relative to the normal phase energy at $σ = 0.05$ eV. (e) Phonon dispersion curve of the distorted phase ( $2 \times 2$ superstructure) in the supercell BZ.
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Figure 4
(a) Electronic band structure and (b) valence and conduction bands of the ${TiSe}_{2}$ monolayer in the distorted phase (relaxed $2 \times 2$ superstructure), represented in the supercell BZ. (c) The corresponding unfolded band structure. (d) Closeup of the area highlighted by the dashed rectangle in (c).
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Figure 5
Schematic evolution of the CDW phase in a ${TiSe}_{2}$ monolayer (top row) along with a schematic band structure (bottom row). (a) In the normal phase, Se $4 p$ -derived hole pockets and Ti $3 d$ -derived electron pockets exist near the $Γ$ and $M$ points, respectively. (b) In a clamped-ion $2 \times 2$ superstructure, the electron and hole pockets are simply replicated (folded) at the $Γ$ and $M$ points without gapping the spectrum. (c) A finite lattice distortion develops when the ions are fully relaxed and a gap simultaneously opens at $E_{F}$ . (a) Undistorted ( $1 \times 1$ structure), (b) Undistorted ( $2 \times 2$ superstructure), and (c) Periodic distortion ( $2 \times 2$ superstructure).
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Physical Review B

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Stable charge density wave phase in a $1 T - {TiSe}_{2}$ monolayer

Bahadur Singh, Chuang-Han Hsu, Wei-Feng Tsai, Vitor M. Pereira, and Hsin Lin

Phys. Rev. B 95, 245136 – Published 28 June 2017

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Physical Review B

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Stable charge density wave phase in a 1T–TiSe2 monolayer

Bahadur Singh, Chuang-Han Hsu, Wei-Feng Tsai, Vitor M. Pereira, and Hsin Lin

Phys. Rev. B 95, 245136 – Published 28 June 2017

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Stable charge density wave phase in a $1 T - {TiSe}_{2}$ monolayer