Electrons and phonons in single layers of hexagonal indium chalcogenides from ab initio calculations

V. Zólyomi, N. D. Drummond, and V. I. Fal'ko

Phys. Rev. B 89, 205416 – Published 14 May 2014

Abstract

We use density functional theory to calculate the electronic band structures, cohesive energies, phonon dispersions, and optical absorption spectra of two-dimensional In $_{2}$ $X_{2}$ crystals, where X is S, Se, or Te. We identify two crystalline phases ( $α$ and $β$ ) of monolayers of hexagonal In $_{2}$ $X_{2}$ , and show that they are characterized by different sets of Raman-active phonon modes. We find that these materials are indirect-band-gap semiconductors with a sombrero-shaped dispersion of holes near the valence-band edge. The latter feature results in a Lifshitz transition (a change in the Fermi-surface topology of hole-doped In $_{2}$ $X_{2}$ ) at hole concentrations $n_{S} = 6.86 \times 10^{13}$ cm $^{- 2}$ , $n_{Se} = 6.20 \times 10^{13}$ cm $^{- 2}$ , and $n_{Te} = 2.86 \times 10^{13}$ cm $^{- 2}$ for X=S, Se, and Te, respectively, for $α$ -In $_{2}$ $X_{2}$ and $n_{S} = 8.32 \times 10^{13}$ cm $^{- 2}$ , $n_{Se} = 6.00 \times 10^{13}$ cm $^{- 2}$ , and $n_{Te} = 8.14 \times 10^{13}$ cm $^{- 2}$ for $β$ -In $_{2}$ $X_{2}$ .

Received 16 March 2014

DOI:https://doi.org/10.1103/PhysRevB.89.205416

Authors & Affiliations

V. Zólyomi, N. D. Drummond, and V. I. Fal'ko

Physics Department, Lancaster University, Lancaster LA1 4YB, United Kingdom

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Vol. 89, Iss. 20 — 15 May 2014

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Figure 1
Structures of the $α$ and $β$ polytypes of monolayer indium chalcogenides In $_{2}$ X $_{2}$ (X=S, Se, or Te). The parameters $a$ , $d_{In - In}$ , and $d_{X - X}$ are the lattice parameter, the In–In bond length, and the vertical distance between X atoms, respectively.
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Figure 2
Phonon dispersion curves for $α$ (top panel) and $β$ (bottom panel) polytypes of In $_{2}$ S $_{2}$ , In $_{2}$ Se $_{2}$ , and In $_{2}$ Te $_{2}$ . The inset shows the low-frequency spectrum of $α$ -In $_{2}$ Se $_{2}$ with several methods. Below we list the DFT-LDA optical-phonon frequencies at $Γ$ , the irreducible representation (irrep.) to which the eigenvectors belong, and the IR and Raman activity. The modes are labeled as longitudinal optical (LO), transverse optical (TO), or out-of-plane optical (ZO). The irreducible representation is given in the conventional molecular notation in which one and two primes indicate $z \to - z$ reflection symmetry and antisymmetry, respectively. For IR activity we indicate the component of electric field involved (out-of-plane, $E_{z}$ , or in-plane, $E_{∥}$ ), while for Raman activity we indicate the components of electric field that are coupled by the Raman tensor.
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Figure 3
HSE06 band structures (solid red lines) for $α$ -In $_{2}$ S $_{2}$ , $α$ -In $_{2}$ Se $_{2}$ , and $α$ -In $_{2}$ Te $_{2}$ . Spin-orbit coupling (SOC) is not included in these results. The zero of energy is taken to be the Fermi level $E_{F}$ and the bottom of the conduction band is marked with a horizontal line. For comparison, the semilocal band structures are also shown, including the effects of SOC. The orbital composition of the $α$ -In $_{2}$ X $_{2}$ states highlighted by $◯$ , $▵$ , and $♦$ are summarized in the table below. Dominant contributions were found to originate from $s$ - and $p$ -type orbitals; the “ $+$ ” and “ $-$ ” subscripts refer to even ( $+$ ) and odd ( $-$ ) states with respect to $z \to - z$ reflection. The LDA spin-orbit splittings $| Δ E_{SO}^{K} |$ of the bands at the K point are also given. The notation “ $p_{x} p_{y}$ ” refers to equal $p_{x}$ and $p_{y}$ contributions as a consequence of symmetry.
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Figure 4
LDA energy contours (with a step of 2 meV) for the valence band of $α$ -In $_{2}$ X $_{2}$ around the $Γ$ point. The contour corresponding to the energy of the saddle point (Lifshitz transition) is highlighted. The table below shows the fitted coefficients $E_{2 i}$ (in units of eV Å $^{2 i}$ ) for the inverted sombrero dispersion near the VBM of $α$ -In $_{2}$ X $_{2}$ in Eq. (1). The zero of energy is set to the VBM. The root mean square of the residuals $σ$ indicates the amount by which the fit is in error. The last column shows the hole density $n_{X}$ for the Lifshitz transition.
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Figure 5
Absorption coefficient of $α$ - and $β$ -In $_{2}$ X $_{2}$ 2D crystals as obtained from the imaginary part of the dielectric function $ɛ$ by normalizing it to absolute units after it was compared to Im $(ɛ)$ evaluated for graphene in the range 0.8–1.5 eV, where monolayer graphene absorbs 2.3% of light. The raw results for Im $(ɛ)$ are indicated on the right-hand axis.
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Figure 6
HSE06, LDA, and PBE DFT band structures for $β$ -In $_{2}$ S $_{2}$ , $β$ -In $_{2}$ Se $_{2}$ , and $β$ -In $_{2}$ Te $_{2}$ . Spin-orbit coupling is taken into account in the case of LDA and PBE. The zero of energy is taken to be the Fermi level $E_{F}$ and the bottom of the conduction band is marked with a horizontal line. The orbital composition of the $β$ -In $_{2}$ X $_{2}$ states highlighted by $◯$ , $▵$ , and $♦$ are summarized in the table below. Dominant contributions were found to originate from $s$ - and $p$ -type orbitals; the “ $+$ ” and “ $-$ ” subscripts refer to even ( $+$ ) and odd ( $-$ ) states with respect to three-dimensional inversion. The notation “ $p_{x} p_{y}$ ” refers to equal $p_{x}$ and $p_{y}$ contributions as a consequence of symmetry.
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