Unfolding the band structure of disordered solids: From bound states to high-mobility Kane fermions

O. Rubel, A. Bokhanchuk, S. J. Ahmed, and E. Assmann

Phys. Rev. B 90, 115202 – Published 4 September 2014

Abstract

Supercells are often used in ab initio calculations to model compound alloys, surfaces, and defects. One of the main challenges of supercell electronic structure calculations is to recover the Bloch character of electronic eigenstates perturbed by disorder. Here we apply the spectral weight approach to unfolding the electronic structure of group III-V and II-VI semiconductor solid solutions. The illustrative examples include formation of donorlike states in dilute Ga(PN) and associated enhancement of its optical activity, direct observation of the valence band anticrossing in dilute GaAs:Bi, and a topological band crossover in ternary (HgCd)Te alloy accompanied by emergence of high-mobility Kane fermions. The analysis facilitates interpretation of optical and transport characteristics of alloys that are otherwise ambiguous in traditional first-principles supercell calculations.

Received 16 May 2014
Revised 14 August 2014

DOI:https://doi.org/10.1103/PhysRevB.90.115202

Authors & Affiliations

O. Rubel^1,2,*, A. Bokhanchuk¹, S. J. Ahmed³, and E. Assmann⁴

¹Thunder Bay Regional Research Institute, 980 Oliver Road, Thunder Bay, Ontario, Canada P7B 6V4
²Department of Physics, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario, Canada P7B 5E1
³Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L8
⁴Institute for Solid State Physics, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria

^*rubelo@tbh.net

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Vol. 90, Iss. 11 — 15 September 2014

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Images

Figure 1
Relation between $k$ mesh in the reciprocal space for a square primitive lattice (a) and that for a $3 \times 2$ supercell (b). The first BZ is outlined on both panels. The overlay of both patterns is shown on the panel (c). The overlapped points form a subgroup of $K$ that represents the Bloch wave vector $k$ . There is a total of six subgroups ( $3 \times 2$ ), which indicates that each $K$ point in the supercell bears information about six $k$ points of the primitive basis.
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Figure 2
Band structure of GaP: 2-atom primitive unit cell (a) and 16-atom $2 \times 2 \times 2$ supercell (b). The supercell band structure is “obfuscated” due to the zone folding. Fat bands on panel (b) illustrate the unfolded band structure where the symbols size and color represent the Bloch spectral weight. Both cells have the Brillouin zone shown on panel (c). Here and in all figures hereafter the Fermi energy is set to zero.
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Figure 3
Band structure of ${Ga}_{64} P_{64}$ (a) and ${Ga}_{64} N_{1} P_{63}$ (b),(c) supercell unfolded to the primitive Bloch representation. A very weak signature of a nitrogen-related conduction band can be seen within the GaP energy gap (b). Panel (c) shows a snapshot of ${Ga}_{64} N_{1} P_{63}$ conduction band with the contrast artificially enhanced by using a nonlinear (square root) intensity scale. The nitrogen band introduces a direct optical transition (b) thereby stimulating radiative optical transitions in Ga(NP) [29].
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Figure 4
Band structure of ${Ga}_{64} {As}_{64}$ (a) and ${Ga}_{64} {As}_{63} {Bi}_{1}$ (b) supercells unfolded to the primitive Bloch representation. Bismuth incorporation leads to perturbations in the valence band, reduction of the band gap, and enhanced spin-orbit splitting.
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Figure 5
Band order and symmetry labels for HgTe and CdTe compounds (a). These two compounds exhibit a band inversion ( $Γ_{6} \leftrightarrow Γ_{8}$ ) and, thus, have different “topology.” As the composition ${Hg}_{1 - x} {Cd}_{x} Te$ varies between two binary compounds, a topological band crossover occurs. The crossover takes place near $x_{c} \approx 0.19$ and is accompanied by emergence of massless Kane fermions at the $Γ$ point (b). Regions of the band structure perturbed by disorder are labeled with arrows. The disorder affects only the electronic states located $1 - 3 eV$ above and below the Fermi energy.
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Figure 6
Evolution of the band structure in ternary (HgCd)Te alloy near the $Γ$ point as a function of the chemical composition: ${Hg}_{25} {Cd}_{2} {Te}_{27}$ (a), ${Hg}_{22} {Cd}_{5} {Te}_{27}$ (b), ${Hg}_{20} {Cd}_{7} {Te}_{27}$ (c), and ${Hg}_{5} {Cd}_{22} {Te}_{27}$ (d). The transition from a semimetal (a) to an insulator (c),(d) occurs by passing through a Kane point (b). The nearly linear dispersion $E \propto | k |$ , which is characteristic of the Kane point, persists in the conduction band after opening of the band gap (c).
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