Exchange-Driven Intermixing of Bulk and Topological Surface States by Chiral Excitons in ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$

Exchange-Driven Intermixing of Bulk and Topological Surface States by Chiral Excitons in ${Bi}_{2} {Se}_{3}$

Bowen Hou, Dan Wang, Bradford A. Barker, and Diana Y. Qiu

Phys. Rev. Lett. 130, 216402 – Published 26 May 2023

Abstract

Topological surface states (TSS) in the prototypical topological insulator (TI) ${Bi}_{2} {Se}_{3}$ are frequently characterized using optical probes, but electron-hole interactions and their effect on surface localization and optical response of the TSS remain unexplored. Here, we use ab initio calculations to understand excitonic effects in the bulk and surface of ${Bi}_{2} {Se}_{3}$ . We identify multiple series of chiral excitons that exhibit both bulk and TSS character, due to exchange-driven mixing. Our results address fundamental questions about the degree to which electron-hole interactions can relax the topological protection of surface states and dipole selection rules for circularly polarized light in TIs by elucidating the complex intermixture of bulk and surface states excited in optical measurements and their coupling to light.

Received 7 October 2022
Revised 17 February 2023
Accepted 1 May 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.216402

Physics Subject Headings (PhySH)

Electronic structure Excitons First-principles calculations Surface states Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Bowen Hou¹, Dan Wang¹, Bradford A. Barker ², and Diana Y. Qiu ^1,*

¹Department of Mechanical Engineering and Material Sciences, Yale University, New Haven, Connecticut 06511, USA
²Department of Physics, University of California, Merced, California 95343, USA

^*diana.qiu@yale.edu

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Vol. 130, Iss. 21 — 26 May 2023

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Figure 1
Electronic band structure at the $G_{0} W_{0}$ (solid red curve) and LDA (dashed blue curve) levels of (a) bulk, (b) 1QL, (c) 3QL, (d) 5QL ${Bi}_{2} {Se}_{3}$ . The gray shadow in (c) shows the extent of a $Γ$ -centered patch with a radius of $0.028 {a_{0}}^{- 1}$ .
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Figure 2
Calculated absorption spectra of (a) 1QL and (f) 3QL ${Bi}_{2} {Se}_{3}$ with(without) electron-hole interactions included at the GW-BSE(GW-RPA) level in red(blue). The gray vertical line indicates the $G_{0} W_{0}$ quasiparticle band gap. (b)–(c) The phase winding of the exciton envelope wave function, $A_{v c k}^{S}$ , of the degenerate bright excitons contributing to peak A of 1QL ${Bi}_{2} {Se}_{3}$ and (g)–(h) degenerate bright excitons contributing to peak $C$ of 3QL ${Bi}_{2} {Se}_{3}$ . The angle of the arrows indicates the phase, and the length of each arrow indicates the amplitude. The index $m$ denotes the winding number around the $Γ$ point. (d)–(e) The winding of the optical interband dipole transition matrix element between the highest valence band and the lowest conduction band for 1QL and (i)–(j) 3QL ${Bi}_{2} {Se}_{3}$ . The index $l_{- (+)}$ denotes the winding number of left(right) circularly polarized light, respectively. A maximally smooth local gauge is imposed following Refs. [60, 61]. The exciton wavefunction and optical interband transitions are plotted in a patch centered around the $Γ$ point in the BZ, marked by gray area in Fig. 1.
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Figure 3
(a) Instantaneous emission intensity of left-hand ( $I_{R L}$ , dark blue) or right-hand ( $I_{R R}$ , orange curve) polarized light after excitation by right-hand polarized light from all exciton states in a patch of radius of $0.028 {a_{0}}^{- 1}$ around $Γ$ in the BZ. The inset shows the amplitudes of the envelope wave function of ${CX}_{1}$ , ${CX}_{3}$ , ${CX}_{8}$ , and ${CX}_{10}$ in the patch in reciprocal space. (b) Contribution of each band to each exciton state. The bands are labeled as either bulklike bands relative to the valence band maximum (VBM) and conduction band minimum (CBM) or as surface states SS1, SS2, and RSS, with the occupied band contributions in blue and unoccupied band contributions in red. Each bulk band is doubly degenerate, while surface states sum contributions from all the occupied or unoccupied surface states. The size of each dot is proportional to $(I_{R R} - I_{R L}) \times \sum_{c k} | A_{v c k}^{S} |^{2}$ for occupied states and $(I_{R R} - I_{R L}) \times \sum_{v k} | A_{v c k}^{S} |^{2}$ for unoccupied states.
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Figure 4
(a)–(b) The modulus of the electronic wave function in 3QL ${Bi}_{2} {Se}_{3}$ for the surface states SS1’ and RSS at the $Γ$ point. (c)–(d) The modulus of the exciton wave function for ${CX}_{3}$ and ${CX}_{10}$ , when the hole (yellow sphere) is fixed. The hole position is set at the location that maximizes the wave function amplitude. The integral of the modulus squared of each wave function over the $x − y$ plane is shown as a side panel in (a)–(d). The percentage label indicates the contribution to the wave function from the central QL. For clarity, the Bi and Se atoms are not explicitly shown. The bonds are depicted as purple lines in the vicinity of Bi and green lines in the vicinity of Se. (e)–(g) Electron-hole exchange matrix elements $⟨ v c k | K^{x} | v^{'} c^{'} k^{'} ⟩$ for $k = k^{'} = Γ$ and $v = v^{'} = SS 1$ for (e) 3QL, (f) 5QL, and (g) 6QL slabs of ${Bi}_{2} {Se}_{3}$ .
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Physical Review Letters

Exchange-Driven Intermixing of Bulk and Topological Surface States by Chiral Excitons in Bi2Se3

Bowen Hou, Dan Wang, Bradford A. Barker, and Diana Y. Qiu

Phys. Rev. Lett. 130, 216402 – Published 26 May 2023

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Exchange-Driven Intermixing of Bulk and Topological Surface States by Chiral Excitons in ${Bi}_{2} {Se}_{3}$