Electric-field-driven exciton vortices in transition metal dichalcogenide monolayers

Yingda Chen, Yongwei Huang, Wenkai Lou, Yongyong Cai, and Kai Chang

Phys. Rev. B 102, 165413 – Published 22 October 2020

Abstract

We predict electric-field-driven exciton vortices in transition metal dichalcogenide monolayers in the Bose-Einstein condensation regime. The Rashba spin-orbit coupling created by perpendicular electric fields couples the bright and dark excitons, behaves like an emerging SU(2) gauge field for excitons, and induces spatially asymmetric distribution of exciton density. We find the interplay between the dipole-dipole interaction among excitons and Rashba spin-orbit coupling leads to the phase transitions containing different vortices, from a single pair of vortices to numerous satellite vortices appearing at the edge of the sample. The exciton condensation at the $K$ and $K^{'}$ valleys shows mirror-symmetric patterns composed of exciton vortices rotating oppositely, which are protected topologically by the winding numbers.

Received 12 February 2020
Revised 2 October 2020
Accepted 6 October 2020

DOI:https://doi.org/10.1103/PhysRevB.102.165413

Physics Subject Headings (PhySH)

Excitons Rashba coupling

Bose-Einstein condensates Monolayer films Transition metal dichalcogenides

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yingda Chen ^1,2, Yongwei Huang ^1,2, Wenkai Lou ^1,2,*, Yongyong Cai^3,†, and Kai Chang^1,2,4,‡

¹SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, People's Republic of China
²CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, People's Republic of China
³Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People's Republic of China
⁴Beijing Academy of Quantum Information Sciences, Beijing 100193, People's Republic of China

^*wklou@semi.ac.cn
^†yongyong.cai@bnu.edu.cn
^‡kchang@semi.ac.cn

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Vol. 102, Iss. 16 — 15 October 2020

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Images

Figure 1
Bright and dark excitons in TMD MLs. (a) Excitons trapped in a circular flake ( $R_{0} = 0.3 μ m$ ) of $W {Se}_{2}$ ML between the electrodes. (b) Optical transitions and the bright/dark $A / B$ exciton states at the $K$ and $K^{'}$ valleys. (c) Spatial separation of bright (left) and dark (right) excitons caused by RSOC, where a pair of bright and dark exciton vortices is also observed. (d) Dispersion of excitons at the $L$ state (down) and $H$ state (up). (e) Valley-selective density (color map) and velocity distributions (green arrows) of bright excitons pumped with $σ_{+}$ , $σ_{-}$ , and linearly polarized laser, with fixed number of bright and dark excitons. The orange solid lines show the density profile along the $x$ axis, while the dashed lines indicate the vortex cores. $|λ_{BR}| = 18 meV Å$ and $n_{b, d} = 10^{10} {cm}^{- 2}$ in (d) and (e).
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Figure 2
Motion of a bright exciton vortex at the $K$ valley in a $W {Se}_{2}$ flake, varied with perpendicular electric fields. (a)–(c) Show the normalized density $η = ρ / N$ (color map) and the unit velocity $\hat{v}$ (white arrows) of the bright excitons at the $K$ valley with fixed average density $n_{b, d} = 10^{10} {cm}^{- 2}$ , at different strengths of the RSOC $|λ_{BR}|$ = (a) $6 meV Å$ , (b) $12 meV Å$ , (c) $36 meV Å$ . (d)–(f) Show the phases $φ$ and the wave vectors $\vec{k} = \nabla φ$ (green arrows) corresponding to (a)–(c). The radius of $W {Se}_{2}$ flake $R_{0} = 0.3 μ m$ .
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Figure 3
(a) The normalized density $η$ (color map) and unit velocity $\hat{v}$ (white arrows) distributions of exciton BEC stripe pattern, for the exciton density $n_{b, d} = 1.5 \times 10^{11} {cm}^{- 2}$ , and the strength of RSOC $|λ_{BR}| = 300 meV Å$ . (b) The zoom-in of (a) at the edge of the BEC pattern, which shows the satellite vortices at the edge and the bulk vortices located at the radical directions. (c) Phase transitions (indicated by different colors) of bright exciton BEC at the $K$ valley with $ℓ = 0$ , as a function of the RSOC. $N_{v}$ is the number of vortices in the sample. (d) The angular momentum distribution $l_{z}$ (color map) of the ground-state bright vortices for different $λ_{BR}$ in the condition of $ℓ \neq 0$ , with their positions and directions indicated by green dashed lines and arrows. $n_{b, d} = 1 \times 10^{10} {cm}^{- 2}$ in (c) and (d).
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Figure 4
Impact of exciton densities on the BEC distributions. (a) $x$ -directional profile of exciton densities $ρ = {| ψ |}^{2}$ at the $K$ valley. (b) Phase diagram (color map) of the ground state varied with Rashba strength $λ_{BR}$ and average density $n_{b, d}$ . (c) The normalized density $η = ρ / N$ (color map) and the unit velocity ${\vec{e}}_{φ} (\hat{v}$ ) (white arrows) distribution of the bright excitons at the $K$ valley, with equal number of bright and dark excitons. (d) Shows the results of the dark excitons in the same condition of (c). (e), (f) Show the distributions of unequally populated bright and dark excitons, and the relocations of the vortices. The radius of $W {Se}_{2}$ flake $R_{0} = 0.3 μ m$ . The strength of the Rashba SOC $|λ_{BR}| = 18 meV Å$ .
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Figure 5
The distribution of normalized density $η$ (color map in the first row), velocity unit velocity $\hat{v}$ (white arrows in the first row), and angular momentum $l_{z}$ (color map in the second row) of bright excitons at the $K$ valley. The influence of random antibonding type impurities on the bulk exciton vortices, and on the satellite vortices, is shown in (a) and (b). While the influence of bonding type impurities are shown (c) and (d), in the same conditions of (a) and (b), respectively. The results are performed at different strengths of the RSOC, $|λ_{BR}|$ = (a) and (c) $36 meV Å$ , and (b) and (d) $300 meV Å$ , in a circular flake of $W {Se}_{2}$ ML with radius $R_{0} = 0.3 μ m$ . The locations (green circles in the second row) and the total number of the impurities are generated randomly.
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Figure 6
The schematic dipole-dipole interactions in our system, where $V_{e i e j}$ represents the Coulomb interaction between electrons, $V_{h i h j}$ between holes, and $V_{e i h j}, V_{h i e j}$ between electrons and holes. $p_{i, j}$ are the dipole moments of exciton $i$ and $j$ , and $r_{i, j}$ are the locations of excitons. $d$ is the field-induced electron-hole separation.
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