Strongly distinct electrical response between circular and valley polarization in bilayer transition metal dichalcogenides

Luojun Du, Mengzhou Liao, Gui-Bin Liu, Qinqin Wang, Rong Yang, Dongxia Shi, Yugui Yao, and Guangyu Zhang

Phys. Rev. B 99, 195415 – Published 10 May 2019

Abstract

We introduce a physical model to describe the influence of a perpendicular electric field on circular polarization (CP) and valley polarization (VP) in bilayer transition metal dichalcogenides. Our results uncover that electric-field-dependent CP and VP are quite distinct from each other. The dependence of CP on the electric field harbors a W pattern and possesses the minimum when the potential energy difference between the two layers is equal to the strength of spin-orbit coupling. Such dependence of CP stems from the modulation of energy cost for interlayer hopping and spin-dependent layer polarization. In contrast, VP is strictly absent in primitive bilayers and increases monotonically with increasing strength of electric field, resulting from the continuous variation of valley magnetic moments and inversion-symmetry breaking. Our model elaborates well the recent experimental observations for which the origin is under debate. Moreover, we demonstrate that the manipulation of layer and valley pseudospin is fully tunable by perpendicular electric fields, paving the way for prospects in electrical control of exotic layer-valleytronics.

Received 17 October 2018
Revised 24 April 2019

DOI:https://doi.org/10.1103/PhysRevB.99.195415

Physics Subject Headings (PhySH)

Spintronics Valleytronics

Transition metal dichalcogenides

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Luojun Du^1,2, Mengzhou Liao¹, Gui-Bin Liu^3,*, Qinqin Wang¹, Rong Yang¹, Dongxia Shi^1,4,5, Yugui Yao³, and Guangyu Zhang^{1,4,5,6,7,†}

¹Institute of Physics and Beijing National Laboratory for Condensed Matter Physics, Chinese Academy of Sciences, Beijing 100190, China
²Department of Electronics and Nanoengineering, Aalto University, Tietotie 3, FI-02150, Finland
³Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology, Beijing 100081, China
⁴School of Physical Sciences, University of Chinese Academy of Science, Beijing 100190, China
⁵Beijing Key Laboratory for Nanomaterials and Nanodevices, Beijing 100190, China
⁶Collaborative Innovation Center of Quantum Matter, Beijing 100190, China
⁷Songshan-Lake Materials Laboratory, Dongguan 523808, Guangdong Province, China

^*Corresponding author: gbliu@bit.edu.cn
^†Corresponding author: gyzhang@iphy.ac.cn

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Issue

Vol. 99, Iss. 19 — 15 May 2019

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Images

Figure 1
(a) Schematics of optical transition selection rules in decoupled bilayer TMDCs with $2 H$ stacking. Then interlayer hopping is introduced shown as the magenta arrows connecting the same spin states between the upper and lower layers. (b) Optical transition selection rules in the $K$ valley of bilayer TMDCs with interlayer hopping. Red (blue) valance band represents $ξ_{+ 1} (ξ_{- 1})$ coupled with pure left (right) circularly polarized light. The thickness of the line indicates the probability. Spin configurations are indicated by $↑$ (spin up) and $↓$ (spin down). $σ^{+}$ ( $σ^{-}$ ) denotes left (right) circularly polarized light (circular arrows).
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Figure 2
The dynamics of photoexcited carriers in the $K$ valley for spin-up (a) and spin-down (b) states. The twofold-degenerate wave functions of the top valence band of the $K$ valley in TMDC bilayers are described in Eq. (2). Left panel shows absorption process by $σ^{+}$ excitation on resonance with the $A$ exciton. The black arrows indicate that the excitation generates holes (green solid ball) with $ξ_{+ 1}$ character in the valence band. Middle panel denotes the relaxation process via LH. Right panel shows the emission process, filling the holes of the valence band. The red (blue) wavy curves with arrows denote the $σ^{+}$ ( $σ^{-}$ ) emission, with amplitude representing the relative intensity of emission.
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Figure 3
(a) CP as a function of spin-orbit coupling $λ$ and interlayer hopping $t_{⊥}$ . (b) CP of TMDC bilayers calculated by our model (red symbols), compared to first-principles calculations (black symbols).
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Figure 4
(a) Schematic illustration of the $2 H$ -TMDC bilayer with an out-of-plane electric field $E$ . (b) Schematic diagram of the valence band structure under a perpendicular electric field. The electric field introduces an energy difference between the upper and lower layer: eEd. Dotted valence band structure is the situation without electric field. (c) Circular polarization as a function of electric field. $λ_{1}, λ_{2}, λ_{3}$ , and $λ_{4}$ are the strength of SOC for ${MoS}_{2}, {MoSe}_{2}, {WS}_{2}$ , and ${WSe}_{2}$ , respectively. (d) Valley polarization versus the electric field. The region highlighted by light blue denotes the potential energy differences allowed by the experimental conditions.
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Figure 5
(a) Electric-field-dependent band structure of $K$ and $K^{'}$ valleys from $k \cdot p$ model for bilayer ${MoS}_{2}$ . $Δ, λ$ , and $t_{⊥}$ are 1.6 eV, 147 meV, and 86 meV. (b) Optical transition energy as a function of electric field. Linewidth is proportional to the population of $σ^{+}$ in corresponding wave function. (c) Schematic diagram of the layer-valley-dependent excitation processes under negative (left), zero (middle), and positive (right) electric field. Positive electric field denotes the direction of electric field from the lower layer to upper layer. (d), (e) Schematic diagram of electrical control of layer-valleytronics for on state (d) and off state (e).
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