Quantitative determination of interlayer electronic coupling at various critical points in bilayer $\mathrm{Mo}{\mathrm{S}}_{2}$

Quantitative determination of interlayer electronic coupling at various critical points in bilayer $Mo S_{2}$

Wei-Ting Hsu, Jiamin Quan, Chi-Ruei Pan, Peng-Jen Chen, Mei-Yin Chou, Wen-Hao Chang, Allan H. MacDonald, Xiaoqin Li, Jung-Fu Lin, and Chih-Kang Shih

Phys. Rev. B 106, 125302 – Published 12 September 2022

Abstract

Tailoring interlayer coupling has emerged as a powerful tool to tune the electronic structure of van der Waals (vdW) bilayers. One example is the usage of the “moiré pattern” to create controllable two-dimensional electronic superlattices through the configurational dependence of interlayer electronic couplings. This approach has led to some remarkable discoveries in twisted graphene bilayers, and transition metal dichalcogenide homo- and heterobilayers. However, a largely unexplored factor is the interlayer distance $d$ , which can impact the interlayer coupling strength exponentially. In this paper, we quantitatively determine the coupling strengths as a function of interlayer spacing at various critical points of the Brillouin zone in bilayer $Mo S_{2}$ . The exponential dependence of the coupling parameter on the gap distance is demonstrated. Most significantly, we achieved a 280% enhancement of $K − valley$ coupling strength with an 8% reduction of the vdW gap, pointing to a strategy for designing a unique electronic system in vdW bilayers.

Received 4 February 2022
Revised 23 August 2022
Accepted 24 August 2022

DOI:https://doi.org/10.1103/PhysRevB.106.125302

Physics Subject Headings (PhySH)

Cellular bilayers Electronic structure

Transition metal dichalcogenides Van der Waals systems

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Wei-Ting Hsu ^1,7,*, Jiamin Quan^1,*, Chi-Ruei Pan ², Peng-Jen Chen^2,3, Mei-Yin Chou², Wen-Hao Chang ^4,5, Allan H. MacDonald¹, Xiaoqin Li¹, Jung-Fu Lin ⁶, and Chih-Kang Shih ^1,7,†

¹Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
²Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan
³Institute of Physics, Academia Sinica, Taipei 11529, Taiwan
⁴Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan
⁵Research Center for Applied Sciences, Academia Sinica, Taipei 11529, Taiwan
⁶Department of Geological Sciences, Jackson School of Geosciences, The University of Texas at Austin, Austin, Texas 78712, USA
⁷Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan

^*These authors contributed equally to this work.
^†shih@physics.utexas.edu

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Vol. 106, Iss. 12 — 15 September 2022

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Figure 1
Control of interlayer electronic coupling using high-pressure diamond anvil cell. (a) Schematic showing the interlayer hybridization of $| ψ_{u} 〉$ and $| ψ_{l} 〉$ . When the interlayer spacing $d$ decreases, the orbital wave function overlap increases, which enhances the coupling strength and leads to larger energy splitting of the hybridized state. (b), (c) DFT band structure of monolayer $Mo S_{2}$ (black) and bilayer $Mo S_{2}$ with interlayer spacing $d = 0.62 nm$ (red) and $0.58 nm$ (blue). At the $Γ_{V}, Q_{C}$ , and $K_{V}$ points, band splitting becomes larger for a bilayer with a smaller $d$ . (d) Schematic showing a diamond anvil cell that can be accessed by optical spectroscopy. (e) An optical image showing the monolayer (1L), bilayer (2L), and bulk $Mo S_{2}$ loaded in the high-pressure chamber. A ruby sphere is used as a pressure calibrant with the Ne medium loaded in the chamber (see Supplemental Material [35]).
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Figure 2
Pressure tuning of interlayer hopping integral at K valley. (a) Schematic showing the spin-allowed optical transitions ( $X_{A}$ and $X_{B}$ ) at the $K^{+}$ valley of monolayer $Mo S_{2}$ . The spin splitting of the conduction band ( $2 λ_{C}$ ) is much smaller than that of the valence band ( $2 λ_{V}$ ). (b) Schematic showing the interlayer hybridization of a $2 H − stacked$ bilayer, in which the valence band splitting is enhanced ( $2 Δ_{V} > 2 λ_{V}$ ). Note that the interlayer hopping is allowed (forbidden) at the valence (conduction) band edge. (c), (d) DR spectra of the monolayer (c) and bilayer (d) $Mo S_{2}$ under applied pressure. The spectra have been shifted vertically for clarity.
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Figure 3
Determination of the spacing-dependent coupling strength. (a) Pressure-dependent $Δ E$ of monolayer and bilayer $Mo S_{2}$ . The dots are experimental data, and the solid lines are a guide to the eye. (b) Coupling strength $t$ as a function of applied pressure, showing a 280% enhancement at 12.7 GPa compared to 0 GPa. (c) Pressure-dependent interlayer spacing $d$ determined by XRD. (d) Comparison of experimentally determined (black dots) and DFT-calculated (blue/red curve) coupling strength as a function of $d$ . In the calculation of DFT-1 (DFT-2), the compression of the in-plane lattice has not (has) been taken into account (see Figs. S1 and S2 in the Supplemental Material [35] for band structures).
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Figure 4
Pressure tuning of the indirect gap. (a) PL spectra of bilayer $Mo S_{2}$ under applied compressive pressure. The indirect gap $X_{I}$ shows a significant redshift at high pressure. (b) A comparison of indirect gap measured by PL (dots) with the predicted evolution of the $E_{Q Γ}$ and $E_{K Γ}$ gap, showing excellent agreement with the energy evolution of the $E_{Q Γ}$ gap (red curve). The spacing dependence of $t^{(Γ v)}$ and $t^{(Q c)}$ extracted from DFT calculation can be found in Fig. S4 in the Supplemental Material [35].
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Physical Review B

covering condensed matter and materials physics

Quantitative determination of interlayer electronic coupling at various critical points in bilayer $Mo S_{2}$

Wei-Ting Hsu, Jiamin Quan, Chi-Ruei Pan, Peng-Jen Chen, Mei-Yin Chou, Wen-Hao Chang, Allan H. MacDonald, Xiaoqin Li, Jung-Fu Lin, and Chih-Kang Shih

Phys. Rev. B 106, 125302 – Published 12 September 2022

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Physical Review B

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Quantitative determination of interlayer electronic coupling at various critical points in bilayer MoS2

Wei-Ting Hsu, Jiamin Quan, Chi-Ruei Pan, Peng-Jen Chen, Mei-Yin Chou, Wen-Hao Chang, Allan H. MacDonald, Xiaoqin Li, Jung-Fu Lin, and Chih-Kang Shih

Phys. Rev. B 106, 125302 – Published 12 September 2022

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Quantitative determination of interlayer electronic coupling at various critical points in bilayer $Mo S_{2}$