Zitterbewegung of Moir\'e Excitons in Twisted ${\mathrm{MoS}}_{2}/{\mathrm{WSe}}_{2}$ Heterobilayers

Zitterbewegung of Moiré Excitons in Twisted ${MoS}_{2} / {WSe}_{2}$ Heterobilayers

I. R. Lavor, D. R. da Costa, L. Covaci, M. V. Milošević, F. M. Peeters, and A. Chaves

Phys. Rev. Lett. 127, 106801 – Published 31 August 2021

Abstract

The moiré pattern observed in stacked noncommensurate crystal lattices, such as heterobilayers of transition metal dichalcogenides, produces a periodic modulation of their band gap. Excitons subjected to this potential landscape exhibit a band structure that gives rise to a quasiparticle dubbed the moiré exciton. In the case of ${MoS}_{2} / {WSe}_{2}$ heterobilayers, the moiré trapping potential has honeycomb symmetry and, consequently, the moiré exciton band structure is the same as that of a Dirac-Weyl fermion, whose mass can be further tuned down to zero with a perpendicularly applied field. Here we show that, analogously to other Dirac-like particles, the moiré exciton exhibits a trembling motion, also known as Zitterbewegung, whose long timescales are compatible with current experimental techniques for exciton dynamics. This promotes the study of the dynamics of moiré excitons in van der Waals heterostructures as an advantageous solid-state platform to probe Zitterbewegung, broadly tunable by gating and interlayer twist angle.

Received 11 February 2021
Accepted 28 June 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.106801

Physics Subject Headings (PhySH)

Dirac fermions Excitons Optoelectronics

2-dimensional systems Bilayer films Heterostructures Honeycomb lattice Transition metal dichalcogenides

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

I. R. Lavor ^1,2,3, D. R. da Costa ¹, L. Covaci³, M. V. Milošević³, F. M. Peeters ³, and A. Chaves ^1,3

¹Departamento de Física, Universidade Federal do Ceará, 60455-760 Fortaleza, Ceará, Brazil
²Instituto Federal de Educação, Ciência e Tecnologia do Maranhão, KM-04, Enseada, 65200-000 Pinheiro, Maranhão, Brazil
³Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium

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Vol. 127, Iss. 10 — 3 September 2021

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Figure 1
(a) Moiré pattern with period $b$ in an ${MoS}_{2} / {WSe}_{2}$ heterobilayer, twisted by 3°. Black diamond represents the supercell. Insets magnify three characteristic locations ( $A$ , $B$ , and $C$ ), where atomic registries resemble lattice-matched bilayers of different $R$ -type stacking. (b) Lateral view of the interlayer distance of the regions $A$ , $B$ , and $C$ (for more details, see Ref. [31]). (c) Corresponding band structures, calculated with the tight-binding model, for the first moiré Brillouin zone with (dashed lines) and without (solid lines) an applied electric field ( $ϵ = ϵ_{0} \approx 0.44 V / nm$ ) for the $K$ (red) and $K^{'}$ (blue) valleys of the crystal. (d) Representation of a honeycomb lattice structure and unit cell (gray region) where sublattice sites $A$ and $B$ correspond to the respective stacking registries labeled in (a) and with lattice constant $a$ . First, second, and third nearest-neighbors hopping parameters are represented by $t_{0}$ (green arrows), $t_{1}$ (blue arrows), and $t_{2}$ (gray arrows), respectively. ${\vec{a}}_{1}$ and ${\vec{a}}_{2}$ are the basis vectors. (e) Color map of the ILE potential landscape in $R$ -type ${MoS}_{2} / {WSe}_{2}$ , as illustrated in (a), where the excitonic potential is tuned by an applied perpendicular electric field $ϵ$ . The inset in each panel shows the potential profile along the high-symmetry points ( $A - B - C - A$ ) of the moiré supercell. For $ϵ = ϵ_{0}$ , the excitonic potential exhibits the same value at regions $A$ and $B$ , whereas for $ϵ = 2 ϵ_{0}$ ( $ϵ = 0$ ), $A$ ( $B$ ) becomes higher in energy than $B$ ( $A$ ).
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Figure 2
ZBW of the expectation values of the position of a moiré exciton in a ${MoS}_{2} / {WSe}_{2}$ vdWHS, considering an initial Gaussian wave packet distribution with $d = 200 Å$ (blue), $d = 300 Å$ (orange), and $d = 500 Å$ (green), and pseudospinors $[C_{1} C_{2}]^{T} = [1 0]^{T}$ and $[C_{1} C_{2}]^{T} = [1 1]^{T}$ , under applied fields (a),(b) $ϵ = 0$ and (c),(d) $ϵ = ϵ_{0}$ . The propagated probability densities for the time instants marked with white and gray circular dots in each panel are shown in Fig. 3.
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Figure 3
Snapshots of the propagated probability density ${| Ψ (\vec{r}, t) |}^{2}$ for an initial Gaussian wave packet with width $d = 500 Å$ and pseudospinors ${[1 0]}^{T}$ and ${[11]}^{T}$ . Top (bottom) row shows results for applied electric field $ϵ = 0$ ( $ϵ = ϵ_{0}$ ). The white (orange) bar corresponds to 500 Å (875 Å) and the small white dot inside each panel represents the center of mass of the wave packet. The profiles of ${| Ψ (\vec{r}, t) |}^{2}$ along the dashed white lines in each panel are shown as insets. The labels (i)–(viii) correspond to different time steps as marked with circular dots in Fig. 2.
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Physical Review Letters

Zitterbewegung of Moiré Excitons in Twisted MoS2/WSe2 Heterobilayers

I. R. Lavor, D. R. da Costa, L. Covaci, M. V. Milošević, F. M. Peeters, and A. Chaves

Phys. Rev. Lett. 127, 106801 – Published 31 August 2021

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Zitterbewegung of Moiré Excitons in Twisted ${MoS}_{2} / {WSe}_{2}$ Heterobilayers