Theory of interlayer exciton dynamics in two-dimensional transition metal dichalcogenide heterolayers under the influence of strain reconstruction and disorder

Monolayers of transition metal dichalcogenides (TMDC) became one of the most studied nanostructures in the last decade. Combining two different TMDC monolayers results in a heterostructure whose properties can be individually tuned by the twist angle between the lattices of the two van der Waals layers and the relative placement of the layers, leading to moiré cells. For small twist angles, lattice reconstruction leads to strong strain fields in the moiré cells. In this paper, we combine an existing theory for lattice reconstruction with a quantum dynamic theory for interlayer excitons and their dynamics due to exciton-phonon scattering using a polaron transformation. The exciton theory is formulated in real space instead of the commonly used quasimomentum space to account for imperfections in the heterolayer breaking lattice translational symmetry. We can analyze the structure of the localized and delocalized exciton states and their exciton-phonon scattering rates for single phonon processes using Born-Markov approximation and multiphonon processes using a polaron transformation. Furthermore, linear optical spectra and exciton relaxation Green's functions are calculated and discussed. A P-stacked ${\mathrm{MoSe}}_2/{\mathrm{WSe}}_2$ heterolayer is used as an illustrative example. It shows excitons localized in the potential generated through the moiré-pattern and strain and a delocalized continuum. The exciton-phonon relaxation times vary depending on the strain and range from subpicoseconds up to nanoseconds.

Figure 1

(a) Illustration of the interlayer exciton formed between a hole in ${\mathrm{WSe}}_2$ and an electron in ${\mathrm{MoSe}}_2$, (b) Illustration of the monolayer Brillouin zone with the special points K, ${\text{K}}^{\prime }$, Q and $\mathrm{\Gamma }$ and the reciprocal-lattice vectors $G_i$.

Figure 2

For the ${\mathrm{WSe}}_2/{\mathrm{MoSe}}_2$ heterolayer example, (a) the displacement $|{\mathrm{u}}_{·}|$ of the bottom layer, (b) the absolute intralayer strain of the bottom layer $|\partial u_x/\partial x|+|\partial u_y/\partial y|$, (c) intralayer hole potential $V_h^{\mathrm{intra}}(·)$, and (d) interlayer hole potential $\frac{1}{2}{V}_h^{\mathrm{inter}}(·)$. The white circle marks the position of state 1 for visual orientation.

Figure 3

For the ${\mathrm{WSe}}_2/{\mathrm{MoSe}}_2$ heterolayer: (a) state index over energy for an $8\times 4$ and $1\times 1$ moiré supercell, (b) size of the exciton states in $x$ and $y$ direction, (c) dipole moment component for ${\sigma }_{+}$ and $\sigma$-heterostructure energy, (d) exciton lifetime at 20 K in polaron transformation including all scattering contributions (all), including only exciton-phonon scattering (phon.), excluding the optical phonon contribution (no opt.), at 5 K, without polaron transformation: standard Born Markov (BM) and only including the radiative contribution. The states are sorted into energy ranges with common properties. This includes the first localized states in area (A), the second localized states in area (B), the higher energy localized states in area (C), and the transition at the mobility edge to the delocalized states in area (D).

Figure 4

For the ${\mathrm{WSe}}_2/{\mathrm{MoSe}}_2$ heterolayer for an $8\times 4$ and $1\times 1$ moiré supercell, (i) state 1 at $-3.57\mathrm{meV}$, (ii) state 98 at 17.38 meV, (iii) state 210 at 30.12 meV, (iv) state 280 at 33.22 meV, (v) state 410 at 39.35 meV, (vi) state 1000 at 49.11 meV, (vii) state 2000 at 64.05 meV, and (viii) state 2320 at 67.88 meV (energy relative to the bound interlayer exciton energy). The white circle marks the position of state 1 for visual orientation.

Figure 5

For the ${\mathrm{WSe}}_2/{\mathrm{MoSe}}_2$ heterolayer example in an $8\times 4$ supercell, the absolute value of dipole element for the (a) ${\sigma }_{+}$ and (b) ${\sigma }_{-}$. Arbitrary units but scale to the ratio of microscopical elements from Ref. [50]. The white circle marks the position of state 1 for visual orientation.

Figure 6

The ${\mathrm{WSe}}_2/{\mathrm{MoSe}}_2$ heterolayer linear spectra: for a single realization of an $8\times 4$ supercell (a) spectrum at 20 K for ${\sigma }_{+}$ polarization in Born-Markov approximation with and without a convolution with Gaussian accounting for a typical spectral resolution of $0.15\mathrm{meV}$ and using a polaron transformation [panels (b)–(d) use polaron transformation], (b) at 20 K for ${\sigma }_{-}$ and ${\sigma }_{+}$ polarization and sum effectively collecting K and ${\text{K}}^{\prime }$ contributions, (c) for ${\sigma }_{+}$ polarization at different temperatures, (d) for ${\sigma }_{+}$ polarization comparison to a $1\times 1$ supercell (scaled), two different random supercells (run A and run B) and averaging over 20 realizations.

Figure 7

The exciton relaxation Green's function ${\mathrm{WSe}}_2/{\mathrm{MoSe}}_2$ heterolayer in an $8\times 4$ supercell using polaron theory at 5 K for a delay time between excitation and luminescence of (i) 100 fs, (ii) 1 ps, (iii) 10 ps, (iv) 40 ps, (v) 100 ps, (vi) 500 ps, (vii) 1 ns, (viii) 10 ns.

Figure 8

The exciton relaxation Green's function ${\mathrm{WSe}}_2/{\mathrm{MoSe}}_2$ heterolayer in an $8\times 4$ supercell using polaron theory at 20 K for a delay time between excitation and luminescence of (i) 100 fs, (ii) 1 ps, (iii) 10 ps, (iv) 40 ps, (v) 100 ps, (vi) 500 ps, (vii) 1 ns, (viii) 10 ns.