Electrical control of excitons in van der Waals heterostructures with type-II band alignment

We investigate excitons in stacked transition-metal dichalcogenide layers under a perpendicularly applied electric field, herein ${\mathrm{MoSe}}_2/{\mathrm{WSe}}_2$ van der Waals heterostructures (vdWHs). Band structures are obtained with density functional theory (DFT), along with electron and hole wave functions in conduction and valence bands, respectively. A minimal continuum model, parametrized by the DFT results, is presented, allowing for the calculation of the excitonic states. Although the type-II nature of the heterostructure leads to a fully charge separated interlayer exciton on the ground states, our results show that moderate values of electric field produce more evenly distributed wave functions along the vdWH, namely, hybrid inter/intralayer exciton states, where both the interlayer exciton binding energy and, most notably, its oscillator strength are enhanced.

Figure 1

Side and top views of the crystal structure of a ${\mathrm{MoSe}}_2/{\mathrm{WSe}}_2$ heterostructure with (a) $AA$ and (e) $A{A}^{\prime }$ stacking orientation. Notice the red (Mo) atoms below the green (Se) ones in the $A{A}^{\prime }$ case (e). Inset: First Brillouin zone and its high symmetry points. The in-plane lattice constant $a$ is 3.33 Å, the Mo-Se distance is 2.54 Å, while interlayer distances are 7.10 and 6.72 Å for the $AA$ and $A{A}^{\prime }$ cases, respectively. The vertical arrow illustrates the positive direction of applied electric field $\vec{E}$. Band structures of an $AA$ stacked ${\mathrm{MoSe}}_2/{\mathrm{WSe}}_2$ heterostructure (b) without applied electric field, and with applied fields (c) $E=0.2$ V/Å and (d) $E=0.3$ V/Å are shown. (f)–(h) The same as (b)–(d), but for $A{A}^{\prime }$ stacking and $E=0.3$ V/Å and $E=0.4$ V/Å. The color scale indicates the relative wave-function projections to the ${\mathrm{MoSe}}_2$ (blue) and ${\mathrm{WSe}}_2$ (pink) layers.

Figure 2

(a) Two lowest conduction band states and highest valence band states (VBM) for an $AA$ stacked ${\mathrm{MoSe}}_2/{\mathrm{WSe}}_2$ heterostructure under a perpendicularly applied electric field at the $K$ point. (b) Valence band extrema at the $\mathrm{\Gamma }$ point as a function of the electric field. The same results, but for $A{A}^{\prime }$ stacking, are shown in (c) and (d), respectively. Color coding is the same as in Fig. 1, thus indicating the relative wave-function projections to the ${\mathrm{MoSe}}_2$ (blue) and ${\mathrm{WSe}}_2$ (pink) layers. (e) Electron and hole probability densities around the ${\mathrm{WSe}}_2$ and ${\mathrm{MoSe}}_2$ layers, for different values of applied electric field $E$, ranging from 0.1 to 0.4 V/Å in steps of 0.025 V/Å.

Figure 3

Normalized wave-function projections on the ${\mathrm{MoSe}}_2$ and ${\mathrm{WSe}}_2$ layers for (a) CBM and (b) VBM at the $K$ point, as well as for (c) VBM at the $\mathrm{\Gamma }$ point, assuming an $AA$ stacked vdW heterostructure. (d)–(f) The same results, but for the $A{A}^{\prime }$ stacking case. Lines and symbols represent results obtained, respectively, from the exciton Hamiltonian matrix in Eq. (2) [in the absence of electron-hole interactions, i.e., for $V_{ij}\left(\rho \right)\equiv 0]$ and from DFT calculations, according to Eq. (1).

Figure 4

(a) Exciton energy states $n=1–4$ and (b) their binding energies $E_b^{\left(n\right)}$ as a function of the applied field, assuming $AA$ stacking. The same results are shown in (c) and (d) for the $A{A}^{\prime }$ stacking case. Color code in (a) represents the electron-hole overlap from zero (gray) to unity (blue).