Mechanisms of interlayer exciton emission and giant valley polarization in van der Waals heterostructures

We perform a theoretical investigation about the valley dynamics of interlayer excitons (IXs) in $\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{\mathrm{S}}_2$ heterostructure with AB stacking configuration grown on ferromagnetic substrate. We find that the interlayer charge transfer process changes dramatically the IX emission as well as its valley polarization (VP). The magnetic proximity effect (MPE) exerted by ferromagnetic substrate, on the other hand, generates a zero-field valley Zeeman splitting, which suppresses the depolarization induced by electron-hole exchange interaction. Remarkably, unlike usual exciton-phonon scattering, which is harmful to the VP, the phonon-assisted intervalley scatterings between two split IX states in the different valleys foster an IX population imbalance in these states, giving rise to a giant VP. A combination of these experimentally tunable physical quantities (interlayer charge transfer rate, MPE intensity and exciton-phonon coupling strength) provides a promising tool for intriguing emerging magneto-optical emissions and their VPs. Considering a large family of layered materials, this study sheds light on the path for development of van der Waals heterostructures with detectable IX emissions and giant VP.

Figure 1

Schematic representation of (a) the relevant band edges of $\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{S}_2$ vdWs heterostructures with AB stacking, formation of DX in $\mathrm{W}{\mathrm{S}}_2$ layer and IX with the electron located in the CB of $\mathrm{Mo}{\mathrm{S}}_2$ and the hole in the VB of $\mathrm{W}{\mathrm{S}}_2$, and (b) the first Brillouin zone. The arrows indicate the spin states. The $\alpha$ valley of the heterostructure is formed by the K valley of $\mathrm{W}{\mathrm{S}}_2$ and the ${\mathrm{K}}^{\prime }$ valley of $\mathrm{Mo}{\mathrm{S}}_2$, while $\beta$ valley comprises the ${\mathrm{K}}^{\prime }$ valley of $\mathrm{W}{\mathrm{S}}_2$ and the K valley of $\mathrm{Mo}{\mathrm{S}}_2$.

Figure 2

(a) Spin and (b) layer resolved band structure of $\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{\mathrm{S}}_2$ heterostructure. The color bar in (a) indicates the spin state, while it displays the layer information in (b).

Figure 3

(a) Comparison of the electronic band structure nearby the K point of $\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{\mathrm{S}}_2$ heterostructure with AB stack obtained by DFT (lines) and the DFM (dots). [(b),(c)] Zoom around the band extremes in $\alpha$ valley. The color codes indicate (b) layer and (c) spin, respectively.

Figure 4

Schematic diagrams of (a) the electronic band structure around energy band extremes and (b) exciton energy levels and the corresponding scatterings of $\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{\mathrm{S}}_2$ heterostructure with AB stacking configuration. The blue and orange rectangles are designated for $\alpha$ and $\beta$ valleys, respectively. Solid (dashed) lines correspond to the energy levels in the presence (absence) of exchange field. The black, green, and orange arrows in (a) represent spin, angular momentum, and valley magnetic moments, respectively. In (b), vertical arrows illustrate intravalley transitions and relaxations, while the slanted arrows indicate intervalley scatterings. The corresponding transition rates are given by ${\gamma }_{ij}$ (nearby the arrows). $|0\rangle$ is the ground state, $|1\rangle$ ($|1^{\prime }\rangle$) and $|3\rangle$ ($|3^{\prime }\rangle$) are intralayer and interlayer exciton states in $\alpha$ ($\beta$) valley, respectively.

Figure 5

Schematic representation of five band model of IX dynamics in the $\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{\mathrm{S}}_2$ heterostructure. The blue and orange rectangles are designated for $\alpha$ and $\beta$ valleys, respectively. Horizontal-solid (dashed) lines represent exciton energy levels with (without) exchange field. Vertical arrows indicate intravalley transitions and relaxation processes, while the slanted arrows indicate intervalley scatterings. The Greek alphabet ${\gamma }_{ij}$ nearby the arrows stand for the corresponding transition rates. $|0\rangle$ is the ground state, $|1\rangle$ ($|1^{\prime }\rangle$) and $|3\rangle$ ($|3^{\prime }\rangle$) are intralayer and interlayer exciton states in $\alpha$ ($\beta$) valley, respectively.

Figure 6

Intervalley scattering rates of (a) ${\gamma }_{{33}^{\prime }}$ and (b) ${\gamma }_{3^{\prime }3}$ as a function of exchange field for $\alpha =5\times {10}^3{\mathrm{ps}}^{-1}{\mathrm{eV}}^{-3}$. $e-h$ and $P$ represent, respectively, the intervalley scattering rates due to the electron-hole exchange interaction and phonon-assisted processes. $T$ stands for the scattering rate for the case involving these two scattering channels.

Figure 7

(a) Single-particle energy at CBM in $\alpha$ (red) and $\beta$ (blue) valleys. Inset shows schematically the formations of the IXs in $\alpha$ (left) and $\beta$ (right) valleys, see the electron-hole pairs insides ellipses. K and ${\mathrm{K}}^{\prime }$ refer to the valleys of the individual monolayer TMDs. (b) Exciton energy in $\alpha$ (red) and $\beta$ (blue) valleys. (c) Valley splitting $VS$, defined by ${\mathrm{\Delta }}_{vs}$ = $|E_{{\sigma }^{+}}-E_{{\sigma }^{-}}|$, where ${\mathrm{E}}_{{\sigma }^{+}}$ and ${\mathrm{E}}_{{\sigma }^{-}}$ denote PL peak positions of ${\sigma }^{+}$ emission in $\alpha$ valley and ${\sigma }^{-}$ emission in $\beta$ valley, respectively. These VSs correspond to an effective $g$ factor of the IX being equal to 13.64.

Figure 8

VP of IX emissions in $\mathrm{Mo}{\mathrm{S}}_2/{\mathrm{MoSe}}_2/\mathrm{Mo}{\mathrm{S}}_2$ heterostructure as a function of out of plane magnetic field for (a) ${\sigma }^{-}$, (b) linearly, and (c) ${\sigma }^{+}$ polarized excitations. Solid-blue lines correspond to our theoretical prediction, while the red circles are experimental data from reference [29].

Figure 9

Exchange field dependence of IX emission spectra in the $\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{\mathrm{S}}_2$ heterostructure grown on ferromagnetic substrate under ${\sigma }^{+}$ polarized excitations at a pumping laser power P = 1 kW/${\mathrm{cm}}^2$ for (a) ${\tau }_{13}=70$ and (b) 0.07 ps, respectively. The blue and red PL peaks refer to the emissions from $\alpha$ valley (${\sigma }^{+}$ detection) and $\beta$ valley (${\sigma }^{-}$ detection), respectively. The PL intensities in (a) are scaled by a factor of 10 for visibility. The dashed lines only guide to eye.

Figure 10

PL intensity of IX emissions in $\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{\mathrm{S}}_2$ heterostructure on insulator magnetic substrate, excited by ${\sigma }^{-}$ laser field, as a function of exchange field for different values of interlayer charge transfer rate (${\tau }_{13}^{-1})$). Solid and dotted lines correspond to the IX emissions in $\alpha$ and $\beta$ valleys, respectively. Inset: ratio of IX to DX PL intensities for different values of ${\tau }_{13}$. Results obtained with ${\alpha }_{ph}=5\times {10}^3{\mathrm{ps}}^{-1}{\mathrm{eV}}^{-3}$ at T = 4.5 K.

Figure 11

PL intensity of IX emissions, excited by (a) ${\sigma }^{+}$, (b) linear, and (c) ${\sigma }^{-}$ lights, as a function of magnetic field at ${\gamma }_{13}={\left(70\text{ps}\right)}^{-1}$ and T = 4.5 K, for different values of ${\alpha }_{ph}$. Solid and dashed curves represent the PL intensities of IX emissions in $\alpha$ and $\beta$ valleys, respectively. ${\alpha }_{ph}$ is in the unit of ${\text{ps}}^{-1}{\mathrm{eV}}^{-3}$.

Figure 12

Valley polarizations of IX emissions as a function of exchange field in $\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{\mathrm{S}}_2$ heterostructure grown on a magnetic substrate at T = 4.5 K and ${\gamma }_{13}={\left(70\text{ps}\right)}^{-1}$. (a) For different excitation laser helicity, at ${\alpha }_{ph}=5\times {10}^3{\mathrm{ps}}^{-1}{\mathrm{eV}}^{-3}$. The blue and red curves correspond to circularly polarized ${\sigma }^{+}$ and ${\sigma }^{-}$ laser excitations, while the black one is obtained by linearly polarized laser excitation. (b) For different ${\alpha }_{ph}$. Green-dotted line separates positive and negative valley polarization, for eye guidance.

Figure 13

Comparison among the DFT (lines) and the $\mathbf{k}·\mathbf{p}$ (dots) bands for $\mathrm{W}{\mathrm{S}}_2$.

Figure 14

Comparison among the DFT (lines) and the $\mathbf{k}·\mathbf{p}$ (dots) bands for $\mathrm{Mo}{\mathrm{S}}_2$.