Momentum-resolved linear dichroism in bilayer ${\mathrm{MoS}}_2$

In solid state photoemission experiments it is possible to extract information about the symmetry and orbital character of the electronic wave functions via the photoemission selection rules that shape the measured intensity. This approach can be expanded in a pump-probe experiment where the intensity contains additional information about interband excitations induced by an ultrafast laser pulse with tunable polarization. Here, we find an unexpected strong linear dichroism effect (up to 42.4%) in the conduction band of bilayer ${\mathrm{MoS}}_2$, when measuring energy- and momentum-resolved snapshots of excited electrons by time- and angle-resolved photoemission spectroscopy. We model the polarization-dependent photoemission intensity in the transiently populated conduction band using the semiconductor Bloch equations. Our theoretical analysis reveals a strongly anisotropic momentum dependence of the optical excitations due to intralayer single-particle hopping, which explains the observed linear dichroism.

Figure 1

(a) TR-ARPES setup: Geometry of the scattering plane with definitions of coordinate system $\{\widehat{\mathit{x}},\widehat{\mathit{y}},\widehat{\mathit{z}}\}$, photoelectron wave vector ${\mathit{k}}_f$, and polarization vectors $\{{\widehat{\mathit{e}}}_p,{\widehat{\mathit{e}}}_s\}$ of the light pulses. The orientations of the BL ${\mathrm{MoS}}_2$ lattice and the Brillouin zone (BZ) are shown for the measurement geometry. The dashed line on the BZ marks the measurement direction. (b) ARPES intensity along the $\overline{\mathrm{\Gamma }}\text{−}{\overline{K}}^{\prime }$ high-symmetry direction before arrival of the pump pulse ($\mathrm{\Delta }t<0$). (c) Intensity difference between the spectrum in (b) and the one obtained at the peak of the optical excitation at $\mathrm{\Delta }t=40$ fs with an $s$-polarized pump pulse. The dispersion of BL ${\mathrm{MoS}}_2$ has been overlaid (orange curves) together with the bulk continuum of Ag(111) [21] (blue hatched area) in (b) and (c). (d)–(f) EDCs of the intensity difference for different pump polarizations, fitted with Gaussian peaks for the CBM at (d) ${\overline{K}}^{\prime }$ and (e) $\overline{K}$, respectively, as well as (f) for the VBM at ${\overline{K}}^{\prime }$. (g) Time dependence of the estimated spectral weight $W$ from (d) fitted with a function composed of an exponential rise and a single exponential decay with the given time constants ${\tau }_i$.

Figure 2

(a) BL ${\mathrm{MoS}}_2$ dispersion around $\overline{K}$ determined within a $\mathit{k}·\mathit{p}$ model. The energy split VB states are labeled $v_1$ and $v_2$ and the CB states are labeled $c_1$ and $c_2$. The direct band gap of $E_g=1.9$ eV is marked by an arrow. (b) Optically induced population in the CB states obtained at $\mathit{q}=(0.16,0){\AA }^{-1}$ [indicated by the green dot in (d)–(f)] by solving the semiconductor Bloch equations for $p$-, $\sigma$-, and $s$-polarized light. (c) Photoemission intensity calculated at $q_y=0$ and $q_x$-averaged from $-0.22$ to $0.22{\AA }^{-1}$, as shown by the dashed black line in (d)–(f), as a function of pump pulse polarization $\theta$ for SL ${\mathrm{MoS}}_2$ (dashed green) and BL ${\mathrm{MoS}}_2$ (solid purple). (d) and (e) Momentum-dependent photoemission intensity calculated for (d) $p$- and (e) $s$-polarized pump pulses. The color scale in (d) also applies to (e). (f) Intensity difference obtained by subtracting the data in (e) from that in (d). The constant energy cuts in (d)–(f) were obtained at $E_q=1.15$ eV as shown via a dashed horizontal line in (a).

Figure 3

Spectral weight $W$ (markers) in the CB as a function of polarization angle $\theta$ measured at ${\overline{K}}^{\prime }$ (red) and $\overline{K}$ (purple). Curves represent fits of the energy- and $k$-integrated photoemission intensity incorporating our model for the excited-state population in Eq. (4) with the given prefactors and amplitude (Amp.).