Revealing Hidden Orbital Pseudospin Texture with Time-Reversal Dichroism in Photoelectron Angular Distributions

We performed angle-resolved photoemission spectroscopy (ARPES) of bulk 2H-${\mathrm{WSe}}_2$ for different crystal orientations linked to each other by time-reversal symmetry. We introduce a new observable called time-reversal dichroism in photoelectron angular distributions (TRDAD), which quantifies the modulation of the photoemission intensity upon effective time-reversal operation. We demonstrate that the hidden orbital pseudospin texture leaves its imprint on TRDAD, due to multiple orbital interference effects in photoemission. Our experimental results are in quantitative agreement with both the tight-binding model and state-of-the-art fully relativistic calculations performed using the one-step model of photoemission. While spin-resolved ARPES probes the spin component of entangled spin-orbital texture in multiorbital systems, we unambiguously demonstrate that TRDAD reveals its orbital pseudospin texture counterpart.

Figure 1

Modulation of the photoemission intensity upon time reversal ($\widehat{T}$). (a) Scheme of the experimental setup: A $p$-polarized femtosecond XUV (21.7 eV) pulse is focused onto a bulk 2H-${\mathrm{WSe}}_2$ crystal at an angle of incidence of 65° with respect to the surface normal, ejecting photoelectrons which are detected by a time-of-flight momentum microscope. (b),(c) Scheme of the experimental geometry and spin-orbital texture: The scattering plane (light purple plane) coincides with the crystal mirror plane ($\mathcal{M}$, green dashed line), which is along the $\mathrm{\Gamma }\text{−}M$ high-symmetry direction. The spin-orbital texture is schematically represented by the orbitals and the up or down spin state at each $K$ or $K^{\prime }$ valley. A 60° azimuthal rotation of the crystal (${\mathcal{R}}_{60°}$) yields the transformation of $K$ to $K^{\prime }$ valley (and vice versa) and is analogous to the action of the time-reversal operator; i.e., ${\mathcal{R}}_{60°}|{\psi }_{\mathbf{k}}^K⟩\equiv \widehat{T}|{\psi }_{\mathbf{k}}^K⟩=|{\psi }_{\mathbf{k}}^{K^{\prime }}⟩$. (d),(e) Constant energy contours for different energies, $E-E_{\mathrm{VBM}}$, measured for the two different crystal orientations described above ($I_{0°}$ and $I_{60°}$). The radius of each constant energy contour corresponds to $1.6{\AA }^{-1}$.

Figure 2

Extraction of the time-reversal dichroism in photoelectron angular distributions (TRDAD). (a),(b) $I_{0°}$ and $I_{60°}$, the CECs for $E-E_{\mathrm{VBM}}=-1.60\mathrm{eV}$, measured for two crystal orientations rotated by 60° with respect to each other. (c) $\mathrm{\Delta }I/I$, the raw normalized difference, i.e., $(I_{0°}-I_{60°})/(I_{0°}+I_{60°})$ between CECs shown in (a) and (b). (d),(e) $A_{0°}^{\mathrm{LDAD}}$ and $A_{\mathrm{LDAD}}^{60°}$, the “left-right asymmetries,” reflect the photoemission intensity asymmetry between the $k_x<0$ and $k_x>0$ hemispheres, for two crystal orientations, respectively, and are calculated using Eq. (1). (f) TRDAD [calculated using Eq. (2)] represents the component of $A_{\mathrm{LDAD}}^{0°/60°}$ which is antisymmetric upon time reversal (i.e., upon 60° azimuthal rotation of the crystal).

Figure 3

Comparison between experimentally measured and theoretically (KKR) calculated TRDAD. In the upper hemisphere of each panel, the raw photoemission intensity, and in the lower hemisphere, TRDAD. (a)–(d) Experimentally measured photoemission intensity and TRDAD. (e)–(i) Calculated photoemission intensity and TRDAD. (a),(e) $E-E_{\mathrm{VBM}}=-0.20\mathrm{eV}$, (b),(f) $E-E_{\mathrm{VBM}}=-0.75\mathrm{eV}$, (c),(g) $E-E_{\mathrm{VBM}}=-1.60\mathrm{eV}$, and (d),(i) $E-E_{\mathrm{VBM}}=-2.07\mathrm{eV}$. The radius of each constant energy contour corresponds to $1.6{\AA }^{-1}$.

Figure 4

Theoretical (KKR) investigations of the microscopic origin of the time-reversal dichroism. In the upper panels, we study the atomic-layer-resolved photocurrent and associated TRDAD. The signal is coming from (a) all layers (all L), (b) first Se atomic layer (Se), (c) first Se and W atomic layers (Se-W), and (d) first Se and W and second Se atomic layers (Se-W-Se). In the lower panels, we investigate the role of spin-orbit coupling on the photocurrent and associated TRDAD, by going from the fully relativistic to the nonrelativistic limit, upon modulating the speed of light in the calculations. (e) For standard speed of light (fully relativistic), which we defined as $1/c^2=1$. (f)–(i) For enhanced speed of light, $1/c^2=0.5$, $1/c^2=0.25$, and $1/c^2={10}^{-3}$ (nonrelativistic limit), respectively. All the constant energy contours are taken at $E-E_{\mathrm{VBM}}=-0.75\mathrm{eV}$, and their radii correspond to $1.6{\AA }^{-1}$.

Figure 5

Linking TRDAD and orbital pseudospin texture using a tight-binding framework. (a) TRDAD for an energy $E-E_{\mathrm{VBM}}=-0.20\mathrm{eV}$ computed using the tight-binding formalism described in Supplemental Material [27]. (b),(c) TRDAD and in-plane orbital pseudospin (black arrows) around the $K$ and $K^{\prime }$ valleys, respectively. The magenta dashed arrow represents the orientation of the orbital vector field, which can be derived from the photoemission matrix elements (more details can be found in Supplemental Material [27]). As shown in Eq. (7) of Supplemental Material [27], when the pseudospin texture is parallel (antiparallel) to the orbital vector field, we get a positive (negative) signal, while TRDAD vanishes if they are orthogonal. This relationship provides a direct link between the experimentally measured TRDAD and the in-plane orbital pseudospin.