Experimental Observation of Hidden Berry Curvature in Inversion-Symmetric Bulk $2H\text{−}{\mathrm{WSe}}_2$

We investigate the hidden Berry curvature in bulk $2H\text{−}{\mathrm{WSe}}_2$ by utilizing the surface sensitivity of angle resolved photoemission (ARPES). The symmetry in the electronic structure of transition metal dichalcogenides is used to uniquely determine the local orbital angular momentum (OAM) contribution to the circular dichroism (CD) in ARPES. The extracted CD signals for the $K$ and $K^{\prime }$ valleys are almost identical, but their signs, which should be determined by the valley index, are opposite. In addition, the sign is found to be the same for the two spin-split bands, indicating that it is independent of spin state. These observed CD behaviors are what are expected from Berry curvature of a monolayer of ${\mathrm{WSe}}_2$. In order to see if CD-ARPES is indeed representative of hidden Berry curvature within a layer, we use tight binding analysis as well as density functional calculation to calculate the Berry curvature and local OAM of a monolayer ${\mathrm{WSe}}_2$. We find that measured CD-ARPES is approximately proportional to the calculated Berry curvature as well as local OAM, further supporting our interpretation.

Figure 1

(a) Side view of bulk $2H\text{−}{\mathrm{WSe}}_2$. W and Se atoms are shown as gray and green balls, respectively. (b) Experimental geometry with the hexagonal BZ and spin-split bands. $K$ and $K^{\prime }$ valleys are color coded in blue and red, respectively. The upper and lower BZs represent the top- and sublayer BZs, respectively. Green dash-dot and brown dashed lines mark the photon incident and cut directions for $K\text{−}K$ and $K^{\prime }\text{−}{K}^{\prime }$ cuts, respectively. The same mirror plane was used for the two cases. ARPES data along $K^{\prime }$ to $K^{\prime }$ valley taken with (c) right- (RCP) and (d) left- (LCP) circularly polarized 94 eV light. (e) Circular dichroism (CD) obtained from the difference between (c) and (d). This cut corresponds to the brown line in panel (b). (f) RCP, (g) LCP, and (h) CD intensities for the $K$ to $K$ cut. (f)–(h) correspond to the cut shown by the green line in panel (b).

Figure 2

(a) Symmetric ($I_{\mathrm{NCD}}^S$) and (b) antisymmetric ($I_{\mathrm{NCD}}^A$) parts of the normalized CD, $I_{\mathrm{NCD}}$, plotted against the momentum of the MDC peak. The plot is made for only half of the momentum range of a cut, between the mirror plane and $K$ or $K^{\prime }$ point. $I_{\mathrm{NCD}}^S$ and $I_{\mathrm{NCD}}^A$ are even and odd functions of $k_{\perp }$. The filled (empty) squares indicate the upper (lower) spin-split bands.

Figure 3

Maps of (a) experimental $I_{\mathrm{NCD}}^S$, (b) calculated Berry curvature from tight binding model calculation, and (c) orbital angular momentum from the first principles calculation. For the map in (a), $I_{\mathrm{NCD}}^S$ is obtained from the upper valence band CD-ARPES data. The original data covers one third of the BZ data which is then symmetrized to cover the whole BZ. (d) $I_{\mathrm{NCD}}^S$ (diamond shape), Berry curvature (circle), and OAM (square) values along the $K^{\prime }\text{−}\mathrm{\Gamma }\text{−}K\text{−}M\text{−}{K}^{\prime }$.