The valley Hall effect has been understood in terms of the Berry curvature ; the symmetries in 1 ML 2H-MX2 cause a sign change in the Berry curvature as one goes from one valley to an inequivalent valley in the BZ . This allows us to understand the valley Hall effect in terms of pseudospins, and provides possibilities to control the pseudo-spins by an external field. On the other hand, the Berry curvature is expected to vanish in the bulk because the bulk TMDCs have an inversion symmetry. However, one can imagine that the valley Hall in each layer could be nonvanishing—only the sum vanishes. This may naturally introduce the concept of “hidden Berry curvature,” a nonvanishing Berry curvature localized in each layer. An analogous case can be found in the case of the hidden spin polarization proposed and measured recently . Existence of hidden Berry curvature implies that the topology could be determined by local field; the local symmetry determines the physics . While experimental verification of a hidden Berry phase in the Bloch state is highly desired, standard measurements such as quantum oscillation cannot reveal a hidden Berry phase because these measurements represent an averaged quantity, with hidden quantity invisible. However, if we use an external field or surface sensitive technique such as angle resolved photo emission, then the direct measurement of such a hidden Berry curvature may be possible. In fact, the surface sensitivity of ARPES has recently been utilized in the measurement of hidden spin polarization. Then, the question is if Berry curvature can be measured by means of ARPES. In this regard, we note a recent proposal, based on a tight-binding model calculation on a simple cubic lattice with s and p orbitals, dutch buckets that the nonAbelian Berry curvature is approximately proportional to the local orbital angular momentum in the Bloch state.

We use a similar approach and derived the relationship between OAM and the Berry curvature by using a three band, tight-binding model for in WSe2. We find that there is a linear relationship between OAM and the Berry curvature . Even though circular dichroism ARPES is not a direct measure of the OAM in the initial state in general, it has been shown that CD-ARPES bears information on the OAM. This fact can provide us a way to observe the existence of hidden Berry curvature by using CD-ARPES. In actual measurements, an important challenge lies in the fact that CD-ARPES has contributions other than the one from OAM. The most notable contribution comes from the geometrical effect, which is caused by a mirror symmetry breaking in the experimental geometry. Therefore, how we separate the Berry curvature and geometrical contributions holds the key to successful observation of the hidden Berry curvature. We exploit the unique valley configurations of TMDCs in the BZ to successfully disentangle the two contributions. Moreover, we find the hidden Berry curvature exists over a wide range in the BZ. These features are consistently explained within the first principles calculations and tight binding description. ARPES measurements were performed at the beam line 4.0.3 of the Advanced Light Source at the Lawrence Berkeley National Laboratory. Data were taken with left and right-circularly polarized 94 eV light, with the circular polarization of the light better than 80%. The energy resolution was better than 20 meV with a momentum resolution of 0.004 Å−1 . Bulk 2H-WSe2 single crystals were purchased from HQ graphene and were cleaved in situ at 100 K in a vacuum better than 5 × 10−11 Torr. All the data were taken at 100 K. Figure 1 shows the crystal structure of 2H-WSe2 for which the inversion symmetry is broken for a ML. In the bulk form of 2H-WSe2, the layers are stacked in a way that inversion symmetry is recovered.

In the actual experiment, the contribution from the top layer to the ARPES signal is more than that from the sublayer, as illustrated by the dimmed color of the sublayer. Figure 1 schematically sketches the electronic structure with the hexagonal BZ of WSe2. The low energy electronic structures of 2H-WSe2 ML was found to be described by the massive Dirac-Fermion model, with hole bands at K and K0 points. The bands are then spin split due to the coupling between the spin and OAM. In the bulk, layers are stacked in a way that K of a layer is at the same momentum position as the K0 of next layer. Consequently, spin and valley symmetries are restored due to the recovered inversion symmetry and any valley sensitive signal should vanish. On the other hand, the in-plane nature of the primary orbital character of the Bloch states around the K and K0 points and the graphenelike phase cancellation as well as the strong spin orbital coupling strongly suppress the interlayer hopping along the c axis and make them quasi-two dimensional. In that case, the valley physics as well as the spin-split nature maybe retained within each layer as illustrated in Fig. 1 by the top- and sublayer spin-split bands . In that case, one may be able to measure the hidden Berry curvature by using ARPES because it preferentially probes the top layer due to its surface sensitivity as, once again, illustrated by the dimmed color of the sublayer. Since the signal is preferentially from the top layer, the situation becomes as if ARPES data are taken from the topmost layer of WSe2, for which the inversion symmetry is broken. As mentioned earlier, it was argued that OAM is directly related to the Berry curvature, which indeed has opposite signs at the K and K0 points as OAM does. Then, the hidden Berry curvature may be measured by using CDARPES, which was shown to be sensitive to OAM. However, CD-ARPES has aforementioned geometrical contribution due to the broken mirror symmetry in the experimental geometry. In order to resolve the issue, we exploit the unique character of the electronic structures of TMDCs. The key idea is that, while the contribution from the geometrical effect is an odd function of k about the mirror plane, we can make the OAM contribution an even function. In that case, the two contributions can be easily isolated from each other.

To make the OAM contribution an even function, we use the experimental geometry illustrated in Fig. 1. The experimental mirror plane, grow bucket which is normal to the sample surface and contains the incident light wave vector, is precisely aligned to cross both K and K0 points. In such experimental condition, the Berry curvature is mirror symmetric about the experimental mirror plane and so is its contribution to the CD-ARPES. Then, the CD-ARPES is taken along the momentum perpendicular to the mirror plane , i.e., from K to K and K0 to K0 as shown in Fig. 1 by green dash-dot and brown dashed lines, respectively. We point out that we kept the same light incident angle for K-K and K0 -K0 cuts [note the color pair for the cut and light incidence in Fig. 1] to prevent any contribution other than those from Berry curvature and experimental chirality. Figures 1–1 show data along the K0 -K0 cut. The dispersion is very symmetric with the minimum binding energy at the K0 point as expected. However, the intensity varies rather peculiarly; there appears to be no symmetry in the CD intensity in Fig. 1. The K-K cut in Figs. 1–1 shows a similar behavior. While the dispersion is symmetric , the CD intensity in Fig. 1 at a glance does not seem to show a symmetric behavior. However, upon a close look of the CD data in Figs. 1 and 1, one finds that the two are remarkably similar; the two are almost exact mirror images of each other if the colors are swapped in one of the images. This is already an indication that the CD data reflect certain aspects of the electronic structure that are opposite at the K and K0 points, most likely the hidden Berry curvature of bulk 2H-WSe2.There are several aspects to be noticed from the figure. First of all, the symmetric part IS NCD for K-K and K0 -K0 cuts have almost identical behavior except their signs are reversed as seen in Fig. 2. As we will show later, this sign changes occurs precisely on the entire Γ-M line. In addition, we find that IS NCD is almost the same for the two spin-split bands . These observations on the behavior of IS NCD are consistent with what we expect from the Berry curvature; it is valley dependent but independent of the spin-split bands. On the other hand, the antisymmetric parts IA NCD from K-K and K0 -K0 cuts shown in Fig. 2 stay very similar to each other over the whole momentum range. The results indicate careful execution of our experiments and trustworthiness of our analysis. To study the behavior of IS NCD better, we expand the range from a cut to a map of IS NCD that covers the BZ. IS NCD of the upper spin-split band is obtained from the CD-ARPES data and plotted in Fig. 3. In addition, in order to understand the connection between IS NCD and Berry curvature as well as OAM, we performed tight binding analysis and first principles calculation for a ML WSe2. For the Berry curvature calculation, we consider the tight binding Hamiltonian based on the hybridization between a W d orbital and a Se p orbital . In the calculation, the parameters are adjusted until the dispersion fits the experimental one and previous TB result . Then, the Berry curvature of the upper band is calculated based on the TKNN formula and its map is plotted in Fig. 3. The momentum dependent local OAM is obtained by density functional theory calculation. The resulting Lz map is depicted in Fig. 3. The in-plane components of the Berry curvature and OAM are also calculated but are found to be negligible over the whole BZ and thus are not presented. One can immediately note that the three plots of experimentally obtained IS NCD, Berry curvature from TB analysis, and local Lz from DFT calculation show remarkably similar behavior; their signs are determined by the valley indices and change only across the Γ − M line. In addition, all of them retain significant values quite far away from the K and K0 points. Our observation shows that IS NCD can be considered as a measure of the OAM and Berry curvature. We also find that IS taken with different photon energies shows no qualitative difference . These observations support the notion that IS reflects an intrinsic property of the state, that is, OAM. For a more quantitative comparison, we plot IS NCD, Berry curvature and OAM along the high symmetry lines . Once again, IS NCD, Berry curvature and OAM show very similar behavior. As the Bloch states at the Γ and M points possess inversion symmetry, IS NCD, and Berry curvature as well as OAM are all zero. One particular aspect worth noting is their behavior near the Γ point. They are approximately zero near the Γ point but suddenly increase about a third of the way to the K or K0 point. Orbital projected band structure from TB calculation shows that this is when the orbital character of the wave function switches from out-of-plane dz2 and pz orbitals to in-plane dxy, dx2−y2 , px, and py orbitals. This behavior can be understood from the fact that the local OAM is formed by in-plane orbitals. These results strongly suggest that IS NCD is indeed representative of the Berry curvature and that the Berry curvature is closely related to the local OAM, at least for TMDCs. Characteristics of electron wave functions in the momentum space often play very important roles in macroscopic properties of solids. For example, topological nature of an insulator is determined by the characteristics of electron wave function at high symmetric points in the momentum space. The Berry curvature which is also embedded in the nature of the electron wave function in the momentum space determines the Berry phase and thus macroscopic properties such as spin and valley Hall effects.