Abstract
It was recently reported that circular dichroism in angleresolved photoemission spectroscopy (CDARPES) can be used to observe the Berry curvature in 2HWSe_{2} (Cho et al. in Phys Rev Lett 121:186401, 2018). In that study, the mirror plane of the experiment was intentionally set to be perpendicular to the crystal mirror plane, such that the Berry curvature becomes a symmetric function about the experimental mirror plane. In the present study, we performed CDARPES on 2HWSe_{2} with the crystal mirror plane taken as the experimental mirror plane. Within such an experimental constraint, two experimental geometries are possible for CDARPES. The Berry curvature distributions for the two geometries are expected to be antisymmetric about the experimental mirror plane and exactly opposite to each other. Our experimental CD intensities taken with the two geometries were found to be almost opposite near the corners of the 2D projected hexagonal Brillouin zone (BZ) and were almost identical near the center of the BZ. This observation is well explained by taking the Berry curvature or the atomic orbital angular momentum (OAM) into account. The Berry curvature (or OAM) contribution to the CD intensities can be successfully extracted through a comparison of the CDARPES data for the two experimental geometries. Thus, the CDARPES experimental procedure described provides a method for mapping Berry curvature in the momentum space of topological materials, such as Weyl semimetals.
Introduction
Angleresolved photoemission spectroscopy (ARPES) is used to directly measure the band structure of solids and is an essential experimental tool for solid state physics research^{1,2,3}. In addition to the band structure, ARPES provides information on other aspects of the electronic structure. For example, ARPES with a spin detector can be used to obtain spin information of the initial states^{4,5,6,7}. Polarization dependent experiments can provide symmetry information on the initial states; initial states from, for example, \(p_x\) and \(p_y\) orbitals can show dramatically different ARPES intensities depending on the polarization of the incident light^{1,8}.
In recent years, there has been much interest in using circular dichroism (CD) in ARPES as a way to measure some aspects of initial states, such as the orbital angular momentum (OAM)^{9,10} or the Berry curvature^{11}. It is well understood that OAM plays an important role in spinsplit phenomena in systems without inversion symmetry^{12,13,14,15,16}, such as surfaces of solids and monolayer (ML) transition metal dichalcogenides^{17,18,19}. CDARPES has been utilized to obtain the crucial information on the electronic structures of such systems^{12,16,20,21}. While the final state of the photoemission process certainly has an effect on the CDARPES intensities^{22,23,24}, experimental results show that CDARPES is a rough measure of the OAM of the initial state^{22,23,24} if the photon energy is not too low^{25}.
Exploiting this feature in CDARPES measurements, information on the OAM and hidden Berry curvature of 2HWSe_{2} was recently obtained using CDARPES^{26}. An important aspect of this research was that the Berry curvature (or OAM) contribution to the CDARPES intensity could be isolated by decomposing the CDARPES intensity map into symmetric and antisymmetric components about the experimental mirror plane, which is perpendicular with respect to the crystal mirror plane of 2HWSe_{2}. The symmetric component was attributed to the OAM or Berry curvature contribution, since the electronic structure should be symmetric about the chosen experimental mirror plane set along K–\(\Gamma \)–\(K'\) in momentum space^{26}.
Results
Experimental geometry, including single crystal orientation, is especially important in this experiment. The crystal structure of the top atomic layer or ML of 2HWSe_{2} is a hexagonal lattice, as shown in Fig. 1a; there is a unique mirror plane in the crystal structure, as indicated in the figure. The experimental mirror plane is defined by the plane defined by the normal of the sample surface and the direction of incident light. The experimental mirror plane was set to be the same as the crystal mirror plane. Two experimental geometries are possible, according to the direction of incident light, as indicated by blue and red arrows in Fig. 1a. The experimental geometries using incident light described by blue and red arrows are regarded as geometryA and geometryB for convention, respectively. Notably, the signals from the top layer of bulk 2HWSe_{2} dominate the CDARPES data due to the surface sensitivity of ARPES^{5,26,27,28,29}; the corresponding momentum space view is shown in Fig. 1b. The mirror plane is oriented along the M–\(\Gamma \)–M direction, and the direction of incident light is indicated by blue and red arrows on the mirror plane in Fig. 1b. This experimental geometry differs from that used in previous work^{26}, in which the experimental mirror plane was rotated by 30° with respect to the crystal mirror plane, such that the experimental mirror plane is oriented along the K–\(\Gamma \)–\(K'\) direction.
We expanded on our previous CDARPES work on 2HWSe_{2} by focusing on a different mirror plane. Here, we report our CDARPES studies on 2HWSe_{2} with the experimental mirror plane parallel to the crystal mirror plane (Fig. 1a) or along the M–\(\Gamma \)–M direction in momentum space (Fig. 1b). Within the experimental constraint, there are two possible experimental geometries based on the incident beam directions, as shown by the blue and red arrows in Fig. 1a,b. The CDARPES values for the two geometries are nearly opposite to each other near the Brillouin zone (BZ) corner, whereas they are almost identical near the \(\Gamma \) point. These observations are well explained by accounting for the Berry curvature (or OAM) contribution to CDARPES. Our results thus indicate that the deviation from the median value between the two experimental geometries can be interpreted as the Berry curvature or OAM.
Figure 1c,d present the constant energy ARPES maps taken by RCP and by LCP incident light in geometryA, respectively. The binding energy (\(E_B\)) of all maps shown in Fig. 1 is 0.5 eV lower than the valence band maximum energy (\(E_{VBM}\)). CD signals, in which the intensity corresponds to the difference in the intensity taken by RCP (\(I_R\)) and that taken by LCP (\(I_L\)), are mapped in the momentum space (Fig. 1e). The antisymmetric function of the CD map for the experimental mirror plane is expected for this experimental geometry, given that the Berry curvature (or OAM) is also antisymmetric with regard to the experimental geometry. Figure 1f–h present the ARPES maps taken with RCP and LCP incident light in geometryB and the corresponding CD map, respectively; the upper left corner corresponds to the \(K'\) point in Fig. 1f–h and the K point in Fig. 1c–e. Remarkably, the CD signals at each corner of the BZ in Fig. 1h are almost opposite to those in Fig. 1e, whereas the CD signals near the center of the BZ are nearly the same. This can be explained by taking the Berry curvatures (or OAM) into account, given that the Berry curvatures (or OAM) are opposite at the K point and \(K'\) point, whereas the Berry curvatures (and OAM) are nearly zero around the \(\Gamma \) point. A detailed analysis of CD data was performed for ARPES cut data along the K–M–\(K'\) and \(K'\)–\(\Gamma \)–K directions in geometryA (blue lines in Fig. 1e) and along the \(K'\)–M–K and K–\(\Gamma \)–\(K'\) directions in geometryB (red lines in Fig. 1h).
Figure 2a,b present ARPES spectra taken by RCP and LCP light, respectively, in geometryA along K–M–\(K'\), as indicated by the dotted line in Fig. 1e. Figure 2d,e present ARPES spectra taken by RCP and LCP light, respectively, in geometryB along the \(K'\)–M–K direction, as indicated by the dotted line in Fig. 1h. Two parallel dispersive bands are evident in the spectra, of which the maxima are located at K and \(K'\). The energy difference between the upper and lower bands originates from atomic spin–orbit coupling of the W atom^{17,18,19}. The spin directions of the two bands are opposite, but the Berry curvature and OAM are the same, as expected from the massive Dirac–Fermion model^{17,18,19}. ARPES intensity clearly depends on the polarization of the incident light. Figure 2c,f present CDARPES intensity distributions for geometryA along K–M–\(K'\) and for geometryB along \(K'\)–M–K, respectively. The CD intensities of the two bands are similar at each momentum point, but the intensities are almost opposite between the CD for geometryA and that for geometryB; this is consistent with the constant energy maps shown in Fig. 1e,h.
Normalized CD intensities (\(I_{NCD}\)) as a function of momentum are shown in Fig. 3a for the upper band and in Fig. 3b for the lower band. \(I_{NCD}\) is obtained by (\(I_RI_L\))/(\(I_R+I_L\)), where \(I_R\) and \(I_L\) correspond to the ARPES intensity taken with RCP and LCP, respectively. \(I_{NCD}\) for the upper band along K–M–\(K'\) in geometryA, as indicated by the filled squares in Fig. 3a, has a positive value toward the K point from the M point. \(I_{NCD}\) exhibits a slight sign change beyond K and \(K'\) points, although it is difficulty to catch the fact in Fig. 2c due to very weak ARPES intensities. \(I_{NCD}\) for the upper band along \(K'\)–M–K (geometryB), indicated by the empty squares in Fig. 3a, exhibits a negative value toward the \(K'\) point from the M point and a positive value toward the K point from the M point, except very close to the M point, as we can also notice in Fig. 2f; sign changes beyond \(K'\) and K were also evident in the data. The \(I_{NCD}\)s in geometryA and B are roughly opposite, but not exactly. The \(I_{NCD}\) for the lower band in geometryA and geometryB are also similar to those of the upper band, but they are slightly weaker.
\(I_{NCD}\) consists of symmetric (\(I^S_{NCD}\)) and antisymmetric functions (\(I^A_{NCD}\)) about the experimental mirror plane (M point). Figure 3c,d present the \(I^S_{NCD}\)s for the upper and lower bands from two geometries, respectively. Figure 3e,f present the \(I^A_{NCD}\)s for the upper and lower bands from two geometries, respectively. As shown in the figures, the \(I^S_{NCD}\)s were close to zero, and \(I^A_{NCD}\)s were dominant components, regardless of the geometry or band. An asymmetric CDARPES distribution about the experimental mirror plane is a usual feature from solids^{23,30,31}, as the inversion symmetry along the surface normal direction is lifted on the surface of solids, which is similar to an oriented CO molecule system^{32,33}. The CDARPES contribution caused by the inversion symmetry breaking in the material surface can be called surface effects. However, it is surprising that the CD was nearly opposite between geometryA and B. Based on this finding, we believe that a substantial portion of \(I^A_{NCD}\) originates from the Berry curvature (or OAM), given that the CD signs follow the Berry curvature (or OAM) direction, as shown in Figs. 1e,h and 2c,f.
It is important to isolate the Berry curvature contribution to \(I^A_{NCD}\) from other contributions. The Berry curvature (or OAM) contribution to CDARPES should be exactly opposite between the normalized CDintensities along K–M–\(K'\) in geometryA and along \(K'\)–M–K in geometryB, because the Berry curvatures (or OAM) themselves are exactly opposite for K and \(K'\) points. We assume that other contributions, mainly the surface effects, are the same, regardless of the geometry. Then, the median values (red dotted lines in Fig. 3e,f) of \(I^A_{NCD}\)s from geometryA and B can be considered from the other contributions to \(I^A_{NCD}\)s. Additionally, this assumption is experimentally justified by CDARPES data near the \(\Gamma \) point, as shown in Figs. 4 and 5. The difference in \(I^A_{NCD}\) with respect to the median value is exactly opposite between the K–M–\(K'\) cut in geometryA and the \(K'\)–M–K cut in geometryB; this difference can be interpreted as the Berry curvature (or OAM) contribution to \(I^A_{NCD}\). Figure 3g presents the differences, along with the theoretical values of the Berry curvature and OAM. The differences are similar to the Berry curvature and OAM, except for the crossing at zero and the changing signs near 0.7 Å\(^{1}\).
The sign change of the difference of \(I^A_{NCD}\) from the median value is mainly due to the change in the final state character as the momentum of the photoelectron varies. We know that the wave function characters of the initial states near the K(\(K'\)) point change gradually and depend on the distance from the K(\(K'\)) point in the massive Dirac–Fermion model^{17,18,19}. The sign of CDARPES data can be reversed for the same initial states by only changing the final states, as indicated in the photon energy dependence of CDARPES^{22,23}.
Figure 4 presents the ARPES cuts and CDARPES data along the \(K'\)–\(\Gamma \)–K in geometryA, and along K–\(\Gamma \)–\(K'\) in geometryB, as indicated in Fig. 1. These cuts are special, in terms of the Berry curvature and OAM of the electronic states near the \(\Gamma \) point being almost negligible, compared with those of states near the K(\(K'\)) point. Therefore, the Berry curvature contribution to CDARPES data is expected to be almost zero near the \(\Gamma \) point. The CDARPES signals in both geometries are quite strong near the \(\Gamma \) point and exhibit a clear node at \(\Gamma \), indicating no symmetric component of the CD intensity. The CDARPES intensities near the K(\(K'\)) point from both geometries are much weaker than those near the \(\Gamma \) point, and the CDARPES intensities near the K(\(K'\)) point from geometryA are even weaker than those from geometryB.
Figure 5a–c present \(I_{NCD}\)s, \(I^S_{NCD}\)s, and \(I^A_{NCD}\)s, respectively. The symmetric components are negligible; the asymmetric components make up the majority of the \(I_{NCD}\)s (Fig. 5b,c). Remarkably, \(I_{NCD}\)s along \(K'\)–\(\Gamma \)–K in geometryA and along K–\(\Gamma \)–\(K'\) in geometryB are the same near the \(\Gamma \) point (Fig. 5a), and \(I^A_{NCD}\)s are, in turn, the same near the \(\Gamma \) point (Fig. 5c). Figure 5d presents the deviations of \(I^A_{NCD}\)s from the median value, along with the theoretical values of the Berry curvature and the OAM. The deviation is almost zero near \(\Gamma \) point and begin to have large value at the momentum at which the Berry curvature and the OAM are also about to increase from almost zero value. This provides experimental evidence that the deviation from the median value of \(I_{NCD}\)s in geometryA and B can be interpreted as the Berry curvature (or OAM) contribution. Although the Berry curvature and the OAM continually increase as they approach the K(\(K'\)) point, the deviation from the median value from CDARPES data seems to be almost constant away from the \(\Gamma \) point.
Discussions
Let us briefly touch upon the possible incident photon energy dependence in CDARPES or the final state effect. This is because one can wonder if the CDARPES pattern we obtained is seen only with the particular photon energy and a different photon energy may give us a different result. In such case, changing the photon energy will also change the CDARPES pattern and the CDARPES may not be related to OAM or the local Berry curvature. We would like to point out that incident photon energy dependent CDARPES has been performed on the same material^{26}. The results showed that CDARPES features related to the local Berry curvature are the same regardless of the photon energy. Even though it was for a different plane of incidence compared to the current one, the photon energy independence of the pattern provides a good reason to believe that the CDARPES pattern is proportional to OAM or the local Berry curvature. In some the other systems such as Bi\(_2\)Te\(_3\)^{24,34}, PtCoO\(_2\)^{16}and Au(111)^{25}, CDARPES results show a sign change. Yet, those results still show that node lines in CDARPES map remain the same except a special resonant channel is involved^{25}. In addition, characteristic patterns of CDARPES map cannot be explained without consideration of OAM^{25}. Therefore, we argue that the interpretation of the CDARPES intensity in this work should be robust although the data was taken only with a single photon energy.
CDARPES data on 2HWSe\(_2\) were taken with the crystal mirror plane set as the experimental mirror plane. Within the experimental constraint, there are two possible experimental geometries. We found that CDARPES data for the two geometries (geometryA and B) are almost opposite to each other near the BZ corners, and nearly the same near the \(\Gamma \) point. The experimental observations are well explained by accounting for the Berry curvature (or OAM) contribution to CDARPES. The Berry curvature (or OAM) contribution to the \(I_{NCD}\)s can be quantitatively extracted through an analysis that compares \(I_{NCD}\)s for the two geometries. Our results provide experimental evidence that the deviation from the median value between the two experimental geometries can be interpreted as the Berry curvature or the OAM. Our work may be applicable to observations of the Berry curvature or the OAM in topological materials, such as Weyl semimetals^{35,36,37}and Berry curvature dipole materials^{38,39,40,41,42,43,44}.
Methods
ARPES measurements were performed at the beam line 4.0.3 of the Advanced Light Source at the Lawrence Berkeley National Laboratory, equipped with a VG Scienta R8000 electron analyzer. The energy resolution was better than 20 meV, with a momentum resolution of 0.004 Å\(^{1}\). The degree of left and rightcircularly polarized (LCP and RCP, respectively) 94 eV light was better than 80%. Singlecrystal bulk 2HWSe\(_2\) was purchased from HQ Graphene (Groningen, Netherlands); the crystal was cleaved in situ at 100 K under high vacuum conditions (\(<1\times 10^{10}\) Torr).
The normalized CDintensity (\(I_{NCD}\)) is defined as the difference between the peak height of the energy distribution curve taken with RCP and LCP, divided by their sum. This normalization can be expressed as \(I_{NCD}\)=(\(I_RI_L\))/(\(I_R+I_L\)), as in previous papers^{9,10,16,20,22,23,25,26}. From this, the symmetric and antisymmetric components can be calculated as a function of the momentum k as \(I^S_{NCD}(k) = [I_{NCD}(k) + I_{NCD}(k)]/2\) and \(I^A_{NCD}(k) = [I_{NCD}(k)  I_{NCD}(k)]/2\), respectively.
The momentumdependent OAM \(L_z\) of ML 2HWSe\(_2\) can be determined from density functional theory calculations. For the calculations, we adopted the structural parameters of 2HWSe\(_2\) from an experiment^{45} to construct the ML 2HWSe\(_2\) structure and removed a WSe\(_2\) layer in the unit cell. We allowed for more than 20 Å spacing (vacuum) between neighboring ML 2HWSe\(_2\) layers to make the interaction between layers negligible. Calculations were performed using OpenMX package^{46,47,48}, which is based on pseudoatomic localized basis functions. The pseudoatomic orbital basis was chosen to be s2p2d1 for both W and Se atoms. The generalized gradient approximation Perdew–Burke–Ernzerhof functional^{49} was applied. We relaxed the electronic structure with the convergence criteria of 10\(^{7}\) Hartree, using an energy cutoff of 120 Ry and a \(10\times 10\times 1\) mesh for kpoint sampling. Spin–orbit coupling was considered. Using the linear combination of atomic orbitals (LCAO) coefficients, we calculated the momentumdependent \(L_z\) values along a certain direction in the BZ.
The tight binding model was applied to obtain the Berry curvature \(\Omega _z\) for the ML 2HWSe\(_2\). The parameters were fitted until the dispersion was consistent with ARPES measurements^{50,51} and previous results^{18,19}. For the Berry curvature calculation, we considered a tightbinding Hamiltonian based on dorbital hybridization of the W atom and the porbitals of the Se atom. The Berry curvature was then calculated using the Thouless–Kohmoto–Nightingale–den Nijs formula. We refer to a previous study^{52} in which the Berry curvature was calculated using a kmesh of \(80 \times 80\).
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Acknowledgements
This work was supported by the research program of Institute for Basic Science (Grant No. IBSR009G2). S. R. P. acknowledges support from the National Research Foundation of Korea (NRF) (Grant No. NRF2017R1D1A1B03036240). The Advanced Light Source is supported by the Office of Basic Energy Sciences of the US DOE under Contract No. DEAC0205CH11231. A preliminary CDARPES experiments at 4A1(PAL) were supported in part by MIST and POSTECH.
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S.C carried out the CDARPES measurements with the help of S.H, W.K, J.D.D and B.G.P; J.H and J.H.S performed the DFT calculations and J.H.P performed the tightbinding analysis; S.C analysed the CDARPES data with S.R.P and C.K; S.C, S.R.P and C.K wrote the manuscript; All authors discussed the results and reviewed the manuscript; S.R.P was responsible for the overall research direction and planning.
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Cho, S., Park, JH., Huh, S. et al. Studying local Berry curvature in 2HWSe_{2} by circular dichroism photoemission utilizing crystal mirror plane. Sci Rep 11, 1684 (2021). https://doi.org/10.1038/s41598020796726
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DOI: https://doi.org/10.1038/s41598020796726
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