Polarization-Modulated Angle-Resolved Photoemission Spectroscopy: Toward Circular Dichroism without Circular Photons and Bloch Wave-function Reconstruction

Angle-resolved photoemission spectroscopy (ARPES) is the most powerful technique to investigate the electronic band structure of crystalline solids. To completely characterize the electronic structure of topological materials, one needs to go beyond band structure mapping and access information about the momentum-resolved Bloch wave function, namely, orbitals, Berry curvature, and topological invariants. However, because phase information is lost in the process of measuring photoemission intensities, retrieving the complex-valued Bloch wave function from photoemission data has yet remained elusive. We introduce a novel measurement methodology and associated observable in extreme ultraviolet angle-resolved photoemission spectroscopy, based on continuous modulation of the ionizing radiation polarization axis. Tracking the energy- and momentum-resolved amplitude and phase of the photoemission intensity modulation upon polarization axis rotation allows us to retrieve the circular dichroism in photoelectron angular distributions (CDAD) without using circular photons, providing direct insights into the phase of photoemission matrix elements. In the case of two relevant bands, it is possible to reconstruct the orbital pseudospin (and thus the Bloch wave function) with moderate theory input, as demonstrated for the prototypical, layered, semiconducting, transition metal dichalcogenide $2\mathrm{H}\text{−}{\mathrm{WSe}}_2$. This novel measurement methodology in ARPES, which is articulated around the manipulation of the photoionization transition dipole matrix element, in combination with a simple tight-binding theory, is general and adds a new dimension to obtaining insights into the orbital pseudospin, Berry curvature, and Bloch wave functions of many relevant crystalline solids.

Popular Summary

Electrons in solids are described by Bloch wave functions and the associated momentum-dependent energies known as the band structure. A large class of materials can be understood solely in terms of their band structure. But for certain materials known as quantum materials, microscopic properties of the wave function can have dramatic effects on macroscopic scales. Photoemission spectroscopy, where electrons are emitted upon absorbing light, is the most direct technique to investigate the band structure of crystalline solids, while accessing wave-function properties is difficult. We tackle this challenge by introducing a novel measurement methodology in photoemission spectroscopy.

We track the modulation of the energy- and momentum-resolved photoemission intensity upon continuous rotation of the light. Complemented with minimal theory input, this new scheme allows for the reconstruction of the Bloch wave function from experimental data, which we demonstrate on a semiconducting transition-metal dichalcogenide. Our work can be seen as the first condensed-matter “complete” photoionization experiments, in which one obtains both the amplitude and the phase of the complex photoemission matrix elements.

The presented methodology provides deep insights into quantum materials, including topological phenomena. We envision that our method can be used to track switching between different topological phases of quantum materials by measuring momentum-dependent wave-function properties.

Figure 1

Experimental setup and measurement protocol. (a) Experimental scheme of polarization-modulated, angle-resolved photoemission spectroscopy. A polarization-axis-tunable, linearly polarized, femtosecond XUV pulse (21.7 eV) is focused onto a bulk $2\mathrm{H}\text{−}{\mathrm{WSe}}_2$ crystal at an angle of incidence of 65° with respect to the surface normal, ejecting photoelectrons that are detected by a time-of-flight momentum microscope, allowing us to measure the energy- and momentum-resolved photoemission intensity as a function of the polarization axis angle $\theta \text{−}I(E,k_x,k_y,\theta )$. (b) Sketch of the first Brillouin zone of $2\mathrm{H}\text{−}{\mathrm{WSe}}_2$. (c) Example of three-dimensional raw data—band structure mapping [$I(E,k_x,k_y)$] using $p$-polarized XUV radiation, an associated cut through high-symmetry directions (${\mathrm{K}}^{\prime }\text{−}\mathrm{\Gamma }\text{−}\mathrm{K}$ and ${\mathrm{M}}^{\prime }\text{−}\mathrm{\Gamma }\text{−}\mathrm{M}$), and a constant energy contour ($E-E_{\mathrm{VBM}}=-0.25\mathrm{eV}$). (d) Two-dimensional cut through the 4D ARPES intensity $I(E,k_x,k_y,\theta )$: at different polarization-axis angles ($\theta$), at $E-E_{\mathrm{VBM}}=-0.25\mathrm{eV}$, and for given K and K’ valleys.

Figure 2

Wave-function properties near the valence band maximum, around $\mathrm{K}/{\mathrm{K}}^{\prime }$ valleys. (a) Sketch of the crystal cell of a monolayer ${\mathrm{WSe}}_2$ and the corresponding coordinate system (used for all three-dimensional plots in this figure). The purple arrows indicate the various polarization directions. (b) Relevant orbitals close to $\mathrm{K}/{\mathrm{K}}^{\prime }$, represented by a constant-value surface of the absolute value. The color coding indicates the real part. The Bloch state $|{\psi }_{\mathbf{k}\alpha }⟩$ in the crystal cell in (a) is well approximated by a superposition of the $|d_{\pm 2}⟩$ and $|d_{z^2}⟩$ orbitals, which defines the hybrid orbital $|{\varphi }_{\mathbf{k}\alpha }⟩$. (c) Plots of the hybrid orbital of the top valence band at selected momentum points close to the K and ${\mathrm{K}}^{\prime }$ [corresponding to the boxes in Fig. 1] valley. There is a one-to-one map of the complex wave-function coefficients $C_{0,\pm 2}(\mathbf{k})$ forming the hybrid orbital and the orbital pseudospin $\mathit{\sigma }(\mathbf{k})$; the corresponding texture is represented by the vector field. The thick gray line is a contour of maximum photoemission intensity for typical binding energy.

Figure 3

Valley-resolved polarization-modulated photoemission. (a,b) Constant energy contours (binding energy $E-E_{\mathrm{VBM}}=-0.25\mathrm{eV}$) for both crystal orientations (averaged over all polarization angles $\theta$). (c,d) Valley-dependent polarization-modulated photoemission signal, integrated over the square boxes shown panels (a) and (b). The $\theta$ dependence follows the generic form (2); the phase shift $\mathrm{\Phi }$ is illustrated by the dotted lines, and the value of $\mathrm{\Phi }$ for the $\mathrm{K}/{\mathrm{K}}^{\prime }$ valleys is given in the corresponding color. (e,f) Calculated polarization-angle modulation of the intensity [analogous to panels (c) and (d)]. (g,h) Analogous to panels (e) and (f), but excluding the $|d_{z^2}⟩$ orbital contribution. The direction of the light incidence is in the $x\text{−}z$ plane.

Figure 4

Energy- and momentum-resolved Fourier analysis of the polarization-modulated ARPES signals. (a,b) Fourier amplitude $|I_2(E,\mathbf{k})|$ (white-to-black color map, right subpanel) and phase $\mathrm{\Phi }(E,\mathbf{k})$ (blue-to-red color map, left subpanel) of the photoemission modulation, for $E-E_{\mathrm{VBM}}=-0.25\mathrm{eV}$. (c) Averaged phase [along a vertical cut and integrated over $k_x$ as indicated by the dashed box in panels (a) and (b)] extracted from both the experimental data in panels (a) and (b) and the theory. (d) Valley-integrated imaginary part $\mathrm{Im}[I_2(E)]$ of the Fourier amplitude (5), comparing experiment and theory at K (red) and ${\mathrm{K}}^{\prime }$ (blue), respectively. The dashed lines represent the corresponding interference contribution $\mathrm{Im}[I_2^{\mathrm{int}}(E)]$.

Figure 5

Circular dichroism without circular photons. (a,b) Absolute value of the CDAD extracted from the experimental data and theory via Eq. (9) at fixed binding energy $E-E_{\mathrm{VBM}}=-0.18\mathrm{eV}$ (a) and $E-E_{\mathrm{VBM}}=-0.25\mathrm{eV}$ (b), as a function of the angle ${\varphi }_k$ tracing the intensity. The inset illustrates how the angle ${\varphi }_k$ is measured along the contour of maximum intensity. (c,d) Theoretical reconstructed [${\tilde{I}}_{\mathrm{CD}}(E,\mathbf{k})$] and calculated CDAD [$I_{\mathrm{CD}}(E,\mathbf{k})$] as in (a,b). The arrows indicate the kink positions that can be used to determine sign changes (indicated by the shaded background).

Figure 6

Reconstruction of orbital pseudospin. (a,b) Pseudospin texture reconstructed from experimental (with emulated CDAD) data. The in-plane pseudospin is represented by the arrows, while the $z$ component is indicated by the color map. We present results for $E-E_{\mathrm{VBM}}=-0.25\mathrm{eV}$, where the reconstruction procedure is most stable. (c,d) Orbital pseudospin from the TB model. The black dashed lines indicate the region in which the intensity $I_{\mathrm{av}}(E,\mathbf{k})>0.1I_{\max }$ ($I_{\max }$ is the maximum intensity).

Figure 7

Valley-integrated orbital-resolved photoemission intensity $I_m(\theta )$ (analogous to Fig. 3) around the K and ${\mathrm{K}}^{\prime }$ points, respectively [same as in Fig. 3], for $m$ denoting the $d_{\pm 2}$ or the $d_{z^2}$ orbital. The intensity has been normalized to the respective maximum.