(TaSe)I has been studied extensively as a model quasi-one-dimensional system undergoing a CDW Peierls transition. As shown in Fig. 1a, (TaSe)I consists of chains of Ta atoms surrounded by Se atoms along the c-axis. The chains are bonded weakly by I atoms, forming a needle-like crystal that naturally cleaves along the (110) plane. The unit cell size of (TaSe)I is (a, b, c) = (9.5373(9), 9.5373(9), 12.770(2)) Å. In Fig. 1b we show the band structure of (TaSe)I, computed using density functional theory (DFT). The left panel shows the bands along high-symmetry lines, and is consistent with known literature. In the right panel, we show the bands along the experimentally relevant path and expressed with respect to the crystallographic axes (k, k, k) = (π/a, 0, k), with the N point at k = π/c = 0.246 Å, and the point at k = -π/c. In Fig. 2a we describe our coordinate system, and how it maps to the crystallographic axes, defining k, k as parallel/perpendicular with respect to the Ta chain direction. We see in the inset that there is a small spin-orbit splitting visible, exposing a KW fermion at the N TRIM point. To facilitate further theoretical calculations of photoemission intensity, we use this DFT input to construct a symmetry-inspired four-band tight-binding model. We use techniques from topological quantum chemistry to ensure this model reproduces the symmetry properties of the four bands closest to the Fermi level, as determined in the Topological Materials Database. Details of the model can be found in the SI. The spectrum of the tight-binding model is shown in Fig. 1c. We see good qualitative agreement with the DFT spectrum, although we have artificially increased the SOC strength in the tight-binding model for clarity.
We first characterized our samples by measuring the four-terminal electrical resistivity ρ as a function of temperature T with the current applied along the c-axis. Consistent with previous work, ρ increases with decreasing T and its logarithmic derivative shows a peak around the expected CDW transition temperature of T ~ 260 K (Fig. 1d). Moreover, the logarithmic derivative saturates around a value of 0.7 (10 K), corresponding to a gap size of 250 meV. This is also consistent with previous transport experiments.
To further characterize the CDW order, we also performed X-ray diffraction (XRD) as a function of temperature above and below T. Similar to observations from other scattering studies, satellite peaks corresponding to q emerge for T < T (see Fig. S5 in ref. ). The intensities of the CDW peaks follow the expected mean-field behavior with T ~ 260 K, as can be seen in Fig. 1e. We also performed synchrotron-based ARPES experiments with a photon energy of 50 eV on the same set of samples to compare with previous such experiments. Figure 1f shows the in-plane band dispersion at room temperature along the chain direction (in this case the ΓZ-direction). We observe the characteristic linearly dispersing valence bands with a minimum at Γ as theoretically predicted and seen by various ARPES experiments. Note that the valence band maximum is significantly below the chemical potential, indicating the intrinsic n-doping of (TaSe)I samples. At these high photon energies, the photoemission intensity for the conduction bands is much weaker compared to that of the valence bands. Nonetheless, the conduction band can be clearly resolved in the ARPES curvature plots, as shown in Fig. 1g and the overall shape of the bands is consistent with our laser ARPES experiments (discussed below).
Our laser ARPES experimental configuration is shown in Fig. 2 along with the bulk 3D BZ. For a photon energy of 6 eV, photoemission primarily originates from the plane corresponding to . This plane is close to the high symmetry N points, as indicated by the shaded plane in Fig. 2a. For our measurements, the sample is aligned such that the Ta chains are parallel to the analyzer slit which gives energy as a function of k for a given scan. The sample is then rotated so that k is close to the momentum of the TRIM point. Changes in k around this point are measured by using electronic deflection without sample rotation. We first show constant energy ARPES maps corresponding to energies E - E = -550 and -500 meV in Fig. 2b, c, respectively. As shown in Fig. 2b, two bands are observed around the KW node at the TRIM point, i.e., around the coordinate (k, k) = (-0.5, 0.5) in units of (-2π/c, ). These bands disperse outward with increasing kinetic energy, i.e., decreasing binding energy, indicating that they correspond to the conduction band of (TaSe)I near the KW node at the TRIM point when compared with first-principles calculations and our tight-binding model (Fig. 1b, c). The almost linearly dispersing 'V-shaped' conduction bands are seen more clearly in the energy vs. momentum (k) cut (see Fig. 2d for a plot along the white lines illustrated in 2c). Similar 'V-shaped' bands with relatively high velocities were also resolved in refs. and for momentum along k. On the other hand, for momentum along k (Fig. 2e, f), the bands have a relatively flat dispersion. These weakly dispersing bands along the k direction are a characteristic feature of (TaSe)I due to its one-dimensional nature and have been observed in a number of previous ARPES studies. We note that in our ARPES data, the spectral weight near the Fermi level is strongly suppressed compared to that of a reference sample (Au or BiSe) due to the incoherent nature of the band. The top of the occupied bands is about 100 meV below μ. As established in early ARPES works, this is due to a strong polaronic effect which makes the spectral weight near the chemical potential incoherent.
Figure 3a presents the band structure near the N points calculated from the DFT, where the binding energy of the N points is set to 0 eV. The band structure across k (Fig. 3b) resembles an hourglass shape. It leaves a periodic wavy projection that originates from the interchain coupling in the constant energy cuts (Fig. 3c-e). Overall, the calculation plots above E = 0 eV are consistent with the upper 'V-shaped' band and a small band splitting in the experiment (Fig. 2).
Having located the conduction bands originating from a predicted KW node in (TaSe)I, we now characterize their spin texture using helicity-dependent laser ARPES measurements. In general, KW nodes can be distinguished from conventional band-inversion Weyl nodes or the Rashba-like surface band splitting that can arise from iodine vacancy on the top surface by their spin texture. Construction of any Fermi surface enclosing a single KW fermion maps onto itself under time-reversal symmetry. Since time-reversal also flips spin, this constrains the electronic states on opposite sides of the Fermi surface around a KW node to have opposite spin. Note that this is in contrast to a conventional Weyl semimetal arising from band inversion, where there are no symmetry constraints on states at a single Fermi surface. The presence of additional rotational symmetries can further constrain the spins of states near the KW node; in the isotropic limit, we expect to see an approximately radial spin texture arising from the dominant k ⋅ σ term in the KW Hamiltonian. This was recently observed in spin-resolved ARPES experiments near the KW points in elemental tellurium. A similar approximately radial spin texture is expected around KW nodes of (TaSe)I, albeit with a stronger anisotropy due to its quasi-one-dimensional band structure.
Due to the nature of the spin texture around the observed KW point, we expect a distinctive asymmetry in the helicity-dependent photoemission from bands on either side of this point. Since the crystal is chiral and therefore lacks simple selection rules for ARPES transition matrix elements, we utilized an effective one-step model of photoemission to calculate the ARPES intensity from a tight-binding model. We used our tight-binding model to construct a large finite slab consisting of one region with zero chemical potential (the "inside" of the material) and one region with a large positive chemical potential (the "vacuum"). As an approximation, we modeled the final state of the photoelectron as a time-reversed low energy electron diffraction (TR-LEED) state, where to a first approximation we treated the material-vacuum interface as a step potential. Note that given the low photon energy (6 eV) used in our photoemission experiment, the quantitative features of the dichroic signal will depend on the final-state which can deviate from that of a simple free electron-like state. Thus, accurately computing the dichroic signal would require advanced Green's function and ab initio calculations of the final electron states in the band-structure. Such a calculation is beyond the scope of the present work. Nevertheless, any observed sign reversal with opposite helicity can provide strong evidence for the KW band, regardless of the details of final electron state. We included a phenomenological decay length for the final state inside the sample to account for scattering processes. Since the final state decays inside the sample, and the initial state decays outside the sample, our large finite slab gives a good approximation to a semi-infinite slab. The experimental geometry was taken into full account for the photoemission intensity. Full details on the calculation are given in the SI.
The results are illustrated in Fig. 4a-d. Figure 4c, d show the calculated photoemission intensity as a function of light-helicity on the left and right side of the point respectively, as shown in Fig. 4a. Note that there is a clear difference in the photoemission intensities for opposite helicities of light on one side of the point but not on the other. We attribute this to a combination of the chirality of the crystal and the incidence angle of the applied light. Since the crystal is chiral, electronic states on the two sides of the point are related by a twofold rotation symmetry and by time-reversal symmetry. Both of these symmetry operations change the polarization and incidence angle of the incoming light, leading to an asymmetry in the photoemission matrix elements.
To study if this is indeed the case, we modulated the light-helicity of the photoemitting beam using a quarter waveplate (angle denoted by θ). We isolate the bands around the point (Fig. 4e) and plot the integrated spectral intensity in each band (region of integration is shown by the dashed contours in Fig. 4f1-h1) as a function of θ. The results are shown in Fig. 4f2-h2. The photoemission helicity dependence on one side of the KW point (Fig. 4f2, g2) is significantly more asymmetric than the other side (Fig. 4h2) in agreement with theoretical predictions (Fig. 4c, d). In addition, the spin-split bands on opposing sides of the point have opposite helicity-dependence. There is a clear difference in the observed intensity between left and right circularly polarized light as shown in Fig. 4f2, g2 (The spin-split bands can clearly be identified in the EDC cuts in Fig. 4f2, g2). Additional observations of circular dichroism ARPES in (TaSe)I, at a photon energy of 47 eV, are reported in ref. . Our observation of helicity-dependence of spin-split bands near the point could potentially also explain the character of the CD-ARPES spectra discussed in that experiment. In addition, each of the spin-split bands on one side of the point has the same helicity dependence whereas that is not the case for the opposite side. There is a clear difference in the observed intensity between left and right circularly polarized light as shown in Fig. 4f2, g2 (The spin-split bands can clearly be identified in the EDC cuts in Fig. 4f2, g2). These observations are a direct consequence of the presence of KW fermions in (TaSe)I and might also explain the observed circular dichroism in other photoemission experiments on (TaSe)I using higher photon energies.
We note here that the θ-dependence of the photoemission intensity near a KW point of (TaSe)I is uniquely related to the radial (pseudo-)spin texture around the KW point and is quite different from other well-studied systems, such as topological insulators and strong Rashba SOC materials with a tangential spin texture. For those systems, circular dichroism (CD-ARPES) experiments are typically performed to measure the photoemission intensity difference between left and right circularly polarized light which in turn gives a measure of the pseudospin texture. In particular, ref. reports the CD-ARPES spectra of (TaSe)I at the iodine-deficient surface where the Rashba-type Dirac band splitting with tangential spin textures are expected. To compare our helicity dependent measurements on bulk bands of (TaSe)I with those on a system with a tangential pseudo spin texture, we studied the prototypical topological insulator BiSe (see Fig. S4 in ref. for more details) with our setup. As observed in previous measurements, we obtain a symmetric θ dependence of the photoemission intensity at points on either side of the Dirac point. This is in contrast to the asymmetric intensity observed in (TaSe)I for bands near the point.
The helicity dependence below the CDW transition is of interest, though experimentally hindered by strong polaronic effects near the Fermi level. KW fermions, protected at TRIM points in chiral space groups, are expected to persist across T, as (TaSe)I remains chiral above and below the transition. However, in contrast to the FSWP case in ref. , no CDW wavevector connecting TRIM points is observed, making the coupling of KW pairs unlikely.