Signatures of a time-reversal symmetric Weyl semimetal with only four Weyl points

Through intense research on Weyl semimetals during the past few years, we have come to appreciate that typical Weyl semimetals host many Weyl points. Nonetheless, the minimum nonzero number of Weyl points allowed in a time-reversal invariant Weyl semimetal is four. Realizing such a system is of fundamental interest and may simplify transport experiments. Recently, it was predicted that TaIrTe4 realizes a minimal Weyl semimetal. However, the Weyl points and Fermi arcs live entirely above the Fermi level, making them inaccessible to conventional angle-resolved photoemission spectroscopy (ARPES). Here, we use pump-probe ARPES to directly access the band structure above the Fermi level in TaIrTe4. We observe signatures of Weyl points and topological Fermi arcs. Combined with ab initio calculation, our results show that TaIrTe4 is a Weyl semimetal with the minimum number of four Weyl points. Our work provides a simpler platform for accessing exotic transport phenomena arising in Weyl semimetals.

A Weyl semimetal is a crystal which hosts emergent Weyl fermions as electronic quasiparticles. In an electronic band structure, these Weyl fermions correspond to accidental degeneracies, or Weyl points, between two bands [1][2][3][4][5] . It is well-understood that Weyl points can only arise if a material breaks either spatial inversion symmetry, I , or time-reversal symmetry, T [6][7][8][9] . At the same time, in a Weyl semimetal, symmetries of the system tend to produce copies of Weyl points in the Brillouin zone. As a result, typical Weyl semimetals host a proliferation of Weyl points. For instance, the first Weyl semimetals observed in experiment, TaAs and its isoelectronic cousins, have an I breaking crystal structure, which gives rise to a band structure hosting 24 Weyl points distributed throughout the bulk Brillouin zone [10][11][12][13][14][15][16][17] . However, most of these Weyl points can be related to one another by the remaining symmetries of TaAs, namely two mirror symmetries, C 4 rotation symmetry and T . In the Mo x W 1−x Te 2 series, which has recently been under intensive theoretical and experimental study as a Weyl semimetal with strongly Lorentz-violating, or Type II, Weyl fermions, mirror symmetry and T relate subsets of the eight Weyl points [18][19][20][21][22][23][24][25] . As another example, according to calculation, the Weyl semimetal candidate SrSi 2 hosts no fewer than 108 Weyl points, copied in sets of 18 by three C 4 rotation symmetries 26 . However, as we review below, it is well-known that the minimal nonzero number of Weyl points allowed is 4 for a T invariant Weyl semimetal. Realizing such a minimal Weyl semimetal is not only of fundamental interest, but is also practically important, because a system with fewer Weyl points may exhibit simpler properties in transport and be more suitable for device applications.
Recently, TaIrTe 4 was predicted to be a Weyl semimetal with only four Weyl points 27 . It was further noted that the Weyl points are associated with Type II Weyl fermions, providing only the second example of a Type II Weyl semimetal after the Mo x W 1−x Te 2 series 18 . Moreover, the Weyl points are well-separated in momentum space, with substantially larger topological Fermi arcs as a fraction of the size of the surface Brillouin zone than other known Weyl semimetals. Lastly, TaIrTe 4 has a layered crystal structure, which may make it easier to carry out transport experiments and develop device applications. All of these desirable properties have motivated considerable research interest in TaIrTe 4 . At the same time, one crucial challenge is that the Weyl points and topological Fermi arcs are predicted to live entirely above the Fermi level in TaIrTe 4 , so that they are inaccessible to conventional angle-resolved photoemission spectroscopy (ARPES).
Here, we observe signatures of Weyl points and topological Fermi arcs in TaIrTe 4 , realizing the first minimal T invariant Weyl semimetal. We first briefly reiterate a well-known theoretical argument that the minimum number of Weyl points for a T invariant Weyl semimetal is four. Then, we present ab initio calculations showing a nearly ideal configuration of Weyl points and Fermi arcs in TaIrTe 4 . Next, we use pump-probe ARPES to directly access the band structure of TaIrTe 4 above the Fermi level in experiment. We report the observation of signatures of

Results
Minimum number of Weyl points under time-reversal symmetry. We first reiterate well-known arguments that four is the minimum number of Weyl points allowed in a T invariant Weyl semimetal. A Weyl point is associated with a chiral charge, directly related to the chirality of the associated emergent Weyl fermion. It can be shown that for any given band the sum of all chiral charges in the Brillouin zone is zero. Further, under T a Weyl point of a given chiral charge at k is mapped to another Weyl point of the same chiral charge at −k. This operation of T on a chiral charge is illustrated in Fig. 1a on the blue Weyl points with +1 chiral charge (the same arrow applies for the red Weyl points but is not drawn explicitly). Now, if an I breaking Weyl semimetal has no additional symmetries which produce copies of Weyl points, then the minimum number of Weyl points is fixed by T symmetry and the requirement that total chiral charge vanish. In the simplest case, T will produce two copies of Weyl points of chiral charge +1, as shown in Fig. 1a. To balance these out, the system must have two chiral charges of −1, also related by T . In this way, the minimum number of Weyl points in a T invariant Weyl semimetal is four. This simple scenario is realized in TaIrTe 4 . The crystal structure of TaIrTe 4 is described by space group 31 (Pmn2 1 ), lattice constants a = 3.77 Å, b = 12.421 Å, and c = 13.184 Å, with layered crystal structure, see Fig. 1b. We note that TaIrTe 4 takes the same space group as Mo x W 1−x Te 2 , but has a unit cell doubled along b. To study where the Weyl points show up in TaIrTe 4 we present the electronic band structure along various high-symmetry directions, see Brillouin zone and ab initio calculation in Fig. 1c, d. Enclosed by the rectangular box along Γ − S is a crossing region between the bulk conduction and valence bands that gives rise to Weyl points. A more detailed calculation shows that the Weyl points have tilted over, or Type II, Weyl cones and that they live above the Fermi level, E F at k z = 0 27 . A cartoon schematic of the resulting constant energy contour at the energy of the Weyl points, E B = E W , is shown in Fig. 1e. The electron and hole pockets form Type II Weyl cones where they touch (red and blue marks). In this way, TaIrTe 4 has four Weyl points, the minimal number allowed in an I breaking Weyl semimetal. The overall electronic structure of TaIrTe 4 near E F is similar to Mo x W 1−x Te 2 , but we note that the role of the electron and hole pockets is reversed in TaIrTe  Unoccupied band structure of TaIrTe 4 by pump-probe ARPES.
Next, we use pump-probe ARPES to directly access the unoccupied band structure of TaIrTe 4 up to E B > 0.2 eV and we find excellent agreement with calculation. In our experiment, we use a 1.48 eV pump laser pulse to excite electrons into low-lying states  above the Fermi level, followed by a 5.92 eV probe laser pulse to perform photoemission 28 . We study E B −k x cuts near Γ, Fig. 2a-c, with key features marked by guides to the eye in Fig. 2d. Above the Fermi level, we see a crossing-like feature near E B~0 .15 eV, labeled 1, and two electron-like bands, 2 and 3, extending out above E F . Below the Fermi level, we observe a general hole-like structure consisting of three bands, labeled 4-6. As we shift k y off Γ, we find little change in the spectrum, suggesting that the band structure is rather flat along k y near Γ. However, we can observe that band 4 moves downward in energy and becomes more intense with increasing k y . We find an excellent match between our ARPES data and ab initio calculation, Fig. 2e-g. Specifically, we identify the same crossing-like feature (green arrow) and top of band 4 (orange arrow). We can also track band 4 in k y in calculation and we find that the band moves down and becomes brighter as k y increases, in excellent agreement with the data. The electron-like structure of bands 2 and 3 and the hole-like structure of bands 5 and 6 are also both captured well by the calculation. Crucially, however, we notice a shift in energy between experiment and theory, showing that the sample is hole-doped bỹ 0.05 eV. Lastly, we plot a constant energy k x −k y cut at E B = E F , where we see again that there is little dispersion along k y near Γ,   Fig. 3, to be discussed below. Our pumpprobe ARPES measurements allow us to directly measure the electronic structure above E F in TaIrTe 4 and we find excellent match with calculation.
Evidence for a Weyl semimetal in TaIrTe 4 . Now we demonstrate that TaIrTe 4 is a Weyl semimetal by directly studying the unoccupied band structure to pinpoint Weyl cones and topological Fermi arcs. Based on calculation, we fix k x near the expected locations of the Weyl points, k x ∼ kW = 0.2 Å −1 and we study E B -k y cuts in ARPES, see Fig. 3a-c, with key features marked by guides to the eye in Fig. 3d. We observe two cone features, labeled 1 and 2, connected by a weak, rather flat arc feature, labeled 3. We find that the cones are most pronounced at k x~kW , but fade for larger k x . Next, we pinpoint the Fermi arc as a small peak directly on the energy distribution curve (EDC) passing through k y = 0, see the blue curve in Fig. 3e, where the dotted black line is a fit to the surrounding features. We further track the arc candidate for k x moving away from k W and we find that the arc disperses slightly upwards, by about~10 meV, see also Supplementary  Fig. 1. This dispersion is consistent with a topological Fermi arc, which should connect the Weyl points and sweep upward with increasing k x 29 . We further pinpoint the upper Weyl cone on a momentum distribution curve (MDC) of the k x~kW cut, Fig. 3f. We find an excellent fit of the Weyl cone peaks to Lorentzians. Using this analysis, we can quantitatively track the dispersions of the Weyl cones and Fermi arc on the k x~kW cut, Fig. 3g and Supplementary Fig. 2. We note that for the upper Weyl cone we track the bands by Lorentzian fits on the MDC. However, for the Fermi arc and lower Weyl cone, the relatively flat dispersion requires us to track the bands in the EDCs. The EDC peak is challenging to fit, in part because the population distribution is strongly dependent on binding energy for a pump-probe ARPES spectrum. As a result, we track the Fermi arc and lower Weyl cone through a naive quadratic fit of the band peaks, again see Fig. 3g. We find that the peak trains are nearly linear, see also Supplementary Fig. 2. Based on our pump-probe ARPES spectra and ab initio calculations, we propose that TaIrTe 4 hosts two pairs of Weyl points of chiral charge ±1 at k x = ±k W , connected by Fermi arcs. This particular structure of two Weyl cones connected by a Fermi arc is arguably the simplest possible, Fig. 3h. We compare our results to calculation in greater detail, Fig. 3i-k. We can easily match the Weyl cones, the Fermi arc and an upper electron-like band, labeled 5 in Fig. 3d. However, we note that from calculation we expect bands 1 and 4 to attach to form a single band, while in our data they appear to be disconnected. We suggest that this discrepancy may arise because photoemission from part of the band is suppressed by low cross-section at the photon energy used in our measurement. In addition, we do not observe good agreement with the lower feature labeled 6 in our calculation, suggesting that this intensity may arise as an artifact of our measurement. At the same time, we consistently observe the broad featureless intensity below the Fermi level in both theory and experiment. Crucially, again we find a mismatch in the Fermi level. In particular, the Weyl points are expected at E B 0.1 eV, but we find the Weyl points at E B~0 .07 eV. We note that this sample was grown in a different batch than the sample of Fig. 2 and a comparison with calculation suggests that the second sample is electron doped by~30 meV, in contrast to a~50 meV hole doping in the first sample. We propose that the difference in doping of the two samples may arise because they were grown under slightly varying conditions. Lastly, we note that the k y position of the Weyl points shows excellent agreement in theory and experiment. In summary, we observe an arc which (1) terminates at the locations of two Weyl points; (2) appears where expected in momentum space, based on calculation; and (3) disperses upward with k x , as expected from calculation. The cones (1) are gapless at a specific k x~kW ; (2) fade for larger k x ; (3) appear where expected, based on calculation; (4) are connected by the arc; (5) show up in pairs only on k x~kW , so that there are ) four in the entire Brillouin zone. This provides strong evidence that TaIrTe 4 is a minimal Weyl semimetal with four Weyl points.

Discussion
We compare TaIrTe Fig. 4a-c. It is also interesting to note that the length of the Fermi arc in TaIrTe 4 is much longer as a fraction of the Brillouin zone than that of TaAs or Mo x W 1−x Te 2 , which can be seen clearly in the projections of the Weyl points on the (001) surface of all three systems, Fig. 4d-f. We see that our discovery of a Weyl semimetal in TaIrTe 4 provides the first example of a minimal I breaking, T invariant Weyl semimetal. One immediate application of our results is that TaIrTe 4 in pump-probe ARPES may provide a platform to observe the time dynamics of carrier relaxation in a Weyl semimetal. More broadly, our results suggest that TaIrTe 4 holds promise as a simpler material platform for studying properties of Weyl semimetals in transport and applying them in devices.

Methods
Pump-probe angle-resolved photoemission spectroscopy. Pump-probe ARPES measurements were carried out using a hemispherical electron analyzer and a modelocked Ti:Sapphire laser system that delivers 1.48 eV pump and 5.92 eV probe pulses at a repetition rate of 250 kHz 28 . The system is state-of-the start, with a demonstrated energy resolution of 10.5 meV, the highest among any existing femtosecond pumpprobe setup to date 31 . The time and energy resolution used in the present measurements were 300 fs and 15 meV, respectively. The spot diameters of the pump and probe lasers at the sample were 250 and 85 μm, respectively. The delay time between the pump and probe pulses was~106 fs. Measurements were carried out at pressures <5 × 10 −11 Torr and temperatures~8 K.
Single crystal growth and characterization. For growth of TaIrTe 4 single crystals, all the used elements were stored in an argon-filled glovebox with moisture and oxygen levels less than 0.1 ppm and all manipulations were carried out in the glovebox. TaIrTe 4 single crystals were synthesized by solid state reaction with the help of Te flux. Ta powder (99.99%), Ir powder (99.999%), and a Te lump (99.999%) with an atomic ratio of Ta/Ir/Te = 1:1:12, purchased from Sigma-Aldrich (Singapore), were loaded in a quartz tube and then flame-sealed under a vacuum of 10 −6 Torr. The quartz tube was placed in a tube furnace, slowly heated up to 1000°C and held for 100 h, then allowed to cool to 600°C at a rate of 0.8°C h −1 , and finally allowed to cool down to room temperature. The shiny, needle-shaped TaIrTe 4 single crystals, see Supplementary Fig. 3a, were obtained from the product and displayed a layered structure, confirmed by the optical micrograph, Supplementary Fig. 3b, and scanning electron microscope images, Supplementary Fig. 3c. The EDX spectrum displays an atomic ratio Ta:Ir:Te of 1.00:1.13(3):3.89 (6), consistent with the composition of TaIrTe 4 , Supplementary  Fig. 3d.
Ab initio band structure calculations. We computed electronic structures using the projector augmented wave method 32,33 as implemented in the VASP [34][35][36] package within the generalized gradient approximation 37 schemes. Experimental lattice constants were used 38 . A 15 × 7 × 7 Monkhorst-Pack k-point mesh was used in the computations with a cutoff energy of 400 eV. The spin-orbit coupling effects were included self-consistently. To calculate the bulk and surface electronic structures, we constructed a first-principles tight-binding model Hamilton, where the tight-binding model matrix elements are calculated by projecting onto the Wannier orbitals [39][40][41] , which use the VASP2WANNIER90 interface 42 . We used Ta d, Ir d, and Te p orbitals to construct Wannier functions, without performing the procedure for maximizing localization.
Data availability. All data is available from the authors on request.