Electronic properties of candidate type-II Weyl semimetal WTe₂. A review perspective

Density functional theory based calculations were performed using the VASP ab initio simulation package [60, 61]. Consistent with previous theoretical reports, the local density approximation within the PBE parametrization for the exchange-correlation potential was used, by expanding the Kohn–Sham wavefunction into plane-waves up to an energy cut-off of 400 eV [62]. The Brillouin zone was sampled on a 24  ×  12  ×  4 regular mesh. Spin–orbit coupling (SOC) was included in a self-consistent manner. Moreover, to go beyond the single particle picture, we used the LDA  +  U to treat electronic correlations within an energy independent Hartree description. Atomic positions of WTe₂ were fully relaxed starting from experimental data in [63].

WTe₂ crystallizes in an orthorhombic crystal structure with space group $Pmn{{2}_{1}}$. The lattice parameters are $a=3.477\,{{\rm }\mathring{\rm A} }$, $b=6.249\,{{\rm }\mathring{\rm A} }$, and $c=14.018\,{{\rm }\mathring{\rm A} }$ such that 4 formula units are incorporated in the unit cell. The $Pnm{{2}_{1}}$ space group is noncentrosymmetric; it has a mirror plane symmetry in its $bc$ plane, and a glide plane symmetry in the $ac$ plane followed by a (0.5, 0, 0.5) translation. It is important to note that the noncentrosymmetric structure is a necessary requirement for the existence of Weyl fermions in a nonmagnetic system like WTe₂ [43]. In addition, the absence of inversion symmetry gives rise to the possibility of the lifting up of spin degeneracy in the electronic structure. The W layer is sandwiched between two Te layers, and strongly linked by covalent bonds. These Te–W–Te layers are then stacked along the crystallographic $c$-direction, and held together via weak van-der Waals interactions. The crystal structure of WTe₂ is shown in figure 1(a). In the specific case of WTe₂, the atoms form a zig-zag chain (Te–W–Te) along the crystallographic $a$-direction, effectively creating an 1D substructure within the layers.

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We begin with the observed electronic structure by high resolution ARPES. Figure 1(b) presents the band dispersion measured along the $X-\Gamma -X$ direction using a photon probe energy of 68 eV and at temperature of 77 K. The main feature of interest, i.e. the existence of electron and hole pockets at the Fermi energy, the key behind the titanic MR of WTe₂, is clearly resolved and in agreement with the other APRES reports [64, 65]. Based on the observation of quantum oscillations [66] and first-principles calculations, there is an almost overlapping pair of electron and hole pockets at both sides of the Brillouin zone center (eight pockets in total). Each of the smaller pockets are completely inside the larger pockets. From the observed spectral weight, we cannot exclude the possibility of a small hole pocket at the Brillouin zone center, as reported in other studies [65]. The areas of the electron and hole pockets are nearly equal in our experimental spectra. In addition, the electron and hole concentration reported by the transport measurement [67] and quantum oscillations measurements [66] show nearly balanced carriers. The perfect electron and hole balance persists up to very high magnetic fields, unlike any other previously known material. In the case of bismuth, for instance, the MR saturates at about 40 T magnetic field [68]. WTe₂ is believed to be the first known system to show a perfectly balanced electron and hole concentration, and almost quadratic MR up to a field as high as 60 T [17] without any tendency of saturation. The closeness between the two pockets and the small overlap between the conduction band minima and valence band maxima might have played important role in the extreme MR of this system [64]. From the Fermi surface map in figure 1(d), it is seen that WTe₂ possesses a highly anisotropic electronic structure, which is reflected in its highly anisotropic magnetotransport properties. The Fermi surface is remarkably unidirectional; extended along the $X-\Gamma -X$ direction while highly restricted along the perpendicular $Y-\Gamma -Y$ direction. The scattering of electron and holes would inherit this unidirectionality; the electron and holes will prefer to flow along the $X-\Gamma -X$ direction i.e. along the direction of Te–W–Te chains.

From the semi-classical theory of MR, it can be shown that in order to obtain non-saturating behavior, perfect electron and hole balance is a necessary condition. The MR in WTe₂ increases quadratically with the applied field up to a very high magnetic field. The quadratic MR behavior is reminiscent of the perfect balance of electron and hole compensation in the material. According to the semiclassical two-band model, the conductivity tensor ($\widehat{\boldsymbol{\sigma }}$) is given by the following expression:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \widehat{\boldsymbol{\sigma }}=e\left[ \frac{n{{\mu }_{{\rm e}}}}{1+{\rm i}{{\mu }_{{\rm e}}}B}+~\frac{p{{\mu }_{{\rm h}}}}{1+{\rm i}{{\mu }_{{\rm h}}}B} \right]\nonumber \end{align} \tag{ 1 }$$

where $n$ and $p$ are the electron and hole carrier densities, and ${{\mu }_{{\rm e}}}$ and ${{\mu }_{{\rm h}}}$ are the carrier mobilities of electrons and holes, respectively. The inverse of the conductivity tensor gives the resistivity tensor ($\widehat{\boldsymbol{\rho }}$). The longitudinal resistivity and the MR are then given by the following expressions:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\rho }_{xx}}\left( B \right)=\operatorname{Re}\left( \widehat{\boldsymbol{\rho }} \right)=\frac{1}{e}~\frac{\left( n{{\mu }_{{\rm e}}}+p{{\mu }_{{\rm h}}} \right)+\left( n{{\mu }_{{\rm h}}}+p{{\mu }_{{\rm e}}} \right){{\mu }_{{\rm e}}}{{\mu }_{{\rm h}}}{{B}^{2}}}{{{\left( n{{\mu }_{{\rm e}}}+p{{\mu }_{{\rm h}}} \right)}^{2}}+{{\left( n-p \right)}^{2}}\mu _{{\rm e}}^{2}\mu _{{\rm h}}^{2}{{B}^{2}}}\nonumber \end{align} \tag{ 2 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle MR=\frac{\sigma {{\sigma }^{\prime }}{{\left( \frac{\sigma }{n}+\frac{{{\sigma }^{\prime }}}{p} \right)}^{2}}{{\left( \frac{B}{e} \right)}^{2}}}{{{\left( \sigma +{{\sigma }^{\prime }} \right)}^{2}}+{{\sigma }^{2}}{{\sigma }^{\prime 2}}{{\left( \frac{1}{n}-\frac{1}{p} \right)}^{2}}{{\left( \frac{B}{e} \right)}^{2}}}\nonumber \end{align} \tag{ 3 }$$

with $\sigma$ and ${{\sigma }^{\prime }}$ being the electrical conductivities of electrons and holes, respectively. From the above equation, in the $n\to p$ limit, we find that the MR is proportional to the square of the magnetic field,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm MR}=\frac{\sigma {{\sigma }^{\prime }}{{\left( \frac{B}{e} \right)}^{2}}}{{{n}^{2}}}=~{{\mu }_{{\rm e}}}{{\mu }_{{\rm h}}}{{B}^{2}}.\nonumber \end{align} \tag{ 4 }$$

The observation of nearly equal electron and hole pocket area qualitatively explains therefore the quadratic increase of MR as a function of the external magnetic field.

Since the MR effect of WTe₂ depends on the precise carriers balance, it is therefore highly dependent on the Fermi surface topography, and affected by the small details of the electronic structure. Such small changes in the Fermi surface can be induced by temperature, pressure, chemical doping or electrical gating, resulting in drastic changes of its MR behavior. It is worthwhile to mention that another type of non-saturating MR has been recently observed in Dirac semimetallic systems [41], where the MR is linear with the field. In the case of the Dirac semimetals, the backscattering of the carriers is prohibited and protected by the time reversal symmetry. As the magnetic field is applied, this protection is gradually lifted resulting in the rise of the electrical resistivity. The Weyl topological properties of WTe₂ might also be involved in this MR behavior.

The MR effect in WTe₂ is significantly reduced above the so-called 'turn on' temperature of 150 K. This 'turn on' behavior is not associated with any charge density wave, structural transition, or other types of electronic instabilities which are commonly observed in other TMD compounds [17]. The 'turn on' temperature is found to shift towards higher temperature with increasing applied magnetic field, which indicates that a scattering mechanism is likely in play behind this phenomenon. A temperature dependent ARPES study reported a reduction in the size of the hole pocket and an increase in the size of the electron pocket with increasing temperature [64]. The imbalance from the perfect electron–hole resonance condition at elevated temperature could be the reason of this 'turn on' behavior. A temperature induced Lifshitz transition, i.e. a change in the Fermi surface topology, is also reported at 160 K [51], around the same temperature range of 'turn on' point. At the onset of the Lifshitz transition, the hole pockets completely disappear and the electron pockets expands in the Brillouin zone. This restructuring of the Fermi surface is caused by the shift of the chemical potential with the temperature. The same study also reported a change in the slope of the temperature dependent thermoelectric power about the same temperature. At lower temperature, the MR scales with the Kohler's exponent of 1.98, very close to the quadratic behavior. However, as the temperature is raised the Kohler's scaling breaks down in the temperature range 70–140 K [51, 69]. These evidences further corroborate that the MR related phenomenon crucially depends on the precise balance of the electron–hole resonance and on the tiny details of the Fermi surface. It has been reported from temperature dependent Hall measurements that the hole carrier density suddenly increases below 160 K. The electron density is found to have a significant reduction below 50 K, leading to a nearly perfect electron and hole resonance at low temperature [67].

Likewise to the temperature induced Lifshitz transition discussed above, a pressure induced Lifshitz transition is also reported in WTe₂ [52], accompanied by the suppression of the MR effect and the appearance of superconductivity. This transition also occurs in the absence of any associated structural phase transition. With increasing applied pressure, the MR effect is found to be gradually suppressed and eventually completely disappears at 10.5 GPa, and the superconductivity arises exactly at the same pressure with a critical temperature ${{T}_{{\rm c}}}$  =  2.8 K. The superconducting temperature reaches a respectable value of 6.5 K at 13.0 GPa, after that the ${{T}_{{\rm c}}}$ decreases with increasing pressure. When a higher pressure is applied, there is a significant reduction in the lattice constant $c$ without exhibiting any structural phase transition. On the other hand, the in-plane lattice constants are not significantly affected by the application of external pressure. Electronically, since the 5p  orbitals of Te and 5d orbitals of W are spatially extended, the band structure is prone to be very sensitive to any change to the lattice constants, and hence to the external pressure. Therefore, the Fermi surface reconstruction could be related to the anisotropic transformation of the Bravais lattice upon change of pressure. From the high-pressure Hall effect measurement, it is found that the density of hole carriers decreases with application of pressure while the carrier density of electrons increases, similar to what is observed in the case of the temperature driven Lifshitz transition. The Hall coefficient in this material is positive at ambient pressure, and gradually decreases with the applied external pressure. A change of the sign of the Hall coefficient is observed at the critical pressure of 10.5 GPa, corroborating the idea of a quantum phase transition associated with the drastic change in Fermi surface topology. Such a quantum phase transition is also in agreement with the observed Shubnikov-de-Haas quantum oscillations [70]. The pressure induced superconductivity is also verified by transport measurements [71], where the pressure is applied without any pressure-transmitting medium.

In order to better understand the observed electronic structure, we have calculated the electronic structure of WTe₂ by first-principles calculations. The analysis is performed by assuming WTe₂ as pure 2D material, i.e. by looking at the bulk band structure for ${{k}_{z}}=0$. We see that our DFT bands well reproduce the electron and hole pockets and other details of the band dispersion [72] (figure 2(a)). However, a closer inspection into the calculated bands reveals some differences with the observed spectra. For instance, the momentum positions of electron and hole pockets in the calculated bands are significantly offset from the Brillouin zone center compared to our experimental spectra. In addition, the maximum binding energy of the electron pocket is larger than the observed spectra by more than a factor of two. Next, we consider the importance of SOC in the better agreement between the experimental and theoretical spectra. WTe₂ consists of two heavy elements, i.e. tungsten and tellurium, therefore strong influence of SOC is expected in the formation of the band structure of the system. In figure 2(b) we show the calculated bandstructure upon the incorporation of SOC as superimposed over the experimental spectra. It is evident that it results in a remarkable better agreement between the electron and hole pocket locations in the Brillouin zone. The changes in the theoretical bandstructure with incorporation of SOC highlights the fact that relativistic effects are significant in this system.

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Despite the fact that we obtain a better agreement between the calculated and experimental spectra after the inclusion of SOC, a quantitative agreement is not achieved yet. It is important to note here that there exists a non-negligible band dispersion in the direction perpendicular to the layers i.e. along the ${{k}_{z}}$ direction. However, the band dispersion along the $X-\Gamma -X$ direction cannot be fully reproduced by considering any specific ${{k}_{z}}$ value. It is also noted that even though the bulk calculation reproduces the details of various Fermi surface features, the calculated Fermi surface turns out to be always much larger than the experimental one [73]. This suggests that constraining ourselves to bulk calculations is not sufficient to explain the electronic properties of this system. We also note that the experimentally observed low energy final states of TMDs may differ from the free-electron dispersions, exhibiting non-parabolic dispersions and incorporating number of plane waves with different k_z values [74].

Next, we discuss the role of dimensionality in the formation of perfect electron and hole balance in this system. After the discovery of graphene and TIs, the important role of dimensionality in the formation of topological states in those systems is seen in a new light. For example, the complete formation of topological surface states in TI requires at least six quintuple layers [31]. In addition, the spin texture of the topological surface states evolves with the number of layers [32]. In particular, dimensionality is also very critical in TMD systems. We have already mentioned that bulk MoS₂ exhibits an indirect bandgap semiconductor behavior, while it becomes a direct bandgap semiconductor in the monolayer. Therefore, we can conjecture that these materials are not truly 2D, nor are they traditional bulk systems; they are 'less than 3D' or in-between 2D and 3D.

In order to elevate the agreement between theory and experiment at the quantitative level, here we have adopted a more realistic modeling where we explicitly take into account the contribution of each individual layer and project them over the surface Brillouin zone. Our model is based on a supercell made by van der Waals-bonded WTe₂ planes stacked along the crystallographic c-direction. We have separated the individual contribution of each Te–W–Te plane from the large number of bands in the supercell calculations by projecting the electronic band structure onto the atoms belonging to a given plane only. It must be noted that the actual spectra measured by ARPES represents a weighted spectral intensity of various layers, with the intensity of the deeper layers attenuated exponentially because of the finite mean free path of the photoemitted electrons inside the material. Such a direct connection of the layer-resolved dispersions with photoemission holds only for purely 2D states, where the interference between the layers vanishes. In figure 2, we have superimposed the supercell bands over the experimental spectra. On panels 2(c)–(e), we present the projected band dispersions corresponding to individual first, second, and third layer, respectively. In this way, we are able to disentangle the role of each individual layer in the formation of various band structure features. In the plots of the supercell electronic states, the size of the circles is proportional to the contribution of that particular plane to the final spectra; the bigger the circle, the more that particular Te–W–Te layer contributes to the final spectral feature. Now comparing the band projections from the different topmost layers, we see that the electron pocket which is located at 0.35 ${{{{\rm }\mathring{\rm A} }}^{-1}}$ from the zone center $\Gamma$ is found already from the top layer, indicating the electron pocket has a strong surface character. On the other hand, the hole pocket at 0.2 ${{{{\rm }\mathring{\rm A} }}^{-1}}$ arises only when we consider the third layer from the top. This indicates that the hole pocket represents a more bulk-like feature. The important point to note here from our layer-resolved density functional theory calculations is that they not only provide for a better agreement with the observed spectra, but they also further suggest that there is a profound connection between the dimensionality and the formation of electron and hole pockets. The electron and hole balance is achieved only when at least three Te–W–Te layers are considered from the top surface, and the balance is maintained within the bulk. Comparing with experiments, recent laser based micro-focused ARPES studies done on encapsulated mono-layer WTe₂ flakes, reported the presence of an electron pocket just touching the Fermi level, while no hole band is observed [75]. The latter only shows up at thicknesses larger than the bilayer system. Moreover, the absence of the hole pocket in single layer WTe₂ and the presence of a full gap in the whole Brillouin zone has been also reported by Scanning Tunneling Microscopy measurements [76].

In accordance to the bulk calculations, there is a non-negligible ${{k}_{z}}$ dispersion of the electronic states in this layered material. We have probed such band dispersion along the direction perpendicular the stacking of the layers by soft-x-ray ARPES. This investigation solidifies our conclusion from the layer resolved DFT calculations that one needs more than three layers to recover the electron–hole balance in this system.

In order to experimentally access the bulk electronic structure of layered WTe₂, we have performed soft-x-ray ARPES measurements. By changing the photon probe energy from 400 eV to 800 eV and exploiting the photon energy dependence of electrons' mean free path, i.e. the depth dependence, we are able to map the electronic structure along the direction perpendicular to the WTe₂ layers over nine Brillouin zones. The ${{k}_{z}}$ dependence of the Fermi surface can be mapped from the photon energy dependence spectra by using the following expression: ${{k}_{z}}={{\left[ \left( \frac{2{{m}_{{\rm e}}}}{{{\hbar }^{2}}} \right)\left( {{E}_{k}}{{\cos }^{2}}\theta -{{V}_{0}} \right) \right]}^{1/2}}+{{\kappa }_{{\rm ph}}}$, where ${{V}_{0}}$ is inner potential of the system [55], and ${{\kappa }_{{\rm ph}}}$ is momentum component of photon along the surface normal. Use of high energy soft-x-ray photon as ARPES probe is very crucial for the better resolution of k_z dispersion. The increase of the photoelectron mean free path in soft-x-ray energy range translates, by the Heisenberg uncertainty principle, to reduced intrinsic uncertainty of k_z allowing thereby more accurate 3D band mapping [77]. In order to obtain the better insight of the 3D fermi surface of WTe₂, both the ${{k}_{x}}$ versus ${{k}_{y}}$ and ${{k}_{x}}$ versus ${{k}_{z}}$ Fermi surfaces are presented in figure 3. Panel (a) shows the ${{k}_{x}}$ versus ${{k}_{y}}$ Fermi surface and the iso-energy cuts at deeper binding energy values. The calculated ${{k}_{x}}$ versus ${{k}_{z}}$ is shown in panel (b), while the experimental panel (c) shows the ${{k}_{z}}$ versus ${{k}_{x}}$ Fermi surface map obtained from soft-x-ray ARPES measurement. We observe a clear 3D evolution of the bandstructure. Several important features are worthy of further consideration.

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The observed periodicity of the ${{k}_{z}}$ evolution matches twice that of the Brillouin zone, i.e. as if only half of the length of the lattice vector along the crystallographic c-direction were considered. Note that the WTe₂ unit cell consists of two non-equivalent Te–W–Te layers. Such a double periodicity is not uncommon in ARPES spectra. It has been already observed in other nonsymmorphic crystal structures such as CrO₂ [78], graphite [79], and in TMDs such as 2H-WSe₂ [80] and 2H-NbSe₂ [81] where the photoemission selection rules determines the final state symmetry. Because of truly free-electron character of high-energy final states, the nonsymmorphicity effects are particularly clear in soft-x-ray ARPES spectra.

Another crucial feature we infer from our soft-x-ray data, differing from more surface sensitive VUV-ARPES, is the absence of any quasiparticle weight at the Brillouin zone center. The extremal orbits estimated from the quantum oscillation study [44] matches well with our VUV-ARPES data, where the areas of the electron and hole pockets are nearly equal. However, the Fermi surface measured using soft-x-ray photons shows that the area of the hole pocket is sensibly larger than that of the electron pocket. Therefore, it is essential to invoke a pronounced ${{k}_{z}}$ dispersion of the electronic band structure to explain the perfect electron–hole compensation in this material. Moreover, this observation supports the magnetotransport results, which bear evidence of three dimensionality in order to explain the extremely large effect [82]. A detailed analysis of our soft-x-ray study is presented in [83].

The investigation of spin-polarization properties of a material is important for two main reasons. Firstly, it gives a quantitative measure of the effect of SOC present in the material. Secondly, materials with large spin polarization are desirable for various spintronics applications. A large spin polarization is reported for the semiconducting TMD WSe₂ [84] as a result of the local asymmetry of the Se–W–Se layers. Despite WSe₂ possesses a global centrosymmetric crystal structure, the alternating layers locally carry a net opposite dipole moment, and therefore, the local spin texture is modulated. In the specific case of WTe₂, the crystal structure does not possess a center of inversion. Spin polarized electronic bands are expected in a non-centrosymmetric system consisting of two heavy elements, foreseeing a strong SOC effect. We have also seen that SOC plays important role in explaining various band features in WTe₂ upon comparing with the theoretical results. Although the magnitude of the spin polarization depends on several material factors, e.g. orbital character, bandgap, crystalline electric field etc, it primarily relies on the strength of SOC. Therefore, it is natural to expect significant spin polarization in this system. We measured the spin resolved EDCs at a number of k-points in the Brillouin zone. The probed k-point locations are indicated in figure 4(d) by colored markers on the Fermi surface; edge of hole pockets on the opposite sides of the Brillouin zone center (with green circle and black triangle, respectively); close to the Brillouin zone center along the $\Gamma -X$ direction (blue square); and close to the Brillouin zone center but away from the $\Gamma -X$ line (red diamond). We observed a strong spin polarization of electronic bands up to 40% at a binding energy value of 0.55 eV (${{S}_{y}}$ component near the edge of hole pocket in figure 4(a)). We also find non-negligible spin polarization near the Fermi energy (figure 4(b)), which is pivotal to any 'device applications' viewpoint. We observe that along the $X-\Gamma -X$ line, the spin texture is oriented in the perpendicular direction i.e. ${{S}_{x}}=0$, ${{S}_{y}}\ne 0$ and ${{S}_{z}}\ne 0$. In other words, the spins are perpendicular to the Te–W–Te chains. The spin vector has both in-plane and out-of-plane non-vanishing components, unlike conventional Rashba systems and TIs, where the spin texture is distinguished by an in-plane orientation. It is also seen that the spins are oppositely oriented in the k-space (i.e. spin orientation is reversed at k and  −k points), respectively, confirming that the spin polarization has a non-magnetic origin. In figures 4(a) and (e), the spin polarization is probed at the positive and negative ${{k}_{x}}$ values, respectively, close to the $X-\Gamma -X$ line, where the spin directs perpendicularly to the $X-\Gamma -X$ direction. We find that the ${{S}_{y}}$ component of the spin direction is indeed reversed at opposite sides of the $\Gamma$ point. The observed opposite spin orientation on the opposite sides of $\Gamma$ might provide a mechanism to protect the electronic backscattering at zero magnetic field as previously suggested [65]. When an external magnetic field is applied, such a protection mechanism is gradually invalidated, with the concomitant increase of the electrical resistivity reminiscent of the MR behavior of WTe₂. The spin texture has been also calculated from first principles, and notable qualitative agreement is seen [73]. The experimental resolution was not enough to reproduce the fine details of the actual spin texture, which is extremely complex and characterized by large fluctuations in both energy and momentum. Our spin resolved ARPES results clearly suggest that the SOC is indeed strong, and plays an active role in the electronic structure of WTe₂, as also seen in the CD-ARPES results [65].

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Notwithstanding the fairly delocalized nature of 5d W orbitals, the inclusion of a modest amount of Coulomb interaction U in the ab initio calculations results in a sizable change of the electronic dispersion in the proximity of the chemical potential. As a function of increasing U, a shift of the electron pocket towards lower momenta and a clear modification of the hole pocket are observed as a general trend (see figures 5(a) and (b)). For U  =  2 eV, i.e. the value that provides a nice comparison between the measured and calculated Fermi surfaces [83], the two pockets exhibit a linear touching ~50 meV above the Fermi energy along the reciprocal direction ${{k}_{x}}$. It is likely that such a correlation-driven trend leads to a change of the topological properties of WTe₂, since the appearance of a type-I Weyl crossing has been recently proposed as a hallmark of topological transitions in non-centrosymmetric TIs [85]. Moreover, the DFT  +  U optical conductivity, as shown in figure 5(c), nicely reproduces the main features of the measured spectrum, including the peak around 60 meV (~500 cm⁻¹) missing in standard DFT calculations, as discussed in [86]. As yet another indication towards an improved description of the optical properties, the shoulder at 150 meV (1400 cm⁻¹) as well as the infrared peak at 20 meV (160 cm⁻¹) are better captured by DFT  +  U.

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The Dirac equation admits three different representations for the Dirac spinor, which in the context of condensed matter translates into three types of relativistic fermions: Dirac, Majorana, and Weyl fermions. These are found to manifest themselves in several condensed matter systems as low energy quasi particle excitations. It has been recently proposed and experimentally observed that the Weyl fermions can exhibit two distinct types of band dispersions. The type-I Weyl points are associated with a point-like Fermi surface and result from a linear crossing of two topologically protected bands at the Fermi level. The type-I Weyl Fermions have been observed in the TaAs family of compounds [2–4, 6–8, 87, 88]. On the contrary, the type-II Weyl points are observed at the frontiers between electron and hole pockets, and the electronic dispersion is characterized by strongly tilted Weyl cones. As a consequence, the Lorentz invariance turns out to be broken in the case of type-II Weyl states. The transport properties of type-II Weyl semimetals are found to be remarkably different from those of the type-I counterparts based upon the differences of their Fermi surfaces. SOC is critical for the actual distribution of Weyl points in WTe₂. In the absence of SOC, there is a total of 16 Weyl points in the Brillouin zone. Once the effect of SOC is considered, half of the Weyl points which were previously present in the ${{k}_{z}}=0$ plane annihilates [43]. Furthermore, the Weyl physics of the system is highly dependent on its lattice constants. A slight change in the lattice constants results in the annihilation of the Weyl points. It has been reported from first principles calculations that the WTe₂ band structure shows eight Weyl points when considering the low temperature lattice parameters in the (0 0 1) plane [43], whereas all the Weyl points annihilates when the slightly larger room temperature lattice constants [47] are considered.

The Weyl points in WTe₂ are predicted to be situated about 50 meV above the Fermi energy [43], making it impossible to observe them by the equilibrium ARPES experiment. However, the signature of the topological Fermi arc connecting the bulk electron and hole pockets can be observed in the (0 0 1) surface projection (see figure 5(a)). From our data, we are unable to resolve the Fermi arcs associated with the Weyl physics of this material. The ARPES studies performed by other groups have reported the signature of the Fermi arcs on the (0 0 1) surface [44–47]. It is moreover reported that the topological states of WTe₂ are not observed with ARPES on certain types of surfaces [44]. Although WTe₂ possesses two non-equivalent surface terminations, the reason for the observation of topological states only on some particular surfaces is not related to the surface termination itself, but it could be rather related to the strain induced to the surface upon cleavage. The topological states in WTe₂ are indeed highly sensitive to strain and pressure as theoretically suggested [43]. It is also found that irrespective of the presence of the Weyl points, surface states connecting the bulk electron and hole pockets are present. The differences in the surface states between the trivial and topologically non-trivial phases are minute, and hardly detectable in experiments. Therefore, it is argued that the detection of surface states alone is insufficient to conclude that WTe₂ belongs to the family of type-II Weyl semimetals [47]. WTe₂ still remains a candidate material for type-II Weyl semimetal, and further experimental proof is necessary to establish this remarkable state. Apart from WTe₂, type-II Weyl Fermions have been observed in MoTe₂ [89, 90] and LaAsGe [91].

Finally, we discuss the non-equilibrium dynamics of the electronic bands of this material near the Fermi energy as investigated by time-resolved ARPES (TR-ARPES). TR-ARPES experiments on WTe₂ crystals were performed to the aim of elucidating the evolution of the electronic bandstructure on a femtosecond timescale, after excitation of the sample with intense near-infrared laser pulses. In this pump-probe spectroscopy, an intense laser pulse (pump) deposits energy into the material and brings it in an out-of-equilibrium condition, while a delayed weaker pulse of appropriate photon energy (probe) is used for the ARPES experiment. ARPES maps are then acquired for different values of the pump-probe delay t. Hence, TR-ARPES provides a direct view on the non-equilibrium electronic band structure; in this context, it is used to reveal how the electronic bands close to the Fermi level are modified after absorption of the pump pulse. In the experiments reported here, the excitation is performed via a 1.55 eV (800 nm) laser beam, with an absorbed fluence of $80\,\mu {\rm J}\,{\rm c}{{{\rm m}}^{-2}}$, while the ARPES probe is performed with a 6.2 eV (200 nm) laser beam, obtained by frequency-mixings processes. The overall time-resolution of the experiment is 120 fs; the energy resolution is of the order 50 meV. The temperature of the sample during the experiments is kept at T  =  120 K. We focus on the electronic band structure in the vicinity of the $\Gamma$ point, in particular we measure a ${{k}_{y}}$ cut along the $\Gamma -Y$ direction. Figure 6(a) shows the ARPES intensity acquired with the 6.2 eV laser probe. The $x$ axis indicates the emission angle along the $\Gamma -Y$, i.e. here ${{k}_{x}}=0$. This map was acquired before the arrival of the pump pulse (t  =  −660 fs). It represents the band structure when probed about 4 µs after excitation (the repetition rate of the laser source being 250 kHz). In this way, the electronic band structure approximates the equilibrium one; however, it takes into account average heating effects introduced by pump pulse. Figure 6(b) shows the photoemission intensity as a function of $E-{{E}_{F}}$, integrated in a  ±1 degree window (dashed white rectangle in figure 6(a)) centered at ${{k}_{y}}=0$ (Angle  =  0). Three main features are found: at ~50 meV, ~170 meV, ~360 meV below the Fermi level (${{E}_{{\rm F}}}$); they are marked by solid black ticks. These features are in good agreement with both band structure calculations (see supplementary information of [43]) and high-resolution ARPES (the same ${{k}_{x}}=0$, ${{k}_{y}}~||~\Gamma Y$ cut is shown in figure 1(c), as acquired by $h\nu =68$ eV photons). In the latter case, similar structures—despite the largely different photon energy used—are found. In particular, each of the three structures correspond to a number of hole-like bands that can only be distinguished by high-resolution ARPES. Figure 6(c) shows the result of the laser excitation on the electronic band structure via a differential ARPES map (blue color: negative intensity; red color: positive intensity; white color: no intensity change). This map is obtained by subtracting the ARPES map reported in figure 6(a) from the ARPES intensity collected at t  =  +330 fs (not shown). We chose the pump-probe delay t  =  +330 fs, for at this delay the photoinduced effect is maximal. Similar to figure 6(b), figure T1(d) shows the differential ARPES intensity as a function of $E-{{E}_{{\rm F}}}$, integrated over a  ±1 degree window centered about ${{k}_{y}}=0$, as indicated by the dashed white rectangle in figure 6(c). The differential ARPES intensity is positive only above ${{E}_{{\rm F}}}$, indicating that the absorption of the pump pulse heats the sample, leading to broadened band structure features. This effect can arise both by an 'heated' Fermi–Dirac distribution (although this band is below the Fermi energy at the temperature considered) or by an intrinsic broadening of the band at $E-{{E}_{{\rm F}}}=-30$ meV. We exclude that we are populating electronic states above the Fermi level, since no bands above ${{E}_{{\rm F}}}$ are expected in this ${{k}_{x}}=0,~{{k}_{y}}\sim 0$ region. Below ${{E}_{{\rm F}}}$, the intensity modification is solely and largely negative. The maximal modification is found in coincidence with the band located 30 meV below ${{E}_{{\rm F}}}$. We can argue that this band is strongly depleted (and possibly broadened, as argued above) since the energy-position of the maximal modification matches the equilibrium position of the band. Note that a band shift would not be compatible with this finding. The depletion, as induced by excitation with a fluence of 80 $\mu {\rm J}\,{\rm c}{{{\rm m}}^{-2}}$, is of the order 18%, which is a remarkably large amount. The structure at $E-{{E}_{{\rm F}}}=-170$ meV shows a similar behavior, although the intensity of the effect is reduced to a few % modification. At variance, the structure at $E-{{E}_{{\rm F}}}=-360$ meV shows a different response, and possibly shifts to higher binding energies after excitation. This can be argued by the fact that the maximal photoemission intensity variation is peaked at a binding energy different (larger) than the one found at equilibrium (the tick positions are the same as in figure 6(b)). Figure 7(a) shows the photoemission energy distribution curves collected at four selected pump-probe delays (and integrated in a window  ±1 degree about ${{k}_{y}}=0$):  −660 fs (black line), same as reported in figure 6(b);  +330 fs (red line);  +1.3 ps (green line);  +4.5 ps (blue line). We can conclude that within 4.5 ps from excitation, the band structure has recovered its equilibrium condition, as indicated by the very good overlap of the energy distribution curves collected at  −660 fs and  +4.5 ps. After 1.3 ps from excitation, a partial depletion of the $E-{{E}_{{\rm F}}}=-30$ meV structure is still present. Detailed information about the relaxation timescales have been obtained by analyzing the time evolution of the photoemission intensity in selected energy-momentum windows. We consider four regions, indicated as colored boxes overlaid on figure 6(c), that have been chosen in order to extract possible band-resolved timescales. The photoemission intensity extracted and averaged from such boxes is reported in figure 7(b) as a function of the pump-probe delay t. The four curves have been analyzed by fitting to them a function composed by one exponential decay with time constant ${{\tau }_{j}}$ $(\,j=A,~B,~C,~D)$ and a positive background, both convoluted with a Gaussian function representing the time resolution of the experiment (pump-probe cross correlation). In the present experiment, the time-resolution is 120 fs FWHM. The constant background at positive delays aims to account for the heat diffusion out of the excited volume. This is a process with nanosecond timescale, and can be safely approximated by a constant in the measured delay range. From the fitting routine (the results of the fit are represented by thin black solid lines) we extract ${{\tau }_{A}}=1090\pm 30$ fs, ${{\tau }_{B}}=970\pm 30$ fs, ${{\tau }_{C}}=650\pm 20$ fs, ${{\tau }_{D}}=1020\pm 30$ fs. These values are in good agreement with the findings by TR-ARPES experiments reported in [92, 93]. The relaxation time of ~1 ps also agrees with (momentum-integrated) results from time-resolved optical spectroscopy experiments [94, 95], and is associated to electron–phonon scattering events. From this timescale, the electron–phonon coupling constant was determined [94]. Note that the electron dynamics measured by TR-ARPES do not show the coherent-phonon features revealed by time-resolved reflectivity experiments [94, 95].

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By analyzing the non-equilibrium band structure of WTe₂ and its ultrafast time-evolution, we can learn important information about its physical properties. In particular, the hole-like bands measured here display a strong depletion which is well beyond what is expected from a simple average heating of the material. In particular, the structure at $E-{{E}_{{\rm F}}}=-30$ meV is depleted by about 18% after photoexcitation with a moderate fluence. This fact indicates that the possibility to photoinduce a fast imbalance of the electron/hole density ratio is realistic. Obviously, a quantitative estimation of this effect would require the investigation of the time-resolved electronic structure over the full Brillouin zone of WTe₂, in particular considering the electron and hole pockets crossing the Fermi level, which are located at ${{k}_{x}}=0.2-0.3\,{{{{\rm }\mathring{\rm A} }}^{-1}}$. The possibility to optically control on ultrafast timescales (less than one ps) the electron-to-hole density ratio, which is considered as the prerequisite for a large and non-saturating MR, will have deep implications for the possible optical control of the MR on timescales faster than what is currently allowed by electrical means.

As a last step of refinement of our review on the properties of WTe₂, we have studied the surface structure of our cleaved single crystals by high resolution STM. We find that the top layer is tilted by 6°. The tilting is independent of samples and bias voltage. We have also simulated the bandstructure by DFT. We find that although the size of electron and hole pocket depends on the tilting, the electron and hole balance is preserved in the bulk after the inclusion of surface distortion. The details of the STM results and the relation between surface distortion and its effect on the electronic structure is discussed elsewhere [73]. We also find that the surface is extremely clean without any significant impurity and defects. It is possible to find large regions without a single defect. The defect density is about one underlying defect per 3000 atoms, corresponding to about 9  ×  10¹¹ defects cm⁻². This is important in the context of excitonic dielectric in the Abrikosov sense [96, 97]. Abrikosov proposed that in the quantum limit, where the electron and hole balance is there, the electron and hole pair might form an exciton, quenching the electrical transport. Three main criterion of an excitonic dielectric, i.e. (i) perfect electron–hole balance, (ii) small overlap of valence and conduction bands at Fermi energy [98], and (iii) extremely high purity i.e. low density of impurities and defects [73], seem to be satisfied in this material.