Disorder-induced coupling of Weyl nodes in WTe 2

The ﬁnite coupling between Weyl nodes due to residual disorder is investigated by magnetotransport studies in WTe 2 . The anisotropic scattering of quasiparticles is evidenced from classical and quantum transport measurements. A theoretical approach using the real band structure is developed in order to calculate the dependence of the scattering anisotropy with the correlation length of the disorder. A comparison between theory and experiments reveals a short correlation length in WTe 2 ( ξ ∼ 5 nm). This result implies a signiﬁcant coupling between Weyl nodes and other bands. Our study thus shows that a ﬁnite intercone scattering rate always exists in weakly disordered type-II Weyl semimetals, such as WTe 2 , which strongly suppresses topologically nontrivial properties.

In Weyl and Dirac semimetals, bulk gapless excitations are described by a Dirac equation, with a linear band dispersion and a crossing at a band degeneracy point close to the Fermi energy.Contrary to Dirac semimetals for which high symmetry Dirac points always gives pairs of Fermions with opposite chiralities, the inversion symmetry breaking in Weyl semimetals splits the position of band degeneracy points in the reciprocal space into two distinct Weyl nodes of opposite chiralities protected by the topology of the overall band structure.This gives rise to topologicaly non-trivial properties, like the presence of Fermi arcs [1] or a chiral anomaly [2][3][4].Nevertheless, the observation of many topological non-trivial properties requires a weak coupling between Weyl nodes or between the Weyl node and some other bands [2,3].
In the type-II Weyl semimetal WTe 2 , the Weyl nodes are tilted such that the Fermi energy crosses the Weyl cones on both the electron and the hole sides [5] (see Fig. 1).This implies the presence of electrons and holes pockets that touch each other when the Fermi energy is at the Weyl node.As electrons and holes are almost perfectly compensated in WTe 2 , this results in the giant magnetoresistance reported in high quality crystals [6].Recently, monolayers of few-layer thin films of WTe 2 attracted attention with the discovery of the quantum spin Hall effect [7][8][9][10][11][12][13][14], the evidence of superconductivity [15][16][17] and the measurement of a non-linear Hall effect [18,19].Although the disorder in WTe 2 plays a key role in the observation of many trivial and non-trivial transport properties and despite some theoretical work [20][21][22], a quantitative and experimental study of the disorder in WTe 2 is still lacking.
In this letter, we investigate the magnetotransport properties of a WTe 2 nanoflake and determine for the first time the correlation length ξ of structural disorder.A comparison with a theoretical model that considers scattering processes within the specific band structure of WTe 2 , using a renormalization method, allows us to infer a rather short ξ (∼ 5 nm) of the disorder.This structural disorder, inherent to transition metal chalcogenides, results in a finite coupling between Weyl nodes that can suppress the transport properties related to the band-structure topology.
In real materials, a particle can be scattered from a Weyl cone to another band due to the presence of impurities.This effect is related to the strength of the disorder (δV in Fig. 1) and to ξ which defines the range of the scattering.In a single band with a Fermi wave vector k, the short-range disorder limit is defined by kξ 1. Scattering is isotropic, i.e. an initial state can be scattered all over the band and the angle θ q between initial and final states is uniformly distributed between 0 and 2π.Instead, in the long-range disorder limit corresponding to kξ 1, the disorder coupled only states close to each other in the reciprocal space and θ q is limited to small values (anisotropic scattering) [23].Similarly, in the multiband case where δk is the distance between an initial state and some other bands (see Fig. 1), the inequality δkξ 1 sets the limit where interband scattering becomes significant.Particularly, for δk W ξ 1 with δk W the distance between two Weyl nodes, disorder induces a finite relaxation rate between two Weyl cones of opposite chirality, preventing the measurement of topologically non-trivial effect.Thus, the correlation length ξ appears as a key parameter for the observation of topologically non-trivial properties in Weyl (or Dirac) semimetals.
In order to measure ξ, we investigated the quantum transport properties of a WTe 2 flake at very low temperature (T 100 mK) and under magnetic field (B < 6 T).A careful study of the Shubnikov-de Haas oscillations allow us to determine both the the quantum life time τ Q of electrons and the Fermi wave vector k F whereas the measurement of the longitudinal and Hall magnetoresistances gives access to the transport time τ tr of electrons.Far from the onset of the bands, our calculations show that the ratio τ tr /τ Q does not depends on the strength of the disorder and is a function of k F and ξ only.Our theoretical approach based on a materials specific band structure (including the spin texture) establishes the correspondence between τ tr /τ Q , k F and ξ in a disordered material and gives hence for the first time a quantitative estimation of ξ.
Bulk single crystals of WTe 2 have been grown by Te flux and were characterized by SEM\EDX mode for compositional analysis and with x-ray diffraction for structural analysis.A WTe 2 flake (about 20 µm × 40 µm × 70 nm) was directly exfoliated onto a Si/SiO x substrate and contacted by standard ebeam lithography and a metal lift-off process.Good ohmic contacts were obtained after in-situ ion beam etching prior to the ebeam evaporation of Cr(10 nm)/Au(100 nm) (see Fig. 2).
When cooled down, the resistance decreases with a residual resistance ratio R 300K / R 4K ∼ 80, indicating the good quality of our nanostructure.At low temperature, a longitudinal magnetoresistance δR/R = (R(B) − R(0))/R(0) exceeding 6000% at B = 6 T is measured (Fig. 2).Hence, a tiny misalignement of Hall voltage probes results in a strong symmetric magnetoresistance which is not related to the anti-symmetric Hall voltage.In order to make a correct treatment, we therefore systematically symmetrized the longitudinal resistances and anti-symmetrized the transverse Hall resistances.As shown in Fig. 2 both the longitudinal and the Hall resistivities can be very well described by the two-band model : with n and p the charge densities of electrons and holes and µ and µ their respective mobilities (results presented in Fig. 2 are almost contact independent).The Hall signal strongly deviates from the linear dependence that is expected fo a fully compensated two-band model (δn = n − p = 0).Nevertheless, as shown in Ref. ? , both the Hall and the longitudinal resistance can be perfectly fitted in a full-compensated four-bands model.Indeed, close to the compensation, four bands are predicted by band calculations [24,25], two spin non-degenerate electron pockets and two spin non-degenerate hole pockets (Fig. 1).Still, the good agreement with the noncompensated two-bands model with |δn| n points at transport properties dominated by two effective bands with an almost perfect compensation between electrons and holes.In this approximation, the two spin nondegenerate electron pockets are considered as one single spin-degenerated pocket and the larger hole pocket is assumed to dominate the hole transport properties.While this approximation is fully justified for electrons, it is less reasonable for the hole pockets.Therefore, we focus below on the properties of electrons instead of holes.
The exact configuration of the current lines is not known in our nanostructure which prevents the use of geometrical parameters to fit the data with Eq.(1).To overcome this difficulty, we fit δR/R(B) = δρ/ρ(B) instead of R(B).Moreover, due to the almost compensation of charges (|δn|/n 1), the term proportional to δn 2 in Eqs.( 1) and ( 2) becomes too small to be reliably fitted.As a result, the fit of the magnetoresistances with Eqs.( 1) and (2) allow us to determine three free parameters from four unknown (n, p, µ and µ ).Therefore, the electron density is determined by the measurement of Shubnikov-de Haas as we will see below.The fit of the magnetotransport data for electrons then leads to τ tr = mµ/e 3.1 ps.
We focus now on the Shubnikov-de Haas oscillations measured down to T 100mK [Fig.3(a)].Close to their onset ( 1 T), the quantum oscillations are periodic in 1/B with a period ∆(1/B) 0.012 T −1 , corresponding to a Fermi wave vector k F 0.5 nm −1 .The temperature dependence of the oscillations indicates a field independent effective mass m * ∼ 0.4 × m e [see Fig. 3(b)].A fit with a Dingle plot gives a quantum life time τ Q ∼ 0.7 ps [Fig.3(c)].It remains now to assign those oscillations to a charge population.
Close to the onset, a single series a periodic maxima with a period ∆(1/B) is observed [blue arrows in Fig. 3(a)].In addition to this first series, a second series of periodic maxima with the same period ∆(1/B) but shifted in 1/B can be measured for B > 1.7 T [red arrows in Fig. 3(a)].To understand the origin of those quantum oscillations, we index the Landau levels separately for the two series of maxima [see Fig. 3(d)].The two series of indices show a clear linear dispersion with very similar slopes and confirms that two charge populations with very similar pocket size are at the origin of the Shubnikov-de Haas oscillations.Following the different band calculations reported so far [5,6,24], we attribute the oscillations to the two electron pockets and we can calculate the charge density associated to a single non-degenerate electron pocket (n s = 2.2 × 10 18 cm −3 ) as well as the total electron density (n = 4 × n s = 8.9 × 10 18 cm −3 ).No oscillations from the hole pockets was measured, eventually due to their larger effective mass.We can now determine the ratio τ tr /τ Q for electrons and we found τ tr /τ Q 5, a typical value also measured in other topological materials [26].
In order to extract the correlation length ξ of the disorder, it is necessary to compare the experimental ratio τ tr /τ Q with the corresponding value calculated from an appropriate theoretical model.We developed therefore a realistic theoretical approach based on a material's specific band structure, including the spin-texture, and a quantitative treatment of the disorder scattering.To achieve our aim, we applied the projective renormalization method (PRM) [27] usually used for many-particles system to a realistic Hamiltonian of WTe 2 with a static disorder.The method will be presented in more details in Ref.? and we describe it briefly below.We first consider the minimal case of WTe 2 in the presence of disorder V where two bands interact with each other within the nodal energy crossings in the Brillouin zone.The corresponding Hamiltonian reads The operator c † k,α creates an electron with momentum k and spin α, i.e. α = {↑, ↓} and the the two-band Hamil-tonian can easily be generalised to more bands by simply adding additional band indices.For a Weyl semimetal, there exists at least one pair of points in the Brillouin zone where the 2 × 2 matrix Ĥk becomes linear in momentum.The scattering by the disorder is described by the V term that relates an initial state (k, α) to a final state (k , β).We take a Gaussian disorder [28] entirely characterized by the Fourier transform of its correlation function where V stands for the strength of the disorder and ξ its correlation length.
In our method, the non-diagonal Hamiltonian H is at first decomposed into a diagonal part H 0 and a nondiagonal part H 1 , i.e.H = H 0 + H 1 .We introduce therefore new fermionic operators a k,α = β [ Dk ] α,β c k,β where the matrix Dk is defined such that the first term in Eq. ( 3) becomes diagonal: All information about the spin texture band structure is now contained in the matrix D. The decomposition (5) of the Hamiltonian allows the application of the PRM to integrate out the scattering term using unitary transformations and to write H as the effective diagonal Hamiltonian H of free fermions with renormalized bands: with X † = −X.Since H is connected to the original Hamiltonian H through a unitary transformation, it has the same eigenvalues than H [29] and it allows access to any quantity of the system.
The parameters Ẽk,α are connected to the initial band structure E k,α and initial scattering matrix elements V a k,k = Dk Vk,k D−1 k through renormalization equations which are derived analytically within the scheme illustrated in Refs.27 and 30.The renormalization equations are solved numerically on a 3d grid of 20×20×20 k points taking as starting values the realistic band structure of WTe 2 from Refs.[31,32] with a disorder described by a momentum distribution according to Eq.( 4).
The numerical solution of the renormalization equation allows to calculate the momentum dependent transport and quantum life times given by the Fermi golden rule, with q = k − k and θ q the angle between k and k .The scattering expectation values a † k,α a k ,β = ã † k,α ãk ,β are calculated within the PRM using the same unitary transformation as for the renormalization of the Hamiltonian ã † k,α = e X a † k,α e −X .Using the described approach we have evaluated numerically the expressions (7) and (8) for fixed k = k F e x and T = 0. We present here the results obtained for k belonging to an electron Fermi pocket but our conclusions are not significantly affected if k belongs to a hole Fermi pocket.We also considered k directed in the other directions e y and e z but, again, the conclusions from numerical results were not affected.Finally, we chose for the strength of the disorder V = 5meV.Nevertheless, the ratio τ tr /τ Q depends on V only for small k F .Hence, the ratio calculated for V = 5 meV and V = 50 meV is the same for the k F 0.35 nm −1 and for the experimental value measured (k F = 0.5 nm −1 ), the value of V has no influence on our conclusions.
In Fig. 4, we plot the calculated ratio τ tr /τ Q as a function of ξ. k F is defined here as the half of the width of the electronic pocket in the e x direction.In the low ξ limit, the short range disorder couple a state to all available states of the different Fermi pockets.Hence, the situation is almost equivalent to an isotropic scattering in spin degenerated bands and the ratio τ tr /τ Q ∼ 1 as expected.In the opposite limit, for large ξ, the long-range disorder induces a strong anisotropic scattering and τ tr /τ Q rises significantly.The limit between the two regimes is given by k F ξ ∼ 1. Importantly, the comparison of the calculated ratio for k F = 0.5 nm −1 with the experimental value we measured give us ξ 5 nm.
We note here that ξ is measured at a Fermi energy which is about 50 meV below the Weyl nodes.Nevertheless, the total charge density should not change significantly at the Weyl nodes and therefore the screening and the value of ξ at a Weyl node.A relevant value of the distance between a Weyl node and some other band in the reciprocal space is given by δk W .This quantity has been calculated in some previous works [5,25,33,34] and fluctuates between 0 and 0.32 nm −1 .According to the value of ξ we measured, we have δkξ ∼ 1, meaning a substantial coupling between a Weyl node and some other band that strongly reduces the signature of topological properties.
Qualitatively, the value we extract for ξ might be not restricted to the case of WTe 2 but could be generalized to any topological materials with a similar charge density and disorder.It highlights the necessity to enhance the crystal quality or to increase the distance in the reciprocal space between Weyl nodes to measure topologically non-trivial properties.This might be achieved in WTe 2 by substitution of tungsten atoms by Molybdenum atoms in the crystal structure of W 1−x Mo x Te 2 [33].
In conclusion, we study the transport properties of a WTe 2 exfoliated flake and measure the anisotropy of the scattering of electrons.We develop a new theoretical method taking into account the real band structure of WTe 2 with the aim to compare our experimental data with the numerical simulations and we determine for the first time the correlation length of the disorder ξ ∼ 5 nm.This value points at a significant coupling between the Weyl nodes and other bands, leading to a strong reduction of topological properties.Our results stress the importance of band structure engineering of Weyl semimetals to enhance the distance δk W between the Weyl nodes in order to investigate topological properties.

FIG. 1 .
FIG.1.Illustration of the band structure of WTe2 with the Weyl nodes indicated by the two arrows.The spin degeneracy of the conduction and the valence bands is lifted by the spin-orbit coupling, leading to two bands slightly shifted in energy (plain and dashed lines).The Fermi energy EF and the related Fermi wave vector kF are indicated for the perfect compensation (n = p), giving rise to two spin non-degenerate electron and hole pockets participating to the transport.An illustration of the disorder potential induced by impurities is given with a graphic definition of its amplitude δV and its correlation length ξ.

FIG. 2 .
FIG. 2. Anti-symmetrized magnetoresistance (left panel) of transverse contacts and symmetrized relative resistance (right panel) up to ±6 T for longitudinal contacts.The black solid lines are the experimental data and the blue dashed lines are the fit to the non-compensated two band model.The current and voltage contact configuration are indicated in the pictures.

FIG. 3 .
FIG. 3. a: Shubnikov-de Haas oscillations without background measured in the longitudinal resistance at T = 100 mK and up to 6 T. The upper inset shows the onset of the Shubnikov-de Haas oscillations and the different attribution of the maximum to the "blue" or the "red" electron pocket.The lower inset shows the fast Fourier transform where a dominant single peak clearly emerge and that does not allow to catch the full information revealed by the Landau level indexing.b: the effective mass for the different extrema observed in δR that leads to m * 0.4×me whith me the mass of a free electron.c: The dingle plot of the quantum oscillations close to their onset.The green dashed line indicates the best fit of the Dingle plot.d: The absolute Landau level indexes for the "blue" and the "red" maxima are indicated by the small blue triangles and the large red squares respectively.The blue straight lines and the red dashed lines are linear fits.

FIG. 4 .
FIG. 4. Ratio τtr/τQ calculated from the PRM where the value of the Fermi momentum kF is fixed and the result is shown as a function of the correlation length ξ.The results for the experimental values kF = 0.5 nm −1 obtained from the quantum oscillation measurements is shown in black color.The dashed grey line indicates the experimental value for τtr/τQ.
S.A. acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) through grant AS 523/4-1 and J.D. acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) through SPP 1666 Topological Insulators program and the ct.qmatExcellence cluster.This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 647276-MARS-ERC-2014-CoG). IVM, SA and BB thank DFG and RSF for financial support in the frame of the joint DFG-RSF project 'Weyl and Dirac semimetals and beyond-prediction, synthesis and characterization of new semimetals'(project number: 405940956).