Elastic and transport properties of topological semimetal ZrTe

Topological semimetal may have substantial applications in electronics, spintronics and quantum computation. Recently, ZrTe is predicted as a new type of topological semimetal due to coexistence of Weyl fermion and massless triply degenerate nodal points. In this work, the elastic and transport properties of ZrTe are investigated by combining the first-principles calculations and semiclassical Boltzmann transport theory. Calculated elastic constants prove mechanical stability of ZrTe, and the bulk modulus, shear modulus, Young's modulus and Possion's ratio also are calculated. It is found that spin-orbit coupling (SOC) has slightly enhanced effects on Seebeck coefficient, which along a(b) and c directions for pristine ZrTe at 300 K is 46.26 $\mu$V/K and 80.20 $\mu$V/K, respectively. By comparing the experimental electrical conductivity of ZrTe (300 K) with calculated value, the scattering time is determined for 1.59 $\times$ $10^{-14}$ s. The predicted room-temperature electronic thermal conductivity along a(b) and c directions is 2.37 $\mathrm{W m^{-1} K^{-1}}$ and 2.90 $\mathrm{W m^{-1} K^{-1}}$, respectively. The room-temperature lattice thermal conductivity is predicted as 17.56 $\mathrm{W m^{-1} K^{-1}}$ and 43.08 $\mathrm{W m^{-1} K^{-1}}$ along a(b) and c directions, showing very strong anisotropy. Calculated results show that isotope scattering produces observable effect on lattice thermal conductivity. It is noted that average room-temperature lattice thermal conductivity of ZrTe is slightly higher than that of isostructural MoP, which is due to larger phonon lifetimes and smaller Gr$\mathrm{\ddot{u}}$neisen parameters. Finally, the total thermal conductivity as a function of temperature is predicted for pristine ZrTe.


I. INTRODUCTION
From topological insulator to semimetal, the recent discovery of new type of topological nontrivial phase has sparked intense research interest in condensed matter physics and material science [1][2][3][4][5][6][7][8][9][10][11] . The representative topological semimetals include Dirac semimetals, Weyl semimetals and nodal line semimetals 3,7,10 , such as Na 3 Bi as a classic Dirac Semimetal 7 , TaAs as a representative Weyl semimetal 8 and ZrSiS as a typical nodal line semimetal 9 , which have been confirmed by angleresolved photoemission spectroscopy (ARPES). In Dirac and Weyl semimetals, four-fold degenerate Dirac point and two-fold degenerate Weyl point can be observed in the momentum space 7,8,10,11 , while two bands cross in the form of a periodically continuous line or closed ring for node-line semimetals 3,9 . Beyond Dirac and Weyl fermions, some new types of topological semimetals are proposed, which are identified by three-, six-or eight-fold band crossings 12 . By the crossing of a double-degeneracy band and a non-degeneracy band, the three-fold degenerate crossing points are predicted in the materials with WCtype structure, such as MoP, WC and ZrTe 13,14 , and in InAs 0.5 Sb 0. 5 15 . Then, the ARPES declares the presence of a triply degenerate point in MoP, and pairs of Weyl points coexist with the three-component fermions 2 . Experimentally, the highly metallic characteristics with remarkably low resistivity and high mobility (2 K) has been found in MoP 16 . The recent ARPES experiments and transport measurements have also shown that WC has the nontrivial topological nature 17,18 . The experimental results of the magnetoresistance, Hall effect, and quantum Shubnikov-de Haas oscillations on single crystals of ZrTe have been reported, indicating ZrTe with low carrier density, high carrier mobility, small cross-sectional area of Fermi surface, and light cyclotron effective mass 19 .
Recently, the elastic and thermal transport properties of some representative topological semimetals have been investigated, such TaAs and MoP [20][21][22][23] . The lattice thermal conductivity of both TaAs and MoP shows obvious anisotropy along the a(b) and c crystal axis 20,21,23 . High thermoelectric performance of TaAs has been predicted, and the maximum thermoelectric figure of merit ZT is up to 0.63 (900 K) in n-type doping along c direction 21 . In this work, the elastic and transport properties of threefold degeneracy topological semimetal ZrTe are investigated by combining the first-principles calculations and semiclassical Boltzmann transport theory. The elastic constants, bulk modulus, shear modulus, Young's modulus and Possion's ratio are predicted with the generalized gradient approximation (GGA). The electronic transport coefficients are also calculated using both GGA and GGA+SOC. It is found that SOC has slight influences on electronic transport coefficients. For pristine ZrTe, the Seebeck coefficient, electrical conductivity, power factor and electronic thermal conductivity are calculated, which can be verified by future experiments. The lattice thermal conductivity as a function of temperature is predicted within GGA, which shows a distinct anisotropic property along the a(b) and c crystal axis. Similar results can be found in topological semimetals TaAs and MoP 20,21,23 . The isotope and size effects on the lattice thermal conductivity are also studied, and the phonon mode analysis is also performed to understand deeply phonon transport of ZrTe. The total lattice thermal conductivity (κ=κ L +κ e ) as a function of temperature is also predicted for pristine ZrTe. This work sheds light on the elastic and transport properties of ZrTe, and could offer valuable guidance for MoP-based nano-electronics devices.
The rest of the paper is organized as follows. In the next section, we shall give our computational details. In the third section, we shall present elastic and transport properties of ZrTe. Finally, we shall give our conclusions in the fourth section.

II. COMPUTATIONAL DETAIL
Within the density functional theory (DFT) 24 , a full-potential linearized augmented-plane-waves method, using GGA of Perdew, Burke and Ernzerhof (GGA-PBE) 34 , is employed to investigate electronic structures of ZrTe, as implemented in the WIEN2k code 26 . The SOC is included self-consistently [27][28][29][30] , which produces observable effects on electronic transport coefficients. The convergence results are determined by using 4000 k-points in the first Brillouin zone (BZ) for the selfconsistent calculation, making harmonic expansion up to l max = 10 in each of the atomic spheres, and setting R mt * k max = 8 for the plane-wave cut-off. The selfconsistent calculations are considered to be converged when the integration of the absolute charge-density difference between the input and output electron density is less than 0.0001|e| per formula unit, where e is the electron charge. Based on calculated energy band structures, transport coefficients of electron part are calculated through solving Boltzmann transport equations within the constant scattering time approximation (CSTA), as implemented in BoltzTrap code 31 . To obtain accurate transport coefficients, the parameter LPFAC is set as 10, and 2772 k-points is used in the irreducible BZ for the calculations of energy band structures. For elastic properties and phonon transport, the firstprinciples calculations are performed within the projected augmented wave (PAW) method, and the GGA-PBE is adopted as exchange-correlation energy functional, as implemented in the VASP code [32][33][34][35] . A planewave basis set is employed with kinetic energy cutoff of 400 eV, and the electronic stopping criterion is 10 −8 eV. The lattice thermal conductivity of ZrTe is performed by solving linearized phonon Boltzmann equation with the single mode relaxation time approximation (RTA), as implemented in the Phono3py code 36 . The lattice thermal conductivity can be expressed as where λ is phonon mode, N is the total number of q points sampling the BZ, V 0 is the volume of a unit cell, and C λ , ν λ , τ λ is the specific heat, phonon velocity, phonon lifetime. The interatomic force constants (IFCs) are calculated by the finite displacement method. The second-order harmonic IFCs are calculated using a 4 × 4 × 4 supercell containing 128 atoms with k-point meshes of 2 × 2 × 2. Using the harmonic IFCs, phonon dispersion of ZrTe can be attained by Phonopy package 37 . The group velocity and specific heat can be attained from phonon dispersion which also determines the allowed three-phonon scattering processes. The third-order anharmonic IFCs are calculated using a 3 × 3 × 3 supercells containing 54 atoms with k-point meshes of 3 × 3 × 3, and the total number of displacements is 508. Based on third-order anharmonic IFCs, the phonon lifetimes can be attained from the three-phonon scattering.
To compute lattice thermal conductivities, the reciprocal spaces of the primitive cells are sampled using the 20 × 20 × 20 meshes.

A. ELASTIC PROPERTIES
The WC-type ZrTe possesses space group P6m2 (No. 187), with Zr and Te atoms occupying the 1d (1/3, 2/3, 1/2) and 1a (0,0,0) Wyckoff positions, respectively. The crystal structure and BZ of ZrTe are shown in Figure 1. All the results attained in the following are from the calculations with experimental lattice constants (a=b= 3.7707Å, c=3.8606Å ) 38 . The most basic physical quantities of elastic properties are elastic constants C ij , which can be used to construct else elastic physical quantities. The elastic constants are a four-rank tensor. However, the elastic constants are reduced to five independent ones: C 11 , C 12 , C 13 , C 33 , C 44 due to the symmetry for hexagonal crystal, and C 66 can be obtained by (C 11 − C 12 )/2. The calculated C ij are shown in Table I. To prove mechanical stability of ZrTe, we use the following mechanical stability criterion for the hexagonal materials 39,40 : By simple calculations, these criteria are satisfied for ZrTe, which means no strong tendency to become unstable with the increasing pressure. The Voigt's, Reuss's and Hill's bulk modulus can be attained by the following equations: The Voigt's, Reuss's and Hill's shear modulus can be calculated by using these formulas: The S ij 's can be obtained by inverting the elastic constants matrix.
The numerical calculated values are E xx =E yy =97.83 GPa and E zz =122.07 GPa, respectively. The Poisson's ratios ν ij can be calculated by:

B. ELECTRONIC TRANSPORT
The energy band structures of ZrTe along highsymmetry paths are shown in Figure 2 using both GGA and GGA+SOC. Our calculated results agree well with previous theoretical ones 13 . It is found that the sixfold degenerated nodal point splits into the two triply degenerate nodal points (TDNPs) along the Γ-A direction, when the SOC is included 13 . However, the six pairs of Weyl nodes appear around K point in its first BZ 13 . Based on calculated energy band structures, the electronic transport coefficients of ZrTe can be attained using CSTA Boltzmann theory. Although the calculated electrical conductivity depends on scattering time τ using CSTA, the Seebeck coefficient is independent of scattering time, directly compared with experimental results. At room temperature, the Seebeck coefficient S, electrical conductivity with respect to scattering time σ/τ , power factor with respect to scattering time S 2 σ/τ and electronic thermal conductivity with respect to scattering time κ e /τ along a(b) and c directions as a function of doping level using both GGA and GGA+SOC are plotted in Figure 3. Within the framework of rigid band approach, the n-or p-type doping can be simulated by simply shifting Fermi level into conduction or valence bands, which is effective in low doping level [42][43][44] .
Calculated results show that SOC has a slightly enhanced effect on n-and p-type Seebeck coefficient (absolute value) along both a(b) and c directions in low doping level. It is found that the Seebeck coefficients are relatively strongly anisotropic along a(b) and c directions. The calculated S along a(b) and c directions for pristine ZrTe is 46.26 µV/K and 80.20 µV/K, respectively, which can be farther verified by the experiment. When the See- beck coefficient vanishes, the doping level along a(b) and c directions is about -0.0065 and -0.0124, respectively. It is found that n-type doping has more excellent Seebeck coefficient than p-type doping. For σ/τ , the slightly reduced influence caused by SOC can be observed along both a(b) and c directions. The electronic thermal conductivity κ e relates to the electrical conductivity σ via the Wiedemann-Franz law: where L is the Lorenz number. So, there are similar dependencies of doping level and SOC between κ e /τ and σ/τ . The τ is attained by comparing experimental electrical conductivity 19 of ZrTe with the calculated value of σ/τ at room temperature, and the τ is found to be 1.59 × 10 −14 s. For pristine ZrTe, the electronic thermal conductivity along a(b) and c directions is 2.37 Wm −1 K −1 and 2.90 Wm −1 K −1 , respectively. Using the calculated τ , the electrical resistivity of pristine ZrTe along a(b) and c directions is 5.26×10 −6 Ωm and 3.66×10 −6 Ωm, respectively. A enhanced SOC effect on power factor can be observed along both a(b) and c directions, which is due to improved S caused by SOC. For pristine ZrTe, the power factor along a(b) and c directions is 3.82 µW/cmK 2 and 17.33 µW/cmK 2 , respectively, showing obvious anisotropy. Next, we obtain the scattering time τ at different temperatures by τ ∝T −145 . For pristine ZrTe, the Seebeck coefficient S, electrical resistivity ρ and electronic thermal conductivity κ e as a function of temperature with GGA+SOC are plotted Figure 4. It is found that the Seebeck coefficient and electrical resistivity along both a(b) and c directions firstly increase with increasing temperature, and then decrease. It is noted that the Seebeck coefficient along c direction changes from positive values to negative ones, when the temperature is larger than 870 K. At high temperature, the electrical resistivity along both a(b) and c directions almost coincides. With in- creasing temperature, the electronic thermal conductivity increases along both a(b) and c directions, showing weak anisotropy.

C. PHONON TRANSPORT
Based on the harmonic IFCs, the phonon dispersion of ZrTe can be attained along high-symmetry path, which along with atom partial density of states (DOS) are shown in Figure 5. Due to each primitive cell containing two atoms, there are six vibrational branches consisting of three acoustic and optical ones, respectively. No imaginary frequencies in the phonon dispersion indicate the thermodynamic stability of ZrTe. The ZrTe belongs to P6m2 space group whose point group is D 3h , and then BZ-centre optical phonon modes of this crystal can be decomposed as The A 2 and E modes are infrared-active, and E mode is also Raman-active. The phonon frequencies of A 2 and E are shown in Table II. It can be clearly seen that there is a well-separated acoustic-optical gap of 0.15 THz at the A point (0, 0, π/2) on the boundary of the BZ. The top phonon band of the gap at the A point is doubly degenerate, while the bottom phonon band is a singlet state. Similar phonon gap can also be found in MoP 23 , which is larger than one of ZrTe. However, the isoelectronic ZrSe shows no acoustic-optical gap, and the presence of the TDNPs of phonon has been predicted 46 . From atom partial DOS, contribution to the acoustic (optical) phonon branches mainly comes from Te (Zr) atoms. The intrinsic lattice thermal conductivity of ZrTe can be attained from harmonic and anharmonic IFCs by solving the linearized phonon Boltzmann equation within single-mode RTA method. The phonon-isotope scattering is also considered, according to the formula proposed by Shin-ichiro Tamura 48 . The lattice thermal conductivities of pure and isotopic ZrTe along a(b) and c directions as a function of temperature are shown in Figure 6. In the considered temperature region, the intrinsic enhancement of phonon-phonon scattering with increasing temperature leads to the decreased lattice thermal conductivity of ZrTe, which typically results as 1/T . From Figure 6, it is clearly seen that the lattice thermal conductivity of ZrTe shows obvious anisotropy, where the lattice thermal conductivity along c direction is very higher than that along a(b)direction. Similar result can be found in MoP 23 , but is different from TaAs 20 . At room temperature, the lattice thermal conductivities of pure (isotopic) ZrTe along a(b) and c directions are 17.56 (16.13) Wm −1 K −1 and 43.08 (37.82) Wm −1 K −1 , which of pure ZrTe are shown in Table II. To measure the anisotropic strength, an anisotropy factor 20 is defined as η = (κ L (cc) − κ L (aa))/κ L (aa), and the calculated value is 145.3%, which is larger than that of MoP, im-plying stronger anisotropy. The lattice thermal conductivity is connected with Young's modulus by the simple relation κ L ∼ √ E 47 . Calculated results show that the orders of Young's modulus and lattice thermal conductivity along a(b) and c directions are identical. It is found that phonon-isotope scattering along c direction produces larger effects on lattice thermal conductivity than that along a(b) direction. With increasing temperature, isotopic effect on lattice thermal conductivity gradually decreases, which is due to improvement of phonon-phonon scattering.
At room temperature, the cumulative lattice thermal conductivities with respect to frequency along with the derivatives along a(b) and c directions are plotted in Figure 7. The cumulative thermal conductivity is defined by: It is clearly seen that the acoustic phonon branches dominate lattice thermal conductivity, up to 96.49% along a(b) direction and 91.12% along c direction. It is found that the optical contribution along c direction is larger than that along a(b) direction. Furthermore, the relative contributions of six phonon branches to the total lattice thermal conductivity along a(b) and c directions at 300K are plotted in Figure 7. Along both a(b) and c directions, longitudinal acoustic (LA) phonon mode has larger contribution than any of two transverse acoustic (TA1 or TA2) phonon modes. It is evident that optical branches along c direction have obvious contribution. The phonon transport of ZrTe can be further understood with the help of the mode level phonon group velocities and lifetimes, which are plotted in Figure 8. The largest phonon group velocity of TA1, TA2 and LA branches in long-wavelength limit is 2.23 kms −1 , 3.86 kms −1 and 4.40 kms −1 , respectively. In the low frequency region, the most of group velocities of TA2 and TA1 branches are lower than those of LA branch, which leads to main contribution to lattice thermal conductivity from LA branch. It is found that the most of both group velocities and phonon lifetimes of acoustic branches are larger than those of optical branches, which lead to the To further understand the size dependence of lattice thermal conductivity of ZrTe, the cumulative lattice thermal conductivity divided by total lattice thermal conductivity (CDT) with respect to MFP (300 K) along a(b) and c directions are plotted in Figure 9. The MFP cumulative lattice thermal conductivity is given by: It can reflect the contribution to total lattice thermal conductivity from individual phonon modes with different MFP, namely it shows how phonons with different MFP make contribution to the thermal conductivity. It is clearly seen that the CDT along both a(b) and c directions approaches one with MFP increasing. The contribution from phonons with MFP larger than 0.60 µm is very little. Phonons with MFP smaller than 0.13 (0.07) µm along a(b) direction and 0.07 (0.05) µm along c direction contribute around 80% (60%) to the lattice thermal conductivity. It is found that phonons dominating the lattice thermal conductivity along a(b) direction have longer MFP than ones along c direction.
Based on calculated electronic and lattice thermal conductivity, the total thermal conductivity can be attained, which is plotted in Figure 10. It is clearly seen that κ firstly decreases with increasing temperature, and then increases. It is because lattice part dominates thermal conductivity at low temperature, while electronic part is predominant at high temperature. The minimum κ along a(b) and c directions is 13.98 Wm −1 K −1 and 23.15 Wm −1 K −1 , respectively, and the corresponding temperature is 650 K and 1000 K. The room-temperature κ is 19.94 Wm −1 K −1 and 45.98 Wm −1 K −1 , respectively. These results are useful for the thermal management of ZrTe-based electronics devices.

IV. CONCLUSION
The elastic properties and phonon transport of the isostructural topological semimetal MoP has been investigated by the same method 23 . It is found that the bulk modulus, shear modulus, Young's modulus of ZrTe are smaller than those of MoP, which means that ZrTe produces easily deformation by applied external force. The lattice thermal conductivity of ZrTe (17.56 Wm −1 K −1 ) along a(b) direction is very close to that of MoP (18.41 Wm −1 K −1 ), while the lattice thermal conductivity of ZrTe (43.08 Wm −1 K −1 ) along c direction is larger than that of MoP (34.71 Wm −1 K −1 ). It is noted that the average lattice thermal conductivity (κ L (av)=(κ L (aa)+κ L (bb)+κ L (cc))/3) of ZrTe is slightly higher than that of MoP. This is because that ZrTe has larger phonon lifetimes and smaller Grüneisen parameters than MoP, which gives rise to higher lattice thermal conductivity for ZrTe than MoP. Phonon transport of another classic topological semimetal TaAs has been investigated 20 . Any of ZrTe, MoP and TaAs shows obvi-ously anisotropic lattice thermal conductivity along a(b) and c directions. However, for TaAs, the lattice thermal conductivity along a(b) direction is larger than one along c direction, but the lattice thermal conductivity of ZrTe or MoP along a(b) direction is smaller than that along c direction.
In summary, the elastic and transport properties of ZrTe are performed by the first-principles calculations and semiclassical Boltzmann transport theory. The elastic tensor components C ij for ZrTe are presented, which confirm the mechanical stability of the structure. The bulk modulus, shear modulus, Young's modulus and Poisson's ratio are also attained by calculated C ij . The electronic transport coefficients are also calculated within CSTA Boltzmann theory. For pristine ZrTe, the Seebeck coefficient, electrical resistivity and electronic thermal conductivity will be of use for comparison with future experimental measurements. The lattice thermal conductivity of ZrTe shows an obvious anisotropy along the a(b) and c crystal axis. It is found that isotope scattering has observable effect on the lattice thermal conductivity, and phonons with MFP larger than 0.60 µm have little contribution to the total lattice thermal conductivity. The higher lattice thermal conductivity of ZrTe than MoP can be explained by larger phonon lifetimes and smaller Grüneisen parameters. The total thermal conductivity is also attained for pristine ZrTe. Our works shed light on elastic and transport properties of ZrTe, and will motivate farther experimental studies of elastic and transport properties of topological semimetals ZrTe.