Electronic Transport in Weyl Semimetals with a Uniform Concentration of Torsional Dislocations

In this article, we consider a theoretical model for a type I Weyl semimetal, under the presence of a diluted uniform concentration of torsional dislocations. By means of a mathematical analysis for partial wave scattering (phase-shift) for the T-matrix, we obtain the corresponding retarded and advanced Green’s functions that include the effects of multiple scattering events with the ensemble of randomly distributed dislocations. Combining this analysis with the Kubo formalism, and including vertex corrections, we calculate the electronic conductivity as a function of temperature and concentration of dislocations. We further evaluate our analytical formulas to predict the electrical conductivity of several transition metal monopnictides, i.e., TaAs, TaP, NbAs, and NbP.

In a WSM, the band structure possesses an even number of Weyl nodes with linear dispersion, where the conduction and valence bands touch.These nodes are monopolar sources of Berry curvature, and hence are protected from being gapped since their charge (chirality) is a topological invariant [7].In the vicinity of these nodes, low energy conducting states behave as Weyl fermions, i.e. massless quasi-particles with pseudo-relativistic Dirac linear dispersion [4][5][6][7].In Weyl fermions, conserved chirality determines the projection of spin over their momentum direction, a condition referred to as "spin-momentum locking".While Type I WSMs are Lorentz covariant, this symmetry is violated in Type II WSMs, where the Dirac cones are strongly tilted [9].
The presence of Weyl nodes in the bulk spectrum determines the emergence of Fermi arcs [8], the chiral anomaly, and the chiral magnetic effect, among other remarkable properties [9].Therefore, considerable attention has been paid to understand the electronic transport properties of WSMs [10][11][12].For instance, there are recent works on charge transport [13] in the presence of spin-orbit coupled impurities [14], electrochemical [15] and nonlinear transport induced by Berry curvature dipoles [16].Somewhat less explored are the effects of mechanical strain and deformations in WSMs.From a theoretical perspective, it has been proposed that different types of elastic strains can be modeled as gauge fields in WSMs [17][18][19].In previous works, we have studied the combined effects of a single torsional dislocation and an external magnetic field on the electronic [20,21] and thermoelectric [20,22] transport properties of WSMs, using the Landauer ballistic formalism in combination with a mathematical analysis for the quantum mechanical scattering cross-sections [23].
In this work, we extend our previous analysis to study the case of a diluted, uniform concentration of torsional dislocations and its effects on the electrical conductivity of type I WSMs.In contrast to our former studies [20][21][22], here we employ the Kubo linear-response formalism at finite temperatures, that therefore requires to explicitly calculate the retarded and advanced Green´s functions for the system, including the multiple scattering events due to the random distribution of dislocation defects in the form of a disorder-averaged self-energy term.For this purpose, we first analyze the phase shift arising from a single torsional dislocation, and obtain the corresponding (retarded and advanced) Green's function in terms of the T-matrix elements by solving analytically the Lippmann-Schwinger equation.We further extend this analysis, incorporating the effect of a random distribution of such dislocations, with a concentration n d , in the form of a disorderaveraged self-energy into the corresponding Dyson's equation.Finally, we analyze the correction to the scattering vertex, and by including this additional contribution we calculate the electrical conductivity from the Kubo formula, as a function of temperature and concentration of dislocations.We present explicit evaluations of our analytical expressions for the electrical conductivity as a function of temperature and concentration of dislocations n d , for several materials in the family of transition metals monopnictides, i.e.TaAs, TaP, NbAs and NbP, where the corresponding microscopic parameters, estimated by ab-initio methods, where reported in the literature [24][25][26].

II. SCATTERING BY A SINGLE DISLOCATION
As a continuum model for a type I WSM under the presence of a single dislocation defect, as depicted in Fig. (1), we consider the Hamiltonian [22] where Here, ξ = ± labels each of the Weyl nodes located at K ± = ±b/2.The expression in Eq. ( 2) is the free-particle Hamilto-nian, whereas the expression in Eq. ( 3) represents the interaction with the dislocation, where torsional strain is described as a pseudo-magnetic field inside the cylinder [21][22][23], as well as the lattice mismatch effect at the boundary of the dislocation, modeled as repulsive delta barrier on its surface [22].The "free" spinor eigenfunctions for the defect-free reference system satisfy where the energy spectrum is given by and λ = ±1 is the band (helicity) index.When projected onto coordinate space, these spinor eigenfunctions have the explicit form and constitute an orthonormal basis for the Hilbert space.
If we now consider the (elastic) scattering effects induced by the torsional dislocation modeled by Eq. ( 3), we need to look for the eigenvectors Ψ λ,k of the total Hamiltonian in Eq. ( 1) with the same energy as in Eq. ( 5).The answer is provided by the solution to the well known Lippmann-Schwinger equation where the free Green's function can be expressed in a coordinate-independent representation form via the resolvent,

Ĝξ
Here, the index R/A stands for retarded and advanced, respectively.As shown in detail in the Appendix A, in the coordinate representation the corresponding free Green's function is given by the explicit matrix form , where r = (x, z) and Here, H (1)  0 (z) and H (1)  1 (z) are the Hankel functions and x = (x, y) is the position vector on any plane perpendicular to the cylinder's axis.
For the scattering analysis, we need the retarded resolvent for the full Hamiltonian, which is defined as the solution to the equation Combining Eq. ( 10) with Eq. ( 8), we readily obtain FIG. 2. Pictorial description of the scattering event on a plane perpendicular to the cylindrical defect axis. Ĝξ where we introduced the standard definition of the T-matrix operator T ξ (E), that can be formally expressed in closed form by Using this definition, along with the property Ĥξ 1 |Ψ k,λ = T ξ |Φ k,λ , we obtain the Lippmann-Schwinger Eq. ( 7) in the coordinate representation As shown in detail in Appendix B, by considering the asymptotic behavior of the Hankel functions, H (1)  ν (x) ∼ 2 πx e i(x− νπ 2 − π 4 ) (for x → ∞.), Eq. ( 13) can be reduced to the x-y plane and takes the explicit asymptotic expression where as we explain in the Appendix, the particles have only momenta perpendicular to the defect's axis, i.e., k = (k x , k y ).
Comparing this last result with our previous reported expression for the scattering amplitude [27] we identify . Therefore, we arrived at an explicit analytical expression for the T-matrix elements in terms of the phase shift δ m (k) for each angular momentum channel m ∈ Z where φ is the angle between k and k , and the analytical expression for the phase shift is given in Appendix B by Eq. (B12).

III. SCATTERING BY A UNIFORM CONCENTRATION OF DISLOCATIONS
FIG. 3. Random distribution of torsional dislocations seen from a plane perpendicular to the cylinders axis.
Let us now consider a uniform concentration n d = N d /A (per unit transverse surface) of identical cylindrical dislocations, as depicted in FIG. 3, represented by the density function where X j is the position of the j-defect's axis.The Fourier transform of this density function is thus given by the expression The operator that plays the role of a scattering potential for this distribution of dislocation defects is where H ξ 1 is defined in Eq. ( 3) as the contribution from a single dislocation.The matrix elements of the scattering operator Eq. ( 19) in the free spinor basis defined by Eq. ( 4) are where Ṽ(k ) is the Fourier transform Then, the matrix elements of the potential in Eq. ( 20) become Let us also introduce the configurational average of some quantity f (X j ) over the distributed dislocations as where P(X j ) is the normalized distribution function for the defects in the sample.In particular, for a uniform distribution we have P(X j ) = 1/A, where A is the area of the plane normal to each cylinder's axis.Now, the full retarded Green's function satisfies for the potential of several dislocations V given in Eq. ( 19) The configurational average, as defined in Eq. ( 23), of the complete Green's function in this last equation can be written as Ĝξ This is the Dyson's equation with the retarded self-energy Σ λ,ξ R (E), that can be explicitly solved to yield The effect of the statistical distribution of dislocations' is entirely dictated by the function ρ(k ).In the perturbative expansion of the complete Green's function, we encounter n thproducts of the form ρ(k 1 ) ρ(k 2 ) • • • ρ(k n ).The configurational average of these products are for a single factor.For the product of two factors, we obtain and we have a similar behavior for higher order products.Now, notice that for d and so on.We define the concentration of defects, i.e., the number of dislocations per unit of area perpendicular to the cylinder's axis as n d = N d /A.As discussed in standard references [28,29], for small concentrations n d 1 the scaling discussed before ensures that the total Green's function in Eq. ( 25) can be calculated accurately by the sequence of diagrams for the retarded selfenergy in momentum space as given in FIG. 4, an approach well known as the non-crossing approximation (NCA).This series of diagrams corresponds to the configurational average of the T -matrix over the random distribution of dislocations after Eq. ( 23) Using the expression in Eq. ( 16) for the T -matrix elements, for k = k then φ = 0 and we have that the real part of the self-energy contains an infinite sum over highly oscillatory terms, that converges to zero.So no contribution comes from the real part of the self-energy; The imaginary part, on the other hand, defines the relaxation time, A. Electrical Conductivity in the linear-response Now, we consider a single Fourier mode for an external electric field in the gauge where A(r, t) = A(r, ω)e −iωt is the vector potential.Then, E = iωA.In the linear response formalism, the current is given by the Kubo expression Here, the conductivity tensor is given by The tensor K αβ is defined, in the Kubo formalism, in terms of the retarded current-current correlator as follows where ρ is the statistical density matrix operator.As shown in detail in the Appendix D, the Fourier transform to the frequency domain of this tensor can be expressed by where f 0 (E) = e (E−µ)/kT + 1 is the Fermi distribution, and we introduced the (disorder-averaged) spectral function that clearly reduces to a Lorentzian distribution whose spectral width is defined by the inverse of the relaxation time.See Appendix C for the details.After some algebraic manipulations, we obtain the conductivity tensor at finite frequency and temperature Using the coordinates representation of the spectral function given in Appendix C, after Eq. (C6), we can read off the Fourier transform to momentum space of the conductivity We are interested in the DC conductivity, so we take the limit q → 0 first and then the limit ω → 0. After a long calculation (details in the Appendix D), the result is

B. Vertex corrections
The self-energy contribution modifies the definition of the retarded and advanced Green´s functions in Eq. ( 40), as depicted by the double lines in Fig. 5(b).However, there are also scattering processes involving links between the two internal Green function lines, as depicted in FIG.5(a).When considering such diagrams with cross-links, as in FIG.5(a), we must include the vertex correction as depicted in FIG.5(b).Taking into account the vertex correction, the conductivity becomes where the vertex function Γ RA (k , E) is given as the solution to the Bethe-Salpeter equation as depicted in FIG. 6.Then, we have The iterative solution of the Eq. ( 42) for Γ RA (k , E) shows that the vertex function must be of the form Then we obtain an integral equation for the scalar function γ(k , E) that in the low concentration limit becomes In the limit of low concentrations, we use the result in Appendix D, Eq.(D13), to obtain At low temperatures, an exact solution is possible since the Fermi distribution derivative takes a compact support at the Fermi energy.Then we can evaluate γ(k) and τ (λ,ξ) (k) at the Fermi momentum k ξ F , to obtain where we defined (for cos (47) After the substitution in Eq. (45) of γ(k , E) given in Eq. ( 44), we get where Li 2 (x) is the polylogarithm of order 2. Here, the total transport relaxation time is defined by Using the form of the T -matrix elements in Eq. ( 16), we get a closed expression in terms of the scattering phase shifts δ m (k) From Eq. ( 48), we can investigate the zero temperature T → 0 and high temperature T v F k F /k B limits, respectively.In the zero temperature limit, we obtain a constant that depends on the miscroscopic material properties (such as v F ), as well as on the concentration of dislocations n d through the relaxation time.
On the other hand, in the high-temperature limit T v F k F /k B , we obtain a quadratic dependence on temperature where the overall constant depends on the microscopic parameters for each material, as well as on the concentration of dislocations through the relaxation time.

IV. RESULTS
In this section, we apply the theory and analytical expressions obtained in the previous section to calculate the electrical conductivity of several materials in the family of transition metals monopnictides, i.e.TaAs, TaP, NbAs and NbP.For an estimation of the concentration of defects n d in real crystal systems, the Ref. [24] reports that the native concentration of dislocations in the lattice of the materials TiO 2 and SrTiO 3 , vary in the range n d ∼ 10 5 − 10 7 cm −2 .These concentrations can be enhanced using different treatments up to 10 13 cm −2 , nearly to the rendering amorphous limit.The microscopic/atomistic parameters involved in our theory are obtained from ab-initio studies for WSM materials, as reported in Ref. [25] and Ref. [26].In particular, the later reference identifies anisotropies in the Fermi velocities and density of charge carries at different Weyl nodes and bands.Using these results for the densities of carriers, we compute the Fermi momentum at each Weyl node, i.e., k ξ F , as displayed in TABLE I.In what follows, for definiteness we shall assume that the axis of the defects is along the crystallographic z-direction and that we are measuring the conductivity along the xdirection.Then, we use the reported x-components of the Fermi velocities [25,26].We have different Fermi velocities v (λ,ξ) F,x , for the conduction band (λ = +1) and for the valence band (λ = −1), and for each of the Weyl nodes (ξ = ±).
Actually, for the valence band Refs.[25,26] report the hole velocity.Their results are presented in the TABLE II.F,x in units of 10 5 m/s, as reported in Ref. [26].Notice that for the valence bands (λ = −1) they report the hole velocity.Now, in order to study the additional effect of the torsional dislocations, we follow our previous work [22].We assume that the dislocations are cylindrical regions along the z-axis with radius a.Here, we further assume that the defects possess an average radius of a = 15 nm.The simple relation between the torsional angle θ (in degrees) and the pseudomagnetic field representing strain is B S a 2 = 1.36 θ φ0 [22], where the modified flux quantum in this material is approx- 1.5 300 • 4.14 × 10 5 T Å2 ≈ 330 T Å2 .In this work, we have chosen a torsion angle θ = 15 • .The lattice mismatch effect at the surface of the dislocation cylinders is modeled by a repulsive delta-potential, with strength V 0 , expressed in terms of the "spinor rotation" angle α = V 0 / v F .According to our previous work [22], a realistic choice is α = 3π/4.With all of these parameters fixed, we can compute the total relaxation time for each material.Our results are presented in TABLE III.Now, we compute the conductivity along the x-direction σ xx .In what follows, we simply call it σ(T ), as a function of temperature.The total conductivity is the sum over nodes and bands where σ (λ,ξ) xx (T ) is given in Eq. ( 48), including the vertex correction.Our results for T = 0 are presented in the TABLE IV.
The conductivity as a function of temperature, for the transition metals monopnictides TaAs, TaP, NbAs and NbP, is presented in FIG.7 for all of them compared, and individually in the pannel FIG. 8 (a)-(d).
Now, let us study the conductivity behavior with respect to the density of dislocations n d .In FIG. 9, we present a plot of the natural logarithm of the conductivity versus temperature for three different concentrations of dislocations.The total < l a t e x i t s h a 1 _ b a s e 6 4 = " T r e 4 t l l 6 F 1 V v U u q u f 3 5 5 X a T X 6 I I j k i x + S U e O S S 1 M g d q Z M m 4 S Q m z + S N v D v K e X J e n N e f a M H J / x y S P 3 I + v g F t N I 8 I < / l a t e x i t > TaAs < l a t e x i t s h a 1 _ b a s e 6 4 = "   conductivity as a function of the concentration of defects and at zero temperature is presented in FIG.10.Finally, a plot of the resistance, defined as the inverse of conductivity, as a function of the dislocations' density is presented in FIG.11.

V. CONCLUSIONS
In this work, we have studied the effect of a distribution of mechanical defects, i.e. torsional dislocations, over the electrical conductivity of the family of transition metals monopnictides TaAs, TaP, NbAs and NbP.Our theory is based on the mathematical analysis of the scattering phase shifts from a single defect, as stated in our previous work [20][21][22][23]27].We extended this previous analysis to develop a Green´s function formalism, in order to represent the scattering due to a finite concentration of randomly distributed defects.Within the non-crossing approximation for the self-energy, we solved explicitly for the disorder-averaged retarded Green´s function, that allows us to calculate the electrical conductivity in the Kubo linear-response formalism.We obtained general analytical expressions in terms of the parameters involved in the low-energy model representing the family of materials, and using the ab-initio estimations for such parameters, we provided a characterization of the conductivity as a function of temperature and concentration of defects for the transition metal monopnictides TaAs, TaP, NbAs and NbP.As a universal feature, we identified a ∼ T 2 temperature dependence for T v F k F /k B , where the pre-factor depends on materialspecific microscopic parameters as well as in the concentration of dislocations n d through the scattering relaxation time.
Our results do not involve the electron-phonon scattering effects, that will presumably contribute at higher temperatures, which is a subject of further study.

Appendix A: Calculation of the retarded free Green's function
The free retarded Green's function (GF) is represented in the coordinate basis as follows In this basis, G ξ R,0 (r, r ; E) satisfies the differential equation where σ 0 is the 2 × 2 unit matrix.Let us introduce the scalar GF, G ξ R,0 (r, r ), by means of the expression Bearing in mind that we are treating the elastic scattering problem, the energy of the out-state must be the same as those of the incident-free-particle state, i.e., E = λξ v F |k|.Then, the scalar GF satisfies the Helmholtz equation Due to the symmetry along the z-axis we can decouple it from its perpendicular plane as follows where G ξ R,0 (x, x ; q z , k) is a reduced GF and x = (x, y) is the position vector on the plane.Then, the Helmholtz equation for the reduced GF on the plane takes the form x +∂ 2 y .As we can be seen from the FIG. 1, the free incident particle's propagation is normal to the cylinder's axis.We assume that the incident particles have negligible momentum along the z-axis, and by momentum conservation, they remain with negligible momentum along that direction during the transport process.Then, we can write k = (k , 0) where k = (k x , k y ).Hence, the system is reduced to an effective two-dimensional description and we can consider the reduced GF on the plane as independent of the Fourier mode q z .Then, from Eq.(A5) we have G ξ R,0 (r, r ; k) = δ(z − z )G ξ R,0 (x, x ; k), and we can expand the reduced GF on the plane in the traverse Fourier space     where q = (q x , q y ).Replacing in the Eq.(A6), we obtain in the traverse Fourier space We perform the integration in Eq.(A7) in polar coordinates where R = |x−x | and we have used the integral representation of the Bessel functions 2π 0 e iz cos φ±inφ dφ = 2πi n J n (z).In order < l a t e x i t s h a 1 _ b a s e 6 4 = " R q w X p p J i T G f N 6 U B o 4 C g n F h j X w m 5 J + Y h p x t H e p 2 T r e 4 t l l 6 F 1 V v U u q u f 3 5 5 X a T X 6 I I j k i x + S U e O S S 1 M g d q Z M m 4 S Q m z + S N v D v K e X J e n N e f a M H J / x y S P 3 I + v g F t N I 8 I < / l a t e x i t > TaAs < l a t e x i t s h a 1 _ b a s e 6 4 = " j W t g t K R 8 y z T j a 8 5 R s f W + + 7 C I 0 T 6 r e W f X 0 9 r R S u 8 w P U S Q H 5 J A c E 4 + c k x q 5 J n X S I J w o 8 k z e y L s T O U / O i / P 6 E y 0 4 + Z 9 9 8 k f O x z e n g Y 6 V < / l a t e x i t > NbP < l a t e x i t s h a 1 _ b a s e 6 4 = " to perform the last integration we need the result together with the relation K n (z) = π 2 i n+1 H (1)  n (iz).The result is This form is adequate because in the asymptotic form for large |x − x | it produces outgoing cylindrical waves as it is desired for the retarded GF.Now, to obtain the final form for the free GF matrix we apply the definition in Eq.( A3) with E = λξ v F |k|, taking into account that we have reduced to a < l a t e x i t s h a 1 _ b a s e 6 4 = " g n F h j X w m 5 J + Y h p x t H e p 2 T r e 4 t l l 6 F 1 V v U u q u f 3 5 5 X a T X 6 I I j k i x + S U e O S S 1 M g d q Z M m 4 S Q m z + S N v D v K e X J e n N e f a M H J / x y S P 3 I + v g F t N I 8 I < / l a t e x i t > TaAs < l a t e x i t s h a 1 _ b a s e 6 4 = "  two dimensional system on the plane x-y In plane polar coordinates where r = |x − x |, ϕ is the angle the vector x − x makes with the x axis and The final form for the retarded Green's function matrix in the coordinates representation is which produces Eq.( 9).We can represent the Lippmann-Schwinger Eq.( 7) in the coordinate basis as follows The form of free spinors in Eq.( 6) with momentum k on the x-y plane is where k x = k cos φ and k y = k sin φ.The incident spinors are assumed to enter the scattering region with momentum along the x axis, and they are represented by Now, Ĥξ 1 is a local potential independent of the z coordinate as can be seen from Eq.(3).Then, the T -matrix is diagonal in the coordinate basis and depends only on vectors x on the plane.Thus, r T ξ (E) r = T ξ (x , E)δ (3) (r − r ), where T ξ (x , E) is a 2 × 2 matrix.The incident spinor is given in Eq.(B3), and using the retarded GF in Eq.(A15), the Lippmann-Schwinger equation in Eq.(B1) is reduced to the x-y plane as follows where G ξ R,0 (x, x ; k) is given in Eq.( 9).The asymptotic form for large argument of the Hankel's functions are where we have used the known limiting form Now, recall the geometry of the scattering process as depicted in FIG. 2. We expand |x − x | for large |x| as follows where r = |x|, r |x | and n is the unit vector the direction of x, i.e., n = x/r.Noting that in this asymptotic form the direction of k coincides with that of x and is practically the same of x − x , i.e., k = k n and that the angle φ the vector k makes with the k incident momentum is approximately the same angle x − x does, i.e., φ ∼ ϕ, we have the asymptotic form for the free Green's function in Eq.( 9) Replacing the asymptotic form in Eq.(B9) in the Eq.(B4) we obatain Eq.( 14) where the T -matrix elements are In order to compute the T -matrix elements we need the phase shifts, whose analytical expression is presented in Eq.(32) of the supplemental material of our previous work Ref. [22].
Here we reproduce the final result where z a = |B ξ |a 2 /2 φ0 (a is the cylinder's radius).
Appendix C: The spectral function The spectral function can be defined as follows in terms of the complete retarded and advanced Green's functions.Then, the spectral function is Hermitian Â ξ (E) † = Â ξ (E).Given the averaged complete retarded Green's function in Eq.( 26), the form of the spectral function in momentum space is Clearly, it takes the form of a Lorentzian distribution with compact support around the free particle's energy where τ (λ,ξ) (k) is the relaxation time and E λ,ξ k = λξ v F k.In the limit of low concentration of defects, i.e., large relaxation time because of Eq.(31), the spectral function becomes a delta distribution Due to its behavior as a Lorentzian, the spectral function has the important property [29] Representing the spectral function Eq.(C1) in the coordinate basis using the complete set of eigenstates of the full Hamiltonian we have (C6) Notice that because k = (k x , k y ), when we perform the integration the spectral function takes the form A ξ (r, r ) = δ(z − z )A ξ (x, x ), which looks similar to the decoupled form of the GF in Eq.(A15).Then, the Fourier transform to the momentum space of the conductivity is the result in Eq.(39).We are computing the DC conductivity, so we take the limit q → 0 first and then the limit ω → 0. The result is where we have used the definition of the spectral function in terms of the retarded and advanced GFs.In the limit of low concentration of defects, i.e., n d → 0, we have that the unique leading contribution to the conductivity are given by the combination where we have used that the self-energy is purely imaginary and its relation with the relaxation time.The other contributions are negligible because they are not singular in n d .For instance, in the low concentration limit, we have a contribution for two retarded GFs of the form The second term is zero and the first term is a function centered at E − λξ v F k, but when integrated over k it gives zero.Something similar occurs with the contribution of two advanced GFs.The result is the diagonal conductivity tensor given in Eq.(40).

FIG. 1 .
FIG. 1. Pictorial description of the scattering of free incident Weyl fermions coming from a left reservoir by a single cylindrical dislocation defect.

FIG. 4 .
FIG. 4. Diagrams contributing to the retarded self-energy Σ R .The solid line corresponds to the free retarded Green's function, the dashed line the scattering perturbation H 1 , and the × a factor of n d .

FIG. 5 .
FIG. 5. (a) A typical diagram contributing to the conductivity in Eq (40), involving the configurational average of the two internal GF with cross-links among them.The upper line corresponds to the retarded GF and the lower to the advanced GF.(b) Diagrammatic representation of the two complete averaged GF (double lines) corresponding to the sum of all diagrams of the kind in (a) with the vertex correction Γ(k ).

Material σ 0 6 TABLE
IV. Computed values for the total conductivity σ 0 = σ(T = 0) at zero temperature for each material.We consider a value of n d = 10 11 cm −2 .
H O k / P i v P 5 E C 0 7 + 5 5 D 8 k f P x D a 8 H j p o = < / l a t e x i t > TaP < l a t e x i t s h a 1 _ b a s e 6 4 = "Q B 7 y U E x h S S A 7 Z D 7 E t M 2 Q X m 9 S Y K k = " > A A A B 6 3 i c b Z C 9 T s M w F I W d 8 l f K X 4 G R x a J C Y q o S h I C x g o U J F Y n + i D a q H P e m t e o 4 k X 2 D q K I + B U w I 2 H g b X o C 3 w S 0 Z o O V M n + 8 5 l u 6 5 Q S K F Q d f 9 c g p L y y u r a 8 X 1 0 s b m 1 v Z O e X e v a e J U c 2 j w W M a 6 H T A D U i h o o E A J 7 U Q D i w I J r W B 0 N f V b D 6 C N i N U d j h P w I z Z Q I h S c o R 3 d d x E e Mb s J 6 p N e u e J W 3 Z n o I n g 5 V E i u e q / 8 2 e 3 H P I 1 8 y z T j a 8 5 R s f W + + 7 C I 0 T 6 r e W f X 0 9 r R S u 8 w P U S Q H 5 J A c E 4 + c k x q 5 J n X S I J w o 8 k z e y L s T O U / O i / P 6 E y 0 4 + Z 9 9 8 k f O x z e n g Y 6 V < / l a t e x i t > NbP < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 d

FIG. 7 .
FIG. 7. A comparison of the total conductivity versus temperature behavior for the transition metals monopnictides TaAs, TaP, NbAs and NbP.We use a value of n d = 10 11 cm −2 .
t e x i t s h a 1 _ b a s e 6 4 = " + f 7 u b B s P x p P x Y r z + W D N G e r N P / s D 4 + A Z O U 5 U E < / l a t e x i t > nd = 10 11 cm 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " S / 8 L 0 n b I B 3 u j 8 P n m r X j A b s Z m 4 7 s = " > A A A B / n i c b V D L S g N B E J y N r x h f U Y + 5 L A b B g 4 b d E N S L E P 1 4 1 m I x D x 0 A 2 n i C M z Z / N s b U 0 + p i e d o j 0 d x p O a 1 6 f I / r R 2 h e 9 6 N u R 9 G C D 7 T F q 2 5 k T A x M K d d m A M u g a G Y a E K Z 5 P p L k 4 2 o p A x 1 Y z k d 3 5 4 P u 0 g a 5 Z J 9 W q r c V I r V y 7 S I L C m Q A 3 J E b H J G q u S a 1 E i d M P J I n s k b e T c e j C f j x X j 9 s W a M 9 G a f / I H x 8 Q 1 P 4 p U F < / l a t e x i t > nd = 10 12 cm 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " O c F 9 P m z J 3

2 FIG. 9 .
FIG. 9. Natural logarithm of the conductivity versus temperature for 3 different concentrations of dislocations.The graphs were computed at zero temperature.
y y 7 D K 2 z q n d R P b 8 7 r 9 S u 8 0 M U y R E 5 J q f E I 5 e k R m 5 J n T Q J J x F 5 J m / k 3 V H O k / P i v P 5 E C 0 7 + 5 5 D 8 k f P x D a 8 H j p o = < / l a t e x i t > TaP < l a t e x i t s h a 1 _ b a s e 6 4 = " Q B 7 y U E x h S S A 7 Z D 7 E t M 2 Q X m 9 S Y K k = " > A A A B 6 3 i c b Z C 9 T s M w F I W d 8 l f K X 4 G R x a J C Y q o S h I C x g o U J F Y n + i D a q H P e m t e o 4 k X 2 D q K I + B U w I 2 H g b X o C 3 w S 0 Z o O V M n + 8 5 l u 6 5 Q S K F Q d f 9 c g p L y y u r a 8 X 1 0 s b m 1 v Z O e X e v a e J U c 2 j w W M a 6 H T A D U i h o o E A J 7 U Q D i w I J r W B 0 N f V b D 6 C N i N U d j h P w I z Z Q I h S c o R 3 d d x E e M b s J 6 p N e u e J W 3 Z n o I n g 5 V E i u e q / 8 2 e 3 H P I 1

FIG. 10 .
FIG.10.Plot of total conductivity versus defects' concentration.The graphs were computed at zero temperature.

y y 7 D
K 2 z q n d R P b 8 7 r 9 S u 8 0 M U y R E 5 J q f E I 5 e k R m 5 J n T Q J J x F 5 J m / k 3 V H O k / P i v P 5 E C 0 7 + 5 5 D 8 k f P x D a 8 H j p o = < / l a t e x i t > TaP < l a t e x i t s h a 1 _ b a s e 6 4 = " QB 7 y U E x h S S A 7 Z D 7 E t M 2 Q X m 9 S Y K k = " > A A A B 6 3 i c b Z C 9 T s M w F I W d 8 l f K X 4 G R x a J C Y q o S h I C x g o U J F Y n + i D a q H P e m t e o 4 k X 2 D q K I + B U w I 2 H g b X o C 3 w S 0 Z o O V M n + 8 5 l u 6 5 Q S K F Q d f 9 c g p L y y u r a 8 X 1 0 s b m 1 v Z O e X e v a e J U c 2 j w W M a 6 H T A D U i h o o E A J 7 U Q D i w I J r W B 0 N f V b D 6 C N i N U d j h P w I z Z Q I h S c o R 3 d d x E e M b s J 6 p N e u e J W 3 Z n o I n g 5 V E i u e q / 8 2 e 3 H P I 1 A I Z f M m I 7 n J u h n T K P g E i a l b m o g Y X z E B t C x q F g E x s 9 m G 0 / o U R h r i k O g s / f v b M Y i Y 8 Z R Y D M R w 6 G Z 9 6 b D / 7 x O i u G F n w m V p A i K 2 4 j 1 w l R S j O m 0 O O 0 L D R z l 2 A Lj W t g t K R 8 y z T j a 8 5 R s f W + + 7 C I 0 T 6 r e W f X 0 9 r R S u 8 w P U S Q H 5 J A c E 4 + c k x q 5 J n X S I J w o 8 k z e y L s T O U / O i / P 6 E y 0 4 + Z 9 9 8 k f O x z e n g Y 6 V < / l a t e x i t > NbP < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 d f K F j e z U S Q H 4 2 V y o 7 0 / R m t L v x I = " > A A A B 7 H i c b Z C 9 T s M w F I W d 8 l f K X 4 G R x a J C Y q o S h I C x w M K E i k R / p D S q H P e 2 t e o 4 k X 2 D q K K + B U w I 2 H g a X o C 3 w S 0 Z o O V M n + 8 5 l u 6 5 Y S K F Q d f 9 c g p L y y u r a 8 X 1 0 s b m 1 v Z O e X e v a e J U c 2 j w W M a 6 H T I D

4 FIG. 11 .
FIG. 11.Total resistance, R = 1/G, as a function of the concentration of defects n d for the family of materials TaAs, TaP, NbAs and NbP.The graphs were computed at zero temperature.

Appendix B:
Scattering by a single cylindrical defect

TABLE II .
Values of the Fermi velocity v(λ,ξ) Material τ [10 −13 s] τ tr[10 −13 s] TABLE III.Computed values for the total relaxation time and the total transport relaxation time for each material.We consider a concentration of dislocations n d = 10 11 cm −2 .