Transversal magnetoresistance and Shubnikov-de Haas oscillations in Weyl semimetals
We explore theoretically the magnetoresistance of Weyl semimetals in transversal magnetic fields away from charge neutrality. The analysis within the self-consistent Born approximation is done for the two different models of disorder: (i) short-range impurties and (ii) charged (Coulomb) impurities. For these models of disorder, we calculate the conductivity away from charge neutrality point as well as the Hall conductivity, and analyze the transversal magnetoresistance (TMR) and Shubnikov-de Haas oscillations for both types of disorder. We further consider a model with Weyl nodes shifted in energy with respect to each other (as found in various materials) with the chemical potential corresponding to the total charge neutrality. In the experimentally most relevant case of Coulomb impurities, we find in this model a large TMR in a broad range of quantizing magnetic fields. More specifically, in the ultra-quantum limit, where only the zeroth Landau level is effective, the TMR is linear in magnetic field. In the regime of moderate (but still quantizing) magnetic fields, where the higher Landau levels are relevant, the rapidly growing TMR is supplemented by strong Shubnikov-de Haas oscillations, consistent with experimental observations.
One of the central research directions in condensed matter physics addresses topological materials and structures. Recently, a novel type of topological materials has received much attention: Weyl and Dirac semimetals. The quasiparticle spectrum near the nodal point of a Dirac semimetal is described by a three-dimensional (3D) Dirac Hamiltonian where excitations close the crossing point of valence and conduction bands disperse linearly. The materials CdAs M. Neupane et al. (2014) and NaBi Liu et al. (2014) represent experimental realizations of Dirac semimetals. For either broken spatial inversion or time-reversal symmetry, the four-component solution of the Dirac equation splits into two independent two-component Weyl fermions of opposite chirality with the Weyl points in the spectrum located at distinct momenta. Recent experiments classify TaAs Lv et al. (2015); Xu et al. (2015a), NbAs Xu Su-Yang et al. (2015), TaP Xu et al. (2015b), and NbP Shekhar Chandra et al. (2015) as Weyl semimetals. Further promising candidates for Weyl semimetals include pyrochlore iridates Wan et al. (2011) and topological insulator heterostructures Burkov and Balents (2011). In the rest of the paper, we will use the term “Weyl semimetal” in a broader sense, including also the degenerate case of Dirac semimetals.
Transport properties of Weyl semimetals are highly peculiar. For recent theoretical studies, see, e.g., Refs. Burkov et al. (2011); Hosur et al. (2012); Vafek and Vishwanath (2014); Burkov (2015a); Ominato and Koshino (2014); Sbierski et al. (2014); Son and Spivak (2013); Biswas and Ryu (2014); Syzranov et al. (2015a); Altland and Bagrets (2015); Burkov (2014, 2015b); Syzranov et al. (2015b); Sbierski et al. (2015); Ominato and Koshino (2015); Rodionov and Syzranov (2015); Wan et al. (2011); Zyuzin (2017) and references therein. An important aspect of the transport properties is the appearance of a disordered critical point within the perturbative analysis. Below the disordered critical point (i.e., for sufficiently weak disorder), the density of states vanishes quadratically in energy around the Weyl point within the perturbation theory. Non-perturbative treatment yields an exponentially small density of states at the Weyl point. In the strong disorder regime, the density of states is finite at the Weyl point already without invoking exponentially small contributions.
The transport in Weyl semimetals reveals a particularly interesting and rich physics when an external magnetic field is applied. One reason for this is the unconventional Landau quantization of Dirac fermions. Further, a single species of Weyl fermions displays the chiral anomaly that gives rise to a possibility of controlling the valley polarization. A strong anomalous Hall effect Vafek and Vishwanath (2014); Burkov (2015a); Vazifeh and Franz (2013) and the longitudinal magnetoresistivity Son and Spivak (2013); Gorbar et al. (2014); Burkov (2014); Lucas et al. (2016); Burkov (2015b); Goswami et al. (2015); Ghimire et al. (2015); Arnold Frank et al. (2016); Shekhar Chandra et al. (2015); Behrends and Bardarson (2017) in Weyl semimetals have been predicted to originate from the chiral anomaly. Furthermore, thermoelectrical effects Gooth Johannes et al. (2017) and induced superconductivity Bachmann et al. (2017) have been studied recently, both theoretically and experimentally.
In this paper, we present a theory of the transversal magnetoresistivity in a Weyl semimetal away from charge neutrality point. (The term “transversal” here means that the magnetic field is perpendicular to the electric field: the relevant resistivity component is , while the magnetic field is along the axis.) The work is motivated by the spectacular experimental observation of a large, approximately linear transversal magnetoresistance (TMR) in Dirac and Weyl semimetals Liang Tian et al. (2015); Feng et al. (2015); Shekhar Chandra et al. (2015); Niemann Anna Corinna et al. (2017); Novak et al. (2015). Theoretically, a linear TMR of a system with Dirac dispersion in the ultra-quantum limit (where only the zeroth Landau level is effective) was obtained by Abrikosov in a seminal paper, Ref. Abrikosov (1998). The crucial ingredient of this result is the dependence of the screening of Coulomb impurities on magnetic field. In a previous work, Ref. Klier et al. (2015), we have carried out a systematic analysis of the magnetoresistivity of a Weyl semimetal at the neutrality point and for different types of disorder. Our results for the case of Coulomb impurities and in strongest magnetic fields yield the linear TMR, in agreement with Ref. Abrikosov (1998). This is not sufficient, however, to explain experimental data since experiments are performed at non-zero electron density. A clear experimental evidence of finite density is provided by Shubnikov-de Haas oscillations (SdHO) superimposed on the background of strong linear TMR in an intermediate range of magnetic fields. It is thus a challenge to understand whether the strong quantum linear TMR and the SdHO may emerge from the theory of disordered Weyl fermions. More generally, our goal is to develop the theory of quantum magnetotransport for systems with Dirac spectrum at non-zero density (chemical potential) of carriers.
Below, we calculate the TMR and the Hall conductivity for arbitrary magnetic field and arbitrary particle density. Depending on their values, the dominant contribution to the TMR comes from the zeroth Landau level (LL), separated LLs, or overlapping LLs. This includes also regimes where the SdHO can be observed. Our analysis has a certain overlap with a recent preprint, Ref. Xiao et al. (2016), where the Born approximation (without self-consistency) was used. We go beyond that work by employing the self-consistent Born approximation (SCBA), analyzing the scaling of conductivities and of TMR in various regimes, and discussing two models of disorder—(i) short-range impurities and (ii) charged (Coulomb) impurities. Further, we study the TMR for two cases—fixed particle density and fixed chemical potential—and find that the results are essentially different.
In the experimentally most relevant case of Coulomb impurities and a fixed particle density, we find a large, linear TMR in the ultra-quantum limit, where only the zeroth Landau level is effective. We show, that even though the analytical result for the resistivity is modified in comparison to that of Ref. Abrikosov (1998) due to a non-zero value of the Hall conductivity, the linear-in- scaling of TMR remains valid. In the regime of moderate (but still quantizing) magnetic fields, where the higher Landau levels are relevant, the TMR curves contain Shubnikov-de Haas peaks whose amplitude grows as a power-law function () of magnetic field. At the same time, the “background” TMR (the envelope of the minima) in such magnetic fields is negligible within the SCBA. Thus, the model with a single type of Weyl nodes does not contain a regime where a strong TMR is supplemented by SdHO, in agreement with the numerical findings of Ref. Xiao et al. (2016).
We further consider a model with Weyl nodes shifted in energy with respect to each other, with the chemical potential corresponding to the total charge neutrality, as illustrated in Fig. 1. Such type of spectrum has been found in various materials both experimentally and by first-principle calculations, see, e.g., Refs. Shekhar Chandra et al. (2015); Niemann Anna Corinna et al. (2017). In this situation, the total Hall conductivity is zero, whereas the shifted pairs of Weyl nodes are characterized by equal carrier (electrons and holes, respectively) densities. For Coulomb impurities, we find in this model a large TMR in a broad range of quantizing magnetic fields. In the ultra-quantum limit, where only the zeroth Landau level is effective, the TMR is again linear in magnetic field. At lower magnetic fields, in the regime of separated LLs, strong SdHO are superimposed on top of a rapidly growing background TMR, in contrast to the case of non-shifted Weyl nodes. Specifically, the envelope of the minima of TMR behaves as , while the maxima evolve as . The overall behavior of the TMR resembles that found in experiments: with increasing magnetic field the (almost linear) TMR shows SdHO and crosses over into a purely linear TMR with no SdHO. Such a behavior emerges when the conductivity in a strong magnetic field is larger (due to the compensation between the shifted nodes) than the total Hall conductivity . This can be realized for shifted Weyl nodes away from the charge neutrality point (where the Hall resistivity is finite), provided that the concentrations of positively and negatively charged impurities are close to each other.
The analysis in this paper is performed in the framework of the SCBA for non-interacting fermions. This discards other possible contributions to the TMR, including the classical memory effects (as discussed in the context of Weyl semimetals in a recent paper, Ref. Song et al. (2015)) and interaction-related mechanisms. We will return to a discussion of such magnetoresistance mechanisms in the end of the paper.
The paper is organized as follows. Section II is devoted to an introduction to the model of impurity scattering. In Sec. III, we calculate the conductivity away from charge neutrality in a finite transverse magnetic field for the model of white-noise disorder. Section IV presents the analysis of the Hall conductivity for the clean case and for the white-noise disorder. In Sec. V, we use the obtained results to calculate and analyze the TMR. In Sec. VI, we extend our analysis to the case of charged impurities. Section VII discusses the TMR at the total charge compensation point for the pairs of Weyl nodes shifted in energy with respect to each other. We summarize our findings and discuss the experimental relations to experiments in Sec. VIII. Throughout the paper we set .
In this section, we introduce the framework Klier et al. (2015) for studying disordered Weyl fermions that will be used throughout the paper. We start from the Hamiltonian for a single Weyl fermion in the presence of a finite magnetic field directed along the axis. The Hamiltonian in the Landau gauge for a clean system is given by
where p is the momentum operator, is the velocity, denotes the Pauli matrices and is the vector potential.
Now we include disorder. The impurity scattering generates a self-energy in the (impurity-averaged) Green’s function, which reads
The Green’s function is a matrix in the pseudospin space (in which the Pauli matrices operate). We will assume that the disorder potential is diagonal in both spin and pseudospin indices and neglect scattering between different Weyl nodes. Clearly, in the absence of internode scattering, the structure in the node space will be trivial for all quantities; the density of states and the conductivities calculated below are those per Weyl node. Under these assumptions, the pointlike impurity potential has the form
where is the unit matrix in the pseudospin space.
In view of the matrix structure of the impurity potential , the impurity correlator becomes a rank-four tensor. Within the self-consistent Born approximation (SCBA), the self-energy reads
For a diagonal impurity potential, the impurity correlator is diagonal as well, which is expressed as
where . We will later generalize the results obtained for white-noise disorder (5) to the case of Coulomb impurities. Similarly to the case of zero magnetic field, we introduce a parameter defined as
where is the ultraviolet energy cutoff for energy (band width). In the following, we will mainly focus on the case of not too strong disorder, .
The self-energy is diagonal in the energy-band space. However, in the presence of magnetic field, the self-energy is no longer proportional to the unit matrix. This asymmetry originates from the asymmetry of states in the zeroth LL. In the clean case, the states of the zeroth LL are only present in one energy band. Note that a strong impurity scattering eliminates this asymmetry. In what follows, it is convenient to switch to the LL representation such that and . The diagonal components of the matrix self-energy determined with the Green’s function (2) read (below ):
Here we introduced the energy scale ,
that combines the disorder coupling and the strength of magnetic field characterized by the distance between the zeroth and first LLs. In general, the self-energy depends on energy and on the LL index, . However, for the white-noise disorder, the dependences on and drop out.
For energies close to the Weyl point, , and for weak disorder, , the asymmetry with respect to the zeroth LL should be taken into account. When the lowest LL is well separated from the others, , the contribution of the sum over is dominated by the term. In this case, we get
Thus, is negligible in the limit of weak disorder.
For energies away from the Weyl point, , and weak disorder, , the asymmetry induced by the zeroth LL is negligible: , where the LL broadening is determined by the self-consistent equation
The solution of Eq. (11) gives a nonsymmetric peak of around the th LL located at with
where marks the energy below which the LLs are fully separated. A detailed analysis of the broadening of LLs reveals that the LLs are separated up to , but for energies in the range the background density of states is larger than the density of states for the particular LL as shown in Fig. 2 (for further details, see Ref. Klier et al. (2015)).
Iii Conductivity away from charge neutrality
Using the introduced model, we calculate now the conductivity of a disordered Weyl semimetal in the presence of magnetic field. We restrict ourselves to the case of weak disorder, . With the use of Kubo formula, the real part of the conductivity reads
where is the bare current operator and is the current vertex dressed by disorder, see Ref. Klier et al. (2015). We first calculate the conductivity without vertex corrections and include them at the final steps of the calculation.
After the evaluation of the trace and using the orthogonality of the wave functions of the different LLs, Eq. (III) transforms into
The Green functions here are written in the LL representation. We distinguish in the following calculations between the zeroth LL and higher LLs because the self-energies for the zeroth LL differ from those of the others. In the following, we will focus on low temperatures, . For small chemical potential, , excitations to higher LLs are exponentially suppressed and the conductivity is dominated by the contribution of the zeroth LL. Note the conductivity in both region match via a narrow window at corresponding to the width of the first LL [cf. the last two lines in Eq.(34)]. In the opposite regime, the conductivity is determined by the position of the chemical potential with respect to separated and overlapping LLs.
iii.1 Small chemical potential, : Zeroth Landau level
We consider first the situation when the zeroth LL gives the dominant contribution to the conductivity. This case is realized under the following two conditions: (i) the zeroth LL is separated from the first one, which is fulfilled under the condition ; (ii) the chemical potential satisfies , while the temperature is close to zero, . Under these conditions, the current vertex corrections are small, , for energies close to the Weyl node. Therefore, we can disregard the difference between quantum and scattering time in the regime of the dominant zeroth LL contribution. The Green function, using and and disregarding the real parts of self-energies (, see Ref. Klier et al. (2015)), reads
where is the number of LLs within the energy band . After the integration over for , we find that the contribution of higher LLs is of the order and therefore negligible compared to the term that is of the order of . For the dominant term coming from the zeroth LL we find
The result is proportional to the magnetic field and disorder strength and is equal to the result of , see Ref. Klier et al. (2015). A finite but small chemical potential, , does not essentially affect : the corrections to Eq. (18) are small in the parameter .
iii.2 Large chemical potential,
For large chemical potentials, , the situation is more subtle. For a given magnetic field, the spectrum is subdivided in three domains: (i) the low-energy part of the spectrum consists of separated LLs, (ii) in the intermediate region LLs are separated, but the background density of states is larger than the height of an individual LL, and, finally, (iii) at higher energies the LLs overlap. At low temperatures, the conductivity will strongly depend on the position of the chemical potential, with the unusual broadening of LLs leading to an unconventional shape of the SdHO.
In view of the structure of the spectrum discussed above, we need to distinguish for the calculation of the conductivity between the three different cases of the position of the chemical potential: (i) fully separated LLs, (ii) separated LLs, but large background, and (iii) fully overlapping LLs. In all three cases the difference between the self-energies can be neglected and the self-energy can be written in terms of LL broadening: . The Green functions take then the form
Substituting these Green functions in the formula for the conductivity (14), we perform the summation over and integration over . (This calculation is analogous to that in the case in Ref. Klier et al. (2015).) The result is given by
In all three cases of the structure of the spectrum near the chemical potential, the conductivity can be expressed by the semiclassical Drude formula, yielding
Here is the transport scattering time that takes into account the vertex corrections in and is related to the quantum time via . In the case of overlapping LLs or of large background density of states compared to the particular LL, , the LL broadening is given by
Using the SCBA relation between the density of states and the scattering time
and the semiclassical expression for the cyclotron frequency in the linear spectrum
we find the conductivity in this region:
In the following, we use Eq. (26) to evaluate the conductivity in all three regimes.
First, we consider the regime of fully separated LLs, when the relevant energies satisfy , assuming that the chemical potential is located within one of the LLs and the temperature is low (smaller than the LL width). The conductivity for a general LL broadening is given by
The broadening of the LLs at the LL center is given by , which yields the conductivity in the center of LLs (in the following denoted by )
The conductivity of the background density of states with a broadening of is denoted by and reads
Next, we turn to the intermediate range of the location of the chemical potential, . In this case, the -term in the denominator of Eq. (26) dominates, yielding
In the last line of Eq. (III.2) we have taken low-temperature limit (here the condition is sufficient).
Finally, for higher chemical potential, , which is the regime of overlapping LLs, we neglect in the denominator of Eq. (26), which leads to
The result coincides with the conductivity in the absence of magnetic field and does not depend on the chemical potential.
Magnetooscillations of the conductivity stem from the oscillations of the density of states and of the transport scattering time , see Ref. Dmitriev et al. (2012). For a Weyl semimetal, the density of states with magnetooscillations is given by
is the Dingle factor determined by the quantum scattering time . Note that in the case of a conventional 3D material with parabolic dispersion (see Ref. Vasko and Raichev (2005)), the frequency of the oscillations is a factor of larger then in the case of Weyl semimetals. A similar behavior is encountered in the 2D case of graphene Briskot et al. (2013) in comparison to conventional 2D materials. The non-equidistant behavior of the LLs for relativistic dispersion relations is expressed via the energy dependent cyclotron frequency . For , which corresponds exactly to the condition of overlapping LLs, the first harmonics, , is the least damped term and hence dominates the oscillations.
where is the smooth part of the conductivity calculated above [Eq. (III.2)]. As usual, the SdHO are exponentially damped in the regime of overlapping LLs, in contrast to the case of separated LLs.
We conclude this section with a summary of the results for the conductivity,
in the different regimes with respect to magnetic field, chemical potential, and disorder strength.
Iv Hall conductivity
In this section, we calculate the Hall conductivity. According to the Kubo-Streda formula Streda (1982), the Hall conductivity is given by
It is convenient to split up the Hall conductivity into a normal, , and an anomalous, , contributions. The normal contribution is determined by states near the Fermi level and can be simplified by using the orthogonality of the wave functions of different LLs. We find
The anomalous contribution reflects the thermodynamic properties of the system in the presence of magnetic field and can be expressed as
Here is the electron density defined as follows:
Below, we will first calculate the Hall conductivity in the clean case, and then will incorporate disorder which is encoded in the density of states .
iv.1 Clean case
We now briefly discuss the Hall conductivity in the clean case. The Green functions in Landau representation for the clean case read
We start with the calculation of the normal part of Hall conductivity and substitute the Green function from Eqs. (39) and (40) in Eq. (IV). After the evaluation of the integral over energy and of sum over energy bands , the normal contribution to the Hall conductivity reads
The evaluation of the integrals for leads to
The normal contribution of the Hall conductivity shows singularities when the chemical potential is at the center of the one particular LL, , see Fig. 3 (a).
The anomalous contribution to the Hall conductivity is obtained from Eq. (37) and the density of states of a Weyl semimetal in clean case,
We evaluate the integral in Eq. (38) for and take the derivative of with respect to magnetic field :
The first term of Eq. (IV.1) also shows singularities when the chemical potential is at the center of the one particular LL opposite of those of the normal contribution from Eq. (42), see Fig. 3 (b). Therefore, these singularities are exactly canceled in the total Hall conductivity. As demonstrated in Appendix A, this cancellation occurs in the clean case in the general case of arbitrary .
The evaluation of the sum over LLs with Euler-Maclaurin formula leads to the leading order to
For , Eq. (45) describes the smoothened part of the Hall conductivity. On top of this background contribution there is an oscillatory part induced by the Landau quantization. The Hall conductivity (normal and anomalous part and the total Hall conductivity) without disorder is visualized in Fig. 3, where the oscillations induced by Landau quantization can be seen clearly in case of fixed chemical potential. Already based on this plot, one can expect that in the presence of disorder the total Hall conductivity is only weakly changed, since the disorder-induced broadening would only smoothen the oscillatory part of the curve.
Further, we can express the Hall conductivity for a fixed particle density instead of a fixed chemical potential, as relevant to experiments. The magneto-oscillations in the chemical potential are then exactly canceled by the oscillations in the particle density:
see inset in Fig. 3 (c). Here the zero level of the density is chosen in such a way that for the chemical potential located in the Dirac point, .
iv.2 Normal Hall conductivity in the presence of disorder
Now, we turn to the Hall conductivity in the presence of disorder and first proceed with the evaluation of the normal contribution. As explained in Sec.II, we distinguish again between the cases when the chemical potential is within the zeroth LL or higher LLs. We focus on low temperatures, , throughout the whole section.
We will start with the calculation of the Hall conductivity under the following conditions: (i) the zeroth LL is separated from higher LLs, ; (ii) excitations to higher LLs are suppressed, . Using the Green functions for energies close to the zeroth LL, Eqs. (15) and (LABEL:G22e0), the formula for normal contribution to the Hall conductivity, Eq. (IV) transforms to
In the following, we will split the summation over the LL index into the term with and the terms with . In contrast to the conductivity , the contribution of the terms with in is of the same order as the term. The evaluation of the terms under the conditions and gives to the leading order
Clearly, this result (linear in disorder) matches the result for a clean system, where the normal contribution for the case of the chemical potential located in the zeroth Landau level is absent. We will see below that the term (48) is negligible in comparison with the anomalous contribution to the Hall conductivity.
Now, we turn to higher chemical potential and analyze the contribution of higher LLs to . For , the difference between the self-energies for the two bands can be neglected and we can use the Green functions (19) and (20) in Eq. (IV). The detailed calculation is presented in Appendix B. The normal contribution to the Hall conductivity reads
The limit of vanishing disorder, , is reproduced in Eq. (B). Similarly to , the normal contribution to the Hall conductivity can be cast in the form of a semiclassical Drude formula:
In the regime where the contribution of the separated LLs to the density of states exceeds the contribution of the background, , the second term in Eq. (IV.2) dominates. In the limit , we get
Next, we evaluate the Hall conductivity for a larger chemical potential, when the LLs are separated but the contribution of the background dominates, or else, the LLs fully overlap. In these cases, the expressions for the density of states, transport scattering time, and cyclotron frequency are given by Eqs. (23), (24), and (25), respectively. In the range , which corresponds to the case of separated LLs with the dominant background density of states, we find
For fully overlapping LLs, , the normal contribution to the Hall conductivity reads
iv.3 Anomalous Hall conductivity in the presence of disorder
In this Section, we calculate the anomalous contribution to the Hall conductivity in the presence of disorder. Furthermore, we subtract the contribution of states below the charge neutrality point since they do not contribute to the Hall conductivity. This is shown explicitly in Appendix A for the clean case and holds for finite disorder in the weak disorder regime, , considered here. The density of states of a disordered Weyl semimetal is given by
In the calculation of the self-energy, we distinguish between the zero LL and the others. For the energy at the zeroth LL the self-energy is given by
which will be used in the regime . The anomalous Hall conductivity in this regime does not depend on weak disorder, :
This result matches the ac anomalous Hall conductivity obtained in Ref. Steiner et al. (2017) in the limit .
For the situation is more subtle. The self-energy depends on the strength of broadening and, for separated LLs, , on the actual position of the chemical potential with respect to the center of a given LL. The shape of the density of states consists of the peak at the center of the LL, the tail of the LL, and the background, see Ref. Klier et al. (2015). For separated LLs with large background and for overlapping LLs, the density of states is dominated by the background contribution. The anomalous Hall conductivity for reads
where is given by Eq. (11).
Under the same approximations as in the calculation of , we obtain the anomalous Hall conductivity in the disordered case, reading
where is defined in Eq. (11). For in Eq. (11), the result (IV.1) obtained in the limit of vanishing disorder is reproduced. Moreover, for non-overlapping LLs, the broadening of LLs in Eq. (IV.3) is only important in the term the sum over LLs that corresponds to the LL where the chemical potential is located; for all other one can replace with , as in Eq. (IV.1). The smoothened part of the Hall conductivity for separated LLs ,
is thus the same as in the limit without disorder. The effects of the oscillations are minor compared to smoothened part of the Hall conductivity. Therefore, we will use Eq. (59) in the following sections to calculate the magnetoresistance. The oscillatory part of the Hall conductivity for fully separated LLs shown in Fig. 4 visualizes the effect of disorder in the Hall conductivity.
For overlapping LLs, the main term in the broadening is given by which is independent of magnetic field and therefore the anomalous Hall conductivity is zero to the leading order. The corrections due to magnetic field in the case of overlapping LLs are proportional to the Dingle factor, as described above. The particle density for zero temperature reads
Since the Dingle factor is exponentially small for overlapping LLs, the anomalous part of the Hall conductivity decays exponentially. The same applies for the TMR. The contributions of overlapping LLs to the TMR will therefore be dominated by effects of finite temperature and will not be discussed here.
V Magnetoresistance for pointlike impurities
We now turn to the evaluation of the TMR,
which quantifies the difference between the resistivity in a finite magnetic field and the resistivity at . Using