Theory for the negative longitudinal magnetoresistance in the quantum limit of Kramers Weyl semimetals

Magnetoresistance measures the change of the electric resistance of a solid due to magnetic fields. In magnetic materials, randomly-oriented ferromagnetic domains suppress the tunneling of electrons and increase the resistance. A magnetic field can align the domains and thus lower the resistance, leading to a negative magnetoresistance. In contrast, negative magnetoresistance is rare in non-magnetic materials Argyres and Adams (1956), because the Lorentz force imposed by the magnetic field prevents electrons from moving forward. One of the mechanisms of negative magnetoresistance is weak localization Lee and Ramakrishnan (1985), which is induced by quantum interference and thus only survives at extremely low temperatures. Recently, negative magnetoresistance at higher temperatures in non-magnetic topological insulators Wang et al. (2012); He et al. (2013); Wiedmann et al. (2016); Wang et al. (2015); Breunig et al. (2017); Assaf et al. (2017) and semimetals Kim et al. (2013, 2014); Li et al. (2016a); Zhang et al. (2016a); Huang et al. (2015); Xiong et al. (2015); Li et al. (2015); Zhang et al. (2017a); Li et al. (2016b); Arnold et al. (2016); Yang et al. (2015a, b); Wang et al. (2016) has attracted tremendous interest. In topological semimetals, the negative magnetoresistance is widely believed to be interpretable as a manifestation of the chiral anomaly, that is, the violation of chiral symmetry by quantum effects Adler (1969); Bell and Jackiw (1969); Nielsen and Ninomiya (1981). In topological insulators, the negative magnetoresistance is found to be related to the anomalous velocity induced by a nontrivial distribution of Berry curvature Dai et al. (2017). Because of the nontrivial mechanism behind each of the cases, a negative magnetoresistance observed in novel systems warrants a detailed study.

In a recent experiment on a single-crystalline silver chalcogenide material $\beta$-Ag₂Se Zhang et al. (2017b), a negative magnetoresistance has been observed when the magnetic field is parallel to the current. $\beta$-Ag₂Se is among the material candidates for a class of systems called Kramers Weyl semimetals, in which Weyl nodes are pinned to time-reversal invariant momenta (TRIMs) in the Brillouin zone because of time-reversal symmetry Bradlyn et al. (2016); Chang et al. (2018). The Kramers Weyl nodes generically appear in all chiral crystals, i.e., is crystals that lack any roto-inversion symmetries and thus have a sense of handedness Chang et al. (2018). Specifically, in symmorphic chiral crystals, every Kramers pair of bands at every TRIM is guaranteed to host a Weyl cone; while in non-symmorphic chiral crystals, it is true for a subset of TRIMs only, which however always includes the $\Gamma$ ($\mathbf{k}=0$) point. $\beta$-Ag₂Se provides an instance of the latter case. In $\beta$-Ag₂Se, a negative magnetoresistance of about $-20\%$ was observed at a magnetic field of about 9 T. At such a strong magnetic field, the system has entered the quantum limit, i.e., only the lowest Landau band crosses the Fermi energy. In the quantum limit, the magnetoresistance depends subtly on scattering mechanisms Lu et al. (2015); Goswami et al. (2015); Zhang et al. (2016b). Besides, since the Kramers Weyl nodes are protected from symmetries in chiral space groups, it provides a new platform for investigating the scattering mechanisms.

In this work, we present a theory for the longitudinal magnetoresistance in the quantum limit of a Kramers Weyl semimetal in strong parallel magnetic fields. We start with a generic model with one Kramers Weyl cone and use the standard Kubo formalism to calculate the conductivity, considering impurity scattering with screened Coulomb potentials and Gaussian potentials. We show that in the quantum limit the resistance has a $1/B$ dependence in the presence of impurities with screened Coulomb potential Dubinskaya (1969), and thus indeed gives rise to a negative magnetoresistance. In many Weyl semimetals that emerge from band inversion, Weyl nodes of opposite chirality are degenerate in energy, e.g. due to some mirror symmetry. This is generically not the case for the Kramers Weyl nodes in chiral crystals, so the inter-valley charge pumping and relaxation is absent in our calculations, suggesting that the negative magnetoresistance in the Weyl semimetal can exist without any apparent link to the chiral anomaly. Although magnetoresistance in the quantum limit has been systematically studied for a number of models and potentials Lu et al. (2015); Goswami et al. (2015); Zhang et al. (2016b), the case we study here has not been addressed before.

To justify our results, we will present the step-by-step details of the calculation. The paper is organized as follows. In Sec. II, we introduce a generic model for Kramers Weyl semimetals. In Secs. III and IV, we present the calculations for the transport time in the presence of the screened Coulomb scattering potential and longitudinal conductivity in the quantum limit, respectively. To justify the isotropic model and screened Coulomb scattering potential we used, we discuss the effect of anisotropy and Gaussian scattering potential in Secs. V and VI, respectively. Finally, we summarize and discuss the results in Sec. VII.

A Kramers Weyl cone can be described by the effective $\bm{k}\cdot\bm{p}$ model proposed in Zhang et al. (2017b),

where we have suppressed the anisotropy of the model parameters $u$ and $v$ present in the original model. We will find that this isotropic model can capture the main physics of negative magnetoresistance. In contrast to the band-inversion Weyl cone, the parabolic term overwhelms the linear term and the energy difference between paired Weyl nodes can be much larger than the temperature scale in Kramers Weyl semimetals. It is reasonable to consider only one Weyl cone as the longitudinal conductivity is dominated by electrons near the Fermi surface.

The energy spectrum of this model has two bands

which are schematically shown in Fig. 1 (b) in the plane $k_{x}=k_{y}=0$. At $\bm{k}=0$, a Kramers Weyl node forms as band $\mathcal{E}_{+}$ touches band $\mathcal{E}_{-}$. A similar Weyl cone is also present in BiTeI Murakami et al. (2003). Weyl nodes always come in pairs because of the fermion-doubling theorem Nielsen and Ninomiya (1983). For Weyl semimetals in which these pairs are (nearly) degenerate in energy, charges can be pumped from one Weyl cone to another Weyl cone of the opposite chirality in an external electric field, and both inter- and intra-cone scattering has to be considered. However, as this kind of energetic degeneracy of Weyl cones is not found in Kramers Weyl semimetals, we can assume that the intra-node scattering dominates. Later, we will show that a negative magnetoresistance can arise in this one-node Kramers Weyl semimetal in the presence of impurities with a screened Coulomb scattering potential.

In a magnetic field, the energy spectrum is quantized into a set of one-dimensional (1D) bands of Landau levels. Because the present model is isotropic, we can assume that the magnetic field is applied along the $z$ direction, that is, $\bm{B}=(0,0,B)$. Along the direction of the magnetic field the wave vector $k_{z}$ is a good quantum number, so the 1D Landau bands disperse with $k_{z}$. We adopt the Landau gauge in which the vector potential is $\bm{A}=(-yB,0,0)$. Under the Peierls replacement, the wave vector becomes $\bm{k}\rightarrow(k_{x}-eyB/\hbar,-i\partial_{y},k_{z})$, and the Hamiltonian in the applied magnetic field becomes

The chosen vector potential breaks the translational symmetry along the $y$-direction but not along the $x$- or $z$-direction, $k_{x}$ and $k_{z}$ are still good quantum numbers. Introducing the ladder operators Shen et al. (2004); Shen (2004)

we can replace the wave vectors $k_{x}^{2}+k_{y}^{2}\rightarrow\omega(a^{\dagger}a+1/2)$, $k_{+}\rightarrow(\sqrt{2}/\ell_{B})a^{\dagger}$, $k_{-}\rightarrow(\sqrt{2}/\ell_{B})a$, where $\ell_{B}=\sqrt{\hbar/eB}$ is the magnetic length, $\omega=2u/\ell_{B}^{2}$, and we have defined $k_{\pm}=k_{x}\pm ik_{y}$. Then the Hamiltonian in Eq. (3) becomes

where $\mathcal{M}^{\pm}_{a}=\omega(a^{\dagger}a+1/2)+uk_{z}^{2}\pm vk_{z}$ and $\eta=\sqrt{2}v/\ell_{B}$. With the trial wave functions $(c_{1}|\nu-1\rangle,c_{2}|\nu\rangle)^{\mathsf{T}}$ for $\nu\geq 1$, and $(0,|0\rangle)^{\mathsf{T}}$ for $\nu=0$, where $\nu$ indexes the Hermite polynomial wave functions given below, the eigenenergies $E$ can be obtained from the secular equation

for $\nu\geq 1$; and $\mathcal{M}^{-}_{\nu}-E=0$ for $\nu=0$, where $\mathcal{M}^{\pm}_{\nu}=\omega\nu\mp\omega/2+uk_{z}^{2}\pm vk_{z}$. The eigenenergies are found as

where $\mathcal{M}=\omega\nu+k_{z}^{2}u$. This gives the spectrum of a set of Landau energy bands ($\nu$ as band index), which is shown in Fig. 1 (c). The eigenstates found for $\nu\geq 1$ are

and the eigenstate for $\nu=0$ is

where $\cos(\theta^{\nu}_{k_{z}}/2)=(k_{z}v-\omega/2)/\sqrt{(k_{z}v-\omega/2)^{2}+\eta^{2}\nu}$. The wave functions in real-space $\psi_{\nu,k_{x},k_{z}}(\mathbf{r})\equiv\langle\mathbf{r}|\nu,k_{x},k_{z}\rangle$ are found to be

for $\nu\geq 1$ and

for $\nu=0$. Here, $C_{\nu}=1/\sqrt{\nu!2^{\nu}\sqrt{\pi}}$ , $L_{x}$ and $L_{z}$ are the lengths of the sample in the $x$ and $z$ directions, respectively, $\xi=(y-y_{0})/\ell_{B}$, $y_{0}=k_{x}\ell_{B}^{2}$ is the guiding center and $\mathcal{H}_{\nu}(\xi)$ are the Hermite polynomials. $k_{x}$ does not appear in the energy spectrum explicitly, because different $k_{x}$ give rise to the Landau degeneracy $N_{\mathrm{L}}=1/2\pi\ell_{B}^{2}$ per unit area in the x-y plane.

In what follows, we only concentrate on the quantum limit, which means only the lowest ($\nu=0$) Landau band crosses the Fermi energy [see Fig. 1 (b)]. The analytical solution of the Landau bands allows us to determine the analytical solution to the longitudinal magnetoresistance in the quantum limit of the Kramers Weyl cone.

To calculate the longitudinal conductivity, we need to know the transport time of carriers Zhang et al. (2016b); Goswami et al. (2015), which sensitively depends on the impurity scattering potential. Following Abrikosov’s treatment when exploring the linear magnetoresistance in the quantum limit of Ag₂Se under perpendicular magnetic fields Abrikosov (1998), we adopt a screened Coulomb scattering potential for the impurities,

with

where $\epsilon$ is the dielectric constant, $e$ is the electron charge, and $1/\kappa$ is the screening length. The screened Coulomb potential is justified because excess silver atoms form clusters doping the rest of the materialAbrikosov (1998). This potential is different from the ionic potential under the Thomas-Fermi approximation Goswami et al. (2015). Following the standard random phase approximation Abrikosov (1998); Bruus and Flensberg (2004),

where $\beta=1/{k_{B}T}$, $w_{m}=(2m+1)\pi/\beta$ are the Matsubara frequencies of fermions, $E_{\mathrm{F}}$ is the Fermi energy, and the energy spectrum $E^{0}_{k_{z}}$ of the $\nu=0$ Landau band has been given in Eq. (II). After summing over the Matsubara frequencies, we have

Substituting this into Eq. (15), we have

where $v_{\mathrm{F}}^{0}$ is the Fermi velocity of the $\nu=0$ Landau band. To calculate the transport time, we need to calculate the scattering matrix elements,

where $U(\mathbf{r})$ is defined in Eq. (13), and $\nu,\nu^{\prime}$ are indices for the Landau bands. In the quantum limit, due to the strong magnetic field, the energy spectrum has split into 1D Landau bands, and the spacing between the Landau bands is large. Moreover, because the Fermi energy crosses only the $\nu=0$ Landau band, the occupation of electrons for $E_{\nu,s}$ $(\nu\geq 1)$ bands vanishes and the $\nu=0$ Landau band is partially filled as shown in Fig. 1 (c). Therefore, we only need to take into account the impurity scatterings between $\nu=0$ Landau states. Using Eq. (13)

where the integral

describes the probability amplitude of charge carriers scattered from state $|0,k_{x},k_{z}\rangle$ to $|0,k^{\prime}_{x},k^{\prime}_{z}\rangle$ by an impurity located at position $\bm{R}_{i}$ in real space. We may then write the impurity scattering potential as

where the Fourier transform is given by

Substituting this into Eq. (III) and integrating along the $x$ and $z$ directions gives

where $\delta$ is the Kronecker symbol. In terms of $\mathcal{I}_{0,0}(\bm{R}_{i})$, the absolute value squared of the scattering matrix element between the states of the $\nu=0$ Landau band can be written as

After averaging over impurity configurations we obtain

where we have used the random impurity approximation

where $n_{\rm imp}$ is the density of impurities over the full sample. Using the energy dispersion of the Landau bands in Eq. (II), the velocity $\hbar v_{z}\equiv\partial E_{\nu}/\partial k_{z}$ is found as

Measured from the band bottom at $k_{z}=v/2u$, the carrier density $n$ of the $\nu=0$ Landau band is given by

Combining the above two relations, we have

so $\hbar v_{\mathrm{F}}^{0}$ is proportional to $1/B$ for a fixed $n$.

The transport time ${\tau^{0,\mathrm{tr}}_{k_{x},k_{z}}}$ of electrons in the $\nu=0$ Landau band is defined as

As shown in Fig. 1 (b), the parabolic $\nu=0$ Landau band crosses the Fermi energy at two points of wave vector, denoted as $k_{z}=\pm k_{\mathrm{F}}^{0}$, from which we can simplify Eq. (III) as

where $\Lambda$ is an extra correction factor introduced to avoid the van Hove singularity at the band edge of the 1D Landau band Lu et al. (2015). Equation (30) suggests that only the backscattering survives, i.e., $q_{z}=2k_{\mathrm{F}}^{0}$. Substituting Eq. (23) into Eq. (30), and considering $\ell_{B}^{2}\kappa^{2}\ll 1$ in strong magnetic fields, we have

By using Eq. (17), we arrive at the transport time of the $\nu=0$ band in strong magnetic fields

With the transport time, we are ready to calculate the longitudinal conductivity of the $\nu=0$ band. Along the $z$-direction, the semiclassical Drude conductivity can be calculated as

where $e$ is the electron charge, $V=L_{x}L_{y}L_{z}$ is the sample volume, $v^{z}_{0}={\partial E^{0}_{k_{z}}}/{\hbar\partial k_{z}}$ is the velocity along the $z$-direction for a state with wave vector $k_{z}$ in the $\nu=0$ Landau band,

are the retarded/advanced Green’s functions with $\tau_{k_{z},k_{x}}^{0}$ the lifetime of a state with wave vector $k_{x}$ and $k_{z}$ in the $\nu=0$ Landau band. In the diffusive regime, $G^{\mathrm{R}}_{0}G^{\mathrm{A}}_{0}$ can be replaced by

Changing the summations into integrals through

where the $k_{z}$ integral covers the entire Brillouin zone and the $k_{x}$ integral is confined by the degeneracy of the Landau levels, the conductivity can be expressed as

By using Eq. (35), we have

where the Landau degeneracy has already been taken into account. Now using the transport time in Eq. (32), we finally arrive at

Considering the $1/\ell_{B}^{4}\propto B^{2}$ in Eq. (39) and the velocity along z-direction obtained in Eq. (28), $\sigma_{zz,0}$ shows a positive linear dependence on magnetic field, i.e.,

In a parallel magnetic field, there is no Hall effect, thus the resistivity $\rho_{zz,0}$ is the inverse of the conductivity. Hence,

which means that the resistivity drops with increasing magnetic field. In other words, we find a negative longitudinal magnetoresistance in the quantum limit of the Kramers Weyl cone in the presence of a screened Coulomb impurity scattering potential.

Another common choice of the scattering potential is the Gaussian one. We can show that there is no negative magnetoresistance in the presence of the Gaussian scattering potential in the quantum limit of the Kramers Weyl cone. The Hamiltonian of the Gaussian scattering reads Goswami et al. (2015); Zhang et al. (2016b)

or in terms of its Fourier transform

where

and $u_{i}$ measures the scattering strength at $\bm{R}_{i}$. A major difference here is that the range of the potential $d$ is not a function of the magnetic field. As $d$ shrinks to zero, the potential reduces to a delta potential.

Following the same procedure that was used to obtain Eq. (32), the transport time for the Gaussian scattering potential is found to be

Inserting Eq. (45) into (38), one obtains the conductivity

According to Eqs. (III) and (28), $k_{\mathrm{F}}^{0}\propto v_{\mathrm{F}}^{0}\propto 1/B$, so $\sigma_{zz,0}^{\rm G}$ will decrease with increasing $B$, which cannot give a negative magnetoresistance. We conclude that the screened Coulomb scattering potential is an essential ingredient for a negative magnetoresistance.

The original model for the Kramers Weyl cone in $\beta$-Ag₂Se is anisotropic. Now we show that the anisotropy does not qualitatively change the negative magnetoresistance. The original model reads Zhang et al. (2017b)

where the model parameters $u_{x}=-14.8$ eVŲ, $u_{y}=-2.97$ eVŲ, $u_{z}=-1.55$ eVŲ, $v_{x}=0.079$ eVÅ, $v_{y}=0.066$ eVÅ, $v_{z}=0.020$ eVÅ. Because of the anisotropy, there is no analytical solution for the Landau bands. We numerically solve the energy spectrum of the Landau bands. Figure 2 shows that the structure of the Landau spectrum does not change qualitatively in the presence of the anisotropy, so one can also expect that a negative magnetoresistance is obtained in the anisotropic case. This justifies our calculation using the isotropic model.

In conclusion, we presented a theory for the negative magnetoresistance observed in the quantum limit of $\beta$-Ag₂Se, a paradigmatic Kramers-Weyl semimetal, in parallel magnetic fields Zhang et al. (2017b). It requires several ingredients: (1) a Kramers Weyl cone; (2) impurities with a screened Coulomb potential; (3) a fixed carrier density; (4) the quantum limit under parallel magnetic fields. These conditions naturally exist in $\beta$-Ag₂Se.

The negative magnetoresistance has been previously studied in the quantum limit of the band-inversion Weyl semimetals in parallel magnetic fields Lu et al. (2015); Goswami et al. (2015); Zhang et al. (2016b). In the presence of the charge-neutral Gaussian potential, the conductivity is linearly proportional to the magnetic field Goswami et al. (2015); Zhang et al. (2016b). In the ionic potential under the Thomas-Fermi approximation, the conductivity is proportional to $B^{2}$ in all cases Goswami et al. (2015). The novel mechanism for the negative magnetoresistance that we present in this work has not been covered in these previous studies. In particular, here the negative magnetoresistance is not related to the chiral anomaly, which has been used to explain the effect in band-inversion induced Weyl semimetals.

In band-inversion Weyl semimetals, two Weyl cones of opposite chirality have to appear in pairs. In a strong magnetic field, the $\nu=0$ Landau bands from the two cones have opposite velocities inherited from the chirality of the Weyl cones. In contrast, in Kramers Weyl semimetals, the energy difference between Kramers Weyl nodes of opposite chirality can be very large, because, e.g., one is located at the $\Gamma$ point and the other at the Brillouin zone corner. As the Fermi energy crosses the vicinity of one of the Weyl nodes, the contributions to transport coming from the other Weyl node can be safely ignored. For instance, the $\nu=0$ band of only one Weyl node in Fig. 1 (b) has no sense of the inherited chirality, although it looks similar to the quantum limit of the band-inversion Weyl semimetal. Therefore, a negative magnetoresistance of the single Kramers Weyl cone has nothing to do with the chiral anomaly.

According to the above result for the hole carriers, the key ingredients of the negative magnetoresistance are the parabolic dispersion of the 0th Landau band and the screened Coulomb scattering potential. According to the DFT band structure in Ref. [26], there is another electron pocket on the Fermi surface and its dispersion looks quite conventional. The conventional electron pocket is expected to have the parabolic dispersion for the 0th Landau band. Therefore, along with the same screened Coulomb scattering potential, the electron pocket is also expected to give the negative magnetoresistance.