Semiclassical magnetotransport including the effects of the Berry curvature and Lorentz force

In topological materials with a nonvanishing Berry curvature, the positive magnetoconductance has been observed experimentally in the presence of parallel electric and magnetic fields KIM2013 ; HUANG2015 ; XIONG2015 ; LI2015 ; LI2016a ; ZHANG2016 ; LI2016b ; ZHANG2016 ; ZHANG2017 ; WANG2012 ; HE2013 ; WANG2015 ; WIEDMANN2016 ; ARNOLD2016 ; BREUNIG2017 ; ASSAF2017 . This is a unique feature to topological materials that does not occur in conventional magnetotransport which only takes classical Lorentz force effect into account.

The prevailing explanation for the positive longitudinal magnetoconductance is the so-called chiral anomaly. In 1983, Nielsen and Ninomiya suggested the chiral anomaly in Weyl fermions under a strong magnetic field regime where the chiral zeroth Landau level creates a one-dimensional conducting channel that pumps electrons from one Weyl node to another NIELSEN1983 . In 2013, Son and Spivak discussed chiral anomaly in Weyl semimetals under weak external magnetic field using the semi-classical Boltzmann approach SON2013 . They argued the positive magnetoconductivity that scales quadratically in magnetic field is due to topological charge pumping. One can expect possible detection of chiral anomaly between the valleys, given that the intervalley scattering is negligible compared to the intravalley scattering SEKINE2017 . However, chiral anomaly cannot be responsible for observed positive magnetoconductivity in topological insulators (TIs) where chiral charges are not well defined WANG2012 ; HE2013 ; WANG2015 ; WIEDMANN2016 ; ARNOLD2016 ; BREUNIG2017 ; ASSAF2017 . In topological materials such as TIs DAI2017 and Weyl semimetals (WSMs) KIM2014 , it is suggested that the anomalous velocity induced by the non-trivial Berry curvature alone can generate an additional contribution to the conductivity that grows with the magnetic field.

To understand magnetotransport properties quantitatively, it is important to consider both the effects of the anomalous velocity due to the non-trivial Berry curvature and the classical Lorentz force effect. Most of the previous studies KIM2014 ; DAI2017 ; NANDY2017 have focused on the Berry curvature effect, while only a few took the Lorentz force into considerations in describing magnetotransport behaviors IMRAN2018 ; JOHANSSON2019 . In this paper, we revisit semiclassical treatment of magnetotransport in topological materials to shed more light on the origin of the observed positive magnetoconductivity. We present a general semiclassical formula for conductivity which fully incorporates the Berry curvature and the Lorentz force. From the Boltzmann transport equation, we obtain a closed-form expression for the nonequilibrium distribution function by solving the corresponding self-consistent equation. We then apply our formula to WSMs and express the magnetoconductivity in terms of dimensionless parameters characterizing magnetic fields associated with the Lorentz force and the Berry curvature, respectively.

The semiclassical Boltzmann transport equation governs the time evolution of a non-equilibrium distribution function $f=f(\mathbf{r},\mathbf{k},t)$ at position $\mathbf{r}$ and momentum $\mathbf{k}$. It states that the time evolution of $f$ equals the probability rate of electrons being scattered in and out of the distribution, called the collision term $\Big{(}\frac{\partial f}{\partial t}\Big{)}_{\text{coll}}$:

In a homogeneous sample with a steady external perturbation, there are no position $\mathbf{r}$ nor time $t$ dependence in the distribution function $f$. Then, $\frac{df}{dt}$ simplifies to

where we have included a subscript $\mathbf{k}$ to the non-equilibrium distribution function $f$ to indicate that it is only a function of $\mathbf{k}$. In a simple relaxation time approximation, the collision integral is replaced by the ratio between the deviation from the equilibrium Fermi-Dirac distribution $f^{(0)}_{\mathbf{k}}$ and the average time $\tau$ between successive collisions. Hence,

where $g_{\mathbf{k}}=f_{\mathbf{k}}-f^{(0)}_{\mathbf{k}}$. Here we assume a constant transport relaxation time $\tau$ in momentum $\mathbf{k}$ and magnetic field $\mathbf{B}$ for a given chemical potential. Note that when we fully consider the collision integral in the system with a nontrivial Berry curvature, the transport relaxation time $\tau$ may show a field dependent anisotropy induced by the coupling between the Berry curvature and magnetic field PARK2021 . However, in weak magnetic field regime, we can assume $\tau$ as a constant. Furthermore, when short-range scattering is dominant or charged impurities are fully screened, we can assume that $\tau$ does not depend on $\mathbf{k}$ XIAO2005 ; Ashcroft1976 ; Ziman1960 .

Here we provide a closed-form expression for magnetoconductivity in the presence of both a non-trivial Berry curvature and the Lorentz force within the semiclassical Boltzmann approach. The semiclassical equations of motion for electrons with a charge $q=-e$ in the presence of the Berry curvature are given by XIAO2010

where $\epsilon_{\mathbf{m}}(\mathbf{k})=\epsilon_{0}(\mathbf{k})-\mathbf{m}_{\mathbf{k}}\cdot\mathbf{B}$, $\epsilon_{0}(\mathbf{k})$ is the unperturbed band energy, $\bf{m}_{\mathbf{k}}$ is an orbital magnetic moment that couple to the magnetic field and $\mathbf{\Omega}_{\mathbf{k}}$ is the Berry curvature. It has been reported that disorder affects not only the carrier distribution but also the semiclassical equations of motion, generating a correction to the velocity proportional to the disorder strength ATENCIA2022 . In this work, we neglect this correction assuming a weak disorder potential for simplicity. In the presence of a magnetic field, $\dot{\mathbf{r}}$ and $\dot{\mathbf{k}}$ in Eq. (4) are coupled through the Lorentz force. Combining these two equations of motion, we have

where $\mathbf{v}_{\mathbf{k}}=\frac{1}{\hbar}\frac{\partial\epsilon_{\mathbf{m}}(\mathbf{k})}{\partial\mathbf{k}}$ and $D_{\mathbf{k}}=1-\frac{q}{\hbar c}(\mathbf{\Omega}_{\mathbf{k}}\cdot\mathbf{B})$ represents the modified density of states in the phase space due to the Berry curvature effect XIAO2005 . The first term in the square bracket in Eq. (5b) corresponds to electric force due to an electric field $\mathbf{E}$ and the second term represents Lorentz force due to a magnetic field $\mathbf{B}$. The last term is anomalous electromagnetic force due to the Berry curvature which leads to positive magnetoconductivity in topological materials.

Replacing $f_{\mathbf{k}}=f^{0}_{\mathbf{k}}+g_{\mathbf{k}}$ according to the relaxation time approximation in Eq. (3), we have

Plugging Eq. (5) into Eq. (6), we get

In previous studies DAI2017 ; KIM2014 ; NANDY2017 of magnetoconductivity and planar Hall conductivity in topological materials, Lorentz force effect has been often neglected. Therefore, in order to better understand positive longitudinal magnetoconductivity and angle-dependent planar Hall conductivity in topological materials, we take all the terms in Eq. (7) into consideration.

We can rewrite Eq. (7) in the following form:

where

Figure 1 shows a schematic picture of $\tilde{\mathbf{v}}_{\mathbf{k}}$ in WSMs in the presence of a magnetic field $\mathbf{B}$. As shown in Fig. 1, $\tilde{\mathbf{v}}_{\mathbf{k}}$ is greater in magnitude than $v_{\mathbf{k}}$ in all $\mathbf{k}$ except when $\mathbf{v}_{\mathbf{k}}$ is parallel to $\mathbf{B}$ giving $\tilde{\mathbf{v}}_{\mathbf{k}}=\mathbf{v}_{\mathbf{k}}$. In general, it is this effect due to Berry curvature that leads to enhanced magnetoconductivity in topological materials.

Our goal now is to solve for $g_{\mathbf{k}}$ in Eq. (8). We assume that a small deviation from the equilibrium distribution takes the form of $g_{\mathbf{k}}=\tilde{\mathbf{v}}_{\mathbf{k}}\cdot\mathbf{G}$ where $\mathbf{G}$ is some arbitrary vector that is independent of $\tilde{\mathbf{v}}_{\mathbf{k}}$. Upon plugging $g_{\mathbf{k}}=\tilde{\mathbf{v}}_{\mathbf{k}}\cdot\mathbf{G}$ into Eq. (8), we obtain

where $\mathds{M}^{-1}_{ij}=\frac{1}{\hbar}\frac{\partial\tilde{\rm v}_{\mathbf{k},j}}{\partial k_{i}}$ is the inverse mass tensor. Factoring $\tilde{\mathbf{v}}_{\mathbf{k}}$ out,

From $g_{\mathbf{k}}=\tilde{\mathbf{v}}_{\mathbf{k}}\cdot\mathbf{G}$ and Eq. (11), we obtain a self-consistent form of $\mathbf{G}$ as

where

The solution for Eq. (12) can be obtained as

where $\mathds{F}_{ij}=\sum_{i}\epsilon_{ijk}B_{k}$ is the magnetic field strength tensor (see Appendix A for a detailed derivation). Thus, we obtain

The current is therefore computed as

where $\alpha$ is the direction in which we measure the current. Let us denote the current associated with $f^{(0)}_{\mathbf{k}}$ as $J^{\text{int}}_{\alpha}=q\int\frac{d^{d}k}{(2\pi)^{d}}D_{\mathbf{k}}f^{(0)}_{\mathbf{k}}{{v}}_{\alpha}$ and that associated with $g_{\mathbf{k}}$ as $J^{\text{ext}}_{\alpha}=q\int\frac{d^{d}k}{(2\pi)^{d}}D_{\mathbf{k}}g_{\mathbf{k}}{{v}}_{\alpha}$. From now on, we will only focus on the extrinsic contribution $J^{\text{ext}}_{\alpha}$ arising from scatterings.

Plugging Eq. (15) into $J^{\text{ext}}_{\alpha}$ and using $J_{\alpha}^{\text{ext}}=\sum_{\beta}\sigma_{\alpha\beta}E_{\beta}$ relation, we finally arrive at

where $\beta$ is the direction of an electric field.

Now let us assume that the mobility tensor $e\tau\mathds{M}^{-1}$ is set to a constant $\mu$ for simplicity. Then $\mathbf{G}$ in Eq. (14) becomes (see Appendix A)

Note that the obtained $\mathbf{G}$ is consistent with the assumption that $\mathbf{G}$ is independent of $\tilde{\mathbf{v}}_{\mathbf{k}}$. Then $g_{\mathbf{k}}$ is given by

Using Eq. (19), we finally obtain the following form for magnetoconductivity

The above form is a general expression of magnetoconductivity for topological materials including the Berry curvature and Lorentz force within the semiclassical regime under the assumption that $\mathbf{G}$ in $g_{\mathbf{k}}=\tilde{\mathbf{v}}_{\mathbf{k}}\cdot\mathbf{G}$ is independent of $\tilde{\mathbf{v}}_{\mathbf{k}}$ and the mobility tensor is a constant.

In this section, we study the magnetotransport properties of WSMs in three dimensions using a closed-form expression for magnetoconductivity Eq. (II) discussed in the previous section. For simplicity, we consider a single Weyl node described by the Hamiltonian $H=\chi\hbar v_{\rm F}\mathbf{k}\cdot\bm{\sigma}$ which has isotropic linear dispersion, where $\chi=\pm 1$ are for the different chiralities of Weyl fermions and $\bm{\sigma}$ are the Pauli matrices.

In this work, we presented a closed-form expression for the magnetoconductivity using the semiclassical magnetotransport theory that fully incorporates the Berry curvature and the Lorentz force effects. We then applied this formula to WSMs and obtained analytic expressions for the longitudinal, planar Hall and Hall conductivities in terms of dimensionless parameters $b_{\mu}$ and $b_{\rm BC}$ which are normalized magnetic fields associated with the Lorentz force and the Berry curvature, respectively. From these results, we showed a non-monotonic field dependence in the longitudinal and planar Hall conductivities depending on $k_{\rm F}l$. Furthermore, we clearly demonstrated that although the Lorentz force effect is manifested in the planar Hall conductivity, its contribution vanishes in the planar Hall resistivity.