Negative Magnetoresistance without Chiral Anomaly in Topological Insulators

An intriguing phenomenon in topological semimetals and topological insulators is the negative magnetoresistance (MR) observed when a magnetic field is applied along the current direction. A prevailing understanding to the negative MR in topological semimetals is the chiral anomaly, which, however, is not well defined in topological insulators. We calculate the MR of a three-dimensional topological insulator, by using the semiclassical equations of motion, in which the Berry curvature explicitly induces an anomalous velocity and orbital moment. Our theoretical results are in quantitative agreement with the experiments. The negative MR is not sensitive to temperature and increases as the Fermi energy approaches the band edge. The orbital moment and $g$ factors also play important roles in the negative MR. Our results give a reasonable explanation to the negative MR in 3D topological insulators and will be helpful in understanding the anomalous quantum transport in topological states of matter.

Figure 1

The comparison between the calculated negative MR and the experiments at three typical temperatures and current directions [17, 18, 19]. In the calculations, the corrections from both the Berry curvature $\mathbf{\Omega }$ and orbital moment $\mathbf{m}$ have been taken into account. The Fermi energy $E_F$ (measured from the bottom of the conduction bands) is a tuning parameter in the numerics. All values of $E_F$ fall in a reasonable range. The current direction and temperature are from the experiments. $x$, $y$, $z$ in the model (4) correspond to [100], [010], and [001] crystallographic directions, respectively. The numerically calculated MR along the $z$ axis is used to approach the experimental MR along the $[11\overline{4}]$ direction by He et al., because the projection of $[11\overline{4}]$ on the $z$ axis is over 94%. Other model parameters are from the $k·p$ calculations [23] and experiments [24], $M_0=-0.169\mathrm{eV}$, $M_z=3.351\mathrm{eV}{\AA }^2$, $M_{\perp }=29.36\mathrm{eV}{\AA }^2$, $V_{\perp }=2.512\mathrm{eV}\AA$, $V_n=1.853\mathrm{eV}\AA$, $C_0=0.048\mathrm{eV}$, $C_z=1.409\mathrm{eV}{\AA }^2$, $C_{\perp }=13.9\mathrm{eV}{\AA }^2$, the $g$ factors $g_z^v=g_z^c=30$ and $g_p^v=g_p^c=-20$.

Figure 2

The energy dispersion (top left $k_y=k_z=0$, top right $k_x=k_y=0$) of the $\mathbf{k}·\mathbf{p}$ model, and the vector plots of the Berry curvature $\mathbf{\Omega }$ (middle) and orbital moment $\mathbf{m}$ (bottom) for the lower conduction band (blue curve). The magnetic field is 5 T. The parameters are the same as those in Fig. 1.

Figure 3

The numerical results for ${\mathrm{MR}}_x(B_x=5\mathrm{T})$ and ${\mathrm{MR}}_z(B_z=5\mathrm{T})$ as functions of the Fermi energy $E_F$ and temperature $T$, in the presence (solid scatters) and absence (hollow scatters) of the orbital moment $\mathbf{m}$. The parameters are the same as those in Fig. 1. $E_F$ is measured from the bottom of the conduction bands. There is no qualitative change if using the parameters from Ref. [42].