Bilinear magnetoresistance and planar Hall effect in topological insulators: Interplay of scattering on spin-orbital impurities and nonequilibrium spin polarization

We have considered theoretically nonlinear transport phenomena known as bilinear magnetoresistance (BMR) and nonlinear planar Hall effect (NPHE) within the effective model describing surface states of a 3D topological insulator. Both phenomena can occur in nonmagnetic materials with strong spin-orbit interaction and reveal a term that depends linearly on the charge current density (external electric field) and in-plane magnetic field. In earlier studies, the physical mechanism of BMR and NPHE was related to scattering on spin-momentum locking inhomogeneities or to the hexagonal warping of Dirac cones. Here, we focus on another mechanism related to scattering on impurities that inherently contain spin-orbit coupling. Using the Green's function formalism and diagramatic method, we have derived analytical results for diagonal and transverse conductivities and determined nonlinear signals. The analytical and numerical results on BMR and NPHE indicate the possibility of determining the material constants, such as the Fermi wave vector and spin-orbit coupling parameter, by simple magnetotransport measurements.

Figure 1

Schematic picture of measurement of the longitudinal, $V$, and transverse, $V_{\text{PH}}$, voltage. The antisymmetric part of the signals determines the bilinear magnetoresistance and bilinear planar Hall effect.

Figure 2

Diagrammatic representation of the conductivity tensor elements ${\sigma }_{\alpha \beta }$. The considered diagrams are the bubble diagram with the renormalized velocity vertex and the side-jump diagrams (a). The renormalized vertex function is calculated within the ladder approximation (b) and calculated self-consistently according to the vertex equation presented in (c).

Figure 3

The longitudinal conductivity ${\sigma }_{xx}$ as a function of the angle $\theta$ defining the orientation of in-plane magnetic field with respect to $\widehat{x}$ direction (a) and Fermi energy ${\varepsilon }_F$ (b)–(d). The angular dependence (a) is plotted for charge current density oriented along the unit vector $\widehat{\mathbf{x}}$; the symmetric and asymmetric parts of the conductivity are also plotted. The contribution to the longitudinal conductivity from ladder diagram and side-jump diagrams is presented in (c) and (d), whereas the contributions from the second and third orders with respect to $\lambda$ to the conductivity are presented in (b). In (a) $\lambda =6{\mathrm{nm}}^2$, whereas in (b)–(d) $\theta =\pi /2, n_i{V}_0^2=1.58\times {10}^{-24}{\mathrm{eV}}^2{\mathrm{m}}^2$.

Figure 4

The off-diagonal conductivity ${\sigma }_{yx}$ as a function of the angle $\theta$, defining the orientation of in-plane magnetic field with respect to the $x$ axis (a) and Fermi energy ${\varepsilon }_F$ (b)–(d). The angular dependence in (a) is plotted for charge current density oriented along $\widehat{\mathbf{x}}$; the symmetric and antisymmetric parts of the conductivity are also plotted there. The contributions to the off-diagonal conductivity from the ladder diagram and side-jump diagrams are presented in (c) and (d), whereas the contributions to conductivity in second and third order with respect to $\lambda$ are presented in (b). In (a) $\lambda =6{\mathrm{nm}}^2$, whereas in (b)–(d) $\theta =0, n_i{V}_0^2=1.58\times {10}^{-24}{\mathrm{eV}}^2{\mathrm{m}}^2$.

Figure 5

(a), (b) Quadratic and bilinear magnetoresistance (QMR, BMR) as well as (c), (d) symmetric and antisymmetric part of the planar Hall angle (${\mathrm{\Theta }}_{\text{PH}}^{\text{S}}, {\mathrm{\Theta }}_{\text{PH}}^{\text{AS}}$) as a function of angle $\theta$ defining the orientation of in-plane magnetic field with respect to $\widehat{x}$ direction. (e), (f) BMR and ${\mathrm{\Theta }}_{\text{PH}}^{\text{AS}}$ as a function of chemical potential and amplitude of magnetic field b for $\theta =\pi /2$ and $\theta =\pi$, respectively. The other parameters, if not indicated, are as follows: $j_x=4\mathrm{A}/\mathrm{m}, \mu =60\mathrm{meV}, \lambda =8{\mathrm{nm}}^2, n_i{V}_0^2=1.58\times {10}^{-24}{\mathrm{eV}}^2{\mathrm{m}}^2$.