Semiclassical theory of anisotropic transport at ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interfaces under an in-plane magnetic field

The unconventional magnetotransport at the interface between transition-metal oxides ${\mathrm{LaAlO}}_3$ ($\mathrm{LAO}$) and ${\mathrm{SrTiO}}_3$ ($\mathrm{STO}$) is frequently related to mobile electrons interacting with localized magnetic moments. However, nature and properties of magnetism at this interface are not well understood so far. In this paper, we focus on transport effects driven by spin-orbit coupling and intentionally neglect possible strong correlations. The electrical resistivity tensor is calculated as a function of the magnitude and orientation of an external magnetic field parallel to the interface. The semiclassical Boltzmann equation is solved numerically for the two-dimensional system of spin-orbit coupled electrons accelerated by an electric field and scattered by spatially correlated impurities. At temperatures of a few Kelvin and densities such that the chemical potential crosses the second pair of spin-orbit split bands, we find a strongly anisotropic modulation of the (negative) magnetoresistance above $10\mathrm{T}$, characterized by multiple maxima and minima away from the crystalline axes. Along with the drop of the magnetoresistance, an abrupt enhancement of the transverse resistivity occurs. The angular modulation of the latter considerably deviates from a (low-field) sinusoidal dependence to a (high-field) step-like behavior. These peculiar features are the consequences of the anisotropy of both intraband and interband scattering amplitudes in the Brillouin zone when the relevant energy scales in the system—chemical potential, spin-orbit interaction, and Zeeman energy—are all comparable to each other. The theory provides good qualitative agreement with experimental data in the literature.

Figure 1

(a) Band dispersion along $k_x$ (at $k_y=0$) near the Lifshitz point. Colors distinguish the orbital character of the electronic states. (b) Equienergetic surfaces at $\epsilon =\mu$ (Fermi level) with on top textures of the the average spin (SAM) and orbital (OAM) angular momentum. Blue (yellow) arrows correspond to states in the outer (inner) band of each pair of subbands. The complete set of parameters used for generating the plots are listed in the first line of Table 1 (Appendix pp2).

Figure 2

Angular modulations of (a–c) the longitudinal magnetoresistance $\mathrm{MR}$ and of (b–d) the transverse resistivity ${\rho }_{xy}$ at several values of the $\mathit{B}$-field in the range $2–20\mathrm{T}$. The temperature is $T=1\mathrm{K}$ and the density $n=2.2\times {10}^{13}{\mathrm{cm}}^{-2}$ (chemical potential at half of the gap between $d_{xz}$ and $d_{yz}$ states at $\mathit{k}=0$). In (a) and (b), scattering amplitudes given by Eq. (4). In (c) and (d), scattering amplitudes given by Eq. (5). Parameters listed in Table 1 (Appendix pp2).

Figure 3

Longitudinal magnetoresistance at $T=1\mathrm{K}$ for different orientations ${\varphi }_B$ (dashed lines) and averaged over all the angles (red solid line). Numerical values on the left axis. Angular-maximum of ${\rho }_{xy}$ as a function of $B$ rescaled by its value at $B=10\mathrm{T}$ (solid gray line) with numerical values on the right scale. A comparison between calculations for scattering models (4) and (5) is shown.

Figure 4

Fermi surfaces at $n=2.2\times {10}^{13}{\mathrm{cm}}^{-2}$ for different configurations of the in-plane magnetic field: (a) $B=0\mathrm{T}$, (b) $B=20\mathrm{T}{\varphi }_B={90}^{\circ }$, (c) $B=10\mathrm{T}{\varphi }_B={45}^{\circ }$, (d) $B=20\mathrm{T}{\varphi }_B={45}^{\circ }$. Magnitude and direction of the vector mean-free-path ${\mathbf{\Lambda }}_{\mathit{k}}$ as a function of the momentum $\mathit{k}$ on the outermost bands (which support all of the total conductivity at any fields) are represented by arrows—colors distinguish the two bands of each pair. The longitudinal conductivity ${\sigma }_{xx}$ is proportional to the average $x$-component of the vector mean-free-path. (At ${\varphi }_B=0,{90}^{\circ } {\rho }_{xx}={\sigma }_{xx}^{-1}$.) The modulation of the vmfp in (c) is mirror-symmetric with respect to the crystalline axes; hence, ${\sigma }_{xy}\left({\rho }_{xy}\right)\approx 0$ because states with opposite velocities compensate each other with equal weights ${\mathrm{\Lambda }}^x$. Instead in (d) the texture of the vmfp clearly violates the symmetry resulting in a finite and large ${\sigma }_{xy}$.

Figure 5

Average spin (SAM) and orbital (OAM) angular momentum around the Fermi surfaces at $B=0$ (top), $B=20\mathrm{T}$ and ${\varphi }_B={90}^{\circ }$ (middle), and $B=20\mathrm{T}$ and ${\varphi }_B={45}^{\circ }$ (bottom). ${\mathrm{\Delta }}_{\mathrm{SO}}>{\mathrm{\Delta }}_{\mathrm{Z}}$ and carrier density $n=2.2\times {10}^{-13}{\mathrm{cm}}^{-2}$, as the calculations in Sec. 4.

Figure 6

$\mathrm{MR}$ and ${\rho }_{xy}/{\rho }_{xy\left(10\mathrm{T}\right)}^{\text{max}}$ at negative $g$-factor (full set of parameters listed in the second line of Table 1).

Figure 7

$\mathrm{MR}$ and ${\rho }_{xy}/{\rho }_{xy\left(10\mathrm{T}\right)}^{\text{max}}$ at different values of the disorder correlation-length $\xi$ (full set of parameters listed in the third line of Table 1).

Figure 8

$\mathrm{MR}$ and ${\rho }_{xy}/{\rho }_{xy\left(10\mathrm{T}\right)}^{\text{max}}$ at lower carrier-densities than the calculations in the main text.