Competition between Kondo screening and magnetism at the ${\mathrm{LaAlO}}_3$/${\mathrm{SrTiO}}_3$ interface

We present a theory of magnetic and magnetotransport phenomena at ${\mathrm{LaAlO}}_3$/${\mathrm{SrTiO}}_3$ interfaces, which as a central ingredient includes coupling between the conduction bands and local magnetic moments originating from charge traps at the interface. Tuning the itinerant electron density in the model drives transitions between a heavy Fermi liquid phase with screened moments and various magnetic states. The dependence of the magnetic phenomena on the electron density or gate voltage stems from competing magnetic interactions between the local moments and the different conduction bands. At low densities only the lowest conduction band, composed of the $d_{xy}$ orbitals of Ti, is occupied. Its antiferromagnetic interaction with the local moments leads to screening of the moments at a Kondo scale that increases with density. However, above a critical density, measured in experiments to be $n_c\approx 1.7\times {10}^{13}{\mathrm{cm}}^{-2}$, the $d_{xz}$ and $d_{yz}$ bands begin to populate. Their ferromagnetic interaction with the local moments competes with the antiferromagnetic interaction of the $d_{xy}$ band leading to eventual reduction of the Kondo scale with density. We explain the distinct magnetotransport regimes seen in experiments as manifestations of the magnetic phase diagram computed from the model. We present data showing a relation between the anomalous Hall effect and the resistivity in the system. The data strongly suggest that the concentration of local magnetic moments affecting the transport in the system is much lower than the carrier density, in accord with the theoretical model.

Figure 1

Schematic phase diagram of the magnetic phases in the LAO/STO interface in the space of itinerant electron density ($n$), local magnetic moment density ($n_{\mathrm{imp}}$), and magnetic field ($H$). The dashed line marks the critical density $n_c$ above which $d_{xz}/d_{yz}$ itinerant electrons are present in the system.

Figure 2

(a) The spatial configuration of the three $t_{2g}$ orbitals of Ti ($d_{xy}$ in blue and $d_{xz}$, $d_{yz}$ in red). (b) The band structure of the Hamiltonian (1) with $B=0$ with bands originating from the different $t_{2g}$ orbitals plotted using the color coding of panel (a). The spin splitting near the avoided crossing points is a result of the spin-orbit coupling, ${\Delta }_{\mathrm{SO}}$, and the hybridization term, ${\Delta }_Z$. The dashed line demarcates a Lifshitz transition point where the two upper bands are first populated. The Fermi surfaces when the chemical potential is slightly above the Lifshitz transition (at $\mu =0$ on this scale) is presented in the inset. Panel (c) shows the density of states (arbitrary units) in this band structure as a function of energy.

Figure 3

Schematic representation of the virtual hopping processes between localized and itinerant electronic states. The local moments (black) are taken to be of $d_{xy}$ symmetry and the itinerant electrons can be of either $d_{xy}$ (blue) or $d_{xz}/d_{yz}$ (red) symmetry. (a) Hopping between the $d_{xy}$ itinerant electron and a $d_{xy}$ localized state with amplitude $t_1$ and hopping between the itinerant $d_{xz}/d_{yz}$ electron and the unoccupied $d_{xz}/d_{yz}$ state on the moment site with amplitude $t_{23}$. Note that the local-moment electron can also hop out of the impurity site to the $d_{xy}$ conduction band with amplitude $t_1$. (b) Processes of type (a) for the $d_{xy}$ result in an effective antiferromagnetic coupling due to Pauli exclusion while the hopping of $d_{xz}/d_{yz}$ is allowed for both spin orientation and results in a ferromagnetic coupling due to the Hund's rule coupling on the local-moment site.

Figure 4

(a) The band structure of the mean-field Hamiltonian (11) in the heavy Fermi liquid phase. Here $\chi \ne 0$ hybridizes the flat impurity band with the $d_{xy}$ conduction band leading to an expanded Fermi surface, marked by $k_F$ on the figure, which includes the total itinerant and local-moment electron density. The blue (red) curves correspond to $d_{xy}$ ($d_{xz}/d_{yz}$) bands, and the dashed line denotes the case with no hybridization field $\chi =0$. (b) Phase diagram for critical field vs itinerant electron density obtained by the mean field and perturbative RG calculations using the band-structure shown in (a). The blue solid curve is the first-order transition line obtained from the mean-field calculation for $J_1=0.9\mathrm{eV}$ and $J_2=-0.625\mathrm{eV}$. It is compared to the critical field extracted from the experimental data in Ref. [13], shown in red (crosses). The green dashed line gives an alternative evaluation of the characteristic field using RG calculation of a single magnetic impurity coupled to the multiple bands. The critical field corresponds to the Kondo scale inferred from this calculation with $J_1=0.55\mathrm{eV}$ and $J_2=-1\mathrm{eV}$. The black dashed-dotted line gives the density of the $d_{xz}/d_{yz}$ conduction electrons, which grows continuously from zero at the Lifshitz point. The Kondo regime has a domelike shape peaked around the Lifshitz point.

Figure 5

The flow diagram of Eqs. (D8) derived in Appendix pp4 for the case of $n=1.3\times {10}^{13}{\mathrm{cm}}^{-2}$. The red line indicates the trajectory of the second stage of the flow for the values of $J_1$ and $J_2$ taken to produce the Kondo scale in the phase diagram Fig. 4 ($J_1=550\mathrm{meV}$ and $J_2=-1\mathrm{eV}$).

Figure 6

The measured normalized resistivity vs the in-plane magnetic field $H$ scaled by the critical field $H_c$, which is plotted in Fig. 4 (red crosses). Here the different sets of data points represent different itinerant electron densities linearly distributed between 2 and 2.6 ($\times {10}^{13}{\mathrm{cm}}^{-2}$). As can be seen, all curves collapse when scaled by $H_c$. The black dashed line is the universal MR curve obtained from the Bethe ansatz for a single Kondo impurity [44] for comparison.

Figure 7

(a) Measurement of the anomalous contribution to the Hall effect. The red curve is the raw data. The blue curve is obtained by subtracting the ordinary linear Hall effect $\sim (H/en)sin\theta$ where $\theta =1^{\circ }$ is the angle between the plane and the tilted magnetic field. (b) The AHE contribution $\Delta {\rho }_{xy}$ extracted for different values of the gate voltage plotted against the longitudinal resistivity $\rho$ at $14\mathrm{T}$ where $\Delta {\rho }_{xy}$ is already saturated in most cases. The deviation from linearity observed at higher resistivity occurs in cases where $\Delta {\rho }_{xy}$ is not fully saturated at $14\mathrm{T}$. The implied linear dependence of the anomalous contribution $\Delta {\rho }_{xy}$ on the scattering rate $\propto {\tau }^{-1}$ suggests that the former stems from the mechanism of skew scattering [45].

Figure 8

Second-order processes contributing to the renormalization of $J$ which involve spin flips on both vertices. The dots represent a vertex of amplitude $J$, the horizontal lines represent the local-moment state, and the solid and dashed lines represent conduction electrons close to Fermi level and from the high-energy shell $D-\left|\delta D\right|<{\epsilon }_{\mathbf{p}}<D$, respectively.

Figure 9

(a)–(e) Third-order processes contributing to the renormalization of $J$. (f) Diagramatic representation of the ground-state energy-shift which contributes to the renormalization procedure in Hamiltonian formalism (see text).