Striped electron fluid on (111) ${\mathrm{KTaO}}_3$

A recent study has revealed that the low carrier density electron gas (2DEG) induced at the interface of EuO and (111) ${\mathrm{KTaO}}_3$ exhibits a broken symmetry phase with a strong in-plane anisotropy of the resistivity. We present a minimal tight-binding model of this (111) 2DEG, including the large spin-orbit coupling from the Ta ions, which reveals a hexagonal Fermi surface with a highly enhanced $2k_F$ electronic susceptibility. We argue that repulsive electronic interactions, together with a ferromagnetic EuO substrate, favor a magnetic stripe instability leading to a partially gapped Fermi surface. Such a stripe state, or its vestigial nematicity, could explain the observed transport anisotropy. We propose a $k·p$ theory for the low energy $j=3/2$ states, which captures the key results from our tight-binding study, and further reveals the intertwined dipolar and octupolar modulations underlying this magnetic stripe order. We conclude by speculating on the relation of this stripe order to the superconductivity seen in this material.

Figure 1

Crystal structure of the top three (111) layers of KTO (Ta, ${\mathrm{KO}}_3$, Ta), with the horizontal axis along ($1,-1,0$) and the vertical axis along ($1,1,-2$). K ions are in purple, O ions in red, Ta ions in the top layer in cyan, and Ta ions in the bottom layer in gold.

Figure 2

(a) Fermi surface of (111) KTO in a $t$-only bilayer model, with $\mathrm{\Gamma }\text{−}K$ along the horizontal axis and $\mathrm{\Gamma }\text{−}M$ along the vertical axis. The bands are labeled by their $t_{2g}$ content, and the hexagon marks the surface Brillouin zone boundary. (b) Electronic band structure including spin-orbit coupling and a small trigonal distortion. The two sets of curves correspond to two chemical potentials adjusted to match the carrier densities reported by Bruno et al. [12] (high density) and Liu et al. [3] (low density). Fermi surface for the (c) high-density and (d) low-density cases. The horizontal arrows indicate the nestings along $\mathrm{\Gamma }\text{−}K$ that were identified by the susceptibility.

Figure 3

(a) Lindhard susceptibility, ${\chi }_0$, along $\mathrm{\Gamma }\text{−}M$ and $\mathrm{\Gamma }\text{−}K$ for the low density case with $T=0.5$ meV. Subscripts 1,3 are band indices (1 for the outer FS, 3 for the inner FS, noting that bands 1,2 and 3,4 are Kramers degenerate). (b) Evolution of the susceptibility with temperature. (Inset) Detail of the cusp in ${\chi }_{11}$ along the $\mathrm{\Gamma }\text{−}K$ direction associated with the outer FS. (c) Lindhard susceptibility including spin matrix elements as defined in Eq. (6). $x$ and $z$ correspond to the spin operators $S_x$ and $S_z$. The strong (11) cusp is only found for the $zz$ component.

Figure 4

Original FS from Fig. 2 (red,blue) as well as the reconstructed outer FS (green) due to a spin density wave potential $V\left({\mathbf{q}}_s\right)$ of strength 4.4 K, with ${\mathbf{q}}_s=(q_s,0)$ and $q_s$ given by the peak in ${\chi }_0$ along $\mathrm{\Gamma }\text{−}K$ for the outer FS.

Figure 5

(a) Outer FS as in Fig. 2, but including a Rashba term which lifts the Kramers degeneracy. (b) Resulting Lindhard susceptibility, ${\chi }_0$, computed with $T=0.5$ meV, including spin matrix elements as defined in Eq. (6) plotted along the ($1,-1,0$) direction. The cusp is again associated with the $zz$ component but now is an interband term between the two Rashba-split bands.

Figure 6

$k·p$ Fermi surfaces for the 2DEG upon including the Rashba SOC and trigonal distortion, shown for chemical potential $\mu =150$ meV ($n\approx 6.5\times {10}^{13}{\mathrm{cm}}^{-2}$) and $\mu =60$ meV ($n\approx 3.5\times {10}^{13}{\mathrm{cm}}^{-2}$). In each case, the spin-split FSs are more clearly visible for the outer pair of bands, with the splitting being slightly more significant along the $k_2\equiv k_{\left[\overline{1}\overline{1}2\right]}$ direction which corresponds to the $\mathrm{\Gamma }\text{−}M$ direction.

Figure 7

Largest eigenvalues of the $3\times 3$ susceptibility matrix truncated to just the dipole operators plotted as a function of momentum along $\mathrm{\Gamma }\text{−}K$.

Figure 8

(a) Maximum eigenvalues of the susceptibility matrix for $j=3/2$ multipoles, showing that the interband instability (thick solid line) slightly dominates over the intraband ones (thin solid lines). (b) Magnitude of the eigenvector components for ${\chi }_{12}+{\chi }_{21}$ at ${\mathbf{q}}_s$ [peak momentum in (a)], showing that the dominant order is composed of intertwined time-reversal breaking dipolar ($\sim J_z$) and octupolar (${\mathrm{\Gamma }}_5$) modulations.

Figure 9

Real and imaginary components of the dominant eigenvector corresponding to the peaks in $\chi$ for [(a) and (b)] intraband ${\chi }_{11}$, [(c) and (d)] intraband ${\chi }_{22}$, and [(e) and (f)] interband ${\chi }_{12}+{\chi }_{21}$. For the leading instability, which corresponds to an interband order, the symmetry breaking pattern involves time-reversal breaking dipolar and octupolar modulations.