Spin-dependent anisotropic electron-phonon coupling in KTaO$_3$

Abstract

KTaO$_3$ (KTO) is an incipient ferroelectric, characterized by a softening of the lowest transverse optical (TO) mode with decreasing temperature. Cooper pairing in the recently discovered KTO-based heterostructures has been proposed to be mediated by the soft TO mode. Here we study the electron coupling to the zone-center odd-parity modes of bulk KTO by means of relativistic Density Functional Perturbation Theory (DFPT). The coupling to the soft TO mode is by far the largest, with comparable contributions from both intraband and interband processes. Remarkably, we find that for this mode, spin-non-conserving matrix elements are particularly relevant. We develop a three-band microscopic model with spin-orbit coupled $t_{2g}$ orbitals that reproduces the main features of the ab initio results. For the highest energy band, the coupling can be understood as a "dynamical" isotropic Rashba effect. In contrast, for the two lowest bands, the Rashba-like coupling becomes strongly anisotropic. The DFPT protocol implemented here enables the calculation of the full electron-phonon coupling matrix projected onto any mode of interest, and it is easily applicable to other systems.

Figures (11)

• Figure 1: Electronic band structure of KTaO$_3$. (a) Full lines are the ab initio results of the three conduction bands along M$-\Gamma-$X. Dashed lines are the tight-binding model [Eq \ref{['eq:H0']}] with the $3\xi=416$ meV SOC gap at the zone center. Gray lines are at Fermi energies $E_{F,1}=40$ meV and $E_{F,2}=240$ meV. The Fermi surface for $E_{F,1}$ with $n=1$ outer band (blue) and $n=2$ inner band (red), in the plane $(k_1,k_2,0)$ perpendicular to (b) [001], (c) [110] and (d) [111]. The angle in panel (b) is used in Fig. \ref{['fig:epc-phi']}, Fig. \ref{['fig:qe-vs-tb-tu']} and Fig. \ref{['fig:qe-vs-tb']}.
• Figure 2: Intraband EPC to zone-center odd-parity modes polarized along $[001]$ computed by QE.$\mathcal{G}_{n,\lambda}(\bm k)$ in Eq. \ref{['eq:QE-intra']} for the electronic band $n=1$ (blue), $n=2$ (red), and $n=3$ (green) for $T_{1u}$ modes (a) $\bar{S}_1$ [Eq. \ref{['eq:S1']}], (b) $\bar{S}_2$, [Eq. \ref{['eq:S2']}], (c) $\bar{S}_3$, [Eq. \ref{['eq:S3']}], and (d) $T_{2u}$ mode $\bar{S}_4$, [Eq. \ref{['eq:S4']}]. The experimental frequencies $\omega_{\bm q=\bm 0, \lambda}$ have been used in Eq. \ref{['eq:LambdaQE']} (Appendix \ref{['app:QEdecomposition']}). All modes show linear-in-$k$ EPC around $\Gamma$ in agreement with Eqs. \ref{['eq:G-sym-T1u']}-\ref{['eq:G-sym-T2u']}.
• Figure 3: Orientation dependence of the $\bm q=\bm 0$ EPC to the $\bar{S}_1$ mode. Intraband $\mathcal{G}_{n,\bar{S}_1} (\bm k,\bm q=\bm 0)$, along the FS of band $n=1$ (blue) and the FS of band $n=2$ (red) vs azimuthal angle $\varphi$ (see Fig. \ref{['fig:FS']}(b)) in the plane $(k_1,k_2,0)$ perpendicular to (a) [111], (b) [110] and (c) [001]. The Fermi energy is $E_{F,1}=40$ meV and the FS of the planes are shown in Figs. \ref{['fig:FS']}(b)-(c). The phonon polarization is perpendicular to the plane examined. The interband matrix elements $\mathfrak{g}_{nm,\bar{S}_1} (\bm{k},\bm q=\bm 0)$ with $m\neq n$ for the same orientations are shown in panels (d)-(f). For interband, we show the matrix element for the scattering of a fermion from the FS of one band to the other band at the same $\bm k$ point (since $\bm q=\bm 0$), which is outside the FS.
• Figure 4: Intraband $\sqrt{\omega_{\bm q 1}}\mathfrak{g}_{11/22,\bar{S}_{1}}(\bm k+\bm q/2,\bm q)$ and interband matrix elements $\sqrt{\omega_{{\bm q}1} }\mathfrak{g}_{12, 1}(\bm k+\bm q/2,\bm q)$ using invariant Eq. \ref{['eq:avrg']}. The polarization of the $\bar{S}_1$ mode is along [001] with $\bm k$ along [100] and $\bm{q}=(q_0,0,0)$ with $q_0=0,\pm 0.1, \pm0.2$ (in units of $\pi /a$).
• Figure 5: Schematic induced odd-parity hopping terms in a structure with a polar phonon along $\hat{\bm n}=\hat{\bm z}$. The phonon is represented by the polar displacement of the oxygen atoms (gray spheres) along $\hat{z}$, with induced hopping channels between NN $t_{2g}$ orbitals along the $\hat{y}$-bond. (a) Spin-independent interorbital $t_{yz,0}(\bm k)\sigma_0$ [Eq. \ref{['eq:tu']}], (b) spin-flip intraorbital $t_{xx,1}(\bm k)\sigma_1$ [Eq. \ref{['eq:tau1']}], (c) spin-flip interorbital $t_{xy,2}(\bm k)\sigma_2$[Eq. \ref{['eq:tau2']}], and (d) spin-dependent interorbital $t_{zx,3}(\bm k)\sigma_3$ [Eq. \ref{['eq:tau3']}]. The spin of the orbitals in the relativistic processes is represented by gray arrows, only shown for $y>0$ for clarity. Notice that the flipping or not of the spin depends on the spin state of the electron considered. We schematically illustrate the processes for spins polarized along the $z$ direction.