Unexpected Competition between Ferroelectricity and Rashba Effects in Epitaxially Strained ${\mathrm{SrTiO}}_3$

The Rashba parameter ${\alpha }_R$ is usually assumed to scale linearly with the amplitude of polar displacements by construction of the spin-orbit interaction. On the basis of first-principles simulations, ferroelectric phases of ${\mathrm{SrTiO}}_3$ reached under epitaxial compressive strain are characterized by (i) large Rashba effects at the bottom of the conduction band near the paraelectric-ferroelectric boundary and (ii) an unexpected suppression of the phenomena when the amplitude of polar displacements increases. This peculiar behavior is ascribed to the inverse dependence of the Rashba parameter with the crystal field ${\mathrm{\Delta }}_{\mathrm{CF}}$ induced by the polar displacements that alleviates the degeneracy of Ti $t_{2g}$ states and annihilates the Rashba effects. Although ${\alpha }_R$ has intrinsically a linear dependance on polar displacements, the latter becomes antagonist to Rashba phenomena at large polar mode amplitude. Thus, the Rashba coefficient may be bound to an upper value.

Figure 1

Phase diagram of ${\mathrm{SrTiO}}_3$ under compressive strain. (a) Sketches of the structural motions associated with the polar mode, the $a^0{a}^0{c}^{-}$ antiferrodistortive, and the combination of polar and antiferrodistortive motions yielding the $P4mm$, $I4/mcm$, and $I4cm$ cells, respectively. Arrows are not representative of the actual displacement amplitude appearing in the compound. (b) Energy difference (in $\mathrm{meV}/\mathrm{f}.\mathrm{u}$) of most stable phases with respect to the high symmetry undistorted $P4/mmm$ cell of ${\mathrm{SrTiO}}_3$ as a function of the strain. (c) Amplitudes (in $\AA /\mathrm{f}.\mathrm{u}$) of polar (blue filled circles) and $a^0{a}^0{c}^{-}$ octahedral rotation distortions (red filled squares) extracted from a symmetry mode analysis with respect to the high symmetry undistorted $P4/mmm$ cell (left scale) and computed spontaneous polarization (in $\mathrm{\mu }\mathrm{C}.{\mathrm{cm}}^{-2}$, green filled diamonds, right scale) for the identified ground states as a function of the strain.

Figure 2

Rashba spin splitting at the bottom of the conduction band in ferroelectric ${\mathrm{SrTiO}}_3$. Bands dispersion along the $\mathrm{\Gamma }\text{−}X$ (a),(b) and $\mathrm{\Gamma }\text{−}Y$ (c),(d) paths projected on the $S_x$ (c),(d) and $S_y$ (a),(b) spin components for states at the bottom of the conduction band for a polarization up (a),(c) or down (b),(d). The ferroelectric state reached with a substrate lattice parameter of 3.80 Å is considered here. High symmetry points are $\mathrm{\Gamma }(0,0,0)$, $\mathrm{X}(½,0,0)$, and $\mathrm{Y}(0,½,0)$ and refers to the $I4cm$ cell Brillouin zone.

Figure 3

Rashba parameter as a function of strain and polar displacements amplitude. Computed Rashba parameter ${\alpha }_R$ (in $\mathrm{meV}.\AA$) as a function of polar displacement amplitude [in $\AA /\mathrm{f}.\mathrm{u}$, panel (a)] and substrate lattice parameter [in Å, panel (b)] in strained ${\mathrm{SrTiO}}_3$ films.

Figure 4

Rashba parameter as a function of lattice distortions in a cubic ${\mathrm{SrTiO}}_3$. (a),(b) Computed Rashba parameter [(a), in $\mathrm{meV}.\AA$, upper panel] and crystal field splitting ${\mathrm{\Delta }}_{\mathrm{CF}}$ [(b), meV, lower panel] between the $d_{xy}$ and $d_{xz}/d_{yz}$ orbitals as a function of the polar mode amplitude $Q_P$ (in $\AA /\mathrm{f}.\mathrm{u}$, lower scale) in a cubic cell of ${\mathrm{SrTiO}}_3$ with (orange unfilled circles) and without (blue filled circles) $a^0{a}^0{c}^{-}$ AFD motions and as a function of AFD motion amplitude $Q_{\mathrm{AFD}}$ (in $\AA /\mathrm{f}.\mathrm{u}$, upper scale, red filled squares) at a fixed polar amplitude. The amplitude of the frozen AFD motion is $0.296\AA /\mathrm{f}.\mathrm{u}$ (orange unfilled circles) and of the frozen polar mode amplitude is $0.104\AA /\mathrm{f}.\mathrm{u}$ (red filled squares). Only the states at the bottom of the conduction band are considered for extracting the Rashba parameter. (c) Normalized Rashba parameter ${\alpha }_R/Q_P$ (meV) as a function of the inverse of the crystal field splitting induced by the polar displacement. ${\mathrm{\Delta }}_{\mathrm{CF}0}$ is the crystal field splitting appearing at $Q_P=0\AA$.