Rashba-like spin-orbit and strain effects in tetragonal ${\mathrm{PbTiO}}_3$

We performed first-principles calculations of the spin-orbit effect appearing in the electronic structure of the well-known ferroelectric perovskite oxide ${\mathrm{PbTiO}}_3$ and analyzed the results within group-theory-derived models. We evidenced some non-negligible linear Rasbha spin splittings of the unoccupied $p$ bands of Pb atoms and occupied $p$ bands of oxygen atoms. Our calculations also show that a cubic spin splitting is present for the unoccupied $d_{xy}$ bands of Ti atoms. All these spin-orbit effects lead to complex spin textures reversible by switching the electric polarization, which could be used for future spinorbitronic applications. Our results also demonstrate how applying epitaxial strain could be envisaged to tune these properties by changing the relative energy of some of the bands or the magnitude of the linear/cubic coefficients describing the spin splitting.

Figure 1

Band structure of ${\mathrm{PbTiO}}_3$ (a) without and (b) with the spin-orbit interaction. The majoritarily $d$-orbital contributions are given in color. The origin of the energy is set at the top of the valence bands, which we also consider as the Fermi level. Some areas of interest around the HS points are highlighted. (c) The spin-moment decomposition (in arbitrary units) is given along the full HS lines Z-R and Z-A, to highlight the spin splitting of the yellow area and (d) the corresponding iso-energy ($E=2.5$ eV) spin texture is given in the Z-R-A plane (for $k_z=\frac{\pi }{c_0}$). The $k_x$ and $k_y$ axis are in units of $\frac{2\pi }{a_0}$.

Figure 2

Spin textures calculated according to the Hamiltonian of Eq. (1) for only one of the two split bands (with only one direction of spin), with (a) a $p_z$ character and (b) a $d_{xy}$ character at the $Z$ point. These spin textures lead to the iso-energy ($E-E_F=2.5$ eV) spin textures given, respectively, in (c) and (d). The parameters $\alpha$, $\gamma$, ${\gamma }^{\prime }$, and ${\gamma }^{″}$ used to solve the Hamiltonian are obtained by fitting the band-energy dispersions calculated with the DFT and are given in Table 1.

Figure 3

(a) Band structure of strained ${\mathrm{PbTiO}}_3$ with spin-orbit interaction, for an in-plane lattice parameter $a=3.8\AA$. (b) Evolution as a function of the strain ${\eta }_{xx}$ of the lowest unoccupied band energies at the $X$ (left panel) and $Z$ (right panel) high-symmetry points.

Figure 4

Evolution as a function of the in-plane strain ${\eta }_{xx}$ of (a) the Rashba parameter and (b) the band character $n$. The results are given for the Ti-$d_{xy}$ (in red) and the Pb-$p_z$ (in black) bands at the $Z$ point.

Figure 5

Spin-projected band structures, with the corresponding iso-energy spin textures for (a) the lowest unoccupied and (b) highest occupied bands. The spin projection for the band structures is done on $m_y$ along $X\text{−}\mathrm{\Gamma }$ direction and $m_x$ along the X-M direction, because only $k_x$ or $k_y$ are, respectively, varying along these directions. The iso-energy spin textures are plotted in the $\mathrm{\Gamma }$-X-M plane (for $k_z=0$) and for energies $E-E_F=2.7$ eV and $-0.3$ eV.

Figure 6

Variation of the linear Rashba parameters ${\gamma }_x$ and ${\gamma }_y$ as a function of the in-plane strain ${\eta }_{xx}$ around the $X$ point.

Figure 7

(a) One-band spin texture calculated from the Hamiltonian of Eq. (1) by using the parameters of Table 1 for the $d_{xy}$ bands around $\mathrm{\Gamma }$ point and (b) the corresponding iso-energy spin texture ($E=1.65$ eV) in the $\mathrm{\Gamma }$-X-M plane (for $k_z=0$).

Figure 8

(a) Calculated out-of-plane lattice parameter $c$ and $c/a$ ratio as a function of the in-plane lattice parameter $a$. (b) Electrical polarization $P$ as a function of the in-plane strain ${\eta }_{xx}$.

Figure 9

Band structure calculated for ${\mathrm{BiScO}}_3$ with the contribution of each atom in color and two spin textures calculated in a ($k_x,k_y$,0) plane, to highlight the spin splittings of the VBM at the $X$ point and the CBM at the $Z$ point.