Structure and ferromagnetic instability of the oxygen-deficient ${\mathrm{SrTiO}}_3$ surface

${\mathrm{SrTiO}}_3$ (STO) is the substrate of choice to grow oxide thin films and oxide heterojunctions, which can form quasi-two-dimensional electronic phases that exhibit a wealth of phenomena, and thus a workhorse in the emerging field of metal-oxide electronics. Hence, it is of great importance to know the exact character of the STO surface itself under various oxygen environments. Using density functional theory within the spin generalized gradient approximation we have investigated the structural, electronic, and magnetic properties of the oxygen-deficient STO surface. We find that the surface oxygen vacancies order in periodic arrays giving rise to surface magnetic moments and a quasi-two-dimensional electron gas in the occupied Ti $3d$ orbitals. The surface confinement, the oxygen-vacancy ordering, and the octahedra distortions give rise to spin-polarized ${\mathit{t}}_{2g}$ dispersive subbands; their energy split near the Brillouin zone center acts as an effective Zeeman term, which, when we turn on a Rashba interaction, produces bands with momentum-spin correlations similar to those recently discovered on oxygen-deficient STO surface.

Figure 1

(a) Four layers of atomically relaxed STO bounded in the (001) direction by an SrO surface on one side and a ${\mathrm{TiO}}_2$ surface on the other side forming a $3\times 1$ supercell with an imposed 1/6 surface OV coverage in a stripe configuration. The presence of the oxygen vacancies (the sites of which are denoted by the two red circles) cause the tetrahedra at the ${\mathrm{TiO}}_2$ surface to buckle and push both the O and the Ti atoms away from the lower layers. The Ti-Ti bond length at the surface is larger by about 0.3 Å  along the $x$ axis as compared to that along the $y$ axis. (b) Top surface head-on view of the superlattice structure of oxygen vacancy ordering in periodic arrays.

Figure 2

(a) Stability graph of the striped vacancy pattern discussed in Fig. 1 with different surface OV densities. All three densities produce similar subsurface $3d_{xy}$ bands near the zone center but differ in the magnitude of the magnetic moments. For an explanation of the free energy and the allowed range of oxygen chemical potential see Appendix pp6. (b) Top view of the various ${\mathrm{Ti}}_{\left(m\right)}{\mathrm{O}}_{\left(n\right)}$ units at the ${\mathrm{TiO}}_2$ surface with one oxygen vacancy per unit cell (u.c.), showing (i) a $2\times 1$ u.c. with one eliminated O atom at $(0.50,0.50)$ creating a stripe of missing oxygen atoms along the $y$ axis (OV surface density $n_v=1/4$) and giving each Ti atom exactly one nearest neighbor vacancy, (ii) a $3\times 1$ u.c. with one eliminated O at $(0.33,0.50)$ causing a similar vacancy stripe but with lower density $(n_v=1/6)$, and (iii) a $4\times 1$ u.c. with one eliminated O at $(0.25,0.50)$, creating vacancy stripes with an OV surface density of $n_v=1/8$. Further vacancy configurations are discussed in Appendix pp1.

Figure 3

The near-$\mathrm{\Gamma }$ spin-GGA band structure of the four STO layer with striped vacancy configuration $(n_v=1/6)$. Up-spin bands are denoted with solid lines and corresponding down-spin bands are denoted with dashed lines. The $3{\mathit{d}}_{xy}$ bands are light and dispersive, while the $3{\mathit{d}}_{xz}$ and the $3{\mathit{d}}_{yz}$ are heavier and less dispersive due to confinement in the $z$ direction, which also produces the observed level quantization of states.

Figure 4

(a) Total up and down DOS of a four STO-layer slab with 1/6 OV surface density at the ${\mathrm{TiO}}_2$ surface. The zero is the Fermi level at the present level of doping. Inset: Integrated up and down DOS, showing the onset of magnetism at $\sim 300$ meV below the Fermi level. (b) Projected DOS of the $3d$ orbitals of a surface Ti atom next to an oxygen vacancy. The occupied polarized states are a mixture of spin-polarized ${\mathit{t}}_{2g}$ and ${\mathit{e}}_g$ located around the $S$ point of the BZ.

Figure 5

(a) The result of fitting the experimentally determined [16] spin $x$ component via a chi-square fit to the form given by Eq. (3) indicating a large planar Rashba coupling near the surface. (b) The spin pair of near-surface GGA bands of the four STO-layer slab after applying a Rashba spin-split term using Eq. (2) as compared to the experimental results [16]. Other occupied ${\mathit{d}}_{xy}$ bands are deeper inside the slab and are omitted in this figure. (c) The same as in (b) using the bands by $\text{GGA}+U$. The black solid line is the result of GGA or $\text{GGA}+U$ without spin relaxation. The green and red solid (dashed) lines are obtained with GGA or $\text{GGA}+U$ after spin relaxation for down (up) with and without the Rashba interaction, respectively.

Figure 6

The planar BZ band structure of the oxygen-deficient four STO-layer slab with $n_v=1/6$. Up spins are denoted with solid lines and down spins are denoted with dashed lines. The lowest occupied pair of spin bands is degenerate near the $\mathrm{\Gamma }$ point and is contributed by the Ti atoms away from the oxygen-deficient ${\mathrm{TiO}}_2$ surface. The second lowest spin-split pair comes from the Ti atoms one pair below the oxygen-deficient ${\mathrm{TiO}}_2$ surface, which are the bands of interest. The majority of the magnetic moment comes from the BZ corner and edge where the states are localized on the doped ${\mathrm{TiO}}_2$ surface as a mixture of $t_{2g}$ and $e_g$ with a considerable spin split.

Figure 7

(a) A four STO-layer slab with $\sqrt{2}\times \sqrt{2}$ (dimerized) vacancies, causing each surface Ti to have one nearest-neighbor vacancy. (b) A $2\times 1$ configuration with chains of vacancies in the $y$ direction. The configuration (b) contains two different types of surface Ti: one with no nearest-neighbor vacancies, and the other with two nearest-neighbor vacancies. Both configurations have doping level $n_v=1/4$ ($25\%$). (c) Stripe configuration, $n_v=1/4$ and (d) stripe configuration, $n_v=1/8$.

Figure 8

Comparison of the near zone-center conduction bands of oxygen-deficient (a) four-layer STO and (b) six-layer STO with $n_v=1/4$. Up-spin bands are denoted by solid lines and down-spin bands by corresponding dashed lines. The spin split is maximum for the bands near the OV surface and reduces as we go deeper into the bulk.

Figure 9

Orbital and site projection of the lowest four spin pairs of bands of a five STO-layer slab $(n_v=1/4)$ around the $\mathrm{\Gamma }$ point and along $k_y$, with band 1 being the deepest pair. The site locations are shown by the crystal unit cell aligned along the horizontal axis. On the graph, red is ${\mathit{d}}_{xy}$ and green is ${\mathit{d}}_{xz}$. Up spin is above the axis and is shown in filled circles, whereas down spin is below the axis and is shown in empty circles. The position of the missing oxygen on the surface is shown with a red circle. Each ${\mathit{d}}_{xy}$ subband can be seen to prefer different ${\mathrm{TiO}}_2$ planes. The deepest pair of bands is the Ti ${\mathit{d}}_{xy}$ near the SrO surface and the next deepest pair is Ti ${\mathit{d}}_{xy}$ next to the ${\mathrm{TiO}}_2$ surface. They are the bound states and are the most localized.

Figure 10

Orbital and site projection of the lowest four spin pairs of bands of a five STO-layer slab $(n_v=1/4)$ around the $\mathrm{\Gamma }$ point and along $k_y$, with band 1 being the deepest pair. A $\text{GGA}+U$ scheme has been used. The site locations are shown by the crystal unit cell aligned along the horizontal axis. On the graph, red is ${\mathit{d}}_{xy}$ and green is ${\mathit{d}}_{xz}$. The position of a missing oxygen on the surface is shown with a red circle. The Coulomb term causes the spins in a pair to move away from each other in real space, but the quantization effect of the confining potential is maintained. Up spin is above the axis and is shown in filled circles, whereas down spin is below the axis and is shown in empty circles.

Figure 11

The near-$\mathrm{\Gamma }$ band structure of the oxygen-deficient four-layer STO $(n_v=1/6)$ with spin-orbit coupling included. Up spins are denoted with solid lines and down spins are denoted with dashed lines. The change in the ${\mathit{d}}_{xy}$ bands close to the $\mathrm{\Gamma }$ point is negligible. The biggest effect of SOC can be seen in lifting the degeneracy of the band crossing at the corners.

Figure 12

DOS of the $3d$ states of a Ti atom one layer below the oxygen-deficient ${\mathrm{TiO}}_2$ surface $(n_v=1/6)$. The occupied states around the zone center are ${\mathit{t}}_{2g}$ in nature. At this GGA level, the ${\mathit{d}}_{xy}$ states carry negligible magnetic moments, and most of the spin polarization is in the ${\mathit{d}}_{xz}$ state.