Strain-induced topological phase transition at (111) ${\mathrm{SrTiO}}_3$-based heterostructures

The quasi-two-dimensional electronic gas at the (111) ${\mathrm{SrTiO}}_3$-based heterostructure interfaces is described by a multiband tight-binding model providing electronic bands in agreement at low energies with photoemission experiments. We analyze both the roles of the spin-orbit coupling and of the trigonal crystal-field effects. We point out the presence of a regime with sizable strain where the band structure exhibits a Dirac cone whose features are consistent with ab initio approaches. The combined effect of spin-orbit coupling and trigonal strain gives rise to nontrivial spin and orbital angular momenta patterns in the Brillouin zone and to quantum spin Hall effect by opening a gap at the Dirac cone. The system can switch from a conducting to a topological insulating state via modification of trigonal strain within a parameter range which is estimated to be experimentally achievable.

Figure 1

(a) Schematic representation of Ti atoms in STO lattice. The red and blue dots represent two different nonequivalent atoms belonging to the two different planes depicted in the figure. (b) Projection on the (111) plane of the Ti atoms with $\tilde{a}=a_0\sqrt{2/3}$; ${\vec{R}}_1$ and ${\vec{R}}_2$ identify our choice of the primitive vectors of the lattice.

Figure 2

(a) Energy band structure and (b) density of states for $\mathrm{\Delta }=-0.005\mathrm{eV}$, $\lambda =0\mathrm{eV}$, and $v=0\mathrm{eV}$. The energy $E=-0.46\mathrm{eV}$ corresponding to the states at the $K$ point is highlighted by a dashed line.

Figure 3

(a) Energy band structure and (b) density of states for $\mathrm{\Delta }=-1\mathrm{eV}$, $\lambda =0\mathrm{eV}$, and $v=0\mathrm{eV}$. The energy $E=-1.14\mathrm{eV}$ corresponding to the Dirac point is highlighted by a dashed line.

Figure 4

Mean values of the out-of-plane components of (a) orbital and (b) spin angular momentum, localized on one of the two Ti layers for $\mathrm{\Delta }=-1\mathrm{eV}$, $\lambda =0.1\mathrm{eV}$, and $v=0\mathrm{eV}$, and with $\mu =-1.14\mathrm{eV}$ (within the gap at the Dirac point).

Figure 5

Cubic structure under a trigonal distortion. The blue sphere is the Ti atom, while the red ones are the O atoms. The pink spheres are the positions of the O atoms in the undistorted structure. The black arrow is the (111) direction. $\theta$ represents the distortion angle used for the parametrization of the strain: When $\theta =arccos(1/\sqrt{3})$ the structure is unstrained; smaller (larger) values of $\theta$ lead to dilatation (contraction) along (111) direction.

Figure 6

(a) $a_g$, $e_g^{\pi }$, and $e_g^{\sigma }$ energy level behavior as a function of the distortion angle $\theta$ using $E_0=2\mathrm{eV}$, $\kappa =4.93$ and $z^{*}=4$. (b) $\mathrm{\Delta }$ parameter as a function of the distortion angle $\theta$ using the same parameters (solid line) and using $\kappa =12.5$, corresponding to the linearized muffin tin approximation (dashed lines).

Figure 7

Schematic representation of a finite-thick zigzag ribbon. The system is confined along the vertical Y axis while is unlimited along the X axis.

Figure 8

(a) Energy dispersion for a finite-thick zigzag ribbon of 20 sites, for $\mathrm{\Delta }=-1\mathrm{eV}$, $\lambda =0.1\mathrm{eV}$, and $v=0\mathrm{eV}$. The colored lines represent the edge states of the system, where there is an energy band inversion within the gap. (b) Energy band structure for the same parameters and the corresponding $Z_2$ invariant for each band. (c) The probability of a carrier occupation of a site for $k_X=\pi /\sqrt{3}$ for the red curve in the (a) panel.

Figure 9

Phase diagram for the first Kramers doublet (a) in the $\lambda -\mathrm{\Delta }$ plane with $v=0\mathrm{eV}$, (b) in the $v-\mathrm{\Delta }$ plane fixing $\lambda =0.1\mathrm{eV}$, and (c) in the $\lambda -v$ plane fixing $\mathrm{\Delta }=-1\mathrm{eV}$. The red region corresponds to $Z_2=1$, while the white one corresponds to $Z_2=0$. Diagram (a) has been computed using an energy step of 0.05 eV, diagram (b) has been computed using an energy step for $\mathrm{\Delta }$ ($v$) of 0.05 (0.001) eV, and diagram (c) using an energy step for $\lambda$ ($v$) of 0.005 (0.001) eV.

Figure 10

Energy gap between the first and the second Kramers doublet as a function of the strain parameter $\mathrm{\Delta }$. The green (red) curve represents the minimum (maximum) energy $E_{\text{min}}^{\text{II}}$ ($E_{\text{max}}^{\text{I}})$ of the upper (lower) Kramers doublet after subtraction of the quantity $\overline{E}=(E_{\text{max}}^{\text{I}}+E_{\text{min}}^{\text{II}})/2$; the dashed line is the chemical potential $\mu -\overline{E}$ with a fixed surface density. The orange section is the region in which the system exhibits QSHE.

Figure 11

Detail of the bottom of the bands near the $\mathrm{\Gamma }$ point, along the line connecting $M$ with $\mathrm{\Gamma }$ ($k_Y$ direction) on the left side and the line connecting $\mathrm{\Gamma }$ with $K$ ($k_X$ direction) on the right side. The dashed line represents the bands using only the TB model while the solid line includes a small trigonal strain parameter: The splitting induced by the latter is of the order of $3/2\mathrm{\Delta }$. The minima of the bands in the two cases have been shifted to coincide for ease of visualization.

Figure 12

Contours of exclusion at $68\%$, $95\%$, and $99.7\%$ confidence level in the $t_2-t_3$ plane: The best fit point is marked in black.

Figure 13

Structure of the lower bands near their intersection at the $K$ point: The bands behave linearly as planes near the $K$ point.

Figure 14

(a) Detail of the band structure near the $K$ point for the pure tight-binding model. (b) Same as above in the presence of a trigonal crystal field $\mathrm{\Delta }=-0.005$ eV: the degeneracy is partially split.

Figure 15

(a) Detail of the band structure near the $K$ point in the presence of a spin-orbit coupling $\lambda =0.01$ eV: the degeneracy is partially split. (b) Same as before, with the simultaneous presence of $\mathrm{\Delta }=-0.005\mathrm{eV}$ and $\lambda =0.01\mathrm{eV}$.

Figure 16

(a) Geometrical representation of the strain $\delta$ and the corresponding change in the bond length between two Ti: the angle ${\theta }_0=arccos(1/\sqrt{3})\sim 54.7^{\circ }$. (b) Dependence of the TB parameters on the distortion angle $\delta$. We show the behavior both of the strain parameter $\left|\mathrm{\Delta }\right|$, for two benchmark values of $\kappa$ (black, solid and dashed), and of the hopping parameter $t_3$ (red) as a function of $\delta$.

Figure 17

Evolution of the band structure for increasing strain applied to the surface. We show the band structure for a distortion angle increasing from $\delta =1^{\circ }$ (a), $\delta =3.2^{\circ }$ (b), $\delta =6^{\circ }$ (c). The dashed line corresponds to the filling needed to reach the intersection of the first two doublets at the $K$ point, namely, $\mu =-0.56\mathrm{eV}$ (a), $\mu =-0.815\mathrm{eV}$ (b), and $\mu =-1.04\mathrm{eV}$ (c). The electric field at the interface is fixed to $v=0$; the spin-orbit coupling is taken as $\lambda =0.01\mathrm{eV}$.

Figure 18

Phase diagram for each of the six Kramers doublets in the $\lambda -\mathrm{\Delta }$ plane with $v=0\mathrm{eV}$. The panels are ordered form lowest to the highest energy doublet. The red region corresponds to $Z_2=1$, while the white one corresponds to $Z_2=0$. Every diagram has been computed using an energy step of 0.05 eV.

Figure 19

Phase diagram for each of the six Kramers doublets in the $v-\mathrm{\Delta }$ plane fixing $\lambda =0.1\mathrm{eV}$ for small values of $v$ in the topologically interesting region. The panels are ordered form lowest to the highest energy doublet. The red region corresponds to $Z_2=1$, while the white one corresponds to $Z_2=0$. Every diagram has been computed using an energy step for $\mathrm{\Delta }$ ($v$) of 0.05 (0.001) eV.

Figure 20

Phase diagram for each of the six Kramers doublets in the $\lambda -v$ plane fixing $\mathrm{\Delta }=-1\mathrm{eV}$. The panels are ordered form lowest to the highest energy doublet. The red region corresponds to $Z_2=1$, while the white one corresponds to $Z_2=0$. Every diagram has been computed using an energy step for $\lambda$ ($v$) of 0.005 (0.001) eV.