Theory of Mott insulator–band insulator heterostructures

A theory of heterostructures comprised of ${\mathrm{LaTiO}}_3$ (a Mott insulator) and ${\mathrm{SrTiO}}_3$ (a band insulator) is presented. The band structure of the $\mathrm{Ti}$ $d$-electrons is treated with a nearest neighbor tight-binding approximation; the electric fields arising from the ${\mathrm{La}}^{3+}∕{\mathrm{Sr}}^{2+}$ charge difference and the carriers are treated within a Hartree approximation; and the on-site interactions are treated by unrestricted Hartree-Fock. The phase diagram as a function of interaction strength and layer number is determined and predictions are made for optical conductivity experiments. A noteworthy finding is that the edges of the heterostructure are generally metallic.

Figure 1

Schematic figure of the model used in the present study. Open and crossed circles show the positions of $\mathrm{Sr}$ and La ions ($\mathrm{La}$ layer number $n=2$). The dots show the positions of $\mathrm{Ti}$ ions. $x,y$ axes are chosen to be parallel to the $\mathrm{La}$ plane, and $z$ axis is perpendicular to the plane.

Figure 2

(Color online) Theoretical ground state phase diagram as a function of the on-site Coulomb interaction $U$ and inverse of the $\mathrm{La}$ layer number $n$. PMM: paramagnetic metallic state, (A) $\mathrm{FM}–\mathrm{OD}$: (anti)ferromagnetic orbitally disordered state, $\mathrm{FM}–\mathrm{OO}$: ferromagnetic state with two-dimensional orbital order, $\mathrm{FM}–\mathrm{G}–\mathrm{OO}$: ferromagnetic state with $\left(\pi ,\pi ,\pi \right)$ orbital order. We take $U^{\prime }=7U∕9$, $J=U∕9$, and $e^2∕\left(\epsilon \mathit{\text{at}}\right)=0.8$. The latter corresponds to $\epsilon \sim 15$ with a lattice constant $a=3.9\mathrm{\AA }$ and transfer intensity $t=0.3\mathrm{eV}$. The triangle is the critical $U$ above which $\left(\pi ,\pi ,\pi \right)$ orbital ordering occurs for $n=\infty$. The broken line shows the expected phase transition to the $\left(\pi ,\pi ,\pi \right)$ antiferromagnetic orbital ordering at finite $n$.

Figure 3

(a) Energy as a function of $U\left(n=1\right)$. Open squares: the spin and orbital-disordered state, filled circles: the spin-order orbital-disorder state, open triangles: the spin and orbital-ordered state, (b) square of magnetization as a function of $U$ in the intermediate coupling region $\left(n=1\right)$.

Figure 4

Calculated total electron density in the small $U$ region $\left(U=6t\right)$. Our results (not shown) for larger $U$ values are almost identical.

Figure 5

Electron density per spin and orbital in the intermediate coupling regime $U=8t$ (a), (b) and the strong coupling regime $U=10t$ (c), (d) for a heterostructure with two La layers $\left(n=2\right)$. Open and filled symbols show the densities of the up and down spin electrons, respectively. Circles and squares in (a), (c) are for the $xz$ and $yz$, respectively. Crossed squares in (a), (b) show the electron density per spin and orbital in a metastable paramagnetic state.

Figure 6

Upper panel: density profile (dashed line) and density in partially occupied bands of heterostructure with $n=6$, $U∕t=10$, and $\epsilon =15$ calculated in orbitally ordered ferromagnetic phase. Lower panel: density profile (dashed lines) and density in unoccupied bands (light and heavy solid lines) for on-site binding to $\mathrm{Ti}$ sites in heterostructure (on site energy for $\mathrm{Ti}$) between two $\mathrm{La}$ planes ${\epsilon }_0$; on site energy for “edge” $\mathrm{Ti}$ sites (between $\mathrm{Sr}$ and $\mathrm{La}$ planes) ${\epsilon }_0∕2$.

Figure 7

Dependence of total charge density on dielectric constant calculated for metastable paramagnetic phase of six layer heterostructure with $U∕t=10$.

Figure 8

(a) Optical conductivity for $n=1$ with $U=6t$ (small $U$ regime) and $10t$ (intermediate $U$ ferromagnetic state). Electric field $E$ is chosen to be parallel to $z$. Delta functions are approximated as Lorentzian with the half-maximum half-width $0.1t$; (b) schematic illustrations of subband energy structure at $U=6t$ (left) and $U=10t$ (right). Shaded areas show the continuum state. Broken lines show the Fermi energy.

Figure 9

Optical conductivity for $n=2$ with $U=8t$ (intermediate $U$ ferrimagnetic state) and $10t$ (large $U$ ferromagnetic orbital ordered state). Schematic illustrations of subband energy structure at $U=8t$ (b) and $U=10t$ (c). Shaded areas show the continuum state. Broken lines are the Fermi energy (band curvature not shown).