First-Principles Modeling of Electrostatically Doped Perovskite Systems

Macroscopically, confined electron gases at polar oxide interfaces are rationalized within the simple “polar catastrophe” model. At the microscopic level, however, many other effects such as electric fields, structural distortions and quantum-mechanical interactions enter into play. Here, we show how to bridge the gap between these two length scales, by combining the accuracy of first-principles methods with the conceptual simplicity of model Hamiltonian approaches. To demonstrate our strategy, we address the equilibrium distribution of the compensating free carriers at polar ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interfaces. Remarkably, a model including only calculated bulk properties of ${\mathrm{SrTiO}}_3$ and no adjustable parameters accurately reproduces our full first-principles results. Our strategy provides a unified description of charge compensation mechanisms in ${\mathrm{SrTiO}}_3$-based systems.

Figure 1

(a) Scheme of the ${\mathrm{TiO}}_2\mathrm{\text{:}}\mathrm{LaO}$ interface, showing the built-in ($P^0$, black arrows) and induced ($\Delta P$, blue arrows) polarization, and the free charge ${\sigma }_{\mathrm{free}}$. (b) Electric displacement as a function of $z$, illustrating Eq. (1). (c) Calculated internal electric fields $\mathcal{E}$ in bulk STO and LAO as a function of $D$. The polar nature of the interface stems from the impossibility of finding a common value of $D$ for which ${\mathcal{E}}_{\mathrm{LAO}}$ and ${\mathcal{E}}_{\mathrm{STO}}$ are both zero.

Figure 2

Conduction charge (a)–(c) and local electric displacement (d) at the ${\mathrm{SrTiO}}_3/{\mathrm{LaAlO}}_3$ interface. In (a)–(c) the solid curves are ${\rho }_{\mathrm{free}}(z)$ and the dashed curves are the nanosmoothed ${\overline{\rho }}_{\mathrm{free}}(z)$. Insets show an approximate diagram of the effective local potential and ${\overline{\rho }}_{\mathrm{free}}$ (shaded areas). In (d) the symbols show the Wannier-based local electric displacement computed from the bound charges. The curves represent ${\int }_{\infty }^z{\overline{\rho }}_{\mathrm{free}}(t)dt+D_{\mathrm{STO}}$.

Figure 3

Bulk properties of ${\mathrm{SrTiO}}_3$ used in the tight-binding model. Upper panels: internal potential (a) and dielectric constant (b) as a function of $D$. Lower panels: $t_{2g}$ conduction-band structure for $D=0$ (c) and $D=-e/2S$ (d).

Figure 4

Electron confinement and spatial distribution. Red curves with square symbols are the first-principles results, black curves (circles) are the results of the model. Parameters are the same as in Figs. 2a, 2b, 2c. (d) shows a thicker 24-cell STO film. The dashed (blue) and dot-dashed (green) curves were obtained by artificially altering selected features of the model (see text). Insets: tight-binding band structures corresponding to the green ($d_1$) and black ($d_2$) curves in (d). The arrow indicates the lowest $d_{yz}/d_{zx}$ state.