Model of two-dimensional electron gas formation at ferroelectric interfaces

The formation of a two-dimensional electron gas at oxide interfaces as a consequence of polar discontinuities has generated an enormous amount of activity due to the variety of interesting effects it gives rise to. Here, we study under what circumstances similar processes can also take place underneath ferroelectric thin films. We use a simple Landau model to demonstrate that in the absence of extrinsic screening mechanisms, a monodomain phase can be stabilized in ferroelectric films by means of an electronic reconstruction. Unlike in the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ heterostructure, the emergence with thickness of the free charge at the interface is discontinuous. This prediction is confirmed by performing first-principles simulations of free-standing slabs of ${\mathrm{PbTiO}}_3$. The model is also used to predict the response of the system to an applied electric field, demonstrating that the two-dimensional electron gas can be switched on and off discontinuously and in a nonvolatile fashion. Furthermore, the reversal of the polarization can be used to switch between a two-dimensional electron gas and a two-dimensional hole gas, which should, in principle, have very different transport properties. We discuss the possible formation of polarization domains and how such configuration competes with the spontaneous accumulation of free charge at the interfaces.

Figure 1

(a) Schematic diagram of the geometry of the system. The sign criterion in all the equations throughout the paper assumes absolute values of both the polarization and the free-charge density, and their relative orientation is the one given in this figure. (b)–(d) Schematic band alignment for a ferroelectric thin film in various configurations. The corresponding value of the relevant gap $\mathrm{\Delta }$; the band offset at the interface ${\varphi }_{CB}$; and the Shottky barrier for electron in the presence of an electrode ${\varphi }_e$ are indicated in each case.

Figure 2

Total polarization (black solid line), zero-field contribution (dashed line), electronic contribution (solid blue line), and surface free charge (solid red line) versus ferroelectric film thickness ($d$). $P_S$ is the spontaneous polarization of the bulk ferroelectric material, $d_c$ is the critical thickness for the onset of ferroelectric metastability, and $d_{\mathrm{GS}}$ is the thickness at which the ferroelectric configuration becomes the ground state of the system. Inset: same plot with ${\chi }_{\eta }=2$ and ${\varepsilon }_{\infty }=1$ to highlight the convergence of $\sigma$ towards $P$ (in this situation $P=P_{\eta }$).

Figure 3

(a) Energy per unit volume as a function of polarization for various thicknesses. The numbers next to the curves indicate the values of the thickness in each case. Dashed line corresponds to a solution with $\sigma =0$ while solid lines are the curves with $\sigma \ne 0$. Curves with $\sigma \ne 0$ possess an equilibrium configuration only for $d\ge d_c$. (b) Graphical solution of the equilibrium condition given by Eq. (11).

Figure 4

Total polarization (black) and free charge (red) versus thickness calculated using different values for the DOS $g$. The different approximations for $g$ are $g\to \infty$ (solid lines), the value corresponding to ${\mathrm{PbTiO}}_3$ obtained from its bulk band structure [30] (dashed lines), and the two extrema of an estimated range of values for this family of materials (top limit in dashed-dotted and bottom limit in dotted lines, respectively).

Figure 5

(a) Energy curves for a ferroelectric thin film as a function of the polarization for different values of the applied electric field (numerical labels next to the curves, in units of $P_S/{\varepsilon }_0$). Red solid (black dashed) sections correspond to solutions with $\sigma =0$ ($\sigma \ne 0$). Energy curves have been shifted vertically for clarity. Light blue dots indicate equilibrium states as $\mathcal{E}$ is swept. Connected (disconnected) dots represent a continuous change (jump) in $P_{\eta }$. (b) Hysteresis loop for $P$ (black line) and $\sigma$ (red line) as a function of the applied electric field. For the polarization, solid lines represent the ferroelectric state with the 2DEG while the dashed lines correspond to the paraelectric phase. Both (a) and (b) are calculated for a ${\mathrm{PbTiO}}_3$ thin film with a thickness of 9 nm ($d\gtrsim d_c\sim 8.3$ nm). Points labeled in the top panel correspond to those indicated in bottom panel (please find a detailed description in the text).

Figure 6

Hysteresis loop for the polarization (black line) and surface free charge (red line) as a function of the applied electric field for a ferroelectric thin film of two different thicknesses: (a) 7.5 nm ($d<d_c\sim 8.3$ nm) and (b) 10 nm (representative of the $d\gg d_c$ situation).

Figure 7

Phase diagram as a function of thickness $d$ and external electric field $\mathcal{E}$ for a ${\mathrm{PbTiO}}_3$ thin film. Circles (crosses) indicate an upward (downward) jump in polarization in forward (backward) sweep. Solid (dashed) lines correspond to transitions from $\sigma \ne 0$ ($\sigma =0$) to $\sigma =0$ ($\sigma \ne 0$). Regions of paraelectricity and ferroelectricity (with 2DEG or 2DHG) coexistence) are shown in dark and light gray, respectively. In white, different states are accessible depending on the sweeping history.

Figure 8

Energy density of a ferroelectric thin film in monodomain (black curve) and polydomain (red curve) states. Curves are calculated for bulk ${\mathrm{PbTiO}}_3$, in the limit of $g\to \infty$, using parameters obtained from first-principles calculations [30]. In this plot $\mathrm{\Delta }=1$ V, realistic for surface redox processes.

Figure 9

Schematic depiction of the electrostatics involved in the case of an insulating thin film on top of a ferroelectric substrate, both (a) before and (b) after an electronic reconstruction. A schematic band diagram of the system is shown at the bottom of each panel. Note that for this geometry, even before the electronic reconstruction, free charges need to be present at the surface to screen the polar discontinuity with the vacuum/air region.

Figure 10

Total polarization (black solid line), zero-field contribution (dashed line), electronic contribution (solid blue line), and surface free charge (solid red line) as a function of thickness. $d_c$ is the one given by Eq. (13) and in this case it does not mark the critical thickness for 2DEG formation.