Intrinsic origin of two-dimensional electron gas at the (001) surface of ${\text{SrTiO}}_3$

It is generally assumed that two-dimensional electron gas (2DEG) recently observed at the (001) ${\text{SrTiO}}_3$ surface can be solely derived by oxygen vacancies introduced during ultrahigh vacuum annealing or through ultraviolet irradiation exposure. However, 2DEG entirely due to defect formation may be at odds with the characteristics of high mobility and easy field-effect manipulation required for applications; to that aim, an intrinsic formation mechanism should be preferred. Using advanced ab initio simulations we give evidence that 2DEG at the (001) ${\text{SrTiO}}_3$ surface may even result from purely intrinsic properties of the pristine surface, provided that the surface is SrO terminated. The key concept is that the SrO termination is electron-attractive as a consequence of both the surface-induced polarity and the specific electronic reconstruction, whereas the ${\text{TiO}}_2$ termination is electron-repulsive. It follows that in vacuum-cleaved samples where both terminations are present, 2DEG can result from the structurally ordered superposition of the two kinds of domain, even in the absence of any extrinsic source. On the other hand, in etching-prepared single-terminated ${\text{TiO}}_2$ samples 2DEG should be assumed as entirely derived by extrinsic factors.

Figure 1

(a) and (b) Calculated structure of the two $(1\times 1)$ surface terminations. Arrows indicate the direction of the Ti-O (blue) and Sr-O (red) off-plane shift (dipole) along the $z$ axis, with arrow lengths roughly proportional to the displacement amplitudes. Insets show a cartoon of the charge dipole $Q$ (green curves) and the corresponding electrostatic potential $V$ (black) seen by the electrons at the most external ${\text{TiO}}_2$ layer of each termination: For SrO, the dipole potential pushes the electrons towards the surface; for ${\text{TiO}}_2$, electrons are repelled from the surface; $\left(\mathrm{a}{}_1\right)$ plane-by-plane cation-oxygen off-plane displacements for the SrO termination (blue diamonds are for Ti-O, red squares for Sr-O). $\left(\mathrm{a}{}_2\right)$ Corresponding electric dipoles for each monolayer $\Delta \mathit{\text{P}}$. $\left(\mathrm{a}{}_3\right)$ Macroscopic-averaged electrostatic potential seen by the electrons. All quantities are plotted vs the $z$ axis orthogonal to the surface, as a function of relative cation position $z/a_0$ $(z/a_0=0$ is for the inmost layer, $z/a_0\sim 8.5$ for the surface layer). $\left(\mathrm{b}{}_1\right)$ is the same as $\left(\mathrm{a}{}_1\right)$ for the ${\text{TiO}}_2$ termination.

Figure 2

(a) Layer- and orbital-resolved DOS calculated for the (001) STO surface with ${\text{TiO}}_2$ termination. (b) Same property for the SrO-terminated surface. Only the relevant orbital contributions are shown: O p (solid orange); Ti 3d $d_{xy}$ (solid gray); $d_{xz},d_{yz}$ (red); $d_{z^2}$ (green). The uppermost panels show the DOS of the surface layers, the lowest panels the DOS of the most bulklike; zero is set at the VBT of the inmost bulklike layers. The dotted vertical lines in correspondence of VBT and CBB bulk values help to visualize the band bending due to the surface dipoles. (c) Band structure for the ${\text{TiO}}_2$-terminated surface; $M=[\pi /a_0,\pi /a_0,0]$, $X=[\pi /a_0$, 0, $0]$. The red dashed line indicates the bulk VBT taken as the energy zero. The orbital character of the most important bands is indicated. (d) Band structure for the SrO-terminated surface.

Figure 3

(a)–(c) Layer-by-layer DOS for the SrO-terminated surface for different doping densities. The energy zero is fixed at the bulk VBT. (a) Undoped surface; (b) surface with 0.08 electrons per unit area $\left(n_{2\text{D}}=5.3\times {10}^{13}{\text{cm}}^{-2}\right)$; (c) surface with 0.1 electrons per unit area $\left(n_{2\text{D}}=6.6\times {10}^{13}{\text{cm}}^{-2}\right)$. Only Ti 3d $d_{xy}$ (solid gray) and $d_{xz},d_{yz}$ (red) orbitals are shown in a small energy interval around the CBB. Dashed vertical lines indicate $E{}_F$. (d)–(f) Layer-by-layer DOS for the ${\text{TiO}}_2$-terminated surface at varying charge density. Labels and doping charges are the same as in (a)–(c), respectively.

Figure 4

(a) Juxtaposed band structures for SrO (red squares) and ${\text{TiO}}_2$ (black circles) terminations, calculated for $n_s=6.6\times {10}^{13}{\mathrm{cm}}^{-3}$ doping charge. $E_F$ of both systems is shifted to zero. (b) Band structure for SrO termination at two different doping values. Black dashed lines indicate $E_F$. $K{}_{100}$ and $K{}_{110}$ indicate $k$-point strings along $\Gamma -X$ and $\Gamma -M$ directions, respectively; $k$ values are in units of $\pi /a_0=0.813{\mathring{\text{A}}}^{-1}$.

Figure 5

Calculated electric resistivity, mobility, and inverse Hall resistivity transport properties for SrO-terminated (a)–(c) and ${\text{TiO}}_2$-terminated (d)–(f) STO surface. Different colors are for different $n$-type doping levels [indicated in the legend of (a)]. (a) Resistivity; (b) mobility; (c) inverse Hall resistivity.