Formation and dynamics of small polarons on the rutile ${\mathrm{TiO}}_2$(110) surface

Charge trapping and the formation of polarons is a pervasive phenomenon in transition-metal oxide compounds, in particular at the surface, affecting fundamental physical properties and functionalities of the hosting materials. Here we investigate via first-principles techniques the formation and dynamics of small polarons on the reduced surface of titanium dioxide, an archetypal system for polarons, for a wide range of oxygen vacancy concentrations. We report how the essential features of polarons can be adequately accounted for in terms of a few quantities: the local structural and chemical environment, the attractive interaction between negatively charged polarons and positively charged oxygen vacancies, and the spin-dependent polaron-polaron Coulomb repulsion. We combined molecular-dynamics simulations on realistic samples derived from experimental observations with simplified static models, aiming to disentangle the various variables at play. We find that depending on the specific trapping site, different types of polarons can be formed, with distinct orbital symmetries and a different degree of localization and structural distortion. The energetically most stable polaron site is the subsurface Ti atom below the undercoordinated surface Ti atom, due to a small energy cost to distort the lattice and a suitable electrostatic potential. Polaron-polaron repulsion and polaron-vacancy attraction determine the spatial distribution of polarons as well as the energy of the polaronic in-gap state. In the range of experimentally reachable oxygen vacancy concentrations, the calculated data are in excellent agreement with observations, thus validating the overall interpretation.

Figure 1

${\mathrm{TiO}}_2$(110) surface structures. Front view (a) of the pristine ($1\times 1$) phase, sketched top view (b) of the reduced ($1\times 1$) phase, and front view (c) of the ($1\times 2$) reconstruction. Reported distances refer to the pristine ($1\times 1$) surface.

Figure 2

FPMD polaron hopping. A representative part of the results for the ($1\times 1$) surface with one oxygen vacancy in the $9\times 2$ slab ($c_{V_{\mathrm{O}}}=5.6\%$) is shown. The gray rectangles highlight data discussed in detail in the main text. (a) $T=0$ DFT+$U$ energies (based on the structures obtained by FPMD at 700 K and fully relaxed at $T=0$ K). The most stable polaronic configuration is taken as a reference. The up-pointing triangles, down-pointing triangles, and squares refer to configurations with the two excess electrons in the slab localized both at the $S0$ layer, both at the $S1$ layer, and one polaron per $S0$ and $S1$ layer, respectively. The circled value indicates the case of a polaronic ${\mathrm{Ti}}_{S1}^B$ site. (b) Polaron positions at every FPMD step. The $y$ axis indicates the polaron positions along each ${\mathrm{Ti}}^A$ and ${\mathrm{Ti}}^B$ [001] row (as sketched in the inset). The thick lines indicate the polaron positions as obtained by the FPMD runs at $T=700$ K, whereas cross-hairs are the corresponding post-FPMD polaron configurations obtained at $T=0$ K. The position of $V_{\mathrm{O}}$ projected along the various [001] rows is shown by dashed circles.

Figure 3

FPMD layer-resolved statistical analysis. (a) Average number of polarons at the various layers per FPMD time step. The dashed line represents the number of polarons (i.e., six polarons) sustained by the $3\times 1$ pattern in our $9\times 2$ slab. (b) Overall occurrences of polaron formation at the various layers. For the $1\times 1$ phase, the histogram bars represent from top to bottom the $S0, S1$, and eventually $S2$ layers, while for the $1\times 2$ phase the topmost bar refers to the reconstructed layer.

Figure 4

FPMD correlation. The radial distribution functions of the correlation between polarons (a), (b), (c) and between a polaron and the nearest oxygen vacancy (d), (e), (f) are shown for the systems with $5.6\%, 11.1\%$, and $16.7\%$ oxygen vacancy concentration. Filled and empty curves refer, in panels (a), (b), (c), to the correlation between two $S1$ polarons in the same and different [001] rows, respectively, and, in panels (d), (e), (f), to the correlation between the oxygen vacancy and the $S1$ and $S0$ polaron, respectively. The positions of the ${\mathrm{Ti}}_{S1}^A$ (filled circles) and $V_{\mathrm{O}}$ (empty circle) sites are sketched. The site-resolved polaron-polaron correlation function along [001] is shown for the ($1\times 2$) and 50% reduced ($1\times 1$) cases (g), (h). Here, the site-resolved correlation at null distance indicates the density of polarons per step on the various types of Ti atoms.

Figure 5

Simulated STM. Post-FPMD DFT+$U$ results for inequivalent polaronic configurations, for a system with $c_{{\mathrm{V}}_{\mathrm{O}}}=16.7\%$. The time-averaged simulated STM images of the in-gap states for the whole FPMD run (a) and for configurations (1168 time steps) with polarons in $S1$ only (b) are shown separately. Empty circles mark the positions of the oxygen vacancies. Ball models representing the $S0$ surface layer are also shown.

Figure 6

Polaron charge density. The side and top views of the ${\mathrm{Ti}}_{S0}^A$ polaron (a), (b), ${\mathrm{Ti}}_{S1}^A$ polaron (c), (d), and ${\mathrm{Ti}}_{S2}^A$ polaron (e), (f) are shown. The inner and outer isosurfaces represent different levels of the charge density of the polaronic states. Faded spheres represent deeper atoms in top-view images $[S0$ and $S1$ atoms not shown in panel (f)].

Figure 7

Polaron formation energy $E_{\mathrm{POL}}$ (a), strain energy $E_{\mathrm{ST}}$ (b), and electronic energy $E_{\mathrm{EL}}$ (c) of a polaron localized at ${\mathrm{Ti}}^A$ sites at various depths (from $S0$ to $S2$). Results of stand-alone DFT+$U$ calculations on an eight-layer-deep, $3\times 2$ slab with one excess electron considering both the cases of one and no $V_{\mathrm{O}}$ in the cell. ${\mathrm{Ti}}^A$ sites closest to the $V_{\mathrm{O}}$ at the $S1$ and $S2$ layers were considered, while the next-nearest neighbor to the vacancy at $S0$ is shown. The electrostatic potential energy for the electrons $E_{\mathrm{pot}}$ (d) was obtained on a neutral, pristine, $3\times 2$, eight-layer-deep slab.

Figure 8

Polaron-polaron interaction. $E_{\mathrm{POL}}$ (in meV per polaron) as a function of the distance between two polarons in the pristine $9\times 2$ slab ($-2$ charged slab). One polaron is fixed at a ${\mathrm{Ti}}_{S1}^A$ site, with the other one sitting in ${\mathrm{Ti}}^A$ sites in $S0$ (up-pointing triangles) and $S1$ (down-pointing triangles). Filled symbols refer to sites lying in the same ($1\overline{1}0$) layer (perpendicular to the surface layers) as the reference polaron fixed at ${\mathrm{Ti}}_{S1}^A$, whereas empty symbols indicate sites in the nearest-neighbor ($1\overline{1}0$) layer formed by ${\mathrm{Ti}}^A$ atoms. The insets show the in-gap DOS polaronic peaks for two particular configurations (energy values in eV units, with respect to the bottom of the conduction band). The spatial distribution of the polaronic charge is also shown for polarons located at two next-nearest-neighbor ${\mathrm{Ti}}_{S1}^A$ sites (two lattice constants apart, $\approx 6$ Å).

Figure 9

Polaron-vacancy interaction. (a) $E_{\mathrm{POL}}$ as a function of the distance between the oxygen vacancy and the polaron, included in the $9\times 2$ slab (i.e., $c_{{\mathrm{V}}_{\mathrm{O}}}=5.6\%, +1$ charged system). The polaron explores the ${\mathrm{Ti}}_{S0}^A, {\mathrm{Ti}}_{S1}^A$, and ${\mathrm{Ti}}_{S1}^B$ sites. Defect states at ${\mathrm{Ti}}_{S0}^B$ sites adjacent to the $V_{\mathrm{O}}$ (not reported) show a large positive $E_{\mathrm{POL}}$ ($\simeq 300$ meV). (b) $E_{\mathrm{pot}}$ on ${\mathrm{Ti}}_{S1}^A$ and ${\mathrm{Ti}}_{S1}^B$ sites as a function of a distance from the oxygen vacancy, in the $+2$ charged slab (one $V_{\mathrm{O}}$ and no excess electrons). In both panels, filled and empty symbols represent polaron positions with the same or different $\left[1\overline{1}0\right]$ coordinate with respect to $V_{\mathrm{O}}$, respectively.

Figure 10

$V_{\mathrm{O}}$-polaron and polaron-polaron combined effect for polarons in ${\mathrm{Ti}}_{S1}^A$ sites. (a)–(e) DOS calculated for two polarons at various distances with opposite (a) and parallel (b)–(e) spins. The total DOS and the projection on the two polaronic sites are shown with filled and dotted curves, respectively. One polaron is fixed at the ${\mathrm{Ti}}_{S1}^A$ closest to $V_{\mathrm{O}}$ while the other one is localized at various sites of the same [001] Ti row. The respective $E_{\mathrm{POL}}$ is indicated on each panel (in meV per polaron). (f)–(i) Spatial charge density of the polaronic states for two adjacent polarons with opposite (f), (g) and parallel (h), (i) spins. Each state is shown separately: spin-up (f) and spin-down (g) charge densities for the antiferromagnetic configuration, and the charge densities of the $-0.90$ eV (h) and $-0.35$ eV (i) energy states for the parallel-spin configuration. The insets show the simulated STM resulting from the in-gap states.