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# Small Polarons in Transition Metal Oxides

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## Abstract

The formation of polarons is a pervasive phenomenon in transition metal oxide compounds, with a strong impact on the physical properties and functionalities of the hosting materials. In its original formulation, the polaron problem considers a single charge carrier in a polar crystal interacting with its surrounding lattice. Depending on the spatial extension of the polaron quasiparticle, originating from the coupling between the excess charge and the phonon field, one speaks of small or large polarons. This chapter discusses the modeling of small polarons in real materials, with a particular focus on the archetypal polaron material TiO_{2}. After an introductory part, surveying the fundamental theoretical and experimental aspects of the physics of polarons, the chapter examines how to model small polarons using first-principles schemes in order to predict, understand, and interpret a variety of polaron properties in bulk phases and surfaces. Following the spirit of this handbook, different types of computational procedures and prescriptions are presented with specific instructions on the setup required to model polaron effects.

## 1 Introduction

Formation of polarons is particularly favorable in polar semiconductors and transition metal oxides owing to the strength of the electron-phonon interaction and is further promoted in the vicinity of a surface, where the crystal lattice is more flexible and the necessary lattice relaxations cost less energy (Deskins et al. 2009, 2011; Kowalski et al. 2010; Setvin et al. 2014a,b). Polarons play a decisive role in a wide range of phenomena, including electron transport (Crevecoeur and De Wit 1970; Moser et al. 2013), optical absorption, and chemical reactivity, and have crucial implications in high-*T*_{c} superconductivity (Salje et al. 2005), colossal magnetoresistance (Teresa et al. 1997; Rønnow et al. 2006), thermoelectricity (Wang et al. 2014a), photoemission (Cortecchia et al. 2017), and photochemistry (Linsebigler et al. 1995).

This chapter is mostly focused on the first-principles modelling of small polarons (Sect. 2) and its application to TiO_{2} (Sect. 3), an archetypal and widely studied polaron material. This introduction is complemented with a brief summary of the fundamental theoretical aspects defining the physics of polarons and with an overview of the experimental techniques used to detect polaron features in materials.

### 1.1 Theoretical Background

The emergence of polaron theory can be traced back to 1933 when Landau elaborated on the possibility for lattice distortions to trap electrons by means of an intrinsic modification of the lattice phonon field induced by the electron itself (Landau 1933). The resulting electron-phonon quasiparticle was later called polaron, a coupled electron-phonon system in which the polarization generated by the lattice distortions acts back on the electron renormalizing the electronic properties, for instance, the effective mass. An analogous discussion is valid for holes rather than electrons. A first formal description of the polaron problem was published in 1946 by Pekar (1946), who considered a free electron interacting with lattice deformations in the continuum approximation. Therefore, Pekar’s study is limited to the case of polarons with a size larger than the lattice constant, so that the atomic discreteness is negligible (Alexandrov and Devreese 2010). With the ionic lattice described as a polarizable dielectric continuum, Landau and Pekar showed that the polaron mass *m*^{∗} is enhanced with respect to the “bare” electron mass (Landau and Pekar 1965), an important result that set the basis for all subsequent theories.

During the 1950s, the second quantization formalism was used to refine the description of the polaron problem in terms of quantum effective Hamiltonians including an electronic term (*H*_{e}), a phonon term (*H*_{ph}) *and*, the electron-phonon coupling term (*H*_{e−ph}). The *e* − *ph* term is of fundamental importance for understanding the polaronic states, since it takes into account the type (e.g., short- or long-range) and intensity (weak to strong) of the mutual interactions between charge carriers and lattice vibrations. However, a complete analytic solution of the polaron Hamiltonian cannot be achieved, and approximations are needed. Fröhlich et al. (1950) and Holstein (1959) separately addressed different aspects of the problem and proposed mathematically more tractable formulations. Fröhlich theory relies on the continuum approximation and assumes long-range forces (large polarons), whereas Holstein theory takes into account explicitly the lattice periodicity and treats the electron-phonon coupling as a short-range interaction (small polaron). The distinction between small and large polaron is defined according to the strength of the electron-phonon coupling (weak/strong) and the extension of the lattice distortion around the electron (large/small) (Alexandrov and Devreese 2010; Rashba 2005; Hahn et al. 2018). The distinct features of small and large polarons are discussed at the end of this section.

#### 1.1.1 Fröhlich Hamiltonian

**k**and a phonon with wave vector

**q**, respectively, and Open image in new window is the electron-phonon coupling function for a system in

*d*dimensions, in three dimensions: Here

*V*is the volume of the system, and

*α*is the dimensionless Fröhlich electron-phonon coupling constant defined in terms of the reduced Plank constant

*ħ*, the charge carrier charge

*e*, the phonon frequency

*ω*of the long-wavelength optical longitudinal phonon, and the material’s static and high-frequency dielectric constants

*ε*

_{0}and

*ε*

_{∞}, respectively (

*ε*

_{0}includes ionic relaxation effects, whereas the ion-clamped

*ε*

_{∞}accounts only for electronic contributions):

*α*ranges between ≈0.02 (InSb) and ≈3.8 (SrTiO

_{3}). In the weak-coupling regime (small

*α*), the Fröhlich Hamiltonian can be solved using perturbation theory, and the Feynmann’s path integral approach provides accurate results for all coupling strengths (Feynman 1955; Rosenfelder and Schreiber 2001). In the large polaron Fröhlich picture, the spatial extension of the large polaron is bigger than the lattice constant, and the dressed electron is accompanied by a phonon cloud, whose density determines the size of the polaron effective mass

*m*

^{∗}and the polaron energy (Devreese 1996; Ermin 2013). Recently, the Fröhlich problem has been recasted within a first-principle perspective by F. Giustino, which allows for predictive nonempirical calculations of material’s specific electron-phonon properties (Giustino 2017).

#### 1.1.2 Holstein Hamiltonian

*t*is the hopping amplitude between neighboring sites,

*c*

_{i}(

*b*

_{i}) and \(c_i^{\dag }\)(\(b_i^{\dag }\)) are fermionic (bosonic) creation and annihilation operators acting on the site

*i*,

*ω*the phonon frequency,

*g*the electron-phonon coupling strength, and \(n_i=c_{i}^{\dagger } c_{i}\) is the fermionic particle number operator. As opposed to the Fröhlich model, the Holstein model only considers electron-phonon interactions at a single lattice site, greatly simplifying the model. In the simplified case of a polaron trapped in a linear molecule, hopping between different molecular sites (Holstein 1959, 2000), the Hamiltonian in Eq. 4 reduces to a two-site Hamiltonian describing the electron hopping between two sites, interacting with an ion placed in between via its vibrational mode (Alexandrov and Devreese 2010):

*M*the ion mass,

*γ*=

*Mω*

^{2}the spring constant and

*x*the linear ion displacement from the equilibrium position. The state |

*ψ*〉 with an extra electron introduced into the unperturbed system |0〉 and localized at the site 1 or 2 with amplitude probability

*u*(

*x*) and

*v*(

*x*), respectively, is written as

*Hψ*=

*Eψ*can be found only in special cases (e.g., ionic vibration perpendicular to the hopping direction), and one has to rely on numerical approximations or restrict the study to the nonadiabatic (

*t*≪

*ω*) or adiabatic (

*t*≫

*ω*) limits. The modelling of small polaron within a first-principles picture is addressed in Sect. 2.

At variance, the physics of an electron attached to a defect is rather different, as it is typically not mobile and its characteristics strongly depend on the type of defect (Freytag et al. 2016).Let us imagine that an electron is momentarily fixed at some point of the crystal. As a result of electron-lattice interaction, the surrounding lattice particles will be displaced to new equilibrium positions […] such as to provide a potential well for the electron. If the well is sufficiently deep, the electron will occupy a bound state, unable to move unless accompanied by the well, that is to say, by the induced lattice deformation.

#### 1.1.3 A Few Additional Considerations on Small and Large Polarons

General properties of large and small polarons

Large polaron | Small polaron |
---|---|

Polaron radius ≫ lattice parameter | Polaron radius ≃ lattice parameter |

Shallow state (∼10 meV below CBM) | In-gap state (∼1 eV below CBM) |

Coherent motion (scattered occasionally by phonons) | Incoherent motion (assisted by phonons) |

Mobility | Mobility |

Decreasing mobility with increasing temperature | Increasing mobility with increasing temperature |

The localization process is also different: a charge carrier introduced into a system can quickly equilibrate with the lattice and form a large polaron, whereas the formation of a small polaron occurs only after overcoming an energy barrier (Mott and Stoneham 1977; Lany 2015). Finally, the two types of polarons are characterized by very different transport properties. The large polaron, a heavy quasiparticle, is weakly scattered by phonons. This weak scattering can compensate the large mass, resulting in a large mobility for the large polaron. A small polaron instead hops between trapping sites with a lower mobility. Since its motion is assisted by phonons, the mobility of small polarons increases with increasing temperature. Both large and small polarons have been observed in several experiments, as reported in the next section, and studied by simulations and computational techniques. Historically, large polarons have been investigated mostly via effective Hamiltonians, in particular by means of variational treatments solved by Feynman’s path integral techniques and by diagrammatic Monte Carlo (Feynman 1955; Prokof’ev and Svistunov 1998; Mishchenko et al. 2000; Rosenfelder and Schreiber 2001; Gerlach et al. 2003; Gerlach and Smondyrev 2008; Hahn et al. 2018) approaches. First-principles techniques are more suitable for the description of the small polaron, but successful attempts to address the large polaron case have been presented in the last few years (Setvin et al. 2014a; Verdi et al. 2017; Miyata et al. 2017).

#### 1.1.4 How First-Principles Modeling Can be Used

The Fröhlich and Holstein model Hamiltonians are simplifications of the more general electron-phonon Hamiltonian that is discussed in great detail in a recent review of Giustino (2017). Giustino also explains how the individual terms can be derived by first-principles density functional theory, as briefly recapitulated below: (i) The first term describes the one-electron band structure, and that can obviously be calculated and parameterized using density functional theory, hybrid functionals or the GW approximation. The original Fröhlich and Holstein Hamiltonian involve only a single band, with a parabolic dispersion in the Fröhlich case, and a tight binding form in the Holstein case. In the case of electron (hole) polarons, one thus needs to parameterize the conduction (valence) band. (ii) The second term describes the vibrational frequencies of the optically active mode. The frequencies as well as the eigenmodes can be calculated by density functional perturbation theory. The Fröhlich Hamiltonian involves only a single dispersion free phonon mode, which couples with a wave vector dependent coupling term *q*. The functional form used by Fröhlich is exact in the long-wavelength limit. The Holstein model involves a single local mode, where the local displacement couples (somehow) to the electronic bands. (iii) The coupling between the electronic and phononic subsystem are described by the last term. The matrix elements in Fröhlich model can be exactly calculated using density functional perturbation theory (Verdi et al. 2017). The Holstein model is somewhat too simplistic and cannot obviously be mapped onto first-principles calculations.

The main issue with both approaches is twofold: (i) In many cases, the restriction to a single band and phonon mode is a too severe simplification, (ii) and even for a single band and a single phonon mode, the solution of the full many body problem can be formidable. This is the main reason why realistic modeling of polarons in real materials has hereto hardly been attempted. Nevertheless, the single band and single phonon model might be very accurate for many simple polar semiconductors and insulators. In this case, all one needs to determine are the effective masses, the phonon frequencies, and the dielectric constants. Then the solution of the Fröhlich model is fairly straightforward and expected to be accurate. Furthermore, there is little reason to invoke first-principles calculations for large polarons, if all materials parameters are experimentally established.

The situation is more complicated for small polarons. As already mentioned, the Holstein model is a too simplistic approach: for small polarons, the lattice distortions are usually large around an atom, and it is not obvious how to map the distortions to a single effective mode; furthermore the implicitly assumed harmonic approximation might not be accurate if the distortions are large. However, small polarons often involve fairly large energy scales around 1 eV. In this case, one might consider to treat the ionic degrees of freedom as classical particles. In many transition metal oxides, this should be a valid approximation, since the typical vibrational frequencies are small compared to the considered energy scales. In other words, quantum fluctuations of the ions can be disregarded. The second crucial approximation is to treat the electronic degrees of freedom using a mean field theory, such as DFT. There is no need to discuss the accuracy of DFT; it often works exceedingly well. Polarons though pose a particular challenge, since they involve localization of the electrons, which many density functionals fail to describe accurately because of self-interaction errors (see Sect. 2). Compared to the model approach, the full first-principles approach is much simpler. One simply choses a large unit cell, adds an electron, and lets the system evolve into the ground state. The disadvantage compared to the model approach is about giving up on the quantum nature of the ions and restricting the treatment of the electrons to a mean field approximation. The advantages are, however, numerous. The model Hamiltonians are obtained by expanding the full electron-phonon Hamiltonian around a reference ground state to second order in the ionic displacements (recall, one can use density functional perturbation theory to calculate the vibrational frequencies, as well as the electron-phonon coupling). Anharmonicities are therefore implicitly neglected in the model approach. In this respect the simplistic, brute force approach certainly excels the model approach.

### 1.2 Experimental Observations of Polarons

List of experimental observations of polarons in oxide materials using different techniques

Type | Material | Source | Exp. technique | Publication |
---|---|---|---|---|

Hole, small polaron | UO | Oxidation | Conductivity measurement | (Nagels et al. 1963) |

Hole, small polaron | MnO | Li doping | Conductivity measurement | (Crevecoeur and De Wit 1970) |

Electron, small polaron | CeO | O vacancies | Conductivity and Seebeck | (Tuller and Nowick 1977) |

Electron, small polaron | BaTiO | Nb doping | EPR | (Possenriede et al. 1994) |

Electron, small and large polaron | a-TiO | Nb doping | Conductivity and optical measurements | (Zhang et al. 2007) |

Electron, small polaron | r-TiO | O vacancies | Resonant photoelectron diffraction | (Krüger et al. 2008) |

Electron, small polaron | r-TiO | O vacancies | EPR | (Yang et al. 2013) |

Electron, large polaron | a-TiO | O vacancies | ARPES | (Moser et al. 2013) |

Electron, small polaron | r-TiO | O vacancies | STM and STS | (Setvin et al. 2014a) |

Electron, small polaron | r-TiO | UV irradiation or H adatom | IR spectroscopy | (Sezen et al. 2015) |

Hole, small polaron | LiNbO | Visible light | IR spectroscopy | (Freytag et al. 2016) |

Electron, small polaron | r-TiO | O vacancies | IR spectroscopy on adsorbates | (Cao et al. 2017) |

Electron and hole, large polaron | Lead halide perovskites | Laser pulse | TR-OKE | (Miyata et al. 2017) |

#### 1.2.1 Conductivity Measurements

^{5+}to U

^{4+}sites upon thermal activation, an interpretation further substantiated by the observation that the activation energy

*ΔE*decreases with increasing oxygen concentration.

Experimental confirmations of Eq. 7 were obtained for a wide range of materials, with charge carriers injected by different types of defects. Examples are given by the hole small polarons observed in Li-doped MnO (Crevecoeur and De Wit 1970) and by the electron small polarons in oxygen-deficient cerium dioxide (Tuller and Nowick 1977). In the latter example, two excess electrons originate from every oxygen vacancy present in the system. Each excess electron localizes at one Ce^{4+} site forming a Ce^{3+} ion. Hopping is activated by temperature, and the conductivity measurements confirmed the expected trend for polarons (Eq. 7). Moreover, by measuring the Seebeck coefficient for thermo-electricity, the number of carriers was found to be temperature independent. This is in contrast with the band model for the conductivity, which predicts an increasing number of charge carriers with raising temperature. On the contrary, according to the polaron hopping model, the charge carriers are introduced in CeO_{2} only by defects, and the mobility increases with raising temperature due to the increased phonon populations.

#### 1.2.2 Electron Paramagnetic Resonance

Electron paramagnetic resonance (EPR) is another experimental technique able to identify lattice ions with unpaired electrons, a distinct feature of the polaron state. In EPR experiments, an external magnetic field splits the energy level of the unpaired electrons (Zeeman effect), thus determining energy levels available by emitting or absorbing a photon at a specific frequency: the sample is illuminated by light, typically at constant frequency, while the magnetic field varies, and the resonance peaks are measured as function of the magnetic field when the conditions for the level transitions are satisfied. The shape, intensity, and energy values of the resonance peaks determine the electronic states of the atoms in the sample where charge trapping can be detected. Small polarons were identified by EPR in Nb-doped BaTiO_{3} samples (Possenriede et al. 1994; Lenjer et al. 2002) and other materials including oxygen-deficient rutile TiO_{2} samples (Yang et al. 2013).

#### 1.2.3 Optical Measurements

The difference between small and large polarons manifests itself prominently in the optical properties of materials. An example is the response of two different polymorphs of TiO_{2}, rutile and anatase, to (electron) doping with Nb, reported by Zhang et al. (2007). Epitaxial thin films of the two different polymorphs were grown on appropriate substrates, SrTiO_{3}(001) or LaAlO_{3}(001) for anatase and Al_{2}O_{3}(r and c-cut) for rutile. By substituting 4-valent Ti with 5% of 5-valent Nb, electrons were added to the system. The two types of films showed contrasting behavior in conductivity measurements: anatase films were metallic while rutile films semiconducting. Together with the similar carrier density observed in both films, this is consistent with the formation of large and small electron polarons, respectively. The optical transmittance of 80% in the visible range for epitaxial anatase films shows that these films can be characterized as transparent conductive oxides (Klein 2012).

#### 1.2.4 Resonant Photoelectron Diffraction

The charge distribution in reduced TiO_{2} rutile (110) single crystals was first determined by Krüger et al. (2008, 2012). Their experiment is based on angle-resolved x-ray photoemission (XPS). Intensity variations in a specific photoelectron peak (here Ti-2*p*) are recorded while changing the polar and azimuthal emission angle. Since forward focusing dominates the scattering of electrons with kinetic energies of a few hundred eV, the configuration of near-range atomic neighbors can be accessed in a rather direct manner and modeled in a simple cluster geometry (Chambers 1992). In their work, the authors took advantage of the fact that the Ti-2*p* XPS peak shows a clear shoulder that is attributed to Ti^{3+}; the intensity of this feature was additionally increased by tuning the photon energy to a resonance condition (Bertel et al. 1983). The photoelectron diffraction pattern of the Ti^{4+} peak (from Ti^{4+} at regular lattice sites) and the reduced Ti^{3+} signature turned out to be drastically different, and the best fit was obtained by attributing the excess electrons to the subsurface Ti atoms. This is true independently of the way the excess electrons are introduced into the lattice, either by creating O vacancies (Krüger et al. 2008) or by adding electrons via Na deposition (Krüger et al. 2012). This points to the fact that the location of polarons in subsurface position is an intrinsic feature of rutile TiO_{2}.

#### 1.2.5 Angle-Resolved Photoemission Spectroscopy

TiO_{2} has been investigated by numerous different experimental techniques in addition to the methods mentioned above. Angle-resolved photoemission spectroscopy (ARPES) (Moser et al. 2013) was used to identify the presence of large polarons in the anatase polymorph of titanium dioxide. The energy dispersion close to the Fermi level measured by ARPES experiment shows satellite (shallow) states below the conduction band, corresponding to large electron polarons, brought about by the oxygen vacancies present in the sample. By tuning the amount of oxygen vacancies (via UV irradiation Locatelli et al. 2007), the density of charge carriers can be controlled. At high vacancy concentrations, the satellite states disappear, due to the overlap of the polaronic wavefunctions, giving rise to a metallic behavior indicated by the crossing of the conduction band with the Fermi energy.

#### 1.2.6 Scanning Tunneling Microscopy and Spectroscopy

A direct view at the polaronic states in TiO_{2} can be achieved by using scanning tunneling microscopy (STM) and spectroscopy (STS) (Setvin et al. 2014a). In-gap states stand clearly from the STM and STS studies on both anatase and rutile TiO_{2} polymorph, due to charge carriers (electrons) induced by oxygen vacancies or Nb doping. However, the two polymorphs present quite different polaronic states: while rutile TiO_{2} is a prototypical small polaron material, anatase was found to host predominantly large polarons. STM is capable to provide information on the spatial distribution of the polaronic charge at the surface, which is helpful for the identification of small and large polarons.

#### 1.2.7 Infrared Spectroscopy

Small polaron states have been detected in TiO_{2} also by absorption infrared (IR) spectroscopy experiments (Sezen et al. 2015). The vibrational energies of the lattice bonds are typically in the infrared regime. Therefore, by exposure of the sample to IR light, resonance peaks appear in the measured transmitted and reflected beams. In the presence of small polarons, the localized charge together with the local lattice distortions contributes to form resonant peaks at characteristic vibrational frequencies. Interestingly, the polaron peaks in TiO_{2} were found to be independent on the source of charge carriers, H adsorption and UV irradiation, a strong evidence of the polaron nature of the peaks. The characteristic vibrational spectra of a defect state, at a donor would, in fact, be susceptible to the type of the donor.

Infrared spectroscopy can also be used to detect polarons indirectly, by inspecting the vibrational properties of bonds in the vicinity of the trapping site. Electron small polarons were found to redshift the stretching frequency of NO molecules adsorbed on reduced TiO_{2} surfaces with respect to the pristine samples (Cao et al. 2017). Analogously, hole small polarons modify the vibrational frequency of the OH impurities at Li vacancies in LiNbO_{3} samples (Freytag et al. 2016). In the latter study, irradiation in the visible region was used to generate hole small polarons at O sites and electron polarons at Nb sites. The hole polarons localize in the vicinity of the Li vacancies, due to the a more favorable electrostatic potential, and therefore strongly contribute to the frequency shift of the OH impurities. Once the source is turned off, hole and electron polarons recombine, since their mobile character, and the original vibrational frequency is restored.

#### 1.2.8 Time-Resolved Optical Kerr Effect

Recently, the time-resolved optical Kerr effect (TR-OKE) was used to investigate polaron formations in CH_{3}NH_{3}PbBr_{3} and CsPbBr_{3} perovskites (Miyata et al. 2017). In a TR-OKE experiment, a laser pulse is sent to the material, and the polarization rotation is detected. With laser pulses of energy larger than the energy gap value, the experiment probes the TR-OKE response upon charge injection. In these lead halide perovskites, the detected signals are compatible with a description of the altered phonon dynamics in terms of the formation of hole and electron, large polarons in the PbBr_{3} sublattice. This TR-OKE study reported also interesting insights on the polaron formation dynamics. In fact, the two materials show different rates for the polaron formation. This is attributed to the reorientations of the cations in CH_{3}NH_{3}PbBr_{3}, which determine a faster polaron formation than in CsPbBr_{3}.

## 2 Modeling Small Polarons by First Principles

The electronic structure community has been quite active in the study of polarons in real materials. Simulations can be helpful to understand and interpret the experimental findings or to predict the formation of polaronic states. This section provides a short overview of the first-principles methods generally used to model polarons in materials and discusses different computational procedures to acquire information on the formation and dynamics of polarons and on their mutual interaction.

### 2.1 Theories and Methods

Density functional theory (DFT) has been largely applied as a starting point to study polarons in materials. However, common schemes used to describe the exchange-correlation functional in DFT simulations, such as the local density and generalized gradient approximations (LDA and GGA, respectively), fail to describe properly the charge localization at atomic sites. By using LDA or GGA, the excess charges are delocalized overall in the lattice, and partial occupations of the available electronic levels are favored against integer occupations. Therefore, the modeling of polarons requires a correction in order to overcome the drawback of standard local and semilocal exchange-correlation approximations.

According to the Janak’s theorem (Janak 1978), variations of the total energy *E* due to the electronic occupation (*n*_{i}) of the state *i* are given in terms of the DFT Hamiltonian eigenstates *ε*_{i}, as d*E*∕d*n*_{i} = *ε*_{i}, independently of the exchange-correlation approximation. The expected behavior of the exact total energy is a piecewise linear function of the electronic occupation, with discontinuities in the first derivative for integer values (*n*_{i} = *N*) (Perdew et al. 1982; Perdew and Levy 1983), that is \({\mathrm {d}^2 E} / {\mathrm {d} n_i^2} = 0\) except at integer occupancies. Therefore, the energy of the state *i* remains constant during electron addition or removal.

*i*upon its own occupation reflects a spurious self-interaction effect, introduced by the type of exchange-correlation approximation. As a consequence of the convexity of the calculated energy, partial occupations are preferred over integer occupations. This leads to well-known failures of DFT simulations including the underestimation of the energy band gaps, the description of strongly correlated insulators as metals, and the difficulty to account for charge localization.

*α*’s (and in some screened hybrids the screening length) can be tuned to match physical quantities determined empirically, such as the energy band gap (He and Franchini 2012). Alternatively, the hybrid functional parameters can be determined ab initio by requiring the linearity of the total energy (Kronik et al. 2012; Elmaslmane et al. 2018) or following a self-consistent procedure (He and Franchini 2017) or fitting the parameters to the dielectric function of the material (Chen et al. 2018; Cui et al. 2018). This is possible since HF theory predicts an opposite result as compared to DFT: The energy change upon electronic occupation is described by a concave function of the electronic occupation, \({\mathrm {d}^2 E} / {\mathrm {d} n_i^2} < 0\); thus, the charge localization is overestimated by HF. This overestimation is due to neglecting screening effects, which are usually not negligible in solids.

*U*method, where an additional term is added to the expression for the total energy. The DFT+

*U*total energy

*E*

_{DFT+U}is given by

*E*

_{DFT}is the energy obtained by standard DFT, while

*E*

_{U}is an on-site correction arising from a local Hubbard-like Coulomb repulsion (

*U*) and a Hund’s parameter (

*J*), including double-counting corrections. Various expressions for

*E*

_{U}have been proposed (Anisimov et al. 1991, 1993; Liechtenstein et al. 1995; Dudarev et al. 1998a,b), such that the integer occupation of electronic states is energetically favored. Clearly, the results depend on the choice of

*U*and

*J*, which is not trivial, and, as a matter of fact, these quantities are typically treated as fitting parameters, by adjusting their values such that a specific quantity (e.g., the band gap) is predicted accurately. In order to maintain the ab initio character of DFT, procedures have been defined to calculate the DFT+

*U*parameters from first principles such as the constrained local density approximation (Gunnarsson 1990; Cococcioni and de Gironcoli 2005), where the interaction parameters are obtained by considering the total energy variation with respect to the occupation number of localized orbitals, and the constrained random-phase approximation (cRPA), which allows for an explicit calculation of the matrix element of the screened interactions

*U*and

*J*(Aryasetiawan et al. 2004). Alternatively, in the specific case of excess charge introduced by impurities, the

*E*

_{U}(

*U*,

*J*) term can be substituted by an on-site angular-dependent potential that does not affect the states of the defect-free system. The potential depends on parameters which are tuned to restore the linearity of the total energy (Lany and Zunger 2009; Lany 2011).

### 2.2 Polaron Formation: Energetics and Structural Distortions

*U*or hybrids), the stability of polarons is analyzed in terms of a set of different energies. Unfortunately, the notion of the polaronic energies is not well standardized, and it varies according to the authors’ preferences. The important ingredients to consider in order to characterize a polaron system are:

- 1.
Type of electronic state: (a) excess charge localized in a lattice site (polaron solution with integer particle occupation on an energy level localized at one lattice site) or (b) delocalized in the entire sample (no polaron, fractional occupation on several lattice sites).

- 2.
Type of structural solution: (a) local distortions at the trapping site (polaron solution) or (b) absence of local distortions (uniform lattice, no polaron). The lattice distortions are obtained fully self-consistently through the minimization of the forces acting on the ions (To avoid any confusion, we specify that also in the non-polaron state, the forces are minimized, but since the excess charge is delocalized in the lattice, the structural changes are uniform and generally very small and become zero for very large supercells.).

*U*and hybrid runs). By selectively switching on or off charge localization and/or lattice distortions, it is possible to compute the total energy of the system in different regimes (see Fig. 4):

- \(E^{\mathrm {loc}}_{\mathrm {dist}}\):
- Total energy of the polaron state: charge localization plus lattice distortions;
- \(E^{\mathrm {deloc}}_{\mathrm {unif}}\):
Total energy of the system with delocalized charge carriers and uniform lattice;

- \(E^{\mathrm {deloc}}_{\mathrm {dist}}\):
Total energy of the system with delocalized charge constrained into the lattice structure of the polaron state. Since that solution is only metastable, it might be difficult to realize it in practice (for instance, by enforcing a non-spin-polarized ground state).

In order to compare these energies with each other, the computational setup must be kept fix (except for the spin degree of freedom) in all type of calculations, in particular the total number of electrons (including excess electrons), the unit cell size, and the value of *U* and *J* (or the value of the mixing parameter in hybrid-DFT).

*E*

_{POL}(polaron formation energy),

*E*

_{ST}(strain energy), and

*E*

_{EL}(electronic energy gain):

*E*

_{POL}stands for stable polarons, i.e., the polaronic solution is energetically more convenient than a system with delocalized charge carriers.

*E*

_{ST}quantifies the structural cost needed to distort the lattice in order to accommodate the excess charge to form a polaron, whereas

*E*

_{EL}is the electronic energy gained by localizing the charge in the distorted lattice via the electron-phonon interaction. The values of

*E*

_{ST}and

*E*

_{EL}depend on the degree of charge localization and the size of the lattice distortion (see the horizontal shift of the parabolas in Fig. 4) and on the curvature of the parabola in Fig. 4. The three polaronic energies are strongly connected. For instance,

*E*

_{POL}can be interpreted as the result of the competition between the structural cost

*E*

_{ST}and the electronic energy

*E*

_{EL}(Setvin et al. 2014a):

*D*around the polaron site to quantitatively measure the degree of local structural distortions (Reticcioli et al. 2018a):

*δ*for the ion site

*i*between the polaronic (distorted lattice) and uniform solutions and

*n*indicates the total number of ions considered in the sum (for small polarons typically first and second nearest neighbors).

### 2.3 Site-Controlled Localization

Typically, charge trapping occurs at different sites of a given material. The various polaronic configurations are characterized by different energies. In general, there is no guarantee that a DFT+*U* or hybrid functional DFT calculation leads to the global minimum of the polaronic system. In fact, the formation of polarons could spontaneously occur at less favorable lattice sites or occur not at all. Therefore, it is important to control the site localization of polarons, by inspecting the formation of polarons at different sites, and compare the relative formation energies. For this purpose it is essential to establish a protocol capable to selectively control charge trapping at specific sites.

*and*structural self-consistent calculation (output). Structural perturbations can be introduced manually by distorting the local structure around a given atomic site, resembling the expected polaron-induced lattice distortions. Alternatively, the initial perturbation can be achieved by chemical substitution, as explained below for the small electron-polaron case:

- Step 1:
Chemical substitution at the selected site where charge trapping should occur with an atom containing one more electron than the original native atom (e.g., Ti → V). Structural relaxation performed at spin-polarized DFT+

*U*level will yield local lattice distortions around the chosen site.- Step 2:
The initial element is reinserted at its original position (e.g., V → Ti). It is often necessary to use a larger value of

*U*at the selected site, while the other atoms keep the original value of*U*. The manual initialization of the local magnetic moment has to take into account the presence of the localized electron at the selected site. A new relaxation is performed starting from the optimized structure obtained in Step 1. The self-consistent run should be able to maintain the polaron solution obtained in Step 1.- Step 3:
A final step is necessary, performed by using the original value

*U*for all atoms. In this case it is recommended to initialize the orbitals with the one obtained in Step 2. In fact, by using a random initialization, it can happen that the self-consistent loop will end up in a different polaron solution (different polaron site) or in a delocalized solution (Papageorgiou et al. 2010; Deskins et al. 2009; Meredig et al. 2010; Morgan and Watson 2009).

*U*) usually helps to localize the electron at the selected site. However, it is possible, in simple problems, to skip this step and obtain charge localization at the selected site by simply performing Steps 1 and 3. This strategy can be extended to the hybrid-DFT level, by using the orbitals and the optimized structure obtained in Step 3 as an input for the hybrid functional calculations. In case more than one electron needs to be localized at selected sites, the Steps 1 to 3 can be performed for every selected site separately, one after the other or, alternatively, at the same time. A systematic use of this strategy allows the identification of the polaronic ground state (global minimum) of the system (Reticcioli et al. 2018a).

### 2.4 Polaron Dynamics

As mentioned in the introduction, polarons are mobile quasiparticles, whose mobility can be activated or increased by temperature. Polaron hopping can be modeled in DFT, by merging first-principles total energies to the Marcus (Marcus and Sutin 1985) and Emin-Holstein-Austin-Mott (EHAM) formalisms (Deskins and Dupuis 2007; Brunschwig et al. 1980; Spreafico and VandeVondele 2014) or, more effectively, by performing first-principles molecular dynamics (FPMD) (Kresse and Hafner 1993; Kowalski et al. 2010; Setvin et al. 2014a) or nudged elastic band (NEB) calculations (Henkelman et al. 2000; Bondarenko et al. 2015; Janotti et al. 2013).

_{3}is given in Fig. 5, showing the temperature-dependent interaction between two polarons and the (immobile) oxygen vacancy (

*V*

_{O}): at T = 700 K one polaron remains anchored to

*V*

_{O}and the second one has a reduced hopping activity; at higher T (T = 1000 K), both polarons become very mobile and start to explore sites far away from

*V*

_{O}, but the strong Coulomb attraction between the negatively charged polaron and the positively charged

*V*

_{O}regularly brings polarons back into the vicinity of

*V*

_{O}, forming polaron-

*V*

_{O}complexes (Hao et al. 2015).

The statistical analysis is usually performed by investigating the occurrences of charge trapping at each lattice site in the simulation cell (site-decomposed population analysis) and by computing different types of correlation functions aiming to describe the interaction of a polaron with other polarons or lattice defects:

*(i) Polaron-polaron site correlation function.*The polaron-polaron site correlation function

*S*

_{pol−pol}can be defined as the distribution of the site distance

*i*along a particular direction in the crystal, between two polarons at a given time-step

*t*, averaged over all the FPMD time steps

*τ*:

*N*is the number of atomic sites and

*ρ*

_{j}(

*t*) indicates the polaronic site density at time

*t*, and it is equal to 1 for the

*j*-th site hosting a polaron and 0 otherwise.

*(ii) Radial correlation function.*Analogously, the polaron-polaron

*R*

_{pol−pol}and defect-polaron

*R*

_{def−pol}radial correlation functions can be defined as a function of the distance

*r*:

*δ*(|

**r**

_{def}−

**r**

_{q}|,

*r*,

*t*) and

*δ*(|

**r**

_{q}−

**r**

_{p}|,

*r*,

*t*) assume the value 1 if, at time

*t*, the polaron

*p*is at distance

*r*from the defect at position

**r**

_{def}or from the polaron

*q*at position

**r**

_{q}, respectively, and are 0 otherwise.

These statistical quantities (*S*_{pol−pol} and *R*_{pol−pol}) furnish information on the spatial distribution of polarons in the lattice, by indicating the statistically more favorable polaron configurations.

The FPMD calculation is also useful to investigate the stability of the polarons under different conditions, as an alternative to the site-controlled localization strategy discussed above. In fact, a FPMD run leads to a set of high-temperature structures, hosting polarons in various configurations, which might be difficult to access by simple structural relaxations. A useful strategy is to use the high-T structures obtained by FPMD as starting structures for relaxations and check their stability by inspecting the corresponding polaron formation energies *E*_{POL} and ultimately determine the global minimum (polaronic ground state) (Setvin et al. 2014a; Reticcioli et al. 2017). This strategy is obviously similar to simulated annealing.

*q*(

*x*) are obtained by a linear interpolation of the ionic positions assumed at the initial (

*A*) and final (

*B*) polaronic states, i.e.,

*q*(

*x*) = (1 −

*x*)

*q*

_{A}+

*xq*

_{B}, which define the polaron pathway from

*A*to

*B*(Deskins and Dupuis 2007), see Fig. 6. At each considered configuration, an electronic self-consistent calculation is performed in order to compute the total energy at each step and to construct an energy diagram of the polaron transition as a function of the reaction coordinate

*x*, as those shown in Fig. 6 (Sun et al. 2017). The form of the energy profile determines whether the polaron transfer is an adiabatic or diabatic (i.e., nonadiabatic) process (Fig. 6a) (Deskins and Dupuis 2007). If the coupling between the initial and final state

*V*

_{AB}is strong (large

*V*

_{AB}, see Fig. 6a), the activation barrier for polaron transfer decreases, and the energy curve exhibits a smooth transition: this corresponds to an adiabatic mechanism which occurs via thermal hopping. The alternative case is when the interaction between the initial and final states is low and the transfer follows a nonadiabatic process. In this case, the energy curve exhibits a cusp at the intermediate state, and the electron transfer occurs via quantum tunneling.

It is important to note that, unfortunately, different functionals can lead to a different description of the polaron transfer process. For instance, DFT+*U* and hybrid functional approaches can lead to qualitative different results, as shown in Fig. 6 (Sun et al. 2017). In DFT+*U*, the partial localization is unfavored, the charge carrier is strongly localized in one site only, the interaction between the initial and final configuration is weak, and the hopping occurs typically diabatically. In this case the energy as a function of the lattice distortions resembles the intersection of two parabola, as shown in Fig. 6c. The cusp is most likely an artifact of the one-center terms in the DFT+U approach, which does not allow for localization of the electron between two lattice sites. Conversely, hybrid functionals predict usually an adiabatic hopping, with a more realistic gradual transfer of the electronic charge from one trapping site to the other one, as shown in Fig. 6b. As a consequence, activation barriers for polaronic hopping are usually smaller in hybrid-DFT compared to DFT+*U*.

## 3 Small Polarons in TiO_{2}

Charge trapping and the formation of polarons is a pervasive phenomenon in transition metal oxide compounds, in particular, at the surface. As already outlined, polaron formation influences the fundamental physical and chemical properties of materials, manifested by a local alteration of the bond lengths, a change of the formal valence at the specific polaronic site, and the emergence of a characteristic peak localized in the gap region (Nagels et al. 1963; Verdi et al. 2017; Sezen et al. 2015; Freytag et al. 2016). These changes affect virtually all functionalities of the material in practical applications. This section is devoted to the presentation of selected results on polaron effects in TiO_{2}, one of the most studied transition metal oxides and a prototypical polaronic material. The results are obtained using the array of first-principles schemes described in the previous section. Considering that small polarons show similar properties among transition metal oxides, the methodologies adopted for TiO_{2} should be representative for this class of compounds.

TiO_{2} has been studied by several experimental techniques, making the basic properties of this material well characterized (Diebold et al. 2010). TiO_{2} is used in many applications, primarily as white pigment in paints and cosmetic products and as a physical blocker for ultraviolet UVA in almost every sunscreen, it is added to cement and tiles (to give the material sterilizing, deodorizing, and antifouling properties), it is utilized in self-cleaning glasses as optical and corrosion-protector (this type of glass is coated in a thin layer of transparent anatase), it is a component of Grätzel cells (in nanostructured form), and it can be used as varistor in electric devices.

- 1.
Introducing defects, such as oxygen vacancies, interstitial Ti atoms, or Nb impurities.

- 2.
By suitably changing (by hand) the number of electrons of the system. In this way, one mimics excess charges injected by UV irradiation in perfect (i.e., defect-free) samples (Sezen et al. 2015) or cases where the donor and excess charge are spatially separated.

- 3.
In surface environments, by means of adsorbates (e.g. hydrogen).

As already mentioned, the stability of the polaron state depends on many factors, in particular, the location of the trapping site (a particularly delicate issue at surfaces where the broken symmetry results in various inequivalent Ti sites), the interaction with defects, and the concentration of charge carriers (polaron-polaron interaction). The system is generally characterized by an energy profile with several local polaronic minima (even the delocalized solution might trap in a local minimum), and there is no guarantee that single-shot runs will lead to the identification of the most favorable configuration; this issue is discussed in Sect. 2.3.

### 3.1 Rutile and Anatase

TiO_{2} exists in various crystal structures (polymorphs). Rutile and anatase, the most common and stable TiO_{2} polymorphs, bear the formation of polarons (Yin et al. 2018), with some important distinctions. The two polymorphs show different properties upon injection of an excess charge into the system, e.g., anatase becomes metallic, while rutile remains semiconducting upon Nb doping (Zhang et al. 2007) (as mentioned in Sect. 1), consistent with the formation of large and small polarons, respectively. Large polarons have been reported in anatase also by ARPES (Moser et al. 2013) and STM (Setvin et al. 2014a) measurements and interpreted by first-principles calculations (Setvin et al. 2014a; Verdi et al. 2017). Conversely, small polarons have been found to localize easily in pristine or defective rutile, while their formation in anatase is more seldom. Small polarons in anatase have been observed only in samples containing chemical or structural defects (e.g., oxygen vacancies and step edges) (Setvin et al. 2014a,b).

*U*(probably unrealistically large) is required to form a small polaron (see Fig. 7b). By considering a realistic

*U*of ≈4 eV, obtained fully ab initio by cRPA, small polarons are formed in rutile but not in anatase (Setvin et al. 2014a). This behavior is explained by inspecting the density of states, shown Fig. 7a. The formation of a polaron involves the perturbation of the conduction band minimum (CBM), which has a different character in the two TiO

_{2}polymorphs: the CB minimum in anatase lies at a lower energy than in rutile, and it exhibits a wider bandwidth, suggesting that a larger

*U*is required to change this favorable configuration (Setvin et al. 2014a).

On the other hand, at reducing conditions, i.e., under conditions where O vacancies (*V*_{O}) are present, small polarons can be formed on the surface of both polymorphs, but their degree of mobility is rather different, as demonstrated by the FPMD analysis shown in Fig. 7c. In rutile, polarons (orange lines) are very mobile and hop preferentially among subsurface and surface sites, while in anatase the polaron is immobile, and it stays attached to the *V*_{O}, forming a polaron-*V*_{O} defect-like complex (Setvin et al. 2014a) (the following sections are devoted to a more detailed discussion of the small polaron properties of rutile TiO_{2}).

The representative examples discussed in this chapter are mostly focused on the rutile phase, in particular on the TiO_{2}(110) surface.

### 3.2 Small Polarons on the Surface of Rutile TiO_{2}

Rutile TiO_{2} crystallizes in a tetragonal symmetry with each Ti atom surrounded by six O atoms in a distorted octahedral configuration (Grant 1959). Neighboring octahedra share one corner and are rotated by 90^{∘}. The titanium atoms are in a nominal 4 + oxidation state (Ti^{4+}), while the oxygens are in a 2 − state (O^{2−}). Rutile can be easily reduced by intrinsic defects, such as interstitial Ti atoms and double-charged oxygen vacancies, and by doping and molecular adsorption. Regardless of the source used to obtain charge carriers in TiO_{2}, excess electrons show similar behaviors (Janotti et al. 2013; Deák et al. 2015; Moses et al. 2016): they are trapped at a titanium site, which becomes a Ti^{3+} site, a hallmark revealed and confirmed by EPR experiments (Yang et al. 2009, 2013).

DFT-based simulations are capable to describe the formation of small polarons in rutile, but the stability of the polaronic states, defined in terms of the polaron formation energy, *E*_{POL}, does depend on the computational approach and on the input parameters (Yin et al. 2018), such as the on-site Coulomb interaction in DFT+*U* (Calzado et al. 2008; Setvin et al. 2014a) and the amount of mixing in hybrid functional DFT (Spreafico and VandeVondele 2014). The calculated *E*_{POL} values are spread in a broad range between 0.1 and 0.8 eV (Yin et al. 2018). Despite these quantitative differences, the predicted properties are qualitatively in good agreement among the various methods and also with the experiments. It has been argued that beyond DFT methods, such as RPA, could result in a more reliable quantitative determination of *E*_{POL} (Spreafico and VandeVondele 2014), but the high computational cost of the RPA method and the difficulty to implement forces at the RPA level (Ramberger et al. 2017) (in existing RPA calculations the lattice has been always relaxed at DFT+*U* or hybrid-DFT level) have limited its applicability. In fact, the large size of the cell, needed to accommodate the polaron together with the induced lattice distortions, prohibits simulations with more advanced quantum chemistry methods, and DFT+*U* and to a lesser extent hybrid-DFT remain the most popular choice.

_{2}(110) surface, consists of a bulk-terminated (1 × 1) surface (see Fig. 8), with large relaxations of the atoms at the surface layers, predominantly along the [110] direction. The surface slab is formed by a sequence of tri-layer building blocks, constituted by a central layer comprising an equal number of Ti

^{4+}and O

^{2−}atoms sandwiched between two layers of oxygen atoms. The topmost (110) layer (the surface layer) is very corrugated since it terminates with undercoordinated (twofold) oxygen atoms (O

_{2c}) arranged in rows along [001]. The layer below the O

_{2c}rows is constituted by trifold coordinated oxygen [001] rows alternated to sixfold coordinated (Ti

_{6c}) and fivefold coordinated (Ti

_{5c}) titanium [001] rows (see Fig. 8). Obviously, the surface breaks the symmetry of the Ti and O atoms and exhibits a higher degree of flexibility and different type of structural distortions compared to the bulk-like layers.

The O_{2c} atoms are relatively easy to remove by thermal annealing, irradiation, electron bombardment, or sputtering, because of their coordinative undersaturation, and this is the main source of excess charge for rutile surfaces. In fact, the propensity to O_{2c} removal reduces the system, and each *V*_{O} results in the injection of two excess electrons, which are eligible to form small polarons. The ability to control the amount of defects and the concentration of polarons is extremely important, as this can be used to tune the chemical and physical properties of the surface. For instance, under extreme reducing conditions, TiO_{2}(110) undergoes a structural reconstruction, from the (1 × 1) to a (1 × 2) phase, doubling the periodicity along [1\(\bar {1}\)0] (Onishi and Iwasawa 1994, 1996; Onishi et al. 1995; Reticcioli et al. 2017). Different models have been proposed to describe the atomic structure of this reconstruction. The debate is still open, but the Ti_{2}O_{3} model proposed by Onishi appears to be the most probable one (Wang et al. 2014b; Reticcioli et al. 2017).

#### 3.2.1 Polaron Configurations and Properties

*Preferable trapping site*. The first question to address while modeling the polaron formation is the determination of the optimal polaron sites. According to FPMD calculations, Ti sites in the subsurface layer (*S*1) are the most favorable trapping centers for excess electrons. Specifically, the site-dependent analysis of *E*_{POL} shows that the most favorable sites are the Ti atoms (Ti\(^A_{S1}\)) at the *S*1 layer below the Ti_{5c} atoms, as indicated by the histogram analysis shown in Fig. 7c. Formation of polarons at the *S*1 (Ti\(^B_{S1}\)) sites below the Ti_{6c} atoms is less probable, due to an unfavorable local structural relaxation. Polaron formation at deeper layers is hindered by the stiffness of the lattice, which rises the energy cost to locally distort the lattice. The FPMD analysis in Fig. 7c also shows that polaron formation at the surface layer (*S*0) is not particularly favorable, mostly due to the reduced electron screening at the surface and large local distortions.

*Polaron orbital symmetry*. A second important aspect is the orbital symmetry of the polaron state. Depending on the hosting site, the excess electron occupies different

*d*orbitals. Which

*d*orbital becomes populated is predominantly determined by the local chemical environment, the crystal field splitting, and the flexibility of the neighboring atoms. Polarons in the Ti

_{5c}sites on the surface layer (\(Ti^A_{S0}\)) are characterized by a

*d*

_{xz}−

*d*

_{yz}orbital symmetry as evidenced by the polaron charge density plot displayed in Fig. 9a, b. On the other hand, below the surface, in the subsurface layer (

*S*1) and in deeper layers, small polarons assume \(d_{z^2}-d_{x^2-y^2}\) and \(d_{x^2-y^2}\) orbital symmetries, alternately along [110] and [1\(\bar {1}\)0] (see Fig. 9c–f). This stacking is the result of the orientation of the local environment (i.e., the orientation of the coordination octahedron around the trapping Ti site). The lattice distortions are larger for the Ti

_{5c}polarons (the Ti site hosting the polaron relaxes outward along [110], and the surrounding O atoms and the nearest-neighbor Ti atoms are pushed away by the excess electron), while the relaxations in deeper layers are smaller (the Ti-O distance in the TiO

_{6}octahedron increases, and the nearest Ti

^{4+}sites relax toward the polaronic site).

*Statistical analysis: polaron-polaron and polaron-vacancy interaction*. Another relevant aspect to understand the physics of polarons is the Coulomb-like polaron-polaron and polaron-defect (in this case

*V*

_{O}) interaction. Being positively charged, a

*V*

_{O}acts as an attractive center for the small polarons and, from the structural point of view, renders the local structure more flexible and reduces the structural cost

*E*

_{ST}to distort the lattice. Conversely, the polaron-polaron interaction is repulsive and particularly effective at small distances. The relative distance among polarons and between polarons and

*V*

_{O}considerably affects the energy and the degree of localization of the characteristic in-gap polaronic states. For instance, at short distance the interaction between polarons can be strong enough to split the polaron levels and lead to the onset of double in-gap peaks well separated in energy (Reticcioli et al. 2018a). As mentioned in the methodology section, a useful tool to decipher these intricate interactions is the statistical analysis of the polaron energies in the explored configuration space. The results for polarons in rutile TiO

_{2}are summarized in Fig. 10.

Figure 10a, b clearly show that the polaron energy decreases with increasing polaron-polaron distance and with decreasing polaron-*V*_{O} distance. Polaron pairs at nearest-neighbor Ti sites along [001] (i.e., a polaron-polaron distance of about 3 Å) is very unstable, due to the strong polaron-polaron repulsion. This Coulomb repulsion is enhanced by the overlap of the polaron charges. In fact, only 70% of the excess charge is confined at the trapping site, while the remaining charge is spread around the surrounding atoms and hinders nearest-neighbor polaronic configurations. For the sake of clarity, it should be noted that the polaron-polaron curve shown in Fig. 10b is obtained for a perfect crystal without any *V*_{O}, in order to decouple the polaron-polaron interaction from the polaron-*V*_{O} interaction. In defective samples, the optimal polaronic configuration is the result of the balance between these two opposite effects: within a simplified picture, one can conclude that polarons try to maximize their mutual distance and to minimize the polaron-*V*_{O} distance.

Clearly the most favorable polaron arrangement and its energetics is concentration dependent. In rutile TiO_{2}, the highest *V*_{O} concentration experimentally achievable is ≈16.7% (Reticcioli et al. 2017). At low concentrations, polarons occupy *S*1 sites. By progressively increasing the reduction level, the excess electrons form increasingly more polarons at the most convenient sites in *S*1, finally forming a characteristic 1 × 3 pattern at the optimal concentration of 16.7% (Reticcioli et al. 2017; Shibuya et al. 2017). This is well reproduced by theory, as exemplified by the peak at a three-site distance in the correlation function shown in Fig. 10d. A pictorial view of the 1 × 3 pattern is shown in Fig. 10c. Larger *V*_{O} concentrations are energetically problematic, since additional polarons in *S*1 would result in strong polaron-polaron repulsion. To solve this instability, the system undergoes a structural reconstruction, as discussed in the following.

*Surface phase diagram (polaron-induced surface reconstruction)*. The existence of a critical concentration of polarons determines a limit for the oxygen vacancy formation via O

_{2c}removal. In fact, rutile TiO

_{2}is able to host up to 16.7%

*V*

_{O}on the surface, which corresponds to a concentration of 33% polarons in the

*S*1 Ti [001] rows (each

*V*

_{O}introduces two excess electrons). However, the system can host larger quantities of excess electrons via a structural reconstruction. The (1 × 1) surface undergoes a (1 × 2) reconstruction, specifically the Ti

_{2}O

_{3}model (Onishi and Iwasawa 1994; Reticcioli et al. 2017), which is able to conveniently handle a larger reduction of the system (50%), as discussed in Reticcioli et al. (2017). This surface reconstruction problem can be described by DFT+

*U*by construction of an appropriate phase stability diagram (see Fig. 11b), by calculating the surface free energy using standard ab initio atomistic thermodynamics (Reuter and Scheffler 2001). The resulting phase diagram is shown in Fig. 11. Close to the critical concentration, the free energy of the reconstructed surface is lower than the corresponding unreconstructed one, thereby marking the structural phase transition. Remarkably, by neglecting polaron effects (panel (a), delocalized setup), DFT+

*U*calculations find the (1 × 2) phase unstable, in disagreement with observations (Reticcioli et al. 2017).

It is important to remark that the structural reconstruction is associated with changes of the in-gap states, as shown in Fig. 11c, d: The localized polaron peak, typical of the (1 × 1) surface (see the corresponding band dispersion in Fig. 2), becomes broader in the reconstructed (1 × 2) phase as a consequence of the higher amount of excess charge localized on the surface.

*Interaction between polarons and adsorbates*. A final paradigmatic case of the importance on polarons on surfaces is the interaction of polarons with adsorbates. This interaction is essential to decode the initial stage of catalytic reactions at defective surfaces. It has been shown that even when adsorbates do not transfer any charge carrier to the substrate, polarons can affect the energetics and configuration of the adsorption process. This is the case for CO adsorption on TiO

_{2}(110) (Reticcioli et al. 2019), where the polaron stability is altered by the presence of this adsorbate, as shown in Fig. 12.

Adsorbed CO shows attractive coupling with polarons on the surface layer and repulsive interaction with polarons in the subsurface. As a result, upon CO adsorption, the most convenient polaron site is not in *S*1 anymore: polarons are attracted onto the surface layer (*S*0) and bind with the CO (Fig. 12c). The effect of polarons on *S*0 stands out clearly from the STM analysis of the TiO_{2} sample. A polaron at *S*0, coupled with the CO, is characterized by a bright filled-state STM signal with two lobes tilted with respect [001] (Fig. 12b), while the *S*1 polaron, right beneath the CO, generates a feeble single spot, due to a lack of polaronic charge transfer toward the adsorbate (Fig. 12a).

## 4 Summary

Small polarons can easily form in oxide materials, in particular near the surface where the increased structural flexibility, molecular adsorption, and the easy formation of surface defects facilitate the transfer of charge and its coupling with the lattice. Small polarons can be effectively studied using first-principles techniques in the framework of DFT. The main issue to be aware of is that small polarons involve the localization of charge on a particular lattice site. Usually this goes in hand with a change of the transition metal oxidation state, for instance, in TiO_{2}, from Ti^{4+} to Ti^{3+}. Such a localization is often not well described by standard semilocal LDA and GGA functionals because of self-interaction errors. Therefore it is compulsory to use functionals beyond the semilocal approximation, for instance, GGA+*U* or hybrid functionals. It goes without saying that such functionals always involve some empiricism, since increasing U or the amount of exact exchange will favor charge localization and, thus, the formation of polarons. Currently no entirely satisfactory approach exists giving an unambiguous answer whether polaron formation is favorable or not. But there are many principles – such as determination of U using some first-principles approaches or inspection that the energy versus electron number shows a straight line behavior – that can guide our choices. Ultimately, comparison with the experiment, however, remains necessary and mandatory.

While modeling small polarons, the most important aspects to address are the trapping process, polaron mobility, and polaron electronic properties. Using the computational prescriptions presented in this chapter, it is possible to explore the configuration space by either guiding the excess charge to localize in selected lattice sites or by performing FPMD calculations. The analysis of the polaron energies (polaron formation energy, structural energy cost, and electronic energy gain) allows to identify the global polaronic minimum and its fundamental energy profile. In order to fully characterize the polaronic state, it is useful to inspect the degree and extent of the local structural distortions, the mutual interaction among polarons, as well as their coupling with defects. This can be done by performing a statistical analysis in terms of the polaron-polaron site correlation function and radial correlation function. A statistical scrutiny of the FPMD data furnishes also important insight on the mobility pattern. An alternative way to study polaron mobility is Markus theory and the Emin-Holstein-Austin-Mott formalism, which model polaron hopping between two sites and permit to discern between adiabatic or nonadiabatic processes. The electronic characteristics of small polarons are defined by the typical band-gap state formed below the conduction band. For an isolated polaron, this state is well localized and typically resides 1 eV below the conduction band. Its specific orbital character (symmetry of the *d* level where the excess charge is trapped) depends on the local structural and chemical environment. For interacting polarons, the situation is more complicated as spin-exchange effects; Coulomb repulsion and hybridization can significantly modify the energy location and the width of the polaron state.

Apart from their intrinsic significance, polarons can induce dramatic changes in the hosting materials and can be used as a mean to drive and control different types of transitions, including metal-to-insulator transitions (not discussed here, an example is given in Franchini et al. 2009), structural transitions, or adsorption processes. Two examples were discussed in this chapter: (i) polaron-induced surface reconstruction (by increasing the polaron concentration the rutile TiO_{2}(110) undergoes a transition from a (1 × 1) to a (1 × 2) surface termination) and (ii) substantial changes of the adsorption energy (the interaction between CO and polarons changes the polaron configuration and affects the CO adsorption scheme).

Finally, it is important to underline that the quality of the theoretical modeling can be assessed by a direct comparison with experimental data, including EPR (by recognizing the change in the formal valence state of the trapping site; for instance in TiO_{2} polaron formation changes the Ti valence state from Ti^{4+} to Ti^{3+}), spectroscopy (energy level and band dispersion), or STM (by a direct comparison between measured and simulated STM images).

## Notes

### Acknowledgements

The authors gratefully acknowledge the support of the Austrian Science Fund (FWF) SFB project VICOM (Grant No. F41) and FWF project POLOX (Grant No. I 2460-N36). The data and analysis presented in this chapter are the results of years-long collaborations on the physics of polarons with the theoretical colleague X. Hao (University of Vienna, now Yanshan University), A. Janotti, and C. Van de Walle (University of California, Santa Barbara) and with the experimental collaborators M. Setvin and M. Schmid (Surface Science Group @ TU Wien). Their invaluable help and stimulating ideas were essential to develop the research on polarons discussed in this work.

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