Efficient Method for Modeling Polarons Using Electronic Structure Methods

s valence electrons in cations of metal oxides serving as descriptors for electron and hole polarons

In some metal oxides, an excess electron can give rise to the formation of a small polaron localized on a single site. However, there are still some metal oxides that exhibit the formation of a large polaron. The underlying mechanism behind this phenomenon remains unclear. In this study, we investigate polaron formation in metal oxides favorable for polaron formation using different functionals and through a review of the literature. Our findings indicate that the s valence electrons in cations could serve as a descriptor to classify the polarons in materials. In metal oxides with cations having ns (n ⩾ 5) valence electrons, excess charges trend to localize on several sites or form a two-dimensional shape, and even a large polaron, as these s electrons are delocalized in nature and have a large effect on p or d state polarons. The delocalized nature of ns (n ⩽ 4) valence electrons in cations is relatively small and does not affect the localization condition of p or d state polarons. Therefore, the excess charges in these metal oxides with ns (n ⩽ 4) valence electrons prefer to form a small polaron localizing on a single site. This work unveils the impact of the s valence in cations on polaron formation and provides a fundamental understanding of various types of polarons in metal oxides.

The Role of Water Molecules on Polaron Behavior at Rutile (110) Surface: A Constrained Density Functional Theory Study

Understanding the behavior of a polaron in contact with water is of significant importance for many photocatalytic applications. We investigated the influence of water on the localization and transport properties of polarons at the rutile (110) surface by constrained density functional theory. An excess electron at a dry surface favors the formation of a small polaron at the subsurface Ti site, with a preferred transport direction along the [001] axis. As the surface is covered by water, the preferred spatial localization of the polarons is moved from the subsurface to the surface. When the water coverage exceeds half a monolayer, the preferred direction of polaron hopping is changed to the [110] direction toward the surface. This characteristic behavior is related to the Ti3d-orbital occupations and crystal field splitting induced by different distorted structures under water coverage. Our work describes the reduced sites that might eventually play a role in photocatalysis for rutile (110) surfaces in a water environment.

Time-Domain View of Polaron Dynamics in Metal Oxide Photocatalysts

The polaron is a fundamental physical phenomenon in transition metal oxides (TMOs), and it has been studied extensively for decades. However, the implication of a polaron on photochemistry is still ambiguous. As such, understanding the fundamental properties and controlling the dynamics of polarons at the atomistic level is desired. In this Perspective, we seek to highlight the recent advances in studying small polarons in TMOs, with a particular focus on nonadiabatic molecular dynamics at the ab initio level, and discuss the implications for photocatalysis from the aspects of the structure, intrinsic physical properties, formation, migration, and recombination of small polarons. Finally, various methods were proposed to advance our understanding of manipulating the small-polaron dynamics, and strategies to design high-performance TMO-based photoelectrodes were discussed.

Lessons Learned from Catalysis to Qubits: General Strategies to Build Accessible and Accurate First-Principles Models of Point Defects

Defects and dopants play critical roles in defining the properties of a material. Achieving a mechanistic understanding of how such properties arise is challenging with current experimental methods, and computational approaches suffer from significant modeling limitations that frequently require a posteriori fitting. Consequently, the pace of dopant discovery as a means of tuning material properties for a particular application has been slow. However, recent advances in computation have enabled researchers to move away from semiempirical schemes to reposition density functional theory as a predictive tool and improve the accessibility of highly accurate first-principles methods to all researchers. This Perspective discusses some of these recent achievements that provide more accurate first-principles geometric, thermodynamic, optical, and electronic properties simultaneously. Advancements related to supercells, basis sets, functionals, and optimization protocols, as well as suggestions for evaluating the quality of a computational model through comparison to experimental data, are discussed. Moreover, recent computational results in the fields of energy materials, heterogeneous catalysis, and quantum informatics are reviewed along with an evaluation of current frontiers and opportunities in the field of computational materials chemistry.

Tuning a small electron polaron in FePO4 by P-site or O-site doping based on DFT+U and KMC simulation

Due to the existence of a small polaron, the intrinsic electronic conductivity of olivine-structured LiFePO₄ is quite low, limiting its performance as a cathode material for lithium-ion batteries (LIBs). Previous studies have mainly focused on improving intrinsic conductivity through Fe-site doping while P-site or O-site doping has rarely been reported. Herein, we studied the formation and dynamics of the small electron polaron in FeP_1-αX_αO₄ and FePO_4-βZ_β by employing the density functional theory with the on-site Hubbard correction terms (DFT+U) and Kinetic Monte Carlo (KMC) simulation, where X and Z indicate the doping elements (X = S, Se, As, Si, V; Z = S, F, Cl), and α and β indicate the light doping at the P position (α = 0.0625) and O position (β = 0.015625), respectively. We confirmed the small electron polaron formation in pristine FePO₄ and its doped systems, and the polaron hopping rates for all systems were calculated according to the Marcus-Emin-Holstein-Austin-Mott (MEHAM) theory. We found that the hopping process is adiabatic for most cases with the defects breaking the original symmetry. Based on the KMC simulation results, we found that the doping of S at the P site changes the polaron's motion mode, which is expected to increase the mobility and intrinsic electronic conductivity. This study attempts to provide theoretical guidance to improve the electronic conductivity of LiFePO₄-like cathode materials with better rate performance.

Fast carrier diffusion via synergistic effects between lithium-ions and polarons in rutile TiO2

Carrier mobility in titanium dioxide (TiO₂) systems is a key factor for their application as energy materials, especially in solar cells and lithium-ion batteries. Studies on the diffusion of Li-ions and polarons in rutile TiO₂ systems have attracted extensive attention. However, how their interaction affects the diffusion of Li-ions and electron polarons is largely unclear and related studies are relatively lacking. By using first-principles calculations, we systematically investigate the interaction between the intercalated Li-ions and electron polarons in rutile TiO₂ materials. Our analysis shows that the diffusion barrier of the electron polarons decreases around the Li-ion. The interaction between the Li-ions and polarons would benefit their synergistic diffusion both in the pristine and defective rutile TiO₂ systems. Our study reveals the synergistic effects between the ions and polarons, which is important for understanding the carrier properties in TiO₂ systems and in further improving the performance of energy materials.

Band versus Polaron: Charge Transport in Antimony Chalcogenides

Antimony sulfide (Sb₂S₃) and selenide (Sb₂Se₃) are emerging earth-abundant absorbers for photovoltaic applications. Solar cell performance depends strongly on charge-carrier transport properties, but these remain poorly understood in Sb₂X₃ (X = S, Se). Here we report band-like transport in Sb₂X₃, determined by investigating the electron-lattice interaction and theoretical limits of carrier mobility using first-principles density functional theory and Boltzmann transport calculations. We demonstrate that transport in Sb₂X₃ is governed by large polarons with moderate Fröhlich coupling constants (α ≈ 2), large polaron radii (extending over several unit cells), and high carrier mobility (an isotropic average of >10 cm² V^-1 s^-1 for both electrons and holes). The room-temperature mobility is intrinsically limited by scattering from polar phonon modes and is further reduced in highly defective samples. Our study confirms that the performance of Sb₂X₃ solar cells is not limited by intrinsic self-trapping.

Strong absorption and ultrafast localisation in NaBiS2 nanocrystals with slow charge-carrier recombination

I-V-VI₂ ternary chalcogenides are gaining attention as earth-abundant, nontoxic, and air-stable absorbers for photovoltaic applications. However, the semiconductors explored thus far have slowly-rising absorption onsets, and their charge-carrier transport is not well understood yet. Herein, we investigate cation-disordered NaBiS₂ nanocrystals, which have a steep absorption onset, with absorption coefficients reaching >10⁵ cm^-1 just above its pseudo-direct bandgap of 1.4 eV. Surprisingly, we also observe an ultrafast (picosecond-time scale) photoconductivity decay and long-lived charge-carrier population persisting for over one microsecond in NaBiS₂ nanocrystals. These unusual features arise because of the localised, non-bonding S p character of the upper valence band, which leads to a high density of electronic states at the band edges, ultrafast localisation of spatially-separated electrons and holes, as well as the slow decay of trapped holes. This work reveals the critical role of cation disorder in these systems on both absorption characteristics and charge-carrier kinetics.

Tin-Substituted Chalcopyrite: An n-Type Sulfide with Enhanced Thermoelectric Performance

The dearth of n-type sulfides with thermoelectric performance comparable to that of their p-type analogues presents a problem in the fabrication of all-sulfide devices. Chalcopyrite (CuFeS2) offers a rare example of an n-type sulfide. Chemical substitution has been used to enhance the thermoelectric performance of chalcopyrite through preparation of Cu1-x Sn x FeS2 (0 ≤ x ≤ 0.1). Substitution induces a high level of mass and strain field fluctuation, leading to lattice softening and enhanced point-defect scattering. Together with dislocations and twinning identified by transmission electron microscopy, this provides a mechanism for scattering phonons with a wide range of mean free paths. Substituted materials retain a large density-of-states effective mass and, hence, a high Seebeck coefficient. Combined with a high charge-carrier mobility and, thus, high electrical conductivity, a 3-fold improvement in power factor is achieved. Density functional theory (DFT) calculations reveal that substitution leads to the creation of small polarons, involving localized Fe2+ states, as confirmed by X-ray photoelectron spectroscopy. Small polaron formation limits the increase in carrier concentration to values that are lower than expected on electron-counting grounds. An improved power factor, coupled with substantial reductions (up to 40%) in lattice thermal conductivity, increases the maximum figure-of-merit by 300%, to zT ≈ 0.3 at 673 K for Cu0.96Sn0.04FeS2.

Ab Initio Prediction of Excited-State and Polaron Effects in Transient XUV Measurements of α-Fe2O3

Transient X-ray and extreme ultraviolet (XUV) spectroscopies have become invaluable tools for studying photoexcited dynamics due to their sensitivity to carrier occupations and local chemical or structural changes. One of the most studied materials using transient XUV spectroscopy is α-Fe₂O₃ because of its rich photoexcited dynamics, including small polaron formation. The interpretation of carrier and polaron effects in α-Fe₂O₃ is currently carried out using a semi-empirical method that is not transferrable to most materials. Here, an ab initio, Bethe-Salpeter equation (BSE) approach is developed that can incorporate photoexcited-state effects into arbitrary material systems. The accuracy of this approach is proven by calculating the XUV absorption spectra for the ground, photoexcited, and polaron states of α-Fe₂O₃. Furthermore, the theoretical approach allows for the projection of the core-valence excitons and different components of the X-ray transition Hamiltonian onto the band structure, providing new insights into old measurements. From this information, a physical intuition about the origins and nature of the transient XUV spectra can be built. A route to extracting electron and hole energies is even shown possible for highly angular momentum split XUV peaks. This method is easily generalized to K, L, M, and N edges to provide a general approach for analyzing transient X-ray absorption or reflection data.

Detection and Correction of Delocalization Errors for Electron and Hole Polarons Using Density-Corrected DFT

Modeling polaron defects is an important aspect of computational materials science, but the description of unpaired spins in density functional theory (DFT) often suffers from delocalization error. To diagnose and correct the overdelocalization of spin defects, we report an implementation of density-corrected (DC-)DFT and its analytic energy gradient. In DC-DFT, an exchange-correlation functional is evaluated using a Hartree-Fock density, thus incorporating electron correlation while avoiding self-interaction error. Results for an electron polaron in models of titania and a hole polaron in Al-doped silica demonstrate that geometry optimization with semilocal functionals drives significant structural distortion, including the elongation of several bonds, such that subsequent single-point calculations with hybrid functionals fail to afford a localized defect even in cases where geometry optimization with the hybrid functional does localize the polaron. This has significant implications for traditional workflows in computational materials science, where semilocal functionals are often used for structure relaxation. DC-DFT calculations provide a mechanism to detect situations where delocalization error is likely to affect the results.

First-principles modelling of the new generation of subnanometric metal clusters: Recent case studies

The very recent development of highly selective techniques making possible the synthesis and experimental characterization of subnanometric (subnanometer-sized) metal clusters (even single atoms) is pushing our understanding far beyond the present knowledge in materials science, driving these clusters as a new generation of quantum materials at the lower bounds of nanotechnology. When the size of the metal cluster is reduced to a small number of atoms, the d-band of the metal splits into a subnanometric d-type molecular orbitals network in which all metal atoms are inter-connected, with the inter-connections having the length of a chemical bond (1-2 Å). These molecular characteristics are at the very core of the high stability and novel properties of the smallest metal clusters, with their integration into colloidal materials interacting with the environment having the potential to further boost their performance in applications such as luminescence, sensing, bioimaging, theranostics, energy conversion, catalysis, and photocatalysis. Through the presentation of very recent case studies, this Feature Article is aimed to illustrate how first-principles modelling, including methods beyond the state-of-the-art and an interplay with cutting-edge experiments, is helping to understand the special properties of these clusters at the most fundamental level. Moreover, it will be discussed how superfluid helium droplets can act both as nano-reactors and carriers to achieve the synthesis and surface deposition of metal clusters. This concept will be illustrated with the quantum simulation of the helium droplet-assisted soft-landing of a single Au atom onto a titanium dioxide (TiO₂) surface. Next, it will be shown how the application of first-principles methods have disclosed the fundamental reasons why subnanometric Cu₅ clusters are resistant to irreversible oxidation, and capable of increasing and extending into the visible region the solar absorption of TiO₂, of augmenting its efficiency for photo-catalysis beyond a factor of four, also considering the decomposition and photo-activation of CO₂ as a prototypical (photo-) catalytic reaction. Finally, I will discuss how the modification of the same material with subnanometric Ag₅ clusters has converted it into a "reporter" of a surface polaron property as well as a novel two-dimensional polaronic material.

Modeling polarons in density functional theory: lessons learned from TiO2

Density functional theory (DFT) is nowadays one of the most broadly used and successful techniques to study the properties of polarons and their effects in materials. Here, we systematically analyze the aspects of the theoretical calculations that are crucial to obtain reliable predictions in agreement with the experimental observations. We focus on rutile TiO₂, a prototypical polaronic compound, and compare the formation of polarons on the (110) surface and subsurface atomic layers. As expected, the parameterUused to correct the electronic correlation in the DFT +Uformalism affects the resulting charge localization, local structural distortions and electronic properties of polarons. Moreover, the polaron localization can be driven to different sites by strain: due to different local environments, surface and subsurface polarons show different responses to the applied strain, with impact on the relative energy stability. An accurate description of the properties of polarons is key to understand their impact on complex phenomena and applications: as an example, we show the effects of lattice strain on the interaction between polarons and CO adsorbates.

Proton-Coupled Defects Impact O-H Bond Dissociation Free Energies on Metal Oxide Surfaces

Proton-coupled electron transfer (PCET) reactions on metal oxides require coupling between proton transfer at the solid-liquid interface and electron transfer involving defects at or near the band edge. Herein, hybrid functional periodic density functional theory is used to elucidate the impact of proton-coupled defects on the bond dissociation free energies (BDFEs) of O-H bonds on anatase TiO₂ surfaces. These O-H BDFEs are directly related to interfacial PCET thermochemistry. Comparison between geometrically similar O-H bonds associated with different defect types, namely conduction d-band electrons or valence p-band holes, reveals that the BDFEs differ by ∼81 kcal/mol (3.50 eV), comparable to the wide TiO₂ band gap. These differences are shown to be determined primarily by differences in electron transfer driving forces, which are analyzed by using band energies and inner-sphere reorganization energies within a Marcus theory framework. These fundamental insights about the impact of proton-coupled defects on PCET thermochemistry at semiconductor surfaces have broad implications for electrocatalysis.

Hole Polaron Migration in Bulk Phases of TiO2 Using Hybrid Density Functional Theory

Understanding charge-carrier transport in semiconductors is vital to the improvement of material performance for various applications in optoelectronics and photochemistry. Here, we use hybrid density functional theory to model small hole polaron transport in the anatase, brookite, and TiO₂-B phases of titanium dioxide and determine the rates of site-to-site hopping as well as thermal ionization into the valance band and retrapping. We find that the hole polaron mobility increases in the order TiO₂-B < anatase < brookite and there are distinct differences in the character of hole polaron migration in each phase. As well as having fundamental interest, these results have implications for applications of TiO₂ in photocatalysis and photoelectrochemistry, which we discuss.

Ag5-induced stabilization of multiple surface polarons on perfect and reduced TiO2 rutile (110)

The recent advent of cutting-edge experimental techniques allows for a precise synthesis of subnanometer metal clusters composed of just a few atoms, opening new possibilities for subnanometer science. In this work, via first-principles modeling, we show how the decoration of perfect and reduced TiO₂ surfaces with Ag₅ atomic clusters enables the stabilization of multiple surface polarons. Moreover, we predict that Ag₅ clusters are capable of promoting defect-induced polarons transfer from the subsurface to the surface sites of reduced TiO₂ samples. For both planar and pyramidal Ag₅ clusters, and considering four different positions of bridging oxygen vacancies, we model up to 14 polaronic structures, leading to 134 polaronic states. About 71% of these configurations encompass coexisting surface polarons. The most stable states are associated with large inter-polaron distances (>7.5 Å on average), not only due to the repulsive interaction between trapped Ti³⁺ 3d¹ electrons, but also due to the interference between their corresponding electronic polarization clouds [P. López-Caballero et al., J. Mater. Chem. A 8, 6842-6853 (2020)]. As a result, the most stable ferromagnetic and anti-ferromagnetic arrangements are energetically quasi-degenerate. However, as the average inter-polarons distance decreases, most (≥70%) of the polaronic configurations become ferromagnetic. The optical excitation of the midgap polaronic states with photon energy at the end of the visible region causes the enlargement of the polaronic wave function over the surface layer. The ability of Ag₅ atomic clusters to stabilize multiple surface polarons and extend the optical response of TiO₂ surfaces toward the visible region bears importance in improving their (photo-)catalytic properties and illustrates the potential of this new generation of subnanometer-sized materials.