Abstract
We present a computationally efficient and predictive methodology for modeling the formation and properties of electron and hole polarons in solids. Through a nonempirical and self-consistent optimization of the fraction of Hartree-Fock exchange (α) in a hybrid functional, we ensure the generalized Koopmans' condition is satisfied and self-interaction error is minimized. The approach is applied to model polaron formation in known stable and metastable phases of TiO2 including anatase, rutile, brookite, TiO2(H), TiO2(R), and TiO2(B). Electron polarons are predicted to form in rutile, TiO2(H), and TiO2(R) (with trapping energies ranging from -0.02 eV to -0.35 eV). In rutile the electron localizes on a single Ti ion, whereas in TiO2(H) and TiO2(R) the electron is distributed across two neighboring Ti sites. Hole polarons are predicted to form in anatase, brookite, TiO2(H), TiO2(R), and TiO2(B) (with trapping energies ranging from -0.16 eV to -0.52 eV). In anatase, brookite, and TiO2(B) holes localize on a single O ion, whereas in TiO2(H) and TiO2(R) holes can also be distributed across two O sites. We find that the optimized α has a degree of transferability across the phases, with α = 0.115 describing all phases well. We also note the approach yields accurate band gaps, with anatase, rutile, and brookite within six percent of experimental values. We conclude our study with a comparison of the alignment of polaron charge transition levels across the different phases. Since the approach we describe is only two to three times more expensive than a standard density functional theory calculation, it is ideally suited to model charge trapping at complex defects (such as surfaces and interfaces) in a range of materials relevant for technological applications but previously inaccessible to predictive modeling.
Highlights
IntroductionCharge trapping in semiconductors and insulators has widespread interest in the fields of physics, chemistry, and materials science and controls performance for important applications, such as photocatalysis, superconductivity, tribocharging, magnetism, and microelectronics.[1−7] The trapping of electrons or holes may take place at pre-existing defects (such as vacancies or impurities) or even in the perfect lattice in some materials (socalled polaronic self-trapping).[8−13] Predictively modeling these effects is vital in order to provide a deeper theoretical understanding and to guide materials optimization for applications
Charge trapping in semiconductors and insulators has widespread interest in the fields of physics, chemistry, and materials science and controls performance for important applications, such as photocatalysis, superconductivity, tribocharging, magnetism, and microelectronics.[1−7] The trapping of electrons or holes may take place at pre-existing defects or even in the perfect lattice in some materials.[8−13] Predictively modeling these effects is vital in order to provide a deeper theoretical understanding and to guide materials optimization for applications
The generalized Koopmans condition (GKC),[31−34] which must be satisfied for an exact Kohn−Sham or generalized Kohn−Sham functional, provides a convenient way to formulate the requirement of piecewise linearity
Summary
Charge trapping in semiconductors and insulators has widespread interest in the fields of physics, chemistry, and materials science and controls performance for important applications, such as photocatalysis, superconductivity, tribocharging, magnetism, and microelectronics.[1−7] The trapping of electrons or holes may take place at pre-existing defects (such as vacancies or impurities) or even in the perfect lattice in some materials (socalled polaronic self-trapping).[8−13] Predictively modeling these effects is vital in order to provide a deeper theoretical understanding and to guide materials optimization for applications. More sophisticated many-body methods that are in principle predictive, such as Møller−Plesset perturbation theory or the GW approximation, are extremely computationally demanding, especially for modeling polaronic charge trapping where full self-consistency for electrons and ions is essential. For these reasons, finding an inexpensive, predictive, and transferable method to model charge trapping in materials is urgently needed. The DFT+U approach introduces a correction that counters the convex error present with (semi)local xc-approximations.[40,41] in principle, with the correct parametrization, both hybrid and DFT+U approaches can result in a functional which may be close to piecewise linear and exhibit much reduced SIE.
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