Density functional theory (DFT) is one of the most popular electronic structure methods due to its good balance between accuracy and speed. Other theoretical methods exist1, but DFT is the most common method for atomistic material simulations, due to its balance of accuracy and computational time. A central problem in modeling polarons with DFT is the delocalization of electrons that may occur due to self-interaction errors. Self-interaction errors2,3,4 occur with many common exchange-correlation potentials, leading to the wrong description of polarons in many materials.
See for example below two different DFT solutions for TiO2. One solution (with self-interaction errors) leads to an unpaired electron being delocalized throughout the simulation cell. The proper solution has the electron localized on a Ti atom, forming a polaron state. If one is to model polarons, then the ‘correct’ method must be chosen in order to avoid an incorrect description of polarons.
While DFT often fails to properly model polarons, methods do exist which enable localization of polarons. The DFT+U method5,6,7,8 is one of the common approaches to enabling charge localization. DFT+U adds a correction term (hence +U) to DFT calculations to help remove self-interaction errors. It requires little more computational time than standard DFT calculations, and is therefore one of the most popular methods for modeling polarons. A different approach uses hybrid functionals9,10,11. Such functionals also enable localization of polarons, but often require considerable more computational time. In any case, an appropriate method must be chosen so that ‘correct’ localization of polarons occurs.
Even if a method is applied that overcomes self-interaction errors, polaron formation is not guaranteed. For a given level of theory there may be several stable or metastable solutions12. The initial structure and wavefunction may all influence whether polarons form. Key characteristics of a polaron are the distortion of bonds between the polaron site and neighboring atoms, as well as the occupancy (or vacancy for holes) of electronic orbitals. The polaron essentially breaks the symmetry of the crystal as the distortions and electrons are centered around the polaron site. Simply using default settings or perfect crystals may not enable polaron formation. The simulation input must be chosen so that initial geometries and/or electronic orbitals mimic a polaron to enable convergence to a stable polaron solution.
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