Department of Physics , University of York , Heslington , York YO10 5DD , United Kingdom.
School of Mathematics and Physics , University of Lincoln , Brayford Pool, Lincoln LN6 7TS , United Kingdom.
J Chem Theory Comput. 2018 Jul 10;14(7):3740-3751. doi: 10.1021/acs.jctc.8b00199. Epub 2018 Jun 21.
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 TiO including anatase, rutile, brookite, TiO(H), TiO(R), and TiO(B). Electron polarons are predicted to form in rutile, TiO(H), and TiO(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 TiO(H) and TiO(R) the electron is distributed across two neighboring Ti sites. Hole polarons are predicted to form in anatase, brookite, TiO(H), TiO(R), and TiO(B) (with trapping energies ranging from -0.16 eV to -0.52 eV). In anatase, brookite, and TiO(B) holes localize on a single O ion, whereas in TiO(H) and TiO(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.
我们提出了一种计算效率高且具有预测能力的方法，用于模拟电子和空穴极化子在固体中的形成和性质。通过对混合泛函中 Hartree-Fock 交换（α）分数进行非经验和自洽优化，我们确保满足广义 Koopmans 条件并最小化自相互作用误差。该方法应用于模型中已知稳定和亚稳相的 TiO 中的极化子形成，包括锐钛矿、金红石、板钛矿、TiO(H)、TiO(R) 和 TiO(B)。预测电子极化子将在金红石、TiO(H)和 TiO(R)中形成（俘获能范围为-0.02 eV 至-0.35 eV）。在金红石中，电子局域在单个 Ti 离子上，而在 TiO(H)和 TiO(R)中，电子分布在两个相邻的 Ti 位上。空穴极化子预测将在锐钛矿、板钛矿、TiO(H)、TiO(R)和 TiO(B)中形成（俘获能范围为-0.16 eV 至-0.52 eV）。在锐钛矿、板钛矿和 TiO(B)中，空穴局域在单个 O 离子上，而在 TiO(H)和 TiO(R)中，空穴也可以分布在两个 O 位上。我们发现优化后的 α 在各相中具有一定的可转移性，α=0.115 可以很好地描述所有相。我们还注意到，该方法可准确预测能带隙，锐钛矿、金红石和板钛矿的结果与实验值相差在 6%以内。我们在不同相的极化子电荷跃迁能级的对齐比较中结束了我们的研究。由于我们描述的方法仅比标准密度泛函理论计算贵两到三倍，因此非常适合于对复杂缺陷（如表面和界面）的电荷俘获进行建模，这些缺陷在一系列与技术应用相关但以前无法进行预测建模的材料中都存在。
