Hubbard U through polaronic defect states

Since the preliminary work of Anisimov and co-workers, the Hubbard corrected DFT+U functional has been used for predicting properties of correlated materials by applying on-site effective Coulomb interactions to specific orbitals. However, the determination of the Hubbard U parameter has remained under intense discussion despite the multitude of approaches proposed. Here, we define a selection criterion based on the use of polaronic defect states for the enforcement of the piecewise linearity of the total energy upon electron occupation. A good agreement with results from piecewise-linear hybrid functionals is found for the electronic and structural properties of polarons, including the formation energies. The values of U determined in this way are found to give a robust description of the polaron energetics upon variation of the considered state. In particular, we also address a polaron hopping pathway, finding that the determined value of U leads to accurate energetics without requiring a configurational-dependent U. It is emphasized that the selection of U should be based on physical properties directly associated with the orbitals to which U is applied, rather than on more global properties such as band gaps and band widths. For comparison, we also determine U through a well-established linear-response scheme finding noticeably different values of U and consequently different formation energies. Possible origins of these discrepancies are discussed. As case studies, we consider the self-trapped electron in BiVO₄, the self-trapped hole in MgO, the Li-trapped hole in MgO, and the Al-trapped hole in 𝛼-SiO₂.

自Anisimov及其合作者的前期工作以来，经哈伯德修正的密度泛函理论+U（DFT+U）泛函，通过向特定轨道施加在位有效库仑相互作用，被用于预测关联材料的各类物性。然而，尽管已提出诸多求解方案，哈伯德U参数（Hubbard U parameter）的确定始终处于激烈讨论之中。在此，我们提出一种基于极化子缺陷态（polaronic defect states）的筛选准则，以强制总能量随电子占据数变化时呈现分段线性特性。我们发现，该准则得到的电子与结构物性（包括形成能），与分段线性杂化泛函（piecewise-linear hybrid functionals）的计算结果吻合良好。以此方式确定的U值，可在所考察的态发生变化时，对极化子的能量学特性给出稳健的描述。特别地，我们还针对极化子跳跃路径（polaron hopping pathway）展开研究，发现所确定的U值能够实现精准的能量学计算，无需采用依赖构型的U（configurational-dependent U）。需要强调的是，U参数的选取应当基于与施加U的轨道直接相关的物理特性，而非诸如带隙、带宽这类全局物性。为开展对比研究，我们还通过一套成熟的线性响应方案（linear-response scheme）求解U参数，结果得到的U值与形成能均存在显著差异。我们对这些差异的可能成因展开了讨论。作为案例研究，我们考察了BiVO₄中的自陷电子、MgO中的自陷空穴与Li束缚空穴，以及α-SiO₂中的Al束缚空穴。