Temperature dependence of electronic eigenenergies in the adiabatic harmonic approximation

The renormalization of electronic eigenenergies due to electron-phonon interactions (temperature dependence and zero-point motion effect) is important in many materials. We address it in the adiabatic harmonic approximation, based on first principles (e.g., density-functional theory), from different points of view: directly from atomic position fluctuations or, alternatively, from Janak's theorem generalized to the case where the Helmholtz free energy, including the vibrational entropy, is used. We prove their equivalence, based on the usual form of Janak's theorem and on the dynamical equation. We then also place the Allen-Heine-Cardona (AHC) theory of the renormalization in a first-principles context. The AHC theory relies on the rigid-ion approximation, and naturally leads to a self-energy (Fan) contribution and a Debye-Waller contribution. Such a splitting can also be done for the complete harmonic adiabatic expression, in which the rigid-ion approximation is not required. A numerical study within the density-functional perturbation theory framework allows us to compare the AHC theory with frozen-phonon calculations, with or without the rigid-ion approximation. For the two different numerical approaches without non-rigid-ion terms, the agreement is better than 7 μeV in the case of diamond, which represent an agreement to five significant digits. The magnitude of the non-rigid-ion terms in this case is also presented, distinguishing specific phonon modes contributions to different electronic eigenenergies.

电子-声子相互作用引发的电子本征能重整化（涵盖温度依赖性与零点运动效应）在众多材料体系中具有关键研究意义。本研究基于第一性原理（如密度泛函理论（density-functional theory）），在绝热简谐近似框架下，从多重视角对该问题展开探讨：既可以直接基于原子位置涨落开展分析，也可借助推广至包含振动熵的亥姆霍兹自由能场景的雅纳克定理（Janak's theorem）进行研究。基于雅纳克定理的常规形式与动力学方程，我们证明了上述两种方法具备等价性。随后，我们将电子本征能重整化的艾伦-海因-卡尔多纳（Allen-Heine-Cardona, AHC）理论纳入第一性原理研究框架。AHC理论依托刚性离子近似，可自然导出自能（Fan）贡献与德拜-沃勒（Debye-Waller）贡献。即便不采用刚性离子近似，完整的简谐绝热表达式也可进行类似的拆分处理。依托密度泛函微扰理论（density-functional perturbation theory）框架开展的数值研究，使得我们能够对比AHC理论与带/不带刚性离子近似的冻结声子（frozen-phonon）计算结果。针对金刚石体系，在不包含非刚性离子项的两种不同数值方法下，两者的偏差小于7 μeV，一致性达到五位有效数字。本研究还给出了该体系中非刚性离子项的量级，并区分了不同声子模式对各类电子本征能的具体贡献。