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.

电子本征能因电子-声子相互作用（涵盖温度依赖性与零点运动效应）引发的重整化现象，在众多材料体系中具有重要研究价值。本文基于第一性原理（如密度泛函理论（density-functional theory）），在绝热简谐近似框架下从两类不同视角开展研究：其一可直接通过原子位置涨落进行分析，其二则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）展开推导。本文基于雅纳克定理的标准形式与动力学方程，证明了上述两种研究视角的等价性。随后，本文将重整化问题的艾伦-海因-卡尔多纳（Allen-Heine-Cardona，AHC）理论纳入第一性原理的研究范畴。AHC理论依托刚性离子近似（rigid-ion approximation），可自然拆分出自能（Fan）贡献与德拜-沃勒（Debye-Waller）贡献。对于无需刚性离子近似的完整简谐绝热表达式，同样可实现此类拆分。基于密度泛函微扰理论（density-functional perturbation theory）框架的数值研究，使得我们能够对比AHC理论与采用/不采用刚性离子近似的冻结声子计算（frozen-phonon calculations）结果。针对两种不含非刚性离子项的数值方法，在金刚石体系中其计算结果的偏差小于7微电子伏特（μeV），一致性达到五位有效数字的精度。本文同时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电子本征能的贡献。