第一性原理计算电荷密度的温度依赖性：应用于硅的(222)禁戒反射
Temperature dependence of the charge density from first principles: application to the (222) forbidden reflection in silicon
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中文总结 AI 辅助
该研究从第一性原理计算硅的温度依赖价电荷密度，结合两种方法得到(222)禁戒反射结果，揭示了价电荷随温度的重新分布规律，为相关理论提供了可靠支撑。
中文摘要 AI 辅助
X射线衍射中的禁戒反射（FR）强度本征很弱，长期以来受到研究，尤其是在硅这类半导体中。它们是对称性破缺、局部应变、杂质和弱电荷重新分布的灵敏探针。尽管实验研究广泛，但其温度依赖性的理论通常依赖简化模型。原子德拜-沃勒因子在允许反射上表现良好，但对于决定Si (222)这类FR强度的原子间价电荷，其适用性存疑，此前理论与实验的吻合依赖特设的德拜-沃勒修正。我们采用两种方法从第一性原理计算硅的温度依赖价电荷密度ρ(r,T)：(i)微扰理论；(ii)非微扰方法中对热畸变超胞取平均。电荷密度的微扰表达式比电子能量的更复杂，因为它依赖波函数本身，且需要对未占据带显式求和。我们利用声学和规则将势的二阶导数表示为一阶导数，使表达式可在现有框架内处理。随后直接通过ρ(r,T)的傅里叶变换得到(222) FR，无需特设因子。两种方法结果相近，与实验合理吻合，热膨胀对温度依赖性有显著影响。该电荷密度解答了测量强度无法确定的问题：价电荷如何随温度实际重新分布。相对于刚性模型，我们发现键区电荷更多、核心区电荷更少，这种重新分布使(222) FR的温度依赖性略弱。
英文摘要
Forbidden reflections (FRs) in X-ray diffraction have inherently weak intensity and have long been studied, in particular in semiconductors like silicon. They serve as sensitive probes of symmetry breaking, local strain, impurities, and weak charge redistribution. Despite extensive experimental work, the theory of their temperature dependence has typically relied on simplified models. While atomic Debye-Waller factors work well on allowed reflections, their applicability to the valence charge between atoms, which determines the intensity of FRs such as Si (222), is questionable, and previous agreement between theory and experiment relied on ad-hoc Debye-Waller corrections. We compute the temperature-dependent valence charge density $ρ(\mathbf{r},T)$ of silicon from first principles, using two methods: (i) perturbation theory, and (ii) averaging over thermally distorted supercells in a non-perturbative approach. The perturbative expression for the charge density is far more demanding than that for electronic energies, since it depends on the wavefunctions themselves and requires an explicit sum over unoccupied bands. We use an acoustic sum rule to express the second derivatives of the potential in terms of first derivatives, making the expression tractable within existing frameworks. The (222) FR then follows directly from the Fourier transform of $ρ(\mathbf{r},T)$, with no ad-hoc factors. Both methods give similar results, in reasonable agreement with experiment, with thermal expansion noticeably affecting the temperature dependence. The charge density answers a question the measured intensities could not settle: how the valence charge actually redistributes with temperature. Relative to the rigid model, we find more charge in the bonds and less in the core regions, a redistribution that shows up in the intensity as a somewhat weaker temperature dependence of the (222) FR.
