自旋-轨道相互作用对晶体热力学性质的影响：铋的specific heat
Effect of the spin-orbit interaction on the thermodynamic properties of crystals: The specific heat of bismuth
摘要：
近年来，人们对绝缘体和半导体的比热$C$越来越感兴趣，这是因为可以获得具有不同同位素质量的样品，并且可以使用电子能带结构作为出发点来执行其温度依赖性$C(T)$的\textit{ab initio}计算。研究的大多数晶体是元素晶体（例如，锗）或二元（例如，氮化镓）半导体。初始的电子计算是在局域密度近似下进行的，不包括自旋轨道相互作用。实验和计算结果之间的一致性通常被发现是好的，除了含有重原子的晶体（例如，PbS），在$C/T^3$的低温最大值处存在20%左右的差异。这种差异是由于忽略了重原子的自旋-轨道相互作用（对于铅原子的$p$价电子，$Δ_0\sim$为1.3eV）。在这里，我们讨论的测量和\textit{ab initio}计算的$C(T)$晶体铋（$Δ_0\sim$ 1.7 eV），严格地说，半金属，但在温度范围内，我们访问（$T >$ 2 K）作为半导体。我们扩展了文献中的实验数据，注意到没有自旋-轨道相互作用的\textit{ab initio}计算在$\sim$ 8 K处表现出最大值，比测量值低约20%。自旋轨道相互作用的加入显著减小了这种差异：$C(T)$的最大值现在只比测量值大7%。如果自旋-轨道哈密顿量减少了$\sim$ 0.8的因子，则得到精确的一致性。
英文摘要：
In recent years, there has been increasing interest in the specific heat $C$ of insulators and semiconductors because of the availability of samples with different isotopic masses and the possibility of performing \textit{ab initio} calculations of its temperature dependence $C(T)$ using as a starting point the electronic band structure. Most of the crystals investigated are elemental (e.g., germanium) or binary (e.g., gallium nitride) semiconductors. The initial electronic calculations were performed in the local density approximation and did not include spin-orbit interaction. Agreement between experimental and calculated results was usually found to be good, except for crystals containing heavy atoms (e.g., PbS) for which discrepancies of the order of 20% existed at the low temperature maximum found for $C/T^3$. It has been conjectured that this discrepancies result from the neglect of spin-orbit interaction which is large for heavy atoms ($Δ_0\sim$1.3eV for the $p$ valence electrons of atomic lead). Here we discuss measurements and \textit{ab initio} calculations of $C(T)$ for crystalline bismuth ($Δ_0\sim$1.7 eV), strictly speaking a semimetal but in the temperature region accessible to us ($T >$ 2K) acting as a semiconductor. We extend experimental data available in the literature and notice that the \textit{ab initio} calculations without spin-orbit interaction exhibit a maximum at $\sim$8K, about 20% lower than the measured one. Inclusion of spin-orbit interaction decreases the discrepancy markedly: The maximum of $C(T)$ is now only 7% larger than the measured one. Exact agreement is obtained if the spin-orbit hamiltonian is reduced by a factor of $\sim$0.8.
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