We discuss surface roughness effects on the conduction of electrons in metals using both the quantal Kubo-Greenwood formalism and the semi-classical Fuchs-Sondheimer method. The main purpose here is to compare these methods and clarify a few subtle conceptual issues. One of such issues is concerned with the conditions under which the broken translation symmetry along a rough surface may be restored. This symmetry has often been presumed in existing work but not always with proper justifications. Another one relates to the physical meaning of a phenomenological parameter (denoted by $p$) intuitively introduced in the semi-classical theory. This parameter, called the specularity parameter or sometimes the \textit{Fuchs} parameter, plays an important role in the experimental studies of surface roughness but has so far lacked a rigorous microscopic definition. The third issue arises as to the domain of validity for the electrical conductivity obtained in those methods. A misplacement of the domain may have resulted in erroneous analysis of surface effects in a variety of electrodynamic phenomena including surface plasma waves.

中文翻译：

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关于具有粗糙表面的金属的导电性

我们使用量子 Kubo-Greenwood 形式主义和半经典 Fuchs-Sondheimer 方法讨论表面粗糙度对金属中电子传导的影响。这里的主要目的是比较这些方法并澄清一些微妙的概念问题。此类问题之一涉及可以恢复沿粗糙表面破坏的平移对称性的条件。这种对称性在现有工作中经常被假定，但并不总是有适当的理由。另一个涉及在半经典理论中直观地引入的现象学参数（由 $p$ 表示）的物理意义。此参数称为镜面反射参数或有时称为 \textit{Fuchs} 参数，在表面粗糙度的实验研究中起着重要作用，但迄今为止缺乏严格的微观定义。第三个问题是关于在这些方法中获得的电导率的有效性范围。域的错位可能导致对包括表面等离子体波在内的各种电动现象中的表面效应的错误分析。