On the electrical conductivity of metals with a rough surface

ABSTRACT

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 and widely misunderstood 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 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.

KEYWORDS:

• Electrical conductivity
• surface roughness
• Kubo–Greenwood formalism
• Boltzmann transport equation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Here the electric field is the sum of an external electric field and the mean field engendered by the electrons themselves. The conductivity to be calculated is thus the bare conductivity excluding many-body effects.

2 In establishing this equation, we have used the fact that $(\mathbf{v}\cdot {\mathrm{\partial }}_{\mathbf{x}}+\frac{{\mathbf{F}}_s}{m}\cdot {\mathrm{\partial }}_{\mathbf{v}})f_0=0$.

3 By coarse-graining of a function $O\left(x\right)$, we mean an average of $O\left(x\right)$ over a region ${\mathrm{\Omega }}_x$ centred about x with a linear dimension ($\sim \mathrm{\Lambda }$) much larger than the length scale for diffusive scattering ($\sim a$), i.e. $\overline{O\left(x\right)}={\int }_{{\mathrm{\Omega }}_x}\mathrm{d}{x}^{\prime }O\left(x^{\prime }\right)/{\int }_{{\mathrm{\Omega }}_x}\mathrm{d}{x}^{\prime }$. This removes any Fourier components of O that vary significantly within that scale.

4 To be more accurate, we may insert here a factor $\alpha \sim 1$ in front of $f_0^{\prime }$, defined by ${\int }_{-z_s/2}^{z}d{z}^{\prime }{e}^{i\frac{\tilde{\omega }(z-z^{\prime })}{v_z}}{\mathbf{E}}_{\mathbf{k}}\left(z^{\prime }\right)f_{0,0}^{\prime }(z^{\prime },\mathbf{v})\approx \alpha f_0^{\prime }{\int }_{-z_s/2}^{z}d{z}^{\prime }{e}^{i\frac{\tilde{\omega }(z-z^{\prime })}{v_z}}{\mathbf{E}}_{\mathbf{k}}\left(z^{\prime }\right).$Here $z>z_s/2$ is understood.

5 One might think that ρ calculated with g can analogously be related to $\tilde{\rho }$ calculated with $g^{\left(b\right)}$ via a phenomenological function, say $w\left(z\right)$. However, on the scale of $\mathrm{\Lambda }\gg z_s$, $w\left(z\right)$ may not degenerate with $\mathrm{\Theta }\left(z\right)$. Rather, $w\left(z\right)$ can be singular like a Dirac delta function. For example, if there are some charges in the surface layer, the density of these charges will appear like a delta function in the limit where the layer appears infinitely thin. This difference between charge and current densities can also be appreciated in terms of the polarisation function $\mathbf{p}$, by which $\rho =-{\mathrm{\partial }}_{\mathbf{x}}\cdot \mathbf{p}$ and $\mathbf{j}=-i\omega \mathbf{p}$. As $\mathbf{p}$ is always finite and smooth, so is $\mathbf{j}$ and hence a smooth profile function must exist. However, ${\mathrm{\partial }}_{\mathbf{x}}\cdot \mathbf{p}$ can be singular.

6 There is no state ${\psi }_{\mathbf{k}q}$ with q<0. Such states cannot satisfy Schrödinger's equation for the system. Electrons with q<0 all stem from ${\psi }_{\mathbf{k}q}^s$ with q>0, in consistency with the fact that the dimension of the Hilbert space for a half space is half that for the whole space. It is also a consequence of physical causality.

7 In Equation (27(27) ${\sigma }_{\omega }^{\mu \nu }(\mathbf{x},{\mathbf{x}}^{\prime })=i\hslash \sum _{\mathbf{k}q,{\mathbf{k}}^{\prime }{q}^{\prime }}\frac{f_0\left({ϵ}^{\prime }\right)-f_0\left(ϵ\right)}{ϵ-{ϵ}^{\prime }}\frac{{\hat{j}}_{\mathbf{k}q,{\mathbf{k}}^{\prime }{q}^{\prime }}^{\mu }\left(\mathbf{x}\right){\hat{j}}_{{\mathbf{k}}^{\prime }{q}^{\prime },\mathbf{k}q}^{\nu }\left({\mathbf{x}}^{\prime }\right)}{\hslash \overline{\omega }+ϵ-{ϵ}^{\prime }},$(27) ), $\overline{\omega }=\omega +i/\tau$. In a clean metal free from impurities and other type of electronic collisions, one may take the limit of $1/\tau$ being a positive infinitesimal. In general, we view $1/\tau$ as the imaginary part of the electron self-energy Σ arising from the collisions, see for instance Ref. [38].

8 We should note that these coefficients are functions of z and $z^{\prime }$. Namely, $p_{\parallel }\left(z\right)=|R_{\mathbf{k}q}{|}^2+\overline{{\mathcal{P}}_{\mathbf{k}q}^{\parallel }\left(z\right)}$ and $p_z\left(z\right)=|R_{\mathbf{k}q}{|}^2+\overline{{\mathcal{P}}_{\mathbf{k}q}^z\left(z\right)}$, as well as ${\alpha }_{\parallel }(z,z^{\prime })=p_{\parallel }\left(z\right)p_{\parallel }\left(z^{\prime }\right)$, ${\alpha }_z(z,z^{\prime })=p_z\left(z\right)p_z\left(z^{\prime }\right)$, ${\alpha }_{\parallel z}(z,z^{\prime })=p_{\parallel }\left(z\right)p_z\left(z^{\prime }\right)$ and ${\alpha }_{z\parallel }(z,z^{\prime })=p_z\left(z\right)p_{\parallel }\left(z^{\prime }\right)$. However, the dependence should be weak if Λ is large.

9 It is useful to see that $p_{\parallel }$ and $p_z$ can also be written as $(p_{\parallel },p_z)=\sum _{{\mathbf{k}}_{\mathbf{s}}}\left|S_K\right(\mathbf{k},{\mathbf{k}}_s\left){|}^2\overline{e^{-i(\delta \mathbf{k}\cdot {\mathbf{k}}_s)/q)z}}\right(1,q_s/q).$Here the sum over ${\mathbf{k}}_s$ includes specularly and diffusively reflected electrons. For the former, ${\mathbf{k}}_s=0$ and $q_s/q=1$. This expression puts them on equal footing. The contributions from diffusively reflected electrons are suppressed by the oscillatory factor, which is one for specularly reflected electrons.