Electrodynamics of Metals

Abstract

There are two aspects of the electrodynamics of metals. The first is linear response theory and the second is the problem of boundary conditions. We have already discussed linear response theory in some detail in Chap. 11. Its application to waves in an infinite medium is fairly straightforward. The problem of boundary conditions is usually much more involved. We shall cover some examples of each type in the rest of this chapter.

Appendices

Problems

14.1

Consider a semi-infinite metal with the surface at $y=0$, in the absence of a dc magnetic field, subject to an electromagnetic wave propagating parallel to the y-axis, which is normal to the surface for the case of polarization in the x-direction (see Fig. 14.13).

Fig. 14.13

The coordinate system for a semi-infinite metallic medium for $y>0$ subject to an electromagnetic wave propagating parallel to the y-axis to the metal surface ($y=0$) for the case of polarization in the x-direction

• (a) The quantum mechanical conductivity tensor is written, in the absence of a dc magnetic field, as

  $$ \underline{\sigma }(q,\omega ) =\frac{\omega _\mathrm{p}^2}{4\pi i\omega }\left\{ \underline{{\mathbf {1}}} +\underline{{\mathbf {I}}}(q,\omega )\right\} , $$

  where

  $$ \underline{{\mathbf {I}}}(q,\omega )= \frac{m}{N}\sum _{{\mathbf {k}}{\mathbf {k}}^\prime }\frac{f_0(\varepsilon _{{\mathbf {k}}^\prime }) -f_0(\varepsilon _{{\mathbf {k}}})}{\varepsilon _{{\mathbf {k}}^\prime } -\varepsilon _{{\mathbf {k}}} -\hbar \omega }<{\mathbf {k}}^\prime |{\mathbf {V}}_q|{\mathbf {k}}> <{\mathbf {k}}^\prime |{\mathbf {V}}_q|{\mathbf {k}}>^*. $$

  Evaluate $\sigma _{xx}(q,\omega )$ at zero temperature.

• (b) Determine the electric field inside the metal with the specular reflection boundary condition. This is the problem of the anomalous skin effect in the absence of a dc magnetic field.

• (c) Show that the surface impedance is written as $\mathcal {Z}=\frac{4\pi i\omega }{c^2}\frac{E(0)}{\partial E/\partial y|_{y=0+}}$, and evaluate $\mathcal {Z}$.

14.2

Consider a helicon wave in a metal propagating at an angle $\theta$ to the direction of a dc magnetic field applied along the z-axis. One may include the effect of the collisional damping by the finite mean collision time $\tau$. Demonstrate that the frequency of the helicon mode is given by

$$ \omega = \frac{\omega _\mathrm{c} c^2 q^2 \cos \theta }{\omega _\mathrm{p}^2 +c^2 q^2} \left( 1+\frac{i}{\omega _\mathrm{c}\tau \cos \theta }\right) . $$

14.3

Investigate the coupling of helicons and plasmons propagating at an arbitrary angle $\theta$ with respect to the applied dc magnetic field $B_0$ in a degenerate semiconductor in which $\omega _\mathrm{p}$ and $\omega _\mathrm{c}$ are of the same order of magnitude. Take $\omega _\mathrm{c}\tau \gg 1$ but let $\omega t$ be arbitrary in the local theory, and study the frequency $\omega$ of the mode as a function of $B_0$.

14.4

Evaluate $\sigma _{xx}$, $\sigma _{xy}$, and $\sigma _{yy}$ from the Cohen–Harrison–Harrison result for propagating perpendicular to ${\mathbf {B}}_0$ in the limit that $w=qv_\mathrm{F}/\omega _\mathrm{c} \ll 1$. Calculate to order $w^2$. See if any modes exist (at cyclotron harmonics) for the wave equation $\xi ^2 = \varepsilon _{xx} +\frac{\varepsilon _{xy}^2}{\varepsilon _{yy}}$ where $\xi =cq/\omega$.

14.5

Consider an electromagnetic wave of ${\mathbf {q}}=(0,q_y, q_z)$ propagating onto a semi-infinite metal of dielectric function $\varepsilon _1$ to fill the space $z >0$, and an insulator of dielectric constant $\varepsilon _0$ in the space $z <0$ (see Fig. 14.14).

• (a) Show that the dispersion relation of the surface plasmon for the polarization with $E_x=0$ and $E_y\ne 0\ne E_z$ is written by $\frac{\varepsilon _1}{\alpha _1} +\frac{\varepsilon _0}{\alpha _0}=0,$ where $\alpha _0$ and $\alpha _1$ are the decay constants in the insulator and metal, respectively.

• (b) Sketch $\frac{\omega }{\omega _\mathrm{p}}$ as a function of $\frac{cq_y}{\omega _\mathrm{p}}$ for the surface plasmon excitation.

Fig. 14.14

The coordinate system of a semi-infinite metallic medium for $z>0$ subject to an electromagnetic wave propagating onto the metal surface ($z=0$)

Summary

In this chapter we study electromagnetic behavior of waves in metals. The linear response theory and Maxwell’s equations are combined to obtain the condition of self-sustaining oscillations in metals. Both normal skin effect and Azbel–Kaner cyclotron resonance are discussed, and dispersion relations of plasmon modes and magnetoplasma modes are illustrated. Nonlocal effects in the wave dispersions are also pointed out, and behavior of cyclotron waves is considered as an example of the nonlocal behavior of the modes. General dispersion relation of the surface waves in the metal–insulator interface is derived by imposing standard boundary conditions, and the magnetoplasma surface waves are illustrated. Finally we briefly discussed propagation of acoustic waves in metals.

The wave equation in metals, in the present of the total current ${\mathbf {j}}_\mathrm{T}(={\mathbf {j}}_\mathrm{0}+{\mathbf {j}}_\mathrm{ind})$, is written as

$$ {\mathbf {j}}_\mathrm{T}=\underline{\varGamma }\cdot {\mathbf {E}}, $$

where $\underline{\varGamma }=\frac{i\omega }{4\pi }\left\{ (\xi ^2-1)\underline{{\mathbf {1}}}-\varvec{\xi }\varvec{\xi }\right\} .$ Here the spin magnetization is neglected and $\varvec{\xi }= \frac{c{\mathbf {q}}}{\omega }$. The ${\mathbf {j}}_\mathrm{0}$ and ${\mathbf {j}}_\mathrm{ind}$ denote, respectively, some external current and the induced current ${\mathbf {j}}_\mathrm{e}=\underline{\sigma }\cdot {\mathbf {E}}$ by the self-consistent field ${\mathbf {E}}$.

For a system consisting of a semi-infinite metal filling the space $z >0$ and vacuum in the space $z<0$ and in the absence of ${\mathbf {j}}_\mathrm{0}$, the wave equation reduces to $[\underline{\sigma }({\mathbf {q}},\omega )-\underline{\varGamma }({\mathbf {q}},\omega )]\cdot {\mathbf {E}}=0$, and the electromagnetic waves are solutions of the secular equation $\mid \underline{\varGamma }-\underline{\sigma }\mid =0.$ The dispersion relations of the transverse and longitudinal electromagnetic waves propagating in the medium are given, respectively, by

$$ c^2q^2 = \omega ^2\varepsilon ({\mathbf {q}},\omega ) \text{ and } \varepsilon ({\mathbf {q}},\omega )=0. $$

In the range $\omega _\mathrm{p} \gg \omega$ and for $\omega \tau \gg 1$, the local theory of conduction ($ql \ll 1$) gives a well-behaved field, inside the metal, of the form

$$ E(z, t)=E_0\mathrm{{e}}^{i\omega t-z/\delta }, $$

where $q =-i\frac{\omega _\mathrm{p}}{c} = -\frac{i}{\delta }.$ The distance $\delta = \frac{c}{\omega _\mathrm{p}}$ is called the normal skin depth. If $l \gg \delta$, the local theory is not valid. The theory for this case, in which the ${\mathbf {q}}$ dependence of $\underline{\sigma }$ must be included, explains the anomalous skin effect.

In the absence of a dc magnetic field, the condition of the collective modes reduces to

$$ (\omega ^2\varepsilon -c^2q^2)^2\varepsilon =0. $$

Using the local (collisionless) theory of the dielectric function $\varepsilon \approx 1-\frac{\omega _\mathrm{p}^2}{\omega ^2},$ we have two degenerate transverse modes of frequency $\omega ^2 =\omega _\mathrm{p}^2+c^2q^2,$ and a longitudinal mode of frequency $\omega =\omega _\mathrm{p}.$

In the presence of a dc magnetic field along the z-axis and ${\mathbf {q}}$ in the y-direction, the secular equation for wave propagation is given by

$$ \left| \begin{array}{ccc} \varepsilon _{xx}-\xi ^2&{}\varepsilon _{xy}&{}0\\ -\varepsilon _{xy}&{}\varepsilon _{yy}&{}0\\ 0&{}0&{}\varepsilon _{zz}-\xi ^2 \end{array}\right| =0. $$

For the polarization with ${\mathbf {E}}$ parallel to the z-axis we have

$$ \frac{c^2q^2}{\omega ^2}=1-\frac{4\pi i}{\omega }\sigma _{zz}({\mathbf {q}},\omega ), $$

where $\sigma _{zz}({\mathbf {q}},\omega )$ is the nonlocal conductivity. For $\omega _\mathrm{p} \gg n\omega _\mathrm{c}$ and in the limit $q\rightarrow 0$, we obtain the cyclotron waves given by $\omega ^2=n^2 \omega _\mathrm{c}^2 + \mathrm{O}(q^{2n})$. They propagate perpendicular to the dc magnetic field, and depend for their existence on the ${\mathbf {q}}$ dependence of $\underline{\sigma }$.

For a system consisting of a metal of dielectric function $\varepsilon _1$ filling the space $z >0$ and an insulator of dielectric constant $\varepsilon _0$ in the space $z <0$, the waves localized near the interface ($z=0$) are written as

$$ \begin{array}{ll} {\mathbf {E}}^{(1)}({\mathbf {r}}, t) &{}= {\mathbf {E}}^{(1)} \mathrm{{e}}^{i \omega t -iq_y y-\alpha _1 z},\\ {\mathbf {E}}^{(0)}({\mathbf {r}}, t) &{}= {\mathbf {E}}^{(0)} \mathrm{{e}}^{i \omega t -iq_y y+\alpha _0 z}. \end{array} $$

The superscripts 1 and 0 refer, respectively, to the metal and dielectric. The boundary conditions at the plane $z=0$ are the standard ones of continuity of the tangential components of ${\mathbf {E}}$ and ${\mathbf {H}}$, and of the normal components of ${\mathbf {D}}$ and ${\mathbf {B}}$. For the polarization with $E_x=0$, but $E_y\ne 0\ne E_z$, the dispersion relation of the surface plasmon is written as

$$ \frac{\varepsilon _1}{\alpha _1} +\frac{\varepsilon _0}{\alpha _0}=0, $$

where $\alpha _1= (\omega _\mathrm{p}^2+q_y^2-\omega ^2)^{1/2}$ and $\alpha _0= (q_y^2-\varepsilon _0\omega ^2)^{1/2}.$

The classical equation of motion of the ionic displacement field $\varvec{\xi }({\mathbf {r}}, t)$ in a metal is written as

$$ M\frac{\partial ^2 \varvec{\xi }}{\partial t^2} = C_{\ell }\nabla (\nabla \cdot \varvec{\xi })-C_\mathrm{t}\nabla \times (\nabla \times \varvec{\xi })+ze{\mathbf {E}} +\frac{ze}{c}\dot{\varvec{\xi }}\times ({\mathbf {B}}_0+{\mathbf {B}})+{\mathbf {F}}. $$

Here $C_{\ell }$ and $C_\mathrm{t}$ are elastic constants, and the collision drag force ${\mathbf {F}}$ is ${\mathbf {F}} = -\frac{zm}{n_0 e\tau }({\mathbf {j}}_\mathrm{e}+{\mathbf {j}}_\mathrm{I}),$ where the ionic current density is ${\mathbf {j}}_\mathrm{I}=n_0e\dot{\varvec{\xi }}$. The self-consistent electric field ${\mathbf {E}}$ is determined from the Maxwell equations ${\mathbf {j}}_\mathrm{T} = \varGamma ({\mathbf {q}},\omega )\cdot {\mathbf {E}}$:

$$ {\mathbf {E}}({\mathbf {q}},\omega ) = \left[ \underline{\varGamma }-\underline{\sigma }\right] ^{-1} \left( i\omega ne\,\underline{{\mathbf {1}}} - \underline{\sigma }\cdot \underline{\varDelta }\right) \cdot \varvec{\xi }. $$

Here a tensor $\underline{\varDelta }$ is defined by $\underline{\varDelta } = \frac{n_0 e i\omega }{\sigma _0}\left\{ \underline{{\mathbf {1}}}-\frac{1}{3}\, \frac{q^2l^2}{i\omega \tau (1+i\omega \tau )} \hat{{\mathbf {q}}}\hat{{\mathbf {q}}}\right\} ,$ where $\hat{{\mathbf {q}}}=\frac{{\mathbf {q}}}{|{\mathbf {q}}|}$. The equation of motion is thus of the form $\underline{{\mathbf {T}}}({\mathbf {q}},\omega )\cdot {\varvec{\xi }}({\mathbf {q}},\omega )=0,$ where $\underline{{\mathbf {T}}}$ is a very complicated tensor. The normal modes of an infinite medium are determined from the secular equation

$$ \mathrm{det} \mid \underline{{\mathbf {T}}}({\mathbf {q}},\omega )\mid =0. $$

The solutions $\omega ({\mathbf {q}})$ have both a real and imaginary parts; the real part determines the velocity of sound and the imaginary part the attenuation of the wave.