(3 pts) Consider a metal in which the conduction current (“free current”) is given by \( \mathbf{j}_\omega=\sigma(\omega) \mathbf{E}_\omega \) (subscript \( \omega \) signals a complex amplitude), while the current due to “bound electrons” is folded into the polarization \( \mathbf{P} \) through dielectric constant \( \epsilon_b (\omega) \). Show that Maxwell’s equations can be reformulated in terms of the full dielectric constant \( \epsilon=\epsilon_b+i\sigma(\omega)/\omega \) while setting the “free current” to zero. (Recall that the polarization is an auxiliary, unobservable quantity, and the separation of the total current in “free” and “bound” parts is an arbitrary choice.) In this context, the word “dielectric” has lost its original meaning.
(1 pts) Note that the low-frequency response of a good metal is dominated by the conduction electrons.
(2 pts) Consider the Drude model: electrons are accelerated by electric field while their momentum is scattered with a relaxation time \( \tau \). Write the equation of motion for an electron (\( \dot{\mathbf{p}}=\cdots \)) taking these two processes into account. Assume there is no magnetic field.
(2 pts) Take monochromatic field with amplitude \( \mathbf{E}_\omega \) and find the complex amplitude \( \mathbf{p}_\omega \) for the electron’s momentum from the equation of motion.
(2 pts) Write the current density in terms of the electron density (number per unit volume), charge, and velocity. Introduce the effective mass \( m^⋆ \) as \( \mathbf{p}=m^⋆ \mathbf{v} \), insert the result of step #4, and work out the frequency-dependent conductivity defined in step #1. Note that the conductivity is real and close to its static limit at \( \omega\tau\ll 1 \), but it becomes purely imaginary at \( \omega\tau\gg 1 \).
(4 pts) The relaxation time for conduction electrons in copper at room temperature is on the order of 30 fs. Estimate the frequency separating the two regimes identified at the end of step #5. What range is it in (radio, microwave, infrared, visible)? Estimate the corresponding wavelength and photon energy (in eV).
(3 pts) Consider the range of frequencies where \( \omega\tau\ll 1 \). Use the wave equation with the dielectric constant found in step #1, a constant magnetic permeability \( \mu \), and the appropriate limiting value for \( \sigma(\omega) \) to find the wavevector of an electromagnetic wave inside the metal. You should end up with a complex number with equal real and imaginary parts.
(2 pts) Define the skin depth \( \delta=1/\mathop{\mathrm{Im}} k \), which gives the depth to which the wave penetrates into a metal. How does it depend on frequency?
(4 pts) Consider a good metal, such as copper or aluminum, with resistivity of about 2 \( \mu\Omega\cdot \)cm. Estimate the skin depth at both ends of the microwave range (300 MHz – 300 GHz, or wavelengths from 1 m to 1 mm). Compare the skin depth with the wavelength in vacuum. What’s the range for the parameter \( \omega\tau \) for these frequencies?
(3 pts) Note that the macroscopic definition of the conductivity (used in step #1) can only be used as long as the skin depth is large compared to the mean-free path of the electrons. (If this condition is violated, we have anomalous skin effect, and a more involved analysis is needed.) The mean-free path \( l \) can be estimated from \( l=v_F \tau \), where \( v_F \) is the Fermi velocity, typically of order of 1/100 of the speed of light. Take \( \tau \) in Cu at room temperature from step #6, estimate \( l \), and compare it with the skin depth found in step #9. Conclusion?
(8 pts) In the limit \( \omega\tau\ll 1 \), use Fresnel equations for normal incidence to show that the reflectance of a good metal in this range is approximately \( R=1-2\omega\delta/c \), where \( \omega\delta/c\ll 1 \).
(6 pts) In contrast to step #7, consider the opposite case \( \omega\tau\gg 1 \). Show that the dielectric constant is approximately real and negative up to a certain threshold frequency, above which is turns positive. This threshold is called the plasma frequency. Consider the wavevector in the metal below and above this threshold. What does it imply about the propagation of electromagnetic waves through the metal?