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PHY 521/321:
Condensed Matter Physics I

Prof. S. Teitel stte@pas.rochester.edu ---- Spring 2010

## Problem Set 3

Due Wednesday, March 22, in lecture

• Problem 1 - surface plasmons

If you went to the Dexter Lecture given by Prof. Capasso on March 3, you heard a lot about surface plasmons. In this problem you will derive the dispersion relation and other properties of the surface plasmon!

Consider a metal with a surface in the xy plane, i.e. z > 0 is metal and z < 0 is the vacuum. The surface plasmon is a solution to Maxwell's equations of the form,

Ex = Aeiqx eKzeiωt,     Ey =0,     Ez = BeiqxeKzeiωt         for z > 0

Ex = Ceiqx eK'zeiωt,     Ey =0,     Ez = DeiqxeK'zeiωt         for z < 0

with q, K, K' real and K, K' positive (so that the amplidue of the wave decays as z moves away from z=0).

Assume that there is no charge density induced in the bulk of the metal, i.e. E=0, (but there may be induced surface charge). By requiring the above expression to satisfy Maxwell's equation, and to satisfy the usual electromagnetic boundary conditions, tangential component of E continuous and normal component of εE continuous, find equations relating the unknown amplitudes A, B, C, D to each other, and relating q, K, K' to each other and to the frequency ω.

Use for the metal the dielectric function ε(ω) = 1+4πiσ(ω)/ω, with σ(ω) the ac Drude conductivity. You may make the assumption that ωτ >> 1 so that ε(ω) ≈ 1 − (ωP/ω)2, where ωP=4πne2/m is the plasma frequency.

Use your results above to solve for q as a function of ω. Plot q2c2 vs ω2, and indicate for what range of ω there is indeed a solution with the desired properties.

Compute the surface charge density induced on the surface of the metal? (Hint: remember your electrostatics!)

What is the dispersion relation ω(q) in the limit of small q? Show that as q gets very large (i.e. qc >> ω) there is a solution at frequency ω = ωP/√2. What is the polarization of this wave when ω = ωP/√2?