Surface Plasmons

Surface Plasmons

At the end of Notes 5-3 we discussed plasma oscillations in conductors, which is a bulk oscilation of the volume charge density ρ together with a longitudinal oscillation of the electric field. In our simple model, this oscillation takes place at the single frequency ω_p, the plasma frequency. When one quantizes this oscillation it is called the plasmon.

But there is another type of plasma oscillation in conductors, where the oscillations are confined to the surface region of the conductor. These are called surface plasma oscillations, and when quantized, are called surface plasmons. Unlike the bulk plasmon, the frequency of the surface plasmon does vary significantly with the wavenumber of the oscillation. We can describe the surface plasmon with the classical Maxwell equations and our simple model for the conductivity of a metal.

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,

E_x = Ae^iqx e^−Kze^−iωt, E_y =0, E_z = Be^iqxe^−Kze^−iωt for z > 0

E_x = Ce^iqx e^K'ze^−iωt, E_y =0, E_z = De^iqxe^K'ze^−iωt 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 D=εE continuous, one can 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.

For the conductor use the simple dielectric function ε(ω) = 1+4πiσ(ω)/ω (so we take ε_b=0 and ignore the bound electrons), with σ(ω) the ac Drude conductivity. Assume that ωτ >> 1 so that ε(ω) ≈ 1 − (ω_P/ω)², where ω_P=4πne²/m is the plasma frequency.

With this simple model you can determine the dispersion relation for these surface plasma solutions. That is, you can determine the wavenumber q as a function of ω, find for what range of ω there is a solution, and find a simple expression for the result in the small q and large q limits. You can also show that this oscillation of the electric field is accompanied by an oscillation of the surface charge density σ. What is the polarization of this wave?