Set of supplementary notes.

14 CHAPTER 1. CLASSICAL MODELS FOR ELECTRONS IN SOLIDS
1
Power reflection coeff. |r| 2
ε ∞ = 5
background permittivity ε ∞ = 1
(ω p /ω)**4
0
*
0 ω p
ω p
Frequency
Figure 1.6: Frequency dependence of the power reflection coefficient R = |r| 2 in the Drude
model.
Moreover, the reflection coefficient at an air/metal interface has an interesting frequency
dependence. Substituting the Drude form for ɛ ω into Eq. 1.4 allows us to analyse this in
some detail (see question on problem sheet 1). We find that the power reflection coefficient R
reaches 1 in the limit ω → 0, then assumes a weakly frequency dependent value less than 1
over a wide frequency range, and drops off as R ∝ ω −4 at high frequency. A finite background
polarisability (caused by the core electrons), which gives rise to ɛ ∞ > 1 causes R to dip to zero
at finite frequency (Fig. 1.6).
1.2.2 Plasma oscillations
These results are directly analogous to some of those seen in the Part 1B Electromagnetism
course on the topic of plasmas. According to the Drude model, electrons in metals behave
like a plasma, i.e. a classical charged gas moving in an oppositely charged environment. The
electrons act in many ways like an ideal gas, but whereas the molecules of an ideal gas are meant
to scatter off each other, we take the electrons as completely non-interacting. Our electrons
do scatter off defects in the solid, however, which includes thermally excited lattice vibrations
(phonons), and this gives them a mean free path l and a scattering rate, which is the inverse of
the relaxation time τ. We assume that a scattering event completely randomises an electron’s
momentum. As will be explained in more detail below, we can identify the damping rate γ in
Eq. 1.5 with the scattering rate τ −1 .
One of the findings of the Part 1B Electromagnetism course was the occurrence of plasma
oscillations. That there should be free oscillations in the plasma is surprising at first, because
we have reduced the restoring force due to the ionic cores to zero (ω T → 0) to obtain the Drude
model. Where could these oscillations come from
Consider probing a slab of material (the sample) by applying an oscillating field (see
Fig. 1.7). The free charges brought into the vicinity of our sample to probe its properties
produce a displacement, or D− field. Because D ⊥ is continuous across the interface, this can
be translated to the electric field, or E−field inside the sample, and the polarisation P can be
determined: