Department of physics, NTNU TFY4340 Mesoscopic Physics Spring 2010

Exercise 6

Assistance: Tuesday March 2.

In the lectures, we showed that within the relaxation time approximation (RTA) to the Boltzmann transport equation (BTE), the distribution function for electrons with effective mass m^{∗} in a uniform electric field E is a shifted Fermi sphere,

ϕ(k) = f(k + eτE/h̄),

assuming a spatially uniform distribution (i.e., ∇ϕ = 0) and stationary conditions (i.e., ∂ϕ/∂t = 0). The distribution function ϕ(k) is related to the number density of electrons in the following way: Since ϕ(k) is the probability of finding an electron with wavenumber k, and D(k) d^{3}k is the number of allowed states in the volume element d^{3}k around k, the product ϕ(k) · D(k) d^{3}k must represent the number of occupied states within d^{3}k, i.e., the number of electrons dN within the volume element d^{3}k. Hence, the contribution dn(k) to the total electron density n is

dn(k) = \frac{dN}{V} = ϕ(k) · \frac{1}{4π^{3}} · d^{3}k,

since D(k) = V/4π^{3}, assuming a three–dimensional system. Here, V is the volume of the system. Now, since the electrons with wavenumber k have velocity v(k) and charge −e, their contribution to the current density must be

dj(k) = (−e) · v(k) · dn(k).

Finally, an integral over k space will provide the total current density j:

j = ∫ dj(k) = ∫ (−e) · v(k) · dn(k).

Use the information given above and below (next page) to show that the resulting conductivity is the same as we obtained within the Drude model, i.e.,

σ = \frac{ne^{2}τ}{m^{∗}} .

Additional information:

j = σE
σ = \frac{j · E}{E^{2}}
ϕ(k) = f(k + eτE/h̄) ≃ f(k) + \frac{eτ}{h̄} E · ∇kf(k)
v(k) = \frac{1}{h̄}∇kE(k) = \frac{h̄k}{m^{∗}}
∇kf(k) = \frac{∂f(k)}{∂E}∇kE(k)
\frac{∂f}{∂E} ≃ −δ(E − E_{F}) = −\frac{1}{|∂E/∂k|} δ(k − k_{F})
k_{F} = (3π^{2}n)^{1/3}

Let e.g. E point along ẑ, and use spherical coordinates (k, θ, ϕ), with polar angle θ between k and ẑ, and azimuth angle φ between x̂ and the projection of k onto the xy plane. (Heinzel uses the opposite notation for φ and θ, pp 44 – 45.) The approximation for ∂f/∂E given above is valid for low temperatures, when the equilibrium (Fermi–Dirac) distribution function f is approximately a (negative) step function, changing rapidly from f = 1 to f = 0 at E = E_{F}.