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ex11 - Solid State Theory Exercise 11 Transport in metals...

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Solid State Theory Exercise 11 FS 11 Prof. M. Sigrist Transport in metals Exercise 11.1 Relaxation time approximation In this exercise we will show that the so-called single-relaxation-time approximation, ∂f ( k ) ∂t coll = - Z d d k 0 (2 π ) d W ( k , k 0 )[ f ( k ) - f ( k 0 )] -→ - f ( k ) - f 0 ( k ) τ , (1) is a true solution to the Boltzmann equation under certain conditions. We consider a homogeneous two-dimensional metal with an isotropic Fermi surface ( ε k = ~ 2 k 2 / 2 m ) at zero temperature. The impurity scattering responsible for a finite resistivity is described by a delta potential in real space, V imp ( r ) = V 0 δ ( r ) . (2) The system is subject to a homogeneous and time-independent electric field along the x -axis. a) Show that the transition rates W ( k , k 0 ) for the impurity potential (2) are constant in k -space. b) Write down the static Boltzmann transport equation for this setup in the form “drift-term” = “collision-integral” (3) and take advantage of the zero-temperature limit and the symmetries of the system to eliminate all but angular variables.

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ex11 - Solid State Theory Exercise 11 Transport in metals...

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