1. Fuchs-Sondheimer model: We used the Boltzmann Transport equation to derive the bulk conductivity of the free electron gas. A thin film is confined by its thickness t in the z-direction and so the occupancy distribution correction g(k) is also z-dependent. a. By evaluating the total derivative dg/dt in the absence of translational symmetry in the z-dir, determine the correction to the relaxation term and write down the new Boltzmann equation for electric field in the x-dir. b. Solve this differential equation and write down two solutions: g + (k) for positive k z , and g-(k) for negative. Apply diffusive scattering at the boundaries, e.g. g(k;z=0,t)=0, to find the undetermined coefficients. c. Write down the z-dependent integral expression for current flow in polar coordinates along the z axis where g + (k) is evaluated for θ=0. .π/2 and g-(k) from π /2. . π . d. Perform the azimuthal integration and evaluate the thickness-averaged value g G ± ²³´µ¶´ G · (still an unresolved integral over the Fermi surface polar angle
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