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ex12 - B The linearized Boltzmann equation in this case...

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Solid State Theory Exercise 12 SS 11 Prof. M. Sigrist Exercise 12.1 Conductivity tensor Calculate explicitly the conductivity tensor σ αβ for a dispersion relation of the form ε k = X α ~ 2 k 2 α 2 m α (1) in the static limit. Exercise 12.2 Residual Resistivity Recapitulate the section 6.3.1 and try to find an explanation for the data in Table 1. What is the major reason for the increase of the resisitivity? Impurity Resistivity (per 1% of impurity atoms) ρ/ (10 - 8 Ω m ) Be 0.64 Mg 0.6 B 1.4 Al 1.2 In 1.2 Si 3.2 Ge 3.7 Sn 2.8 As 6.5 Sb 5.4 Table 1: Residual resistivity of Cu for different impurities (From Landolt-B¨ ornstein Ta- bles, Vol 15, Springer, 1982) Hint: In analogy to doped semiconductors, assume that an impurity atom will adjust itself corresponding to its neighbourhood, i.e. it rejects all electrons from those shells which are not occupied by the surrounding atoms such that an effective charge of the nucleus remains.
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Exercise 12.3 Magnetoresistance and Hall effect We want to consider the electrical resistivity in the presence of a stationary magnetic field
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Unformatted text preview: B . The linearized Boltzmann equation in this case reads-e v k · E ∂f ( ε k ) ∂ε k =-g ( k ) τ ( ε k ) + e ~ ( v k × B ) ∂g ( k ) ∂ k (2) where g ( k ) = f ( k )-f ( k ). a) Why is in (2) the magnetic field in first order not a driving field, contrary to the electric field? b) Calculate for the case of B = (0 , ,B ) the resistivity tensor ˆ ρ = ˆ σ-1 and show that (i) the Hall resistance becomes independent of the scattering time τ and that (ii) the transverse magnetoresistance defined by Δ ρ xx ( B ) = ρ xx ( B )-ρ xx (0) (3) is equal to zero. Hint: Consider independently the cases where the electric field points in the x , y and z direction, respectively, and use the ansatz g ( k ) = ak x + bk y . (4) Office hour: Monday, May 23th, 2011 - 9:00 to 11:00 am HIT K 31.3 Tama Ma...
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