9780030839931, Chapter 16, Problem 3P

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Chapter 16, Problem 3P Problem A metal temperature and in a uniform static magnetic field perturbed a uniform static electric field (a) Making the relaxation time approximation solve the Boltzmann uation (16. 13) to linear order in the electric field (treating the magnetic field under the assumption &(k) (16.35) 2m* Verify that your solution the form (b) Construct the conductivity tensor from your and verify that agrees with what one finds by evaluating (13.69) and (13 70) for single free electron Step solution Step The limitations of the relaxation time approximation 1. The energy must depend only on the magnitude of the vector k 2 The probability of scattering between two levels k and k' must vanish unless and must depend only on the common value of their energies and on the angle between k and Step (a) The total rate at which the distribution function changing due to collisions defined as dg The change in the number of electrons per unit volume with wave vectors in the volume dg(k) element about the time dt due dt (2π) Since electrons can be scattered either into out of by simply the dg and dg dg dt dt Substitute for ag and dt for in the above expression and (2π) dg(k) (1) dt Step The above equation due relaxation time approximation simplifies as, dg(k) dt r(k) Ignore the possibility of collisions taking place between dt collisions r and coordinates every electron would evolve according the semi-classical equations motion as and F(r,k) Since dt infinitesimal the explicit solution to these equations to inear order in found electron r.k at time must have been at time the absence of only point electrons at r.k could have come and every electron at this point will reach the point Step To take collisions into account one must add two correction terms to equation (2). The right- side wrong because assumes that all electrons are obtained from the time dt. ignoring the fact that some are deflected by collisions also wrong because fails count those electrons that are found at r.k at time not result their unimpeded semi classical motion since time the result of a collision between and Add the corrections In the limit of dt to the above equation (3) reduces to The above equation known as Boltzmann equation approximation then the relaxation time approximation the non- equilibrium distribution function the presence of static uniform electric field and temperature gradient is Step Here, and the vector function depends on k only through its magnitude 2m* When the scattering elastic impurity scattering and approximations (1.) and (2.) are hold It can be shown that the solution to the Boltzmann equation in the relaxation-time approximation has the form equation then also solution the full Boltzmann Step (b) The electrical current density and thermal current density from the distribution function are defined as (4) Here, the matrix element and defined terms of (5) and The structure of these results simplified by defining In terms To evaluate equation (5) for one can exploit the fact that negligible except within of Since the integrands in and have factors that vanish when evaluate them one must retain the first temperature correction the Somerfield Once an accuracy of order be found out as given by the following equations (6) (7) Step Here, (9) Step Equation (4) and equation (6) equation (9) are the basic results theory of electronic contributions to the thermoelectric Instead the zero field result must replaced by Here, weighted average the velocity passing through k (10) the low limit the orbit traverse very slowly Only points the immediate vicinity of k contribute appreciably to the average equation (10) and the zero- field result recovered