9780030839931, Chapter 12, Problem 2P

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Chapter 12, Problem 2P Problem For electrons near band minimum (or maximum) (k) has the form = constant + 2 (12.64) where the matrix M independent (Electrons in semiconductors are almost always treated in this approximation.) (a) Calculate the cyclotron effective mass from (12. 44) and show that is independent and and given by M \1/2 m* = (cyclotron) (12.65) where the determinant of the matrix M (b) Calculate the electronic specific heat (2 80) resulting from the band structure (12 64), and by comparing with the corresponding free electron show that the band structure contribution to the specific heat effective mass (page 48) given by m* M (specific heat) (12.66) Step by -step solution Step The band dispersion for electrons near the band minimum given as, The term is the inverse of the effective mass Now, the surfaces of constant energy are the orbits in the uniform magnetic field are It is convenient to define the matrix made up of the upper left 2x2 block of the full matrix given Step Let us define vector given by q the two component vector the equation describing the ellipse formed by the intersection of the surface and a plane of constant k. has the q and let us complete the square. This is done by shifting according a q constant Finally, we rotate in the xy plane to the principal axes of the equation describing M2 The area of the ellipse is Step of 4 The product of and is the determinant of which in turn is related to the component of the full matrix M given by the following relation Thus, M The cyclotron effective mass given by m*= Substitute in the above equation, a 2π de 2 2π = Therefore, the cyclotron effective mass is M The electronic specific heat of the Fermi gas at low temperature is, Boltzmann constant and temperature of the Fermi gas is T For fermions which has mass m and Fermi wave vector the density of states at the Fermi Thus, the density of states is proportional to the mass for a fixed number of For the given problem, the density of states has the form given by Put Step of 4 Rotating to the principle axes of the matrix and rescaling the rotated components of k according to Here, The above equation reduces M2 and M2 are the Eigen values of the matrix M Thus, The integral is the same as the integral for the density of states for free fermions having mass 1. The result is Comparing this with the density of states for free fermions we see that the only difference is that the mass m has been replaced by the specific heat effective is It follows Here, Fermi wave vector corresponding to free fermions with the same number density as the actual system considered here.