In this chapter, we shall derive the DC and AC conductivities for the Bloch electrons by using a relaxation time approximation. 1. Relaxation Time Approximation In our treatment, we consider a nonequilibrium distribution function g n (r,k,t) defined so that g n (r,k,t)drdk/4 3 is the number of electrons in the n th band at time t in the semiclassical phase space volume drdk about the point r, k. At equilibrium, g n (r,k,t) reduces to the Fermi function, (6b.1) However, in the presence of applied fields and/or temperature gradients, g n (r,k,t) will differ from the equilibrium form. In this chapter, we shall adopt a picture of collisions as that discussed in the Drude Model, i.e., an electron has a probability of dt/ experiencing a collision in an infinitesimal time interval dt. But in here, we also allow for the possibility that the collision rate depends on the position, wave vector and band index of the electron, i.e., = n (r,k). We further assume the following: 1. The distribution of electrons emerging from collisions does not depend on the structure of the distribution g n (r,k,t) (equilibrium or not) just before the collision. 2. If the electrons in a region about r have the equilibrium distribution g n 0 (r,k) appropriate to a local temperature T(r), collisions will not alter the form of the distribution, i.e., g n (r,k,t) = g n 0 (r,k) = f( n (k), (r),t(r)). Assumption 1 asserts that collisions are completely effective in obliterating any information about any nonequilibrium distribution the electrons may be carrying. This almost certainly overestimates the efficacy of collisions in restoring equilibrium (see Chapter 16 of A&M). Assumption 2 asserts that the role of collisions is to maintain thermodynamic equilibrium at whatever local temperature is imposed to the system. Based on these assumptions, below we determine the form of dg n (r,k,t), the distribution function describing just those electrons that have emerged from a collision near point r in the time interval dt about t. By Assumption 1, dg n (r,k,t) cannot depend on the distribution, g n (r,k,t), just before the collision. So, it suffices to determine dg n (r,k,t) for any g. The simplest case is when g n (r,k,t) = g n 0 (r,k,t), the local equilibrium distribution. With this, assumption 2 suggests that the role of collisions is to keep g n (r,k,t) (= g n 0 (r,k,t)) unchanged. But in time dt, a fraction dt/ n (r,k) of the electrons in band n with wave vector k and position r will suffer a collision that does alter their wavevector or band index. So, if g n (r,k,t) is to be unaltered, dg n (r,k,t) must precisely compensate for this loss. So, (6b.2) by OKC Tsui based on A&M 1
Equation (6b.2) is the precise mathematical formulation of the relaxation time approximation. 2. Nonequilibrium Distribution due to External Fields & Temperature Gradients The number of electrons in the n th band at time t in the volume element drdk about r, k is: (6b.3) We shall compute this number by grouping them according to when and where they had their last collision. Let r n (t ), k n (t ) be the electron trajectory, as determined by the semiclassical equation of motion, passing through the point r, k at t = t. That r n (t ), k n (t ) denotes the trajectory determined by the equation of motion means that the electron must have its last collision prior to or right at time t. Suppose the electron had the last collision during the time interval dt at time t, it must have emerged from the volume element dr dk at r n (t ), k n (t ). The number of such electrons is: (6b.4) where use has been made of Liouville s theorem that Of the number given in eqn. (6b.4), only a fraction P n (r, k, t; t ) (to be determined below) survives from time t to time t without suffering any further collision. Combining this fact with eqn. (6b.4), we may write: Comparing this result with eqn. (6b.3), we have: (6b.5) (6b.6) The simplicity of equation eqn. (6b.6) is obscured by the state labels, n, r and k. To obliterate that, we temporarily adopt an abbreviated notation as follows: With these, eqn. (6b.6) can be written: (6b.7) (6b.8) by OKC Tsui based on A&M 2
Next, we compute P(t, t ). One may notice that the fraction of electrons surviving from time t to t is less than the fraction surviving from an infinitesimally later time t + dt to t by the factor [1 dt / (t )], which gives the probability of an electron surviving a collision between t and t + dt. Mathematically, In the limit where dt 0, we have (6b.9) (6b.10) whose solution subject to the boundary condition is (6b.11) (6b.12) Next, we use eqn. (6b.10) to rewrite the distribution function (6b.8) in the form (6b.13) It is convenient to integrate this equation by parts, using eqn. (6b.11) and the physical condition, P(t,- ) = 0. The result is (6b.14) which expresses the distribution function as the local equilibrium distribution plus a correction. To evaluate the time derivative of g 0, note that it depends on time only through n (k(t )), T(r n (t )) and (r n (t )) so that (6b.15) If we use v = dr n (t )/dt and ħdk n /dt = -e(e + v B) to replace, respectively, dr n (t )/dt and dk n /dt in eqn. (6b.15), eqn. (6b.14) can be written as (6b.16) by OKC Tsui based on A&M 3
where f is the Fermi function evaluated at the local temperature and chemical potential, and all the quantities depend on t via r n (t ) and k n (t ). Notice that although B does not appear explicitly in eqn. (6b.16) (since the Lorentz force is perpendicular to v), it appears implicitly through the time dependence of r n (t ) and k n (t ). 3. Simplification of g in Special Cases Equation (6b.16) gives the distribution function in the relaxation approximation under general conditions. But in many cases, special circumstances permit considerably simplification: 1. Weak E field and temperature gradients. The E field and temperature gradients T commonly applied to metals are usually weak enough to permit calculation of the induced currents to linear order. Since the second term of eqn. (6b.16) is linear in E and T, the t -dependence of the integrand can be calculated at zero electric field and constant T. 2. Spatially uniform EM fields and temperature gradients, and position-independent relaxation times. In this case, the entire integrand in eqn. (6b.16) will be independent of r n (t ). The only t dependence (aside from explicit dependence of E and T on time) will be through k n (t ), which will be time-dependent if a B field is present. Since the Fermi function depends on k only through n (k), which is conserved in a B field, the t dependence of the integrand in eqn. (6b.16) will be entirely contained in P(t,t ), v(k n (t )) and (if they are time-dependent) E and T. 3. Energy-dependent relaxation time. If depends on k only through n (k), then since n (k) is conserved in a B field, (t ) will not depend on t and eqn. (6b.12) reduces to: (6b.17) As discussed by A&M, most calculations in the relaxation-time approximation adopt the 3 rd assumption, and often even use a constant (energy-independent). Since the distribution in eqn. (6b.16) contains f/, which is negligible except within k B T of the Fermi energy, only the energy dependence of near F is significant in metals. Under these conditions, we can rewrite eqn. (6b.16) as (6b.18) In the following, we shall apply this equation to compute the dc and ac electrical conductivities. by OKC Tsui based on A&M 4
4. DC Electrical Conductivity If B = 0, the k(t ) appearing in eqn. (6b.18) can be approximated by k (since typically experimental E does not cause k to change significantly within a relaxation time), and the time integration is elementary for static E and T. If the temperature is uniform, we find: (6b.19) Recall that the current density in a band is (6b.20) Each partially filled band makes such a contribution to the current density; the total current density is the sum of these contributions over all the bands. Based on eqns. (6b.19) and (6b.20), it can be written as j = E, where the conductivity tensor is a sum of the contributions from each band: (6b.21) (6b.22) It is noteworthy that this result is consistent with the idea poised earlier that the particle and hole pictures are equivalent in metals. In a metal, to an accuracy of order (k B T/ F ) 2, we can evaluate (6b.22) at T = 0. Since (- f/ ) = ( F ), the relaxation time can be evaluated at F and taken outside the integral. Furthermore, since one may integrate by parts to find (6b.23) (6b.24) Since M -1 (k) is the derivative of a periodic function, its integral over the entire primitive cell must vanish, and we may write (6b.24) in the alternative form by OKC Tsui based on A&M 5
(6b.25) Comparing the two forms (6b.24) and (6b.25), one finds that the contribution to the current can be regarded as coming from the unoccupied rather than the occupied levels, provided that the sign of the effective mass tensor is changed. This result had been discussed before. We repeat it here to emphasize that it emerges from a more formal analysis as well. Next, we show that eqn. (6b.24) recovers the free electron result when the effective mass description is valid. Specifically, if M -1 = (1/m*) independent of k for all the occupied levels in the band, then eqn. (6b.24) reduces to the Drude form: (6b.26) On the other hand, if M -1 = -(1/m*) independent of k for all the unoccupied levels in the band, then eqn. (6b.24) reduces to (6b.27) where n h is the number of unoccupied levels per unit volume. Equation (6b.27) states that the conductivity of the band is of the Drude form, with m replaced by m* and the electronic density replaced by the density of holes. 5. AC Electrical Conductivity If the electric field is not static, but has the time dependence E(t) = Re[E( )exp(-i t) ] (6b.28) The derivative of the conductivity from eqn. (6b.18) proceeds as in the DC case, except for an additional factor exp(-i t) in the integrand. One finds where j(t) = Re[j( )exp(-i t) ] (6b.29) (6b.30) by OKC Tsui based on A&M 6
and (6b.31) Thus, as in the free-electron case, the ac conductivity is just the dc conductivity divided by 1 i, except that we must now allow for the possibility that the relaxation time may differ from band to band. The form of eqn. (6b.31) permits a simple direct test of the validity of the semiclassical model in the limit >> 1, where it reduces to (6b.32) Or, equivalently, as derived in the dc case (6b.33) - end - by OKC Tsui based on A&M 7