Conductivity and the Fermi Surface

It was stated previously that the expressions for conductivity and mobility obtained from the simple Drude theory also held in the quantum mechanical treatment of electrons in simple metals. Since only the electrons near the Fermi surface are able to respond to an applied electric field, we must integrate the available states over the Fermi surface. We can write the current density as 

When an electric field, Eap, is applied in the a -direction, the symmetry is disturbed and we must expand the integrand 

The first term of the expansion integrates to zero by symmetry. We can write the incremental change in momentum in the a -direction resulting from the applied field as Akx = mv lh = —er Eap/fi. We can also write dS k)/dkx = fiVxdS k)/dE and d k = dE/li v. Putting all of this back into Equation 19.27, we can write the conductivity as 

For a simple metal, such as Cu where the s-electrons are the primary carries, the Fermi surface is nearly spherical and the integral over its siu-face is 4iTfcp. The total volume in fc-space included in the integral is 4iTfcp. If the system has volume I , the number of states is therefore which can accommodate 8tt1 /(2 ii/L) electrons or 

8TTfc /(2Tr) electrons per unit volume. Remembering from Equation 19.4 A /3it = JXe, the electron density of the Fermi sphere. Equation 19.30 can be written for simple metals as 

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