Free Electrons 1

exercise 3_4930

Consider the conduction electrons of a one-dimensional metal (length a ) and a two-dimensional metal (length a , width b ) at absolute zero. Approximate the electrons as free particles.

(a) Assuming periodic boundary conditions, find the allowed values of k x for the one-dimensional metal, and of k x and k y for the two-dimensional metal.

(b) Write the expression for the total energy of a single electron for the two cases in terms of, ħ , m , and the components of the wavevector .

(c) Given that there are N electrons, calculate k F and E F for the two cases at T = 0 in terms of N, a , b , ħ , and m .

(d) Derive expressions for the density of states in energy, g ( E ), for the two cases.

Draw a qualitatively correct picture of g ( E ) vs E to illustrate the contrast between g ( E ) for one-, two-, and three-dimensional metals.

(e) Use g ( E ) to find the average electron energy at absolute zero ( T = 0) in terms of E F for the one- and two-dimensional solids.