10.1

Derive the expression for the Fermi energy for a two-dimensional free electron gas analogous to the corresponding one- and three-dimensional Fermi energies, (10.25) and (10.26), given in the text.

10.2

Use the two-dimensional density-of-states derived in Problem 10.1 above.

1. Show on the basis of Eq. (10.29) that the chemical potential of a free-electron gas in two dimensions is given by:

   
   

   for N _e electrons per unit area.

2. Plot ?( T) /E _F as a function of kT/E _F as in Figure 10.6(b).

10.3

For a typical 1-D energy band, sketch graphs of the relationships between the wave vector, k, of an electron and its:

1. energy,

2. group velocity, and

3. effective mass.

4. Sketch the approximate density-of-states D ⁽¹⁾( E) for the energy band of part (a).

10.4

The E( k _x) vs. k _x dependence for an electron in the conduction band of a one-dimensional semiconductor crystal with lattice constant a = 4 is given by:

1. Sketch E( k _x) for this band in the reduced and periodic zone schemes.

2. Find the group velocity of an electron in this band and sketch it as a function of k _x.

3. Find the effective mass of an electron in this band as a function of k _x and sketch it in the reduced-zone scheme. A uniform electric field E _x is applied in the x direction, in what direction will an electron...