4.1 A particle of mass m and fixed energy E is confined to a two-dimensional box. The x and y side-lengths of the box are a and 4 respectively. Also Vex, y) = constant everywhere inside the box. Assuming the side-lengths of the box are much larger than atomic dimensions, de¬rive an expression for the density of states (g2D) for the given particle in a two-dimensional box. Record all steps in obtaining your answer.
4.4 The conduction band minima in GaP occur right at the first Brillouin zone boundary along the < 100 > directions in k-space. Taking the constant energy surfaces to be ellipsoidal with mYme = 1.12 and WM% = 0.22, determine the density of states effective mass for electrons in Gap.
4.6 In Si, where ?SAO = 0537 and mib/nto = 0.153, what fraction of the holes are heavy holes? (For simplicity, assume that the quoted 4 K effective masses can be employed at any temperature.)
4.0 Because m; and thus Nc are relatively small in GaAs, the donor doping at which the materi¬al becomes degenerate is also relatively small. Determine the ND doping at the non-degen¬erate/degenerate transition point in GaAs at 300 K
4.14 Determine the equilibrium electron and hole concentrations inside a uniformly doped sam¬ple of Si under the following conditions:
(a) T = 300 K, NA « N0, No = 1014/cm3.
(b) T = 300 IC, NA = 10151=3. No « Na.
(c) T = 300 K, NA = 9 x 10"/cm', No = 10"Icm'.
(d) T = 470 K, NA = 0, N, = 1014/cms.
(e) T = 645 K, NA = 0, ND = 1014/ems•
4.16 (a) A set of five Si samples with decade values of donor doping ranging from 1013/cnu to 10"'/cm' are maintained at T = 545 K In all cases, N0 >> NA. What are the electron and hole concentrations inside the five Si samples?
(b) Determine the position of E,, compute E, . Ei and draw a carefully dimensioned en-ergy band diagram for each of the Si samples.
4.17 Given a Si sample phosphorus-doped with No = 1015/CM3 » NA. calculate niNo for tem-peratures T = 25 IC, 50 IC, 75 IC, and 100 K_ Assume 910 = 2 and E, - ED = 0.045 eV. Com¬pare your calculated results with Fig. 4.18(a). ii