 Calculate the packing density of the body centered cubic, the face centered cubic and the diamond lattice, listed in example 2.1
 At what temperature does the energy bandgap of silicon equal exactly 1 eV?
 Prove that the probability of occupying an energy level below the Fermi energy equals the probability that an energy level above the Fermi energy and equally far away from the Fermi energy is not occupied.
 At what energy (in units of kT) is the Fermi function within 1 % of the MaxwellBoltzmann distribution function? What is the corresponding probability of occupancy?
 Calculate the Fermi function at 6.5 eV if E_{F} = 6.25 eV and T = 300 K. Repeat at T = 950 K assuming that the Fermi energy does not change. At what temperature does the probability that an energy level at E = 5.95 eV is empty equal 1 %.
 Calculate the effective density of states for electrons and holes in germanium, silicon and gallium arsenide at room temperature and at 100 °C. Use the effective masses for density of states calculations.
 Calculate the intrinsic carrier density in germanium, silicon and gallium arsenide at room temperature (300 K). Repeat at 100 °C. Assume that the energy bandgap is independent of temperature and use the room temperature values.
 Calculate the position of the intrinsic energy level relative to the midgap energy
E_{midgap} = (E_{c} + E_{v})/2
in germanium, silicon and gallium arsenide at 300 K. Repeat at T = 100 °C
 Calculate the electron and hole density in germanium, silicon and gallium arsenide if the Fermi energy is 0.3 eV above the intrinsic energy level. Repeat if the Fermi energy is 0.3 eV below the conduction band edge. Assume that T = 300 K.
 The equations (2.6.37) and (2.6.38) derived in section 2.6 are only valid for nondegenerate semiconductors (i.e. E_{v} + 3kT < E_{F} < E_{c}  3kT). Where exactly in the derivation was the assumption made that the semiconductor is nondegenerate?
 A silicon wafer contains 10^{16} cm^{3} electrons. Calculate the hole density and the position of the intrinsic energy and the Fermi energy at 300 K. Draw the corresponding band diagram to scale, indicating the conduction and valence band edge, the intrinsic energy level and the Fermi energy level. Use n_{i} = 10^{10} cm^{3}.
 A silicon wafer is doped with 10^{13} cm^{3} shallow donors and 9 x 10^{12} cm^{3} shallow acceptors. Calculate the electron and hole density at 300 K. Use n_{i} = 10^{10} cm^{3}.
 The resistivity of a silicon wafer at room temperature is 5 Wcm. What is the doping density? Find all possible solutions.
 How many phosphorus atoms must be added to decrease the resistivity of ntype silicon at room temperature from 1 Wcm to 0.1 Wcm. Make sure you include the doping dependence of the mobility. State your assumptions.
 A piece of ntype silicon (N_{d} = 10^{17} cm^{3}) is uniformly illuminated with green light (l = 550 nm) so that the power density in the material equals 1 mW/cm^{2}. a) Calculate the generation rate of electronhole pairs using an absorption coefficient of 10^{4} cm^{1}.b) Calculate the excess electron and hole density using the generation rate obtained in (a) and a minority carrier lifetime due to ShockleyReadHall recombination of 0.1 ms. c) Calculate the electron and hole quasiFermi energies (relative to E_{i}) based on the excess densities obtained in (b).
 A piece of intrinsic silicon is instantaneously heated from 0 K to room temperature (300 K). The minority carrier lifetime due to ShockleyReadHall recombination in the material is 1 ms. Calculate the generation rate of electronhole pairs immediately after reaching room temperature. (E_{t} = E_{i}).If the generation rate is constant, how long does it take to reach thermal equilibrium?
 Calculate the conductivity and resistivity of intrinsic silicon. Use n_{i} = 10^{10} cm^{3}, m_{n} = 1400 cm^{2}/Vsec and m_{p} = 450 cm^{2}/Vsec.
 Consider the problem of finding the doping density, which results the maximum possible resistivity of silicon at room temperature. (n_{i} = 10^{10}, m_{n} = 1400 cm^{2}/Vsec and m_{p} = 450 cm^{2}Vsec.)
Should the silicon be doped at all or do you expect the maximum resistivity when dopants are added?
If the silicon should be doped, should it be doped with acceptors or donors (assume that all dopants are shallow).
Calculate the maximum resistivity, the corresponding electron and hole density and the doping density.
 The electron density in silicon at room temperature is twice the intrinsic density. Calculate the hole density, the donor density and the Fermi energy relative to the intrinsic energy. Repeat for n = 5 n_{i} and n = 10 n_{i}. Also repeat for p = 2 n_{i}, p = 5 n_{i} and p = 10 n_{i}, calculating the electron and acceptor density as well as the Fermi energy relative to the intrinsic energy level.
 The expression for the Bohr radius can also be applied to the hydrogenlike atom consisting of an ionized donor and the electron provided by the donor. Modify the expression for the Bohr radius so that it applies to this hydrogenlike atom. Calculate the Bohr radius of an electron orbiting around the ionized donor in silicon. ( e_{r} = 11.9 and m_{e}^{*} = 0.26 m_{0})
 Calculate the density of electrons per unit energy (in electron volt) and per unit area (per cubic centimeter) at 1 eV above the band minimum. Assume that m_{e}^{*} = 1.08 m_{0}.
 Calculate the probability that an electron occupies an energy level, which is 3kT below the Fermi energy. Repeat for an energy level which is 3kT above the Fermi energy.
 Calculate and plot as a function of energy the product of the probability that an energy level is occupied with the probability that that same energy level is not occupied. Assume that the Fermi energy is zero and that kT = 1 eV
 The effective mass of electrons in silicon is 0.26 m_{0} and the effective mass of holes is 0.36 m_{0}. If the scattering time is the same for both carrier types, what is the ratio of the electron mobility and the hole mobility.
 Electrons in silicon carbide have a mobility of 1000 cm^{2}/Vsec. At what value of the electric field do the electrons reach a velocity of 3 x 10^{7} cm/s? Assume that the mobility is constant and independent of the electric field. What voltage is required to obtain this field in a 5micron thick region? How much time do the electrons need to cross the 5 micron thick region?
 A piece of silicon has a resistivity which is specified by the manufacturer to be between 2 and 5 Ohm cm. Assuming that the mobility of electrons is 1400 cm^{2}/Vsec and that of holes is 450 cm^{2}/Vsec, what is the minimum possible carrier density and what is the corresponding carrier type? Repeat for the maximum possible carrier density.
 A silicon wafer has a 2inch diameter and contains 10^{14} cm^{3} electrons with a mobility of 1400 cm^{2}/Vsec. How thick should the wafer be so that the resistance between the front and back surface equals 0.1 Ohm?
 The electron mobility is germanium is 1000 cm^{2}/Vsec. If this mobility is due to impurity and lattice scattering and the mobility due to lattice scattering only is 1900 cm^{2}/Vsec, what is the mobility due to impurity scattering only?
 An atomic system consists of three energy levels with energy 0, 10 and 20 meV, which can contain a maximum of 1000, 2000 and 1000 electrons, respectively. The total energy of the system in thermal equilibrium is 25 eV and the total number of electrons is 2000. Calculate the Fermi energy and the temperature. (Challenge problem)
 A 20 mm thin piece of gallium arsenide consists of two regions with a different carrier life time, namely t = 20 ns for 0 £ x £ 10 mm and t = ¥ for 10 £ x £ 20 mm. The material is illuminated with light so that 10^{22} cm^{2}s^{1} electronhole pairs are created at x = 20 mm. Calculate the steady state electron and hole density at x = 0, 10 and 20 mm. (Assume m_{n} = m_{p} = 1000 cm^{2}/Vs and T = 300 K)
 A 10micron thin piece of ntype silicon (N_{d} = 10^{17} cm^{3}) is uniformly illuminated with light, resulting in an electronhole pair generation rate of 10^{23} cm^{3}s^{1}. The silicon is contacted only on one side with an ideal Ohmic contact. Assume there is no recombination in the semiconductor. (Use m_{n} = 1000 cm^{2}/Vs and m_{p} = 300 cm^{2}/Vs)
a) What is the total current measured at the contact. Justify your answer.
b) What properties of the minority carrier density can you identify without actually solving the problem.
c) How would you solve for the minority carrier density.
d) Calculate the minority carrier density throughout the silicon.
e) Calculate the hole current density at the Ohmic contact.
 Consider a semiconductor with two parabolic conduction bands having conduction band minima E_{c1} and E_{c2} with effective masses m_{e1} = 0.06 m_{0} and m_{e2} = 0.4 m_{0} and mobility m_{n1} = 8000 cm^{2}/Vs and m_{n2} = 1000 cm^{2}/Vs. The conduction band minimum E_{c2} is 30 meV higher than E_{c1}.
Find an expression as well as the numeric value for the effective density of states, N_{c}^{*}, taking into account both conduction bands, so that the usual expressions for the intrinsic carrier concentration and the intrinsic Fermi energy still hold, namely:
Where the energy bandgap, E_{g}, equals E_{c1} – E_{v} and E_{v} is the top of the valence band.
Find an expression as well as the numeric value for the effective electron mobility m_{n}^{*}, again taking into account both conduction bands so that the usual expression for the conductivity still holds, namely s_{n} = q n m_{n}^{*} where n is the combined density of electrons in both conduction bands.
 An intrinsic piece of GaAs (n_{i} = 2 x 10^{6} cm^{3} ) of length L (= 1 mm) is uniformly illuminated with light yielding an electronhole pair generation rate of 10^{22} cm^{3} s^{1}. On both ends (at x = 0 and x = L) the material is contacted with “ideal” ohmic contacts. Calculate the maximum value of the steadystate excess hole density in the material under illumination. The hole diffusion length, L_{p}, equals L/ln(2) and the hole mobility is 500 cm^{2}/Vs. Assume that there is no electric field in the semiconductor and T = 300 K.
 An intrinsic piece of GaAs (n_{i} = 2 x 10^{6} cm^{3} ) of length L (= 1 mm) is uniformly illuminated with light yielding an electronhole pair generation rate of 10^{22} cm^{3} s^{1}. On one end (at x = 0) the material is contacted with an “ideal” ohmic contact, while on the other end (at x = L) the current is limited by a finite recombination velocity, s, (s = 10^{5} cm/s), as described by the following relation between the excess hole density, d_{p}, and the hole current density, J_{p}:
Calculate the maximum value of the steadystate excess hole density in the material under illumination. The hole diffusion length, L_{p}, equals L/ln(2) and the hole mobility is 500 cm^{2}/Vs. Assume that there is no electric field in the semiconductor and T = 300 K.
