Chapter 2: Semiconductor Fundamentals

1. Calculate the packing density of the body centered cubic, the face centered cubic and the diamond lattice, listed in example 2.1.
2. At what temperature does the energy bandgap of silicon equal exactly 1 eV?
3. 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.
4. At what energy (in units of kT) is the Fermi function within 1 % of the Maxwell-Boltzmann distribution function? What is the corresponding probability of occupancy?
5. 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 %.
6. 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.
7. 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.
8. 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.
9. 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.
10. The equations (2.6.34) and (2.6.35) derived in section 2.6 are only valid for non-degenerate semiconductors (i.e. E_v + 3kT < E_F < E_c - 3kT). Where exactly in the derivation was the assumption made that the semiconductor is non-degenerate?
11. A silicon wafer contains 10¹⁶ 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 = 10¹⁰ cm^-3.
12. A silicon wafer is doped with 10¹³ cm^-3 shallow donors and 9 x 10¹² cm^-3 shallow acceptors. Calculate the electron and hole density at 300 K. Use n_i = 10¹⁰ cm^-3.
13. The resistivity of a silicon wafer at room temperature is 5 Wcm. What is the doping density? Find all possible solutions.
14. How many phosphorus atoms must be added to decrease the resistivity of n-type silicon at room temperature from 1 Wcm to 0.1 Wcm. Make sure you include the doping dependence of the mobility. State your assumptions.
15. A piece of n-type silicon (N_d = 10¹⁷ cm^-3) is uniformly illuminated with green light (l = 550 nm) so that the power density in the material equals 1 mW/cm². a) Calculate the generation rate of electron-hole pairs using an absorption coefficient of 10⁴ cm^-1. b) Calculate the excess electron and hole density using the generation rate obtained in (a) and a minority carrier lifetime due to Shockley-Read-Hall recombination of 0.1 ms. c) Calculate the electron and hole quasi-Fermi energies (relative to Ei) based on the excess densities obtained in (b).
16. A piece of intrinsic silicon is instantaneously heated from 0 K to room temperature (300 K). The minority carrier lifetime due to Shockley-Read-Hall recombination in the material is 1 ms. Calculate the generation rate of electron-hole 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?
17. Calculate the conductivity and resistivity of intrinsic silicon. Use n_i = 10¹⁰ cm^-3, m_n = 1400 cm²/V-sec and m_p = 450 cm²/V-sec.
18. Consider the problem of finding the doping density which results the maximum possible resistivity of silicon at room temperature. (n_i = 10¹⁰, m_n = 1400 cm²/V-sec and m_p = 450 cm²V-sec.)

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 dopant are shallow).

Calculate the maximum resistivity, the corresponding electron and hole density and the doping density.

19. 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.
20. The expression for the Bohr radius can also be applied to the hydrogen-like 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 hydrogen-like atom. Calculate the Bohr radius of an electron orbiting around the ionized donor in silicon. ( e_r = 11.9 and m_e^* = 0.26 m₀)
21. 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 m₀.
22. 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.
23. 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
24. The effective mass of electrons in silicon is 0.26 m₀ and the effective mass of holes is 0.36 m₀. If the scattering time is the same for both carrier types, what is the ratio of the electron mobility and the hole mobility.
25. Electrons in silicon carbide have a mobility of 1000 cm²/V-sec. At what value of the electric field do the electrons reach a velocity of 3 x 10⁷ cm/s? Assume that the mobility is constant and independent of the electric field. What voltage is required to obtain this field in a 5 micron thick region? How much time do the electrons need to cross the 5 micron thick region?
26. 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 cm²/V-sec and that of holes is 450 cm²/V-sec, what is the minimum possible carrier density and what is the corresponding carrier type? Repeat for the maximum possible carrier density.
27. A silicon wafer has a 2 inch diameter and contains 10¹⁴ cm^-3 electrons with a mobility of 1400 cm²/V-sec. How thick should the wafer be so that the resistance between the front and back surface equals 0.1 Ohm.
28. The electron mobility is germanium is 1000 cm²/V-sec. If this mobility is due to impurity and lattice scattering and the mobility due to lattice scattering only is 1900 cm²/V-sec, what is the mobility due to impurity scattering only?
29. A piece of n-type silicon is doped with 10¹⁷ cm^-3 shallow donors. Calculate the density of electrons per unit energy at kT/2 above the conduction band edge. T = 300 K. Calculate the electron energy for which the density of electrons per unit energy has a maximum. What is the corresponding probability of occupancy at that maximum?
30. Phosphorous donor atoms with a concentration of 10¹⁶ cm^-3 are added to a piece of silicon. Assume that the phosphorous atoms are distributed homogeneously throughout the silicon. The atomic weight of phosphorous is 31.
    1. What is the sample resistivity at 300 K?
    2. What proportion by weight does the donor impurity comprise? The density of silicon is 2.33 gram/cm³
    3. If 10¹⁷ atoms cm^-3 of boron are included in addition to phosphorous, and distributed uniformly, what is the resulting resistivity and type (i.e., p- or n-type material)?
    4. Sketch the energy-band diagram under the condition of part c) and show the position of the Fermi energy relative to the valence band edge.
    
31. Find the equilibrium electron and hole concentrations and the location of the Fermi energy relative to the intrinsic energy in silicon at 27 ^oC, if the silicon contains the following concentrations of shallow dopants.
    1. 1 x 10¹⁶ cm^-3 boron atoms
    2. 3 x 10¹⁶ cm^-3 arsenic atoms and 2.9 x 10¹⁶ cm^-3 boron atoms.
    
32. The electron concentration in a piece of lightly doped, n-type silicon at room temperature varies linearly from 10¹⁷ cm^-3 at x = 0 to 6 x 10¹⁶ cm^-3 at x = 2 mm. Electrons are supplied to keep this concentration constant with time. Calculate the electron current density in the silicon if no electric field is present. Assume m_n = 1000 cm²/V-s and T = 300 K.