# Chapter 2: Semiconductor Fundamentals

## Examples

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 Example 2.1 Calculate the maximum fraction of the volume in a simple cubic crystal occupied by the atoms. Assume that the atoms are closely packed and that they can be treated as hard spheres. This fraction is also called the packing density. Example 2.2 Calculate the energy bandgap of germanium, silicon and gallium arsenide at 300, 400, 500 and 600 K. Example 2.3 Calculate the number of states per unit energy in a 100 by 100 by 10 nm piece of silicon (m* = 1.08 m0) 100 meV above the conduction band edge. Write the result in units of eV-1. Example 2.4 Calculate the effective densities of states in the conduction and valence bands of germanium, silicon and gallium arsenide at 300 K. Example 2.4b Calculate the intrinsic carrier density in germanium, silicon and gallium arsenide at 300, 400, 500 and 600 K. Example 2.5 Calculate the ionization energy for shallow donors and acceptors in germanium and silicon using the hydrogen-like model. Example 2.6a A germanium wafer is doped with a shallow donor density of 3ni/2. Calculate the electron and hole density. Example 2.6b A silicon wafer is doped with a shallow acceptor doping of 1016 cm-3. Calculate the electron and hole density. Example 2.6c 4H-SiC is doped with 2 x 1017 cm-3 nitrogen donor atoms (Ec – Ed = 90 meV). Use Nc = 4 x 1020 cm-3. Calculate the electron density at 300 K. Calculate the hole density at 300 K after adding 2 x 1018 cm-3 aluminum acceptor atoms (Ea – Ec = 220 meV) Use Nv = 1.6 x 1020 cm-3. Example 2.7 A piece of germanium doped with 1016 cm-3 shallow donors is illuminated with light generating 1015 cm-3 excess electrons and holes. Calculate the quasi-Fermi energies relative to the intrinsic energy and compare it to the Fermi energy in the absence of illumination. Example 2.8 Electrons in undoped gallium arsenide have a mobility of 8,800 cm2/V-s. Calculate the average time between collisions. Calculate the distance traveled between two collisions (also called the mean free path). Use an average velocity of 107 cm/s. Example 2.9 A piece of silicon doped with arsenic (Nd = 1017 cm-3) is 100 mm long, 10 mm wide and 1 mm thick. Calculate the resistance of this sample when contacted one each end. Example 2.10 The hole density in an n-type silicon wafer (Nd = 1017 cm-3) decreases linearly from 1014 cm-3 to 1013 cm-3 between x = 0 and x = 1 mm. Calculate the hole diffusion current density. Example 2.11 Calculate the electron and hole densities in an n-type silicon wafer (Nd = 1017 cm-3) illuminated uniformly with 10 mW/cm2 of red light (Eph = 1.8 eV). The absorption coefficient of red light in silicon is 10-3 cm-1. The minority carrier lifetime is 10 ms. Example 2.12 A 1cm long piece of undoped silicon with a lifetime of 1ms is illuminated with light, generating Gopt = 2x1019cm-2s-1 electron-hole pairs in the middle of the silicon. This bar silicon has ideal Ohmic contacts on both sides. Find the excess electron density throughout the material using the simple recombination model and assuming that un = up = 1000 cm2/V-s. Also find the resulting electron current density throughout the material.