Gallium Arsenide (GaAs) is a semiconductor used for light emitting diodes in your CD player and microwave amplification in your cell phone. The energy gap between the valence band and the conduction band is Δ = 1.43 eV. In the first part of this problem, we will see that the number of free carriers in a pure crystal depends strongly on temperature. (Here, "free" means "free to move," and "carriers" means particles that carry electric charge.) In the second part, we will see how introducing free carriers via impurities affects the electron and hole densities. (k = 8.617×10^-5 eV/K)

At T = 0 K the valence band has all the electrons. At T > 0 K (shown), electrons are thermally excited across the gap into the conduction band, leaving an equal number of holes behind.

1)

Calculate the density of free electrons (number per cubic meter) in a pure crystal of GaAs at T = 3°C. The quantum density for carriers in GaAs at 3°C is 1.108 x 10²⁵m^-3(for this problem we assume this is the value for both the electrons and holes).

n_e =

2)

We now increase the temperature to T=98 °C, where the quantum density increases to 1.727 x 10²⁵m^-3. What is the density ofholesat this temperature?

n_h(hot)=

3)

10¹⁴- 10¹⁶carriers per cubic meter may sound large, but it is really a very small density of carriers. For comparison, a good conductor like copper has many more conduction electrons per cubic meter. Using the fact that the density of copper is 8960 kg/m³, that its atomic weight is 64 g/mole, and that each atom contributes about 1 free electron, estimate the electron carrier density in copper.

n_e,Cu =

4)

To make a semiconductor more conducting, impurities are added in the growth of the crystal or implanted by various means. Consider "doping" GaAs with 1 ppm (part-per-million) selenium atoms, i.e., we will replace one out of every million As atoms with a Se atom. When selenium is added to a crystal of GaAs, it provides an additional electron which is not needed for the bonding, i.e., it is only weakly bound. This electron can be free to move through the crystal. Thus, it provides a mechanism for electrical conductivity. Such a system is called an "n-type" semiconductor, where n stands for negative because the carrier of electricity is the negatively charged electron.

The ionization energy required to remove one electron from the Selenium and allow it roam freely through the lattice is only 0.006 eV. Since kT is substantially greater than this here, and since there are many more states available to the electron if it is not bound to the Se nucleus, we can assume that at T = 98°C each selenium atom essentially contributes one free electron to the GaAs. Estimate the new electron density of the doped GaAs. Note: The density of GaAs is 5317 kg/m³, and the molecular weight of GaAs is ~145 g/mole.

n_e,doped =

5)

Use the Law of Mass Action, n_en_h= n_i², where n_iis the intrinsic electron density - the electron density that you calculated in 2) when no impurities were present - to calculate the density of holes at T = 98°C.

n_h,doped =