PHT.301 Physics of Semiconductor Devices

Extrinsic semiconductors

In the section on intrinsic semiconductors we found that the conductivity of an intrinsic semiconductor depends exponentially on temperature and that at room temperature intrinsic semiconductors are rather poor conductors. It is possible to dope semiconductors with impurity atoms that improve the conductivity dramatically and makes the conductivity nearly constant as a function of temperature near room temperature. Doped semiconductors are called extrinsic semiconductors.

Reading: Singh 2.6 - 2.9 or Thuselt 2.3

For the exam:
• Be able to explain under what conditions donor and acceptor atoms become ionized.
• You need to know how to calculate the concentration of electrons and holes for an extrinsic semiconductor.
• You need to know how to calculate the Fermi energy in an extrinsic semiconductor as a function of doping and temperature.
• Know what a degenerate semiconductor is.

Problems
• Temperature dependence of a doped semiconductor
• Dopant separation
• Carriers in a doped semiconductor
• Carrier generation
• Ionized donors

Additional material

The donor energies are the differences of the donor levels E_d to the bottom of the conduction band E_c. The acceptor energies are the differences of the acceptor levels E_a to the top of the valence band E_v.

Doped semiconductors

For a doped semiconductor, the density of electrons in the conduction band is,

the density of holes in the valence band is,

the density of ionized donors is,

and the density of ionized acceptors is,

The factor of 4 is valid in the formula for the acceptors if the semiconductor has a light hole and a heavy hole band as Si and Ge do.

The four quantities n, p, N_d, and N_a can only be determined if the Fermi energy, E_f, is known. Typically, E_f must first be determined from the charge neutrality condition,

n + N_a^- = p + N_d⁺.

The Fermi energy can be found by solving the charge neutrality condition numerically. One way to do this is to program the formulas for n, p, N_d⁺, and N_a^- in a spreadsheet. Then choose a temperature and calculate n, p, N_d⁺, N_a^- for every value of the Fermi energy between E_v and E_c. For one of these E_f values, the charge neutrality condition will be satisfied.

When n + N_a^- and , p + N_d⁺ are ploted as a function of E_F, the Fermi energy is where the two lines cross. A plot like the one below can be generated by pressing the button below the plot.

Make new plot

The Fermi energy can be calculated as a function of temperature by determining where n + N_a^- = p + N_d⁺ for every temperature. A plot like the one below can be generated by pressing the button below the plot.

Make new plot

Normally it is not necessary to determine E_f numerically and the following approximation is sufficient.

n-type N_d > N_a:

n = N_d - N_a
p = n_i²/n
E_f = E_c - k_BTln(N_c/(N_d - N_a))

p-type N_a > N_d:

p = N_a - N_d
n = n_i²/p
E_f = E_v + k_BTln(N_v/(N_a - N_d))