Determine the amount (in grams) of boron required to be substitutionally incorporated into 1.000 kg of germanium in order to achieve a charge carrier density of 3.091 x 1017/cm3. Assume intrinsic carriers are negligible.
What type of conduction will be present in this material?
ionic
intrinsic
p-type
n-type
metallic
1.04e-3
p type
To determine the amount of boron required to be incorporated into the germanium, we need to use the concept of charge carrier density.
Charge carrier density (n) is defined as the number of charge carriers per unit volume. In this case, the charge carrier density given is 3.091 x 10^17/cm^3.
The formula to calculate the charge carrier density for a semiconductor is:
n = N * exp(-Eg / (2 * k * T))
where:
n is the charge carrier density
N is the effective density of states in the conduction or valence band
Eg is the energy gap between the conduction and valence bands
k is the Boltzmann constant (8.617333262145 × 10^-5 eV/K)
T is the temperature in Kelvin
In this problem, we are assuming that intrinsic carriers are negligible, meaning that the charge carriers are solely from the addition of boron. This makes it a p-type semiconductor.
For germanium, the energy gap (Eg) is approximately 0.67 eV, and the effective density of states (N) can be approximated as:
N = 2 * (2 * pi * (2 * m * kb * T / h^2)^(3/2))
where:
m is the effective mass of the charge carrier
kb is the Boltzmann constant
h is the Planck constant
The effective mass for boron in germanium is approximately 0.31 times the free electron mass.
By rearranging the equation for charge carrier density and solving for N, we can find the effective density of states required for the given charge carrier density.
Once we have the effective density of states, we can calculate the amount of boron required to achieve this density by multiplying it by the volume of germanium (1.000 kg).
Finally, to convert the mass of boron to grams, we can multiply it by 1000.
Based on the amount of boron required and the fact that we added boron to germanium to achieve a charge carrier density, the type of conduction present in this material would be p-type.