Volume Density of states (electrons)

[orangeincup]

Homework Statement

a)Find the densities of states 0.08 eV above the conduction band edge and 0.08 eV below the valence band edge for germanium. Be careful with units and be sure to give the units for your answer.

b) Find the volume density of states (i.e. number of states per unit volume) with energies between the conduction band edge and 0.4 eV above the conduction band edge for germanium.

Homework Equations

1/(1+exp(e-ef/kt))

The Attempt at a Solution

a) So I found density of states in conduction band is meant to be 1.04*10^19, and in valence band it's meant to be 6.0*10^18.
above conduction band edge:
1.04*10^-19

holes=.044 ih
holes= .28 hh

Assuming it's 300k my solution would be

Nc*exp(.08/0.0259)=0.044
Nc=0.002000

Nv*exp(.08/0.0259)=0.28
Nv=0.0128

for b, I have two values so do I need an integral, or should I average between conduction band and .04?

 

[anathema]

a) To find the densities of states above and below the band edges, you can use the formula given in the homework equations:
N(E) = 1/(1+exp((E-Ef)/kT))
Where N(E) is the density of states, E is the energy, Ef is the Fermi energy, k is the Boltzmann constant, and T is the temperature.
For germanium, the conduction band edge is at 0 eV and the valence band edge is at -0.66 eV (relative to the Fermi energy). Therefore, for E=0.08 eV above the conduction band edge, we have:
N(E) = 1/(1+exp((0.08-0)/0.0259)) = 1/(1+exp(3.09)) = 1/(1+21.1) = 0.045
The units for N(E) are states/eV.
Similarly, for E=-0.08 eV below the valence band edge, we have:
N(E) = 1/(1+exp((-0.08-(-0.66))/0.0259)) = 1/(1+exp(22.0)) = 1/(1+1.4*10^9) = 7.1*10^-10
The units for N(E) are states/eV.

b) To find the volume density of states, we need to multiply the density of states by the volume of the material. For germanium, the volume is approximately 5.32*10^-23 cm^3 per atom. Therefore, the volume density of states for energies between the conduction band edge and 0.4 eV above the conduction band edge would be:
N(E)*5.32*10^-23 = 0.045*5.32*10^-23 = 2.4*10^-24 states/cm^3
Note that the units for N(E) are states/eV, and the units for the volume are cm^3, so the resulting units are states/cm^3.