Calculating Densities and Width of Depletion Region in Ge

[NaN089]

Homework Statement

a. The densities of electrons and holes required for the Fermi levels in
both the p- and n-type Ge to enter the bands. That is, in n-type Ge the Fermi
level coincides with the bottom of the conduction band and for p-type Ge the
Fermi level coincides with the top of the valence band. [[I know how to do this but I have given this question so that question (b) is comprehensible.]]

b. Hence calculate the fraction of ionised dopants and the total density of
dopants required for (a) above.

c. For the situation depicted in Esaki’s figure 2 below calculate the
densities of electrons and holes on either side of the junction as well as the
total densities of dopants.

d. Estimate the width of the depletion region and compare your
calculation with Esaki’s estimate

Homework Equations

for a:
n(E_c)= N_c.e^(-((E_c-E_f))/(k.T))
p(E_v)= N_v.e^(-((E_f-E_v))/(k.T))

for b:
N⁺_D = (N_D)/(1+ g_D*exp((E_f-E_D)/(k.T))
N^-_A = (N_A)/(1+ g_A*exp((E_A-E_f)/(k.T))

The Attempt at a Solution

for a:
Putting E_c-E_f= 0, we get n_c=N_c and the similar for holes.

for b:
I have used the equations for N⁺_D and N⁺_A to find the dopants, but I do not know how to find the fraction of them. Next, the density is found for the condition E_c-E_f= 0.

for c:
I believe that we need the Fermi energy to find the solutions.
E_f==KT*ln{[(1/4)*[e^(E_d/KT)]*([(1+(8N_d/N_c)e^(deltaE_d/KT))^(1/2)]-1)}
Is this the correct formula to find the Fermi energy?


Thanks

 

[mark726]

for your question. It seems like you have a good understanding of the concepts and equations involved in solving this problem. Here are some tips to help you with the remaining parts:

b. To calculate the fraction of ionized dopants, you can use the equations for N+D and N+A and set them equal to each other. This will give you an equation with Ef as the only unknown, which you can then solve for.

c. To find the densities of electrons and holes on either side of the junction, you can use the equations for n(Ec) and p(Ev) that you mentioned in part a. For the total density of dopants, you can use the equations for N+D and N+A and plug in the values for the fraction of ionized dopants that you calculated in part b.

d. To estimate the width of the depletion region, you can use the equation for the electric field in the depletion region, which is given by E= (qND + qNA)/(εε0). You can then integrate this equation to find the width of the depletion region. Compare your result with Esaki's estimate to see how accurate your calculation is.

I hope this helps and good luck with your calculations!