Conduction band, valence band and Fermi energy

[Pushoam]

Homework Statement

Homework Equations

The Attempt at a Solution

The probability of getting a state with energy $E_v$ is $\frac { N_v } { N_v +N_c } = \frac1{ e^{-(E_v – E_f)/k_BT} +1}$ ………….(1)

Since, $E_v < E_f, e^{-(E_v – E_f)/k_BT}>>1$ as $E_f – E_v>> k_BT$……….(2)

So, $\frac { N_v } { N_v +N_c } = \frac1{ e^{-(E_v – E_f)/k_BT} }$ ……….(3)

Similarly, probability of getting a state with energy $E_c$ is $\frac { N_c} { N_v +N_c } = \frac1{ e^{-(E_c – E_f)/k_BT} +1}$...(4)

Dividing (1) by (4) gives,

$\frac { N_v } {N_c } =$ $\frac{ e^{-(E_c – E_f)/k_BT} +1}{ e^{-(E_v – E_f)/k_BT} }$ $= \frac{ e^{-(E_c – E_f)/k_BT} }{ e^{-(E_v – E_f)/k_BT} }$

$k_BT \ln \frac { N_v } {N_c } = -(E_c – E_f) +(E_v – E_f) =E_v – E_c$

Is this correct?

 

[mjc123]

Correct or not, it's not an answer to the question.
And why do you say e^{-(E_c-E_f)/kT} + 1 = e^{-(E_c-E_f)/kT} ?

 

[Pushoam]

Correct or not, it's not an answer to the question.

I meant : Is this correct so far?
As I doubted the correctness of what I had done in the original post.

 

[Pushoam]

And why do you say e-($E_c-E_f$)/kT + 1 = e-($E_c-E_f$)/kT ?

I said about $E_v$, not $E_c$.
According to the question, $E_f - E_v >>k_B T$. So, $exp\{(E_f - E_v )/k_BT \}>> 1$.
Hence, ignoring 1 to make calculation simple,$exp\{(E_f - E_v )/k_BT\} +1 \approx exp\{(E_f - E_v )/k_BT\}$.

 

[mjc123]

But you say it implicitly in the penultimate line of your calculation.

 

[Pushoam]

But you say it implicitly in the penultimate line of your calculation.

It is said implicitly in eqn(2).

 

[Pushoam]

I need to know whether my approach so far is correct. So, please help me here.

 

[mjc123]

It is said implicitly in eqn(2).

No, that's E_v. I'm talking about E_c. Look at the line after "dividing 1 by 4 gives"
I have my suspicions that the question is not correct. The formula gives P_v << 1, which does not look sensible. Should the minus sign before (E-E_f)/kT be a plus sign?

 

[mjc123]

OK, I've had a closer look at it. It should be a plus sign in that expression, i.e. P = 1/(e^(E-E_f)/kT +1). This is not the "probability of getting a state", but the probability of an individual state being occupied. It is not a function of N_c and N_v.
Now take the statement that at equilibrium the number of electrons in the conduction band is equal to the number of holes in the valence band. How would you express that mathematically in terms of the quantities you are given?