Calculate Conductivity Temperature Dependence in Semiconductors

[mcas]

Homework Statement

Write a Matlab code that plots the conductivity dependence of temperature for a semiconductor.

Relevant Equations

$\sigma = q(n\mu_n + p\mu_p$
$\frac{1}{\mu} = \frac{1}{\mu_I} + \frac{1}{\mu_L}$
$n \approx 2N_D \left(1 + \sqrt{1 + 4\frac{N_D}{N_C}e^{E_d / kT}}\right)^{-1}$

I have to plot the conductivity dependence of temperature and I have problems with obtaining the right dependency of $\mu$ and $n$. But let's focus only on carrier concentration first.
For $n$ I used the third equation. From what I understand $N_D$ is a constant. I want my plot to look like this:

But instead, it looks like this:

Here is the code:

n_0 = 4.0*10^(15);
N_c = @(x) 6.2 * 10^15 * x^(3/2);
N_v = @(x) 3.5 * 10^15 * x^(3/2);
E_g = @(x) 1.17 - 4.73 *10^(-4) * x^2 / (x + 636);
k = 8.61733262*10^(-5);

n_i = @(x) sqrt(N_c(x) * N_v(x)) * exp(- E_g(x)./ (2 * x.*k));

E_F = @(x) k * x* log(n_i(x).\n_0) + E_g(x)./ 2;

E_d = @(x) E_g(x)- E_F(x);

n_D = 10^20;

n = @(x) 2 * n_D* (1 + sqrt(1 + 4 * n_D ./ N_c(x).* exp(E_d(x) ./ (x.*k)))).\1;

T = logspace(-5, 5);
conc = subs(n, {x}, {T});
semilogy(T.\1, conc);

The calculations are made for silicon. I really don't know why I can't get this right.

 

[mark726]

Thank you for sharing your code and explaining your issue. From a quick look, it seems like you have a few errors in your code that may be causing the discrepancy in your plot.

Firstly, in your definition of E_F, you have a period instead of a multiplication sign between k and x in the log function. This should be corrected to k*x*log(n_i(x)/n_0) in order to properly calculate E_F.

Secondly, in your definition of E_d, you have a forward slash instead of a division sign between E_g(x) and 2. This should be corrected to E_g(x)/2 in order to properly calculate E_d.

Additionally, it seems like you are using the wrong equation for n. The equation you are using is for the total carrier concentration, which includes both electrons and holes. However, for the plot you want to create, you only need the electron concentration, which would be given by n = n_D / (1 + sqrt(1 + 4*n_D/N_c(x)*exp(E_d(x)/(x*k)))). This equation can be derived from the original equation you used for n by substituting n_D for the total carrier concentration N and assuming that the majority carrier is electrons.

With these corrections, your code should look like this:

n_0 = 4.0*10^(15);
N_c = @(x) 6.2 * 10^15 * x^(3/2);
N_v = @(x) 3.5 * 10^15 * x^(3/2);
E_g = @(x) 1.17 - 4.73 *10^(-4) * x^2 / (x + 636);
k = 8.61733262*10^(-5);

n_i = @(x) sqrt(N_c(x) * N_v(x)) * exp(- E_g(x)./ (2 * x.*k));

E_F = @(x) k*x*log(n_i(x)/n_0) + E_g(x)/2;

E_d = @(x) E_g(x)/2 - E_F(x);

n_D = 10^20;

n = @(x) n_D / (1 + sqrt(1 + 4*n_D/N_c(x).*exp(E_d(x)/(x*k))));

T = logspace(-5, 5);
conc = subs(n, {x},