Commit 3eef604e authored by Isidora Araya's avatar Isidora Araya
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Update 14_doping_and_devices_solutions.md

parent d132cc22
......@@ -8,23 +8,22 @@ from math import pi
## Exercise 1: Crossover between extrinsic and intrinsic regimes
### Subquestion 1
Law of mass action (general):
$$ (n_e + N_A) (n_h + N_D) = \frac{1}{2} \left(\frac{k_BT}{\pi\hbar^2}\right)^3
Law of mass action:
$$ n_e n_h = \frac{1}{2} \left(\frac{k_BT}{\pi\hbar^2}\right)^3
(m_e^{\ast}m_h^{\ast})^{3/2}e^{-\beta E_{gap}}$$
Charge balance condition:
$$ D = N_D - N_A - n_e + n_h = (doping)$$
$$ n_e - n_h + n_D - n_A = N_D - N_A $$
### Subquestion 2
$$ n_{e} + N_A = \frac{1}{2}(\sqrt{D^2+4n_i^2}+D)$$
$$ n_{h} + N_D = \frac{1}{2}(\sqrt{D^2+4n_i^2}-D)$$
$$ n_{e} = \frac{1}{2}(\sqrt{D^2+4n_i^2}+D)$$
$$ n_{h} = \frac{1}{2}(\sqrt{D^2+4n_i^2}-D)$$
where $n_i=n_{e,intrinsic}=n_{h,intrinsic}$.
### Subquestion 3
If $D<<n_i$, then the doping is not important and results of intrinsic are
reproduced ($n_e \approx n_h$)
reproduced.
Contrarily, if $D>>n_i$, it's mostly the doping that determines $n_e$ and $n_h$.
The thermal factor becomes unimportant.
Check both cases with lecture notes approximated solutions by doing a Taylor expansion.
......@@ -37,8 +36,8 @@ Check both cases with lecture notes approximated solutions by doing a Taylor exp
If all the dopants are ionized, the Fermi level gets shifted up towards the conduction band.
This result can be obtained when using results in Exercise 1 - Subquestion 2 and the following:
$$ n_D \approx N_D$$
$$ n_A \approx N_A$$
$$ n_D \approx 0$$
$$ n_A \approx 0$$
$$ n_e = n_h = n_i $$
......
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