 ### 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\$, 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 \$\$ ... ...
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!