Commit 9bfea0be authored by Isidora Araya's avatar Isidora Araya
Browse files

Update 13_semiconductors_solutions.md

parent 6e02a9a4
......@@ -80,13 +80,17 @@ dispersion(10, 2, 8)
```
### Subquestion 1
Apply the following to max an min of valence and conduction band, respectively:
Apply the following to valence and conduction band, respectively:
$$v=\hbar^{-1}\partial E(k)/\partial k$$
$$m_{eff} = \hbar^2\left(d^2 E(k)/dk^2\right)^{-1}$$
Be careful with hole calculations.
These results assume $t_{cb}$ and $t_{vb}$ positive.
$$v_e = \frac{2at_{cb}}{\hbar}sin(ka) $$
$$m_e = \frac{\hbar^2}{2a^2t_{cb}cos(ka)}$$
$$v_h = -\frac{2at_{vb}}{\hbar}sin(ka) $$
$$m_h = \frac{\hbar^2}{2a^2t_{vb}cos(ka)}$$
### Subquestion 2
This approximation indicates the chemical potential is "well bellow" the conduction band and "well above"
......@@ -96,12 +100,14 @@ approximate Fermi statistics by Boltzmann's.
For electrons, do a taylor expansion around $k=0$ (min) of $E_{cb} = E_G - 2 t_{cb} [\cos(ka)-1],$.
For holes, do a taylor expansion around $k=0$ (max) of $E_{vb} = 2 t_{vb} [\cos(ka)-1]$.
Assume $t_{cb}$ and $t_{vb}$ positive.
### Subquestion 3
### Subquestion 4
$$n = \int_{E_{cb}}^{E_{cb}+4t_{cb}} f(\varepsilon)g_c(\varepsilon)d\varepsilon$$
$$p = \int_{-4t_{vb}}^{0} (1-f(\varepsilon))g_h(\varepsilon)d\varepsilon$$
$$n_e = \int_{E_{cb}}^{E_{cb}+4t_{cb}} f(\varepsilon)g_c(\varepsilon)d\varepsilon$$
$$n_h = \int_{-4t_{vb}}^{0} (1-f(\varepsilon))g_h(\varepsilon)d\varepsilon$$
### Subquestion 5
For an intrinsic semiconductor the number of electrons excited into the conduction band must be equal
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment