semic

Revamp lectures 13 and 14.

src/14_doping_and_devices.md

@@ -98,6 +98,8 @@ ax.plot(E, g_A, label='$g_A$')
 ax.legend()
 ax.set_xticks([E_V, E_C, E_A, E_D])
 ax.set_xticklabels(['$E_V$', '$E_C$', '$E_A$', '$E_D$'])
+ax.set_ylabel('$g$')
+ax.set_xlabel('$E$')
 draw_classic_axes(ax, xlabeloffset=.2)
 ```
 
@@ -119,6 +121,13 @@ As a result, the impurity concentration is bounded to $N_D \lesssim (1/4\textrm{
 
 ## Number of carriers
 
+| Symbol | Meaning |
+| - | - |
+| $n_e$ | Concentration of electrons in the conduction band |
+| $n_h$ | Concentration of holes in the valance band |
+| $n_D$ | Concentration of electrons in the donor bound state|
+| $n_A$ | Concentration of holes in the acceptor bound state|
+
 We now have the necessary tools to determine how the Fermi level changes with doping. 
 The algorithm to determine the Fermi level of a semiconductor was outlined in the previous lecture and we continue to use it here.
 The process is the same up until the third step - charge conservation. 
@@ -167,7 +176,7 @@ $$E_F = kT\ln[N_V/(N_A-N_D)], \textrm{ for } N_A > N_D$$
 ## Temperature dependence of the carrier density and Fermi level
 
 It is instructive to consider how $E_F$, $n_e$ and $n_h$ depend on carrier concentrations.
-In this case, we consider an $n-doped$ semiconductor, however, the same logic applies to $p-doped$ semiconductors.
+In this case, we consider an n-doped semiconductor, however, the same logic applies to p-doped semiconductors.
 
 ![](figures/E_F_and_carrier_density.svg)
 
@@ -176,7 +185,7 @@ There are several relevant temperature limits:
 * **Intrinsic limit** . If the temperature is sufficiently large, then $n_i \gg |N_D-N_A|$ and therefore $n_e = n_h = n_i$. Additionally, if holes are heavier than electrons, then $E_F$ has an upturn in this limit.
 * **Extrinsic limit**. If we decrease the temperature, we decrease the number of intrinsic carriers to the point where most of the charge carriers come form the fully ionized donors. As a result, the number of carriers stays approximately constant in this temperature range.
 * **Freeze-out limit**. Once the temperature is sufficiently low $kT \ll E_G - E_D$, we expect the electrons to "freeze away" from the conduction band to the donor band. The charge carriers still come from the donors, however, not all donors are ionized now.
-* Zero temperature. There are no charge carriers in neither conduction nor valance bands. The highest energy electrons are in the donor band and therefore $E_F$ should match the donor band. 
+* **Zero temperature**. There are no charge carriers in neither conduction nor valance bands. The highest energy electrons are in the donor band and therefore $E_F$ should match the donor band. 
 
 !!! check "Exercise"
     check that you can reproduce all the relevant limits in a calculation.