How do we count the number of allowed eigenstates using the k-space graphic description?
What assumptions are used in the free electron model?
What is the physical significance of the Fermi energy and Fermi k vector?
Why is the Fermi temperature so high?
How do we find the density of electrons in the conduction band?
The density of electrons <math>n_c</math> in the conduction band is given by
<math>n_c=\int_
^
g_c (\epsilon) \cdot f(\epsilon)d\epsilon</math>
where <math>g_c (\epsilon) = \sqrt
\frac{m_c^{*
}}
\propto \sqrt
</math> is the density of states function, and <math>f(\epsilon) = \frac
{e^{\frac
{k_B T}}+1}</math> is the probability of having an electron with a certain energy.
<math>\mu</math> is the chemical potential.
How do we find the carrier density for an intrinsic material?
<math>if \begin
\epsilon_c - \mu >> k_B T
\mu - \epsilon_v >> k_B T \end
</math>
then <math>f(\epsilon) = \frac
{e^{\frac
{k_B T}}+1}</math> can be simplified.
<math>n_c \simeq \int_
^
d \epsilon \cdot g_c(\epsilon) \cdot e^{\frac{-(\epsilon - \mu)}{k_B T}}
= \int_
^
d \epsilon \cdot g_c(\epsilon) \cdot e^{\frac{(\epsilon - \epsilon_c)}{k_B T}} \cdot e^{\frac{(\epsilon_c - \mu)}{k_B T}}
= N_c (T) \cdot e^{\frac{-(\epsilon_c - \mu)}
</math>
<math>N_c (T) = \frac
\left ( \frac
\right)^{\frac
{2}}</math> from the integral
We can do a similar derivation with the valence band:
<math>p_v (T) \simeq P_v (T) \cdot e^{\frac{-(\mu - \epsilon_v)}
</math>
<math>P_v (T) = \frac
\left ( \frac
\right)^{\frac
{2}}</math>
Law of Mass Action
<math>n_c \cdot p_v = N_c \cdot P_v \cdot e^{\frac
{k_B T}} = N_c \cdot P_v \cdot e^{\frac{-E_g}
</math>
Note: Dependent only on temperature and bandgap
For an Intrinsic (undoped) Semiconductor
<math>n_c = p_v = n_i
</math>
<math>n_i^2 = N_c \cdot P_v \cdot e^{\frac{-E_g}
</math>
<math>n_i = \sqrt
e^{\frac{-E_g}
</math>
<math>N_c (T) \cdot e^{\frac{-(\epsilon_c - \mu)}
}= \sqrt
e^{\frac{-E_g}
</math>
<math>n_i</math> for silicon is 1.12 eV (know this number)