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h1.  How do we count the number of allowed eigenstates using the k-space graphic description? 
h1.  What assumptions are used in the free electron model? 
h1.  What is the physical significance of the Fermi energy and Fermi k vector? 
h1.  Why is the Fermi temperature so high? 
h2.  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_{\epsilon_c}^{\infty}  g_c (\epsilon) \cdot f(\epsilon)d\epsilon</math>

where  <math>g_c (\epsilon) = \sqrt{2(\epsilon - \epsilon_c)} \frac{m_c^{*{3/2}}}{\pi^2 \hbar^3} \propto \sqrt{\epsilon}</math> is the density of states function, and <math>f(\epsilon) = \frac{1}{e^{\frac{\epsilon - \mu}{k_B T}}+1}</math> is the probability of having an electron with a certain energy.

<math>\mu</math> is the chemical potential.

h2. How do we find the carrier density for an intrinsic material?

<math>if \begin{cases} \epsilon_c - \mu >> k_B T  \\ \mu - \epsilon_v >> k_B T \end{cases}
</math>

then <math>f(\epsilon) = \frac{1}{e^{\frac{\epsilon - \mu}{k_B T}}+1}</math> can be simplified.

<math>n_c \simeq \int_{\epsilon_c}^{\infty} d \epsilon \cdot g_c(\epsilon) \cdot e^{\frac{-(\epsilon - \mu)}{k_B T}}
= \int_{\epsilon_c}^{\infty} 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)}{k_B T}
</math>



<math>N_c (T) = \frac{1}{4} \left ( \frac{ 2 m_c^* k_B T }{ \pi \hbar^2  } \right)^{\frac{3}{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)}{k_B T}</math>

<math>P_v (T) = \frac{1}{4} \left ( \frac{ 2 m_h^* k_B T }{ \pi \hbar^2  } \right)^{\frac{3}{2}}</math>


h2. Law of Mass Action

<math>n_c \cdot p_v = N_c \cdot P_v \cdot e^{\frac{\epsilon_v - \epsilon_c}{k_B T}} = N_c \cdot P_v \cdot e^{\frac{-E_g}{k_B T}</math>

Note: Dependent only on temperature and bandgap


h2. 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}{k_B T}</math> 

<math>n_i = \sqrt {N_c \cdot P_v} e^{\frac{-E_g}{2 k_B T}</math>

<math>N_c (T) \cdot e^{\frac{-(\epsilon_c - \mu)}{k_B T} }= \sqrt {N_c \cdot P_v} e^{\frac{-E_g}{2 k_B T}
</math>

<math>n_i</math> for silicon is 1.12 eV (know this number)

h1.  What are the basic steps used to derive the Fermi-Dirac distribution and where di the fermionic properties of the electrons enter in the derivation? 
h1.  What electrons participate in determining the thermodynamic and transport properties of metals?