How to derive an approximate analytical expression for bandgaps
<math>V(\mathbf
) = \sum_{\mathbf{K}}{V_{\mathbf{K}}e^{i \mathbf
\cdot\mathbf
}}</math>
where <math>\mathbf
= m_1 \mathbf
_1 + m_2 \mathbf
_2 + m_3 \mathbf
_3 </math> for any set of integers <math>(m_1, m_2, m_3)</math>.
How is the electronic distribution related to the energy of a solution?
The wavefunction <math>\Psi
</math> piles up electronic charge on the cores of positive ions, thereby lowering the potential energy in comparison with the average potential energy seen by a traveling wave. The wavefunction <math>\Psi
</math> piles up charge in a region between the ions, thereby raising the potential energy in comparison with that seen by a traveling wave.
Also, one can think of the electronic distribution in this way: increasing atomic size means increasing number of shells and subshells, increasing the electronic screening, resulting in a decreased potential felt by the outermost electrons.
How does the bandgap scale with the potential parameters?
The bandgap increases in magnitude with larger potential
Bandgap trends in the periodic table
Smaller bandgaps correspond to larger atoms. The energy gaps span <math>0.08 - 5.4 eV</math>. The wider the gap, the heavier the electron
How are bandgaps measured
Band gap is measured by illuminating a sample with light and measuring absorption. The wavelength corresponding to the first point that light is absorbed corresponds to the energy of the band gap.
How is the group velocity calculated from the Bloch wave form and from the band diagram?
Is there an average momentum associated with the Bloch wave? Momentum measurements would be any of the <math>\hbar (k-G)</math> momentum values (momentum eigenvalues), and these would correspond to probabilities of <math>|C_
|^2</math>. Assign an average value to the momentum of the Bloch state:
<center>
<br>
<math>\overline
_
=<u_
|\hat P|u_
></math>
<br>
<math>\overline
_
=\int_\infty^\infty u_
^* \frac
\frac
u_
dx</math>
<br>
<math>\overline
_
=...+|C_
|^2 \hbar (k-2g) + |C_k|^2 \hbar k + |C_{k+g|^2 \hbar {k+g)+...</math>
<br>
</center>
The average momentum per eigenfunction can also be calculated from the band diagram, and it is below.
<center>
<br>
<math>v = \frac{\overline
_{n,k}}
</math>
<br>
<math> \vec v_g = \frac
\vec \nabla_k \epsilon (\vec k) </math>
<br>
</center>
How does the group velocity change from mid-band to band edge?
At mid-band (i.e., k is between zero and pi/a), the group velocity (the slope of the E-k curve) is nonzero. But as one approaches the band edge (k = pi/a), the slope approaches zero and the group velocity approaches zero, implying that solutions with this k-number are standing waves.
How is the effective mass found from the band diagram?
<math>\epsilon_c = E_g + \frac
</math>
<math>\epsilon_v = \frac{-\hbar^2 k^2}
</math>
Modeling as parabolas
The effective mass is the inverse of the bandgap curvature (upward-facing parabolas have positive curvature, like a smile; downward-facing parabolas have negative curvature, like a frown). Furthermore, the magnitude of the effective mass is directly proportional to bandgap width.
How to derive an approximate expression for the effective mass of the electrons and holes
What periodic boundary conditions are, and how they loead to discretization of the Bloch wave vector
How many states exist per band?
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)