What is the partition function, and what is its significance?
The partition function
<math>Z = e^{-\beta F}</math> links the micro (Z) with the macro (F). We get the thermodynamic quantity<math> F = - \frac
\ln Z</math> from just looking at the microstates of our system.
From Recitation
<math>P_S=\frac{e^{\frac{-E_S}
}}{ \sum_S e^{\frac{-E_S}{k_B T}} }
</math>
<math>\sum_S e^{\frac{-E_S}{k_B T}}</math> is the partition function that normalizes the probability <math>P_S</math> such that <math>\sum_S P_S = 1</math>
<math>\frac
{P_{S'}} = \frac{N e^{\frac{-E_S}
}}{N e^{\frac{-E_S'}
}} = e^{\frac{-(E_S-E_S')}{k_B T}}
</math>
<math>\sum_S P_S = \frac
\sum_S e^{\frac{-E_S}{k_B T}} = \frac
Z = 1 </math>
What is the physical insight at the basis of the parabolic band edge approximation?
Near the band edges (extremum) the dispersion curve follows the expression below. The term <math>E_g</math> is the band gap and the effective masses are <math>m^*</math>, which is given by the curvature of the conduction and valence band). Model as free electrons with different mass (effective mass).
<center>
<br>
<math>\epsilon_c ( \vec k ) = E_g + \frac
</math>
<br>
<math>\epsilon_v ( \vec k ) = - \frac
</math>
<br>
</center>
What happends near the BZ edge?
The edge of the BZ is defined by <math>k= \pm \frac
</math>
The master equation is <math>\left ( \frac
-E \right ) C(k) + \sum_G V(G)C(k-G)=0 </math>
What is the meaning of holes?
An empty state in a band filled with electrons is called a hole. This vacant state behaves in many waves as if it were a charge carrier of positive sign <math>+e</math> and equal mass to that of the missing electron.
What role does the chemical potential play in determining the properties of a semiconductor?
The number of charge carriers per unit volume at a given temperature is the most important property of any semiconductor. The values of these are highly dependent on the number of impurities. In the equations below, <math>n_c</math> is the number of electrons in the conduction band and <math>p_v</math> is the number of holes in the valence band. Impurities affect the values of <math>n_c</math> and <math>p_v</math> through their effect on the chemical potential.
<center>
<br>
<math>n_c(T) = \int_
^\infty d\epsilon g_c (\epsilon) \left ( \frac
{e^{\frac
{k_B T}}+1} \right )</math>
<br>
<math>p_v(T) = \int_{-\infty}^
d\epsilon g_v (\epsilon) \left ( 1 - \frac
{e^{\frac
{k_B T}}+1} \right )</math>
<br>
<math>p_v(T) = \int_{-\infty}^
d\epsilon g_v (\epsilon) \left (\frac
{e^{\frac
{k_B T}}+1} \right )</math>
<br>
</center>
Control the chemical potential control the carrier type. The chemical potential <math>\mu</math> is a "handle" (to be pulled up or down) for tuning the type and density of charge carriers.
How does the position of the chemical potential relative to the band edge affect the type and density of charge carriers?
To calculate the position of the chemical potential in extrinsic semiconductors, we use charge neutrality:
<p>
</p>
To figure out how many states are populated, multiply the Fermi function <math>f(\epsilon)</math> with the density of states <math>g(\epsilon)</math>. The product is the density of occupied states.
<p>
</p>
Based on the Fermi function, when <math>\frac
\gg 1</math> the number of electrons promoted to the conduction band is small and will occupy the lowest energy levels in that band.
<p>
</p>
Consider the following question. What is the electron and heavy hole density as a function of position of the quasi-Fermi level? Compare the exact calculation using the Fermi-Dirac distribution and the Boltzmann approximation.
<p>
</p>
The following analysis describes the electrons in the conduction band. It is also valid for the holes in the valence band replacing the electron effective mass with the hole effective mass and changing the sign of the energy difference.
<p>
</p>
In the parabolic approximation for the band edges of a bulk semiconductor, the density of states as a function of the energy is written as below.
<center>
<br>
<math>\rho_c (E) = \frac
\left ( \frac
\right ){3/2) (E - E_c)
</math>
<br>
</center>
The term <math>m_c^*</math> is the effective mass related to the parabolicity of the band-edge, and <math>E_c</math> is the energy of the bottom of the conduction band.
<p>
</p>
Electrons and holes inside the material (both fermions) obey to the Fermi-Dirac statistics representing the occupation of the particles over the energy levels of the system depending on the Fermi energy <math>E_F</math>.
<center>
<br>
<math>f(E) = \frac
{1+ e^{\frac
{kT}}</math>
<br>
</center>
For an intrinsic semiconductor at thermal equilibrium the Fermi energy lies in the middle of the band-gap and is the same for electron and holes. When a non-equilibrium situation is considered (for example under optical or electrical pumping, with carrier densities strongly exceeding the thermal values), the electron and hole distribution can be described separately through two different quasi-Fermi levels if intraband relaxation processes are much faster than interband recombination, so that quasi-thermal equilibrium within the bands can be assumed.
<p>
</p>
The product of the quasi-Fermi distribution and the density of states in the conduction band will represent the carrier distribution over the given energy levels. If we integrate this distribution over all the available energy levels we will obtain the density of electrons in the band.
<center>
<br>
<math>n = \int_
^
\rho_c (E) f(E) dE</math>
<br>
<math>n = N_c F_
\left (\frac
\right )</math>
<br>
<math>N_c = \frac
\left ( \frac
\right )^
</math>
<br>
<math>F_
(u) = \frac
{\pi^{1/2}} \int_0^
\frac{x^{1/2}}{1+e^{(x-u)}}dx</math>
<br>
</center>
If the number of particles in the system is known, one can extract from the expression of <math>n</math> the value of the quasi-Fermi energy at a given temperature. The integral in the previous relation cannot be calculated analytically and requires a numerical calculation.
<p>
</p>
If the condition <math>E - E_F \gg kT</math> is valid, the Fermi distribution can be replaced by the Boltzmann distribution and there is an analytical solution of the density of particles.
<center>
<br>
<math>n = N_c e^{- \frac
</math>
<br>
<math>E_F^c = E_c - kT \ln \left ( \frac
\right )</math>
<br>
</center>
How is the chemical potential engineered?
By adding impurities.
<math>\mu_i = \epsilon_v + \frac
E_g + \frac
k_B T \ln{\frac
{m_c}}</math>
<math>\sigma = n_e e \frac
+ n_h e \frac
</math>
Electron mobility <math>\mu_e = \frac
</math>
Hole mobility <math>\mu_n = \frac
</math>
Impurities at the ppm level drastically change the conductivity (5-6 orders of magnitude).
Note that impurity states are localized near the impurity atom (whereas the Bloch states we looked at before were delocalized), and decay exponentially as we move away from the location.
Adding impurities to determine carrier type:
- <math>n_i^2</math>for Si: <math>\sim 10^6 cm^{-3}</math>
- Add <math>10^
Unknown macro: {16}</math> dopant atoms per <math>cm^{-3}</math>
cm^{-3} </math> (~1ppm) phosphorus (donors) to Si: <math>n_c \sim N_d</math>
- <math>n_c \sim 10^16 cm^{-3}</math>, <math>p_v ~ 10^4 (n_i^2/N_d)</math>
Adding impurities to change carreir density:
- 1 part in <math>10^6</math> impurity in a crystal <math> (\sim 10^
Unknown macro: {22}/ 10^6</math> = <math>10^
cm^{-3}</math> atom density)
- <math>10^
- Conductivity is proportional to the number of carriers leading to 6 orders of magnitude change conductivity!
What is the Fermi Energy?
In physics and Fermi-Dirac statistics, the Fermi energy of a system of non-interacting fermions is the smallest possible increase in the ground state energy when exactly one particle is added to the system. It is equivalent to the chemical potential of the system in its ground state at absolute zero. It can also be interpreted as the maximum energy of an individual fermion in this ground state. The Fermi energy is one of the central concepts of condensed matter physics.
<p>
</p>
The position of the chemical potential is obtained from the expressions of the charge carrier density. In intrinsic semiconductors the number of electrons in the conduction band is equal to the number of holes in the valence band.
<center>
<br>
<math>n_c(T) = N_c (T) e^{-(\epsilon_c - \mu )/k_B T}</math>
<br>
<math>n_c(T) = e^{-(\mu - \epsilon_v)/k_B T}</math>
<br>
<math>n_c(T) = p_v(T)</math>
<br>
<math>n_c(T) = n_i(T)</math>
<br>
<math>n_i(T) = \sqrt
e^{\frac
</math>
<br>
</center>
There is then an expression of the chemical potential.
<center>
<br>
<math>\mu = \epsilon_v + \frac
E_g + \frac
k_B T \ln \left ( \frac
\right )</math>
<br>
</center>
The position of the chemical potential is close to the gap center in an intrinsic semiconductor. Note that the only assumption we used was that of non-degeneracy.
What are degenerate and non-degenerate semiconductors?
In non-denerate semiconductors, the chemical potential satisfies the following relations:
<center>
<br>
<math>\epsilon_c - \mu \gg k_B T</math>
<br>
<math>\mu - \epsilon_v \gg k_B T</math>
<br>
</center>
The following approximation can be used under the above conditions. Note that if the electron energy is far from the Fermi energy, the FD distribution approximates a MB distribution, which means the electrons behave like a classical gas.
<center>
<br>
<math>\epsilon > \epsilon_c</math>
<br>
<math>e^{\frac{-(\epsilon - \mu)}
\approx \frac
{e^{\frac{-(\epsilon - \mu)}
+ 1}</math>
<br>
<math>\epsilon < \epsilon_v</math>
<br>
<math>e^{\frac{-(\mu - \epsilon)}
\approx \frac
{e^{\frac{-(\mu - \epsilon)}
+ 1}</math>
</center>
The expressions derived for the carrier density are approximated in relations below.
<center>
<br>
<math>n_c \approx \int_
^
d \epsilon \cdot g_c(\epsilon) \cdot e^{\frac{(\epsilon - \epsilon_c)}{k_B T}} e^{\frac{-(\epsilon_c - \mu)}{k_B T}} </math>
<br>
<math>p_v \approx \int_{-\infty}^
d \epsilon \cdot g_v(\epsilon) \cdot e^{\frac{(\epsilon_v - \epsilon)}{k_B T}} e^{\frac
{k_B T}} </math>
<br>
</center>
What is the physical significance of the exponential-like behavior of the Fermi-function in non-degenerate semiconductors?
The Fermi function shows that for <math>\frac
\gg 1</math> the number of electrons promoted to the conduction band is small and occupies the lowest energy levels in the band.
<center>
<math>n_c (T) = \int_
^
d \epsilon g_c ( \epsilon ) e^{- (\epsilon - \epsilon_c ) / k_B T} e^
</math>
<br>
<math>p_v (T) = \int_
^
d \epsilon g_v ( \epsilon ) e^{- (\epsilon_v - \epsilon ) / k_B T} e^
</math>
<br>
</center>
Because of the rapidly decaying function in the integrand only energies that within <math>k_B T</math> of the band edge contribute significantly. Assume a quadratic density of states of the form below.
<center>
<br>
<math>N_c(T) = \frac
\left ( \frac
\right )^
</math>
<br>
<math>P_v(T) = \frac
\left ( \frac
\right )^
</math>
<br>
</center>
It is not possible to calculate carrier density without specific knowledge of the chemical potential. The product of the hole and electron carrier densities does not depend on the chemical potential.
What is the density of filled states function and how does it effect the type of semiconductor?
<center>
<math>n_c (T) = \int_
^
d \epsilon g_c ( \epsilon ) e^{- (\epsilon - \epsilon_c ) / k_B T} e^
</math>
<br>
<math>p_v (T) = \int_
^
d \epsilon g_v ( \epsilon ) e^{- (\epsilon_v - \epsilon ) / k_B T} e^
</math>
<br>
</center>
Because of the rapidly decaying function in the integrand only energies that within <math>k_B T</math> of the band edge contribute significantly. Assume a quadratic density of states of the form below.
<center>
<br>
<math>N_c(T) = \frac
\left ( \frac
\right )^
</math>
<br>
<math>P_v(T) = \frac
\left ( \frac
\right )^
</math>
<br>
</center>
It is not possible to calculate carrier density without specific knowledge of the chemical potential. The product of the hole and electron carrier densities does not depend on the chemical potential.
What is the Law of Mass Action? How is it calculated
<center>
<br>
<math>n_c p_v = N_c P_v e^
= N_c P_v e^{-E_g/k_B T}</math>
<br>
</center>
At a given temperature, the density of one carrier type can be calculated from knowledge of the other. One of the results of the law of mass action is that the product <math>np</math> is constant. By adding a large number of carriers of one type, the carrier concentration of the other type is caused to decline.
<p>
</p>
A crystal with negligible contribution of the impurities to the charge density is classified as an intrinsic semiconductor.
<center>
<br>
<math>n_x (T) = p_v (T) = n_i (T)</math>
<br>
<math>n_i = \sqrt
</math>
<br>
<math>n_i = \sqrt
e^{-E_g / 2k_BT}</math>
<br>
<math>n_c (T) = N_c (T) e^{-(\epsilon_c - \mu)/k_B T}</math>
<br>
<math>p_v (T) = P_v (T) e^{-(\mu - \epsilon_v)/k_B T}</math>
<br>
</center>
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
It is valid for both intrinsic and extrinsic semiconductors.
How many charge carriers does a SC have at temperature T?
<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>
Calculating the position of the chemical potential in extrinsic semiconductors
<center>
Unable to render embedded object: File (Fermi_level_in_silicon_versus_T_and_doping_conc..PNG) not found.
<math>p_v - n_c + N_d^+ - N_a^- = 0</math>
<br>
<math>P_v e^
- N_c e^
+ \frac
{1 + 2 e^{(\mu - \epsilon_d)/k_B T}} - N_c e^
+ \frac
{1 + 2 e^{(\mu - \epsilon_d)/k_B T}}</math>
<br>
</center>
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)
The number of electrons in the conduction band <math>n_c = \int_
^
f(E) \cdot g(E) dE</math>
Impurity states modelled as modified hydrogen atoms
- Consider the weakly bound 5th electron in phosphorous as a modified hydrogen atom
- For hydrogenic donors or acceptors, think of the electron or hole, respectively, as an orbiting electron around a net fixed charge
- Estimate the energy to free the carrier into the conduction band or valence band by using a modified expression of the energy of an electron in the hydrogen atom.
<center>
<br>
<math>E_n = \frac
</math>
<br>
<math>E_n = - \frac
eV</math>
<br>
<math>E_n = \frac
\frac
{\epsilon_r^2</math>
<br>
<math>E_n = - \frac
\frac
\frac
</math>
<br>
</center>
- In the ground state <math>n=1</math>, <math>\epsilon</math> is on the order of <math>10</math>. The binding energy of the carrier to the impurity atom is <math><0.1 eV</math>.
- Expect that many carriers are ionized at room temperature.
References
*Fermi Energy http://en.wikipedia.org/wiki/Fermi_energy