Chapter 3 - Carriers in Semiconductor
ECE 4339
Han Q. Le
(copyrighted) U. of Houston
Part 2
0. Physical constants or frequently used formulas
6. Fundamental relations in carrier concentration
6.a Review: Donors and acceptors
Te
donor in GaAs
http://www.mse.berkeley.edu/groups/weber/research/stm.html
http://demonstrations.wolfram.com/DopedSiliconSemiconductors/
Contributed by: S. M. Blinder
6.b Review: Thermal excitation
Discussion of atomic or molecular gas thermal
behavior.
Maxwell-Boltzmann distribution: 
(see demo/discussion of the statistics in class using phet
simulation).
We use 
,
Boltzmann’s constant, in unit of eV/K: 
(eV/K)
http://phet.colorado.edu/en/simulation/gas-properties
http://physics.weber.edu/schroeder/md/
App Demo: Electron-Hole Thermal Excitation
6.1 Fermi-Dirac statistics
- Instead of classical Maxwell-Boltzmann distribution (which is for distinguishable particles or classical particles), electrons and other fundamental particles obey quantum statistics for indistinguishable particles.
- For even-spin particles (S=0,1,2..), they obey Bose-Einstein statistics. For odd-spin particles, (S=1/2, 3/2,...) (per Pauli exclusion principle), they obey Fermi-Dirac statistics.
Here, we deal with electrons, and hence, we use Fermi-Dirac statistics:
which specifies the probability that states of
energy 
are occupied at a given temperature.
App Demo: Fermi-Dirac distribution
The quantity 
is a parameter that is a charactistic of a particular system and
referred to as Fermi energy level or, for short, just Fermi level.
Note the asymptotic approximation of Fermi-Dirac statistics to
Maxwell-Boltzmann for energy far above the Fermi level and much
larger than 
:
for
6.2 Density of states
Concept analogy: How many apartments are available for occupancy at a certain height level?
How many quantum states are available for an electron to occupy at a certain energy level?
Density of state D[E] is a function that determines
the number of quantum states available for electrons to occupy at
energy level E per unit volumn.
It is a function of the band structure. For isotropic parabolic
band, the DoS function is:
where 
is the band edge energy. Usually it is the band gap if we take the
energy origin as the valence band. We'll learn more about this in
Chapter 4.
To express the function above in its natural unit:
where 
and 
We
substitute: 
We can do a similar conversion, except that we use electron rest
mass rather than effective mass. Then:
We see that the unit of density of states is indeed 
For numerical calculation, we can use the
unit 
Below is an example:
Density of states
App Demo: 3D Density-of-state of isotropic parabolic bands
Analogy of different e and h effective mass: tall skinny apartment building vs. wide large apartment building.
Also, the number of units may vary vs. floor level.
6.3 Carrier concentration and Fermi level
Having a building with many floors and apartment
doesn’t mean that all units are occupied. What is the total number
of occupant units for a building? To answer, we must know which
apartment is rented and which one is not.
Suppose someone provides us the following information:
The number of units for the 
floor is 
.
The percentage of occupied units for the 
floor 
Example: the building has 5 floors. The information
is as follow:
What is the formula to calculate the total number
of occupied units?
6.3.1 Carrier distribution and density - electrons in conduction band
We obtain the carrier distribution vs. energy by
taking the product of DoS function and Fermi-Dirac statistics:
For a parabolic band in a 3-D system: 
Hence, carrier concentration is:
Below, you will plot the conduction band density of state, the Fermi distribution, and their product to obtain the electron energy distribution
App Demo: Electron Distribution in Conduction Band
6.3.2 Carrier distribution and density - electrons in valence band
Below, you will plot the valence band density of state, the Fermi distribution, and their product to obtain the electron energy distribution
App Demo: Electron Distribution in Valence Band
6.3.3 Carrier distribution and density - holes (absence of electrons) in valence band
Below, you will plot the valence band density of state, the Fermi distribution, and their product to obtain the hole energy distribution
We see that for the valence band, it is usually
full of electrons. That’s not what we are interested in. We are
interested in the absence of electrons, in other words, holes.
Hence, instead of plotting valence band electron density, we want
to plot the hole density. To do that, we subtract 
as the distribution hole, because the probability of hole=1- prob
of electron:
App Demo: Electron-Void/Hole Distribution in Valence Band
6.3.4 Carrier distribution and density - both electrons and holes in both conduction and valence bands
Below, you will plot both bands density of state, the Fermi distribution, and their products to obtain the electron and hole energy distribution
App Demo: Electron and Hole Distribution at Thermal Equilibrium
6.4 Calculation and approximation for carrier density and Fermi level
In the above calculation, we obtain the carrier
density with the formula:
This a bit inconvenient because we have to do numerical
integration. Fortunately, we can use some approximation of the
integral that makes it easier and with sufficient accuracy.
The approximated relation between carrier concentration (in general,
not just intrinsic) and the Fermi level is given
by:
and 
where 
and 
are two constants, called density-of-state function that are NOT
the DoS functions we have above, but ONLY some values that allow
us to do the approximation. They are specific to the semiconductor
of interest:
Which comes first? carrier concentration or Fermi level?
Conceptually, neither. Each determines the other depending on the case! (If carriers are doped, for example, the concentration determines the Fermi level. If E field is applied to cause band bending, such as in MOSFET, the Fermi level determines the carrier concentration).
Example: Consider the ratio of the two types of carrier density
We can chose the orgin of the energy axis to be right in the middle of the band gap
Now we can solve for Ef
Hence: 
Or: 
where 
is the intrinsic Fermi level as we see below.
What is the Fermi level?
It is an energy level in a semiconductor that defines and is defined by the energy distribution of the carrier population. That's why we are interested in this parameter: knowing Fermi level is the same as knowing how the carrier density. Their relationship is constitutional, not defining, in which one quantity causally determines the other.
7. Intrinsic and doped carrier concentration
7.1 Intrinsic carriers and intrinsic Fermi level
We learn that at a finite temperature, even in undoped semiconductors, there are some carriers thermally excited from the valence band to conduction band:
App Demo: Electron-Hole Thermal Excitation
If all carriers are intrinsic, then:
and 
and: 
Can we find what the intrinsic Fermi level is? Just using the
result above with 
.
But here, we use notation 
,
to indicate that they are intrinsic density, not by doing.
7.1.1 Intrinsic Fermi level
Line by line illustration for the calculation of Fermi level (tutorial in Mathematica):
At T=0:
So, indeed, at absolute 0 temperature, the Fermi level is right at the middle of the bandgap as expect. We can find the intrinsic Fermi level and simplify the results somewhat:
A better-looking formula:
What is the Fermi level?
Where is the intrinsic Fermi level at T=0? Plot the intrinsic Fermi level for GaAs as a function of temperature from 0 to 300 K.
For GaAs: 
;
7.1.2 Intrinsic carrier concentration
Because:
and 
and: 
,
Thus, the intrinsic carriers are given by:
Another useful formula is:
We can find both the intrinsic Fermi level and intrinsic carrier concentration for any semiconductor at any temperature, if we know the semiconductor intrinsic properties.
7.2 Charge neutrality condition and equilibrium condition
What if the semiconductor is doped?
A key condition is charge neutrality:
.
Also: at thermal equilibrium
(and low doping density), the rate of carrier spontaneous
recombination has to be equal to that of generation. Thus:
Hence, these two equations allow solving for carrier densities.
