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Mobile Carriers and the Fermi Function

Module by: Chris Winstead. E-mail the author

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Summary: The Fermi-Dirac distribution, also called the "Fermi function," is a fundamental equation expressing the behavior of mobile charges in solid materials. This module explains the Fermi function and the Fermi energy level, and shows how they relate to the density of mobile carriers in solid-state semiconducting materials.

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In solid materials, the behavior of charges depends on their energy. To understand the electronic characteristics of a material, we need to know how charges are distributed among the available energy levels. This distribution is described by two components:

  • The Fermi function gives the probability that an electron acquires a given energy, as a function of temperature.
  • The Density of States gives the distribution of energy levels that electrons are allowed to have.

The Fermi function is completely general, and applies to any solid material in thermal equilibrium. The Density of States is specific to a particular type of material. By combining these functions, we can determine the total density of charges that are available for electrical activity at a given temperature for a particular piece of material.

The Fermi function.

The Fermi function , FEFE , gives the probability that a state SS at energy EE is occupied by an electron, given that EE is an allowed energy level. The Fermi function has units of electrons per state. The Fermi energy level , E F E F , is the energy at which the probability of occupancy is exactly 1/21/2 for temperatures greater than zero. The Fermi function is given by

FE=1 e E-E F /kT +1,FE=1 e E-E F /kT +1, (1)
where kk is the Boltzmann constant and TT is the temperature in Kelvin.

Throughout nature, particles seek to occupy the lowest energy state possible. Therefore electrons in a solid will tend to fill the lowest energy states first. Electrons fill up the available states like water filling a bucket, from the bottom up. At T=0T=0 , every low-energy state is occupied, right up to the Fermi level, but no states are filled at energies greater than E F E F .

Figure 1: Illustration of the Fermi function for zero temperature. All electrons are stacked neatly below the Fermi level.
Figure 1 (Ef-zero-temp.png)

For T>0T>0 , some electrons can be excited into higher-energy states. This is similar to a bucket of hot water. Most of the water molecules stick around the bottom of the bucket. The Fermi level is like the water line. A fraction of water molecules are excited and drift above the water line as vapor , just as electrons can sometimes drift above the Fermi level.

Figure 2: Illustration of the Fermi function for temperatures above zero. Some electrons drift above the Fermi level. Their density at higher energies is proportional to the Fermi function.
Figure 2 (Ef-positive-temp.png)

Recall that an energy level may contain several sublevels, all with the same energy. Each sublevel is called a "state," and can be occupied by exactly one electron. Suppose there are NN allowed states at energy EE . Then the probability of finding an occupied state at energy EE is N×FEN×FE . As discussed in the next subsection, in continuous-band theory we represent NN as a density of states. The density of states reveals how the allowed sublevels are spread out across energy bands in a specific material.

Density of states.

In a semiconductor, not every energy level is allowed. For example, there are no allowed states within the forbidden gap, as illustrat ed in Figure 3. To make use of the Fermi function, we need another function that has units of states per energy level per volume. In a solid with numerous atoms, a large number of states appear at energy levels very close to each other. We approximate these states as a continuous "band" and imagine that an "energy level" is a vanishingly small energy interval of width dEdE .

Figure 3: The density of electrons in a semiconductor, showing how the Fermi function is modulated by the density of allowed states (which is zero inside the forbidden gap).
Figure 3 (Fermi-semi-electrons.png)

The density of states , NENE , is the fraction of all allowed states that lie within EE and E+dEE+dE . This is a density function, meaning ∫ 0 ∞ NE=1∫ 0 ∞ NE=1 . The Fermi function tells us the probability that a state is occupied. The density of states complements the Fermi function by telling us how many states actually exist in a particular material. We can multiply NENE and FEFE together, resulting in units of electrons per energy level per unit volume:

e - s t a t e × s t a t e s l e v e l × v o l u m e =e - l e v e l × v o l u m e .e - s t a t e × s t a t e s l e v e l × v o l u m e =e - l e v e l × v o l u m e . (2)

By integrating over all energy levels, we obtain the total number of electrons. By integrating over the conduction band only , we obtain the total number of mobile electrons (i.e. electrons that can participate in electric current). Let nn be the total concentration (per volume) of mobile carriers in the conduction band. Then nn is given by

n=∫ E C E 0 FENEdE,n=∫ E C E 0 FENEdE, (3)
where E 0 E 0 is the top of the conduction band, called the vacuum level. An electron with energy greater than E 0 E 0 is no longer confined to the solid, and can fly off into space. Under normal circumstances, E 0 →∞E 0 →∞ is a sufficient approximation when integrating carrier densities.

In the conduction band, the density of available states has the form

NE=2 πN C E-E C kT 3/2 ,NE=2 πN C E-E C kT 3/2 , (4)
where
N C =η2πm n kT/h 2 . 3/2 N C =η2πm n kT/h 2 . 3/2 (5)
For Si, η=12η=12 and for GaAs, η=2η=2 .

Electrons vs holes.

In a solid semiconductor at thermal equilibrium, every mobile electron leaves behind a hole in the valence band. Since holes are also mobile, we need to account for the density of "hole states" that remain in the valence band. Because a hole is an unoccupied state, the probability of a mobile hole existing at energy EE is 1-FE1-FE , as indicated in Figure 4.

Figure 4: The density of mobile electrons is shown in the conduction band. The corresponding density of mobile holes is shown in the valence band.
Figure 4 (Fermi-electrons-holes.png)

Because holes can move, we should count them when considering the total concentration of mobile charges. Let pp be the total concentration of mobile charges in the valence band. Then

p=∫ 0 E V 1-FENEdE.p=∫ 0 E V 1-FENEdE. (6)

In the conduction band, the density of states for holes is

NE=2 πN V E V -E kT 3/2 ,NE=2 πN V E V -E kT 3/2 , (7)
where
N V =22πm p kT/h 2 . 3/2 N V =22πm p kT/h 2 . 3/2 (8)

Mobile carrier density.

Combining the results for nn and pp , we can determine the total density of mobile carriers in a semiconductor at equilibrium. In the conduction band, where E-E F >3kTE-E F >3kT , we use the approximation

FE≈e -E-E F /kT .FE≈e -E-E F /kT . (9)

With a change of variables x=E-E C /kTx=E-E C /kT , the integral in Equation 3 takes the form

n=2 πN C exp-E C -E F /kT∫ 0 ∞ x 1/2 e -x dx.n=2 πN C exp-E C -E F /kT∫ 0 ∞ x 1/2 e -x dx. (10)
This integral is in standard form and can be found on many integral tables. The integral evaluates to π/2π/2 , yielding
n=N C exp-E C -E F /kT.n=N C exp-E C -E F /kT. (11)

For holes, we use the valence-band approximation

1-FE≈e -E F -E/kT .1-FE≈e -E F -E/kT . (12)
Integrating Equation 6 with Equation 7 and Equation 12 yields
p=N V exp-E F -E V /kT.p=N V exp-E F -E V /kT. (13)

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