Carrier Concentration and Fermi Level

The electronic properties of semiconductor depend on the number of free electrons and holes available for current conduction. It should be noted that electron-hole pair generation at room temperature only contributes to a relatively small number of free carriers for current conduction. The electron density in the conduction band can be obtained if the density-of-state function Ne(E) and its distribution function fe(E) are known. The density-of-state function describes the number of states that could be occupied by electrons and is given by

where h is the Planck’s constant (h = 6.626 x 10-34 J ? s), me is the effective mass of electron which is equal to 1.18 x meo for silicon, meo is the electron mass which is equal to 9.11 x 10-31 kg, and Ec is the conduction band energy level. The probability that an energy level E is occupied by an electron is described by the Fermi-Dirac distribution function

where k is the Boltzmann’s constant (k = 1.38 06 x 10-23 J ? K-1), Ef is the Fermi level which is the energy level whereby the probability of finding an electron is 50% at any temperature. This is also known as the electron occupancy probability. For energy levels higher than 3 kT above Ef, the Fermi-Dirac distribution function can be approximated by the Maxwell-Boltzmann distribution function

as the exponential term is very much larger than unity if (E-Ef) is greater than 3 kT. The electron density in the conduction band can be found by integrating the product of the density of states function and the occupancy probability

This is shown pictorially in Fig. 2.7. Performing the integration, the electron density in the conduction band is

where

is known as the effective density of states in the conduction band (in units of cm-3) for silicon. At 300 K, it is 2.86 x 1019 cm-3. By analogy, the density of allowed energy states in the valence band is given by Density of states and the occupancy probability functions on energy state diagram

Fig. 2.7. Density of states and the occupancy probability functions on energy state diagram.

where mh is the effective mass of hole which is equal to 0.5 x meo for silicon. The hole occupancy probability is simply the probability that a level is not occupied by an electron, and is given by

Thus, the density of holes in the valence band can be obtained by integrating the product of the density of states for holes and its occupancy probability

The hole density in the valence band is

where

is known as the effective density of states in the valence band for silicon. At 300 K, it is 1.04 x 1019 cm-3. The product of the electron and hole densities is expressed as

where (Ec-Ev) is the bandgap energy Eg. It should be noted that the bandgap energy is a function of temperature and for silicon it can be expressed by Eg = 1.1785 — 9.025 x 10-5 x T — 3.05 x 10-7 x T2. At room temperature

of 293 K, the bandgap energy is 1.126 eV. Under a constant temperature, the n(T)p(T) product in Eq. (2.12) is a constant in equilibrium and it is independent of the Fermi-level position. In an intrinsic semiconductor, the electron density is exactly equal to the hole density due to electron-hole pair generation. Thus,

This is known as the mass-action law and is valid for both intrinsic (or called pure) and extrinsic (or called doped) semiconductors under thermal equilibrium. Extrinsic semiconductors are those semiconductors that are doped with impurities (called dopants) to increase their free carrier concentrations.

Table 2.2. Effective densities of states and intrinsic carrier concentrations.

Germanium (cm 3)

Silicon (cm 3)

Gallium arsenide (cm 3)

Ne

1.04 x 1019

2.86 x 1019

4.7 x 1017

Ny

6.0 x 1018

1.04 x 1019

7.0 x 1018

ni

2.4 x 1013

1.45 x 1010

9 x 106

Table 2.2 shows the typical values of effective densities of states and the intrinsic carrier concentrations in germanium, silicon, and gallium arsenide.

 
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