Charge Carrier Statistics

Density of States

The density of states is given in general by the equation:

The term g(E) is the number of states with E between E and E + dE per unit volume (crystal volume) per dE:

Applying this to the conduction and valence band in general gives:

��

where �and �depend on the semiconductor:

GaAs:��

Si:��

Ge:��

where �and �are the longitudinal and tranverse effective mass respectively (see Pierret page 79) and �and �are the light and heavy hole effective mass respectively.

Fermi Function

�The Fermi function f(E) is a probability distribution function that tells one the ratio of filled to total allowed states at a given energy E� (Pierret pg. 96) and is given by:

where K is Boltmann�s constant, T is the temperature of the system, and E_F is termed the Fermi Energy that will be investigated in more detail later.

The graph of f(E) at several temperatures is shown below:

Donors and Acceptors

Consider a silicon crystal and a phosphorus atom as a substitutional impurity.� The valency five phosphorus atom will form four bonds with the surrounding Si atoms but will have one valence electron left over.� This valence electron is weakly bound to the phosphorus atom and can easily escape to the conduction band of Si to aid in conduction.� This phophorus atom is therefore called a donor since it donates one electron to the conduction band.�

Likewise, consider a boron atom as a substitutional impurity in a Si crystal.� The valency three boron atom will form four bonds with the surrounding Si atoms but needs to acquire an additional electron to reach a valency of eight.� The boron atom can easily acquire this electron from a neighboring Si atom producing a hole in the valence band that can then travel throughout the crystal and contribute to conduction.� The boron �accepts� an electron and hence produces a hole in the Si and is therefore called an acceptor.

This donor and acceptor concept is schematically represented by the following figure (Pierret Fig. 4.13):

The term intrinsic is used to denote values for pure or undoped semiconductors.� A intrinsic Fermi energy level E_i will be defined in later sections that represents the Fermi energy level of undoped semiconductors.� Also, intrinsic electron and hole concentrations will be defined as n_i and p_i well.

Equilibrium Concentration Relationships

The following definitions will be used:

n � Electron concentration.� Total number of electrons per cm³ in the conduction band.

p � Hole concentration.� Total number of holes per cm³ in the valence band.

n_i � Intrinsic carrier concentration.� Both electron and hole concentrations in an intrinsic semiconductor.

�- Number of ionized donors per cm³

�- Total donor concentration

�- Number of ionized acceptors per cm³

�- Total acceptor concentration

The conduction band electron concentration and the valence band hole concentration are given by:

For nondegenerate semiconductors (where the fermi energy in more than 3KT away from both the valence and conduction band), the equations for n and p simplify to:

where N_C and N_V are the effective density of states in the conduction and valence bands respectively and are:

�When the semiconductor is undoped, n=p=n_i and E_F= E_i which leads to the following equations for n and p:

Intrinsic carrier concentration n_i, Intrinsic Fermi energy level and the np product

If the semiconductor is not doped, then the concentration of electrons in the conduction band n is equal to the concentration of holes in the valence band p, hence n=p=n_i where n_i is defined as the intrinsic carrier concentration. Using the equations above, it is easy to show that the product np is:

where E_gap=E_C-E_V. Thus we have an equation for the intrinsic carrier concentration:

Besides the product np, we can equate n and p to obtain the intrinsic Fermi energy level:

Carrier concentration as a function of temperature and determination of E_F

The Fermi energy level is a function of the carrier concentration, thus the carrier concentration of the semiconductor will be analyzed as a function of temperature.

The ratio of ionized donor atoms to total donors and the ratio of ionized acceptor atoms to total acceptor atoms is given by:

��

where E_D is the donor energy level, E_A is the acceptor energy level, g_D and g_A are the degeneracy factors.

Now armed with all of these equations describing carrier concentrations as a function of multiple variables, it is now valuable to analyze the carrier concentration and Fermi energy level as a function of temperature.

The plot of the carrier concentration versus temperature is shown below:

�

There are three regions of the above figure:� Freeze out, Extrinsic and Intrinsic regions.� The carrier concentration and Fermi energy level will be derived for each region.

Freeze out and low tempertature region

Consider a donor doped semiconductor doped with a concentration of N_D donors.� For very low temperatures close to zero Kelvin, all the donor atoms retain their electrons which do not have enough energy to be excited to the conduction band.� All the states in the valence band are occupied by electrons which do not have enough energy to be excited across the relatively large bandgap (relative to the gap required by the donor atom�s electrons).� At slightly higher but still low temperatures, the donor atom�s electrons will start to be excited to the conduction band according to the equation for �.� Since all the valence band states are happily occupied by electrons, the conduction band carrier concentration n is given by:

Using the relation , we can express the above equation as:

where .� The carrier concentration n is then easily solve for:

The Fermi energy level is now easily solved for using �to get:

Extrinsic Region

For moderate temperatures, practically all the donor and acceptor atoms will be ionized.� Writing the charge neutrality equation:

and using the equation , we have:

when the semiconductor is donor-doped such that >>n_i, then we have:

and the Fermi energy level is easily shown to be:

when the semiconductor is acceptor-doped such that >>n_i, then we have:

and the Fermi energy level is easily shown to be:

Intrinsic Region

For high temperatures, all the donor and acceptor atoms will be ionized yet the intrinsic carrier concentration grows to the point where it is the primary source of conduction and valence band charge carriers.� Hence, we have: