variation of fermi level with carrier concentration

Band Structure and Effective Mass

The basic description of a semiconductor is its bandstructure, i.e. the variation of energy E with wave-vector k . The most important bands are:

Valence band - the last filled energy level at T=0 K

Conduction band - the First unfilled energy level at T=0 K

The valence band maximum is at k =0, is known as the gamma point . Where the conduction-band minimum also occurs at k =0, the semiconductor is said to be a direct band semiconductor. At non-zero k =0, the semiconductor is an indirect-band semiconductor . In addition to these two main conduction bands other bands may also be present. In III-V semiconductors, Ge and Si there are 3 valence bands with maxima at k =0. These are the light-hole , heavy-hole and spin-orbit split-off band .

The bands in a semiconductor material are approximated by parabolic functions of k close to the bandedges.

Conduction-band:

(2)

Valence-band:

(3)

The expression for the effective mass is found from the dynamics of a wave-packet, which represents a localised particle. The wave packet is a modulation envelope, with a carrier-wave running through it. The packet is made up of a small spread of frequencies w around a central value w 0 ; these are superimposed on each other. The wave packet moves at the group velocity v g

(4)

If an electric field E f is applied, so that the wave packet moves a distance dx in time dt. The change in energy of the wave packet is

(5)

 

This change corresponds to a change dk of the central k value k 0 ; it is given by

(6)

We convert this to a time derivative:

(7)

But

, so

(8)

The equation for the acceleration, can be calculated from

(9)

(10)

Substituting for from our first major result,

(11)

Comparing these forms, we see,

(12)

The dynamics of the holes is more complicated. It is necessary to consider one unfilled state in the otherwise filled valence band. The result is that the hole mass acts like a particle with positive charge + e and mass m h given by

(13)

The Fermi Level and Intrinsic Semiconductors

Electrons are Fermions , and thus follow Fermi-Dirac distribution function

(14)

where μ is the Fermi Energy often denoted E f or chemical potential in semiconductor physics is the energy at which there would be a fifty percent chance of finding an electron, if all energy levels were allowed. In order to apply the statistics, we need the density of states in the conduction and valence bands. These are derived from the basic principle that the density of states is constant in k-space. In the conduction band the density of states is given by:

(15)

 

and the valence band,

(16)

 

where E is measured from the top of the valence band.

(17)

The density of electrons in the conduction band is

(18)

 

In the valence band, the probability of a hole is

(19)

and can be approximated

(20)

A similar calculation yields the hole density

(21)

Calculation of the Fermi level given the carrier concentration is useful in the calculation of laser gain, but since the function is not invertable, there is no analytical method for achieving this. However numerous approximations have been forumulated to calculate the Fermi level.

The value of μ depends on N a and N d . However μ can be eliminated between (18) and (21) to give the important relation

(22)

 

where N c and N v are the prefactors in (18) and (21) .

(23)

 

As stated, (22) holds for all T and independent of the values of N a and N d . In the intrinsic region, the extrinsic density is negligible, and then n=p since each electron excited to the conduction band leaves a hole behind it. In the intrinsic region, therefore

(24)

 

If we substitute into (24) the values of n and p from (18) and (21)

(25)

This gives the value of μ in the intrinsic region, simple manipulation leads to

(26)

 

That is, μ is displaced from the middle of the band gap by a temperature dependent term that depends on the ratio of the effective masses.

The Fermi Level and Extrinsic Semiconductors

What happens to μ with temperature when donors and acceptors are present? The charge neutrality condition governs the numbers of carriers.

(27)

 

where N a - and N d + are the number of ionised acceptor and donor sites. As the sketch shows, the probability of finding an electron on a donor site. The number of sites that are ionised is:

(28)

A similar argument shows that

(29)

The four terms in (27) are given in terms of μ (18) , (29) , (21) and (28) respectively, so μ can in fact be determined from (19). The general case has to be dealt with numerically;

We take the case of n-type doping but with some counter-doping:

and

(30)

at T=0, N a electrons move off donor sites to occupy the acceptor sites. Thus

(31)

The donor sites are partially occupied. This is only possible at T=0 if the Fermi-level is at the donor-site energy:

(32)

This will not change for very low temperatures, , so substitution of the value of μ into (18) gives

for

(33)

It is seen that (29) is definitely a low-temperature result. For p-type doping, the result corresponding to (33) is

for

(34)

The important technical region in the n-type material is the temperature range in which all the donors are ionised and the extrinsic electron density is higher than the intrinsic density. Full ionisation means:

(35)

Since N a electrons are required for occupation of the acceptor sites. Comparison of (24) and (7) gives

(36)

The corresponding results for p-type doping are

(37)

(38)

Note that in this technical region if the counter doping is negligible, or , (35) and (37) simplify to

(39)

(40)

which is what we tell the engineers. courtesy;britneyspears.ac/physics/basics/basics.htm


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Thu, 05/28/2009 - 15:03

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