# 2.5 Carrier density and the fermi energy

In this Section

1. Introduction
2. 3-D Density of states
3. Fermi function
4. General expression
5. Approximation for non-degenerate semiconductors
6. Approximation for degenerate semiconductors

## 2.5.1 Introduction

The density of electrons in a semiconductor is related to the density of available states and the probability that each of these states is occupied. The density of occupied states per unit volume and energy is simply the product of the density of states and the Fermi-Dirac probability function (also called the Fermi function):

(f46)

Since holes correspond to empty states in the valence band, the probability of having a hole equals the probability that a particular state is not filled, so that the hole density per unit energy equals:

(f47)

The density of carriers is then obtained by integrating the density of carriers per unit energy over all possible energies within a band. The general expression is derived as well as an approximate analytic solution which is valid for non-degenerate semiconductors. In addition we also present the Joyce-Dixon approximation.

## 2.5.2 3-D Density of states

The density of states in a semiconductor is obtained by solving the Schrödinger equation for the particles in the semiconductor. Rather than using the actual and very complex potential in the semiconductor, one frequently uses the simple particle-in-a box model, where one assumes that the particle is free to move within the material. The boundary conditions which express the fact that the particles can not leave the material, force the density of states in k-space to be constant for this and also other more complex models. If the E-k relation is known, one can then find the corresponding density of states. The reader is refered to the section on density of states for a detailed derivation in one, two and three dimensions.

For an electron which behaves as a free particle with effective mass, m*, the density of states in three dimensions is given by:

(f24a)

where Ec is the bottom of the conduction band below which the density of states is zero. The density of states for holes in the valence band is given by:

(f24b)

## 2.5.3 The Fermi function

The Fermi-Dirac probability density function provides the probability that an energy level is occupied by a fermion which is in thermal equilibrium with a large reservoir. Fermions are by definition particles with half-integer spin (1/2, 3/2, 5/2 ...). A unique characteristic of fermions is that they obey the Pauli exclusion principle which states that only one fermion can occupy a state which is defined by its set of quantum numbers n,k,l and s. The definition of fermions could therefore also be particles which obey the Pauli exclusion principle. All such particles also happen to have a half-integer spin.

Electrons as well as holes have a spin 1/2 and obey the Pauli exclusion principle. As these particles are added to an energy band, they will fill the available states in an energy band just like water fills a bucket. The states with the lowest energy are filled first, followed by the next higher ones. At absolute zero temperature (T = 0 K), the energy levels are all filled up to a maximum energy which we call the Fermi level. No states above the Fermi level are filled. At higher temperature one finds that the transition between completely filled states and completely empty states is gradual rather than abrupt. The Fermi function which describes this behavior, is given by:

(f18)

This function is plotted in the figure below for an ambient temperature of 150 K (red curve), 300 K (blue curve) and 600 K (black curve).

fermi.tcl - fermi.gif

Fig.2.5.1 The Fermi-Dirac distribution function at T = 150 K (red curve), 300 K (blue curve) and 600 K (black curve).

The Fermi function has a value of one for energies which are more than a few times kT below the Fermi energy, equals 1/2 if the energy equals the Fermi energy and decreases exponentially for energies which are a few times kT larger than the Fermi energy. While at T = 0 K the Fermi function equals a step function, the transition is more gradual at finite temperatures and more so at higher temperatures.

A more detailed description of the Fermi function and other distribution functions can be found in the section on distribution functions

## 2.5.4 General expression

The general expression which provides the carrier density in a semiconductor is obtained by integrating the product of the density of states with probability density function over all possible states. For electrons in the conduction band the integral is taken from the bottom of the conduction band, labeled Ec to the top of the conduction band as expressed in the equation below:

(f48)

Where Nc(E) is the density of states in the conduction band and f(E) is the Fermi function.

This general expression is illustrated with the figure below for a parabolic density of states function with Ec = 0. The figure shows the density of states function (green curve), the Fermi function (blue curve) as well as the product of both which is the density of electrons per unit volume and per unit energy, n(E) (red curve). The integral corresponds to crosshatched area under the red curve.

density.tcl - density.gif

Fig.2.5.2 The carrier density integral. Shown are the density per unit energy, n(E), (green curve) and the probability of occupancy, f(E), (blue curve). The carrier density equals the cross-hatched area under the red curve.

Since the Fermi function reduces to almost zero for energies which are more than a few kT above the fermi energy EF, one finds that the actual top of the conduction band does not need to be known and the upper limit can be infinity without affecting the integral, yielding:

(f53)

For a three-dimensional semiconductor this integral becomes:

(f54)

While this integral can not be solved analytically we can either obtain a numeric solution or an approximate analytical solution. Similarly for holes one obtains:

(f56)

and

(f57)

These equations were used to obtain the figure 2.5.2.

## 2.5.5 Approximate expressions for non-degenerate semiconductors

Non-degenerate semiconductor are defined as semiconductors for which the Fermi energy is at least 3kT away from either band edge. The reason we restrict ourselves to non-degenerate semiconductors is that this definition allows the Fermi function to be replaced by a simple exponential function. The carrier density integral can then be solved analytically yielding:

(f55)

with

(f5)

where Nc is the effective density of states in the conduction band. Similarly for holes one can approximate the hole density integral as:

(f58)

with

(f6)

where Nv is the effective density of states in the valence band.

distrib1.tcl - carrier.gif

Fig.2.5.3 The density of states and carrier densities in the conduction and valence band. Shown are the electron and hole density per unit energy, n(E) and p(E), the density of states in the conduction and valence band, gc(E) and gc(E) and the probability of occupancy, f(E), (green curve). The electron (hole) density equals the red (blue) cross-hatched area.

## 2.5.6 Approximate expressions for degenerate semiconductors

A useful approximate expression which enables to relate the carrier density to the Fermi energy of degenerate semiconductors was obtained by Joyce and Dixon and is given by:

(f51)

for electrons and by:

(f52)

for holes.

2.4 ¬ ­ ® 2.6

© Bart J. Van Zeghbroeck, 1996, 1997