Question

give the equation for current density in form of fermi dirac distribution function and graph it

Answer 1

To evaluate the current density, we must specify how the scattering solutions are statistically weighted.   Let us follow through the derivation of the electron density in a spatially uniform three-dimensional semiconductor in equilibrium to calculate the current density. We approximate the conduction band structure by a simple parabolic dispersion relation:

where is the wavevector. The probability that each state will be occupied by an electron is given by the Fermi-Dirac distribution function:

where is the Fermi level or chemical potential and , T being the absolute temperature. (To avoid confusion with the transmission probability which is also denoted by T, the absolute temperature will always be shown multiplied by Boltzmann's constant . Let us now make anassumption that the semiconductor crystal is a cube with each side of length L, and apply periodic boundary conditions. Then the stationary quantum states are plane waves (normalized to unit amplitude) of the form

Due to the periodic boundary conditions, must assume discrete values:

where , , and are integers. The total number of electrons in the crystal N is just the sum over all of the states of the probability that each state is occupied

where the factor of 2 comes from the two spin states. Now,L is large, So the allowed values of are very closely spaced, and the sum over can be well approximated by an integral:

We can now write an expression for the density of electrons n:

In order to evaluate densities using expressions such as (23) it is usually more convenient to transform the integration variable to E. By expressing in spherical coordinates and manipulating the dispersion relation :

We will also have occasion to use the corresponding expressions for integrals over two- and one-dimensional vectors. For the two-dimensional case (still assuming a parabolic dispersion relation):

For integration over a one-dimensional k, the definition of the group velocity may be used to obtain an expression valid for any dispersion relation:

Inserting (24) into (23) leads to the usual expression for the electron density in a semiconductor .

The expectation value of the density for a state is simply

The expectation value of j is simple, though the operator itself often is not. (Link to manuscript with full details.) If the dispersion relation is not parabolic and independent of position, the form of the operator j is not given by the simple textbook expression . The current density operator is instead whatever remains of the kinetic energy term of the Hamiltonian after the application of Green's identity, and this obviously depends upon the form of the Hamiltonian itself. For unit-incident-amplitude scattering states

  

Of course, in equilibrium, these two currents cancel each other (by the principle of detailed balance) and there is no net current flow.

To investigate the transport properties of a quantum system one must generally evaluate the current flow through the system, and this requires that one examine systems that are out of thermal equilibrium. A common situation, in both experimental apparatus and technological systems, is that one has two (or more) physically large regions densely populated with electrons in which the current density is low, coupled by a smaller region through which the current density is much larger. It is convenient to regard the large regions as ``electron reservoirs'' within which the electrons are all in equilibrium with a constant temperature and Fermi level, and which are so large that the current flow into or out of the smaller `` device'' represents a negligible perturbation. The reservoirs represent the metallic contacting leads to discrete devices or experimental samples, or the power-supply busses at the system level. Consequently, the electrons flowing from a reservoir into the device occupy that equilibrium distribution which characterizes the reservoir. In a simple one-dimensional system with two reservoirs, the electrons flowing in from the left-hand reservoir have k > 0 and those flowing from the right-hand reservoir have k < 0.

Within this picture, the current that is injected from the left-hand reservoir is

and the current injected from the right-hand reservoir is

In order to simplify the calculation of J further, we must invoke some special properties of the system. The most useful such property is that symmetry which permits the separation of the spatial variables. The separation of variables is possible if the Hamiltonian can be separated into two parts:

(Here the notation and is defined with respect to the direction of current transport.) Then the wavefunction separates into a product of two factors:

and the energy can be separated into a product of two terms:

The expression for the total current density J can now be simplified to

where is the larger of the two asymptotic potentials (minimum energy for a propagating state) and F is the Fermi-Dirac distribution function summed over the transverse states:

The form of the sum over depends upon the spatial configuration of the tunneling system. Note that the velocity factor does not appear in (35) because it was canceled by the density of states.

If the system in question is macroscopically large in its transverse dimensions, the form a two-dimensional continuum, and . Then using the two-dimensional analogue of (22) and (25) F can be analytically evaluated:

The current density can now be written in the form usually given for the tunnelling current density.

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