# Physical models¶

## Concentrations of charges in thermal equilibrium¶

Probability that an electronic state with energy is occupied is given by the Fermi-Dirac distribution:

with denoting the Fermi energy, denoting the Boltzmann constant and denoting the temperature.

The states in the conduction band are distributed in energy according to the density of states function . The total concentration of electrons is given by integral

In the valence band, almost all states are occupied by the electrons. It is therefore useful to track unoccupied states, holes, instead of the occupied states. The concentration of holes is given by:

Noting that

and changing the integration variable, both concentrations can be written in a common form as:

(1)

Practical note: The SI base unit of energy is Joule (J), however the energies such as are very small and should be expressed in electronvolts (eV). The SI basic unit of concentration is , although is often encountered. The value of at the room temperature is approximately 26 meV. The SI unit of temperature is Kelvin, .

## Band energies¶

In an idealized case, the energies and of the conduction and valence bands are

(2)

where is the electron affinity energy, and is the bandgap energy. is the electrostatic potential. q is the elementary charge.

## Electrostatic potential¶

The electric field is related to the elestrostatic potential as

(3)

In linear, isotropic, homogeneous medium the electric displacement field is

with permittivity

where is the vacuum permittivity, and is the relative permittivity of the material.

The electric displacement field satisfies the electric the Gauss’s equation

(4)

where is the density of free charge

with denoting the elementary charge. Above, denotes other charges, such as ionized dopants.

Combining the above equations gives the usual Poisson’s equation for electrostatics:

(5)

The SI unit of electrostatic potential is Volt, and the unit of electric field is Volt/meter. The unit of permittivity is Farad/meter. The unit of charge density is .

## Approximation for low concentrations¶

If the concentration of charge carriers is low enough, only states on the edge of band gap are important. In such case, the density of states can be assumed as a sharp energetic level,

in case of electrons and

in case of holes. Substiting into (1) gives

At low charge carrier concentrations, Fermi-Dirac distribution is simplified as

The approximation is considered valid when .

Approximate charge carrier concentrations are

(6)

## Gaussian density of states¶

In the case of Gaussian DOS, the density of states shape function is the Gaussian distribution function scaled by total density of states :

Concentrations of species are given by integral

## Conservation equation¶

The conservation equation is:

where denotes the concentration, is time, and is the flux density. denotes source term, which is positive for generating particles, and negative for sinking particles of type i. The SI unit of source term is .

The conservation equation must be satisfied for each species separately. In the case of transport of electrons and holes, this gives

(7)

where the source S term contains for example generation and recombination terms

The conservation of electric charge must be satisfied everywhere. Therefore, the source terms acting at given point must not create a net electric charge. In the case of system of electron and holes, this requires

## Current density¶

Current density is related to the density flux by the charge of single particle . Obviously, for electrons and for holes , therefore

(8)

Note that a convention is adopted to denote the electric current with uppercase letter , and the flux density with lowercase letter . The SI unit of density flux is , while the unit of electric current density is .

## Equilibrium conditions¶

In the equilibrium conditions, Fermi level energy has the same value everywhere. The electrostatic potential can vary, and the density of free charge does not need to be zero. Equations (1), (2), (4) are satisfied simultaneously. The current flux, the source terms, and the time dependence are all zeros, so conservation (7) is trivially satisfied.

## Nonequilibrium conditions¶

In the non-equilibrium conditions, the transport is introduced as a perturbation from equilibrium. The Fermi energy level is replaced with quasi Fermi level, which is different for each species. In (1), the equilibrium Fermi level for electrons is replaced with a quasi Fermi level . Similarly,, the equilibrium Fermi level for holes is replaced wuth quasi Fermi level for holes , giving

(9)

Quasi Fermi levels have associated quasi Fermi potential according to the formula for energy of an electron in electrostatic field :

(10)

The transport is modeled by approximating electric current density as

(11)

where denotes the respective mobilities. The SI unit of mobility is , although is often used.

Equations (2), (5), (9), (11), (7) are simultaneously satisfied in non-equilibrium conditions.

## Drift-diffusion system¶

Standard form of density fluxes in the drift-diffusion system is

(12)

or more generally, allowing arbitrary charge per particle

(13)

is the diffusion coefficient, with SI unit .

## Drift-diffusion system: low concentration limit¶

To obtain the conventional drift-diffusion formulation (12), the the low concentration approximation (6) should be used. After introducing quasi Fermi levels, as it is done in (9), one obtains

From that, the quasi Fermi energies are calculated as

Using (2), and assuming constant ionization potential , bandgap , constant total densities of states , and constant temperature , substituting into (11), and using (3)

In terms of density flux (8), this reads

(14)

where thermal voltage

## Einstein’s relation¶

Equation (14) is written in the standard drift-diffusion form (12) when the diffusion coefficient satisfies

(15)

This is called Einstein’s relation.

## Drift-diffusion system: general case¶

Using functions defined in (1), bands (2) and approximation (9)

current densities under assumptions are

(16)

## Generalized Einstein’s relation¶

In equation (16), assuming

In order to express equation (16) in the standard drift-diffusion form (12), the diffusion coefficient must satisfy

This is so called generalized Einstein’s relation .

## Intrinsic concentrations¶

Intrinsic concentrations , , and intrinsic Fermi level satisfy electric neutrality conditions

## Direct recombination¶

Direct recombination introduces source term

where can be chosen freely.

## Unidimensional form¶

By substituting and , the equations (5), (7), (12) of the basic drift-diffusion device model are

## Total electric current density¶

Total electric current is a sum of currents due to transport of each species and the displacement current

Total electric current satisfies the conservation law

This can be verified by taking time derivative (4), using (7) and considering that the sum of all charge created by the source terms must be zero.

## Electrode current¶

Current passing through a surface of electrode is

(17)

## Metal¶

In metal, the relation between the electrostatic potential , the workfunction energy and the Fermi level is

On the other hand, the Fermi potential corresponds to the applied voltage

This leads to electrostatic potential at metal surface

## Ohmic contact¶

Ohmic contact is an idealization assuming that there is no charge accumulation at the contact, and the applied voltage is equal to quasi Fermi potentials (10) of charged species

Above three conditions uniquely determine the charge concentrations , , and the electrostatic potential at the contact.

## Electrochemical transport¶

Electrochemical potential for ionic species is

It should be noticed that so defined “potential” has the unit of energy, unlike the electrostatic potential and quasi Fermi potentials. Above denote corrections, for example due to steric interactions. Electrochemical potential should not be confused with mobility .

Density flux is approximated as

yielding the standard form (13) using Einstein’s relation (15).

Electrochemical species should be included in Poisson’s equation, by including proper source terms of form . A variant of Poisson’s equation (5) where are free charges are ions can be written as

## Steric corrections¶

To account for finite size of ions, the electrochemical potential in the form introduced in [LE13] is useful

where denotes volume of particle of type . is the unoccupied fraction of space

where summing is taken over all species occupying space, including solvent.

 [LE13] Jinn-Liang Liu and Bob Eisenberg. Correlated Ions in a Calcium Channel Model: A Poisson–Fermi Theory. The Journal of Physical Chemistry B, 117(40):12051–12058, 2013. PMID: 24024558. URL: https://doi.org/10.1021/jp408330f, arXiv:https://doi.org/10.1021/jp408330f, doi:10.1021/jp408330f.