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:
and changing the integration variable, both concentrations can be written in a common form as:
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, .
In an idealized case, the energies and of the conduction and valence bands are
where is the electron affinity energy, and is the bandgap energy. is the electrostatic potential. q is the elementary charge.
The electric field is related to the elestrostatic potential as
In linear, isotropic, homogeneous medium the electric displacement field is
where is the vacuum permittivity, and is the relative permittivity of the material.
The electric displacement field satisfies the electric the Gauss’s equation
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:
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
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
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
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 is related to the density flux by the charge of single particle . Obviously, for electrons and for holes , therefore
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 .
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.
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
Quasi Fermi levels have associated quasi Fermi potential according to the formula for energy of an electron in electrostatic field :
The transport is modeled by approximating electric current density as
where denotes the respective mobilities. The SI unit of mobility is , although is often used.
Standard form of density fluxes in the drift-diffusion system is
or more generally, allowing arbitrary charge per particle
is the diffusion coefficient, with SI unit .
Drift-diffusion system: low concentration limit¶
From that, the quasi Fermi energies are calculated as
In terms of density flux (8), this reads
where thermal voltage
This is called Einstein’s relation.
Drift-diffusion system: general case¶
current densities under assumptions are
Generalized Einstein’s relation¶
In equation (16), assuming
This is so called generalized Einstein’s relation .
Intrinsic concentrations , , and intrinsic Fermi level satisfy electric neutrality conditions
Direct recombination introduces source term
where can be chosen freely.
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
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 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 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
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
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.|