Fermi-Dirac distribution function

Fermi-Dirac distribution function

[¦fer·mē di¦rak ‚dis·trə′byü·shən ‚fəŋk·shən]
(statistical mechanics)
A function specifying the probability that a member of an assembly of independent fermions, such as electrons in a semiconductor or metal, will occupy a certain energy state when thermal equilibrium exists.
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where [f.sub.FD]([epsilon], T) = [{exp[[beta]([epsilon] - [mu],)] + 1}.sup.-1] is the Fermi-Dirac distribution function that gives the average occupation of the single-particle energy state [epsilon] at absolute temperature T.
The current through the system is calculated using the Landauer-Buttiker formula: I = (2e/h) [integral] T(E,V)[[f.sub.L](E - [[mu].sub.L]) - [f.sub.r](E - [[mu].sub.r])]dE [46, 47], where e is the electron charge, h is Planck's constant, T(E,V) is the transmission probability through the junction for an electron at energy E and an applied bias voltage V, [f.sub.L/R] (E - [[mu].sub.L/R]) is the Fermi-Dirac distribution function, [mu].sub.L] and [[mu].sub.R] is the electrochemical potential of the left and right electrode, respectively.
where {[mu], v} indicate Cartesian components, T shows temperature, and f(E, T) refers to Fermi-Dirac distribution function, f(E, T) = [[1 + exp(E/T)].sup.-1].
where [[mu].sub.1] and [[mu].sub.2] are the electrochemical potentials of the left and right electrodes, V is the external bias, and f(E) is the Fermi-Dirac distribution function. T(E, V) is the transmission coefficient at energy E and bias voltage V and represents the quantum mechanical transmission probability for electrons.
where [phi] is the phase of the scattered electrons and the factor (-[partial derivative][f.sub.FD]/[partial derivative]E) is the first derivative of the Fermi-Dirac distribution function and it is given by:
where [[f.sub.0](E)= {1 + exp[(E-[E.sub.F])/[k.sub.B]T]}.sup.-1] is the Fermi-Dirac distribution function, [E.sub.F] is the Fermi energy in eV, [[rho].sub.c](E) and [[rho].sub.v](E) are, respectively, the electron density of states for the conduction band and the hole density of states for the valence band, [k.sub.B] is the Boltzmann constant, and T is the temperature in kelvins.
where [f.sub.0](E) = 1/(1 + exp[(E - [mu])/[k.sub.B]T]) is the Fermi-Dirac distribution function, [k.sub.B] is Boltzmann's constant, and [mu] is the chemical potential.
From the kinetic energy principle, [absolute value of v] = [square root of 2E/m], DOS(E) is density of states, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Fermi-Dirac distribution function which gives the probability of occupation of a state at any energy level, and n is carrier concentration [33].
where [f.sub.FD] is the Fermi-Dirac distribution function. Now, substituting Eq.(8) into Eq.(9), we get a complete expression for conductance which depends on the angles 0,9, and on the barrier height, [V.sub.0], and its width, the gate voltage, [V.sub.g], and the photon energy, [??][omega].
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] stands as Fermi-Dirac distribution function which yields the probability of occupation of a state at any energy level.
where [GAMMA](E) is the photon-assisted tunneling probability, [f.sub.FD(s/d)] are the Fermi-Dirac distribution function corresponding to the source (s) and drain (d) electrodes, while e and h are electronic charge and Planck's constant respectively.
Carrier concentration in a band is achieved by integrating the Fermi-Dirac distribution function over energy band as n = [integral] D(E)[f.sub.F] (E)dE, where D(E) and [f.sub.F](E) = [(1 + exp((E - [E.sub.F])/[k.sub.B]T)).sup.-1] are available energy states (density of states) and Fermi-Dirac distribution function, respectively.