Fermi-Dirac function

At this point, it is interesting to extend the theory to include temperature effects, as was first done by Brako and Newns . When the temperature is nonzero, the Fermi-Dirac function... 

Figure 3.3 Fermi Dirac function [Eq. (3.10)] with h, — 1 and several values of a.

For tunneling between two equilibrium leads distribution functions are simply Fermi-Dirac functions (48) and current can be finally written in the well known form (To do this one should multiply the integrand on 1 = f 6(E-Eq)dE.)... 

Dirac Statistics involve exponential functions, their form is not log-normal or Gaussian. Fermi-Dirac Statistics apply directly and specifically to the absorption spectra of chromophores. No arbitrary terms need be added to the Fermi-Dirac Function of well behaved crystalline and liquid crystalline absorbers. 

As discussed in Section 2.6.2 electrons in a solid in thermal equilibrium obey Fermi-Dirac statistics in which the probability F ) that a state of energy is occupied is given by the Fermi-Dirac function... 

Figure 7. Sketch of the density of electron states as a function of energy for a typical intrinsic semiconductor or insulator, (a) 0 K (b) T > 0 K, some electrons from the valence band are thermally excited into the conduction band in agreement with the Fermi-Dirac function. The electrochemical potential or Fermi level is located in the middle of the gap.

Reducing device temperature leads to increase in transient EL intensity, but the EL decay dynamics does not change, as shown in Figure 7.7. The temperature dependence of EL peak intensity was found to be the same for light emission at turnon and at turn-off (inset of Figure 7.7). Such luminescence intensity dependence on temperature is not uncommon,14-16 and falls into a pattern modeled with Fermi -Dirac function... 

The condnction band of a metal possesses a continuum of allowed electronic states, whose effective density-of-states (i.e. the number of states per atom per unit energy interval available to the redox species) may be taken to be independent of energy. The probabihties pXE) and p (E) that a state of energy E is occupied by an electron or hole, respectively, are given by the Fermi-Dirac function... 

P E) is the Fermi-Dirac function giving the probability of an electron being in the conduction band. 

What happens for T > 0 K In this case the electrons can take up additional thermal energy. Some orbitals above the Fermi energy get occupied. The same number of orbitals below the Fermi energy becomes unoccupied. The overlap of occupied and unoccupied levels is given by the Fermi-Dirac function ... 

Figure 2.22 Fermi-Dirac function (after KitteF) the widths of the distribution around Ef is k T.

Fig. 10 Illustration of density distribution in real block copolymer micellar systems. The data correspond to a core with a constant density profile convoluted by Gaussian function. The density profile of the corona grafted to the core is calculated using a Fermi-Dirac function n(r) (1 -I- exp[(r - Rm)/ (

The Fermi-Dirac function of the electron energy distribution is given by 1... 

The charge distribution in energy follows a quasi-equilibrium distribution and can be described as the DOS multiplied by a Fermi-Dirac function. 

The Fermi-Dirac function can be approximated by the Boltzmann function if Ep - E) is much greater than kpT. The Fermi energy Ep determines n and p, the concentrations of... 

In the approach to trapped states described in Sect. 2.6 above, the traps are occupied according to a steady state occupation probability that resembles a Fermi-Dirac function, which allows one to define a quasi-Fermi level for the trapped charge. In transient simulations, the occupation function evolves as a function of time through capture, emission and recombination, which means that the differential equations... 

To find the occupation statistics for a trap - the Shockley-Read-Hall statistics [232,233] - we need to consider the four processes shown in Fig. 10. A single trap can capture and emit an electron and capture and emit a hole. If the same trap captures a hole and an electron, one recombination event happens. If a trap captures and emits an electron or a hole, the trap will have slowed down transport only. Table 3 summarizes the four rates that we need to consider. However, the four rates are not independent of each other in quasi-equilibrium. Because in equilibrium, detailed balance between inverse processes must be obeyed, the capture and emission processes must be connected. In addition, in thermal equilibrium the occupation function for all charge carriers (free or trapped, electrons or holes) must be the Fermi-Dirac function in thermal equilibrium, i.e. 

In this approximatiOTi, the Fermi—Dirac function is a unity step function at the Fermi level. Therefore, displacing the Fermi level by d p simply fills with carriers a slice of the DOS dn = g Ep)dEp. is also denoted a thermodynamic density of states [138]. 

In thermal equilibrium (no electric field, no thermal gradient) the distribution function is the well-known Fermi-Dirac function... 

By replacing (3.24) into (3.23) and using the definition of the Fermi-Dirac function we directly obtain for the particular terms... 

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