Richardson constant

Fig. 30. Barrier height A measured by current-voltage (/-V) characteristics of (Ga,Mn)As/GaAs diodes. A shown by closed circles is the barrier height between the Fermi energy of (Ga,Mn)As and the valence band top of GaAs as shown in the inset. Open circles depict the effective Richardson constants. (Ohno et al. 2001).

As = surface area of a semiconductor contact [A ] = concentration of the reduced form of a redox couple in solution [A] = concentration of the oxidized form of a redox couple in solution A" = effective Richardson constant (A/A ) = electrochemical potential of a solution cb = energy of the conduction band edge Ep = Fermi level EF,m = Fermi level of a metal f,sc = Fermi level of a semiconductor SjA/A") = redox potential of a solution ° (A/A ) = formal redox potential of a solution = electric field max = maximum electric field at a semiconductor interface e = number of electrons transferred per molecule oxidized or reduced F = Faraday constant / = current /o = exchange current k = Boltzmann constant = intrinsic rate constant for electron transfer at a semiconductor/liquid interface k = forward electron transfer rate constant = reverse electron transfer rate constant = concentration of donor atoms in an n-type semiconductor NHE = normal hydrogen electrode n = electron concentration b = electron concentration in the bulk of a semiconductor ... 

For liquids, the only major difference in the kinetic equations lies in the analog of the Richardson constant. The probability of interfacial charge transfer at a semiconductor/liquid... 

Thermionic emission. The number of electrons which escape from the metal surface increases rapidly with temperature (thermionic emission). In general, the higher the temperature and the lower the work function, the higher is the electron emissivity. The current density can be calculated by the Richardson-Dushman equation (in the absence of an external electrical field), according to i — AT exp(—rp/kT), where A is the Richardson constant (A cm K ), T is the temperature (K), and

work function (eV). For pure tungsten A — 60.2 (A cm K ) [1.91]. The thermionic current (A cm ) can then be calculated as i — 60.2r exp(—52230/T) [1.37]. 

By treating surface recombination as a hopping process in the image charge potential, Scott and Malliaras [140] have derived a very simple equation that describes the injected current as a function of electric field, temperature, and measurable parameters of the organic, namely the dielectric constant, the site density, and the drift mobility. The current has the usual form of thermionic emission, but with an effective Richardson constant that is several orders of magnitude lower than that in inorganic semiconductors. The results of the model are in... 

This is the effective Richardson constant for thermionic emission. In the case of isotropic effective mass one can rewrite Eq. (2.15) as... 

Electrode area Richardson constant Activity coefficient Differential Helmholtz capacity Differential space charge capacity Concentration of species j in solution... 

Temperature dependent I-V measurements in the temperature range 300-500 K, which allowed determination of the activation energy, were also performed, and the results are listed in Table 6.2. The barrier heights calculated from the Richardson plot using Equation (6.2) are consistent with the room temperature values for each sample. However, the calculated effective Richardson constant is much smaller than the theoretically expected value which is commonly reported in the literature for GaN [2]. As we can see in Table 6.2, the effective Richardson constant is also related to the crystal quality, and obviously further work is required to shed light on the discrepancy between the measured and theoretical values endemic to GaN. 

Here, A = Anem k lh is the effective Richardson constant, which is proportional to the effective mass m of the charge carriers and otherwise contains only physi-... 

Thermionic emission over the metal/semiconductor interface yields a current of the form [9,14,15], I = A T exp(—B /kT) where A is known as the effective Richardson constant. The field dependence is included in the effective barrier height, which is reduced compared to the triangular... 

The Richardson constant /I = 120 amp/cm-K for free electrons A will differ from this value by the effective mass ratio m /m in a semiconductor, and for nearly all useful semiconductors one has m lm< and thus /4 < 120 amps/cm -K. One must have a barrier height to have a true... 

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