Gartner equation

The Gartner equation (equation (8.6)) provides a satisfactory description of the photocurrent voltage curves measured at many semiconductor electrodes, but it contains no information about electron transfer kinetics because its derivation involves an a priori assumption about the boundary conditions. 

Fig. 8.6. Components of the current response of a n-type semiconductor electrode to an illumination step. ch is the charging current, is the current due to interfacial electron transfer and rc,. is the current due to electrons recombining with holes via surface states. The total current, given by the sum of and j , is equal to qg - /rec, where g is the flux of minority carriers given by the Gartner equation. The dimensionless normalised time axis is ( lr + k,c<.)t. The dimensionless normalised current axis is jlqg.

It should be mentioned here that Eq. (41) had already been derived by Gartner [86] for the photocurrent of a semiconductor-metal junction in reverse bias, assuming P = 0. In the derivation of Reichman [85], however, P was obtained in a manner consistent with the interface boundary conditions. Although the derivation of Reichman is much more general, most scientists only applied the Gartner-equation (see e.g. [87]). Later on, Wilson extended the Gartner model by including recombination via surface states [88]. 

Expansion of the Gartner equation (eq. 12.2) for small values of 6T shows that the photoeurrent is a linear funetion of the absorption eoeffieient near the band edge. The bandgap ean therefore be obtained using the plot illustrated in Fig. 12.8. 

Photocurrent spectroscopy can be used to determine the minority-carrier diffusion length of semiconductors. The Gartner equation (eq. 12.2) can be written in the linear form... 

Gartner equation [25]. Later on Wilson extended the Gartner model by including recombination via surface states [26],... 

The Gartner equation provides a useful theoretical basis for the analysis of the photocurrent response of illuminated semiconductor electrodes. It can be rewritten in the form... 

Gartner [130] for the boundary condition that the concentration of minority carriers tends to zero in the depletion layer. The Gartner equation can be written in the normalised form... 

The ratio g/I(0) defines the photocurrent efficiency . In the absence of surface recombination, qg corresponds to the photocurrent density Jphow measured in the external circuit. The Gartner equation has been used successfully to explain the photocurrent-potential characteristics of many semiconductor electrodes under conditions where surface recombination is absent. Plots of ln — O) against dsc (which according to the Mott-Schottky relationship is proportional to (1/ — have... 

The rate constants for electron transfer and recombination are readily separated because in the limit ca 0), Eq. 4.17 tends to ktr/ kir + krec), and the maximum of the semicircle occurs when ca = Inf = ktr + krec- In the absence oi RC attenuation effects (see section 4.2), the high frequency intercept of the IMPS plot d>((u —> oo) is unity, so that the ac photocurrent can be used to determine gac, the Gartner flux of minority carriers. Measurements of gac as function of potential (band bending) can be used to determine the minority carrier lifetime and absorption coefficient [44]. The main advantage of using the IMPS data rather than dc measurements of the photocurrent is that the high frequency limit of the IMPS effectively freezes out the effects of surface recombination which are responsible for substantial deviations of the dc photocurrent from the Gartner equation. 

If krec = 0, the photocurrent simply follows the illumination step and contains no information about the rate of charge transfer at the interface. The comparison shows that, unlike IMPS and PEIS, light modulated microwave reflectivity measurements still provide kinetic information at high band bending where recombination is negligible and the steady state photocurrent is described by the Gartner equation. 

EQE spectra can be analyzed to obtain the bandgap of absorber materials. The photocurrent response (Jphoto) of the semiconductor electrolyte junction is described by the Gartner equation [148] as... 

In general, both the drift and diffusion components contribute additively to the photocurrent. The steady-state efficiency of the photogeneration process can be found by solving the drift and diffusion equations and adding the currents. When assuming that the built-in field is located primarily in either the n-type or p-type side of the junction, the Gartner equation for monochromatic photocurrent generation applies, which reads... 

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