Shockley Queisser Formulation

Shockley and Queisser's approach removed the need to keep track of the recycled photons by switching attention to what was happening outside the cell. They realised that an efficient cell would have to be an efficient absorber of solar photons and hence have properties related to a black-body, at least for energies above the cell's bandgap.

By assuming that an ideal cell did have black-body properties, they were able to very simply calculate the net rate of recombination in the cell taking into account photon recycling effects. By considering such a cell in thermal equilibrium (no external light on it and no voltage applied to it), they realised it would have to be emitting Planckian black-body radiation at these energies, of the same type as treated in Chap. 2.

By attributing all this radiation for energies above the semiconductor's bandgap to band-to-band recombination, they were able to very simply calculate the net rate of recombination events occurring in the cell. This is given by A times the hemispherical black-body emission rate (Eq. (2.9)) for photons of energy higher than the bandgap. A is the area of the cell from which light can be emitted (this is ideally only the front surface area since a reflector can be placed on the rear).

Moreover, From Eqs. (4.2) and (4.7), all radiative recombination in the cell should increase exponentially with applied voltage. Since photon recycling effects would remain proportionately the same, this meant Shockley and Queisser could now calculate the net recombination rate at any voltage. This leads instantly to the ideal solar cell equation with unity ideality factor (N = 1) and Io given by:

I0 = qAN(egPo) « qA() \eg2 + 2(kT)EG + 2(kT)2]e~EG/kT (4.9)

The expression on the right can be derived if the -1 term in the denominator of Eq. (2.9) is neglected, a good assumption for EG >> kT (the resulting integral is then readily integrated by parts).

The interesting feature of this expression is that I0 no longer depends on cell volume. Although the total number of radiative recombination events increases with volume, photon recycling becomes more effective so that the net number of recombination events stays the same. With the infinite mobility assumption, all photogenerated carriers will be collected. This means no photogenerated carrier will be lost, regardless of how large the cell becomes.

Although no semiconductor has infinite mobility, the parallel multijunction structure of Fig. 4.2 allows the infinite mobility case to be approached arbitrarily

4.2 Shockley-Queisser Formulation 39

Fig. 4.2: Parallel multijunction solar cell.

closely for materials with finite mobility. The thickness of each multilayer region has to be a lot less than a quantity of the order of (kTn¡1 /J) where ¡1 is the mobility of minority carriers of concentration, n, and J is the current density perpendicular to the surface. The lateral extent between contacts has to be less than a similar term for majority carriers, with J the lateral current density in this case.

If the sun is modelled as a black-body at temperature Ts, the current-voltage curve can be expressed as:

I = qAfsN(EG,~,Ts)- qA fcN(EG,~,Tc )(eqV/kT -1) (4.10)

where fs and fc are geometrical factors to be discussed. Equation (4.10) shows that cell properties are described solely in terms of the bandgap in this analysis. For the non-concentrating system analysed by Shockley and Queisser, the solar intercept fraction fs = 2.1646 x 10-5 while fc = 1, leading to a peak efficiency of 31.0% for Eg = 1.3 eV for Ts = 6000 K and Tc = 300 K.

For a concentrating system, fs = 1. From Eq. (4.10), this means that a larger V is required to reduce I to zero, i.e., a larger Voc is obtained in this case. This increases the peak efficiency to 40.8% for EG = 1.1 eV. The same value is also obtained for the limiting efficiency conversion of direct sunlight at other concentration levels, including non-concentration. In this case, f-factors, fs and fc would apply to both sun and cell. The former would be determined by the concentration level and the latter by the angular selectivity of the cell. To collect all direct sunlight, fc > fs. In principle, it is always possible to design a system with fc = fs, regardless of f-factor value between the extremes mentioned. This means that the same Voc and hence efficiency is obtained regardless of concentration level, for direct light.

The work of Shockley and Queisser represented a major conceptual advance and provided a new set of tools for analysing solar cell performance. One limitation is that it applies only to non-degenerate situations as determined by (EG - qV) >> kT. This is not a serious limitation for single junction cells but could be for more general photovoltaic converters.

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