LESSONS OF EASTER ISLAND
March 17th, 2016
In Section 3.6.2 we have discussed nonradiative transitions between the bands and an impu – rity level. Impurities with energies for electrons in the middle of the energy gap were found to greatly enhance recombination, which is detrimental for the efficiency. In the analysis, generation of electronhole pairs by optical transitions was neglected. Now we do just the opposite. Our model now permits only radiative transitions between the bands and to and from an impurity level. Thermalization of free charge carriers is considered, but not impact ionization or nonradiative recombination.[32]
The model, as shown in Figure 8.11, has states in the valence band with £e < Ey, at the impurity level £jmp and in the conduction band with £e > £c – In order to optimally utilize the incident spectrum, the photons will be distributed over the different transitions in such a way that photons capable of a higher energy transition, e. g., bandband, are not wasted in lower energy transitions. For the impurity energy £jmp in the lower half of the energy gap (not in the middle), photons having energies £imp — £y < ho < £c — Єішр are exclusively absorbed in transitions from the valence band to the impurity. Photons having Єс — Єішр < ho < £c — £у are exclusively absorbed in transitions from the impurity to the conduction band, and photons having ho > £c — £y provide for the bandband transitions.
The absorption properties of the impurities are characterized by optical crosssections, Суд for transitions from the valence band to the impurity and G{,c for transitions from the impurity to the conduction band. Optical crosssections are of the same order of magnitude as geometrical crosssections, 10“15 cm2 is a typical value. Although optical crosssections vary with energy, we assume them to be constant over the energy range of absorbable photons.
Figure 8.11: In addition to radiative band – band transitions with the rates G and R. radiative transitions between the bands and the impurity are taken into account. Nonradiative transitions are excluded.
The electrons and holes are assumed to have a high mobility resulting in their homogeneous distribution, even though they are generated inhomogeneously. The steady state concentrations belonging to a given value of the charge current, follow from the continuity equations for the particle densities, in which in addition to generation and recombination we consider the contribution to the charge current by the divergence of the electron and hole currents. From the concentrations of electrons and holes, the sum of their electrochemical potentials and thus the voltage can be derived. The continuity equations are
= Gbb + <jj, c ~ Rvb – Ri. c ~ div Л = 0 (8.12)
In the last equation the divergence of the particle current is missing, since electrons in the impurities are considered immobile, so they do not contribute to the current.
Since these three equations are not independent, we need the charge neutrality as an additional equation, as in Section 3.6.2. As a result of the high absorption required, however, the impurity concentration щmp is now no longer negligible compared with the densities of the electrons and holes. For this reason, most impurities must be electrically neutral, that is either occupied, if they are donorlike, or unoccupied if they are acceptorlike. Both situations are unfavourable for the desired impurity absorption, since for transitions from the valence band to the impurity they must be unoccupied, and for transitions from the impurity to the conduction band they must be occupied. The smaller of the two transition rates will determine
The generation rates, averaged over the thickness d, are given by
1 [°°
Gbb = 7 / abb djy(Rm)
a 7 єc Ey
Gi, c — 7 [ aic d/y(/iO)) (8,16)
й ^£CE. mp
——————————— 1— ҐС gimp——————– ;———————————————————————————————————————————–
Gv, i — «уд djy(Hoy)
“ ”’Є, тр – Є
The absorptivity for the bandband transitions is assumed to be Abb = 1, whereas the ab sorptivities for the impurity transitions depend on the concentration and occupation of the impurities,
flj. c = 1 — exp(Oi, c<f) «v, i — 1 — exp(av. i^) ,
where the absorption coefficients follow from Eq, (3.96)
0^i, C — t^i. C ajmp (/і — /с) and OCv, i — Ov, i wimp (/v ~ fi)
These equations determine the generation rates due to illumination, but also in the dark state with its incident 300 К background radiation. According to the Principle of Detailed Balance, in this state of chemical equilibrium with the background radiation, the recombination rates must have the same value as the generation rates. The Principle of Detailed Balance is ac – counted for, if we write the recombination rates in terms of the generalized Planck radiation law. The rate of downward transitions per energy from a level j to a level t, averaged over the thickness d is, from Eq. (3.101),









This rate is then integrated over the energy range associated with the band to band, valence band to impurity, and impurity to conduction band transitions, to give the actual recombination rates.
The charge current Jq finally results from the integral of the divergence of the electron current (or the hole current) over the thickness d of the cell and, because of the assumed homogeneous distribution of the electrons and holes, it is given by
Jq = —e divy‘e d = — e div y‘h d. (8,18)
For a given charge current the continuity equations (8,12)(8.14) are solved for the positions of
sG/eV Figure 8.12: Efficiency as a function of the energy gap Єс – £v for radiative bandband transitions and radiative transitions between the bands and an impurity level at Ejmp. Nonradiative transitions are excluded. The numbers at the curve give the optimal position of the impurity level with regard to the valence band for selected band gaps. 
1Г u Figure 8.13: Equivalent circuit for a solar cell with an impurity level between valence and conduction bands as shown in Figure 8.11. 
The efficiency reaches a maximum value of r = 0.46 for an energy gap Єс — Єу = 2.4 eV and an impurity level at £imp — £y = 0.93 eV.
parallel to a series connection of two solar cells representing the transitions involving the impurity level. Variations of the absorptivities in the course of the currentvoltage characteristic can, however, not be treated in an equivalent circuit model.
As for tandem cells, it is expected that the efficiency increases when more than one impurity level is present and the incident spectrum is divided into smaller portions over more transitions. Ensuring good absorption properties for all transitions, however, is a problem. Moreover, it must be emphasized that nonradiative recombination has been excluded. Al – though the optimal position of the impurity level for optical transitions is not in the middle of the energy gap, where nonradiative recombination is most probable, including nonradiative recombination will certainly reduce the improvement expected from impurity transitions.
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