2.2 “To First Order”: Salient Features
2.5 Application to Direct Band Gap Semiconductors
in anticipating performance limits in practical devices. With Au-catalyzed Si wires, our group has measured minority carrier diffusion lengths of ≈2 μm, [55] which is therefore a diffusion length worth considering in more detail also. Setting τn = 57 ns throughout the cell, so that Ln = 2 μm, the maximal efficiency observed in the simulations was η = 10.0
%, when the wire radiusR= 1.5μm, and wire lengthL= 30μm. In this, depletion region recombination dominated case, surface recombination effects are much less important, and the maximum efficiency was unchanged for S ranging from 100 cm s−1 up to 1 × 105 cm s−1. In this case, Jsc = 0.0324 A cm−2, Voc = 0.453 V, and F F = 0.684. Under these conditions, increasing the wire length to 100μm resulted in a drop in efficiency to 9.6 %.
In the case of homogeneous trap distributions, optically thick wires (i.e. with L ≈ 100 μm) were optimal only whenLn10μm, i.e. τn 100 ns. For lower lifetime material, the penalty associated with increased junction and surface area associated with increased wire length outweighed the benefits of increased light absorption. Insofar as we believe that the homogeneous trap density analysis is a more realistic model for what we are likely to see with solid state junctions in grown wire arrays, these results set targets for material quality as well as aspect ratio for this device geometry to be of relevance as a PV technology.
Namely, in growing Si wires for use in a radial pn junction geometry, one would like to grow wires with microscopic rather than nanoscopic dimensions, and with diffusion lengthsLn 10 μm (equivalently, lifetimes of100 ns).
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Figure 2.17. Short-circuit current density Jsc vs. cell thickness L and quasineutral region minority-electron diffusion length Ln for (a) a conventional planar pn junction GaAs cell and (b) a radial pn junction GaAs cell. In both cases the short-circuit current density is unaffected by decreasing the trap density in the depletion region. In the radial pn junction case, the cell radius R is set equal toLn, a condition that was found to be near optimal.
Nd=1×1017 cm−3, Na =1×1017 cm−3,
μp =320 cm2 V−1 s−1,
⇒Dp =kBT
q μp = 8.27 cm2 s−1, μn=5000 cm2 V−1 s−1,
⇒Dn=kBT
q μn= 129 cm2 s−1, d1 =x1+x2 ≥1×10−6 cm, Sp =Sn= 1×105 cm s−1, σp =σn= 1×10−15 cm2,
vth =1×107 cm s−1, (2.59)
As for Si, the maximum doping level was limited by the need to have a nondegenerate material, the recombination in which is not dominated by Auger processes. [57]
The results forJsc,Voc, and overall cell efficiencyηfor GaAs are presented in Figs. 2.17,
Figure 2.18. Open-circuit voltageVoc vs. cell thicknessLand quasineutral region minority- electron diffusion length Ln for (a) a conventional planar pn junction GaAs cell and (b) a radial pn junction GaAs cell. In both cases the top surface shown in the plot has a depletion region trap density fixed at 1014cm−3, while the bottom surface has a depletion-region trap density equal to the trap density in the quasineutral region, at each value of Ln. In the radial pn junction case, the cell radius R is set equal to Ln, a condition that was found to be near optimal.
Figure 2.19. Efficiency η vs. cell thickness L and quasineutral region minority-electron diffusion length Ln for (a) a conventional planar pn junction GaAs cell and (b) a radial pn junction GaAs cell. In both cases the top surface shown in the plot has a depletion region trap density fixed at 1014cm−3, while the bottom surface has a depletion-region trap density equal to the trap density in the quasineutral region, at each value of Ln. In the radial pn junction case, the cell radius R is set equal to Ln, a condition that was found to be near optimal.
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2.18, and 2.19, respectively. Clearly, the performance difference between the planar and radial pn junction structures for GaAs is not nearly as dramatic as that for Si. This is due simply to the change in the relative length scales involved with the direct band-gap material.
As noted above, the optical thickness of GaAs is 891 nm. Therefore, severely collection- limited GaAs planar cells must have diffusion lengths much less than 891 nm, i.e., on the order of 100 nm or less. Recall also that the dopant density determines the depletion region width, with higher doping leading to thinner depletion regions. Nondegenerate doping implies that depletion regions in GaAs are on the order of hundreds of nanometers (191 nm for the planar case in our simulation). In the radial geometry the depletion region width depends upon the value of x4, with 191 nm setting a lower bound. Making a radial pn junction with a radius equal to the minority-electron diffusion length, that is, less than ∼ 100 nm for severely collection-limited GaAs, would thus result in fully depleted pn junctions.
This is a problem for several reasons. Firstly, a fully depleted cell cannot attain its full built- in voltage as given in Eq. 2.48, and the built-in voltage drops rapidly as the cell radius decreases further. Secondly, one might expect majority carrier transport to become more complicated in a fully-depleted wire because in this case the majority carriers are always in the presence of a strong electric field. One might expect that this could dramatically increase the effects of surface and junction recombination, and/or lead to other majority carrier transport issues not dealt with here. Thirdly, a fully depleted cell by definition has no quasineutral region; therefore, the concept of keeping the trap density in the depletion region low while the trap density in the quasineutral regions increases is meaningless. If the cell is fully depleted, then the lifetime in the depletion region, which needs to be high, is in fact the lifetime in the whole wire, and this defeats the whole concept of using lower lifetime material in the radial pn junction geometry. Finally, a fully depleted wire can no longer be adequately modeled by assuming purely radial minority carrier transport, because in this case the minority carriers are never swept into a quasineutral region in which they are majority carriers, and so ignoring majority carrier transport effects, as we have in the preceding analysis, may no longer be valid.
Note, however, that, as for the Si case, Jsc in the radial pn junction cell does not decrease with increasing trap density. This is not true in the planar geometry. Thus, it may be possible that degenerately doped GaAs wires with radial pn junction could exhibit some of the performance advantages seen in nondegenerately doped Si. This, however, is
left to the subject of a future study that deals explicitly with the transport and statistics of degenerately doped systems.