2.2 “To First Order”: Salient Features
2.4 Numerical Assessment of Device Behavior in Si
2.4.4 Homogeneous Trap Distribution Analysis
Figure 2.13. Efficiency η vs. cell thickness L and quasineutral region minority-electron diffusion length Ln for (a) a conventional planar pn junction Si cell and (b) a radial pn junction Si cell. Depletion region trap density is held fixed at a relatively low level,Nr= 1014 cm−3, so that depletion region lifetimeτn0= 1μs, leading to quasineutral region dominated recombination. In the radial pn junction case, the cell radiusRis set equal toLn, a condition that was found to be near optimal.
that τn ≈ 1.4 ns) in the quasineutral regions and τn0 = 1 μs in the depletion region, the maximal efficiency of the radial pn junction geometry was 13%, compared with 5% in the planar geometry. This maximal efficiency occurred for a radial pn junction cell between 100 and 240 μm thick, whereas the efficiency saturated in the planar geometry when thickness exceeded 5μm.
40
Figure 2.14. Open-circuit voltageVoc vs. cell thicknessLand quasineutral region minority- electron diffusion length Ln for (a) a conventional planar pn junction Si cell and (b) a radial pn junction Si cell. Depletion region trap density is set equal to quasineutral region recombination density, leading to depletion region dominated recombination at low values of Ln. In the radial pn junction case, the cell radiusR is set equal to Ln, a condition that was found to be near optimal.
case (Fig. 2.14). In the planar case, the open-circuit voltage is independent of cell thickness and decreases as the quasineutral region electron diffusion length decreases - slowly until Ln dropped to∼2 μm, and more rapidly asLn dropped below this value. In the radial pn junction case, the open-circuit voltage decreases as the cell thickness increases because the junction area increases. Also, the open-circuit voltage decreases as the electron diffusion length decreases in the quasineutral region - slowly untilLndropped to a critical value, and more rapidly as Ln dropped below this value. What this critical value was depended on the thickness of the cellL. WithL = 100μm, this value was∼5 μm. For a 100-μm-thick Si solar cell, Voc in the radial pn junction geometry dropped from 0.58 to 0.01 V as τn
decreased from 1.4 μs to 14 ps. In the planar geometry, Voc dropped from 0.59 to 0.24 V over the same range.
In this regime, the fill factor was nearly constant at ∼ 0.8 as a function of L and Ln
for both planar and radial geometries except forLn≤5μm (i.e., forτn 40 ns), when the fill factor began to drop, steeply for the radial pn junction cell and more gradually for the planar cell (Fig. 2.15).
Taken together, the factors discussed above imply that depletion-region trap density
Figure 2.15. Fill factor F F vs. cell thicknessL and quasineutral region minority-electron diffusion length Ln for (a) a conventional planar pn junction Si cell and (b) a radial pn junction Si cell. Depletion region trap density is set equal to quasineutral region recombi- nation density, leading to depletion region dominated recombination at low values 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 2.16. Efficiency η vs. cell thickness L and quasineutral region minority-electron diffusion length Ln for (a) a conventional planar pn junction Si cell and (b) a radial pn junction Si cell. Depletion region trap density is set equal to quasineutral region recombi- nation density, leading to depletion region dominated recombination at low values 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.
42
limits the ability of the radial geometry to ameliorate the negative effects of low diffusion length. Specifically, for a radial junction Si cell of thickness L = 100 μm, a change in slope appears in the efficiency vs. Ln relationship at Ln ≈ 5 μm (Fig. 2.16). AsLn drops below this value, depletion region recombination dominates, and efficiency drops much more rapidly with decreasing minority carrier lifetime.
For a Si solar cell with τn set to 14 ps throughout the cell, so that Ln = 100 nm, the maximal efficiency of the radial pn junction geometry was 1%, compared with 0.5% in the planar geometry. In this case, the maximal efficiency occurred for a radial pn junction cell between 1 and 30μm thick, whereas the efficiency saturated for a planar cell thicker than 1 μm. For a Si cell with τn set to 1.4 ns throughout the cell, so that Ln = 1 μm, the maximal efficiency of the radial pn junction geometry was 7%, compared with 4% in the planar geometry. In this case, the maximal efficiency occurred for a radial pn junction cell between 5 and 100 μm thick, whereas the efficiency saturated for a planar cell thicker than 5μm. For a Si cell withτnset to 36 ns throughout the cell, so thatLn= 5μm, the maximal efficiency of the radial pn junction geometry was 13%, compared with 10% in the planar geometry. For a Si cell withτn set to 140 ns throughout the cell, so that Ln = 10 μm, the maximal efficiency of the radial pn junction geometry was 14%, compared with 12% in the planar geometry.
Note that these simulations were performed with a surface recombination velocity S = 1 ×105 cm s−1 assumed throughout, and this limits the maximum efficiency attainable in the radial geometry when quasi-neutral region recombination dominates depletion region recombination. Also, settingR =Ln was only approximately optimal. As seen in Fig. 2.9, the maximum efficiency observed when τn was set to 140 ns throughout the cell, so that Ln = 10μm, with S = 1000 cm s−1, was η = 16.8 %, when the wire radius R = 5.5 μm, and wire length L = 90 μm. In this case, Jsc = 0.0369 A cm−2, Voc = 0.588 V, and F F
= 0.775. Decreasing S to 100 cm s−1 only increased the efficiency to 16.9 %. On the other hand, with S = 1 × 104 cm s−1, the maximum efficiency was 16.2 %, and with S = 1 × 105 cm s−1, the maximum efficiency dropped further to 14.9 %, all at optimal radiiR = 5.5μm.
Our group has recently reported the ability to grow Si wires with minority carrier diffusion lengths of ≈ 10 μm when using Ni as a growth catalyst, [54] by techniques that will be discussed in Ch. 4, which makes the above simulations potentially of great relevance
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).