2.2 “To First Order”: Salient Features
2.4 Numerical Assessment of Device Behavior in Si
2.4.1 Preliminary Observations
The behavior of the cells was first investigated as a function of doping levels in the emitter and base, emitter thickness, and wire radius. Some general conditions for an optimal cell thus became apparent. At a given value of the minority-electron diffusion length, radial junction cells favored high doping. Furthermore, smaller wire radii necessitated high doping to prevent full depletion of the wire core. Carrier mobility is coupled to the doping in a well known fashion in Si, [52] the lifetime is related to the trap density in (2.49) and (2.50) above, and the mobility, lifetime, and diffusion length are related in (2.53)-(2.55). At a fixed trap density, increasing the doping will decrease the mobility and hence decrease the diffusion length. On the other hand, increasing the doping level will increase the built-in voltage, through (2.48). And because carriers travel a mean distance of one diffusion length through a quasineutral region before recombining, setting the wire radius approximately equal to the minority-electron diffusion length allows carriers to traverse the cell radially even if the diffusion length is low, provided that the trap density is relatively low in the depletion region.
Further insight into the differences between the two cell geometries was obtained by evaluating the efficiency η, Voc, and Jsc, versus L and R for a radial pn junction cell (Figs. 2.7, 2.8, and 2.9). Jsc increased with increasing wire length, plateauing when the
length of the wire became much greater than the optical thickness of the material. Also,Jsc
was essentially independent of wire radius, provided that the radius was less thanLn. The value ofJsc decreased steeply for R > Ln. Jsc was essentially independent of trap density in the depletion region.
The open circuit voltage Voc decreased with increasing wire length, and increased with increasing wire radius. The extent to which Voc decreased with increasing wire length depended strongly on the trap density in the depletion region: as the trap density became high (>∼3×1015 cm−3 for Si) in the depletion region, theVoc declined rapidly. The trap density in the quasineutral regions, on the other hand, had relatively less effect on Voc.
We can thus identify two regimes. In the regime of low depletion region trap density (3× 1015 cm−3 for Si), in which quasineutral region recombination dominates depletion region recombination, Voc is lost through the geometric increase in pn junction interface area γ, and the subsequent decrease in light-generated current relative to dark current, per unit junction area, as described above in Ch. 2.2, as well as in [49]. In the regime in which depletion region recombination dominates (Nr 3×1015 cm−3 for Si), the high trap density effectively greatly increases the dark current, in addition to the geometrical effects.
The optimal wire dimensions are obtained when the wire radiusR is between about 0.5 and 1 times the minority carrier diffusion length in the core of the wires (Ln for a p-type core) and a length that is determined by the specific trade-off between the increase in Jsc
and the decrease in Voc with length (Figs. 2.7, 2.8, and 2.9). If the trap density in the depletion region is relatively low (i.e., <∼3×1015 cm−3 for Si), the maximum efficiency occurs for wires having a length approximately equal to the optical thickness (∼ 125 μm for Si). For higher trap densities in the depletion region, shorter wire lengths are optimal.
Because majority carrier transport issues are neglected, this model shows that in the planar case the efficiency reaches a limiting value as the thickness increases. In contrast, the efficiency of the radial pn junction cell attains a maximum as a function of thickness - if the thickness is increased further, the efficiency is reduced. This behavior can be understood by realizing that the light-generated current density goes as
Jl∼(1−e−αL), (2.56)
whereα is the absorption coefficient of the material, andLis the cell thickness, while dark
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Figure 2.7. Example of (a) 3D and (b) plan views of efficiency η, and 3D views of (c) Jsc, and (d) Voc, versus cell thickness L and wire radius R for a radial pn junction cell. This particular example is for a Si wire withLn= 1μm in the p-type core, with depletion region trap density fixed at 1014 cm−3, and with surface recombination velocity S of 105 cm s−1 at the external surface of the wires. Note that peak efficiency is obtained when L ≈ 100 μm and R ≈0.5 - 1 ×Ln.
Figure 2.8. Example of (a) 3D and (b) plan views of efficiency η, and 3D views of (c) Jsc, and (d) Voc, versus cell thickness L and wire radius R for a radial pn junction cell. This particular example is for a Si wire withLn = 1μm in the p-type core, with a homogeneous trap distribution, and with surface recombination velocity S of 105 cm s−1 at the external surface of the wires. Note that in this case of depletion region dominated recombination, optimal efficiency occurs at cell thicknesses of far less than 100 μm.
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Figure 2.9. Example of (a) 3D and (b) plan views of efficiency η, and 3D views of (c) Jsc, and (d) Voc, versus cell thickness L and wire radius R for a radial pn junction cell. This particular example is for a Si wire withLn= 10μm in the p-type core, with a homogeneous trap distribution, and with surface recombination velocityS of 1000 cm s−1 at the external surface of the wires. Note that peak efficiency is obtained when L ≈100 μm and R ≈0.5 - 1× Ln.
current density goes as
J0 ∼L. (2.57)
The competition between these two effects determines the optimum thickness of the radial pn junction cell for maximum energy conversion efficiency.