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June 16th, 2019

Minority carrier recombination in polycrystalline semiconductor materials occurs in the volume of the grains, at the grain boundaries and at the surface of the cell. The quantum efficiency model has therefore to account for all three locations of recombination. Three-dimensional modeling is necessary.

Referring to the cell geometry shown in Figure 2.28 on p. 46, we calculate the quantum efficiency under monochromatic illumination with light that has the absorption coefficient as. The spatial carrier generation rate

g(x, y,z) = asexp(-asz) (C. 13)

decays exponentially into the depth Z. The reduced absorption coefficient as is measured in units of inverse grain size G~l here. The position z = Z/G, and similarly x and y, are also scaled by the grain size. The incident photon flux is unity. With this normalization, the quantum efficiency reads

1/2 1/2 wb,,

IQE{as)= j’dx jdy jdzr]c(x, y,z)g(z)

*=-1/2 >>=-1/2 z=0

Applying the reciprocity theorem [37] that we discuss in section 3.2 on p. 57, the collection efficiency

T]c(x, y,z)= An(x, y,z) (C.15)

at any position (x, y9 z) equals the excess minority carrier concentration An(x, y,z) at this position in the dark with unity excess carrier concentration at the injecting junction. Hence, the internal quantum efficiency is

1/2 1/2 "has wbas __

IQE(as)= Jdx jdy jdzAn(x, y,z)asexp(-asz) = as jdzA«(z)exp(-a5z) (C.16)

We assume a cell with zero back surface reflectance. Integrating the carrier concentration An{z) under forward bias as given by Eq. (B.57) on p. 235, we find the internal quantum efficiency

IQE(as)=Z {a^-P. ak,

ij ds %•

[a,2 cosh(*()+ as Л. sinh(A:ywbm)] (C‘1 ?)

+ e*-’* p0 [as2 sinh(A:. )+a, k:j cosh(ktJwbas)]}

for which the coefficients are defined in Eqs. (B.54) through (B.56) on p. 234. This expression for the quantum efficiency of a thin-film cell applies for grains with a square cross-section and with the grain boundaries oriented perpendicularly to the junction. For the case of a thick cell (Wbas/G = wbas со), we find Ду = 1, and hence the expression for the quantum efficiency

is considerably simplified.

Example

With the above result for the internal quantum efficiency of poly crystalline thin-film cells, we calculate the IQE~l as a function of the optical absorption length La. Figure C.6 shows the inverse quantum efficiency IQE~l from Eq. (C.17) for a Si thin-film cell with a thickness Wf – 5 pm, a diffusion length L – 10 pm that exceeds the thickness by a factor of two, a medium quality back surface passivation with Sb= 103 cm s’1, a grain size G = 5 pm that is equal to the film thickness, and a typical grain boundary recombination

ABSORP. LENGTH La[[im]

Figure C.6. Simulated inverse internal quantum efficiency of a thin-film cell of base thickness Wbas = 5 pm with grain size G = 5 pm, diffusion length L = 10 pm, back surface recombination velocity Sb = 103 cm s”1, grain boundary recombination velocity Ssrb = 104 cm s_1, and diffusion coefficient Dn = 20 cm2 s”1.

velocity of Sgrb = 104 cm s"1. The inverse quantum efficiency IQE~l is not linear in La. Instead, the curve IQE~l(La) exhibits two linear regions, one with a smaller slope at small La that defines LQ, and the other with a larger slope for large La that defines Lc.

For other examples the slope for large La may also be less than the slope at small La.

Quantum efficiency diffusion length Lq for thin films

From the definition of the quantum efficiency diffusion length Lq in Eq. (3.4) on p. 55 and the expression for the quantum efficiency (C.17), we derive

for polycrystalline thin-film cells. We proved on p. 65 that a thin-film cell with effective diffusion length Lq has the diode saturation current jQ = qnJDn! LQ (recombination other than in the base being neglected). Thus, the diode saturation current of the polycrystalline thin-film cell equals that of an infinitely thick monocrystalline cell that has the base diffusion length L = Lq.

Quantum efficiency diffusion length Lq for thick films

Figure C.7 shows lines of the constant reduced effective diffusion length lQ = Lq /G for the case of an infinitely thick cell (Wbas > L). Then, we find Ду = 1, and thus

Z У/гА

The convergence of this series is poor for large grain boundary recombination velocities in Figure C.7. Here the calculation of 29 terms (i, j = 0,1,2, …511) is required for a precision of 1% in lQ. Figure C.7 shows that lQ is almost independent of the grain boundary recombination velocity Sgrb for diffusion lengths L much smaller than grain size G, i. e. /« 1. In contrast, the effective diffusion length lQ only depends on the grain boundary recombination velocity sgrb for very large intra-grain diffusion lengths I» 1. In the upper right-hand comer of Figure C.7 at sgrb = / = 103 we calculate an effective diffusion length lQ = 0.073. The effective diffusion length LQ is more than one order of magnitude smaller than the grain size G.

Simplified calculation of LQfor thick cells

In this section, we derive an approximate analytical equation for Iq since Eq. (C.20) requires some computational effort. The derivation of the approximate expression bases on the summation rule veJfl = Tbufl + zgrb~l for the effective lifetime rejfl := Dn /Lq2, the bulk lifetime tbulk1 = A* !L2, and the grain boundary lifetime Tgrb~l = Dn !Lgrb. Here, the grain boundary-controlled diffusion length Lgrb describes the minority carrier current into the grain

as a recombination current into a surface of recombination velocity /2 and area 4 GLgrb. We calculate a grain boundary lifetime xgrb’ = DILgrb2 = 2 SgrbD„/G. With the above summation rule for the lifetimes тф tgrb, and xgrb, we find

^J"’[3] +0.072, |

This relation also follows immediately from Eq. (C.20) by expansion to the first order in sgrb. Equation (C.22) is thus accurate only for small sgrb. To account for the limiting behavior we found for large sgrb, we use the expression

Comparison of lQ from Eqs. (C.20) and (C.23) reveals an agreement better than 30% in the range / < 103 and sgrb < 102.

Simplified calculation ofbQfor thin cells

Equation (C.22) holds only for thick cells. In order to find a simple analytic expression for Lq that also holds for thin cells with the bulk diffusion length L, the grain boundary recombination velocity Sgrb, the back surface recombination velocity Sb, the base thickness WbaS9 and grain size G, we first assume that the grain boundary recombination velocity is Sgrb = 0* This cell has a quantum efficiency diffusion length Ьд<топо given by Eq. (C.10). Now consider an infinitely thick polycrystalline cell that has an intra-grain diffusion length equal to Ьд>топо. Let this thick cell have the grain boundary

recombination velocity Sgrb of the thin polycrystalline cell. We may then use the approximate expression (C.23). We thus find an easy-to-handle, approximate expression

( ShLvMWkJL)+DncoMWbJLY |
-2 ( 1 |
gd^ |
1/2 Л + 0.072 G J |
-2> |
-1/2 |

[ SbL cosh(^„ / L)+ Z>„ sinh(ffte /L)J V |
1 |
л2 Srb J |
J |

(C.24) |

for the quantum efficiency diffusion length Lq of polycrystalline thin-film cells. Please note that the derivation of the expression assumes an excess carrier concentration at the junction that is spatially constant. For large Sgrb this assumption may become incorrect.

Collection diffusion length Lc for thin films

From the defining Eq. (3.5) for Lc and from the quantum efficiency expressed in Eq. (C.17), we erive the collection diffusion length

Lc = YjXj Pi i1 “ cosh(V^»))+ sinh(^HVs) ] (C.25)

for polycrystalline thin-film solar cells. This expression also holds for cells with light trapping and no parasitic absorption (absorption that does not create electron-hole pairs), since we use the internal quantum efficiency to define Lc. In general, solar cells exhibit parasitic absorption for very weakly absorbed light, and thus the value of Lc depends in practice on the optics of the device.

The physical interpretation of Lc is as follows: the collection diffusion length Lc is the diffusion length that a thick monocrystalline cell has to have in order to yield the same short-circuit current density as a thin-film cell with no light trapping under spatially homogeneous photogeneration. Evidence for this interpretation of Lc was first given for the special case of an infinitely thick cell with either zero or infinite grain boundary recombination by Donolato [412].

Collection diffusion length Lcfor thick films

For thick cells, which means base layer thickness Wbas» L, Eq. (C.22) simplifies to

:I У’У’Т

We neglect the emitter here. This series for lc converges much faster than the series for Iq and 25 terms are sufficient for a numerical precision better than 1% in the parameter range /< 103 and sgrb < 103.

Iso-lines of the reduced collection diffusion length lc = Lc IG are shown in Figure C.8. The behavior of /cis similar to that of lQ in Figure C.7. However, in the upper right – hand comer of Figure C.8 at sgrb = 103 and / = 103 we find lc = 0.172, a value 2.4 times greater than lQ. For the effective diffusion length lc we find numerically the limiting value

253

Figure C.8. Iso-lines of the effective diffusion length /c, which depends on the grain boundary recombination velocity s^b= Sgrb G/D and the intra-grain diffusion length / = L/G. The behavior is similar to that of lc depicted in Figure C.7. However, the numerical values are different for large sgrh and large /.

We first reported this limiting value without a derivation in Eq. 4 of Ref. [197], a result that was confirmed by Donolato [412]. A poly crystalline material with grain size G and negligible volume recombination (that is, / = oo), and a worst-case grain boundary recombination (that is, sgrb = °°), collects current from weakly absorbed light as well as a material with a volume diffusion length L = 0.17 G and no grain boundary recombination.

Simplified calculation ofLcfor thick cells

A slight modification of the empirical relation (C.22)

(C.28)

of polycrystalline thin-film cells. As noted in the corresponding section on Lq, the applicability of this equation may be limited to small values of Sgrb.

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