The *Z* calculator determines the optical pathlength enhancement of a solar cell.

Version 1.1.1, 20-Nov-2013.

# BACKGROUND

The object of any solar cell is to absorb photons and to convert their energy into electricity.

The number of absorbed photons can be increased by (i) guiding light at an oblique path within the solar cell,
and (ii) reflecting light at internal surfaces. In each case, the light travels further within the active region
of the solar cell, increasing the likelihood that every photon is absorbed.

One way to quantify how well a solar cell absorbs light is to define its ‘optical width’. The optical width
represents the width that the solar cell would have if it did not divert the light at an angle or reflect the
light at internal surfaces, but still absorbed the same amount of light.

The optical pathlength enhancement *Z* is then defined as the ratio of the optical width *W*_{opt} to
the actual width *W*:

*Z* = *W*_{opt} / *W*.

To put these definitions into context, Figure 1 shows a light ray traversing a solar cell with (a) *Z* = 1, (b) *Z*
= 1/cos*θ*, and (c) *Z* > 1/cos*θ*.

In Figure 1a, the light travels perpendicularly to the plane of the solar cell. It is not reflected at the rear surface.
In this case, the optical thickness is the same as the actual thickness and therefore *Z* = 1.

In Figure 1b, the light travels at an angle *θ* to the plane of the solar cell. It is not reflected at the rear surface.
In this case, the optical thickness is 1/cos*θ* times greater than the actual thickness. E.g., when *θ*
is 60^{o}, *W*_{opt} = 2⋅*W*, and therefore *Z* = 2.

In Figure 1c, the light travels at an angle *θ* to the plane of the solar cell and some fraction is reflected at the rear surface.
Some proportion of this might be reflected by the front surface as well.

**Figure 1**

*Z* CALCULATOR—GENERAL

The *Z* calculator is the equivalent of tracing a single ray through a solar cell.

For each pass the ray travels through the solar cell, the user defines the angle of propagation *θ* and the fraction of
the ray that is reflected at the next interface *R*. OPAL 2, can be
used to determine reasonable reflection values for a given interface morphology and angle of incidence.

The calculator then determines the infinite sum as described by the equations below.

As its outputs, the calculator gives *Z* and *W*_{opt}. It also quantifies the losses as fractions of the incident
intensity.

# USES FOR *Z*

In conventional silicon solar cells, *Z* is independent of *W* and depends only on the surfaces of the solar cell.
This makes it a useful metric for comparing the light trapping ability of solar cell surfaces.

In standard screen-printed solar cells, *Z* is between about 2 and 5 [Ref?]. It is low because internal reflection at the rear Al
is low and free-carrier absorption is high. In high-efficiency rear-contact solar cells, *Z* has been measured at ~6 [McI04];
in these cells, internal reflection at the rear is high but free-carrier absorption is significant. And in an ‘ideal’ solar cell—where
light enters from one side with unity and Lambertian transmission, and where the other side has unity reflection—it can be shown
that *Z* = 4*n*^{2}, where *n* is the refractive index of the active material [Cam87, Section II].
(If *Z* were infinite, no light could escape the structure, which means that no light could enter it either, and the structure
would not deserve to be called a solar cell. Be wary of trying to maximise *Z*!)

*Z* can also be used to quantify the generation current *J*_{G} in a solar cell:

where *q* is the charge of an electron, *I*_{T} is the photon flux transmitted into the solar cell, *α* is the
absorption coefficient, and the integral is performed over all wavelengths *λ*.

Thus, with knowledge of *I*_{T}, one can observe how *Z* and *W* affect the generation current, and hence the
efficiency of a solar cell.

Note that *Z* is typically dependent on *λ* because it cannot exceed 1 without either refraction or reflection,
both of which are wavelength-dependent. In many cells, however, *Z* is only relevant in a small wavelength range (900–1200 nm
for conventional silicon solar cells) because effectively all shorter-wavelength photons are absorbed within *W*,
and effectively no longer-wavelength photons are absorbed within *W*_{opt}.

OPAL 2 uses the above equation to calculate *J*_{G}
but assumes a single *Z* that is independent of *λ*.

# MORE ON *Z*

*Z* is a simplification of the ‘light trapping’ ability of a solar cell. It does not contain any information about where the
light is absorbed in the cell.

Consequently, if the collection of carriers within a solar cell depends on where they are generated, it cannot be construed from *Z*.
Put differently, one might be able to determine the generation current *J*_{G} from *Z*, but more information is needed
to determine the collection current *J*_{L}.

# ASSUMPTIONS IN THE *Z* CALCULATOR

The *Z* calculator makes two assumptions in determining *Z*. The first is a major approximation that can be difficult to justify;
the second is minor.

The first assumption is that all light transmitted into a solar cell can be represented by a single ray. This is only true when both surfaces
of a sample are planar, parallel, and specular. For textured surfaces in particular, the use of a single ray is a gross approximation.

Nevertheless, by investigating the results of full ray tracing studies, one can approximate their results with a single ray. The dropdown box
on the calculator page provides a few examples that we deduced from Figures 6 and 10 of Campbell and Green [Cam87]. In this case, we set *θ*
for each pass and found *R*_{intF} that matched their results (since in [Cam87], *R*_{intR} was set to 1 and FCA was
neglected). One can then use the *Z* calculator to vary *R*_{intR} and FCA to see how they affect *Z* for the various structures.
Note that the ray tracing in this study also contains approximations (like rays cannot exit one pyramid and re-enter another.)

Another useful approximation is that after Lambertian reflection, the ‘average’ angle of transmission is 60^{o}, since the
average pathlength of light across the substrate is 2⋅*W* [Cam87, Section II].

We note that a full ray tracer, like Sunrays or Raysim, will always be more accurate than this single-ray *Z* calculator, but they also
take longer to compute. We aim to offer a multi-ray *Z* calculator in 2013.

The second assumption entailed by the *Z* calculator is that any free-carrier absorption occurs only at the surfaces and in a layer of zero
thickness.

Having described the assumptions, it is also worth emphasising that, in general, the efficiency of a solar cell does not depend critically on
*Z* or the location of the carrier generation. Consequently, approximate values of *Z* are sufficient for many investigation (such
as finding the optimum ARC using OPAL 2).

# FREE-CARRIER ABSORPTION (FCA)

Not all absorbed photons are beneficial to a solar cell. To be beneficial, they need to generate free carriers by ejecting an electron from
the valence band to the conduction band; i.e., band-to-band absorption. But some photons, particularly those of low energy, can be absorbed
by carriers that are already ‘free’ and this energy is wasted. This process is referred to as free-carrier absorption (FCA).

Typically, FCA is only significant in the ‘surface diffusions’ of a solar cell, like in the emitter or the back-surface field. In these regions,
the doping concentration is high and there are many more free carriers than in the bulk of the cell (by several orders of magnitude).

The *Z* calculator gives the user the option to account for FCA. In this case, the user defines the fraction of photons that are absorbed
in a single pass through the surface diffusion. This fraction can be determined by the
FCA calculator.

Note that the fraction of light absorbed by FCA per pass depends on *θ*, but not by the ratio 1/cos*θ* because it is not
linearly related to the free-carrier concentration. Be sure to insert the appropriate FCA fractions for every *θ* used.

Also note that FCA depends on wavelength.

# EQUATIONS NEGLECTING FCA

The optical pathlength enhancement *Z* is the ratio of the optical width *W*_{opt} to the actual width *W*,

where

.

In Equation 2, *f*_{Lp} is the fraction of the ray intensity that is lost (i.e., not reflected) at the end of pass *p*,
and *d*_{p} is the distance the light has traveled after pass *p*; this distance depends on the sample width *W* and
the angle of propagation *θ* by the expression,

.

The fraction of the ray intensity that is lost at the end of pass *p* is

*f*_{Lp} = *f*_{p} ⋅ (1 – *R*_{p}),

where *R*_{p} is the internal reflection at the end of pass *p*, and *f*_{p} is the fraction of the ray intensity
remaining just prior to that reflection, given by

*f*_{p} = *f*_{p–1} ⋅ *R*_{p–1}

Finally, the fractions lost through the front and rear surfaces are given by

,

,

In fact, the calculator does not perform the infinite sum, as suggested by Equations 2 and 6, but approximates the infinite sum by summing
until the fraction of remaining light is decreased to *f*_{p} <
Computation time: 0.001%, or *p* = Computation time: 10000, which ever comes first.

# EQUATIONS INCLUDING FCA

When FCA is included the equations are more complicated. They are difficult to follow, as evident here, and any improvement on this representation
will be gratefully received, uploaded, and acknowledged.

Anyway, to include FCA in the equations, replace Equation 2 with

.

where *f*_{FCA_Ap} is the fraction of the ray intensity absorbed by FCA at the surface where the pass commenced,
and *f*_{FCA_Bp} is the fraction of the ray intensity absorbed by FCA at the surface where pass ends. It is assumed
that the FCA occurs within a layer of zero thickness.

Thus, for a pass where the ray is travelling downwards from front to rear, the subscript *A* represents *F* (front),
and *B* represents *R* (rear). When the ray is travelling upwards from rear to front, *A* represents *R*
and *B* represents *F*.

Note also that *d*_{p=0} = 0.

Replace Equation 4 with

*f*_{FCA_Ap} = *f*_{p–1} ⋅ (1 – *R*_{p–1}) ⋅ *FCA*_{Ap},

*f*_{FCA_Bp} = *f*_{p–1} ⋅ (1 – *R*_{p–1}) ⋅ *FCA*_{Ap} ⋅ *FCA*_{Bp},

*f*_{Lp} = *f*_{p} ⋅ (1 – *R*_{p}) ⋅ *FCA*_{Ap} ⋅ *FCA*_{Bp},

where *FCA*_{Ap} and *FCA*_{Bp} are the fractions of light absorbed by FCA in the surface diffusion
as given by the FCA calculator.
More specifically, *FCA*_{Ap} and *FCA*_{Bp} are the fractions of light
absorbed by light passing solely through the surface diffusion in one pass.

Finally, replace Equation 5 by

*f*_{p} = *FCA*_{Ap} ⋅ *FCA*_{Bp},

*f*_{p} = *f*_{p–1} ⋅ *R*_{p–1} ⋅ *FCA*_{Ap} ⋅ *FCA*_{Bp},

The total fraction of light lost by not reflecting at the surfaces remains the same as that given by Equation 6.
The total fraction of light lost to FCA at the front and rear surfaces is given by

,

.

# FEEDBACK

Please email corrections, comments or suggestions to support@pvlighthouse.com.au.

# REFERENCES

| |

[Cam87] | P. Campbell and M.A. Green, "Light trapping properties of pyramidally textured surfaces," *Journal of Applied Physics* **62**, pp. 243–249, 1987. |

[McI04] | K.R. McIntosh, N. Shaw and J.E. Cotter, "Light trapping in SunPower’s rear-contact solar cells," *Proc. 19th EC PVSEC*, Paris, pp. 844–847, 2004. |