Comparison between distributed circuit simulations and

In this chapter the results obtained by distributed circuit simulations, which were presented in chapter 7.3.2 (constant distance between two adjacent semiconductor fingers of 512.5 µ m), are compared to the ones obtained by using the area weighted mean IV characteristic Jmean(V), which is calculated by the means of the local, series resistance free IV characteristics Jlocal,without Rs(V) according to

( )

J A A

V

J _{localwithoutRs}

i i total

mean ,

4

1

) 1

(

∑

=

= (7.1)

with A_total: area of the whole symmetry element

Ai: areas of the regions with different local IV characteristics according to Fig. 7.7.

Fig. 7.7: Schematic of the symmetry element. In case of contact pattern B, the four areas have different local IV characteristics, in case of contact pattern A, area 1 and area 3 have the same local IV characteristics

Fig. 7.8 shows the relative difference between the IV characteristic parameter calculated by distributed circuit simulations P_dcs and the one obtained by the area weighted mean P_awm according to rel. difference = (P_dcs – P_awm)/P_dcs.

Effect on short circuit current density (Fig. 7.8 a)

All short circuit current density-characteristics obtained by distributed circuit simulations agree quite well with the accordant results obtained by the area weighted mean as the short circuit current density is not much influenced by series resistance effects.

Effect on open circuit voltage (Fig. 7.8 b)

The relative differences between the Voc determined by distributed circuit simulations and the ones calculated with the area weighted mean are very low (less than 0.12%).

Therefore the chosen resolution of the underlying local IV characteristics with 0.5 mV and of the distributed circuit simulations with 5 mV around Voc is comparatively high, which results in the discontinuous behavior of the characteristics especially for low distances between two adjacent metal fingers.

Nevertheless it can be detected, that the open circuit voltage is overestimated using the area weighted mean IV characteristic without considering the series resistances compared to the distributed circuit simulations in all cases. This is caused by the internal current flows from the illuminated regions into the ones beneath the metallization, which also occur at open circuit conditions ([77] and chapter 5). These are taken into account in the distributed circuit simulations but non in the area weighted mean IV characteristic.

The overestimation increases with increasing distance between two adjacent metal fingers due to the increasing series resistance.

For distances between two adjacent fingers up to 2 mm an emitter sheet resistance of 150 Ohm/sq and the according local IV characteristics in the lightly doped region (closed black squares) results in a smaller overestimation of V_oc than using an emitter sheet resistance of 350 Ohm/sq and the according local IV characteristics (closed red circles and green triangles) – using in both cases contact pattern A (full contact between metal finger and emitter). But the first characteristic has a steeper slope than the one of the 350 Ohm/sq emitter. As the difference in V_oc between area weighted mean and distributed circuit simulations depends on the product of current and resistance [77], the lower overestimation of Voc using the 150 Ohm/sq emitter compared to using the 350 Ohm/sq one is due to the lower emitter sheet resistance.

The steeper slope of the characteristic using the 150 Ohm/sq emitter in the lightly doped region is due to the higher absolute value of local current density in the lightly doped region if the 150 Ohm/sq emitter is used compared to the 350 Ohm/sq emitter.

This is shown in Fig. 7.9 a), b) and c) exemplarily for a distance between two adjacent metal fingers of 2 mm.

In case of contact pattern B (point contacts between metal finger and emitter), (open symbols) the difference in V_oc is lower compared to the one using contact pattern A (full contact between metal finger and emitter), (according closed symbols). The reason for this is analyzed regarding the differences in the local voltage maps under external open circuit conditions between contact pattern A and contact pattern B for the same symmetry element dimensions as analyzed just above (see Fig. 7.9 d). In this case the local voltages using contact pattern B are between 0.35 mV and 0.61 mV higher than if contact pattern A is used, which results in lower absolute values of local currents (Fig. 7.9 c) and therefore in a lower voltage difference between area weighted mean and distributed circuit simulations.

0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2

relative difference in J sc [%]

distance between two adjacent metal fingers [mm]

relative difference in V mpp [%]

distance between two adjacent metal fingers [mm]

relative difference in J mpp [%]

distance between two adjacent metal fingers [mm]

*: lowly doped region

**: width of semiconductor finger on the symmetry element

a) b)

c) d)

Contact pattern B

(point contacts between metal finger and emitter)

R_sheet* ρ_cont w_sf**

(full contact between metal finger and emitter)

R_sheet* ρ_cont w_sf**

[Ohm/sq] [Ohm cm²] [µm]

150 0.0001 15 350 0.0001 15 350 0.003 15

relative difference in V oc [%]

distance between two adjacent metal fingers [mm]

Relative difference between results of

distributed circuit simulations and area weighted mean

Distance between two adjacent semiconductor fingers: 512.5 µm

Fig. 7.8: Comparison between IV characteristic parameters obtained by distributed circuit simulations and using the area weighted mean IV characteristic calculated by the means of the local, series resistance free IV characteristics. Given is the relative difference between the IV characteristic parameter calculated by distributed circuit simulations Pdcs and the one obtained by the area weighted mean Pawm according to rel. difference = (P_dcs – P_awm)/P_dcs.

Fig. 7.9: a), b) Local voltages calculated by distributed circuit simulations, which are achieved when the symmetry element is operated under external open circuit conditions. In a) an emitter sheet resistance and according local IV characteristics of 350 Ohm/sq is used in the lightly doped region, in b) of 150 Ohm/sq. In both cases a distance between two adjacent metal fingers of 2 mm, a distance between two adjacent semiconductor fingers of 512.5 µm and contact pattern A (full contact between metal finger and emitter) are used.

c) Local IV characteristics of the lightly doped region. Shown is the voltage range, which occurs in the V@V_oc maps shown in a) and b).

d) Difference in local voltages under external open circuit conditions between contact pattern B (point contacts between metal finger and emitter) and contact pattern A (full contact between metal finger and emitter) for the same symmetry element dimensions as used in a).

Effect on current density at maximum power point (Fig. 7.8 c)

The differences between J_mpp obtained by distributed circuit simulations and J_mpp calculated using the series resistance free area weighted mean is low as J_mpp is not much influenced by series resistance effects. Only the characteristic using contact pattern B (point contacts between metal finger and semiconductor), a contact resistance of 0.003 Ohm cm² and a width of the semiconductor finger of 15 µm (open green triangles) shows a greater slope than the other characteristics which is due to the high series resistance effect in this case, which is not taken into account in the area weighted mean.

Effect on voltage at maximum power point (Fig. 7.8 d)

The differences between V_mpp obtained by distributed circuit simulations and V_mpp obtained using the series resistance free area weighted mean are the ones, which are major within this comparison as V_mpp is the parameter within this comparison, which is influenced most by series resistance effects.

To be able to predict V_mpp by means of an area weighted mean IV characteristic the effective series resistance has to be known. However, the analytical calculation of the effective series resistances of solar cells with semiconductor fingers is not straight forward as the current flows in the symmetry element with semiconductor fingers are two-dimensional in comparison to a symmetry element without semiconductor fingers, which has one dimensional current flow directions. This is shown in Fig. 7.10. The figure shows the current densities within the emitter resistances of the distributed circuit model in x- and y-direction and within the contact resistances for a symmetry element without semiconductor fingers (Fig. 7.10 a) and f); The y-direction of the current flow within the emitter resistances is not shown in this case as there is no current flow in this direction.) and a symmetry element with semiconductor fingers using contact pattern A (full contact between metal finger and emitter), (Fig. 7.10 b), d), and g)) and using contact pattern B (point contacts between metal finger and emitter), (Fig. 7.10 c), e) and h)).

An emitter sheet resistance of 350 Ohm/sq was used in the lightly doped region, a contact resistance of 0.0001 Ohm cm², a width of the semiconductor finger of 15 µm on the symmetry element and a distance between two adjacent semiconductor fingers of 512.5 µm. The metal finger has a width of 25 µm on the symmetry element. The distance between two adjacent metal fingers is 2 mm.

While without semiconductor fingers the current flows only in x-direction within the emitter, it flows in y- and x-direction when a semiconductor finger is implemented.

Beneath the metal finger the current flows in opposite direction than in the lightly doped region in case of contact pattern A (full contact between metal finger and emitter), which results in a two-dimensional problem to calculate the effective contact resistance in this case (Fig. 7.10 d).

Because of these two-dimensional effects analytical effective series resistances are not calculated for the semiconductor finger structure within this work, which makes distributed circuit simulations a very useful tool to analyze this structure and to predict IV characteristics.

Fig. 7.10: Current densities within the emitter and contact resistances calculated by distributed circuit simulations. Details are given on page 128.

7.3.4 Comparison between simulated efficiencies of solar cells with and