Solution to Exercise 8.1-3

Characteristics of Real Solar Cells

This is the starting point for doing the exercise:

Equivalent circuit diagram of a solar cell

I	=	I₁·	æ ç è	exp	eU_eff kT	– 1	ö ÷ ø	+	I₂·	æ ç è	exp	eU_eff nkT	– 1	ö ÷ ø	+	U_eff R_SH	– I_Ph

U_eff	=	U – I · R_SE

Discuss qualitatively the influence of the two resistors (and, as a more minor point, the idealiy factor n) on the IV characteristics.

We will get to this, but here we will actully discuss the questions first quantitatively. As input parameters we need j₁, j₂ and the ideality factor n, which we take as (see also exercise 8.1-5)

j₁ = 10^–9 A/cm².
j₂ = 10^–7 A/cm².
n = 2.

All we have to do is to solve the the equation from above numerically for various values of the parameters:

Series resistance R_SE.
Shunt resistance R_SH.
Diode ideality factor n.
Pre-exponential factors I₁ and I₂.
And possibly the whole thing as a function of temperature T.

This is a big program, but it is not too difficult to see some major points.

Series resistance R_SE

Here is a plot of the IV characteristics of a typical solar cell with 5 different series resistances R_SE.

Everything else has been kept "ideal". This means that the shunt resistance R_SH is very large ("infinity"), the ideality factor of the second diode is n = 2, and the two pre-exponential factors are I₁ = 0,1 µA and I₂ = 10 µA. The photo current is 3 A.

Even without looking a the (numerically) calculated figure, we can deduce qualitatively a few facts from our basic equation above, aw asked in the exercise.

For I = 0 A, we have U_eff = U. That means that all IU-characteristics must run through U_OC, no matter what kind of serial resistance we might have.

For large negative U (reverse direction) , the current I is simply constant. We loose a part of the applied voltage in the serial resistance, but that does not effect the current. The characteristics in the 3rd quadrant thus does not depend on R_SE if |U| is large enough.

For large positive U (forward direction), the diode by itself will admit large currents for voltages above about 0.5 V, i.e. the diode resistance becomes very low. The IU-characteristics then must be dominated by R_SE; it will simply turn into an ohmic straight line with a slope given by 1/R_SE

In the fourth quadrant for voltages below U_OC some of the voltage drops a the series resistor. The magnitude of the current thus can only be lower than in the case without a series resistor.

All of this is exactly what the calculated figure shows- taking into account the "Ohmic" straight line we could have derived most of the graph above without any quantitative calculations

We now can draw several conclusions:

1. The efficiency h is proportional to the product of I_Sc · U_OC · FF. As long as R_SE is not too large (e.g. R_SE < 100 mW for the example given), series resistances primarily decrease the fill factor FF and thus reduce the efficiency h.

2. While in normal "electrical" life, "milliohms" hardly count, a few mW serial resistance are enough to make your solar cell measurable worse.

3. Given the specific resistivity of good metals of r » 2 µWcm, a Cu wire of 1 cm length and 1 mm² cross section has a resistance of R = 2 mW. The cross sectional area of the grid metallization on a solar cell is < 1 mm², which means we have a real and unavoidable problem with series resistances of real solar cells!

Shunt resistance R_SH

Here is a plot of the IV characteristics of a typical solar cell with 4 different shunt resistances R_SH.

IV characteristics of a solar cell with shunt resistance

Everything else has been kept "ideal". This means that the series resistance R_SE is now close to zero.

Again, without looking a the (numerically) calculated figure, we can easily deduce qualitatively what is going to happen.

U_eff = U is always true. For U_eff = U = 0 V all characteristics must run through I_SC since the term U_eff/R_SH is zero.

Otherwise, for any voltage U in reverse and forward direction we have a current I_SH = U_eff/R_SH that must be added to the diode current and thus shifts the total current upwards (towards larger values (-1 is larger than -2!) and thus decreases its magnitude in the fourth quadrant by just _Ueff/R_SH. The flat part of the ideal characteristic thus turns into a straight line with slope 1/R_SH

In the fourth quadrant, where it counts, we will loose voltage and fill factor and thus severely reduce the efficiency h.

We also have a reverse current increasing linearly with the reverse voltage - very bad in a module!

All of this is exactly what the calculated figure shows- taking into account the "Ohmic" straight line centered at I_SC we could have derived most of the graph above without any quantitative calculations

We now can draw several conclusions:

1. The efficiency h is proportional to the product of I_Sc · U_OC · FF. As long as R_SH is not too small (>» 1W) for the example given, shunts are not too bad. Real short circuits < » 1W, however, are disastrous for the efficiency.

2. Since the pn-junction is very large and extends all the way out to the edge of the solar cell, we must expect that local short circuits happen. The rather difficult question coming up now is how a few local short circuits affect the global solar cell.

	We have now answered the exercise questions.
		However, we will go on and discuss a few more points.

Ideality factor n

Here is a plot of the IU characteristics of a typical solar cell with deviations from ideality expressed in the ideality factor n and the pre-exponential factors I₁ and I₂

IV characteristics of a solar cell and ideality factor

Everything else has been kept simple - no shunt or series resistors. This means that the shunt resistance R_SH is very large ("infinity"), R_SE is zero. The ideality factor of the second diode is n₂ = 2 or n = 3 (the ideality factor of the first diode is always n₁ = 1 by definition), and the two pre-exponential factors are I₁ = 0,1 µA and I₂ = 10 µA as starting values once more, but also 10 times and 100 times that number. The photo current is 3 A.

Without looking a the (numerically) calculated figure, we cannot easily deduce what is going to happen.

Well, looking a the figure, we see that changing the ideality factor of the second diode from n = 2 to n = 3 does not produce a noticeable change in the characteristics. The simple reason for this is that in reverse direction the exponentials in the I(U) equation don't matter, and that in the forward characteristics the ideal diode always "wins" except for small positive voltages.

However, the relation between the two diodes is also influenced by the pre-exponential factors. Divided by the cell area, they were abbreviations for the following current densities:

j₁ =	æ ç è	e · L · n_i ² t · N_A		+	e · L · n_i ² t · N_D	ö ÷ ø

j₂ =	æ ç è	e · n_i · d(U) t	ö ÷ ø

Why did we pick j₂ so much larger than j₁? We have, in fact, already discussed the relation j₂ / j₁ for pn-junctions, even so you probably forgot it all, and found that j₂ >> j₁ is unavoidable for Si and other semiconductors with bandgaps <» 1 eV. This is why the values chosen for the pre-exponential factors and given above have the relation they have.

On the other hand, both factors are functions of variables like the diffusion length L or the recombination time t, i.e. of crystal perfection; of the doping N_Dop; and of the temperature T (via the intrinsic carrier concentration n_i).

I₁ and I₂ are thus variables up to a point, and we want them as small as possible because the currents they cause diminish the photo current and the open circuit voltage. The figure shows that clearly. Increasing I₁ or I₂ substantially, decreases U_OC and, for the case of I₂, also the fill factor FF.

However, they should not be too small, either. If they would be zero, we would just have a constant photo current and no voltage ever builds up. The values chosen are rather optimal, that's why we called them "ideal".

Temperature T

Our basic equation on top contains the temperature explicitly in the two exponentials and implicitly in the two pre-exponential factors j₁ and j₂.

The two equations right above for these two factors contain n_i(T), the intrinsic carrier density, which grows exponentially with increasing temperature.

On top of that, the lifetime t might be temperature dependent as well as the series and parallel resistors, but we will neglect that here.

So what it the total effect of temperature? This is shown below for negligible resistances and an ideality factor n = 2.

Temperature dependence of solar cell IV characteristics

What we have is quite clear: As long as I_Ph does not depend on temperature (e.g. because we have a very good solar cell where all photo generated carriers are turned into photo current), the influence of the temperature comes from the exponents of our basic equation and from the (exponentially; via n_i)) temperature dependent j₁ and j₂

The major effect is that the open circuit voltage decreases a lot (which is bad).

Taking into account that I_Ph might be somewhat temperature dependent too (via the temperature dependence of the diffusion length, for example), that the series and shunt resistors most likely will be temperature dependent like most everything else, the situation can become quite complicated.

However, the total effect is practically always that the efficiency comes down quite a bit with increasing temperature - high temperatures are bad for solar cells!

This gives at least some comfort to cold and sun-deprived areas like Schleswig-Holstein. We may not have as much sun as the people in Spain or Sicily, but we don't have to worry as much about keeping our solar cells cool!

Final Conclusion

	We now can draw some conclusions:

	If you want to understand solar cells at the most fundamental non-trivial (= University) level, you better make damn sure that you understand the basic equation above and all its connotations by heart!
		That's not as difficult as it may appear! It's all in "Introduction to Materials Science II" - see chapter 6! That's why we spend so much time on the pn-junction and its finer details, which are not usually covered in standard undergraduate text books.