Basic IU-Characteristics of Solar Cells

Foreword

	This module is at once the solution of an exercise and a detailed discussion of the UI-characteristics of a theoretical and practical ideal solar cell.
		If you are not already familiar with the diode equation and what exponential terms can do, you will profit very much by going through it in detail.
		For easy readability parts of the data given in the exercise will be repeated her in a somewhat modified form.

Starting Point

	The diode equation with generation and recombination in the space charge region part describes a pn-junction made from a semiconductor like Si that has well defined properties, for example the doping concentration and the diffusion length / life time. It thus also describes an "ideal" solar cell. We will now try to see what we can do with this equation with respect to solar cells.
		We use the abbreviations j1 and j2 for the current densities in the brackets and d(U) as the "abbreviation " for the equation shown below that gives us the width of a space charge region in a pn-junction.
		For the current density j(U) and the (less important) space charge region width d(U) we have
j(U)  =  æ ç è e · L · ni 2 t  · NA + e · L · ni 2 t  · ND ö ÷ ø  ·  æ ç è exp  eU kT  –  1 ö ÷ ø   +  æ ç è e · ni · d(U) t  ö ÷ ø  ·   æ ç è exp eU 2kT  –  1 ö ÷ ø  –  jPh                                                 ( j1 )           ( j2 )             d(U)   =  æ ç è 2 · eSi · e0 · (DEF  – e ·Uex) e2 · ND ö ÷ ø ½

	In evaluating this equations for solar cells as schematically given in the drawing, we must first find numerical values for important parameters (all others have their usual meaning and values). We have:
		L = diffusion length = (Dt)½ average distance a minority carrier travels between its birth by a generation event (mostly caused by light in a "working" solar cell) and its death by recombination. A suitable value for good bulk Si is L = 100 µm. D is the diffusion coefficient and t the (minority carrier) life time. A good enough value for the life time going with a diffusion length of 100 µm is t = 1 ms. ni is the intrinsic carrier concentration. It increases exponentially with temperature T. A good value for Si at room temperature (RT) is ni(RT) = 1010 cm–3. NA and ND are the acceptor and donor concentrations in the p-part called base, the usually several 100 µm thick part of a bulk Si solar cell, and the n- part, called emitter, the thin "layer" on top of the solar cell, respectively. The base is lightly doped (otherwise the diffusion length suffers) whereas the emitter is heavily doped (good conductivity is important). NA = 1016 cm–3 and ND = 1019 cm–3 are good round numbers for the purpose here. We could also calculate the width of the space charge region with the equation given but take it as d(U) = 1 µm as a first simple approximation.
		Now we consider a real but still "ideal" solar cell under "standard" illumination. This gives us the following (simplified) second set of numbers: Area of the Si bulk solar cell = 100 cm2. It's actually more like 200 cm2 in 2008 but let's stay with easy numbers. Photo current density jPh = 30 mA/cm2 at "AM 1.5" the condition of maximum solar intensity on earth ("High noon in the tropics") The photo current here is thus jPh = 3 A.

	The first question was:
		1a: Using only the first term in the bracket for j1 as a sufficient approximation, give an equation for the relation of j2/j1 and some numbers for these current densities.
		1b: Does the result imply that you can neglect one of the ji terms in the equation above in the forward direction? How about the reverse direction?

	For the numerical values of j1 and  j2 we obtain
		j1  =  e · L · ni 2 t  · Ndot   =   1.6 · 10–19 · 10–2 · 1020 10–3 · 1016 C  s · cm2  =  1.6 · 10–14 A/cm2                  j2  =  e · ni · d(U) t    =   1.6 · 10–19 · 1010 · 10–4 10–3 C  s · cm2   =   1.6 · 10–10 A/cm2
	For the relation of j1 to j2 (and using NDot instead of NA, D) we obtain
		j1 j2  =  e · (ni)2 · L t · NDot e · ni · d(U)  t  =  ni · L NDot · d(U)
		Inserting the numbers from above yields once more
		j1 j2  =  1010 · 10–2 1016 · 10–4  =  10–4
	Now we can address the second part of the first question. Since j2 is so much larger than j1, can we simply neglect the ji terms in the diode equation? We should know the answers from before:
		For biasing in the reverse direction, we have jrev » j1 + j2 » j2 and we can indeed neglect j1.
		For the forward direction - which is the one of interest to us - we have approximately
		jfor  »   j1 · exp eU kT   +   j2 · exp eU 2 · kT
		No, we cannot neglect the j1 term "just so", we also have to consider what the exponential terms will do. That will become very clear as soon as we look at the third part of the question.

	The prelude to the second question was: If we now measure the actual UI characteristics of a good real solar cell and fit the curve obtained to our equation from above, we find values for the current densities j1 and j2 like
		j1 = 10–9 A/cm2. j2 = 10–7 A/cm2
	The second question was:
		2a: Do the measured values of j1 and j2 and their relation meet your expectations based on your results from question 1?
		2b: If not, what could be reasons for the discrepancy?

	Let's compare what we have in a table:
		Calculated Measured j1 1.6 · 10–14 A/cm2 10–9 A/cm2 j2 1.6 · 10–10 A/cm2 10–7 A/cm2 j1 j2 104 102
		We neglected the second term for j1, which we will now call j1E for the time being; i.e. the reverse current flowing from the heavily doped thin n-emitter into the lightly doped p-base. If we would naively calculate jE1, we would get an even smaller value than what we already have for j1B because the doping concentration ND of the emitter is larger than NA of the base and appears in the denominator of the equation for j1E.
		However, we would commit a grave error in doing this because the diode "master" equation from above is only valid for one-dimensional junctions in "infinitely" long semiconductors, meaning that the semiconductor must extend at least a few diffusion lengths in both directions as seen from the junction. This is clearly not the case here.
		More advanced theory teaches us that in the case of "thin" semiconductors we have to replace the diffusion length L by the thickness d of the layer.. This makes sense because the diffusion length came into the equation as the dimension over which carriers are collected that could make it across the junction.
		If we use this insight, however, j1E gets even smaller because L is found in the nominator of the equation.
	A first but wrong conclusion could be the discrepancy between theory and experiment cannot come from the emitter part of the reverse current.
		However, we forgot the life time t, which we find in the denominator of the equation, and now we must take into account that heavy doping simply "kills" the life time, i.e. makes it very small. The diffusion length L gets smaller, too, but the combined effect is that L/t µ t½ so heavy doping always increases the j1 part coming from the heavily doped region, and this increase can be substantial
	As a first insight we note that a heavily doped thin emitter can indeed lead to a substantial increase in j1. But there are more reasons for this.
		At the most elementary level of deriving j1 we simply equated it with diffusion length L times the generation rate G; and G was equal to the recombination rate R = nmin/t. An increased j1 thus demands for an increase in generation -we simply need more charges to have larger currents.
		In not-so-perfect Si we might have generation of carriers at grain boundaries or at the huge surfaces in excess of what just thermal generation can produce in a perfect lattice. To be sure, the recombination rate R must still be equal to G in equilibrium, but j1 will go up with increasing generation anyway.
	We see that there are several reason why we have the discrepancy. We simply must accept that the experimental j1 and j2 values are essentially fitting parameters of something called "solar cell" that do not fall within the range of a simple theory but still allow to describe the solar cell by the "simple" ideal theory if one accepts these empirical "fitting parameters" instead of the theoretical constructs in the equation.

	Now to the third question: Given the measured ji values from above and the jPh value given, we now can consider the short circuit current ISC and the open circuit voltage UOC
	Question 3:
		3a: What do you get for ISC? Does it depend on ji and j2? If not, what determines its value?
		3b: What can you say about the open circuit voltage UOC?

	The first part is easy: ISC is what we get for U = 0 and that is simply –jPh.
		We have used –jPh as a constant in the master equation, it thus does not depend on the values of j1 or j2 or on the variables determining their numerical values.
		This is not generally correct, of course. For example, if the diffusion length L increases, more carriers generated by light deep in the volume of the solar cell can reach the junction and |jPh| should increase with L.
		However, we have assumed good solar cells along, and this means that practically all carriers generated by light end up in the photo current. This simply implies that for diffusion lengths good enough not much can be gained anymore by increasing L. In other word, if the longest distance between a generation event and the next contact is 20 µm, it just doesn't matter much if your carriers could go 200 µm or 500 µm.

	The second part is tough: If we try to solve the master equation for UOC, i.e. setting j = 0, we realize that it can't be done.
		There is no analytical expression for UOC that we can gain from the master equation. Short of going numerical, we need to consider other ways of gaining some insight, including approximations.
		One way is to go for a graphical solution of the problem. We actually have done that already, but probably not recognized what we can learn for solar cells from this. All we have to do is to draw the master equation in a log j - eU plot. This is actually a very good exercise and you should do - at least look at the solution - and learn how it's done.
		Exercise 8_1_6 log j - eU
		The result looks like this:
	From looking at the graph we can learn a lot of things
	1. The "-1" term in the master equation is only noticeable for currents <» j2.
		We can safely neglect it for solar cells as long as we have a photo currents in just the µA/ cm2 region, i.e 1 000 times smaller than the maximum photo current density on earth.
	2. The open circuit voltage UOC depends only on j1 for reasonable photo currents. Even so j2 is much larger, the exponential term going with j1 always "wins" for voltages above 0.3 V - 0.4 V.
		If we neglect the j2 term in the master equation, we can solve it for UOC and obtain for eUOC as measured in eV:
		eUOC   =   kT · ln  jPh – j1 j1   »   kT · ln jPh j1
		This means that j1 is the decisive term for UOC, one of the prime properties of a solar cell.
	Now we look at the temperature dependence of UOC. From the solution of the exercise we take only one curve here:
		Increasing the temperature has the following effects:
		The slope of both exponentials decreases. This would lead to a higher UOC. The ji increase exponentially because their defining equations contain the intrinsic carrier density ni, which increases exponentially with T. The total effect is a decrease of UOC with T
	We can see that also in the equation for UOC. Inserting  j1 = c1 · ni2 = j1' · exp–(Eg/kT) yields
		eUOC =  Eg  +  kT   · ln  jPh j1'   =   Eg  –  kT  · ln j1' jPh
		It first looks like we add something to eUOC with increasing T; increasing eUOC. However, it is important to realize that  j1' >> jPh - even if that is counterintuitive - and ln(jPh/ j1') thus is a negative number! That the open circuit voltage indeed decreases is better seen in the final formulation where we subtract a positive number from Eg.
	If you made it to this point, you learned quite a bit about basic solar cell characteristics. You also learned another thing:
		Don't rely on "feeling" if exponentials are involved!

With frame

8.1.1 Basic Solar Cell Topics

Exercise 8.1-5 IV Characteristics of Real Solar Cells

Solution to Exercise 8.1-5 IV Characteristics of Real Solar Cells

Exercise 8.1-6 Constructing Quantitative Logarithmic IV Characteristics

Solution to Exercise 8.1-6 Constructing Quantitative Logarithmic IV Characteristics

© H. Föll (Semiconductor Technology - Script)