On the improvement of tandem solar cell conversion efficiency

Renewable Energy Vol, 2, No. I, pp. 41-46. 1992 Printed in Great Britain.

0960- 1481/92 $5.00+.00 Pergamon Press Ltd

ON THE I M P R O V E M E N T OF T A N D E M SOLAR CELL CONVERSION EFFICIENCY M. A. EL-KOSHEIRY a n d A. H. M. SHOUSHA Electronics and Electrical Communications Department, Cairo University, Cairo, Egypt (Received 31 May 1991 ; accepted 23 July 1991) Abstract--The main objective of this paper is to investigate how lhe physical and structural parameters of the tandem series-connected solar cell structure can be chosen to yield higher conversion efficiencies.A quick suboptimal tandem cell design method based on the criterion of photocurrent matching in the individual cells forming the tandem structure is presented. In this method, the algorithm does not involve detailed calculations in order to obtain a suboptimal tandem cell design. In this respect, the method offers a significant speed advantage compared to the previous tandem cell efficiencycalculations. To further enhance the tandem cell efficiency, a constrained optimization program is developed and merged with the tandem solar cell model. Here, the valuesof all the relevant parameters are allowed to change simultaneously within the limits dictated by the cell's physics and fabrication technology. The resulting boost in efficiency is assessed by comparison with the common suboptimal design based on tandem cell modelling alone, and is shown to be significantin the cases considered.

1. INTRODUCTION

been attempted before. Thus, we propose a method where tandem solar cell modelling and optimization are systematically integrated in order to optimize the tandem cell design. The merit of this technique will become evident by assessing the increase in efficiency compared with the suboptimal tandem design based on devising a model and adjusting some of its parameters while fixing the others. It is important to emphasize at the outset that the specific model used to describe the tandem solar cell is not under investigation. This can be a simple model that concentrates on some few key parameters which are controlling the tandem cell operation, such as bandgaps and thicknesses. The model, on the other hand, can be more elaborate by accounting for other material and device structure parameters, as well as the possible added optical and electrical losses which would degrade the tandem cell performance. Our interest focuses on the underlying concepts of (a) applying the photocurrent matching criterion to obtain a quick suboptimal tandem cell design, and (b) optimizing an arbitrary tandem cell model for the purpose of maximizing the tandem efficiency. For the sake of definiteness, we have opted for a simple tandem cell model that has been previously applied to both crystalline and amorphous cells ill. This model can be applied for the general N-cell tandem structure, but our calculations have been limited to two- and three-cell tandems. Moreover, our calculations are for amorphous tandem solar cells, but

The advantages offered by tandem solar cells over single-junction cells have been previously investigated [1, 4, 7-12]. The most widely studied tandem structure is the one where the component cells are series-connected forming a two-terminal device requiring one external load. The photocurrent drawn from the series-connected cells will be then limited by the cell with the lowest current. The optimum condition of operation would be the one where the cell photocurrents are matched. This optimality condition has been widely asserted in the tandem cell literature and is the starting point of our proposed design method. The photocurrent matching can be achieved, in principle, by adjusting the band gaps and active layer thicknesses of the component cells. This photocurrent matching algorithm does not involve at each step detailed efficiency calculations, and will thereby offer a significant speed advantage over conventional tandem cell calculations [1]. All tandem solar cell models developed so far did not yield truly optimum structures because they adjust few of the cell parameters and fix all the others at a time. The simultaneous interaction between all of the parameters needs a comprehensive optimization program that supplements the tandem solar cell model for yielding a design with the highest possible conversion efficiency. Such merging of the tandem solar cell modelling and optimization, to our knowledge, has not 41

42

M. A. EL-KOSHEIRYand A. H. M. SHOUSHA

can be used for crystalline cells as well, in the context of the simple model adopted here.

2. TANDEM SOLAR CELL MODEL The tandem cell model relies basically on the following assumptions :

The fill factor of each cell can then be determined in the standard way, and from which the cell conversion efficiency will be calculated from r/% -

(a) The ~ V characteristics of each cell is modelled with the familiar superposition principle, as a diode in parallel with a current source that represents the photogenerated (short-circuit) current density. (b) The diode dark current is modelled using conventional closed-form expressions based on the prevailing current transport process in the junction considered (e.g. recombination, diffusion, etc.). (c) For a N-cell tandem structure, the short-circuit photocurrent density in each cell (i = 1, 2 . . . . , N) can be written as J~c~ = q

Nph(2) exp

(1)

where Nph (2) is the spectral photon distribution function derived from the solar spectrum data [2]. Optical losses are not considered in this expression for simplicity. (d) The optical absorption coefficient of the tandem cell material is assumed to follow an empirical expression, which for the case of amorphous semiconductor is [3]

~/(~)

7.516 x 1052 [1.242 1.245 ,~

1.242~ 2 ' 2~, J

(2)

where ~(2) is expressed in cm ~ and )~g~is the cutoff wavelength of the ith cell material, as determined from its hand gap, in #m. In view of assumption (b), the dark current density of a cell in the tandem structure is assumed to increase exponentially with the cell's terminal voltage at a rate determined by the junction's ideality factor n, according to the standard from

Ja = Jo(ev:"v~- 1)

(3)

where the dark saturation current density is assumed to be a simple recombination current [2]

Jo-

qn~L 2z"

100,

(6)

where P~ is the solar constant determined from the incident solar spectrum. The tandem cell conversion efficiency at the matched condition can be calculated as the sum of the individual cell conversion efficiencies. Thus, the efficiency for the case of a three-cell tandem (N = 3) will be

q = J~c

[Vo~,FF' + VocfF2+ VocfF3] Pi '

(7)

where Jsc is the matched photocurrent in the tandem cell

-- ~ 09(2)L i

x [ l - e x p (-c~j(2)L/) ] d2,

J~cV,,cFF × Pi

(4)

The open-circuit voltage for each cell in the tandem is then given by

J~c = Lc, = Lc, = Z~=~.

(8)

3. SUBOPTIMAL TANDEM CELL CALCULATIONS The algorithm [1] that makes use of the above set of equations is described in the flowchart shown in Fig. 1(a), for the case of a two-cell tandem structure (N = 2). The two subcell bandgaps are systematically adjusted so as to yield the highest efficiency under a specific solar spectrum. The cell thicknesses are assumed fixed. As illustrated, the calculations are made by assuming a bandgap value for the top cell and computing the efficiency for a series of bottom cell bandgaps. The bottom cell bandgap at the maximum efficiency is the desired unknown. This value, in conjunction with the trial value of the top bandgap, form one suboptimum combination. As it is clear, the efficiency calculations are part of the loop, and this makes the calculations expensive in terms of execution time. In the case of a three-cell tandem (N = 3), the same algorithm still applies, but a further assumption should be made for the mid-cell bandgap. Figure l(b) shows our proposed algorithm for the suboptimum tandem cell design for the case of a twocell structure (N = 2). This algorithm is based on the fact that for maximum tandem efficiency, the photocurrents in the two cells making up the tandem structure should be matched. This has been widely treated in the tandem cell literature (e.g. by Gee [4]). Here, the loop involves photocurrent calculations only and

Tandem solar cell conversion efficiency

43

(b)

(a)

Start )

Start )

t

t

Initialize

Egl"~2

t

I

t !

I

I Compute Jscland Jsc2

Compute

Jse 1, Jse 2, and 11

[

I UpdateEg21

1 Yes

S

Searchfor ~lm~ T

Optimumtandem Eg1 and Eg2

Optimumtandem I EgI and Eg2

I

t

'

Compute~q

t

( stop )

( stop )

Fig. 1. Flowchart for a two-cell tandem suboptimum design : (a) detailed calculations algorithm, and (b) matched photocurrent algorithm.

the adjustment of the relevant parameters until the matching condition is attained. The tandem efficiency calculation is then carried out once to determine the optimum tandem cell performance. Execution time on the computer is much reduced because the timeconsuming calculations of efficiency are not part of the loop. 4. OPTIMUM TANDEM CELL DESIGN The general problem for the optimum design of a tandem solar cell structure resolves into deriving an objective efficiency function in several material and structural design parameters to be maximized for optimum performance. There are constraints imposed on the material and structural parameters upon which the efficiency function depends. These constraints are limits imposed on the parameters by either the current technology or the performance specifications. The tandem cell model is inserted into a constrained optimization cycle for the purpose of enhancing the

tandem efficiency through a reproducible design that uses real-life parameters. Figure 2 shows a flowchart where both the tandem cell optimization and modelling are merged. Starting from some initial key cell parameters, the model is solved and the conversion efficiency calculated iteratively in the optimization cycle to maximize the tandem cell efficiency. The cells bandgaps and thicknesses are both taken into consideration. The design problem for maximizing the conversion efficiency of an N-cell tandem under a specific AM solar spectrum can be formulated as follows : maximize:

q(LJ, Eqi),

such as

E~,~<2.0eV;

L~ > 0 ;

i = 1,2 .....

N

(9)

E~i>0

Li ~< 2.2 #m.

The general constrained optimization problem is a hard one that needs a sophisticated method in order

44

M. A. EL-KOSHEIRYand A. H. M, SHOUSHA

Start )

f

"@. I

Set initial solar cell parameters

!

Modify parameters

Yes

Device model (analytic/ numerical) efficiency computed I

]

t

I

!

Modify parameters I

I Optimizationprogram I

No@ to maximizeefficiency

I

Optimum

I

solar cell design I

t

( stop)

results obtained from this technique are compared with those obtained from the standard method. Close agreement between both sets of results will prove the credibility of our method in suboptimal tandem cell design. All the results presented here are for small carrier life-times, which are representative of the a-Si material. Figure 3 presents the results of a two-cell tandem structure calculated in two different ways; namely, (a) matched photocurrent method and (b) detailed method. The plots show the maximum tandem efficiency variations as function of both the top and bottom cell bandgaps. These results show that the highest possible efficiency under AM 1 solar spectrum is around 15.5% and is achieved with a tandem having Eg~ = 1.75 eV and Eu2 = 1.25 eV. The results show that the proposed simple method, when compared to the detailed one, gives nearly compatible results that differ by less than 1%. Table ! shows the complete results of the highest efficiency two-cell tandem, where all the cell pertinent performance parameters (J~c, Voc,FF, and 7) are contrasted in the two methods of calculations. Again, agreement between the two sets of values in the first two columns of the table is remarkable. It is worth noting that eq. (7) should not be used in the tandem efficiency calculations when the cells

Fig. 2. Flowchart for the tandem cell optimization process.

1.7

19

(a) to obtain a practical solution [5, 6]. Intensive research on constrained optimization is still progressing at a rapid pace, and the lack of a universal method has led to the present availability of numerous algorithms. In principle, a strong constrained nonlinear optimization algorithm is needed in order to enhance the rate of convergence, which in turn will reduce the total number of computations of the tandem cell modelling undertaken. In this work, we have elected to apply a direct search method; namely, that of Hooke and Jeeves (see [5]). This unconstrained optimization procedure has been modified in this work to take account of the constraints on the tandem solar cell parameters. The details of the optimization algorithm are covered in the optimization literature [6]. 5. RESULTS AND DISCUSSION In this section we present the results obtained using the proposed method to design a tandem solar cell. First, the photocurrent matching technique is applied to two- and three-cell tandem suboptimal design. The

.

i "°

1.6

18

1.5

17

~ 1.4 =~ I.M 1.3

15

1.2

14

1.1

1

1.0

1~.6

13 1~.7

1~.8 1~9 Eg1 (e~

210

2~j

Fig. 3. Two-cell tandem maximum efficiencyas function of the top and bottom cell band gaps calculated using: (a) matched photocurrent calculations, (b) detailed efficiency calculations. The results are for AM 1 solar spectrum and the following parameter values: • = 1 0 - 9 s , L t = L2 = 0.5 #m, and n = 2.

45

Tandem solar cell conversion efficiency Table 1. Two-cell tandem design under AM1 and z = 10 9s Parameter

Eq~, eV Eq2, eV L~, ,um L2, #m J,c,, mAcm ~ J,¢~, mAcm ~' V,>~,,V V,~., V

FF~ FF, q~, % r/2, % ~/, %

Detailed ~ calculations

Matching photocurrents

Optimized values

1.750 1.240 0.50 0.50 14.603 15.718 0.940 0.434 0.794 0.658 11.173 4.599 15.746

1.750 1.268 0.50 0.50 14.603 14.595 0.940 0.458 0.794 0.669 11.173 4.583 15.759

1.91 1.36 2.00 2.00 16.49 16.49 1.032 0.486 0.807 0.680 13.22 5.59 18.81

are not photocurrent-matched, by assuming that J~c is the smallest photocurrent in the individual cells [l]. Such assumption gives lower efficiency, since the smallest photocurrent will override. In the unmatched case, the tandem cell efficiency calculation should be based on the tandem solar cell characteristic curve, which is obtained by adding the individual J - V component cell curves on a point-by-point basis. At each current level, the voltages are assumed since the cells are in series. Once the tandem cell maximum power point is identified on the composite curve, the efficiency can be computed directly. Figure 4 shows the calculated efficiency of a two-cell tandem using both eq. (7) and the general method of searching for the maximum power point on the

composite J - V curve. The results indicate that eq. (7) gives erroneous results, except at the maximum efficiency point, where the individual cell photocurrents become matched. Using the suboptimal two-cell tandem design results as the initial starting values for the optimization cycle, we can further improve the conversion efficiency. The third column of Table 1 gives the results derived from the two-cell tandem optimization. The efficiency increase is appreciable and this clarifies the importance of integrating tandem cell modelling and optimization in a single computing environment. The resulting optimized tandem design clearly shows that the photocurrent matching condition is fulfilled, and this result adds credibility to our suboptimal design technique. The proposed method is also applied to the threecell tandem structures. The suboptimal design results are shown in Fig. 5 for the two methods of calculations. It is seen from the figure that the highest computed efficiency (b) is again in close agreement with our proposed method (a). Here, the three cell photocurrents should be equal, and this requires the adjustment of the three bandgaps. Figure 5 shows the maximum tandem efficiency variations as a function of the three cell bandgaps. It is noted that the detailed three-cell tandem calculations require an extra assumption for the middle cell band gap, while the

2.0

(a) (b) . . . . .

1.8

19

"%~1/#Eg2

18

j"

20

1.6

(a)

/" .

(b) . . . . .

t="

# Eg3

17

#"

16

u.~ 1 4

16

12

1.2

15

1.0

8

o I

1.0

I

1.1

I

I

1,2 1.3 Eg1 (eV)

I

1.4

I

1.5

Fig. 4. Two-cell tandem efficiency calculation details using : (a) search for maximum power point on the composite J V curve, and (b) using eq. (7). Same solar spectrum and parameter values as in Fig. 3.

~

1

Eg2

4 " 1

~3

I

I

1.7

1,9

14

I

I

I

2.1

2.3

2,s

13

Eg1 (e~ Fig. 5. Three-cell tandem maximum efficiency as a function of the three cell band gaps using (a) matched photocurrent algorithm, and (b) detailed efficiency calculations algorithm. Same solar spectrum and parameter values as in Fig. 3. All cells are assumed to have the same thickness.

46

M.A. EL-KOSHEIRYand A. H. M. SHOUSHA

Table 2. Optimum design of a three-cell tandem under AM 1 and for z = 10-9 s Parameter Egt, eV Eq2, eV Eg3, eV Lt,/~m L2,/~m L~, ~m Jsc,, mAcm- 2 J~c~, mAcm 2 J~c, mAcm 2 V,,c,, V V,,c:, V V~,c~,V FFt FF2 FF 3 ql, % q2, % q3, % q, %

Initial value

Optimized value

1.99 1.563 1.279 0.5 0.5 0.5 9.587 9.591 9.608 1.167 0.736 0.447 0.823 0.756 0.664 9.446 5.473 2.664 17.832

2.108 1.552 1.289 0.624 0.292 0.372 9.601 9.602 9.601 1.265 0.749 0.473 0.833 0.759 0.675 10.380 5.59 3.14 19.11

top and bottom cell gaps are varied. Figure 5 shows the results of such calculations under the following assumption Eg2 = E ~ , - 0 . 4 5 which yielded the highest tandem efficiency in the proposed method. Table 2 gives the results of optimization for a threecell tandem, starting from the suboptimal design that emerged from the modelling approach alone. The increase in the tandem cell efficiency is noted. The matching between the three photocurrents further confirms the proposed suboptimal design technique. In general, more cells can be put in tandem to yield higher maximum efficiency. Our proposed method can be generalized to such N-cell tandem structures. However, for tandems with more than three cells, the efficiency increase is too small to justify their fabrication cost and complexity.

6. CONCLUSION A new method for the suboptimal tandem solar cell design has been presented. A general optimization algorithm has also been merged with the tandem cell model in order to assess the possible improvement in efficiency, The proposed suboptimal method is based on the photocurrent matching in the individual cells and has been applied to two- and three-cell tandem structures. The results obtained from this proposed method are in close agreement with the detailed method of tandem efficiency calculations. The method is simpler and faster. It should be pointed out, however, that both methods of calculations do not yield truly-optimum tandem structures because not all the relevant parameters come into play. A general optimization program that permits the concurrent adjustment of all relevant tandem cell parameters under constraints is thus adopted to enhance the tandem efficiency. In this context, all the cell parameters are allowed to change simultaneously within the a-Si physics and technology constraints imposed, and a boost in efficiency emerged. Modelling techniques alone cannot match the scope of such optimization process due to the implicit limitation of fixing some parameters and varying others. REFERENCES 1. J. C. C. Fan and B. J. Palm, Sol. Cells 10, 81 (1983). 2. A. L. Fahrenbrunch and R. H. Bube, Fundamentals of Solar Cells--Photovoltaic Solar Energy Conversion. Academic Press (1983). 3. G. D. Cody, C. R. Wronski, B. Abeles, R. B. Stephens and B. Brooks, Sol. Cells 2, 227 (1980). 4. J. M. Gee, Sol. Cells 24, 147 (1988). 5. P. E. Gill et al., Practical Optimization. Academic Press (1982). 6. B. D. Bunday and G. R. Garside, Optimization Methods in Pascal. Edward Arnold, London (1987). 7. S. M. Bedair, M. F. Lamorte and J. R. Hauser, Appl. Phys. Lett. 34, 38 (1979). 8. J. Burdick and T. Glatfelter, Sol. Cells 18, 301 (1986). 9. I. Chambouleyron, Sol. Cells 12, 393 (1984). 10. M. F. Lamorte and D. Abbott, Solid State Electron. 22, 467 (1979). 11. M. E. Nell and A. M. Barnett, IEEE Trans. Electron Devices ED-34, 257 (1987). 12. R. R. Potter and J. R. Sites, J. Appl. Phys. 53, 5269 (1982).