Solid-Stale Eleclronics, 1977, Vol. 20, pp. 119-123. Pergamon Press. Printed in Great Briti

ANALYSIS OF SILICON SOLAR CELLS WITH STRIPE GEOMETRY JUNCTIONS

CHENMING Hut and JOHN EDELBERG Center for Materials Science and Engineering, Department of Electrical Engineering and Computer Science,

Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.

(Received 4 March 1976; in reoisedfom 2 July 1976)

Abstract-Silicon solar cells with stripe geometry junctions are analyzed. The base region collection efficiency is found to be insensitive to the transmissivity of the electrode and/or the diffused layer but quite sensitive to the width and separation of the stripe junctions. The additional losses of carriers are mainly due to the increased bulk recombination rather than the surface recombination. In one case analyzed the collection etliciency decreases by 22% when the junctions are separated by about one sixth of the diffusion length with the surface recombination velocity at 3OOcmlsec. Possible uses of the stripe-junction design in p-n junction cells and Schottky barrier cells are re-examined in the light of the new calculation and found to be less attractive than previously suggested.

1. INTRODUCTION

Loferski et al. [l] have reported theoretical investigation of “grating” type photovoltaic cells made by forming a fine grating pattern of either p-type diffusions (Fig. 1) or alloyed-metal junctions into n-silicon wafers. Samples of the latter type were actually fabricated and subjected to experimental study. Much of the area of such cells consists of virgin base material; the surface recombina- tion velocity of this material should be considerably lower than the values encountered in the case of diffused surfaces. This would result in higher blue response.

Such stripe geometry junction designs can also be applied to the Schottky barrier photovoltaic solar cells to enhance the light absorption in the semiconductor and to reduce the saturation current [2]. In p-n junctions cells, there may also be an advantage in using stripe geometry diffused retions in that the junction shunt current can perhaps be reduced by reducing the junction area. This would be helpful in achieving the high conversion efficiencies predicted for the low resistivity cells [3].

When narrow diffused junctions are employed, trans- parent semiconducting films such as Sn02/In203 may be suitable for making electrical contact to the diffused regions. (Although heterojunction solar cells[4] can be made by depositing these films directly on silicon substrates, these cells currently have lower VX’s than the p-n junction cells.) It seems reasonable to expect that these films will form low resistivity ohmic contacts with the heavily doped silicon.

A similar device geometry has been analytically treated by Mukherjee and Das[5] as a model for polycrystalline CdS solar cells. Their analysis was based on a one- dimension approximation.

In this report the two dimensional stripe junction cells are analyzed using the method of variation of parameters. The analytic model employed and the governing equations are introduced first. Sample calculations of the base region carrier collection efficiencies as well as data pertaining to overall cell efficiencies are next presented, and the effects of the material and device parameters on

tCurrent address: Dept. of Electrical Engineering and Computer Science, University of California, Berkeley, CA 94720, U.S.A.

the collection efficiencies are discussed. Both cells having diffused junctions and cells whose junction regions are covered with opaque electrodes are considered. The saturation current density and the power conversion efficiency are also discussed.

2. ANALYSIS

A schematic representation of a typical stripe geometry junction cell is shown in Fig. l(a). Specifically, the base region is considered to be n-type silicon. To determine the short-circuit current of the base region, the minority carrier continuity equation must be solved. To avoid the problem caused by the very thin embedded p-n junctions, we have chosen to analyze a model cell whose geometry permits easier computation of the excess carrier concentration. The model cell is shown in Fig. l(b). Its dimension in the z direction is taken to be one unit length.

The continuity equation governing the distribution of excess carriers within the base region is

-L2(a2p/ax2+aZp/ay2)+p =g7 (1)

where p is the excess hole concentration in cm-‘, L the diffusion length for holes in cm, 7 the hole lifetime in seconds, and g the number of electron-hole pairs photon-generated per unit time and volume. The as- sociated boundary conditions are:

ap/ay(x,O)=apldy(x,2b)=O (2)

P(t,Y)=O (3)

Here s is the base region surface recombination velocity at the silicon-oxide interface in cm . SC’. Equation (2) reflects the periodic symmetry of the cell geometry; eqn (3) is a consequence of the ohmic contact on the bottom surface, and eqn (4) reflects both the excess carrier sink provided by the p-n junction and surface recombination beneath the oxide.

119

120 CHENMING Hu and JOHN EDELBERC

Fig. l(a). Schematic representation of stripe junction p-n solar cell.

Cb)

Fig. l(b). Model of stripe junction cell adopted for mathematical analysis.

The solar light spectrum is divided into eight wavelength intervals with constant absorption coeffi- cients (a condensed version of tables presented in[6,7]). For each interval the generation rate is

gi = eNi ema? ; a<y<(2b-a)

= cyiNL eC~(*+d); otherwise. (5)

Here (Y~ in cm-’ is the absorption coefficient and N, in cm’s_’ is the number of incident photons associated with the ith wavelength interval. When d is made very large, eqn (5) becomes appropriate for opaque-electrode cells.

The boundary conditions (eqns 2-4) are such as to permit the superposition of the eight solutions obtained by successively solving eqn (1) for each of the eight wavelength intervals. Knowledge of the excess carrier profile, p (x, y) permits the determination of the recombi- nation rates and the hole current across the junction. A description of the method of solving eqn (1) and finding the hole current is given in the Appendix.

When needed, the electron current from the diffused p-regions can be determined using the familiar formula[6] of solar cell analysis. It is assumed that the collection

efficiency of these embedded regions of the stripe junction cell is the same as that of a conventional cell’s diffused region. (A more exact solution, however, is available [8].) The short-circuit current density, J,, is the sum of the currents collected from the p and the n regions in one cell section divided by 2b.

The method of determining the saturation current density J,,, is the following. If the doping levels of both the base and diffused regions are know, the excess minority carrier concentrations at both boundaries of the junction depletion region are known functions of an applied voltage V. The electron current injected into the diffused region is that of a short base planar diode. The hole current on the base side may be determined by solving eqn (1) with g = 0 (no illumination). Boundary conditions (2) and (3) remain unchanged; condition (4) is changed to reflect the excess hole concentration along the junctions.

If the sum of the electron and hole currents so determined at the junction in one cell section is I, J,,, is given by

J,,, = 1/[2b(e4”“T- l)] (6)

3. COLLECTION EFFICIENCY

This section will focus on the collection efficiency in the base region. Unless otherwise specified, the cells are assumed to have a diffused region thickness of d = 0.4 pm and a cell thickness of t = 150 pm. Three sets of substrate material parameters are used: r = 20 psec, L=200pm; 7=3ysec, L=90pm; and r=lpsec, L = 30 ym. They approximately correspond to n-type silicon with resistivities of 10, 1 and 0.1 R cm, respec- tively. At the assumed diffused region thickness, 18.6% of the incident photons are absorbed in the diffused layer in a conventional cell.

Table 1 compares the carrier losses in the conventional cells and the stripe-junction cells. In the IORcm cells (larger V- and L) the major loss mechanism is the recombination loss at the back surface. This loss component increases rapidly as 2b increases. In the 0.1 R-cm cells the major loss mechanism is the bulk loss. In all cases the recombination at the oxide interface is not the largest loss component. Apparently an important consequence of the stripe junction geometry is to cause an increased accumulation of carriers in the bulk, thus increasing both the bulk recombination and the flow toward the back surface. Since the oxide loss is already

Table 1. Effects of material and device parameters on the losses of base region carriers. The oxide, bulk, and back losses, and the base region collection efficiency, n8 represent carrier recombination at the oxide-silicon interface, within the bulk base region, and at the back contact, and carrier collection at the junction, respectively; they are expressed as percentages of carriers generated in

the base period, 2a = 15 pm

Oxide 10s Bulk loss Back.loss tlB

S =300 L =30pm s =2000 7=20*sec L =200pm cmlsec r=lpsec cmlsec

Conventional 2b = 50 pm 26 = 100 pm Conventional 2b =50&m - 1.6 4.6 - 12.8 1.5 3.4 6.9 23.3 39.0 9.4 14.9 26.1 2.2 2.1

89.0 80.1 61.8 74.4 46.0

Solar cells with stripe geometry 121

small at s = 300 cm/see, further reduction in s should not produce significant improvement in the collection effi- ciency. Typically, a reduction of s from 300cmlsec to 50cmlsec results in an improvement in g, by 1%. Increasing s from 3OOcmlsec to 2OOOcmlsec typically decreases 178 by 5%.

Table 2 summarizes the dependence of 7B on material quality and cell geometry. It is seen that vB can have a stronger dependence on the junction separation than one might expect. In the first column, for example, ve drops by 30% when 2b -2a (=85 pm) is still smaller than L/2. This drop is mainly due to the increased back surface loss rather than the recombination at the front surface (see Table 1). It can be seen in Table 2 that such dependence on the junction separation is even stronger in cells with narrower diffusion regions.

It should be noted that the diffused region width can significantly affect qs, too. Consider the row with 2b - 2a = 35 f~ m in Table 2. When 2a is reduced from 15 to 5 to 2 pm, Q decreases from 80.1 to 74.9 to 69.7%. The cells with narrower diffused regions accumulate significantly greater amounts of carriers in the base bulk and hence a corresponding increase in the carrier losses-particularly back loss.

In the alloyed grating cells or the Schottky barrier cells, the junction areas are covered with opaque electrodes. Table 2 also shows the qs’s of some of these cells. Notice that they have qs’s strikingly close to those of their counterparts with transparent electrode.

This suggests that the long diffusion length and the narrow junction width make Q insensitive to the exact spatial distribution of the illumination. Therefore, Q’S given in Table 2 can be taken as very good estimates for Schottky barrier cells having electrodes of arbitrary transmissivities and for p-n junction cells having diffused layers of arbitrary thicknesses.

4. POWER CONVERSION EFFICIENCY

(a) p-n Junction C.&S Some sample calculations for p-n junction cells are

shown in Table 3. Two sets of material parameters that correspond roughly to 10 and 0.1 &cm silicon are used. The cells are assumed to have transparent electrodes; The average short-circuit current density, Js,, is, of course, related to the collection effciencies discussed in the previous section. The average reverse saturation current density, J,,, is calculated by the method outlined in Section 2; note that it includes the contribution from the cell’s diffused regions, which are characterized by doping

Table 2. Carrier collection efficiency of the base region, Q (per cent). Unless otherwise specified, the cells have transparent electrodes, 7 = 20 ~1 sec. L = 200 fim, s = 300 cmlsec. a7 = 3 CL set, L = 90 pm; “7 = 1 psec, L = 30 pm; ‘opaque electrode

Table 3. Silicon p-n junction solar cell. 2a = 15pm, base s = 300 cm/set. 10 n-cm base: doping density = 4.6 X lOI cm-‘, 7 = 20 ps, L = 200p.m. 0.1 R-cm base: doping density =

7.8x10’6cm~3,~=1~s,L=30pm(~=20~s,L=200~m)

2b -2a J,, J ra, Grn) (mA/cm’) (PA/cm2) ($i (2)

0 39.5 107.6 0.510 13.2 10Scm 15 40.1 113.5 0.509 13.3

35 37.6 107.8 0.508 12.6 0 34.1 1.33 0.619 14.2 (17.4)

0.1 n-cm 15 31.8 1.18 0.621 13.3 (17.8) 35 24.9 0.91 0.621 10.5 (16.6)

density of 1OL9 cm-‘, carrier lifetime of 0.1 p set, diffusion length of 5 pm, and surface recombination velocity of lO’cm/sec. The carrier collection efficiency of the diffused regions is 61%.

One might expect that the saturation current is roughly proportional to the junction area and that a stripe-junction cell has significantly smaller J,,, and hence higher open-circuit voltage, V,,. Table 3 indicates that this is not true. As 2b increases, Jsal tends to decrease, but does so only modestly as long as 26 is small compared with L. This is because the injected minority carriers, through diffusion, cover the x =0 plane almost uniformly, resulting in an effectively enlarged junction area. J,,, of a model cell actually exceeds that of a conventional cell in the case of the lO&cm cells due to the added surface recombination. Note that for a given resistivity, V,, is nearly constant, indicating that the ratio J,,/J,,, is roughly the same for the conventional cells and the various stripe junction cells. Since V,, is nearly constant, J,, (or the collection efficiency) determines the relative conversion efficiencies of cells within one resistivity group.

(b) Schottky barrier cells

J,,, of Schottky barrier diodes, a majority-carrier device, does vary roughly in proportion to the junction area. As a result, stripe-junction cells can have higher open-circuit voltages. Green[2] recently calculated the conversion efficiency of stripe-junction Schottky cells with opaque electrodes assuming J,,, to be proportional to a lb and 7~ to be unity, and found that efficiencies beyond the “semi-empirical limit”[2] (the efficiency of an ideal cell having non-absorbing metal electrode and unity collection-efficiency) were possible if the ratio b/a is large. Since the present analysis has shown that the dependence of 77~ on a and b is significant, some sample calculations that include the variations of Q have been made and are shown in Fig. 2 together with Green’s calculation.

Green’s efficiency curve is a universal one. The present calculation (for 7=20psec, L =2OOpm, S= 3OOcmlsec.) indicates that the efficiency depends on a and b (and the material parameters) and peaks out at values lower than the semi-empirical limit unless the junction width is extremely small.

Although a stripe junction cell with opaque electrode will not likely have a higher efficiency than an ideal cell, the stripe-junction design can usually improve the efficiency of a real Schottky barrier cell.

2b-2a =Opm 5gm

15em 35em 85gm

122 CHENMING Hu and JOHN EDELBERG

(b-a)/a

Fig. 2. Conversion efficiency of stripe-junction Schottky barrier cells. Calculations of the present study are for 10 ohm-cm silicon.

5. CONCLUSIONS

Silicon colar cells with stripe geometry junctions are analyzed using the method of variation of parameters. Both cells that have opaque electrodes over the junction areas and cells that do not are considered. It is shown that the collection efficiencies presented are also excellent estimates for cells having partially transmitting elec- trodes.

The base region collection efficiency Q of a stripe junction cell has stronger dependence on the separation between the stripe junction than commonly assumed. A 20% decrease in qB is possible for a separation about one sixth of the carrier diffusion length. qB also decreases markedly when the stripe width is made small with the separation between stripes held constant. The decrease in qs is due much more to the increased bulk recombination and back surface losses than to the added surface recombination. These indicate that the lower Q in stripe junction cells is mainly due to the higher density of excess minority carriers in the base region. Little is to be gained by reducing the surface recombination velocity below lo3 cmlsec.

By reducing the light blockage by the metal electrodes of the Schottky barrier cells, the stripe-junction design can usually increase the short-circuit current. Further- more, the saturation current of a Schottky barrier cell varies roughly in proportion to the actual junction cell area. The smaller junction area of a stripe-junction results in a smaller saturation current and, in turn, a larger open-circuit voltage. For given diffusion length, carrier lifetime and the transmissivity of the electrode, a particular set of the width and the separation of the junctions will maximize the conversion efficiency.

Due to the lateral diffusion of the minority carriers, the saturation current of a stripe-p-n -junction cell is only slightly smaller (if any) than that of a conventional cell. Also, one must not expect a stripe-junction cell to have substantially higher short circuit current than a state-of- the-art conventional cell, whose diffusion region already has a respectable collection efficiency.

If long diffusion lengths can be achieved in low resistivity materials, stripe junction cells as well as

conventional cells can be made more efficient. Since stripe junction cells have smaller junction areas, the number of junction defects and the shunt current can perhaps be proportionally reduced. T’his would be helpful in achiev- ing the predicted high efficiencies. The area and perhaps the periphery of the diffused regions can be further reduced by replacing the stripe geometry with a two- dimensional array of squares or circles. However, as is the case with the stripe junction geometry, a reduction in junction area would tend to bring about a decrease in ?s. On the other hand, a more carefully designed array geometry should result in an increase in q~ due to a decrease in the mean distance a generated carrier must travel to be collected at the nearest junction.

REFERJNCES

1. J. J. Loferski, E. E. Crisman, W. Armitage and L. Y. Chen, Proc. IEEE Photoooltaic Specialists Conf. 58 (1973).

2. M. A. Green, appl. Phys. Lett. 27, 287 (1975). 3. H. W. Brandhorst, Jr., Proc. IEEE Photouoltaic Specialists

Conf. 37 (1972). 4. R. L. Anderson, NSF Photouoftaic Conf. Philadelphia, 276

(1975). 5. M. K. Mukhergee and S. N. Das, IEEE Trans. on Elec. Deuices

ED-21, 379 (1974). 6. M. Wolf, Proc. of IRE 48, 1246 (1960). 7. M. B. Prince and M. Wolf, .r. Bit. IRE 583 (1958). 8. M. K. Mukherjee and S. N. Das, Solid-St. Electron. 18, 716

(1975).

APPENDIX

Method of solving eqn (1) and finding the hole current The following discussion concerns the solution of eqn (1) within

a (monochromatic) spectral interval. The complete solution within this interval is the sum of any particular solution pp that satisfies eqns (l)-(3) and the homogeneous solution ph that satisfies the resulting related homogeneous problem. A convenient solution pI, chosen such that pp (0, y) = 0, is given by

where L is diiusion length in centimeters, A, = (mvlb), u0 = 1 and u, =0 for m#O.

c 2-3C 2 sin(aA,) 0 0 -___

b ’ Z.=- b A, (A2)

tv,,, = emuX -cash p,,,x + cash /?,,,t -e--l

sinh Pmt sinh &x (A3)

pm = (l/Lz+ &,,2)1’*.

The related homogeneous problem can be solved by the technique of eigenfunction expansion.

ph = 2 A, sinh fi,,, (t -x) cos h,y /sinh &t (A4) m-cl

where the A,,, ‘s can be seen to be the Fourier Series coefficients of the excess hole concentration along fhe boundary x = 0. (The particular solution does not contribute to the overall solution along this boundary, since p,(O, y) = 0). Since the sum of ps and ph must satisfy eqn (4), the Am’s are constrained to satisfy the

Solar cells with stripe geometry 123

following relationship:

A, = C,,, {A& + Qo[e-“d + G(1 - e-‘?]]

+(l-U,)(cO+) [A,R,,, +(I-e-“d)Q&,,l

j% (C,+i+C,~,)[A,R,+(l-e-“d)QiC,l (A9

it,

where

Here the law of surface recombination, the fact that the excess carrier concentration along the p-n junctions is zero, and the fact that p,(O, y) = 0 are used to evaluate F,,.

Similarly, expressions for FbaEk and F,, can be derived by integrating D(dp/dx) along the back surface and by integrating p/7 over the section volume, respectively, Again, A, is the only Fourier coefficient that appears in the final expressions.

Finally, F,, the number of holes flowing across the junctions of a cell section per second, can be seen to be given by

R, = -(D/s)&, coth Pmt (‘46) F, = F,.. - Fox - F.., - F,.,,

Qm = aNL cash pm! -em”’

l-LZaz+LZA_2 sinhp,t -o ’ > 3 647)

The infinite set of eqns (A5) was modeled as a system of M simultaneous linear equations to be solved for the Jr4 lowest-order Fourier Series coefficients A,. At least m = 70 coefficients were solved for numerically. Through (Al) and (A4) these coefficients provided p(x, y).

Interestingly, only the lowest-order Fourier coefficient, A, is needed to determine the recombination rates and the hole current across the junction. Let F..., F,.,, F,,, and F,.,, be, respectively, the number of holes (per second per cell section) generated in the base, recombining in the bulk, lost to surface recombination under the oxide, and lost at the back contact. The carrier-loss processes can be described by appropriate integrals of the excess carrier concentration p or its partial derivatives. For example:

26--o

F,,=s I PK’. y)dy = s (1 I

2b

pra (0, Y) dy = Z&b L48) 0

2oL2Nb [e__d =_ 1 -LZa2

+ C,( 1 - eP)]

(A9)

The first term of eqn (A9) will reduce to the expression for F, in a conventional cell[6] (in which case C, = 0 and Au = 0). The second term shows the effect of A,, which represents the “average” excess hole concentration along the boundary x = 0. This method of evaluating F, circumvents the evaluation of D(dp/dx) along the junctions, which requires the retention of many high-order terms of the slowly converging sequence dpldx.

Summing up the contributions from all the wavelength intervals, the short circuit hole current and the base region collection efficiency can be easily computed.

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