Solar Cells, I0 (1983) 211 - 222 211

MONTE CARLO SIMULATION OF SOLAR CELLS

C. MANFREDOTTI and M. MELIGA

Istituto di Fisica Superiore, Corso Massimo D'Azeglio 46, 10125 Turin (Italy)

(Received December 8, 1982; accepted May 10, 1983)

Summary

The Monte Carlo method was applied for the first time to calculate the performance of a solar cell. It was proved that the method is self-consistent even for a relatively low number of histories and, in the simple case of a Schottky barrier, the results are in good agreement with the existing simple theory. The possible application of this method is relatively straightforward and could give much more information than existing numerical methods.

1. Introduction

Even though solar cells are semiconductor devices that are easy to pro- duce, a complete analysis which takes into account the morphology of the cell and the influence of various parameters is rather complex. Generally it is necessary, for a complete understanding of the problem, to solve Poisson's equation and the continuity equation simultaneously for electrons and holes. Under illumination, these are non-linear partial differential equations. A problem of this kind may be solved essentially in two ways: by numerical analysis and by the Monte Carlo method (MCM).

Numerical analysis, which has many examples in the literature [1 - 3], presents the advantage of short computation times, but it cannot avoid problems due to boundary conditions, discontinuities, complex models or structures. The MCM, which differs from numerical methods by its simplic- ity and versatility, can also overcome very easily the most complicated prob- lems connected with structural complexity, electrical potential behaviour, discontinuities etc. without increasing the program complexity and, by tak- ing the appropriate precautions, without even a marked increase in computa- tion time. In fact this time can be kept to the same order of magnitude as for numerical analysis; moreover, it is not proportional to the complexity of the problem. Finally, the MCM, being generally based on a microscopic treat- ment, leads to an easy visualization of the physical processes occurring in solar cells and for this reason it can give various results without modification of the program itself.

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212

In our work, a Schottky barrier n-type Au-Si solar cell was chosen because of both structural simplicity and lack of literature of previous work in this field. However, the program is quite general and with slight modifica- tions it may be applied to other kinds of solar cells, such as p -n junction cells, metal- insulator-semiconductor (MIS) cells or heterojunction cells, and it can easily be used for more complex structures, e . g . tandem cells.

2. The method

The MCM is very extensively used in various fields of physics, whenever processes or experiments have to be simulated. The best advantages result in cases where a few types of process are repeated several times (cooling down of neutrons, electromagnetic or nucleonic cascades etc.). In these cases each particle is followed until it has lost all its energy and the various processes that occur are interpreted on a probability basis. Since the sum of the prob- abilities of the various processes at each step is of course equal to uni ty, a random number between zero and uni ty is consequently chosen in order to decide the kind of process which should occur. In our case, for instance, in order to select a photon from the solar spectrum, the integral of its energy distribution is normalized to uni ty and the random number chosen by the program is compared with the normalized partial integrals of the energy dis- tribution. For the subsequent process, light absorption in the cell, the prob- lem can be solved analytically since the probability of absorption between x and x + dx is given by

d/ p ( x ) d x = - - = o~ exp(--ax) dx

I0

which is obviously normalized to uni ty , and thus the random number R is given by the following expression:

d

f exp(--ax) dx = 1 -- exp(--~d) = 1 - - R 0

Therefore the penetration depth of the involved photon can be obtained from the equation

In R d - - - - ~(~,)

a(~) being the absorption coefficient at the selected photon wavelength.

3. The model

In the present work, the program describes an n-type Au-Si Schottky barrier solar cell with a surface area of 1 cm 2. However, the various physical

213

ant i ref /~ coating

Parameter Symbol Value

Metal thickness SM Solar cell thickness SL Barrier height qb B

{lowered by Schottky effect} Energy gap Eg(T)

(as a function of temperature) Resistivity p Electron lifetime T e Hole lifetime Tp Hole diffusion length Lp

100 A 300 pm 0.78 eV

1.2 - - (3 X 10-4T) eV

1 ~ c m 3.95 x 10 -4 s 2.62 x 10-Ss 150 pm

mE

0.766

! barrier

tal

W

0.780.

0.778

0.774

uJ

r base 0.770

20 40 60 8'0

TABLE 1

Symbols and parameters used in the Monte Carlo method

$L d(A}

Fig. 1. Geometry of the solar cell with the various zones considered by the program. The figure is not to scale.

Fig. 2. The metal-semiconductor potential barrier used in the program as a function of depth. The Schottky effect is included and the lowest energy corresponds to a tunnelling transmission of 3%.

and geometrical parameters are given as external data and can be varied according to the characteristics o f the device under investigation. In Table 1 the parameters which were used in this work are given.

From a geometrical point of view, the model structure may be divided into four different parts (Fig. 1).

In the ant i re f lec t ion coat ing and m e t a l e lec t rode z o n e only the trans- mission of incident light is taken into account [4]. In fact, for typical metal thicknesses of a Schot tky barrier solar cell the photoexci ta t ion of electrons from the metal to the semiconductor is negligible.

The barrier z o n e thickness is evaluated in the program [5]. The shape of the potential barrier is shown in Fig. 2: it was obtained by considering

214

TABLE 2

Variation in Iasc and Voc as functions of the time interval used in the calculation of the current

At Number o f Iasc Voc boxes (mA) (V)

(1/30)Tp 8 24.28 0.281 (1/100)Tp 14 26.52 0.283 (1/150)Tp 17 27.38 0.284 (1/200)rp 20 27.63 0.284 (1/250)Tp 22 27.86 0.284

a barrier height of 0.8 eV and determining the lowering due to the Schottky effect. In any case, it is given as an external vector and therefore it is possible to consider barriers of any shape and height. The barrier zone is divided into boxes in such a way as to obtain a constant electrical field in each box.

In the base zone, carrier diffusion takes place and therefore the char- acteristic parameter will be the minori ty {hole) diffusion length Lp. The base zone is also divided into boxes and the box width depends on the time interval At chosen to calculate the instantaneous current. The diffusion length {given in terms of the number of boxes) is determined by simulating a semi-infinite crystal with the same physical characteristics as the base and by considering hole injection at each time interval (see Table 2) and thus the number of boxes necessary for the base zone is obtained.

The back electrode is important because it gives rise to photon reflec- t ion and it is characterized by a reflectivity coefficient that depends on the metal of the electrode and varies with the photon wavelength.

4. The program

The flow chart of the program is shown in Fig. 3. The MCM, if rigorous- ly applied to our program, would take a lot of central processing unit (CPU) time. In fact, under air mass {AM) 1 conditions about 3 × 101~ photons should impinge in 1 s on a cell of area 1 cm 2. Four of five carrier lifetimes are enough for the semiconductor to reach a stationary state. In fact it was proved that the cell reaches stationary conditions in a t ime shorter than the lifetime and therefore the calculation is limited to an interval equal to one lifetime. A further reduction in the number of photons can be obtained by considering a number of histories sufficient to give a good representation of the solar spectrum and of the absorption coefficient. Usually, 10 4 histories are enough from this point of view and therefore a multiplication coefficient is applied to the number of carriers reaching the barrier zone. This point will be discussed further below. The photon source considered exhibits an AM 1 solar spectrum which, for simplicity, is restricted to a wavelength interval

215

®

Fig. 3. Flow chart of the program (for nomenclature see Appendix A).

TABLE 3

Comparison of the total energy and the average energy of photons in the real and in the s i m u l a t e d s o u r c e

Data for the real source Data for the simulated source

Total number ofphotons 2.49 × 1017 cm -2 s -1 104 histories Totalpower density (mW cm -2) 72.62 72.6171 Average energy o f photons (eV) 1.8235 1.8230

between 3000 and 12 000 A, corresponding to a total intensity o f 72.62 mW cm -2. Table 3 shows that the simulated source (104 histories) gives a total intensity and an average p h o t o n e n e r g y almost coincident with those of the real source.

The dark current is simulated by considering the potent ia l barrier illus- t ra ted in Fig. 2, taking for electrons in the metal a density of states with a normal distr ibution and including the effect o f the Fermi-Dirac statistics. Of course, only thermionic emission of electrons in the metal is taken into account. The energy interval fo r electrons is restricted in order to neglect, on the high energy side (at about 1 eV above the Fermi level), t oo low an

216

o.sj/ O

0.3

0.1

0.I O.2 0/3 AE (eV)

Fig. 4. Q u a n t u m m e c h a n i c a l reflection coefficient Q as a f u n c t i o n of t he energy differ- ence between electrons a n d t h e t o p o f t h e barr ier .

occupat ion probabil i ty and, on the low energy side, too low a tunnelling probabil i ty (approximately 3%). For each history, both the tunnelling prob- ability and, for E > ~B, the quantum mechanical reflection coefficient [6] are taken into account (Fig. 4). The dark current is then calculated as the drift current of electrons injected into the semiconductor.

The dark electric field is evaluated as a derivative of the electrical potential, and the carrier mobil i ty is calculated as a function of the doping and of the electric field according to ref. 7. The simulation of the short- circuit current Isc is obtained with 104 histories; for each history, the wave- length of an incident photon and its path length d = --(ln R)/o~(~) in the cell (where ~(~) is the absorption coefficient and R is a random number) are determined. If d ~< w the pho ton will be absorbed in the depletion layer (the quantum efficiency is taken as equal to unity); if w < d <~ SL the photon will be absorbed in the base zone; finally, if d > SL the photon will be reflected by the back electrode, according to the relative reflectivity coef- ficient. After each event, the charge parameters C(I) and G(I), which rep- resent the photogenerated carrier numbers in box I o f the depletion zone and of the base zone respectively, are updated.

In the calculation of the stationary short-circuit current, the diffusion process f rom each box of the base is fol lowed at each t ime interval At, which is expressed as a fraction of the minority carrier lifetime. This parameter is quite important since it implicitly determines, as previously shown, the num- ber of boxes in the base zone. Subsequently, after a t ime lag equal to the lifetime rp the drift current and diffusion current are calculated, their sum being Isc.

As shown in Fig. 5 (where At = (1/30)rp) I,c increases rapidly with time, reaching for t = Tp a value within 2% of the stationary value. Moreover, it turns out that the average current Ia,~ calculated in the interval 0 - Tp is also within 3% of the stationary value. The rapid increase in Is¢ can be under-

217

15: E

l{:t

I~J/30 20/30 30/30 t(Vp}

Fig. 5. Short-circuit current transient calculated at time intervals At = (I/30)~p.

stood if it is considered that the current transient is affected not only by the minority carrier lifetime but also by the drift time in the depletion layer, which is much shorter than the lifetime. Even the diffusion time in the base zone is practically "weighted" according to the exponential distribution of the photogenerated carriers. As far as At is concerned, it is shown in Table 2 that it is not necessary to choose the shortest At since, for example by changing from At = (1/100)rp to At = (1/150)rp, only a 3% increase in the average Isc value is obtained. Neverthless, in our calculation At = (1/250)rp was used.

The present version of the program calculates the short-circuit current, the dark current, the open~ircuit voltage, the current-voltage (I-V) charac- teristics, the fill factor FF and the conversion efficiency; the CPU time for 104 histories is about 5 rain. Moreover, the program also provides the pos- sibility of evaluating all these data at various temperatures.

5. Results and discussion

Since in the model the number of histories is necessarily low, some checks are needed in order to confirm that this number is sufficient to account for the various physical processes involved with good precision. Among the possible tests, the solar spectrum and the absorption coefficient at different wavelengths have been considered. The minority carrier diffusion (since it is simulated in a semi-infinite crystal) and the carrier transport in the deplet ion layer are clearly less dependent on the total number of events.

218

!

0.084

!

i

0.034

'I ii i I--

~' iii I Iii

I i-i I

J

I --If - - 'I-- II i

-] ........ I.~ I ~I I I I

-T I-~ I E-

l ] I- J'i-

I ~, -i ~ -z - r I

I I ( r -, I I

I - i

I

3' I

T [

0.3 0.7 I~

I

II I {

1

Fig. 6. Comparison between the real and simulated solar spectrum in the wavelength range 0.3 - 1.2 pro. On the y axis the ratio of the total number of photons in the real source to the total number of photons in the simulated source is reported. The spectra have been shifted for comparison.

TABLE 4

Comparison between the simulated and the real absorption coefficient at three wave- lengths and for two thicknesses

Thickness X Numberof ~sinmla~d ~m~ (~m) (A) absorbed photons

1 4000 9876 4.39 x 104 6000 3429 4.199 x 103 8000 952 1.0004 x 103

10 4000 10000 -- 6000 9850 4.1997 × 103 8000 6320 0.99967 × 103

4.4 x 104 4.2 x 103 1.0 × 103

4.2 x 103 1.0 x 10 3

Figure 6 shows that the solar spectrum in the interval 0.3 - 1.2 pm is very well reproduced by simulation. It must be remembered (see Table 3) tha t the total solar energy simulated is, within 0.004%, coincident with that delivered to the cell by taking into account the real number of photons.

The absorpt ion coeff icient ~(k) is also simulated at various wavelengths; it is derived f rom the decrease in the number o f photons due to the absorp- t ion in the various boxes and is compared with the real absorption coeffi- cient. As shown in Table 4, when the absorption in a crystal either 1 or 10 /~m thick is considered the maximum possible error in ~(k) is 0.2%.

Two examples of simulated pho t on a t tenuat ion at 4000 • for a thick- ness of 1 ~m and at 8000 A for a thickness of 10 #m are shown in Fig. 7

219

..~ 4,

Z _.= ~i

i+

Nixie l e

w

N

m

" o . +

l e e

50+

| - [--

Jr-*

l-++

I~+.. [+.

I - ] . . .

[ - - .

J - - ] - -

10 30 50 10 30 50

~x *box Fig. 7. Attenuation of 4000 A photons in a thickness of 1 pm (logarithmic scale).

Fig. 8. Attenuation of 8000 A photons in a thickness of 10 pm (linear scale).

TABLE 5

Comparison between the solar cell performances obtained by simulation and by theory

MCM results

With Schottky effect b Without Schottky effect

Average values Stationary values

Theoretical results a

Isc (A) 27.86 x 10 -3 27.86 x 10 -3 28.67 x 10 -3 28.84 x 10 -3 I s (A) 5.0 x 10 -7 2.2 x 10 -7 2.2 x 10 -7 2.253 × 10 -7 Voc (V) 0.284 0.304 0.304 0.304 FF 0.71 0.72 0.72 0.72 7) (%) 7.695 8.39 8.66 8.69

aThe theory does not include the Schottky effect. The values given are stationary values. b Average values.

and Fig. 8 respectively: a truly exponential photon attenuation is clearly evident.

The results obtained in the present work are summarized in Table 5, where I~, Is, Voc, FF and W are reported and compared with theoretical results obtained by using equations quoted in the literature [8]. In order to make the comparison more understandable, the results of the simulation are also reported for a barrier without the lowering due to the Schottky effect, since theoretical results can be obtained for this case only. For com- pleteness, results are also presented for both the average and the stationary values of the parameters. In the theoretical calculation, an average value of the quantum mechanical reflection coefficient has been used [6]. The agreement between the Monte Carlo simulation and theoretical calculation is extremely good, the larger differences being 2% for I, (stationary values) and 3% for Isc and ~ (average values).

220

10

5

V (v} 0.1 0.2 0.3

J / 1

20

30 l Fig. 9. I-V characteristics in AM 1 conditions as obtained by the MCM (o) (average values) and by the theory (4). The theory does not include the Schottky effect.

0.6

~ 0.4

0.2

0 O. 5 0.7 O. 9 1.1

k('/~m)

Fig. 10. Responaivity R ( m A m W - 1 ) as a function of wavelength as obtained by the MCM (e ) and by the theory (4) (reflection from the back electrode was not included). The theory does not take into account the Schottky effect.

The comparison was extended to the I - V characteristics (Fig. 9) and to the responsivity (Figs. 10 and 11). In Fig. 9 the differences can be attributed to the Schottky effect which lowers Voe. The differences which appear in Fig. 10 can be explained by looking at Fig. 11, which shows the drift current in the depletion layer and the diffusion current in the base zone. In the former case the disagreement is a maximum at 4000 A, corresponding to a penetration depth comparable with the depletion layer thickness, while in the latter case the disagreement corresponds to a deep penetration of light in the base zone (about 30 pm). It is therefore probable that the former differ- ence is due to the lower surface electric field in the MCM and the latter re- sults because the diffusion current is calculated by the theory under the assumption o f complete depletion at the end o f the base zone.

221

~ i o n

0.3 0.5 0.7 0.9 1.1

Fig. 11. The drift current and the diffusion current as functions of wavelength as calcu- lated by the MCM (o) and by the theory (A). Monochromatic l ight wi th a power of 10 mW was incident on the cell.

6. Conclusions

Our work may be considered as a first example of the possibility of using the MCM in the simulation of solar cells. It was proved that the MCM can work with good precision, even taking into account a reasonable number of photons and considering the diffusion process in microscopic detail. Since reliable or usable experimental results are not available, our results were compared with those of the usual theory.

The good agreement which was obtained can be considered as proof of the validity of the MCM in a very simple physical case. However, as has been noted in the introduction, the MCM can easily be extended to more complex structures such as tandem cells, heterojunct ion cells, graded band gap cells and MIS cells and to transient effects in solar cells. This extension will be carried ou t in future work, particularly if more experimental data are avail- able. From our point o f view, the MCM is probably the simplest me thod by which to s tudy the influence of the various parameters (lifetime, temper- ature, diffusion profiles, thickness of the various zones, absorption coef- ficient etc.) on the performance o f the cell, particularly in cases where these parameters are also dependent variables. Efforts will be made in future work to investigate all these possibilities o f the MCM.

We are also confident that a two- or three<limensional extension of our program, in order to deal with complex problems such as series resistance or vertical cells, may be attained wi thout severe complications.

References

1 P.M. Dumbar and R. J. Hauser, Solid-State Electron., 19 (1976) 95. 2 G.J . Fossum, Solid-State Electron., 19 (1976) 269.

222

3 J. Michel and A. Mircea, Acta Electron., 18 (4) (1975) 331. 4 M. V. Schneider, Bell Syst. Tech. J., 45 (1966) 1611. 5 S. M. Sze, Physics o f Semiconductor Devices, Wiley-Interscience, New York, 1960,

Chap. 8, p. 363. 6 C. R. Crowell and S. M. Sze, J. Appl. Phys., 37 (7) (1966) 2683. 7 S. Sharfetter and A. Gummel, IEEE Trans. Electron Devices, 16 (1969) 64. 8 H. J. Hovel, in R. K. Willardson and A. C. Beer (eds.), Semiconductors and Semi-

metals, Vol. 11, Solar Cells, Academic Press, New York, 1975, pp. 112 - 126.

Appendix A: nomenclature for Fig. 3

E(I) electric field FF fill factor Iasc average current Id diffusion current /dr drift current Iisc instantaneous current Is dark current Isc short-circuit current I - V I -V characteristic Nbb number of boxes in the base Nbw number of boxes in the barrier Ndt number of time intervals Np number of time intervals NB number of histories V(I) potential Voc open,circuit voltage w barrier width At time interval

efficiency Pe electron mobility #h hole mobility