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Detailed Balance Limit of Efficiency of *p‐n* Junction Solar Cells

### Abstract

In order to find an upper theoretical limit for the efficiency of *p‐n* junction solar energy converters, a limiting efficiency, called the *detailed balance limit* of efficiency, has been calculated for an ideal case in which the only recombination mechanism of hole‐electron pairs is radiative as required by the principle of detailed balance. The efficiency is also calculated for the case in which radiative recombination is only a fixed fraction *f*_{c} of the total recombination, the rest being nonradiative. Efficiencies at the matched loads have been calculated with band gap and *f*_{c} as parameters, the sun and cell being assumed to be blackbodies with temperatures of 6000°K and 300°K, respectively. The maximum efficiency is found to be 30% for an energy gap of 1.1 ev and *f*_{c} = 1. Actual junctions do not obey the predicted current‐voltage relationship, and reasons for the difference and its relevance to efficiency are discussed.

© 1961 The American Institute of Physics

Received 03 May 1960
Revised 31 October 1960
Published online 11 June 2004

/content/aip/journal/jap/32/3/10.1063/1.1736034

1.

1.D. M. Chapin, C. S. Fuller, and G. L. Pearson, J. Appl. Phys. 25, 676 (1954).

2.

2.W. G. Pfann and W. van Roosbroeck, J. Appl. Phys. 25, 1422 (1954).

3.

3.M. B. Prince, J. Appl. Phys. 26, 534 (1955).

4.

4.J. J. Loferski, J. Appl. Phys. 27, 777 (1956).

5.

5.P. Rappaport, RCA Rev. 20, 373 (1959).

6.

6.M. Wolf, Proc. I.R.E. 48, 1246 (1960).

7.

7.A treatment of photovoltage, but not solar‐cell efficiency free of such limitations, has been carried out by A. L. Rose, J. Appl. Phys. 31, 1640 (1960).

8.

8.H. A. Müser, Z. Physik 148, 380 (1957),

8.and A. L. Rose (see footnote 7) have used the second law of thermodynamics in their treatments of photovoltage.

9.

9.W. van Roosbroeck and W. Shockley, Phys. Rev. 94, 1558 (1954).

10.

10.A preliminary report of the analysis of this paper was presented at the Detroit meeting of the American Physical Society: H. J. Queisser and W. Shockley, Bull. Am. Phys. Soc. Ser. II, 5, 160 (1960).

11.

11.This discrepancy appears to have been first emphasized by Pfann and van Roosbroeck (see footnote 2), who point out that the forward current varies as with values of A as large as three.

12.

12.Once a photon exceeds about three times the energy gap the probability of producing two or more hole‐electron pairs becomes appreciable: V. S. Vavilov, J. Phys. Chem. Solids 8, 223 (1959),

12.and J. Tauc, J. Phys. Chem. Solids 8, 219 (1959). These authors interpret this result in terms of a threshold of about for an electron to produce a pair. However, the data can be well fitted up to quantum yields greater than two by assuming a threshold of only slightly more than and assuming the energy divides equally between the photohole and the photoelectron. This effect would slightly increase the possible quantum efficiency; however, we shall not consider it further in this article.

12.See also W. Shockley, Solid State Electronics 2, 35 (1961).

13.

13.For example: I. M. Ryshik and I. S. Gradstein, Tables of Series, Products and Integrals (Deutscher Verlag d. Wissensch., Berlin, 1957), pp. 149, 413;

13.E. Jahnke and F. Emde, Tables of Functions, (Dover Publications, New York, 1945) 4th ed., pp. 269, 273.

14.

14.For the calculations, numerical tables of the integrals involved were used as given by K. H. Böhm and B. Schlender, Z. Astrophysik 43, 95 (1957).

14.We are indebted to A. Unsöld who directed our attention to this publication. A convenient aid to such calculations is a slide rule, manufactured by A. G. Thornton, Ltd., Manchester, England. It is described by W. Makowski, Rev. Sci. Instr. 20, 884 (1949).

15.

15.Similar conclusions have been reached by H. A. Müser, Z. Physik, 148, 385 (1957), who estimates approximately 47% for the maximum of but does not show a curve. Results similar to those described above have also been derived by W. Teutsch, in an internal report of General Atomic Division of General Dynamics, and by H. Ehrenreich and E. O. Kane, in an internal report of the General Electric Research Laboratories. A curve which is quantitatively nearly the same has also been published, since the submission of this article, by M. Wolf (see footnote 6) who defines the ordinate as “portion of sun’s energy which is utilized in pair production,” a definition having the same quantitative significance but a different interpretation from our quantity

16.

16.For example: W. Shockley, Electrons and Holes in Semiconductors (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1950) p. 308; the product of Eqs. (18) and (19).

17.

17.See footnote 16, p. 305;

17.also W. Shockley, Bell System Tech. J. 28, 435 (1949), Sec. 5.

18.

18.Equations like (3.16) occur in published treatments of solar‐cell efficiency. The difference is that the term in due to which is small but required by the principle of detailed balance, is included, and the coefficient of is related to the fundamental minimum reverse saturation current rather than to a semi‐empirical value.

19.

19.As for Eq. (3.16), factors like υ have been introduced by various authors, most recently by M. Wolf (see footnote 6). However, the forms are dependent upon additional semiempirical quantities so that they cannot be used for the purposes given in the introduction.

20.

20.Similar maximization of the output power has been carried out in terms of the maximum area rectangle on the I‐V plot by various authors, in particular W. G. Pfann and W. van Roosbroeck (See footnote 2). The results do not, however, appear to have been published in analytic form in which the matching factor m is shown to be a function solely of the variable

21.

21.P. T. Landsberg, Proc. Inst. Elec. Engrs. (London) 106, Pt. II, Suppl. No. 17, 908 (1959).

22.

22.V. M. Tuchkevich and V. E. Chelnokov, J. Tech. Phys. (U.S.S.R.) 28, 2115 (1958).

23.

23.C. T. Sah, R. Noyce, and W. Shockley, Proc. I.R.E. 45, 1228 (1957).

24.

24.A. E. Bakanowski and J. H. Forster, Bell System Tech. J. 39, 87 (1960).

25.

25.M. Wolf (footnote 6, p. 1252) reports agreement with this model, but his data is apparently not available in the literature.

26.

26.A. G. Chynoweth and K. G. McKay, Phys. Rev. 106, 418 (1957).

27.

27.M. Wolf and M. B. Prince, Brussels Conference 1958, in Solid State Physics (Academic Press, Inc., New York, 1960) Vol. 2, Part 2, p. 1180.

28.

28.A. Goetzberger and W. Shockley, Structure and Properties of Thin Films, edited by C. A. Neugebauer, J. B. Newkirk, and D. A. Vermilyea (John Wiley & Sons, Inc., New York, 1959), p. 298;

28.Bull. Am. Phys. Soc. Ser. II, 4, 409 (1959).

29.

29.A. Goetzberger, Bull. Am. Phys. Soc. Ser. II, 5, 160 (1960);

29.A. Goetzberger and W. Shockley, J. Appl. Phys. 31, 1821 (1960).

30.

30.H. Reiss, C. S. Fuller, and F. J. Morin, Bell System Tech. J. 35, 535 (1956).

31.

31.W. Shockley and J. L. Moll, Phys. Rev. 119, 1480 (1960).

32.

32.M. Wolf, Proc. I.R.E. 48, 1259 (1960).

33.

33.L. Pincherle, Proc. Phys. Soc. (London) B68, 319 (1955);

33.L. Bess, Phys. Rev. 105, 1469 (1957);

33.A. R. Beattie and P. T. Landsberg, Proc. Roy. Soc. (London) A249, 16 (1958).

34.

34.H. K. Kroemer, Proc. I.R.E. 45, 1535 (1957);

34.W. Shockley, U.S. Patent 2,569,347, issued September 25, 1951.

35.

35.H. J. Queisser, Bull. Am. Phys. Soc. 6, 106 (1961).

36.

36.M. Wolf and M. B. Prince (see footnote 27), p. 1186.

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