JOURNAL OF APPLIED PHYSICS
`
`VOLUME 32, NUMBER 3
`
`MARCI, 1961
`
`Detailed Balance Limit of Efficiency of p-n Junction Solar Cells*
`
`Witi1AM SHOCKLEY AND Hans J. QUEISSER
`Shockley Transistor, Unit of Clevite Transistor, Palo Alto, California
`(Received May 3, 1960; in final form October 31, 1960)
`
`In order to find an upper theoretical limit for the efficiency of p-# 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 frac-
`tion f. of the total recombination, the rest being nonradiative. Efficiencies at the matched loads have been
`calculated with band gap and /, as parameters, the sun and cell being assumed to be blackbodies with tem-
`peratures 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.=1. Actual junctions do not obey the predicted current-voltage relationship, and reasons for
`the difference and its relevance to efficiency are discussed.
`
`
`
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`
`and compared with the semiempirical limit in Fig. 1.
`Actually the two limits are not extremely different, the
`detailed balance limit being at most higher by about
`50% in the range of energy gaps of chief interest. Thus,
`to some degree, this article is concerned with a matter
`of principle rather than practical values. The difference
`is much moresignificant, however, insofar as estimating
`potential for improvement is concerned. In fact, the
`detailed balance limit may lie more than twice as far
`above the achieved values as does the semiempirical
`limit, thus suggesting much greater possible improve-
`ment(see Fig. 1).
`The situation at present may be understood by
`analogy with a steam powerplant. If the second law of
`thermodynamics were unknown,there mightstill exist
`quite good calculations of the efficiency of any given
`configuration based on heats of combustion, etc. How-
`ever, a serious gap would still exist since it would be
`impossible to say how much the efficiency might be
`improved by reduction of bearing friction, improving
`heat exchangers, etc. The second law of
`thermody-
`namics provides an upperlimit in terms of more funda-
`mental quantities such as the temperature of the ex-
`othermic reaction and the temperature of the heatsink.
`The merit of a given powerplant can then be appraised
`in terms of the limit set by the second law.
`A similar situation exists for the solarcell, the missing
`theoretical efficiency being, of course, in no way com-
`parable in importance to the second law of thermo-
`dynamics. Factors such as series resistance and reflec-
`tion losses correspondto friction in a power plant. There
`are even two temperatures, that of the sun T, and that
`of the solar cell T,. The efficiency of a solar converter
`can in principle be brought
`to the thermodynamic
`limit (7,—T.)/T. by using reflectors, etc. However, a
`planar solar cell, without concentrators of radiation,
`cannot approach this limit. The limit it can approach
`depends on its energy gap and certain geometrical
`factors such as the angle subtended by the sun and the
`
`1, INTRODUCTION
`
`ANY papers have been written about the effi-
`ciency of solar cells employing -n junctions in
`semiconductors, the great potential of the silicon solar
`cell having been emphasized by Chapin, Fuller, and
`Pearson! in 1954. Also in 1954, Pfann and van Roos-
`broeck? gave a more detailed treatment including ana-
`lytic expressions optimizing or matching the load. A
`further treatment was given by Prince® in 1955,
`in
`which the efficiency was calculated as a function of the
`energy gap. Loferski‘ has attempted to predict the de-
`pendenceof efficiency upon energy gap in more detail.
`Review papers have recently appeared in two journals
`in this country.*-
`The treatmentsof efficiency presented in these papers
`are based on empirical values for the constants de-
`scribing the characteristics of the solar cell.7 They are
`in general
`in fairly good agreement with observed
`efficiencies, and predict certain limits. These predic-
`tions have become generally accepted as theoretical
`limits (see, for example, the review articles by Rappa-
`port® and Wolf®).
`It is the view of the present authors that the ac-
`ceptance of this previously predicted limiting curve
`of efficiency vs energy gap is not theoretically justified
`since it is based on certain empirical valuesoflifetime,
`etc, Weshall refer to it as the semiempirical limit.
`There exists, however, a theoretically justifiable upper
`limit. This limit is a consequence ofthe nature of atomic
`processes required by the basic laws of physics, par-
`ticularly the principle of detailed balance. In this paper
`this limit, called the detailed balance limit, is calculated
`
`* Research supported by Wright Air Development Center.
`1D, M. Chapin, C. S. Fuller, and G. L. Pearson, J. Appi. Phys.
`25, 676 (1954).
`“sb G. Pfann and W.van Roosbroeck, J. Appl. Phys. 25, 1422
`3M. B. Prince, J. Appl. Phys. 26, 534 (1955).
`4J. J. Loferski, J. Appl. Phys. 27, 777 (1956).
`5 P. Rappaport, RCA Rev.20, 373 (1959).
`®M. Wolf, Proc. ILR.E. 48, 1246 (1960).
`7A 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).
`
`8H. A. Miiser, Z. Physik 148, 380 (1957), and A, L. Rose (see
`footnote 7) have used the second law of thermodynamicsin their
`treatments of photovoltage.
`510
`
`HANWHA1027
`
`HANWHA 1027
`
`

`

`EFFICIENCY OF p-» JUNCTION SOLAR CELLS
`
`Sil
`
`[%]
`30F
`
`|
`
`y
`
`20
`
`10
`
`LO
`
`20
`
`[volts]
`40
`
`3.0
`
`Detailed Batance
`
`pm
`
`‘.
`i+
`[Best experimental,
`7 efficiency for
`f Si- cells
`i
`/
`
`\
`
`s
`
` 0.
`
`s
`Semi-Empiricat
`Limit
`4
`
`6
`
`%
`
`2
`
`
`
`Obbb:LZpZOZAineLL
`
` 17 July 2024 21:44:46
`
`Fic. 1. Comparison of the ‘semiempirical limit” of efficiency
`of solar cells with the “detailed balance limit,” derived in this
`paper. + represents the ‘‘best experiment efficiency to date” for
`silicon cells. (See footnote 6.)
`
`defined as
`
`t,=the probability that a photon with ky>£,
`incident on the surface will produce a hole-
`electron pair.
`
`(1.6)
`
`For the detailed balance efficiency limit to be reached,
`t, must be unity.
`Other parameters involving transmission of radiative
`recombination out of the cell and the solid angle sub-
`tended by the sun enter as factors in a quantity f
`discussed in Eq. (3.20). The value of f for the highest
`efficiency, corresponding to the detailed balance limit,
`is determined by the solid angie subtended by the sun,
`the other factors related to material properties being
`given their maximum values, which are unity.
`To a very good approximation the efficiency is a
`function {%,, %e, ts, f) of four variables just discussed.
`It can be expressed in termsof analytic functions based
`on the Planck distribution and other known functions.
`The development of this relationship is carried out in
`Secs. 2~5, Section 6 compares calculations of the de-
`tailed balance limit with the semiempirical limit.
`
`2. ULTIMATE EFFICIENCY: u(x,)
`
`There is an ultimate efficiency for any device em-
`ploying a photoelectric process which hasa single cut-
`off frequency p,.
`Weshall consider a cell in which photons with energy
`greater than hy, produce precisely the same effect as
`photons of energy 4v,, while photons of lower energy
`will produce no effect. We shall calculate the maximum
`efficiency which can be obtained from such a cell sub-
`jected to blackbody radiation.
`Figure 2 (a) illustrates an idealized solar cell model
`which we shall consider in this connection, It repre-
`sents a p-n junction at temperature T,=0, surrounded
`by a blackbody at temperature T,. In a later discussion
`
`angle of incidence of the radiation, and certain other
`less basic degrading factors, which in principle may
`approach unity, such as the absorption coefficient for
`solar energy striking the surface.
`Among the factors which may approach unity, at
`least so far as the basic laws of physics are concerned,
`is the fraction of the recombination between holes and
`electrons which results in radiation. Radiative recom-
`bination sets an upper limit to minority carrier life-
`time. The lifetimes due to this effect have been calcu-
`lated using the principle of detailed balance. It is this
`radiative recombination that determines the detailed
`balance limit for efficiency.” If radiative recombina-
`tion is only a fraction f, of all the recombination, then
`the efficiency is substantially reduced below the de-
`tailed balance limit.
`How closely any existing material can approach the
`desirable limit of unity for f, is not known, Existing
`silicon solar cells fail to fit the current-voltage charac-
`teristics predicted on the basis of any of the existing
`recombination models." The extent of this discrepancy
`and one suggested explanation are discussed in Sec. 6.
`In determining the detailed balance limit of effici-
`ency, the efficiency 7 calculated below is defined in the
`usual way as the ratio of power delivered to a matched
`load to the incident solar power impinging on the cell.
`The following sections present a step-by-step calcula-
`tion of this efficiency as a function of the essential
`variables, including several which may reducetheeffi-
`ciency below the detailed balance limit. Three of these
`variables have the dimensions of energy and can be
`expressed as temperatures, voltages or
`frequencies.
`These variables are: the temperature of the sun 7,,
`
`kT.=9V3
`
`the temperature of the solar cell T.,
`
`kT.=@V03
`and the energy gap E,,
`Ey= hrg=qVa,
`
`(1.1)
`
`(1.2)
`
`(1.3)
`
`where & is Boltzmann’s constant, g=|q! is the elec-
`tronic charge, and # is Planck’s constant. The effici-
`ency is found to involve only the two ratios
`
`xg E,/kT,
`
`xe= TfT
`
`(1.4)
`
`(1.5)
`
`The efficiency also depends strongly upon #,, which is
`
`(ios) van Roosbroeck and W. Shockley, Phys. Rev. 94, 1558
`0 A preliminary report of the analysis of this paper was pre-
`sented at the Detroit meeting of the American Physical Society:
`H.J. Queisser and W. Shockley, Bull. Am. Phys. Soc. Ser. II, 5,
`160 (1960).
`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 exp(gV/ART) with values of A as
`large as three.
`
`

`

`512
`
`WwW.
`
`SHOCKLEY AND H. J. QUEISSER
`
`
`
`in which the symbol x, is that of Eq. (1.4),
`
`xgkTs=hvg=qVg.
`
`(2.3)
`
`is seen to be a function of the form T,° times a
`Qs;
`function of 2,.
`If the surface subject to the radiation in Fig. 2 has
`an area A, then in accordance with the ultimate effi-
`ciency hypothesis, the output power will be given by:
`
`output power= hy,AQ,.
`
`(2.4)
`
`The incident power, due to the radiation at 7, falling
`upon the device of Fig. 2, will evidently be:
`
`(a)
`
`(b)
`
`incident power= 4 P,.
`
`(2.5)
`
`the solar battery con-
`Fic. 2. Schematic representation of
`sidered. (a) A spherical solar battery surrounded by a blackbody
`of temperature T,;
`the solar battery is at temperature T,=0.
`(b) A planarcell irradiated by a spherical sun subtending a solid
`angle w, at angle of incidence @.
`
`P, is the total energy density falling upon unit area in
`unit time for blackbody radiation at temperature T,,.
`In accordance with well-known formulas for the Planck
`distribution, P, is given by
`
`
`
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`
` 17 July 2024 21:44:46
`
`P,=2nh/ef. vidv/Lexp(hv/kT.)—1]
`=2n(erye[ xidx/(e*— 1)
`
`00,
`
`0
`
`= 205 (RT,)4/15h3C?.
`
`(2.6)
`
`It is instructive to compare P, with the total number
`of incident photons per unit time Q(0,7;,) so as to ob-
`tain the average energy per photon:
`
`P||f. as/(e—1)|/J. sde/(e—1)|
`
`XLRT.0(0,T.) ]
`=L3'§(4)/2'¢(3)TRT.00,T.)]
`=[ (304/90)/ (w2/25.794- --)[AT.0(0,T.)]
`
`=2.701---kT,0(0,T:).
`
`(2.7)
`
`The integrals in Eqs. (2.6) and (2.7), one of which is
`the limiting form for x=0 in Eq.
`(2.2), may be ex-
`pressed by products of the gamma function and the
`Riemann zeta function. The mathematicalrelations in-
`volved and numerical values are found in standard
`references,"
`In accordance with the above definitions, the ulti-
`mate efficiency is a function only of x, andis
`
`u(Xy) = hvQ./Ps
`
`-[=f dal(e?—v|/f x8da/(e7—1).
`
`g
`
`(2.8)
`
`3 For example: I. M. Ryshik and I. S. Gradstein, Tables of
`Series, Products and Integrals (Deutscher Verlag d. Wissensch.,
`Berlin, 1957), pp. 149, 413; E. Jahnke and F. Emde, Tables of
`pees (Dover Publications, New York, 1945) 4th ed., pp.
`269, 273.
`
`we shall allow a finite 7, and replace the surrounding
`body at temperature 7, by radiation coming from the
`sun at a small solid angle w, as represented in Fig. 2 (b).
`Weshall assume that some means not indicated in the
`figure are present for maintaining the solar cell at
`temperature 7.=0 so that only steady state conditions
`need be considered. According to the ultimate effici-
`ency hypothesis":
`
`than
`Each photon with energy greater
`hv, produces one electronic charge g at
`a voltage of V,=hyv,/q.
`
`(2,1)
`
`from the solar
`The number of photons incident
`radiation in Fig. 2 is readily calculated in accordance
`with the formulas of the Planck distribution. We de-
`note by Q, the number of quanta of frequency greater
`than v, incident per unit area per unit time for black-
`body radiation of temperature 7. For later purposes
`we shall also introduce the symbol Q(v,,7',) in order to
`be able to represent situations for different values of
`the limiting frequency. In accordance with this nota-
`tion and well-known formulas, we have
`
`0.=0(v,,T.)= (2n/2)f [exp(ku/RT,)—1}-s%dv
`=prceroyire)f xdx/{e*—1),
`
`(2.2)
`
`22 Once a photon exceeds about three times the energy gap F,,
`the probability of producing two or more hole-electron pairs
`becomes appreciable: V.
`S$. Vavilov, J. Phys. Chem. Solids 8,
`223 (1959), and J. Tauc, J. Phys. Chem. Solids 8, 219 (1959).
`These authors interpret this result in terms of a threshold of
`about 2, 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 pho-
`toelectron. This effect would slightly increase the possible quan-
`tum efficiency; however, we shall not consider it further in this
`article. See also W. Shockley, Solid State Electronics 2, 35 (1961).
`
`

`

`EFFICIENCY OF p-nx JUNCTION SOLAR CELLS
`
`513
`
`F.9, where
`
`Fo=AtQ-=AtQO(»,,T.).
`
`(3.1)
`
`In this expression, é, represents the probability that an
`incident photon of energy greater than £, will enter
`the body and produce a hole-electron pair. 4 is the
`area of the body.
`The total rate of generation of, hole-electron pairs
`due to the solar radiation falling upon the body is
`given by
`
`F.=A folsOs;
`
`(3:2)
`
`
`
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`
` 17 July 2024 21:44:46
`
`in which the factor f. is a geometrical factor, taking
`into account
`the limited angle from which the solar
`energy falls upon the body. ¢, is the probability that
`incident photons will produce a hole-electron pair and
`may differ from f, because of
`the difference in the
`spectral distribution of
`the blackbody radiation at
`temperature 7, and 7, and the dispersion of the re-
`flection coefficient or transmission coefficients for the
`surface of the battery.
`is dependent upon the
`The geometrical factor jf,
`solid angle subtended by the sun and the angle of
`incidence upon the solar battery. The solid angle sub-
`tended by the sun is denoted by w,, where
`
`we=n(D/L)?/4 =x (1.39/149)?2/4
`(3.3)
`= 6.8510"sr,
`and D, L are, respectively, the diameter and distance
`of the sun, taken as 1.39 and 149 million km.
`If the solar cell is isotropic (i.e.,
`is itself a sphere)
`then it is evident that f,, should be simply the fraction
`of the solid angle about the sphere subtended by the
`sun, so that
`
`fo=a/4dr.
`
`(3.4)
`
`If the cell is a flat plate with projected area A », thenit
`is more natural to deal with incident energy on the
`basis of the projected area A, rather than the total
`
`Energy Gap Vg —
`{
`2
`
`3 [er]
`
`0
`
`[%
`40
`
`30
`
`20
`
`1o
`
`1
`u(x)
`
`4
`
`5
`
`6
`
`0
`
`1
`
`2
`
`3
`hu
`"9" Ts —
`Vic, 3, Dependence of the ultimate efficiency #(«,) upon the
`energy gap V, of the semiconductor.
`
`It is evident that «(x,) has a maximum value, since
`the numerator in Eq. (2.8) is finite and vanishes both
`as “, approaches zero and as it approaches infinity.”
`Figure 3 shows the dependence of u(x,) as a function
`of x,.15 It is seen that the maximum efficiency is ap-
`proximately 44% and comes for an x, value of 2.2 in
`terms of a temperature of 6000°K for the sun. This
`corresponds to an energy gap value given by Eq. (2.3)
`of 1.1 ev.
`
`3. CURRENT-VOLTAGE RELATIONSHIP
`FOR A SOLAR CELL
`
`In this section we shall consider a solar cell sub-
`jected to radiation from the sun, which is considered
`to subtend a small solid angle as represented in Fig.
`2 (b). The treatment will be based upon determining
`the steady state current-voltage condition which pre-
`vails on the basis of requiring that hole-electron pairs
`are eliminated as rapidly as they are produced. In
`order to carry out the calculation, five processes must
`be considered: (1) generation of hole-electron pairs by
`the incident solar radiation,
`the rate for the entire
`device being F,;
`(2)
`the radiative recombination of
`hole-electron pairs with resultant emission of photons,
`the rate being /.; (3) other nonradiative processes
`which result in generation and (4) recombination of
`hole-electron pairs; and (5) removal of holes from the
`p-type region and electrons from the m-type region in
`the form of a current 7 which withdraws hole-electron
`pairs at a rate [/g. The steady state current-voltage
`relationship is obtained by setting the sum of
`these
`five processes equalto zero.
`Consider first
`the net rate of generation of hole-
`electron pairs for the solar battery of Fig. 2 (a) under
`the condition in which it is surrounded by a blackbody
`at its own temperature, 7.340, Under these conditions
`photons with frequencies higher than », will be incident
`per unit area per unit time on the surface at a rate
`Q., where O-=Q(v,,1'-) as given by Eq. (2.2). Evidently
`Q. is a function of the form 7? times a function of
`%q/%-. The number of these photons which enter the
`cell and produce hole-electron pairs is represented by
`
`tables of the integrals in-
`‘For the calculations, numerical
`volved were used as given by K. H. Béhm andB. Schlender, Z.
`Astrophysik 43, 95 (1957). We are indebted to A. Unsdéld 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).
`18 Similar conclusions have been reached by H. A. Miiser, Z.
`Physik, 148, 385 (1957), who estimates approximately 47% for the
`maximum of w#(x,), 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 submissionof 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 quan-
`titative significance but a different
`interpretation from our
`quantity “(2,).
`
`

`

`514
`
`W. SHOCKLEY AND H. J.
`
`QUEISSER
`
`
`
`Ob:bb:12ZvZOZAineLL
`
` 17 July 2024 21:44:46
`
`for the steady state condition. This leads to
`
`0=F,—F,(V)+R(0)—R(V)~I/q
`=F,—Fot+(Fa-FAV)+R(0)—R(V)J~1/¢.
`
`(3.10)
`
`In Eq. (3.10) the quantity in square brackets repre-
`sents the net rate of generation of hole-electron pairs
`whenthe cell is surrounded by a blackbody at tempera-
`ture T,. If the cell is so surrounded, it is evident that
`the term F,—F) vanishes. The steady state condition,
`under these circumstances, gives the current-voltage
`characteristic of the cell in the absence of a disturbance
`in the radiation field. On the other hand, if the cell is
`surrounded by cold space,
`it will generate a small
`open-circuit reverse voltage due to the —F.o term.
`In order to describe the current-voltage character-
`istics of the cell we introduce the quantity f,, which
`represents the fraction of the recombination-generation
`current which is radiative. This leads to
`
`(3.11)
`
`For the particularly simple case, which occurs in ger-
`manium #-% junctions, that the nonradiative recom-
`bination fits the ideal rectifier equation, we can write
`
`R(V)=R(O) exp(V/V.).
`
`(3.12)
`
`For this condition f, is a constant
`voltage, and is given by
`
`independent of
`
`Fo= Foo/[PaotR(0)].
`
`(3.13)
`
`Under these conditions the current-voltage character-
`istic for the cell in the absence of radiative disturbance
`is given by
`
`where
`
`T=Iofi1—exp(V/V.)],
`
`To=q(FotR(O)]
`
`(3.14)
`
`(3.15)
`
`is the reverse Saturation current.
`It is noted that this equation differs in sign from the
`usual rectifier equation, the convention chosen in this
`paper being that current flowing into the cell in what
`is normally the reverse direction is regarded as positive,
`and voltage across the cell
`in the normally forward
`direction is also regarded as positive, These are the
`polarities existing when the illuminated cell is furnish-
`ing power to an external load, so that positive values
`of I and V correspond to the cell acting as a power
`source,
`For an energy gap of 1.09 ev and a temperature of
`300°K, QO, is 1.7108 cmsec~!, Thus per cm? of sur-
`face the recombination current is 2.7X10-'* amp, For
`a planar cell with ¢.=1 radiating from both sides this
`leads to a contribution to fp of 5.4% 107% amp/cm?of
`junction area, according to Eq. (3.8). As discussed in
`Sec. 6, actual cells have currents larger by about 10
`orders of magnitude, so that f.= 107".
`In the event that R(V} does not obey Eq. (3.12),
`then the quantity f, must be regarded as a function of
`
`area of both sides, which is 24,. In terms of A, the
`total power falling on thecell is:
`incident power= A »Pyw, cos0/7,
`
`(3.5)
`
`where @ is the angle between the normal to the cell and
`the direction of the sun. This expression integrates as
`it should to A,P. when w, is mtegrated over a hemi-
`sphere (2m steradians) since cos@ has an average value
`of %. For normal incidence, the incident poweris thus:
`
`where
`
`incident power=A Pfu,
`
`fom @/Ww= 2.18% 10-5.
`
`(3.6)
`
`(3.7)
`
`It is evident that the rate of generation of hole-electron
`pairs by solar photons involves the same factor so that
`this value of f., should be used in Eq. (3.2). The black-
`body radiation from the cell comes from an area of
`2A, so that
`
`F o=2A gtQu.
`
`(3.8)
`
`The rate of recombination, with resultant radiation,
`of hole-electron pairs depends upon the disturbance
`from equilibrium. For the case in which the battery is
`in equilibrium, and is surrounded by a blackbody at
`temperature T',, the rate of emission of photons due to
`recombination must be exactly equal to the rate of
`absorption of photons which produce recombination.
`As discussed above, this is given by Fo in Eq. (3.1).
`To begin with, we shall consider that the only radiative
`recombination of importance is direct recombination
`between free holes and electrons and is accordingly
`proportional to the product of the hole and electron
`density, i.e., to the product 2g. When this product is
`equal to the thermal! equilibrium value ,?, the rate of
`recombination will be #,. Accordingly we may write
`for F., the rate of radiative recombination throughout
`the cell,
`
`FAV) =F onp/n2=F « exp(V/V.),
`
`(3.9)
`
`in which V represents the difference in imrefs or quasi-
`Fermi levels for holes and electrons, and the product
`np is proportional
`to the Boltzmann factor for this
`difference expressed as a voltage.’® V is evidently the
`voltage between the terminals connected to the p- and
`n-regions of the solar cell”; V. stands for kT./g.
`The net rate of increase of hole-electron pairs in-
`volves, in addition to generation, corresponding to F,,
`and recombination, corresponding to F,, nonradiative
`processes and removal! of hole-electron pairs by current
`to the external circuit. The nonradiative recombination
`and generation processes are represented by R(V) and
`R(Q) respectively. They will be equal for V=0, the
`thermal equilibrium condition. The algebraic sum of
`the rates of increase of hole-electron pairs must vanish
`
`\6 For example: W. Shockley, Hlectrons and Holes in Semicon-
`ductors (D. Van Nostrand Company, Inc., Princeton, NewJersey,
`1950) p. 308; the product of Eqs. (18) and (19).
`17 See footnote 16, p. 305; also W. Shockley, Bell System Tech.
`J. 28, 435 (1949), Sec. 5.
`
`

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`EFFICIENCY OF p-x JUNCTION SOLAR CELLS
`
`515
`
`the voltage, so that 7) must be regarded as voltage
`dependent.
`The current-voltage relationship for the cell when
`subjected to radiant energy may be obtained by solv-
`ing Eq. (3.10) for J. This leads to’:
`IT=q(F.—F0) + (qFe0o/f.)[1—exp (V/V) ]
`=I,t+Iol1—exp(V/V-) ],
`
`(3.16)
`
`in which the symbol J,, represents the short circuit
`current corresponding to V=0, and for the case of a
`planarcell of projected area A, is given by
`
`Tn=Q(F.— Foo) = GA pl false 2600)
`=F= 9A p(fulsQs).
`
`(3.17)
`
`Thelast form in Eq. (3.17) corresponds to the approxi-
`mation that in most conditions of interest the solar
`energy falling upon the body produces hole-electron
`pairs at a rate that is so much larger than would black-
`body radiation at the cell’s temperature that the latter
`term can be neglected in comparison with the former.
`The open-circuit voltage V,, which the cell would
`exhibit is obtained by solving Eq.
`(3.16) for the case
`of 7=0. This leads to
`
`Vop= Ve MLan/Lo) +1]
`=V.In[(f-F./Foo)— fet1].
`
`(3.18)
`
`This particular solution is valid for the case in which
`R depends upon voltage as given in Eq. (3.12). Other-
`wise, Eq. (3.18) will contain the open-circuit voltage
`in the term fo on the right side of the equation.
`As discussed in connection with Eq. (3.17), for most
`cases of interest the solar energy falling on the cell will
`be very large compared to blackbody radiation at the
`temperature of the cell, and accordingly the terms
`which do not involve F, in Eq. (3.18) can be neglected
`in comparison, as long as f, is not too small, This leads
`to the approximateresult
`
`Vop=Ve In(fefutsOs/2leOc) = Ve n(fQ;/Q-),
`
`(3.19)
`
`in whichit is seen that the geometrical and transmission
`factors together with the effect of excess recombination
`over radiative recombination may be lumped together
`in the single expression f, where
`
`SH fefats[Ue.
`
`(3.20)
`
`The factor 2 comes from the fact that sunlight falls on
`only one of
`the two sides of
`the planar cell.
`[See
`Eq. (3.8).]
`It is thus evident that as far as open-circuit voltage
`is concerned,similar results are produced by any of the
`four following variations: (1) reducing the efficiency of
`transmission of solar photonsinto the cell; (2) reducing
`
`18 Equations like (3.16) occur in published treatments of solar-
`cell efficiency. The difference is that the term in 7,, due to Feo,
`which is small but required by the principle of detailed balance,
`is included, and the coefficient of Jo is related to the fundamental
`minimum reverse saturation current rather than to a semi-
`empirical value.
`
`the solid angle subtended by the sun, or (3) its angle of
`incidence upon the solar cell; or (4) introducing addi-
`tional nonradiative recombination processes, thus mak-
`ing smaller the fraction of the recombination which is
`radiative.
`The maximum open-circuit voltage which may be
`obtained from the cell, in accordance with the theory
`presented,
`is the energy gap V,. This occurs as the
`temperature of the cell is reduced towards zero. Under
`these circumstances the quantity Q,
`tends towards
`zero and the logarithm in Eq.
`(3.19) to large values.
`The limiting behavior can be understood by noting
`that in accordance with Eq. (2.2) we have
`
`—InQ.=hy,/kT-+order of InT,
`Gah)
`Vig ¥-4-order of nF.
`The terms which are of the order V.InT, vanish as
`T,. and V, approach zero in Eq. (3.19), so that
`
`lim(V.— 0) Vop=V7p.
`
`(3.22)
`
`At higher temperatures the voltage is only a fraction
`of V,. This fraction”? denoted by » may be expressed as
`a function of three of the four variables discussed in
`the introduction for the important case in which the
`last two terms in the In term of Eq. (3.18) can be neg-
`lected. The necessary manipulations to establish this
`relationship are as follows:
`
`U(%9,%e,f= Vop/ Vo=(V./ V,) In ({Q0/Qe)
`
`= (x,/x,) function of (x,,7, and f)
`
`(3.23)
`
`0./Q.<0e4f- -/f. A,
`
`(3.24)
`
`where the integrands are each that of Eq. (2.2). [For
`cases of very low illumination in which the approxi-
`mation of Eq. (3.19) would not hold, v also depends
`explicitly on f.. |
`In the following two sections we shallconsider expres-
`sions for the output power in terms of the open-circuit
`voltage and short circuit currents just discussed.
`
`4. NOMINAL EFFICIENCY
`
`the geometrical configuration represented in
`For
`Fig. 2 (b),
`the incident power falling from the sun
`upon the solar cell may evidently be written in the
`form
`
`Pine= fod Ps=A fulvgQo/Uulxy);
`
`(4.1)
`
`(2.8) has been used to introduce the
`in which Eq,
`ultimate efficiency function #(x,) for purposes of sim-
`plifying subsequent manipulations.
`a
`A nominal efficiency can be defined in terms of the
`
`(3.16), factors like » have been introduced by
`19 As for Eq.
`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.
`
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`516
`
`Ww.
`
`SHOCKLEY AND H. J. QUEISSER
`
`1.0
`
`™
`
`0.
`
`10
`
`{0
`
`(00
`
`(900
`
`Fic. 4. Relationship between the impedance matching factor
`m and the open circuit voltage of a solar cell.
`
`the open-circuit
`incident power and the product of
`voltage V., and the short circuit current 7,4. The actual
`efficiency will be somewhat lower since the current-
`voltage characteristic is not perfectly rectangular. We
`shall consider the problem of matching the impedance
`in the following section.
`‘The nominalefficiency in terms of open-circuit volt-
`age and short circuit current is evidently given by
`
`VoplhfPine= VopA QfutsOs/LA Fok O./u(%y) J]
`= (Von/ Vg)u(xg)ts
`= Y (Ag,Xeyfu (Xe)tss
`
`(4.2)
`
`in which the symbol ¢ is the ratio of Eq. (3.23) of the
`open-circuit voltage V.,
`to the ultimate voltage V,
`that could be obtained if
`the battery were at zero
`temperature.
`
`5. DETAILED BALANCE LIMIT OF
`EFFICIENCY AND 9 (xg)Xatof)
`
`The maximum power output from the solar battery
`is obtained by choosing the voltage V so that
`the
`product (V is a maximum. In accordance with the
`current-voltage relationship, Eq. (3.16), and Eq. (3.18)
`for the open-circuit voltage, the current-voltage rela-
`tionship may be rewritten in the form
`
`(5.2). Substituting the symbols introduced in Eq. (5.3)
`into Eq. (5.2) leads readily to the relationship
`
`Sop=tntin(i+sz,).
`
`(5.4)
`
`This gives the functional relation between the open-
`circuit voltage and the voltage at which maximum
`power is obtained. In effect it establishes a functional
`relationship between z,, and z.,, and thus between zm
`and the variables /, x, and xy.
`It is seen that the open-circuit voltage is always
`larger than the voltage for maximum output, and when
`both voltages are small compared to thermal voltage
`V., then Eq. (5.4) leads to a maximum power voltage
`equal to one-half the open-circuit voltage, the situation
`corresponding to a battery with an ohmic internal
`resistance. On the other hand, when either 2.) or 2m
`is large compared to unity, then the ratio between the
`two approaches unity.”
`The maximum power is smaller than the nominal
`power [,,V., by the impedance matching factor m,
`where m is given by
`
`m=ICV (max) [V(max)/TnVop
`= Bm2/ (Lam — 6?) [tmtle (A+ em) |
`=m (v04/0) = M(Xp,Xeyf).
`
`(5.5)
`
`Figure 4 shows the dependence of m upon 2.) obtained
`by computing pairs of values of m(2m) and Zop(Zm) for
`various values of zm. The limits of m are 0.25 and 1.0
`for small and large values of 205.
`In terms of m the efficiency 7 can nowbe expressed as
`a function of the four variables x,, x,, f,, and f intro-
`duced in Sec. 1. The detailed balance limit corresponds
`to setting !,=1 and f=/f,/2. The efficiency » may be
`written as
`
`n(%y,Xeybs,f) =LEV (max) [V (max)/Pine
`= (ou)Vf,eX q)M(VX_/%Xe).
`
`(5.6)
`
`f=[,,+1o—Lyexp(V/V.)
`= Iylexp(Vop/ Ve) —exp(V/V-)|.
`The maximum power occurs when”:
`UV
`Tofexp(Vop/Ve)-LV +V)/Ve] exp(V/V.)} =0.
`This equation may be conveniently rewritten by intro-
`ducing the symbols
`
`5.1
`
`iSh)
`
`rl)
`
`d(IV)/dV =0
`
`(5.3)
`Zop=Vop/Ve= 0%4/Xey Sm=V(max)/V,,
`in which V(max)
`is the voltage which satisfies Eq.
`
`* Similar maximization of the