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The potentialities of silicon and gallium arsenide solar batteries
Solid-Slate
THE
Electronics
Pergamon
Press
1961.
Vol.
POTENTIALITIES ARSENIDE
2, pp. 222-231.
OF
Printed
SILICON
SOLAR
in Great
Britain
AND
GALLIUM
BATTERIES
T. S. MOSS Royai Aircraft (Received
Establishment,
21 October 1960;
Farnborough,
in revised
Hants.
form13 December
1960)
Abstract-The
theory of the spectral response of a p-n junction solar battery unit is given, and detailed comparisons are made of the expected performance of silicon and gallium arsenide units. Data now available on absorption in GaAs show that it is a particularly favourable material because it has a very steep absorption edge. The low effective masses and correspondingly low densities of states in GaAs are also advantageous. The efficiency of practical Si units is unlikely to exceed 15 per cent even nith intensive development, whereas GaAs units with efficiencies of about 20 per cent should be achievable in the near future and the performance may ultimately approach 25 per cent, even if (as assumed) the carrier lifetimes stay well below those for Si.
RBsumB-La
theorie de la reponse spectrale d’un element de batterie solaire a jonction p )I est demontree, et des comparaisons detaillees du rendement attendu des elements de silicium ct d’arseniure de gallium sont faites. Les don&es de l’absorption dans 1’AsCa qui sent maintenant disponibles demontrent que c’est un materiau particulieremcnt favorable parce qu’il comporte une at-&e d’absorption tres raide. Les faibles masses effectives ct les faibles densites de condition correspondantes dans 1’AsGa sont aussi avantageuses. Le rendement des elements pratiques de Si ne depasscra probablement pas 15 pour cent meme avec un developpement intensif tandis que les elements d’AsGa ayant des rendements d’enyiron 20 pour cent seraient realisables t&s prochainement et le rendement pourrait m&e atteindre 25 pour cent, m&me si (comme il est suppose) les durees de vie du porteur seraient plus courtes que celles du Si.
Zusammenfassung-Nach Entwicklung der Theorie der spektralcn Empfindlichkeit ciner Sonnenbatterie mit p-n-ubergang, werden die zu erwartenden Leistungen yen Siliziumund Galliumarsenid-Elementen im einzelnen verglichen. Die jetzt verfiigbaren Angaben iiber die Absorption van GaAs zeigen, dass es ein besonders geeignetes Material ist, da es eine sehr steile Absorptionskante besitzt. Die niedrigen Effektivmassen und die entsprechend niedrige Besetzung dcr Energiezustande in GaAs sind ebenfalls van Vorteil. Die Leistung der praktischen Siliziumzellen wird such bei intensiver Entwicklung im Bereich van 15 Prozent bleiben, wahrend sich bei den GaAs-Zellen bald Leistungen bis zu 20 Prozent erzielen lassen sollten. Die endgiiltige Leistung kijnnte sogar bis zu 2.5 Prozent gesteigert werden, such wenn. wie angenommen wird, die Lebensdauer der Trager betrachtlich unterhalb der yen Silizium liegt.
obtained (with the latest cells, 13 per cent for appreciable numbers and 14 per cent for a few selected units) is only about half that estimated in early theoretical work. It is, therefore, timely to investigate in some detail the theoretical behaviour of Si solar batteries in order to find the fundamental causes of this rather disappointing performance, and at the same time apply the theory to gallium arsenide-for
1. INTRODUCTION THE first publication
describing the use of largearea p-n junctions as solar-power converters was by CHAPIN et al. (1) These workers used silicon, and nearly all the development work carried out since that time has been on this same semiconductor. It has become increasingly evident over the past year or two that the efficiency of silicon units is approaching a limit, although the performance 222
THE
i POTENTIALITIES
OF SILICON
AND
which we have recently carried out the necessary measurements of absorption behaviour (MOSS and \
GALLIUM
ARSENIDE
SOLAR
BATTERIES
223
Consideration is restricted to single-component of some svstems. The nossible performance “multi-layer” cells has been discussed by WOLF@). I
I
2. NOTATION Thicknesses Electron and hole currents Electron and hole currents at the junction Electron and hole short-circuit currents Integrated short-circuit current Junction parameter JO Electron concentrations n, no, nd, fli Hole concentrations P, PO, Pa Concentrations of photo-carriers An, AP Functions of carrier concentration N P Constants of integration A, B, R, S Lifetimes of minority carriers Te, 7h Diffusion constants of minority carriers De, Dh Diffusion lengths of minority carriers L 1 Absorption coefficient K X exp (-Kt) Functions of K, L and 1 F, G Surface recombination parameters s, a, a’ Photon fluxes 2 Qo Energy gap Boltzman’s constant k Absolute temperature T Junction voltages v, Power from the junction w, Wo*t x Wavelength Junction parameter Y I? Unit charge Wave vector II Circumference/diameter ms, men, ~22, rrr*h Carrier masses Efficiencies ?,17,
(4 (Unit charge (Unit charge (Unit charge (Unit charge (Unit charge (cmm3) (cm-3) (cm-3)
(set) (cm2 set-I) (tL) (cm-l)
(cmea set-l)
(eV) (eV/W (“W (V)
vm,
3. THEORY
OF THE
SPECTRAL
RESPONSE
OF A
p-n JUNCTION
The configuration treated and the notation used are similar to those of MOSS(~) (Chap. 4). AS shown diagrammatically in Fig. 1, the junction is assumed
1photons&C
flow) flow) flow) flow) flow)
m
(PI (Cl
tion at y = 0 so that I photons/set are absorbed per unit area. The n-layer extends from y = t to y = d. The behaviour of the minority carriers in the p- and n-regions will be considered in turn. The problem is taken to be one-dimensional (i.e. edge effects are ignored) and, to simplify the notation, currents will be expressed as flows of unit charges. 3.1 In the p-region, for the electrons The continuity equation 4.15) is
(corresponding
dJ-ldy = An/r, - KI exp ( - Ky)
I1__-y.d FIG. 1. Diagrammatic
equation
representation
of solar battery.
to consist of a thin p-layer extending from y = 0 to the junction at y = t, irradiated in the y-direc-
to
(1)
where re is the lifetime of electrons in thep-region. The current equation (4.13) is replaced by J-
= De dAn/dy
(2)
224
T.
S.
MOSS
if we ignore the field component of current compared with the diffusion current. This is a good approximation when the field is due only to the Dember voltage, which is very small in heavily doped material. The solution of these equations is
giving
An = A coshy/L + B sinhy/L
is the hole diffusion
- (FL/D,):exp( - Ky)
Ap = R coshy/l+
(10) where 1 = (Dh Th)‘/’
(3) where
the diffusion
length
The boundary
conditions
1) s
Y = f,
nf = no exp eV/‘lhT
= sAn:
ZVL/De = no (exp eV/kl’-
1)
(7) For maximum efficiency it is thus clearly necessary to make a as small as possible, and t/L as small as is compatible with the requirements of absorption in the layer and transverse-layer resistance. 3.2 Zn the n-region, for the holes are similar
to (1) and (2)
dJ+/dy = KI exp ( - Ky) - Ap/rh J+ = -Da
dApjdy
J: = P-
(121
thick II layer.
G(Kl-
1) exp (- Kt)
J&
= IKl/( 1+ K/)
(14)
which shows that the full current of I electrons per second will he obtained if Kl $ 1, i.e. if the absorption takes place well within ;I diffusion length in the ?l-region. In the practical solalbattery, photoholes produced in the n-region will give an important contribution to the total currelIt. particularly at the longer wavelengths where the absorption is such that the p-layer is relativeI>transparent (Kt < 1) but Kl> 1. For typical parameters, i.e. x = 0.1 and t// = t/L = 0.3, the current contribution of the n-layer exceeds that of the p-layer when KL > 1.5, i.e. for h > 0.75~ for silicon. 3.3 Current at thejunction Writing for convenience exp( -33) = -Ti, the currents at the junction [equations (6) and (13) 1 become J;; = FKLX-
F(M+ KL)/(cosh t$L+cc sinh t/L) +(N+XF)tanh(a’+t/L)
(8) (9)
(13)
It is interesting to evaluate this expression for thr condition t -+ 0, i.e. the p-layer is so thin that then can be no absorption in it. ‘The short-circliit current (i.e. V = 0 and hence P = 0) becomes
(6)
JL~/Z= (cash t/L + CC sinh t/L)-’ - exp (- Kt)
equations
Ap = 0
Following from the LISA of a thick layer, WC put exp(- Kd) = 0, and coth d/l = I and obtain folthe current at the junction due to holes
This expression is readily shown to be finite at KL= lwhereF+a. For short wavelengths where K is large (i.e. KL >> 1) and for typical conditions where M < t/L < 1, the short-circuit current (I’ = 0 = N) b ecomes
The primary namely
are
(5)
(- Kt)] tanh (CC’+t/L)!
a relatively
(11)
assuming
Jr = FKL exp( - Kt) - F(a + KL)/(cosh t/L +
x’ = a: = rs/L and
1)
(4)
where no is the electron concentration in the dark, V the voltage across the junction, and s the surface recombination velocity. Evaluating A and B we find the current at the junction (y = t) to be
where tanh
conditions
y = t+d,
J- = De dAn/dy
and
y = f, pt= POexp eV/kT
are
y = 0,
a sinh t/L) + [N+Fexp
length
G = KII/(KY”The boundary
L = (De.re)l/2and F =[KIL/(K2Lz-
(Gl/Dh) exp (- Ky)
S sinhyll-
JI’=
-G(Kl-l)+P
= P-KIlX/(l+KZ)
(15) (16)
THE
POTENTIALITIES
The total short-circuit N = P = 0)isthus
-Jac = F
OF SILICON
junction
current
GALLIUM
ARSENIDE
SOLAR
BATTERIES
225
(i.e.
more mobile majority carriers and lower transverse resistance. As the relative magnitudes of I and L have little effect, it is convenient to put them equal in equation (17) and obtain
(17)
-JseP =
U-!-JCL cash t/L+ u sinh t/L
KIlX X tanh (cl’+ t/L) +- l+K2
- KLX-
AND
In discussing this expression, it will be seen that the useful terms are the first and last, the second and third terms represent losses. For reasonably high absorption conditions the second and last
KL[a+KL-(l+a)
exp(t/L-Kt)] (18)
(KzLz- 1) (cash t/L+a sinh t/L)
This expression, which is readily seen to be finite even when KL = 1, is plotted in Fig. 2 as a function of the dimensionless parameters KL and t/L. It has a maximum (for any absorption level
$ 2
E E
a”
80 0
O-i
0.2 Surface
0.4
D3 layer
thickness,
t/L
FIG. 2. Calculated collection efficiencies for various absorption levels.
terms largely cancel-showing that what radiation is transmitted through the p-layer is used effectively in the n-layer provided that it is absorbed within a diffusion length. Clearly the first term should be maximized and the third minimized. Both trends result from reducing tc, so it is clearly advisable to keep the surface recombination small, although the effect is not very marked. Increase in L is always advantageous as it reduces t/L, whilst increases in either diffusion length extend the waveband over which absorption is effective. Equation (17) shows only slight dependence on the relative magnitudes of 1 and L. It is not, therefore, automatically preferable to have the higher mobility minority carriers in the surface layer, as is common design practice, but on the contrary it may be advantageous to have a surface n-layer with
such that KL> 1) when t is given by exp Kt = coth t/L exp t/L- l/(KL sinh t/L) or approximately
(19)
by
Kt (expKt--1)
= KL-1
(20)
If the useful range of K is thus 2500-10,000 cm-l and L N 8~, the surface layer should be 1-3~ thick. The curves however are very flat, and a value of t/L = 0.25 will be very near the optimum for KL = 2-8. For this value of t/L the “collection efficiency” given by equation (18) will exceed 92 per cent for KL > 5. More generally there is also a maximum when L # I, i.e. for equation (17). In the practical design of a solar battery it may be advisable to have the surface layer somewhat thicker than this optimum in order to reduce the
T.
226
S.
transverse resistance through which the output current has to flow. The total short-circuit current (Jo) will be obtained by integrating equation (18) over all wavelengths, i.e. 00
Jo =
-
Jse dA electrons/set
(21)
I
0
MOSS
carrier reaching the junction provided (Y- 1) kll joules of energy to the external circuit. Ways of increasing JSc and (hence Jo) in order to have the maximum number of carriers reaching the junction have already been discussed. To increase the energy yield per carrier it is necessary to make Y as large as possible, and hence, since r _ log Joho, to make js as small as possible. Non ju
=
p0
j0
=
pO&h+rlOf/T?
From equations (15) and (16) the expression for the junction current is J=jo(expeV/kT-I)-Jo
general
(22)
where
Or as K’ should be considerably Now PO and no are the carriers, given by po = n&
jo = poDh/l+(noD&) The open-circuit eV,,
tanh (CL+ t/L)
photo-voltage
(23)
is thus given by
= kT log (I+ Jo/‘jo) = kT log Joho (24)
as Joijo is extremely large. The power delivered by the junction equation (22) PTJ= V(jo exp eV/kT- Jo) This can readily value when
be shown
is, from
(25)
to have its optimum
V opt = r(kT/e) and Jopt = -
Jor/(r+ 1) (26)
where (y+ 1) er = Jo/j0 The optimum
power is then W opt’ = (y- 1) JokT/e
and the efficiency
(at optimum
7 = (r-l)
tanh (L+t/L)
J!jh/l+(noou/L)
(29)
3.4 Voltage and power output
power)
kTJo/l.4&
(27) is given by (28)
where Qs is the total rate of arrival in the semiconductor of solar photons and 1.4 eV is their average energy. 3.5 Conditions for high efficiency Under optimum matching conditions the solar battery is seen to behave as though each photo-
where ni is the intrinsic p, are the concentrations acceptors respectively. Thus
less than t/I,. densities of mlllorit\
120= ni’/p[,cl concentration of ionized
(30)
and ~(1 anti donors and
The main feature of equation (31) is the rapid increase with ni and thus activation energy. Il\ contrast the other parameters will probably var\ little from one semiconductor to another, particularly if the technologies of preparing the scmiconductors are at similar states of dcvelopmcnt. As the efficiency will be seen to depend roughly OII the logarithm of jo, and only fall about 3 per cent for a 3 :l increase in ja, high accuracy in estimating je is not needed. In order to reduce js it is thus clearly necessary to use high doping levels. With increasing doping, however, the lifetimes will fall and at high concentrations there will be a tendency for 7h?lCiand T& to become constant and roughly equal. Also, as it is necessary to maintain long diffusion lengths in order to keep JO large, the doping cannot he too high. It will generally be found that the first term in equation (31) is several times smaller than the second, so that the latter is the important one to minimize. For Si, for example, where typically pa = nd = 2x 1018, t = &L, Dh =: 7cm2/sec and 71, rzd E asp, = 1011 cm-s set we have j&z2 = (2+6)10-1s
orjs
_ 10 -1-1~2:
(32)
THE
POTENTIALITIES
4. DISCUSSION AND
OF SILICON
AND
OF THE PROPERTIES OF SILICON GALLIUM
ARSENIDE
The main compromise in the design of a solar battery is between materials of low energy gap, where most of the solar photons are sufficiently energetic to produce photoelectrons so that Ja and the short-circuit current are large, and high energy-gap materials where ni and js are low so that the open-circuit voltage is high. In the simple
FIG. 3.
4.1 Absorption characteristics that in a material of given energy gap (and hence given EQ) the absorption should be high for wavelengths right up to those equivalent to the energy gap, i.e. the material should have a steep absorption edge, and in this respect GaAs is markedly superior to silicon. Results of recent measurements of the absorption coefficient for pure single-crystal GaAs, made in this laboratory, are shown in Fig. 3 along with silicon data from DASH and NEWMANc5) and P
ARSENIDE
SOLAR
BA4TTERIES
227
BRAUNSTEIN et al.@) The edge in GaAs is seen to be very steep, the fundamental reason being that the transitions are vertical as both band extrema are at k = 0. From these curves the spectral distribution of the short-circuit current has been computed and plotted in Fig. 4. On multiplying these curves by the distribution of solar radiation (taking the latter to be that of a black-body at 6OOO’C) and integrat-
Absorption in GaAs and Si.
case where all photons of energy greater than the energy gap are assumed to produce useful photocarriers and all other parameters are taken to be the same for all materials, it may be shown that there is an optimum energy gap, and that it is about 1.4 eV.t4)
It is clearly very important
GALLIUM
ing, we can determine the collection efhciency in terms of short-circuit electron flow per photon, i.e. Jo/@. The results are given in Table 1 for a silicon unit with the usual surface p-layer and for GaAs units with the surface layer of either p- or n-type. It should be noted that no atmospheric Table 1. Collection efficiency in electrons per solar photon Dimensions (VL) t=2,L=l=8 t=3,L=l=8 t = 2, L = 10,l = 4 t=1*6,L=4,1=10
UdQd
Silicon
‘g 38 42 39
22x
T.
co-
S.
MOSS
.-.--.
_/----
4. Collection
FIC.
efficiencies
for
advantage
of the GaAs
unit
Tuble 2. Solar I
Parameter Surface doping” Hulk doping Surface thickness D in bulk D in surface 7hlld = rep
solar hatterirs.
where 7’ is the absolute opcratiqg temperatut?, taken as %jo”I<. This expression is thus - 105 times lower for GaAs than Si. The intrinsic carrier concentration
4.2 Intrinsic carrier concentration significant
Garls
that this somewhat higher collection eficiency is achieved with a room-tempcraturc energy gap of 13 :-= 142 eV, compared with I:’ = 1 .1 CV for silicon, and hence: with a far smallerj,) value. The intrinsic carrier conccntratioli x-arics exponentially with cnergv, __ i.c.
absorption has heen included and no allowance has yet hcen made for surface reflection, i.e. the tabulated results are in terms of photons entering the semiconductor. (The diffusion lengths taken arc considered to he about the best obtainable for the doping levels _ used in Tahlc 2.) It will be seen from the table that the aerformante of the GaAs units is slightly superior.
The
Si and
is
battery pammeters I
Silicon 3 x lo’*/, 1w*rr 2 or 3~ 7.5 cm”/sec 25 cm”/sec 101’ cm-3 set
I
/
(;;I&
;
(normal)
Ga.4~
(rcverscd)
x 101: p
5 x 101;,r
I
2P 7.5 cm”/sec 6U cmz/sec 10’0 cn-3 set
:
jo
I * Heavier
surface doping
I has been assumed for the Si unit to obtain
low transverse
resistance.
THE
POTENTIALITIES
also depends the complete
OF SILICON
AND
on the effective masses of the carriers, expression being (at 300°K)
?zt”= (2?Tm,&T/ha)a = 6.3 x 1038 (me&s)3
exp ( - E/KT) exp (- E/kT)
cm-s
(33)
where merr is an effective density-of-states mass. For silicon, allowance for the six minima in the conduction band gives(T)
GALLIUM
ARSENIDE
SOLAR
BATTERIES
229
gives very low reflectivity,(ls) and about 95 per cent of the radiation enters the semiconductor. (2) Imperfect electrical characteristics of the unit. According to equation (26) the “junction efficiency” defined as the ratio of the optimum output power to the product of the open-circuit voltage and short-circuit current should be V-l VJ
For GaAs, we have simply (m,&?zs)a
= (??zZm$/m;)s”
(34)
where
m:/ms = 0.072 and mg/ms = 0.68 (EHRENREICH)@) giving (rn,&?~s)~ = 0.011. Thus this density-of-states term is -50 times smaller in GaAs than in Si, mainly as a consequence of the low effective-electron mass. Hence at 300”K, we have for Si, nf = 4x 1019 cme6, and, for GaAs, n: = 4x 1012 cm-6 with an improvement in favour of GaAs of N 107:l. The value ofjs for Si is given in equation (32). For GaAs, with the present state of technology, lifetimes arc worse than in silicon and they may well continue to be inferior because of the importance of direct recombination in GaAs.(s) We will, therefore, assume for GaAs Thtid = T& = 1010 cm-s set, i.e. a factor 10 times lower than for Si, as calculated by MAYBURGcg). The parameters taken for solar batteries of the two materials and the values of js computed from equation (31) are given in Table 2. 4.3 Expected efficiencies for Si and GaAs solar cells For solar radiation outside the earth’s atmosphere (i.e. as for satellites) where the intensity is 0.13 Wjcms, the photon flux is Qs = 5.8~ 1017 photon/set per ems. Hence for silicon, from Tables 1 and 2, Jo/j0 = 4.8 x 1012 and, from equation (26), r = 24. For GaAs, Jo/j, = (11 or 4.7) x 1017 and Y = 38 or 37. Thus from equation (28) the efficiencies are Si = 16 per cent, GaAs = 28 per cent (normal) and 26 per cent (reversed). In practice these values will be reduced by two loss factors. (1) Reflexion at the front surface. The method of forming the p-layer in silicon junctions
=
ivOt/,pt/_hVOC = r+log
(Y+ 1)
(35)
= 0.84 for Y = 24 as above. By comparison, the figure obtained from the current/voltage curves of the latest “gridded” Hoffman solar cells (of 13 per cent efficiency) is O-72. Hence the junction efficiency, including contact losses, is about 86 per cent of theoretical. Combining these figures, the resultant efficiency expected for the best silicon solar cells is v = 16x0.95
x0.86
= 13 per cent
This value is in excellent agreement with the performance obtained by the latest Hoffman cells. The limit of development of the silicon cell would seem to be little better than the above figure. It is unlikely that the absorption can be reduced significantly below 5 per cent already assumed, while a junction plus contact efficiency exceeding 90 per cent of theoretical seems unlikely. Thus the limit of the silicon solar battery is probably an efficiency of about 15 per cent, with production units inevitably somewhat worse than this. The loss factors may be estimated also for GaAs units: (1) It may be difficult to obtain quite such low reflectivities as for silicon, so we will assume the more pessimistic surface transmission factor of 90 per cent. (2) The technology of GaAs is not yet as highly developed as for Si, so that at the present time pn junctions in this material are far from ideal. However, there seems no reason why intensive development work should not remedy this situation, so that in the not too distant future junction efficiencies about 80 per cent of theoretical may be achieved for “normal” GaAs junctions. With these parameters a realistic figure for the
T.
230
s.
expected efficiency of a normal GaAs solar batterJ is 7 = 28 x09x0.X = 20 per cent unit with a surface rr-layer ITor the “reversed” it should he possible to reach very high junction efficiencies, as the electron mobility is 2500 cm’/\‘set even in material with 101s cm-3 free carriers.(s) \I;ith good development AVOI-k it would SWIII efficiency of I-casonal~lc to hope for a junction 90 per cult of thwrctical, ,$virig 7, == 20 x 0.9 x 0.0 == 21 per cult ‘I’hc best GaAs units reported co date(ll. I%Jhaw efficiencies of 7 per cent. It has been suggested by TAFERSKI et ul. 113)that recombination and regcneration of carriers in the depletion region is a major source of ineticiency. However, the results reported t)>- both the ahove ,groups sho\v that the open-circuit wltages are quite high (> 0.8 V) which is nearly 90 per cent of that expected from the present theor)-, showing that the junctions arc in fact rather good. The same data show that the major failing is in the short-circuit current, which is only half the expected value. This low collection efficiency stems from low diffusion lengths, which, in a 5 per cent efficient cell(13)? wcrc only -0.2 p and in the 7 per cent 1~11~~1)had been increased to -0.6.
MOSS and that the limiting resist.incc transverse . . c efficiency achicvablc 111 the future will be ahout 15 per cellt. iz similar conclllsion has hecn rcachcd lw \~olJ:(:J). _1%~contrast, (;a& units, with nioderatv ticvelopment, should reach 20-21 per cent cficicrlq~. the higher figltrc. being obtained for ;I reversed configuration, i.e. the surface layer n-type and the. bulk-material P-type. ‘lYhe ultimate development limit shr)ulcl approach 25 1x1- cent. 11) Order t(J dlicv~
thCW
reSlll&
it
will
k
I1CcCS-
sarv to ad\-aiicr the state of Gai4.s tcchnologv to appro~m~atel!the Icvcl of present clay \vcjl-k on silicoil in respect of: (1) 1)oping techlliqucs (2) I~ahricatioll of sharp junctions (3) I,oW resistance COlltilCtS (4) Reflectivity and surface rccomhination Irr addition it is very importaut to obtain carrier lifetimes as long as possible, hut it has lwl I)eell assumed in the theory. that thcsc will ever approach the \-alues ohtaincd for Si. On the contrary they have been taken as only 10 per cent of the values for Si of the same degree of doping. Also the rcflectivity of GaAs has 1~~11 assumed tc) 1~ some’what higher than Si. The fundamental advantages of GaBs over Si arr : (1)
5. CONCLUSIONS Detailed analysis of the photoelectric processes in p-n junctions has been used to estimate the probable efficiencies of silicon and gallium arsenide solar batteries. The theory has shown that for any given absorption constant there is an optimum thickness of surface layer. In the past,(s) it has been considered that this layer would be indefinitely thin if it were not for the effects of lateral resistance in the layer. The optimum thickness is roughly the reciprocal of the absorption constant, averaged over the effective waveband involved. Surface recombination is not a very important factor, particularly when the condition of optimum thickness is realized, or approached. The numerical results obtained show that silicon units cannot be expected to be much better than 13 per cent efficiency at the present time, even using the “gridded” structures to reduce the
(&As has a very sharp absorption edge, as :I of direct transitions from consequence valcncc to conduction band. (2~1) Activation energy of G;L%s is considerably higher. (2h) I,ower effective masses and only ow cow diictiori hand minimiml, giving Ion,-densit\ states.
Very high mobility giving low transverse resistance in the cast of the “rrversed” design. The result of items (2a) and (21~) is that the minority-carrier concentration for il given doping lercl is - 107 times lower than in Si.
(3)
‘I’he only fundamental disadvantage of GaAs is that the lifetimes will probably he shorter than in Si because of the influence of direct recombination. Values only l/l0 of those for Si have been assumed in the analysis, and the effect of this is only to reduce the factor in (2) above by 5 or 10 time.;,
THE
POTENTIALITIES
OF SILICON
AND
still leaving a net advantage to GaAs of N 106:l. In general terms, a good GaAs solar battery should give a slightly higher short-circuit current than a Si unit, and about 50 per cent greater voltage. The state of GaAs technology at the present day is, of course, well below that of silicon, but the above figures show such superior performance for GaAs solar batteries that the necessary development work is clearly called for. As, in addition, GaAs is proving a very promising material for transistors and various types of diodes, there seems little doubt that devoting intensive effort to the technology of this material is justified.
REFERENCES 1. D. M. CIIAPIN, C. S. FULLER and G. L. PEARSON, J. Appl. Phys. 25, 676 (1954). 2. T. S. Moss and T. D. F. HAWKIKS, Infrnred Phys. In press (1961). 3. M. WOLF, Proc. I.R.E. 48, 1246 (1960).
GALLIUM
6. 7. 8. 9. 10.
11.
12. 13.
SOLAR
BATTERIES
231
S. MOSS, Optical Properties of Semiconductors. Butterworths, London; Academic Press, New York (1959-61). W. C. DASH and R. NEWMAN, Phys. Rev. 99, 1151 (1955). R. BRAUNSTEIN,A. R. MOORE and F. HERMAN, Phys. Rev. 109, 695 (1958). H. BROOKS, Advanc. Electron. Electron Phys. 7, 85 (1955). H. EHRENREICH,Phys. Rev. 120, 1951 (1960). S. MAYBURG, Solid State Devices Conference, Carnegie Tech., Pittsburgh, June, 1960. J. F. HALL, J. Opt. Sot. Amer. 50, 717 (1960); M. WOLFE and M. B. PRINCE, Brussels Solid State Congress p. 1180. Academic Press, New York (1960). J. J. LOFERSKI, P. RAPI APORT and J. J. WYSOCKI, Proceedings of 13th Annual Power Conference. U.S. Army Signal Research and Development Laboratory (1959). E. G. BYLANDER, A. J. HODGESand J. -4. ROBERTS, J. Opt. Sot. Amer. 50, 983 (1960). J. J. WYSOCKI, J. J. LOFER~KI and P. RAPPAPORT, Proceedings of 14th Annual Power Conference. U.S. Army Signal Research and Development Laboratory (1960).
4. T.
5.
ARSENIDE
THE
Electronics
Pergamon
Press
1961.
Vol.
POTENTIALITIES ARSENIDE
2, pp. 222-231.
OF
Printed
SILICON
SOLAR
in Great
Britain
AND
GALLIUM
BATTERIES
T. S. MOSS Royai Aircraft (Received
Establishment,
21 October 1960;
Farnborough,
in revised
Hants.
form13 December
1960)
Abstract-The
theory of the spectral response of a p-n junction solar battery unit is given, and detailed comparisons are made of the expected performance of silicon and gallium arsenide units. Data now available on absorption in GaAs show that it is a particularly favourable material because it has a very steep absorption edge. The low effective masses and correspondingly low densities of states in GaAs are also advantageous. The efficiency of practical Si units is unlikely to exceed 15 per cent even nith intensive development, whereas GaAs units with efficiencies of about 20 per cent should be achievable in the near future and the performance may ultimately approach 25 per cent, even if (as assumed) the carrier lifetimes stay well below those for Si.
RBsumB-La
theorie de la reponse spectrale d’un element de batterie solaire a jonction p )I est demontree, et des comparaisons detaillees du rendement attendu des elements de silicium ct d’arseniure de gallium sont faites. Les don&es de l’absorption dans 1’AsCa qui sent maintenant disponibles demontrent que c’est un materiau particulieremcnt favorable parce qu’il comporte une at-&e d’absorption tres raide. Les faibles masses effectives ct les faibles densites de condition correspondantes dans 1’AsGa sont aussi avantageuses. Le rendement des elements pratiques de Si ne depasscra probablement pas 15 pour cent meme avec un developpement intensif tandis que les elements d’AsGa ayant des rendements d’enyiron 20 pour cent seraient realisables t&s prochainement et le rendement pourrait m&e atteindre 25 pour cent, m&me si (comme il est suppose) les durees de vie du porteur seraient plus courtes que celles du Si.
Zusammenfassung-Nach Entwicklung der Theorie der spektralcn Empfindlichkeit ciner Sonnenbatterie mit p-n-ubergang, werden die zu erwartenden Leistungen yen Siliziumund Galliumarsenid-Elementen im einzelnen verglichen. Die jetzt verfiigbaren Angaben iiber die Absorption van GaAs zeigen, dass es ein besonders geeignetes Material ist, da es eine sehr steile Absorptionskante besitzt. Die niedrigen Effektivmassen und die entsprechend niedrige Besetzung dcr Energiezustande in GaAs sind ebenfalls van Vorteil. Die Leistung der praktischen Siliziumzellen wird such bei intensiver Entwicklung im Bereich van 15 Prozent bleiben, wahrend sich bei den GaAs-Zellen bald Leistungen bis zu 20 Prozent erzielen lassen sollten. Die endgiiltige Leistung kijnnte sogar bis zu 2.5 Prozent gesteigert werden, such wenn. wie angenommen wird, die Lebensdauer der Trager betrachtlich unterhalb der yen Silizium liegt.
obtained (with the latest cells, 13 per cent for appreciable numbers and 14 per cent for a few selected units) is only about half that estimated in early theoretical work. It is, therefore, timely to investigate in some detail the theoretical behaviour of Si solar batteries in order to find the fundamental causes of this rather disappointing performance, and at the same time apply the theory to gallium arsenide-for
1. INTRODUCTION THE first publication
describing the use of largearea p-n junctions as solar-power converters was by CHAPIN et al. (1) These workers used silicon, and nearly all the development work carried out since that time has been on this same semiconductor. It has become increasingly evident over the past year or two that the efficiency of silicon units is approaching a limit, although the performance 222
THE
i POTENTIALITIES
OF SILICON
AND
which we have recently carried out the necessary measurements of absorption behaviour (MOSS and \
GALLIUM
ARSENIDE
SOLAR
BATTERIES
223
Consideration is restricted to single-component of some svstems. The nossible performance “multi-layer” cells has been discussed by WOLF@). I
I
2. NOTATION Thicknesses Electron and hole currents Electron and hole currents at the junction Electron and hole short-circuit currents Integrated short-circuit current Junction parameter JO Electron concentrations n, no, nd, fli Hole concentrations P, PO, Pa Concentrations of photo-carriers An, AP Functions of carrier concentration N P Constants of integration A, B, R, S Lifetimes of minority carriers Te, 7h Diffusion constants of minority carriers De, Dh Diffusion lengths of minority carriers L 1 Absorption coefficient K X exp (-Kt) Functions of K, L and 1 F, G Surface recombination parameters s, a, a’ Photon fluxes 2 Qo Energy gap Boltzman’s constant k Absolute temperature T Junction voltages v, Power from the junction w, Wo*t x Wavelength Junction parameter Y I? Unit charge Wave vector II Circumference/diameter ms, men, ~22, rrr*h Carrier masses Efficiencies ?,17,
(4 (Unit charge (Unit charge (Unit charge (Unit charge (Unit charge (cmm3) (cm-3) (cm-3)
(set) (cm2 set-I) (tL) (cm-l)
(cmea set-l)
(eV) (eV/W (“W (V)
vm,
3. THEORY
OF THE
SPECTRAL
RESPONSE
OF A
p-n JUNCTION
The configuration treated and the notation used are similar to those of MOSS(~) (Chap. 4). AS shown diagrammatically in Fig. 1, the junction is assumed
1photons&C
flow) flow) flow) flow) flow)
m
(PI (Cl
tion at y = 0 so that I photons/set are absorbed per unit area. The n-layer extends from y = t to y = d. The behaviour of the minority carriers in the p- and n-regions will be considered in turn. The problem is taken to be one-dimensional (i.e. edge effects are ignored) and, to simplify the notation, currents will be expressed as flows of unit charges. 3.1 In the p-region, for the electrons The continuity equation 4.15) is
(corresponding
dJ-ldy = An/r, - KI exp ( - Ky)
I1__-y.d FIG. 1. Diagrammatic
equation
representation
of solar battery.
to consist of a thin p-layer extending from y = 0 to the junction at y = t, irradiated in the y-direc-
to
(1)
where re is the lifetime of electrons in thep-region. The current equation (4.13) is replaced by J-
= De dAn/dy
(2)
224
T.
S.
MOSS
if we ignore the field component of current compared with the diffusion current. This is a good approximation when the field is due only to the Dember voltage, which is very small in heavily doped material. The solution of these equations is
giving
An = A coshy/L + B sinhy/L
is the hole diffusion
- (FL/D,):exp( - Ky)
Ap = R coshy/l+
(10) where 1 = (Dh Th)‘/’
(3) where
the diffusion
length
The boundary
conditions
1) s
Y = f,
nf = no exp eV/‘lhT
= sAn:
ZVL/De = no (exp eV/kl’-
1)
(7) For maximum efficiency it is thus clearly necessary to make a as small as possible, and t/L as small as is compatible with the requirements of absorption in the layer and transverse-layer resistance. 3.2 Zn the n-region, for the holes are similar
to (1) and (2)
dJ+/dy = KI exp ( - Ky) - Ap/rh J+ = -Da
dApjdy
J: = P-
(121
thick II layer.
G(Kl-
1) exp (- Kt)
J&
= IKl/( 1+ K/)
(14)
which shows that the full current of I electrons per second will he obtained if Kl $ 1, i.e. if the absorption takes place well within ;I diffusion length in the ?l-region. In the practical solalbattery, photoholes produced in the n-region will give an important contribution to the total currelIt. particularly at the longer wavelengths where the absorption is such that the p-layer is relativeI>transparent (Kt < 1) but Kl> 1. For typical parameters, i.e. x = 0.1 and t// = t/L = 0.3, the current contribution of the n-layer exceeds that of the p-layer when KL > 1.5, i.e. for h > 0.75~ for silicon. 3.3 Current at thejunction Writing for convenience exp( -33) = -Ti, the currents at the junction [equations (6) and (13) 1 become J;; = FKLX-
F(M+ KL)/(cosh t$L+cc sinh t/L) +(N+XF)tanh(a’+t/L)
(8) (9)
(13)
It is interesting to evaluate this expression for thr condition t -+ 0, i.e. the p-layer is so thin that then can be no absorption in it. ‘The short-circliit current (i.e. V = 0 and hence P = 0) becomes
(6)
JL~/Z= (cash t/L + CC sinh t/L)-’ - exp (- Kt)
equations
Ap = 0
Following from the LISA of a thick layer, WC put exp(- Kd) = 0, and coth d/l = I and obtain folthe current at the junction due to holes
This expression is readily shown to be finite at KL= lwhereF+a. For short wavelengths where K is large (i.e. KL >> 1) and for typical conditions where M < t/L < 1, the short-circuit current (I’ = 0 = N) b ecomes
The primary namely
are
(5)
(- Kt)] tanh (CC’+t/L)!
a relatively
(11)
assuming
Jr = FKL exp( - Kt) - F(a + KL)/(cosh t/L +
x’ = a: = rs/L and
1)
(4)
where no is the electron concentration in the dark, V the voltage across the junction, and s the surface recombination velocity. Evaluating A and B we find the current at the junction (y = t) to be
where tanh
conditions
y = t+d,
J- = De dAn/dy
and
y = f, pt= POexp eV/kT
are
y = 0,
a sinh t/L) + [N+Fexp
length
G = KII/(KY”The boundary
L = (De.re)l/2and F =[KIL/(K2Lz-
(Gl/Dh) exp (- Ky)
S sinhyll-
JI’=
-G(Kl-l)+P
= P-KIlX/(l+KZ)
(15) (16)
THE
POTENTIALITIES
The total short-circuit N = P = 0)isthus
-Jac = F
OF SILICON
junction
current
GALLIUM
ARSENIDE
SOLAR
BATTERIES
225
(i.e.
more mobile majority carriers and lower transverse resistance. As the relative magnitudes of I and L have little effect, it is convenient to put them equal in equation (17) and obtain
(17)
-JseP =
U-!-JCL cash t/L+ u sinh t/L
KIlX X tanh (cl’+ t/L) +- l+K2
- KLX-
AND
In discussing this expression, it will be seen that the useful terms are the first and last, the second and third terms represent losses. For reasonably high absorption conditions the second and last
KL[a+KL-(l+a)
exp(t/L-Kt)] (18)
(KzLz- 1) (cash t/L+a sinh t/L)
This expression, which is readily seen to be finite even when KL = 1, is plotted in Fig. 2 as a function of the dimensionless parameters KL and t/L. It has a maximum (for any absorption level
$ 2
E E
a”
80 0
O-i
0.2 Surface
0.4
D3 layer
thickness,
t/L
FIG. 2. Calculated collection efficiencies for various absorption levels.
terms largely cancel-showing that what radiation is transmitted through the p-layer is used effectively in the n-layer provided that it is absorbed within a diffusion length. Clearly the first term should be maximized and the third minimized. Both trends result from reducing tc, so it is clearly advisable to keep the surface recombination small, although the effect is not very marked. Increase in L is always advantageous as it reduces t/L, whilst increases in either diffusion length extend the waveband over which absorption is effective. Equation (17) shows only slight dependence on the relative magnitudes of 1 and L. It is not, therefore, automatically preferable to have the higher mobility minority carriers in the surface layer, as is common design practice, but on the contrary it may be advantageous to have a surface n-layer with
such that KL> 1) when t is given by exp Kt = coth t/L exp t/L- l/(KL sinh t/L) or approximately
(19)
by
Kt (expKt--1)
= KL-1
(20)
If the useful range of K is thus 2500-10,000 cm-l and L N 8~, the surface layer should be 1-3~ thick. The curves however are very flat, and a value of t/L = 0.25 will be very near the optimum for KL = 2-8. For this value of t/L the “collection efficiency” given by equation (18) will exceed 92 per cent for KL > 5. More generally there is also a maximum when L # I, i.e. for equation (17). In the practical design of a solar battery it may be advisable to have the surface layer somewhat thicker than this optimum in order to reduce the
T.
226
S.
transverse resistance through which the output current has to flow. The total short-circuit current (Jo) will be obtained by integrating equation (18) over all wavelengths, i.e. 00
Jo =
-
Jse dA electrons/set
(21)
I
0
MOSS
carrier reaching the junction provided (Y- 1) kll joules of energy to the external circuit. Ways of increasing JSc and (hence Jo) in order to have the maximum number of carriers reaching the junction have already been discussed. To increase the energy yield per carrier it is necessary to make Y as large as possible, and hence, since r _ log Joho, to make js as small as possible. Non ju
=
p0
j0
=
pO&h+rlOf/T?
From equations (15) and (16) the expression for the junction current is J=jo(expeV/kT-I)-Jo
general
(22)
where
Or as K’ should be considerably Now PO and no are the carriers, given by po = n&
jo = poDh/l+(noD&) The open-circuit eV,,
tanh (CL+ t/L)
photo-voltage
(23)
is thus given by
= kT log (I+ Jo/‘jo) = kT log Joho (24)
as Joijo is extremely large. The power delivered by the junction equation (22) PTJ= V(jo exp eV/kT- Jo) This can readily value when
be shown
is, from
(25)
to have its optimum
V opt = r(kT/e) and Jopt = -
Jor/(r+ 1) (26)
where (y+ 1) er = Jo/j0 The optimum
power is then W opt’ = (y- 1) JokT/e
and the efficiency
(at optimum
7 = (r-l)
tanh (L+t/L)
J!jh/l+(noou/L)
(29)
3.4 Voltage and power output
power)
kTJo/l.4&
(27) is given by (28)
where Qs is the total rate of arrival in the semiconductor of solar photons and 1.4 eV is their average energy. 3.5 Conditions for high efficiency Under optimum matching conditions the solar battery is seen to behave as though each photo-
where ni is the intrinsic p, are the concentrations acceptors respectively. Thus
less than t/I,. densities of mlllorit\
120= ni’/p[,cl concentration of ionized
(30)
and ~(1 anti donors and
The main feature of equation (31) is the rapid increase with ni and thus activation energy. Il\ contrast the other parameters will probably var\ little from one semiconductor to another, particularly if the technologies of preparing the scmiconductors are at similar states of dcvelopmcnt. As the efficiency will be seen to depend roughly OII the logarithm of jo, and only fall about 3 per cent for a 3 :l increase in ja, high accuracy in estimating je is not needed. In order to reduce js it is thus clearly necessary to use high doping levels. With increasing doping, however, the lifetimes will fall and at high concentrations there will be a tendency for 7h?lCiand T& to become constant and roughly equal. Also, as it is necessary to maintain long diffusion lengths in order to keep JO large, the doping cannot he too high. It will generally be found that the first term in equation (31) is several times smaller than the second, so that the latter is the important one to minimize. For Si, for example, where typically pa = nd = 2x 1018, t = &L, Dh =: 7cm2/sec and 71, rzd E asp, = 1011 cm-s set we have j&z2 = (2+6)10-1s
orjs
_ 10 -1-1~2:
(32)
THE
POTENTIALITIES
4. DISCUSSION AND
OF SILICON
AND
OF THE PROPERTIES OF SILICON GALLIUM
ARSENIDE
The main compromise in the design of a solar battery is between materials of low energy gap, where most of the solar photons are sufficiently energetic to produce photoelectrons so that Ja and the short-circuit current are large, and high energy-gap materials where ni and js are low so that the open-circuit voltage is high. In the simple
FIG. 3.
4.1 Absorption characteristics that in a material of given energy gap (and hence given EQ) the absorption should be high for wavelengths right up to those equivalent to the energy gap, i.e. the material should have a steep absorption edge, and in this respect GaAs is markedly superior to silicon. Results of recent measurements of the absorption coefficient for pure single-crystal GaAs, made in this laboratory, are shown in Fig. 3 along with silicon data from DASH and NEWMANc5) and P
ARSENIDE
SOLAR
BA4TTERIES
227
BRAUNSTEIN et al.@) The edge in GaAs is seen to be very steep, the fundamental reason being that the transitions are vertical as both band extrema are at k = 0. From these curves the spectral distribution of the short-circuit current has been computed and plotted in Fig. 4. On multiplying these curves by the distribution of solar radiation (taking the latter to be that of a black-body at 6OOO’C) and integrat-
Absorption in GaAs and Si.
case where all photons of energy greater than the energy gap are assumed to produce useful photocarriers and all other parameters are taken to be the same for all materials, it may be shown that there is an optimum energy gap, and that it is about 1.4 eV.t4)
It is clearly very important
GALLIUM
ing, we can determine the collection efhciency in terms of short-circuit electron flow per photon, i.e. Jo/@. The results are given in Table 1 for a silicon unit with the usual surface p-layer and for GaAs units with the surface layer of either p- or n-type. It should be noted that no atmospheric Table 1. Collection efficiency in electrons per solar photon Dimensions (VL) t=2,L=l=8 t=3,L=l=8 t = 2, L = 10,l = 4 t=1*6,L=4,1=10
UdQd
Silicon
‘g 38 42 39
22x
T.
co-
S.
MOSS
.-.--.
_/----
4. Collection
FIC.
efficiencies
for
advantage
of the GaAs
unit
Tuble 2. Solar I
Parameter Surface doping” Hulk doping Surface thickness D in bulk D in surface 7hlld = rep
solar hatterirs.
where 7’ is the absolute opcratiqg temperatut?, taken as %jo”I<. This expression is thus - 105 times lower for GaAs than Si. The intrinsic carrier concentration
4.2 Intrinsic carrier concentration significant
Garls
that this somewhat higher collection eficiency is achieved with a room-tempcraturc energy gap of 13 :-= 142 eV, compared with I:’ = 1 .1 CV for silicon, and hence: with a far smallerj,) value. The intrinsic carrier conccntratioli x-arics exponentially with cnergv, __ i.c.
absorption has heen included and no allowance has yet hcen made for surface reflection, i.e. the tabulated results are in terms of photons entering the semiconductor. (The diffusion lengths taken arc considered to he about the best obtainable for the doping levels _ used in Tahlc 2.) It will be seen from the table that the aerformante of the GaAs units is slightly superior.
The
Si and
is
battery pammeters I
Silicon 3 x lo’*/, 1w*rr 2 or 3~ 7.5 cm”/sec 25 cm”/sec 101’ cm-3 set
I
/
(;;I&
;
(normal)
Ga.4~
(rcverscd)
x 101: p
5 x 101;,r
I
2P 7.5 cm”/sec 6U cmz/sec 10’0 cn-3 set
:
jo
I * Heavier
surface doping
I has been assumed for the Si unit to obtain
low transverse
resistance.
THE
POTENTIALITIES
also depends the complete
OF SILICON
AND
on the effective masses of the carriers, expression being (at 300°K)
?zt”= (2?Tm,&T/ha)a = 6.3 x 1038 (me&s)3
exp ( - E/KT) exp (- E/kT)
cm-s
(33)
where merr is an effective density-of-states mass. For silicon, allowance for the six minima in the conduction band gives(T)
GALLIUM
ARSENIDE
SOLAR
BATTERIES
229
gives very low reflectivity,(ls) and about 95 per cent of the radiation enters the semiconductor. (2) Imperfect electrical characteristics of the unit. According to equation (26) the “junction efficiency” defined as the ratio of the optimum output power to the product of the open-circuit voltage and short-circuit current should be V-l VJ
For GaAs, we have simply (m,&?zs)a
= (??zZm$/m;)s”
(34)
where
m:/ms = 0.072 and mg/ms = 0.68 (EHRENREICH)@) giving (rn,&?~s)~ = 0.011. Thus this density-of-states term is -50 times smaller in GaAs than in Si, mainly as a consequence of the low effective-electron mass. Hence at 300”K, we have for Si, nf = 4x 1019 cme6, and, for GaAs, n: = 4x 1012 cm-6 with an improvement in favour of GaAs of N 107:l. The value ofjs for Si is given in equation (32). For GaAs, with the present state of technology, lifetimes arc worse than in silicon and they may well continue to be inferior because of the importance of direct recombination in GaAs.(s) We will, therefore, assume for GaAs Thtid = T& = 1010 cm-s set, i.e. a factor 10 times lower than for Si, as calculated by MAYBURGcg). The parameters taken for solar batteries of the two materials and the values of js computed from equation (31) are given in Table 2. 4.3 Expected efficiencies for Si and GaAs solar cells For solar radiation outside the earth’s atmosphere (i.e. as for satellites) where the intensity is 0.13 Wjcms, the photon flux is Qs = 5.8~ 1017 photon/set per ems. Hence for silicon, from Tables 1 and 2, Jo/j0 = 4.8 x 1012 and, from equation (26), r = 24. For GaAs, Jo/j, = (11 or 4.7) x 1017 and Y = 38 or 37. Thus from equation (28) the efficiencies are Si = 16 per cent, GaAs = 28 per cent (normal) and 26 per cent (reversed). In practice these values will be reduced by two loss factors. (1) Reflexion at the front surface. The method of forming the p-layer in silicon junctions
=
ivOt/,pt/_hVOC = r+log
(Y+ 1)
(35)
= 0.84 for Y = 24 as above. By comparison, the figure obtained from the current/voltage curves of the latest “gridded” Hoffman solar cells (of 13 per cent efficiency) is O-72. Hence the junction efficiency, including contact losses, is about 86 per cent of theoretical. Combining these figures, the resultant efficiency expected for the best silicon solar cells is v = 16x0.95
x0.86
= 13 per cent
This value is in excellent agreement with the performance obtained by the latest Hoffman cells. The limit of development of the silicon cell would seem to be little better than the above figure. It is unlikely that the absorption can be reduced significantly below 5 per cent already assumed, while a junction plus contact efficiency exceeding 90 per cent of theoretical seems unlikely. Thus the limit of the silicon solar battery is probably an efficiency of about 15 per cent, with production units inevitably somewhat worse than this. The loss factors may be estimated also for GaAs units: (1) It may be difficult to obtain quite such low reflectivities as for silicon, so we will assume the more pessimistic surface transmission factor of 90 per cent. (2) The technology of GaAs is not yet as highly developed as for Si, so that at the present time pn junctions in this material are far from ideal. However, there seems no reason why intensive development work should not remedy this situation, so that in the not too distant future junction efficiencies about 80 per cent of theoretical may be achieved for “normal” GaAs junctions. With these parameters a realistic figure for the
T.
230
s.
expected efficiency of a normal GaAs solar batterJ is 7 = 28 x09x0.X = 20 per cent unit with a surface rr-layer ITor the “reversed” it should he possible to reach very high junction efficiencies, as the electron mobility is 2500 cm’/\‘set even in material with 101s cm-3 free carriers.(s) \I;ith good development AVOI-k it would SWIII efficiency of I-casonal~lc to hope for a junction 90 per cult of thwrctical, ,$virig 7, == 20 x 0.9 x 0.0 == 21 per cult ‘I’hc best GaAs units reported co date(ll. I%Jhaw efficiencies of 7 per cent. It has been suggested by TAFERSKI et ul. 113)that recombination and regcneration of carriers in the depletion region is a major source of ineticiency. However, the results reported t)>- both the ahove ,groups sho\v that the open-circuit wltages are quite high (> 0.8 V) which is nearly 90 per cent of that expected from the present theor)-, showing that the junctions arc in fact rather good. The same data show that the major failing is in the short-circuit current, which is only half the expected value. This low collection efficiency stems from low diffusion lengths, which, in a 5 per cent efficient cell(13)? wcrc only -0.2 p and in the 7 per cent 1~11~~1)had been increased to -0.6.
MOSS and that the limiting resist.incc transverse . . c efficiency achicvablc 111 the future will be ahout 15 per cellt. iz similar conclllsion has hecn rcachcd lw \~olJ:(:J). _1%~contrast, (;a& units, with nioderatv ticvelopment, should reach 20-21 per cent cficicrlq~. the higher figltrc. being obtained for ;I reversed configuration, i.e. the surface layer n-type and the. bulk-material P-type. ‘lYhe ultimate development limit shr)ulcl approach 25 1x1- cent. 11) Order t(J dlicv~
thCW
reSlll&
it
will
k
I1CcCS-
sarv to ad\-aiicr the state of Gai4.s tcchnologv to appro~m~atel!the Icvcl of present clay \vcjl-k on silicoil in respect of: (1) 1)oping techlliqucs (2) I~ahricatioll of sharp junctions (3) I,oW resistance COlltilCtS (4) Reflectivity and surface rccomhination Irr addition it is very importaut to obtain carrier lifetimes as long as possible, hut it has lwl I)eell assumed in the theory. that thcsc will ever approach the \-alues ohtaincd for Si. On the contrary they have been taken as only 10 per cent of the values for Si of the same degree of doping. Also the rcflectivity of GaAs has 1~~11 assumed tc) 1~ some’what higher than Si. The fundamental advantages of GaBs over Si arr : (1)
5. CONCLUSIONS Detailed analysis of the photoelectric processes in p-n junctions has been used to estimate the probable efficiencies of silicon and gallium arsenide solar batteries. The theory has shown that for any given absorption constant there is an optimum thickness of surface layer. In the past,(s) it has been considered that this layer would be indefinitely thin if it were not for the effects of lateral resistance in the layer. The optimum thickness is roughly the reciprocal of the absorption constant, averaged over the effective waveband involved. Surface recombination is not a very important factor, particularly when the condition of optimum thickness is realized, or approached. The numerical results obtained show that silicon units cannot be expected to be much better than 13 per cent efficiency at the present time, even using the “gridded” structures to reduce the
(&As has a very sharp absorption edge, as :I of direct transitions from consequence valcncc to conduction band. (2~1) Activation energy of G;L%s is considerably higher. (2h) I,ower effective masses and only ow cow diictiori hand minimiml, giving Ion,-densit\ states.
Very high mobility giving low transverse resistance in the cast of the “rrversed” design. The result of items (2a) and (21~) is that the minority-carrier concentration for il given doping lercl is - 107 times lower than in Si.
(3)
‘I’he only fundamental disadvantage of GaAs is that the lifetimes will probably he shorter than in Si because of the influence of direct recombination. Values only l/l0 of those for Si have been assumed in the analysis, and the effect of this is only to reduce the factor in (2) above by 5 or 10 time.;,
THE
POTENTIALITIES
OF SILICON
AND
still leaving a net advantage to GaAs of N 106:l. In general terms, a good GaAs solar battery should give a slightly higher short-circuit current than a Si unit, and about 50 per cent greater voltage. The state of GaAs technology at the present day is, of course, well below that of silicon, but the above figures show such superior performance for GaAs solar batteries that the necessary development work is clearly called for. As, in addition, GaAs is proving a very promising material for transistors and various types of diodes, there seems little doubt that devoting intensive effort to the technology of this material is justified.
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GALLIUM
6. 7. 8. 9. 10.
11.
12. 13.
SOLAR
BATTERIES
231
S. MOSS, Optical Properties of Semiconductors. Butterworths, London; Academic Press, New York (1959-61). W. C. DASH and R. NEWMAN, Phys. Rev. 99, 1151 (1955). R. BRAUNSTEIN,A. R. MOORE and F. HERMAN, Phys. Rev. 109, 695 (1958). H. BROOKS, Advanc. Electron. Electron Phys. 7, 85 (1955). H. EHRENREICH,Phys. Rev. 120, 1951 (1960). S. MAYBURG, Solid State Devices Conference, Carnegie Tech., Pittsburgh, June, 1960. J. F. HALL, J. Opt. Sot. Amer. 50, 717 (1960); M. WOLFE and M. B. PRINCE, Brussels Solid State Congress p. 1180. Academic Press, New York (1960). J. J. LOFERSKI, P. RAPI APORT and J. J. WYSOCKI, Proceedings of 13th Annual Power Conference. U.S. Army Signal Research and Development Laboratory (1959). E. G. BYLANDER, A. J. HODGESand J. -4. ROBERTS, J. Opt. Sot. Amer. 50, 983 (1960). J. J. WYSOCKI, J. J. LOFER~KI and P. RAPPAPORT, Proceedings of 14th Annual Power Conference. U.S. Army Signal Research and Development Laboratory (1960).
4. T.
5.
ARSENIDE