- Email: [email protected]
Theoretical efficiency of realistic solar cells intended for thermophotovoltaic applications
Solar Cells, 19 (1986 - 1987) 123 - 130
123
T H E O R E T I C A L EFFICIENCY OF REALISTIC S O L A R CELLS INTENDED F O R T H E R M O P H O T O V O L T A I C APPLICATIONS A. CARUSO and G. PIRO
Universitd di Napoli, Dipartimento di lngegneria Elettronica, 80125 Napoli (Italy) (Received August 16, 1 9 8 5 ;a c c e p te d April 7, 1986)
Summary The ideal and theoretical efficiencies of one-, two- and three-cell systems using GaAs, silicon and germanium semiconductor materials for thermophotovoltaic applications have been calculated to select the most suitable combination for these applications. It has been verified that, although the G a A s - S i - G e system is better than the Si-Ge system in the ideal case, this is no longer true in the theoretical case.
1. Introduction It is well known that tandem-cell converters are particularly useful in concentration applications and therefore in the thermophotovoltaic (TPV) conversion systems. The efficiency of tandem-cell systems that present no losses and collect all photogenerated carriers, called the ideal efficiency, has been calculated by many researchers. It has been found that, b y selecting sets of materials with suitable band gaps, the efficiency increases with the number of cells [1 - 3]. However, the result can be different if one refers to the actual characteristics of the system and, in particular, to the actual physical and geometrical characteristics of each cell and of the radiation source. In this work we deal with tandem-cell systems obtained with the more usual semiconductor materials, namely GaAs, silicon and germanium, to achieve a TPV system utilising a b l a c k b o d y radiation source (2500 K) and spectrum splitting b y dichroic filters. The theoretical efficiency has been c o m p u t e d using a simple photovoltaic model. In this model the effect of the back-surface field (BSF) is n o t taken into account for the sake of simplicity. In any case, the BSF, if the cell is suitably modelled, can increase the efficiency of the cells made from different materials of comparable quantity, so that the relative effect can be neglected. 0379-6787/86/$3.50
© Elsevier Sequoia/Printed in The Netherlands
124
The theoretical efficiency evaluation is made by assuming no structural losses for the cells. Values for the physical and geometrical parameters were chosen by fitting the measured or calculated data from various published papers [ 4 - 8]. The goal of this paper is to select the most suitable set of cells for the TPV system. In fact, it will be shown that the theoretical efficiency sometimes decreases if the number of cells increase, so that the most suitable configuration differs from that found by using the ideal efficiency. It has been verified that the theoretical efficiency of the Si-Ge system is greater than that of the G a A s - S i - G e system, although its ideal effficiency is lower. 2. Calculation procedure A computer program has been developed to calculate the single-cell efficiency of a TPV system: ,ff ( % ) -
Vmplmp
- -
X 100
(1)
PiCR where CR is the concentration ratio, Pi is the total blackbody radiation
power density incident on the system (Fig. 1), Vmp and Imp are the voltage and current respectively at m a x i m u m power, obtained as a function of the reverse saturation current I0 and the short-circuit current Isc. By referring to the system in Fig. 2 with (n + 1) cells and n filters, Isc for the Jth cell is calculated from the relation
/,oj
~cj
= CR ,,f
J-
Iscj(~k)Tj(X)
0
1
[I R i ( X ) d X
(2)
1
r-
2010 TM
E E
Filter 1
20 15.10~6
Cell1
co
ab
Z
15
a 10.10TM ~
F.2 %
. . . .
~F.(n-1)
C.(n-1)
,,
e
% 5"10TM
o
C.2
F.n
C,(n+l)
zo
1.1016 0,5 1 1,5 WAVELENGTH, ~, (pro)
C,n
Fig. 1. Curve a, spectral irradiance of the 2500 K blackbody radiation source; curve b, number of photons emitted us. wavelength. Fig. 2. Configuration of the spectrum splitting system.
125 TABLE 1 Parameters used in the efficiency calculation Parameter
GaAs
Silicon
Germanium
Energy band gap (eV) Acceptor density (cm -3) Donor density (cm -3) Electron mobility (cm 2 V -1 s-1) Hole mobility (cm 2 V -1 s- l ) Electron lifetime (s) Hole lifetime (s) Electron diffusion constant (cm 2 s-1) Hole diffusion constant (cm 2 s - l ) Electron diffusion length (pro) Hole diffusion length (btm) Junction depth (pro) Recombination velocity at front (cm s-1) Recombination velocity at back ( c m s - 1 ) B (cm-6) a
1.42 2 × 1017 1019 3700 62 2.5 × 10 -6 0.5 × 10 -9 95.6 1.6 15.5 0.29 0.04 106
1.11 2 × 1017 1019 600 31 20 X 10 -6 10 -8 15.5 0.8 176 0.9 0.1 103
0.42 2 × 1017 1019 2500 115 3 X 10 -4 3 × 10 - s 64.5 2.97 1390 3 0.3 103
oo
oo
o¢
0.224 X 1037
0.137 × 1040
0.697 X 1038
ani2 = B e x p ( - - E G / k T
).
where kcj is the cut-off wavelength of the Jth cell and Tj(k) in the Jth filter transmittance considered equal to unity for J = (n + 1); the ith filter reflectance R~(k) is R~(X) = 1 - - T i ( k )
(3)
The p r o d u c t J-1
II R~(),)
(4)
1
has to be considered equal to unity for J = 1. The overall efficiency of the tandem system can be simply c o m p u t e d as the sum of the single-cell efficiencies. In this study the assumptions have been made that the cells present: (a) no reflection losses; (b) infinite shunt resistance; (c) no series resistance; (d) Jdiode due to diffusion currents only. Moreover, for the ideal and theoretical case we assume the following relations:
(5)
Jsc(ideal) -~ CRqNo
J~(theoretical) =
CRq[t
oL. 1t e-ottoh(K ÷
t
(SLp/Dp) + O~Lp 1 odV°Lne-at] + cosh(t/Lp) + (SLp/Dp) sinh(t/Lp) --c~Lpe-at + ~ n + l ]
(6)
126
u# iO0 ~
80 t
20
~.t ~ - 0,1 --'I WAVE L ENGTH.~,(~4m)
Fig. 3. Spectral transmittance of the filter.
where tanh K -
SLp Dp
(7)
and q is the electronic charge, cz is the absorption coefficient per centimetre and No is the number of incident photons (cm -2 s -1 #m-l). The other parameters are described in Table 1. The transmittance characteristics of the selective filters used in both efficiency calculations [9] are shown in Fig. 3. Finally, the effect of the k t shift (kt is the wavelength at which the filter starts to cut off) owing to a possible mismatch of the splitting mirror, has been analysed. The parameter set assumed in our calculation is reported in Table 1. 3. Discussion In Figs. 4 - 6 the theoretical efficiencies of the GaAs, silicon and germanium solar cells vs. the source temperature are reported for different concentration ratios. We observe that for all temperatures in the useful range ( 1 0 0 0 - 2500 K) the theoretical efficiency of the germanium cell is the highest: 77 = 15.4% compared with 8.2% for the silicon cell and 4.8% for the GaAs cell. Moreover, the efficiency of the germanium cell shows the greatest sensitivity to a change in concentration. In fact, assuming a fill factor of unity and that the spectral distribution of the radiation is n o t altered by the concentrator, we have A~7
In CR2--1n CRI
77
In CR 1 + ln(Isc/Io)
(8)
and the germanium cell, owing to its very high reverse saturation current I0, presents the lowest I,c:Io ratio. We have calculated the theoretical efficiency for the three possible two-cell combinations, GaAs-Si, G a A s - G e and Si-Ge, as a function of ),t for different concentration ratios. The results are shown in Fig. 7.
127
30 CR:100( CR: CR:500 I0(~ CR:I
>: zu
2c
>: oz uJ
CR:~00~ CR CR:IO0 : 50C
20
c..)
CR:I
,.=
w cu
O ~
10
tu v¢
10
0 uJ "T
I
L
i
1000 2000 3000 4000 5000 6000 SOURCE TEMPERATURETs(K)
1000 2000 3000 4000 5000 6000 SOURCE TEMPERATURE,T~(K) Fig. 4. Theoretical e f f i c i e n c y o f the GaAs cell d i f f e r e n t c o n c e n t r a t i o n ratios.
us.
b l a c k b o d y source temperature for
Fig. 5. Theoretical e f f i c i e n c y o f t h e silicon cell different c o n c e n t r a t i o n ratios.
vs.
b l a c k b o d y source temperature for
Si-Ge
30
20
_~
~16
~,, 020 z
Z 17 0z 16
"
,9, ~0<~ 10
FILTERCUTOFF WAVELENGTH, SYSTEM, Zt(#m~ 0,957 1,057 1,157 1,257
"
~
~
C:5R00CR cR:1C:1R00:t00
OwCCF~-m 141 w85!I' "'"....i~I~''-'-'''"~". I--
~.\ " ."...~R SO~ , CR
1000 2000 3000 4000 50006000 SOURCE TEMPERATURE,Ts (K) Fig. 6. T h e o r e t i c a l e f f i c i e n c y o f the g e r m a n i u m cell for different c o n c e n t r a t i o n ratios.
IC~=-,
0,723 0,823 0,923 1,023 FILTERCUTOFFWAVELENGTH,GaAs-Si AND GaAs-Ge SYSTEM,Zt(pm) vs.
b l a c k b o d y source temperature
Fig. 7. T h e o r e t i c a l e f f i c i e n c y o f the G a A s - S i , G a A s - G e and S i - G e s y s t e m s v s . filter c u t - o f f w a v e l e n g t h for different c o n c e n t r a t i o n ratios: - - , Si-Ge; .... , GaAs-Ge; -- • --, G a A s - S i .
128
25
p
i!THEORY-BC J
--
10
. 1
L
10 CONCENTRATION
100 RATIO,
1000
CR
Fig. 8. I d e a l a n d t h e o r e t i c a l e f f i c i e n c y o f t h e S i - G e a n d G a A s - S i - G e s y s t e m s v s . c o n c e n t r a t i o n r a t i o in t h e b e s t c a s e ( B . C . ) a n d w o r s t c a s e ( W . C . ) w h i c h r e f e r e n c e t o t h e k t shift : -- -- --, GaAs-Si-Ge; , Si-Ge.
It can be seen that the o p t i m u m configuration is Si-Ge with a maxim u m conversion efficiency equal to 19.1% {with optimised ~t) for a concentration ratio of 500, compared with 17.5% for GaAs-Ge and 7.1% for GaAs-Si. The effect of the Xt shift can be explained by considering the distance between the cut-off wavelengths ~¢ for the two cells of the system. With reference to the transmittance characteristics of the filter (Fig. 3), it is clear that if we take kt to be less than the lowest cut-off wavelength, we subtract a small b u t important portion of the incident spectrum from the higher band-gap cell, while we send a greater b u t less important portion of the lower band-gap cell. In this case the efficiency drop increases as the distance between the t w o cut-off wavelengths increases. However, if we take kt > ~¢, we subtract one portion of the incident spectrum from the lower band-gap cell, b u t this portion is n o t useful for the higher band-gap cell. In this case the efficiency drop increases as the distance between the two cut-off wavelengths decreases. This is the reason for the very great efficiency drop in the GaAs-Si system. Finally, the conversion efficiency of the system consisting of three cells, G a A s - S i - G e , has been calculated. In Fig. 8 the ideal and the theoretical efficiencies of this system are plotted as a function of the concentration ratio, and compared with that of the Si-Ge system in the best and in the worst case with reference to variations in hr. Two general conclusions may be drawn. The first is that while in the best case, the ideal efficiency of the GaAs--Si-Ge system is better than that o f Si-Ge (7? = 24.5% compared with 23.6%; C R = 500), in the worst case the situation is the opposite, 07 = 20.8% compared with 20.9%). The reason is that in the G a A s - S i - G e system we have two filters with a double error pos-
129 sibflity. Also, the GaAs-Si combination is more sensitive to the kt shift. Thus the result depends on the transmission characteristics of the filters in a critical way. The second and more important conclusion is that the theoretical efficiency of the Si-Ge system is always greater than that of the G a A s - S i Ge system (for C R = 500, ~ = 19.1% compared with 18.6% in the best case and 16.7% - 15.6% in the worst case). This behaviour can be explained if we consider that in the ideal case (unity collection efficiency) the short-circuit current of the silicon cell in the Si-Ge system is practically equal to the sum of the short-circuit currents of the GaAs and the silicon cells in the G a A s - S i - G e system. Then the overall conversion efficiency of this latter system is greater than that of the Si-Ge cell because the open-circuit voltage of the GaAs cell is greater than that of the silicon cell. N o w theoretical efficiencies have been calculated to take into account the real physical characteristics of each cell in the short-circuit current determination. In this case, even if the open-circuit voltage of the GaAs cell is greater than that of the silicon cell, the efficiency of the G a A s - S i - G e system is lower than that of Si-Ge because the short-circuit current of the GaAs cell is very small due to the very small lifetime and the particular spectrum considered. It is shown in Fig. 8 that the Si-Ge system is also more sensitive to the concentration ratio. Moreover, it is interesting to note that in the previous analysis we have considered a source temperature of 2500 K. Now, if we operate at a lower source temperature, and consider the correct functionality of each cell efficiency with the source temperature (Figs. 4 - 6), we expect an increase in the theoretical efficiency o f the Si-Ge system compared with that of GaAs-Si-Ge.
4. Conclusions The ideal and theoretical efficiencies of one-, two- and three-cell systems using GaAs, silicon and germanium semiconductor materials for TPV applications have been calculated. The purpose of this study is to select the most suitable combination for these applications. We have verified that, although the G a A s - S i - G e combination is slightly better than the Si-Ge system in the ideal case, this is no longer true in the theoretical case. Moreover, the Si-Ge system becomes more advantageous with respect to the G a A s - S i - G e system when the concentration ratio increases, and we expect the same result if the source temperature decreases. Therefore the G a A s - S i - G e tandem-cell system is n o t only more expensive and more difficult to realize, b u t is also less efficient than the Si-Ge system.
130
Acknowledgment The author would like to thank Professor P. Spirito for very helpful discussions. This work was supported by Comitato Nazionale per la Ricerca e per lo Suiluppo dell'Energia Nucleare e delle Energie Alternative (ENEA, Italy).
References 1 N. A. Gokeen and J. J. Loferski, Efficiency of tandem solar cell systems as a function of temperature and solar energy concentration ratio, Sol. Energy Mater., 1 (1979) 271 - 286. 2 A. Benett and L. C. Olsen, Analysis of multiple-cell concentrator photovoltaic systems, Proc. 13th IEEE Conf. Photovoltaic Specialists, Washington, DC, June 5 - 8, 1978, IEEE, New York, 1978, pp. 868 - 873. 3 G. W. Masden and C. E. Backus, Increased photovoltaic conversion efficiency through use of spectrum splitting and multiple cells, Proc. 13th IEEE Conf. Photovoltaic Specialists, Washington, DC, June 5 - 8, 1978, IEEE, New York, 1978, pp. 853 - 858. 4 C. Goradia, M. G. Goradia and H. Curtis, Near-optimum design of GaAs-based concentrator space solar cells for 80 °C operation, Proc. 1 7th IEEE Conf. Photovoltaic Specialists, Kissimmee, FL, May 1 - 4, 1984, IEEE, New York, 1984, pp. 5 6 - 62. 5 E. S. Vera, J. J. Loferski and M. Spitzer, Theoretical limit efficiency of two junction tandem silicon-germanium solar cells intended for thermophotovoltaic application, Proc. 15th IEEE Conf. Photovoltaic Specialists, Orlando, FL, May 12- 15, 1981, IEEE, New York, 1981, pp. 872 - 882. 6 M. A. Green, A. W. Blakers, J. Shi, E. M. Ketper and S. R. Wenham, High efficiency silicon solar cells, IEEE Trans. Electron Devices, ED-31 (1984) 679 - 683. 7 C. T. Sah, K. A. Yamakawa and R. Lutwack, Effect of thickness on silicon solar cell efficiency, IEEE Trans. Electron Devices, ED-29 (1982) 923 - 908. 8 R. C. Knechtli, R. Loo and G . Sanjivkamath, High efficiency GaAs solar cells, IEEE Trans. Electron Devices, ED-31 (1984) 577 - 588. 9 F. Demichelis, E. Minetti-Mezzetti, M. Agnello and V. Pesotto, Band-pass filters for thermophotovoltaic conversion systems, Sol. Cells, 5 (1982) 135 - 141.
123
T H E O R E T I C A L EFFICIENCY OF REALISTIC S O L A R CELLS INTENDED F O R T H E R M O P H O T O V O L T A I C APPLICATIONS A. CARUSO and G. PIRO
Universitd di Napoli, Dipartimento di lngegneria Elettronica, 80125 Napoli (Italy) (Received August 16, 1 9 8 5 ;a c c e p te d April 7, 1986)
Summary The ideal and theoretical efficiencies of one-, two- and three-cell systems using GaAs, silicon and germanium semiconductor materials for thermophotovoltaic applications have been calculated to select the most suitable combination for these applications. It has been verified that, although the G a A s - S i - G e system is better than the Si-Ge system in the ideal case, this is no longer true in the theoretical case.
1. Introduction It is well known that tandem-cell converters are particularly useful in concentration applications and therefore in the thermophotovoltaic (TPV) conversion systems. The efficiency of tandem-cell systems that present no losses and collect all photogenerated carriers, called the ideal efficiency, has been calculated by many researchers. It has been found that, b y selecting sets of materials with suitable band gaps, the efficiency increases with the number of cells [1 - 3]. However, the result can be different if one refers to the actual characteristics of the system and, in particular, to the actual physical and geometrical characteristics of each cell and of the radiation source. In this work we deal with tandem-cell systems obtained with the more usual semiconductor materials, namely GaAs, silicon and germanium, to achieve a TPV system utilising a b l a c k b o d y radiation source (2500 K) and spectrum splitting b y dichroic filters. The theoretical efficiency has been c o m p u t e d using a simple photovoltaic model. In this model the effect of the back-surface field (BSF) is n o t taken into account for the sake of simplicity. In any case, the BSF, if the cell is suitably modelled, can increase the efficiency of the cells made from different materials of comparable quantity, so that the relative effect can be neglected. 0379-6787/86/$3.50
© Elsevier Sequoia/Printed in The Netherlands
124
The theoretical efficiency evaluation is made by assuming no structural losses for the cells. Values for the physical and geometrical parameters were chosen by fitting the measured or calculated data from various published papers [ 4 - 8]. The goal of this paper is to select the most suitable set of cells for the TPV system. In fact, it will be shown that the theoretical efficiency sometimes decreases if the number of cells increase, so that the most suitable configuration differs from that found by using the ideal efficiency. It has been verified that the theoretical efficiency of the Si-Ge system is greater than that of the G a A s - S i - G e system, although its ideal effficiency is lower. 2. Calculation procedure A computer program has been developed to calculate the single-cell efficiency of a TPV system: ,ff ( % ) -
Vmplmp
- -
X 100
(1)
PiCR where CR is the concentration ratio, Pi is the total blackbody radiation
power density incident on the system (Fig. 1), Vmp and Imp are the voltage and current respectively at m a x i m u m power, obtained as a function of the reverse saturation current I0 and the short-circuit current Isc. By referring to the system in Fig. 2 with (n + 1) cells and n filters, Isc for the Jth cell is calculated from the relation
/,oj
~cj
= CR ,,f
J-
Iscj(~k)Tj(X)
0
1
[I R i ( X ) d X
(2)
1
r-
2010 TM
E E
Filter 1
20 15.10~6
Cell1
co
ab
Z
15
a 10.10TM ~
F.2 %
. . . .
~F.(n-1)
C.(n-1)
,,
e
% 5"10TM
o
C.2
F.n
C,(n+l)
zo
1.1016 0,5 1 1,5 WAVELENGTH, ~, (pro)
C,n
Fig. 1. Curve a, spectral irradiance of the 2500 K blackbody radiation source; curve b, number of photons emitted us. wavelength. Fig. 2. Configuration of the spectrum splitting system.
125 TABLE 1 Parameters used in the efficiency calculation Parameter
GaAs
Silicon
Germanium
Energy band gap (eV) Acceptor density (cm -3) Donor density (cm -3) Electron mobility (cm 2 V -1 s-1) Hole mobility (cm 2 V -1 s- l ) Electron lifetime (s) Hole lifetime (s) Electron diffusion constant (cm 2 s-1) Hole diffusion constant (cm 2 s - l ) Electron diffusion length (pro) Hole diffusion length (btm) Junction depth (pro) Recombination velocity at front (cm s-1) Recombination velocity at back ( c m s - 1 ) B (cm-6) a
1.42 2 × 1017 1019 3700 62 2.5 × 10 -6 0.5 × 10 -9 95.6 1.6 15.5 0.29 0.04 106
1.11 2 × 1017 1019 600 31 20 X 10 -6 10 -8 15.5 0.8 176 0.9 0.1 103
0.42 2 × 1017 1019 2500 115 3 X 10 -4 3 × 10 - s 64.5 2.97 1390 3 0.3 103
oo
oo
o¢
0.224 X 1037
0.137 × 1040
0.697 X 1038
ani2 = B e x p ( - - E G / k T
).
where kcj is the cut-off wavelength of the Jth cell and Tj(k) in the Jth filter transmittance considered equal to unity for J = (n + 1); the ith filter reflectance R~(k) is R~(X) = 1 - - T i ( k )
(3)
The p r o d u c t J-1
II R~(),)
(4)
1
has to be considered equal to unity for J = 1. The overall efficiency of the tandem system can be simply c o m p u t e d as the sum of the single-cell efficiencies. In this study the assumptions have been made that the cells present: (a) no reflection losses; (b) infinite shunt resistance; (c) no series resistance; (d) Jdiode due to diffusion currents only. Moreover, for the ideal and theoretical case we assume the following relations:
(5)
Jsc(ideal) -~ CRqNo
J~(theoretical) =
CRq[t
oL. 1t e-ottoh(K ÷
t
(SLp/Dp) + O~Lp 1 odV°Lne-at] + cosh(t/Lp) + (SLp/Dp) sinh(t/Lp) --c~Lpe-at + ~ n + l ]
(6)
126
u# iO0 ~
80 t
20
~.t ~ - 0,1 --'I WAVE L ENGTH.~,(~4m)
Fig. 3. Spectral transmittance of the filter.
where tanh K -
SLp Dp
(7)
and q is the electronic charge, cz is the absorption coefficient per centimetre and No is the number of incident photons (cm -2 s -1 #m-l). The other parameters are described in Table 1. The transmittance characteristics of the selective filters used in both efficiency calculations [9] are shown in Fig. 3. Finally, the effect of the k t shift (kt is the wavelength at which the filter starts to cut off) owing to a possible mismatch of the splitting mirror, has been analysed. The parameter set assumed in our calculation is reported in Table 1. 3. Discussion In Figs. 4 - 6 the theoretical efficiencies of the GaAs, silicon and germanium solar cells vs. the source temperature are reported for different concentration ratios. We observe that for all temperatures in the useful range ( 1 0 0 0 - 2500 K) the theoretical efficiency of the germanium cell is the highest: 77 = 15.4% compared with 8.2% for the silicon cell and 4.8% for the GaAs cell. Moreover, the efficiency of the germanium cell shows the greatest sensitivity to a change in concentration. In fact, assuming a fill factor of unity and that the spectral distribution of the radiation is n o t altered by the concentrator, we have A~7
In CR2--1n CRI
77
In CR 1 + ln(Isc/Io)
(8)
and the germanium cell, owing to its very high reverse saturation current I0, presents the lowest I,c:Io ratio. We have calculated the theoretical efficiency for the three possible two-cell combinations, GaAs-Si, G a A s - G e and Si-Ge, as a function of ),t for different concentration ratios. The results are shown in Fig. 7.
127
30 CR:100( CR: CR:500 I0(~ CR:I
>: zu
2c
>: oz uJ
CR:~00~ CR CR:IO0 : 50C
20
c..)
CR:I
,.=
w cu
O ~
10
tu v¢
10
0 uJ "T
I
L
i
1000 2000 3000 4000 5000 6000 SOURCE TEMPERATURETs(K)
1000 2000 3000 4000 5000 6000 SOURCE TEMPERATURE,T~(K) Fig. 4. Theoretical e f f i c i e n c y o f the GaAs cell d i f f e r e n t c o n c e n t r a t i o n ratios.
us.
b l a c k b o d y source temperature for
Fig. 5. Theoretical e f f i c i e n c y o f t h e silicon cell different c o n c e n t r a t i o n ratios.
vs.
b l a c k b o d y source temperature for
Si-Ge
30
20
_~
~16
~,, 020 z
Z 17 0z 16
"
,9, ~0<~ 10
FILTERCUTOFF WAVELENGTH, SYSTEM, Zt(#m~ 0,957 1,057 1,157 1,257
"
~
~
C:5R00CR cR:1C:1R00:t00
OwCCF~-m 141 w85!I' "'"....i~I~''-'-'''"~". I--
~.\ " ."...~R SO~ , CR
1000 2000 3000 4000 50006000 SOURCE TEMPERATURE,Ts (K) Fig. 6. T h e o r e t i c a l e f f i c i e n c y o f the g e r m a n i u m cell for different c o n c e n t r a t i o n ratios.
IC~=-,
0,723 0,823 0,923 1,023 FILTERCUTOFFWAVELENGTH,GaAs-Si AND GaAs-Ge SYSTEM,Zt(pm) vs.
b l a c k b o d y source temperature
Fig. 7. T h e o r e t i c a l e f f i c i e n c y o f the G a A s - S i , G a A s - G e and S i - G e s y s t e m s v s . filter c u t - o f f w a v e l e n g t h for different c o n c e n t r a t i o n ratios: - - , Si-Ge; .... , GaAs-Ge; -- • --, G a A s - S i .
128
25
p
i!THEORY-BC J
--
10
. 1
L
10 CONCENTRATION
100 RATIO,
1000
CR
Fig. 8. I d e a l a n d t h e o r e t i c a l e f f i c i e n c y o f t h e S i - G e a n d G a A s - S i - G e s y s t e m s v s . c o n c e n t r a t i o n r a t i o in t h e b e s t c a s e ( B . C . ) a n d w o r s t c a s e ( W . C . ) w h i c h r e f e r e n c e t o t h e k t shift : -- -- --, GaAs-Si-Ge; , Si-Ge.
It can be seen that the o p t i m u m configuration is Si-Ge with a maxim u m conversion efficiency equal to 19.1% {with optimised ~t) for a concentration ratio of 500, compared with 17.5% for GaAs-Ge and 7.1% for GaAs-Si. The effect of the Xt shift can be explained by considering the distance between the cut-off wavelengths ~¢ for the two cells of the system. With reference to the transmittance characteristics of the filter (Fig. 3), it is clear that if we take kt to be less than the lowest cut-off wavelength, we subtract a small b u t important portion of the incident spectrum from the higher band-gap cell, while we send a greater b u t less important portion of the lower band-gap cell. In this case the efficiency drop increases as the distance between the t w o cut-off wavelengths increases. However, if we take kt > ~¢, we subtract one portion of the incident spectrum from the lower band-gap cell, b u t this portion is n o t useful for the higher band-gap cell. In this case the efficiency drop increases as the distance between the two cut-off wavelengths decreases. This is the reason for the very great efficiency drop in the GaAs-Si system. Finally, the conversion efficiency of the system consisting of three cells, G a A s - S i - G e , has been calculated. In Fig. 8 the ideal and the theoretical efficiencies of this system are plotted as a function of the concentration ratio, and compared with that of the Si-Ge system in the best and in the worst case with reference to variations in hr. Two general conclusions may be drawn. The first is that while in the best case, the ideal efficiency of the GaAs--Si-Ge system is better than that o f Si-Ge (7? = 24.5% compared with 23.6%; C R = 500), in the worst case the situation is the opposite, 07 = 20.8% compared with 20.9%). The reason is that in the G a A s - S i - G e system we have two filters with a double error pos-
129 sibflity. Also, the GaAs-Si combination is more sensitive to the kt shift. Thus the result depends on the transmission characteristics of the filters in a critical way. The second and more important conclusion is that the theoretical efficiency of the Si-Ge system is always greater than that of the G a A s - S i Ge system (for C R = 500, ~ = 19.1% compared with 18.6% in the best case and 16.7% - 15.6% in the worst case). This behaviour can be explained if we consider that in the ideal case (unity collection efficiency) the short-circuit current of the silicon cell in the Si-Ge system is practically equal to the sum of the short-circuit currents of the GaAs and the silicon cells in the G a A s - S i - G e system. Then the overall conversion efficiency of this latter system is greater than that of the Si-Ge cell because the open-circuit voltage of the GaAs cell is greater than that of the silicon cell. N o w theoretical efficiencies have been calculated to take into account the real physical characteristics of each cell in the short-circuit current determination. In this case, even if the open-circuit voltage of the GaAs cell is greater than that of the silicon cell, the efficiency of the G a A s - S i - G e system is lower than that of Si-Ge because the short-circuit current of the GaAs cell is very small due to the very small lifetime and the particular spectrum considered. It is shown in Fig. 8 that the Si-Ge system is also more sensitive to the concentration ratio. Moreover, it is interesting to note that in the previous analysis we have considered a source temperature of 2500 K. Now, if we operate at a lower source temperature, and consider the correct functionality of each cell efficiency with the source temperature (Figs. 4 - 6), we expect an increase in the theoretical efficiency o f the Si-Ge system compared with that of GaAs-Si-Ge.
4. Conclusions The ideal and theoretical efficiencies of one-, two- and three-cell systems using GaAs, silicon and germanium semiconductor materials for TPV applications have been calculated. The purpose of this study is to select the most suitable combination for these applications. We have verified that, although the G a A s - S i - G e combination is slightly better than the Si-Ge system in the ideal case, this is no longer true in the theoretical case. Moreover, the Si-Ge system becomes more advantageous with respect to the G a A s - S i - G e system when the concentration ratio increases, and we expect the same result if the source temperature decreases. Therefore the G a A s - S i - G e tandem-cell system is n o t only more expensive and more difficult to realize, b u t is also less efficient than the Si-Ge system.
130
Acknowledgment The author would like to thank Professor P. Spirito for very helpful discussions. This work was supported by Comitato Nazionale per la Ricerca e per lo Suiluppo dell'Energia Nucleare e delle Energie Alternative (ENEA, Italy).
References 1 N. A. Gokeen and J. J. Loferski, Efficiency of tandem solar cell systems as a function of temperature and solar energy concentration ratio, Sol. Energy Mater., 1 (1979) 271 - 286. 2 A. Benett and L. C. Olsen, Analysis of multiple-cell concentrator photovoltaic systems, Proc. 13th IEEE Conf. Photovoltaic Specialists, Washington, DC, June 5 - 8, 1978, IEEE, New York, 1978, pp. 868 - 873. 3 G. W. Masden and C. E. Backus, Increased photovoltaic conversion efficiency through use of spectrum splitting and multiple cells, Proc. 13th IEEE Conf. Photovoltaic Specialists, Washington, DC, June 5 - 8, 1978, IEEE, New York, 1978, pp. 853 - 858. 4 C. Goradia, M. G. Goradia and H. Curtis, Near-optimum design of GaAs-based concentrator space solar cells for 80 °C operation, Proc. 1 7th IEEE Conf. Photovoltaic Specialists, Kissimmee, FL, May 1 - 4, 1984, IEEE, New York, 1984, pp. 5 6 - 62. 5 E. S. Vera, J. J. Loferski and M. Spitzer, Theoretical limit efficiency of two junction tandem silicon-germanium solar cells intended for thermophotovoltaic application, Proc. 15th IEEE Conf. Photovoltaic Specialists, Orlando, FL, May 12- 15, 1981, IEEE, New York, 1981, pp. 872 - 882. 6 M. A. Green, A. W. Blakers, J. Shi, E. M. Ketper and S. R. Wenham, High efficiency silicon solar cells, IEEE Trans. Electron Devices, ED-31 (1984) 679 - 683. 7 C. T. Sah, K. A. Yamakawa and R. Lutwack, Effect of thickness on silicon solar cell efficiency, IEEE Trans. Electron Devices, ED-29 (1982) 923 - 908. 8 R. C. Knechtli, R. Loo and G . Sanjivkamath, High efficiency GaAs solar cells, IEEE Trans. Electron Devices, ED-31 (1984) 577 - 588. 9 F. Demichelis, E. Minetti-Mezzetti, M. Agnello and V. Pesotto, Band-pass filters for thermophotovoltaic conversion systems, Sol. Cells, 5 (1982) 135 - 141.