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Interference filters for thermophotovoltaic solar energy conversion
Solar Cells, 10 (1983) 273 - 286
273
INTERFERENCE FILTERS FOR THERMOPHOTOVOLTAIC SOLAR ENERGY CONVERSION
H. HOFLER, H. J. PAUL, W. RUPPEL and P. WI]RFEL Institut f~'r Angewandte Physik, Universitti't Karlsruhe, D-7500 Karlsruhe (F.R.G.) (Received March 9, 1983; accepted June 14, 1983)
Summary In a thermophotovoltaic (TPV) energy converter the necessary selectivity of the radiation incident on the solar cells may be obtained with a filter. Filters consisting o f a metal layer and a sequence of dielectric layers with given indices of refraction are optimized with respect to the layer thicknesses and the number of layers. No sensible improvement is obtained by increasing the number o f layers to more than seven. The c o m p u t e d performance of a solar TPV converter equipped with a black intermediate absorber, an optimized filter and silicon solar cells is expected to be superior by a factor of 1.4 to that of direct conversion b y silicon cells. A TPV converter equipped with germanium cells and an optimized filter yields even an improvement by a factor o f 2.1 over direct conversion by germanium cells and an improvement by a factor of 1.8 over direct conversion by silicon cells. No further improvement is achieved b y the combination of a filter and a selective emitter instead of a filter and a black emitter.
1. Introduction Thermophotovoltaic (TPV) energy conversion has an advantage with respect to direct conversion by solar cells in that the spectral distribution offered to the solar cells m a y be tailored according to their sensitivity. This tailoring can be achieved by two means. The first is selectivity in the emission of the intermediate emitter. The second is selectivity obtained by passing the radiation through a filter. The first possibility is discussed in the accompanying paper [ 1 ]. The second is the subject of this paper. The filter in a TPV converter is to transmit photons whose energy ~¢o is only slightly larger than the band gap eg of the cells. High energy (/i¢o >> eg) and low energy (rico < eg) p h o t o n s cannot be converted by solar cells effectively and must be reflected back to the emitter. This condition is less stringent for high energy photons, especially with respect to silicon solar cells (% = 1.1 eV), since for realistic absorber temperatures their intensity is
274
small. In this case a mirror on the back side of the solar cell suffices. The solar cell itself acts as the desired filter, provided that it does not absorb photons which it does not convert [2]. If this condition is not sufficiently fulfilled, the filter must be placed in front of the solar cells. For the design of a filter, criteria must be developed according to which the performance of a filter is to be evaluated. A high transmission of useful p h o t o n s is not sufficient. A better criterion, which also includes a high reflectivity for unwanted photons, is the efficiency itself of the TPV converter as a whole. This concept is a little more complicated than the optimization of transmission, but it ensures that the filter is properly adjusted to other factors influencing the overall efficiency, such as the spectrum of the emitter or the spectral q u a n t u m efficiency of the solar cells.
2. Evaluation o f filter performance In ref. 1 a figure of merit is defined and used selective emitters in TPV conversion. It also allows filters. This figure f of merit relates the solar energy direct conversion to the solar energy current input IE, for the same power o u t p u t with the same solar cells: f -
IE, dir IE,
for the evaluation of for the evaluation of current input IE, ~ in WeVin TPV conversion
(1)
TPV
For identical solar cells in direct and in TPV conversion the equality of power outputs can be replaced by the equality of short-circuit currents, which are easier to calculate. The solar energy current IE, TPV incident on a TPV converter can be split up into three parts. The energy current IE, lo~s is emitted back by the intermediate absorber through the entrance aperture. It is lost to the outside. The energy current IE. f is absorbed by the filter and also lost. The energy current IE, s is transmitted through the filter and incident on the solar cells. The sum is (2)
IE, T e V = IE,,oss + IE, f + IE, s
It is shown in ref. 1 t h a t the energy current IE, lo~ is at least IE, T P v ( T i n t / T s ) 4 where Tint is the temperature of the intermediate absorber and Ts that of the Sun. The insertion of this expression into eqn. (2) yields IE, f + IE, s IE, TPV = 1 - - ( T i n t / T s ) 4
(3)
With the assumption that the intermediate absorber emits black radiation also from the surface o f area B exposed towards the filter, it is B
~o A(~co)(h~) 3 d(~/oJ)
IE, r -- 41r2~h3c 2 oj
exp(~/kTint)
-- 1
(4)
275
and B IE, s -
~
T(/t¢o)(/t¢o) 3 d(~i¢o)
47r2~3c2 0J
exp(~co/kTint) -- 1
(5)
where A(~co) is the spectral absorptivity of the filter and T(~co) its spectral transmissivity. The p h o t o n s which are transmitted by the filter generate a short-circuit current IQ, TeV in the solar cells / ~(~t¢o)T(fico)(/f¢o) 2 d(~v)) 47r2~3C2 0 exp(~co/kTi,,t) -- 1 eB
IQ, T P V
--
(6)
where/3(t/c0) is the q u a n t u m efficiency of the solar cells. The q u a n t u m efficiency for germanium shown in Fig. 1 b y the broken line is taken from ref. 3. The q u a n t u m efficiency for silicon shown in Fig. 1 by the chain line was measured on A E G - T e l e f u n k e n solar cells t y p e H E C / B S F [4]. Both curves give the number of electrons contributing to the short-circuit current per incident photon. The larger values of ~(~co) for silicon are due to an antireflection coating which is missing on the germanium cells, thereby allowing more p h o t o n s to enter the solar cen. The filter in a TPV converter is best placed directly on t o p of the solar cell. The interface to the solar cell then influences the filter properties and must therefore be accounted for in the calculation of its reflectivity and transmissivity. The transmitted energy current IE, s in eqn. (5) then is the current which enters the solar cell. The q u a n t u m efficiency to be used in eqn. (6) for the short-circuit current must therefore be modified so as to relate to the n u m b e r o f p h o t o n s entering the solar cell instead of to the n u m b e r of p h o t o n s that are incident. If the reflectivity for silicon as we have measured it on the A E G - T e l e f u n k e n cells and the reflectivity for germanium as taken
08
06
O~
02
1~w(ev} Fig. 1. Q u a n t u m efficiency ~] as a f u n c t i o n o f p h o t o n energy for g e r m a n i u m and silicon solar cells: - - - --, ~ per incident p h o t o n according to ref. 3 ; - - • - - , ~ per incident p h o t o n according to ref. 4; - - , /~ per p h o t o n entering the solar cells, calculated f r o m the broken and chain lines.
276 from ref. 8 are taken into account, corrected quantum efficiencies per entering p h o t o n are found, which are given by the full lines in Fig. 1. For direct illumination of a solar cell the short-circuit current IQ, caused by the incident energy current IE, ~ is calculated quite analogously to eqn. (6), as discussed in ref. 1. The figure of merit can then be written as f -
(7)
IE, ~ II0,
TPv
with the condition that the short.circuit currents IQ, d~ and IQ, TPV be equal. Written in this form, f depends only on the transmissivity T(/ico) and the absorptivity A(#ico) of the filter, on the q u a n t u m efficiency fl(/ico) of the solar cells and on the temperature Trot of the black intermediate absorber. For the procedure to evaluate eqn. (7), the reader is referred to ref. 1. If TPV conversion is to outperform direct conversion with the same solar cells, the figure f of merit of the converter must be greater than unity. In the following the figure f of merit will be calculated for various filters as a function of their transmissivity T(#t¢o) and absorptivity A (/iw) for silicon and germanium solar cells.
3. Method for filter optimization For a given multilayer filter the transmissivity T(/~co) and the absorptivity A(fi~0) are determined by m e t h o d s outlined in ref. 6. With these results the figure of merit for a TPV converter with this filter is calculated from eqns. (2) - (7) for a given solar cell, characterized by 13(~co), and a given absorber temperature Trot. The m a x i m u m values of f are f o u n d by varying the thickness of individual layers in a multilayer interference filter. For this purpose a m e t h o d which was developed by Nelder and Mead [7] to find an absolute m i n i m u m o f an n-dimensional function is used. The function to be minimized is l/f, and the n independent variables are the thicknesses of the n layers of the filter. The number n of the layers, their sequence and their optical constants are n o t subject to variations. The procedure requires starting values for the individual layer thicknesses. The result of the minimization of 1/f is not sensitive to the starting values, as long as the starting values are within an order of magnitude of the final values.
4. Results o f filter optimization calculation The filters to be optimized consist of a metal layer embedded in a sequence of dielectric layers with alternating high (H) and low (L) indices of refraction. The arrangement is shown in Fig. 2. A high refiectivity for all p h o t o n s with flu) < eg c a n n o t be achieved with an all~lielectric filter. For a high reflectivity the filters must contain a metallic layer. The transmission for useful p h o t o n s (/~o) ~> %) is achieved by a suitable dielectric antireflection coating.
277 vacuum
H : ZnS. n H = 2~$2 L : MgFz. nL. = 'L38
H M : metal
lag
or Au)
H
L H
$ubstrote
Fig. 2. Basic structure of a multilayer interference filter consisting of a metal layer embedded in stacks of dielectric layers with alternating high (H) and low (L) indices of refraction.
The optical constants used in the calculation are those for ZnS (nil = 2.32), MgF: (n L = 1.38) and the metals silver or gold for which the data are taken from ref. 8. The filters were optimized for substrates of glass (n = 1.5) and of germanium and silicon, for which the optical constants are taken from refs. 8 and 9 respectively. The absorber temperature Trot is taken to be 2500 K b u t the dependence of TPV performance on absorber temperature will also be investigated. The calculated properties of the filters are listed in Tables 1 and 2. 4.1. Filter for silicon solar cells Demichelis et al. [10] have published details of a filter for use with silicon solar cells in a TPV converter. It consists o f 11 layers on a glass substrate, five alternating layers o f ZnS and MgF2 on each side of a silver film. Since Demichelis et al. give only the spectral dependence o f the transmissivity T(~¢o) for the filter and not its absorptivity, it is n o t possible to evaluate h o w well it qualifies with respect to TPV conversion. With the data for layer thicknesses and optical constants from ref. 10, we recalculated the properties of this filter, as listed in Table 1, first column. The results are shown in Fig. 3(a). The transmissivity T(/~¢o) is identical with the results given in ref. 10. T(~f¢o) is quite large in the p h o t o n energy range rico greater than 1.1 eV, for which it was obviously optimized. Its maximum, however, appears to lie at t o o small a p h o t o n energy with respect to the slow rise in the q u a n t u m efficiency for silicon cells as shown in Fig. 1. The absorptivity o f this filter is considerable with a maximum of 0.22 at 1.08 eV, in the vicinity of the band gap of silicon. These deficiencies are even more pronounced in the transmitted and absorbed energy currents generated by an intermediate black emitter at Trot = 2500 K, as can be seen in Fig. 3(b). A comparison with the q u a n t u m efficiency ~(r~0) of silicon reveals that a large portion o f the energy current IE,~ at around 1.1 eV, which is transmitted by the filter and falls onto the solar cells, will n o t be converted. We determined the figure fsi of merit for
278 TABLE 1 Layer t h i c k n e s s e s o f various filters for silicon solar cells at Tint = 2 5 0 0 K Layer
L a y e r t h i c k n e s s ( n m ) f o r the f o l l o w i n g filters Filter I a
Filter 2 b
Vacuum H L H L H L H M H L H L H L H L H Substrate
Glass (n = 1.5)
fsi
0.83
7.2 153.9 86.2 144.9 86.2 20.0 (Ag) 86.2 144.9 86.2 303.7 163.4
15.9 184.5 143.8 205.1 38.8 25.8 (Ag) 59.6 155.5 108.3 220.9 153.5
1.24
Filter 3 c
30.2 276.0 125.9 201.3 30.7 28.8 ( A u ) 56.2 145.4 135.2 207.4 165.2
Filter 4 d
133.9 204.2 29.9 28.5 ( A u ) 55.6 161.9 138.7
Filter 5 e
68.6 10.6 88.9 138.3 125.1 210.0 26.9 28.4 ( A u ) 58.0 151.8 133.0 396.4 24.0 68.3 151.3 38.3 23.1
Filter 6 f
126.3 250.9 4.3 27.5 ( A u ) 51.5 194.5 205.6
Si 1.34
1.32
1.39
1.32
H, Z n S , n H = 2.32; L, MgF 2, n L = 1.38; M, m e t a l layer, optical c o n s t a n t s f r o m ref. 7. a Filter 1 is a n l l - l a y e r filter as d e s c r i b e d in ref. 10, o p t i m i z e d for m a x i m u m t r a n s m i s s i o n (see Fig. 3). b F i l t e r 2 is a n l l - l a y e r filter, o p t i m i z e d f o r m a x i m u m figure o f m e r i t (see Fig. 4). e F i l t e r 3 is t h e same as filter 2, b u t w i t h a gold layer instead o f a silver layer (see Fig. 5). d F i l t e r 4 is a seven-layer filter with a gold layer. e F i l t e r 5 is a 17-layer filter w i t h a gold layer. f F i l t e r 6 is t h e s a m e as filter 4 b u t o n a silicon s u b s t r a t e i n s t e a d o f a glass s u b s t r a t e .
this filter in combination with silicon cells to be 0.83. This result shows that it would be better to operate the silicon cells with direct solar irradiation instead of in a TPV converter with this filter. It shows furthermore that optimizing the transmissivity alone for f i e > eg does not guarantee good filter performance in a TPV converter. We tried to improve this filter b y varying only the thicknesses of the individual layers, until a m a x i m u m is obtained for fsi- The resulting curves for transmissivity T(ff¢o) and absorptivity A ( ~ ) are shown in Fig. 4(a). The corresponding transmitted and absorbed energy currents from a 2500 K black b o d y source are shown in Fig. 4(b). The figure fsi of merit for this improved filter, whose data are listed in Table 1, is 1.24, which makes TPV conversion favourable. The improvement over the filter proposed in ref. 10 is a little surprising, since its transmissivity in the ~¢~ range greater than 1.1 eV is smaller. This drawback, however, is more than compensated for by
279 TABLE L a y e r t h i c k n e s s e s o f various filters for g e r m a n i u m solar cells at Tint = 2 5 0 0 K Layer
L a y e r thickness ( n m ) for the following filters Filter 1 a
Filter 2 b
Filter 3 c
Filter 4 d
Filter 5 e
Filter 6 f
45.2 76.5 129.1 269.8 102.8 18.1 ( A u ) 133.4 172.9 186.7 486.7 64.7
218.3 217.6 114.5 19.0 ( A u ) 114.5 263.2 228.5
207.0 207.3 119.5 19.8 ( A u ) 127.5 284.9 254.2
Vacuum H L H L H L H M H L H L H L H Substrate
113.7 123.3 82.2 103.6 82.2 64.9 226.4 183.8 129.3 163.8 217.3 201.2 129.3 130.4 20.0 (Ag) 25.7 (Ag) 129.3 134.1 217.3 243.3 129.3 165.4 226.4 217.2 82.2 66.2 82.2 116.8 82.2 105.6 Glass (n = 1.5)
134.0 110.1 65.7 170.9 173.6 207.5 125.3 19.9 ( A u ) 123.4 235.3 200.9 206.1 59.6 113.7 99.8
fGe
1.88
2.16
2.08
Ge 2.07
2.02
1.97
H, ZnS, n H = 2.32; L, M g F 2, n L = 1 . 3 8 ; M, m e t a l layer, o p t i c a l c o n s t a n t s f r o m ref. 7. a Filter 1 is a 15-layer filter as d e s c r i b e d in ref. 10, o p t i m i z e d for m a x i m u m t r a n s m i s s i o n (see Fig. 8). b F i l t e r 2 is a 15-layer filter, o p t i m i z e d for m a x i m u m figure o f m e r i t (see Fig. 9). c Filter 3 is t h e same as filter 2, b u t w i t h a gold layer i n s t e a d o f a silver layer (see Fig. 10). d F i l t e r 4 is an l l - l a y e r filter w i t h a gold layer. e F i l t e r 5 is a seven-layer filter w i t h a gold layer. f F i l t e r 6 is t h e same as filter 5 b u t o n a g e r m a n i u m s u b s t r a t e i n s t e a d o f a glass s u b s t r a t e .
a smaller absorptivity, a higher reflectivity for photons with ~co < eg and a better adjustment of the peak transmissivity to the silicon solar cell. For further improvement in this filter the reflectivity for small energy photons (~co < Q) must be increased, in order to reduce the energy losses in this energy range. An improvement in this respect is achieved when the silver layer is replaced by a gold layer, as was also noted by Demichelis e t al. [11]. An l l - l a y e r filter, the layer thicknesses o f which are again optimized to yield a m a x i m u m figure o f merit, is thereby improved to fsi = 1.34. The data for this filter are listed in Table 1. Its transmissivity T(~co), absorptivity A ( ~ w ) and the corresponding energy currents are shown in Fig. 5. Because o f the higher reflectivity for ~co < 1.1 eV, the absorptivity and transmissivity of the filter are reduced and so are the losses. The large peak in the absorptivity, reaching 0.35 at 2.3 eV, is not a disadvantage because the absorber at Tin t = 2500 K does not emit much at this "high" p h o t o n energy.
280
IF
T,A
,oo
°'
S0
O6
60
I~..l~,(J.:
0l, /
2O
02
ii
~
x
05
1
15
2
"-I
25
05
]
1
1B
-
2
25
1~W(eV)
~~w (eV)
(b)
(a)
Fig. 3. (a) Transmissivity T ( ) and absorptivity A ( - - - - - - ) as a function of p h o t o n energy for an l l - l a y e r interference filter containing a silver layer as proposed by Demichelis e t al. [ 10 ] for use with silicon solar cells; (b) energy currents transmitted (IE,s, --) and absorbed (IE, f, - - - - - ) by the filter with T ( f i ~ ) and A ( ~ ) as s h o w n in (a). The energy currents are given per unit area of a black emitter at 2 5 0 0 K.
%001
O.S
(a)
I
15
2
2.5
05
3
1~w/ev/
W
(b)
I
15
2
25
hw (ev)
Fig. 4. (a) Transmissivity T ( ) and absorptivity A (-- -- --) for an l l-layer filter containing a silver layer o p t i m i z e d for a m a x i m u m figure fsi of merit; (b) energy currents transmitted (IE, s, ) and absorbed (IE, f, - - - - - - ) by the filter with T(~i~) and A(~¢o) as s h o w n in (a). The energy currents are given per unit area o f a black emitter at 2 5 0 0 K.
An improvement in filter performance can also be expected from altering the number o f layers of the filter. The crosses in Fig. 6 show h o w the figure fsi of merit for silicon cells depends on the number of layers. All filters have essentially the same structure, i.e. a gold layer, with an antireflection coating on both sides. Each filter is optimized to yield the maximal figure o f merit attainable for the number of layers in this structure. It can be seen from Fig. 6 that the figure of merit increases with the number of layers and saturates for more than about seven layers. For a seven-layer filter with fsl = 1.32 and a 17-layer filter with fsi = 1.39 the individual layer thicknesses are listed in Table 1. Table 1, last column, gives the data for a seven-layer filter on a silicon substrate with optical constants taken from ref. 9. This
281 1
w ioo k~.lu~(~,~.v)
T A
80
O6
0.5
]
1
15
2
"hw (ev)
(a)
2.5
3
hu~ (eV)
(b)
Fig. 5. (a) Transmissivity T ( ) and absorptivity A ( - - - - --) for an 1 l - l a y e r filter containing a gold layer o p t i m i z e d for a m a x i m u m figure fsi of merit; (b) energy currents transmitted (IE,s, ) and absorbed (IE,f, - - -- --) by the filter with T(~tc0) and A(~i~o) as shown in (a). The energy currents are given per unit area of a black e m i t t e r at 2500 K.
f
15
( I X 0.5
I s
I io
i is
I 20
number of foyers
0
I 1ooo
i 2000
I ~ooo
I ~.ooo
T,n+(K)
Fig. 6. Figure f o f merit o f o p t i m i z e d interference filters for silicon (x) and g e r m a n i u m (e) solar cells as a f u n c t i o n of their n u m b e r o f layers, operating with black b o d y radiation of 2500 K. Fig. 7. Figure o f merit o f seven-layer interference filters for silicon ( ) and g e r m a n i u m (-- -- --) solar cells as a f u n c t i o n of the temperature Tint o f the i n t e r m e d i a t e emitter.
filter has fsi = 1.32, identical with the performance o f the seven-layer filter on glass. In the above calculations the spectrum incident onto the filter is emitted by a 2 5 0 0 K black body source. In order to see h o w much is gained or lost by altering the temperature o f the intermediate absorber, a sevenlayer filter was optimally adapted to several different emitter temperatures. The full line in Fig. 7 shows the figures fsi of merit for these different sevenlayer filters, each optimized for a particular temperature Tint of the intermediate emitter. The m a x i m u m fsi value of 1.39 occurs at T ~ = 3 0 0 0 K.
282
The decrease in fsi for higher temperatures is caused by the increasing loss of radiation from the absorber to the Sun through the entrance aperture. As a consequence, f goes to zero a s Tin t approaches Ts, the temperature of the Sun.
4.2. Filter for germanium solar cells Optimized filter details with a maximum figure of merit were calculated for germanium in the same way as described for silicon in Section 4.1. We started with the 15-layer filter on a glass substrate (n = 1.5) containing a silver layer that was proposed for a TPV converter with germanium cells by Demichelis et al. [10]. As only the transmission is given in ref. 10, we had to recalculate the filter properties. The transmissivity T(/lco), which was very well reproduced, and the absorptivity A(/~co) are shown in Fig. 8(a). Figure 8(b) shows the corresponding energy current IE.f absorbed by the filter and the corresponding energy current IE,, transmitted through the filter and incident on the germanium solar cells. Like the filter for silicon cells, this filter had been optimized for m a x i m u m transmission. It, too, exhibits excessive absorption at the band gap, namely A = 0.33 at 0.8 eV. The figure fGe of merit is, nevertheless, quite good with a value of 1.88. I~- T.A
200
0.8
1
o6
0~ o2
o
(a)
05
1
15
LL-,
w IE~, IE,~(~w,~v)
160
120
BO
&O
25
3
O5
I
A 15
2
25
(b)
Fig. 8. (a) Transmissivity T ( ) a n d absorptivity A ( - - - - - - ) of a 15-layer filter containing a silver layer as proposed by Demichelis e t a l . [10] for use with g e r m a n i u m solar cells; (b) energy currents transmitted (IE,s, ) and absorbed ( I E , f , - - - - - - ) by the filter with T(hco) a n d A(~og) as shown in (a). The energy currents are given per unit area of a black emitter at 2500 K.
By varying the thickness of the individual layers with respect to maximum TPV conversion efficiency, even this filter could be improved. The data for both filters are listed in Table 2. The optical properties of the improved filter are shown in Fig. 9. This filter has fGe = 2.08. The improvement is mainly due to the reduction in t h e absorption at 0.8 eV and to the reduction in absorption and transmission at 0.5 eV. A further slight improvement is achieved when the silver layer in the middle of the stack o f dielectric layers is replaced b y a gold layer, resulting
283 I
20o IE~,IE. ~ ( ~w)
T.A
o8 o6 OZ, 02 0
160
1200 80 _ ~ z,C 05
1
15
2
25
(a)
3
I~WCeV)
15
2
25
(b)
3
I~W(eV)
Fig. 9. (a) Transmissivity T ( ) and absorptivity A ( - - - - - - ) of a 15-layer filter containing a silver layer o p t i m i z e d for a m a x i m u m figure fGe o f merit; (b) energy currents transmitted (IE,s, ) and absorbed (IE, f, -- -- --) by the filter with T(fic0) and A(fic~) as s h o w n in (a). The energy currents are given per unit area o f a black emitter at 2 5 0 0 K.
1
T,A
200
08
IE,s' IE,,¢(cm-~eV)
160
120
80
02
4C
0
• •
(a)
2
i
I
25
3
1%o(eV)
C~
(b)
05
L 1
1.$
2
,
L
25
3
~hw(eV)
Fig. 10. (a) Transmi~ivity T ( ) and absorptivity A ( - - - - --) of a 15-layer filter containing a gold layer optimized for a m a x i m u m figure fGe of merit; (b) energy currents transmitted (IE,s, ) and absorbed (IE,f, --) by the filterwith T(~co) and A(~io9) as s h o w n in (a). The energy currents are given per unit area of a black emitter at 2500 K.
in fGe = 2.16. The optical properties o f this filter are shown in Fig. 10. The data are listed in Table 2. The dependence o f fGe on the number of layers of filters on glass substmtes is shown by full circles in Fig. 6. Like fsi for silicon cells, fGe saturates at about seven layers with a Value of 2.02. When a seven-layer filter is optimized on a germanium substrate for which the optical constants are taken from ref. 8, a figure fG~ of merit o f 1.97 is obtained. The data for this filter are given in Table 2, last column. We c o m p u t e d fG, for seven-layer filters optimized for different values of Trot, the temperature of the intermediate emitter. The result is shown by the broken line in Fig. 7. A m a x i m u m fG~ value of 2.07 appears at 2 1 0 0 K.
284
5. Selective emission in addition to filtering So far in this paper we have assumed the intermediate emitter to be black. The selectivity of the radiation impinging onto the solar cell is achieved solely by a filter. A query arises whether the filtering process could be enhanced by selectivity of the intermediate emitter. Let e(ho~) denote the emissivity of the intermediate emitter, i.e. the ratio of its intensity emitted at a frequency co to that emitted by a black emitter. Together with the reflectivity R(titv) of the filter, multiple reflections between emitter and filter lead to a ratio e' for the resulting radiation incident on the filter: e e'(~o)
(8)
-
1 --R(I --e)
In computing the figure f of merit for the combination of filterswith selective emitters, the energy currents in eqns. (4) and (5) need to be redefined. Using the effective emissivity e' in eqn. (8), instead of eqns. (4) and (5) the following equations are obtained: ,
B
IE' f = 4 ~ 2 ~ 3 C 2
ff e'(hco)A(t$co)(ti¢~) a d(/t¢o) 0j
exp(~co/kTint)
-- 1
(o)
and I
,_ E, s
B 41r2tI3c2
~ e ' ( ~ c o ) T ( ~ ) ( ~ ¢ o ) 3 d(~¢o) oj
e x p ( ~ o ~ / k T i n t ) -- 1
(10)
Two examples will serve to illustrate how the spectral composition of the radiation incident on the filter is modified by the simultaneous presence of the filter with a reflectivity R(/f¢0) and a selective emitter with an emissivity e(~o0). As the first example, y t t e r b i u m oxide (Yb:O3) as a selective emitter (as has been proposed by several workers [5, 12]) in a TPV converter with silicon solar cells is considered. Figure 11 shows the emissivity e(~io~) of Yb:O 3 at 1598 K according to ref. 5. The reflectivity R in Fig. 11 refers to a seven-layer filter that is optimized with respect to e'(ticv) shown in the same figure. The value for e'(t/~o) that results from eqn. (8) shows the resulting radiation to be nearly black at p h o t o n energies below the band gap of silicon. The use of a selective emitter instead of a black emitter is in fact of no advantage. This is reflected by the values o f the figure fsi of merit. For the combination of the filter and Yb203, "as shown in Fig. 11, fsi is 1.14 while, for a black emitter and a filter optimized for it, fsi is 1.33. The reason for the actual degradation by the introduction of selectivity is that the value of e is smaller than unity in the sensitive spectral range o f the silicon solar cell. Since R has to be small in this spectral range, the silicon solar cell receives less radiation in this range from the selective emitter Yb203 than from a black emitter.
285 EThOI+W,R, E
Eybzo3,R,E'
i
"~,
J i
"1¸-'
vj
~i
,x
' i ii
,
~i ii
i ,i '
,!* !
i! d
i
2
215
02
ti
~Y
o!5
;
115
~
,t5
13
0
015
1
115
I
3
Fig. 11. Emissivity eYb~O 3 of Yb203 (-- - - - - ) at Tint = 1 5 9 8 K as proposed by G u a z z o n i [5] as a selective emitter for silicon solar cells and reflectivity R ( - - . - - ) o f a seven-layer interference filter o p t i m i z e d to the degree e' ( ) o f blackness of radiation incident on the filter after multiple reflection b e t w e e n the Yb203 emitter and the filter. Fig. 12. Emissivity eThO2+W of a ThO2 layer on tungsten ( - - - - - - ) at Tint = 1 6 6 5 K as proposed as a selective emitter for germanium solar cells [1 ] and reflectivity R (-- • --) of a seven-layer filter o p t i m i z e d to the degree e' ( ) o f blackness o f radiation incident on the filter after multiple reflection b e t w e e n the ThO2/W emitter and the filter.
• As the second example, germanium solar cells with thorium oxide (ThO2) on a tungsten substrate as the selective emitter are considered. The emissivity of the ThO2, as shown in Fig. 12, is taken from ref. 1. The reflectivity R is that of a 15-layer filter optimized with respect to the emissivity e'(/~co) shown. According to eqn. (8), a spectral distribution results between the selective emitter and filter, as characterized by e'(~co) in Fig. 12. The figure fGe of merit for this radiation impinging on germanium cells is 2.00. This is a larger value than the value o f 1.52 obtained for ThO2 on tungsten without a filter [1] but, in comparison with the combination of a black emitter and an optimized filter which yield fGe = 2.16, there is again no improvement.
6. Conclusions The composition of a filter in between the intermediate emitter and the solar cells in a TPV converter may be optimized with respect to a maximal value of the figure f of merit o f the TPV converter. An f value of more than or less than unity indicates whether the TPV converter system is superior or inferior respectively to direct conversion o f solar energy by solar cells. Filter details are c o m p u t e d which yield for TPV conversion with silicon solar cells a figure fsi of merit of 1.39, and for germanium solar cells even fc~ = 2.16. TPV conversion with germanium cells may thus be more than twice as effective as direct conversion by germanium cells. Compared with direct conversion by silicon cells, TPV conversion with germanium cells may also be superior, although by a smaller factor. As shown in ref. 1, the figure fGe/Si of merit that relates TPV conversion with germanium cells to direct
286
conversion by silicon cells is 0.84fGe. Hence for the optimal filter in a TPV converter with germanium cells an optimal value for fGe/Si of 1.82 results, which shows TPV conversion with germanium cells to be an attractive possibility for solar energy conversion. This result is obtained with a black intermediate emitter. The use of a selective intermediate emitter has no advantage over the use of a black emitter, as is shown by the selective emitter Yb203 for silicon cells and by the selective emitter ThO: on tungsten for germanium cells. Once the filter is optimized with respect to the intermediate emitter, a black intermediate emitter appears to be superior to any real selective emitter.
Acknowledgments We gratefully acknowledge support by the Deutsche Forschungsgemeinschaft. Further, one of us (H. H.) is indebted to the Krupp F o u n d a t i o n for a scholarship.
References 1 H. HSfler, P. Wiirfel and W. Ruppel, Selective emitters for thermophotovottaic solar energy conversion, Sol. Cells, 10 (1983) 257. 2 R. M. Swanson, A proposed thermophotovoltaic solar energy conversion system, Proc. IEEE, 67 (1979) 446. 3 H. Melchior and W. T. Lynch, Signal and noise response of high-speed germanium avalanche photodiodes, IEEE Trans. Electron Devices, 13 (1966) 829. 4 K. Roy, AEG-Telefunken, 7100 Heilbronn, personal communication, May 1980. 5 G. E. Guazzoni, High temperature spectral emittance of oxides of erbium, samarium, neodymium and ytterbium, Appl. Spectrosc., 26 (1972) 60. 6 A. Thelen, Design of multilayer interference filters, Phys. Thin Films, 5 (1969) 47. 7 J. A. Nelder and R. Mead, A simplex method for function minimization, Comput. J., 7 (1965) 308. 8 G. Haas and L. Hadley, American Institute o f Physics Handbook, McGraw-Hill, New York, 3rd edn., 1972. 9 W. C. Dash and R. Newman, Intrinsic optical absorption in single crystal germanium and silicon, Phys. Rev., 99 (1955) 1151. 10 F. Demichelis, E. Minetti-Mezzetti, M. AgneUo and V. Perotto, Bandpass filters for thermophotovoltaic conversion systems, Sol. Cells, 5 (1982) 135. 11 F. Demichelis, E. Minetti-Mezzetti and V. Perotto, Optical studies of multilayer dielectric-metal-dielectric coatings as applied to solar cells, Sol. Cells, 6 (1982) 323. 12 E. S. Vera, J. J. Loferski, J. Severns and M. Spitzer, Operating characteristics of thin thermophotovoltaic cells, Proc. 4th Commission o f the European Communities Conf. on Photovoltaic Solar Energy, Stresa, 1982, Reidel, Dordrecht, 1982, p. 659.
273
INTERFERENCE FILTERS FOR THERMOPHOTOVOLTAIC SOLAR ENERGY CONVERSION
H. HOFLER, H. J. PAUL, W. RUPPEL and P. WI]RFEL Institut f~'r Angewandte Physik, Universitti't Karlsruhe, D-7500 Karlsruhe (F.R.G.) (Received March 9, 1983; accepted June 14, 1983)
Summary In a thermophotovoltaic (TPV) energy converter the necessary selectivity of the radiation incident on the solar cells may be obtained with a filter. Filters consisting o f a metal layer and a sequence of dielectric layers with given indices of refraction are optimized with respect to the layer thicknesses and the number of layers. No sensible improvement is obtained by increasing the number o f layers to more than seven. The c o m p u t e d performance of a solar TPV converter equipped with a black intermediate absorber, an optimized filter and silicon solar cells is expected to be superior by a factor of 1.4 to that of direct conversion b y silicon cells. A TPV converter equipped with germanium cells and an optimized filter yields even an improvement by a factor o f 2.1 over direct conversion by germanium cells and an improvement by a factor of 1.8 over direct conversion by silicon cells. No further improvement is achieved b y the combination of a filter and a selective emitter instead of a filter and a black emitter.
1. Introduction Thermophotovoltaic (TPV) energy conversion has an advantage with respect to direct conversion by solar cells in that the spectral distribution offered to the solar cells m a y be tailored according to their sensitivity. This tailoring can be achieved by two means. The first is selectivity in the emission of the intermediate emitter. The second is selectivity obtained by passing the radiation through a filter. The first possibility is discussed in the accompanying paper [ 1 ]. The second is the subject of this paper. The filter in a TPV converter is to transmit photons whose energy ~¢o is only slightly larger than the band gap eg of the cells. High energy (/i¢o >> eg) and low energy (rico < eg) p h o t o n s cannot be converted by solar cells effectively and must be reflected back to the emitter. This condition is less stringent for high energy photons, especially with respect to silicon solar cells (% = 1.1 eV), since for realistic absorber temperatures their intensity is
274
small. In this case a mirror on the back side of the solar cell suffices. The solar cell itself acts as the desired filter, provided that it does not absorb photons which it does not convert [2]. If this condition is not sufficiently fulfilled, the filter must be placed in front of the solar cells. For the design of a filter, criteria must be developed according to which the performance of a filter is to be evaluated. A high transmission of useful p h o t o n s is not sufficient. A better criterion, which also includes a high reflectivity for unwanted photons, is the efficiency itself of the TPV converter as a whole. This concept is a little more complicated than the optimization of transmission, but it ensures that the filter is properly adjusted to other factors influencing the overall efficiency, such as the spectrum of the emitter or the spectral q u a n t u m efficiency of the solar cells.
2. Evaluation o f filter performance In ref. 1 a figure of merit is defined and used selective emitters in TPV conversion. It also allows filters. This figure f of merit relates the solar energy direct conversion to the solar energy current input IE, for the same power o u t p u t with the same solar cells: f -
IE, dir IE,
for the evaluation of for the evaluation of current input IE, ~ in WeVin TPV conversion
(1)
TPV
For identical solar cells in direct and in TPV conversion the equality of power outputs can be replaced by the equality of short-circuit currents, which are easier to calculate. The solar energy current IE, TPV incident on a TPV converter can be split up into three parts. The energy current IE, lo~s is emitted back by the intermediate absorber through the entrance aperture. It is lost to the outside. The energy current IE. f is absorbed by the filter and also lost. The energy current IE, s is transmitted through the filter and incident on the solar cells. The sum is (2)
IE, T e V = IE,,oss + IE, f + IE, s
It is shown in ref. 1 t h a t the energy current IE, lo~ is at least IE, T P v ( T i n t / T s ) 4 where Tint is the temperature of the intermediate absorber and Ts that of the Sun. The insertion of this expression into eqn. (2) yields IE, f + IE, s IE, TPV = 1 - - ( T i n t / T s ) 4
(3)
With the assumption that the intermediate absorber emits black radiation also from the surface o f area B exposed towards the filter, it is B
~o A(~co)(h~) 3 d(~/oJ)
IE, r -- 41r2~h3c 2 oj
exp(~/kTint)
-- 1
(4)
275
and B IE, s -
~
T(/t¢o)(/t¢o) 3 d(~i¢o)
47r2~3c2 0J
exp(~co/kTint) -- 1
(5)
where A(~co) is the spectral absorptivity of the filter and T(~co) its spectral transmissivity. The p h o t o n s which are transmitted by the filter generate a short-circuit current IQ, TeV in the solar cells / ~(~t¢o)T(fico)(/f¢o) 2 d(~v)) 47r2~3C2 0 exp(~co/kTi,,t) -- 1 eB
IQ, T P V
--
(6)
where/3(t/c0) is the q u a n t u m efficiency of the solar cells. The q u a n t u m efficiency for germanium shown in Fig. 1 b y the broken line is taken from ref. 3. The q u a n t u m efficiency for silicon shown in Fig. 1 by the chain line was measured on A E G - T e l e f u n k e n solar cells t y p e H E C / B S F [4]. Both curves give the number of electrons contributing to the short-circuit current per incident photon. The larger values of ~(~co) for silicon are due to an antireflection coating which is missing on the germanium cells, thereby allowing more p h o t o n s to enter the solar cen. The filter in a TPV converter is best placed directly on t o p of the solar cell. The interface to the solar cell then influences the filter properties and must therefore be accounted for in the calculation of its reflectivity and transmissivity. The transmitted energy current IE, s in eqn. (5) then is the current which enters the solar cell. The q u a n t u m efficiency to be used in eqn. (6) for the short-circuit current must therefore be modified so as to relate to the n u m b e r o f p h o t o n s entering the solar cell instead of to the n u m b e r of p h o t o n s that are incident. If the reflectivity for silicon as we have measured it on the A E G - T e l e f u n k e n cells and the reflectivity for germanium as taken
08
06
O~
02
1~w(ev} Fig. 1. Q u a n t u m efficiency ~] as a f u n c t i o n o f p h o t o n energy for g e r m a n i u m and silicon solar cells: - - - --, ~ per incident p h o t o n according to ref. 3 ; - - • - - , ~ per incident p h o t o n according to ref. 4; - - , /~ per p h o t o n entering the solar cells, calculated f r o m the broken and chain lines.
276 from ref. 8 are taken into account, corrected quantum efficiencies per entering p h o t o n are found, which are given by the full lines in Fig. 1. For direct illumination of a solar cell the short-circuit current IQ, caused by the incident energy current IE, ~ is calculated quite analogously to eqn. (6), as discussed in ref. 1. The figure of merit can then be written as f -
(7)
IE, ~ II0,
TPv
with the condition that the short.circuit currents IQ, d~ and IQ, TPV be equal. Written in this form, f depends only on the transmissivity T(/ico) and the absorptivity A(#ico) of the filter, on the q u a n t u m efficiency fl(/ico) of the solar cells and on the temperature Trot of the black intermediate absorber. For the procedure to evaluate eqn. (7), the reader is referred to ref. 1. If TPV conversion is to outperform direct conversion with the same solar cells, the figure f of merit of the converter must be greater than unity. In the following the figure f of merit will be calculated for various filters as a function of their transmissivity T(#t¢o) and absorptivity A (/iw) for silicon and germanium solar cells.
3. Method for filter optimization For a given multilayer filter the transmissivity T(/~co) and the absorptivity A(fi~0) are determined by m e t h o d s outlined in ref. 6. With these results the figure of merit for a TPV converter with this filter is calculated from eqns. (2) - (7) for a given solar cell, characterized by 13(~co), and a given absorber temperature Trot. The m a x i m u m values of f are f o u n d by varying the thickness of individual layers in a multilayer interference filter. For this purpose a m e t h o d which was developed by Nelder and Mead [7] to find an absolute m i n i m u m o f an n-dimensional function is used. The function to be minimized is l/f, and the n independent variables are the thicknesses of the n layers of the filter. The number n of the layers, their sequence and their optical constants are n o t subject to variations. The procedure requires starting values for the individual layer thicknesses. The result of the minimization of 1/f is not sensitive to the starting values, as long as the starting values are within an order of magnitude of the final values.
4. Results o f filter optimization calculation The filters to be optimized consist of a metal layer embedded in a sequence of dielectric layers with alternating high (H) and low (L) indices of refraction. The arrangement is shown in Fig. 2. A high refiectivity for all p h o t o n s with flu) < eg c a n n o t be achieved with an all~lielectric filter. For a high reflectivity the filters must contain a metallic layer. The transmission for useful p h o t o n s (/~o) ~> %) is achieved by a suitable dielectric antireflection coating.
277 vacuum
H : ZnS. n H = 2~$2 L : MgFz. nL. = 'L38
H M : metal
lag
or Au)
H
L H
$ubstrote
Fig. 2. Basic structure of a multilayer interference filter consisting of a metal layer embedded in stacks of dielectric layers with alternating high (H) and low (L) indices of refraction.
The optical constants used in the calculation are those for ZnS (nil = 2.32), MgF: (n L = 1.38) and the metals silver or gold for which the data are taken from ref. 8. The filters were optimized for substrates of glass (n = 1.5) and of germanium and silicon, for which the optical constants are taken from refs. 8 and 9 respectively. The absorber temperature Trot is taken to be 2500 K b u t the dependence of TPV performance on absorber temperature will also be investigated. The calculated properties of the filters are listed in Tables 1 and 2. 4.1. Filter for silicon solar cells Demichelis et al. [10] have published details of a filter for use with silicon solar cells in a TPV converter. It consists o f 11 layers on a glass substrate, five alternating layers o f ZnS and MgF2 on each side of a silver film. Since Demichelis et al. give only the spectral dependence o f the transmissivity T(~¢o) for the filter and not its absorptivity, it is n o t possible to evaluate h o w well it qualifies with respect to TPV conversion. With the data for layer thicknesses and optical constants from ref. 10, we recalculated the properties of this filter, as listed in Table 1, first column. The results are shown in Fig. 3(a). The transmissivity T(/~¢o) is identical with the results given in ref. 10. T(~f¢o) is quite large in the p h o t o n energy range rico greater than 1.1 eV, for which it was obviously optimized. Its maximum, however, appears to lie at t o o small a p h o t o n energy with respect to the slow rise in the q u a n t u m efficiency for silicon cells as shown in Fig. 1. The absorptivity o f this filter is considerable with a maximum of 0.22 at 1.08 eV, in the vicinity of the band gap of silicon. These deficiencies are even more pronounced in the transmitted and absorbed energy currents generated by an intermediate black emitter at Trot = 2500 K, as can be seen in Fig. 3(b). A comparison with the q u a n t u m efficiency ~(r~0) of silicon reveals that a large portion o f the energy current IE,~ at around 1.1 eV, which is transmitted by the filter and falls onto the solar cells, will n o t be converted. We determined the figure fsi of merit for
278 TABLE 1 Layer t h i c k n e s s e s o f various filters for silicon solar cells at Tint = 2 5 0 0 K Layer
L a y e r t h i c k n e s s ( n m ) f o r the f o l l o w i n g filters Filter I a
Filter 2 b
Vacuum H L H L H L H M H L H L H L H L H Substrate
Glass (n = 1.5)
fsi
0.83
7.2 153.9 86.2 144.9 86.2 20.0 (Ag) 86.2 144.9 86.2 303.7 163.4
15.9 184.5 143.8 205.1 38.8 25.8 (Ag) 59.6 155.5 108.3 220.9 153.5
1.24
Filter 3 c
30.2 276.0 125.9 201.3 30.7 28.8 ( A u ) 56.2 145.4 135.2 207.4 165.2
Filter 4 d
133.9 204.2 29.9 28.5 ( A u ) 55.6 161.9 138.7
Filter 5 e
68.6 10.6 88.9 138.3 125.1 210.0 26.9 28.4 ( A u ) 58.0 151.8 133.0 396.4 24.0 68.3 151.3 38.3 23.1
Filter 6 f
126.3 250.9 4.3 27.5 ( A u ) 51.5 194.5 205.6
Si 1.34
1.32
1.39
1.32
H, Z n S , n H = 2.32; L, MgF 2, n L = 1.38; M, m e t a l layer, optical c o n s t a n t s f r o m ref. 7. a Filter 1 is a n l l - l a y e r filter as d e s c r i b e d in ref. 10, o p t i m i z e d for m a x i m u m t r a n s m i s s i o n (see Fig. 3). b F i l t e r 2 is a n l l - l a y e r filter, o p t i m i z e d f o r m a x i m u m figure o f m e r i t (see Fig. 4). e F i l t e r 3 is t h e same as filter 2, b u t w i t h a gold layer instead o f a silver layer (see Fig. 5). d F i l t e r 4 is a seven-layer filter with a gold layer. e F i l t e r 5 is a 17-layer filter w i t h a gold layer. f F i l t e r 6 is t h e s a m e as filter 4 b u t o n a silicon s u b s t r a t e i n s t e a d o f a glass s u b s t r a t e .
this filter in combination with silicon cells to be 0.83. This result shows that it would be better to operate the silicon cells with direct solar irradiation instead of in a TPV converter with this filter. It shows furthermore that optimizing the transmissivity alone for f i e > eg does not guarantee good filter performance in a TPV converter. We tried to improve this filter b y varying only the thicknesses of the individual layers, until a m a x i m u m is obtained for fsi- The resulting curves for transmissivity T(ff¢o) and absorptivity A ( ~ ) are shown in Fig. 4(a). The corresponding transmitted and absorbed energy currents from a 2500 K black b o d y source are shown in Fig. 4(b). The figure fsi of merit for this improved filter, whose data are listed in Table 1, is 1.24, which makes TPV conversion favourable. The improvement over the filter proposed in ref. 10 is a little surprising, since its transmissivity in the ~¢~ range greater than 1.1 eV is smaller. This drawback, however, is more than compensated for by
279 TABLE L a y e r t h i c k n e s s e s o f various filters for g e r m a n i u m solar cells at Tint = 2 5 0 0 K Layer
L a y e r thickness ( n m ) for the following filters Filter 1 a
Filter 2 b
Filter 3 c
Filter 4 d
Filter 5 e
Filter 6 f
45.2 76.5 129.1 269.8 102.8 18.1 ( A u ) 133.4 172.9 186.7 486.7 64.7
218.3 217.6 114.5 19.0 ( A u ) 114.5 263.2 228.5
207.0 207.3 119.5 19.8 ( A u ) 127.5 284.9 254.2
Vacuum H L H L H L H M H L H L H L H Substrate
113.7 123.3 82.2 103.6 82.2 64.9 226.4 183.8 129.3 163.8 217.3 201.2 129.3 130.4 20.0 (Ag) 25.7 (Ag) 129.3 134.1 217.3 243.3 129.3 165.4 226.4 217.2 82.2 66.2 82.2 116.8 82.2 105.6 Glass (n = 1.5)
134.0 110.1 65.7 170.9 173.6 207.5 125.3 19.9 ( A u ) 123.4 235.3 200.9 206.1 59.6 113.7 99.8
fGe
1.88
2.16
2.08
Ge 2.07
2.02
1.97
H, ZnS, n H = 2.32; L, M g F 2, n L = 1 . 3 8 ; M, m e t a l layer, o p t i c a l c o n s t a n t s f r o m ref. 7. a Filter 1 is a 15-layer filter as d e s c r i b e d in ref. 10, o p t i m i z e d for m a x i m u m t r a n s m i s s i o n (see Fig. 8). b F i l t e r 2 is a 15-layer filter, o p t i m i z e d for m a x i m u m figure o f m e r i t (see Fig. 9). c Filter 3 is t h e same as filter 2, b u t w i t h a gold layer i n s t e a d o f a silver layer (see Fig. 10). d F i l t e r 4 is an l l - l a y e r filter w i t h a gold layer. e F i l t e r 5 is a seven-layer filter w i t h a gold layer. f F i l t e r 6 is t h e same as filter 5 b u t o n a g e r m a n i u m s u b s t r a t e i n s t e a d o f a glass s u b s t r a t e .
a smaller absorptivity, a higher reflectivity for photons with ~co < eg and a better adjustment of the peak transmissivity to the silicon solar cell. For further improvement in this filter the reflectivity for small energy photons (~co < Q) must be increased, in order to reduce the energy losses in this energy range. An improvement in this respect is achieved when the silver layer is replaced by a gold layer, as was also noted by Demichelis e t al. [11]. An l l - l a y e r filter, the layer thicknesses o f which are again optimized to yield a m a x i m u m figure o f merit, is thereby improved to fsi = 1.34. The data for this filter are listed in Table 1. Its transmissivity T(~co), absorptivity A ( ~ w ) and the corresponding energy currents are shown in Fig. 5. Because o f the higher reflectivity for ~co < 1.1 eV, the absorptivity and transmissivity of the filter are reduced and so are the losses. The large peak in the absorptivity, reaching 0.35 at 2.3 eV, is not a disadvantage because the absorber at Tin t = 2500 K does not emit much at this "high" p h o t o n energy.
280
IF
T,A
,oo
°'
S0
O6
60
I~..l~,(J.:
0l, /
2O
02
ii
~
x
05
1
15
2
"-I
25
05
]
1
1B
-
2
25
1~W(eV)
~~w (eV)
(b)
(a)
Fig. 3. (a) Transmissivity T ( ) and absorptivity A ( - - - - - - ) as a function of p h o t o n energy for an l l - l a y e r interference filter containing a silver layer as proposed by Demichelis e t al. [ 10 ] for use with silicon solar cells; (b) energy currents transmitted (IE,s, --) and absorbed (IE, f, - - - - - ) by the filter with T ( f i ~ ) and A ( ~ ) as s h o w n in (a). The energy currents are given per unit area of a black emitter at 2 5 0 0 K.
%001
O.S
(a)
I
15
2
2.5
05
3
1~w/ev/
W
(b)
I
15
2
25
hw (ev)
Fig. 4. (a) Transmissivity T ( ) and absorptivity A (-- -- --) for an l l-layer filter containing a silver layer o p t i m i z e d for a m a x i m u m figure fsi of merit; (b) energy currents transmitted (IE, s, ) and absorbed (IE, f, - - - - - - ) by the filter with T(~i~) and A(~¢o) as s h o w n in (a). The energy currents are given per unit area o f a black emitter at 2 5 0 0 K.
An improvement in filter performance can also be expected from altering the number o f layers of the filter. The crosses in Fig. 6 show h o w the figure fsi of merit for silicon cells depends on the number of layers. All filters have essentially the same structure, i.e. a gold layer, with an antireflection coating on both sides. Each filter is optimized to yield the maximal figure o f merit attainable for the number of layers in this structure. It can be seen from Fig. 6 that the figure of merit increases with the number of layers and saturates for more than about seven layers. For a seven-layer filter with fsl = 1.32 and a 17-layer filter with fsi = 1.39 the individual layer thicknesses are listed in Table 1. Table 1, last column, gives the data for a seven-layer filter on a silicon substrate with optical constants taken from ref. 9. This
281 1
w ioo k~.lu~(~,~.v)
T A
80
O6
0.5
]
1
15
2
"hw (ev)
(a)
2.5
3
hu~ (eV)
(b)
Fig. 5. (a) Transmissivity T ( ) and absorptivity A ( - - - - --) for an 1 l - l a y e r filter containing a gold layer o p t i m i z e d for a m a x i m u m figure fsi of merit; (b) energy currents transmitted (IE,s, ) and absorbed (IE,f, - - -- --) by the filter with T(~tc0) and A(~i~o) as shown in (a). The energy currents are given per unit area of a black e m i t t e r at 2500 K.
f
15
( I X 0.5
I s
I io
i is
I 20
number of foyers
0
I 1ooo
i 2000
I ~ooo
I ~.ooo
T,n+(K)
Fig. 6. Figure f o f merit o f o p t i m i z e d interference filters for silicon (x) and g e r m a n i u m (e) solar cells as a f u n c t i o n of their n u m b e r o f layers, operating with black b o d y radiation of 2500 K. Fig. 7. Figure o f merit o f seven-layer interference filters for silicon ( ) and g e r m a n i u m (-- -- --) solar cells as a f u n c t i o n of the temperature Tint o f the i n t e r m e d i a t e emitter.
filter has fsi = 1.32, identical with the performance o f the seven-layer filter on glass. In the above calculations the spectrum incident onto the filter is emitted by a 2 5 0 0 K black body source. In order to see h o w much is gained or lost by altering the temperature o f the intermediate absorber, a sevenlayer filter was optimally adapted to several different emitter temperatures. The full line in Fig. 7 shows the figures fsi of merit for these different sevenlayer filters, each optimized for a particular temperature Tint of the intermediate emitter. The m a x i m u m fsi value of 1.39 occurs at T ~ = 3 0 0 0 K.
282
The decrease in fsi for higher temperatures is caused by the increasing loss of radiation from the absorber to the Sun through the entrance aperture. As a consequence, f goes to zero a s Tin t approaches Ts, the temperature of the Sun.
4.2. Filter for germanium solar cells Optimized filter details with a maximum figure of merit were calculated for germanium in the same way as described for silicon in Section 4.1. We started with the 15-layer filter on a glass substrate (n = 1.5) containing a silver layer that was proposed for a TPV converter with germanium cells by Demichelis et al. [10]. As only the transmission is given in ref. 10, we had to recalculate the filter properties. The transmissivity T(/lco), which was very well reproduced, and the absorptivity A(/~co) are shown in Fig. 8(a). Figure 8(b) shows the corresponding energy current IE.f absorbed by the filter and the corresponding energy current IE,, transmitted through the filter and incident on the germanium solar cells. Like the filter for silicon cells, this filter had been optimized for m a x i m u m transmission. It, too, exhibits excessive absorption at the band gap, namely A = 0.33 at 0.8 eV. The figure fGe of merit is, nevertheless, quite good with a value of 1.88. I~- T.A
200
0.8
1
o6
0~ o2
o
(a)
05
1
15
LL-,
w IE~, IE,~(~w,~v)
160
120
BO
&O
25
3
O5
I
A 15
2
25
(b)
Fig. 8. (a) Transmissivity T ( ) a n d absorptivity A ( - - - - - - ) of a 15-layer filter containing a silver layer as proposed by Demichelis e t a l . [10] for use with g e r m a n i u m solar cells; (b) energy currents transmitted (IE,s, ) and absorbed ( I E , f , - - - - - - ) by the filter with T(hco) a n d A(~og) as shown in (a). The energy currents are given per unit area of a black emitter at 2500 K.
By varying the thickness of the individual layers with respect to maximum TPV conversion efficiency, even this filter could be improved. The data for both filters are listed in Table 2. The optical properties of the improved filter are shown in Fig. 9. This filter has fGe = 2.08. The improvement is mainly due to the reduction in t h e absorption at 0.8 eV and to the reduction in absorption and transmission at 0.5 eV. A further slight improvement is achieved when the silver layer in the middle of the stack o f dielectric layers is replaced b y a gold layer, resulting
283 I
20o IE~,IE. ~ ( ~w)
T.A
o8 o6 OZ, 02 0
160
1200 80 _ ~ z,C 05
1
15
2
25
(a)
3
I~WCeV)
15
2
25
(b)
3
I~W(eV)
Fig. 9. (a) Transmissivity T ( ) and absorptivity A ( - - - - - - ) of a 15-layer filter containing a silver layer o p t i m i z e d for a m a x i m u m figure fGe o f merit; (b) energy currents transmitted (IE,s, ) and absorbed (IE, f, -- -- --) by the filter with T(fic0) and A(fic~) as s h o w n in (a). The energy currents are given per unit area o f a black emitter at 2 5 0 0 K.
1
T,A
200
08
IE,s' IE,,¢(cm-~eV)
160
120
80
02
4C
0
• •
(a)
2
i
I
25
3
1%o(eV)
C~
(b)
05
L 1
1.$
2
,
L
25
3
~hw(eV)
Fig. 10. (a) Transmi~ivity T ( ) and absorptivity A ( - - - - --) of a 15-layer filter containing a gold layer optimized for a m a x i m u m figure fGe of merit; (b) energy currents transmitted (IE,s, ) and absorbed (IE,f, --) by the filterwith T(~co) and A(~io9) as s h o w n in (a). The energy currents are given per unit area of a black emitter at 2500 K.
in fGe = 2.16. The optical properties o f this filter are shown in Fig. 10. The data are listed in Table 2. The dependence o f fGe on the number of layers of filters on glass substmtes is shown by full circles in Fig. 6. Like fsi for silicon cells, fGe saturates at about seven layers with a Value of 2.02. When a seven-layer filter is optimized on a germanium substrate for which the optical constants are taken from ref. 8, a figure fG~ of merit o f 1.97 is obtained. The data for this filter are given in Table 2, last column. We c o m p u t e d fG, for seven-layer filters optimized for different values of Trot, the temperature of the intermediate emitter. The result is shown by the broken line in Fig. 7. A m a x i m u m fG~ value of 2.07 appears at 2 1 0 0 K.
284
5. Selective emission in addition to filtering So far in this paper we have assumed the intermediate emitter to be black. The selectivity of the radiation impinging onto the solar cell is achieved solely by a filter. A query arises whether the filtering process could be enhanced by selectivity of the intermediate emitter. Let e(ho~) denote the emissivity of the intermediate emitter, i.e. the ratio of its intensity emitted at a frequency co to that emitted by a black emitter. Together with the reflectivity R(titv) of the filter, multiple reflections between emitter and filter lead to a ratio e' for the resulting radiation incident on the filter: e e'(~o)
(8)
-
1 --R(I --e)
In computing the figure f of merit for the combination of filterswith selective emitters, the energy currents in eqns. (4) and (5) need to be redefined. Using the effective emissivity e' in eqn. (8), instead of eqns. (4) and (5) the following equations are obtained: ,
B
IE' f = 4 ~ 2 ~ 3 C 2
ff e'(hco)A(t$co)(ti¢~) a d(/t¢o) 0j
exp(~co/kTint)
-- 1
(o)
and I
,_ E, s
B 41r2tI3c2
~ e ' ( ~ c o ) T ( ~ ) ( ~ ¢ o ) 3 d(~¢o) oj
e x p ( ~ o ~ / k T i n t ) -- 1
(10)
Two examples will serve to illustrate how the spectral composition of the radiation incident on the filter is modified by the simultaneous presence of the filter with a reflectivity R(/f¢0) and a selective emitter with an emissivity e(~o0). As the first example, y t t e r b i u m oxide (Yb:O3) as a selective emitter (as has been proposed by several workers [5, 12]) in a TPV converter with silicon solar cells is considered. Figure 11 shows the emissivity e(~io~) of Yb:O 3 at 1598 K according to ref. 5. The reflectivity R in Fig. 11 refers to a seven-layer filter that is optimized with respect to e'(ticv) shown in the same figure. The value for e'(t/~o) that results from eqn. (8) shows the resulting radiation to be nearly black at p h o t o n energies below the band gap of silicon. The use of a selective emitter instead of a black emitter is in fact of no advantage. This is reflected by the values o f the figure fsi of merit. For the combination of the filter and Yb203, "as shown in Fig. 11, fsi is 1.14 while, for a black emitter and a filter optimized for it, fsi is 1.33. The reason for the actual degradation by the introduction of selectivity is that the value of e is smaller than unity in the sensitive spectral range o f the silicon solar cell. Since R has to be small in this spectral range, the silicon solar cell receives less radiation in this range from the selective emitter Yb203 than from a black emitter.
285 EThOI+W,R, E
Eybzo3,R,E'
i
"~,
J i
"1¸-'
vj
~i
,x
' i ii
,
~i ii
i ,i '
,!* !
i! d
i
2
215
02
ti
~Y
o!5
;
115
~
,t5
13
0
015
1
115
I
3
Fig. 11. Emissivity eYb~O 3 of Yb203 (-- - - - - ) at Tint = 1 5 9 8 K as proposed by G u a z z o n i [5] as a selective emitter for silicon solar cells and reflectivity R ( - - . - - ) o f a seven-layer interference filter o p t i m i z e d to the degree e' ( ) o f blackness of radiation incident on the filter after multiple reflection b e t w e e n the Yb203 emitter and the filter. Fig. 12. Emissivity eThO2+W of a ThO2 layer on tungsten ( - - - - - - ) at Tint = 1 6 6 5 K as proposed as a selective emitter for germanium solar cells [1 ] and reflectivity R (-- • --) of a seven-layer filter o p t i m i z e d to the degree e' ( ) o f blackness o f radiation incident on the filter after multiple reflection b e t w e e n the ThO2/W emitter and the filter.
• As the second example, germanium solar cells with thorium oxide (ThO2) on a tungsten substrate as the selective emitter are considered. The emissivity of the ThO2, as shown in Fig. 12, is taken from ref. 1. The reflectivity R is that of a 15-layer filter optimized with respect to the emissivity e'(/~co) shown. According to eqn. (8), a spectral distribution results between the selective emitter and filter, as characterized by e'(~co) in Fig. 12. The figure fGe of merit for this radiation impinging on germanium cells is 2.00. This is a larger value than the value o f 1.52 obtained for ThO2 on tungsten without a filter [1] but, in comparison with the combination of a black emitter and an optimized filter which yield fGe = 2.16, there is again no improvement.
6. Conclusions The composition of a filter in between the intermediate emitter and the solar cells in a TPV converter may be optimized with respect to a maximal value of the figure f of merit o f the TPV converter. An f value of more than or less than unity indicates whether the TPV converter system is superior or inferior respectively to direct conversion o f solar energy by solar cells. Filter details are c o m p u t e d which yield for TPV conversion with silicon solar cells a figure fsi of merit of 1.39, and for germanium solar cells even fc~ = 2.16. TPV conversion with germanium cells may thus be more than twice as effective as direct conversion by germanium cells. Compared with direct conversion by silicon cells, TPV conversion with germanium cells may also be superior, although by a smaller factor. As shown in ref. 1, the figure fGe/Si of merit that relates TPV conversion with germanium cells to direct
286
conversion by silicon cells is 0.84fGe. Hence for the optimal filter in a TPV converter with germanium cells an optimal value for fGe/Si of 1.82 results, which shows TPV conversion with germanium cells to be an attractive possibility for solar energy conversion. This result is obtained with a black intermediate emitter. The use of a selective intermediate emitter has no advantage over the use of a black emitter, as is shown by the selective emitter Yb203 for silicon cells and by the selective emitter ThO: on tungsten for germanium cells. Once the filter is optimized with respect to the intermediate emitter, a black intermediate emitter appears to be superior to any real selective emitter.
Acknowledgments We gratefully acknowledge support by the Deutsche Forschungsgemeinschaft. Further, one of us (H. H.) is indebted to the Krupp F o u n d a t i o n for a scholarship.
References 1 H. HSfler, P. Wiirfel and W. Ruppel, Selective emitters for thermophotovottaic solar energy conversion, Sol. Cells, 10 (1983) 257. 2 R. M. Swanson, A proposed thermophotovoltaic solar energy conversion system, Proc. IEEE, 67 (1979) 446. 3 H. Melchior and W. T. Lynch, Signal and noise response of high-speed germanium avalanche photodiodes, IEEE Trans. Electron Devices, 13 (1966) 829. 4 K. Roy, AEG-Telefunken, 7100 Heilbronn, personal communication, May 1980. 5 G. E. Guazzoni, High temperature spectral emittance of oxides of erbium, samarium, neodymium and ytterbium, Appl. Spectrosc., 26 (1972) 60. 6 A. Thelen, Design of multilayer interference filters, Phys. Thin Films, 5 (1969) 47. 7 J. A. Nelder and R. Mead, A simplex method for function minimization, Comput. J., 7 (1965) 308. 8 G. Haas and L. Hadley, American Institute o f Physics Handbook, McGraw-Hill, New York, 3rd edn., 1972. 9 W. C. Dash and R. Newman, Intrinsic optical absorption in single crystal germanium and silicon, Phys. Rev., 99 (1955) 1151. 10 F. Demichelis, E. Minetti-Mezzetti, M. AgneUo and V. Perotto, Bandpass filters for thermophotovoltaic conversion systems, Sol. Cells, 5 (1982) 135. 11 F. Demichelis, E. Minetti-Mezzetti and V. Perotto, Optical studies of multilayer dielectric-metal-dielectric coatings as applied to solar cells, Sol. Cells, 6 (1982) 323. 12 E. S. Vera, J. J. Loferski, J. Severns and M. Spitzer, Operating characteristics of thin thermophotovoltaic cells, Proc. 4th Commission o f the European Communities Conf. on Photovoltaic Solar Energy, Stresa, 1982, Reidel, Dordrecht, 1982, p. 659.