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Solar Energy Materials & Solar Cells 87 (2005) 757–769
www.elsevier.com/locate/solmat
Theoretical analysis of the optimum energy band
gap of semiconductors for fabrication of solar
cells for applications in higher latitudes locations
T. Zdanowicza,, T. Rodziewiczb, M. Zabkowska-Waclawekb
a
Faculty of Microsystem Electronics and Photonics, SolarLab,Wroclaw University of Technology,
ul. Janiszewskiego 11/17, 50-372 Wroclaw, Poland
b
University of Opole, ul. Oleska 48, 45-052 Opole, Poland
Received 15 May 2004; received in revised form 22 July 2004; accepted 26 July 2004
Available online 2 December 2004
Abstract
In this work some results of theoretical analysis on the selection of optimum band gap
semiconductor absorbers for application in either single or multijunction (up to five junctions)
solar cells are presented. For calculations days have been taken characterized by various
insolation and ambient temperature conditions defined in the draft of the IEC 61836 standard
(Performance testing and energy rating of terrestrial photovoltaic modules) as a proposal of
representative set of typical outdoor conditions that may influence performance of
photovoltaic devices. Besides various irradiance and ambient temperature ranges, these days
additionally differ significantly regarding spectral distribution of solar radiation incident onto
horizontal surface. Taking these spectra into account optimum energy band gaps and
maximum achievable efficiencies of single and multijunction solar cells made have been
estimated. More detailed results of analysis performed for double junction cell are presented to
show the effect of deviations in band gap values on the cell efficiency.
r 2004 Elsevier B.V. All rights reserved.
Keywords: Multijunction solar cell; Solar spectrum; Conversion efficiency; Absorber; Energy gap
Corresponding author. Tel./fax: +48 71 355 48 22.
E-mail address: [email protected] (T. Zdanowicz).
0927-0248/$ - see front matter r 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.solmat.2004.07.049
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0. Introduction
The key property of each solar cell is its capability to absorb effectively wide
spectrum of photons contained in solar radiation reaching its active surface. This
feature depends on intrinsic optical and electronic properties of semiconductor
material used as an absorber layer in a cell and is described by wavelength dependent
value of the absorption coefficient, the parameter being directly related to
semiconductor’s energy band gap and energy band structure.
As it is well known photons with energy lower than the absorber band gap cannot
be absorbed and so they do not contribute to energy conversion. On the other hand
one photon, even if its energy exceeds doubled value of that of the band gap, cannot
generate more than single electron-hole pair, dissipating all its excess energy as a heat
in the cell. With these limitations the role of the absorber in the conventional solar
cell may be briefly explained as follows.
When using wide band semiconductor light absorption becomes limited only to
high energy photons while for the sub-band gap photons solar cell practically
remains ‘‘transparent’’. This results in lower photocurrent of such cell but the
advantages in this case are more efficient energy conversion of the absorbed high
energy part of solar spectrum—due to the fact that higher fraction of photons energy
is being converted into electricity—and higher value of the output voltage of the cell.
Contrary to that, solar cells made of narrow band semiconductors—though capable
of absorbing larger part of solar spectrum and hence exhibiting higher photocurrent
values—have lower energy conversion efficiency in the range of high energy photons
and exhibit lower output voltage.
The above discussion leads to obvious conclusion that to achieve maximum
conversion efficiency for a specified solar spectrum absorber material with optimum
band gap should be used for any solar cell fabrication.
Proper matching of the solar cell absorber band gap to light spectrum is so much
fundamental for efficient energy conversion that first numerical analyses approaching this problem for single junction cells were done already in the late fifties [2] and in
spite of growing number of semiconductor materials being applied in solar cell
technology, results of those calculations basically still become valid with at most
some minor quantitative corrections. During recent two decades apart from single
junction solar cells with optically uniform absorber more and more complex
structures consisted of two or more absorber layers characterized by different energy
gaps—i.e., heterojunction and/or multijunction solar cells—have been developed
and many of them appeared on the market. Good example of the heterojunction
structure, i.e., having single junction but two various absorber regions, is thin-film
CdS/CdTe cell already being in mass production. Examples of multijunction devices
already available as commercial products are thin-film CIS and CIGS cells [3] and
wide range of single, double and triple junction structures based on amorphous
silicon (a-Si) [4–6]. Apparently such structures are promising and cost effective
solutions for more efficient use of solar spectrum mainly in standard terrestrial
applications. Other group of highly advanced solar cells are extremely sophisticated
and costly multijunction devices for space and/or concentrator applications based on
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GaAs and other III–V compounds [7]. In this group besides more common double
and triple junctions devices, also structures with up to five junctions have been
demonstrated.
Other options, though still rather futuristic today, are related to the so called
band-gap engineering. They include application of graded gap semiconductors and
third generation solar cells aiming to obtain structures with higher than one
quantum efficiency due to the introducing to the band gap multiple intermediate
bands (IB) [8,9]. It is expected that in such structures photons may generate more
than one electron–hole pair however experiments proving the principle of operation
of these type of a cell have not been demonstrated yet.
Wide overview of the current status of novel solar cell structures including also all
aspects mentioned above may be found in [10].
1. Selection of outdoor operating conditions affecting solar cell performance
To evaluate performance of photovoltaic solar cells or modules to be operated in
outdoor conditions so called standard testing conditions (STC) with specified values
of irradiance level (1000 W/m2), ambient temperature (25 1C) and reference solar
spectrum (AM1.5G ) have been recommended as IEC standard [11]. Unfortunately
spectrum corresponding to ideal AM1.5G can be rarely met in real operating
conditions due to the fact that spectral distribution of the natural solar light is
continuously submitted both to regular seasonal and daily changes as well as to
random fluctuations due to weather conditions. The importance of spectral effects
on PV modules performance became non-disputable especially since thin-film
devices, single and soon after multijunction, have been widely used in terrestrial PV
installations. A lot of research work has been done during the last decade or more to
show the impact of permanently changing spectral distribution of the natural solar
light on the performance of various PV modules [12–16] and it has become quite
obvious that spectral effects must be taken into account when predicting solar cell or
module performance to be operated in realistic conditions.
Unfortunately, so far a unique parameter has not even been yet established which
would allow to quantify properly solar spectrum. One method is using Air Mass
factor [11,13,14] which is parameter merely reflecting a change in the solar spectrum
due to actual position of the Sun in respect to specified location on the Earth. AM
factor does not reflect, however, any effects of actual weather conditions on the solar
spectrum. Another parameter that has been introduced to quantify solar spectrum is
‘‘spectral factor’’ [13,16]. It relates optical properties of a specific solar cell with
energy distribution of photons in the incoming solar light. Unfortunately,
determination of ‘‘spectral factor’’ requires uneasy and troublesome measurements
of real solar spectrum which is a severe disadvantage of this approach.
Different though simplified approach has been proposed in the draft of the IEC
61853 standard [1] intended to unify procedures applied to evaluate energy rating
performance of photovoltaic modules manufactured in different technologies. In the
draft six various days have been specified as typical for outdoor conditions
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corresponding to different locations and seasons. These days can be shortly
described as follow:
(1) HIHT (high irradiance, high temperature)—typical very hot and sunny summer
day in a desert location with peak irradiance 1100 W/m2 and peak ambient
temperature as high as 45 1C;
(2) MIHT (medium irradiance, high temperature)—typical very hot but cloudy and
humid summer day with medium irradiance below 600 W/m2 and ambient
temperature Tamb ¼ 27–33 1C.
(3) HILT (high irradiance, low temperature—day typical for early spring season
with high value of irradiance exceeding 1000 W/m2 and low ambient temperature
Tamb ¼ 1–+3 1C;
(4) LILT (low irradiance, low temperature)—typical winter day in Central Europe
with low irradiance GmaxE260 W/m2 and Tamb ¼ 0.6–0 1C;
(5) MIMT (medium irradiance, medium temperature)—warm but cloudy day
with peak irradiance Gmax ¼ 350 W/m2 and ambient temperature Tamb ¼
+71C–+14 1C.
(6) NICE (normal irradiance, cool environment)—typical summer day in a cool
coastal region with peak irradiance 1000 W/m2 and ambient temperature 18 1C.
For each day the authors of the draft provide tables containing set of hourly data
defining most probable spectral distributions of the solar light regarding both angle
of incidence of solar radiation as well as typical weather and climatic conditions.
For the purpose of this work spectra corresponding to the noon time have been
taken.
This is well known that both irradiance level as well as weather conditions such as
ambient temperature and wind speed have an effect on the operating temperature of
PV module. Also construction of the module—related to value of module’s nominal
operating cell temperature (NOCT) [17]—and way of its installation in the real PV
system additionally affect its temperature [18]. To estimate operating temperature of
the module the following equation was used [19]:
T m ¼ T amb þ G 0
NOCT 20 C
;
800 W=m2
(1)
where Go is global irradiance in the plane of array.
In Table 1 operating conditions assumed for calculations have been summarized.
2. Approach to optimum absorber band gap selection [20]
2.1. Conversion efficiency of a single junction solar cell
Assuming superposition principle holds the output current of the illuminated ideal
solar cell may be described by simplified equation:
I ¼ I SC I dark ¼ I SC I o ðeU=j 1Þ;
(2)
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Table 1
Conditions taken to calculate module temperature corresponding to various days [1]
Type of a
day
Sunrise/sunset
hours
Time
Tamb
(1C)
Go (W/m2)
NOCT
(1C)
Tm
(1C)
AOI at
12.00 (deg)
HIHT
MIHT
HILT
LILT
MIMT
5.00/19.00
5.00/19.00
6.00/18.00
7.00/17.00
6.00/19.00
12.00
12.00
12.00
12.00
12.00
45
30
1
0
14
1100
450
1032
260
350
42
42
42
42
42
75
42
29
7
24
27.6
27.7
19.0
24.2
18.1
AOI is angle of incidence of direct solar beam at noon and Tm is cell operating temperature estimated
using Eq. (1).
where U is cell voltage and Idark is current flowing through cell without illumination.
Here Io is dark saturation current described by
E g =gkT
Io ¼ Ke
;
(3)
where Eg is energy gap of the semiconductor material of the cell absorber region and
ISC is cell’s short circuit current, in most practical cases just equal to photogenerated
current related to spectrum of the incoming light as follows:
Z 1
I SC ¼ Qq
N ph ðE ph ÞdE ph :
(4)
Eg
Parameter j is called thermal voltage and is related to cell temperature T
j ¼ nkT=q:
(5)
Here k and q are Boltzmann’s constant and electron charge, respectively. Parameter
n is called diode ideality factor and typically it has value in the range of 1–2.
In Eq. (4) Q is overall cell quantum efficiency and Nph is number of photons which
have energies in the range (Eph+dEph).
For calculation purposes constant K has been taken as equal to 1.5 105 [9] and
both constants g and n have been assumed equal to unity.
The maximum output power of the illuminated solar cell corresponds to the point
on the current–voltage curve where product of voltage and current reaches
maximum value which means that first derivative of the IV–V curve, i.e., d(IV)/
dV, must be equal to zero, i.e.:
dP dðIUÞ
d½ðI SC I o ðeU=j 1ÞÞU
¼
¼
¼ 0:
(6)
dU U¼U m
dU U¼U m
dU
U¼U m
Solving Eq. (6) gives an implicit relation from which value of voltage Um
corresponding to maximum power point may be derived
Um
I SC
þ 1 eU m =j 1 ¼
(7)
j
Io
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which with an obvious assumption that ISCbIo Eq. (7) may be further simplified to
the form:
eU m =j ¼
I SC
1
:
I o U m=j þ 1
(8)
Using definition of the solar cell conversion efficiency it may be written
Z¼
Pm ðI SC I o ðeU m= j 1ÞÞU m
¼ R1
;
Go
0 ðE ph N ph ðE ph ÞÞ dE ph
(9)
where Pm is maximum power of the cell and Go described by integral in the
denominator in the second part of the equation is the value of the total power—
global irradiance—of the incident photon flux.
By using Eqs. (3)–(5) and taking eU m =j 1 ffi eU m =j , which is true for UmbnkT,
Eq. (9) can be written as follows:
R1
R1
U 2m
q E g N ph ðE ph ÞdE ph
q E g N ph ðE ph ÞdE ph
j
ffi QR1
U m:
(10)
ZðE g Þ ¼ Q R 1
Um
0 ðN ph ðE ph ÞE ph ÞdE p 1 þ j
0 ðN ph ðE ph ÞE ph ÞdE p
Here Um comes from numerical solution of the Eq. (11) which is obtained by taking
logarithms of both sides of Eq. (8)1
!
Z
U m U m =j
q E g =kT 1
e
ln
N ph ðE ph ÞdE ph
(11)
¼ ln Q e
K
j
Eg
2.2. Conversion efficiency of a multijunction solar cell
Conversion efficiency of a multijunction cell is calculated as a simple sum of the
efficiencies corresponding to particular cells, i.e.,
ZðE g Þ ¼
n
X
ZðE gi Þ;
(12)
i¼1
where index i denotes consecutive cell in the stack with Egi being energy gap of its
absorber region. Calculation is made using Eqs. (9) and (10) starting from the front
structure, where absorber is characterized by the highest value of energy gap. It is
assumed that overall quantum efficiency Q of each cell is equal to unity, i.e., Q ¼ 1,
which means that each next cell can reach only those photons that had energy too
low to be absorbed in the preceding cell. It means that integration limits in the
expression in the numerator of Eq. (9) change for the consecutive cells as follow:
Eg1-N, Eg2-Eg1, y., Egn-Egn1.To obtain the optimum energy band gaps Eq. (12) is
iteratively solved until maximum value of Z is achieved.
1
Taking logarithm instead of direct form of the Eq. (8) enabled to avoid dealing with very large numbers
during multiple iterations when solving Eqs. (10) and (11) what resulted in significant reduction of the
calculation process.
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3. Results
3.1. Optimum absorber’s energy band gap for single junction solar cells
Fig. 1 shows dependence of the maximum achievable solar cell efficiency made of
various semiconductor materials characterized by different values of the energy band
gap. The curves were calculated using Eqs. (10) and (11) for various solar spectra and
ambient temperatures corresponding to days as specified in Table 1. Calculated
values of efficiency were normalized to maximum value that may be theoretically
achieved for the cell with absorber having energy band gap of about 1.39 eV.
Possible change of the cell temperature has been taken into account using Eq. (1).
For high latitude regions like Central or Northern Europe, especially for seasons
beyond summertime, spectra with higher values Air Mass factors and significant
diffuse component are more typical rather than reference spectrum AM1.5. Fig. 2
shows results of calculations similar to those from Fig. 1 but performed for wide
range of various spectra defined by AM value. To calculate spectral distributions
software spectral2, available as shareware from the NREL website [21], was used. To
show ‘‘pure’’ effect of the actual shape of solar spectrum on efficiency unlike to Fig. 1
cell temperature was assumed to be constant in this case. As can be seen increasing
number of the AM factor, meaning higher relative contents of low energy photons in
the spectrum, affects mainly the efficiency of those cells which absorber is
characterized by wide energy band gap like CdSe or a-Si.
3.2. Selection of optimum energy band gap for absorbers in tandem and multijunction
solar cells
Zn3P2
GaAs
CdTe
CdS
c-Si
0.96
HIHT
HILT
MIHT
LILT
MIMT
0.88
a-Si
0.92
CIS
Efficiencynorm
1.00
InP
Cu2S, Cu2Se
Fig. 3 shows the effect of the absorbers selection on the efficiency for the case of a
tandem junction solar cell. Calculations have been done using Eq. (12) for solar
0.84
0.80
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Eg [eV]
Fig. 1. Maximum efficiency of single junction solar cell versus absorber energy band gap calculated for
solar spectra corresponding to various days as specified in Table 1; cell temperature was calculated
according to Eq. (1).
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Zn3P2
InP
GaAs
CdTe
AM-1
AM-1,5
AM-2
AM-3
AM-4
AM-6
AM-8
AM-10
CdSe
a-Si
c-Si
CIS
0.8
0.7
AlAs
Efficiencynorm
0.9
Cu2S;
Tm=const=25˚C
1.0
Cu2Se
1.1
0.6
0.5
0.4
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Eg [eV]
Fig. 2. Maximum efficiency of single junction solar cell versus absorber energy band gap of the calculated
for solar spectra corresponding to various Air Mass factors; to show only the effect of solar spectrum cell
temperature was assumed constant; spectral distributions were calculated using spectral2 software
downloaded from the NREL homepage [21].
spectra corresponding to various days as specified in Table 1 and assuming that cell
temperature changes according to Eq. (1). Some interesting result here may be the
fact that highest efficiencies for this kind of a cell may be obtained during MIMT
type of day (Fig. 1e). Quite likely this is the result of the moderate cell temperature—
it was estimated as 24 1C as can be seen from Table 1—which compensates less
efficient usage of the solar spectrum. This conclusion seems to be justified if one will
notice that for HILT type day (Fig. 1b) efficiency of the cell is almost as high as that
for MIMT while for HIHT (Fig. 1a) day cell exhibits very poor performance.
The other quite logical conclusion from results presented in Fig. 3 is that efficiency
drop of the tandem cell is fastest when band gap of the bottom cell—i.e., long
wavelength absorber—is increasing while band gap of the upper cell—i.e., short
wavelength absorber—is decreasing at the same time. Examples of similar results for
tandem junction cells, though plotted only for one spectrum in two-dimensional
graph, may be also found in [22].
Summarized results of calculations performed for single and multijunction (up to
five junctions) structures with use of Eq. (12) have been presented in Figs. 4 and 5.
Here Fig. 4a–c show optimum energy band gaps of the absorbers that should be
selected to achieve maximum efficiencies while Fig. 5 presents maximum values of
efficiencies that may be expected if optimum absorbers, as from Fig. 4, are used.
Calculations were made for various solar spectra corresponding to days specified in
Table 1. Cell temperature was estimated using Eq. (1).
An important conclusion that may be formulated when analysing results presented
in Fig. 5 is that significant increase of cell efficiency when compared with single
junction cell may be expected mainly for tandem and triple junction structures.
Further increase of the number of junctions seems to be unjustified when taking into
account possible complexity of the technological process and resulting fabrication
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Fig. 3. Efficiency of double junction solar cell versus absorber’s energy band gap of the particular cells
calculated for solar spectra corresponding to various days as specified in Table 1; cell temperature was
calculated according to Eq. (1).
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1.400
Eg [eV]
1.390
HIHT
MIHT
HILT
LILT
MIMT
1.88
1.86
1.84
1.82
Eg [eV]
1.395
1.90
HIHT
MIH T
HILT
LILT
MIMT
1.385
1.380
1.80
1.78
1.76
1.74
1.375
1.72
1.70
1.370
1.14
1.12
1.10
2
1
(a)
(b)
junction number
2.5
2.2
2.1
1.9
HIHT
MIHT
HILT
LILT
MIMT
2.3
2.2
2.1
1.8
Eg [eV]
Eg [eV]
2.4
HIHT
MIHT
HILT
LILT
MIMT
2.0
1.7
1.6
2.0
1.9
1.8
1.7
1.5
1.6
1.4
1.5
1.0
1.4
0.9
1.0
0.8
0.9
1
(c)
junction number
2
3
1
2
(d)
junction number
3
4
junction number
2.8
HI HT
MIHT
HI LT
LILT
MIMT
2.6
2.4
2.2
Eg [eV]
2.0
1.8
1.6
1.4
1.2
1.0
0.8
1
(e)
2
3
4
5
junction number
Fig. 4. Optimum energy band gaps of the absorbers in single and multijunction solar cells calculated for
solar spectra corresponding to various days as specified in Table 1.
cost of such structures. This conclusion may be of particular significance when
regarding multijunction cells for terrestrial applications where reasonable balance
between manufacturing costs and efficiency is required.
3.3. Effect of solar spectrum on the performance of real solar cells
Fig. 6 shows the effect of strongly changing solar spectrum on the efficiency of
several real solar cells. The applied range of AM factor corresponds practically to
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58
HIHT
MIHT
HILT
LILT
MIMT
56
Efficiency [%]
54
52
50
48
46
44
42
34
32
30
1
2
3
4
5
number of junctions
Fig. 5. Maximum efficiencies of single and multijunction solar cells calculated for absorber band gaps and
solar spectra as shown in Fig. 4; cell temperature was calculated according to Eq. (1).
full day of operation. All plots have been normalized to the value of efficiency
obtained for AM1.5 spectrum. In the insert figure the same plots are shown with
efficiency scaled in real values. Cell temperature has been assumed constant.
As an example of triple junction device a-Si solar cell with band gap values taken
from [23] has been taken to calculations. For comparison plot for optimised triple
junction cell with band gaps as presented in Fig. 4 has been added to the graph.
As can be seen a-Si cell single junction cell is most affected by the increase of AM
factor. Since increase of AM factor means that relative contents of the high energy
photons in the spectrum is decreasing this must be clear effect of the wide band gap
material being poor absorber for low energy photons from long wavelength range of
the spectrum. Similar effect may be observed also for both a-Si triple junction cells
which is result of decreasing input coming from the front cell to the overall efficiency
of the cell with increasing AM factor.
Very similar plots to those presented in Fig. 6 have been also reported by King and
co-workers in [24]
4. Conclusions
The IEC 60904-3 standard with recommended AM1.5 solar spectrum distribution
seems to be nonsuitable for performance prediction of multijunction solar cells for
terrestrial applications. It is as well nonsuitable in the case of single junction devices
with energy band gap significantly different from the optimum 1.39 eV value, e.g.,
a-Si cells. This is especially true for Central and Northern Europe locations where
spectra characterized by higher values of AM meaning higher contents of low energy
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1.0
Efficiencynorm
0.9
0.8
0.7
0.6
1
2
3
4
5
6
7
8
9
10
Air Mass
Fig. 6. Dependence of the normalized efficiency of several types of single junction solar cells calculated for
solar spectra corresponding to various Air Mass factors; in the insert figure the same curves are shown
in the absolute scale. Theoretical efficiency of triple junction a-Si cell was calculated both for the case
where absorbers have optimum band gaps (see Fig. 4c) as well as for real cell with band gap values taken
from [23].
photons and with usually higher contents of the diffused component are more
common. On the other hand, single junction cells with energy band gap close to
optimum value are much less sensitive to energy distribution of the incoming
spectrum and prediction of their performance in real operating conditions based on
the STC calibration should be more reliable.
For heterojunction or multijunction structures situation is more complicated.
With band gaps of the stacked absorbers optimised for a specified light spectrum
they are relatively more sensitive to shape of the spectrum but theoretically
achievable efficiencies are much higher than for even the best single junction
structures.
The higher is the number of junctions in the cell the lower is the relative increase in
cell efficiency. This is particularly true for the cells with four and more junctions.
Effect of solar spectrum on solar cell performance cannot be analysed without
taking into account natural effect as is heating up of the cell. However, for complex
thin-film structures correct estimation of the cell temperature in real operating
conditions may be a very difficult task.
One should be aware that calculations done in this work have been based both on
simplified cell’s model and not always valid superposition principle, Eq. (2), as well
as on simple formula for the dark saturation current Io, Eq. (3). Hence presented
results may merely serve only as a useful guide to the range of band gap values of
interest. For more accurate predicting of the efficiency of thin-film multijunction
cells calculations assuming more realistic I–V models and overall quantum efficiency
should be applied.
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