Spectrally selective reflector surfaces for heat
reduction in concentrator solar cells:
modeling and applications of
TiO
2
:Nb-based thin films
Christopher M. Maghanga,
1,2,
* Gunnar A. Niklasson,
1
Claes G. Granqvist,
1
and Mghendi Mwamburi
3
1
Department of Engineering Sciences, The Ångström Laboratory, Uppsala University,
P.O. Box 534, SE-75121 Uppsala, Sweden
2
Permanent address: Kabarak University, Department of Mathematics and Computing Sciences,
P.O. Private Bag 20157, Kabarak, Kenya
3
Physics Department, Moi University, P.O. Box 1125, Eldoret, Kenya
*Corresponding author: c.maghanga@yahoo.com
Received 28 February 2011; accepted 25 April 2011;
posted 6 May 2011 (Doc. ID 143258); published 29 June 2011
The energy conversion efficiency of a conventional pn junction solar cell decreases as the temperature
increases, and this may eventually lead to failures in the photovoltaic system, especially if it uses con-
centrated solar radiation. In this work, we show that spectrally selective reflector (SSR) surfaces can be
important for reducing the heat buildup on passively cooled solar cells. We outline a computational
scheme for optimizing DC magnetron-sputtered TiO
2
:Nb-based SSRs tailored for silicon solar cells
and find good agreement of the reflectance with an experimental realization of the optimal SSR. A figure
of merit for SSRs has also been derived and applied to the experimental data. © 2011 Optical Society of
America
OCIS codes: 310.3840, 310.4165, 310.6188, 310.6805, 310.6860, 310.7005.
1. Introduction
It is well known that the energy conversion efficiency
of pn junction solar cells decreases as their tempera-
ture increases [1], and elevated temperatures may
also have detrimental effects on other components
of photovoltaic (PV) systems and lead to thermal
stress, which may result in failures. High solar cell
temperatures are particularly serious for concentrat-
ing solar cell systems, and cooling is then necessary.
Different cooling strategies have been reviewed
recently [ 2], and it was found that passive cooling has
a large potential in cases where coproductio n of hot
water is not desirable or economically viable. Min
et al. showed that a large heat sink area, approxi-
mately equal to the concentration ratio times the cell
area, would be needed in order to keep the cell tem-
perature at 50 °C [3]. Surfaces of this size may be un-
wieldy or simply not available, and alternatives are
important. One of these is offered by spectrally selec-
tive reflectors (SSRs), whose ideal integrated reflec-
tance is unity below the wavelength corresponding to
the bandgap of the absorber (λ
c
) and zero for λ > λ
c
[4–9]. By integrating selective reflection of the radia-
tion with known conventional radiative and convec-
tive cooling techniques [2,3,10,11], the heat buildup
on solar cells can be further reduced, thus cutting
back on the temperature-induced drop of the effi-
ciency of the cells. Jiang et al. have proposed a
0003-6935/11/193296-07$15.00/0
© 2011 Optical Society of America
3296 APPLIED OPTICS / Vol. 50, No. 19 / 1 July 2011
similar optical system for combined PV and photo-
thermal conversion by using a spectral beamsplitting
filter [12]. Despite the fact that a number of papers
on SSR coatings have been published, it appears that
their effect on solar cell performance has not been
previously assessed.
SSRs can be devised by overcoating reflecting
layers with transparent conducting films. Our initial
work used SnO
x
:F [4–7], and more recently we have
investigated SSRs based on anatase TiO
2
:Nb [8,9].
This latter material was discovered as lately as
2005 [13] and has attracted much interest recently
as a high-performance low-cost transparent conduc-
tor [14–18]. Our earlier work demonstra ted that
TiO
2
:Nb backed by Al leads to a reflector with
pronounced optical selectivity [8,9]. Specifically, the
experimental integrated reflectance was 77% and
28% in the ranges 300 < λ < 1100 nm and 1100 <
λ < 2500 nm, respectively, for a TiO
2
:Nb film contain-
ing 3:7 at: % of Nb. We also showed that, in order to
fabricate a good TiO
2
:Nb-based SSR on Al, an inter-
mediate dielectric layer of Al
2
O
3
is required to
suppress the deep interference fringes that would
otherwise compromise the selectivity of the
surface [6,7].
The purpose of the present paper is twofold. First,
we perform heat balance calculations for concentrat-
ing solar cell systems in order to evaluate the effect of
the SSR technology on cell efficiency. We find that the
use of an SSR material makes a contribution toward
lowering the temperature of the concentrator PV
cells and may have a niche for systems employing
compound para bolic concentrators with concentra-
tion ratios up to a factor of ten. Second, we outline
a methodology for optimizing the properties of SSR
coatings. We analyze the dependence of the optical
properties of TiO
2
:Nb coatings on the film thick-
nesses of the active layer and an intermediate alumi-
na layer. We also derive a figure of merit for SSRs
and apply it to our experimental results.
2. Solar Cell Efficiency and Operating Temperature
A. Ideal SSR
The ideal properties of an SSR for use with crystal-
line silicon solar cells are shown in Fig. 1. The wave-
length λ
c
for switching from high to low reflectance
lies at 1100 nm and corresponds to the silicon band-
gap. R
cell
and R
therm
are the integrated reflectance
values, which for a Si solar cell, are given by
R
cell
¼
R
1100
300
GðλÞRðλÞdλ
R
1100
300
GðλÞdλ
; ð1Þ
R
therm
¼
R
2500
1100
GðλÞRðλÞdλ
R
2500
1100
GðλÞdλ
; ð2Þ
where GðλÞ is the air mass (AM) 1.5 solar spectrum
[19]. For an SSR with ideal properties, R
cell
¼ 1 and
R
therm
¼ 0 will ensure that solar radiation in the
range 300 < λ < 1100 nm is reflected toward the so-
lar cell while the rest is absorbed by the SSR and
hence does not contribute to the heating of the solar
cell.
B. Solar Cell Efficiency
The dependence of solar cell efficiency η is often ap-
proximated by a linear relation [20,21] according to
η ¼ η
ref
½1 − βðτ − τ
0
Þ; ð3Þ
where η
ref
is the efficiency at a PV cell temperature τ
0
corresponding to the ambient (25 °C), and when the
solar irradiance on the cell is equal to 1000 Wm
−2
, β is
the temperature coefficient of efficiency, and τ is the
PV cell temperature. In this paper we consider the
most common solar cells, namely those made of
crystalline Si, which are characterized by β ∼
0:005 K
−1
[21].
C. Heat Balance of Solar Cells
The operating temperature can be estimated from
the energy balance equation for a concentrator solar
cell, which in our case can be expressed as
RαA
0
Cq
0
− ηRαA
0
Cq
0
− A
r
εσ
B
ðτ
4
− τ
0
4
Þ
− A
c
hðτ − τ
0
Þ¼0; ð4Þ
where the first term denotes the solar power re-
flected to the cell from mirrors with solar reflectance
R. Here the cell’s surface absorptivity is denoted α,
the cell area is A
0
, the geometric concentration ratio
is C, and the solar energy density is q
0
. The second
term in Eq. (4) is the electric power delivered to t he
external load with conversion efficiency η; it should
be noted that the efficiency is defined with respect to
the total absorbed solar power. The third term repre-
sents the power dissipated through radiation from
the surface area A
r
with surface emissivity ε where
the Stefan–Boltzmann constant is denoted σ
B
. The
last term, finally, characterizes the power dissipated
Fig. 1. Ideal spectral reflectance of an SSR appropriate for use
with a silicon solar cell. A normalized AM 1.5 solar spectrum is
included (shaded).
1 July 2011 / Vol. 50, No. 19 / APPLIED OPTICS 3297
through convection, which depends on the surface
area A
c
and the convective heat transfer coefficient h.
If the reflector material in the concentrator cell has
SSR properties, the cell absorbs only the useful ra-
diation, i.e., energy corresponding to the wave-
lengths λ < λ
c
, while in the absence of the SSR
property the wavelength range of absorption spans
the whole solar spectrum, say 300 < λ < 2550 nm.
Therefore, we express the general heat balance equa-
tion in terms of contributions from the radiation
below and above λ
c
as
R
cell
αA
0
Cq
cell
ð1 − ηÞþR
therm
αA
0
Cq
therm
ð1 − ηÞ
− A
r
εσ
B
ðτ
4
− τ
0
4
Þ − A
c
hðτ − τ
0
Þ¼0; ð5Þ
where q
cell
and q
therm
are the energy densities in the
ranges 300 < λ < 1100 nm and 1100 < λ < 2500 nm,
respectively. They can be estimated from the solar ir-
radiation data [19] as being approximately 0:8q
0
and
0:2q
0
, respectively. For a reflector with ideal SSR
properties according to Fig. 1, one finds
R
cell
αA
0
Cq
cell
ð1 − ηÞþA
r
εσ
B
ðτ
4
− τ
0
4
Þ
− A
c
hðτ − τ
0
Þ¼0: ð6Þ
Table 1 contains the parameter values used in our
model calculations. The absorptivity of the solar cell
varies with wavelength and also depends on surface
engineering and can be as high as 0.95 [22]. Even in
the long wavelength region below the bandgap, a Si
solar cell can have an absorptivity of the order of 0.8
[22]. For simplicity, we used an average value of 0.85
in our calculations as in [3] for the entire solar
wavelength range, and for the convective heat trans-
fer coefficient, we also take the value given by Min
et al. [3]. Employing the parameters in Table 1 in
Eq. (6), we computed the solar cell temperature as
a function of concentration for a solar cell with and
without an SSR. The results are plotted in Fig. 2 for
two different cases. At low geometric concentration,
the cell temperatur e is low, and the difference in tem-
perature between the two cases is small. However,
the effect of the SSR becomes evident as concentra-
tion increases, and the temperature of the cell
without the SSR is significantly higher than with
the SSR.
We now discuss two possible cases where SSR coat-
ings may be useful. First we consider a compound
parabolic concentrator of the type depicted in Fig. 3.
This design has a low concentration of ∼2 to 5. We
consider the case of minimum extra passive cooling
from convection and radiation by assuming a small
convective and radiative area, such that A
r
¼
A
c
¼ 4A
o
. Figure 2(a) shows that the temperature
rise can be kept at 35 to 60 °C above the ambient
if an SSR is used. In order to lower the temperatures
further, radiativ e and convective cooling can be
Table 1. Parameters Used in Calculations of
Solar Cell Temperature
Parameter Description Value
α Surface absorptivity of the cell 0.85
η Efficiency of the cell 0.2
τ
0
Ambient temperature 300 K
H Convective heat transfer 5 W=m
2
K
C Concentration factor Variable
ε Emissivity of the cell 0.85
Fig. 2. (Color online) Solar cell temperature versus concentration
calculated for constructions with and without SSR for (a) A
r
¼
A
c
¼ 4A
0
, (b) A
r
¼ 4 A
0
and A
c
¼ 10A
0
.
Fig. 3. Integration of an SSR surface and a PV cell in a compound
parabolic concentrator.
3298 APPLIED OPTICS / Vol. 50, No. 19 / 1 July 2011
enhanced by increasing the con vective and radiative
surface areas. However, the temperatures mentioned
above correspond to reasonable cell efficiencies of
14% t o 16.5%, given an efficiency of 20% at ambient
temperature as in our example. It is, however, more
interesting to consider the gain in efficiency that is
attainable by using an SSR. In the concentration
range that we consider, the temperature is lowered
by 10°C to 20 °C when an SSR is used, which leads
to efficiency gains of 0.8% to 1.8% for the solar cell.
This is certainly not negligible, and whether or not
an SSR is of practical interest will probably depend
on economic considerations.
Figure 2(b) considers parameters appropriate for a
parabolic trough concentrator employing the elabo-
rate passive cooling schemes reviewed by Royne
et al. [2]. It is seen that, already at a concentration of
15, the cell temperature has risen to almost 100 °C
above the ambient, which leads to a cell efficiency of
only half the value at ambient temperature. We
think that this is not tolerable under most circum-
stances and that active cooling with PV/photother-
mal cogeneration is more appropriate in the case
of concentration ratios in this range.
D. Figure of Merit for SSRs
For a PV concentrator employing a practical SSR,
the heating power is proportional to R
cell
q
cell
þ
R
therm
q
therm
, while in the absence of an SSR we as-
sume that R ¼ R
cell
in the whole solar wavelength
range. In the latter case, the heating power is propor-
tional to R
cell
q
0
. We now define a dimensionless
quantity, which we refer to as a figure of merit
(FOM), of an SSR as
FOM ¼
R
cell
q
cell
þ R
therm
q
therm
R
cell
q
0
≈ 0:8 þ 0:2
R
therm
R
cell
:
ð7Þ
The FOM attains a minimum value of 0.8 for an
ideal SSR and should approach this value as closely
as possible for real SSR coatings. For a nonselective
reflecting surface, the FOM is equal to 1.0.
3. Experiments and Calculations
Thin films of TiO
2
:Nb were made by dual-target
reactive DC magnetron sputtering in an Ar þ O
2
plasma onto substr ates of alumina-coated Al and of
Si, following procedures described elsewhere [9,14].
Doping of TiO
2
with Nb was achieved by setting
the Ti target power at a constant value and varying
the Nb target power. A small amo unt of H
2
was
added to the sputter plasma in order to avoid target
poisoning and allow stable sputtering conditions
[23,24]. The compositions of TiO
2
:Nb films backed
by Si were determined by ion beam techniques in the
range of atomic weights from 1 (H) to 41 (Nb).
Rutherford backscattering spectrom etry and time-
of-flight elastic recoil detection analysis were em-
ployed using facilities of the Uppsala University
Tandem Laboratory. Spectral normal transmittance
TðλÞ and near-normal reflectance RðλÞ were mea-
sured in the 300 < λ < 2500 nm range by use of a
Perkin–Elmer Lambda 900 double-beam spectro-
photometer equipped with an integrating sphere.
A barium sulfate film served as reflectance standard.
The results of the film composition and of the spec-
trophotometric data analysis for obtaining optical
constants have been presented elsewhere [14].
The optical properties of the SSRs were modeled
using the characteristic matrix formalism for an as-
sembly of thin films [25, 26 ]. For light of wavelength λ
incident on an assembly of N layers, the characteris-
tic matrix is the product of individual matrices for
each interface, I
m−1;m
, and for each layer, L
m
,
expressed as
E
þ
ð0
−
Þ
E
−
ð0
−
Þ
¼
Y
N
m¼1
I
m−1;m
L
m
I
N;Nþ1
E
þ
ðz
þ
Þ
0
¼
S
11
S
12
S
21
S
22
E
þ
ðz
þ
Þ
0
: ð8Þ
Here E
þ
and E
−
are the complex amplitudes of for-
ward- and backward-tra velling plane waves. The
front interface toward air of the assembly is denoted
0
−
and the back interface toward the substrate is
denoted z
þ
. The interface matrix components can
be obtained from the Fresnel relations as shown in
[26]. The layer matrix is obtained [26] from the phase
factor
δ
m
¼ð2π=λÞdN
m
cos θ
m
; ð9Þ
where θ
m
represents the direction of propagation in
the mth layer and can be obtained from the incident
angle using Snell’s law, N
m
¼ n
m
þ ik
m
is the com-
plex refractive index, and d denotes the layer thick-
ness. The reflectivity amplitude of this assembly may
then be found from
rðθ; λÞ¼E
−
ð0
−
Þ=E
þ
ð0
−
Þ¼S
21
=S
11
: ð10Þ
In our case m ¼ 3, 2, and 1 represent the substrate
(Al), a dielectric (Al
2
O
3
), and the transparent
Fig. 4. Optical constants, n and k, of an Nb:TiO
2
film containing
3:7 at: % Nb, from [9].
1 July 2011 / Vol. 50, No. 19 / APPLIED OPTICS 3299
conducting oxide (TiO
2
:Nb) layer, respectively. Lit-
erature data for the optical constants of Al [27]
and Al
2
O
3
[28] were employed in our calculations.
The Al
2
O
3
is an intermediate oxide layer that was
used to suppress strong optical interference effects
[6–8]. It has been shown before that a TiO
2
:Nb film
containing 3:7 at: % Nb exhibits very good selective
reflectance [9] and is close to the optimum composi-
tion [8]. The optical constants for this material, from
[9,14], are shown in Fig. 4 and were used in our
model calculations.
4. Results
A. Calculated Reflectance
Figure 5 illustrates the roles of the Al
2
O
3
and
TiO
2
:Nb layer thicknesses on the reflectance. It is
found that the layer thicknesses influence both the
wavelength at which the reflectance is at a minimum
(λ
min
) and the reflectance value at that wavelength. A
shift of λ
min
to longer wavelengths occurs with in-
creasing thickness while at the same time the reflec-
tance is lowered. Increasing TiO
2
:Nb film thickness,
as demonstrated in Fig. 5(b), enhances interference
fringe depth, i.e., the distan ce between the maxima
and minima of an interference fringe, but has the
advantage of lowering the reflectance at λ
min
. A shift
to longer wavelengths is also seen.
The optimum thicknesses of the Al
2
O
3
and
TiO
2
:Nb layers can be determined from the contour
plots in Figs. 6(a) and 6(b). The optimal properties
are attained by combining a high value of R
cell
with
a low value of R
therm
. Inspection of the plots indicates
that the lowest values of R
therm
are achieved for
Al
2
O
3
thicknesses between 60 and 150 nm and a
TiO
2
:Nb thickness exceeding 80 nm. The highest va-
lues of R
cell
, however, require a thickness of TiO
2
:Nb
Fig. 5. (Color online) Calculated influence of the thickness of
(a) the Al
2
O
3
layer and (b) the TiO
2
:Nb film on the normal spectral
reflectance of TiO
2
:Nb=Al
2
O
3
=Al SSRs. In (a) the thickness of the
TiO
2
:Nb film is 90 nm, while in (b) the Al
2
O
3
thickness is 90 nm.
Fig. 6. (Color online) Contour plots showing the variation of
(a) R
cell
and (b) R
therm
as a function of the thicknesses of TiO
2
:
Nb and Al
2
O
3
in a TiO
2
:Nb=Al
2
O
3
=Al stack.
3300 APPLIED OPTICS / Vol. 50, No. 19 / 1 July 2011
