journal of applied physics cuprous photovoltaic cells
TRANSCRIPT
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8/10/2019 Journal of Applied Physics Cuprous photovoltaic Cells
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Cuprous oxide indium tin oxide thin film photoYoltaic cells
Masaharu Fujinaka
Department
of
Electronics Tokyo
Denki
University Chiyoda-Ku Tokyo 101 Japan
Alexander
A
Berezin
Department
of
Engineering Physics McMaster University Hamilton Ontario Canada L8S 4M1
(Received 25 June 1982; accepted for publication 14 January 1983)
We studied thin films of cuprous oxide deposited on glass coated with transparent conducting
ind ium-tin oxide (ITO) films. The deposition
of
both
Cu
2
0 and
ITO
was made by rf sputtering in
an Ar/02 gas mixture.
For
the deposition ofCu
2
0 a pure copper target was used and
ITO
films
were deposited from a disk target, the halves
of
which were made ofSn and In, respectively. This
allows variation of the stoichiometry of the deposited
ITO
film by changing the position of the
substrate glass beneath the Sn/ln target. X-ray diffraction
of
Cu
2
0 films indicates the typical
pattern
of
amorphous material. We were able to produce Cu
2
0 films
of
different stoichiometry by
varying the O
2
to Ar ratio during rf sputtering. The maximum resistivity of the films corresponds
to an ideal stoichiometry ofCu
2
0.
An
activation energy of 0.55 eV found from thermostimulated
conductivity is related to excess
Cu
vacancies. The band gap found from the spectral dependence
of the photovoltaic effect is 2.0 eV. The composition of
ITO
films was studied by Auger analysis
and can be described as a variable composition mixture ofSn0
2
+ x and
In
2
0
3
+
y To
produce an
Ohmic electrode, gold was evaporated on the top
of
the
Cu
2
0
film
and hence the resulting
structure
of
the photocell could be specified as
ITO-Cu
2
0-Au,
for which we propose a barrier
band diagram. We studied the photovoltaic characteristics of the fabricated photocells under an
incandescent lamp with 100mW/c m
2
output. The open-circuit voltage and short-circuit
current
of
our
cells were about 20-90 mV and 50 uA/cm2 respectively, and some dependence of
the outpu t characteristics on the composition ofITO film was observed. Conversion efficiency for
thin films Cu
2
0/ITO
cells was found to be substantially lower than for Cu
2
0/Cu Schottky
barrier cells. This is tentatively attributed to small diffusion lengths and/or presence of interface
recombination centers.
PACS numbers: 85.60.Dw, 72.40. + w, 73.40.Lq,
8U5.Cd
I INTRODUCTION
Cuprous oxide (Cu2
0
is
considered to be a useful mate
rial for photo voltaic energy conversion because it has a prop
er band gap ~ 2 . 0 eV), high efficiency
of
generation
of
pho
tocarriers and also because of its potential low cost. Cuprous
oxide thin films can be also produced
by the same sputtering
technique as indium-tin oxide (ITO) films considered ear
lier.
with an oil diffusion pump system. The water cooled target
was 12.7 cm in diameter and the distance between the upper
and the lower cathodes was 9.2 cm.
Most of the earlier studies
of Cu
2
0 photocells included
Schottky barrier, MIS, and heterojunction structures based
on thermally prepared CU20
or
electrodeposited Cu
2
0
see,
e.g., Refs.
2-9
and references therein).
t
is the purpose
of
this paper to report on the study of
the photocells
of
Cu
2
0 deposited on glass coated with trans
parent conducting
ITO
films
of
different compositions.
The
ITO
can form a heterojunction with
Cu
2
0 film
4
5
as well as
be used as a window material for solar radiation.
The deposition of both
Cu
2
0 and
ITO
was made by
radio-frequency
rf)
sputtering of metals in an Ar/02 gas
mixture. Some dependence of the output characteristics on
the composition of
ITO
film was observed.
II. EXPERIMENT
The schematic diagram
of
the apparatus used in this
work is shown in Fig. I. The basic sputtering system was a
Materials Research Corporation type
8551 rf
sputter unit
After pumping down to a background pressure of
6 X 10
Torr
in the sputtering chamber,
Ar
and O
2
gases,
controlled by separate flowmeters, were mixed together in
the gas mixture chamber, and then introduced into the sput
tering chamber through a gas leak valve. Then,
we
raised the
total pressure to the desired value of2.5 X 10
3
Torr
in order
to keep the discharge continuous and the plasma stable. The
target was sputter cleaned for a few minutes with a shutter
covering the substrates. Finally, the deposition was started
by moving the shutter.
III. ITO PREPARATION AND CHARACTERIZATION
The transparent conducting
ITO
films were deposited
by rfsput tering in the gas mixture of Ar flow rate: 100 cm
3
/
min) and O
2
flow rate: 20 cm
3
/min) on cleaned micro slide
glass (Sargent-Welch Scientific Company). The disk target
consisted of two semicircular parts which were made of Sn
and In, respectively 0.5
mm
thick, 99.99% purity). This im
purity level is probably not very important for the overall
resistivity of he film (it is basically dominated by the
In
to Sn
ratio), but it may be essential for the recombination processes
at the Cu
2
0/ITO interface. This finally can have a detrimen-
3582 J. Appl. Phys. 54 6), June 1983
0021-8979/83/063582-07 02.40
1983 American Institute of Physics
3582
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-
8/10/2019 Journal of Applied Physics Cuprous photovoltaic Cells
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pow r
_ ..... ..:..:.. supply
tal effect of a conversion efficiency (see Sec. VI). The deposi
tion time was
18
min at an rf power level
of
50 W.
The position
A,B
F
of
the glass substrates and the
relative sizes
of
Sn
and In
are shown in Fig.
2.
Figure
2(a)
is
for the case when the disk target consists of two equal halves
ofSn
and
In, respectively. Figure
2(b)
is for the case when the
In side is substantially larger than the Sn side [see insert in
Fig. 2(b)]. We measured the electrical resistance, the Auger
signal, and the optical transmit tance to evaluate the compo
sition of the fabricated
ITO
films.
1
To measure sheet resistance
of
ITO films we used two
parallel evaporated gold electrodes
of
5-mm length with 0.3-
mm
gap between them (see the insert to Fig. 5 for the similar
arrangement). The measured sheet resistances are shown in
Fig. 2.
The resistance changes rapidly just beneath boundary
of
Sn and In on the target. By moving the position of sub
strate glasses toward In side, the resistance can be decreased.
The observation of the energy spectra
of
Auger elec
trons emitted from the
ITO
films provides an effective tool
M
FIG. I. Schematic diagram of the apparatus
for the deposition of thin
films by
the radio
frequency sputtering system.
FM
= flow
meters).
for the chemical analysis
of
the composition. The Auger
spectra
of
samples
A, C,
and
E of
Fig. 2 are shown in Fig. 3.
In the case of 50% Sn-50 In target [Fig. 2(a)], the main
spectrum (Sn
430
) of Sn at 430 eV was stronger than the main
spectrum (In
404
) ofIn at 404 eV. In the case of he target with
a larger indium side [Fig. 2(b)] the line In
404
was stronger
than Sn
430
in contrast with first target.
In
each case, the
atomic concentration
of
Sn and In in
ITO
films was deter
mined by measuring the peak-to-peak amplitude of the cor
responding element (Sn
430
or
In
404
)
with elemental sensitiv
ity factors,1O and was shown in Fig. 2. Due to the partial
overlap
of
Sn
430
and
In
404
lines the accuracy
of
determina
tion
of
concentrations is relatively low. An indirect indica
tion of this accuracy may be drawn from the position of
resistivity minimum in Fig.
2(b).
We think that this accuracy
is sufficient for the purpose of our semiqualitative analysis.
Our
Auger
data
relate this minimum with approximately
1:9
ratio ofSn to In. A similar minimum found in 11 corresponds
to 4-6 wt. %
of
Sn02 in Sn:In
2
0
3
mixture.
I t is obvious from Fig. 2 that the resistance correlates
with the relative concentration ofSn and In in the ITO film.
Position of Snlln target
I d ~ ~ ~ = T ~ l n C = = = ~ 1
d
___
.......... In
C o n ~ i - .
I ---.
I
I
I
I
I
I
I
Sn
I
80.-
60Q
~
a:
4 ffi
)
z
8
2
1
A
B
D E F
Ia)
lO'---..;..t.;....--,----.J:-L.:L-....:L......i..--..IO
Posit
on of subs tr
ate
glass
3583 J. Appl. Phys., Vol. 54, No.6, June 1983
4
1
:
~ l
z
~
(J)
ID
2
::1
I
~ c I
Sn
In
,0
5mm
I
102
mm
r
1 --
__
I d l ~ _ A ~ ~ B ~ _ C ~ S ~ n _ D ~ _ E ~ - ~ F ~ ~
Position
of substrate
glass
8
o b)
FIG.
2.
Room temperature sheet resis
tances of transparent conducting ITO
films
with respect to the configuration
of
Sn/In target and the position of substrate
glass. Electrodes length 5 mm, gap
between the two paralIel Au electrodes is
0.3
mm. (a) Symmetrical In/Sn target; (b)
nonsymmetrical In/Sn target. Insert to
Fig.
2(b):
The configuration of the non
symmetrical In/Sn target. 1-1-1: resis
tance;
. - . - . :
concentration.
M.
Fujinaka and
A. A.
Berezin
3583
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A
C
: e.
1.0
C )
C )
1.0
C )
o t
:d
;d
:d
:d
:d
n
Sn
430
n
0
04
b)
n
404
The overall resistance
of
samples in Fig.
2 b)
are higher than
those in Fig.
2 a)
in spite
of
increasing the concentration
of
In. The deposition rate was changed by increasing the area
of
In target because In has a relatively lower sputtering rate. As
a result, under the same sputtering conditions (power, sput
tering time, and constant
flow
rate: Ar: 100 cm
3
/min; O
2
:
20
cm
3
/min), the thickness
of
ITO film
is smaller in the case
of
In>
Sn
target than for the case when 50 Sn-50
In
target
was used.
The relative concentration
of
oxygen in the ITO films
was constant regardless
of
the position of the substrate as
follows from almost constant Auger signal (0
52 0
,
about
50 ). As a result,
we
can describe our ITO films as a vari
able composition mixture
of Sn 0
2
+
x
and In203+y
lxi,
Iyl < ), where x and y reflect the excess
or
deficit
of
a lattice
3584
J. Appl. Phys. Vol. 54 No.6, June 1983
E
1.0
:0 :d
n
404
oxygen relatively to the ideal stoichiometry.
Sn 0
2
was a
dominant participant when the symmetrical
SnlIn
target
was used [Fig.
2 a)],
whereas In20 3dominated in ITO films
when the second, In-dominated target was used [Fig.
2 b)].
Then,
we
observed the spectral dependence
of
the opti
cal transmittance in the visible region which
is
shown in Fig.
4 a)
for the case
of
Fig.
2 a)
target. The case
of
Fig.
2 b)
target
is shown in Fig.
4 b).
The relative transmittance
of
In-richer
ITO films was
80 -90
and rather spectrally unsensitive
in the visible region. At the same time the transmittance of
Sn-richer areas could be substantially
imprOVed at
a shorter
wavelength by increasing In concentration. The samples
and in Fig.
4 a)
have spectral characteristics similar to
those shown in Fig.
4 b)
due to the increasing proportion
of
In
as could be seen in the Auger signal.
M. Fujinaka and
A. A.
Berezin
3584
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-
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100
80
---
~
~ 6 0
Z
~
t::
~ 4 0
C/)
Z
~
I-
20
F.
...... . . ...-
. : ; ; a ~
.......... ,.,. ............ ( : 4 ~
...-.
/E / %. .,.; ............
/ . ,eh./AI'
~
/ :.,
..... ..,
~ C
... ... .
.'_.,.
-..................
BC 0 E F
100
80
FIG. 4.
Optical transmittance for
ITO
films.
(a)
For
the target of Fig.
2(a);
(b)
for the tar
get of Fig. 2(b).
O ~ - - - - - - ~ - - - - - - ~ - - - - ~
O L - - - - - - - ~ - - - - - - ~ - - - - - ~
0.4
0.5
0.6 0.7
0.4
0.5
0.6 0.7
la)
WAVELENGTH ,u.
m)
WAVELENGTH C)L m
b)
IV. Cu
2
0 PREPARATION AND CHARACTERIZATION
We studied thin films of Cu
2
0 deposited by
rf
sputter
ing in constant
Ar
flow (flow rate 100
cm
3
/min and various
Oz flow rates (5, 10,20,30, and 50 cm
3
/min on glass. The Cu
target used for sputtering had 99.9 purity and was 3 mm in
thickness. All Cu
2
0
films for the electrical conductivity
measurements were made by supplying an
rf
power level of
150 W for 20 min, and the films obtained had a thickness of
the order
of
4500 A. Thickness
of
all CUzO films were mea
sured by a Taylor-Hobson (TALYSURF4) probe.
Gold electrodes were evaporated on top
of
CUzO films
as shown in Fig. 5. The electrical conductivity of the films
was measured at room temperature and the results are also
shown in Fig. 5.
The obtained CUzO films exhibited a minimum in con
ductivity (0 min - 10- 3 n 1 cm - 1) at an oxygen flow rate of
approximately 20 that of Ar (100 cm
3
/min . The film
which corresponds to this ratio of
flow
rates probably corre
sponds to an ideal stoichiometry of
CUzO. At
lower
Oz
per
centage than this minimum point, the conductivity of the
films increases because
of
the larger copper content than in
ideal CUzO. Higher levels
of
O
2
flow
result in an increase of
the conductivity presumably because of the admixture of
CuO to CUzO. We also measured the thermostimulated con
ductivity
of
films fabricated
at
various
Oz
flow
rates (10 ,
20 , 30 , and 50 of Ar flow rate) in the region from
-
50C
to + 85 C. The results are shown in Fig. 6. The
conductivity at a relative Oz flow rate
of
5% is not shown
because the obtained film behaves like metallic Cu. The acti
vation energy
of
the thermostimulated current
of
the films
corresponding to the oxygen flow rate of20% was calculated
from linear log I
vs
T - 1 plot (labeled in Fig. 6 as Ozo ), and
is equal to 0.55
eV
[assuming the conductivity -exp( - acti
vation energy/2kT)].
Based on these results we have chosen the Oz and
Ar
flow rates of20 cm
3
/min and 100 cm
3
/min, respectively, for
the subsequent fabrication of CUzO photocells.
3585
J.
Appl. Phys., Vol. 54, No.6, June 1983
V.
PROPERTIES OF ITO/Cu20 HETEROJUNCTION
CELLS
To produce an ohmic electrode for the cells, gold was
evaporated on the top of the CUzO films (20 Oz flow rate),
10
~ ~ ~ ~ i l l Z i I ~ r H + - - t h i n
film
5
10
-6
5X10
r ) I J I ,
o 10 20 30
40
50
02 to
Ar
( )
FIG.
5. Electric cur rent for the films ofCu
2
0 deposited for various O
2
to Ar
flow ratios,
at
rf power level 150 W for 20 min, at bias of 5 V.
M.
Fujinaka and A. A. Berezin
3585
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200
250
T K)
300
350
400
FIG.
6. Electric
current
vs reciprocal te mperature for
Cu
2
0
films
of
Fig. 5.
0
, 020
0
, and
0'0
are 10, 20, 30, and
50 of 0
in the gas mixture,
respectively. Bias is 5
V.
which are deposited on glass coated with ITO films of differ
ent composition. Hence the resulting structure of the photo
cell could be specified as ITOICuzOI
Au as illustrated in the
insert to Fig. 8.
In the case when the sputtering power level was 150
W,
we could not obtain samples having a rectifying1-
V
charac
teristics, possibly because of the presence of many recombin
ation centers
at
the junction interface of ITOICuzO formed
due to the particle bombardment during the discharge.
However, by decreasing the sputtering power level, we could
observe the characteristics such as shown in Fig.
7.
The ITO
film used for the samples represented in Fig. 7 was a sub
strate of the type C from Fig.
2 a).
For
CUzO
films sputtered
at a power level of75 W rectification can be obtained for all
types of
composition
of
ITO films i.e., A-F from Fig.
2).
The 1- characteristics for the cells were measured un
der dark conditions and under illumination
of
about 100
mW
cm-
2
calibrated by light meter LI-I85A of LAMBDA
Instruments Corporation) with an incandescent lamp Syl-
vania Co.)
of
750 W placed 20 em from the face of the ITO
films. For the ITO films from Fig.
2 a),
we were able to find
the dependence of the photo-output on the composition of
ITO films, but we could not find such pronounced depen-
3586 J. AppL Phys., Vol.
54, No.6,
June
1983
FIG. 7.1 V
characteristics for the ITO/Cu,O/Au cells
at
different sputter
ing power levels. The substrate is the
ITO
film at the position C in Fig. 2 a).
rf
power level
W)
150
100
75
thickness A)
4500
2500
2000
sputter ing time min.)
20
25
30
dence when the ITO films from Fig. 2 b) were used. This
difference may be related with the overall high resistivity
of
ITO films from Fig. 2 b) to that from Fig.
2 a)
and with possi
ble differences in conduction mechanisms of In
z
0
3
:Sn and
Sn 0
2
:In films which are not yet fully understood. II
For the cells
of
the type A and F from Fig. 2 a), we did
not show their characteristics in Fig.
8,
because of the possi
ble influence
of
the end effects from the sputter ing Cu target
and also because of the decrease of the photo-output.
PhotovoItaic output was studied only for ITO films of
the composition with Sn > In and the maximum open circuit
voltage was obtained for the ratio of n : I n ~ 8 : 15.
For
the cells deposited on substrates
of
the types B, C,
D, and E, the spectral distribution of output open circuit
lighti>
+
Au 2.
5m
rt
cu
2
0
dia.)
ITO
substrate glass
-0.2 -0.1
60
0.2
V v)
FIG. 8. Dark
solid lines) and iIluminated dotted lines)
I V
characteristics
for the cells fabricated on
ITO
substrates of different composition in Fig.
2 a). rf power level used: 75
W;
sputtering time: 30 min; thickness
of
film:
2000
A
M. Fujinaka and
A.
A. Serezin
3586
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1.0
w
rJ
z
0 0 5
Cl.
rJ)
w
a:
:I:
Cl.
.........
....... .........
/ .
/0 .
.
_.......
.
.......
. ,..
.
. - ........ .
.' E . . ~ ....
.
.
q,:4 0.5 0.6 .7
WAVELENGTH CJL m)
FIG. 9.
Spectral distributions for the cells (of Fig. 8) made from
ITO
films
of
Fig.2(a).
voltage was measured as the function
of
the wavelength as
shown in Fig. 9. The maximum photovoltaic peaks were es
sentially similar to those found for
Cu/Cu
2
0 heterojunction
solar cells
4
or MIS-type CU/
CU
2
0
cells.
6
VI. DISCUSSION
ND
CONCLUSIONS
The
output characteristics for the ITO/Cu
2
0 photo
cells depended on the composition of both the ITO film and
the
Cu
2
0 film.
The
CuzO cells deposited by sputtering on the
ITO
films
ofB
and C type (Sn:In=85:
15)
exhibit
I V
characteris
tics as shown in Fig. 8, and the polarity
of
photovoltage is
always positive on the gold contact side. The activation ener
gy of 0.55 eV for CuzO film was found from thermostimulat
ed conductivity whereas the band gap of 2.0 eV was found
from the spectral dependence
of
the photovoltaic effect.
Therefore, we can propose a barrier band diagram for
ITO/
CuzO cells such as shown in Fig. 10.
The photovoltaic characteristics of ITO/Cu
2
0 cells
were measured under an incandescent lamp with 100
ITO
Au
FIG.
10. Energy band diagram for
ITO/Cu O
cells.
3587
J. Appl.
Phys . Vol. 54. No.6. June 1983
mW cm-
z
.
The
open-circuit voltage and the short-circuit
current were in the region 20 - 90 m V and about 50 /-LA/cm
2
,
respectively, which is unfortunate ly insufficient for practical
solar cells.
For
our data we estimated the standard photovoltaic
conversion parameters
see,
e.g., 5 of Ref. 3).
For
the illumi
nated curve labeled as B,C in Fig. 8 we obtained
oc =90
mY;
jsc
=
45/-LA/cm2,
V
mp
=48
mY;
jmp=23/-LA/cm
2
,
F F ~ O . 2 7 ;
77=10 .
This undoubtedly very low value
of
the conversion effi-
ciency
of
CU20/ITO for our thin film photocell does not,
however, finally discard the Cu
2
0 as a feasible photovoltaic
material.
For
single crystal
Cu
2
0/Cu
Schottky barrier pho
tocells conversion efficiencies
of
= 1% have been demon
strated under
AMI
illumination.
7
Even higher values
of
(up to 3%) were obtained for a monochromatic illumination
A ~ 6 7 5 mm, illumination level ~ 6 0
/ L
W/c m
2
) in our pre
vious study?
The
study
2
was carried out on single crystal
Cu
2
0/Cu
photocells. At the same time, the study of thin films CuzO/
Sn0
2
cells prepared by electrodeposition
8
again resulted in
very low values of
oc andjsc (V C ~ 5
m V
andjsc =50
-LA/
cm
2
for
AMI
conditions) comparable with our present val
ues. These results allow us to suggest that low conversion
efficiencies
of
thin films Cu
2
0 cells are related with deficien
cies
of
the presently used thin film techniques for Cu
2
0,
rather than with the cuprous oxide itself. The chief problem
here is, in
our
opinion, the high resistivity of deposited Cu
2
0
films (which limits jsc rather than limitations related with
the low barrier height.
At the same time, we have to note that our Cu
2
0 films
show the typical x-ray diffraction pattern of amorphous ma
terial see Fig. 11), and probably have a high concentration
of
recombination centers in the interfacial layer (these centers
may be related with trace impurities in ITO) which tends to
lower the barrier height
of
the junction. These two reasons
are likely to have a detrimental effect of the photovoltaic
output characteristics.
In this connection we should mention that attempts
have been made to explain the efficiency plateau of Cu
2
0
cells on the basis
of
poor barrier quality.
9
In this respect, the
special study
of
diffusion lengths in Cu
2
0 thin films may
provide a better insight into the future prospectives of cu
prous solar cells.
Figure 11 a) shows the x-ray diffraction pattern for a
Cu
2
0 film
of
5500-A thickness deposited during sputtering
at an
rf
power level
of
150 W for 30 min.
For
comparison, in Fig. 11 b)
we
show the diffraction
pattern
Cu
Ka line; A
=
1.54
A of
CuzO powder prepared
chemically. In contrast with a thin film pattern [Fig. l1(a)]
this pattern shows a more pronounced crystalline structure.
From Fig.
11
b) we determined the Miller index (1.1.1) at the
Interplanar spacing
of
d = 2.46 A assuming the cubic struc
ture.
In this study, the oxygen flow was kept constant during
the fabrication
of
ITO films. The amounts of oxygen during
M. Fujinaka and
A.
A. Sarazin
3587
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------.--- r--
46
44
4
4
38
the deposition as well as during the heat annealing after de
position are important for the improving of the photovoltaic
output, since electrical resistance
is
affected strongly by both
of
them.
So,
in terms
of
preparation characteristics, the low
efficiencies
of our
cells can be reasoned by insufficient opti
mization
of
oxygen
flow
parameters. As illustrated by Figs. 5
and
6,
the
Oz
to Ar ratio has a very pronounced effect on the
conductivity of ITO and
CUzO
films.
In conclusion, we can stress that the height
of
the poten
tial barrier at junction interface is strongly influenced by the
amount
of
available oxygen during the various stages
of
the
fabrication process.
One can also tentatively attribute the very low conver
sion efficiencies observed for thin films Cu
2
0/ITO
cells to
either small diffusion length or to interface recombination
centers. Both these reasons can also act jointly and eventual
ly could be related to the above-mentioned excessive
or
defi
cient oxygen.
Keeping in mind the fact that both thin
film
compo
nents of our cells (Cu
2
0 and ITO) were produced by the same
rf
sputtering process, the system Cu
2
0/ITO could be inter
esting from the viewpoint
of
mass production, provided the
photo voltaic output could be substantially increased by the
proper control
of
the preparation conditions. As one
of
the
probable ways to increase the conversion efficiency, we can
mention here the temperature annealing
of
ITO films as well
as the use of higher purity starting materials.
3588
J. Appl. Phys. Vol. 54. No.6 June 1983
T
A
...
(
CKNOWLEDGMENTS
FIG.
II.
X-ray
diffraction patterns;
a)
for
the Cu
2
0
film of5500 A hickness deposit
ed by supplying rf power level of 150 W for
30 min;
b)
for the powder
of Cu,O
made
chemically.
This work was supported by the Natural Sciences and
Engineering Research Council of Canada. The authors also
thank N. Dalacu for technical assistance on thin film prep
aration and
B.
Diacon for help during the measurement
of
Auger characteristics.
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Fujinaka and
A. A. Berezin,
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3588