1
Supporting Information for:
Excitonic Effects in Emerging Photovoltaic Material
s: A Case
Study in Cu
2
O
Stefan T. Omelchenko
†§
, Yulia Tolstova
†
, Harry A. Atwater
†§||
, Nathan S. Lewis
*
‡§||
⊥
†
Division of Engineering and Applied Sciences, Calif
ornia Institute of Technology, Pasadena,
CA 91125
‡
Division of Chemistry and Chemical Engineering, Cal
ifornia Institute of Technology, Pasadena,
CA 91125
§
The Joint Center for Artificial Photosynthesis, Cal
ifornia Institute of Technology, Pasadena, CA
91125
||
Kavli Nanoscience Institute, California Institute o
f Technology, Pasadena, CA 91125
⊥
Beckman Institute, California Institute of Technolo
gy, Pasadena, CA 91125
*Corresponding Author:
nslewis@caltech.edu
This is a 15 page document with 7 sections (Experim
ental Methods, S1, S2, S3, S4, S5, S6) and 4 figure
s
(S1-S4)
2
Experimental Methods
Substrate Growth:
Cuprous oxide substrates were grown by two methods.
For photoluminescence
measurements, single crystalline Cu
2
O wafers were prepared by the floating zone techniq
ue.
Feed and seed rods were grown by the thermal oxidat
ion of high-purity Cu rods (Alfa Aesar,
99.999%) in a vertical tube furnace (Crystal System
s Inc.) in air for 100 h at 1050 °C. The rods
were then cooled in N
2
at 120 °C/h. Prior to growth, the rods were clean
ed in acetone and etched
using dilute nitric acid (0.1 M) for 60 seconds. T
he rods were suspended by either Cu or Pt wire.
Single crystals were grown in an optical floating z
one furnace (CSI FZ-T-4000-H-VII-VPO-PC).
Crystals were grown in air with the seed and feed r
ods counter-rotating at 7 rpm. The resulting
single crystalline boules were diced into wafers al
ong the growth axis and mechanically polished
to a specular finish using diamond grit.
Polycrystalline Cu
2
O substrates were grown for solar cell fabrication.
High-purity Cu foil
(Alfa Aesar, 99.9999%, 0.5 mm thick) was heated in
a quartz tube under N
2
(g) to 1025 °C at
1000 °C/h. The foils were then oxidized in air for
24 h and cooled under N
2
(g) to room
temperature. The resulting substrates were ~ 0.8 mm
thick and had carrier concentrations ~10
13
cm
-3
.
Solar Cell Fabrication:
The photovoltaic device in this work was fabricated
on a ~ 0.8 mm thick Cu
2
O wafer
grown by thermal oxidation of copper foil. The grai
n size of wafers grown by this process is
typically of the order of several millimeters and i
n some cases almost the size of the entire wafer
(~1 cm
2
). The photovoltaic cells in this study were fabric
ated using a circular shadow mask
3
resulting in an ultimate cell size ~0.02 cm
2
so that individual solar cells were generally isol
ated
to only 1 or 2 grains.
Prior to fabrication, the polycrystalline Cu
2
O substrates were cleaned with isopropanol
and loaded into a magnetron sputtering system with
a base pressure of 1.7x10
-7
Torr. The Cu
2
O
wafers were heated in vacuum for 90 min at 100 °C.
A 45 nm layer of Zn(O,S) was co-sputtered
from ZnO and ZnS targets at a working pressure of 5
mTorr Ar. The power on the ZnO target
was 100W and the power on the ZnS target was 85W.
After deposition, the samples were cooled
to room temperature in vacuum and removed from the
chamber. A shadow mask was placed
over the samples and a 60 nm ITO layer was sputtere
d at 50 W in an Ar atmosphere with a
working pressure of 3 mTorr at room temperature. A
100 nm Au back-contact was then sputter
deposited on the back of the sample.
Characterization:
A Ti:sapphire laser (Libra, Coherent Inc.) with a f
undamental 800 nm laser pulse, 120 fs
pulse width, and 10 kHz repetition rate was used to
pump an optical parametric amplifier (Opera
Solo, Coherent Inc.) and generate visible light. S
ingle crystalline Cu
2
O wafers grown by the
float-zone method were illuminated with wavelengths
ranging from 400 to 550 nm. The time-
averaged photoluminescence spectra were collected u
sing a time-correlated single-photon-
counting method using a streak camera (Hamamatsu In
c.) with 20 ps time resolution. The
spectral response measurements were performed using
a Xe arc lamp and slit monochromator
(Newport Inc.), and a calibrated reference Si photo
diode (Thor Labs Inc.) with a known spectral
responsivity.
4
S1. Device Model and Model Parameters for Cu
2
O
The Cu
2
O-based photovoltaic was modeled as a simplified
p-n
+
solar cell, using the following
assumptions adopted by Ref. 1: (1) the depletion ap
proximation; (2) the drift and diffusion
currents are opposite and equal in magnitude within
the depletion region; (3) recombination is
neglected in the depletion region; (4) the solar ce
ll is operating in low-level injection; (5) in the
bulk, minority carriers flow by diffusion. The incl
usion of excitons requires the modification of
the “free carrier” model to include an additional t
erm that accounts for the exchange between the
excitons and free-carrier populations. In this cas
e, the excess minority- carrier (
n
e
) and excess
exciton (
n
x
) concentrations are governed by the following coup
led differential equations:
∆
=
∆
−
+
∆
− ∆
∗
(1)
∆
=
∆
−
−
∆
− ∆
∗
(2)
where
D
is the diffusion coefficient,
τ
is the lifetime, and
G
is the wavelength-dependent
generation rate.
1
The subscripts
e
and
x
refer to electrons and excitons, respectively. Th
e third
term in Equation S1 and S2 is the net rate at which
electrons and holes bind to form excitons, and
is derived from the law of mass action, where
b
is the coefficient for binding free carriers into
excitons,
N
A
is the
p
-type doping density and
n
*
is the equilibrium constant for the exchange
between excitons and free carriers (in equilibrium
∗
=
). Further, free carriers and
excitons were assumed to be in quasi-equilibrium at
the edge of the depletion region. The
coupled differential equations yield analytical sol
utions for the dark saturation current density,
J
0
, and the short-circuit current density,
J
sc
,:
=
+
!
" +
#
+
!#
"
(3)
$%
=
&
'
+
!
&
'
" +
#
&
'
+
!#
&
'
"
(4)
where
e
is the fundamental unsigned charge on an electron;
n
0
and the
n
x
0
are the equilibrium
concentrations of electrons and excitons, respectiv
ely; and
α
is the wavelength-dependent
absorption coefficient.
2
Additionally:
( =
!
)
−
*
+
&
,
-
-
)
√
/
(5)
0 =
!
)
+
*
+
,
-
-
)
√
/
(6)
1
!
=
!
√
2
(7)
5
1
)
=
!
√
2
(8)
3
!
=
!
)
4
!!
+ 4
))
−
√
5
(9)
3
)
=
!
)
4
!!
+ 4
))
+
√
5
(10)
4
6
= 4
!!
− 4
))
(11)
5 = 4
6
)
+ 44
!)
4
)!
(12)
4
!!
=
!
+
"
!
8
(13)
4
))
=
!
+
∗
"
!
8
(14)
4
!)
= −
:
∗
8
(15)
4
)!
= −
:;
<
8
(16)
The “free carrier” solutions for the dark saturatio
n and short-circuit current densities,
respectively, for an
p-n
+
solar cell are given by:
,>?
= −
8
∆
@
(17)
$%,>?
=
A
&
'
(18)
Equation S17 and S18 were used to compare the perfo
rmance of the excitonic model to that of
the traditional “free carrier” model. The major ef
fect of excitons, effecting a coupling between
the electron and hole population and exciton popula
tion, alters the diffusion characteristics of
both free carriers and excitons, as can be seen fro
m Equation S3-S4 and S17-S18. Thus, a
fraction
γ
of photogenerated electrons move with a diffusion
length
L
1
and the remaining
photogenerated electrons (1-
γ
) diffuse with a diffusion length
L
2
, where
L
1
and
L
2
are effective
diffusion lengths that account for the interactions
between the exciton and free carrier
populations.
Similarly, a portion of photogenerated excitons
ζ
and the remaining exciton fraction
(1-
ζ
) have diffusion lengths
L
1
and
L
2
,
respectively.
Equation S3 and S4 are fundamentally dependent on t
he experimentally measured parameters
D
and
τ
, which are affected by temperature and doping dens
ity. The performance of the
p-n
+
Cu
2
O
solar cell can then be evaluated a function of temp
erature and
N
A
. The dependence of the exciton
binding energy on doping density can be estimated a
ssuming that the exciton binding energy
falls off to zero as the doping density approaches
the Mott density:
6
B
= B
C
D1 −
F
;
<
,GHH
I
)
(19)
where
E
x
∞
is the unscreened exciton binding energy, 150 meV i
n Cu
2
O.
3-4
The Mott density was
estimated using the value for Si as a function of t
emperature:
*JKK
= 10
!M
N
O
PQ
N
O
RS
T
2
PQ
2
RS
T
U
(20)
where
a
B
is the exciton Bohr radius and
ε
is the dielectric constant. The superscripts Si and
Cu
2
O
refer to silicon and Cu
2
O, respectively. The unscreened exciton binding ene
rgy is assumed to be
independent of temperature.
The electronic band gap
E
g
of Cu
2
O was measured down to 4 K using the threshold ener
gy of the
free exciton peak in the photoluminescence spectrum
. The temperature dependence of
E
g
was fit
using an oscillator model that accounts for exciton
-phonon coupling:
B
V
U
= B
V
0
+ WℏY − WℏY coth
ℏ^
)_
O
`
"
(21)
where
E
g
(0) = 2.173 eV is the electronic band gap at
T
= 0 K,
S
=1.89 is a material specific
constant, and
ℏY
= 13.6 meV is the phonon energy of the phonon (
a
!)
emitted during exciton
luminescence.
5
The electron mobility was estimated from majority-
carrier data in literature. The effect of
temperature and doping density on the majority carr
ier mobility was estimated as:
!
b
c
=
!
b
d
+
!
b
e
(22)
where
f
`
= 8511 × 10
.Mkl`
(23)
is the mobility caused by lattice vibrations, with
the value determined from as-grown Cu
2
O
crystals.
6-7
Empirical data for the mobility as a function of h
ole concentration due to Na doping
was used as an interpolating function in the model
for
I
.
8
The minority-carrier lifetime is an important mater
ials property that plays a significant
role in determining the performance of solar cells
in the free carrier model. Generally, the
electron lifetime is estimated from the electron di
ffusion length fit from the external quantum
efficiency, and varies from ~100 ns for undoped samp
les to on the order of ~ 1 ns for doped
samples.
9-10
As such, we have estimated the electron lifetime a
s:
m
=
!
!&!
' n
;
<
(24)
The lowest lying exciton states in Cu
2
O are the spin singlet “paraexciton” and spin tripl
et
“orthoexciton”, which are split by a spin exchange,
with the paraexciton lying 12 meV lower
than the orthoexciton. The paraexciton transition i
s dipole- and quadrupole-forbidden, and the
7
transition is dipole-forbidden for the orthoexciton
due to inversion symmetry of the Cu
2
O crystal.
This behavior leads to long-lived exciton states; t
he paraexciton lifetime, for example, has been
measured to be > 14
s at low temperatures.
11
The small energetic splitting between the two
states causes the orthoexciton to decay into the pa
raexciton state on the picosecond time scale,
while paraexcitons up-convert to orthoexcitons at t
he same rate.
12-13
Consequently, for
temperatures relevant to photovoltaic operation, th
e ortho- and paraexciton lifetimes are the
same, given by the most rapid recombination pathway
. Thus, the temperature-dependent
orthoexciton lifetime data from Ref. [14] was used
for
τ
x
(implemented as an interpolating
function in our code), assuming that the exciton li
fetime is < 1
s.
14
The mean time for excitons to form is given by
m
:
=
!
:
∗
, where
n
*
is found by treating
the exciton and electron-hole system as an ideal ga
s mixture and neglecting exciton-exciton
interactions:
∗
=
@
@c
@
U
l/)
p
/_
O
`
(25)
with the density of states,
q
=
V
Q
)rs
Q
_
O
t/
t
(26)
where
u
q
is the degeneracy term,
m
i
is the translational mass and
h
is Planck’s constant. For
Cu
2
O
u
= 2,u
= 2
and
u
= u
u
= 4
and
w
= 0.99w
,w
= 0.58w
,
and
w
= 3.0w
,
where
m
0
is the fundamental electron mass. The exciton bind
ing coefficient
b
has not been
measured in Cu
2
O, so we have used the variation of
b
with temperature for Si:
= 10
l
U
)
+ 2.5 × 10
M
U
!/)
+ 1.5 × 10
z
(27)
in units of cm
3
⋅
s
-1
.
15
This is likely an underestimation of
b
in Cu
2
O, because the exciton binding
energy in Cu
2
O is approximately an order of magnitude larger tha
n that in Si.
S2. Effect of Excitons on Diffusion Length
The diffusion lengths of electrons and excitons wer
e calculated by use of:
1
=
{
m
(28)
1
=
{
m
(29)
The diffusion coefficient of electrons in Cu
2
O has yet to be measured, so
D
e
was estimated using
the Einstein relation:
=
!
s
c
s
f
|
}
U
(30)
8
where the electron mobility was estimated by weight
ing the hole mobility by the ratio of the
electron and hole translational masses. This appro
ach yields values of ~2
m for
L
e
, which
agrees well with measured values from the literatur
e.
9-10, 16-17
Similarly, the exciton diffusion
length was calculated using:
= f
|
}
U
(31)
The exciton mobility
x
has been measured accurately down to low temperatur
es. Above 10 K,
the following expression was found to be in good ac
cord with the experimentally measured
exciton mobility:
f
=
)
√
)r
ℏ
~
l
s
/
|
}
U
l/)
(32)
where,
= 6.11
g
⋅
cm
-3
is the mass density of the Cu
2
O crystal,
= 4.5 × 10
m
⋅
s
-1
is the
thermal velocity, and
= 1.2
is the deformation potential.
18
The principal effect of excitons on solar cell perf
ormance is to modify the diffusion
characteristics of photogenerated species by effect
ively coupling the motion of free carriers and
excitons.
1-2
Figure S1 shows the effect of temperature and dopi
ng density on the diffusion
lengths
L
e
,
L
x
,
L
1
, and
L
2
. The exciton diffusion length is approximately co
nstant with
temperature, and is almost an order of magnitude gr
eater than
L
e
, which varies significantly with
temperature. For low-to intermediate doping densiti
es, this behavior causes
L
1
to approach
L
e
,
especially at high temperatures. At low temperature
s, where excitons dominate,
L
1
tends towards
L
x
.
L
2
is substantially lower than
L
1
for all temperatures.
9
Figure S1.
The simulated electron, exciton and effective diff
usion lengths for Cu
2
O for doping
densities a) 10
12
, b) 10
14
, and c) 10
16
.
10
S3. Equilibrium concentration of excitons and free
carriers for calculation of
J
0
A fundamental assumption in our model is that excit
ons and free carriers are in
equilibrium in the Cu
2
O bulk up to the edge of the depletion region, such
that:
q
)
=
=
∗
(33)
where,
n
0
,
p
0
,
and
n
0
x
are the electron, hole concentrations, respectivel
y. Here, we assume that
the hole concentration is given by the ionized dopa
nt density
N
A
and thus, in equilibrium, the
ratio of excitons to free electrons is given by:
@
@
=
;
<
∗
(34)
The equilibrium exciton ratio is shown in
Figure S2
. As expected, the excitonic fraction of the
photogenerated population increases with decreasing
temperature and increasing doping density.
The exciton density is greater than the free carrie
r population at room temperature for large
doping densities. From, Equation S32 and S33 the eq
uilibrium exciton concentration is given by:
=
Q
∗
(35)
and the equilibrium electron concentration is given
by the typical expression:
=
Q
;
<
(36)
The intrinsic carrier concentration can be calculat
ed from the effective density of states in the
valence and conduction bands, respectively:
?
U
= 2
)rs
_
O
`
"
t
= 4.75 × 10
!
U
l/)
(37)
U
= 2
)rs
c
_
O
`
"
t
= 2.14 × 10
!
U
l/)
(38)
q
)
U
=
?
d
O
d
= 1.014 × 10
l!
U
l/)
(39)
in cm
-3
.
19
Using these values and the parameters outlined abo
ve, the dark saturation current
density
J
0
can be calculated.
11
Figure S2:
The equilibrium ratio of excitons to free carriers
as a function of doping density for
temperatures ranging from 200 to 500 K.
S4. Absorption, generation and calculation of J
sc
The calculation of
J
sc
requires knowledge of the generation rate of excit
ons and free
carriers, as well as the absorption coefficient in
the visible spectrum. We have calculated
J
sc
using the full wavelength dependence of
G
e
and
α
. The experimentally determined absorption
coefficient was used, and the absorption coefficien
t of Cu
2
O was assumed to not vary
substantially over the temperature range evaluated
in this study (200 – 500 K).
20
The
wavelength-dependent electron and exciton generatio
n rate were estimated by:
=
(40)
12
where
is the photon flux from the global AM 1.5 solar sp
ectrum. To differentiate between
exciton and free carrier generation, the free carri
ers were assumed to be generated for absorption
of a photon with energy above the electronic band g
ap, and excitons were assumed to be
generated only in the case for photon excitation wi
th an energy between the electronic band gap
and the excitonic band edge (
E
g
-
E
x
). Thus, the short-circuit current density is the s
ummation of
two parts, one free-carrier and one excitonic:
$%
=
+ 1
!
!
+
1 −
+ 1
)
)
C
p
+
0
+ 1
!
!
+
1 − 0
+ 1
)
)
p
p
p
(41)
This approach is an oversimplification, as we have
demonstrated experimentally in the main text.
Even above-band-gap illumination leads to excitonic
generation that is observable as a free-
exciton peak in the photoluminescence spectrum.
S5. Fill Factor and Efficiency
The current density and voltage at the maximum powe
r point,
J
mpp
and
V
mpp
, were
determined by generating an
J
-
V
characteristic by numerically solving Equation 1 f
or an array of
voltages. The fill factor (
FF
) was then calculated using the formula:
=
G
(42)
and the efficiency (
η
) is then given by:
=
G
>>
Q
(43)
where
P
in
is the power incident on the Cu
2
O cell, in this case, the standard AM 1.5 solar
spectrum. The
FF
is depicted in
Figure S3
. The fill factors were in close agreement between
the
excitonic and FC models, except for the highest dop
ing density.
13
Figure S3:
The simulated fill-factor for the excitonic and FC
models. The fill factors were in
close agreement except at the highest doping densit
ies, where large enhancement in the
J
sc
dominates and leads to an increase in the solar-cel
l efficiency relative to the FC case.
References
S6. Characteristics of Polycrystalline Cu
2
O wafers
Room temperature photoluminescence spectra were als
o collected for the polycrystalline Cu
2
O
wafers grown by thermal oxidation, which were used
in device fabrication. The spectra were
collected using a 514 nm excitation for powers rang
ing from 85
W to 5.4 mW. The
photoluminescence spectrum near the orthoexciton lu
minescence peak is shown in Figure S4.
The unusual peak shape is due to the convolution of
the orthoexciton peak and the phonon-
assisted exciton peak, which broadens and grows at
higher temperatures where the absorption
probability of phonons by excitons is large.
5, 21-22
The peak was evident even at the lowest of
excitation powers (0.85
W). Thus, excitons are generated under visible exci
tation even in
polycrystalline Cu
2
O substrates used in photovoltaic fabrication.
14
Figure S4: Photoluminescence spectrum of the free e
xciton peak in thermally oxidized,
polycrystalline Cu
2
O wafers at room temperature using a 2.4 mW, 514 nm
excitation.
References
1.
Corkish, R.; Chan, D. S. P.; Green, M. A., Excit
ons in silicon diodes and solar cells: A
three
‐
particle theory.
Journal of Applied Physics
1996,
79
(1), 195-203.
2.
Zhang, Y.; Mascarenhas, A.; Deb, S., Effects of
excitons on solar cells.
Journal of
Applied Physics
1998,
84
(7), 3966-3971.
3.
Kane, D. E.; Swanson, R. M., The effect of excit
ons on apparent band gap narrowing and
transport in semiconductors.
Journal of Applied Physics
1993,
73
(3), 1193-1197.
15
4.
Moskalenko, S. A.; Snoke, D. W.,
Bose-Einstein Condensation of Excitons and
Biexcitons and Coherent Nonlinear Optics with Excit
ons
. Cambridge University Press: New
York, New York, 2000.
5.
Ito, T.; Masumi, T., Detailed Examination of Rel
axation Processes of Excitons in
Photoluminescence Spectra of Cu
2
O.
Journal of the Physical Society of Japan
1997,
66
(7),
2185-2193.
6.
Shimada, H.; Masumi, T., Hall Mobility of Positi
ve Holes in Cu
2
O.
Journal of the
Physical Society of Japan
1989,
58
(5), 1717-1724.
7.
Biccari, F. Defects and Doping in Cu
2
O. Universita di Roma, 2009.
8.
Minami, T.; Nishi, Y.; Miyata, T., Impact of inc
orporating sodium into polycrystalline p-
type Cu
2
O for heterojunction solar cell applications.
Applied Physics Letters
2014,
105
(21),
212104.
9.
Xiang, C.; Kimball, G. M.; Grimm, R. L.; Brunsch
wig, B. S.; Atwater, H. A.; Lewis, N.
S., 820 mV open-circuit voltages from Cu
2
O/CH3CN junctions.
Energy & Environmental
Science
2011,
4
(4), 1311-1318.
10. Biccari, F.; Malerba, C.; Mittiga, A., Chlorine
doping of Cu
2
O.
Solar Energy Materials
and Solar Cells
2010,
94
(11), 1947-1952.
11. Mysyrowicz, A.; Hulin, D.; Antonetti, A., Long
Exciton Lifetime in Cu
2
O.
Physical
Review Letters
1979,
43
(15), 1123-1126.
12. Snoke, D. W.; Shields, A. J.; Cardona, M., Phon
on-absorption recombination
luminescence of room-temperature excitons in Cu${}_
{2}$O.
Physical Review B
1992,
45
(20),
11693-11697.
13. Snoke, D. W.; Lin, J. L.; Wolfe, J. P., Coexist
ence of Bose-Einstein paraexcitons with
Maxwell-Boltzmann orthoexcitons in Cu
2
O.
Physical Review B
1991,
43
(1), 1226-1228.
14. Koirala, S.; Naka, N.; Tanaka, K., Correlated l
ifetimes of free paraexcitons and excitons
trapped at oxygen vacancies in cuprous oxide.
Journal of Luminescence
2013,
134
, 524-527.
15. Nolle, E., Recombination Through Exciton States
in Semiconductors.
Soviet Physics -
Solid State
1967,
9
(1), 90-94.
16. Trivich, D.; Wang, E. Y.; Komp, R. J.; Weng, K.
; Kakar, A., Cuprous oxide photovoltaic
cells.
Journal of the Electrochemical Society
1977,
124
(8), C318.
17. Olsen, L. C.; Addis, F. W.; Miller, W., Experim
ental and theoretical studies of Cu
2
O
solar cells.
Solar Cells
1982,
7
(3), 247-279.
18. Trauernicht, D. P.; Wolfe, J. P., Drift and dif
fusion of paraexcitons in Cu
2
O:
Deformation-potential scattering in the low-tempera
ture regime.
Physical Review B
1986,
33
(12), 8506-8521.
19. Sze, S. M.; Ng, K. K.,
Physics of semiconductor devices
. 3rd ed.; Wiley-Interscience:
Hoboken, N.J., 2007; p x, 815 p.
20. Malerba, C.; Biccari, F.; Leonor Azanza Ricardo
, C.; D’Incau, M.; Scardi, P.; Mittiga, A.,
Absorption coefficient of bulk and thin film Cu
2
O.
Solar Energy Materials and Solar Cells
2011,
95
(10), 2848-2854.
21. Petroff, Y.; Yu, P. Y.; Shen, Y. R., Luminescen
ce of Cu
2
O-Excitonic Molecules, or Not?
Physical Review Letters
1972,
29
(23), 1558-1562.
22. Petroff, Y.; Yu, P. Y.; Shen, Y. R., Study of p
hotoluminescence in Cu
2
O.
Physical
Review B
1975,
12
(6), 2488-2495.