J. Appl. Phys.
125
, 064103 (2019);
https://doi.org/10.1063/1.5083632
125
, 064103
© 2019 Author(s).
Photovoltaic effect in multi-domain
ferroelectric perovskite oxides
Cite as: J. Appl. Phys.
125
, 064103 (2019);
https://doi.org/10.1063/1.5083632
Submitted: 29 November 2018 . Accepted: 24 January 2019 . Published Online: 13 February 2019
Ying Shi Teh
, and
Kaushik Bhattacharya
Photovoltaic effect in multi-domain ferroelectric
perovskite oxides
Cite as: J. Appl. Phys.
125
, 064103 (2019);
doi: 10.1063/1.5083632
View Online
Export Citation
CrossMar
k
Submitted: 29 November 2018 · Accepted: 24 January 2019 ·
Published Online: 13 February 2019
Ying Shi Teh
and Kaushik Bhattacharya
AFFILIATIONS
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA
ABSTRACT
We propose a device model that elucidates the role of domain walls in the photovoltaic effect in multi-domain ferroelectric
perovskites. The model accounts for the intricate interplay between ferroelectric polarization, space charges, photo-generation,
and electronic transport. When applied to bismuth ferrite, results show a signi
fi
cant electric potential step across both 71
and
109
domain walls, which in turn contributes to the photovoltaic (PV) effect. We also
fi
nd a strong correlation between polariza-
tion and oxygen octahedra tilts, which indicates the nontrivial role of the latter in the PV effect. The domain wall-based PV
effect is further shown to be additive in nature, allowing for the possibility of generating the above-bandgap voltage.
Published under license by AIP Publishing.
https://doi.org/10.1063/1.5083632
I. INTRODUCTION
In conventional photovoltaics, electron-hole pairs are
created by the absorption of photons that are then separated
by an internal
fi
eld in the form of heterogeneous junctions
such as p-n junctions. Less than a decade ago, Yang
et al.
1
reported large photovoltages generated in thin
fi
lms of multi-
domain bismuth ferrite (BFO) and suggested a new mecha-
nism where electrostatic potential steps across the ferroelec-
tric domain walls drive the photocurrent. This discovery has
since revitalized the
fi
eld of photoferroics. Among many fer-
roelectric oxides, BFO has particularly attracted considerable
interest due to its high ferroelectric polarization and a rela-
tively small bandgap.
Many novel experiments were subsequently devised to
investigate the role of domain walls in the observed photovol-
taic (PV) effect in ferroelectric perovskites. Alexe and Hesse
2
performed measurements of the local photoelectric effect
using atomic force microscopy (AFM). They found that the
photocurrent is essentially constant across the entire
scanned area, hence indicating the absence of the domain
wall (DW) effect. The nanoscale mapping of generation and
recombination lifetime using a method combining photoin-
duced transient spectroscopy (PITS) with scanning probe
microscopy (SPM) points to a similar conclusion.
3
This led to
the hypothesis that the bulk photovoltaic (BPV) effect, which
arises from the noncentrosymmetry of perovskites, is the
key mechanism instead.
4
,
5
However, further recent studies
focused on characterizing both the BPV and DW effects show
that the latter effect is much more dominant.
6
,
7
The lack of
clear consensus among the scienti
fi
c community on the key
mechanism in the PV effect in perovskites as well as on the
role of domain walls could be understood from the inherent
dif
fi
culties in the experimental techniques. The nanoscale
order of ferroelectric domain walls makes it dif
fi
cult to probe
into and separate the effects from the bulk domains and the
domain walls. Other issues such as defect formation and grain
boundaries in perovskite crystals further complicate the
analysis.
First-principles calculations have provided a detailed
understanding of the structure of domain walls
8
–
10
and have
established the drop in electrostatic potential across it.
However, they are limited to a few nanometers and cannot
examine the interaction of domain walls with other features.
On the other hand, models at the device scale provide under-
standing at the scale
11
but assume
a priori
the polarization
and other aspects of the domain wall. Finally, various phase
fi
eld models provide understanding of the domain pattern,
12
,
13
but in the absence of space charge and photocurrent. Thus,
there is a gap in our modeling of the intricate interplay
between space charge, ferroelectric polarization, and elec-
tronic transport.
This paper seeks to
fi
ll this gap by building on the prior
work of Xiao
et al.
14
,
15
and Suryanarayana and Bhattacharya
16
who developed a continuum theory of semiconducting ferro-
electrics including electron and hole transport. We extend
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their work to include photogeneration due to illumination
and study the photovoltaic effect in ferroelectric perovskite
oxides. We investigate the photovoltaic response of BFO
fi
lms
with different domain wall con
fi
gurations by solving the
model at the device scale. At a 71
or 109
domain wall, we
observe a change in the component of polarization perpen-
dicular to the domain wall. This in turn results in a relatively
large electrostatic potential step across the wall, which allows
for separation of photogenerated electron-hole pairs. Thus,
this model supports the hypothesis of domain walls contrib-
uting to the photovoltaic effect.
We emphasize that this model does not
a priori
assume
the domain wall structure or the electrostatic potential step
across it. Instead, this is a prediction of the model that is based
on well-established Devonshire-Landau models of ferroelec-
trics and lumped band models of semiconductors. Similarly,
this modeling framework is not limited to BFO or to 71
or 109
domain walls. In fact, we argue that the features presented
here are generic for any non-180
domain wall in any ferro-
electric; of course, the extent depends on the details.
The rest of the paper is organized as follows: In Sec.
II
,
we brie
fl
y review the structure of BFO and discuss the classi-
fi
cations of ferroelectric domain walls in BFO. We develop the
theoretical framework in Sec.
III
. We apply the theory to
examine PV effect in ferroelectrics with domain walls in Sec.
IV
. We conclude with a brief discussion in Sec.
V
.
II. BISMUTH FERRITE
In this work, we focus on bismuth ferrite (BFO), though
we note that the same framework can be applied to other fer-
roelectrics and the results are expected to be similar
qualitatively.
At room temperature, BFO has a rhombohedral phase
with space group
R
3
c
[
Fig. 1(a)
]. The displacements of the
atoms from the ideal cubic structure in this phase lead to a
spontaneous polarization pointing in the [111] pseudocubic
direction. Another distinctive feature is the network of O
6
octahedra surrounding the Fe ions that rotate or tilt
out-of-phase about the polarization axis. This is also com-
monly known as the antiferrodistortive (AFD) mode and has
been found to play an important role in the ferroelectric
phase of the material.
10
Electric polarization in rhombohedral BFO can take one of
the eight variants of the [111] pseudocubic direction, which
gives possible domain wall orientations of 71
,109
, and 180
[
Figs. 1(b)
and
1(c)
]. On each domain, there can be two possible
orientations of oxygen octahedra. We follow Lubk
et al.
’
s
method of classifying oxygen octahedra tilts (OTs) across
domain walls as either
continuous
or
discontinuous
.
9
In the
continuous case, the direction of the oxygen octahedra tilt
remains the same along the polarization vector
fi
eld. In the dis-
continuous case, the direction reverses across the domain wall.
III. THEORY
We consider a metal-perovskite-metal (MPM) structure
that is connected to an external voltage source to form a
closed electrical circuit (see
Fig. 2
). The multi-domain ferro-
electric perovskite
fi
lm occupying the space
Ω
is subjected to
light illumination. The two metal-perovskite interfaces are
denoted by
@
Ω
1
,
@
Ω
2
[
@
Ω
. All the processes are assumed to
occur at constant temperature
T
. We present the equations
and their physical meanings here. Readers may refer to
Appendix A
for the thermodynamically consistent derivation.
A. Charge and electrostatic potential
The total charge density (
x
[
Ω
) is given by
ρ
¼
q
(
p
v
n
c
þ
z
d
N
þ
d
z
a
N
a
),
(1)
where
q
is the electronic charge,
n
c
is the density of electrons
in the conduction band,
p
v
is the density of holes in the
valence band,
N
þ
d
is the density of ionized donors,
N
a
is the
density of ionized acceptors,
z
d
is the valency number of
donors, and
z
a
is the valency number of acceptors. The polar-
ization and space charge in the ferroelectrics together
FIG. 1.
(a) Crystal structure of bulk BFO. At the rhombohedral phase, the two O
6
octahedra rotate out-of-phase about the polarization axis marked by the dotted line.
(b) and (c) show domain walls with orientations of 71
and 109
, respectively. The red arrow in each domain points in the direction of polarization.
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125,
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125,
064103-2
Published under license by AIP Publishing.
generate an electrostatic potential. This is determined by the
Gauss equation
r
(
ε
0
r
f
þ
p
)
¼
ρ
,
(2)
where
ε
0
is the permittivity of free space, subject to appropri-
ate boundary conditions.
B. Transport equations
In the presence of light illumination, an incident photon
may be absorbed in the semiconductor to promote an electron
from the valence band to the conduction band, thus generating
an electron-hole pair in the process of photogeneration. The
reverse may also occur such that an electron and a hole
recombine. Electrons and holes may also move from one point
to another point, as represented by the electron and hole
density
fl
ux terms,
J
n
and
J
p
. With conservation of electrons
and holes, we can relate the time derivatives of densities of
electrons and holes to the aforementioned processes via the
following transport equations:
_
n
c
¼r
J
n
þ
G
R
,
(3)
_
p
v
¼r
J
p
þ
G
R
,
(4)
where
G
is the rate of photogeneration, which can be taken to
be proportional to the intensity of light illumination, while
R
is
the recombination rate. Here, we assume that the only form of
recombination present is radiative recombination, which
involves the transition of an electron from the conduction
band to the valence band along with the emission of a photon.
R
takes the form of
B
(
n
c
p
v
N
2
i
) with the intrinsic carrier
density being given by
N
i
¼
ffiffiffiffiffiffiffiffiffiffiffiffi
N
c
N
v
p
exp (
E
c
E
v
2
k
B
T
).
17
The radiative
recombination coef
fi
cient
B
is a material property, indepen-
dent of the carrier density.
The electron and hole
fl
uxes
J
n
,
J
p
are taken to be
proportional to the gradient in its electro-chemical
potential
17
via
J
n
¼
1
q
ν
n
n
c
r
μ
n
,
(5)
J
p
¼
1
q
ν
p
p
v
r
μ
p
,
(6)
where
ν
n
and
ν
p
are the electron and hole mobilities,
respectively.
In this work, the diffusion of donors and acceptors is
neglected.
C. Free energy
The free energy of the ferroelectric is postulated to be of
the form
W
¼
W
DL
(
p
,
θ
)
þ
W
G
(
r
p
,
r
θ
)
þ
W
n
c
(
n
c
)
þ
W
p
v
(
p
v
)
þ
W
N
d
(
N
þ
d
)
þ
W
N
a
(
N
a
)
:
(7)
The various terms are currently explained.
W
DL
refers to the Devonshire-Landau free energy of bulk
ferroelectrics. In addition to the typical primary order param-
eter of electric polarization
p
, we include a second order
parameter
—
oxygen octahedral tilts
θ
. We adopt the following
energy form for BFO.
12
The corresponding coef
fi
cients can be
found in
Table I
.
W
DL
¼
a
1
(
p
2
1
þ
p
2
2
þ
p
2
3
)
þ
a
11
(
p
4
1
þ
p
4
2
þ
p
4
3
)
þ
a
12
(
p
2
1
p
2
2
þ
p
2
2
p
2
3
þ
p
2
1
p
2
3
)
þ
b
1
(
θ
2
1
þ
θ
2
2
þ
θ
2
3
)
þ
b
11
(
θ
4
1
þ
θ
4
2
þ
θ
4
3
)
þ
b
12
(
θ
2
1
θ
2
2
þ
θ
2
2
θ
2
3
þ
θ
2
1
θ
2
3
)
þ
c
11
(
p
2
1
θ
2
1
þ
p
2
2
θ
2
2
þ
p
2
3
θ
2
3
)
þ
c
12
[
p
2
1
(
θ
2
2
þ
θ
2
3
)
þ
p
2
2
(
θ
2
1
þ
θ
2
3
)
þ
p
2
3
(
θ
2
1
þ
θ
2
2
)]
þ
c
44
(
p
1
p
2
θ
1
θ
2
þ
p
1
p
3
θ
1
θ
3
þ
p
2
p
3
θ
2
θ
3
)
:
(8)
The energy stored in the ferroelectric domain walls
is accounted for through the gradient or Ginzburg
energy term
W
G
, which includes the energy cost associ-
ated with rapid change in polarization and octahedral
tilts.
W
G
(
r
p
,
r
θ
)
¼
1
2
a
0
jr
p
j
2
þ
1
2
b
0
jr
θ
j
2
:
(9)
Here, we assume that the gradient terms are isotropic
for simplicity but can easily be modi
fi
ed.
FIG. 2.
Schematic of a device model in a metal-perovskite-metal con
fi
guration.
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W
n
c
,
W
p
v
,
W
N
d
, and
W
N
a
in Eq.
(7)
are the free energies of
electrons in the conduction band, holes in the valence band,
donors, and acceptors, respectively. The explicit expressions
of these energies can be determined by considering each
system as a canonical ensemble in the framework of statistical
mechanics
16
W
n
c
(
n
c
)
¼
n
c
E
c
þ
k
B
T
[
N
c
log
N
c
þ
n
c
log
n
c
þ
(
N
c
n
c
) log (
N
c
n
c
)],
(10)
W
p
v
(
p
v
)
¼
(
N
v
p
v
)
E
v
þ
k
B
T
[
N
v
log
N
v
þ
p
v
log
p
v
þ
(
N
v
p
v
) log (
N
v
p
v
)],
(11)
W
N
d
(
N
þ
d
)
¼
(
N
d
N
þ
d
)
E
d
(
N
d
N
þ
d
)
k
B
T
log (2
z
d
)
þ
k
B
T
[
N
d
log
N
d
þ
N
þ
d
log
N
þ
d
þ
(
N
d
N
þ
d
) log (
N
d
N
þ
d
)],
(12)
W
N
a
(
N
a
)
¼
N
a
E
a
(
N
a
N
a
)
k
B
T
log (2
z
a
)
þ
k
B
T
[
N
a
log
N
a
þ
N
a
log
N
a
þ
(
N
a
N
a
) log (
N
a
N
a
)]
:
(13)
D. Polarization and tilt equations
The polarization and tilt evolve according to the (time-
dependent) Landau-Ginzburg equations
1
ν
p
_
p
¼r
@
W
G
@
r
p
@
W
DL
@
p
r
f
,
(14)
1
ν
θ
_
θ
¼r
@
W
G
@
r
θ
W
DL
@
θ
,
(15)
where
ν
p
,
ν
θ
are the respective mobilities. They are subject to
natural boundary conditions
^
n
@
W
G
@
r
p
¼
0,
(16)
^
n
@
W
G
@
r
θ
¼
0
:
(17)
E. Electrochemical potentials
The electrochemical potentials are obtained from the
energy to be
μ
n
¼
@
W
n
c
@
n
c
q
f
,
(18)
μ
p
¼
@
W
p
v
@
p
v
þ
q
f
,
(19)
μ
N
þ
d
¼
@
W
N
d
@
N
þ
d
þ
qz
d
f
,
(20)
μ
N
a
¼
@
W
N
a
@
N
a
qz
a
f
:
(21)
At thermal equilibrium,
μ
n
¼
μ
p
¼
μ
N
þ
d
¼
μ
N
a
¼
E
F
m
, where
the magnitude of
E
F
m
is the workfunction of the metal elec-
trode. Furthermore, using Eqs.
(10)
to
(13)
, we can invert the
relations to obtain
n
c
¼
N
c
1
þ
exp
E
c
E
F
m
q
f
k
B
T
,
p
v
¼
N
v
1
þ
exp
E
F
m
E
v
þ
q
f
k
B
T
,
N
þ
d
¼
N
d
1
1
1
þ
1
2
z
d
exp
E
F
m
þ
E
d
q
f
z
d
k
B
T
2
4
3
5
,
N
a
¼
N
a
1
þ
2
z
a
exp
E
F
m
þ
E
a
q
f
z
a
k
B
T
1
,
(22)
consistent with the Fermi-dirac distribution.
18
Finally, assum-
ing
N
c
n
c
and
N
v
p
v
, Eqs.
(5)
and
(6)
become
J
n
¼
ν
n
k
B
T
q
r
n
c
þ
ν
n
n
c
r
f
,
(23)
J
p
¼
ν
p
k
B
T
q
r
p
v
ν
p
p
v
r
f
:
(24)
Equations
(23)
and
(24)
show that each of
J
n
and
J
p
can be
resolved into two contributions: (1) a diffusion current, driven
by the concentration gradient of carriers and (2) a drift
current, driven by an electric
fi
eld. By applying the Einstein
relation, which relates diffusion constant
D
to mobility
ν
via
D
¼
ν
k
B
T
=
q
, we recover the equations that are typically
written to describe the
fl
ow of electrons and holes in
solar cells.
TABLE I.
Coefficients of the Laudau-Devonshire energy for BFO.
Symbols
Values
Units
a
1
1
:
19
10
9
VmC
1
a
11
9
:
93
10
8
Vm
5
C
3
a
12
3
:
93
10
8
Vm
5
C
3
b
1
1
:
79
10
10
Vm
3
C
b
11
1
:
14
10
11
Vm
3
C
b
12
2
:
25
10
11
Vm
3
C
c
11
1
:
50
10
10
VmC
1
c
12
7
:
50
10
9
VmC
1
c
44
1
:
60
10
1
VmC
1
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J. Appl. Phys.
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064103-4
Published under license by AIP Publishing.
F. Ohmic boundary conditions
We prescribe ohmic boundary conditions at the contacts
with metal electrodes following
19
for convenience. We have
n
c
¼
N
c
e
(
E
c
E
f
m
)
=
k
B
T
p
v
¼
N
v
e
(
E
f
m
E
v
)
=
k
B
T
on
@
Ω
1
<
@
Ω
2
:
This is equivalent to assuming that the Fermi level in the
semiconductor aligns with that of the metal, hence giving rise
to electron and hole densities that are independent of the
applied voltage.
G. Steady-state model
At steady state, all the
fi
elds of interest do not vary with
respect to time. Furthermore, we are interested in domain
walls and, therefore, can assume that things are invariant par-
allel to the domain wall. This means that we have one inde-
pendent space variable, which we denote
r
. We denote the
components of polarization and tilt parallel (respectively per-
pendicular) to the domain wall to be
p
s
,
θ
s
(respectively
p
r
,
θ
r
).
With
z
d
¼
z
a
¼
1, we have a coupled system of differential
equations for region
x
[
(0,
L
), where
L
is the length of
the
fi
lm.
a
0
d
2
p
r
dr
2
@
W
DL
@
p
r
d
f
dr
¼
0,
(25)
a
0
d
2
p
s
dr
2
@
W
DL
@
p
s
¼
0,
(26)
b
0
d
2
θ
r
dr
2
@
W
DL
@
θ
r
¼
0,
(27)
b
0
d
2
θ
s
dr
2
@
W
DL
@
θ
s
¼
0,
(28)
ε
0
d
2
f
dr
2
þ
dp
r
dr
¼
q
(
p
v
n
c
þ
N
þ
d
N
a
),
(29)
dJ
n
dr
þ
G
B
(
n
c
p
v
N
2
i
)
¼
0,
(30)
dJ
p
dr
þ
G
B
(
n
c
p
v
N
2
i
)
¼
0,
(31)
J
n
¼
ν
n
k
B
T
q
dn
c
dr
þ
ν
n
n
c
d
f
dr
,
(32)
J
p
¼
ν
p
k
B
T
q
dp
v
dr
ν
p
p
v
d
f
dr
,
(33)
where
N
þ
d
¼
N
d
1
1
1
þ
1
2
exp
E
F
m
þ
E
d
q
f
k
B
T
2
4
3
5
,
N
a
¼
N
a
1
þ
2 exp
E
F
m
þ
E
a
q
f
k
B
T
1
,
N
i
¼
ffiffiffiffiffiffiffiffiffiffiffiffi
N
c
N
v
p
exp
E
c
E
v
2
k
B
T
,
with boundary conditions
dp
r
dr
(
r
¼
0)
¼
dp
r
dr
(
r
¼
L
)
¼
0,
dp
s
dr
(
r
¼
0)
¼
dp
s
dr
(
r
¼
L
)
¼
0,
d
θ
r
dr
(
r
¼
0)
¼
d
θ
r
dr
(
r
¼
L
)
¼
0,
d
θ
s
dr
(
r
¼
0)
¼
d
θ
s
dr
(
r
¼
L
)
¼
0,
f
(
r
¼
0)
¼
0,
f
(
r
¼
L
)
¼
0,
n
c
(
r
¼
0)
¼
n
c
(
r
¼
L
)
¼
N
c
e
(
E
c
E
f
m
)
=
k
B
T
,
p
v
(
r
¼
0)
¼
p
v
(
r
¼
L
)
¼
N
v
e
(
E
f
m
E
v
)
=
k
B
T
:
H. Numerical issues
The model derived above comprises differential equa-
tions that are nonlinear and coupled, which can prove trou-
blesome numerically. So, we non-dimensionalize the problem
as in
Appendix B
. Furthermore, we notice that the coupling
between the
fi
rst
fi
ve governing equations and the rest of the
model is weak. This is especially so when the length of the
simulated device is much smaller than the Debye length or
TABLE II.
Material and simulation parameters.
Parameters
Symbols Values
Units
Electron mobility
μ
n
2
10
5
m
2
V
1
s
1
Hole mobility
μ
p
1
10
5
m
2
V
1
s
1
Energy of the conduction band
E
c
3
:
3eV
Energy of the valence band
E
v
6
:
1eV
Donor level
E
d
3
:
7eV
Acceptor level
E
a
5
:
8eV
Effective density of states for the conduction band
N
c
1
10
24
m
3
Effective density of states for the valence band
N
v
1
10
24
m
3
Donor concentration
N
d
1
10
20
m
3
Acceptor concentration
N
a
0m
3
Polarization gradient coefficient
a
0
9
10
10
Vm
3
C
1
AFD gradient coefficient
b
0
2
10
9
Vm
1
C
Rate of photogeneration
G
1
10
27
m
3
s
1
Radiative recombination coefficient
B
1
10
9
m
3
s
1
Thickness of the film
L
100
nm
Temperature
T
300
K
Work function of Pt
E
Fm
5
:
3eV
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when the dimensionless quantity
δ
is small, which is generally
the case in the simulations in this paper. Therefore, we treat
them as two subproblems, which are then solved self-
consistently until convergence occurs. Each subproblem is
constructed within the
fi
nite difference framework, and the
resulting system of nonlinear equations is solved using the
trust-region dogleg method.
IV. APPLICATION TO BISMUTH FERRITE
A. Material constants
The coef
fi
cients of the Devonshire-Landau energy for
BFO in Eq.
(8)
are presented in
Table I
. They are derived to
match the values of spontaneous polarization, tilt angles, and
FIG. 3.
Spatial variation of
fi
eld quantities (polarization components, OT tilt angles, electric potential, and carrier densities) along the 71
DW device with either continuous
or discontinuous OT at short circuit under light illumination.
TABLE III.
Device models with different types of domain walls. [
]
!
[
]and
h
i
!h
i
denote the directions of electric polarization and oxygen octahedra tilt (OT), respectively, on
two neighboring domains.
J
sc
and
V
oc
are the short-circuit current density and open-circuit
voltage obtained from our device model simulations.
E
DW
refers to the domain wall energy
calculated at thermal equilibrium in the absence of light illumination.
71
domain wall
109
domain wall
Polarization: [111]
!
[11
1]
Polarization: [111]
!
[1
1
1]
(a)
(b)
(c)
(d)
Continuous OT
Discontinuous OT
Continuous OT Discontinuous OT
h
111
i
!h
11
1
ih
111
i
!h
1
11
ih
111
i
!h
1
1
1
ih
111
i
!h
111
i
E
DW
¼
0
:
53 J m
2
E
DW
¼
0
:
63 J m
2
E
DW
¼
0
:
53 J m
2
E
DW
¼
0
:
45 J m
2
J
sc
¼
0
:
22 A m
2
J
sc
¼
0
:
84 A m
2
J
sc
¼
0
:
98 A m
2
J
sc
¼
1
:
0Am
2
V
oc
¼
6
:
8mV
V
oc
¼
29 mV
V
oc
¼
70 mV
V
oc
¼
38 mV
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dielectric constant.
10
,
20
,
21
Other material parameters including
band structure information
22
and carrier mobility values
23
are
listed in
Table II
. The values of
a
0
and
b
0
are chosen to match
a ferroelectric domain wall width of 2 nm. Typically, BFO
exists as a n-type semiconductor due to oxygen vacancies. It
can also become p-type with the Bi de
fi
ciency. Here, we
restrict our simulations to n-type semiconductors.
B. Two-domain ferroelectrics
1. 71
and 109
domain walls
We consider a device comprising a BFO
fi
lm with two
ferroelectric domains separated by either a 71
or 109
domain
wall, with continuous or discontinuous oxygen octahedra
rotations across the DW. This gives a total of four different
cases, as illustrated in
Table III
.
Figures 3
and
4
show the variation of various
fi
eld quanti-
ties when the perovskite
fi
lm is exposed to light illumination
and shorted. Notice that in all cases, the perpendicular
component of the polarization
p
r
is not constant in the vicin-
ity of the domain wall. In other words, the polarization is not
divergence free, and we see a voltage drop across the domain
wall. The polarization pro
fi
le (i.e.,
p
r
)of71
and 109
domain
walls with continuous OT is qualitatively similar to those
obtained from
fi
rst-principles calculations.
9
This voltage drop
across the domain wall leads to charge separation of photo-
generated electron-hole pairs and a non-zero photocurrent.
This is evident in the current-voltage plots shown in
Fig. 5
and is consistent with the mechanism proposed by Yang
et al.
1
Figure 5
shows that the magnitude and direction of the
photocurrent due to the domain wall effect hinge greatly
upon the changes in the crystallographic structure across the
domain wall. The case with 109
DW and continuous OT gives
a positive short-circuit current, which is in the same direction
as net polarization in the
fi
lm, while the rest show negative
currents. Importantly, this DW-based photovoltaic effect
shows that the direction of the current
fl
ow does not
FIG. 4.
Spatial variation of
fi
eld quantities (polarization components, OT tilt angles, electric potential, and carrier densities) along the 109
DW device with either continu-
ous or discontinuous OT at short circuit under light illumination.
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necessarily correlate with the direction of net polarization
consistent with experimental observations.
6
Furthermore, we observe strong coupling between polari-
zation and oxygen octahedra tilt (OT). This is evident from the
vastly different results (including the change in the current
direction) obtained when changing the OT pro
fi
les without
changing the type of domain walls. In actual experiments, we
may only observe one type of oxygen octahedra rotation for
each domain wall type. We compute the domain wall energy
for each case as in
Table III
and
fi
nd that it is energetically
more favorable to have a 71
domain wall with continuous OT
and a 109
domain wall with discontinuous OT. This is consis-
tent with previous
fi
rst-principles calculations.
10
,
12
As the perovskite
fi
lm is
fi
rst exposed to light illumina-
tion, which is simulated in terms of an increase in the photo-
generation rate of electron-hole pairs, the magnitude of the
FIG. 5.
Current-voltage plot in dark (
G
¼
0) and under light illumination (
G
¼
10
27
m
3
s
1
) for BFO devices with a 71
or 109
DW with continuous or
discontinuous OT.
FIG. 6.
Short-circuit current density versus photogeneration rate for a two-
domain ferroelectrics separated by a 71
DW with continuous OT.
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short-circuit current density generated increases rapidly ini-
tially as shown in
Fig. 6
. The increase slows down at higher
illumination (or photogeneration rate) due to recombination
of the excited electrons and holes.
Finally, we consider changing the order of the domains in
a two-domain device.
Figure 7
shows that doing so does not
pose any difference to the pro
fi
les of other quantities such as
p
r
and
f
. This implies that current
fl
ows in a single direction
irrespective of the order of the domains. If we were to stack
the different domains to form a device with the periodic
domain pattern (i.e., alternating domains), the photovolatic
effect would be additive and would not cancel out. This is
exactly what we observe in Sec.
IV C
.
2. 180
domain walls
In the case of the 180
domain walls, with either continu-
ous or discontinuous OTs, there is no visible disturbance to
the polarization component normal to the domain wall at the
domain wall. With a lack of symmetry breaking, the photovol-
taic effect fails to be generated. The
fi
gures are omitted for
brevity.
C. Ferroelectrics with multiple domain walls
We now examine the case with multiple domain walls.
We keep the width of the perovskite
fi
lm constant and uni-
formly place a number of domain walls parallel to the metal
electrodes within the
fi
lm. The distribution of polarization,
oxygen octahedra tilts (OTs), and other
fi
eld quantities for a
shorted device with ten 71
domain walls is presented in
Fig. 8
. Polarization and OTs are periodic and electric potential
varies in a zig-zag manner but with a slope.
Figure 9
shows that the magnitudes of both short-circuit
current and open-circuit voltage increase with the density of
domain walls in the device. The additive effect becomes
FIG. 7.
Spatial variation of
fi
eld quantities along a two-domain device (71
DW, continuous OT) at short circuit. The two cases are identical except the order of the two
domains is reversed.
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smaller at a higher domain wall number as it is in
fl
uenced by
the boundary.
D. Effect of varying doping and width of the
ferroelectric
fi
lm
Next, we investigate the effect of doping and width on
the ferroelectric response using a two-domain example. All
the previous simulations are run using a small donor doping
density of
N
d
¼
10
20
m
3
and a width of 100 nm, which corre-
sponds to the state of complete depletion. Typically, a deple-
tion layer forms at a metal-semiconductor interface and the
width of the depletion layer is related to the Debye length,
which is dependent on the dopant density and dielectric
constant. Complete depletion occurs when the Debye length
of the material is much larger than the width of the device.
Otherwise, there is partial or local depletion.
Figure 10
FIG. 8.
Spatial variation of
fi
eld quantities for ferroelectrics with ten 71
DWs and continuous OTs at short circuit.
FIG. 9.
Current-voltage plot for multi-domain BFO devices with 71
DWs and
continuous OTs.
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