of 11
Two
-
dimen
sional
Halide
Perovskite
s
:
Tuning
Electronic Activit
ies of
D
efects
Yuanyue Liu
,
*
1,2
Hai Xiao
1
and William A. Goddard III
*
1
1
Materials and Process Simulation Center
2
the Resnick Sustainability Institute
California Institute of Technology,
Pasadena, CA 91125, USA
*
Corresponden
ce to
:
yuanyue.liu.microman@gmail
.com
and
wag@wag.caltech.edu
Keywords:
halide perovskites; two
-
dimensional materials
;
defects;
first
-
principle
calculations
Abstract:
Two
-
dimensional (2D)
halide
perovskites
are
emerg
ing
a
s promising candidates for
nano
-
electronics and optoelectronics. To
realize their full potential
, it is important to understand
the role of
those
defects
that
can
strong
ly
impact material properties
.
In contrast to
other popular
2D semiconductors (e.g. transition metal dichalcogenides MX
2
)
for
which defects typically
induce
harmful traps
,
we show that
the electronic activities of defects
in 2D perovskites
are
significantly tunabl
e.
For example,
even with a fixed
lattice
orientation,
one can
change
the
synthesis conditions to convert
a
line defect
(edge
or
grain boundar
y
)
from electron accept
or
to
inactive
site
without
deep
gap states
.
We show that this difference
originate
s
from the
enhanced
ionic bonding
i
n
these
perovskites
compared with MX
2
.
The donors tend to have high formation
energies
,
and
the harmful
defects
are difficult to form at
a low
halide
chemical potential
.
Thus
we
unveil unique properties of defects in 2D perovskites and suggest practical routes to improve
them.
Main text
:
Halide perovskite
s have attracted great interest
due to their low cost and high
efficiency for solar cell applications
1
. Recently two
-
dimensional
(2D)
halide
perovskites
have
been r
ealized
experimentally and
demonstrated
to have
attractive properties
. These materials
have
thicknesses of just one
to
few
unit
-
cell(s)
,
with
an
A
2
BX
4
stoichiometry
(where X = Halides,
B = group
-
14 elements, and A = long
-
chain
organic
molecules
such as C
4
H
9
NH
3
)
in contrast to
ABX
3
for
3D
perovskites
2
-
5
.
The
excellent properties
of 2D
perovskites
combined with
their
ease
of fabrication render them promising for nano
-
device applications. For example, they
exhibit
strong light absorption and photoluminescence at room temperature
2, 5
,
making them interesting
for photovoltaic
s
and
light emitters
6
-
10
. In addition, the
hi
gh
mobility of charge carriers
11
-
15
in
thin film perovskites
render
s them promising can
didates for solution
-
process
ed
field
-
effect
transistors
12, 13, 15
.
To optimize the 2D perovskites, it is important to understand the
impact
of defe
cts
on the
material properties and device performance.
Although
defects in 3D perovskites
16
-
19
and
in other
2D materials (graphene
20, 21
, boron nitride
22, 23
, transition metal dichalcogenides
24
-
26
,
black
phosphorous
27
)
have been studied extensively
,
little is known
about defects in
the emerging 2D
perovskites.
H
ere we
report
first
-
principles
studies
to
a
nswer
such
questions
as
:
what are the
electronic
properties of defects in
2D perovskites
?
how are they differe
nt from other 2D semiconductors (especially transition metal
dichalcogenides, which are also hetero
-
elemental semiconductors)
and 3D perovskites
?
how
can we
control
defects
to
optimize the device performance
?
We
performe
d
Density functional theory (DFT)
using
the Vienna Ab
-
initio Simulation Package
(VASP)
28, 29
with projector augmented wa
ve (PAW) pseudopotentials
30, 31
.
We employed the
Perdew
-
Burke
-
Ernzerhof (PBE) exchange
-
correlation functional
32
in most systems. For
comparison,
we
also calculated
the band gap using
the
HSE functional
33
with spin
-
orbit coupling
(SOC)
.
The
plane
-
wave cut
-
off energy is 400 eV, and
the systems
are fully relaxed until the final
force on each atom becomes less than 0.01 eV/Å.
In order to reduce computational cost
s
, we use
Rb
to represent
the
long
-
chain organic molecules
(A)
.
This is based on the
considerations that the
main role of A in the electronic structures of 3D perovskites is to donate one electron
into the
host
34
. Although Rb has a smaller size than A and hence leads to
a different lattice parameter, it
does
not affect our main conclusions about the defect properties, as explained
below
.
Figure 1
.
Atomic structure (left) of 2D perovskite and charge density distributions (middle and
right) of the band edge states, shown in both top (upper panels) and side (lower panels) views;
blue: Pb; reb:
I
; grey:
Rb
.
The band gap is calculated to be ~ 2.2 eV with
both PBE and
HSE+SOC flavors of DFT.
Fig
.
1 show
s
the atomic structure of 2D Rb
2
PbI
4
. The octahedr
a
are tilted,
along
both in
-
pl
ane
and out
-
of
-
plane directions
.
Using the PBE functional without SOC, we calculate a
band gap
of
2.2
2
eV, which is consistent with
the band gap of
2.2
1
eV
that
we
obtain from the more accurate
HSE + SOC
method
. This suggests that PBE
is acceptable
for
study
ing
defect properties, as
previously
noted
for 3D perovskites
16
-
19
.
T
he spatial distri
butions of the band edge states
show
that
the
valence b
and maximum (VBM)
is mainly composed of
Pb and I
states
,
while
the
conduction band minimum (CBM
)
is dominated by Pb
states,
with
Rb no
t contribut
ing
to the
band edge
s. Th
is
absence of Rb component
s near the band edges
further
valida
tes our choice of
Rb
to
mimic
A for studying
the
defect
electronic
properties
.
These features are similar to
those of
3D
perovskites
34
, indicating a similar electronic origin despite the app
arently different
stoichiometry
.
On the other hand, these band edge compositions
are very different from 2D MX
2
,
whose
VBM and CBM are
both
dominated
by M d states split in a
ligand
field
35
.
The spatial
separation
of VBM and CBM onto
anions and cations
suggests that
the 2
D pero
vskites possess
more ionic
bonding than MX
2
.
Figure 2
.
Edges in 2D perovskite and their electronic structures. ‘
A
indicate
s ‘armchair’
orientation
and
Z
indicates ‘zigzag’. T
he suffix denotes the specific structure:
-
p
indicates that
the edge
creates acceptor levels
, and ‘
-
N
means that the edge is inactive (
‘neutral’
)
. Spin
-
polarized states are shown in different colors in the band structures, and charge density
distributions of the states indicated by arrows are shown in the inset.
(a) shows
the cases of
A
p
and
Z
p
edges, and (b) shows the rest.
See SI for more edge structures.
D
efects in 2D MX
2
(point defects, edges, grain boundaries)
typically create deep
electronic
level
s inside the band gap
24
-
26, 36
, which could trap/scatter/recombine charge carriers
making
them generally
harmful for many (opto)electronic applications.
These deep levels are difficult to
el
iminate
by local structural variations without introducing
additional
chemical species
26, 37
-
39
,
due to the
difficulty in
restoring
the original ligand field
. However
,
for
2D perovskites
, it is
possible to recover the charge transfer
characteristics
of the ionic bonding by
manipulating the
ratio of
cations and anions at the defect sites
, thereby
tuning their electronic levels
.
Indeed, our study of the edges
an important type of line defects
validates
this
speculation
.
Figure 2 shows tw
o representative edge
orientation
s
: armchair
(
A
) and
zigzag (
Z
)
direction
s
.
The
A
edge orientation is along the axis of the primitive cell, and the
Z
is along the diagonal direction.
Each edge orientation can have various structures, denoted by the suffix (e.g.
p
,
N+,
N
-
).
The
A
p
edge, which
ha
s
the
same
coordination of Pb and Rb
as in the lattice
(i.e. f
our I
atoms
close to
Pb, with Rb atoms
up
and down
in the centers of the polygons
)
,
creates
shallow acceptor
levels located along the edge, as shown by the band structure and the charge density distribution
in Fig.
2a. These
edge
states can be
partially
occupied by thermal
ly
ionized electrons from the
lattice
valance band,
generating free
holes in the lattice
(hence denoted as
A
p
)
.
T
he acceptor states
originate from
the
non
-
fully filled
valence bands
created by the
surplus
I
atoms at the
A
p
type
edge
.
This can be understood by counting the
charges
for the
local
stoichiometry
.
The I are distributed in three layers (Fig. 1). In the top and bottom layers
of the
ideal lattice, each I
receives
¼
electron per neighboring
Rb
from four
Rb neighbors
, thus the
charge
s
are balanced. This is different from
the middle layer,
where
each I
receives
1/2 electron
per neighboring Pb from two Pb neighbors,
neutralizing
the middle layer
.
However,
A
t the
A
p
edge,
although the top and bottom layers are charge balanced,
the
outmost I atoms
in the middle
layer
lack
½
charge
per I
due to the missing Pb
(Fig 2a),
which gives rise to the acceptor states.
Although there are other ways to count the charges, they all should
lead to the same conclusion.
T
he above analysis
suggests
that adding one Rb atom
to the
A
p
edge
might
saturate
the tw
o
outmost
I.
Indeed,
our calculations of
band structure and
charge density distribution
(Fig. 2b,
A
N+
, where
N
denotes
neutral
, and + means adding atoms to the previous
A
p
edge
)
show that
the acceptor levels disappear
from
the
band gap,
leading to
the absence of edge states
. Therefore,
this edge is relatively inactive
with regard to
the lattice
electronic properties
.
Alternatively,
remov
ing
the u
n
s
aturated I atoms, also results in an electronically inactive edge
A
N
-
(Fig. 2b
;
-
means removing atoms from the previous
A
p
edge
).
We can
construct
edge
s
with
even more
cations
or
fewer
anions
(see the SI for structures),
that
would
create donor levels
(hence denoted
as
A
n
)
to
generate free electrons in
the
lattice
conduction band
. H
owever, as shown
below
,
we
find that
these edges
are very unstable (
very high formation energies
)
.
Similarly, the
Z
edge
provides
opportunities
,
to stabilize
either electron acceptor
(Fig. 2a,
Z
p
)
or
inactive
(Fig. 2b,
Z
N+
,
Z
N
-
)
states
, depending on
the
stoichiometry
at the edge
. It is
also
unlikely to
be donor
due to the
high formation energies of the
Z
edge
structure
s that
could
create
donor levels.
T
hese
edge
properties are
very
different
from
those of
MX
2
, which
always
exhibit
deep levels
independent
of structural variations
36
, demonstrating the unique electronic
structure
of 2D perovskites
.
These
observations
suggest that
,
even
for fixed
edge direction
s
,
the
electronic
activit
y
can still
be
tuned
by
varying
synthesis conditions
.
Figure 3
.
Gain boundaries in 2D perovskite and their electronic structures. The dashed line in
the left panel shows the periodic length. The charge density distributions of the acceptor levels
are shown in the right panel. See SI for more grain boundary structures.
Grain boundar
ies
provide
another
common type of line defects, usually formed when the edges
of two mis
-
orientated
grains
join
together during growth.
F
igure 3
shows a
n example
grain
boundary
constructed
by connecting two edges with a shared I atom
.
We choose the kinked
edges
that
contain both
A
and
Z
segments to represent a general
case.
B
y varying the number of
Rb atoms, shallow acceptor levels can be created or eliminated
. It is energet
ically unfavorable to
have surplus
cations
(as shown
below
),
so we expect
the
grain boundary
is unlikely to
provide
electron donor
states
.
T
hese
grain boundary properties are
very different from
those of
MX
2
,
which always
render
deep levels regardless of structural variations
24
.
We
find
a
similar
charge
-
balance
-
controlled electronic activity
for point defects in 2D
perovskites
, as shown in Fig.
4
. A
lthough a
Rb vacancy (V
Rb
)
creates
an acceptor level,
a
neighboring
V
I
(hence con
v
erting it to V
RbI
)
could eliminate this gap state.
Similarly
,
V
PbI2
does
not exhibit
deep levels
.
Such
defects have also been found to be
electroni
c
ally inactive in
3D
perovskites
19
.
Most of
point de
fects
have
the
electronic
behavior
expect
ed
for
a
typical ionic
semiconductor. For example,
cation vacancies/anion interstitials usually generate
acceptor
levels
,
while
anion vacancies/cation interstitials
typically create
donor
states
.
These point defect
properties are
very different from those of MX
2
,
I
n
the latter case
, a cation vacancy
(V
M
)
generate
s deep acceptor levels,
while
the stoichiometric vacancies
(V
MX2
)
produce
more gap
states
25
.
Figure 4
. Electronic levels
of point defects in
2D perovskite
.
A
long bar
denote
s two degenerate
states, while a short bar stands for a single state. Spin polarized states are shown in
different
colors, and
the occupied states are marked by arrows.
In order to identify optimal conditions
for
growth
of
material
s
that
suppress harmful defects, we
examine
the
formation energies
(
E
f
)
following the method used for 3D perovskites
16
.
T
hermodynamic equilibrium condition
requires
:
2
μ
Rb
+
μ
Pb
+
4
μ
I
=
μ
Rb2PbI4
(1)
where
μ
is the chemical potential. To avoid phase separation,
the following con
straint
s
must
be
satisfied:
μ
Rb
<
μ
Rb
-
bulk
(2)
μ
Pb
<
μ
Pb
-
bulk
(3)
μ
I
<
μ
I2
/2
(4)
μ
Rb
+
μ
I
<
μ
RbI
(5
)
μ
Pb
+
2
μ
I
<
μ
PbI2
(6
)
Substituting (1) into (2)
and (5
)
,
we
get:
μ
Pb
+ 4
μ
I
>
μ
Rb2PbI4
-
2
μ
Rb
-
bulk
(7)
μ
Pb
+
2
μ
I
>
μ
Rb2PbI4
-
2
μ
RbI
(8
)
where
μ
Rb2PbI4
,
μ
PbI2
,
μ
RbI
,
μ
Rb
-
bulk
,
μ
Pb
-
bulk
and
μ
I2
can be approximated by the internal energies of
the
corresponding condensed phases.
We find that (7)
or (2)
is
automatically
satisfied when (3),
(4), (6) and (8)
are met
. Hence (3), (4), (6) and (8)
together define
a range of (
μ
Pb
,
μ
I
)
where
the
2D
perovskite
is
thermodynamic
ally stable. Since the
E
f
depend
s
linear
ly
on
μ
, the maximum
and
minimum
of
E
f
should fall on the corners of the phase boundaries
.
T
herefore
Fig. 5
shows
E
f
along the
two
boundary
lines
:
μ
Pb
+
2
μ
I
=
μ
PbI2
, and
μ
Pb
+
2
μ
I
=
μ
Rb2PbI4
-
2
μ
RbI
(
or
μ
Rb
+
μ
I
=
μ
RbI
)
.
T
he
thermodynamic equilibrium
concentration of defects (
n
) in the materials
can be estimated by
:
n
~ e
^
(
-
E
f
/k
B
T
) /
S
(9)
where
S
is
the
area of the
primitive cell, and
T
is
temperature.
The experimentally grown 2D
per
o
v
skites
typically
exhibit sizes
less than 10
μ
m, and T is usually below
100
o
C
2
.
Based on (9),
we estimate that defects with
E
f
<
0.62
eV would
likely form
under
the
se
experimental
growth
cond
i
tions.
Therefore
we
use
this value
as a criterion to judge if
E
f
is
high
or
low
.
Although
V
RbI
and V
PbI2
generally
have a low
E
f
(Fig. 5a),
the
y are electronically inactive and hence have
limited
impact o
n the lattice
properties
.
The dominating defects
at high
μ
I
are
those with
surplus
anion
s
or
deficient
cation
s
, such as I
Rb
, I
i
, and I
Pb
(Fig. 5a)
. These defects create deep levels (Fig.
4) and hence are harmful to many applications.
F
ortunately, their
E
f
increase
to a high level
as
μ
I
decreases.
On the other hand, the
E
f
for defects with surplus
cations/deficient anions still remains
high at low
μ
I
. T
herefore
using
synthesis
conditions that
lower
μ
I
, should
reduce
the total
concentration of harmful defects.
Figure
5
.
Formation energies of point defects (a), and line defects (b) in 2D perovskite, as a
function of
I
chemical potential
(with respect to that of I
2
molecule)
along phase boundaries (see
the text). For line defects, edges
along
A
orientation
are shown here as an example, and the rest
can be found in the SI; the energies are referred to that of
A
N+
. Shadowed regions mark the
point defects that would likely form in a 10μm size square sheet grown at 100
o
C in
thermodynamic equilibrium
.
We
find
that
the line defects
exhibit
a similar behavior for
E
f
. Fig. 5b shows the
E
f
for
various
edge
structure
s along
A
orientation
,
with respect to that of
A
N+
(
the
E
f
for edges along
Z
orientation
and grain
boundaries
can be found in the SI).
At high
μ
I
, t
he
edge
that creates
acceptor levels
,
with
surplus
I
(
A
p
)
, possesses
a
n
E
f
comparabl
e with those of inactive edges
(
A
N+ and
A
N
-
)
.
However, it becomes
unfavorable at low
μ
I
. In contrast, t
he
edges
that create
donor
levels with
surplus
cations/deficient anions (
A
n
+
and
A
n
-
, see structures in the SI)
,
exhibit
a high
E
f
in
the whole range of
μ
I
, and therefore are unlikely to form
(
as mentioned
above
).
To check whether the trends of
E
f
can be generalized to other 2D perovskites with different
chemical compositions, we calculate the
E
f
for point defects in (CH
3
NH
3
)
2
SnBr
4
as a test
example. As shown in Fig. S6, we find
again
that a low
μ
Br
can
decrease the total concentration
of harmful d
efects
, therefore confirming the
general
ity of the
trends
.
T
he
behavior
of
E
f
in 2D perovskites is different from that
in 3D
perovskites
,
where
point
defects
with
surplus
cat
ions/
deficient anions
can have a low
E
f
at
low
μ
I
, rendering n
-
dop
ing of
the host
16, 17
. This n
-
doping is unlikely to exist in 2D case, because of the high
E
f
for donors across the
whole range of chemical potential.
Note that the same calculation methods were used to study
the
defects in
3D case, i.e. PBE functional with plane
-
wave basis set
s
,
allowing for direct
comparison.
Fo
r grain
boundaries
in 3D perovskites
,
theoretical
analyses suggested
that the
y
do
not create
deep
levels
, due to the strong coupling between
Pb s orbitals and
I
p orbitals
and the
large atomic size of
Pb
17, 18
.
W
e show
h
ere
that
these previous results ar
o
se
because the
grain
boundary models
chosen
we
re all
neutral
(
charge balanced
)
,
mak
ing
them electronically
inactive
as explained
above
for 2D cases
.
Considering that both donor
-
and acceptor
-
like point defects
can form in 3D perovskites, we anticipate that the grai
n boundaries can also have
surplus
/deficient cations/anions, making them
donors/acceptors
depending on the
μ
I
. This is
different from the
grain boundaries in
our
2D case, which
are
unlikely
to be donors
.
In
addition
,
theoretical analyses suggested that
deep
-
level defects are difficult to from in 3D perovskites and
the dominating defects all have shallow states
16, 17
;
t
his contrast
s
with defects in 2D perovskites,
where
deep
-
level defects (e.g. I
i
, I
Rb
) can form
easily
at high
-
μ
I
conditions.
This study demonstrates that
defects in
ionic semiconductor
s
can be tuned to be less harmful in
general, providing a
guideline to design
new 2D semiconductors.
It also explains the
experimental observation of
relatively
high q
uantum efficiency
in 2D perovskites, and suggests
ways to further improve it.
A common way
to adjust the chemical p
otential is to change the
concentration of reactants
.
For example, recent experiments use PbX
2
and Cs
-
oleate to
synthesize CsPbX3 nanostructures, creating a Pb
-
rich (or I
-
poor) environment
40, 41
. Be
sides the
intrinsic
defects
which are the focus of this work
,
extrinsic defects
would also play an
important
role in the electronic properties
. A major source is the solvent residues adsorbed on the surface.
The ionic contaminant
s could
induce
n
-
or
p
-
doping
, while neutral adsorbates
should have less
impact.
Given that 3D perovskite is not very stable
in the ambient conditions,
one would expect a
similar issue for 2D perovskite.
Particularly, humidity
could
have a strong effect on
the material.
This could be mitigated by using encapsulation techniques
(e.g. using h
-
BN to seal
the
material/device
42
)
, or
choosing hydrophobic organic cation
s
43
.
In summary,
we use
first
-
princ
iples calculations
to
predict
unique properties of
defects in
2D
perovskites
. The
line defects
with fixed orientation
can be tuned from
electron
acceptors
to
inactive
sites
by varying synthesis conditions
,
while
donors
are energetically unfavorable
.
This
is
consistent with the trends of point defects formation.
The
optimal
synthesis conditions
are
also
identified
.
Supporting Information:
Computational details, more line defect structures, energies of
Z
edges and grain boundaries,
energies of point defects in 2D MA
2
SnBr
4
.
Acknowledgements:
YL
thanks discussions with Prof. Wan
-
Jian Yin, and
acknowledges the support from Resnick
Prize Postdoctoral Fellowship at Caltech.
HX
and WAG were
support
ed
by the Joint Center for
Artificial Photosynthesis, a DOE Energy Innovation Hub, supported through the Office of
Scienc
e of the U.S. DOE
under Award No. DE
-
SC0004993.
This research was also supported by
NSF (CBET
-
1512759,
p
rogram manager: Robert McCabe), DOE (DE FOA 0001276, program
manager: James Davenport)
. This work used
computational resources of
National Energ
y
Research Scientific Com
puting Center, a DOE
Office of
Science User Facility supported
by the
Office of Science of the
US DOE under Contract DE
-
AC02
-
05CH11231
, and
the Extreme
Science and Engineering Discovery Environment (XSEDE)
, which is supported by NSF
grant
number ACI
-
105357
5
.
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