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A DFT Study of Mechanical Properties and Stability of Cubic Methyl Ammonium Lead Halide Perovskites (CH3NH3PbX3, X = I, Br, Cl) Mahdi Faghihnasiri, Morteza Izadifard, and Mohammad Ebrahim Ghazi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b07129 • Publication Date (Web): 08 Nov 2017 Downloaded from http://pubs.acs.org on November 9, 2017
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The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
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ABSTRACT: In this study, using the density functional theory, the mechanical properties of
3
methyl ammonium lead halide perovskites (CH3NH3PbX3, X = I, Br, Cl) were investigated.
4
Young’s modulus, bulk modulus, and shear modulus, Poisson’s ratio, and many other parameters
5
were calculated using the PBEsol and vdW approximations. Also, in this work, utilizing a new
6
accuracy in calculating the elastic constants, the intense conflict between the previous theoretical
7
results and the experimental data were fixed. Moreover, for the first time, through combination of
8
the PBEsol and vdW methods, the effect of the interaction between methyl ammonium and PbX3
9
scaffold on the mechanical properties of lead halide perovskites was well cleared. In continuation,
10
using the PBEsol+vdW method, a phase transition appeared for the MAPbBr3 and MAPbCl3
11
structures, which proved more stability of MAPbBr3 and MAPbCl3 in comparison with MAPbI3. In
12
what follows, by studying these materials under an applied strain beyond the harmonic region, the
13
transition zone to the plastic area in the strain region of 5.5% and smaller was identified, and the
14
small values of the aforementioned applied strains were found to be the reason for the instability of
15
these materials at room temperature and above.
16 17 18 19 20 21
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INTRODUCTION
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Hybrid halide perovskites are a group of crystalline materials possessing the ABX3 chemical
3
formula. In these crystal structures, an organic cation such as methyl ammonium (CH3NH3 = MA)
4
is located in the A positions and plays the role of a carrier for positive charges. On the other hand,
5
a dication such as (Pb2+) or (Sn2+) is placed at the B sites, bonding with three X atoms (X = I, Br,
6
Cl), and creates the main crystal framework 1.
7
One of the most promising materials in this group is CH3NH3PbI3, which has been widely studied
8
in the past few years for its use as an absorbance layer in solar cells 2. Above room temperature,
9
more than 327 K, this material has a cubic crystal structure; through decreasing the temperature
10
from 327 K to 162 K, it changes to a tetragonal phase; and with a further decrease in temperature
11
under 162 K, it undergoes another transition to the orthorhombic phase
12
power-conversion efficiency (PCE) of these new generation of solar cells have grown up
13
significantly from 3% to above 22.1% 7.
14
Electron-hole diffusion length as long as 1 micron 8-9, low effective mass 10, high charge mobility
15
11
, high optical absorbance coefficient, narrow optical absorbance edge, large dielectric constant 12-
16
13
, proper optical band gap in the visible range
17
features that make MAPbI3 one of the most promising materials to use as an absorbance layer in
18
the solar cells 16-19.
19
The current challenge in the development of hybrid perovskite solar cells is its low stability, so that
20
great efforts have been made to overcome this issue using a protective layer on the surface of the
21
absorbance layer to promote its stability against the external factors like humidity 20-21. Different
22
mechanical properties between protective and absorbance layers could cause the interface to be
14
3-6
. In recent years, the
, and strong dipole moment
15
are some of the
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subjected to the external stress (strain), and consequently, apply a corruptive influence on the
2
material properties, absorb process, and its efficiency
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knowledge about the mechanical properties of MAPbI3 and their responses to the external stresses
4
should be a great deal in the fabrication and application of these layers 23.
5
To date, there have been scant studies on the mechanical properties of these hybrid perovskite
6
structures. However, there is not a good compliance in their computed results with the experimental
7
studies 23-27. Therefore, it is necessary to continue our efforts to achieve the accurate elastic constant
8
for MAPbX3 structures, understand their behavior, and know which variables would affect their
9
elastic response and how these variables would affect those constants.
22
. Therefore, having a comprehensive
10
In this study, the mechanical properties of MAPbX3 (X = I, Br, Cl) were studied. Some mechanical
11
parameters such as the Young’s, bulk and shear moduli were calculated by fitting the higher order
12
polynomial functions with the energy-strain curve using the second-order linear elastic constants
13
(SOECs) 28-30. In order to study the elastic constants of the system, we used Lagrangian deformation
14
to apply strain to the crystalline structure. The total energy of the systems after structure relaxation
15
was calculated with good accuracy using the density functional theory (DFT) at any applied strain.
16
Structures including MAPbI3, MAPbBr3 and MAPbCl, undergo a phase transition from
17
orthorhombic to tetragonal phase around 162, 149 and 172 K and also from tetragonal to cubic
18
phase around 327, 236.9 and 178 K, respectively
19
about using these materials in solar cells as an absorber layer is their stability at the working
20
temperature. From the viewpoint of structural stability, all calculations were done for the cubic
21
phases. Then a comparison was made between the results obtained and the reported experimental
22
results.
23, 31-32
. One of the most important challenges
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METHODS
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Theory. Continuum mechanics is a branch of mechanics that investigates the behavior and
3
properties of the materials. Although in this field it is assumed that considered structure is
4
homogeneous and anisotropic elastic medium, works have shown that the results of the continuum
5
theories are accurate enough in order to investigate the mechanical properties of materials in atomic
6
scales that include cavities and vacant spaces. Based upon the non-linear elastic continuum
7
mechanics, calculating the elastic constants is possible through using the energy and stress
8
approaches. In the energy approach, which has been explained in details by Stadler 33, the variation
9
in total energy versus applied strain is used, while, in the stress approach, which is based upon the 34
10
Hook’s law and has been suggested by Nielsen
11
investigated. As it has been explained in details about the cubic structures by Wang 35, the applied
12
strain, which is defined by the deformation tensor, is expressed by Eq. (1)
𝐽𝐽𝑖𝑖𝑖𝑖 =
13
𝜕𝜕𝑥𝑥𝑖𝑖 𝜕𝜕𝑎𝑎𝑗𝑗
, the stress changes with applied strain is
(1)
14
Based upon this equation, the deformation tensor represents the first derivation of the configuration
15
of a structure point in the system after deformation to the initial configuration at the equilibrium
16
state, where i and j (= 1, 2, 3) represent the Cartesian coordinates.
17
Using the Lagrangian tensor, deformation of the structure is defined as follows: 1 𝜂𝜂 = 𝜖𝜖 + 𝜖𝜖 2
18 19 20
(2)
where 𝜼𝜼 and 𝝐𝝐 are the Lagrangian and physical strains, respectively. Since the relation between
𝑱𝑱 and 𝝐𝝐 is (𝑱𝑱 = 𝟏𝟏 + 𝝐𝝐 ), thus:
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𝜂𝜂 =
1 2
1 𝑇𝑇 (𝐽𝐽 𝐽𝐽 − 𝐼𝐼) (3) 2
Based on Eqs. (1) - (3), Eq. (4) is as follows:
1 𝐽𝐽𝑖𝑖𝑖𝑖 = 𝛿𝛿𝑖𝑖𝑖𝑖 + 𝜂𝜂𝑖𝑖𝑖𝑖 − � 𝜂𝜂𝑘𝑘𝑘𝑘 𝜂𝜂𝑘𝑘𝑘𝑘 + ⋯ 2
3
𝑘𝑘
(4)
4
Through applying the deformation tensor to each one of the three crystal lattice vectors, the
5
deformed structure is modeled using 𝑹𝑹𝒊𝒊 = 𝑱𝑱𝒊𝒊𝒊𝒊 𝒓𝒓𝒋𝒋 . In addition, the relation between stress tensor (𝝉𝝉),
6
physical stress (𝝈𝝈), and physical strain (𝝐𝝐) can be written as:
𝜏𝜏 = det(1 + 𝜖𝜖) (1 + 𝜖𝜖)−1 𝜎𝜎(1 + 𝜖𝜖)−1
7
(5)
8
Considering Eq. (5) and the Hook’s law 36, elements of the Lagrangian stress tensor are expanded
9
based on the elements of the Lagrangian strain, as below: 3
𝜏𝜏𝑖𝑖𝑖𝑖 = � 𝐶𝐶𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝜂𝜂𝑘𝑘𝑘𝑘
10
(6)
𝑘𝑘.𝑙𝑙=1
12
In Eq. (6), the 𝐶𝐶 coefficients are the elastic stiffness constants of the crystal structure. Eq. (7)
13
parameters:
11
demonstrates the relation between the total energy of the crystal and the Lagrangian strain
3
𝐸𝐸(𝜂𝜂) = 𝐸𝐸(0) + 𝑉𝑉0 �
14
𝑖𝑖.𝑗𝑗=1
(0) 𝜏𝜏𝑖𝑖𝑖𝑖 𝜂𝜂𝑖𝑖𝑖𝑖
3
𝑉𝑉0 + � 𝐶𝐶𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝜂𝜂𝑖𝑖𝑗𝑗 𝜂𝜂𝑘𝑘𝑘𝑘 2! 𝑖𝑖.𝑗𝑗.𝑘𝑘.𝑙𝑙=1
(7)
16
where 𝐸𝐸(0) and 𝑉𝑉0 depict the energy and volume of the crystal unit cell under the equilibrium state
17
In order to simplify the indexing, the Voigt notation is used, in which each pair of ij index is
18
replaced by index I, as follows:
15
(without strain).
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�
1 2 3 4 5
𝑖𝑖𝑖𝑖 𝐼𝐼
11 1
8 9
22 2
33 3
13 23� 33
23 4
13 5
12 6
Based upon this indexing, Eqs. (6) and (7) are converted to Eqs. (8) and (9): 3
𝜏𝜏𝐼𝐼 = � 𝐶𝐶𝐼𝐼𝐼𝐼 𝜂𝜂𝐽𝐽
6
7
11 12 22
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𝐸𝐸(𝜂𝜂𝐼𝐼 ) = 𝐸𝐸(0) +
3
𝐽𝐽=1
(0) 𝑉𝑉0 � 𝜏𝜏𝐼𝐼 𝜂𝜂𝐼𝐼 𝐼𝐼=1
3
(8)
𝑉𝑉0 + � 𝐶𝐶𝐼𝐼𝐼𝐼 𝜂𝜂𝐼𝐼 𝜂𝜂𝐽𝐽 2! 𝐼𝐼.𝐽𝐽=1
(9)
(0)
When the reference structure is under the free stress state (equilibrium state), the values for 𝜏𝜏𝛼𝛼 are zero and could be neglected. Based on these equations, the elastic constants (𝐶𝐶𝐼𝐼𝐼𝐼 ) are calculated
10
using the derivation in Eqs. (8) and (9).
11
𝐶𝐶𝐼𝐼𝐼𝐼 = 𝐶𝐶𝐼𝐼𝐼𝐼 =
12
𝜕𝜕𝜏𝜏𝐼𝐼 � ; SOEC 𝜕𝜕𝜂𝜂𝐽𝐽 𝜂𝜂=0
1 𝜕𝜕 2 𝐸𝐸 � ; SOEC 𝑉𝑉0 𝜕𝜕𝜂𝜂𝐼𝐼 𝜕𝜕𝜂𝜂𝐽𝐽 𝜂𝜂=0
(10) (11)
14
In Eqs. (10) and (11), 𝜂𝜂𝐼𝐼 and 𝜂𝜂𝐽𝐽 are equal to each other since the figures of energy or stress should
15
Since all the structures studied in this paper are cubic, considering the crystal symmetry of cubic
16
structures, there are three independent second-order elastic constant terms (SOECs)
17 18
(𝐶𝐶11 . 𝐶𝐶12 . 𝐶𝐶44 ) 35, 37. Thus in order to extract these parameters, some deformation tensors were
19
Based on the existing crystal symmetries, Jamal et al.
20
tensors that lead to decrease in the computational error in order to calculate the bulk modulus. In
13
be extracted versus a unique 𝜂𝜂.
defined. Using the deformation tensors and Eqs. (8) and (9), the elastic constants were calculated. 37
have suggested appropriate deformation
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addition, in this study, the tensor of the Lagrangian conversion correction was added to the
2
deformation tensor, which was introduced by Eq. (4) to calculate the elements of the deformation
3
tensor. Thus the Lagrangian tensors were defined as follow:
4
Dortho
η 0 0 −η =� 0 0
0 0
η2
1−η2
�, Dcubic
0 η η 0 0 η 0 = �0 η 0�, Dmonoc = � 0 0 η 0 0
0 0
η2
1−η2
�
5
In order to compute SOECs based on the energy approach, the total energy of the optimized
6
structure was calculated per each strain using DFT, and then through fitting the suitable equations
7
with the energy-strain curve, all SOECs were extracted.
8 9
Computational Details. In these calculations, the cubic unit cell including 12 atoms for the
10
MAPbX3 (X = I, Br, Cl) structures was considered. The total energy of all the studied systems,
11
forces applied on each atom, the stress and stress-strain relation in each deformed structure were
12
computed using DFT. All the DFT calculations were performed through the open source Quantum
13
ESPRESSO input code 38. For more accuracy, all-electron-like projector augmented wave (PAW)
14
pseudo-potentials
15
correlation functional were used. Also the non-local correlation functional for van der Waals
16
interactions (vdW) and vdW-DF2-B86R functional were used for description of the wide range
17
interactions 41.
18
Calculations were performed with a cut-off energy of 45 Ry, while the integration of the Brillouin
19
zone was performed with 888 k-points using the standard Monkhorst-Pack
20
Geometry optimizations were performed by employing the Broyden-Fletcher-Goldfarb-Shanno
39
with Perdew−Burke−Ernzerh of revised for solids (PBEsol)
42
40
exchange
special grids.
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(BFGS) algorithm (for stress minimization), and the total forces acting on each ion were minimized
2
to reach less than 110-5 Ry/bohr by movement of the ionic positions.
3
In the MAPbX3 structures, the CH3NH3 (MA) organic cation is suspended in scaffolding of the
4
PbX3 structure, and can have van der Waals interactions (vdW) with this scaffolding. Since the
5
vdW approximation is much more appropriate for long-range interactions considering the terms in
6
its equations, and is not accurate enough to investigate the structural properties
7
prepare logical results about the mechanical properties. In addition, regarding that the PBEsol
8
approximation considers the gradient of charge density variations in its equation and is sensitive to
9
those variations, it may lead to some errors in order to calculate the long-range interactions of
10
materials that have large gaps between different parts of their structures. In order to rectify these
11
defects that are solely in each of the vdW and PBEsol approximations, a new idea that is a
12
combination of these two approximations was used to investigate the mechanical properties of the
13
hybrid structures. In order to do so, in addition to the calculations based on the vdW and PBEsol
14
approximations separately, a simple model named PBEsol+vdW was used. In this model, the part
15
of the energy related to interaction between Pb and three X atoms in the PbX3 structure and also the
16
one related to the interaction between C, N, and six H atoms in the CH3NH3 structure was calculated
17
separately using the PBEsol approximation. Moreover, the part of energy related to the interaction
18
between the CH3NH3 organic cation and the PbX3 scaffold were calculated by vdW approximation
19
separately, and finally, they were added to each other as follows:
20
CH NH PbX
CH NH
PbX
CH NH3 −PbX3
3 3 3 3 3 3 3 EPBEsol+vdW = EPBEsol + EPBEsol + EvdW
CH NH PbX
CH NH
PbX
CH NH3 −PbX3
43
, it would not
(12)
22
3 3 3 3 3 3 3 In the above formula, EPBEsol+vdW , EPBEsol , EPBEsol , and EvdW
23
energy of the PbX3 scaffold based on PBEsol, and the interaction energy between CH3NH3 cation
21
express the total energy
of the system based on PBEsol+vdW, total energy of the CH3NH3 cation based on PBEsol, total
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and PbX3 scaffold based on vdW, respectively. In order to compute these four terms, first of all,
2
each one of the structures was optimized through PBEsol approximation. Then using each one of
3
the PBEsol and vdW models, the total energy of the optimized structure was computed
4
3 3 (EPBEsol
5 6 7 8
CH NH PbX3
CH3 NH3 PbX3 . EvdW ). Subsequently, through separating the CH3NH3 cation and PbX3
scaffold in two different input codes and calculating the total energy of each one of them CH NH
PbX
CH NH3
3 3 3 independently (EPBEsol . EPBEsol . EvdW3
PbX3 . EvdW ) and using the following equations, the CH NH −PbX3
3 3 interaction energy between CH3NH3 cation and PbX3 scaffold (EPBEsol
gained.
CH NH −PbX3
3 3 EPBEsol
9
CH NH3 −PbX3
3 EvdW
10
CH NH PbX3
3 3 = EPBEsol
CH NH3 PbX3
3 = EvdW
CH NH
PbX
3 3 3 − �EPBEsol + EPBEsol �
CH NH3
3 − �EvdW
PbX
+ EvdW3 �
CH3 NH3 −PbX3 . EvdW ) was
(13) (14)
11
Finally, through substitution of the interaction energy between CH3NH3 cation and PbX3 scaffold
12 13
3 3 using vdW approximation in Eq. (14) with EPBEsol
14
model, all the calculations for each one of the three structures were performed corresponding to all
15
mechanical strains, and the energy-strain diagram was used to analyze the mechanical properties.
16
Egger
17
investigating the structural properties of organic−inorganic halide perovskites by the
18
Tkatchenko−Scheffler (TS) pairwise dispersion 45.
CH NH −PbX3
in Eq. (13), the total energy of the system
was gained. In order to compute the mechanical properties of the material using the PBEsol+vdW
44
has earlier made such an attempt to integrate approximation of PBE and vdW for
19 20 21
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RESULTS AND DISCUSSION
2
The unit cell parameters for the cubic phase of the MAPbX3 (X = I, Br, Cl) crystal structures after
3
optimization by all-electron-like projector augmented wave (PAW) pseudo-potentials with
4
Perdew−Burke−Ernzerh of revised for solids (PBEsol) were: a = 6.28, b = 6.22, c = 6.37 (X = I), a
5
= 5.92, b = 5.92, c = 5.92 (X = Br), and a = 5.68, b = 5.68, c = 5.68 (X = Cl) angstrom, respectively,
6
which are in good agreement with the experimental and theoretical data 3, 5, 27, 44, 46-50.
7
In addition, results given by vdW approximation have been also added to the Table 1. As it can be
8
seen, in MAPbI3 and MAPbBr3, the value of unit cell vector based on the vdW is about 1% more
9
than the one based on the PBEsol and for MAPbCl3, the value of unit cell parameter gained using
10
vdW is around 0.5% more than the one based on the PBEsol approximation. It is noteworthy to
11
mention that in present work, vdW-DF2-B86R functional known as rev-vdW-DF2 has been used.
12
The results of vdW-DF2-B86R are much closer to experimental data and have lower overestimate
13
in comparison to other vdW approximations regarding the revised B86b exchange functional
14
(B86R) in order to be used with the nonlocal correlation functional of vdW-DF2. Also, as Egger et
15
al.44 mentioned before, PBE approximation because of weaker energy-volume dependence leads to
16
overestimation in calculating the structure geometry and has poor agreement with experimental
17
results. Considering that the results of computed geometry using vdW-DF2-B86R is in good
18
agreement with experimental results, so utilizing this approximation in order to calculate the
19
mechanical properties of MAPbX3 compounds could be important as it matches the experimental
20
results and PBEsol approximation well from the point of geometry and it also could consider the
21
long range interactions of MA and the scaffold of PbX3. So, in present work, we compared the
22
results of this approximation with PBEsol.
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The results obtained are reported in Table 1, and the schematic representation of the crystal
2
structures of MAPbX3 (X = I, Br, Cl) are shown in Figure 1.
3 4
Table 1. Calculated structural parameters (a, b, and c) of cubic lead−halide perovskites compared to experimental and theoretical data. Sample 5
Exp. Exp. 3 Exp. 46 PBE 44 PBE 47 PBE 48 vdW (DFT-D3) 49 PBEsol 50 PBEsol * vdW * Exp. 5 Exp. 27 PBE 44 PBE 48 PBEsol * vdW * Exp. 5 PBE 44 PBE 48 PBEsol * vdW *
MAPbI3
MAPbBr3
MAPbCl3
a (Å) 6.33 6.27 6.311 6.49 6.54 6.45 6.32 6.29 6.28 6.34 5.90 5.91 6.08 6.08 5.92 5.97 5.68 5.81 5.81 5.68 5.70
b (Å) 6.33
c (Å) 6.33
6.311 6.49
6.316 6.50
6.48
6.45
6.23 6.22 6.28 5.90
6.37 6.37 6.43 5.90
5.92 5.97 5.68
5.92 5.97 5.68
5.68 5.70
5.68 5.70
* present work 5
6
Figure 1. Schematic representation of crystal structures of MAPbX3 (X = I, Br, Cl).
7
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Page 12 of 36
1
Elastic Constants and Mechanical Properties. In order to calculate the elastic constants, the
2
energy approach was used in this study. Through applying the Lagrangian strain between -3% and
3
+3% called harmonic region, and therefore, this range is an appropriate one to calculate the elastic
4
coefficients. Mechanical properties like the Young’s, bulk and shear moduli, Poisson’s ratio, sound
5
velocity, and Debye temperature could be gained by fitting the second-order equation with the
6
variation of strain energy versus Lagrangian strain (see Theory section). Our results and other
7
theoretical ones (for comparison) are reported in Table 2.
8
Table 2. SOECs of MAPbX3, X = I, Br, Cl (all units are in GPa). C11 C12 C44
PBEsol 30.9 7.9 3.2
vdW 34.3 9.7 3.7
PBEsol+vdW 35.4 10.0 6.1
PBE 26 27.1 11.1 9.2
C11
34.9
48.6
41.8
47.2
C12
6.0
9.8
6.8
10.3
C44
3.2
2.5
3.7
8.1
C11 C12 C44
39.5 5.8 2.9
46.3 10.0 4.2
45.9 5.2 2.0
-
Sample MAPbI3
MAPbBr3
MAPbCl3
Exp. Repots ~32.2 25 ~35.9 27 ~9.0 25 ~11.2 27 ~3.4 25 ~ 3.9 27 -
9 10
In order to make sure about the accuracy of the results, all the elastic constants were checked with
11
the Born-Huang elastic stability criteria 28, which are as follow for the cubic systems:
12
𝐶𝐶11 > 0 ,
𝐶𝐶11 − 𝐶𝐶12 > 0 ,
𝐶𝐶11 + 2𝐶𝐶12 > 0 ,
𝐶𝐶44 > 0
(15)
13
As the results show, most of the SOECs based on the PBEsol approximation are smaller than the
14
corresponding coefficients based on the vdW and PBE 26 approximations. The results of parameters
15
C11 and C44 for MAPbBr3 that were calculated by PBEsol (C11 = 34.9, C44 = 3.2 GPa) and
16
PBEsol+vdW (C11 = 41.8, C44 = 3.7 GPa) are in very good agreement with the experimental data
17
(C11 = 32.2, C44 = 3.4 GPa), and (C11 = 35.9, C44 = 3.9 GPa) that were obtained by the laser ACS Paragon Plus Environment
13
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The Journal of Physical Chemistry
25
ultrasonic technique
2
respectively.
3
Young’s, bulk and shear moduli of the system were calculated from the elastic constants.
4
Considering the Voigt approach 51, the bulk and shear moduli are as follow:
5 6 7
8 9 10 11 12 13
G=
, and by combining coherent neutron, Raman, and Brillouin scattering
27
1
1 B = [(C11 + C22 + C33 ) + 2(C12 + C13 + C23 )] 9
(16)
1 [(C + C22 + C33 ) − (C12 + C13 + C23 ) + 3(C44 + C55 + C66 )] 15 11
,
(17)
For the cubic structure, considering the crystal symmetries, Eqs. (16) and (17) are expressed as: 1 B = [C11 + 2C12 ] 3
1 G = [C11 − C12 + 3C44 ] 5
(18)
(19)
Using the bulk modulus and shear modulus, the Young’s modulus (E) and Poisson ration (ν) were calculated as follow: E=
ν=
9BG 3B + G
3B − 2G 2(3B + G)
(20)
(21)
14
In Table 3, the values for the Young’s modulus, bulk modulus, and shear modulus and Poisson’s
15
ratio of the studied structures were reported. Other theoretical and experimental reports were also
16
depicted in this table. As one can see, the acquired results based on the PBEsol approximation are
17
in good agreement with the experimental values. On the other hand, the results obtained based on
18
the vdW correlations are not in appropriate agreement with the experimental results. In addition,
19
the behavior and sequence of Young’s modulus, bulk modulus, and shear modulus of all three
20
samples were the same (𝐸𝐸𝐼𝐼 < 𝐸𝐸𝐵𝐵𝐵𝐵 < 𝐸𝐸𝐶𝐶𝐶𝐶 , 𝐵𝐵𝐼𝐼 < 𝐵𝐵𝐵𝐵𝐵𝐵 < 𝐵𝐵𝐶𝐶𝐶𝐶 , 𝐺𝐺𝐼𝐼 < 𝐺𝐺𝐵𝐵𝐵𝐵 < 𝐺𝐺𝐶𝐶𝐶𝐶 ), although the bulk
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Page 14 of 36
1
modulus of MAPbBr3 (22.7 GPa), calculated by vdW, was slightly larger than MAPbCl3 (22.1
2
GPa).
3 4 5
Table 3. Values for Young’s modulus, bulk modulus, and shear modulus (in GPa), Poisson ratio, and B/G of studied structures. E BV GV ν B/G
PBEsol 17.2 15.6 6.5 0.31 2.48
vdW 19.0 17.9 7.1 0.32 2.48
PBEsol+vdW 22.8 18.5 8.7 0.22 2.10
PBE 26 22.2 16.4 8.7 0.28 1.88
E
20.3
24.5
23.8
29.1
BV
15.9
22.7
18.5
22.6
GV ν B/G E BV GV ν B/G
7.8 0.28 2.03 21.9 17.0 8.5 0.28 2.01
9.3 0.32 2.45 25.6 22.1 9.8 0.30 2.25
9.2 0.28 2.00 24.1 18.8 9.3 0.28 1.99
10.4 0.29 2.17 -
Sample
MAPbI3
MAPbBr3
MAPbCl3
Exp. [100] 17.7-15.6 23 19.6 24 28.3 25 15.6 24 16.8 25 7.6 24 0.22, 0.37 25 2.05 24 19.8-17.4 23 -
6 7
There are some factors that affect the mechanical properties of these materials including Pb-X bond
8
strength, relative density, structural density, and van der Waals interactions of MA with PbX3
9
scaffold. Considering the bulk modulus, applied strain to the material is three dimensional and
10
volumetric, and X atoms around Pb are subjected to the strain in all directions, having the strength
11
of Pb-X bonds can help a better understanding of the bulk modulus. Based on the enthalpy of Pb-
12
X bonds 52, given in Table 4, it is expected that due to the stronger Pb-Cl bonds in MAPbCl3, it has
13
greater bulk modulus in comparison to those of the two other materials. From the PBEsol
14
approximation, it is equal to 17 GPa for MAPbCl3, which is greater than those for MAPbBr3 and
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1
MAPbI3. In addition, based on the values reported in Table 4, Pb-Cl bonds are 64 KJ/mol stronger
2
than Pb-Br bonds but Pb-Br bonds are 42 KJ/mol stronger than Pb-I bonds, which clarify the 1.1
3
GPa difference in bulk modulus of MAPbCl3 and MAPbBr3, and 0.3 GPa in MAPbBr3 and MAPbI3,
4
respectively. This behavior in vdW method is not established just for MAPbBr3 but for MAPbCl3
5
and MAPbI3, it is well confirmed. However, the bulk modulus calculated by the vdW method is
6
slightly worse than the PBE result found in ref. 26.
7
The bond strength is similarly effective in calculating the Young’s modulus. It should be noted that
8
in order to study the Young’s modulus, the applied strain to the structure should be uniaxial. In
9
PbX3 scaffold, only one of the Pb-X bonds are subjected to strain and the other two bonds tolerate
10
no forces. Thus regarding this issue, to investigate the role of bond enthalpy in Young’s modulus,
11
the third column of Table 4 should be used (Pb(X)2-X bond enthalpy) 52 and represents the bond-
12
breaking energy of a Pb-X bond under the situation that the other two bonds are free of any changes.
13
Similar to the bulk modulus, changing the procedure of Young’s modulus reduces with increase in
14
the radii of Cl, Br, and I atoms. As a result, considering the PBEsol approximation, MAPbCl3 has
15
the most value of Young’s modulus, equal to 21.9 GPa, and then MAPbBr3 and MAPbI3 have the
16
Young’s modulus equal to 20.3 and 17.2 GPa, respectively. According to the reported magnitude
17
of the enthalpy of Pb(X)2-X, the small difference between the enthalpies of Pb(Cl)2-Cl and Pb(Br)2-
18
Br, equal to 11 KJ/mol, yields to a little difference of 1.6 GPa in Young’s modulus of MAPbCl3
19
and MAPbBr3, and subsequently, the enthalpy difference of 37 KJ/mol between Pb(Br)2-Br and
20
Pb(I)2-I yields to a 3.1 GPa difference in Young’s modulus of MAPbBr3 and MAPbI3. This trend
21
in vdW approximation has also been preserved and well confirmed but the exact amount of Young’s
22
modulus calculated using this approximation does not match the experimental results. However, it
23
has fewer errors in comparison to the previous theoretical reports 26.
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Table 4. Bond dissociation energy (BDE) or enthalpy of Pb(X)y-X (X = I, Br, Cl) (in KJ/mol) 52. Sample Pb(I)y-I
BDE (y=0) 194±38
BDE (y=2) 75±80
Pb(Br)y-Br
236±42
112±80
Pb(Cl)y-Cl
301±50
123±84
2 3
Shear modulus also has a similar decreasing trend with increase in the radius of halide atoms (R Cl
1.75)
20
indicates a tendency for ductility; otherwise, the material behaves in a brittle manner. For example,
21
the B/G ratio for the kesterite-type copper zinc tin sulfide (CZTS) is 2.28, which indicates that the
22
kesterite-type CZTS is more prone to ductility
23
MAPbCl3 based on the PBEsol approximation are 2.48, 2.03, and 2.01, respectively (Table 3),
58
57
to predict a
. The B/G ratios for MAPbI3, MAPbBr3, and
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1
which show the high level of ductility of these compounds. Besides, it proves that despite the high
2
Young’s modulus of MAPbCl3 in comparison to the two other compounds, it has a lower ductility,
3
and between these three compounds, MAPbI3 represents the highest ductility ratio due to its lowest
4
shear modulus (6.5 GPa). CZTS with a B/G ratio of 2.28 58 has lower ductility ratio compared to
5
the MAPbI3 compound. In addition, the trend of changes in B/G by the vdW and PBEsol+vdW
6
approximations are the same as PBEsol, and MAPbI3 has the most and MAPbCl3 has the least
7
values. Moreover, the amount of B/G for MAPbBr3 and MAPbCl3, which were calculated by the
8
PBEsol+vdW approximation, are 2% lower than the PBEsol values, except for MAPbI3 which is
9
16% lower than the PBEsol approximation.
10
As it was mentioned earlier, the interaction of MA with PbI3 scaffolds using vdW approximation is
11
greater than the PBEsol method. This stronger interaction reduces the tilting probability of PbI3
12
octahedral and leads to the increase in the shear modulus of the material. Increase in the shear
13
modulus, and subsequently, decrease in B/G represent the reduction in ductility of the material.
14
Since the difference value of B/G for the PBEsol and PBEsol+vdW approximations is more
15
significant in MAPbI3, the ductility of MAPbI3 is intensely dependent on the interaction between
16
MA and PbI3 scaffold.
17
In addition to the information reported in Table 3, other parameters including sound velocity and
18
Debye temperature represent the strength of the average chemical bonds and so on, which were
19
calculated using SOECs. The Debye temperature has a close relation with the elastic constants,
20
specific heat, and melting point. At low temperatures, the Debye temperature is computed utilizing
21
the elastic constants. One of the common methods used for calculating the Debye temperature is
22
based on estimation of the average sound velocity (𝜈𝜈𝑚𝑚 ) in the specific material, which is calculated
23
as follows:
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1 2 3 4 5
θ𝐷𝐷 =
(22)
where h, k, NA, n, M, and 𝜌𝜌 represent the Planck’s constant, Boltzmann’s constant, Avogadro’s number, atoms in the organic cation, molecular mass, and density, respectively. The average sound
velocity (𝜈𝜈𝑚𝑚 ) can be calculated from νt and νl , which are the shear and compressional velocities, respectively. The relation between 𝜈𝜈𝑚𝑚 versuse 𝜈𝜈𝑡𝑡 and 𝜈𝜈𝑙𝑙 is as follows: ν𝑡𝑡 = �G/ρ
6
(23)
4 ν𝑙𝑙 = �(B + G)/ρ 3
7
8
1
h 3n NA ρ 3 � � �� νm k 4π M
Page 20 of 36
1 2 1 ν𝑚𝑚 = � � 3 + 3 �� 3 νl νt
−
1 3
(24) (25)
9
The Debye temperature of these perovskite compounds ranged from 180 to 281 K by different
10
approximations in Table 5. The values are much lower than most of the inorganic compounds and
11
close to the stronger organic compounds. The calculation of these parameters will be used in
12
analyses related to charge carriers scattering in hybrid halide perovskites by phonons. The similar
13
argument applies to sound velocity. In sum, the elastic properties are dependent on the building
14
elements of the B2+ and X− ions 26.
15 16 17 18 19 20
Table 5. Values for Debye temperature, average sound velocity, and shear and compressional velocities of studied structures.
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The Journal of Physical Chemistry
sample MAPbI3
MAPbBr3
MAPbCl3
θ𝐷𝐷 (K) νt (m/s) νl (m/s) νm (m/s) θ𝐷𝐷 (K) νt (m/s) νl (m/s) νm (m/s) θ𝐷𝐷 (K) νt (m/s) νl (m/s) νm (m/s)
PBEsol 180 1257 2425 1670 216 1434 2626 1883 260 1651 3014 2166
vdW 190 1320 2581 1759 238 1556 3029 2072 281 1768 3352 2341
PBEsol+vdW 208 1459 2705 1922 234 1553 2836 2038 241 1541 2759 2012
PBE 26 175 1455 2612 1620 219 1699 3099 1894 -
1 2
Crystal Phase Transition. Considering the fact that the mechanical properties resulting from the
3
PBEsol approximation were in very good agreement with the experimental results, through
4
investigating the energy-strain figures, the crystal phase transition based on this approximation was
5
studied. Figure 3 shows the changes in energy with strain of the perovskite based on the three
6
deformation tensors including Dcubic, Dortho, and Dmono. In the case of Dortho, for the MAPbBr3 and
7
MAPbCl3, the crystal phase transition is perceived in 4% and 3.5% tensional strain, respectively.
8
While, for the MAPbI3, no phase transition could be seen. Table 6 demonstrates the primary and
9
secondary strain percentages for the compound including I, Br, and Cl.
10 11 12 13 14
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Page 22 of 36
b) Dortho
a) Dcubic
MAPbI3
MAPbBr3
MAPbCl3 1 2
Figure 3. Total energy of MAPbX3 (X= I, Br, Cl) structures as a function of strain under the influence of two types of deformation tensor, a) Dcubic and b) Dortho, within PBEsol approximation.
3 4
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1 2
The Journal of Physical Chemistry
Table 6. Phase transition point of MAPbX3 (X = I, Br, Cl) under tensional and compressive strains within PBEsol and vdW approximations.
MAPbI3
Strain of phase transition (PBEsol) -
MAPbBr3
ortho, 4%
MAPbCl3
ortho, 3.5%
Sample
Strain of phase transition (PBEsol+vdW) cubic, -1% ortho, 4% cubic, -0.5 % ortho, 3.5%
3 4
In addition to energy-strain variations based on PBEsol, the energy change versus strain was studied
5
based on the PBEsol+vdW approximation, which is in good agreement with the PBEsol results. As
6
expected, Dortho for the compound containing Br depicts the phase transition in 4% tensional strain,
7
and for the compound containing Cl shows a phase transition occur between states separated by
8
small differences in energy in 3.5% tensional strain. Furthermore, Phase transition has happened
9
for both PBEsol and PBEsol+vdW approximations under the applied compressive strain based on
10
cubic deformation (Dcubic) for MAPbBr3 and MAPbCl3. This phase transition is not clear in energy-
11
strain diagram related to the PBEsol, but in PBEsol+vdW, energy change on transition point has
12
been exaggerated and is clear to see. Figure 4 demonstrates the energy-strain relation for
13
PBEsol+vdW. Table 6 also includes the primary and secondary strain percentages for the
14
compounds including I, Br, and Cl based on the PBEsol+vdW approximation.
15 16 17 18 19
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a) Dcubic
Page 24 of 36
b) Dortho
MAPbI3
MAPbBr3
MAPbCl3 1 2 3
Figure 4. Total energy of MAPbX3 (X = I, Br, Cl) structures as a function of strain under the influence of two types of deformation tensor, a) Dcubic and b) Dortho, within PBEsol+vdW approximations.
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The Journal of Physical Chemistry
1
For as much as the strain is equivalent to linear expansion and also to the volume expansion under
2
the temperature change, these phase transitions in 4% strain for compounds including Br and Cl
3
lead to a reduction in the energy level of the structures as a result of phase transitions between states
4
separated by small differences in energy (from one phase to another with a lower energy level).
5
When the material is placed in these local minima, it helps the stability of the material as the
6
temperature increases. Presence of the phase transition in -1% and -0.5% compressive strain for the
7
compounds containing Br and Cl will stabilize them more against the likely positive and negative
8
temperature changes. Therefore, for this reason and in order to increase the stability of solar cells,
9
some compounds like IxBr1-x and IxCl1-x are used in order to create the aforementioned increase in
10
the stability of solar cells.
11 12
Stress-Strain Relationships and Stability. Figure 5 depicts the changes in Second Piola-Kirchhoff
13
(PK2) stress with Lagrangian strain, which has been calculated through Cauchy or true stresses.
14
PK2 related to the Cauchy stresses is as 59:
15 16
Σ = 𝐽𝐽𝐹𝐹 −1 𝜎𝜎(𝐹𝐹 −1 )𝑇𝑇
(26)
where 𝐽𝐽 is the determinant of the deformation gradient tensor 𝐹𝐹.
17
Cauchy stresses were calculated by the Quantum ESPRESSO based on the DFT in the framework
18
of the PBEsol approximation. The results of the PBEsol approximation are in good agreement with
19
the experimental values. In order to gain the PK2 stress-strain figures, symmetric stress due to the
20
deformation tensor Dcubic was applied to the materials. The maximum tension of the material can
21
withstand, while being stretched is called maximum resistance and the strain corresponding to
22
maximum (ultimate) stress is called ultimate strain. The ultimate strain of the pristine ideal material
23
is always more than the maximum tolerable strain by the material under the non-ideal situations
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Page 26 of 36
1
like thermal fluctuations and/or crystal defects. Therefore, the material is not stable under the
2
situations beyond the ultimate strain, and will collapse under the factors like crystal defect, vacancy,
3
and thermal fluctuations. It can be concluded that only the information within the ultimate strain
4
region has a physical definition, and so, based on this assumption, in order to calculate the elastic
5
constants, only this region should be used. Moreover, it is noteworthy to mention that the ultimate
6
strain shows the inherent strength of bonds in the structure.
MAPbI3
MAPbBr3
MAPbCl3
7 8 9
Figure 5. Stress–strain responses of MAPbX3 (X = I, Br, Cl) under cubic deformation tensor. Maximum strain with maximum stress is defined as critical strain; with more than that, structure goes to plastic region.
10
The amounts of ultimate stress and strain for compounds were gathered and compared in Table 7
11
from the PK2 stress figure and its slope of changes. It can be understood that the compressive strain
12
possesses an increase in tension greater than the tensile strain. Furthermore, the stress exerted on
13
material in the compressive strain, -5%, is nearly twice the stress in the tensional strain, +10%. It
14
should be mentioned that considering the steep slope stress changes, a compressive stress more than
15
10% is not reachable physically, and the structure is unstable under such applied stress. As
16
mentioned in Table 7, the ultimate strain corresponding to the ultimate applied stress for MAPbBr3
17
(~5.5%) in c direction is more than the other compounds.
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The Journal of Physical Chemistry
1
Since the critical strain of the ideal pristine material is always more than the maximum tolerable
2
strain by the material under non-ideal situations like thermal changes and presence of the crystal
3
defects, this small elastic region is the reason for the low stability of compounds with thermal
4
changes.
5 6 7
Table 7. Ultimate stresses and ultimate strain of MAPbX3 (X = I, Br, Cl) in a, b, and c directions under cubic deformation tensor.
Σ𝑢𝑢 (Ry/bohr3)
8 9
Σ𝑢𝑢𝑎𝑎 𝜂𝜂𝑢𝑢𝑎𝑎 Σ𝑢𝑢𝑏𝑏 𝜂𝜂𝑢𝑢𝑏𝑏 Σ𝑢𝑢𝑐𝑐 𝜂𝜂𝑢𝑢𝑐𝑐
MAPbI3
MAPbBr3
MAPbCl3
7.262e-5 3.2 % 1.0799e-4 5% 1.0396e-4 5%
8.7788e-5 3.5 % 1.2504e-4 4.5 % 9.6152e-5 5.5 %
8.2796e-5 2% 1.3719e-4 5% 7.3186e-5 5%
10 11 12 13 14 15 16 17 18 19 20 21
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Page 28 of 36
1
CONCLUSION
2
In this paper, a wide range of mechanical properties related to the cubic structures MAPbX3 (X =
3
I, Br, Cl) were calculated by PBEsol, vdW, and a new method called PBEsol+vdW approximation.
4
The computed values were compared with the experimental values and also the previous theoretical
5
results. It was found that the PBEsol approximation was in a very good agreement with the
6
experimental results. Using the PBEsol approximation, Young’s modulus, as the most important
7
mechanical property, was found to be equal to 17.2, 20.3, and 21.9 GPa for MAPbI3, MAPbBr3,
8
and MAPbCl3, respectively. Values of Young’s modulus are in good agreement with previous
9
experimental results for MAPbBr3 and MAPbCl3, being equal to 19.6
24
and 19.8
23
GPa,
10
respectively. It should be noted that till now, no experimental values for cubic phase of MAPbI3
11
have been reported, and all results are related to the tetragonal phase of this material.
12
Besides, although vdW-DF2-B86R approximation is close to experimental and PBEsol outcomes
13
geometrically, it still not accurate enough as PBEsol has to obtain mechanical properties.
14
Considering the utilized calculation method in this paper and the good agreement of our results with
15
the experimental ones, it is possible to summarize the reason for the mismatch between the previous
16
theoretical data 26 and the experimental results in the framework of two points. Firstly, selecting the
17
type of approximation used to calculate the mechanical properties of these structures is very
18
important since the previous results utilized the PBE approximation and took advantage of norm-
19
conserving pseudo-potentials so have not had appropriate accuracy in order to estimate the
20
interaction energy between PbX3 scaffold and MA. However, the PBEsol approximation and using
21
the PAW pseudo-potentials were so effective to increase the accuracy of energy calculation.
22
Secondly, the reason for increasing the accuracy of computations is the modification of deformation
23
tensors based on the work of Jamal et al. 37 to calculate the elastic coefficients of cubic structures, ACS Paragon Plus Environment
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1
which led to a decrease in the measurement error of the elastic coefficient calculations.
2
Furthermore, correction of the elements of this deformation tensor in this study using Eq. (4) (see
3
Theory section) was so influential in increasing the accuracy of the results for computing the elastic
4
properties and subsequently mechanical properties, which showed the compatibility of the results
5
with the experimental results.
6
Among the experimental results reported, the results reported by Rakita et al.
7
structure, were obtained through using the nanoindentation method, have the best agreement with
8
our results. As Sun et al.
9
properties of perovskite, the difference between these two synthesis methods of crystal structures
10
should be mentioned, and it should be emphasized that the Rakita method has been more successful
11
to produce more pure crystals with the lowest level of crystal defect.
12
Considering the local minima and phase transitions of compounds containing Bromine and
13
Chlorine, the more stability of compounds containing Bromine and Chlorine in comparison to the
14
ones containing Iodine was clear. Compounds containing Iodine have more suitable band gaps,
15
much better ability to absorb light, and as a result, have much greater PCE in comparison to the
16
compounds containing Bromine and Chlorine, and also are more appropriate to fabricate solar cells.
17
Some more targeted tries and studies are necessary to increase the ultimate tolerable strain of I-
18
based compounds in order to increase their stability.
23
25
for MAPbBr3
have also used nanoindentation in order to calculate the mechanical
19 20 21 22 23
AUTHOR INFORMATION ACS Paragon Plus Environment
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1
Corresponding Author
2
* E-mail:
[email protected], Tel: 0098 9126733601, Fax: 0098 23 32395270, Post
3
Code :3619995161.
4
ORCID
5
Mahdi Faghihnasiri: 0000-0001-8821-2128
6
Morteza Izadifard: 0000-0002-7626-8383
7
Mohammad Ebrahim Ghazi: 0000-0002-4496-1757
8
Notes
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The authors declare no competing financial interests.
Page 30 of 36
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ACKNOWLEDGMENTS
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The authors gratefully acknowledge the support of the Nano Research and Training Center of Iran
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(NRTC.ir) under Grant No.7.1396.06.01. Computational resources were provided by the Salehi
13
Computing Cluster belong to Shahrood University of Technology which is supported by the Nano
14
Research and Training Center of Iran (NRTC.ir).
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REFERENCES
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(1) Kim, H.-S.; Mora-Sero, I.; Gonzalez-Pedro, V.; Fabregat-Santiago, F.; Juarez-Perez, E. J.; Park, N.-G.; Bisquert, J. Mechanism of Carrier Accumulation in Perovskite Thin-Absorber Solar Cells. Nature communications 2013, 4. (2) Brenner, T. M.; Egger, D. A.; Kronik, L.; Hodes, G.; Cahen, D. Hybrid Organic—Inorganic Perovskites: Low-Cost Semiconductors with Intriguing Charge-Transport Properties. Nature Reviews Materials 2016, 1, 15007. (3) Baikie, T.; Fang, Y.; Kadro, J. M.; Schreyer, M.; Wei, F.; Mhaisalkar, S. G.; Graetzel, M.; White, T. J. Synthesis and Crystal Chemistry of the Hybrid Perovskite (Ch 3 Nh 3) Pbi 3 for Solid-State Sensitised Solar Cell Applications. Journal of Materials Chemistry A 2013, 1, 56285641. (4) Weller, M. T.; Weber, O. J.; Henry, P. F.; Di Pumpo, A. M.; Hansen, T. C. Complete Structure and Cation Orientation in the Perovskite Photovoltaic Methylammonium Lead Iodide between 100 and 352 K. Chemical Communications 2015, 51, 4180-4183. (5) Poglitsch, A.; Weber, D. Dynamic Disorder in Methylammoniumtrihalogenoplumbates (Ii) Observed by Millimeter‐Wave Spectroscopy. The Journal of chemical physics 1987, 87, 63736378. (6) Whitfield, P. S.; Herron, N.; Guise, W. E.; Page, K.; Cheng, Y. Q.; Milas, I.; Crawford, M. K. Structures, Phase Transitions and Tricritical Behavior of the Hybrid Perovskite Methyl Ammonium Lead Iodide. Scientific reports 2016, 6, 35685. (7) Kurtz, S.; Levi, D. National Renewable Energy Laboratory (NREL) 2017.
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
(8) Noh, J. H.; Im, S. H.; Heo, J. H.; Mandal, T. N.; Seok, S. I. Chemical Management for Colorful, Efficient, and Stable Inorganic–Organic Hybrid Nanostructured Solar Cells. Nano letters 2013, 13, 1764-1769. (9) Park, N.-G. Organometal Perovskite Light Absorbers toward a 20% Efficiency Low-Cost SolidState Mesoscopic Solar Cell. The Journal of Physical Chemistry Letters 2013, 4, 2423-2429. (10) Umari, P.; Mosconi, E.; De Angelis, F. Relativistic Gw Calculations on Ch3nh3pbi3 and Ch3nh3sni3 Perovskites for Solar Cell Applications. Scientific Reports 2014, 4, 4467. (11) Ponseca Jr, C. S.; Savenije, T. J.; Abdellah, M.; Zheng, K.; Yartsev, A.; Pascher, T. r.; Harlang, T.; Chabera, P.; Pullerits, T.; Stepanov, A. Organometal Halide Perovskite Solar Cell Materials Rationalized: Ultrafast Charge Generation, High and Microsecond-Long Balanced Mobilities, and Slow Recombination. Journal of the American Chemical Society 2014, 136, 5189-5192. (12) De Wolf, S.; Holovsky, J.; Moon, S.-J.; Löper, P.; Niesen, B.; Ledinsky, M.; Haug, F.-J.; Yum, J.-H.; Ballif, C. Organometallic Halide Perovskites: Sharp Optical Absorption Edge and Its Relation to Photovoltaic Performance. The Journal of Physical Chemistry Letters 2014, 5, 1035-1039. (13) Stroppa, A.; Quarti, C.; De Angelis, F.; Picozzi, S. Ferroelectric Polarization of Ch3nh3pbi3: A Detailed Study Based on Density Functional Theory and Symmetry Mode Analysis. The journal of physical chemistry letters 2015, 6, 2223-2231. (14) Yin, W.-J.; Yang, J.-H.; Kang, J.; Yan, Y.; Wei, S.-H. Halide Perovskite Materials for Solar Cells: A Theoretical Review. Journal of Materials Chemistry A 2015, 3, 8926-8942. (15) Yin, W. J.; Shi, T.; Yan, Y. Unique Properties of Halide Perovskites as Possible Origins of the Superior Solar Cell Performance. Advanced Materials 2014, 26, 4653-4658. (16) Kojima, A.; Teshima, K.; Shirai, Y.; Miyasaka, T. Organometal Halide Perovskites as VisibleLight Sensitizers for Photovoltaic Cells. Journal of the American Chemical Society 2009, 131, 6050-6051.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
Page 32 of 36
(17) Chung, I.; Lee, B.; He, J.; Chang, R. P. H.; Kanatzidis, M. G. All-Solid-State Dye-Sensitized Solar Cells with High Efficiency. Nature 2012, 485, 486-489. (18) Lee, M. M.; Teuscher, J.; Miyasaka, T.; Murakami, T. N.; Snaith, H. J. Efficient Hybrid Solar Cells Based on Meso-Superstructured Organometal Halide Perovskites. Science 2012, 338, 643-647. (19) Burschka, J.; Pellet, N.; Moon, S.-J.; Humphry-Baker, R.; Gao, P.; Nazeeruddin, M. K.; Grätzel, M. Sequential Deposition as a Route to High-Performance Perovskite-Sensitized Solar Cells. Nature 2013, 499, 316-319. (20) Zhao, Y.; Wei, J.; Li, H.; Yan, Y.; Zhou, W.; Yu, D.; Zhao, Q. A Polymer Scaffold for SelfHealing Perovskite Solar Cells. Nature communications 2016, 7. (21) Yang, S.; Wang, Y.; Liu, P.; Cheng, Y.-B.; Zhao, H. J.; Yang, H. G. Functionalization of Perovskite Thin Films with Moisture-Tolerant Molecules. Nature Energy 2016, 1, 15016. (22) Li, H.; Castelli, I. E.; Thygesen, K. S.; Jacobsen, K. W. Strain Sensitivity of Band Gaps of SnContaining Semiconductors. Physical Review B 2015, 91, 045204. (23) Sun, S.; Fang, Y.; Kieslich, G.; White, T. J.; Cheetham, A. K. Mechanical Properties of Organic–Inorganic Halide Perovskites, Ch 3 Nh 3 Pbx 3 (X= I, Br and Cl), by Nanoindentation. Journal of Materials Chemistry A 2015, 3, 18450-18455. (24) Rakita, Y.; Cohen, S. R.; Kedem, N. K.; Hodes, G.; Cahen, D. Mechanical Properties of Apbx 3 (a= Cs or Ch 3 Nh 3; X= I or Br) Perovskite Single Crystals. MRS Communications 2015, 5, 623-629. (25) Lomonosov, A. M.; Yan, X.; Sheng, C.; Gusev, V. E.; Ni, C.; Shen, Z. Exceptional Elastic Anisotropy of Hybrid Organic–Inorganic Perovskite Ch3nh3pbbr3 Measured by Laser Ultrasonic Technique. physica status solidi (RRL)-Rapid Research Letters 2016, 10, 606-612. (26) Feng, J. Mechanical Properties of Hybrid Organic-Inorganic Ch3nh3bx3 (B= Sn, Pb; X= Br, I) Perovskites for Solar Cell Absorbers. APL Materials 2014, 2, 081801. (27) Letoublon, A.; Paofai, S.; Ruffle, B.; Bourges, P.; Hehlen, B.; Michel, T.; Ecolivet, C.; Durand, O.; Cordier, S.; Katan, C.; Even, J. Elastic Constants, Optical Phonons, and Molecular Relaxations in the High Temperature Plastic Phase of the Ch3nh3pbbr3 Hybrid Perovskite. J Phys Chem Lett 2016, 7, 3776-3784. (28) Born, M.; Huang, K. Dynamical Theory of Crystal Lattices. Clarendon press: 1954. (29) Wallace, D. C. Thermodynamics of Crystals Dover. New York 1998. (30) Ansari, R.; Malakpour, S.; Ajori, S. Structural and Elastic Properties of Hybrid Bilayer Graphene/H-Bn with Different Interlayer Distances Using Dft. Superlattices and Microstructures 2014, 72, 230-237. (31) Chen, Q.; De Marco, N.; Yang, Y. M.; Song, T.-B.; Chen, C.-C.; Zhao, H.; Hong, Z.; Zhou, H.; Yang, Y. Under the Spotlight: The Organic–Inorganic Hybrid Halide Perovskite for Optoelectronic Applications. Nano Today 2015, 10, 355-396. (32) Kim, H.-S.; Im, S. H.; Park, N.-G. Organolead Halide Perovskite: New Horizons in Solar Cell Research. The Journal of Physical Chemistry C 2014, 118, 5615-5625. (33) Stadler, R.; Wolf, W.; Podloucky, R.; Kresse, G.; Furthmüller, J.; Hafner, J. Ab Initio Calculations of the Cohesive, Elastic, and Dynamical Properties of Cosi 2 by Pseudopotential and All-Electron Techniques. Physical Review B 1996, 54, 1729. (34) Nielsen, O. H.; Martin, R. M. First-Principles Calculation of Stress. Physical Review Letters 1983, 50, 697. (35) Wang, H.; Li, M. Ab Initio Calculations of Second-, Third-, and Fourth-Order Elastic Constants for Single Crystals. Physical Review B 2009, 79, 224102.
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The Journal of Physical Chemistry
(36) Nye, J. F. Physical Properties of Crystals: Their Representation by Tensors and Matrices. Oxford university press: 1985. (37) Jamal, M.; Asadabadi, S. J.; Ahmad, I.; Aliabad, H. A. R. Elastic Constants of Cubic Crystals. Computational Materials Science 2014, 95, 592-599. (38) Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G. L.; Cococcioni, M.; Dabo, I. Quantum Espresso: A Modular and Open-Source Software Project for Quantum Simulations of Materials. Journal of physics: Condensed matter 2009, 21, 395502. (39) Blöchl, P. E. Projector Augmented-Wave Method. Physical review B 1994, 50, 17953. (40) Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E.; Constantin, L. A.; Zhou, X.; Burke, K. Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. Physical Review Letters 2008, 100, 136406. (41) Hamada, I. Van Der Waals Density Functional Made Accurate. Physical Review B 2014, 89, 121103. (42) Monkhorst, H. J.; Pack, J. D. Special Points for Brillouin-Zone Integrations. Physical review B 1976, 13, 5188. (43) Hermann, J.; DiStasio Jr, R. A.; Tkatchenko, A. First-Principles Models for Van Der Waals Interactions in Molecules and Materials: Concepts, Theory, and Applications. 2017. (44) Egger, D. A.; Kronik, L. Role of Dispersive Interactions in Determining Structural Properties of Organic–Inorganic Halide Perovskites: Insights from First-Principles Calculations. The journal of physical chemistry letters 2014, 5, 2728-2733. (45) Tkatchenko, A.; Scheffler, M. Accurate Molecular Van Der Waals Interactions from GroundState Electron Density and Free-Atom Reference Data. Physical review letters 2009, 102, 073005. (46) Stoumpos, C. C.; Malliakas, C. D.; Kanatzidis, M. G. Semiconducting Tin and Lead Iodide Perovskites with Organic Cations: Phase Transitions, High Mobilities, and near-Infrared Photoluminescent Properties. Inorganic chemistry 2013, 52, 9019-9038. (47) Saidi, W. A.; Choi, J. J. Nature of the Cubic to Tetragonal Phase Transition in Methylammonium Lead Iodide Perovskite. The Journal of chemical physics 2016, 145, 144702. (48) Giorgi, G.; Fujisawa, J.-I.; Segawa, H.; Yamashita, K. Cation Role in Structural and Electronic Properties of 3d Organic–Inorganic Halide Perovskites: A Dft Analysis. The Journal of Physical Chemistry C 2014, 118, 12176-12183. (49) Bechtel, J. S.; Seshadri, R.; Van der Ven, A. Energy Landscape of Molecular Motion in Cubic Methylammonium Lead Iodide from First-Principles. The Journal of Physical Chemistry C 2016, 120, 12403-12410. (50) Brivio, F.; Frost, J. M.; Skelton, J. M.; Jackson, A. J.; Weber, O. J.; Weller, M. T.; Goni, A. R.; Leguy, A. M. A.; Barnes, P. R. F.; Walsh, A. Lattice Dynamics and Vibrational Spectra of the Orthorhombic, Tetragonal, and Cubic Phases of Methylammonium Lead Iodide. Physical Review B 2015, 92, 144308. (51) Voigt, W. Lehrbuch Der Kristallphysik (Mit Ausschluss Der Kristalloptik). Springer-Verlag: 2014. (52) Luo, Y.-R. Comprehensive Handbook of Chemical Bond Energies. CRC press: 2007. (53) Nagabhushana, G. P.; Shivaramaiah, R.; Navrotsky, A. Direct Calorimetric Verification of Thermodynamic Instability of Lead Halide Hybrid Perovskites. Proceedings of the National Academy of Sciences 2016, 113, 7717-7721. (54) Da Silva, E. L.; Skelton, J. M.; Parker, S. C.; Walsh, A. Phase Stability and Transformations in the Halide Perovskite Cssni 3. Physical Review B 2015, 91, 144107.
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(55) Popovich, V. A.; Yunus, A.; Janssen, M.; Richardson, I. M.; Bennett, I. J. Effect of Silicon Solar Cell Processing Parameters and Crystallinity on Mechanical Strength. Solar Energy Materials and Solar Cells 2011, 95, 97-100. (56) Kim, T.; Kim, J.-H.; Kang, T. E.; Lee, C.; Kang, H.; Shin, M.; Wang, C.; Ma, B.; Jeong, U.; Kim, T.-S. Flexible, Highly Efficient All-Polymer Solar Cells. Nature communications 2015, 6. (57) Pugh, S. F. Xcii. Relations between the Elastic Moduli and the Plastic Properties of Polycrystalline Pure Metals. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 1954, 45, 823-843. (58) He, X.; Shen, H. First-Principles Study of Elastic and Thermo-Physical Properties of KesteriteType Cu 2 Znsns 4. Physica B: Condensed Matter 2011, 406, 4604-4607. (59) Peng, Q.; Ji, W.; De, S. Mechanical Properties of the Hexagonal Boron Nitride Monolayer: Ab Initio Study. Computational Materials Science 2012, 56, 11-17.
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