DFT Study of Mechanical Properties and Stability of Cubic

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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

1

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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.

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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.

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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.

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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

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, high optical absorbance coefficient, narrow optical absorbance edge, large dielectric constant 12-

16

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, proper optical band gap in the visible range

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features that make MAPbI3 one of the most promising materials to use as an absorbance layer in

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the solar cells 16-19.

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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

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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

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should be a great deal in the fabrication and application of these layers 23.

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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

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studies 23-27. Therefore, it is necessary to continue our efforts to achieve the accurate elastic constant

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for MAPbX3 structures, understand their behavior, and know which variables would affect their

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elastic response and how these variables would affect those constants.

22

. Therefore, having a comprehensive

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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.

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Structures including MAPbI3, MAPbBr3 and MAPbCl, undergo a phase transition from

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orthorhombic to tetragonal phase around 162, 149 and 172 K and also from tetragonal to cubic

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phase around 327, 236.9 and 178 K, respectively

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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

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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

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homogeneous and anisotropic elastic medium, works have shown that the results of the continuum

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theories are accurate enough in order to investigate the mechanical properties of materials in atomic

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scales that include cavities and vacant spaces. Based upon the non-linear elastic continuum

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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

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in total energy versus applied strain is used, while, in the stress approach, which is based upon the 34

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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)

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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.

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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:


5

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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)

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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

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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


7

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1

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.

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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 888 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 110-5 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


9

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1

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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1

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

Page 13 of 36

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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


14


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

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


15

Page 15 of 36

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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.


16


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1

Page 16 of 36

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


19

Page 19 of 36

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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:


20


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

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.


21

Page 21 of 36

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


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


22


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

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


23

Page 23 of 36

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1 2


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


24


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

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.


25

Page 25 of 36

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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


26


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

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.


27

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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


28


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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

29

Page 29 of 36

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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

30


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

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

9

The authors declare no competing financial interests.

Page 30 of 36

10

ACKNOWLEDGMENTS

11

The authors gratefully acknowledge the support of the Nano Research and Training Center of Iran

12

(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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DFT Study of Mechanical Properties and Stability of Cubic

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