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Exploration of direct band gap double
perovskites A2AgIrCl6 (A= Cs, Rb, K): a DFT
study
Md. Abu Rayhan
ID No: 20PPHY002P
Session: 2020-21
Department of Physics
Chittagong University of Engineering & Technology (CUET)
Chittagong-4349, Bangladesh
2
Supervisor
Prof. Dr. Md. Ashraf Ali

• Introduction
• Literature review
• Motivation
• Objectives
• Methodology
• Results and discussions
• Conclusions
Outlines
3

Introduction
• Perovskite materials: Crystalline compounds with unique structure (ABX3). High
potential for solar cells, LEDs, due to excellent properties. Example- CaTiO3
• Types- i) single perovskites, ii) double perovskites
Single Perovskite (ABX3)
• Oxide perovskite (ABO3)
• Halide perovskite (ABF3)
• Nitrides perovskite
(ABN3)
Double perovskite (A2BBʹX6)
• Oxide
• Halide
4

Why double Perovskites Halide?
 Tunable properties
 Optoelectronic devices
 Energy conversion
 Advanced electronics
 Environmental considerations
 Magnetism and spintronics
Double perovskite:
A2BBʹX6
A= Cs+, Rb+, K+
B= Ag+
Bʹ= Ir3+
X= F-/Cl-/Br-
5

Literature review
Author Compounds Property Study Journal
E. Greul et al. Cs2AgBiBr6 Structural and optoelectronic J. Mater. Chem. A, vol. 5, p.
19972, 2017.
N. Guechi et al. Cs2AgBiX6 (X= Cl, Br) Elastic, optoelectronic and
thermoelectric
J. Electron. Mater., vol. 47,
pp. 1533–1545, 2018.
W. Shi et al. Cs2MBiCl6 (M= Ag, Cu,
Na, K, Rb, and Cs)
Structural and opto-electronic J. Chem. Phys., vol. 153, p.
141101, 2020.
M. Nabi et al. Cs2CuMCl6 (M= Sb, Bi) Structural stability, electronic, elastic,
thermoelectric and optical
Sci. Rep., vol. 11, p. 12945,
2021.
Shruthi Nair et
al.
Cs2TlBiI6 Structural, electronic and optical J. Phys.: Condensed Matter,
vol. 31, p. 445902, 2019
T. Saha et al Cs2AgAsCl6 structural, mechanical, electronic,
thermodynamic, phonon and optical
Phys. Chem. Chem. Phys.,
vol. 24, p. 26609, 2022.
T. Y. Tang et al. A2CuSbX6 (A= Cs, Rb,
K; X= Cl, Br, I)
Physical and optoelectronic Chemical Physics, vol. 570,
p. 111897, 2023.
M. Caid et al. Cs2CuIrF6 Structural stability and optoelectronic J. Molecular Modeling, vol.
29, p. 178, 2023.
6

Motivation
 These Double perovskites compounds are (A2AgIrCl6 (A= Cs,
Rb, K) composed of abundant and non-toxic elements, making
them environmentally friendly along with the cost effective and
time efficient.
 No theoretical or practical work has been done on these
substances in order to determine physical properties qualities via
DFT computation.
 The comprehensive study of these double perovskites also
contributes to fundamental scientific understanding.
7

Objectives
 To study the structural stability such as tolerance factor, octahedral factor, new
tolerance factor.
 To check the thermo-dynamical stability such as formation energy, binding energy
and decomposition energy and dynamical stability of the compounds.
 To study the mechanical stability and elastic behavior.
 To find out the energy gap of band structure and electron density of states.
 To find the optical properties of the compounds for using in solar cells.
 To find the suitability for use in solar cells and/or thermoelectric devices
8

Why Density Functional Theory?
9
Computational Methodology
DFT (Density Functional Theory)
The calculation of physical and chemical properties of multi- particle
systems (atoms, molecules or solids) require the exact determination of
electronic structure and total energy of these systems.
 Schrödinger equation successfully explains the electronic structure of
simple systems and numerically exact solutions are found for small no. of
atoms and molecules.
 This n-electron problem was solved when Kohn and Sham in 1965
formulated a theory concerning 3-dimensional electron density and energy
functionals.
Electron density n(r) plays central role instead of wave function ψ(r).
The problem of many-interacting particles system in static potential is
reduced to non-interacting single particle system in an effective potential.

Many body problem:
 For large interacting system, we first need to consider a many particle wave
function.
 Many body Hamiltonian for electron and nucleus is of the form given below
Hѱ (r,R,t) =E ѱ (r,R,t)
Innocent look of wave equation
Hѱ
=
M
m
e
Hѱ
=
=
Ѱ
=
ѱ
=
ѱ 10

Since the total Hamiltonian for electron and nucleus is:
then the hamiltonian for the electronic part will be
Approximations for solving many body problem
 The Born-Oppenheimer approximation
 Hartree approximation
 Hartree-Fock method
 Hohenberge- Kohen
 Kohn-Sham approach (Walter Kohn and Lu.J.Sham)
 The nuclei are much heavier than electrons.
 They move much more slowly and hence neglect the nuclear kinetic energy.
 The wave function separated into electronic and nuclear part and determine
motion of electrons with nuclei held fixed.
Hѱ =
=
Hѱ
=
11

Hartree approximation: One electron model
 Reduce the complexity of electron-electron interactions.
 Electrons are independent and interacts with others in an averaged way.
 For an n-electron system, each electron does not recognize other as single entities but
as a mean field.
 Hence, n-electron system becomes a set of non-interacting one-electron system where
each electron moves in the average density of rest electrons.
Self-consistent field procedure to solve the wave equation:
Vext = electron and
nuclei interaction
potential
VH = Hartree potential
(e-e interaction)
( )
+VH +Vext Ѱ(r) = EѰ(r)
E = E1+E2+E3+…..+En
R-nuclear
r- electron
12

Hartree method produced crude estimation of energy due to two
oversimplifications:
 Hartree method does not follow two basic principles of quantum
mechanics: the antisymmetry principle and Pauli’s exclusion principle.
 Does not count the exchange and correlation energies coming from n-
electron nature.
The Hartree method, therefore, was soon refined into the Hartree-Fock method.
Hartree-Fock method
Based on the one-electron and mean-field approach by Hartree, V.A. Fock enhanced the
methods to higher perfection. Fock and J.C. Slater in 1930 generalized the Hartree's theory to
take into account the antisymmetry requirement.
 In HF method, the n-electron wave function approximated as a linear
combination of non-interacting one-electron wave function in the form of
Slater determinant.
Slater determinant
13

VH = Vij Hartree or Coulomb interaction
energy of two electrons
Ex = Exchange energy comes from the
antisymmetric nature of wave function in
the Slater determinant.
Difficulties with Hartree-Fock Theory:
A new approach has been developed known as Density Functional Theory
(DFT).
 In 1964 Hohenberg and Kohn showed that schrodinger equation (3N dimensional e.g. 10 electrons
require 30 dimensions) could be reformulated in terms of electron density n(r) with non-interacting n
separate 3-dimensional ones.
 The main objective of DFT is to replace the many-particle electronic wavefunction with the
electron density as the basic quantity.
 The electron density n(r), the central player in DFT decides everything in an n-electron quantum state
where there is no individual electron density but a 3-dimensional density of electrons.
 The addition of all the electron densities over the whole space naturally return to the total number of
electrons in the system.
 The knowledge of overlapping of atomic electron density, roughly generate the electron density of the
solids.
 This theory gives approximate solutions to both Exchange and Correlation Energies.
 Correlation energy and
 Problem of dealing 3N dimensional .
)ѱ(r) = E ѱ(r)
E = Ekin+ EH +Eext + Ex
14

The Fundamental Pillars of DFT
First Hohenberg Kohn (HK) theorem: The ground-state energy is a unique functional
of the electron density n(r).
 This theorem provides one to one mapping between ground state wave function and
ground state charge density.
 The ground state charge density can uniquely describe all the ground state
properties of system.
 The fundamental concept behind density functional theory is that charge density (3-
Dimensional) can correctly describe the ground state of N-particle instead of using a
wave function (3N-Dimensional).
Second Hohenberg Kohn (HK) theorem: The electron density that minimizes the
energy of the overall functional is the true electron density.
 If the true functional form of energy in terms of density gets known, then one could
vary the electron density until the energy from the functional is minimized, giving us
required ground state density.
 This is essentially a variational principle and is used in practice with approximate
forms of the functional.
 The simplest possible choice of a functional can be a constant electron density all
over the space.
15

Kohn- Sham Approach (1965):
 KS replace the interacting n-electron system with a system of one-electron (non-
interacting) system in effective potential having the same ground state.
since the kinetic energy; E= Ekin+ Eext+EH +Ex+ Ec
int
non
non int
Ekin = Ekin + Ekin
where
E = Ekin + Ekin + Eext + EH +Ex + Ec
int
non int
E = Ekin + Eext + EH +Exc = F [n(r)] + Eext
non
16

Hence final KH equation has the form:
DFT in Practice: Kohn-Sham Self Consistency loop
17

1. Local density approximation (LDA)
Exchange-correlation approximation
 Approximation used to find out exchange-
correlation function.
 Exchange-correlation energy functional is
purely local.
 Ignores corrections to the exchange-
correlation energy at a point r due to nearby
inhomogeneities in the electron density.
2. Generalized Gradient Approximation (GGA)
 Depends on local density and its gradient.
 GGA uses information about the local electron density and also the local gradient in the
electron density. Though GGA includes more physical information than LDA. It is not
necessary that it must be more accurate. There are large number of distinct GGA functionals
depending on the ways in which information from the gradient of the electron density can
be included in a GGA functional.
18

Results & Discussion
Structural parameter
 Cubic structure
 Space group (Fm-3m, 225)
 Lattice parameter (a = b = c;
α = β = γ = 90⁰)
 Cs/Rb/K (0.25, 0.25, 0.25),
Ag(0.5, 0.5, 0.5), Ir (0, 0, 0),
and Cl (0.25, 0, 0).
 Four formula unit (2:1:1:6)
Fig.2: Unit cell of Cs2AgIrCl6
19

Structural entity & stability
Table1: lattice parameter
Compound Lattice
constant
Cs2AgIrCl6 10.19
Rb2AgIrCl6 10.09
K2AgIrBr6 10.03
• Tolerance factor
• Octahedral factor
• New Tolerance factor
• Formation energy
• Binding energy
• Decomposition energy
𝑡𝐺 =
𝑅𝐴+𝑅𝑋
√2(
𝑅𝐵′+𝑅𝐵′′
2
+𝑅𝑋)
[1] 𝜏 =
𝑅𝑋
𝑅𝐵
− 𝑛𝐴 𝑛𝐴 −
𝑅𝐴 𝑅𝐵
ln 𝑅𝐴 𝑅𝐵
[3]
µ =
𝑅𝐵′+𝑅𝐵′′
2𝑅𝑋
[1]
𝐸𝑓 =
𝐸𝐴2𝐴𝑔𝐼𝑟𝐶𝑙6−𝑛𝐴×
𝐸𝐴
𝑘
−𝑛𝐴𝑔×
𝐸𝐴𝑔
𝑙
− 𝑛𝐼𝑟×
𝐸𝐼𝑟
𝑚
− 𝑛𝐶𝑙×
𝐸𝐶𝑙
𝑝
𝑁
[2]
𝐸𝑏 = 𝐸𝐴2𝐴𝑔𝐼𝑟𝐶𝑙6
− 𝑛𝐴 × 𝜇𝐴 − 𝑛𝐴𝑔 × 𝜇𝐴𝑔 − 𝑛𝐼𝑟 × 𝜇𝐼𝑟 − 𝑛𝐶𝑙 × 𝜇𝐶𝑙 [2]
∆𝐻𝐷 = 2𝐸 𝐴𝐶𝑙 + 𝐸 𝐴𝑔𝐶𝑙 + 𝐸 𝐼𝑟𝐶𝑙3 − 𝐸 𝐴2𝐴𝑔𝐼𝑟𝐶𝑙6
1. Liu, XiangChun; Hong, Rongzi; Tian, Changsheng (24 April 2008). "Tolerance factor and the stability discussion of ABO3-type ilmenite". Journal of Materials
Science: Materials in Electronics. 20 (4): 323–327
2. X. Du, D. He, H. Mei, Y. Zhong, and N. Cheng, “Insights on electronic structures, elastic features and optical properties of mixed-valence double perovskites
Cs2Au2X6 (X= F, Cl, Br, I),” Phys. Lett. A, vol. 384, no. 8, p. 126169, 2020.
3. C.J. Bartel, C. Sutton, B.R. Goldsmith, R.H. Ouyang, C.B. Musgrave, L. M. Ghiringhelli, M. Scheffler, New tolerance factor to predict the stability of perovskite
oxides and halides, article eaav0693, Sci. Adv. 5 (2) (2019)
20

Table.2: Different stability factor
DP Ionic radius of cations
(Å)
Ionic
radius of
anion (Å)
Tolerance
factor (t)
Octahedral
factor (𝝁)
New
tolerance
factor (𝝉)
Binding
energy, 𝑬𝒃
(eV/atom)
Formation
enthalpy,
𝑬𝒇
(eV/atom)
Decomp
osition
energy,
∆𝐻𝐷
(meV/at
om)
Cs2AgIrCl6 𝑟𝐶𝑠
1.88
(𝑟𝐴𝑔+𝑟𝐼𝑟)/2
0.92
𝑟𝐶𝑙
1.81 0.96 0.51 3.83 -4.29 -2.10 73.55
Rb2AgIrCl6 𝑟Rb
1.72
0.92
𝑟𝐶𝑙
1.81 0.91 0.51 3.93 -4.28 -2.06 50.66
Cs2AgIrBr6 𝑟K
1.64
0.92
𝑟𝐶𝑙
1.81 0.89 0.51 4.04 -4.28 -2.03 0.19
0.81 ≤ 𝑡 ≤ 1.1; 0.41 ≤ µ ≤ 0.89; τ < 4.18
21

Dynamical stability
Fig.3: Phonon dispersion with total and
partial DOS of A2AgIrCl6 ; A= Cs, Rb, K
22

Mechanical stability
C11 > 0, C11 - C12 > 0; C11 + 2C12 > 0; C44 > 0 and C11 > B > C12 [4]
Table.3: elastic parameter
Parameters Cs2AgIrCl6 Rb2AgIrCl6 K2AgIrCl6
Born
stability
127.79 90.86 78.25
35.70 19.55 17.47
16.22 18.28 12.83
92.09 71.31 60.78
199.19 129.96 113.19
19.48 1.27 4.64
Bulk modulus, B (GPa) 66.39 43.32 37.73
Shear modulus, G (GPa) 25.02 23.97 18.27
Young modulus, Y (GPa) 66.69 60.70 47.19
0.33 0.27 0.29
Pugh’s ratio, B/G 2.65 1.81 2.06
0.35 0.51 0.42
4. M. Born, K. Huang, and M. Lax, “Dynamical theory of crystal lattices,” Am. J. Phys., vol. 23, no. 7, p. 474, 1955.
Y > B > G
0.26 < ductile
1.75 < ductile
23

Electronic properties
• Band structure
• DOS (TDOS and PDOS)
• Direct band gap nature
• Effective mass of electrons and holes
Table.4: Band gap values of compounds
Compounds Band
gap
(eV)
Functional Band gap
nature
Effective mass
of electron, 𝒎𝒆
∗
Effective mass
of hole, 𝒎𝒉
∗
Cs2AgIrCl6 0.34
1.43
GGA PBE
Tb-mBJ
Direct
Direct
0.13 me
0.19 me
1.10 me
1.43 me
Rb2AgIrCl6 0.36
1.50
GGA PBE
Tb-mBJ
Direct
Direct
0.13 me
0.18 me
1.10 me
1.70 me
K2AgIrCl6 0.38
1.55
GGA PBE
Tb-mBJ
Direct
Direct
0.13 me
0.18 me
1.64 me
1.14 me
24

Fig. 4: Energy band structure of
Cs2AgIrCl6, Rb2AgIrCl6, K2AgIrCl6 by
GGA and TB-mBJ method.
25

Fig 5: Total and partial DOS of
Cs2AgIrCl6, Rb2AgIrCl6,
K2AgIrCl6 by TB-mBJ method
26

Fig. 6 Charge density of A2AgIrCl6 (A = Cs, Rb, K) compounds
27

Optical properties
Fig.7: Real and imaginary part of dielectric constant, refractive index, extinction coefficient of
the studied compound
28

Optical properties
Fig.8: Absorption coefficient, optical conductivity, reflectivity, and loss function of the studied
compound
29

Fig. 9 Calculated thermoelectric properties including a) Electrical conductivity, b)
Electronic portion of thermal conductivity, c) Lattice thermal conductivity, d)
Seebeck coefficient, e) Power factor (PF), and f) Figure of merit (ZT) of
Cs2AgIrCl6 (red color), Rb2AgIrCl6 (blue color), and K2AgIrCl6 (green color)
double perovskite.
31

Table 5: The calculated values of electrical conductivity (σ), electronic
conductivity (Ke), Seebeck coefficient (S), power factor (PF), and figure of merit
(ZT) for A2AgIrCl6 (A=Cs, Rb, K) at room temperature.
Material property Cs2AgIrCl6 Rb2AgIrCl6 K2AgIrCl6
Transport
properties (300 K)
𝜎 × 105
Ω ∙ m 0.66 0.67 0.68
Ke (Wm-1K-1) 0.97 0.85 0.91
S (μV/K) 189.30 176.23 182.20
PF (×10-3 Wm-1K-2) 2.40 2.09 2.25
ZT 0.74 0.74 0.83
32

Conclusions:
• The suggested compounds exhibit stability in dynamic, thermodynamic, and
mechanical aspects.
• The compounds exhibit a characteristic of ductility.
• The A2AgIrCl6 possesses a direct band gap.
• The energy gap corresponds to the visible range of electromagnetic waves, making it
applicable for utilization in solar cells, renewable energy technology, and
photocatalytic applications.
• Effective mass of electrons are small compared to effective mass of holes indicating
higher carrier mobility.
• The high absorption coefficients of 105 order and other optical constant, making their
suitability for opto-electronic application.
• The low reflectivity values (less than 13%) also indicates their high absorption
ability.
• A2AgIrCl6 (A = Cs/Rb/K) are desirable choice for potential candidates for use in
thermoelectric devices.
33

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