Computational methods and vibrational properties applied to materials modeling

Computational methods and vibrational
properties applied to materials modeling
Dr. Federico Brivio
Workshop BTHA - March 26, 2019
.Raffaello Sanzio - La scuola di Atene

1
Material Simulation
Solve theoretical equation to
predict Material Properties
(often) Cheaper, safer,
cleaner, faster, ...
|

2
Computer evolution
top500.org
Exponential growth
Limit Moore’s law
Better description
New type of simulations
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3
Material Design
Stage I (Past):
Reproduction of speciﬁc cases
Few "home-brew" software
Stage II (Today):
Prediction of Properties of materials
Accessible/commercial software
Stage III (Future):
Prediction of Materials with speciﬁc
properties
Network, cloud, data-mining
REAL material design
|

4
Material Design
Stage I (Past):
Reproduction of speciﬁc cases
Few "home-brew" software
Stage II (Today):
Prediction of Properties of materials
Accessible/commercial software
Stage III (Future):
Prediction of Materials with speciﬁc
properties
Network, cloud, data-mining
REAL material design
|

5
DFT in pills
GROUND-STATE energy and density of the electronic system
Function of e−
n(r) coordinates. Nuclei are parameters
(Born-Oppenheimer)
A-Thermal, slightly different from 0 K.
Zero-Point-Energy missing
Static picture, different statistics, ...
DFT is a exact-theory, but it has an "imperfect" implementation:
Exchange-correlation functional unknown (accuracy)
Numerical noise (precision)[1, 2]
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6
DFT Errors vs Experiment
Accuracy
Taken from [2].
Accuracy is independent from PRECISION. All the calculation has to
be converged against the parameters of interest.
Minimization of NOISE.
|

7
Beyond DFT
We can intrinsically improve DFT:
Electronic description: GW, post-HF, TD-DFT, ...
We can obtain Dynamics from Thermodynamic:
Jacob’s Ladder of Phonons:
Phonon - Harmonic app. (Q2
)
Quasi-Harmonic
approximation (Q2
)
Phonon-Phonon scattering
Anharmonic (Q3
)
Anharmonicity++ (Q4
, ...)
G. Doré|

8
Phonon
Phonon:Quasi-particle: coordinate collective motions of atoms that is
described in a single particle formalism: momentum, energy.
Molecular Vibrations:
Molecules atom forms normal
modes.
Real space coordinates.
Crystal Vibrations:
The vibrations have a phase
component.
Brillouin space: Dispersion
curves, similar to e−
band
structure
|

9
Harmonic model
Easier approach is the Harmonic approach based on Hooke’s law.
F = −kx
U = 1
2 kx2
Pros:
Analytic solution
Good description of many
properties
Limits:
Immortal modes
Non-physical (lack of thermal
expansion and other
anharmonic effects)
|

10
3D Harmonic Crystal
The harmonic model can be applied to a classic system and to a
quantum oscillator (QO) model (discrete solutions). The QO model
can be extended to periodic systems:
Periodic potential
Solutions are planewaves
Pair-Forces Dij(R − R ) = ∂2
U
∂ui(R)∂uj (R)
Dynamical matrix D(k) = R D(R)eik·R
Mω2
= D(k)
us(R, t) = Re[ s(k)ei(k·R−ωs(k)t)
]
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11
Finited Displacement Method
GOAL: Calculate D(R) force matrix.
Molecules have 3N-3 DOF. Starting from a converged structure (no
forces acting on atoms):
1. Symmetry analysis
2. Displace every atom
3. Collection of Forces
4. Dynamical Matrix and phonon
properties
5. Eventual correction
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12
Phonon dispersion
If we apply the FDM to a large supercell, we can extract information
on the whole Brillouin space.
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13
Phonon dispersion
The vibration could originate local long-range polarization, then it is
necessary to apply an analytical correction.
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14
Phonon Density of States
From the Phonon dispersion it is possible to obtain the Density of
vibrational States.
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15
Spectra
From the frequency at Gamma it is possible to obtain IR and Raman
Spectra:
Symmetry analysis for
selection rule
Polarizability variation ∂α/∂Q
Raman
Polarization (BEC) IR
|

16
MAPI - approach to complex materials
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18
Thermal properties
We can introduce T with basic thermodynamics at ﬁxed volume,
harmonic regime:
n = 1
exp( ω(qν)/kBT )−1
E = qν ω(qν) 1
2 + 1
exp( ω(qν)/kBT )−1
Z = exp(−ϕ/kBT) qν
exp(− ω(qν)/2kBT )
1−exp(− ω(qν)/kBT )
F = −kBT ln Z
S = −∂F
∂T
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19
Example Silicon
Speciﬁc heat at constant V: CV = ∂E/∂T
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20
Anharmonicity
Thermal expansion can not happen in harmonic regime and generally
we work in a constant P environment.
Quasi-Harmonic-Approximation (QHA) consist in a set of harmonic
phonon calculations at different constrained volumes.
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21
Anharmonicity
The calculation for QHA requires different set of basic phonon
calculation.
This typically makes calculation 5 times more expensive.
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22
QHA - Al
G(T, p) = minV [U(V ) + Fphonon(T; V ) + pV ] This give access to the
full spectra of thermodynamic quantities:
Bulk modulus (GPa) EOS
Gibbs free energy (eV) vs T
Volumetric expansion
Cp as −T∂2
G/∂T2
Cp from Maxwell relations
F vs V
|

23
Grüneisen - Si
The anharmonicity of the system is caught by the Grüneisen
parameter that express how the vibrations change in function of the
Temperature.
γ = V
dP
dE V
=
αKT
CV ρ
=
αKS
CP ρ
=
αv2
s
CP
|

25
Higher order anharmonicity
The QHA is still an harmonic theory[3]. To go beyond we need to
include phonon-phonon scattering. The cost of calculations goes as
( ˆN)2
Lattice thermal conductivity
Phonon lifetime/linewidth
Imaginary part of self energy
Joint density of states (JDOS)
and weighted-JDOS
|

26
Bibliography
[1] Min-Cheol Kim, Eunji Sim, and Kieron Burke. “Understanding and reducing errors in density functional calculations”. In: Physical
review letters 111.7 (2013), page 073003.
[2] Kurt Lejaeghere, Veronique Van Speybroeck, Guido Van Oost, and Stefaan Cottenier. “Error estimates for solid-state
density-functional theory predictions: an overview by means of the ground-state elemental crystals”. In: Critical Reviews in Solid
State and Materials Sciences 39.1 (2014), pages 1–24.
[3] Atsushi Togo, Laurent Chaput, and Isao Tanaka. “Distributions of phonon lifetimes in Brillouin zones”. In: Physical Review B 91.9
(2015), page 094306.
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27
Acknowledgement
Thank you for your attention! I also want to thanks:
- Prof. Nachtigall and the whole research group
- EU- European structural and investing funds and the MSMT
- Dr. Jonathan Skelton
|

28
Soft Modes
If the system is not STABLE, vibration can DECREASE the energy!
∆E = 1
2 mω2
x < 0 → ω ∈ C
This is the ﬁngerprint for a phase transition. During a PT, the T lowers
the energy of a vibrations, until that becomes negative and triggers
the rearrangement of atoms!
[doi:10.1007/s100510070248]
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29
It’s always the k-space
DFT potential depends on the k-point sampling.
Phonon potential is Harmonic! k-dependence is a FT! Supercell are
improving the force description.
Gamma soft mode = Unstable phase
Off-Gamma = Phase modulation -
need of a new unit cell
[10.1103/PhysRevB.91.144107]
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30
T spectra
Double set QHA can obtain properties at different V, hence T.
0 20 40 60 80 100
VibrationalDOS[a.u.]
Frequency [THz]
Compound 3 - Space group C2221
0 K
50 K
150 K
350 K
500 K
|

32
Higher order anharmonicity
The QHA is still an harmonic theory[3]. To go beyond we need to
include phonon-phonon scattering. The cost of calculations goes as
( ˆN)2
Lattice thermal conductivity
Phonon lifetime/linewidth
Imaginary part of self energy
Joint density of states (JDOS)
and weighted-JDOS
|

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