Crystal Relaxation, Elasticity, and Lattice Dynamics
Pasquale PavoneHumboldt-Universität zu Berlin
http://exciting-code.org
PART I: Structure Optimization
Pasquale PavoneHumboldt-Universität zu Berlin
http://exciting-code.org
Outline Part I
Structure optimization
Cell optimization
Internal degrees of freedom
Relaxing molecules
HoW exciting! 2016
Structure Optimization
(a): Cell shape
(b): (Relative) atomic positions
stress
force
Energy Minimization
parabola paraboloid
Energy Minimization
Lattice (Cell) Optimization
𝐸 = 𝐸(𝒂, 𝒃, 𝒄, 𝛼, 𝛽, 𝛾)
𝐸 = 𝐸(𝑽, 𝒃, 𝒄, 𝛼, 𝛽, 𝛾)
Equation of State (EOS)
𝐸 = 𝐸(𝑽)
Murnaghan EOS
Birch-Murnaghan EOS
Vinet EOS
Polynomial EOS
X
Equation of State of Silver
Lattice Optimization in
Tool: OPTIMIZE-lattice.sh
Example 𝑬 = 𝑬 𝑽, 𝒄
– STEP1: opt. 𝑽 at fixed 𝒄: get 𝑽𝟏
– STEP2: opt. 𝒄 at fixed 𝑽𝟏: get 𝒄𝟐
– STEP3: opt. 𝑽 at fixed 𝒄𝟐: get 𝑽𝟑
– ...
Energy Minimization: Relaxation
Internal degree of freedom: atomic positions
Relaxation methods in
newton
harmonic
bfgs
harmonic
A parabola has a constant 2nd derivative
bfgs
Extension to N-degrees of freedom:
– Similar to harmonic
– Hessian matrix vs. 2nd derivative
– Very efficient if close to minimum
– Default in exciting
<input>
…
<structure … />
<groundstate … />
<relax method=´´bfgs´´>
<relax/>
</input>
input.xml
Relaxation of Pyridine
for Molecules
carbon dioxide
water
pyridinesexithiophene
for Molecules
In exciting always 3D periodicity:
for Molecules
‘‘Isolated‘‘ 3D periodical molecules:
<input>
…
<structure cartesian=´´true´´>
…
</structure>
…
<groundstate ngridk=´´1 1 1´´ …>
…
</groundstate>
<relax/>
</input>
input.xml
Relaxation of Pyridine
Molecular Orbitals of CO2
Molecular Orbitals of CO2
HOMO
Molecular Orbitals of CO2
LUMO
PART II: Lattice Dynamics
Pasquale PavoneHumboldt-Universität zu Berlin
http://exciting-code.org
Phonons: Atoms Move Together
A phonon in a crystal isa coherent collection of atomic displacements.
Phonons: Atoms Move Together
Phonons Oscillations
Oscillation in time:
𝒒 = wavevector
Oscillation in space:
𝝎 = frequency (→ energy)
Why Are Phonons Interesting
𝑹𝒖(𝑹)
𝑹′𝒖(𝑹′)
Atomic Displacements
= set of atomic displacements
})({uEu
E
}{u
E F Φ
'
2
uu
E
Energy, Forces, and Phonons
Force Constants & Dynamical Matrix
det 𝑫 𝒒 − 𝝎𝟐 𝒒 𝐈 = 0
𝜱(𝑹 − 𝑹′) =𝜕𝟐𝑬
𝜕𝒖(𝑹)𝜕𝒖(𝑹′)
𝑫 𝒒 = Fourier transform of 𝜱(𝑹)
Equilibrium
q ≠ 0
q = 0
Phonon Periodicity
Equilibrium
q ≠ 0
q = 0
Phonon Periodicity
Phonon at Γ in Diamond Structure
Equilibrium
q ≠ 0
q = 0
Phonon Periodicity
Phonon at X in Diamond Structure
Phonon at X in Diamond Structure
X
<input>
…
<phonons ngridq=´´2 2 2´´ … >
…
</phonons>
…
</input>
input.xml
Reciprocal and Real-Space Cells
Brillouin zone sampling
(Super)cell
Reciprocal and Real-Space Cells
Brillouin zone sampling
(Super)cell
Reciprocal and Real-Space Cells
Brillouin zone sampling
(Super)cell
<input>
…
<phonons ngridq=´´2 2 2´´ … >
…
<phonondos … > … </phonondos>
…
</phonons>
…
</input>
input.xml
<input>
…
<phonons ngridq=´´2 2 2´´ … >
…
<phonondos … > … </phonondos>
…
<phonondispplot … > … />
…
</phonons>
…
</input>
input.xml
Phonon Dispersion: Diamond
small ngridk + small ngridq
Phonon Dispersion: Diamond
larger ngridk + larger ngridq
PART III: Elasticity
Pasquale PavoneHumboldt-Universität zu Berlin
http://exciting-code.org
What Is Elasticity?
Description of distorsions of rigid bodies and ofthe energy, forces, and fluctuations arising fromthese distorsions.
Describes mechanics of extended bodies fromthe macroscopic to the microscopic.
Generalizes simple mechanical concepts
Force → Stress
Displacement → Strain
Strain: State of deformation
• Equilibrium:Zero strainZero forcesZero stressZero displacements
Uniaxial strainShear strain
Homogeneous strain
𝒓 =
𝒓𝐬 =
unstrained position
strained position
𝒓𝐬 = 𝑭 · 𝒓 = 𝟏 + 𝜺 · 𝒓
𝑭 = Deformation Matrix
𝜺 = Physical Strain Matrix
Voigt notation
Voigt indices:
Representative vector:
ε= (𝜀1, 𝜀2, 𝜀3, 𝜀4, 𝜀5, 𝜀6)
𝑖, 𝑗 = 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑦𝑧 or 𝑧𝑦 𝑥𝑧 or 𝑧𝑥 𝑥𝑦 or 𝑦𝑥
α = 1 2 3 4 5 6
Strain definitions
Physical strain:
Lagrangian strain:
𝒓 𝒓𝐬𝒓𝐬 = 𝟏 + 𝜺 · 𝒓
𝜟𝜟𝐬
𝜟𝐬2 − 𝜟 2 = 𝜟 · 2𝜼 · 𝜟
𝜼 = 𝜺 +1
2𝜺 · 𝜺
Linear elastic response
Low pressure expansion in terms of Lagrangian strain 𝜼 :
𝐸0, 𝑉0= Reference (equilibrium) energy and volume
𝐸 𝜼 = 𝐸0 +𝑉02!𝜼 · 𝑪 𝟐 · 𝜼 + ⋯
𝑪(𝟐) =1
𝑉0
𝜕2𝐸(𝜼)
𝜕𝜼𝜕𝜼𝜼=0
Linear elastic constant (2nd order):
Diamond
𝑪𝟏𝟏, 𝑪𝟏𝟐, 𝑪𝟒𝟒
Numerical derivatives
Fitting a polynomial to the calculated points
Animated Phonon Modes
... and let‘s listen now to Andris!
The Last Slide: We are so Excited!