crystal relaxation, elasticity, and lattice...

Crystal Relaxation, Elasticity, and Lattice Dynamics

Pasquale PavoneHumboldt-Universität zu Berlin

http://exciting-code.org

PART I: Structure Optimization



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

newton

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

Visualization

Molecular Orbitals of CO2


HOMO


LUMO

Tutorials

PART II: Lattice Dynamics



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

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

<input>

…


…

<phonondos … > … </phonondos>

…

</phonons>

…

</input>

input.xml

<input>

…


…

<phonondos … > … </phonondos>

…

<phonondispplot … > … />

…

</phonons>

…

</input>

input.xml

Phonon Dispersion: Diamond

small ngridk + small ngridq

Phonon Dispersion: Diamond

larger ngridk + larger ngridq

PART III: Elasticity



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

Tutorials

Animated Phonon Modes

... and let‘s listen now to Andris!

The Last Slide: We are so Excited!

crystal relaxation, elasticity, and lattice...

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