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Intro to Phonons 2012 1 / 57
Introduction to phonon and spectroscopy calculations in CASTEP
Keith RefsonSTFC Rutherford Appleton Laboratory
May 30, 2012
Motivations for ab-initio lattice dynamics I
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 2 / 57
Motivations from experimental spectroscopy:
• Vibrational spectroscopy is sensitive probe ofstructure and dynamics of materials.
• All experimental methods (IR, raman, INS,IXS) provide incomplete information.
• IR and raman have inactive modes• Hard to distinguish fundamental and
overtone (multi-phonon) processes inspectra
• No experimental technique provides com-plete eigenvector information ⇒ modeassignment based on similar materials,chemical intuition, guesswork.
• Hard to find accurate model potentials to de-scribe many systems
• Fitted force-constant models only feasible forsmall, high symmetry systems. 0 1000 2000 3000 4000
Frequency (cm-1
)
0
0.2
0.4
0.6
0.8
Abs
orpt
ion
IR spectrum
Motivations for ab-initio lattice dynamics II
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 3 / 57
Motivations from predictive modelling
• Lattice dynamics calculation can establish stability or otherwise of putativestructure.
• LD gives direct information on interatomic forces.• LD can be used to study phase transitions via soft modes.• Quasi-harmonic lattice dynamics can include temperature and calculate
ZPE and Free energy of wide range of systems.• Electron - phonon coupling is origin of (BCS) superconductivity.
Lattice Dynamics of Crystals
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ReferencesMonatomicCrystal in 1d (I)
MonatomicCrystal (II)
Diatomic Crystal- Optic modes
Characterizationof Vibrations inCrystals
Formal Theory ofLattice Dynamics
Quantum Theoryof Lattice ModesFormal Theory ofLattice DynamicsII
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 4 / 57
References
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ReferencesMonatomicCrystal in 1d (I)
MonatomicCrystal (II)
Diatomic Crystal- Optic modes
Characterizationof Vibrations inCrystals
Formal Theory ofLattice Dynamics
Quantum Theoryof Lattice ModesFormal Theory ofLattice DynamicsII
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 5 / 57
Books on Lattice Dynamics
• M. T. Dove Introduction to Lattice Dynamics, CUP. - elementaryintroduction.
• J. C. Decius and R. M. Hexter Molecular Vibrations in Crystals - Latticedynamics from a spectroscopic perspective.
• Horton, G. K. and Maradudin A. A. Dynamical properties of solids (NorthHolland, 1974) A comprehensive 7-volume series - more than you’ll need toknow.
• Born, M and Huang, K Dynamical Theory of Crystal Lattices, (OUP, 1954)- The classic reference, but a little dated in its approach.
References on ab-initio lattice dynamics
• K. Refson, P. R. Tulip and S. J Clark, Phys. Rev B. 73, 155114 (2006)• S. Baroni et al (2001), Rev. Mod. Phys 73, 515-561.• Variational DFPT (X. Gonze (1997) PRB 55 10377-10354).• Richard M. Martin Electronic Structure: Basic Theory and Practical
Methods: Basic Theory and Practical Density Functional Approaches Vol 1
Cambridge University Press, ISBN: 0521782856
Monatomic Crystal in 1d (I)
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ReferencesMonatomicCrystal in 1d (I)
MonatomicCrystal (II)
Diatomic Crystal- Optic modes
Characterizationof Vibrations inCrystals
Formal Theory ofLattice Dynamics
Quantum Theoryof Lattice ModesFormal Theory ofLattice DynamicsII
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 6 / 57
Monatomic Crystal (II)
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ReferencesMonatomicCrystal in 1d (I)
MonatomicCrystal (II)
Diatomic Crystal- Optic modes
Characterizationof Vibrations inCrystals
Formal Theory ofLattice Dynamics
Quantum Theoryof Lattice ModesFormal Theory ofLattice DynamicsII
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 7 / 57
Diatomic Crystal - Optic modes
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ReferencesMonatomicCrystal in 1d (I)
MonatomicCrystal (II)
Diatomic Crystal- Optic modes
Characterizationof Vibrations inCrystals
Formal Theory ofLattice Dynamics
Quantum Theoryof Lattice ModesFormal Theory ofLattice DynamicsII
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 8 / 57
More than one atom per unit cell gives rise to optic modes with differentcharacteristic dispersion.
Characterization of Vibrations in Crystals
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ReferencesMonatomicCrystal in 1d (I)
MonatomicCrystal (II)
Diatomic Crystal- Optic modes
Characterizationof Vibrations inCrystals
Formal Theory ofLattice Dynamics
Quantum Theoryof Lattice ModesFormal Theory ofLattice DynamicsII
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 9 / 57
• Vibrational modes in solids take form of waves with wavevector-dependentfrequencies (just like electronic energy levels).
• ω(q) relations known as dispersion curves
• N atoms in prim. cell ⇒ 3N branches.• 3 acoustic branches corresponding to sound propagation as q → 0 and
3N − 3 optic branches.
Formal Theory of Lattice Dynamics
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ReferencesMonatomicCrystal in 1d (I)
MonatomicCrystal (II)
Diatomic Crystal- Optic modes
Characterizationof Vibrations inCrystals
Formal Theory ofLattice Dynamics
Quantum Theoryof Lattice ModesFormal Theory ofLattice DynamicsII
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 10 / 57
• Based on expansion of total energy about structural equilibrium co-ordinates
E = E0 +∑
κ,α
∂E
∂uκ,α.uκ,α +
1
2
∑
κ,α,κ′,α′
uκ,α.Φκ,κ′
α,α′ .uκ′,α′ + ...
where uκ,α is the vector of atomic displacements from equilibrium and
Φκ,κ′
α,α′ (a) is the matrix of force constants Φκ,κ′
α,α′ (a) =∂2E
∂uκ,α∂uκ′,α′
• At equilibrium the forces − ∂E∂uκ,α
are all zero so 1st term vanishes.
• In the Harmonic Approximation the 3rd and higher order terms areassumed to be negligible
• Assume Born von Karman periodic boundary conditions and substitutingplane-wave uκ,α = εmκ,αqexp(iq.Rκ,α − ωt) yields eigenvalue equation:
Dκ,κ′
α,α′ (q)εmκ,αq = ω2m,qεmκ,αq
where frequencies are square roos of eigenvalues. The dynamical matrix
Dκ,κ′
α,α′ (q) =1
√
MκMκ′
Cκ,κ′
α,α′ (q) =1
√
MκMκ′
∑
a
Φκ,κ′
α,α′ (a)e−iq.Ra
is the Fourier transform of the force constant matrix.
Quantum Theory of Lattice Modes
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ReferencesMonatomicCrystal in 1d (I)
MonatomicCrystal (II)
Diatomic Crystal- Optic modes
Characterizationof Vibrations inCrystals
Formal Theory ofLattice Dynamics
Quantum Theoryof Lattice ModesFormal Theory ofLattice DynamicsII
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 11 / 57
• The classical energy expression canbe transformed into a quantum-mechanical Hamiltonian for nuclei.
• In harmonic approximation nuclearwavefunction is separable into prod-uct by mode transformation.
• Each mode satisfies harmonic oscil-lator Schroedinger eqn with energylevels Em,n =
(
n+ 12
)
~ωm for modem.
• Quantum excitations of modesknown as phonons in crystal
• Transitions between levels n1 andn2 interact with photons of energy(n2 − n1) ~ωm, ie multiples of fun-
damental frequency ωm.• In anharmonic case where 3rd-order
term not negligible, overtone fre-quencies are not multiples of funda-mental.
-2 -1 0 1 20
1
2
3
Formal Theory of Lattice Dynamics II
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ReferencesMonatomicCrystal in 1d (I)
MonatomicCrystal (II)
Diatomic Crystal- Optic modes
Characterizationof Vibrations inCrystals
Formal Theory ofLattice Dynamics
Quantum Theoryof Lattice ModesFormal Theory ofLattice DynamicsII
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 12 / 57
• The dynamical matrix is a 3N × 3N matrix at each wavevector q.
• Dκ,κ′
α,α′ (q) is a hermitian matrix ⇒ eigenvalues ω2m,q are real.
• 3N eigenvalues ⇒ modes at each q leading to 3N branches in dispersioncurve.
• The mode eigenvector εmκ,α gives the atomic displacements, and itssymmetry can be characterised by group theory.
• Given a force constant matrix Φκ,κ′
α,α′ (a) we have a procedure for obtaining
mode frequencies and eigenvectors over entire BZ.• In 1970s force constants fitted to experiment using simple models.• 1980s - force constants calculated from empirical potential interaction
models (now available in codes such as GULP)• mid-1990s - development of ab-initio electronic structure methods made
possible calculation of force constants with no arbitrary parameters.
ab-initio Lattice Dynamics
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
TheFrozen-phononmethodThe Finite-DisplacementmethodThe SupercellmethodFirst and secondderivativesDensity-FunctionalPerturbationTheory
FourierInterpolation ofdynamicalMatrices
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 13 / 57
The Frozen-phonon method
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
TheFrozen-phononmethodThe Finite-DisplacementmethodThe SupercellmethodFirst and secondderivativesDensity-FunctionalPerturbationTheory
FourierInterpolation ofdynamicalMatrices
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 14 / 57
The frozen phonon method:
• Create a structure perturbed by guessed eigenvector• evaluate ground-state energy as function of amplitude λ with series of
single-point energy calculations on perturbed configurations.
• Use E0(λ) to evaluate k = d2E0dλ2
• Frequency given by√
k/µ. (µ is reduced mass)• Need to use supercell commensurate with q.• Need to identify eigenvector in advance (perhaps by symmetry).• Not a general method: useful only for small, high symmetry systems or
limited circumstances otherwise.• Need to set this up “by hand” customised for each case.
The Finite-Displacement method
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
TheFrozen-phononmethodThe Finite-DisplacementmethodThe SupercellmethodFirst and secondderivativesDensity-FunctionalPerturbationTheory
FourierInterpolation ofdynamicalMatrices
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 15 / 57
The finite displacement method:
• Displace ion κ′ in direction α′ by small distance ±u.• Use single point energy calculations and evaluate forces on every ion in
system F+κ,α and F+
κ,α for +ve and -ve displacements.• Compute numerical derivative using central-difference formula
dFκ,α
du≈F+κ,α − F−
κ,α
2u=
d2E0
duκ,αduκ′,α′
• Have calculated entire row k′, α′ of Dκ,κ′
α,α′ (q = 0)
• Only need 6Nat SPE calculations to compute entire dynamical matrix.• This is a general method, applicable to any system.• Can take advantage of space-group symmetry to avoid computing
symmetry-equivalent perturbations.• Like frozen-phonon method, works only at q = 0.
The Supercell method
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
TheFrozen-phononmethodThe Finite-DisplacementmethodThe SupercellmethodFirst and secondderivativesDensity-FunctionalPerturbationTheory
FourierInterpolation ofdynamicalMatrices
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 16 / 57
The supercell method is an extension of the finite-displacement approach.
• Relies on short-ranged nature of FCM; Φκ,κ′
α,α′ (a) → 0 as Ra → ∞.
• For non-polar insulators and most metals Φκ,κ′
α,α′ (a) decays as 1/R5 or faster.
• For polar insulators Coulomb term decays as 1/R3
• Can define “cut off” radius Rc beyond which Φκ,κ′
α,α′ (a) is treated as zero.
• In supercell with L > 2Rc then Cκ,κ′
α,α′ (q = 0) = Φκ,κ′
α,α′ (a).
• Method:
1. choose sufficiently large supercell and compute Cκ,κ′
α,α′ (qsupercell = 0)
using finite-displacement method.
2. This object is just the real-space force-constant matrix Φκ,κ′
α,α′ (a).
3. Fourier transform using
Dκ,κ′
α,α′ (q) =1
√
MκMκ′
∑
a
Φκ,κ′
α,α′ (a)e−iq.Ra
to obtain dynamical matrix of primitive cell at any desired q.
4. Diagonalise Dκ,κ′
α,α′ (q) to obtain eigenvalues and eigenvectors.
• This method is often (confusingly) called the “direct” method.
First and second derivatives
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
TheFrozen-phononmethodThe Finite-DisplacementmethodThe SupercellmethodFirst and secondderivativesDensity-FunctionalPerturbationTheory
FourierInterpolation ofdynamicalMatrices
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 17 / 57
Goal is to calculate the 2nd derivatives of energy to construct FCM or Dκ,κ′
α,α′ (q).
• Energy E = 〈ψ| H |ψ〉 with H = ∇2 + VSCF
• Force F = − dEdλ
= −⟨
dψdλ
∣
∣
∣H |ψ〉 − 〈ψ| H
∣
∣
∣
dψdλ
⟩
− 〈ψ| dVdλ
|ψ〉
where λ represents an atomic displacement perturbation.• If 〈ψ|represents the ground state of H then the first two terms vanish
because 〈ψ| H∣
∣
∣
dψdλ
⟩
= ǫn 〈ψ|∣
∣
∣
dψdλ
⟩
= 0. This is the Hellman-Feynmann
Theorem.• Force constants are the second derivatives of energy
k = d2Edλ2 = − dF
dλ=
⟨
dψdλ
∣
∣
∣
dVdλ
|ψ〉+ 〈ψ| dVdλ
∣
∣
∣
dψdλ
⟩
− 〈ψ| d2Vdλ2 |ψ〉
• None of the above terms vanishes.• Second derivatives need linear response of wavefunctions wrt perturbation
(⟨
dψdλ
∣
∣
∣).
• In general nth derivatives of wavefunctions needed to compute 2n+ 1th
derivatives of energy. This result is the “2n+ 1 theorem”
Density-Functional Perturbation Theory
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
TheFrozen-phononmethodThe Finite-DisplacementmethodThe SupercellmethodFirst and secondderivativesDensity-FunctionalPerturbationTheory
FourierInterpolation ofdynamicalMatrices
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 18 / 57
• In DFPT need first-order KS orbitals φ(1), the linear response to λ.• λ may be a displacement of atoms with wavevector q (or an electric field E.)• If q incommensurate φ(1) have Bloch-like representation:
φ(1)k,q
(r) = e−i(k+q).ru(1)(r) where u(1)(r) has periodicity of unit cell.
⇒ can store u(1)(r) in computer rep’n using basis of primitive cell.• First-order response orbitals are solutions of Sternheimer equation
(
H(0) − ǫ(0)m
) ∣
∣
∣φ(1)m
⟩
= −Pcv(1)
∣
∣
∣φ(0)m
⟩
Pc is projection operator onto unoccupied states. First-order potential v(1)
includes response terms of Hartree and XC potentials and therefore dependson first-order density n(1)(r) which depends on φ(1).Finding φ(1) is therefore a self-consistent problem just like solving theKohn-Sham equations for the ground state.
• Two major approaches to finding φ(1) are suited to plane-wave basis sets:
• Green’s function (S. Baroni et al (2001), Rev. Mod. Phys 73, 515-561).• Variational DFPT (X. Gonze (1997) PRB 55 10377-10354).
CASTEP uses Gonze’s variational DFPT method.• DFPT has huge advantage - can calculate response to incommensurate q
from a calculation on primitive cell.• Disadvantage of DFPT - lots of programming required.
Fourier Interpolation of dynamical Matrices
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
TheFrozen-phononmethodThe Finite-DisplacementmethodThe SupercellmethodFirst and secondderivativesDensity-FunctionalPerturbationTheory
FourierInterpolation ofdynamicalMatrices
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 19 / 57
• DFPT formalism requires self-consistent iterative solution for every separateq.
• Hundreds of q’s needed for good dispersion curves, thousands for goodPhonon DOS.
• Can take advantage of short-range nature of real-space FCM Φκ,κ′
α,α′ (a).
• Compute Dκ,κ′
α,α′ (q) on a Monkhorst-Pack grid of q vectors.
• Approximation to FCM in p× q × r supercell given by Fourier transform ofdynamical matrices on p× q × r grid.
Φκ,κ′
α,α′ (a) =∑
q
Cκ,κ′
α,α′ (q)eiq.Ra
• Fourier transform using to obtain dynamical matrix of primitive cell at any
desired q, Exactly as with Finite-displacement-supercell method
• Diagonalise mass-weighted Dκ,κ′
α,α′ (q) to obtain eigenvalues and eigenvectors.
• Longer-ranged coulombic contribution varies as 1/R3 but can be handledanalytically.
• Need only DFPT calculations on a few tens of q points on grid to calculate
Dκ,κ′
α,α′ (q) on arbitrarily dense grid (for DOS) or fine (for dispersion) path.
Lattice Dynamics in CASTEP
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEPMethods inCASTEPA CASTEPcalculation I -simple DFPT
Example output
CASTEP phononcalculations II -Interpolation
CASTEP phononcalculations III -Supercell
Running a phononcalculation
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 20 / 57
Methods in CASTEP
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEPMethods inCASTEPA CASTEPcalculation I -simple DFPT
Example output
CASTEP phononcalculations II -Interpolation
CASTEP phononcalculations III -Supercell
Running a phononcalculation
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 21 / 57
CASTEP can perform ab-initio lattice dynamics using
• Primitive cell finite-displacement at q = 0• Supercell finite-displacement for any q
• DFPT at arbitrary q.• DFPT on M-P grid of q with Fourier interpolation to arbitrary fine set of q.
Full use is made of space-group symmetry to only compute only
• symmetry-independent elements of Dκ,κ′
α,α′ (q)
• q-points in the irreducible Brillouin-Zone for interpolation• electronic k-points adapted to symmetry of perturbation.
Limitations: DFPT currently implemented only for norm-conservingpseudopotentials and non-magnetic systems.
A CASTEP calculation I - simple DFPT
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEPMethods inCASTEPA CASTEPcalculation I -simple DFPT
Example output
CASTEP phononcalculations II -Interpolation
CASTEP phononcalculations III -Supercell
Running a phononcalculation
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 22 / 57
Lattice dynamics assumes atoms at mechanical equilibrium.Golden rule: The first step of a lattice dynamics calculation is a
high-precision geometry optimisation
• Parameter task = phonon selects lattice dynamics calculation.• Iterative solver tolerance is phonon_energy_tol. Value of 1e− 5 ev/ang**2
usually sufficient. Sometimes need to increase phonon_max_cycles
• Need very accurate ground-state as prerequisite for DFPT calculationelec_energy_tol needs to be roughly square of phonon_energy_tol
• Internal defaults in CASTEP are very adequate, but check that MaterialsStudio has not chosen too large a value for elec_energy_tol.
• Dκ,κ′
α,α′ (q) calculated at q-points specified on STANDARD (coarse) set
• Set may either a symmetry-reduced and weighted M-P grid for DOS or ak-space path for dispersion.
• But it is cheaper and almost always better to use an inperpolation
calculation for DOS and dispersion.
Example output
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEPMethods inCASTEPA CASTEPcalculation I -simple DFPT
Example output
CASTEP phononcalculations II -Interpolation
CASTEP phononcalculations III -Supercell
Running a phononcalculation
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 23 / 57
=====================================================================+ Vibrational Frequencies ++ ----------------------- ++ ++ Performing frequency calculation at 10 wavevectors (q-pts) ++ ++ Branch number Frequency (cm-1) ++ ================================================================= ++ ++ ----------------------------------------------------------------- ++ q-pt= 1 ( 0.000000 0.000000 0.000000) 0.022727 ++ q->0 along ( 0.050000 0.050000 0.000000) ++ ----------------------------------------------------------------- ++ ++ 1 -4.041829 0.0000000 ++ 2 -4.041829 0.0000000 ++ 3 -3.927913 0.0000000 ++ 4 122.609217 7.6345830 ++ 5 122.609217 7.6345830 ++ 6 165.446374 0.0000000 ++ 7 165.446374 0.0000000 ++ 8 165.446374 0.0000000 ++ 9 214.139992 7.6742825 ++ ----------------------------------------------------------------- +
N.B. 3 Acoustic phonon frequencies should be zero by Acoustic Sum Rule.Post-hoc correction if phonon_sum_rule = T.
CASTEP phonon calculations II - Interpolation
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEPMethods inCASTEPA CASTEPcalculation I -simple DFPT
Example output
CASTEP phononcalculations II -Interpolation
CASTEP phononcalculations III -Supercell
Running a phononcalculation
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 24 / 57
To select set phonon_fine_method = interpolate
Specify “standard” grid of q-points. DFPT Dκ,κ′
α,α′ (q) will be calculated
explicitly on this grid.Golden rule of interpolation: Always include the Γ point (0,0,0) in theinterpolation grid. For even p, q, r use shifted gridphonon_fine_kpoint_mp_offset 0.125 0.125 0.125 to shift one point to Γ
Dκ,κ′
α,α′ (q) interpolated to q-points specified on FINE set, which may be grid for
DOS or a dispersion path.Real-space force-constant matrix is stored in .check file. All fine_kpointparameters can be changed on a continuation run. Interpolation is very fast. ⇒can calculate fine dispersion plot and DOS on a grid rapidly from one DFPTcalculation.Two methods of mapping large supercell dynamical matrix to force constantmatrix.
cumulant method of K. Parlinski (default)cutoff Parameter phonon_force_constant_cutoff applies real-space cutoff to
Φκ,κ′
α,α′ (a). Default is chosen according to MP grid.
CASTEP phonon calculations III - Supercell
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEPMethods inCASTEPA CASTEPcalculation I -simple DFPT
Example output
CASTEP phononcalculations II -Interpolation
CASTEP phononcalculations III -Supercell
Running a phononcalculation
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 25 / 57
• To select set phonon_fine_method = supercell
• Set supercell in .cell file, eg 2× 2× 2 using
%BLOCK phonon_supercell_matrix2 0 00 2 00 0 2%ENDBLOCK phonon_supercell_matrix
• Dκ,κ′
α,α′ (q) interpolated to q-points specified in cell file by one of same
phonon_fine_kpoint keywords as for interpolation.• Cumulant or Cutoff method of interpolation applies as for Interpolation
calculation.• Real-space force-constant matrix is stored in .check file.• As with interpolation, all fine_kpoint parameters can be changed on a
continuation run. Interpolation is very fast. ⇒ can calculate fine dispersionplot and DOS on a grid rapidly from one DFPT calculation.
• Convergence: Need very accurate forces to take their derivative.• Need good representation of any pseudo-core charge density and
augmentation charge for ultrasoft potentials on fine FFT grid. Usually needlarger fine_gmax (or fine_grid_scale) than for geom opt/MD to get goodresults.
Running a phonon calculation
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEPMethods inCASTEPA CASTEPcalculation I -simple DFPT
Example output
CASTEP phononcalculations II -Interpolation
CASTEP phononcalculations III -Supercell
Running a phononcalculation
Examples
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 26 / 57
• Phonon calculations can be lengthy. CASTEP saves partial calculationperiodically in .check file if keywords num_backup_iter n orbackup_interval t . Backup is every n q-vectors or every t seconds.
• Phonon calculations have high inherent parallelism. Because perturbationbreaks symmetry relatively large electronic k -point sets are used.
• Number of k-points varies depending on symmetry of perturbation.• Try to choose number of processors to make best use of k-point parallelism.
If Nk not known in advance choose NP to have as many different primefactors as possible - not just 2!
Examples
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
DFPT withinterpolation -α-quartz
Supercell method- SilverConvergenceissues for latticedynamics
“Scaling” andother cheats
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 27 / 57
DFPT with interpolation - α-quartz
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
DFPT withinterpolation -α-quartz
Supercell method- SilverConvergenceissues for latticedynamics
“Scaling” andother cheats
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 28 / 57
Γ K M Γ A0
200
400
600
800
1000
1200
1400
Freq
uenc
y [c
m-1
]
Supercell method - Silver
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
DFPT withinterpolation -α-quartz
Supercell method- SilverConvergenceissues for latticedynamics
“Scaling” andother cheats
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 29 / 57
Γ X Γ L WK0
1
2
3
4
5
6
ω (
TH
z)
Ag phonon dispersion3x3x3x4 Supercell, LDA
Convergence issues for lattice dynamics
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
DFPT withinterpolation -α-quartz
Supercell method- SilverConvergenceissues for latticedynamics
“Scaling” andother cheats
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 30 / 57
ab-initio lattice dynamics calculations are very sensitive to convergence issues.A good calculation must be well converged as a function of
1. plane-wave cutoff2. electronic kpoint sampling of the Brillouin-Zone (for crystals)
(under-convergence gives poor acoustic mode dispersion as q → 03. geometry. Co-ordinates must be well converged with forces close to zero
(otherwise calculation will return imaginary frequencies.)4. For DFPT calculations need high degree of SCF convergence of
ground-state wavefunctions.5. supercell size for “molecule in box” calculation and slab thickness for
surface/s lab calculation.6. Fine FFT grid for finite-displacement calculations.
• Accuracies of 25-50 cm−1 usually achieved or bettered with DFT.• need GGA functional e.g. PBE, PW91 for hydrogenous and H-bonded
systems.• When comparing with experiment remember that disagreement may be due
to anharmonicity.• Less obviously agreement may also be due to anharmonicity. There is a
“lucky” cancellation of anharmonic shift by PBE GGA error in OH stretchmodes!
“Scaling” and other cheats
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
DFPT withinterpolation -α-quartz
Supercell method- SilverConvergenceissues for latticedynamics
“Scaling” andother cheats
LO/TO Splitting
Modelling ofspectra
Intro to Phonons 2012 31 / 57
• DFT usually gives frequencies within a few percent of experiment.Exceptions are usually strongly-correlated systems, e.g. sometransition-metal Oxides where DFT description of bonding is poor.
• Discrepancies can also be due to anharmonicity. A frozen-phononcalculation can test this.
• In case of OH-bonds, DFT errors and anharmonic shift cancel each other!• In solid frquencies may be strongly pressure-dependent. DFT error can
resemble effective pressure. In that case, best comparison with expt. maynot be at experimental pressure.
• Hartree-Fock approximation systematically overestimates vibrationalfrequencies by 5-15%. Commpn practice is to multiply by ”scaling factor”≈ 0.9.
• Scaling not recommended for DFT where error is not systematic. Over- andunder-estimation equally common.
• For purposes of mode assignment, or modelling experimental spectra tocompare intensity it can sometimes be useful to apply a small empirical shifton a per-peak basis. This does not generate an “ab-initio frequency”.
LO/TO Splitting
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Two similarstructuresZincblende anddiamonddispersion
LO/TO splitting
DFPT withLO/TO splittingin NaClLO/TO splittingin non-cubicsystems
The non-analyticterm and BornCharges
Electric Fieldresponse inCASTEPIonic andElectronicDielectricPermittivity
Example: NaHF2
Intro to Phonons 2012 32 / 57
Two similar structures
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Two similarstructuresZincblende anddiamonddispersion
LO/TO splitting
DFPT withLO/TO splittingin NaClLO/TO splittingin non-cubicsystems
The non-analyticterm and BornCharges
Electric Fieldresponse inCASTEPIonic andElectronicDielectricPermittivity
Example: NaHF2
Intro to Phonons 2012 33 / 57
Zincblende BN Diamond
Zincblende and diamond dispersion
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Two similarstructuresZincblende anddiamonddispersion
LO/TO splitting
DFPT withLO/TO splittingin NaClLO/TO splittingin non-cubicsystems
The non-analyticterm and BornCharges
Electric Fieldresponse inCASTEPIonic andElectronicDielectricPermittivity
Example: NaHF2
Intro to Phonons 2012 34 / 57
Phonon dispersion curves
Γ X Γ L W X0
500
1000
1500ω
(cm
-1)
BN-zincblende
Γ X Γ L W X0
500
1000
1500
2000
ω (
cm-1
)
diamond
Cubic symmetry of Hamiltonian predicts triply degenerate optic mode at Γ inboth cases.2+1 optic mode structure of BN violates group theoretical prediction.Phenomenon known as LO/TO splitting.
LO/TO splitting
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Two similarstructuresZincblende anddiamonddispersion
LO/TO splitting
DFPT withLO/TO splittingin NaClLO/TO splittingin non-cubicsystems
The non-analyticterm and BornCharges
Electric Fieldresponse inCASTEPIonic andElectronicDielectricPermittivity
Example: NaHF2
Intro to Phonons 2012 35 / 57
• Dipole created by displacement of charges of long-wavelength LO modecreates induced electric field.
• For TO motion E ⊥ q ⇒ E.q = 0• For LO mode E.q 6= 0 and E-field adds additional restoring force.• Frequency of LO mode is upshifted.
• Lyddane-Sachs-Teller relation for cubic case:ω2LO
ω2TO
= ǫ0ǫ∞
• LO frequencies at q = 0 depend on dielectric permittivity
DFPT with LO/TO splitting in NaCl
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Two similarstructuresZincblende anddiamonddispersion
LO/TO splitting
DFPT withLO/TO splittingin NaClLO/TO splittingin non-cubicsystems
The non-analyticterm and BornCharges
Electric Fieldresponse inCASTEPIonic andElectronicDielectricPermittivity
Example: NaHF2
Intro to Phonons 2012 36 / 57
LO modes can be seen in infrared, INS, IXS experiments, but not raman.
Γ X Γ L W Xq
0
50
100
150
200
250
300ω
(cm
-1)
Exper: G. Raunio et al
NaCl phonon dispersion
(∆) (Σ) (∆) (Q) (Z)
LO/TO splitting in non-cubic systems
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Two similarstructuresZincblende anddiamonddispersion
LO/TO splitting
DFPT withLO/TO splittingin NaClLO/TO splittingin non-cubicsystems
The non-analyticterm and BornCharges
Electric Fieldresponse inCASTEPIonic andElectronicDielectricPermittivity
Example: NaHF2
Intro to Phonons 2012 37 / 57
Γ K M Γ A0
500
1000
1500
2000
ω (
cm-1
)
In systems with unique axis(trigonal, hexagonal, tetragonal)LO-TO splitting depends on di-
rection of q even at q = 0.
LO frequency varies with anglein α-quartz
0 30 60 90 120 150 180Approach Angle
480
490
500
510
520
530
540
550
Zon
e C
ente
r Fr
eque
ncy
(1/c
m)
The non-analytic term and Born Charges
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Two similarstructuresZincblende anddiamonddispersion
LO/TO splitting
DFPT withLO/TO splittingin NaClLO/TO splittingin non-cubicsystems
The non-analyticterm and BornCharges
Electric Fieldresponse inCASTEPIonic andElectronicDielectricPermittivity
Example: NaHF2
Intro to Phonons 2012 38 / 57
LO-TO splitting at q → 0 is automatically included in DFPT. At q = 0 exactly,need to add additional non-analytic term to dynamical matrix
Cκ,κ′
α,α′ (q = 0)(NA) =4π
Ω0
∑
γ qγZ∗κ,γα
∑
γ′ qγ′Z∗κ′,γ′α′
∑
γγ′ qγǫ∞γγ′
qγ′
ǫ∞γγ′
is the dielectric permittivity tensor
Z∗κ,βα is the Born Effective Charge tensor
Z∗κ,β,alpha = V
∂Pβ
∂xκ,α=∂Fκ,α
∂Eβ
Z∗ is polarization per unit cell caused by displacement of atom κ in direction αor force exerted on ion by macroscopic electric field.Z∗κ,βα and ǫ∞
γγ′can both be computed via DFPT response to electric field
perturbation.(Fourier interpolation procedure uses a dipole-dipole model for Coulombic tailcorrection of FCM which also depends on Z∗ and ǫ∞. Not included in mostsupercell calculations ⇒ be suspicious of convergence in case of polar systems).
Electric Field response in CASTEP
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Two similarstructuresZincblende anddiamonddispersion
LO/TO splitting
DFPT withLO/TO splittingin NaClLO/TO splittingin non-cubicsystems
The non-analyticterm and BornCharges
Electric Fieldresponse inCASTEPIonic andElectronicDielectricPermittivity
Example: NaHF2
Intro to Phonons 2012 39 / 57
• Can also apply DFPT to electric field perturbation.• Need trick to evaluate position operator. Evaluate ∇kφ. See NMR lecture
for details.• set task = efield or task = phonon+efield
• Convergence controlled by efield_energy_tol
• Computes dielectric permittivity for crystals and polarisability (formolecules)
• Includes lattice contribution for ω → 0 response.• Writes additional file seedname.efield containing ǫγγ′ (ω) in infrared
region.
===============================================================================Optical Permittivity (f->infinity) DC Permittivity (f=0)---------------------------------- ---------------------3.52475 0.00000 0.00000 10.00569 0.00000 0.000000.00000 3.52475 0.00000 0.00000 10.00569 0.000000.00000 0.00000 3.52475 0.00000 0.00000 10.00569
===============================================================================
===============================================================================Polarisabilities (A**3)
Optical (f->infinity) Static (f=0)--------------------- -------------
5.32709 0.00000 0.00000 19.00151 0.00000 0.000000.00000 5.32709 0.00000 0.00000 19.00151 0.000000.00000 0.00000 5.32709 0.00000 0.00000 19.00151
===============================================================================
Ionic and Electronic Dielectric Permittivity
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Two similarstructuresZincblende anddiamonddispersion
LO/TO splitting
DFPT withLO/TO splittingin NaClLO/TO splittingin non-cubicsystems
The non-analyticterm and BornCharges
Electric Fieldresponse inCASTEPIonic andElectronicDielectricPermittivity
Example: NaHF2
Intro to Phonons 2012 40 / 57
• Pure covalent system - Si Z∗ = −0.009,• Ionic system, NaCl. ǫ0 = 6.95, ǫ∞ = 2.70. Z∗(Na) = 1.060,
Z∗(Cl) = −1.058• α-quartz, SiO2:
ǫ∞ =2.44 0.00 0.000.00 2.44 0.000.00 0.00 2.49
ǫ0 =5.01 0.00 0.000.00 4.71 0.150.00 0.15 4.58
(exp: ǫ0 = 4.64/4.43)
Z∗(O1) =
−2.16 −0.03 0.81−0.08 −1.04 0.140.00 0.00 −1.71
Z∗(Si) =
2.98 0.00 0.000.00 3.67 0.260.00 −0.30 3.45
• Note anisotropic tensor character of Z∗.
Example: NaHF2
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Two similarstructuresZincblende anddiamonddispersion
LO/TO splitting
DFPT withLO/TO splittingin NaClLO/TO splittingin non-cubicsystems
The non-analyticterm and BornCharges
Electric Fieldresponse inCASTEPIonic andElectronicDielectricPermittivity
Example: NaHF2
Intro to Phonons 2012 41 / 57
• Unusual bonding, nominally ionic Na+ FHF−,with linear anion.
• Layer structure, R3 space group.• Phase transition to monoclinic form at ≈ 0.4 GPa.
• Phonon spectrum measured at ISIS on TOSCA
• high-resolution neutron powder spectrometer.
• No selection rule absences• Little or no anharmonic overtone contamina-
tion of spectra• No control over (q, ω) path - excellent for
molecular systems but spectra hard to inter-pret if dispersion present.
Dynamical charges in NaHF2
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Two similarstructuresZincblende anddiamonddispersion
LO/TO splitting
DFPT withLO/TO splittingin NaClLO/TO splittingin non-cubicsystems
The non-analyticterm and BornCharges
Electric Fieldresponse inCASTEPIonic andElectronicDielectricPermittivity
Example: NaHF2
Intro to Phonons 2012 42 / 57
• highly anisotropic Z∗
xy z0.34 2.02 (H)
−0.69 −1.52 (F)1.06 1.04 (Na)
• charge density n(1)(r) from DFPTshows electronic response underperturbation
• Charge on F ions moves in responseto H displacement in z direction.
n(1)(r) response to Hx(blue) and Hz (green)
n(1)(r) response to Ex(blue) and Ez (green)
Modelling of spectra
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 43 / 57
Interaction of light with phonons
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 44 / 57
Energy conservation Es = hνs = Ei + δEphonon = hνi + hνphonon
Momentum conservation
• ks = ki + δkphonon
• Typical lattice parameter a0 ≈ 5 ⇒ 0 ≤ kphonon < 0.625 1/A
• For a 5000A photon, k = 0.00125 1/A• =⇒ optical light interacts with phonons only very close to 0 (the “Γ” point).
Powder infrared
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 45 / 57
In a powder infrared absorption experiment, E field of light couples to electric
dipole created by displacement of ions by modes. Depends on Born effectivecharges and atomic displacements.
Infrared intensity Im =∣
∣
∣
∑
κ,b1√Mκ
Z∗κ,a,bum,κ,b
∣
∣
∣
2
α-quartz
400 600 800 1000 1200
Phonon Frequency (cm-1
)
0
0.05
0.1
0.15
IR I
nten
sity
(A
rb. U
nits
)
• Straightforward to compute peak areas.• Peak shape modelling depends on sample and experimental variables.• Multiphonon and overtone terms less straightforward.
Accurate Modelling of powder IR
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 46 / 57
Need to include optical cavity effects of light interation with crystallitesize/shape. Size/shape shift of LO modes by crystallites.See E. Balan, A. M. Saitta, F. Mauri, G. Calas. First-principles modeling of the
infrared spectrum of kaolinite. American Mineralogist, 2001, 86, 1321-1330.
Single-crystal infrared
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 47 / 57
Prediction of reflectivity of optically flat single crystal surface given as functionof q - projected permittivity ǫq(ω)
R(ω) =
∣
∣
∣
∣
∣
ǫ1/2q (ω)− 1
ǫ1/2q (ω) + 1
∣
∣
∣
∣
∣
2
with ǫq defined in terms of ǫ∞ andmode oscillator strength Sm,αβ
ǫq(ω) = q.ǫ∞.q +4π
Ω0
∑
m
q.S.q
ω2m − ω2
= q.ǫ(ω).q
ǫ(ω) is tabulated in the seed.efield
file written from a CASTEP efield re-sponse calculation.Example BiFO3 (Hermet et al, Phys.Rev B 75, 220102 (2007)
Non-resonant raman
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 48 / 57
Raman scattering depends on raman activity tensor
Iramanαβ =
d3E
dεαdεβdQm=dǫαβ
dQm
i.e. the activity of a mode is the derivative of the dielectric permittivity withrespect to the displacement along the mode eigenvector.CASTEP evaluates the raman tensors using hybrid DFPT/finite displacementapproach.Raman calculation is fairly expensive ⇒ and is not activated by default (thoughgroup theory prediction of active modes is still performed)Parameter calculate_raman = true in a task=phonon calculation.Spectral modelling of IR spectrum is relatively simple function of activity.
dσ
dΩ=
(2πν)4
c4|eS .I.eL|
2 h(nm + 1)
4πωm
with the thermal population factor
nm =
[
exp
(
~ωm
kBT
)
− 1
]−1
which is implemented in Materials Studio “raman” calculation.
Raman scattering of t-ZrO2
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 49 / 57
Symmetry Analysis
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 50 / 57
+ -------------------------------------------------------------------------- ++ q-pt= 1 ( 0.000000 0.000000 0.000000) 0.000765 ++ q->0 along ( 0.003623 0.000000 0.000000) ++ -------------------------------------------------------------------------- ++ Acoustic sum rule correction < 0.949591 THz applied ++ N Frequency irrep. ir intensity active raman active ++ (THz) ((D/A)**2/amu) ++ ++ 1 -0.000975 a 0.0000000 N N ++ 2 -0.000719 b 0.0000000 N N ++ 3 0.065030 c 0.0000000 N N ++ 4 2.533116 d 0.0000000 N N ++ 5 3.100104 a 150.2988903 Y N ++ 6 3.534759 b 92.9297454 Y N ++ 7 3.818109 e 0.0000000 N Y ++ 8 10.158077 c 1.7805122 Y N ++ 9 11.124650 b 10.9933170 Y N ++ 10 11.421874 d 0.0000000 N N ++ 11 12.097042 f 0.0000000 N Y ++ 12 12.844543 c 2.5718809 Y N ++ 13 13.673018 g 0.0000000 N Y ++ 14 13.673018 g 0.0000000 N Y ++ 15 14.179285 b 22.0325348 Y N ++ 16 17.922800 e 0.0000000 N Y ++ 17 23.317934 c 121.6365963 Y N ++ 18 24.081857 f 0.0000000 N Y ++ .......................................................................... +
Symmetry Analysis
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 51 / 57
+ .......................................................................... ++ Character table from group theory analysis of eigenvectors ++ Point Group = 15, D4h ++ (Due to LO/TO splitting this character table may not contain some symmetry ++ operations of the full crystallographic point group. Additional ++ representations may be also be present corresponding to split LO modes. ++ A conventional analysis can be generated by specifying an additional null ++ (all zero) field direction or one along any unique crystallographic axis ++ in %BLOCK PHONON_GAMMA_DIRECTIONS in <seedname>.cell.) ++ .......................................................................... ++ Rep Mul | E 2 2 2 I m m m ++ | -------------------------------- ++ a B3u 2 | 1 -1 -1 1 -1 1 1 -1 ++ b B2u 4 | 1 -1 1 -1 -1 1 -1 1 ++ c B1u 4 | 1 1 -1 -1 -1 -1 1 1 ++ d Au 2 | 1 1 1 1 -1 -1 -1 -1 ++ e Ag 2 | 1 1 1 1 1 1 1 1 ++ f B3g 2 | 1 -1 -1 1 1 -1 -1 1 ++ g Eg 1 | 2 0 0 -2 2 0 0 -2 ++ -------------------------------------------------------------------------- +
Inelastic Neutron Scattering
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 52 / 57
Thermal neutrons have both energies and momenta of same magnitude asphonons (unlike photons). ⇒ Can interact with phonons at any q.To model spectra need to treat scattering dynamics of incident and emergentradiation.In case of INS interaction is between point neutron and nucleus - scalarquantity b depends only on nucleus – specific properties.
d2σ
dEdΩ=kf
kib2S(Q, ω)
Q is scattering vector and ω is frequency - interact with phonons at samewavevector and frequency.Full measured spectrum includes overtones and combinations and instrumentalgeometry and BZ sampling factors.Need specific spectral modelling software to incorporate effects aspostprocessing step following CASTEP phonon DOS calculation.A-Climax : A. J. Ramirez-Cuesta Comput. Phys. Comm. 157 226 (2004))
The TOSCA INS Spectrometer
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 53 / 57
• ISIS pulsed neutron source at Rutherford Appleton Laboratory, UK• TOSCA - high-resolution powder spectrometer at ISIS• Little or no anharmonic overtone contamination of spectra• No selection rule absences• Signal is weighted DOS integral over BZ
Dispersion, DOS and Gamma
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 54 / 57
Phonons represented across the Brillouin Zone by
Dispersion Curve ωm(k) on high-symmetry linesPhonon Density of States g(ω) =
∫
dk∑
m δ(ωm(k)− ω)
Γ X R Z Γ M A Z0
200
400
600
800
ω (
cm-1
)
g(r)
INS of Ammonium Fluoride
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 55 / 57
• NH4F is one of a series of ammonium halidesstudied in the TOSCA spectrometer. Collab.Mark Adams (ISIS)
• Structurally isomorphic with ice ih• INS spectrum modelled using A-CLIMAX
software (A. J. Ramirez Cuesta, ISIS)• Predicted INS spectrum in mostly excellent
agreement with experiment• NH4 libration modes in error by ≈ 5%.• Complete mode assignment achieved.
Inelastic X-Ray scattering
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 56 / 57
IXS spectrometers at ESRF, Spring-8, APS synchrotrons.
IXS of Diaspore
Motivations forab-initio latticedynamics I
Motivations forab-initio latticedynamics II
Lattice Dynamicsof Crystals
ab-initio LatticeDynamics
Lattice Dynamicsin CASTEP
Examples
LO/TO Splitting
Modelling ofspectra
Interaction oflight withphonons
Powder infraredAccurateModelling ofpowder IR
Single-crystalinfraredNon-resonantramanRaman scatteringof t-ZrO2SymmetryAnalysis
SymmetryAnalysis
Inelastic NeutronScattering
The TOSCA INSSpectrometer
Intro to Phonons 2012 57 / 57
B. Winkler et el., Phys. Rev. Lett 101 065501 (2008)
wave vector [A -1]
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
ener
gy
[meV
]
355
360
365
370
375
380(0.5, 0, 0) to (0, 0, -0.33)