introduction to phonon and spectroscopy calculations in...

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


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






Examples

LO/TO Splitting

Modelling ofspectra


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




ReferencesMonatomicCrystal in 1d (I)

MonatomicCrystal (II)

Diatomic Crystal- Optic modes

Characterizationof Vibrations inCrystals

Formal Theory ofLattice Dynamics

Quantum Theoryof Lattice ModesFormal Theory ofLattice DynamicsII



Examples

LO/TO Splitting

Modelling ofspectra


References












Examples

LO/TO Splitting

Modelling ofspectra


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)












Examples

LO/TO Splitting

Modelling ofspectra


Monatomic Crystal (II)












Examples

LO/TO Splitting

Modelling ofspectra


Diatomic Crystal - Optic modes












Examples

LO/TO Splitting

Modelling ofspectra


More than one atom per unit cell gives rise to optic modes with differentcharacteristic dispersion.

Characterization of Vibrations in Crystals












Examples

LO/TO Splitting

Modelling ofspectra


• 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












Examples

LO/TO Splitting

Modelling ofspectra


• 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












Examples

LO/TO Splitting

Modelling ofspectra


• 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












Examples

LO/TO Splitting

Modelling ofspectra


• 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





TheFrozen-phononmethodThe Finite-DisplacementmethodThe SupercellmethodFirst and secondderivativesDensity-FunctionalPerturbationTheory

FourierInterpolation ofdynamicalMatrices


Examples

LO/TO Splitting

Modelling ofspectra


The Frozen-phonon method








Examples

LO/TO Splitting

Modelling ofspectra


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








Examples

LO/TO Splitting

Modelling ofspectra


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








Examples

LO/TO Splitting

Modelling ofspectra


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








Examples

LO/TO Splitting

Modelling ofspectra


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








Examples

LO/TO Splitting

Modelling ofspectra


• 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








Examples

LO/TO Splitting

Modelling ofspectra


• 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





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


Methods in CASTEP






Example output




Examples

LO/TO Splitting

Modelling ofspectra


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






Example output




Examples

LO/TO Splitting

Modelling ofspectra


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






Example output




Examples

LO/TO Splitting

Modelling ofspectra


=====================================================================+ 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






Example output




Examples

LO/TO Splitting

Modelling ofspectra


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






Example output




Examples

LO/TO Splitting

Modelling ofspectra


• 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






Example output




Examples

LO/TO Splitting

Modelling ofspectra


• 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






Examples

DFPT withinterpolation -α-quartz

Supercell method- SilverConvergenceissues for latticedynamics

“Scaling” andother cheats

LO/TO Splitting

Modelling ofspectra


DFPT with interpolation - α-quartz






Examples




LO/TO Splitting

Modelling ofspectra


Γ K M Γ A0

200

400

600

800

1000

1200

1400

Freq

uenc

y [c

m-1

]

Supercell method - Silver






Examples




LO/TO Splitting

Modelling ofspectra


Γ X Γ L WK0

1

2

3

4

5

6

ω (

TH

z)

Ag phonon dispersion3x3x3x4 Supercell, LDA

Convergence issues for lattice dynamics






Examples




LO/TO Splitting

Modelling ofspectra


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






Examples




LO/TO Splitting

Modelling ofspectra


• 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






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


Two similar structures






Examples

LO/TO Splitting


LO/TO splitting




Example: NaHF2


Zincblende BN Diamond

Zincblende and diamond dispersion






Examples

LO/TO Splitting


LO/TO splitting




Example: NaHF2


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






Examples

LO/TO Splitting


LO/TO splitting




Example: NaHF2


• 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






Examples

LO/TO Splitting


LO/TO splitting




Example: NaHF2


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






Examples

LO/TO Splitting


LO/TO splitting




Example: NaHF2


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






Examples

LO/TO Splitting


LO/TO splitting




Example: NaHF2


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






Examples

LO/TO Splitting


LO/TO splitting




Example: NaHF2


• 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






Examples

LO/TO Splitting


LO/TO splitting




Example: NaHF2


• 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






Examples

LO/TO Splitting


LO/TO splitting




Example: NaHF2


• 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






Examples

LO/TO Splitting


LO/TO splitting




Example: NaHF2


• 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






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


Interaction of light with phonons






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




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






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




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






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




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






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




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






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




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






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




Symmetry Analysis






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




+ -------------------------------------------------------------------------- ++ 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






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




+ .......................................................................... ++ 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






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




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






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




• 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






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




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






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




• 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






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




IXS spectrometers at ESRF, Spring-8, APS synchrotrons.

IXS of Diaspore






Examples

LO/TO Splitting

Modelling ofspectra




SymmetryAnalysis




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)

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