Lattice Dynamics - University of Yorkmijp1/teaching/grad_FPMM/lecture_n… · Supercell needs to be...

Lattice Dynamics

Peter Byrne

October 2019

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LatticeDynamics

Peter Byrne

Motivation

LatticeDynamicsof Crystals

Ab initioLatticeDynamics

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LatticeDynamicsin CASTEP

PhononExamples

QuantitiesfromPhonons

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Outline

1 Motivation

2 Lattice Dynamics of Crystals

3 Ab initio Lattice Dynamics

4 Break

5 Lattice Dynamics in CASTEP

6 Phonon Examples

7 Quantities from Phonons

Motivation

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Spectroscopy

Experiments measure response of a system to a perturbationProbes dynamic properties of crystalNot ground state directly!

Spectroscopic techniques provideincomplete information

IR and Raman have inactive modesHard to distinguish fundamental andovertones processes in spectra

Little information on which atomsinvolved means that mode assignment isdifficult

Would like a predictive technique thatdoes not rely on intuition to calculatevibrational responses within a crystal.

0 1000 2000 3000 4000

Frequency (cm-1

)

0

0.2

0.4

0.6

0.8

Abso

rpti

on

IR spectrum

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Why ab initio?

ab initio methods give us a highly transferable, parameter-free probe ofthe experimental results.

Calculate vibrational properties on the same theoretical basis as electronicproperties.Can probe whether a structure is stable wrt perturbationsCan compute zero point energy and phonon entropy contributions to freeenergy.Predict Raman and IR peaksCaptures the effects of electron-phonon interactions

Lattice Dynamics of Crystals

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LatticeDynamics

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1d Chain of Atoms – I

Start with infinite 1d chain of atoms connected by springs (force constant K )

Equilibrium separation is a. un is the displacement of an atom from equilibriumposition.Assuming only nearest neighbours interact, the force between neighbors i and i + 1 is

Fi,i+1 = �K (un+1 � un)

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1d Chain of Atoms – II

The total force on an atom i is the sum of both nearest neighbours which pull inopposite directions

Fi = Fi,i+1 � Fi�1,i = �K (un+1 � un) + K (un � un�1) = K (2un � un+1 � un�1)

We use Newton’s second law, F = ma so,

Md2ui

dt2 = K (2un � un+1 � un�1)

A known solution of this differential equation is a travelling wave

un,q(t) = eun,qei(qx�!q t)

where q = 2⇡� is a wavenumber and !q is an angular frequency. eun,q is a vector

representing the motion of atom n.

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1d Chain of Atoms – III

Substituting this into the differential equation gives us

M!2q = 2K [1 + cos(qa)]

This leads to the dispersion relation:

!q =

r4K

M|sin(qa/2)|

Single solution or branch foreach value of q.

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Diatomic Crystal – Optic modes I

If instead, we attempt to determine the modes for a crystal with two differentmasses of atoms:

We find that there are now multiple values for ! that will satisfy thedifferential equation for each q

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Diatomic Crystal – Optic modes II

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

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Diatomic Crystal – Optic modes III

Reminder from earlier: eun,q is a vector representing the motion of atom n.If we examine these for both modes at the � = (0, 0, 0) point, we see

Accoustic Modes Atoms move in-phase (same direction at the same time)Optical Modes Atoms move anti-phase

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Characterization of Vibrations in 3D Crystals

Vibrational modes in solids take form of waves withwavevector-dependent frequencies (just like electronic energy levels).!(q) relations known as dispersion curves

N atoms in prim. cell ) 3N branches.3 acoustic branches corresponding to sound propagation as q ! 0 and3N � 3 optic branches.

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Setting up the 3d problem

Firstly, we will fill space with periodically repeating unit cells in 3dimensionsTake one as an (arbitrary) originLabel the rest with respect to thisLabel the unit cells

a = (a1, a2, a3)

where l1, l2 and l3 are integers

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Coordinate axes

The shape of each unit cell is defined by 3 linearly independent vectorsa1, a2 and a3The origin of the a

th unit cell can be defined as

ra = a1a1 + a2a2 + a3a3

relative to the origin a=(0,0,0)This co-ordinate system gives us the origin of any of the periodicallyrepeating unit cells throughout space

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Placement of Atoms

Within each unit cell we will place n atomsWe will label these atoms = 1, 2, . . . , nRelative to the origin of a given unit cell (position rl ) the atoms will beplaced at positions rSo that each atom will have position

r,a = ra + r

relative to our “origin” of the unit cells

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Atomic Motion

Now we’ve defined the equilibrium positions of the atoms, we need tomove them to describe thermal motionThe atoms move an amount

u,a = (ux,,a, uy,,a, uz,,a)

from their equilibrium position ra,

The actual position of atom (a,) under thermal motion is then

R,a = r,a + u,a

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Diagram of coordinate system

ra

rκ

uκa

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Formal Theory of Lattice Dynamics

Based on expansion of total energy about structural equilibrium co-ordinates

E = E0 +@E

@u.u +

12!@2E

@u2 .u2 +13!@3E

@u3 u3 + ...

At equilibrium the forces F↵, = � @E

@uare all zero so 1st term vanishes.

E = E0 +12

Xu↵,,a.�

,0

↵,↵0 .u0,↵0,a + ...

where u↵,,a is the displacement of atom in unit cell a in Cartesian direction ↵.

In the Harmonic Approximation the 3rd and higher order terms are assumed tobe negligible

�,0

↵,↵0 (a) is the matrix of force constants

�,0

↵,↵0 (a) =@2E

@u↵,@u0,↵0

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The Force Constant Matrix (FCM)

This matrix represents all the effective3d spring constants between atoms

�,0

↵,↵0(a) =@2

E

@u↵,@u0,↵0

= �@Fu↵,,a

@u0,↵0,a

Alternative view is change on force onatoms due to displacing an atom

Rc

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The Dynamical Matrix (DM)

Solution in 1d can be reused with a few modifications for 3d:

u↵, = "m↵,qeiq.R↵,�!t

Taking the derivative of the total energy equation to get the force, F and substitutingthis trial solution, we have

D,0

↵,↵0 (q)"m↵,q = !2m,q"m↵,q

where

D,0

↵,↵0 (q) =1

pMM0

C,0

↵,↵0 (q) =1

pMM0

X

a

�,0

↵,↵0 (a)e�iq.Ra

κ

κ'

q=a*/8

The dynamical matrix D,0

↵,↵0 (q) isthe Fourier transform of the forceconstant matrix.The solutions of the eigenvalueequation correspond to vibrationalmodesMode frequency is square root ofcorresponding eigenvalue !m,q .

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Formal Theory of Lattice Dynamics II

The dynamical matrix is a 3N ⇥ 3N matrix at each wavevector q.

D,0

↵,↵0(q) is a hermitian matrix ) eigenvalues !2m,q are real.

3N eigenvalues ) modes at each q leading to 3N branches indispersion curve.The mode eigenvector "m↵, gives the atomic displacements, and itssymmetry can be characterised by group theory.

Given a force constant matrix �,0

↵,↵0(a) we have a procedure forobtaining mode frequencies and eigenvectors over entire BZ.

Ab initio Lattice Dynamics

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The Finite-Displacement method

The finite displacement method:Displace ion 0 in direction ↵0 by small distance ±u.Use single point energy calculations and evaluate forces on every ion insystem F

+,↵ and F

+,↵ for +ve and -ve displacements.

Compute numerical derivative using central-difference formula

dF,↵

du⇡

F+,↵ � F

�,↵

2u=

d2E0

du,↵du0,↵0

Have calculated entire row k0,↵0 of D

,0

↵,↵0(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 computingsymmetry-equivalent perturbations.Works only at q = 0.

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Non-diagonal supercell method

New method by J. Lloyd-Williams and B. Monserrat, Phys Rev B, 92,184301 (2015).Extension of finite displacement methodOld "Direct" Supercell method calculates the FCM for each atom

Construct supercell big enough that we can ignore periodicitySupercell needs to be big enough that interactions fall to zeroOften requires very large calculations with lots of atoms

Non-diagonal supercell method takes advantage of periodicity ofsystem

Calculates response at q by constructing a minimal supercell.Supercells are much smaller than those in the supercell method.Very efficient and can calculate DM on arbitrary grid.

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Second derivatives in QM

Goal is to calculate the 2nd derivatives of energy to construct FCM or D,0

↵,↵0 (q).

Energy E = h | H | i with H = � 12r

2 + VSCF

Forces (first derivative) can be shown to be

F = �dE

d�= �h |

dV

d�| i

Force constants are the second derivatives of energy

k =d2E

d�2 = �dF

d�=

⌧d

d�

��dV

d�| i+ h |

dV

d�

��d

d�

�� h |

d2V

d�2 | i

None of the above terms vanishes.

Need linear response of wavefunctions wrt perturbation (ieD

d d�

��).

In general nth derivatives of wavefunctions needed to compute 2n + 1th

derivatives of energy. This result is the “2n + 1 theorem”.

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Density-Functional Perturbation Theory

In DFPT need first-order KS orbitals �(1), the linear response to �.� may be a displacement of atoms with wavevector q (or an electric field E.)

If q incommensurate �(1) have Bloch-like wavefunction:�(1)k,q(r) = e�i(k+q).r u(1)(r) where u(1)(r) has periodicity of unit cell.

First-order density n(1)(r) and potential v (1) have similar Bloch representation.

First-order response orbitals are solutions of Sternheimer equation⇣

H(0) � ✏

(0)m

⌘ ��(1)m

E= �v

(1)��(0)m

E

First-order potential v (1) includes response terms of Hartree and XC potentialsand therefore depends on first-order density n(1)(r) which depends on �(1).

Finding �(1) is therefore a self-consistent problem just like solving theKohn-Sham equations for the ground state.

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Variational and Green function approaches

Two major approaches to finding �(1) are suited to plane-wave basis sets:

Variational DFPT (X. Gonze (1997) PRB 55 10377-10354).Conjugate-gradient minimization of variational 2nd-order energyexpression subject orthogonality constraint

D�(1)n |�(0)m

E= 0

Green function (S. Baroni et al (2001), Rev. Mod. Phys 73, 515-561).Solve Sternheimer equation in self-consistent loop with1st-order density mixing.

CASTEP implements both DFPT methods (phonon_dfpt_method).Variational DFPT implemented for insulators only, Green function/DM for bothinsulators and metals.DFPT has huge advantage - can calculate response to incommensurate q froma calculation on primitive cell.Disadvantages of DFPT:

Needs derivatives for the XC functional – only works for some functionals(LDA, PBE, etc)Not implemented for ultrasoft pseudopotentials – have to use NCP

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Fourier Interpolation of dynamical Matrices

FCM decays quickly

Approximate by reverse Fouriertransform of DM

Use forward transform to get DM atarbitrary qHandle Coulomb analytically

Φ(r)

r

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Acoustic Sum Rule

At the � point, the 3 lowest energy modes should be exactly zeroThis corresponds to the 3 translational symmetries in a periodic crystalAtomic motion becomes more and more like a rigid shift as q ! 0

Insufficient convergence may lead to this not being trueNumerical noise can affect thisInsufficient sampling in real or reciprocal space

We can "fix" solution to enforce this sum ruleREALSPACE Correct the FCM in real space.

RECIPROCAL Correct the DM at q = 0 and then apply this correctionto all DMs.

Select method by phonon_sum_rule_method

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Lattice Dynamics in CASTEP

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LatticeDynamics

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Methods in CASTEP

CASTEP can perform ab initio lattice dynamics usingPrimitive cell finite-displacement at q = 0Supercell finite-displacement for any qDFPT at arbitrary q.DFPT on MP grid of q with Fourier interpolation to arbitrary fine set of q.Finite displacements using non-diagonal supercells with Fourierinterpolation.

Full use is made of space-group symmetry to only compute only

symmetry-independent elements of D,0

↵,↵0(q)q-points in the irreducible Brillouin-Zone for interpolationelectronic k -points adapted to symmetry of perturbation.

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k-points and q-points

For phonons we have two sets of points in the Brillouin zone which are bothdefined in the .cell file

k-points These are the points where we solve the Kohn-Sham equations toobtain wavefunctions and total energies. These are specified by:kpoint_<tag>spectral_kpoint_<tag>supercell_kpoint_<tag>

q-points These are the points that we calculate the phonons modes on. Theyare specified by:phonon_kpoint_<tag> phonon_fine_kpoint_<tag>

where <tag> is one ofmp_grid A Monkhurst-Pack grid specification (nx , ny , nz )

mp_offset An offset to apply to the above gridlist A list of points to sample

spacing Use a grid with at most this spacingpath Generate a path between this list of points

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A CASTEP calculation I - simple DFPT

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 10�5

eV/Ang**2 usually sufficient. Sometimes need to increasephonon_max_cycles

Need very accurate ground-state as prerequisite for DFPT calculationelec_energy_tol needs to be roughly square ofphonon_energy_tol

D,0

↵,↵0(q) calculated at q-points specified in cell file byphonon_kpoint_<tag>

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Example Phonon Output – Si2

==============================================================================

+ Vibrational Frequencies +

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

+ +

+ Performing frequency calculation at 3 wavevectors (q-pts) +

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

+ +

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

+ q-pt= 1 ( 0.000000 0.000000 0.000000) 0.1250000000 +

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

+ Acoustic sum rule correction < 11.519522 cm-1 applied +

+ N Frequency irrep. ir intensity active raman active +

+ (cm-1) ((D/A)**2/amu) +

+ +

+ 1 -0.026685 a 0.0000000 N N +

+ 2 -0.026685 a 0.0000000 N N +

+ 3 -0.026685 a 0.0000000 N N +

+ 4 514.731729 b 0.0000000 N Y +

+ 5 514.731729 b 0.0000000 N Y +

+ 6 514.731729 b 0.0000000 N Y +

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

+ Character table from group theory analysis of eigenvectors +

+ Point Group = 32, Oh +

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

+ Rep Mul | E 4 2 2’ 3 I -4 m_h m_v -3 +

+ | ---------------------------------------- +

+ a T1u 1 | 3 1 -1 -1 0 -3 -1 1 1 0 +

+ b T2g 1 | 3 -1 -1 1 0 3 -1 -1 1 0 +

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

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CASTEP phonon calculations II - Fourier Interpolation

Specify grid of q-points using phonon_kpoint_mp_grid p q r.To select interpolation phonon_fine_method = interpolate

Golden rule of interpolation: Always include the � point (0,0,0) in theinterpolation grid. For even p, q, r use shifted gridphonon_fine_kpoint_mp_offset

12p

12q

12r

to shift one point to �

D,0

↵,↵0(q) interpolated to q-points specified in cell file byphonon_fine_kpoint_<tag>

Can calculate fine dispersion plot and DOS on a grid rapidly from oneDFPT calculation.

Real-space force-constant matrix is stored in .check file.All fine_kpoint parameters can be changed on a continuation run.Interpolation is very fast.

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CASTEP phonon calculations III - Non-diagonal Supercell

To select set both phonon_method = finite_displacement andphonon_fine_method = interpolation

Specify grid of q-points using phonon_kpoint_mp_grid p q r - as forDFPT. CASTEP will automatically determine supercells to use - noneed to explicitly set supercell in .cell file.K-points for supercell set using spacing or grid keywordssupercell_kpoint_mp_spacing

CASTEP automatically chooses a series of non-diagonal (skew)supercells and performs FD phonons and computes D

,0

↵,↵0(q) on grid ofq-points specified in cell file by one of same phonon_kpoint

keywords.From there calculation proceeds exactly as for supercell or DFPTinterpolation.

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Running a phonon calculation

Phonon calculations can be lengthy. CASTEP saves partial calculationperiodically in .check file:

num_backup_iter n – Backup every n q-vectorsbackup_interval t – Backup every t seconds

Phonon calculations have high inherent parallelism. Becauseperturbation breaks symmetry relatively large electronic k-point sets areused.Number of k-points varies depending on symmetry of perturbation.Try to choose number of processors to make best use of k -pointparallelism. If Nk not known in advance choose NP to have as manydifferent prime factors as possible - not just 2!

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DFPT with interpolation for Au

Γ X W L Γ K W0

50

100

150

ω (

cm-1

)

0 0.02 0.04 0.06DOS (cm)

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↵-quartz

Γ K M Γ A0

200

400

600

800

1000

1200

1400

Fre

quen

cy [

cm-1

]

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MoS2 – Bulk vs monolayer

M GK HG0

100

200

300

400

500

ω (

cm-1

)

Monolayer MoS2

Bulk MoS2

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Convergence issues for lattice dynamics

ab initio lattice dynamics calculations are very sensitive to convergence issues. Agood 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.

Accuracy 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 toanharmonicity.

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Nb – Imaginary Phonon Modes/Negative Frequencies

Γ H N P Γ N

0

50

100

150

200

ω (c

m-1

)

Quantities from Phonons

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ThermodynamicsGiven the phonon frequencies, the phonon density of states, g(!) canbe obtainedThis is straightforward: just count the number of frequencies in therange w to ! + d!

This histogram is the phonon density of statesFinite temperature information is obtained by manipulating g(!)

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Thermal Energy

The total thermal energy due to atomic motion is given by

Evib(T ) = E + kBT

X

q

log

2 sinh

✓~!(q)2kBT

◆�

Many thermodynamic properties can be calculated from this such asentropy, specific heat, etc. which you are free to investigate on your own

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LO/TO Splitting

Two similar structures

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BN and Diamond

Note degeneracies of optical modes at �?

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LO/TO Splitting

Dipole created by displacement of charges of long-wavelength LOmode creates induced electric field.For TO motion, E?q so E .q = 0For LO motion, E k q and E .q adds additional restoring forceEnergy (and so frequency) of LO mode is increasedThis is included in the formalism of DFPT.

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Powder infra-red

IR absorption is governed by the change in polarisation due to aphonon at the � point.Can calculate which modes are IR active and how strongly they willinteract.

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Raman Spectroscopy

Raman scattering depends on raman activity tensor

Iraman↵� =

d3E

d"↵d"�dQm

=d✏↵�

dQm

Can do this by displacing the atoms and doing an e-field calculation ateach point or by the 2n + 1 theorem

Both methods give similar results but the perturbative method is muchfaster (ice13 – 16,000 s vs 80,000 s)

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Raman Spectra of ZrO2

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Conclusion

Phonons can be calculated by eitherFinite displacement with

Primitive cell at q = 0Non-diagonal supercell on MP gridDirect supercell to calculate FCM

Density functional perturbation theoryAt arbitrary q

Interpolation is very useful for finely sampling phonons.Acoustic sum rule can help correct frequencies at q = 0Thermodynamic properties available!Raman and and IR can be calculated

Thanks for listening!

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References

Books on Lattice DynamicsM. 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

(North Holland, 1974) A comprehensive 7-volume series - more thanyou’ll need to know.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 dynamicsK. 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

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The Supercell method

The supercell method is an extension of the finite-displacement approach whichdepends on short-ranged nature of FCM: �,0

↵,↵0 (a) ! 0 as Ra ! 1.

Φ(r)

r

In non-polar insulators and mostmetals�,0

↵,↵0 (a) decays as 1/R5

or faster.In polar insulators Coulomb termdecays as 1/R3

Define radius Rc beyond which�,0

↵,↵0 (a) is negligibleFor supercell with L > 2Rc thenC

,0

↵,↵0 (sc) ⇡ �,0

↵,↵0 (a).

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The Supercell method

1 Choose sufficiently large supercell and compute C,0

↵,↵0 (sc) usingfinite-displacement method.

2 Non-periodic real-space force-constant matrix directly mapped from periodicsupercell FCM. �,0

↵,↵0 (a) ⌘ C,0

↵,↵0 (SC)

3 Fourier transform using definition of D to obtain dynamical matrix of primitive cellat any desired q.

4 Diagonalise D,0

↵,↵0 (q) to obtain eigenvalues and eigenvectors.

This method is often (confusingly) called the “direct” method.

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Overlap Errors

Size of supercell limits Rc . Too small a supercell means that �,0

↵,↵0(a) can

not be cleanly extracted from C,0

↵,↵0(SC) and dispersion curves will containerror.

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