Lattice Dynamics and Spectroscopy from DFT
Dominik Jochym (Keith Refson)
STFC Rutherford Appleton Laboratory
31st May 2012
ISIS Neutron Training Course 2012
1 / 42
Lattice Dynamics of
Crystals
References
Formal Theory of Lattice
Dynamics
Quantum Theory of
Lattice Modes
Formal Theory of Lattice
Dynamics II
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Lattice Dynamics of Crystals
Examples
LO/TO Splitting
Modelling of spectra
ISIS Neutron Training Course 2012
2 / 42
References
Lattice Dynamics of
Crystals
References
Formal Theory of Lattice
Dynamics
Quantum Theory of
Lattice Modes
Formal Theory of Lattice
Dynamics II
Books on Lattice Dynamics
•
•
•
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Examples
LO/TO Splitting
Modelling of spectra
•
M. T. Dove Introduction to Lattice Dynamics, CUP. - elementary introduction.
J. C. Decius and R. M. Hexter Molecular Vibrations in Crystals - Lattice
dynamics from a spectroscopic perspective.
Horton, G. K. and Maradudin A. A. Dynamical properties of solids (North
Holland, 1974) A comprehensive 7-volume series - more than you’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 dynamics
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ISIS Neutron Training Course 2012
S. Baroni et al (2001), Rev. Mod. Phys 73, 515-561.
Variational DFPT (X. Gonze (1997) PRB 55 10377-10354).
K. Refson, P. R. Tulip and S. J Clark, Phys. Rev B. 73, 155114 (2006)
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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Formal Theory of Lattice Dynamics
Lattice Dynamics of
Crystals
References
Formal Theory of Lattice
Dynamics
Quantum Theory of
Lattice Modes
Formal Theory of Lattice
Dynamics II
•
E = E0 +
Examples
∂E
1
.uκ,α +
∂uκ,α
2
′
uκ,α .Φκ,κ
.uκ′ ,α′ + ...
α,α′
X
κ,α,κ′ ,α′
where uκ,α is the vector of atomic displacements from equilibrium and
′
′
κ,κ
Φκ,κ
(a)
is
the
matrix
of
force
constants
Φ
(a) =
′
α,α
α,α′
•
•
LO/TO Splitting
Modelling of spectra
X
κ,α
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Based on expansion of total energy about structural equilibrium co-ordinates
•
∂E
At equilibrium the forces − ∂u
κ,α
∂2 E
∂uκ,α ∂uκ′ ,α′
are all zero so 1st term vanishes.
In the Harmonic Approximation the 3rd and higher order terms are assumed to
be negligible
Assume Born von Karman periodic boundary conditions and substituting
plane-wave uκ,α = εmκ,αq exp(iq.Rκ,α − ωt) yields eigenvalue equation:
κ,κ′
Dα,α′ (q)εmκ,αq
2
= ωm,q
εmκ,αq
where frequencies are square roos of eigenvalues. The dynamical matrix
κ,κ′
Dα,α′ (q)
1
= p
Mκ Mκ′
κ,κ′
Cα,α′ (q)
1
= p
Mκ Mκ′
X
κ,κ′
Φα,α′ (a)e−iq.Ra
a
is the Fourier transform of the force constant matrix.
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Quantum Theory of Lattice Modes
Lattice Dynamics of
Crystals
References
Formal Theory of Lattice
Dynamics
Quantum Theory of
Lattice Modes
Formal Theory of Lattice
Dynamics II
ab-initio Lattice
Dynamics
3
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•
•
Lattice Dynamics in
CASTEP
Examples
•
LO/TO Splitting
Modelling of spectra
•
•
The classical energy expression can
be transformed into a quantummechanical Hamiltonian for nuclei.
In harmonic approximation nuclear
wavefunction is separable into product
by mode transformation.
2
Each mode satisfies harmonic oscillator Schroedingereqn with energy levels
Em,n = n + 21 ~ωm for mode m.
Quantum excitations of modes known
as phonons in crystal
Transitions between levels n1 and 1
n2 interact with photons of energy
(n2 − n1 ) ~ωm , ie multiples of fundamental frequency ωm .
In anharmonic case where 3rd -order
term not negligible, overtone frequencies are not multiples of fundamental. 0
-2
ISIS Neutron Training Course 2012
-1
0
1
2
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Formal Theory of Lattice Dynamics II
Lattice Dynamics of
Crystals
References
Formal Theory of Lattice
Dynamics
Quantum Theory of
Lattice Modes
Formal Theory of Lattice
Dynamics II
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Examples
LO/TO Splitting
Modelling of spectra
•
•
•
The dynamical matrix is a 3N × 3N matrix at each wavevector q.
κ,κ′
2
Dα,α
′ (q) is a hermitian matrix ⇒ eigenvalues ωm,q are real.
3N eigenvalues ⇒ modes at each q leading to 3N branches in dispersion curve.
•
The mode eigenvector εmκ,α gives the atomic displacements, and its symmetry
can be characterised by group theory.
′
Given a force constant matrix Φκ,κ
(a) we have a procedure for obtaining
α,α′
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.
•
•
•
•
ISIS Neutron Training Course 2012
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Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
The Frozen-phonon
method
The Finite-Displacement
method
The Supercell method
First and second
derivatives
Density-Functional
Perturbation Theory
Fourier Interpolation of
dynamical Matrices
ab-initio Lattice Dynamics
Lattice Dynamics in
CASTEP
Examples
LO/TO Splitting
Modelling of spectra
ISIS Neutron Training Course 2012
7 / 42
The Frozen-phonon method
Lattice Dynamics of
Crystals
The frozen phonon method:
ab-initio Lattice
Dynamics
The Frozen-phonon
method
The Finite-Displacement
method
The Supercell method
First and second
derivatives
Density-Functional
Perturbation Theory
Fourier Interpolation of
dynamical Matrices
•
•
Lattice Dynamics in
CASTEP
•
•
•
•
•
•
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.
2
Use E0 (λ) to evaluate k = ddλE20
p
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.
Examples
LO/TO Splitting
Modelling of spectra
ISIS Neutron Training Course 2012
8 / 42
The Finite-Displacement method
Lattice Dynamics of
Crystals
The finite displacement method:
ab-initio Lattice
Dynamics
The Frozen-phonon
method
The Finite-Displacement
method
The Supercell method
First and second
derivatives
Density-Functional
Perturbation Theory
Fourier Interpolation of
dynamical Matrices
•
•
Lattice Dynamics in
CASTEP
Examples
LO/TO Splitting
•
Displace ion κ′ in direction α′ by small distance ±u.
Use single point energy calculations and evaluate forces on every ion in system
+
+
Fκ,α
and Fκ,α
for +ve and -ve displacements.
Compute numerical derivative using central-difference formula
+
−
Fκ,α
− Fκ,α
dFκ,α
d2 E0
≈
=
du
2u
duκ,α duκ′ ,α′
′
•
•
•
•
Modelling of spectra
•
ISIS Neutron Training Course 2012
κ,κ
Have calculated entire row k′ , α′ of Dα,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.
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The Supercell method
Lattice Dynamics of
Crystals
The supercell method is an extension of the finite-displacement approach.
ab-initio Lattice
Dynamics
The Frozen-phonon
method
The Finite-Displacement
method
The Supercell method
First and second
derivatives
Density-Functional
Perturbation Theory
Fourier Interpolation of
dynamical Matrices
•
•
•
•
′
•
•
Relies on short-ranged nature of FCM; Φκ,κ
(a) → 0 as Ra → ∞.
α,α′
For
For
κ,κ′
non-polar insulators and most metals Φα,α′ (a)
polar insulators Coulomb term decays as 1/R3
decays as 1/R5 or faster.
′
Can define “cut off” radius Rc beyond which Φκ,κ
(a) can be treated as zero.
α,α′
In supercell with L > 2Rc then
κ,κ′
Cα,α′ (q
= 0) =
κ,κ′
Φα,α′ (a).
Method:
′
Lattice Dynamics in
CASTEP
1.
Examples
2.
3.
LO/TO Splitting
Modelling of spectra
κ,κ
choose sufficiently large supercell and compute Cα,α
′ (qsupercell = 0) using
finite-displacement method.
′
This object is just the real-space force-constant matrix Φκ,κ
(a).
α,α′
Fourier transform using
′
κ,κ
Dα,α
′ (q) = p
4.
•
ISIS Neutron Training Course 2012
1
Mκ Mκ′
X
′
Φκ,κ
(a)e−iq.Ra
α,α′
a
to obtain dynamical matrix of primitive cell at any desired q.
κ,κ′
Diagonalise Dα,α′ (q) to obtain eigenvalues and eigenvectors.
This method is often (confusingly) called the “direct” method.
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First and second derivatives
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
The Frozen-phonon
method
The Finite-Displacement
method
The Supercell method
First and second
derivatives
Density-Functional
Perturbation Theory
Fourier Interpolation of
dynamical Matrices
Goal is to calculate the 2nd derivatives of energy to construct FCM or
•
•
•
Lattice Dynamics in
CASTEP
Examples
•
LO/TO Splitting
Modelling of spectra
•
•
•
ISIS Neutron Training Course 2012
κ,κ′
Dα,α′ (q).
Energy E = hψ| Ĥ |ψi with Ĥ = ∇2 + VSCF
Force
dψ
dV
dE
dψ =−
−
hψ|
|ψi
Ĥ
|ψi
−
hψ|
Ĥ
F =−
dλ
dλ
dλ dλ
where λ represents an atomic displacement perturbation.
If hψ|represents the ground state of Ĥ then the first two terms vanish because
dψ
dψ
= ǫn hψ| =0
hψ| Ĥ dλ
dλ
. This is the Hellman-Feynmann Theorem.
Force constants are the second derivatives of energy
dF
dψ dV
dV dψ
d2 V
d2 E
=−
=
|ψi + hψ|
− hψ|
|ψi
k=
dλ2
dλ
dλ dλ
dλ dλ
dλ2
None of the above terms vanishes.
Second
D derivatives need linear response of wavefunctions wrt perturbation
).
( dψ
dλ 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
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
The Frozen-phonon
method
The Finite-Displacement
method
The Supercell method
First and second
derivatives
Density-Functional
Perturbation Theory
Fourier Interpolation of
dynamical Matrices
•
•
•
•
Lattice Dynamics in
CASTEP
Examples
LO/TO Splitting
Modelling of spectra
•
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:
(1)
φk,q (r) = e−i(k+q).r u(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
E
E
(0) (1)
(1) (0)
(0)
H
− ǫm φm = −Pc v φm
Pc is projection operator onto unoccupied states. First-order potential v (1)
includes response terms of Hartree and XC potentials and therefore depends on
first-order density n(1) (r) which depends on φ(1) .
Finding φ(1) is therefore a self-consistent problem just like solving the
Kohn-Sham equations for the ground state.
Two major approaches to finding φ(1) are suited to plane-wave basis sets:
•
•
•
•
ISIS Neutron Training Course 2012
Green’s function (S. Baroni et al (2001), Rev. Mod. Phys 73, 515-561).
Variational DFPT (X. Gonze (1997) PRB 55 10377-10354).
CASTEP has implementations of both methods.
DFPT has huge advantage - can calculate response to incommensurate q from
a calculation on primitive cell.
Disadvantage of DFPT - lots of programming required.
12 / 42
Fourier Interpolation of dynamical Matrices
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
The Frozen-phonon
method
The Finite-Displacement
method
The Supercell method
First and second
derivatives
Density-Functional
Perturbation Theory
Fourier Interpolation of
dynamical Matrices
Lattice Dynamics in
CASTEP
•
•
•
•
•
Modelling of spectra
′
κ,κ
Compute Dα,α
′ (q) on a Monkhorst-Pack grid of q vectors.
Approximation to FCM in p × q × r supercell given by Fourier transform of
dynamical matrices on p × q × r grid.
X κ,κ′
κ,κ′
Φα,α′ (a) =
Cα,α′ (q)eiq.Ra
q
•
Examples
LO/TO Splitting
DFPT formalism requires self-consistent iterative solution for every separate q.
Hundreds of q’s needed for good dispersion curves, thousands for good Phonon
DOS.
′
Can take advantage of short-range nature of real-space FCM Φκ,κ
(a).
α,α′
•
•
•
ISIS Neutron Training Course 2012
Fourier transform using to obtain dynamical matrix of primitive cell at any
desired q, Exactly as with Finite-displacement-supercell method
κ,κ′
Diagonalise mass-weighted Dα,α
′ (q) to obtain eigenvalues and eigenvectors.
Longer-ranged coulombic contribution varies as 1/R3 but can be handled
analytically.
Need only DFPT calculations on a few tens of q points on grid to calculate
κ,κ′
Dα,α
′ (q) on arbitrarily dense grid (for DOS) or fine (for dispersion) path.
13 / 42
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Methods in CASTEP
A CASTEP calculation I
- simple DFPT
Example output
CASTEP phonon
calculations II - Fourier
Interpolation
CASTEP phonon
calculations III Supercell
Running a phonon
calculation
Lattice Dynamics in CASTEP
Examples
LO/TO Splitting
Modelling of spectra
ISIS Neutron Training Course 2012
14 / 42
Methods in CASTEP
Lattice Dynamics of
Crystals
CASTEP can perform ab-initio lattice dynamics using
ab-initio Lattice
Dynamics
•
•
•
•
Lattice Dynamics in
CASTEP
Methods in CASTEP
A CASTEP calculation I
- simple DFPT
Example output
CASTEP phonon
calculations II - Fourier
Interpolation
CASTEP phonon
calculations III Supercell
Running a phonon
calculation
Examples
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
′
•
•
•
κ,κ
symmetry-independent elements of Dα,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-conserving
pseudopotentials.
LO/TO Splitting
Modelling of spectra
ISIS Neutron Training Course 2012
15 / 42
A CASTEP calculation I - simple DFPT
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Methods in CASTEP
A CASTEP calculation I
- simple DFPT
Example output
CASTEP phonon
calculations II - Fourier
Interpolation
CASTEP phonon
calculations III Supercell
Running a phonon
calculation
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 calculation
elec energy tol needs to be roughly square of phonon energy tol
κ,κ′
Dα,α′ (q) calculated at q-points specified in cell file by one of
•
•
Examples
LO/TO Splitting
•
Modelling of spectra
•
ISIS Neutron Training Course 2012
%BLOCK phonon kpoint list for the explicitly named points
%BLOCK phonon kpoint path to construct a path joining the nodal points
given. Spacing along path is phonon kpoint path spacing
phonon kpoint mp grid p q r and possibly
phonon kpoint mp offset 0.125 0.125 0.125 to explicitly specify a M-P
grid for a DOS.
phonon kpoint mp spacing δq 1/ang to generate a M-P grid of a specified
linear spacing
16 / 42
Example output
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Methods in CASTEP
A CASTEP calculation I
- simple DFPT
Example output
CASTEP phonon
calculations II - Fourier
Interpolation
CASTEP phonon
calculations III Supercell
Running a phonon
calculation
Examples
LO/TO Splitting
Modelling of spectra
=====================================================================
+
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.
ISIS Neutron Training Course 2012
17 / 42
CASTEP phonon calculations II - Fourier Interpolation
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Methods in CASTEP
A CASTEP calculation I
- simple DFPT
Example output
CASTEP phonon
calculations II - Fourier
Interpolation
CASTEP phonon
calculations III Supercell
Running a phonon
calculation
Examples
To select set phonon fine method = interpolate
Specify grid of q-points using phonon kpoint mp grid p q r .
Golden rule of interpolation: Always include the Γ point (0,0,0) in the
interpolation grid. For even p, q, r use shifted grid phonon fine kpoint mp offset
0.125 0.125 0.125 to shift one point to Γ
κ,κ′
Dα,α
′ (q) interpolated to q-points specified in cell file by one of
•
•
•
•
LO/TO Splitting
Modelling of spectra
%BLOCK phonon fine kpoint list for the explicitly named points
%BLOCK phonon fine kpoint path to construct a path joining the nodal points
given. Spacing along path is phonon fine kpoint path spacing
phonon fine kpoint mp grid p q r and possibly
phonon fine kpoint mp offset 0.125 0.125 0.125 to explicitly specify a M-P
grid for a DOS.
phonon fine kpoint mp spacing δq 1/ang to generate a M-P grid of a
specified linear spacing
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. ⇒ can
calculate fine dispersion plot and DOS on a grid rapidly from one DFPT calculation.
′
Real-space cutoff is applied to Φκ,κ
(a) using cumulant method of Parlinski by
α,α′
default. Alternative spherical cutoff may be specified using parameters
phonon force constant method=SPHERICAL and
phonon force constant cutoff=R.
ISIS Neutron Training Course 2012
18 / 42
CASTEP phonon calculations III - Supercell
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Methods in CASTEP
A CASTEP calculation I
- simple DFPT
Example output
CASTEP phonon
calculations II - Fourier
Interpolation
CASTEP phonon
calculations III Supercell
Running a phonon
calculation
•
To select set phonon fine method = supercell
•
%BLOCK phonon_supercell_matrix
2 0 0
Set supercell in .cell file, eg 2 × 2 × 2 using 0 2 0
0 0 2
%ENDBLOCK phonon_supercell_matrix
•
κ,κ′
Dα,α′ (q)
•
•
•
•
Examples
LO/TO Splitting
Modelling of spectra
•
•
•
ISIS Neutron Training Course 2012
interpolated to q-points specified in cell file by one of same
phonon fine kpoint keywords as for interpolation.
Kpoints for supercell set using block or grid keywords supercell kpoint...
Cumulant/spherical real-space cutoff applies exactly as in 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 dispersion plot
and DOS on a grid rapidly from one DFPT calculation.
%BLOCK phonon_supercell_matrix
1 1 -1
1 -1 1
-1 1 1
%ENDBLOCK phonon_supercell_matrix
Tip. For fcc primitive cells use non-diagonal matrix
to make cubic supercell.
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 need larger fine gmax
(or fine grid scale) than for geom opt/MD to get good results.
19 / 42
Running a phonon calculation
Lattice Dynamics of
Crystals
•
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Methods in CASTEP
A CASTEP calculation I
- simple DFPT
Example output
CASTEP phonon
calculations II - Fourier
Interpolation
CASTEP phonon
calculations III Supercell
Running a phonon
calculation
•
•
•
Phonon calculations can be lengthy. CASTEP saves partial calculation
periodically in .check file if keywords num backup iter n or backup interval
t. Backup is every n q-vectors or every t seconds.
Phonon calculations have high inherent parallelism. Because perturbation
breaks 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 prime factors
as possible - not just 2!
Examples
LO/TO Splitting
Modelling of spectra
ISIS Neutron Training Course 2012
20 / 42
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Examples
DFPT with interpolation
- α-quartz
Supercell method - Silver
Barium Rhenium
Hydride
Convergence issues for
lattice dynamics
“Scaling” and other
cheats
Examples
LO/TO Splitting
Modelling of spectra
ISIS Neutron Training Course 2012
21 / 42
DFPT with interpolation - α-quartz
Lattice Dynamics of
Crystals
1400
ab-initio Lattice
Dynamics
1200
LO/TO Splitting
1000
-1
Examples
DFPT with interpolation
- α-quartz
Supercell method - Silver
Barium Rhenium
Hydride
Convergence issues for
lattice dynamics
“Scaling” and other
cheats
Frequency [cm ]
Lattice Dynamics in
CASTEP
Modelling of spectra
800
600
400
200
0
Γ
ISIS Neutron Training Course 2012
K
M
Γ
A
22 / 42
Supercell method - Silver
Lattice Dynamics of
Crystals
Ag phonon dispersion
3x3x3x4 Supercell, LDA
ab-initio Lattice
Dynamics
6
Lattice Dynamics in
CASTEP
4
ω (THz)
Examples
DFPT with interpolation
- α-quartz
Supercell method - Silver
Barium Rhenium
Hydride
Convergence issues for
lattice dynamics
“Scaling” and other
cheats
5
3
2
LO/TO Splitting
Modelling of spectra
1
0
ISIS Neutron Training Course 2012
Γ
X
K
Γ
L
W
23 / 42
Barium Rhenium Hydride
Lattice Dynamics of
Crystals
•
ab-initio Lattice
Dynamics
•
Lattice Dynamics in
CASTEP
Examples
DFPT with interpolation
- α-quartz
Supercell method - Silver
Barium Rhenium
Hydride
Convergence issues for
lattice dynamics
“Scaling” and other
cheats
with S. F. Parker, ISIS facility, RAL., Inorg. Chem.
45, 19051 (2006)
BaReH9 with unusual ReH2+
9 ion has very high molar hydrogen content.
INS spectrum modelled using A-CLIMAX software
(A. J. Ramirez Cuesta, ISIS)
Predicted INS spectrum in mostly excellent agreement with experiment
LO/TO splitting essential to model INS.
Librational modes in error (c.f NH4 F)
Complete mode assignment achieved.
•
•
•
•
•
LO/TO Splitting
300
Modelling of spectra
200
-1
ω (cm )
250
150
100
50
0
Γ
ISIS Neutron Training Course 2012
K
M
Γ
A
H
L
A
24 / 42
Convergence issues for lattice dynamics
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Examples
DFPT with interpolation
- α-quartz
Supercell method - Silver
Barium Rhenium
Hydride
Convergence issues for
lattice dynamics
“Scaling” and other
cheats
LO/TO Splitting
Modelling of spectra
ab-initio lattice dynamics calculations are very sensitive to convergence issues. A
good calculation must be well converged as a function of
1.
2.
3.
4.
5.
6.
•
•
•
•
ISIS Neutron Training Course 2012
plane-wave cutoff
electronic kpoint sampling of the Brillouin-Zone (for crystals)
(under-convergence gives poor acoustic mode dispersion as q → 0
geometry. Co-ordinates must be well converged with forces close to zero
(otherwise calculation will return imaginary frequencies.)
For DFPT calculations need high degree of SCF convergence of ground-state
wavefunctions.
supercell size for “molecule in box” calculation and slab thickness for surface/s
lab calculation.
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 stretch modes!
25 / 42
“Scaling” and other cheats
Lattice Dynamics of
Crystals
•
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
•
Examples
DFPT with interpolation
- α-quartz
Supercell method - Silver
Barium Rhenium
Hydride
Convergence issues for
lattice dynamics
“Scaling” and other
cheats
•
•
•
•
LO/TO Splitting
Modelling of spectra
•
ISIS Neutron Training Course 2012
DFT usually gives frequencies within a few percent of experiment. Exceptions
are usually strongly-correlated systems, e.g. some transition-metal Oxides where
DFT description of bonding is poor.
Discrepancies can also be due to anharmonicity. A frozen-phonon calculation
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. may not be at
experimental pressure.
Hartree-Fock approximation systematically overestimates vibrational frequencies
by 5-15%. Common practice is to multiply by ”scaling factor” ≈ 0.9.
Scaling not recommended for DFT where error is not systematic. Over- and
under-estimation equally common.
For purposes of mode assignment, or modelling experimental spectra to
compare intensity it can sometimes be useful to apply a small empirical shift on
a per-peak basis. This does not generate an “ab-initio frequency”.
26 / 42
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Examples
LO/TO Splitting
Two similar structures
Zincblende and diamond
dispersion
LO/TO splitting
DFPT with LO/TO
splitting in NaCl
LO/TO splitting in
non-cubic systems
LO/TO Splitting
Modelling of spectra
ISIS Neutron Training Course 2012
27 / 42
Two similar structures
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Examples
LO/TO Splitting
Two similar structures
Zincblende and diamond
dispersion
LO/TO splitting
DFPT with LO/TO
splitting in NaCl
LO/TO splitting in
non-cubic systems
Modelling of spectra
Zincblende BN
ISIS Neutron Training Course 2012
Diamond
28 / 42
Zincblende and diamond dispersion
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Phonon dispersion curves
BN-zincblende
diamond
2000
1500
Lattice Dynamics in
CASTEP
1500
Modelling of spectra
-1
ω (cm )
1000
-1
LO/TO Splitting
Two similar structures
Zincblende and diamond
dispersion
LO/TO splitting
DFPT with LO/TO
splitting in NaCl
LO/TO splitting in
non-cubic systems
ω (cm )
Examples
500
1000
500
0
0
X
X
Γ
Γ L WX
Γ
Γ L WX
Cubic symmetry of Hamiltonian predicts triply degenerate optic mode at Γ in both
cases.
2+1 optic mode structure of BN violates group theoretical prediction.
Phenomenon known as LO/TO splitting.
ISIS Neutron Training Course 2012
29 / 42
LO/TO splitting
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Examples
LO/TO Splitting
Two similar structures
Zincblende and diamond
dispersion
LO/TO splitting
DFPT with LO/TO
splitting in NaCl
LO/TO splitting in
non-cubic systems
Modelling of spectra
•
•
•
•
•
•
ISIS Neutron Training Course 2012
Dipole created by displacement of charges of long-wavelength LO mode creates
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.
Lyndane-Sachs-Teller relation for cubic case:
2
ωLO
2
ωT
O
=
ǫ0
ǫ∞
LO frequencies at q = 0 depend on dielectric permittivity
30 / 42
DFPT with LO/TO splitting in NaCl
Lattice Dynamics of
Crystals
LO modes can be seen in infrared, INS, IXS experiments, but not raman.
NaCl phonon dispersion
ab-initio Lattice
Dynamics
300
Lattice Dynamics in
CASTEP
Exper: G. Raunio et al
Examples
200
-1
ω (cm )
LO/TO Splitting
Two similar structures
Zincblende and diamond
dispersion
LO/TO splitting
DFPT with LO/TO
splitting in NaCl
LO/TO splitting in
non-cubic systems
250
150
100
Modelling of spectra
50
0
ISIS Neutron Training Course 2012
Γ
(∆)
X
(Σ)
q
Γ
(∆)
L
(Q)
W
(Z) X
31 / 42
LO/TO splitting in non-cubic systems
2000
ab-initio Lattice
Dynamics
1500
-1
Lattice Dynamics in
CASTEP
ω (cm )
Lattice Dynamics of
Crystals
1000
Examples
500
LO/TO Splitting
Two similar structures
Zincblende and diamond
dispersion
LO/TO splitting
DFPT with LO/TO
splitting in NaCl
LO/TO splitting in
non-cubic systems
0
Γ
K
M
Γ
A
In systems with unique axis (trigonal, hexagonal, tetragonal) LOTO splitting depends on direction
of q even at q = 0.
550
Modelling of spectra
LO frequency varies with angle in
α-quartz
Zone Center Frequency (1/cm)
540
530
520
510
500
490
480
ISIS Neutron Training Course 2012
0
30
60
90
Approach Angle
120
150
180
32 / 42
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Examples
LO/TO Splitting
Modelling of spectra
Inelastic Neutron
Scattering
INS of Ammonium
Fluoride
Single-crystal infrared
Powder infrared
Modelling of powder IR
Non-resonant raman
Raman scattering of
t-ZrO2
Inelastic X-Ray
scattering
IXS of Diaspore
ISIS Neutron Training Course 2012
Modelling of spectra
33 / 42
Inelastic Neutron Scattering
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
To model spectra need to treat scattering dynamics of incident and emergent
radiation.
In case of INS interaction is between point neutron and nucleus - scalar quantity b
depends only on nucleus – specific properties.
kf 2
d2 σ
=
b S(Q, ω)
dEdΩ
ki
Examples
LO/TO Splitting
Modelling of spectra
Inelastic Neutron
Scattering
INS of Ammonium
Fluoride
Single-crystal infrared
Powder infrared
Modelling of powder IR
Non-resonant raman
Raman scattering of
t-ZrO2
Inelastic X-Ray
scattering
IXS of Diaspore
Q is scattering vector and ω is frequency - interact with phonons at same
wavevector and frequency.
Full measured spectrum includes overtones and combinations and instrumental
geometry and BZ sampling factors.
Need specific spectral modelling software to incorporate effects as postprocessing
step following CASTEP phonon DOS calculation.
A-Climax : A. J. Ramirez-Cuesta Comput. Phys. Comm. 157 226 (2004))
ISIS Neutron Training Course 2012
34 / 42
INS of Ammonium Fluoride
Lattice Dynamics of
Crystals
•
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
•
•
Examples
LO/TO Splitting
Modelling of spectra
Inelastic Neutron
Scattering
INS of Ammonium
Fluoride
Single-crystal infrared
Powder infrared
Modelling of powder IR
Non-resonant raman
Raman scattering of
t-ZrO2
Inelastic X-Ray
scattering
IXS of Diaspore
•
•
•
ISIS Neutron Training Course 2012
NH4 F is one of a series of ammonium halides
studied 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.
35 / 42
Single-crystal infrared
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Prediction of reflectivity of optically flat single crystal surface given as function of q
- projected permittivity ǫq (ω)
1/2
ǫ (ω) − 1 2
q
R(ω) = 1/2
ǫ (ω) + 1 Lattice Dynamics in
CASTEP
q
Examples
LO/TO Splitting
Modelling of spectra
Inelastic Neutron
Scattering
INS of Ammonium
Fluoride
Single-crystal infrared
Powder infrared
Modelling of powder IR
Non-resonant raman
Raman scattering of
t-ZrO2
Inelastic X-Ray
scattering
IXS of Diaspore
with ǫq defined in terms of ǫ∞ and mode oscillator strength Sm,αβ
∞
ǫq (ω) = q.ǫ
4π X q.S.q
= q.ǫ(ω).q
.q +
2 − ω2
Ω0 m ω m
ǫ(ω) is tabulated in the seendname.efield file written from a CASTEP
efield response calculation.
Example BiFO3 (Hermet et al, Phys. Rev
B 75, 220102 (2007)
ISIS Neutron Training Course 2012
36 / 42
Powder infrared
Lattice Dynamics of
Crystals
α-quartz
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
0.15
LO/TO Splitting
Modelling of spectra
Inelastic Neutron
Scattering
INS of Ammonium
Fluoride
Single-crystal infrared
Powder infrared
Modelling of powder IR
Non-resonant raman
Raman scattering of
t-ZrO2
Inelastic X-Ray
scattering
IXS of Diaspore
IR Intensity (Arb. Units)
Examples
0.1
0.05
0
•
•
•
ISIS Neutron Training Course 2012
400
600
800
-1
Phonon Frequency (cm )
1000
1200
Straightforward to compute peak areas.
Peak shape modelling depends on sample and experimental variables.
Multiphonon and overtone terms less straightforward.
37 / 42
Modelling of powder IR
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
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.
Spectral shape determined by optical effects and shifted LO modes in specific
size/shape of crystallites.
Examples
LO/TO Splitting
Modelling of spectra
Inelastic Neutron
Scattering
INS of Ammonium
Fluoride
Single-crystal infrared
Powder infrared
Modelling of powder IR
Non-resonant raman
Raman scattering of
t-ZrO2
Inelastic X-Ray
scattering
IXS of Diaspore
ISIS Neutron Training Course 2012
38 / 42
Non-resonant raman
Lattice Dynamics of
Crystals
Raman scattering depends on raman activity tensor
ab-initio Lattice
Dynamics
raman
Iαβ
Lattice Dynamics in
CASTEP
Examples
LO/TO Splitting
Modelling of spectra
Inelastic Neutron
Scattering
INS of Ammonium
Fluoride
Single-crystal infrared
Powder infrared
Modelling of powder IR
Non-resonant raman
Raman scattering of
t-ZrO2
Inelastic X-Ray
scattering
IXS of Diaspore
dǫαβ
d3 E
=
=
dεα dεβ dQm
dQm
i.e. the activity of a mode is the derivative of the dielectric permittivity with respect
to the displacement along the mode eigenvector.
CASTEP evaluates the raman tensors using hybrid DFPT/finite displacement
approach.
Raman calculation is fairly expensive ⇒ and is not activated by default (though
group 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
(2πν)4
2 h(nm + 1)
=
|e
.I.e
|
S
L
dΩ
c4
4πωm
with the thermal population factor
nm = exp
~ωm
kB T
−1
−1
which is implemented in dos.pl using the -raman flag.
ISIS Neutron Training Course 2012
39 / 42
Raman scattering of t-ZrO2
Lattice Dynamics of
Crystals
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Examples
LO/TO Splitting
Modelling of spectra
Inelastic Neutron
Scattering
INS of Ammonium
Fluoride
Single-crystal infrared
Powder infrared
Modelling of powder IR
Non-resonant raman
Raman scattering of
t-ZrO2
Inelastic X-Ray
scattering
IXS of Diaspore
ISIS Neutron Training Course 2012
40 / 42
Inelastic X-Ray scattering
Lattice Dynamics of
Crystals
IXS spectrometers at ESRF, Spring-8, APS synchrotrons.
ab-initio Lattice
Dynamics
Lattice Dynamics in
CASTEP
Examples
LO/TO Splitting
Modelling of spectra
Inelastic Neutron
Scattering
INS of Ammonium
Fluoride
Single-crystal infrared
Powder infrared
Modelling of powder IR
Non-resonant raman
Raman scattering of
t-ZrO2
Inelastic X-Ray
scattering
IXS of Diaspore
ISIS Neutron Training Course 2012
41 / 42
IXS of Diaspore
Lattice Dynamics of
Crystals
B. Winkler et el., Phys. Rev. Lett 101 065501 (2008)
ab-initio Lattice
Dynamics
380
Lattice Dynamics in
CASTEP
375
LO/TO Splitting
Modelling of spectra
Inelastic Neutron
Scattering
INS of Ammonium
Fluoride
Single-crystal infrared
Powder infrared
Modelling of powder IR
Non-resonant raman
Raman scattering of
t-ZrO2
Inelastic X-Ray
scattering
IXS of Diaspore
ISIS Neutron Training Course 2012
energy [meV]
Examples
(0.5, 0, 0) to (0, 0, -0.33)
370
365
360
355
0.00
0.02
0.04
0.06
0.08
0.10
wave vector [A
-1
0.12
0.14
0.16
]
42 / 42