Phonon band structures and thermodynamic properties within the harmonic approximation

1
Phonon band structures and thermodynamic
properties within the harmonic approximation
International School on Vibrational
Spectroscopies Queretaro 2008
P. Boulanger Université Catholique de
Louvain Université de Montréal boulanger_at_pcpm.ucl.
ac.be
2
Outline

• Introduction to Interatomic Force Constants
• Fourier Interpolation scheme
• IFC decay in materials
• Results
• Thermodynamic properties
• The quasi-harmonic approximation
• Conclusion

3
Interatomic force constants
Consider a crystal with small deviations of
atomic positions with respect to the equilibrium
configuration
is the position of the origin of unit cell a
is the equilibrium position of atom k with
respect to the origin of its unit cell
are small deviations for atom k in unit cell
a
4
Interatomic force constants
Born-Oppenheimer approximation
In the harmonic approximation, the total energy
of a crystal with small atomic position
deviations is
where the matrix of IFCs is defined as
5
Physical Interpretation of the Interatomic Force
Constants
The force conjugate to the position of a nucleus,
can always be written
We can thus rewrite the IFCs in more physically
descriptive fashion
The IFCs are the rate of change of the atomic
forces when we displace another atom in the
crystal.
6
Relation between the IFCs and the dynamical
matrix
The Fourier transform of the IFCs is directly
related to the dynamical matrix,
The phonon frequencies are then obtained by
diagonalization of the dynamical matrix or
equivalently by the solution of this eigenvalue
problem
masses
phonon displacementpattern
square ofphonon frequencies
7
Phonon band structure of a-Quartz
SiO2 9 atoms per unit cell
Nb. of phonon bands
Nb. of acoustic bands 3
Nb. of optical bands
Polar crystal LO non-analyticity
Directionality !
X.Gonze, J.-C.Charlier, D.C.Allan, M.P.Teter,
PRB 50, 13055 (1994)
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Outline

• Introduction to Interatomic Force Constants
• Fourier Interpolation scheme
• IFC decay in materials
• Results
• Thermodynamic properties
• The quasi-harmonic approximation
• Conclusion

9
Usefulness of the IFCs Fourier Interpolation
From DFPT, it is quite straightforward, although
lengthy to compute, for one wavevector
However, a full band structure needs values for
many wavevectors ...
X.Gonze, J.-C.Charlier, D.C.Allan, M.P.Teter,
PRB 50, 13055 (1994)
10
Usefulness of the IFCs Fourier Interpolation
From DFPT, it is quite straightforward, although
lengthy to compute, for one wavevector
However, a full band structure needs values for
many wavevectors ...
Supposing the IFCs are available, the same
information could be obtained in a much faster
way, for any number of wavevectors
and IFCs can be generated by
This is equivalent to Fourier interpolation of
the dynamical matrices.
11
Usefulness of the IFCs Fourier Interpolation
The key of the interpolation is to replace the
integral,
by a summation on a few Q-points.
periodic cells
points
Grid of
IFCs in box of
Fourier
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Usefulness of the IFCs Silicon
Real space IFCs calculated with 10 Q-points
Real space IFCs calculated with 18 Q-points
13
Usefulness of the IFCs Fourier Interpolation
The key of the interpolation is to replace the
integral,
by a summation on a few Q-points.
Problem non-analyticity in polar
semiconductors for ? 0 long-range
character of IFCs
Trick Treat separately the analytic and
non-analytic parts of the IFCs
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Model for the non-analytic term
When a ion with charge Z is displaced from its
equilibrium position, a dipolar electric field is
created. Its effect on other ions is described by
a dipole - dipole interaction appearing in IFCs.
Suppose homogeneous material with isotropic
dielectric tensor , ions with charges Zk
and Zk , then
Where,
-


The expression presents a long range decay of the
IFCs 1/d3!!
-
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Model for the non-analytic term
The Fourier transform of this dipole-dipole term
exhibits the following non analytical behavior
Directionality !!!
Exemple
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Model for the non-analytic term
In the general case, the Fourier transform
exhibit a non-analytical behavior, mediated by
the long-wavevector electric field
Where,
Born effective charge tensor for atom k
(Proportionality coefficient between dipole and
displacement)
high frequency static dielectric tensor
(electronic contribution to the screening of the
charges)
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Fourier Interpolation Trick
The key of the interpolation is to replace the
integral,
by a summation on a few Q-points. But since
is not well behaved, we replace it
by
which is local and analytic. Thus,
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Fourier Interpolation Trick
We can now use the real-space IFCs to interpolate
at any Q point
But, we must now add the non-analytic term
corresponding to that Q-point
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Outline

• Introduction to Interatomic Force Constants
• Fourier Interpolation scheme
• IFC decay in materials
• Results
• Thermodynamic properties
• The quasi-harmonic approximation
• Conclusion

20
Decay of Interatomic Force Constants
In metals Z 0 (electrostatic
interactions are screened completely)
Friedel oscillations cause a long-ranged decay
of IFCs cos 2kFd / k3F d3 In elemental
insulators, with sufficient symmetry, it may
happen that Z 0 Si , Ge ,
rare gases, but not Se IFCs ?1/d5
1/d7 In all other insulators Z?0, and
the IFCs decay as 1/d3
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Decay of IFCs Silicon
As expected, the IFCs are short range, i.e.
falling to zero after the nearest-neighbors.
total
NN
X. Gonze, Advances in Quantum Chemistry 33, 225
(1999)
22
Decay of IFCs a-Quartz
Quartz 3 Si 6 O
NN
Si
O
NN
X. Gonze, Advances in Quantum Chemistry 33, 225
(1999)
23
Decay of IFCs Stishovite
Stishovite 2 Si 4 O
Si
O
NN
X. Gonze, Advances in Quantum Chemistry 33, 225
(1999)
24
Z Charge neutrality sum rule Acoustic sum rule
Invariance of all physical quantities to a rigid
translation of the crystal,
same for all atoms
In practice, slightly broken by finite mesh size!!
Energy is invariant
Action-Reaction Principle
Polarization is invariant
Z charge neutrality sum-rule
Force on one atom is invariant
acoustic sum-rule
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Z Charge neutrality sum rule Acoustic sum rule
The sum-rules can be enforced by subtracting the
excess effective charge,
And distributing it evenly on all atoms,
Other weighting schemes can be designed.
Similarly, we can impose the acoustic sum-rule
with
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Interpolation Scheme
ifcflag
Use Abinit to calculate on a few
Q-point.
Abinit
Calculate and .
chneut
asr
Enforce sum-rules
dipdip
Subtract the non-analyticity .
Fourier Transform to obtain .
Anaddb
Use the real space IFCs to interpolate at any
Q-point.
Add the non-analytic part for that Q-point
Diagonalize dynamical matrix to find phonon
frequencies
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Outline

• Introduction to Interatomic Force Constants
• Fourier Interpolation scheme
• IFC decay in materials
• Results
• Thermodynamic properties
• The quasi-harmonic approximation
• Conclusion

28
Experiment vs. Theory
Elemental and III-V semiconductors
IFCs are long long-ranged in the 110 direction
for zinc-blende structure. Related to bond chains
propagating in that direction.
Forces in Ha/bohr in silicon
Really good fit to experiment !!
1 error
Giannozzi, P., S. de Gironcoli, P. Pavone, and S.
Baroni, 1991, Phys. Rev. B 43, 7231.
29
Experiment vs. Theory
Simple Metals
In all cases the calculated dispersion curves are
in good agreement with experiments if an
appropriate smearing technique is used.
FCC BZ
0.3 eV smearing width 0.7 eV smearing width
Good phonon dispersions are necessary for
electron-phonon coupling Eliashberg function
MacMillan equation for Tc
BCC BZ
de Gironcoli, S., 1995, Phys. Rev. B 51, 6773.
30
Experiment vs. Theory
Theory successfully applied to

• Simple semiconductors nitrides, silicon carbide,
  graphite, etc.
• Simple metals and superconductors
• Insulators
• Ferroelectrics
• Phonons in semiconductor alloys and superlattices
• Lattice vibrations at surfaces

31
LO-TO splitting
We have seen that for polar semiconductors and
insulators, there is an non-analyticity in the
IFCs due to dipole-dipole interactions.
For transverse modes we have
Thus, the non-analytic term does not contribute
to these modes. Only the longitudinal modes are
affected. This leads to the LO-TO splitting and
the Lyddane-Sachs-Teller relation.
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LO-TO splitting
Wrong behaviour
Calculated phonon dispersions of ZrO2 in the
cubic structure at the equilibrium lattice
constant a0 5.13 Å.
From Parlinski K., Li Z.Q., and Kawazoe
Y.,Phys. Rev. Lett. 78, 4063 (1997)

• DFPT (Linear-response)with 5.75
• -2.86
• and 5.75
• LO - TO splitting 11.99 THz
• Non-polar mode is OK

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Stability Issues
Calculation of the phonon frequencies should be
done with respect to the fully optimized atomic
positions. If this is the case, the energy is
minimal for zero displacements and the structure
is stable. The harmonic approximation is
justified.
If this is not the case, we have an energy curve
that has a minimum for a finite displacement. The
phonon frequencies are thus imaginary.
a-quartz
Baroni, S., and P. Giannozzi, 1998, in High
Pressure Materials Research, edited by R. M.
Wentzcovitch et al., Mater. Res. Soc. Symp. Proc.
No. 499 (Materials Research Society,
Pittsburgh),p. 233.
34
Spin-Orbit interaction

• Relativistic corrections (Schrödinger to Dirac)
• -Mass-velocity Darwin term
• gt modification of kinetic energy and potential
• Spin-orbit interaction
• gt new term

Important for heavier atoms and d and f shells.
M. Verstraete, M. Torrent, F. Jollet, G. Zerah,
X.G, unpublished
35
Bismuth phonon band structure
Full line DFPT without spin-orbit
Only very old experimental data available
Yarnell et al, IBM J.Res. Dev. 1964 Smith,
internal report Los Alamos 1967
10-15 change due to SO
Full line DFPT with spin-orbit
LE. Diaz-Sanchez, A.H. Romero, XG, Phys. Rev. B
76, 104302 (2007)
36
Outline

• Introduction to Interatomic Force Constants
• Fourier Interpolation scheme
• IFC decay in materials
• Results
• Thermodynamic properties
• The quasi-harmonic approximation
• Conclusion

37
Thermodynamic properties
In the harmonic approximation, the phonons can be
treated as an independent boson gas. They obey
the Bose-Einstein distribution
The total energy of the gas can be calculated
directly using the standard formula
Phonon DOS
Energy of the harmonic oscillator
thmflag
All thermodynamic properties can be calculated in
this manner.
Note
38
Phonon DOS
The phonon density of states,
-quartz
(cm)
stishovite
is calculated using the histogram method in
Anaddb. The prefactors where chosen to normalize
the distribution
(cm-1)
Changyol Lee X.Gonze, Physical Review B 51,
8610 (1995)
39
Phonon DOS
The phonon density of states,
-quartz
(cm)
stishovite
Quartz
Stishovite
Si
O
(cm-1)
Changyol Lee X.Gonze, Physical Review B 51,
8610 (1995)
40
Internal Energy
The internal energy is simply the sum of the
total electronic energy and the energy of the
phonon gas
The zero-point motion energy is
Quartz ?F0 ?E0 30.0 kJ/molStishovite
?F0 ?E0 31.5 kJ/mol
Changyol Lee X.Gonze, Physical Review B 51,
8610 (1995)
41
Important quantities can be calculated from the
derivative of the total energy
The total pressure
The specific heat
42
Specific heat
J/mol.K
The relation between the constant pressure
specific heat and the constant volume specific
heat is given be
?-quartz
stishovite
K
Changyol Lee X.Gonze, Physical Review B 51,
8610 (1995)
43
Helmholtz free energy
The Helmholtz free energy is used to calculate
every thermodynamic property, it is defined by
the equation
The phonon contribution is
Changyol Lee X.Gonze, Physical Review B 51,
8610 (1995)
44
Atomic Temperature Factors
The X-ray diffraction peaks are given, neglecting
the atomic motion, by the structure factor of the
crystal
For G wave vector
Scattering amplitude of atom
Attenuation of X-ray diffraction intensities due
to thermal motion of the atoms is included using
the atomic temperature factors
where
mean-square displacement matrix
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Atomic Temperature Factors
mean-square displacement matrix
Generalized density of states
If only one kind of atom with sufficient local
symmetry, all the W(k,T) are identical
Debye-Waller factor
The atomic temperature factors of Si and O in
a-quartz (solid line) and stishovite (dashed
line) for the diffraction with scattering vector
G (2p/c).
46
Outline

• Introduction to Interatomic Force Constants
• Fourier Interpolation scheme
• IFC decay in materials
• Results
• Thermodynamic properties
• The quasi-harmonic approximation
• Conclusion

47
The quasi-harmonic approximation
The harmonic approximation is not sufficient to
model thermal expansion and the temperature and
volume dependence of the elastic constants, width
of Raman Peaks, etc.
We directly assume that the cell parameters can
change with temperature.
We also assume the potential is harmonic, leading
to the same formalism presented thus far. The
only difference is that we now calculate the
phonon band structure for different cell
parameters.
48
The quasi-harmonic approximation
Contracted by 0.06 bohr
Si
equilibrium
Stretched by 0.06 bohr
49
Grüneisen Parameters
The Grüneisen parameters are defined as
The thermal expansion can be written
50
Quasi-harmonic approximation and thermal expansion
Free energy is calculated using above formalism
as a function of structural parameters and the
structure is obtained by free energy minimization
and the bulk modulus from the curvature of the
Helmholtz free energy.
Si
Volume dependence of the Helmholtz free energy F
of silicon for four different temperatures.
G.-M. Rignanese, J.-P. Michenaud, and X. Gonze,
Physical Review B 53, 4488 (1996)
51
Quasi-harmonic approximation and thermal expansion
Si
Si
Crosses represent experimental results.
G.-M. Rignanese, J.-P. Michenaud, and X. Gonze,
Physical Review B 53, 4488 (1996)
52
Outline

• Introduction to Interatomic Force Constants
• Fourier Interpolation scheme
• IFC decay in materials
• Results
• Thermodynamic properties
• The quasi-harmonic approximation
• Conclusion

53
Conclusion

• We have seen the usefulness of the IFCs we
  can cheaply compute the
• dynamical matrix at any Q point (Fourier
  Interpolation).
• Importance of convergence studies with respect
  to parameters Q point
• sampling, smearing for metals, etc.
• Special care to be taken with certain materials
  LO-TO splitting, spin-
• orbit coupling and equilibrium conditions.
• Thermodynamic properties are easily calculated
  using anaddb.
• Quasi-harmonic approximation enables calculation
  of thermal expansion,
• bulk modulus and related properties.