Fonones y Elasticidad bajo presión
ab initio .
Alfonso Muñoz
Dpto. de Física Fundamental II
Universidad de La Laguna. Tenerife.
MALTA Consolider Team
Canary Islands, SPAIN
Plan de la charla:
Introducción : Ab initio methods
Fonones. Propiedades dinámicas. Ejemplos
 Elasticidad estabilidad mecánica bajo presión
Conclusiones.
Ab initio methods
 State of the art Ab Initio Total Energy Pseudopotential calculations are useful
to study many properties of materials.
 No experimental input required (even the structure). Only Z is required. They
can provide and “predict” many properties of the material if the
approximations are correct!
 DFT is the standar theory applied, it is “exact” but one need to use
approximations, XC functional (LDA, GGA etc…), BZ integration with k-special
points, etc. (some problems in high correlated systems, f-electrons etc..). DFPT
also available, allows to study phonons, elastic constants etc…
 More elaborated approximations are also available, like LDA + U, MD, etc..
 Many computer programs available, some times free (Abinit, quantum
espresso, VASP, CASTEP, etc…)
 Ab initio methods provide and alternative and complimentary technique to the
experiments under extreme conditions.
"Those who are enamoured of
Practice without Theory are like
a pilot who goes into a ship
without rudder or compass and
never has any certainty of where
he is going. Practice should
always be based upon a sound
knowledge of Theory.“
Leonardo da Vinci,
(1452-1519 )
Well tested:
.
“Prediction is very difficult,
especially about the future”.
Niels Bohr (1885-1962)
Fonones y espectroscopía,
¿para que?
Thermal Expansion
Elasticity - deformation
Superconductivity
Thermal Conductivity
Lattice Dynamics
Ti  Vi  Ee ( R)  Ti  U ( R)  E
 Lattice Potential:
  2U ( R) 
  3U ( R) 
1
1
 ul ul ' ' 

 ul ul ' 'ul '' ''  ...
U  U R0    

2 l ,l ' '  ul ul ' ' 0
3! l ,l ' ',l '' ''  ul ul ' 'ul '' '' 0
 Harmonic approximation:
1
U ( R )      ,  l , l '  ' u (l )u  (l '  ' )
2 l ,l ' '  , 
F  mx   kx
Hooke’s law!
m u (l )      ,  l , l '  ' u  (l '  ' )
l ' ', 
IFC
u (l ) 
1
m
 u~ (q |  ) e 
i qr l t 
q
 2u~ (q |  )   D ,   ' | q  u~ (q |  ' )
 ', 
D ,   ' | q    2   '  0
Phonons: linear chain of atoms
Harmonic approximation:
Ansatz:
=>
Phonons: linear chain of atoms
Linear chain of atoms
K
 qa 
ω  q = 2
sin 

M
2


q=0
π
q=
2a
π
q=
a
4K
M
Linear chain with two different
"spring constants"
two atoms per unit cell
Ansatz:
Phonons
Linear chain with two different "spring constants"
Two solutions:
K+G 1
ω(q) =
±
K2 + G2 + 2KGcos  qa 
M
M
2
2 K + G 
M
2G
M
acoustic (-) and optic (+) branches
Γac
Γop
Lac
Lop
2K
M
Γ
L
3D Phonon Dispersion Relations
3THz ~ 100 cm-1 ; 1meV ~ 8 cm-1
THz
Si
j = 3N branches
cm-1
3C-SiC
LO
TO
G. Nilsson and G. Nelin, PRB 6, 3777 (1972)
W. Weber, PRB 15, 4789 (1977)
J. Serrano et al., APL 80, 23 (2002)
Polar crystals: LST relation
 Ionic crystals: Macroscopic electric field
Lyddane-Sachs-Teller relation
1
~
~
C  ' | q    C l ,  ' l 'e iqR l R l' 
N ll '
P ,mac

*
Born effective
Z   
4 u q  0 |   charges
2
 LO
0

2
TO  
       
4

C~q  0  M 
2
Z   

  
*
'
'
2
2



q

0




 m
m  q  0  
m
' '
*
Z
 ' ', '  '  '
X. Gonze and C. Lee, PRB 55, 10355 (1997)

4

1
 M

*
*
q
Z

Z


  
   '  q

q   q


Anisotropy: crystal field
GaN
T. Ruf et al., PRL 86, 906 (2001)
Anisotropy: Selection Rules
 Not all modes are visible with the same technique! B1:
SILENT modes
 Not all allowed modes are visible at the same time!
J.M. Zhang et al., PRB 56, 14399 (1997)
Elasticity
h-BN
 V 2  0
 ij 
1

C ilmj nl nm
Christoffel
equations
A. Bosak et al., PRB 73 041402(R) (2006)
Vibrational spectroscopies
 Probes:
• Brillouin spectroscopy
• Raman spectroscopy
• Infrared absorption spect.
• Inelastic X-ray Scattering
 Light: photons
 Particles

electrons: High Resolution e- Energy Loss

He: He atom scattering

neutrons
• Time-of-flight spectroscopy
• Inelastic Neutron Scattering
Vibrational spectroscopies
2 K + G 
M
2K
M
2G
M
Vibrational spectroscopies
Brillouin spec.
• Excitations of 2meV-0.6meV
• Acoustic phonon branches at low q
(sound waves)
• Information: E  Vs (sound speed)
linewidth Atenuation
X-ray scattering
• Excitations ~ meV, ~ Å-1
• whole BZ available
àNo kinematics restrictions
à Dispersion + ()
• Energy resolution ~1meV
Optic b.
• 1meV-eV Excitations
• Optic phonons at the center of
the Brillouin zone
• High resolution
• Different selection rules
Acoustic branches
Raman spec.
Neutron scattering
• Excitations ~ meV, ~ Å-1
• whole BZ available
àDispersion + ()
Kinematical limit: vs < 3000 m/s
Infrared Spectroscopy
 Absorption spectroscopy: dipolar selection rules
 Target: polar molecular vibrations, determination of
functional groups in organic compounds, polar modes in
crystals
BILBAO CRYSTAL… SERVER
SAM
ISOTROPY PACKAGE
(STOKES ET AL.)
SMODES, FINDSYM, ETC……..
Interatomic force constants
Born-Oppenheimer approximation
In the harmonic approximation, the total energy of a crystal with small
atomic position deviations is:

1
(0)
Etot ({u})  Etot    Ck ,k ' (a, b)uka ukb'
ak bk '  2
where the matrix of IFC’s is defined as:
 2 Etot
Ck ,k ' (a, b)  a b
uk uk '
25
Physical Interpretation of the Interatomic
Force Constants
The force conjugate to the position of a nucleus, can always be written:
a
Etot
Fk    a
rk
We can thus rewrite the IFC’s in more
physically descriptive fashion:
Fka
Ck ,k ' (a, b)   b
uk '
The IFC’s are the rate of change of the
atomic forces when we displace another
atom in the crystal.
26
Dynamical properties under pressure..
The construction of the dynamical matrix at gamma
point is very simple:
Phonon dispersion, DOS, PDOS requires
supercell calculations. Also DFPT allows to
include T effects, Thermod. properties, etc…
Relation between the IFC’s and the dynamical
matrix
The Fourier transform of the IFC’s is directly related to the dynamical matrix,



~
iq R b
Ck ,k ' (q )   Ck ,k ' (0, b)e
 M k M k ' Dk ,k ' (q )
b
The phonon frequencies are then obtained by diagonalization of the dynamical
matrix or equivalently by the solution of this eigenvalue problem:

~
2
 (k '  )  M     (k )
C
(
q
)

 k ,k ' mq
k mq mq
k '
phonon displacement
pattern
masses
square of
phonon frequencies
28
Phonon band structure of α-Quartz

~
C
(
q
 k ,k ' ) mq (k '  )  M km2q mq (k )
k '
SiO2
9 atoms per unit cell
Nb. of phonon bands:
3nat  27
Nb. of acoustic bands: 3
Nb. of optical bands:
3nat  3  24
Polar crystal : LO non-analyticity
Directionality !
[X.Gonze, J.-C.Charlier, D.C.Allan, M.P.Teter, PRB 50, 13055 (1994)]
29
LO-TO splitting
Wrong
behaviour
High - temperature : Fluorite structure
( Fm3m , one formula unit per cell )
 Supercell calculation
+ interpolation
! Long-range dipole-dipole
interaction not taken into account
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 Z Zr
= 5.75
*
Z0 = -2.86
and   = 5.75
LO - TO splitting 11.99 THz
Non-polar mode is OK
30
Thermodynamic properties
In the harmonic approximation, the phonons can be treated as an independent
boson gas. They obey the Bose-Einstein distribution:
1
n( ) 
e

k BT
1
The total energy of the gas can be calculated directly using the standard formula:
max
1

U phon     n( )   g ( )d
2

0
Energy of the harmonic oscillator
Phonon DOS
All thermodynamic properties can be calculated in this manner.
Note:
n( ) 
2 April 2008
1

2
1
e

k BT

k BT
1 1 e 1 1



coth(
)

2 2 kT
2
2k BT
B
1
e 1

ISVS 2008: Phonon Bands and Thermodynamic Properties
31
1. Even with f electrons
(PRB 85, 024317 (2012)
TbPO4 , DyPO4
ZnS,
Phys. Rev B 81 075207 (2010)
Cardona, …Muñoz . et al.
Spin-orbit, phonon dispersión, temperature effects, etc….
DFPT
Inverted s-o interaction.
Contribution of the
negative splitting of 5d G15
states of Hg wich
overcompesate the
positive splitting of the S
3p.
CuGaS2 electronic and phononic properties
Eficient photovoltaic materials.
(Phys.Rev. B 83,195208 (2011) )
Chalcopyrite tetragonal SG I-42d
Few it is know about this compounds. We did a structural ,
electronic and phononic study of the thermodynamical properties
Two main groups of chalcopyrites:
•I-III-VI2 derived from II-VI zb compounds (CuGaS2, AgGaS2..)
•II-IV-V2 derived from III-V zb comp.ounds (ZnGaAs2,….)
Two formula units per primitive cell
We will focus on the study of some
thermodynamics properties, like the
specific heat with emphasis in the low-T
region where appear strong desviation
of the Debye T3 law, phonons, etc…
silicon
zincblende, ZnSS
chalcopyrite, CuGaS2
Elastic Constants Cij (no experimental data available)
5
Phonons
Starting from the electronic structure we calculate the phonon dispersion
relations with density functional perturbation theory. We compare them
with Raman and IR measurements at the center of the BZ (see Figure).
CuGaS2
comparison with
Raman and IR
measurements ()
shows good
agreement
inelastic neutron
scattering data
are not available
as yet
7
Phonon Density of States
PDOS (states / formula unit)
Through BZ integration of the phonon dispersion relations the
phonon density of states (total or projected on the individual atoms
Cu, Ga, S)) are obtained (see Figure).
sulphurlike
CuGa
GaCu
below 120 cm-1:
essentially Cu- and
Ga-like phonons
above 280 cm-1:
essentially S-like
phonons
midgap feature at
~180 cm-1: Ga-,
Cu-like
The partial density of states are
useful for calculating the effect of
isotope disorder on the phonon
linewidths
Two-phonon
No second-order Raman
spectra
available.
The
calculated
sum
an
difference
densities will
help to interpret future
measured spectra.
It is posible to establish a
correspondence
between
the calculated two-phonon
Raman spectra of CuGaS2
and other two-phonon
measured spectra of binary
compounds.
The effect of phonons the on Vo(T) for a (cubic) crystal can be
expressed in terms of mode Grüneisen parameters γqj :
Due to the large number of phonons bands, a first approximation is
to use only the values at the Zone center for the evaluation of the
termal expansion coefficient.
The temperature dependence of Vo for q= 0 is:
Or from thermodyn…
using S(P,T)
Heat capacity
The phonon DOS allows to calculate the Free Energy F(T), and the
specific heat at constant volume
And the constant pressure Cp can be obtained:
Comparison of calculated and measured specific heat
peak at ~ 20 K in
CP /T 3
representation from
Cu/Ga like phonons
(ratio 1:6 to lowfrequency peak in
phonon DOS)
3
4
Cp/T (mJ/molK )
CuGaS2 versus AgGaS2
our data CuGaS2
500
Abrahams and Hsu,
J. Chem. Phys. 63, 1162 (1975)
400
ABINIT LDA
VASP GGA PBE
VASP GGA PBEsol
300
ABINIT LDA
reproduces peak
position, but
absolute value at
peak ~20% lower
200
100
0
qDebye(0) = 355 K
0
20
40
60
T (K)
80
8
100
VASP GGA
reproduces peak
position and
magnitude
Comparison of calculated and measured specific heat
9
3
4
Cp/T (mJ/molK )
Extension to AgGaS2 and AgGaTe2
our data AgGaTe2
4000
our data AgGaS2
our data CuGaS2
3000
Abrahams and Hsu,
J. Chem. Phys. 63, 1162 (1975).
ABINIT LDA
VASP GGA PBE
VASP GGA PBEsol
2000
1000
0
0
20
40
60
T (K)
80
100
lattice softening
by Cu  Ag
replacement
lattice softening
by S  Te
replacement
Even more properties?
Inclusion of Temperature effects is computationally very expensive,
e-ph interaction,…
Many experimental results of T dependence of the gap in binary and
ternary compounds.
The degree of cation-anion hybridization on the electronic an
vibrational properties, leads to anomalous dependence of the band
gaps with temperature.
The presence of d-electrons in upper VB lead to anomalies, like
negative s-o splitting. For example in Cu or Ag chalcopyrite, the
other constituents correspond to decrease the gap, but Cu or Ag
tends to increase. The sum of boths effects generates a non
monotonic dependence of gaps with T.
It can be fitted using two Einstein oscillator according to:
E0 is the zero-point un-renormalized gap energy, A1 is the
contribution to the zero-point renormal., nB is the Bose-Eisntein
function .
energy (eV)
12
S 3p4
4
Se 4p
10
10
3d
Te 5p4
8
10
4d
1
Cu
6
4s
1
Ag
5s
The admixture of p and d electrons in the
valence bands produces anomalies e.g. in the
temperature dependence of the energy gap: at
low T the gap increases with T (up to~100K)
presumably because of the presence of delectrons. Above 100K it decreases. Detailed
theoretical explanation not yet available.
Ramdas et al.
2.70
AgGaS2
2.69
accoustic
phonons
(d-electrons)
2.68
optic
phonons
2.67
CuGaS2 Ramdas and Bhosale
2.51
CuGaS2
2.50
2.49
gap energy (eV)
energy gap (eV)
2.71
gap energy (eV)
4
2.48
2.47
2.504
2.502
2.500
2.498
0
50
100
150
T (K)
2.66
2.46
0
100
200
300
T (K)
Temperature dependence of the energy gap of
AgGaS2 with two-phonon fit.
0
100
200
300
T (K)
Temperature dependence of the energy gap of
CuGaS2 with two-phonon fit.
ELASTICITY
Elasticity - deformation
σij =
VOIGT’S
NOTATION
Cijkl εkl (only two index)
Some examples of elasticity under pressure
Mechanical stability criteria
Pressure 0 GPa Born Criteria
Pressure P ≠ 0 GPa
Born Generalized Criteria
YGa5O12 garnet
unit cell)
(160 atoms
CONCLUSIONS:
 Ab initio methods can provide interesting and
useful information of the physics and chemistry of
materials properties under high pressure, from small
system to big systems. Phonons and elastic properties
provide interesting info, dynamical and mechanical
stability
Temperature, S-O etc…T effects can be included.
These techniques can help to design and to
understand problems in experimental interpretations.
But remember!!!!! WE USE APPROXIMATIONS
“An expert is a person who has
made all the mistakes that can
be made in a very narrow field”.
Niels Bohr (1885-1962)
Physics is to mathematics like sex is
to masturbation.”
—Richard Feynman, (1918-1988)
Thank you for your
attention!