fonones y elasticidad bajo presión ab initio. alfonso muñoz dpto. de física fundamental ii...
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Fonones y Elasticidad bajo presión ab initio .
Alfonso MuñozDpto. de Física Fundamental II
Universidad de La Laguna. Tenerife.MALTA Consolider TeamCanary Islands, SPAIN
Plan de la charla:
Introducción : Ab initio methods Fonones. Propiedades dinámicas.
Ejemplos Elasticidad estabilidad mecánica bajo presiónConclusiones.
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)
Thermal Expansion Superconductivity
Elasticity - deformationThermal Conductivity
Fonones y espectroscopía, ¿para que?
March 31st, 2008 ISVS: A hands-on introduction to ABINIT
Lattice Dynamics
Lattice Potential:
Harmonic approximation:
...)(
!3
1)(
2
1''''''
'''','', 0''''''
3
'''', 0''
2
0
lll
lll lllll
ll ll
uuuuuu
RUuu
uu
RURUU
ERUTREVT ieii )()(
0|'
)'|(~|')|(~
)|(~1)(
'2
,
,',
2
q
qqq
q rq
q
D
uDu
eum
lu tli
)''('',)(
)''()('',2
1)(
,'',
'', ,,
lulllum
kxxmF
lulullRU
l
ll
Hooke’s law!
IFC
March 31st, 2008 ISVS: A hands-on introduction to ABINIT
Ansatz:
Harmonic approximation:
=>
Phonons: linear chain of atoms
Phonons: linear chain of atoms
qaK
ω q =2 sinM 2
q=0
πq=
2aπ
q=a
Linear chain of atoms 4KM
March 31st, 2008 ISVS: A hands-on introduction to ABINIT
two atoms per unit cell
Ansatz:
Linear chain with two different "spring constants"
March 31st, 2008 ISVS: A hands-on introduction to ABINIT
2 K+G
M
2KM
2GM
Linear chain with two different "spring
constants"
Phonons
2 2 2K+G 1ω(q) = ± K +G +2KGcos qa
M M
Two solutions:
acΓ
opΓ
acL
opL
Γ L
acoustic (-) and optic (+) branches
March 31st, 2008
3D Phonon Dispersion Relations3THz ~ 100 cm-1 ; 1meV ~ 8 cm-1
3C-SiC
J. Serrano et al., APL 80, 23 (2002)
cm-1
LO
TO
SiTHz
G. Nilsson and G. Nelin, PRB 6, 3777 (1972)W. Weber, PRB 15, 4789 (1977)
j = 3N branches
March 31st, 2008
Polar crystals: LST relationIonic crystals: Macroscopic electric field
0
2
2
TO
LO
Born effectivecharges
|04
,*
qu
PZ mac
'0~
4
*'',''
2
' ''
*'
ZMqC
Z
'
'',~1
|'~
ll
i l'lellCN
C RRqq
2 2
* * '4 1
m mm
q Z Z q
M q q
q 0 q 0
X. Gonze and C. Lee, PRB 55, 10355 (1997)
Lyddane-Sachs-Teller relation
March 31st, 2008
Anisotropy: crystal fieldGaN
T. Ruf et al., PRL 86, 906 (2001)
March 31st, 2008
Anisotropy: Selection RulesNot all modes are visible with the same
technique! B1: SILENT modesNot all allowed modes are visible at the same
time!
J.M. Zhang et al., PRB 56, 14399 (1997)
March 31st, 2008
Elasticity
A. Bosak et al., PRB 73 041402(R) (2006)
02 V
mlilmjij nnC
1
Christoffelequations
h-BN
March 31st, 2008 ISVS: A hands-on introduction to ABINIT
Probes:
Light: photons
Particles
Vibrational spectroscopies
• Brillouin spectroscopy• Raman spectroscopy• Infrared absorption spect.• Inelastic X-ray Scattering
electrons: High Resolution e- Energy Loss
He: He atom scattering
neutrons • Time-of-flight spectroscopy• Inelastic Neutron Scattering
March 31st, 2008
Vibrational spectroscopies
2 K+G
M
2KM
2GM
March 31st, 2008
Brillouin spec.• Excitations of 2eV-0.6meV• Acoustic phonon branches at low q (sound waves)
• Information: Vs (sound speed) linewidth Atenuation
Op
tic b
.
Acou
sti
c b
ran
ch
es
Neutron scattering• Excitations ~ meV, ~ Å-1
• whole BZ availableDispersion + ()Kinematical limit: vs < 3000 m/s
X-ray scattering• Excitations ~ meV, ~ Å-1
• whole BZ availableNo kinematics restrictions Dispersion + ()• Energy resolution ~1meV
Vibrational spectroscopiesRaman spec.• 1meV-eV Excitations• Optic phonons at the center ofthe Brillouin zone• High resolution • Different selection rules
March 31st, 2008 ISVS: A hands-on introduction to ABINIT
Absorption spectroscopy: dipolar selection rules Target: polar molecular vibrations,
determination of functional groups in organic compounds, polar modes in crystals
Infrared Spectroscopy
BILBAO CRYSTAL… SERVER SAM
ISOTROPY PACKAGE (STOKES ET AL.)
SMODES, FINDSYM, ETC……..
25
bk
ak bk
akkktottot uubaCEuE
'
'',
)0( ),(2
1})({
where the matrix of IFC’s is defined as:
bk
ak
totkk uu
EbaC
'
2
', ),(
Interatomic force constants
In the harmonic approximation, the total energy of a crystal with small atomic position deviations is:
Born-Oppenheimer approximation
26
Physical Interpretation of the Interatomic Force Constants
ak
totak r
EF
bk
ak
kk u
FbaC
'', ),(
The force conjugate to the position of a nucleus, can always be written:
We can thus rewrite the IFC’s in more physically descriptive fashion:
The IFC’s are the rate of change of the atomic forces when we displace another atom in the crystal.
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…
28
Relation between the IFC’s and the dynamical matrix
)()'()(~ 2
'',
kMkqC qmqmkqm
kkk
)(),0()(~
','',', qDMMebCqC kkkkb
Rqikkkk
b
The Fourier transform of the IFC’s 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:
phonon displacementpattern
massessquare ofphonon frequencies
29
SiO2 9 atoms per unit cell
[X.Gonze, J.-C.Charlier, D.C.Allan, M.P.Teter, PRB 50, 13055 (1994)]
Nb. of phonon bands: 273 atn
Nb. of acoustic bands: 3
Nb. of optical bands:
Polar crystal : LO non-analyticity
)()'()(~ 2
'',
kMkqC qmqmkqm
kkk
2433 atn
Directionality !
Phonon band structure of α-Quartz
30
LO-TO splitting
High - temperature : Fluorite structure( , one formula unit per cell )Fm3m
Supercell calculation+ interpolation
! Long-range dipole-dipoleinteraction not taken into account
Calculated phonon dispersions of ZrO2 in the cubicstructure at the equilibrium lattice constant a0 = 5.13 Å.
DFPT (Linear-response)with = 5.75
= -2.86and = 5.75LO - TO splitting 11.99 THz
Non-polar mode is OK
ZZr*
Z0*
Wrongbehaviour
[From Parlinski K., Li Z.Q., and Kawazoe Y.,Phys. Rev. Lett. 78, 4063 (1997)]
2 April 2008 ISVS 2008: Phonon Bands and Thermodynamic Properties 31
Thermodynamic properties
In the harmonic approximation, the phonons can be treated as an independent boson gas. They obey the Bose-Einstein distribution:
1
1)(
TkBe
n
The total energy of the gas can be calculated directly using the standard formula:
dgnU phon )(2
1)(
max
0
Energy of the harmonic oscillator Phonon DOS
Note: )2
coth(2
1
1
1
2
1
2
1
1
1
2
1)(
Tke
e
e
nBTk
Tk
Tk B
B
B
All thermodynamic properties can be calculated in this manner.
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…. Inverted s-o
interaction. Contribution of the negative splitting of d states of Hg wich overcompesate the positive splitting of the S 3p.
DFPT
CuGaS2 electronic and phononic properties
Eficient photovoltaic materials. (Phys.Rev. B 83,195208 (2011) )
Chalcopyrite tetragonal SG I-42dFew 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-Tregion where appear strong desviationof the Debye T3 law, phonons, etc…
silicon
zincblende, ZnSS
chalcopyrite, CuGaS2
Elastic Constants Cij (no experimental data available)
5PhononsStarting 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).
comparison with Raman and IR measurements () shows good agreement
inelastic neutron scattering data are not available as yet
CuGaS2
PD
OS
(st
ate
s /
form
ula
un
it)
Phonon Density of StatesThrough 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).
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
7
Ga-Cu
Cu-Ga
sulphur-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 capacityThe 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:
0 20 40 60 80 1000
100
200
300
400
500
Cp/T
3 (J
/mol
K4)
T (K)
our data CuGaS2
Abrahams and Hsu, J. Chem. Phys. 63, 1162 (1975)
ABINIT LDA VASP GGA PBE VASP GGA PBEsol
Comparison of calculated and measured specific heat
CuGaS2 versus AgGaS2peak at ~ 20 K in CP /T 3 representation from Cu/Ga like phonons (ratio 1:6 to low-frequency peak in phonon DOS)
ABINIT LDA reproduces peak position, but absolute value at peak ~20% lower
VASP GGA reproduces peak position and magnitude
8
Debye(0) = 355 K
0 20 40 60 80 1000
1000
2000
3000
4000
Cp/T
3 (J
/mol
K4)
T (K)
our data AgGaTe2
our data AgGaS2
our data CuGaS2
Abrahams and Hsu, J. Chem. Phys. 63, 1162 (1975).
ABINIT LDA VASP GGA PBE VASP GGA PBEsol
Comparison of calculated and measured specific heat
Extension to AgGaS2 and AgGaTe2
lattice softening by Cu Ag replacement
lattice softening by S Te replacement
9
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 .
AgGaS2
0 100 200 3002.66
2.67
2.68
2.69
2.70
2.71
optic phonons
T (K)
ener
gy g
ap (
eV) Ramdas et al.
accoustic phonons(d-electrons)
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 d-electrons. Above 100K it decreases. Detailed theoretical explanation not yet available.
Temperature dependence of the energy gap of AgGaS2 with two-phonon fit.
4
6
8
10
12
Ag
3d10
4s1
Cu
5p4
4p4
Te
Se
ener
gy
(eV
) S 3p4
4d10
5s1
CuGaS2
Temperature dependence of the energy gap of CuGaS2 with two-phonon fit.
0 100 200 3002.46
2.47
2.48
2.49
2.50
2.51
0 50 100 1502.498
2.500
2.502
2.504
gap
ener
gy (
eV)
T (K)
CuGaS2 Ramdas and Bhosale
gap
ener
gy (
eV)
T (K)
Elasticity - deformation
ELASTICITY
σij = Cijkl εklVOIGT’S NOTATION (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 (160 atoms unit cell)
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 stabilityTemperature, 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!
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