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 - deformation Thermal Conductivity

Fonones y espectroscopía, ¿para que?

March 31st, 2008 ISVS: A hands-on introduction to ABINIT

Lattice Dynamics

Lattice Potential:

Harmonic approximation:

...)(!3

1)(21

'''''''''','', 0''''''

3

'''', 0''

2

0

lll

lll lllll

ll ll

uuuuuu

RUuuuuRURUU

ERUTREVT ieii )()(

0|'

)'|(~|')|(~

)|(~1)(

'2

,

,',

2

q

qqq

q rq

q

D

uDu

eum

lu tli

)''('',)(

)''()('',21)(

,'',

'', ,,

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

qaKω q =2 sinM 2

q=0

πq=2aπq=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+GM

2KM

2GM

Linear chain with two different "spring constants"

Phonons

2 2 2K+G 1ω(q) = ± K +G +2KGcos qaM 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

,*

quP

Z 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+GM

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

Opt

ic b

.Ac

oust

ic b

ranc

hes

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( ),(21})({

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 uFbaC

'', ),(

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 kkkk

b

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 atnNb. 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 splittingHigh - 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 )(21)(

max

0

Energy of the harmonic oscillator Phonon DOS

Note: )2

coth(21

1

121

21

1

121)(

Tke

e

en

BTk

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 propertiesEficient 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

PDO

S (s

tate

s / f

orm

ula

unit)

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 (e

V) 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

TeSe

ener

gy (e

V) 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 (e

V)

T (K)

CuGaS2 Ramdas and Bhosale

gap

ener

gy (e

V)

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!