Presentation on theme: "Dpto. de Física Fundamental II Universidad de La Laguna. Tenerife."— Presentation transcript:
1 Dpto. de Física Fundamental II Universidad de La Laguna. Tenerife. Fonones y Elasticidad bajo presión ab initio .Alfonso MuñozDpto. de Física Fundamental IIUniversidad de La Laguna. Tenerife.MALTA Consolider TeamCanary Islands, SPAIN
2 Plan de la charla:Introducción : Ab initio methodsFonones. Propiedades dinámicas. EjemplosElasticidad estabilidad mecánica bajo presiónConclusiones.
3 Ab initio methodsState 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.
4 "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,( )
12 Linear chain with two different "spring constants" two atoms per unit cellAnsatz:March 31st, 2008ISVS: A hands-on introduction to ABINIT
13 Phonons Linear chain with two different "spring constants" Two solutions:acoustic (-) and optic (+) branchesMarch 31st, 2008ISVS: A hands-on introduction to ABINIT
14 3D Phonon Dispersion Relations 3THz ~ 100 cm-1 ; 1meV ~ 8 cm-1j = 3N branchesSiTHzG. Nilsson and G. Nelin, PRB 6, 3777 (1972)W. Weber, PRB 15, 4789 (1977)3C-SiCJ. Serrano et al., APL 80, 23 (2002)cm-1LOTOMarch 31st, 2008
15 Polar crystals: LST relation Ionic crystals: Macroscopic electric fieldLyddane-Sachs-Teller relationBorn effectivechargesX. Gonze and C. Lee, PRB 55, (1997)March 31st, 2008
16 Anisotropy: crystal field GaNT. Ruf et al., PRL 86, 906 (2001)March 31st, 2008
17 Anisotropy: Selection Rules Not 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, (1997)March 31st, 2008
18 Elasticity h-BN Christoffel equations A. Bosak et al., PRB (R) (2006)March 31st, 2008
19 Vibrational spectroscopies Probes:Light: photonsParticlesBrillouin spectroscopyRaman spectroscopyInfrared absorption spect.Inelastic X-ray Scatteringelectrons: High Resolution e- Energy LossHe: He atom scatteringneutronsTime-of-flight spectroscopyInelastic Neutron ScatteringMarch 31st, 2008ISVS: A hands-on introduction to ABINIT
21 Vibrational spectroscopies Raman spec.1meV-eV ExcitationsOptic phonons at the center ofthe Brillouin zoneHigh resolutionDifferent selection rulesX-ray scatteringExcitations ~ meV, ~ Å-1whole BZ availableNo kinematics restrictionsDispersion + r()Energy resolution ~1meVOptic b.Neutron scatteringExcitations ~ meV, ~ Å-1whole BZ availableDispersion + r(w)Kinematical limit: vs < 3000 m/sAcoustic branchesBrillouin spec.Excitations of 2meV-0.6meVAcoustic phonon branches at low q(sound waves)Information: E Vs (sound speed)linewidth AtenuationMarch 31st, 2008
22 Infrared Spectroscopy Absorption spectroscopy: dipolar selection rulesTarget: polar molecular vibrations, determination of functional groups in organic compounds, polar modes in crystalsMarch 31st, 2008ISVS: A hands-on introduction to ABINIT
24 ISOTROPY PACKAGE (STOKES ET AL.) SMODES, FINDSYM, ETC……..
25 Interatomic force constants Born-Oppenheimer approximationIn the harmonic approximation, the total energy of a crystal with small atomic position deviations is:where the matrix of IFC’s is defined as:
26 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 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.
27 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 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 displacement patternmassessquare of phonon frequencies
29 Phonon band structure of α-Quartz SiO2 9 atoms per unit cellNb. of phonon bands:Nb. of acoustic bands: 3Nb. of optical bands:Polar crystal : LO non-analyticityDirectionality ![X.Gonze, J.-C.Charlier, D.C.Allan, M.P.Teter, PRB 50, (1994)]
30 LO-TO splitting High - temperature : Fluorite structure ( , one formula unit per cell )Wrongbehaviour Supercell calculation + interpolation! Long-range dipole-dipole interaction not taken into accountCalculated phonon dispersions of ZrO2 in the cubicstructure 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 ==and =LO - TO splitting THzNon-polar mode is OK
31 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 DOSEnergy of the harmonic oscillatorAll thermodynamic properties can be calculated in this manner.Note:2 April 2008ISVS 2008: Phonon Bands and Thermodynamic Properties
32 Even with f electrons (PRB 85, 024317 (2012) TbPO4 , DyPO4
33 ZnS, Phys. Rev B (2010)Cardona, …Muñoz . et al.
35 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.
36 CuGaS2 electronic and phononic properties Eficient photovoltaic materials. (Phys.Rev. B 83, (2011) )Chalcopyrite tetragonal SG I-42dFew it is know about this compounds. We did a structural , electronic and phononic study of the thermodynamical propertiesTwo 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 cellWe will focus on the study of somethermodynamics properties, like thespecific heat with emphasis in the low-Tregion where appear strong desviationof the Debye T3 law, phonons, etc…
39 Elastic Constants Cij (no experimental data available)
40 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 agreementinelastic neutron scattering data are not available as yetCuGaS2
41 Phonon Density of States 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).7PDOS (states / formula unit)sulphur-likebelow 120 cm-1: essentially Cu- and Ga-like phononsabove 280 cm-1: essentially S-like phononsmidgap feature at ~180 cm-1: Ga-, Cu-likeCu-GaGa-CuThe partial density of states are useful for calculating the effect of isotope disorder on the phonon linewidths
42 Two-phononNo 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.
43 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)
48 Comparison of calculated and measured specific heat 8Comparison of calculated and measured specific heatCuGaS2 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% lowerVASP GGA reproduces peak position and magnitudeqDebye(0) = 355 K
49 Comparison of calculated and measured specific heat 9Comparison of calculated and measured specific heatExtension to AgGaS2 and AgGaTe2lattice softening by Cu Ag replacementlattice softening by S Te replacement
50 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 .
51 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.AgGaS2CuGaS2Temperature dependence of the energy gap of AgGaS2 with two-phonon fit.Temperature dependence of the energy gap of CuGaS2 with two-phonon fit.
60 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
61 “An expert is a person who has made all the mistakes that can be made in a very narrow field”. Niels Bohr ( )
62 Physics is to mathematics like sex is to masturbation.” —Richard Feynman, ( )