1、Phonons & electron-phonon coupling,Claudia Ambrosch-DraxlDepartment fr Materialphysik, Montanunversitt Leoben, AustriaInstitut fr Physik, Universitt Graz, Austria,Aspects of e-ph Coupling,Charge transport Heat transport Thermal expansion Electron (hole) lifetimes Superconductivity,Some important phe
2、nomena,effective electron-electron interaction,?,Outline,The frozen phonon approach Lattice dynamics Atomic forces,Basics,Phonons and electron-phonon coupling,Symmetry Vibrational frequencies Normal vectors Raman scattering Linear-response theory Comparison with experiment LAPW / WIEN2k specific asp
3、ects and examples,The Frozen-Phonon Approach,1D case:,Calculate energies Fit expansion coefficients,Harmonic approximation,The Harmonic Approximation,General case:,Force constant:,change of the force acting on atom a in unit cell n in direction i, when displacing atom b in unit cell m in direction j
4、.,Displacement wave:,The Harmonic Approximation,Equation of motion:,Vibrational frequencies w:,by diagonalization of the dynamical matrix D,N atoms per unit cell 3N degrees of freedom Set of 3N coupled equations,Computational Effort,N atoms per unit cell,Harmonic case only! Interpolation only no fit
5、!,The Hellmann-Feynman Theorem,Many particle Schrdinger equation,electronic coordinates,ionic coordinates,groundstate wavefunction with respect to fixed ions,The Hellmann-Feynman Force,Hellmann-Feynman force:,total classical Coulomb force acting on the nucleus a stemming from all other charges of th
6、e system =electrostatic force stemming from all other nuclei + electrostatic force stemming from the electronic charges,component of the electric field caused by the nuclear charge,Forces in DFT,Atomic force:,Pulay corrections,Total energy:,Forces in the LAPW Basis,Hellmann-Feynman force: classical
7、electrostatic force excerted on the nucleus by the other nuclei and the electronic charge distribution,IBS force: incomplete basis set correction due to the use of a finite number of position-dependent basis functions,Core correction: contribution due to the fact that for core electrons only the sph
8、erical part of the potential is taken into account,The LAPW Method,Atomic spheres: atomic-like basis functions,Interstitial: planewave basis,site-dependent!,Example: YBa2Cu3O7,O(1),Cu(1),Ba,Y,Cu(2),O(4),O(3),O(2),Orthorhombic cell: Pmmm,Symmetry,Bilbao Crystallographic Server: http:/www.cryst.ehu.es
9、/,Factor group analysis:,5 Ag + 8 B1u + 5 B2g + 8 B2u + 5 B3g + 8 B3u,Dynamical matrix,Raman-active,Infrared-active,q=0,Forces in YBa2Cu3O7,Force contributions for a mixed distortion:,-O(2), -O(4),YBa2Cu3O7: Phonon Frequencies,Ag modes,YBa2Cu3O7: Normal Vectors,Ag modes,YBa2Cu3O7: Lattice Vibrations
10、,oxygen modes,Ba / Cu modes,Raman Active Phonons,Theory,Raman Scattering Intensities,CAD, H. Auer, R. Kouba, E. Ya. Sherman, P. Knoll, M. Mayer, Phys. Rev. B 65, 064501 (2002).,Experiment,Probing e-ph Coupling Strength,O(4) mode,Probing e-ph Coupling Strength,Resonance:,Peak at 2.2 eV All oxygen mod
11、es O(4) displacement!,Probing Normal Vectors,Ba-Cu modes:,Experiment: site-selective isotope substitution,Isotope Substitution,Raman scattering intensities:,Influence of mass and eigenvectors,Isotope Substitution,Raman scattering intensities:,Change of e-ph coupling strength through normal vectors,C
12、AD, H. Auer, R. Kouba, E. Ya. Sherman, P. Knoll, M. Mayer, Phys. Rev. B 65, 064501 (2002).,Relevant for superconductivity,E. Ya. Sherman and CAD, Eur. Phys. J. B 26, 323 (2002) .,q-dependent Phonons,Equilibrium,q 0,q = 0,Supercells vs. Perturbation Theory,Unit cell commensurate with the q-vector (su
13、percell) Computationally very demanding,Supercell method:,Linear response theory:,Starting point: undisplaced structure Treat q-dependent displacement as perturbation Self-consistent linear-response theory Keep single cell Computational effort nearly independent of q-vector Anharmonic effects neglec
14、ted,N. E. Zein, Sov. Phys. Sol. State 26, 1825 (1984).S. Baroni, P. Gianozzi, and A. Testa, Phys. Rev. Lett. 58, 1861 (1987).,Linear Response Theory,Atomic displacement:,small polarization vector,Superposition of forward and backward travelling waveStatic first-order perturbation within density-func
15、tional perturbation theory (DFPT)Determine first-order response on the electronic charge, effective potential and Kohn-Sham orbitals,Linear Response Theory,Iterative solution of three equations:,Determine q-dependent atomic forces and dynamical matrix Alternatively compute second order changes (DM)
16、directly,e-ph Matrix Elements,Electron-phonon matrix element:,Need to evaluate matrix elements like:,Scattering process of an electron by a phonon with wavevector q,Matrix elements including Pulay-like terms:,S. Y. Savrasov and D. Y. Savrasov, Phys. Rev. B 54, 16487 (1996) .,R. Kouba, A. Taga, CAD,
17、L. Nordstrm, and B. Johansson, Phys. Rev. B 64, 184306 (2002).,e-ph Coupling Constants,Coupling constant for a phonon branch n:,bcc S,R. Kouba, A. Taga, CAD, L. Nordstrm, and B. Johansson, Phys. Rev. B 64, 184306 (2002).,J. K. Dewhurst, S. Sharma, and CAD, 68, 020504(R) (2003);H. Rosner, A. Kitaigor
18、odotsky, and W. E. Pickett, Phys. Rev. Lett. 88, 127001 (2002).,What Can We Learn?,Comparison with experiment .,P. Puschnig, C. Ambrosch-Draxl, R. W. Henn, and A. Simon, Phys. Rev. B 64, 024519-1 (2001).,helps to analyze measured data contributes to assign modes,Theory can .,predict superconducting transition temperatures,predict phase transitions (phonon softening) much more .,Thank you for your attention!,
