The electron-phonon coupling in ABINIT

Transcription

1 The electron-phonon coupling in ABINIT Matthieu J. Verstraete University of Liège, Belgium May /43

2 Outline 1 Motivation 2 EPC introduction 3 A bit of theory 4 Transport 5 ABINIT 2/43

3 Impact of phonons I Electrical resistivity Heat transport Superconductivity Thermoelectricity el-ph is 1 contribution + anharm, impurities, isotope... 3/43

4 Impact of phonons II Inelastic/relaxation mechanism: IR ESR (w/ SO interaction) X Ray widths Raman Reznik, Adv Cond Matt, (2010) 4/43

5 Raman linewidth M. Lazzeri and F.Mauri, PRL 90, (2003) Wang et al. PRB (2006) Linewidth contribution from el-ph processes 5/43

6 Outline 1 Motivation 2 EPC introduction 3 A bit of theory 4 Transport 5 ABINIT 6/43

7 Bare phonons Rigid motion of ions (no screening) Completely unrealistic phonon frequencies (usu. too hard) 7/43

8 Screened phonons Screening of ion motion by electrons (SC DFT / DFPT) Very realistic phonon frequencies (max 10% error) 8/43

9 Independent electrons... Perfect crystal and indep. e no resistivity Electron-ion interaction is periodic: Renormalized e energies but still ideal quasi-particles KS states are in this category 9/43

10 ... & things that perturb them In real system, perturbations add to Hamiltonian H = H0 + H1 give finite lifetimes for indep. part. eigenstates interaction only full MB Ψ has lifetime Perturbations = Coulomb, photons, defects, impacts and phonons In many cases (± high T) phonons dominate 1 What is the system (and the external perturbation)? 2 Are the particles independent? 10/43

11 Bare electrons Self consistent screening of static el charge Decent BS within DFT treatment of Coulomb interaction 11/43

12 Perturbed electrons Finite lifetime for KS states EP coupling constants usually quite good NB: Phonon eigenstates are Bloch-like Figure shows localized phonon wavepacket 12/43

13 Energy scales Ion vibrations have low energy (0-20 max few 100 mev) photons in IR Electronic excitations: Metals: ' 0 ev: gold κt = 318 W/mK Semiconductors (> Egap ' O(1) ev) no EPC with low ω e.g. diamond κt = W/mK Need strong coupling and large e DOS 13/43

14 Temperature According to preceding energy scale: Low T: only FS electrons contribute ABINIT: = neglect ω wrt electronic energies High T: more phonon ω insulators have EPC too T Debye freq classical Boltzmann stat. Hypothesis: phonons remain harmonic and lifetime 14/43

15 FS effects Phonon q connects k, k FS Nesting: many k-k give same q And energy/frequency dependency Huge change in electron screening Kohn anomaly in e bands (PRL (1959)) 15/43

16 Superconductivity Effective e e interaction w/ phonons can be attractive superconducting instability at low T New mixed quasiparticles have no resistance from EPC 16/43

17 Outline 1 Motivation 2 EPC introduction 3 A bit of theory 4 Transport 5 ABINIT 17/43

18 Eliashberg Theory Coupling Hamiltonian (Frölich): X Hep = < ~k + ~q δv ~k > ~dq ck +q ck kq Single phonon scattering (Migdal) Perturbed (electronic) potential δv from DFPT 18/43

19 Eliashberg Theory II Displacement operator: ~dq = X ( j ~ )~uqj (aqj + a qj ) 2NMωqj gkk matrix elements: ~ ~uqj < k~0 δv ~k > gkqj0 k = p 2NMωqj 19/43

20 Eliashberg Theory III Perturbation theory in Hep Use spectral representations of Green s functions for e, ph e-p and e-e self-energies Full phonon self-energy is not perturbational But pre-screened DFPT is ok 20/43

21 Spectral function Eliashberg spectral functions: X qj α2 F (Ω) = NF gk 0 k 2 δ(ω ωqj ) kk 0 j k-point sums over Fermi Surface Related to imaginary part of e self-energy 21/43

22 Spectral function II Closely linked to phonon DOS F (Ω) = P qj δ(ω ωqj ) Savrasov2 PRB 54, (1996) 22/43

23 EP quantities - Link to experiments starts here Superconducting coupling strength Z dω 2 λ=2 α F (Ω) Ω Phonon lifetime from scattering with e : X ~q m γ~qphm = 2πω~q m g ~0 0~ 2 δ( ~k i F )δ( k +qi ~ 0 F ) k i ki ~k ii 0 and electron lifetime due to phonons: X ~q m γ~keli = 2π g 0 ~0 ~ 2 {[fk~0 i 0 + n~q m ]δ( ~k i k +qi ~ 0 + ω~q m ) ~q mi 0 i k ik + [1 fk +qi ~ 0 + n~q m ]δ( ~k i k +qi ~ 0 + ω~q m )} 23/43

24 Further approximations Neglect band energy dependence in α2 F : ω Bands are not too narrow and DOS varies slowly Shouldn t be true for localized bands or low D!! 24/43

25 Coulomb interactions Most already in ωq and nk (DFT is a good start) Remaining e-e repulsion in Retarded coulomb parameter : Z c Vkk µ = NF 0 /(1 + log(ωel /ωd )) FS 2 ωel Coulombic frequency 10eV Approximate: only need the change btw SC and N states McMillan formula for Tc (W. L. McMillan, Phys. Rev. 167, 331 (1968)) 25/43

26 Outline 1 Motivation 2 EPC introduction 3 A bit of theory 4 Transport 5 ABINIT 26/43

27 Transport equations Classical statistical mechanics Boltzmann equations for flow of e and phonons Coupled by precisely the e-ph matrix elements gkk 0 27/43

28 Boltzmann equations 2π X fk = gkk +q 2 {fk (1 fk +q )[ t ~Nc q + (1 fk )fk +q [ (Nq + 1) δ( k k +q ~ωq ) Nq δ( k k +q + ~ωq )] (Nq + 1) δ( k k +q + ~ωq ) + Nq δ( k k +q ~ωq )]} Nq 4π X = gkk +q 2 fk (1 fk +q )[ t ~Nc Nq δ( k k +q + ~ωq ) k (Nq + 1) δ( k k +q ~ωq )] e.g. J.M. Ziman Electrons and Phonons Oxford U Press (1960) 28/43

29 Steady state solutions Relate κph, σe to coupling btw fluxes of ph and e Steady state transport under T or E gradient fk / t = diffusion terms Linearize Boltzmann eqs and simplify k dependency 29/43

30 Transport spectral function: Generalization by Allen: 2 F (ω) αin(out) αtr2 F (ω) = X 1 N0 < vx2 > ν = 2 αout F (ω) X 0 gkqν0 n0 kn vx (~k )vx (~k ( ) )δ(ω ωqν ) knk 0 n0 FS 2 αin F (ω) Average velocity < vx2 > 30/43

31 e- and ph- Resistivity Z dω x 2 πωcell kb T αtr2 F (ω) ρ(t ) = N0 < vx2 > 0 ω sinh2 x Z dω x 2 6Ωcell w(t ) = πkb N0 < vx2 > 0 ω sinh2 x 2x 2 2 4x αtr F (ω) + 2 αout F (ω) + 2 αin F (ω) π π where x = ω/2kb T Only lowest order approx. to full Boltzmann eq. Only electronic contribution to thermal resistance no lattice thermal conductivity 31/43

32 Outline 1 Motivation 2 EPC introduction 3 A bit of theory 4 Transport 5 ABINIT 32/43

33 Workflow (NEW!) 1 run ABINIT GS + phonons with minimal k-grid 2 run ABINIT for _GKK matrices on dense k-grid _DDB files = E (2) and _GKK = < k~0 δv ~k > 3 mrgddb pastes all E (2) into one file for ANADDB mrggkk pastes all < k~0 δv ~k > into one file 4 3 Natom perts grouped by qpoint, and complete! 5 Run anaddb with telphon = 1 and additional file names 33/43

34 Getting EP matrix elements SCF phonon calculation yields n(1) and hence H (1) Now do non-scf, 1 step calculation of < k~0 H (1) ~k > Can use any k-grid we want converge EPC integration NB: still need all perturbations for each ~q use prepgkk 1 in SCF phonon run 34/43

35 Symmetries Symmetry operations complete E (2) does not work for < k~0 δvqj ~k > (phase interference) < k~0 δvqj ~k >< ~k δvqj 0 k~0 > eliminates gauge need all 3 Natom perturbations! Q-points completed by symmetry k 0 = k + q so q-grid must be consistent with k-grid 35/43

36 Transport outputs Electrical resistivity/conductivity (_RHO) Thermal conductivity (_WTH) Lorentz coefficient (_LOR) Transport α2 Ftr (_A2F_TR) 36/43

37 Transport outputs Electrical resistivity (_RHO) Spin polarized Fe (MJ Verstraete JPCM 25, (2013)) 37/43

38 Electrical resistivity (_RHO) High pressure Fe in Earth s core (Gomi et al. PEPI 224, 88 (2013)) 38/43

39 Conclusions Widely useful quantities Strong links to experiment Small cost beyond phonon calculation Many other processes involve EPC Should work in PAW as well (ASAP!) 39/43

40 Collaborators Lots of input along the way Bin Xu, Momar Diakhate Matteo Giantomassi Jean-Paul Crocombette + the CEA boys Xavier Gonze, Samuel Poncé, Yannick Gillet 40/43

41 References: General: P.B. Allen and B. Mitrovic Theory of Superconducting Tc, Sol. State Phys., 37 (Academic Press, New York, 1982) J.M. Ziman Electrons and Phonons Oxford U Press (1960) G. Grimvall The electron phonon interaction in metals (North-Holland, Amsterdam, 1981) L. Hedin, S. Lundqvist Sol. Stat. Phys. 23 ed. Ehrenreich, Seitz, Turnbull (1969) Implementation: Savrasov2 PRB 54, (1996) 41/43

42 Beyond Eliashberg: Issues Strong coupling Anharmonic phonons Strong e-e correlation ( beyond Migdal) High Tc superconductivity 42/43

43 Beyond Eliashberg: Formalism Gross / Van Leeuwen formalism: Quantum ionic Density matrix No Born-Oppenheimer approx. in principle In practice: no external fitting of µ 43/43