M. Verstraete The electron-phonon coupling in ABINIT

Transcription

1 The electron-phonon coupling in ABINIT M. Verstraete 5 April /67

2 Outline 1 Motivation 2 EPC introduction 3 A bit of theory 4 Transport 5 ABINIT 6 Examples 2/67

3 Impact of phonons I Electrical resistivity Heat transport Superconductivity 3/67

4 Impact of phonons II Inelastic/relaxation mechanism: Raman IR ESR (w/ SO interaction) X Ray M. Lazzeri and F.Mauri, PRL 90, (2003) 4/67

5 Outline 1 Motivation 2 EPC introduction 3 A bit of theory 4 Transport 5 ABINIT 6 Examples 5/67

6 Bare phonons Rigid motion of ions (no screening) Completely unrealistic phonon frequencies (usu. too hard) 6/67

7 Screened phonons Screening of ion motion by electrons Very realistic phonon frequencies (max 10% error) 7/67

8 Independent electrons... Perfect crystal and indep. e no resistivity Electron-ion interaction is periodic, gives Renormalized e energies, but still ideal QP KS states are in this category 8/67

9 ... & things that perturb them In real system, perturbations add to Hamiltonian H = H0 + H1 give finite lifetimes for indep. part. eigenstates interaction only MB Ψ has lifetime Perturbations = Coulomb, photons, defects, impacts and phonons 1 What is the system (and external perturbation)? 2 Are the particles independent? 9/67

10 Bare electrons Self consistent screening of static el charge Decent BS within treatment of Coulomb interaction 10/67

11 Perturbed electrons Finite lifetime for KS states EP coupling constants usually quite good (Phonon eigenstates are Bloch like Localized phonon wavepacket in fig) 11/67

12 Energy scales Low energy pert. = ion vibrations (0-20+ 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 coupling and large DOS (both phonon and e ) 12/67

13 Temperature According to preceding energy scale: Low T only FS electrons contribute (ABINIT approx too) High T more phonon ω insulators too T Debye freq classical Boltzmann stat. Remember phonons remain harmonic and lifetime 13/67

14 FS effects Phonon q connects k, k FS Nesting: many k-k give same q And energy/frequency dependency Kohn anomaly 14/67

15 Superconductivity Effective e e interaction in presence of phonons can be attractive, pairing +k -k superconducting instability at low T New mixed quasiparticles have no resistance. 15/67

16 Outline 1 Motivation 2 EPC introduction 3 A bit of theory 4 Transport 5 ABINIT 6 Examples 16/67

17 BCS Theory Effective electron theory with pairing term P c c c Vc Functional relationships for weak coupling Lesson: if you know the physics and H1 the rest is easy J. Bardeen +CS Phys. Rev (1957) 17/67

18 More general H Must include phonons and electrons + ep interaction + Coulomb interaction H = Hel + Hph + Hep P Hel = nk cnk cnk P Hph = ωqj bqj bqj 18/67

19 Eliashberg Theory Coupling Hamiltonian (Frölich type): X Hep = < ~k + ~q δv ~k > ~dq ck +q ck kq 19/67

20 Eliashberg Theory Coupling Hamiltonian (Frölich type): X Hep = < ~k + ~q δv ~k > ~dq ck +q ck kq Perturbed (electronic) potential δv = SCF potential in phonon-distorted geometry 19/67

21 Eliashberg Theory Coupling Hamiltonian (Frölich type): X Hep = < ~k + ~q δv ~k > ~dq ck +q ck kq Displacement operator: ~dq = X ( j ~ )~uqj (aqj + a qj ) 2NMωqj ~u is phonon eigenvector (Help Jorge!) 19/67

22 Eliashberg Theory Coupling Hamiltonian (Frölich type): X Hep = < ~k + ~q δv ~k > ~dq ck +q ck kq Useful def of gkk matrix elements: ~ ~uqj < k~0 δv ~k > gkqj0 k = p 2NMωqj 19/67

23 Eliashberg Theory II Perturbation theory in Hep Use spectral representations of Green s functions for e, ph and for self-energies Full ph S.E. is not perturbational (remember bare phonons) 20/67

24 Spectral function Eliashberg spectral functions: X qj α2 F (Ω, k, k 0 ) = NF gk 0 k 2 δ(ω ωqj ) j α F (Ω, k ) = NF X α2 F (Ω) = NF X 2 gkqj0 k 2 δ(ω ωqj ) k 0j gkqj0 k 2 δ(ω ωqj ) kk 0 j kpoint sums over Fermi Surface Related to spectral representation of e S.E. 21/67

25 Spectral function II Closely linked to phonon DOS F (Ω) = P qj δ(ω ωqj ) Savrasov2 PRB 54, (1996) dash line = α2 22/67

26 Related EP quantities Superconducting strength (or mass renormalization factor): Z dω 2 λ=2 α F (Ω) Ω Phonon linewidth = lifetime from scattering with electrons: X qj gk 0 k 2 γqj = k FS Link to experiments starts here 23/67

27 Migdal Need to solve for SE and then calculate useful quantities Migdal theorem : only 1 phonon interaction is needed Could be invalid, e.g. strong e correlations (e.g. Grimaldi) C. Grimaldi, et al. PRL 75, 1158 (1995) 24/67

28 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!! 25/67

29 Coulomb interactions Most already in ωq and nk Remaining e-e repulsion in Cooper pair interaction Retarded coulomb parameter: Z c µ = NF Vkk 0 /(1 + log(ωel /ωd )) FS 2 ωel Coulombic resp freq 10eV Can be approx because = change btw SC and N states 26/67

30 Eliashberg equations Combine Coulomb, ep and other self energies Solve for e Green s functions self-consistently Or for gap (k ) ( supercond part of S.E.) Isotropic energy-only formalism (ω) 27/67

31 Outline 1 Motivation 2 EPC introduction 3 A bit of theory 4 Transport 5 ABINIT 6 Examples 28/67

32 Transport equations Classical statistical mechanics Boltzmann equations for flow of e and phonons Coupled by precisely the e-ph matrix elements gkk 0 29/67

33 Boltzmann equations 2π X fk = gkk 0 2 {fk (1 fk 0 )[ t ~Nc q (Nq + 1) δ( k k 0 ~ωq ) Nq + (1 fk )fk 0 [ δ( k k 0 + ~ωq )] (Nq + 1) δ( k k 0 + ~ωq ) Nq + δ( k k 0 ~ωq )]} 0 k =k +q Nq 4π X = gkk 0 2 fk (1 fk 0 )[ t ~Nc Nq δ( k k 0 + ~ωq ) k (Nq + 1) δ( k k 0 ~ωq )] e.g. J.M. Ziman Electrons and Phonons Oxford U Press (1960) 30/67

34 Steady state solutions Relate κph, σe to coupling btw fluxes of ph and e e.g. constant difference in T and V constant currents: fk / t and Nq / t Linearize Boltzmann eqs and simplify k dependency 31/67

35 Technicalities Incorporate mat elem of velocity or ~p ( optical conductance...) Implementation in Abinit as in Savrasov Eliashberg-like spectral functions 32/67

36 Transport spectral functions: 2 αout F (ω) 2 αin F (ω) αtr2 F (ω) = X 1 2 N0 < vx > ν = X 1 2 N0 < vx > ν = X gkqν0 n0 kn vx (~k )vx (~k )δ(ω ωqν ) knk 0 n0 FS X gkqν0 n0 kn vx (~k )vx (k~0 )δ(ω ωqν ) knk 0 n0 FS 2 2 αout F (ω) αin F (ω) Where q = k 0 k, < vx2 > is average on FS of x velocity. Cubic symmetry case. 33/67

37 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 34/67

38 Outline 1 Motivation 2 EPC introduction 3 A bit of theory 4 Transport 5 ABINIT 6 Examples 35/67

39 Getting EP matrix elements Phonon calculation evolves ψ (1) and n(1) to find E (1), E (2) We need < k~0 δv ~k > and ωqj, ~uqj First order Hamiltonian (δv ) is screened in phonon calculation full convergence of n(1) (not bare phonons) Adding a new electron-phonon coupling some double counting, but perturbation is much smaller (use in Σel ) 36/67

40 Symmetries Symmetry operations complete E (2), but phase differences preclude using them for < k~0 δvqj ~k > < k~0 δvqj ~k >< ~k δvqj 0 k~0 > eliminates gauge freedom need all 3 Natom perturbations! Qpoints completed by symmetry k and k 0 = k + q on FS so qpt grid must be consistent with kpt 37/67

41 Workflow 1 In ABINIT: _DDB files contain E (2) and _1WF files contain ψ (1) along with our precious < k~0 δv ~k > Use prtgkk 1 to get GKK file w/o saving huge 1WF 2 mrgddb pastes together all E (2) into one file for phonons mrggkk extracts < k~0 δv ~k > and/or pastes them into one 3 file 4 3 Natom perturbations must be grouped by qpoint, and complete! 5 Run anaddb with telphon = 1 and additional file names 38/67

42 FCC Al band structure BS defines Fermi Surface need many kpoints Gives GS wf for < k~0 δv ~k > matrix elements 39/67

43 GS input file nshiftk 1 shiftk ngkpt tolwfr1 1.0d-14 acell 3*7.5 rprim iscf 7 kptopt1 3 ecut 4.0 nband 10 occopt 7 tsmear natom 1 typat 1 xred nstep 800 ntypat 1 znucl 13 40/67

44 Phonon BS Phonon frequencies and eigenvectors in whole BZ + electron-phonon interaction Get all perturbations explicitly: 41/67

45 Calculate the phonons rfatpol3 1 1 rfdir rfphon3 1 tolvrs3 1.0e-5 getwfk3 1 getwfq3 2 kptopt3 3 nqpt3 1 qpt rfatpol4 1 1 rfdir rfphon4 1 tolvrs4 1.0e-5 getwfk4 1 getwfq4 2 kptopt4 3 nqpt4 1 qpt rfdir /67

46 mrgddb/mrggkk input files telphon_2o.ddb.out Total ddb for Al FCC 9 telphon_1o_ds3_ddb telphon_1o_ds4_ddb telphon_1o_ds5_ddb telphon_1o_ds7_ddb telphon_1o_ds8_ddb telphon_1o_ds9_ddb telphon_1o_ds11_ddb telphon_1o_ds12_ddb telphon_1o_ds13_ddb telphon_3o_gkk.bin 0 telphon_1o_ds1_wfk telphon_1o_ds3_1wf1 telphon_1o_ds4_1wf2 telphon_1o_ds5_1wf3 telphon_1o_ds7_1wf1 telphon_1o_ds8_1wf2 telphon_1o_ds9_1wf3 telphon_1o_ds11_1wf1 telphon_1o_ds12_1wf2 telphon_1o_ds13_1wf3 43/67

47 Elphon features - basics Integration over FS with weights from Gaussian smearing (telphint = 1) with width elphsmear Tetrahedron method (telphint = 0) needs input of kptrlatt For interpolations and tetrahedra, need Γ and special points for k and q grids Flags for disk paging of gkk internal arrays: gkqexist,gkqwrite gkk_rptexist,gkk_rptwrite Set XXwrite 1 for paging, XXexist 1 to read in previous run s output. 44/67

48 Qpoint grid in anaddb elphflag 1 nqpath 7 qpath /2 1/ /2 1/2 1/2 1/2 1/2 3/4 1/2 1/ / /4 1/2 mustar ngqpt nqshft 1 q1shft asr 2 dipdip 1 brav 1 ifcflag 1 ifcana 1 natifc 0 atifc dieflag 0 eivec 1 nph1l 1 qph1l /67

49 Tetrahedron method in anaddb ngqpt nqshft 1 q1shft elphflag 1 telphint 0 kptrlatt elphsmear 0.01 nqpath 7 qpath... mustar asr 2 dipdip 1 brav 1 ifcflag 1 ifcana 1 natifc 0 atifc dieflag 0 eivec 1 nph1l 1 qph1l /67

50 Using the transport features Set in anaddb input ifltransport 1 do transport tkeepbands 1 keep band resolution in gkk doscalprod 1 do scalar prod before interpol Need extra ddk input run 3 1WFx files Extra line in anaddb files file with: ddk files WFK Read in and directly added to sums over FS to get α2 Ftr etc... 47/67

51 Transport features II Huge arrays use disk paging: set gkqwrite 1 Need to remove files before each run (not squashed by default) Reading in _tr files for next run not functional yet 48/67

52 Transport outputs Electrical resistivity/conductivity (_RHO) Thermal conductivity (_WTH) Lorentz coefficient (_LOR) Transport α2 F (_A2F_TR) 49/67

53 Transport outputs Electrical resistivity/conductivity (_RHO) 50/67

54 Transport outputs Electrical resistivity/conductivity (_RHO) 50/67

55 Transport outputs Thermal /conductivity (_WTH) Linear high T behavior is simplified (LOVA) 51/67

56 Transport outputs Thermal /conductivity (_WTH) Linear high T behavior is simplified (LOVA) 51/67

57 General performance Phonon calculation is prohibitive step (nkpt, nqpt) Most systems in principle need better Coulomb, or anisotropy... but in practice? Electron-phonon interaction well modeled in general Max no atoms? 52/67

58 Outline 1 Motivation 2 EPC introduction 3 A bit of theory 4 Transport 5 ABINIT 6 Examples 53/67

59 3D Al phonons FCC Al has strongest modes between Γ and X or Γ and L 54/67

60 Phonon linewidths for FCC Al lwd crossing needs doscalprod 0 FT interpolation gives wiggles 55/67

61 Monoatomic wire of Al Al monowire electronic BS Get _WFK and _DEN 1D gives simple BS and interaction w/ phonons 56/67

62 Wire phonons transverse modes (not 1D! spirals) unstable at X Thickness = longitudinal character Slight Kohn anomaly + gapless Peierls 57/67

63 Phonon linewidths for monowire Wire phonon linewidths: transverse modes = no coupling 58/67

64 α2 F : unstable modes have little coupling to electrons. Small overall coupling (100 times smaller than fcc) Verstraete PRB 74, (2006) 59/67

65 What is not implemented yet? Anisotropic quantities (ie. not FS averaged) Resistivity/conductivity for non cubic systems Band resolution (present but not used in standard elphon) 60/67

66 MgB2 : a literature case study Kortus et al. PRL (2001) Nagamatsu et al. Nature 410, 63 (2001) Layered compound FS = cylinder+sheet: highly anisotropic 61/67

67 MgB2 Shukla et al. PRL (2003) Choi et al. Nature (2002) E2g phonon with strong EPC, + anharmonicity 2 gaps for σ and π FS layers 62/67

68 Beyond Eliashberg: Issues Strong coupling Anharmonic phonons Strong e-e correlation ( beyond Migdal) High Tc superconductivity 63/67

69 Beyond Eliashberg: Formalism Gross / Van Leeuwen formalism: Quantum ionic Density matrix No Born-Oppenheimer approx. in principle In practice: no external fitting of µ 64/67

70 Conclusions Widely useful quantities Strong links to experiment Small cost beyond phonon calculation Many other processes could use EPC Worry about extensions... 65/67

71 Collaborators and Funding Help Matteo Giantomassi Nicole Helbig Jean-Paul Crocombette + the CEA boys Xavier Gonze They pay you to do this? Marie Curie MANET project funding Nanoquanta NOE funding WRG/HPCX/CISM computer time 66/67

72 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) 67/67