Density-functional theory of superconductivity

Transcription

1 Density-functional theory of superconductivity E. K. U. Gross MPI for Microstructure Physics Halle CO-WORKERS: HALLE A. Sanna C. Bersier A. Linscheid H. Glawe A. Floris S. Kurth N. Lathiotais M. Lüders M. Marques CAGLIARI S. Massidda C. Franchini L AQUILA A. Continenza G. Profeta P. Cudazzo

2 What do we want to describe? Resistance drops to zero at T c Meissner-Ochsenfeld-Effect

3 Isotope effect T c 1 A Specific heat

4 [ mev] Energy gap in excitation spectrum MgB T [ K]

5 First steps towards BCS theory Isotope effect (1950) implies that interaction with ions plays a role Fröhlich shows (1950) that electron-phononinteraction treated in nd -order perturbation theory leads to an effective attraction between electrons Cooper demonstrates (1956) that, given an attractive interaction, the electron-gas ground state is unstable with respect to pair formation

6 Ansatz: Product of N/ identical spin-singlet pair wavefunctions ( Bose condensate ) N 1 N N 1 N,r r,r r,r r  r r ~ Antisymmetrizer For uniform system: singlet r i ir F e e g r,r i.e. vac â â g creation operator of plane wave with momentum 1 N N N 1 1 N 1 N â â â â â â g g g ~ N ~ Note: has well-defined particle number vac

7 all BCS vac â â v u all BCS vac â â u v 1 u all vac â â g 1 C u C u v g where, vac â â g 1 â â g C ,,4, N N N ~ * N N BCS BCS N 1 O ~ Bˆ ~ Bˆ * * Consequence: N * N N

8 r u r d r v r r, r u r v m 3 eff eff eff r v r v r v m r d r u r, r eff 3 * eff short-hand: eff eff eff eff v u v u ] [ĥ ˆ ˆ ] ĥ [ Standard theory: Bardeen Cooper Schrieffer (BCS) For inhomogeneous superconductors, BCS taes the form of the Bogoliubov- de Gennes equations:

9 Mean fields v eff e r r r v r r lattice d 3 r Coulomb interaction density r ˆ rˆ r r 1, r Wmod el r1, r, r1, r r1, r rd 3 3 eff d r BCS model interaction (from Fröhlich) order parameter anomalous density r, r 1 r 1 r

10 BCS theory describes the universal features that all (phonon-driven) superconductors have in common, e.g. universal ratio (gap at 0K) / ( B T c ) BCS theory is not able to mae predictions of material-specific properties such as T c.

11 BCS theory describes the universal features that all (phonon-driven) superconductors have in common, e.g. universal ratio (gap at 0K) / ( B T c ) BCS theory is not able to mae predictions of material-specific properties such as T c. Improved version of BCS: Eliashberg theory Still not fully ab-initio: Adjustable parameter μ*

12 Finite- temperature DFT Hohenberg-Kohn-type 1-1 correspondence: ρ(r) [v(r) μ] (Mermin, 1965) where ρ(r) is the density in thermal equilibrium KS equations: r' 3 ( T ) vnuc r d r' v [ ] r r r xc j j j m r r' r r f T ( j) j Fermi distribution j

13 Finite- temperature DFT Hohenberg-Kohn-type 1-1 correspondence: ρ(r) [v(r) μ] (Mermin, 1965) where ρ(r) is the density in thermal equilibrium KS equations: r' 3 ( T ) vnuc r d r' v [ ] r r r xc j j j m r r' r r f T ( j) j Fermi distribution j Finite-temperature LDA available

14 DENSITY-FUNTIONAL THEORY OF SPIN-POLARIZED SYSTEMS Quantity of interest: Spin magnetization m(r) Include m(r) as basic variable in the formalism, in addition to the density ρ(r).

15 DFT for spin-polarized systems ˆ ˆ ˆ ˆ 3 3 H T W r v r d r m r B r d r v,b ee ˆ KS scheme m v( r) v ( r) v xc (r) B( r) B xc (r) ( r) ( r) H o j j j B 0 limit These equations do not reduce to the original KS equations for B 0 if, in this limit, the system has a finite m(r).

16 DENSITY-FUNTIONAL THEORY OF THE SUPERCONDUCTING STATE BASIC IDEA: Include order parameter, χ, characterising superconductivity as additional density L.N. Oliveira, E.K.U.G., W. Kohn, PRL 60, 430 (1988) Include N-body density matrix, Γ, of the nuclei as additional density T. Kreibich, E.K.U.G., PRL 86, 984 (001)

17 General (model-independent) characterization of superconductors: Off-diagonal long-range order of the -body density matrix: xx',yy' ˆ x' ˆ xˆ yˆ y' XY ˆ x' ˆ x ˆ y ˆ y' x x' y x,x' y,y' y' r,r' ˆ rˆ r' order parameter of the N-S phase transition

18 DENSITY-FUNTIONAL THEORY FOR SUPERCONDUCTORS

19 Hamiltonian Ĥ e Tˆ e Ŵ ee ˆ rv( r)d r - d r d r' ˆ r,r' *(r,r') H.c.

20 ANALOGY S N F B P m x x proximity effect

21 Hamiltonian Ĥ Ĥ e Tˆ e Ŵ ee Nn Tˆ d R ˆ(R W(R) n n ) ˆ rv( r)d r - d r d r' ˆ r,r' *(r,r') H.c. Ĥ Ĥ e Ĥ n Û en 3 densities: r ˆ r ˆ r (r,r') (R) ˆ ˆ r ˆ ˆ r' R R R 1 1 R ˆ electron density order parameter ˆ diagonal of nuclear N n -body density matrix

22 Hohenberg-Kohn theorem for superconductors 1-1 [v(r),(r,r ),W(R)] [(r),(r,r ),(R)] Densities in thermal equilibrium at finite temperature

23 Electronic KS equation 3 v s [,,](r) ur s [,,](r,r ) vr' d r' Eur 3 u r' d r' v s [,,](r) v r Ev r s *[,,](r,r ) Nuclear KS equation N n 1 M W s [,,](R) (R) E(R) Exactification of BdG mean-field eqs. KS theorem: There exist functionals v s [,,], s [,,], W s [,,], such that the above equations reproduce the exact densities of the interacting system

24 In a solid, the ions remain close to their equilibrium positions: W R s W s R U 0 W R W U s 0 s R (because forces vanish at equilibrium positions) n i jw(r) s Ui Uj ij N R o Ĥn,KS qbˆ qbˆ q q 3 O U 3

25 (r) F r' r ) (r' r' d R r N(R) R d Z v v v v v xc 3 3 xc H ee H en ext s o ) *(r,r' F r' r ) r,r' ( xc xc H ext s o (R) F r d R r (r) R R Z Z 1 W W W W W xc N 3 N xc H en nn ext s n n o

26 Ĥ Ĥ o Ĥ CONSTRUCTION OF APPROXIMATE F xc : 1 Ĥ o d ˆ 3 r (r) d 3 r' ˆ (r) ˆ v s (r, R ) o (r') * s ˆ (r)d 3 (r,r') H.c. develop diagrammatic many-body perturbation theory on the basis of the H o -propagators: G s normal electron propagator (in superconducting state) r q q bˆ q bˆ q 3 F s F s * D s anomalous electron propagators phonon propagator Immediate consequence: ph el F xc = F xc + F xc all diagrams containing D s all others diagrams

27 Phononic contributions First order in phonon propagator: F ph xc n,, 1 1 ij ij d d 1 i F ij( ) E E F ( ) ij 1 i E E i i j j * j j I(E, E, ) I(E,E, ) i i E E i j j j I(E i, E j, ) i I(E i,e j, ) j Input to ph F xc : Full, resolved Eliashberg function F n,n'' q g n,n'' q q

28 Purely electronic contributions F ee xc ρ, F GGA [ ρ ] xc RPA-screened electron-electron interaction Crucial point: NO ADJUSTABLE PARAMETERS

29 To separate the normal (band-structure) energy scale from the superconducting energy scale, the Bogoliubov-KS equations are decomposed into: n 1 v s ( r ) n'' w eff n ( r ) n, n' ' n n ( r ) tanh n'' n'' n'' n'' n'' w eff n,n' ' d r d r' d x d x' nr nr' w eff r,r',x,x' n'' x n' ' x' w eff (r,r',x,x' ) w el xc (r,r',x,x' ) w ph xc (r,r',x,x' ) with w ph xc el ph el FHxc, *(r,r' ) *(x,x' ) 0

30 Transition temperatures from DFT calculation Al Nb Ta Pb Cu DFT <0.01 Experimental Gap at zero temperature Al Nb Ta Pb Cu DFT Experimental M. Lüders et al, PRB 7, (005), M. Marques et al, PRB 7, (005)

31 Isotope effect: Tc M Calculations Experiment Pb Mo The deviations from BCS value 0.5 are correctly described

32 Jump of specific heat at Tc

33 MgB

34 MgB T c = 39.5 K -D -bonding hole pocets 3-D p and p Fermi surfaces p p p A

35 Fermi Surface of MgB

36 C el / T Specific heat of MgB T / T c A. Floris et al, Phys. Rev. Lett. 94, (005)

37 [ mev] MgB T [ K] A. Floris et al, Phys. Rev. Lett. 94, (005)

38

39 n 1 v s ( r ) n'' w eff n ( r ) n, n' ' n n ( r ) tanh n'' n'' n'' n'' n'' w eff n,n' ' d r d r' d x d x' nr nr' w eff r,r',x,x' n'' x n' ' x' w eff (r,r',x,x' ) w el xc (r,r',x,x' ) w ph xc (r,r',x,x' )

40 n 1 v s ( r ) n'' w eff n ( r ) n, n' ' n n ( r ) tanh n'' n'' n'' n'' n'' w eff n,n' ' d r d r' d x d x' nr nr' w eff r,r',x,x' n'' x n' ' x' w eff (r,r',x,x' ) w el xc (r,r',x,x' ) w ph xc (r,r',x,x' ) w ph eff n, n' ' strongly attractive, short-ranged

41 n 1 v s ( r ) n'' w eff n ( r ) n, n' ' n n ( r ) tanh n'' n'' n'' n'' n'' w eff n,n' ' d r d r' d x d x' nr nr' w eff r,r',x,x' n'' x n' ' x' w eff (r,r',x,x' ) w el xc (r,r',x,x' ) w ph xc (r,r',x,x' ) w ph eff w el eff n, n, n' ' n' ' strongly attractive, short-ranged repulsive, very long-ranged

42 Simple metals? Li and Al under high pressure Li T c increases under P Li Al T c decreases under P Al

43 P(GPa) Calculated and experimental critical temperatures for fcc-al as a function of pressure. Blue circles represent the ab initio SCDFT values, green squares are the semi-empirical McMillan results (with μ * = 0.13).

44 Calculated and experimental critical temperatures for fcc-li as a function of pressure. Blue circles: SCDFT results (dashed part: fcc structure unstable); Green full squares: McMillan's formula with * = 0.13; Green empty squares: McMillan s formula with * = 0.. Vertical dashed lines indicate the structural transition pressures for Li (experimental). Inset: e-ph coupling constant vs pressure in GPa. G. Profeta et al, Phys. Rev. Lett. 96, (006)

45 Upper panel: phonon dispersion of Li along the X-K-Γ line, at several different pressures, for the lower frequency mode. (Frequencies below the zero axis denote imaginary values.) Lower panel: electron-phonon coupling q;1 and phonon line-width q;1.

46 Prediction: T c rises with pressure Potassium (fcc phase) A. Sanna, C. Franchini, A. Floris, G. Profeta, N.N. Lathiotais, M. Lüders, M.A.L. Marques, E.K.U. Gross, A. Continenza and S. Massidda, Phys. Rev. B 73, (006).

47 CaC 6 History: Graphite doped with K (1965), Na (1986), Li (1989): T c 1K Graphite doped with Ca, Yb (005): CaC 6 : T c = 11.5K, YbC 6 : T c = 6.5K

48 Δ

49 Calculated T c : 9.5 K

50 How does hydrogen behave under extreme pressure? Metallic phase? High-T c superconductor? N.W. Ashcroft (1968); C.F. Richardson and N.W. Ashcroft, PRB (1996), K.A. Johnson and N.W. Aschcroft, Nature (000); N.W. Ashcroft, J. Phys. C (004)

51 The idea comes from Jupiter: 90% of Jupiter s mass is hydrogen extremely high magnetic field (suggesting large circulating currents)

52 In search for the metallic phase, experiments (on earth) showed so far only insulating phases

53 In search for the metallic phase, experiments (on earth) showed so far only insulating phases hcp: rotational disorder

54 In search for the metallic phase, experiments (on earth) showed so far only insulating phases hcp: rotational disorder rotational degrees of freedom frozen

55 In search for the metallic phase, experiments (on earth) showed so far only insulating phases hcp: rotational disorder rotational degrees of freedom frozen still molecular crystal, increased IR activity

56 In search for the metallic phase, experiments (on earth) showed so far only insulating phases hcp: rotational disorder Metallic phase?? rotational degrees of freedom frozen still molecular crystal, increased IR activity

57 ΔH (ev) Calculated phase diagram of hydrogen

58 Calculated phase diagram of hydrogen ΔH (ev) Molecular phase (Cmca) stable until ~ 500 GPa C.J. Picard R.J. Needs, Nature Physics 3, 473 (007)

59 Calculated phase diagram of hydrogen ΔH (ev) Molecular phase (Cmca) stable until ~ 500 GPa C.J. Picard R.J. Needs, Nature Physics 3, 473 (007)

60

61 Band structure Band-overlap metallization found at P=400 GPa M. Städele, R.M. Martin, PRL 84, 6070 (000)

62 DOS at 414 GPa FS at 414 GPa

63 Three types of vibrational modes: phononic libronic vibronic

64 α F(Ω) at 414 (solid line) and 46 (dashed line) GPa.

65 α F(Ω) at 414 (solid line) and 46 (dashed line) GPa. phononic

66 α F(Ω) at 414 (solid line) and 46 (dashed line) GPa. phononic libronic

67 α F(Ω) at 414 (solid line) and 46 (dashed line) GPa. phononic libronic vibronic

68 Hydrogen under extreme pressure 414 GPa Predictions: Three-gap superconductivity Increase of T c with increasing P until T c ~ 4K at 450 GPa P. Cudazzo, G. Profeta, A. Sanna, A. Floris, A. Continenza, S. Massidda, E.K.U.G., PRL 100, (008)

69 Correlation of T c with bonding properties (localization of σ charges) CaBeSi: CaBeB: MgB : LiBC: T c = 0.4 K (experiment and calculation) T c = 3.1 K (calculation) T c = 39.5 K (experiment and calculation) T c = 75 K (calculation: Picet et al)

70 Fermi Surface of CaBeSi

71 charge in CaBeSi vs MgB C. Bersier, A. Floris, A. Sanna, G. Profeta, A. Continenza, EKUG, S. Massidda, Phys. Rev. B 79, (009) Si Be in plane B p bonding charge out of plane CaBeSi MgB In MgB much stronger charge localization than in CaBeSi

72 SUMMARY of DFT for Superconductors Coulomb and el-ph interactions enter the theory on the same footing no adjustable parameters, such as *, are used TRUE AB-INITIO PREDICTION OF TC AND