Impurity states and marginal stability of unconventional superconductors

Transcription

1 Impurity states and marginal stability of unconventional superconductors A.V.Balatsky(LANL) local =atomic or coherence length scale impurity or defect Impurity states inconventional supercondcutors Impurity states in unconventional superconductors Impurity states in pseudogap state Rev Mod Phys, v 78, p 373 (2006)

2 See also Peter Hischfeld lectures

3 Motivation 1 Impurities as a local probe of the symmetry of the superconducting order parameter: 30 Å logarithmic intensity scale 0 Å b -1.2 mv a Zn atoms in Bi 2 Sr 2 Ca(Cu 1-x Ni x ) 2 O 8+δ Four-fold symmetry, aligned with the gap nodes S.H. Pan et al, Nature (2000)

4 Motivation 2 Impurities as a probe of the Pseudogap state of high-tc superconductors

5 Motivation 3 Most of the theory and experiment on conventional superconductors was done *before* Local probes, like Scanning Tunneling Microscopy were Invented. A lot of views have had calcified into standard Textbook wisdom. I will show some examples where local probes bring in New and unexpected perspective, sometime new results.

6 1911 H. Kamerlingh-Onnes: superconductivity in mercury, Tc = 4K Brief history 1933 Meissner effect: expulsion of magnetic field 1957 Bardeen-Cooper-Schrieffer (BCS) theory 1962 Josephson effect 1986 first high-tc superconductor, LBCO Tc = 35K 1987 YBCO, Tc = 93K warmer than liquid N 1995 Hg 0.8 Tl 0.2 Ba 2 Ca 2 Cu 3 O 8.33 Tc = 138K

7 Superconductivity primer Metal Superconductor Cooper pairs (k & -k) ε n ε E n E EF N(ε) 2 EF N(E) 2 k = 2 m ε k E 2 2 F Ek = ± ε k + Electrons: c, c k k Bogoliubov q.particles: γ, γ ( γ u + kc v k k k + = kc k k )

8 Local Probes:Scanning tunneling microscopy/spectroscopy ε z ε F + ev I t LDOS sample (ε)dε ε F ev I t di t dv LDOS sample (ε) Tip DOS sample LDOS Spectroscopic map (LDOS map) can be obtained.

9 Bogoliubov-Nambu Hamiltonian background Bogoliubov angle measures the relative weight between particle And hole component in Bogoliubov degennes quasiparticle ( ) ks ks ( k ks k s ) H = ε k c c + c c + h c γ = uc + vc * k k k k k 2 ( ) 1/2 ( uk = 1 ε ( k)/ Ek ( )) 2 ( ) 1/2 ( vk = 1 + ε ( k)/ Ek ( )) True excitations in SC state γ k That are quantum mechanical γ Mixture of particle and hole k Ψ = 0 Ψ 0; = Ψ 0 1 Important! Dual nature of qp insc γ Ψ 0 = Ψ1 c ( k) k Ψ1 and at the same time c ( k) Ψ 1

10 Bogoliubov Quasiparticles = + + = = i i i i i i i i i i i i v u v u v u ψ ψ γ ψ ψ γ mixing particles and holes 0 ) ( 0 0 > = Ψ i i i i v i u ψ ψ GS: Excited State: + > Ψ = Ψ Ψ = Ψ + = Ψ = Ψ , 0 ) ( u v v u i i i i i ψ ψ ψ ψ ψ γ Positive bias Negative bias

11 Tunneling into SC Particle and hole component in tunneling sample the same Bogoliubov Nambu quasiparticle. The only difference is relative weight of these two processes. Ψ Ground state 0 Bogoliubov excitation γ n i Ψ 0 = vnc ( ui+ vc i ic i)0 Electron spin down Tunneling out

12 Tunneling into SC Particle and hole component in tunneling sample the same Bogoliubov Nambu quasiparticle. The only difference is relative weight of these two processes. Ψ Ground state 0 Physically one is left with the same state (~1/N) As a result of tunneling. Amplitudes (u vs v) are different Bogoliubov excitation γ n i n Ψ 0 = unc ( ui+ vc i ic i)0 Electron spin up Tunneling in

13 Mind trick

14 Mind trick

15 Mind trick

16 Mind trick

17 Lets do it again

18 Lets do it again

19 Lets do it again

20 Lets do it again

21 You noticed! Few points on overall approach in this lecture: Impurities and defects are interesting. They are telling us a story about host They can be useful as a markers of new physics Controlled impurity physics is at the core of multibillion $ semiconducting industry Impurity states enable the new functionality, that is used in devices. Case; my computer that allows me to project this lecture. Impurities can be placed deliberately to destroy the state we are trying to understand From response of unknown state to impurities we can infer what it is made of Not average description but one at a time. Averaged quantities can be useful. But they can be misleading.

22 Conventional, s-wave superconductors (BCS)

23 T-matrix approximation ) ; ( ) ( ) ; ( ) ; ( ) ;, ( ω ω ω ω ω r r r r r r + = G T G G G Single impurity of strength V at r = 0 (Yu Lu, Shiba, ): r r r r r r 0 U G G 0 r r 0 U 0 U = ) (0, 1/ 1... ) (0, ) (0, ) ( ω ω ω ω G U G U G U U T = = Impurity-induced resonance: = = = k k r ), ( 1 ) 0, ( / ω ω G N G U

24 Bound state in 2D metal due to weak attarctive potential energy size Local DOS

25 Same prcedure is applied to SC case, except Green s functions are now matrices

26 Theory of impurity states in s- wave SC Yu Lu(65), Shiba(68) H = H BCS + JSimp σ N ω0 1 = ± 1 + ( JN ( JN 0 0 S) S) Ψ ( r) = exp( r/ ξω ) 0 r ξ ω = ξ N ξ ( ω ) / 0 isotropic gap E Magn imp -ω ω E Isotropic wave function

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28 Imp S + Cooper pairs continuum imp> Ground state 0> Bound state formed due to magnetic gain Bound state is FULLY spin polarized!

29 Intrinsic pi junction

30 Remark about Abrikosov-Gorkov theory N N Weak impurity E Strong impurity E

31 N N E Weak impurity N At finite imp concentration Strong impurity N E Imp band growth in Abrikosov-Gorkov theory; one parameter: Nimp N(0) J S(S+1) E General case: Gapless SC much sooner Lifshitz tails and gapless Balatsky, Trugman, PRL, 97 E

32 Predicted behavior only occurs for weak impurities: N E

33 Predicted behavior only occurs for weak impurities: N E

34 Predicted behavior only occurs for weak impurities: Gapless After a large concentration is added N E

35 Gapless SC region

36 In Reality Imurity states can be at any energy Energy of a single impurity state is a center of impurity band Gd Cluster Bare Nb Votage[V] A.Yazdani, Science,97

37 A.Yazdani et al, Science(1997) Impurity intragap States on Nb Superconductor. Gold has no effect on SC state Wave function of imp state Gd, Mn atoms on Nb surface (Tc = 7K)

38 Lifshitz Tails and gapless supercoductivity in s-wave Case. Rare fluctuations are the ones that are nominally very unlikely. However is the mean exectation of some observable is zero and Fluctuaion will render this expectation nonzero then no matter how low probability is this fluctuation is important (hugely important in fact) = /0-> infty Rare fluctuations in the impurity distribution will make any s-wave superconductor gapless, regardless how low impurity Concentration is. It is gapless albeit with exponentially small Density of States. Rev Mod Phys, v 78, p 373 (2006)

39 Average theory of impurity scattering depend on 2 2 Only scattering rate Γ=nimpJ S Band in case of strong impurity scattering, J~1 Band in case of weak impurity scattering J <<1 Γ= const n imp small JS >> 1 ( unitary) Averaged theory From textbooks does not capture this ditsinction Γ= const n large imp JS << 1 ( Born)

40 Lifshits tails or rare fluctuations

41 Lifshitz tails in SC make it gapless s wave superconductivity: rare but important fluctuations are the ones where local concentration of impurities exceed critical and Locally we have a puddle of normal metal inside SC. These regions always exist no matter how small probability is. Tail at E=0 grows with doping Gapless SC region

42 Unconventional Superconductors: d-wave (High Tc supercondcutors)

43 Experimental Phase Diagram of HTSC T AF T cr AF antiferromagnet SC d-wave superconductor T cr onset of spin fluctuations T * opening of pseudogap T * SC n = 1 Doping, x

44 YBCO structure

45 F i, j θ j Earlier work: d-sc Impurity Resonances On-site potential U>0 On-site LDOS LDOS Image at Ω Ψ 2 for impuritystate - Rev. Mod Phys, 78, p 373, (2006) Differential Conductance (ns) typical region on center of Zn atom Sample Bias (mv) E E 0 1 = 2N U ln F 8 Cross shaped state 1 ( N U ) F Impurities can be a useful tool to investigate the nature of an unknown state

46 Theory of impurity resonance in d-wave SC Balatsky, Salkola, Rosengren (95) Hint = U n(0) N E Nonmagnetic And magnetic imp

47 Theory of impurity resonance in d-wave SC Balatsky, Salkola, Rosengren (PRB,95) Hint = U n(0) N ω 1 = πn U 0 ln 1 ( πn U )[1 + iπ ln 0 1 ( πn U )] 0 N E Nonmagnetic And magnetic imp ω E

48 Theory of impurity resonance in d-wave SC Balatsky, Salkola, Rosengren (PRB,95) Hint = U n(0) N ω 1 = ln πn U 0 1 ( πn U )[1 + iπ ln 0 1 ( πn U )] 0 N E Nonmagnetic And magnetic imp Differential Conductance (ns) ω typical region on center of Zn atom E Sample Bias (mv)

49 Theory of impurity resonance in d-wave SC Balatsky, Salkola, Rosengren (95) Hint = U n(0) N ω 1 = ln πn U 0 1 ( πn U )[1 + iπ ln 0 1 ( πn U )] 0 N Cross shaped state E Nonmagnetic And magnetic imp Universal:Similar states will be created in p,f,g states with gap nodes. ω Splitting due to exchange J E

50 Zn Impurity-State ~20 Zn atoms Å! Å SI-STM Image at E= 1.5mV Bi 2 Sr 2 Ca(Cu 1-x Zn x ) 2 O 8+d : x 0.2% Cross shaped Impurity state (due to 4 fold gap Nature 403, 746 (2000).

51 Data by A. Matsuda NTT Research, Japan

52 N(r) ~ 1/r Remains to be seen experimentally if this is true in 2

53 Motivation 2: magnetic vs nonmagnetic impurities Impurities as a probe of the high-tc mechanism Differential Conductance (ns) typical region on center of Zn atom ε = -1.5 mev Differential Conductance (ns) ε = 9 & 18 mev typical region on center of Ni atom Sample Bias (mv) Zn, non-magnetic, S = 0 suppresses superconductivity Sample Bias (mv) Ni, magnetic, S = 1, does not suppress superconductivity

54 Impurity states in ANY Dirac point materials

55 Impurity states in ANY Dirac point materials

56 Impurity states in ANY Dirac point materials

57 Impurity states in ANY Dirac point materials

58 Local Density of states in Dirac Materials: Nonmagnetic impurity in d-wave SC and in graphene will have similar r dependence Nr (, ω)~im Grr (,, ω)~ Re[ G( rr,, ω) Gr (, r, ω)]im T( ω) 0 imp imp Grr (,, ω) = G( rr,, ω) + G( rr,, ω) T( ω) G( r, r, ω) T( ω) = U /(1 G ( ω) U) 0 0 imp 0 2 = Ψ( rr, imp ) Zimpδω ( Eimp ) 1 0 Eimp = U LogU / W pole 1/U = G 0( ω) sin( kfr + δ ) Ψ( r, ω) ~ Re G(, 0 r ω)~ r 0 2 LDOS N(r) ~ 1/ r for all Dirac propagators nature of local scattering center is not important: G ( ω) = ω[ Log( ω/ W ) + i sign( ω)] Kondo, potential scattering site, spin orbit impurity etc. regardless being d-wave SC or graphene or anythng else result of power counting imp T. Wehling, PRB, 75, (2007) C. Bena, PRL 100, (2008) T. Pereg-Barnea, A. H. MacDonald, PRB 78, (2008)

59 H H + H H Model P.J. Stamp (1987) Byers, Flatte, Scalapino (1993) Balatsky, Salkola, Rosengren (1995) Salkola, Balatsky, Schrieffer (1997) t t = 0 H imp 0 = t + ijciσ c jσ V c c c c + h.c. i j j i i, j, σ i, j, σ imp = Vimp( n + n ) Simp( n n ) V Hopping: Attraction: nn: t = 400 mev nnn: t = 0.3 t = 120 mev nn: V = t = mev

60 Ni,Theory vs. Expt Images Theory: V imp = t ; S imp = 0.4 t; doping 16% Experiment: E.W. Hudson, et.al. 9 mv - 9 mv

61 Zn: Strong potential imp Theory V imp = -10 t = -4 ev doping 16% Experiment S.H. Pan et al Impurity site has to be bright!!!

62 Ni, weak potential + spin imp Theory: V imp = t ; S imp = 0.4 t; doping 16% Experiment: Impurity level splitting caused by Ni spin Differential Conductance (ns) typical region on center of Ni atom ε = 9 & 18 mev ε 1 =0.052 t (21 mv), ε 2 =0.037 t (15 mv) Sample Bias (mv)

63 Potential impurity in BSCCO (0.16 doping) Ni Zn Ni weak repulsive 8 electrons in 3d Zn strong attractive 10 electrons in 3d (filled shell)

64 Gap is suppressed on the atomic scale Not only on scale of coherence length as GL would Lead us to believe.

65 Impurity as a probe of Pseudogap (PG) Regime Pseudogap (Loram) Vanishing of density of states at the fermi level is sufficient for formation of impurity state. H. Kruis, I. Martin and AVB PRB, 2002

66 T = U/(1-G0(Ω)U) Hence the pole at G0(Ω) = 1/U H. Kruis et al, PRB 64,

67 Role of the Interlayer Tunneling Ψ ij value impurity state wavefunction on site (i,j) Intensity in CuO layer: A ij = Ψ ij 2 Martin, Balatsky, Zaanen (2000) Intensity in BiO layer: A ij = Ψ ij-1 + Ψ ij+1 - Ψ i-1j - Ψ i+1j 2 ~ cos kx cos ky Direct tunneling (tip to CuO) is exponentially suppressed: t ~ exp(-r/d), d~0.5a

68 Bi Zn tip Cu Hopping via virtual states on Bi and Zn(Ni) ev One uses the easiest available path. In this case direct tunneling into Cu-O plane is exponentially suppressed. r, ω = dφ t φ N r, φ, ω - N eff ( ) ( ) ( ) surface seen Density of States

69 BiO vs. CuO plane image Experiment!

70 Zn: Strong potential imp Theory V imp = -10 t = -4 ev doping 16% Experiment S.H. Pan et al Impurity site has to be bright!!!

71 BiO vs. CuO plane image Experiment!

72 Why local nanoscale probes are useful

73 Examples ( incomplete list) MIT in VO2 Basov Science, v 318, 1759 (2007) A. Yacobi J. Stroscio Gap inhomogeneity in novel FeAs superconductors, J. Hoffman et al Gap inhomogeneity In high Tc oxides J.C. Davie, S.H. Pan, A. Yazdani URu2Si2inhomo geneity in di/dv at 35 mev JC Davis et al

74 J.C. Davis Group Gap Map, BSCCO (0.5%Ni) 55 mev mev 512 Å Frequency of Observation Counts gap, Gap magnitude Magnitude (mv) (mv)

75 Cren, et al, PRL (2000) Bi2212 film

76 YBCO - BaO plane YBCO BaO vs BiO BSCCO - BiO plane 300 Å 280 Å Do equivalent nanoscale electronic structure variations exist in YBCO? Probably not. If they do they are probably weaker and more ordered.

77 Impusity states and STM on Graphene Differential Conductance (ns) typical region on center of Zn atom Sample Bias (mv) HOPG Y. Niimi et. al., PRL 97,

78 Conclusion Impurity and defects in correlated states Are important in that enable new properties Or make identification of an unknown state easier One would need right set of tools: experiment Theory: investigate local features, no average lifetime Average scattering potential etc. A lot of correlated materials do show variety of inhomogeneous competing states.

79 Thanks M. Salkola I. Martin J.X. Zhu I. Vekhter J.R.Schrieffer D. Scalapino P. Kumar J. Zaanen J. Smakov J.C. Davis and Co D. Morr A. de Lozanne