Disorder and quasiparticle interference in high-t c superconductors

Transcription

1 Disorder and quasiparticle interference in high-t c superconductors Peter Hirschfeld, U. Florida P. J. Hirschfeld, D. Altenfeld, I. Eremin, and I.I. Mazin}, Phys. Rev. B92, (2015) A. Kreisel, P. Choubey, T. Berlijn, B. M. Andersen and P. J. Hirschfeld, PRL114, (2015) P. Choubey, T. Berlijn, A. Kreisel, C. Cao, and P. J. Hirschfeld, Phys. Rev. B 90, (2014) U. Tennessee, 8 February 2016

2 Collaborators from rest of world: from U. Florida Dept. of Physics: Niels Bohr Inst., Copenhagen Peayush Choubey Yan Wang (now U. Tenn.) Brian Andersen Maria N. Gastiasoro Tom Berlijn (now ORNL) Andreas Kreisel Wei Ku, BNL

3 Outline Unconventional superconductivity STM Quasiparticle interference Bogoliubov-de Gennes + Wannier method Applications 1. Zn impurity in BSCCO 2. QPI in BSCCO QPI as a qualitative tool

4 How can two electrons attract each other? Dance analogy: coherent pairs J. Robert Schrieffer: By dancing they lower their energy or mae themselves happier Another analogy:

5 How Cooper pairs form in conventional superconductors the glue : electron-phonon interaction Effective residual e-e interaction including Coulomb ( Jellium model ) a b Realistic system: a b! Depends on details Screened Coulomb Electron-phonon (attraction) Note: electrons avoid Coulomb repulsion in time (interaction is retarded)

6 Attraction from repulsion: Kohn-Luttinger 1965 Walter Kohn Quinn Luttinger Also: Landau and Pitaevsii KL: an electron gas with no phonons and only repulsive Coulomb interactions can be a superconductor! A new paradigm: electrons avoid repulsive part of Coulomb interaction in space rather than time!

7 Kohn-Luttinger 1965 Friedel: screened Coulomb interaction V ( r) = cos 2 r / r F 3 At finite distances, screened Coulomb interaction becomes attractive: finite-l pairing

8 Kohn-Luttinger 1965 U effective pairing interaction bare interaction (repulsive) screening terms (attractive in some L-channels) Example: short range U>0 for rotationally invariant system ( 3 He ) T c E F exp( 2.5L 4 ) Best calculation in 1965: Bruecner Soda Anderson Morel PR 1960 : predicted L=2 for 3 He T c ~ K But had they taen L=1 they would have gotten T c ~ 1 mk!

9 Higher L pair wavefunctions translated into crystalline environment: Symmetry of order parameter () in 1-band superconductor + () () () y Fermi surface y Fermi surface x x s-wave, L=0 no nodes conventional pair state d-wave, L=2 nodes unconventional pair state

10 Gap symmetry vs. structure: Important clue to pairing mechanism! A 1g B 1g Fe-based superconductors

11 2 paradigms for superconductivity according to how pairs choose to avoid Coulomb interaction conventional : isotropic s-wave pair wave fctn, interaction retarded in time Overall effective interaction attractive unconventional : anisotropic or sign-changing pair wave fctn, Overall effective interaction repulsive A. Chubuov and P.H. Phys. Today 2015

12 Scanning tunnelling microscopy Tunnelling current: STM tip sample e.g. SC J. Hoffman 2011 Rep. Prog. Phys (2011) Local Density Of States (LDOS) of sample at given energy J. Tersoff and D. R. Hamann, PRB 31, 805 (1985) Conductance di/dv of FeSe T C =8 K Topograph of Fe centered impurity in FeSe at V=6 mv LDOS and conductance map: Zn impurity in BSCCO at V=-2 mv Song et al., Science 332, 1410 (2011) Can-Li Song, et al. PRL 109, (2012) Pan et al., Nature 403, 746 (2000)

13 Quasiparticle interference ( QPI ) experiments P.T. Sprunger et al, Science 1997 Silver 111 surface 2 F LDOS ρ(, r ω) i ρ( q, ω) = e q r ρ( r, ω) L L

14 Quasiparticle interference ( QPI ) experiments can use a real space probe (STM) to give info about momentum space electronic structure ε n can probe symmetry and structure of superconducting gap function relies on disorder to provide a signal! 2 F

15 Anderson s theorem P. W. Anderson, J.Phys. Chem. Solids 11, 26 (1959) In the presence of dirt one can still pair time-reversed members of Kramer s doublet: thermodynamics (T c, gap, sp. ht., ) are not affected by nonmagnetic impurities

16 Balian-Werthamer: p-wave superconductivity Nonmagnetic impurities are pairbreaing in unconventional superconductors

17 Strong magnetic impurity creates bound state in s-wave SC Yu Lu, Acta Physica Sinica 21, 75 (1965) see also H. Shiba, Prog. Theor. Phys. 40, 435 (1968). A. I., Rusinov, 1969, Zh. Esp. i Teor. Fiz. 56, 2047, [Sov. Phys. JETP 29, 1101 (1969)].

18 Bound states of nonmagnetic impurity in d-wave SC 1 ρ(, rω) = Im Grr (,; ω) π δρ imp see also Stamp, 1986 (p-wave)

19 Cuprates T p~0.1 p~0.2 doping d-wave SC: T c is too high for electron-phonon glue to wor! What holds pairs together? 0 = cos cos 2 ( ) x y

20 BiO SrO CuO 2 Impurities in cuprates Probe the response of SC to a spin/charge local perturbation Ca CuO 2 SrO BiO Ba Dilute Cu in-plane substitutions Ni 2+ 3d 8 spin 1 BiO SrO Cu Y O Zn 2+ 3d 10 no spin Li + no spin CuO 2 Ca (Cu?) vacancies CuO 2 SrO BiO BSCCO-2212 Out-of-plane dopants: O interstitial, cation switching,

21 T = 4.2 K 200 pa, -200 mv ~20 Zn atoms Å Bi 2 Sr 2 Ca(Cu 1-x Zn x ) 2 O 8+d : x 0.3% LDOS map at 1.5mV! Å Pan et al, Nature 403, 746 (2000).

22 Zn On-site LDOS spectrum: W 0 =-2 mev 2.5 Differential Conductance (ns) Sample Bias (mv) Pan et al, Nature 403, 746 (2000)

23 Compare Zn STM LDOS pattern with simple theory of nonmagnetic impurity in dsc

24 Compare Zn STM LDOS pattern with simple theory of nonmagnetic impurity in dsc

25 Theories of impurity resonance spatial pattern Chemistry : M.E. Flatté et al. 2001, 03. Assume generalized extended impurity potential. Filter : C.S. Ting et al. 2001, Martin & Balatsy STM probes LDOS of neighboring Cu s due to -dependent tunnel matrix elements. Correlations : Pololniov et al 2001, account for Kondo screening of correlation-induced local moment

26 Tunneling involves orbitals that extend out of the planes, such as 4s Cu. These orbitals are symmetric in the Cu-O plane and hence couple to the neighboring metal 3 d x 2 y 2 orbitals through the d-wave-lie for

27 Bogoliubov-de Gennes (BdG) equations for Cu lattice Applying Bogoliubov transformation leads to BdG equations u, v and E n are obtained by solving BdG equations self-consistently for a given filling.

28 Local Density of States Lattice Green s function and Lattice LDOS Local continuum Green s function and LDOS (Wannier function for one band system)

29 Tight-binding band downfolded From WIEN2K Wannier transformation rn> = e -ir U nj () ν> ν + d-wave superconductivity ()= 0 (cos x cos y )

30 Results*: Wannier d x 2 y 2 orbital Cut through w(r) 5Å above plane Cu- d x 2 y 2 Wannier function * A. Kreisel, P. Choubey, T. Berlijn, B. M. Andersen and P. J. Hirschfeld, PRL 114, (2015)

31 Results: Lattice and Continuum LDOS Cont. LDOS (5 Å above BiO surface)) Expt. (Pan et al) Lattice LDOS (CuO2 plane)

32 Height dependence Exponential limit

33 Quasi Particle Interference (QPI) Simple metal Fermi surface Cuprate superconductor: Fermi surface and gap F 2 F 2 F

34 (π,π) q 1 (π,0) q 7 (π,π) q 1 (π,0) q 7 (π,π) Hoffman et al (2002), McElroy (2003)

35 Octet analysis Hoffman et al (2002), McElroy (2003) qp const. energy contours q 5 q 4 q 7 q 7 q 6 q 3 q 7 q 2 q 1 -space 1 n( E) = E( = ) E E( ) d q-space

36 wea impurities Capriotti et al 2003 ρ( q) Im Λ( q) uq ( ) 1 Λ e G rg r π iq r 0 0 (q) ( ) ( ) r L L

37 Critique of 1-impurity and wea scattering analyses: McElroy et al Neither can explain pea widths and weights: (100) peas too small Octet pea positions alone give no insight into origins of disorder potential? Why are peas so broad in expt.? (no broadening in Capriotti et al analysis)

38 LDOS FTDOS EXPT 8% wea potential scatterers, V 0 =2t, range λ=a, 0.2% unitary scatterers, V 0 =30t, Zhu Atinson PH 2004

39 QPI simulation of continuum LDOS 5Å above plane no intra-unit cell information BSCCO: wea potential scatterer atomic scale local density of states at STM tip position spots from octet model Fourier transform full information for all scattering vectors no information beyond first BZ

40

41 Comparison to experiment no large q information relative conductance map, Fourier transformation energy integrated relative conductance maps K Fujita et al. Science 344, 612 (2014)

42 QPI as a model-free phase-sensitive tool in unconventional superconductors PH, D. Altenfeld, I.I. Mazin, I. Eremin PRB92, (2015) 1. QPI is not a quantitative tool a low-energy (near-fermi) property proportional to coherence factors 2. But QPI may be a qualitative tool, if one can see qualitatively different behavior in different cases of interest (e.g., s ± vs. s ++ )

43 1. QPI is not a quantitative tool e.g. Häne et al. PRL 2012 proportional to coherence factors Maltseva & Coleman, 2009

44 QPI as a model-free phase-sensitive tool Chi et al, PRB 2014 Does superconductivity enhance the large q transitions and suppress the small q ones? coherence factors? LiFeAs intensity q y q x Qualitative probe of gap sign change?

45 Chi et al 2014 Assuming inter s +- intra s ++ s +- s +- most sensitive test s ++ Theory: PH, Altenfeld, Eremin, Mazin PRB 2015

46 Conclusions 1. Simple method of using discarded Wannier function information to calculate local STM conductance in inhomogneous SC: enhances resolution, preserves local symmetries, allows calculation of true surface properties. 2. Application to BSCCO: resolution of old Zn and Ni paradoxes Dramatic improvement of QPI calculations 3. Suggestions to determine gap signs: (i) s/c-normal differences and (ii) symmetized/antisymmetrized combinations are more informative than just QPI. (iii) monitor T-dependence at a given bias

47 Coherence factors Hanaguri et al 2008, Maltseva and Coleman 2009 Chi et al 2014

48 FTLDOS for single impurity small q integrated weight: Independent sums: use Similarly for interband terms

49 Determine sign change by measuring symmetrized and antisymmetrized conductances at large and small q T-dependence as one enters SC state should be most sensitive measure!

50 Remar on Hanaguri method T. Nunner et al., Phys. Rev. B 73, (2006): QPI with Andreev τ 0 pointlie scatterers T. Pereg-Barnea and M. Franz, Phys. Rev. B 78, (2008): proposal to use disordered vortex lattice as source of controlled disorder T. Hanaguri et al, Science 323, 923 (2009): BSCCO: sign-changing q-peas M. Maltseva and P. Coleman, PRB 80, : formalism with coherence factors T. Hanaguri et al Science 328, 474 (2010): same for Fe(Se,Te) Pereg-Barnea and Franz 2008 No real theoretical understanding of why field suppresses +- q vectors

51 Emergent defect states: theory 1. SDW state Gastiasoro, PJH, and Andersen, PRB 2013 Mean field

52 Nematogens grow as T lowered: magnetization M. Gastiasoro et al PRB 2013

53 Emergent defect states: theory 2. Nematic state T N <T<T S tx/ty =1.05 V imp =6eV Ishida et al 2013

54 Scattering rate anisotropy Self-consistent BdG with without impurity 5% band structure anisotropy 250% anisotropy in scattering rate! spin fluctutation enhancement of impurity potential anisotropy

55 Scattering rate in b direction diverges at T N Gastiasoro et al axv:

56 Summary of results ortho tetragonal SDW T s T n small fixed strain T controlled strain (tetragonal phase)

57 Summary of results ortho tetragonal SDW T s T n small fixed strain T controlled strain (tetragonal phase)

58 Summary of results ortho tetragonal SDW T s T n small fixed strain T controlled strain (tetragonal phase)

59 Generalization to strong scatterers (Zhu et al 2003) t 1 iφ δρ( q) = t ( ) Im[ e α π α q Λα( q)] α = t-matrix for 1 zero range impurity = phase of 1-imp. t-matrix ( q) t e ; Λ ( q) = G (, ω) τ G ( + q, ω) iqr i 0 0 α α i α α response function α=3: potential scatt. α=0: magnetic scatt. δρ still = (octet peas) (noise) No broadening in q-space unless scattering is not wea and not zero range

60 Why QPI is not a quantitative tool? bul right at the surface 5A above

61 Ab initio evidence for wea normal state filter: (Wang, Cheng, PH PRB 2004) 1.5Å above surface Cu-O plane min max Q: How to include SC?

62 Ni impurity in BSCCO: expt.* Ω = +9 mev Ω = -9 mev *Hudson et al., Nature 403, 786 (2000)

63 Magnetic impurity in BdG Ni has 3d 8 configuration which leads to magnetic moment on the impurity site (S=1). Approximating it as a classical spin: => Electrons with spin up and down see effective impurity potentials V imp + J and V imp J respectively. V imp =.625 ev Spin-up resonance Ω = ±2.4 mev Spin-down resonance Ω = ±7.2 mev

64 Results*: Lattice and Continuum LDOS Ω = 2.4 mev Ω = -2.4 mev * Manuscript under preparation

65 Results: Lattice and Continuum LDOS Experiment BdG+W Impurity NN NNN BdG Far away

66 V- vs U- shape LDOS spectra Underdoped cuprates show clean V-shape d-wave lie spectrum Optimal-overdoped cuprates show U-shaped spectrum why? Pan et al. Nature, 413, 282 (2001) Kohsaa et al. Nature, 454, 1072 (2008) Alldredge et al. Nature Physics, 4, 319 (2008) BdG+W LDOS gives spectra resembling U-shape. What will happen if strong correlations are included?

67 = = RR R R RR r w G r w r w G r r G 2 ' ' ' ) ( ), ( )* ( ) ( ) ( ) ;, ( ω ω ω = R R i RR e G G ') ( ' ), ( ) ( ω ω where... ] sin cos 2 sin )[cos 2 ( ] cos 2 )[cos 2 ( ] cos )[cos ( ) ( ) ( ) ( = x y y x y x y x R i R r a r a r a r a e r w r w NN NNN Wannier function above surface Nonlocal contributions to local continuum Green s function homogeneous case:

68 ... ] sin cos 2 sin )[cos 2 ( ] cos 2 )[cos 2 ( ] cos )[cos ( ) ( ) ( ) ( = x y y x y x y x R i R r a r a r a r a e r w r w NN NNN... ) ( ) ( ) (... ) cos 2 (cos 2 ) ( ), ( ) cos (cos ) ( ), ( ) ( ), ( ) ( ), ( ) ;, ( = ω ω ω ω ω ω ω ω O r a r a r a G r a G r a G r w G r r G y x y x (interference terms vanish) Wannier analysis: implications for filter mechanism Conclude: linear-ω contribution to LDOS comes from local piece of w(r) Any purely NN filter tunneling mechanism (Balatsy, Ting) yields ω 3 only Effective Wannier function range may shrin with correlations

69 Iron-based superconductors Recent reviews: Stewart RMP 2012; Paglione & Greene Nat Phys 2010 T c =28K T c =38K T c =18K T c =8K (55K for Sm) Kamihara et al JACS (2008) Ren et al Chin. Phys. Lett. (2008) Rotter et al. PRL (2008) Ni et al Phys. Rev. B 2008 (single xtals) Wang et al Sol. St. Comm Hsu et al PNAS 2008 No arsenic!

70 STM: emergent defect states those shown believed to be Fe vacancies or substituents (J.E. Hoffman) 1111 (LaFeAsO) 111 (LiFeAs) 11 (FeSe) 10nm Zhou, PRL 106, (2011) Hanaguri, unpublished 10nm Song, Science 332, 1410 (2011)

71 STM: exotic local defect states 1. Geometric dimer FeSe on graphite, Song et al., PRL 109, (2012) 2. Electronic dimer

72 Results*: Wannier orbitals d xy Wannier orbitals on two Fe atoms in unit cell *P. Choubey, T. Berlijn, A. Kreisel, C. Cao, and P. J. Hirschfeld, Phys. Rev. B. 90, (2014)

73 Results FeSe: Lattice LDOS LDOS far from impurity site, at impurity sites and at NN and NNN sites to impurity. DOS in homogeneous system Bound states at ω = ±8 mev for 5 mev impurity potential

74 Results FeSe: Lattice & continuum LDOS maps Real space patterns of lattice LDOS at 8 mev xy- cuts through continuum LDOS(x,y,z; ω) at different heights z from Fe plane. ω=8mev ω=8mev C4 Csymmetry 4 intact intact on in Fe Fe plane plane Dimer lie structures obtained above Se plane

75 Topograph Comparison of experimental topograph at 6 mev set-point bias (a) with BdG only (b) and BdG+W LDOS (c)