Enhancement of superconducting Tc near SIT

Transcription

1 Enhancement of superconducting Tc near SIT Vladimir Kravtsov, ICTP (Trieste) Collaboration: Michael Feigelman (Landau Institute) Lev Ioffe (Rutgers) Emilio Cuevas (University of Murcia) KITP, Santa Barbara, September 13, 010

2 The Hamiltonian 3d lattice tight-binding model with diagonal disorder Local attraction f ( ε i ) H H int = ε δ δ + + V + r r ', r r, r aˆ H int λ + + = ψ ( r) ψ ( r) ψ ( r) ψ ν 0 r W / +W / ( r) ε i NO Coulomb interaction

3 NO Coulomb interaction? Where can it happen? Cold fermionic atoms in disordered optical traps: atoms are neutral In solids with a certain band structure: large background dielectric constant

4 Q: What does the strong disorder do to superconductivity? A1: weak disorder do not do anything; strong disorder kills superconductivity A: eventually disorder kills superconductivity but before killing it enhances it

5 Weak and strong disorder Anderson transition disorder W /V L Extended states l Critical states ~ λ F Localized states

6 Direct superconductor to insulator transition FAQ:Why superconductivity is possible even when single-particle states are localized Single-particle conductivity: only states in the energy strip ~T near Fermi energy contribute ξ R(T) R( T ) = 1 ν 0T 1/3 Superconductivity: states in the energy strip ~Δ Τc near the Fermi-energy contribute Interaction sets in a new scale Δ Τc which stays constant as T->0 ξ > R( T c ) ξ < R T ) R(Τc) ( c

7 Multifractality of critical and off-critical states r n 1 Ψi ( r) = d ( n 1) L n Multifractal insulator Multifractal metal

8 Matrix elements Interaction comes to play via matrix elements λm nm = λv d d r Ψ n ( r) Ψ m ( r)

9 Ideal metal and insulator M nm = Vd d r Ψ n ( r) Ψ ( r) m 1 1 Metal: V V = 1 V V Small amplitude 100% overlap ξ 1 ξ 1 ξ d ξ V d Insulator: V = 1 d d Large amplitude but rare overlap

10 Critical enhancement of correlations E E' 1+ d / d Amplitude higher than in a metal but almost full overlap States rather remote (δ<<\e-e <E0) in energy are strongly correlated

11 Simulations on 3D Anderson model Ideal metal: ξ< l 0 E 0 d d E E E / 1 0 ' Multifractal metal: ξ> l 0 ν λ ξ W W W c c F = Ψ Ψ = m n m n m n m n m n d E E E E E E E E r r r d V E E C,, ') ( ) ( ') ( ) ( ) ( ) ( ') ( δ δ δ δ ( ) W c D E ~ = l ν ( ) 1 0 = d ξ ν δ ξ Critical power law persists δ ξ W=10 W=5 W= Wc=16.5 1/3 0 ~ c λ F W l

12 L ω < ξ Multifractal metal Multifractal metal Multifractal metal Anderson transition disorder Wave function does not occupy all the available space: enhanced amplitude (normalization) Regions with enhanced amplitude are strongly correlated for different wave 1/3 functions as long as L <

13 l ~ 1/3 0 λ F W c Is a pixel of fractal pattern E 0 ~ E W F c ~ E F 16

14 Enhancement of matrix element λ M nm = λv d d r Ψ n ( r) Ψ m ( r) E 0

15 What to do at strong disorder? Δ(r) cannot be averaged independently of the MF kernel K(r,r ): No Anderson theorem Fock space instead of the real space H eff = ε isi U M i z i j ij ( S + i S j + S i S + j Single-particle states, strong disorder included ) E F S + + = a a Superconducting phase S z = ( a a + a a E F Normal phase 1), y x, y S 0 = 0 x i S i

16 Why the Fock-space mean field is better than the real-space one? x H ε S λ M ( S S + eff = z i i i i j ij x i j S y i S y j ) M ij = dr Ψ i ( r ) Ψ j ( r ) i j Infinite or large coordination number for extended and weakly localized states: supports mean field approximation Δ i = λ Δ j j tanh( E E j j / T ) M ij Mij is energy dependent

17 MF critical temperature close to critical disorder T c Δ( E) = λ de' M( E E') = 1/(1 d / d) 0 ~ Δ( E') E0 E E' ~ E λ E λ 0 tanh( E'/ T ) E' 1 d 1.78 At a small λ parametrically large enhancement of Tc / d M( E E') Critical enhancement of correlations >> [ ] ω exp 1/ λ D M.V.Feigelman, L.B.Ioffe, V.E.K. and E.Yuzbshyan, Phys.Rev.Lett. v.98, (007);

18 Isotop effect T c ~ E λ E ~ E / W ~ E 0 F c F /16 No isotop effect

19 Annals of Physics 35, 1368 (010) The phase diagram Tc I λ BCS = 0. Extended states M I Localized states S S F Mobility edge

20 Relevant experiment?

21 Li-doped ZrNCl Cl Zr N double honeycom layer Cl Intercalated Li atoms, provide electrons to ZrN layer, source of disorder? Without Li ZrNCl is a band insulator with a gap of 1.6 ev

22 Why natural in this compound? Initial motivation for our theory: direct SIT in amorphous films of InO, TiN ZrN layer Superconductivity survives until disorder is strong enough to cause Anderson localization. Why Coulomb interaction does not destroy it well before? Ti Zr 40 Hf 7 In Mendeleev s periodic Table Li-doped HfNCl is a similar SC with higher Tc~5K

23 Electron-phonon attraction constant ~0.5 follows from band-structure calculations Tc is too large Eliashberg theory + band structure calculations Isotop effect is too small

24 Does the band structure support large background dielectric constant?

25 ε ~ σ ν iν f ω ω ω 0 0 Ef ν ~ 0.4 states / ev ~ 0. ν i ν ~ 10 states / Ry ~ 3ν ~ 0.5eV ~ 0. 1 f ω 0 E FGold Gold Gold DoS in good metals (gold) ~ states/ev (per atom) ~ 30 states/ry <10 ω 0 = 0.5 ev Ry=13.6 ev Fermi surface occupies a very small part of the Brillouin zone F<<a: clear separation between low-energy and highenergy physics High peaks in DoS close to low DoS conduction band

26 Al N ev No high peaks in the DoS close to conduction band ε 0 ~ 6 ω 0 ~ 6 ev

27 ε 0 ~ 50 For large-distance screening Does not seem to be unreasonable as long as the high DoS peak at ~0.6 ev persists Small-distance (large k~1/a) screening may be much weaker because of the effect of dispersion ω ( k) =ω k / m 0 0 +

28 Conclusions Enhancement of superconductive Tc by nearcritical disorder: combined action of holes in the single-particle wavefunction amplitude and correlation in positions of peaks for different wavefunctions. Power-law dependence of Tc on the attractive interaction constant λ at near-critical disorder. The exponent depends on the critical exponent: the fractal dimension d. Possible experimental system with weak longrange Coulomb interaction

29 d weak multifractality: the Ovchinnikov- Mayekawa-Fukuyama-Finkelstein effect The diffuson diagrams ΔT λ 1 c = ln 3 T g c Tc τ For long-range Coulomb interaction λ = 1 V 1 1 = V 0 ( q) Π ν 0 The cooperon diagrams For short-range attraction one has effectively ln^ as λ=1/ln

30 Cold atoms trapped in an optical lattice Fermionic atoms trapped in an optical lattice speckles Disorder is produced by: One, two and three-dimensional tight-binding systems with disorder Other trapped atoms (impurities)

31 One-dimensional Anderson localization in systems of cold atoms Alain Aspect group Massimo Inguscio group J.Billy et al., Nature 453, 891 (008); Roati et al., Nature 453, 895 (008) Work on two- and three-dimensional localization is in progress

32 BUT

33 Sweet life is only possible without Coulomb interaction e r

34 Mean-field approximation and the Anderson theorem Δ( r) = dr' Δ( r') K( r, r'; T ) K( r, r'; T ) * = U η ( T ) ij 1/V Ψi ( r) Ψj ( r) Ψj ( δr ') Ψi ij ij * ( r') η ( T ) ij tanh( Ei = / T ) + tanh( E E i + E j j / T ) Wavefunctions drop out of the equation Anderson theorem: Tc does not depend on properties of wavefunctions

35 Tc at the Anderson transition: MF vs virial expansion M.V.Feigelman, L.B.Ioffe, V.E.K. and E.Yuzbshyan, Phys.Rev.Lett. v.98, (007); γ E E ' 1/ T crit = c E λ γ γ = 1 d / d γ 0, MF result Virial expansion

36 Superconducting transition temperature Virial expansion on the 3d Anderson model metal insulator Anderson localization transition (disorder)

37 Conclusion: Enhancement of Tc by disorder metal insulator Maximum of Tc in the insulator BUT Fragile superconductivity: Direct superconductor to insulator transition Small fraction of superconducting phase Critical current decreasing with disorder

38 Two-eigenfunction correlation in 3D Anderson model (insulator) NC( ω) ln d 1 δξ ω Mott s resonance physics Ideal insulator limit only in one-dimensions E E' + d / d 1 δ ξ critical, multifractal physics

39 Superconductor-Insulator transition: percolation without granulation Coordination number K>>1 T c (crit) > δ ξ SC c{ T } δξ Coordination number K=0 T c (crit) < δ ξ INS T{c Only states in the strip ~Tc near the Fermi level take part in superconductivity δξ

40 First order transition? c c c T U T M T K ~ ) ( ) ( T Τ T T f T T T T E T f T KMU d d = ) / ( ) / ( ~ ~ ) / ( ) / ( ) / ~ ( / ξ ξ γ ξ ξ δ λδ δ δ ξ δ Tc f(x)->0 at x<<1

41 Conclusion Fraclal texture of eigenfunctions persists in metal and insulator (multifractal metal and insulator). Critical power-law enhancement of eigenfunction correlations persists in a multifractal metal and insulator. Enhancement of superconducting transition temperature due to critical wavefunction correlations.

42 T K ( T ) M ( T ) U ~ c c T c BCS limit M SC SC IN SC or IN disorder KMU Anderson localization transition δ ξ T Τ KMU γ ~ ( T / δ ξ )( E / T ) λδξ ~ 0 T 0 ( T / T 0 ) d / d

43 Corrections due to off-diagonal terms Δ ij = U Δ kl ηkl ( T ) kl M ijkl Δ i = U l Δ l η + l ( T ) M il U η k, k m km M iikm M llkm Average value of the correction term increases Tc Average correction is small when λ 3 d d d 4 d << 1

44 Melting of phase by disorder In the diagonal approximation Δ( r) = Δijηij Ψi ( r) Ψj ( r) Δ ( r ) ij = i Δ i tanh( E / T ) Ψ The sign correlation <Δ(r)Δ(r )> is perfect : solutions Δi>0 do not lead to a global phase destruction i i ( r ) Beyond the diagonal approximation : Δ i = U l Δ l η l ( T ) M il + U η k, k m km stochastic term destroys phase correlation M iikm M llkm

45 How large is the stochastic term? Stochastic term: Q il = U η k, k m km M iikm M llkm d<d/ Q il 1 d / d ~ N >> Qil ~ N QQ ili d d > >d/ weak oscillations d d < d/ strong / oscillations but still too small to support the glassy solution

46 More research is needed

47 Conclusions Mean-field theory beyond the Anderson theorem: going into the Fock space Diagonal and off-diagonal matrix elements Diagonal approximation: enhancement of Tc by disorder. Enhancement is due to sparse single-particle wavefunctions and their strong correlation for different energies Off-diagonal matrix elements and stochastic term in the MF equation The problem of cold melting of phase for d <d/