Superconductivity: modelling impurities and coexistence with magnetic order Collaborators: Pedro R Bertussi (UFRJ) André L Malvezzi (UNESP/Bauru) F. Mondaini (UFRJ) Richard T.Scalettar (UC-Davis) Thereza Paiva (UFRJ) Financial support: Brazil-India Workshop on Theoretical Condensed Matter Physics Brazilian Academy of Sciences, April 2008 Raimundo R dos Santos

Layout: A) Disordered Superconductors 1.Motivation 2.The disordered attractive Hubbard model 3.Quantum Monte Carlo 4.Ground state properties 5.Finite-temperature properties 6.Conclusions B) Coexistence of Superconductivity and Magnetism 1.Motivation 2.Model 3.DMRG 4.Results 5.Conclusions C) Overall Conclusions

Disordered superconducting films F Mondaini et al.

Sheet resistance: R  at a fixed temperature can be used as a measure of disorder Disorder on atomic scales: Sputtered amorphous films CRITICAL TEMPERATURE T c (kelvin) Mo 77 Ge 23 film J Graybeal and M Beasley, PRB 29, 4167 (1984) t ℓ ℓ  independent of the size of square SHEET RESISTANCE AT T = 300K (ohms) Disorder is expected to inhibit superconductivity

How much dirt (disorder) can take a super- conductor before it becomes normal (insulator or metal)? Question even more interesting in 2-D (very thin films): superconductivity is marginal  Kosterlitz-Thouless transition metallic behaviour also marginal  Localization for any amount of disorder in the absence of interactions (recent expts: MIT possible?) A M Goldman and N Marković, Phys. Today, Page 39, Nov 1998 Issues

Metal evaporated on cold substrates, precoated with a-Ge: disorder on atomic scales. D B Haviland et al., PRL 62, 2180 (1989) Superconductor – Insulator transition at T = 0 when R  reaches one quantum of resistance for electron pairs, h/4e 2 = 6.45 k   Quantum Critical Point Bismuth (evaporation without a-Ge underlayer: granular  disorder on mesoscopic scales. 1 ) SHEET RESISTANCE R  (ohms) TEMPERATURE (K) Behaviour near QCP will not be discussed here

Our focus here: interplay between occupation, strength of interactions, and disorder on the SIT; fermion model.

B. Berche et al. Eur. Phys. J. B 36, 91 (2003) XY 2D Stinchcombe JPC (1979) T c (p)/T c (1) p Heisenberg 3D Yeomans & Stinchcombe JPC (1979) Ising 2D Dilute magnets: fraction p of sites occupied by magnetic atoms: T c  0 at p c, the percolation concentration (geometry)

The disordered attractive Hubbard model [Paiva, dS, et al. (04)] Homogeneous case ◊ particle-hole symmetry at half filling ◊ strong-coupling in 2D: half filling: XY (SUP) + ZZ (CDW)  T c  0 away from half filling: XY (SUP)  T KT  0

Disordered case  particle-hole symmetry is broken Heuristic arguments [Litak + Gyorffy, PRB (2000)] : f c  as U  The disordered attractive Hubbard model c  1- f mean- field approx’n

Quantum Monte Carlo Calculations carried out on a [square + imaginary time] lattice: x NsNs M Absence of the “minus-sign problem” in the attractive case

For given temperature 1/ , concentration f, on-site attraction U, system size L  L etc, we calculate the pairing structure factor, averaged over 50 disorder configurations. N.B.: half filling from now on

Ground State Properties Spin - wave – like theory (two- component order parameter) Huse PRB (88) : zero-temperature gap

We estimate f c as the concentration for which   0; can plot f c (U )... normalized by the corresponding pure case For 2.5 < U < 6, a small amount of disorder seems to enhance SUP ~~

f c increases with U, up to U ~ 4; mean-field behaviour sets in above U ~ 4? transition definitely not driven solely by geometry (percolative): f c = f c (U ) (c.f., percolation: f c = 0.41)

Finite-temperature properties Finite-size scaling for Kosterlitz-Thouless transitions KT usual line of critical points (  = ∞) Barber, D&L (83)  cc L1/1L1/1 L2/2L2/2 L1/1L1/1 L2/2L2/2   KT

Finite-size scaling at T > 0: KT transition For infinite-sized systems one expects

T c initially increases with disorder: breakdown of CDW-SUP degeneracy

Conclusions (half-filled band) A small amount of disorder seems to initially favour SUP in the ground state. f c depends on U  transition at T = 0 not solely geometrically driven; quantum effects; correlated percolation? Two possible mechanisms at play: MFA: as U increases, pairs bind more tightly  smaller overlap of their wave functions, hence smaller f c. QMC: this effect is not so drastic up to U ~ 4  presence of free sites allows electrons to stay nearer attractive sites, increasing overlap, hence larger f c. QMC: for U > 4, pairs are tightly bound and SUP more sensitive to dirt. A small amount of disorder allows the system to become SUP at finite temperatures; as disorder increases, T c eventually goes to zero at f c.

Coexistence between superconductivity and magnetic order PR Bertussi et al.

Motivation Competition between exchange interaction and electronic correlations, as, e.g., in: - Magnetic superconductors (attractive correlations) * heavy fermions (FM; AFM) - bulk * borocarbides (AFM) - layers - Diluted magnetic semiconductors (repulsive correlations). In this work: attractive correlations

Borocarbides [Canfield et al., (1998)] Coexistence of magnetic order and superconductivity

Borocarbides [Lynn et al., (1997)] ErTmTb Rare earth 4f electrons order (AF) magnetically Conduction electrons form Cooper pairs R = Pr, Dy, Ho

Model Electronic correlations  Attractive Hubbard Model Exchange interaction between conduction electrons and local moments  Kondo term

Method DMRG  approximate ground state Up to 60 sites Density n=1/3 Open boundaries  consider only sites away from the boundaries (~5 sites) Analysis of ground state properties through correlation functions (pairing, magnetic and charge) and their respective structure factors

Density Matrix Renormalization Group: Obtain the ground state by using, for example, Lanczos

Density Matrix Renormalization Group: Obtain the ground state by using, for example, Lanczos Use density matrix to select the states of the system (environment) that are the most important to describe the ground state of the universe  truncation System S Environment E Superblock

Density Matrix Renormalization Group: Obtain the ground state by using, for example, Lanczos Use density matrix to select the states of the system (environment) that are the most important to describe the ground state of the universe  truncation Add sites to create a new system (environment) System S Environment E Superblock S E S’E’

Results

Electron-spin  localized-spin correlations S · s (U = 8t ) - Non-exhausted singlet states (Kondo) above (J/U) c

Electron spin-spin correlations s z ( i ) · s z ( j ) (U = 8t) - Rapidly decaying correlations: electrons on different sites are not magnetically ordered

Localized spin-spin correlations S z ( i ) · S z ( j ) (U = 8t) - SDW correlations for small J/U - FM for large J/U S x ( i ) · S x ( j ) (U = 8t)

Localized spin-spin correlations structure factor (U = 8t) - maximum at k = 0 indicates FM and at k = π, SDW (U = 6t) - maximum at intermediate k  ISDW, incommensurate with lattice spacing - Gradual transition from maximum at k = π to k = 0 (U = 4t)

Comparison: S(k) peaks  No significant finite-size effects

Pairing correlations - Superconductivity possible only below (J/U)c (U = 8t)

Comparison: P s (r)

P s fit  Ps ~ 1 / r β

Phase Diagram

Conclusions Conduction electrons never order magnetically Coexistence of Superconductivity with magnetic ordering of the local moments (SDW or ISDW) below (J/U)c Kondo effect (singlets between local moments and conduction electrons) with a tendency of spiral ferromagnetism of the local moments

Overall conclusions Use of simple attractive Hubbard model allows investigation of “real-space” phenomena in superconductors BCS model: hard to extract info in similar contexts  need to learn how to incorporate finite-size effects (in progress)

Collaborators: Antônio José Roque da Silva (IFUSP) Adalberto Fazzio (IFUSP) Luiz Eduardo Oliveira (IFGW/UNICAMP) Tatiana G Rappoport (IF/UFRJ) Materials for Spintronics: Diluted Magnetic Semiconductors