BrazilIndia Workshop on Theoretical Condensed Matter Physics Brazilian

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

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

Issues How much dirt (disorder) can take a superconductor 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

Metal evaporated on cold substrates, precoated with a-Ge: disorder on atomic scales. Quantum Critical Point (evaporation without a-Ge underlayer: granular disorder on mesoscopic scales. 1) SHEET RESISTANCE R�(ohms) Superconductor – Insulator transition at T = 0 when R�reaches one quantum of resistance for electron pairs, h/4 e 2 = 6. 45 k Bismuth Behaviour near QCP will not be discussed here D B Haviland et al. , PRL 62, 2180 (1989) TEMPERATURE (K)

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

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

The disordered attractive Hubbard model ◊ particle-hole symmetry at half filling ◊ strong-coupling in 2 D: • half filling: XY (SUP) + ZZ (CDW) Tc 0 • away from half filling: XY (SUP) TKT 0 Homogeneous case [Paiva, d. S, et al. (04)]

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

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

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 (twocomponent order parameter) Huse PRB (88): zero-temperature gap

~ U~ For 2. 5 < < 6, a small amount of disorder seems to enhance SUP We estimate fc as the concentration for which 0; can plot fc (U ). . . normalized by the corresponding pure case

fc increases with U, up to U ~ 4; mean-field behaviour sets in above U ~ 4? transition definitely not driven solely by geometry (percolative): fc = fc ( U ) (c. f. , percolation: fc = 0. 41)

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

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

Tc initially increases with disorder: breakdown of CDW-SUP degeneracy

Conclusions (half-filled band) üA small amount of disorder seems to initially favour SUP in the ground state. üfc depends on U transition at T = 0 not solely geometrically driven; quantum effects; correlated percolation? üTwo possible mechanisms at play: • MFA: as U increases, pairs bind more tightly smaller overlap of their wave functions, hence smaller fc. • 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 fc. • QMC: for U > 4, pairs are tightly bound and SUP more sensitive to dirt. ü A small amount of disorder allows the system to become SUP at finite temperatures; as disorder increases, Tc eventually goes to zero at fc.

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

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: System S Environment E Superblock • 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

Density Matrix Renormalization Group: System S Environment E S’ Superblock E’ S E • 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)

Results

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

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

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

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

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

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

Comparison: Ps(r)

Ps fit Ps ~ 1 / rβ

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)

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