1. Disordered two-dimensional superconductors Financial support: Collaborators: Felipe Mondaini (IF/UFRJ) [MSc, 2008] Gustavo Farias (IF/UFMT) [MSc, 2009] Thereza Paiva (IF/UFRJ) Richard T Scalettar (UC-Davis) Raimundo Rocha dos Santos IF/UFRJ

3. †

4. UFRJ

5. Rio-Niterói bridgeSugar Loaf

6. Outline Motivation The model: disordered attractive Hubbard model Quantum Monte Carlo Ground state properties Finite-temperature properties Conclusions

7. Motivation: disorder on atomic scales Sputtered amorphous films Low coverage: isolated incoherent islands High coverage: islands “percolate”  film thickness tracks disorder

8. How much dirt (disorder) can a superconductor take 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

9. CRITICAL TEMPERATURE T c (kelvin) SHEET RESISTANCE AT T = 300K (ohms) Mo 77 Ge 23 film J Graybeal and M Beasley, PRB 29, 4167 (1984) Sheet resistance: R  at a fixed temperature can be used as a measure of disorder t ℓ ℓ  independent of the size of square Disorder is expected to inhibit superconductivity Issues:  Why does T c behave like that with disorder?  Is the transition at T = 0 percolative (i.e., purely geometrical)?  If not, how does it depend on system details?

10. 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 )

11. Our purpose here: to understand the interplay between occupation, strength of interactions, and disorder on the SIT, through a fermion model. First task: reproduce, at least qualitatively, T c vs. R □ Generic phase diagram for Quantum Critical Phenomena: T disorder (mag field, pressure, etc.) SUC METAL INS QCP TcTc TT

12. Interlude: Phase transitions and critical phenomena Long-ranged correlations at a phase transition  singular behaviour of thermodynamic quantities  e.g., order parameter of the transition:  magnetic transition: magnetization (1, 2 or 3-component)  superconducting transition: gap (complex; macroscopic wave function)  2-component  ( T)  (0) T/TcT/Tc s-wave  isotropic specific heat susceptibility and so forth

13. Universality (expt. and theory; 1970’s): main features of phase transitions (including critical exponents) do not depend on details (magnitude of interactions, etc.); they depend on: symmetry of order parameter (# of components) dimensionality of lattice  determine nature and number of excitations,  can be so overwhelming to the point of depressing T c to zero (Mermin-Wagner th m ) magnets, superconductors, superfluids, fluids, etc., share common main features! 2-component order parameter in 2 space dimensions:  M-W th m forbids long range order at T > 0  but phase with quasi--long-range order possible below T KT : the Kosterlitz-Thouless transition

14. 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)

15. The homogeneous 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

16. 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

17. 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 non-local updates: MN s 2 Green’s functions

18. Typical behaviour for  →  : correlations limited by finite system size 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 =1

19. Ground State Properties 0.000.050.100.15 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 f= 0 1/16 f= 2/16 f= 3/16 f= 4/16 f= 5/16 U=4 P S /L 2 1/L Spin - wave – like theory (two- component order parameter) Huse PRB (88) : zero-temperature gap n = 1 F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )

20. 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 ~~ n = 1 F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )

21. 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) n = 1 F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )

22. Results for n = 0.875 GJ Farias, MSc thesis, UFMT, (2009)

23. n = 0.875

24. GJ Farias, MSc thesis, UFMT, (2009) n = 0.875

25. GJ Farias, MSc thesis, UFMT, (2009) U = 6 For n = 1: f =0  CDW+SUP f >0  SUP  dirt initially enhances SUP For n < 1: f =0  only SUP f >0  still only SUP  dirt always tend to suppress SUP

26. GJ Farias, MSc thesis, UFMT, (2009) ? f c  with n  for fixed U (at least for U  4): less electrons to propagate attraction

27. 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

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

29. Kosterlitz-Thouless transition: Curves should cross/merge at β c for η(T c )=1/4: Repeat this for other values of f...

30. f Pure case at half filling:

31. n = 1 F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )

32. Somewhat arbitrary  Check against independent method Superfluid density,  s, (  helicity modulus) measures the system response to a twist in the order parameter [M.E. Fisher et al. PRA 8, 1111 (1973)]  need current-current correlation function [DJ Scalapino et al. PRB 47, 7995 (1993)]  very costly: imaginary-time integrals At T KT -,  s has universal jump-discontinuity [D.R. Nelson and J.M. Kosterlitz, PRL 39, 1201 (1977)] :

33. n = 1 2T/2T/ ss F Mondaini, et al. PRB 78, 174519 (2008 )

34. n = 1 F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )

35. n = 1 F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )

36. Results for n = 0.875 GJ Farias, MSc thesis, UFMT, (2009)

37. F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 ) Different concavities? Need to refine:  s Possibly concavity changes with n ? Possibly non-linear relation between R □ and f ?

38. Conclusions At half filling, small amount of disorder seems to initially favour SUP in the ground state; not for other fillings. f c depends on U  transition at T = 0 not solely geometrically driven; quantum effects; correlated percolation? for given U, f c  with n  (need more calcn’s for U < 4) 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. n=1: 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. n <1: T c decreases steadily with f. Concavity???