Superconductivity in Systems with Diluted Interactions XXV encontro nacional de física da matéria condensada Superconductivity in Systems with Diluted Interactions Thereza C. de L. Paiva (IF/UFRJ),† Grzegorz Litak (TU-Lublin), Carey Huscroft (HP Co., Roseville), Raimundo R. dos Santos (IF/UFRJ),† Richard T. Scalettar (UC Davis) †Supported by Talk downloadable from www.if.ufrj.br/~rrds/rrds.html

Outline Motivation The Model Quantum Monte Carlo Results Conclusions

Motivation1 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 (recent expts: MIT possible?) 1 A M Goldman and N Marković, Phys. Today, Page 39, Nov 1998

t Mo77Ge23 film Disorder on atomic scales: sputtered amorphous films2 Sheet resistance: R at a fixed temperature can be used as a measure of disorder t ℓ Mo77Ge23 film CRITICAL TEMPERATURE Tc (kelvin)  independent of the size of square SHEET RESISTANCE AT T = 300K (ohms) Tc decreases with increasing disorder: screening of Coulomb repulsion is weakened 2 J Graybeal and M Beasley, PRB 29, 4167 (1984)

Quantum Critical Point Metal evaporated on cold substrates, precoated with a-Ge: disorder on atomic scales.3 Bismuth System undergoes a Super-conducting - Insulator transition at T = 0 when R reaches one quantum of resistance for electron pairs, h/4e2 = 6.45 k Quantum Critical Point (evaporation without a-Ge underlayer: granular  disorder on mesoscopic scales.1) 3 D B Haviland et al., PRL 62, 2180 (1989)

Typical phase diagram for Quantum Critical Phenomena (any d): disorder (mag field, pressure, etc.) SUC METAL INS QCP Tc T Our purpose here: to understand the interplay between occupation, strength of interactions, and disorder on the SIT, through a fermion model.

The Model Use attractive Hubbard model to describe superconductivity  real-space pairing Band (hopping) energy: favours fermion motion U < 0 favours on-site pairing of electrons with opposite spins Chemical pot’l: controls fermion density n Schematic phase diagram for the clean model (2D).4 Note Tc = 0 at half filling, due to coexistence of SUC with CDW. 4 RT Scalettar et al., PRL 62, 1407 (1989)

defects Disordered attractive Hubbard model  U < 0 with concentration c  U = 0 with concentration f 1c defects Simple random walk arguments + time-energy uncertainty relations yield an estimate for the critical concentration of defects, f0, above which the system is an insulator:5 Dynamical, instead of purely percolative. What about dependence with filling factor? 5 G Litak and BL Gyorffy, PRB 62, 6629 (2000)

Quantum Monte Carlo6  = M   x  For the attractive model, auxiliary variables For the attractive model, det O = det O so there is no “minus-sign problem” Ns M x 6 W von der Linden, Phys. Rep. 220, 53 (1992); RR dos Santos, BJP (2002)

Calculated quantities, for a given realization of disorder, on an L x L lattice: These quantities are then averaged over ~ 20 - 40 disorder configurations We also need to extrapolate: T  0 L  

Results Site occupations and Pair-field structure factor 8  8, U =  4t, T = t/(8 kB) ~ 100 K n c = 0.875 occ’n of free sites n c = 0.625 n c = 0.375 disorder Occupation of attractive sites increases with disorder Pair-field structure factor decreases with disorder (note how cdw dip at half filling disappears with increasing disorder)

Size dependence of Pair-field Structure Factor: extrapolation to the thermodynamic limit Expectation from spin-wave theory: Collecting extrapolated data for several concentrations of disorder, SUC Sp/N allows one to extract a critical fc for given temperature, n, and U.

Results are equivalent so far A technical remark: fixed  such that the desired n emerges after a sequence of disorder runs (faster but leads to fluctuations in n), or for each disorder realization choose  such that the desired n emerges (slower but leads to much smaller fluctuations in n) ? Conclusion: Results are equivalent so far

Disorder-temperature section of the phase diagram Repeating the above procedures for different temperatures yields a phase diagram fc (T ), which can be inverted to yield Tc (f ) [for fixed U and n ] The theory reproduces the sharp decrease of Tc with disorder, as observed experimentally

Conclusions Description of disordered SUC’s through random-U attractive Hubbard model seems promising For more accurate quantitative predictions: need to consider other fillings and interaction strengths; need to use other quantities (e.g., current-current correlation functions) QCP: need to go to much lower temperatures Superconducting gap: need to examine density of states