1. 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

2. Outline
Motivation
The Model
Quantum Monte Carlo
Results
Conclusions

3. 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

4. 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)

5. 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)

6. 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.

7. 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)

8. 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)

9. 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)

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

11. 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)

12. 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.

13. 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

14. 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

15. 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