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

†

UFRJ

2009?in oldnewccRio-Niterói bridge

Sugar Loaf

Outline• Motivation• The model: disordered

attractive Hubbard model• Quantum Monte Carlo• Ground state properties• Finite-temperature properties• Conclusions

Motivation: disorder on atomic scales Sputtered amorphous films

Low coverage: isolated incoherent islands

High coverage: islands “percolate”

film thickness tracks disorder

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

CR

ITIC

AL

TE

MP

ER

AT

UR

E T

c (ke

lvin

) SHEET RESISTANCE AT T = 300K (ohms)

Mo77Ge23 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

ℓℓ

ttAR

independent of the size of

square

Disorder is expected to inhibit superconductivity

Issues: Why does Tc 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?

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/4e2 = 6.45 k

Quantum Critical Point

Bismuth

(evaporation without a-Ge underlayer: granular disorder on mesoscopic scales.1)

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, Tc vs. R□

Generic phase diagram for Quantum Critical Phenomena:

T

disorder (mag field, pressure, etc.)

SUC

METAL

INS

QCP

Tc

T

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)

ieTT |)(|)( kk 2-component

( T

)(0

)

T/Tc

s-wave isotropic

TTC

CTTC

specific heat

susceptibility CTT

and so forth

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 Tc to zero (Mermin-Wagner thm)

magnets, superconductors, superfluids, fluids, etc., share common main features!

2-component order parameter in 2 space dimensions: M-W thm forbids long range order at T > 0 but phase with quasi--long-range order

possible below TKT: the Kosterlitz-Thouless transition

B. Berche et al. Eur. Phys. J. B 36, 91 (2003)

XY 2D

Stinchcombe JPC (1979)

Tc(

p)/T

c(1)

p

Heisenberg 3D

Yeomans & Stinchcombe JPC (1979)

Ising 2D

Dilute magnets: fraction p of sites occupied by magnetic atoms:Tc 0 at pc, the percolation concentration (geometry)

†( . ) ( )i j i i i i iìj i i

H t c c h c U n n n n

The homogeneous attractive Hubbard model

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

k BT

C/t

< n >[Paiva, dS, et al. (04)]

Homogeneous case

◊ particle-hole symmetry at half filling◊ strong-coupling in 2D:

• half filling: XY (SUP) + ZZ (CDW) Tc 0• away from half filling: XY (SUP) TKT 0

sites of 1fraction aon

sites of fraction aon 0

fU

fU i

Disordered case

particle-hole symmetry is broken

Heuristic arguments [Litak + Gyorffy, PRB (2000)] : fc as U

†( . ) ( )i j i i i i iìj i i

H t c c h c U n n n n

The disordered attractive Hubbard model

c 1- f

mean-field approx’n

Quantum Monte CarloCalculations carried out on a [square + imaginary time] lattice:

x

Ns

M

1M

T

Absence of the “minus-sign problem” in the attractive case

non-local updates: MNs2 Green’s functions

0 5 10 15 20 25

0

1

2

3

4

5

6

7

8

8X8 10X10 12X12 14X14 16X16

U=3 f=1/16

Ps

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,

iii

rrii ccPs with ,

averaged over 50 disorder configurations.

n =1

Ground State Properties

0.00 0.05 0.10 0.150.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

f= 0 f= 1/16 f= 2/16 f= 3/16 f= 4/16 f= 5/16

U=4P S

/L2

1/L

Spin-wave–like theory (two-component order parameter) Huse PRB (88):

zero-temperature gap

2

2sP C

L L

n = 1

F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )

0.0 0.1 0.2 0.3 0.40.0

0.5

1.0

1.5

2.0

U=2 U=2.5 U=3 U=4 U=6

f

We estimate fc as the concentration for which 0;

can plot fc (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 )

0 1 2 3 4 5 60.0

0.1

0.2

0.3

0.4

f c

U

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)

n = 1

F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )

Results for n = 0.875

GJ Farias, MSc thesis, UFMT, (2009)

GJ Farias, MSc thesis, UFMT, (2009)

n = 0.875

GJ Farias, MSc thesis, UFMT, (2009)

n = 0.875

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

GJ Farias, MSc thesis, UFMT, (2009)

?

fc with n for fixed U(at least for

U 4):less

electrons to

propagate attraction

Finite-temperature propertiesFinite-size scaling for Kosterlitz-Thouless transitions

KTusual

line of critical points ( = ∞)

Barber, D&L (83)

Lg

LL )(

c

L1/1 L2/2L1/1 L2/2

KT

2

0

2 1~

with ,

Lr

rd

ccP

L

s iiir

rii

Finite-size scaling at T > 0: KT transition

For infinite-sized systems one expects

KT

KT

TTTT

A,exp~ 21

LfLLPs 2),(

0 2 4 6 8 10 12 14 16 18 200.00

0.01

0.020.03

0.04

0.05

0.06

0.07

0.080.09

0.10

0.11

8 10 12 14

U = 4 f = 2/16

Ps/L

2-

Kosterlitz-Thouless transition: Curves should cross/merge at βc for η(Tc)=1/4:

Repeat this for other values of f...

0,0 0,2 0,4 0,6 0,8 1,00,00

0,05

0,10

0,15

0,20

0,25

TC

< n > f

Pure case at half filling:

0.0 0.1 0.2 0.3 0.40.00

0.05

0.10

0.15

0.20

0.25

Tc

f

n = 1

F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )

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 TKT-, s has universal jump-discontinuity [D.R. Nelson and J.M.

Kosterlitz, PRL 39, 1201 (1977)]: KTs T

2

determined is ,2

when

)(plot

KTs

s

TT

T

0.00

0.05

0.10

0.15

0.20

0.00

0.05

0.10

0.15

0.20

0.00

0.05

0.10

0.15

0.20

0.0 0.1 0.20.00

0.05

0.10

0.15

0.20

L=12 U = 4(a) f =1/16

(b) f =2/16

s (c) f =3/16

(d) f =4/16

T

n = 1 2T/s

F Mondaini, et al. PRB 78, 174519 (2008 )

0.0 0.1 0.2 0.3 0.40.00

0.05

0.10

0.15

0.20

0.25

Tc

f

n = 1

F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )

0.0 0.1 0.2 0.3 0.40.00

0.05

0.10

0.15

0.20

0.25T

c

f

U = 3 Ps

U = 4 Ps

U = 4 s

U = 6 Ps

n = 1

F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )

Results for n = 0.875

GJ Farias, MSc thesis, UFMT, (2009)

GJ Farias, MSc thesis, UFMT, (2009)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 ?

ConclusionsAt half filling, small amount of disorder seems to initially favour SUP in the ground state; not for other fillings.

fc depends on U transition at T = 0 not solely geometrically driven; quantum effects; correlated percolation?

for given U, fc 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 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.

n=1: A small amount of disorder allows the system to become SUP at finite temperatures; as disorder increases, Tc eventually goes to zero at fc.

n <1: Tc decreases steadily with f. Concavity???