Superconductivity: modelling impurities and coexistence with

magnetic orderCollaborators: Pedro R Bertussi (UFRJ) André L Malvezzi (UNESP/Bauru) F. Mondaini (UFRJ) Richard T.Scalettar (UC-Davis) Thereza Paiva (UFRJ)

Financial support:

Brazil-India Workshop on Theoretical Condensed Matter PhysicsBrazilian Academy of Sciences, April 2008

Raimundo R dos Santos

Layout:A) Disordered Superconductors

1. Motivation2. The disordered attractive Hubbard model3. Quantum Monte Carlo4. Ground state properties5. Finite-temperature properties6. Conclusions

B) Coexistence of Superconductivity and Magnetism1. Motivation2. Model3. DMRG4. Results5. Conclusions

C) Overall Conclusions

Disordered superconducting films

F Mondaini et al.

Sheet resistance:

R at a fixed temperature can be used as a measure of disorder

Disorder on atomic scales: Sputtered amorphous films

CR

ITIC

AL

TE

MP

ER

AT

UR

E T

c (ke

lvin

)

Mo77Ge23 film

J Graybeal and M Beasley, PRB 29, 4167 (1984)

t

ℓℓ

ttAR

independent of the size of

square SHEET RESISTANCE AT T = 300K (ohms)

Disorder is expected to inhibit superconductivity

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 in the absence of interactions (recent expts: MIT possible?)

A M Goldman and N Marković, Phys. Today, Page 39, Nov 1998

Issues

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)

SHE

ET

RE

SIST

AN

CE

R (

ohm

s)

TEMPERATURE (K)

Behaviour near QCP will not be discussed here

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

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 disordered 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

TC

< 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 Carlo

Calculations carried out on a [square + imaginary time] lattice:

x

Ns

M

1M

T

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

0 5 10 15 200

1

2

3

4

5

6

7

8

9

10

8 10 12 14

U=3 f=1/16

Ps

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.B.: half filling from now on

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/L

2

1/L

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

2

2sP C

L L

zero-temperature gap

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

~~

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)

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

21exp~KTTT

A

LfLLPs 2),(

0 2 4 6 8 10 12 14 16 18 200.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

8 10 12 14

U =3 f=2/16

Ps/L

2-

0.0 0.1 0.2 0.3 0.40.00

0.05

0.10

0.15

0.20

0.25

0.30

U=3 U=4 U=6

Tc

f

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

Conclusions (half-filled band)

A small amount of disorder seems to initially favour SUP in the ground state.

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

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.

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

[Lynn et al., (1997)]

Er TmTb

• Rare earth 4f electrons order (AF) magnetically• Conduction electrons form Cooper pairs

R = Pr, Dy, Ho

Model

• Electronic correlations Attractive Hubbard Model

• Exchange interaction between conduction electrons and local moments Kondo term

H †

,

( . .) ( )i j i ii j i

t C C H c U n n

i

iiJ sS

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:

• 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

System S

EnvironmentE

Superblock

Density Matrix Renormalization Group:

• 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)

System S

EnvironmentE

Superblock

S E

S’ E’

Results

Electron-spinlocalized-spin correlations

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

J / U

< S

· s

>

S · s (U = 8t )

- Non-exhausted singlet states (Kondo) above (J/U)c

Electron spin-spin correlations

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 3 6 9 12 15

| i - j |

< s

z (

i )

· s

z (

j )

>

J/U = 0.20J/U = 0.55J/U = 0.60J/U = 0.80

sz ( i ) · sz ( j ) (U = 8t)

- Rapidly decaying correlations: electrons on different sites are not magnetically ordered

Localized spin-spin correlations

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 3 6 9 12 15

| i - j |

< S

x (

i )

· Sx (

j )

>

Sz ( i ) · Sz ( j ) (U = 8t)

- SDW correlations for small J/U - FM for large J/U

Sx ( i ) · Sx ( j ) (U = 8t)

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 3 6 9 12 15

| i - j |

< S

z (

i ) ·

Sz

( j )

>

J/U = 0.20J/U = 0.55J/U = 0.60J/U = 0.80

0

0.5

1

1.5

2

2.5

3

3.5

4

0 5 10 15 20

k (2 π / Ns)

S (

k )

J/U = 0.20J/U = 0.55J/U = 0.60J/U = 0.80

Localized spin-spin correlations structure factor

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20

k (2 π / Ns)

S (

k )

J/U = 0.20J/U = 0.55J/U = 0.60J/U = 0.80

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20

k (2 π / Ns)

S (

k )

J/U = 0.05J/U = 0.35J/U = 0.40J/U = 0.45J/U = 0.50J/U = 0.55J/U = 0.65

(U = 8t)

- maximum at k = 0 indicates FM and at k = π, SDW

(U = 6t)

- maximum at intermediate k ISDW, incommensurate with lattice spacing

- Gradual transition from maximum at k = π to k = 0

(U = 4t)

Comparison: S(k) peaksU = -8 t

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

| J / U |

S(k

) p

eak

/ π

24 sites30 sites

U = -6 t

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

| J / U |

S(k

) p

eak

/ π

24 sites30 sites

U = -4 t

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

| J / U |

S(k

) p

eak

/ π

24 sites30 sites

U = -2 t

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

| J / U |

S(k

) p

eak

/ π

24 sites30 sites

No significant finite-size effects

Pairing correlations

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 3 6 9 12 15

| i - j |

< P

ζ (

i , j

) > J/U = 0.20

J/U = 0.55J/U = 0.60J/U = 0.80

..)()(),( cHjijiP sss

- Superconductivity possible only below (J/U)c

(U = 8t)

iis cci)(

Comparison: Ps(r)

10 20 30 40 50 60

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

PS(r

)

r

60sitios 30sitios 24sitios

Ps fit

0.1 1 10

1E-3

0.01

0.1

= 0.98511 ± 0.0228

PS(r

)

r

Ps ~ 1 / rβ

Phase Diagram

0.0 0.2 0.4 0.6 0.80

2

4

6

8

U

/ t

| J / U |

FMSDW + SC

ISDW + SC ISDW

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)

Collaborators: Antônio José Roque da Silva (IFUSP) Adalberto Fazzio (IFUSP) Luiz Eduardo Oliveira (IFGW/UNICAMP) Tatiana G Rappoport (IF/UFRJ)

Materials for Spintronics: Diluted Magnetic Semiconductors