superconductivity: modelling impurities and coexistence with magnetic order collaborators: pedro r...
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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
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