Superconductivity: modelling impurities and coexistence with magnetic order

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Superconductivity modelling impurities and
coexistence with magnetic order
Brazil-India Workshop on Theoretical Condensed
Matter Physics Brazilian Academy of Sciences,
April 2008
Raimundo R dos Santos

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

Financial support
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• Layout
• A) Disordered Superconductors
• Motivation
• The disordered attractive Hubbard model
• Quantum Monte Carlo
• Ground state properties
• Finite-temperature properties
• Conclusions
• B) Coexistence of Superconductivity and Magnetism
• Motivation
• Model
• DMRG
• Results
• Conclusions
• C) Overall Conclusions

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Disordered superconducting films
F Mondaini et al.
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Disorder on atomic scales Sputtered amorphous
films
Sheet resistance R at a fixed
temperature can be used as a measure of disorder
Mo77Ge23 film
CRITICAL TEMPERATURE Tc (kelvin)
independent of the size of square
SHEET RESISTANCE AT T 300K (ohms)
Disorder is expected to inhibit superconductivity
J Graybeal and M Beasley, PRB 29, 4167 (1984)
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Issues
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 Markovic, Phys. Today, Page 39,
Nov 1998
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Metal evaporated on cold substrates, precoated
with a-Ge disorder on atomic scales.
Bismuth
Superconductor Insulator transition at T 0
when R reaches one quantum of resistance for
electron pairs, h/4e2 6.45 k?
SHEET RESISTANCE R (ohms)
?Quantum Critical Point
(evaporation without a-Ge underlayer granular ?
disorder on mesoscopic scales.1)
Behaviour near QCP will not be discussed here
D B Haviland et al., PRL 62, 2180 (1989)
TEMPERATURE (K)
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Dilute magnets fraction p of sites occupied by
magnetic atoms
Tc ? 0 at pc, the percolation concentration
(geometry)
Yeomans Stinchcombe JPC (1979)
Ising 2D
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The disordered attractive Hubbard model

• ? 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

Homogeneous case
Paiva, dS, et al. (04)
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The disordered attractive Hubbard model
? particle-hole symmetry is broken
Heuristic arguments Litak Gyorffy, PRB (2000)
fc ? as U?
Disordered case
mean- field approxn
c ? 1- f
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Quantum Monte Carlo

• Calculations carried out on a square imaginary
  time lattice

Absence of the minus-sign problem in the
attractive case
x
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For given temperature 1/?, concentration f,
on-site attraction U, system size L ? L etc, we
calculate the pairing structure factor,
averaged over 50 disorder configurations. N.B.
half filling from now on
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Ground State Properties
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We estimate fc as the concentration for which ? ?
0 can plot fc (U )...
normalized by the corresponding pure case
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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)
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Finite-temperature properties
Finite-size scaling for Kosterlitz-Thouless
transitions
L2/?2
L1/?1
KT
L2/?2
L1/?1
usual
?
?c
?
?KT
Barber, DL (83)
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For infinite-sized systems one expects
Finite-size scaling at T gt 0 KT transition
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Tc initially increases with disorder breakdown
of CDW-SUP degeneracy
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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 gt 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.

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Coexistence between superconductivity and
magnetic order
PR Bertussi et al.
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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
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Borocarbides
Canfield et al., (1998)
Coexistence of magnetic order and
superconductivity
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Borocarbides
Er
Tm
Tb
R Pr, Dy, Ho
Lynn et al., (1997)

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

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Model

• Electronic correlations ? Attractive Hubbard
  Model
• Exchange interaction between conduction
  electrons and local moments ? Kondo term

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Method

• DMRG ? approximate ground state
• Up to 60 sites
• Density n1/3
• 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

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Density Matrix Renormalization Group

• Obtain the ground state by using, for example,
  Lanczos

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Density Matrix Renormalization Group
Superblock
System S
Environment E

• 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

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Density Matrix Renormalization Group
Superblock
System S
Environment E
S
E
S
E

• 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)

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Results
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Electron-spin?localized-spin correlations
S s (U 8t )
- Non-exhausted singlet states (Kondo) above
(J/U)c
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Electron spin-spin correlations
sz ( i ) sz ( j ) (U 8t)
- Rapidly decaying correlations electrons on
different sites are not magnetically ordered
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Localized spin-spin correlations
Sz ( i ) Sz ( j ) (U 8t)
Sx ( i ) Sx ( j ) (U 8t)
- SDW correlations for small J/U
- FM for large J/U
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Localized spin-spin correlations structure factor
(U 8t)
(U 6t)
- maximum at k 0 indicates FM and at k p, SDW
(U 4t)
- maximum at intermediate k ? ISDW,
incommensurate with lattice spacing
- Gradual transition from maximum at k p to k
0
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Comparison S(k) peaks
? No significant finite-size effects
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Pairing correlations
(U 8t)
- Superconductivity possible only below (J/U)c
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Comparison Ps(r)
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Ps fit
? Ps 1 / rß
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Phase Diagram
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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

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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)
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Materials for Spintronics Diluted Magnetic
Semiconductors
Collaborators Antônio José Roque da Silva
(IFUSP) Adalberto Fazzio (IFUSP) Luiz
Eduardo Oliveira (IFGW/UNICAMP) Tatiana G
Rappoport (IF/UFRJ)
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