Magnetoresistance of granular superconductors at low temperatures

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Nov 15, 2013 (4 years and 1 month ago)

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Igor Beloborodov
Magnetoresistance
of granular
superconductors at
low temperatures
Rev Mod Phys 79, 469 (2007)
Rev Mod Phys 79, 469 (2007)
Collaboration:
K. Efetov, A.I. Larkin,
A. Lopatin, Ya. Fominov, V. Vinokur
Materials Science Division, Argonne
University of Chicago
2
Motivation:
Motivation:
New materials paradigm:
Artificial Nanosolids
metals, semiconductors, superconductors,
ferromagnets, hybrid nanostructures
Fundamental physical parameters for
Fundamental physical parameters for
nanosolids
nanosolids
i) Nanograinsizes –electron confinement
ii) Coupling between grains –electron tunneling
iii) Electron interaction –Coulomb blockade
iv) Dimensionality –multiple tunneling pathways
3
Properties of a single grain
Properties of a single grain
-
single grain conductance
single grain conductance
0T
gE
δ
=
0.1meV
δ

10
T
E
meV≈
15 nm Al grain:
δ
mean energy
level spacing
confinement
2
1
T
R
ED
τ
==
diffusive
disorder
in grain
ballistic
imperfect
shape
1
TF
R
Ev
τ
==
R
4
Single superconducting grain
Single superconducting grain
δ
Δ>
condition for
-superconductivity
in a single grain
Δ
-superconducting energy gap
~
δ
Δ
-
big fluctuations of destroy
superconductivity
Δ
Anderson’59
Focus on grain sizes > 7 nm
5
Nanosolids
Nanosolids
characterized by two
characterized by two
conductances
conductances
0
g
-
grain conductance
T
g
-
tunneling conductance
0T
gg

0T
g
g>
-
-
granular
granular
-homogeneous
0
g
T
g
6
Array of superconducting grains
Array of superconducting grains
Assumptions:
,,CT
tEE
δ
<
Δ<
()R
ξ
<
good metal
small grains
R
-single grain size
ξ
-coherence length for
corresponding bulk sample
7
lll
BCST
H
HH=+
l
l
0
,,,','
,,'
||
BCS
ikikikik
ikk
HHaaaa
λ
++
−−
=−

Model Hamiltonian
Model Hamiltonian
e-e attraction
free e + disorder
tunneling
l
..
T
ijij
ij
H
taacc
+
=+

8
or
CC
Z
C
BR
BR

Effect of magnetic field on superconductivity
Effect of magnetic field on superconductivity
Zeeman
Zeeman
Δ
0B
=
B
Z
BC
μ
Β
≈Δ
Orbital
Orbital
B
2
0
or
C
B
ξ
≈Φ
bulk -
0
or
C
BR
ξ
≈Φ
grain -
0
C
R
D
Φ
=
Δ
-critical sample size
diffusion coefficient
C
RR>
C
-orbital
R
R
<
-Zeeman
Beloborodov et al., PRB 61, 9145 (2000)
9
Magnetoresistance
Magnetoresistance
of granular superconductors
of granular superconductors
Gerber et al’97
-grain size 120 A
-B up to 17 T
1.58
C
TK

(B = 0)
Experiment
Experiment
B
10
Magnetoresistance
Magnetoresistance
of granular superconductors
of granular superconductors
123
σ
σσσ
=
++
single electron
tunneling
virtual Cooper
pair tunneling
1
()()
ij
σ
νενε

ε
F
ε
ν
no interaction
ε
F
ε
ν
disorder & interaction
suppression of conductivity
2
2,3
T
σ

C
TT
Cooper pairs localized
no current contribution
at
interference
11
Magnetoresistance
Magnetoresistance
of granular superconductors
of granular superconductors
1)
Superconducting fluctuations at low
temperaturesand high magnetic fields
lead to density of states suppression
2) All Cooper pairs are localized
conductivity reduction
B
R
T << TC
Beloborodov,
Efetov,
Larkin
PRB 61, 9145 (2000)
12
Insulating state of granular superconductors
Insulating state of granular superconductors
SUPERCONDUCTOR
EXPERIMENT:
Gantmakher et al.’00
Sambandamurthyat al.’04
Baturinaet al.’04
Steiner et al.’05
Paalanenet al.’92
B
R
METAL
INSULATOR
~ (3 –5) T
perturbation
theory
13
Insulating state of granular superconductors
Gantmakher, et al ‘2000
G. Sambandamurthy et el ’2004
In-O, perpendicular field
perpendicularparallel
Experiment:
Experiment: stronggrains coupling
14
Superconductor
Superconductor


Insulator transition
Insulator transition
in granular metals
in granular metals
Efetov’80
E ~ E -SI transition
c
J
experiment:
g > 1,E E~ Δ/g
c
c
eff
E ~ g Δ >> E
J
c
eff
superconducting state
Insulating state possible for E > E
c
J
We need different model !
15
Insulating state of granular superconductors
Insulating state of granular superconductors
B
no magnetic fieldapplied magnetic field
grains of slightly different sizes
magnetic field

change relative fraction of
superconducting and normal grains
In 2D exist concentrations of sites where
simultaneously neither black no white
sites percolate
16
Insulating state: theoretical description
Insulating state: theoretical description
NS
N
S
S
N
S
N
N
SN
S
SN
S
N
g >> g , g
nsnn
ss
Due to magnetic field :
i
cns
SSS=+
for g >> 1
ns
22
~()
|
()
|
cq
SddqEq
τ
τ
Φ
∫∫
i
11
()([1]),
ccq
EqEBE
−−
=+−
0
~/,Bg
Δ
1
cos
2
q
a
Eqa=

17
B
Insulating state of granular superconductors
Insulating state of granular superconductors
applied field
gapin electron spectrum
()
(
)
00
ln
C
ggEΔΔΔ∼
0
Δ
-superconducting gap at B=0
g
-tunneling conductance,
C
E
-charging energy
Conductivity :
(
)
expT
σ
−Δ∼
What is the applicability of this result ?
R
B
Metal
Insulator
superconductor
Beloborodov at el., Phys Rev B 74, 014502 (2006)
18
Stability of insulating state
Stability of insulating state
Insulating state is stable for :
Insulating state is stable for :
()
1
3
0
g
δ

Electron tunneling via virtual state
Cooper pair tunneling
with respect to
formation
of normal
state
with respect to
formation of
superconducting
state
N
1
N
2
S
S
1
N
S
2
N
1
N
2
S
SN
S1
S
2
N
small for
0
g
δ

small for
(
)
13
0
g
δ

19
B
Summary
Summary
magnetic field
(
)
(
)
00
ln
C
ggE
Δ
ΔΔ∼
(
)
expT
σ
−Δ∼
B
R
METAL
INSULATOR
superconductor
stable for :
()
1
3
0
g
δ

perturbation
theory