B'Spivak

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Quantum (T0) superconductor-metal?
(insulator?) transitions.

• B.Spivak
• with A. Zuyzin

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Quantum superconductor-metal (insulator)
transitions take place at T0 as functions of
external parameters such as the strength of
disorder, the magnetic field and the film
thickness.
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Examples of experimental data
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Experiments suggesting existence of
quantum superconductor-insulator transition
Insulator
Rc 4 h/e
Bi layer on amorphous Ge. Disorder is varied by
changing film thickness.
Superconductor
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Experiments suggesting existence of
quantum superconductor-metal transition
Insulator

Ga layer grown on amorphous Ge
A metal?
Superconductor
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Superconductor glass (?) transition in a film
in a parallel magnetic field
W. Wo, P. Adams, 1995
H
The mean field theory The phase transition is of
first order.
There are long time (hours) relaxation processes
reflected in the time dependence of the
resistance.
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T0 superconductor-metal transition in a
perpendicular magnetic field
N. Masson, A. Kapitulnik
There are conductors whose T0 conductance is
four order of magnitude larger than the Drude
value.
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The magneto-resistance of quasi-one-dimensional
superconducting wires
P. Xiong, A.V.Hertog, R.Dynes
The resistance of superconducting wires is much
smaller than the Drude value The
magneto-resistance is negative and giant (more
than a factor of 10)
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A model Superconducting
grains embedded into a normal metal
S
N
R

• Theoretically one can distinguish two cases
• R gt x and R lt x
• x is superconducting coherence length at T0

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A mean field approach
The parameters in the problem are grain radius
R, grain concentration
N , interaction constants lN lt0, lS gt0.

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If Rlt x there are no superconducting solutions
in an individual grain. In this case a system of
grains has a quantum (T0) superconductor-metal
transition even in the framework of mean field
theory!
At lN lS the critical concentration of
superconducting grains is of order

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An alternative approach to the same problem in
the same limit R lt x
S
Jij
S
Si is the action for an individual grain I.
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Perturbation on the metallic side of the
transition R lt x
The metallic state is stable with respect to
superconductivity if
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Quantum fluctuations of the order parameter in an
individual grain (T0, Rlt )
The action describing the Cooper instability of
a grain
At Rlt Rc the metallic state is stable
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Rltx
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At T0 in between the superconducting and
insulating phases there is a metallic phase.
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Properties of the exotic metal near the quantum
metal-superconductor transition

• Near the transition at T0 the conductivity of
  the metal is enhanced.
• b. The Hall coefficient is suppressed.
• c. The magnetic susceptibility is enhanced.
• d. There is a large positive magneto-resistance.

In which sense such a metal is a Fermi
liquid? For example, what is the size of
quasi-particles ? Is electron focusing at work
in such metals ?
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Are such exotic metals localized in 2D?
There is a new length associated with the Andreev
scattering of electrons on the fluctuations of D
S
This length is non-perturbative in the electron
interaction constant
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The case of large grains Rgtgtx there is a
superconducting solution in an individual
grain Fluctuations of the amplitude of the order
parameter can be neglected.
Ggt1 is the conductance of the film.
Caldeira, Legget, Ambegaokar, Eckern, Schon
Does this action exhibit a superconductor-metal
or superconductor-insulator transition ?
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Correlation function of the order parameter of an
individual grain
Kosterlitz 1976
t
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At T0 in between the superconducting and
insulating phases there is a metallic phase.
However, in the framework of this model the
interval of parameters where it exists is
exponentially narrow.
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A superconducting film in a perpendicular
magnetic field (T0)
Mean field solution mean field treatment of
disorder.
Mean field solution without inappropriate
averaging over disorder.
Solution taking into account both mesoscopic and
quantum fluctuations.
H
a. The mean field superconducting solution exists
at arbitrary magnetic filed. b. Classical and
quantum fluctuations destroy this solution and
determine the critical magnetic field.
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Theoretically in both cases (Rgt x and Rlt x )
the regions of quantum fluctuations are too
narrow to explain experiments of the Kapitulnik
group.
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Is it a superconductor-glass transition in a
parallel magnetic field?
W. Wo, P. Adams, 1995
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In the pure case the mean field T0 transition
as a function of H is of first order.
Hc
In 2D arbitrarily disorder destroys the first
order phase transition. (Imry, Wortis).
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Phase 1
A naive approach (HHc).
R
Phase 2
L
b. More sophisticated drawing
x
is independent of the scale L This
actually means that
The surface energy vanishes and the first order
phase transition is destroyed!
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The critical current and the energy of SNS
junctions with spin polarized electrons changes
its sign as a function of its thickness L
Bulaevski, Buzdin
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Does the hysteresis in a perpendicular magnetic
field indicate glassiness of the problem in a
perpendicular magnetic field?
N.Masson, A. Kapitulnik
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Why is the glassy nature more pronounced in the
case of parallel magnetic field? Is this
connected with the fact that in the pure case
the transition is second order in perpendicular
magnetic field and first order in parallel field?
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Negative magneto-resistance in 1D wires
P. Xiong, A.V.Hertog, R.Dynes
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Negative magneto-resistance in quasi-1D
superconducting wires.
A model of tunneling junctions
A
B
At small T the resistance of the wire is
determined by the rare segments with small
values of Ic .
The probability amplitudes of tunneling paths A
and B are random quantities uniformly
distributed, say over an interval (-1,1).
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An explanation of experiments The probability
for Ic to be small is suppressed by the magnetic
field. Consequently, the system exhibits giant
magneto-resistance. A problem in the experiment
Ggtgt 1. A question
P(G)
e2/h
G
1
GD
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Do we know what is the resistance of an
SNS junction ?
s
s
N
The mini-gap in the N-region is of order d(f)D/L2
te is the energy relaxation time
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Conclusion
In slightly disordered films there are quantum
superconductor-metal transitions. Properties of
the metallic phase are affected by the
superconducting fluctuations
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Superconductor-disordered ferromagnetic
metal-superconductor junctions.
S
F
S
r
r
Mean field theory (A.Larkin, Yu.Ovchinnikov,
L.Bulaevskii, Buzdin).
Here I is the spin splitting energy in the
ferromagnetic metal.
The quantity J(r,r) exhibits Friedel
oscillations. At LgtgtLI only mesoscopic
fluctuations survive.
Ic
In the case of junctions of small area the
ground state of the system corresponds to a
nonzero sample specific phase difference
. Ic is not exponentially small
and exhibits oscillations as a function of
temperature.
T
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SFS junctions with large area
S F S
r r
The situation is similar to the case of FNF
layers, which has been considered by
Slonchevskii.
The ground state of the system corresponds to

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A qualitative picture.
The amplitudes of probabilities Ai have random
signs. Therefore, after averaging of ltAAAAgt
only blocks ltAi2gt1/R survive.
r4
R
r2
l
r1
Ai
r3
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Friedel oscillations in superconductors (mean
field level)
The kernel is expressed in terms
of the normal metal Green functions
and therefore it exhibits the Friedel
oscillations.
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