B.Spivak
with A. Zuyzin
Quantum (T=0) superconductor-metal? (insulator?) transitions.
Quantum superconductor-metal (insulator) transitions take place at T=0 as functions of external parameters such as the strength of disorder, the magnetic field and the film thickness.
Examples of experimental data
Rc = 4 h/e
Insulator
Superconductor
Bi layer on amorphous Ge.Disorder is varied by changing film thickness.
Experiments suggesting existence of quantumsuperconductor-insulator transition
Ga layer grown on amorphous Ge
Insulator
Superconductor
Experiments suggesting existence of quantumsuperconductor-metal transition
A metal?
Superconductor – glass (?) transition in a film in a parallel magnetic field
There are long time (hours) relaxation processes reflected in the time dependence of the resistance.
H||
The mean field theory:The phase transition is of first order.
W. Wo, P. Adams, 1995
T=0 superconductor-metal transition in a perpendicular magnetic field
There are conductors whose T=0 conductance is four order of magnitude larger than the Drude value.
N. Masson,A. Kapitulnik
The magneto-resistance of quasi-one-dimensional superconducting wires
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)
P. Xiong, A.V.Hertog, R.Dynes
A model:Superconducting grains embedded into a normal metal
S N
Theoretically one can distinguish two cases:
R > and R <
is superconducting coherence length at T=0
R
A mean field approach
)2(|'|
1)',(
||)'()',(')()(
2
2
d
rrrrK
arrrKrdrr
The parameters in the problem are: grain radius R, grain concentration N ,interaction constants S >0.
If R< there are no superconducting solutions in anindividual grain.In this case a system of grains has a quantum (T=0) superconductor-metal transition even in the framework of mean field theory!
2
1
RNc
At =S the critical concentration of superconducting grains is of order
*)()(
)(
ji
ji iji i JdSS
An alternative approach to the same problem in the same limit R <
Si is the action for an individual grain “I”.
2N
2
2
)ln||21(
1
R
r r
RJ
ijij
ij
SS Jij
Perturbation on the metallic side of the transition R <
)()0()(
,1)(
* ttPwhere
tPdtJi
ij
The metallic state is stable with respectto superconductivity if
i ijijij
ij J
R
r r
RJ ;
)ln||21(
1
2N
2
2
F
c
vR
RR
diS
1
000
20
,1
;|)(|)1
||(
Quantum fluctuations of the order parameterin an individual grain (T=0, R< )
The action describing the Cooper instability of a grain
At R< Rc the metallic state is stable
t
RR
CC
C
ettP
tttP
iP
)()0()(
1)()0()(
)1
||(
1)(
*
2*
0
)(tPtd
R<
)(tPdtJi
ij
At T=0 in between the superconducting and insulating phases there is a metallic phase.
Properties of the exotic metal near the quantum metal-superconductor transition
a. Near the transition at T=0 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 ?
Are such “exotic” metals localized in 2D?
There is a new length associated with the Andreev scattering of electrons on the fluctuations of
)|(|;1
||
10
2022
20
2 cccc
RRRNRNR
L
This length is non-perturbative in the electron interaction constant
S
The case of large grains R>>: there is a superconducting solution in an individual grainFluctuations of the amplitude of the order parametercan be neglected.
Caldeira, Legget, Ambegaokar, Eckern, Schon
;)'(
)]'()([41
sin'
8)cos(
2
22
2
2
ijiij ij ddGd
d
e
CJ
dtS
Does this action exhibit a superconductor-metal or superconductor-insulator transition ?
G>1 is the conductance of the film.
)exp(|| jjj i
Gt /1/1
Get *
2/1 t
t
1)(,1 GetPtdG
Correlation function of the order parameterof an individual grain
))0()(()( tietP
Kosterlitz 1976
h
eofunitsinmeasuredisG
ddGS
2
2
2
)'(
)]'()([41
sin'
)(tPdtJi
ij
At T=0 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.
A superconducting film in a perpendicular magnetic field (T=0)
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.
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
Theoretically in both cases (R> and R< ) the regions of quantum fluctuations are too narrow to explain experiments of the Kapitulnik group.
Is it a superconductor-glass transition in a parallel magnetic field?
W. Wo, P. Adams, 1995
||H
Hc
In the pure case the mean field T=0 transition as a function of H|| is of first order.
cH
In 2D arbitrarily disorder destroys the first order phase transition. (Imry, Wortis).
)2())((2 2/12/12 ARRARES
LLAEALxxLAL
xE SS 3/13/43/22/1
2
)/(,)(
3/13/4 A
.0ln LwhenandL
A naive approach (H=Hc).R
b. More sophisticated drawing:Lx
is independent of the scale LThis actually means that
The surface energy vanishes and the first order phase transitionis destroyed!
Phase 2
Phase 1
The critical current and the energy of SNS junctions with spin polarizedelectrons changes its sign as a function of its thickness L
2/1
)(
cos)exp(;sin
FFFI
IIcc
kkk
DmL
L
L
L
LIII
Fk
Fk
Bulaevski, Buzdin
Does the hysteresis in a perpendicular magnetic field indicate glassiness of the problem in a perpendicularmagnetic field?
N.Masson, A. Kapitulnik
Why is the glassy nature more pronouncedin 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 fieldand first order in parallel field?
Negative magneto-resistance in 1D wires
P. Xiong, A.V.Hertog, R.Dynes
Negative magneto-resistance in quasi-1D superconducting wires.
A model of tunneling junctions:
cccc
c Ih
eEIF
T
EdI );()exp(
At small T the resistance of the wire is determined by the rare segments with small values of Ic .
.,,|)exp(| 02
0
quantafluxtheisHSiBAIc
The probability amplitudes of tunneling paths A and B are random quantities uniformly distributed, say over an interval (-1,1).
A
B
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 G>> 1.
A question:
P(G)
GGD1
e2/h0H
Do we know what is the resistance of an SNSjunction ?
s sN
The mini-gap in the N-region is of order )~D/L2
GGVdtdn
dt
dS 222
)/()(
is the energy relaxation time
Conclusion:
In slightly disordered films there are quantum superconductor-metal transitions. Properties of the metallic phase are affected by the superconducting fluctuations
Superconductor-disordered ferromagnetic metal-superconductor junctions.
))'()(cos()',(' rrrrJdrdrE RL
The quantity J(r,r’) exhibits Friedel oscillations. At L>>LI only mesoscopic fluctuations survive.
Here I is the spin splitting energy in the ferromagnetic metal.
RL0
ijJ
S F S
r r’Mean field theory (A.Larkin, Yu.Ovchinnikov, L.Bulaevskii, Buzdin).
;)/();exp(........sin 2/1IDLL
LJ I
Ic
LIDrrrrJ c /1)2(;|'|/1)',( 232
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.
Ic
T
SFS junctions with large area2))'()(cos()',(' ssRL VdrNrrrrJdrdrE
The situation is similar to the case of FNF layers, which has been considered by Slonchevskii.
The ground state of the system corresponds to 2/ RL
)(r
S F Sr r’
**4321
2 ),(),(;||)',( lkjijkl
ii i AAAArrnrrnArrn
The amplitudes of probabilities Ai have random signs. Therefore, after averaging of <AAAA> only blocks <Ai
2>~1/R survive.
l
R
r1r2
r3
r4
A qualitative picture.
Ai
)',()',()',(
..)'()',(')(
rrGrrGdrrK
termslnrrrKdrr
The kernel is expressed in terms of the normal metal Green functions and therefore it exhibits the Friedel oscillations.
)' , (r r K) (r G
62
3
|'|/1
,|'|/)|'|
exp(|,'|/)|'|
exp(........
rrK
rrL
rrKrr
l
rrG
H
Friedel oscillations in superconductors (mean field level):