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):