Quantum (T=0) superconductor-metal transitions in highly conducting films

B.SpivakUniversity of Washington

with S. Kivelson, and P. OretoStanford University

Quantum superconductor-metal (insulator) transitions take place at T=0 as functions of external parameters

Theoretical possibilities

a. Quantum S-wave superconductor-metaltransition in an external magnetic field

b. Quantum S-wave superconductor-metaltransition as a function of disorder

c. Quantum D-wave superconductor-metaltransition as a function of disorder

d. If the electron-electron interaction constanthas random sign the system may exhibitquantum superconductor-metal transition

Examples of experimental data

Experiments suggesting existence of quantumsuperconductor-insulator transition

Insulator

Superconductor

Rc = 4 h/e

Bi layer on amorphous Ge.Disorder is varied by changing film thickness.

Ga layer grown on amorphous Ge

Insulator

Superconductor

Experiments suggesting existence of quantumsuperconductor-metal transition

A metal?

T=0 superconductor-metal transition in a perpendicular magnetic field

N. Masson,A. Kapitulnik

There are conductors whose T=0 conductanceis four order of magnitude larger than the Drude value.

Superconductor – glass transition in amagnetic field parallel to the film

W. Wu, P. Adams,

∆

There are long time (hours) relaxation processes reflected in the time dependence of the resistance.

H||Hc

The mean field theory:The phase transition is of first order.

The magneto-resistance of quasi-one-dimensional superconducting wires

P. Xiong, A.V.Hertog, R.Dynes

The resistance of superconducting wires is muchsmaller than the Drude value The magneto-resistance is negative and giant (more than a factor of 10)

A model:Superconducting grains embedded into a normal metal

S N

.

var,

).(

)'(|)'()(|'||||)(

20

*

2

2

20

42

grainiofvolumetheisdLV

iablesrandomareandV

ccJ

dddS

thii

iiiii

jiij ij

iii

ii

ii

i

=

∆≈≈

+∆∆+

+⎭⎬⎫

⎩⎨⎧

−∆−∆

+⎥⎦

⎤⎢⎣

⎡∆∆

+∆−

=

∑

∑ ∫ ∫

αβναγ

ττττττβ

γγγτα

Effective action:

Does this action exhibit a superconductor-metalor superconductor-insulator transition ?

Correlation function of the order parameterof an individual grain

( ) 0>− icaseThe γγ

))(exp()()0()(

1)()0()(

))(||(1)(

0*

2*

0

iRR

CC

iC

tttPt

ttP

iP

γγ

γγωντω

−∆−∝>∆∆∆∆

Correlation function of the order parameterof an individual grain

Γ− )/(11

it γγ

)(* iet γγ −Γ∝

2/1 t

)(tPc

( )( )γγ

γγ

−∆=∆

Γ>−

ii

icaseThe20

2||

/1

Kosterlitz 1976

t

Superconducting susceptibility of an individual grain

∞

r- dependence of the Josephson coupling

),,(||

12 Trrfrr

J jiji

ij −∝

Both in D-wave case and in the case when superconductivity is suppressed by themagnetic field the exchange energyhas random sign (or phase)

2

2)(

)( Σ−

−∝

ii

eF iγγ

γ

Quantities ri and γi are randomly distributed

The distribution function of γi

Criterion for superconductivity:

1∝iijJ χThere is a superconductor-metal quantum phase

transition. It is of a quantum percolation type.

1

)exp(;40

22

40

2220

2

>>∆Σ

∆Σ∝∆Σ≈−

β

ββγγ Rc

The case of S-superconducting film in a perpendicular magnetic field

2

22222

2

2

2

2

)((int))((int)

220

*

2

2

22

,1;

)exp(

c

ccccc

c

c

c

c

H

clHH

clHHH

H

H

Hc

cc

Hi

LA

G

GR

GH

HH

LL

∝≈+=

≈

≈−

=

σσσσσ

σ

σξ

σ

is the variance of the critical magnetic field on the grains

G is the conductance of the film

A comparison with a theory of Larkin Feigelman and Skvortsov; and Larkin and Galitskii

4/111

)(2,

2 GG

ifvalidareresultsOur

SkvortsovFeigelmanLarkinDe

cH

G

The case of D-wave disordered superconductor

pathfreemeanelasticcriticaltheislGd

eR

Gll

DL

c

c

c

opti

>≈

Γ≈

−

∆∝

Γ

020

00*

2/10

/

νξ

ξξ

ξ

A mean field approach

)2(|'|

1)',(

||)'()',(')()(

2

2

=−

∝

∆∆+∆=∆ ∫d

rrrrK

arrrKrdrr

rr

rrrrrr λ

The parameters in the problem are: grain radius R, grain concentration N ,interaction constants λΝ 0.

If R< ξ 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!

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

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 Andreevscattering of electrons on the fluctuations of ∆

S

)|(|;1||

10

2022

20

2 cccc

RRRNRNR

L

Is it a superconductor-glass transition in a parallel magnetic field?

W. Wo, P. Adams, 1995

∆

||HHc

In the pure case the mean field T=0 transition as a functionof H|| is of first order.

cHδ

( )GH

H

c

c 12

2

∝δ

In the absence of the spin-orbit scattering spatial fluctuationsof the critical magnetic field is entirely due to interference effects

In 2D arbitrarily disorder destroys the first order phase transition. (Imry, Wortis).

)2())((2 2/12/12 ππσπσπ ARRARES −=−=

δσσδσσδ LLAEALxxLALxE SS −=−=⇒≈−=

3/13/43/22/12

)/(,)(

3/13/4 σδσ A=

.0ln ∞→→−∝ LwhenandL σδσ

Phase 2

Phase 1 A naive approach (H=Hc).R

Lx

b. More sophisticated drawing:

is independent of the scale LThis actually means that

The surface energy vanishes and the first order phase transitionis destroyed!

The critical current and the energy of SNS junctions with spin polarizedelectrons changes its sign as a function of its thickness L

Bulaevski, Buzdin↓Fk

↑Fk

2/1

)(

cos)exp(;sin

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

−∝=

↓↑ FFFI

IIcc

kkkDmL

LL

LLIII ϕ

Does the hysteresis in a perpendicular magnetic field indicate glassiness of the problem ?

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?

Do we know what is the resistance of an SNSjunction ?

ss N

The mini-gap in the N-region is of order δ(φ)~D/L2

εεε

ττϕτδ

∝⇒∝∝∝ GGVdtdndtdS 222 )/()(

τε is the energy relaxation time

Conclusion:

In slightly disordered films there arequantum superconductor-metaltransitions. Properties of the metallicphase are affected by the superconductingfluctuations

Superconductor-disordered ferromagnetic metal-superconductor junctions.

))'()(cos()',(' rrrrJdrdrE RL ϕϕ −= ∫

The quantity J(r,r’) exhibits Friedel oscillations. At L>>LI only mesoscopicfluctuations survive.

Here I is the spin splitting energy in the ferromagnetic metal.

πϕϕ ∝<

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 notexponentially small and exhibits oscillations as a function of temperature.

Ic

T

SFS junctions with large areaS F Sr r’

2))'()(cos()',(' ssRL VdrNrrrrJdrdrE ∫∫ +−= ϕϕThe situation is similar to the case of FNF layers, which has beenconsidered by Slonchevskii.

δϕϕϕ +>==−< RL

A qualitative picture.

>>=

Negative magneto-resistance in quasi-1D superconducting wires. A model of tunneling junctions:

ΦA

B

cccc

c IheEIF

TEdI =−∝ ∫ );()exp(ρ

At small T the resistance of the wire is determined by the rare segments withsmall values of Ic .

.,,|)exp(| 02

0

quantafluxtheisHSiBAIc Φ=ΦΦΦ

+∝

The probability amplitudes of tunneling paths A and B are random quantitiesuniformly distributed, say over an interval (-1,1).

Friedel oscillations in superconductors (mean field level):

)',()',()',(

..)'()',(')(

rrGrrGdrrK

termslnrrrKdrr

εεε

λ

−∫∫=

+∆=∆

The kernel is expressed in terms of the normal metal Green functions and therefore it exhibits the Friedel oscillations.

62

3

|'|/1

,|'|/)|'|exp(|,'|/)|'|exp(........

rrK

rrL

rrKrrl

rrGH

−>∝<

−−

−>∝∝

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)

G

e2/h0≠H

1 GD

Negative magneto-resistance in 1D wires

P. Xiong, A.V.Hertog, R.Dynes

The case of large grains:The effective action for a single tunnel junction• There is a nonzero mean field solution for the order para- meter

on a single grain: amplitude fluctuations can be neglected• The effective action for a single tunnel junction contains a

dissipative term (Ambegaokar, Eckern and Schön, 1982)

which reproduces the action of Caldeira and Leggett (1981) asa special case.

• The total action is

where is the charging energy.

[ ] ∫∫ −−

=ββ

ττ

τϑτϑττ

πτϑ

02

2

0 )'(

))'()((41sin

'2)( ddGSD

∫++=β

ττθ0

)( )(cos dESSS JDi

)(iS

The system of large superconducting grainsembedded in a normal metal

(Feigel’man and Larkin, 1998)

• The integral of the correlation function converges due to the fact that at largest times

• Notice:If the time during which the phase undergoes a variation of 2 π is finite, the asymptotic behaviorof the corresponding retarded correlation function is

0))0()(()( 〉〈≡ −θθ tietC

2/1)( ttC ∝

2

/2/))0()((

/1)(

2

ttCee tt

∝

∝〉〈 −−−

τθθ)(tθ

τ

The time during which the phase undergoes a variation of 2π, in the model of Feigelman and Larkin

• can be found estimating the limits of validity of linearized (in 1/G, G>>1) RG equations for the (inverse-square) X-Y model (Kosterlitz, 1976).

τ

2/1 t

Gt /1/1

τ≈ t

)(tC

)exp(0 0

))0()(( Gedt ti ≈≈〉〈∫∞ − τθθ

The large grain results of Feigel’manand Larkin

• The perturbation theory in the interaction between grains (taking account of Coulomb repulsion in the metal,

) breaks down for

(where )• As a result

– There is a quantum superconductor – metal transition;– Critical concentration is exponentially small in dimensionless film conductance G

]exp[~)( 10 0

))0()(( GedtJ ti −〉〈≈ −∞ −∫ θθ

||)ln21)(/( 222 jiijij

ijij rrrRr

rRJ rr −=+∝ − , || Nλν

)/ln(/ 2/1 RNNJ −∝

CN

An alternative approach to the same problem in the same limit R < ξ

SJijS*)()(

)(ji

ji iji iJdSS ωωω −≠ ∆∆+= ∑ ∫∑

Si is the action for an individual grain “I”.

2N

2

2

)ln||21(

1

Rr

rRJ

ijijij

λν+∝

Perturbation on the metallic side of the transition R < ξ

The metallic state is stable with respectto superconductivity if

>∆∆=<

Quantum fluctuations of the order parameterin an individual grain (T=0, R< )

The action describing the Cooper instability of a grain

F

c

vRRR

diS

ξτττ

ωωτ

ωντ

=∆=−

=

∆+−=

−

∫1

000

20

,1

;|)(|)1||(

At R< Rc the metallic state is stable

∞