transition in an external magnetic field b. Quantum S-wave superconductor-metal
transition as a function of disorder c. Quantum D-wave superconductor-metal.
Quantum (T=0) superconductor-metal transitions in highly conducting films B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University
Quantum superconductor-metal (insulator) transitions take place at T=0 as functions of external parameters
Theoretical possibilities a. Quantum S-wave superconductor-metal transition in an external magnetic field b. Quantum S-wave superconductor-metal transition as a function of disorder c. Quantum D-wave superconductor-metal transition as a function of disorder d. If the electron-electron interaction constant has random sign the system may exhibit quantum superconductor-metal transition
Examples of experimental data
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
Experiments suggesting existence of quantum superconductor-metal transition Insulator
Ga layer grown on amorphous Ge
A metal? Superconductor
T=0 superconductor-metal transition in a perpendicular magnetic field
N. Masson, A. Kapitulnik
There are conductors whose T=0 conductance is four order of magnitude larger than the Drude value.
Superconductor – glass transition in a magnetic field parallel to the film W. Wu, P. Adams,
∆
H|| Hc 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.
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)
A model: Superconducting grains embedded into a normal metal S
N
Effective action:
⎧ ⎡ (γ − γ i ) | ∆ i |4 ⎤ | ∆ i (τ ) − ∆ i (τ ' ) |2 ⎫ 2 S = ∑i⎨αi ∫ dτ ⎢ | ∆ i | + 2 ⎥ + β i ∫ dτ dτ ' ⎬+ 2 ∆0 ⎦ (τ − τ ' ) ⎣ γ ⎩ ⎭ + ∑ij J ij ( ∆ i ∆*j + c.c )
γ i , αi ≈ νVi and
βi ≈
αi ∆0
are random var iables
Vi = L2i d is the volume of ith grain.
Does this action exhibit a superconductor-metal or superconductor-insulator transition ?
Correlation function of the order parameter of an individual grain
The case (γ − γ i ) > 0 Dynamics determined by the Cooper phenomenon
1 ⇒ PC (ω ) = ντ 0 ( −i | ω | + (γ − γ i ))
∫ dt P (t )
1 PC (t ) = < ∆(0) ∆ (t ) >C ∝ 2 t PR (t ) = < ∆ (0) ∆* (t ) > R ∝ exp( −t∆ 0 (γ − γ i )) *
c
1 / Γ
| ∆ i |2 = ∆20 (γ i − γ )
Pc (t )
1 t
1 /( γ −γ i ) Γ
1/ t 2 t ∝e *
Γ ( γ −γ i )
Kosterlitz 1976
t
∫ dt P (t ) c
Superconducting susceptibility of an individual grain
χ i ∝ exp[ Γi (γ − γ i )], Γi ≈ Viν∆ 0
> 1
R ∝ exp(Σ 2 β 2 ∆40 )
The case of S-superconducting film in a perpendicular magnetic field Li = LH c 2 H c2 − H c2 ≈ Gσ H2 c 2 H c2 R* ≈ ξ0 exp(σ H2 c 2 G 2 )
σH
c2
G
is the variance of the critical magnetic field on the grains is the conductance of the film
σ H = σ H(int) + σ H( cl ) ; c2
c2
c2
σ H(int) ≈ c2
A 1 , σ H( clc 2) ∝ G LH c 2
A comparison with a theory of Larkin Feigelman and Skvortsov; and Larkin and Galitskii
χ ∝e
G
, D = 2 ( Larkin Feigelman Skvortsov )
Our results are valid if 1 1 < σ H c 2 < 1/ 4 G G
The case of D-wave disordered superconductor Li ∝ D / ∆ opt
ξ 0 − lc lc
Γ1 / 2 ≈ G
R* ≈ ξ0 e Γ >> ξ0 Γ ≈ νξ02d∆ 0 < G lc is the critical elastic mean free path
A mean field approach r r r r r r ∆ ( r ) = λ ( r ) ∫ dr 'K ( r , r ' ) ∆ ( r ' ) + a | ∆ |2 ∆ r r K (r , r ' ) ∝
1 2 | r − r' |
( d = 2)
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 an individual 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 Andreev scattering of electrons on the fluctuations of ∆ S
1 ∆20 1 2 Lϕ = ≈ ∆ 0 ντ 0 ; 2 2 2 NRc NRc
(| R − Rc | ∝ 1 / | r − r ' |3 ; ( D = 2) → < I c2 >∝ 1 / L In the case of junctions of small area the ground state of the system corresponds to a nonzero sample specific phase difference 0 < ϕ L − ϕ R < π . Ic is not exponentially small and exhibits oscillations as a function of temperature.
Ic
T
SFS junctions with large area E = ∫ drdr ' J (r , r ' ) cos(ϕ L (r ) − ϕ R (r ' )) + N s ∫ drVs2
S F S r r’
The situation is similar to the case of FNF layers, which has been considered by Slonchevskii.
ϕ (r ) =< ϕ > +δϕ
The ground state of the system corresponds to < ϕ L − ϕ R >= π / 2
A qualitative picture.
n(r , r ' ) =| ∑i Ai |2 ; < n(r1 , r2 )n(r3 , r4 ) >=< ∑ Ai A*j Ak Al* > ijkl
The amplitudes of probabilities Ai have random signs. Therefore, after averaging of only blocks ~1/R survive.
r4 R
r1
r2
l
Ai
r3
Negative magneto-resistance in quasi-1D superconducting wires. A model of tunneling junctions:
Φ
A B
Ec ρ ∝ ∫ dI c exp( − ) F ( I c ); T
e Ec = I c h
At small T the resistance of the wire is determined by the rare segments with small values of Ic .
Φ 2 I c ∝| A + B exp(i ) | , Φ = HS , Φ 0 is the flux quanta. Φ0
The probability amplitudes of tunneling paths A and B are random quantities uniformly distributed, say over an interval (-1,1).
Friedel oscillations in superconductors (mean field level):
∆(r ) = λ ∫ dr ' K (r , r ' )∆(r ' ) + n.l. terms K (r , r ' ) = ∫ dεGε (r , r ' )G−ε (r , r ' ) The kernel is expressed in terms of the normal metal Green functions and therefore it exhibits the Friedel oscillations.
< G >∝ ........ exp(−
| r − r '| ) / | r − r ' |, l
< K 2 >∝ 1 / | r − r ' |6
< K >∝ exp(−
| r − r '| ) / | r − r ' |3 , LH
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)
e2/h H ≠0
G
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)
1 sin (ϑ (τ ) − ϑ(τ ' )) 2 4 SD [ϑ (τ )] = G∫ dτ ∫ dτ ' 2 ( τ − τ ' ) π which reproduces the action of 0 Caldeira and Leggett (1981) as 0 a special case. • The total action is β
S(i)
β
S = S + SD + ∫ EJ cosθ (τ )dτ 0 is the charging energy. (i )
where
2
β
The system of large superconducting grains embedded in a normal metal (Feigel’man and Larkin, 1998) i(θ (t )−θ (0)) C ( t ) ≡ 〈 e 〉0 • The integral of the correlation function converges due to the fact that at largest times C(t) ∝1/ t 2
• Notice: If the time τ during which the phase undergoes a variation of 2 π is finite, the asymptotic behavior of the corresponding retarded correlation function is θ (t)
〈e
− (θ ( t ) −θ ( 0 )) 2 / 2
〉∝e
C (t ) ∝ 1 / t
2
− t /τ
The time during which the phase undergoes a variation of 2π, in the model of Feigelman and Larkin τ •
C (t )
can be found estimating the limits of validity of linearized (in 1/G, G>>1) RG equations for the (inverse-square) XY model (Kosterlitz, 1976).
∫
∞
0
1/ t1/ G
dt〈ei (θ (t )−θ (0)) 〉 0 ≈ τ ≈ exp(G)
1/ t ≈τ
t
2
The large grain results of Feigel’man and Larkin • The perturbation theory in the interaction between grains rij −2 repulsion r r 2 2 (taking account of Coulomb J ij ∝ ( R / rij )(1 + 2 ν | λN | ln ) , rij =| ri − rj | in the metal, R ) breaks down for ∞
J ≈ (∫ dt〈ei (θ (t )−θ (0)) 〉 0 )−1 ~ exp[−G] 0
J ∝ N / ln(N−1/ 2 / R) (where ) • As a result – There is a quantum superconductor – metal transition; NC – Critical concentration is exponentially small in dimensionless film conductance G
An alternative approach to the same problem in the same limit R < ξ S
S = ∑iSi + ∑( i ≠ j ) ∫ dωJ ij ∆ ∆
Jij
( i ) ( j )* ω −ω
Si is the action for an individual grain “I”.
R2 J ij ∝ 2 rij
1 rij 2 (1 + 2 ν | λN | ln ) R
S
Perturbation on the metallic side of the transition R < ξ
The metallic state is stable with respect to superconductivity if
∑ J × ∫ dt P(t ) < 1, ij
i
where P(t ) =< ∆(0) ∆* (t ) > R2 J ij ∝ 2 rij
1 rij 2 (1 + 2 ν | λN | ln ) R
;
∑J
