anisotropic superconductors - roma tre universityanisotropic superconductors anisotropic...

Anisotropic superconductors


YBa2Cu

3O

7-x

Bi2Sr

2CaCu

2O

8+x

LaFeAsO0.89

F0.11

O/F

Cuprates

Pnyctides


Artificial multilayers

S

N, I, F, ...

S

N, I, F, ...

S

N, I, F, ...

S

N, I, F, ...

A few examples:

Superconductor/insulator:

Al/Ge

Nb/Ge

Nb/Si

Pb/Ge

Nb0.52Ti0.48/Ge

NbN/AIN

Superconducting/normal-metal:

Nb/Cu

Nb/Ti

Nb/Zr

Nb/AI

Nb/Ta

V/Ag

V/Mo

Pb/Bi

Superconductor/magnetic-metal:

V/Fe

V/Cr

V/Ni

Mo/Ni

Nb/rare earth

Motivation

1. the modulation wavelength may be made comparable with various length

scales which characterize the superconducting state;

2. study of interaction of different orderings (magnetism and

superconductivity);

3. can be grown atomically, the control of the material is very good;

... that’s for metallic multilayers...

but:

4. High-Tc superconductors are anisotropic, extremely anisotropic in:

– structural properties;

– normal state properties;

– superconducting state properties.

for HTC you have to deal with anisotropy.

Theoretical tools

Many of the observed properties may be described by the

phenomenological GL theory.

However, GL theory is usually limited to temperatures close to

the bulk (zero-field) transition temperature.

In general, for quantitative results, one needs various extensions

of the BCS theory to inhomogeneous systems.

For the sake of simplcitity,

we will use GL theory.

We will always limit ourselves to uniaxial superconductors

(isotropic “planes” along two coordinates + perpendicular anisotropy axis).

Isotropic vs. anisotropic materials

● Isotropic material:

assume a “source” vector (e.g. the current field in a conductor, J), and an “effect” vector (e.g., the electric field E).

⇒ effect // source, by means of a scalar response function.

In our example: E = ρJ

● Anisotropic material: the effect is in general not // to the source

● Assume {x,y,z} coincide with principal axes (and no Hall effect). Then

scalar resistivity

tensor resistivity.

diagonal matrix

E

J

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Example: resistance measurementsAssume simple 2D anisotropy:

Isotropic:

external

measured

Measurements along the principal axes: direct

measures of single elements of the response tensor

not trivial, complex analytical probelm

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m11

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mab

> 1

Effective mass tensor● Drude model

● Anisotropic material, principal axes:

● Uniaxial anisotropic [super]conductors:

x=a

y=b

z=c

anisotropy factor

mass tensor

(τ is assumed to be scalar)

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How anisotropic are HTC superconductors?

YBa2Cu

3O

7-xBi

2Sr

2CaCu

2O

8+x

“Moderate” anisotropy Extreme anisotropy

Hagen etal, PRB 37, 7928 (1988)

Martin etal, PRL 60, 2194 (1988)

resistivity in the normal

state

Anisotropy in the mixed state: GL theory

References:

K. Fossheim , A. Sudbø,“Superconductivity - Physics and Applications”,

John Wiley and Sons, 2004, Chapter 7.5

B.Y. Jin, J.B. Ketterson (1989)Artificial metallic superlattices

Advances in Physics, 38:3, 189-366

The general procedure

The anisotropic properties reside in the relevant lengths: ξ and λ⇒ upper and lower critical field.

General procedure:

• Write down the appropriate GL free energy

• Minimize with respect to ψ* ⇒ 1st GL equation.

• Linearize: H≈Hc2 ⇒ high order terms in ψ are irrelevant.

• Find lowest eigenvalue of the resulting equation: it gives Hc2.

• Include the orientation of B, i.e. the components of A in the

free energy to find out the angular dependence of Hc2.

−α =

(

n+1

2

)

!2eB

mBc2 = −α

m

!e

Bc2 = −αm

!e=

Φ0

2πξ2

−α =!2

2mξ2

Bc2 ∝ (1− t)

Hc2: isotropic case

H ≈ Hc2 : linearized GL free energy density. The term |ψ|2 is dropped.

F = F0 +1

2α |ψ |2 +

1

2m

∣

∣

∣

(−i!∇ − 2eA)2ψ

∣

∣

∣

2

Minimization of (integrated) free energy yields

∂F

∂ψ= αψ +

1

m(−i!∇ − 2eA)2ψ = 0 (7.36)

1

2m(−i!∇ − 2eA)2ψ = −αψ

Identical to a Schrödinger equation for a particle of mass m, charge -2e, in a field B. Eigenvalue is α.

n=0 yields the lowest eigenvalue, corresponding to B = Bc2.

(close to T/Tc=t=1)

G G0

G

αψ + β|ψ|2ψ −!2

2m

∣

∣

∣

∣

(

∇− i2e

!A

)

ψ

∣

∣

∣

∣

= 0

←→m

−1=

1

m//0 0

01

m//0

0 01

m⊥

Continuous anisotropic superconductors

αψ + β |ψ |2 ψ −

!2

2

(

∇ − i2e

!A

)

·↔−1m ·

(

∇ − i2e

!A

)

ψ = 0 (7.47)

The 1st GL equation:

is written in anisotropic, continuous form with the introduction of the inverse effective mass tensor:

where (principal values, uniaxial superconductor):

←→m

−1=

1

m//0 0

01

m//0

0 01

m⊥

ξi ∝1

√mi

λi ∝√mi

Continuous anisotropic superconductors

αψ + β |ψ |2 ψ −

!2

2

(

∇ − i2e

!A

)

·↔−1m ·

(

∇ − i2e

!A

)

ψ = 0 (7.47)

whence different, anisotropic GL coherence length:

The order parameter changes differently along the different axes.

ξ2

i =

!2

2miα(T )

using (GL) and the fact that Bc has no anisotropy!!"!#

$!

"%"& "#Bc ! condensation energy!

λ has opposite anisotropy with respect to ξ:

along principal axes

Note on anisotropy of λ

λi concerns decay of currents flowing along the i-th direction, i.e. magnetic fields perpendicular to the ith direction.

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λbx=b

y=c

z=a

B//a

J//b J//c

z=b

x=cy=a

λc

B//b

µ0Hc2(0) =Φ0

2πξ2ab

µ0Hc2(90◦) =

Φ0

2πξabξc

µ0Hc2(90◦) = γ

Φ0

2πξ2ab

µ0Hc2 ∝ (1− t)

Anisotropy: fluxons and Bc2

y=b

z=c

x=a

y=b

B(0)//c B(90°)//a

core cross section

Circular section (like isotropic case)

Ellyptical section


(close to T/Tc=t=1)Note: in both orientation, one still has

µ0Hc2,⊥ =Φ0

2πξ2‖

µ0Hc1,‖ =Φ0

4πλ‖λ⊥

γ =

(

m⊥

m‖

)1/2

=

ξ‖

ξ⊥=

λ⊥

λ‖=

Hc2,‖

Hc2,⊥=

Hc1,⊥

Hc1,‖

Anisotropy ratio

The anisotropy ratio γ can be written alternatively as:

ξi ∝1

√mi

λi ∝√mi

Note: in this model (continuous), the anisotropy ratio is temperature independent

µ0Hc2,‖ =Φ0

2πξ‖ξ⊥

µ0Hc1,⊥ =Φ0

4πλ2

‖

−α =

(

n+1

2

)

!ωc

ωc(θ) = (−2e)B

(

sin2 θ

m‖m⊥+

cos2 θ

m2

‖

)1/2

µ0Hc2(θ) =Φ0

2π(

ξ2‖ξ2

⊥ sin2 θ + ξ4‖ cos2 θ

)1/2=

µ0Hc2,⊥(

γ−2 sin2 θ + cos2 θ)1/2

Anisotropy: fluxons and Bc2

The angular dependence

1

2m(−i!∇ − 2eA)2ψ = −αψ

Identical to a Schrödinger equation for a particle of mass m, charge -2e, in a field B. Solution (eigenvalues) can be written in terms of the cyclotron angular frequency of the particle:

Back to the isotropic case, linearized GL 1st equation:

using the Newton’s equation of motion:ma = (-2e) v × B one finds ellyptical orbits

whence finally:

z B

)

θ

or equivalently: Exercise: plot this function, highlight important features.

Hc2(θ) =Hc2,⊥

(

γ−2 sin2 θ + cos2 θ)1/2

∝ (1− t)

γ =

(

m⊥

m‖

)1/2

=

ξ‖

ξ⊥=

λ⊥

λ‖=

Hc2,‖

Hc2,⊥=

Hc1,⊥

Hc1,‖∼ const

Anisotropy in the continuous modelThe temperature dependence

z B

)

θ

This is predicted by the continuous model, where

or equivalently:

Exercise: plot this function, highlight important features.

(with T)

µ0Hc2,⊥ =Φ0

2πξ2∝ (1− t)

µ0Hc2,‖ =√12

Φ0

2πξd∝ (1− t)1/2

γ =Hc2,‖

Hc2,⊥=∝ (1− t)−1/2

The thin slab (“Tinkham’s formula”)

z B

)

θ

d

Thin slab of isotropic superconductor: d << ξ

The angular dependence of Hc2 has a cusp in parallel orientation (compare with the round maximum in the continuous model).

Reference: B.Y. Jin, J.B. Ketterson (1989)Artificial metallic superlattices

Advances in Physics, 38:3, 189-366

The anisotropy ratio depends on Tand diverges at Tc.

Exercise: plot this function, highlight important features.

ξ⊥ → ∞, T → Tc

ξ⊥(T∗) = s/

√

2Hc2(θ →‖)Hc2,‖(T )

∼ (1− t)

∼ (1− t)1/2

Multilayers: the dimensional crossover

⇒ 2D-3D crossover

at T* such that

One finds:

YBCO: T*/Tc ~ 0.84, 3D above T* ~ 78 K

BSCCO: T*/Tc = 0.999, 3D only ~0.1 K below T

c !!

d

s

ξ⊥(T∗) ∼ s

3D:

2D: cusp

round

Experiments

Artificial multilayers

S

S

S

S

Syntetic multilayers: crossover 2D – 3D

- Nb: (bulk: Tc=9.25 K, ξ(0)=38 Å)- Ge: insulator (amorphous)

DNb

DGe

Ruggiero et al, PRL 45, 1299 (1980)3D

2D

T*

Nb/Ge

Nb

Nb

Nb

- V (bulk: Tc=5.4 K)

Kanoda etal, PRB 33, 2052 (1986)

T*

T*

V/Ag

T*


V

V

V

3D2D

Chun etal, PRB 29, 4915 (1984)

Nb/Cu

-10 40 90 130

-10 40 90 130


3D2D

The scaling approach

Blatter etal, PRL 68, 875 (1992)Hao e Clem, PRB 46, 5853 (1992)

Angular scaling.● It is possible to demonstrate that, when:

–– κ≫1 (cuprates: κ∼100 )

–– H≫Hc1 (cuprates: μ0Hc1∼10−100G)

the Gibbs free energy and all derived thermodynamic quantities depend on θ and B

only through the ratio B/μ0Hc2(θ)

● Writing μ0Hc2(θ)= μ0Hc2⟘ ƒ(θ), one introduces the angular scaling function ƒ.

● For an observable, Q(B, θ; T) = Q(B/μ0Hc2⟘ ƒ(θ); T): the anisotropy can be

studied through ƒ alone.

● One finds that also transport properties obey the angular scaling, in appropriate

geometric conditions (e.g., no variation of the Lorentz force on fluxons)

Is it relevant?

● remember the numerical values of Hc2 in cuprates...

Scaling properties: specific heat

YBCO crystal

cH // c

b

a

Roulin et al, Physica C 260, 257 (1996)

H _|_ c

Scaling properties: quasi-upper-critical field

YBCO crystal

Naughton et al, Phys. Rev. B 38, 9280 (1988)

Angular dependence

c

b

a

)

H

Scaling properties: irreversibility field

BSCCO film

Fastampa et al, Phys. Rev. Lett. 67, 1795 (1991)

Angular dependence

dimensional crossover

c

b

a

)

H

Scaling properties: ρ(H,θ) in d.c.

YBCO filmsθ

cH

Jb

aNote: different choice for θ

determine ƒsuch that:

S. Sarti et al, PRB 55, 6133 (1997)

Extremely anisotropic superconductors: Consequences on vortex matter

References:

K. Fossheim , A. Sudbø,“Superconductivity - Physics and Applications”,

John Wiley and Sons, 2004, Chapter 7.5

W.Buckel, R. Kleiner,“Superconductivity - Fundamentals and Applications”,

Wiley, 2004, Chapter 4.7

Vortices

Different competing length:

• London penetration depth: currents• GL coherence length: vortex cores.• Interlayer separation!

Atomically small _|_ to planes!

in HTCS:

Most of peculiarities go beyond a continuum approach,

especially in BiSrCaCuO:extreme “anisotropy”, short

perpendicular GL coherence length.

“Layered superconductor”BiO

Ca

SrO

SrO

SrO

BiO

BiO

CuO2

CuO2

BiO

Figure 2.14 Structure of Bi2Sr2CaCu2O8 crystal.

Correlated planes (typically YBCO)

Fig. 4.33 Possible phases of the so-called vortex matter in a spatially

homogeneous superconductor in the case where thermal fluctuations

play an important role [68]. On the left we see schematically different

vortex configurations: (a) vortex lattice, (b) a liquid phase where the

vortices still remain separated, and (c) a liquid phase with entangled

vortices. On the right a schematic phase diagram is shown [69].

In the liquid phase a certain amount of short-range order is possible

(“hexatic vortex liquid”). One must note also that the liquid phase is

inserted between the Meissner phase and the vortex lattice.

Uncorrelated planes ⇒ “Pancakes”

(a) (b) (c)

Figure 7.8 Stacks of 2D pancake vortices: (a) straight stack, the configuration of lowest

energy at zero temperature, and (b) disordered stack, which occurs at higher temperatures.

(c) Sketch of a tilted stack of pancake vortices in successive superconducting layers, con-

nected by interlayer Josephson strings.

BiO

Ca

SrO

SrO

SrO

BiO

BiO

CuO2

CuO2

BiO


H

perpendicular field

BiSrCaCuO

Uncorrelated planes: phase diagram

Little energy to shift individual pancakes ⇒ at finite temperatures a

number of different vortex phases are possible:• “crystalline” state: pancakes form a triangular flux-line lattice;• “flux-line liquid”: pancakes still form flux lines, freely mobile: small critical current, small irreversible magnetization;• quasi-2D vortex solid: pancakes form a triangular lattice within a plane, but lattices in different planes are freely shifted relative to each other;• “pancake gas”: pancakes are freely mobile within a plane and also are not ordered any more perpendicular to it: completely reversible, zero critical current.

Fig. 4.35 Vortex phase diagrams of Bi2Sr2CaCu2O8+x single crystals

BiO

Ca

SrO

SrO

SrO

BiO

BiO

CuO2

CuO2

BiO


H

perpendicular field

BiSrCaCuO

In-plane field: Josephson vortices

SC

“insulating”BiO

Ca

SrO

SrO

SrO

BiO

BiO

CuO2

CuO2

BiO


H

parallel field

Inclined field ⇒ “Staircase vortices”

BiO

Ca

SrO

SrO

SrO

BiO

BiO

CuO2

CuO2

BiO


H

inclined field

Fig. 4.37 Staircase pattern of flux lines in a magnetic field applied at

an angle θ to the superconducting layers. The flux lines pass across

the planes in the form of pancake vortices joined together by short

segments (of length L) of Josephson vortices.

BiSrCaCuO

anisotropic superconductors - roma tre universityanisotropic superconductors anisotropic...

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