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Outline1. Introduction

a) type I and type II superconductorsb) phase diagram of type II superconductors

2. Basic measurements testing phase diagrama) magnetic measurements

m-H (T=const)critical state models, hysteresis loop in real superconductors: fishtail anomaly, thin films

m-T(H=const)diamagnetic and Meissner effects, irreversibility line, paramagnetic Meissner effect

ac susceptibilityb) transport and magnetotransport

technical details measurement geometry, the role of microstructure

vortex lattice meltingvortex liquid-vortex glass transition

scaling predictions and methodology 3. Superconductor-ferromagnet heterostructures4. Superconductor-insulator transition

Meissner effect, 1933Meissner effect, 1933

Walther Meissner Robert Ochsenfeld

F.& H. London, 1935: shielding currents

2s

sn eJ Bm

∇× = −

0µ∇× =B J}

2

2

1 0zz

d B Bdx λ

− =Maxwell’s law

penetration depth

20

λµ

=Ls

mn e

λ x

B0/−= 0

xB(x) B e λ

www.physics.leidenuniv.nl paintings by nephew Harm

Heike Kamerlingh Onnes, 1911, Leiden

Zero resistanceZero resistance

0.7ξλ >0.7ξλ <

3.41 KIndium (In)3.72 KTin (Sn)4.15 KMercury (Hg)4.47 KTantalum (Ta)4.88 KLanthanum (La)

7.196 KLead (Pb)

<κ 1/ 2

λκ=ξ

15 K (nanotubes)9.25 K7.80 K5.40 K

CNbTcV

Nb3Ge 19 KNb3Si 18.1 KNb3Sn 18 K

MgB2 39 KOver 50 different high-Tc superconductors, Tc: 30-187K extreme type II

YBa2Cu3O7: ξ(0)~0.4-3 nm; λ(0)~150nm

Nb: ξ(0)~ 38nm λ(0)~ 39 nm

Pairing mechanism still unknown

>κ 1/ 2

List of superconductors: superconductors.org

Phenomenological Ginzburg-Landau theory,1950

( )

( )

2 2

Free energy expansion in powers of

expansion coefficients

-1 2

2

2

ψ

→α β

∇ − ψ +β ψ ψ = −αψ

⎡ ⎤= ψ − ∇ − ψ +⎣ ⎦s

,

,

i eAm

eJ i eA c.c .m

Complex order parameter: ( )( ) ( ) ie ϕΨ = Ψ rr r2 ( )( ) sn

nΨ =

rra measure of the superfluid density

theory of phase transitions: macroscopic properties of superconductors

2

0c

c0c

Φ TH = =H (0 ) 1 -T2π µ 2 ξ

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

λ ⎢ ⎥⎝ ⎠⎣ ⎦( )2

0 c n s

Thermodynamic critical field: 1µ H T = f-f2given by a free-energy difference between the normal and superconducting states(i.e. condensation energy)

c

ccT

dH 4π∆CSpecific heat jump ∆C: dT T

⎛ ⎞ =⎜ ⎟⎝ ⎠

λκ=ξ

GL parameter:

F0

cB

BCS coherence length, mean-free pathhvξ 0.18 -k T

= l

1/2c

0 0c

1/2c

0 0c

(clean limit)

(dirty limit)

Tξ(T)=0.74ξ , ξT-T

Tξ(T)=0.85 ξ , ξ T -T

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

l

l l

coherence length penetration depth,ξ(T) - (T) - λ

1/2c

0c

1/2c0

0c

(clean limit)

(dirty limit)

1 T(T)= , ξT-T2

ξ T(T)=0.64 , ξ T -T

⎛ ⎞λ λ ⎜ ⎟

⎝ ⎠

⎛ ⎞λ λ ⎜ ⎟

⎝ ⎠

L

L

l

ll

cand diverge asξ T Tλ →

0.7ξλ >0.7ξλ <

Triangular (Abrikosov) vortex lattice (1957)

ξλ

3.41 KIndium (In)3.72 KTin (Sn)4.15 KMercury (Hg)4.47 KTantalum (Ta)4.88 KLanthanum (La)

7.196 KLead (Pb)

<κ 1/ 2

⎛ ⎞⎜ ⎟⎝ ⎠

1/20

02Φa =3B

λκ=ξ

-15 20Φ = =2.07x10 Tm

2he

15 K (nanotubes)9.25 K7.80 K5.40 K

CNbTcV

Nb3Ge 19 KNb3Si 18.1 KNb3Sn 18 K

MgB2 39 KOver 50 different high-Tc superconductors, Tc: 30-187K extreme type II

YBa2Cu3O7: ξ(0)~0.4-3 nm; λ(0)~150nm

Nb: ξ(0)~ 38nm λ(0)~ 39 nm

Pairing mechanism still unknown

>κ 1/ 2

02

0c2

c1

c2

ΦH = lnκ4πΦH = = 2κH

2π ξ

λ

cc2

cc2

T Y PE I

T Y PE I I

H H H H

<

>

List of superconductors: superconductors.org M. Tinkham, “Introduction to superconductivity”

Thermal fluctuations

Ginzburg number:

relative size of the thermal energy and the condensadion energy within the coherence volume; ε – anisotropy parameter

HighTc, small ξ : Gi~10-2 (low-Tc: Gi~10-8)

2cB

i 2 3c

εk TG / 2H (0)ξ(0)

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Pinning of the vortex by disorder (defects)

Vortex of length L removes condensation energy ~

It is energetically favorable to place vortex in the defect site, with locally suppressed order parameter.

Small ξ weak pinning - but artificial pinning sites may be created (irradiation, doping, thickness modulation,…).

( )2 2c0µ H π ξ L / 2

Melting of the vortex latticeMelting of the vortex lattice

Lindeman criterion: , cL ~ 0.1-0.4

u - thermal displacement of the vortices from equilibrium positions

Clean superconductor: melting is the 1-st order transition

Disorder: vortex glass instead of lattice; transition is continuous

Note: in liquid vortices may remain pinned !!!

24L

m c2i c

5.6 c TB (T ) H (0 ) 1G T

⎛ ⎞ ⎛ ⎞≈ −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

2 2 2m L 0th

u (T ) ~ c a

large Gi small Bm: large region of the vortex liquid

1-st order

continuous

Thermal depinning of vortices (crossover)

u - thermal displacement of the vortices from equilibrium positions

Correlated disorder (twin boundaries, screw dislocations, artificial columnar defects) with mean distance dr < a0:

depinning in case of collective pinning

Vortex dynamics Experiments:

Flux creep at j<jc: disordered landscape hops of vortex segments and bundles over the energy barriers U

FF Flux flow: U << kT (unpinned liquid) ohmic resistance (j-independent)

TAFF Thermally assisted flux flow: U >> kT (pinned liquid)small j: ohmic ; large j: power-law

Vortex glass: U diverges as j 0 ρ = 0 in the low-j limit

2

i c2dpc

TB (T ) 8 G H (0 )T

⎛ ⎞≈ ⎜ ⎟

⎝ ⎠

2 2m th

u (T ) ~ ξ

Theoretical descriptions:

TAFF - Anderson-Kim model: wrong large-j limit (exponential)

Vortex glass-vortex liquid transition model (M.P.A.Fisher, PRL 62 (1989); M.P.A.Fisher, D.S.Fisher, and D.A.Huse, PRB 43 (1991)).

2 2m rth

u (T ) ~ d

G. Blatter et al., Rev. Mod. Phys. 66 (’94); Y. Yeshurun et al., RMP 68 (’96)

TAFF

FF

Vortex glass

High T

Low T

isotherms

0

BM= -Hµ

Magnetic measurements in type II superconductor:

Thermodynamic evidence of the 1st order magnetic phase transitions

But: challenging to interpret (sometimes)

Magnetometers:

SQUID, VSM, etc…

local magnetometry: magnetooptics, Hall (effect) probe

Ac susceptibility

-4 -2 0 2 4-1

0

1

CA

4πM

Ha

M vs H, T=const

Hysteresis loop

Tc*T

M vs T, H=const

Meissner and diamagnetic effects

Shielding current in type II superconductor: critical state

B0

B0

2a

z 0

0

y0 0

B (x) = n(x) Φ ; n(x): vortex per unit area Φ : flux per vortex

dΦ n(x) = µ J (x)dx

Maxwell Equation: ∇ 0×B = µ J → z y0d B (x) = µ J (x)dx

Lorentz Force density acting on vortex: F=J×B

Pinning Force: pF = F = J×B

zpx y z z

0

1 dBF = J B = Bµ dx

z y pB (x), J (x), F (x)

=cy z c

z

JJ (B ) = ; J(B )

constfCritical state models:

Ref.: Charles P. Poole, Jr., Superconductivity, Farach, Creswick, Academic Press, 1995

Shielding current in type II superconductor: critical state

+

⎡ ⎤+⎣ ⎦

c

cp

k

c

k

c

k

J(B) = JJJ(B) = (F=const);

B(x) / BJJ(

Bean model, Bean (1962,1964)

Fixed pinning model, Ji et al. 1989

Kim modeB) = 1 B(x) / B

JJ(B) = 1 B

l, Kim et al., 1962,1963

Generalized model, Lam(x) / B

et alβ ., Xu et al.,1990

c0

0

0

0BJ =µ (a- )

B=' µa a

When a’=0

Magnetization in the Bean model

a0µ M= B -B

→ =aac1 0

0

B <H B 0, µ M=-Bµ

→* *a a a0

1 1 3 3B = B B = B µ M=- B =- B2 4 4 8

→* * *a 0

1 1B =B B = B µ M=- B2 2→ →c c

0

+ -B* M -Ma'=0 J = high B: J =µ a a

0c

0

BJ =µ (a-a')

High field range: B-dependent Jc

-4 -2 0 2 4-1

0

1

4πM

Ha

Hysteresis loop in real bulk type II superconductor

Fietz et al., Phys. Rev.136 (’64): Kim-like model description of niobium

M. F. Schmidt et al., PRB48 (1993)

Nb film 500 nm thick

c1H (T)

Peak effect and fishtail anomaly

Peak effect: amorphization of the vortex matter at high and low H

Banerjee et al., PRB62 (2000)

Bi2Cr2CaCu2O8 -local (Hall probe)

magnetization measurementssample center

Bsp : second-order transition from vortex lattice (low H) into a strongly pinned state (high H)

Kayakovich et al., PRL76 (1996)

enhanced pinning in the glassy state due to disorder –better adaptation of the vortices to disordered landscape

Angst et al., PRB65 (2002)

Fishtail anomaly:

-100 -50 0 50 100

-1.0x10-3

0.0

1.0x10-3

Sample: Si(3nm)/Nb(78nm) Tc=8.55K

M[e

mu]

H[Oe]

T=8K; T/Tc=0.94

Film is almost always in the mixed state - flux enters the film at the edge as soon as the field is applied.

The maximum of M is always in close vicinity of H=0.

λ~200nm d < λ

Thin films

( )

0

z x xy

0 0

×B = µ J1 H H 1 HJ x =- - - -µ x z µ z

∇

∂ ∂ ∂⎛ ⎞ ⎛ ⎞≈⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠

Critical state differs from the bulk

Films: Jy depends on ∂Hx/∂z which has a large change across the film thicknessBulk: H=(0,0,Hz)

In thin films currents flow all the way to the middle of the film;

The B(x) in always nonlinear, even for the Bean model.

Zeldov et al., Phys. Rev. B 49 (‘94)Bean model for slab & solution for thin film (Bz=0 inside)

Slab

Film

c1edge appliedWB B H<H not accesibled

W-sample width, d-thickness

≈ →

W~1 mm, and d~100 nm x100

Films: large demagnetization factor increase of the external

magnetic field at the film edge

Meissner effect and diamagnetic effect

Tc

Ceramic YBa2Cu3O7 +Au

G. Xiao et al., PRB38, 776 (1988)

FC = Meissner effectmeasurement during cooling in the field

FC less than ZFC :

flux pinning by voids between grains

FC ZFCM < M

→B=0 M=-H

ZFC = diamagnetic (shielding) effect

1) cooling sample without field2) measurement during warming

Ideal superconductor:

≈

ZFCVM =- H

1-n1 r

1-n t

V – sample volumen - demagnetization factor

2r

tB