1

Superconductivity

𝑇

r

pure metal

metal with impurities

0.1 K

𝑇 c

Electrical resistance

… critical temperature

const.c Tmαa

a is a material constant (isotopic shift of the critical temperature)

Superconductivity

2

The superconductivity was discovered in 1911 by Heike Kamerlingh Onnes at the Leiden University. At 4.2 K (-296°C), he observed a disappearance of resistivity in mercury. His experiments were made possible by the condensation of helium (1908).

Heike Kamerlingh Onnes

1913 Nobel prize in physics

3

Superconductivity

Superconducting elements

T [K]Al 1.19Cd 0.56Ga 1.09Hg 4.00In 3.40Ir 0.14La 5.00Mo 0.92Nb 9.13Os 0.65Pb 7.19Re 1.70

T [K]Ru 0.49Sn 3.72Ta 4.48Tc 8.22Th 1.37Ti 0.39Tl 2.39U 0.68V 5.30Zn 0.87Zr 0.55

4

Isotopic Shiftconst.c Tmα

a

Material T [K] a

Zn 0.87 0.45±0.05Cd 0.56 0.32±0.07Sn 3.72 0.47±0.02Hg 4.00 0.50±0.03Pb 7.19 0.49±0.02Tl 2.39 0.61±0.10

Material T [K] a

Ru 0.49 0.00±0.05Os 0.65 0.15±0.05Mo 0.92 0.33Nb3Sn 18 0.08±0.02

Mo3Ir 0.33±0.03

Zr 0.55 0.00±0.05

5

Superconductivity

Superconductor in a magnetic field

T

Hc

normal state

superconducting state

Tc

2

2

0 1c

c T

THH

Temperature dependence of the critical magnetic field

1

0

)/(104);( 700

HM

MHB

AmVsMHµB

Superconductor: Meissner effect

Otherwise: -10-6

Meissner-Ochsenfeld effect

6

Magnetic levitation train

7

8

Superconductor in a magnetic field

2

0)(

2

00

0

0

e

BB

e

e

ee

BBdBBdMW

HM

MHµB

HµB

ee

External field:Inner field:

Magnetization:

Work per unit of volume

(magnetization direction of a superconductor is opposite to the magnetic field direction)

Energy of a superconductor within an magnetic field is higher than without an magnetic field

This is caused by the “superconducting” electrons

9

Transition between normal and superconducting state

STGUB

BGU

ST

TSB

UG

TSUG

ce

ec

e

2

2

1

22

2

Thermodynamic consideration

… Gibbs free energy … enthalpy … temperature … entropy … external magnetic field

: (and ) small for SC state, therefore the SC state is stable: bigger in normal state (less order), therefore the normal

state is stable: free Gibbs energy is smaller, if is bigger (normal state)

10

Superconductivity

Material [K]

NbC 14

NbN 16

Nb3Al 18

Nb3Ge 23

Nb3Sn 18

SiV3 17

La2-xBaxCuO4 30

MgB2 40

YBa2Cu3O7-d 110S.L. Bud’ko and P.C. Canfield: Temperature-dependent Hc2 anisotropy in MgB2 as inferred from measurements on polycrystals, Phys. Rev. B 65 (2002) 212501.

Crystal structures of La2-xBaxCuO4 and YBa2Cu3O7-x

11

YBa2Cu3O7-x

Space group: PmmmLattice parameters:a = 3.856(2) Åb = 3.870(2) Å c = 11.666(3) Å

a b c/3

La2-xBaxCuO4

Space group: BmabLattice parameters:a = 5.33915(9) Åb = 5.35882(9) Å c = 13.2414(2) Å

a b a/2 < c/3 < a

12

SuperconductivityType I superconductors• Transition to normal state

after exceeding

Type II superconductors• Superconductivity

decreases gradually between und

• Transition to normal state after exceeding

𝐻

−𝑀

normal state

superconducting

𝐻 c 𝐻

−𝑀

𝐻 c 1 𝐻 c 2𝐻 c

13

Theories of Superconductivity

Super electrons :

• No scattering• Entropy of the system is zero

(the system is perfectly ordered)

• Large coherence length

14

London Theory (Meissner Effect)Ej

Ohm: BArot;Aλµ

jL

12

0

London:

2

0

0

0000

20

rotdivgradrotrot

rotrot

1rot

L

L

λ

BB

jµBBB

jµBt

EµjµB

Bλµ

j

London:

Maxwell:

(static conditions)

Meissner effect:

Solution:

Lλ

xBxB exp0

B

x

… London penetration depth

15

Consequences of the London Theory

describes the penetration depth of the magnetic field into a material. Inside the material at a distance to the surface the intensity of the magnetic field falls to of its original value.

An external magnetic field penetrates completely homogeneous a thin layer, if the thickness is much smaller than . In such a layer the Meissner effect isn’t complete.

The induced field (in the material) is smaller than , therefore the critical magnetic field, which is oriented parallel to the thin layers is very high.

16

Coherence LengthThe distance in which the width of the energy gap, in a spatial variable magnetic field, doesn’t change essentially.

rBrA

rAλµ

rjL

rot

12

0London:

17

BCS Theory of SuperconductivityJ. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 106 (1957) 162.J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175.1. Interactions between electrons can cause a ground state,

which is separated from the electronically excited states by an energy gap. However: there are also superconductors without an energy gap!

𝐸

𝐸

18

BCS Theory of Superconductivity2. The energy gap is caused by the interaction between

electrons via lattice vibrations (phonons). One electron distort the crystal lattice, another electron “sees” this and assimilate his energy to this state in a way, which reduces the own energy. That’s how the interaction between electrons via lattice vibrations work.

19

BCS Theory of Superconductivity3. The BCS theory delivers the London penetration depth

for the magnetic field and the coherence length. Thereby the Meissner effect is explained.

rBrArAλµ

rjL

rot;

12

0

London:

Meissner:

LL λ

xBxB

λ

BB exp0

2

Coherence length: g

F

πE

v0