SOLID STATE PHYSICS MNT-510

SUPERCONDUCTORS

Physics of Metals - Introduction •  Atoms form a periodic lattice

•  Know (!) electronic states key for the behavior we are interested in

•  Solve the Schro …

… in a periodic potential

K is a Bravais lattice vector

Physics of Metals – Bloch’s Theorem

•  Bloch’s theorem tells us that eigenstates have the form …

… where u(r) is a function with the periodicity of the lattice … Free particle Schro

Physics of Metals – Drude Model •  Model for electrons in a metal ▫  Noninteracting, inertial gas ▫  Scattering time τ

•  Apply Fermi-Dirac statistics

damping term

http://www.doitpoms.ac.uk/tlplib/semiconductors/images/fermiDirac.jpg

Physics of Metals – Drude Model Comments

•  Wrong! ▫  Lattice, e-e, e-p, defects, ▫  τ ~ 10-14 seconds MFP ~ 1

nm

•  Useful! ▫  DC, AC electrical conductivity

▫  Thermal transport   Lorenz number κ/σT

▫  Heat capacity of solids Electronic contribution

Lattice

Zero Resistance •  Metallic R vs T ▫  e-p scattering (lattice interactions) at high temperature ▫  Impurities at low temperatures

R

Temperature

Residual Resistance (impurities)

Electrical resistance

R0

Lattice (phonon) interactions

TD/3

Zero Resistance, cont.

•  Superconducting R vs T

R

Temperature

R0

Tc “Transition temperature”

What's a superconductor? Superconductors have two outstanding features: 1). Zero electrical resistivity. •  This means that an electrical current in a

superconducting ring continues indefinitely until a force is applied to oppose the current.

2). The magnetic field inside a bulk sample is zero (the Meissner effect).

•  When a magnetic field is applied current flows in the outer skin of the material leading to an induced magnetic field that exactly opposes the applied field.

•  The material is strongly diamagnetic as a result. •  In the Meissner effect experiment, a magnet floats

above the surface of the superconductor

What's a superconductor?

• Most materials will only superconduct, at very low temperatures, near absolute zero.

• Above the critical temperature, the material may have conventional metallic conductivity or may even be an insulator.

• As the temperature drops below the critical point,Tc, resistivity rapidly drops to zero and current can flow freely without any resistance.

Superconducting elements

• Ferromagnetic elements are not superconducting • The best conductors (Ag, Cu, Au..) are not superconducting • Nb has the highest TC = 9.2K from all the elements

What's a superconductor? •  Linear reduction in resistivity as temperature is

decreased: ρ = ρo (1 + α(T-To)) where ρ: resistivity and α: the linear temperature coefficient of resistivity.

•  Resistivity: ρs ~ 4x10-23 Ω cm for superconductor.

•  Resistivity: ρm ~ 1x10-13 Ω cm for nonsuperconductor metal.

Meissner Effect •  When a material makes the transition from the normal to

superconducting state, it actively excludes magnetic fields from its interior; this is called the Meissner effect.

•  This constraint to zero magnetic field inside a superconductor is distinct from the perfect diamagnetism which would arise from its zero electrical resistance.

•  Zero resistance would imply that if we tried to magnetize a superconductor, current loops would be generated to exactly cancel the imposed field (Lenz’s Law).

Non-superconductor

Bint = Bext

Superconductor

Bint = 0

Bext

15

Penetration depth

λ depicts the distance where B(x) is e-time smaller than on the surface

Temperature

Pene

trat

ion

dept

h λ

Superconductor

Magnetic Levitation • Magnetic fields are actively excluded from

superconductors (Meissner effect). •  If a small magnet is brought near a

superconductor, it will be repelled becaused induced supercurrents will produce mirror images of each pole.

•  If a small permanent magnet is placed above a superconductor, it can be levitated by this repulsive force.

Resistivity at low temperatures- pure mercury (could repeatedly distilled producing very pure samples).

• Repeated resistivity measurements indicated zero resistance at the liquid-helium temperatures. Short circuit was assumed! • During one repetitive experimental run, a young technician fall asleep. The helium pressure (kept below atmospheric one) slowly rose and, therefore, the boiling temperature. As it passed above 4.2 K, suddenly resistance appeared.

From: Rudolf de Bruyn Ouboter, “Heike Kamerlingh Onnes’s Discovery of Superconductivity”, Scientific American March 1997

Hg TC=4.2K

1895 William Ramsay in England discovered helium on the earth 1908 H. Kamerlingh Onnes liquefied helium (boiling point 4.22 K)

Superconductivity- discovery I

Superconductivity- discovery II

• Liquid Helium (4K) (1908). Boiling point 4.22K.

• Superconductivity in Hg TC=4.2K (1911)

„Mercury has passed into a new state, which on account of its extraordinary electrical properties may be called the superconducting state“ H. Kamerlingh Onnes 1913 (Nobel prize 1913)

Resistivity R=0 below TC; (R<10-23 Ω⋅cm, 1018 times smaller than for Cu)

20

Further discoveries

1986 (January): High Temperature Superconductivity (LaBa)2 CuO4 TC=35K

K.A. Müller und G. Bednorz (IBM Rüschlikon) (Nobel prize 1987)

1987 (January): YBa2Cu3O7-x TC=93K

1987 (December): Bi-Sr-Ca-Cu-O TC=110K,

1988 (January): Tl-Ba-Ca-Cu-O TC=125K

1993: Hg-Ba-Ca-Cu-O TC=133K

(A. Schilling, H. Ott, ETH Zürich)

1911-1986: “Low temperature superconductors” Highest TC=23K for Nb3Ge

Critical temperature, current density, and magnetic field boundary separating superconducting and normal conducting states (schematic).

Type I and Type II Behavior •  Type I ▫  Material Goes Normal

Everywhere at Hc

•  Type II –  Material Goes Normal Locally at Hc1,

Everywhere at Hc2

Complete flux exclusion up to Hc, then destruction of superconductivity by the field

Complete flux exclusion up to Hc1, then partial flux penetration as vortices

Current can now flow in bulk, not just surface

Superconductor type I (λ/ξGL<0.71) in a magnetic field

Bi=Ba+µ0M

Superconductor Bi=0

Normal conductor Bi=Ba

Negative units !

The field inside the superconductor

The field created on the surface of the superconductor compensating the outside field

Outside field

Outside field Ba Outside field Ba

Insi

de f

ield

Bi

Mag

neti

zati

on –

µ 0M

Superconductor type II in a magnetic field

Vortex-lattice in superconductor type II. Magnetic flux of a vortex is quantized: Φ0=h/2e≅2.07·10-15Tm2

Bi=Ba+µ0M

Outside field Ba

Outside field Ba

Mag

neti

zati

on –

µ 0M

Ave

rage

insi

de f

ield

Bi Meissner

phase

Mixed phase

Normal condu-ctor

Types I Superconductors •  There are 30 pure metals which exhibit zero

resistivity at low temperature. •  They are called Type I superconductors (Soft

Superconductors). •  The superconductivity exists only below their

critical temperature and below a critical magnetic field strength.

Mat. Tc (K)

Be 0 Rh 0 W 0.015 Ir 0.1 Lu 0.1 Hf 0.1 Ru 0.5 Os 0.7 Mo 0.92 Zr 0.546 Cd 0.56 U 0.2 Ti 0.39 Zn 0.85 Ga 1.083

Mat. Tc (K)

Gd 1.1 Al 1.2 Pa 1.4 Th 1.4 Re 1.4 Tl 2.39 In 3.408 Sn 3.722 Hg 4.153 Ta 4.47 V 5.38 La 6.00 Pb 7.193 Tc 7.77 Nb 9.46

Type I Superconductors

Types II Superconductors •  Starting in 1930 with lead-bismuth alloys, were

found which exhibited superconductivity; they are called Type II superconductors (Hard Superconductors).

•  They were found to have much higher critical fields and therefore could carry much higher current densities while remaining in the superconducting state.

Type II Superconductors

The Critical Field • An important characteristic of all superconductors

is that the superconductivity is "quenched" when the material is exposed to a sufficiently high magnetic field.

•  This magnetic field, Bc, is called the critical field. •  Type II superconductors have two critical fields. •  The first is a low-intensity field, Bc1, which

partially suppresses the superconductivity. •  The second is a much higher critical field, Bc2,

which totally quenches the superconductivity.

The Critical Field • Researcher stated that the upper critical field of

yttrium-barium-copper-oxide is 14 Tesla at liquid nitrogen temperature (77 degrees Kelvin) and at least 60 Tesla at liquid helium temperature.

•  The similar rare earth ceramic oxide, thulium-barium-copper-oxide, was reported to have a critical field of 36 Tesla at liquid nitrogen temperature and 100 Tesla or greater at liquid helium temperature.

The Critical Field •  The critical field, Bc, that destroys the

superconducting effect obeys a parabolic law of the form:

where Bo = constant, T = temperature, Tc = critical temperature.

•  In general, the higher Tc, the higher Bc.

33

Superconductor type II. B-T-Diagram

Mixed phase

Meissner phase

Normal state

Temperature T

Mag

neti

c in

duct

ion

B

STM (Scanning Tunneling Microscopy). Abrikosov-lattice in NbSe2

H. Hess, R.B. Robinson, and J.V. Waszczak, Physica B 169 (1991) 422

Classical model of superconductivity

The lattice deformation creates a region of relative positive charge which can attract another electron.

An electron on the way through the lattice interacts with lattice sites (cations). The electron produces phonon.

1957 John Bardeen, Leon Cooper, and John Robert Schrieffer

During one phonon oscillation an electron can cover a distance of ~104Å. The second electron will be attracted without experiencing the repulsing electrostatic force .

John Bardeen, Leon Neil Cooper, John Robert Schrieffer

Nobel Prize in Physics 1972 "for their jointly developed theory of superconductivity, called the BCS-theory”

e-

e-

Phonon

Coherence length ξ

Cooper pair model

Fermi and Bose-Statistic

• Total spin of C-P is zero. C-P are bosons. Pauli-Principle doesn’t obey.

• All C-P can have the same quantum state with the same energy.

Cooper-Pairs are created with electrons with opposite spins.

• Fermions- elemental particles with 1/2 spin (e.g. electrons, protons, neutrons..)

• Pauli-Principle –every energy level can be occupied with maximum two electrons with opposite spins.

Energy

Density of states

Energy

Density of states

A movement of the C-P when a supercurrent is flowing, is considered as a movement of a centre of the mass of two electrons creating C-P.

Creation of a C-Pairs diminishes energy of electrons. Breaking a pair (e.g. through interaction with impurity site) means increase of the energy.

All the C-P are in the same quantum state with the same energy. A scattering by a lattice imperfection (impurity) can not change quantum state of all C-P at the same time (collective behaviour).

e-

e-

Phonon

38

Good electrical conductors are showing no superconductivity

In case of good conductors is the interaction of carriers with the lattice very week. This is, however, important for superconductivity.

BCS Theory: some consequences

Isotope effect

The Cooper-Pairs are created (“glued”) by the electron-phonon interaction. Energy of the phonons (lattice vibrations) depends on the mass of the lattice site . Superconductivity (Tc) should depend on the mass of the ions (atoms) creating the lattice.

TC~M-α For most of the low-

temperature superconductors α=0.5

39

What destroys superconductivity?

High temperatures: strong thermal vibration of the lattice predominate over the electron-phonon coupling. Magnetic field: the spins of the C-P

will be directed parallel.

(should be antiparallel in C-P)

A current: produces magnetic field which in turn destroys superconductivity.

Current density

Temperature

Magnetic field

JOSEPHSON EFFECT •  JOSEPHSON EFFECT, the flow of electric current, in

the form of electron pairs (called Cooper pairs), between two superconducting materials that are separated by an extremely thin insulator.

•  A steady flow of current through the insulator can be induced by a steady magnetic field.

•  The current flow is termed Josephson current, and the penetration ("tunneling") of the insulator by the Cooper pairs is known as the Josephson effect.

•  Named after the British physicist Brian D. Josephson, who predicted its existence in 1962.

"for his theoretical predictions of the properties of a supercurrent through a tunnel barrier, in particular those phenomena which are generally known as the Josephson effects".

Nobel Prize in Physics 1973

Brian David Josephson Josephson discovered in 1963 tunnelling effect being 23-years old PhD student

The superconducting tunnel Josephson) junction (superconductor–insulator–superconductor tunnel junction (SIS) — is an electronic device consisting of two superconductors separated by a very thin layer of insulating material

I SL SL

x< ξGL

SC SC

Alexei A. Abrikosov, Vitaly L. Ginzburg, Anthony J. Leggett

Nobel Prize in Physics 2003

"for pioneering contributions to the theory of superconductors and superfluids".

“In a phenomenological level, superfluid can flow through narrow capillaries or slits without dissipating energy. Superfluid does not possess the shear viscosity.”

“Liquid helium can become superfluid, that is, its viscosity vanishes at low temperatures. Atoms of the rare isotope 3He have to form pairs analogous with pairs of electrons in metallic superconductors.”

43

Type II Type I

Superconductor Ceramics

•  The ceramic materials used to make superconductors are a class of materials called perovskites.

•  One of these superconductor is an yttrium (Y), barium (Ba) and copper (Cu) composition.

•  Chemical formula is YBa2Cu3O7. •  This superconductor has a critical transition

temperature around 90K, well above liquid nitrogen's 77K temperature.

High Temperature Superconductor (HTS) Ceramics

•  Discovered in 1986, HTS ceramics are working at 77 K, saving a great deal of cost as compared to previously known superconductor alloys.

•  However, as has been noted in a Nobel Prize publication of Bednortz and Muller, these HTS ceramics have two technological disadvantages: ▫  they are brittle and ▫  they degrade under common environmental influences.

HTS CERAMICS • HTS materials the most popular is orthorhombic

YBa2Cu3O7-x (YBCO) ceramics. • Nonoxide/intermetallic solid powders including

MgB2 or CaCuO2 or other ceramics while these ceramics still have significant disadvantages as compared to YBCO raw material.

Table I: Transition temperatures in inorganic superconductors

Compound Tc (K) PbMo6S8 12.6 SnSe2(Co(C5H5)2)0.33 6.1 K3C60 19.3 Cs3C60 40 (15 kbar applied pressure) Ba0.6K0.4BiO3 30 Lal.85Sr0.l5CuO4 40 Ndl.85Ce0.l5CuO4 22 YBa2Cu3O7 90 Tl2Ba2Ca2Cu3O10 125 HgBa2Ca2Cu3O8+d 133

APPLICATIONS: Superconducting Magnetic Levitation

The track are walls with a continuous series of vertical coils of wire mounted inside. The wire in these coils is not a superconductor.

As the train passes each coil, the motion of the superconducting magnet on the train induces a current in these coils, making them electromagnets.

The electromagnets on the train and outside produce forces that levitate the train and keep it centered above the track. In addition, a wave of electric current sweeps down these outside coils and propels the train forward.

The Yamanashi MLX01MagLev Train

APPLICATIONS: Medical

The superconducting magnet coils produce a large and uniform magnetic field inside the patient's body.

MRI (Magnetic Resonance Imaging) scans produce detailed images of soft tissues.

APPLICATIONS: Power

Superconducting Transmission Cable From American Superconductor

The cable configuration features a conductor made from HTS wires wound around a flexible hollow core. Liquid nitrogen flows through the core, cooling the HTS wire to the zero resistance state.

The conductor is surrounded by conventional dielectric insulation. The efficiency of this design reduces losses.

LHC - Largest New Accelerator Project

•  Nb-Ti at 1.9 K at CERN France/Switzerland

•  5000 Superconducting Magnets in 27 km tunnel

•  Beam-steering dipole magnets reach 8.36 T (1.9 K)

http://lhc.web.cern.ch/lhc/general/gen_info.htm

1500 tonnes of SC cables

27 km Tunnel

3286 HTS Leads

Large Hadron Collider

15000 MJ of magnetic energy

1232 SC Dipoles

Switzerland

France