Poole Ch.P., Jr. Handbook of Superconductivity
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6 Chapter 1: Introduction
Table 1.1.
(continued)
Superconducting state expressions
Coherence length
Density of states
Electron-phonon coupling constant
Energy gap ratio
Isotope effect exponent
Specific heat jump
Transition temperature
BCS factors
= 2h v F/~:Eg
Ds(E ) = Dn(0)/[1 -
(A/E)2]I/2
)~- #* : VoDn(EF)
Eg/kB
Tc = 2A/kB Tc = 3.528
1
(ToM ~'
= const), aBcs =
(C s -
7Tc)/~T c
= 1.43
T c = 1.13| exp[--1/(2 -- #*)]
Expressions based on models and theories
Amprre law
Bean model critical current
Critical current
Josephson equations
London equations
Lorentz force
Penetration depth (London)
Penetration depth (Josephson)
Shielding current
V • B = #o J
Jc = 2AMId
A/m 2
Jc -- Bc/#O'~
d4~
J = Jc sin ~b; ~ =
2eV/h
1st E =
#o22dJ/dt
2nd B = -#02~V x J
F/L = J x 9 o
2 L =
(m/~one2) 1/2
2j = [Oo/2rC#OJc(22 +
d)] 1/z
Jsh = V • M
Vortex expressions
Critical field, lower
Critical field, thermodynamic
Critical field, upper
Surface sheath field
Ginzburg-Landau parameter
Vortex field
Bcl = O01n
K/4TC~, 2
B c = O0/2~/-2rt2 {
Bc2
--(I)0/2~ 2 ~
1.83T c
Bc3
= 1.695Bc2
= 2/~
B(r) ,~ e-r/'~/(r/2) 1/2, r >
Anisotropies and shape dependence
Characteristic lengths
Axial symmetry
High-temperature superconductors
Demagnetization factor
Normalization
Flat disk
Long rod
Sphere
Susceptibility
~a,~a --- ~b,~b --- ~c/].c
~ab/~c -- '~c/'~ab ---
(mc/mab) 1/2
~c (( ~ab (( /]'ab (( /]'c
NBin
+ lt0(1 -N)Hin : Bap p
Sa +Sb +Nc
=N~
+eN• = 1
S~ : 1 -~, S~ :
89
Nil -- 6, N_L = 89 -- 6)
N=89
Z = Zexp/( 1 - NZexp)
and conversion factors involving them. In most cases we adopt the SI (Systrme
International d'Unitrs) or mks (meter-kilometer-second) system of units, and for
the convenience of workers who feel more comfortable with cgs (centimeter-
gram-second), some expressions will be repeated in cgs notation. There are also
other units in common use for particular quantities such as pressure and energy,
D. Several Miscellaneous Tables 7
and some of the spectroscopies have their own characteristic units, so conversion
factors will be given where appropriate.
Table 1.2 lists some of the symbols that appear commonly in the super-
conductivity literature, together with their meaning and sometimes with the
conventional units used for them. Table 1.3 lists most of the elements alphabet-
ically by symbol and gives the transition temperatures of the ones that super-
conduct. Table 1.4 provides ionic radii of many of the elements in their more
important valence states. The chapter ends with a long glossary of terms. A
shorter glossary was published in an earlier work [Owens and Poole, 1996].
Table 1.2.
List of symbols that appear commonly in the superconductivity literature. Values and units are given
for some of the entries in this table.
A, amp or ampere, unit of electric current, coulomb/second, C/sec
A, angstrom unit, 1A = 0.1 nm
B, magnetic field symbol, tesla, T
Be, critical magnetic field (some authors write Ho)
Bcl, lower critical field (some authors write He1)
Bc2,
upper critical field (some authors write He2)
c, speed of light in a vacuum, 2.9979 x 108 m/sec
~ degrees Celsius (centrigrade), unit of temperature
cm, centimeter
e, charge of an electron, 1.6022
x 10-19 C
E, electric field, V/m
Eg, energy gap
(Eg--2A)
F, force, newton, N
h, Planck's constant, 6.6261
x 10
TM
J-sec
J, joule, unit of energy
J, electric current density, A/cm 2
J~, critical current density. A/cm 2
K, kelvin, unit of temperature
kB, Boltzmann's constant, 1.3807 x 10 -23 joule/kelvin
m, meter, unit of length
Are, number of valence electrons in an atom
R, resistance, ohm, f~
sec, second
T, tesla, unit of magnetic field B
T~, critical temperature
V, volt
4, coherence length, radius of a vortex core (Greek xi)
~c, Ginzburg Landau parameter, ~c = 2/4 (Greek kappa)
2, penetration depth, electron phonon coupling constant (Greek lambda)
2e, London penetration depth
;(, susceptibility (Greek chi)
O, magnetic flux, with units Tm 2 (Greek phi)
Oo, flux quantum, with value
h/2e--2.0678
x 10 -15 Tm 2 (Greek phi)
8
Chapter 1: Introduction
Table 1.3.
Symbols for most of the elements together with their atomic numbers and the transition temperatures
Tc of those that superconduct. Most of the radioactive elements are not included.
Ag, silver, 47
A1, aluminum, 13; Tc = 1.18 K
Am, americium, 95; Tc = 1.0 K
Ar, argon, 18
As, antimony, 33
Au, gold, 79
B, boron, 5
Ba, barium, 56
Be, beryllium, 4; Tr
Bi, bismuth, 83
Br, bromine, 35
C, carbon, 6
Ca, calcium, 20
Cd, cadmium 48; T~ =0.5 K
Ce, cerium, 58
C1, chlorine, 17
Co, cobalt, 27
Cr, chromium, 24
Cs, cesium, 55
Cu, copper, 29
Dy, dysprosium, 66
Er, erbium, 68
Eu, europium, 63
F, fluorine, 9
Fe, iron, 26
Ga, gallium, 31; T~ = 1.1 K
Gd, gadolinium, 64
Ge, germanium, 32
H, hydrogen, 1
He, helium, 2
Hf, hafnium, 72, Tc = 0.1 K
Hg, mercury, 80; Tc = 4.15 K
Ho, holmium, 67
I, iodine, 53
In, indium, 49; T~ = 3.4 K
Ir, iridium, 77; Tc=0.1K
K, potassium, 19
Kr, krypton, 36
La, lanthanum, 57; Tr = 6.1 K
Li, lithium, 3
Lu, lutetium, 71, Tc = 0.1 K
Mg, magnesium, 12
Mn, manganese, 25
Mo, molybdenum, 42; Tc = 0.9 K
N, nitrogen, 7
Na, sodium, 11
Nb, niobium, 41; Tc = 9.3 K
Nd, neodymium, 60
Ne, neon, 10
Ni, nickel, 28
O, oxygen, 8
Os, osmium, 76; Tc = 0.7
P, phosphorus, 15
Pa, protactinium, 91; Tc = 1.4 K
Pb, lead, 82; Tr = 7.2 K
Pd, palladium, 46
Pm, promethium, 61
Pr, praseodymium, 59
Pt, platinum, 78
Re, rhenium, 75, Tc- 1.7 K
Rb, rubidium, 37
Rh, rhodium, 45
Ru, ruthenium, 44, To = 0.5 K
S, sulfur, 16
Sb, antimony, 51
Sc, scandium, 21, Tr -- 0.5 K
Se, selenium, 34
Si, silicon, 14
Sm, samarium, 62
Sn, tin, 50; Tc = 3.7 K
Sr, strontium, 38
Ta, tantalum, 73; Tc = 4.5 K
Tb, terbium, 65
Tc, technetium, 43; Tc = 7.8 K
Te, tellurium, 52
Th, thorium, 90; Tc = 1.4 K
Ti, titanium, 22; Tc = 0.4 K
T1, thallium, 81; T~ -- 2.4 K
Tm, thulium, 69
U, uranium, 92
V, vanadium, 23; T~ = 5.4 K
W, tungsten (wolfram), 74; Tc=0.02K
Xe, xenon, 54
Y, yttrium, 39
Yb, ytterbium, 70
Zn, zinc, 30; Tc- 0.85 K
Zr, zirconium, 40; Tr = 0.6 K
D. Several Miscellaneous Tables 9
Table 1.4.
Ionic radii in angstroms of selected elements for various positive charge states a [from Poole
et al.,
1988, p. 79]
z Element + 1 + 2 + 3 + 4 + 5 + 6
3 Li 0.68
11 Na 0.97
19 K 1.33
37 Rb 1.47
55 Cs 1.67
4 Be 0.44
12 Mg 0.82
20 Ca 1.18
38 Sr
56 Ba 1.53
5 B 0.35
13 A1
31 Ga 0.81
49 In
81 T1 1.47
6 C
14 Si
32 Ge
50 Sn
82 Pb
15
33
51
83
16
34
52
21
22
23
24
25
26
P
As
Sb
Bi
S
Se
Te
Sc
Ti
V
Cr
Mn
Fe
0.65
0.89
0.98
0.66
0.82
0.96
0.81
Alkali
Alkaline earths
0.35
0.66
0.99
1.12
1.34
Group III
Group IV
0.23
0.51
0.62
0.81
0.95
0.73
0.93
1.20
Group V
0.44
0.58
0.76
0.96
Chalcogenides
First transition series (3dn)
0.81
0.94 0.76
0.88 0.74
0.89 0.63
0.80 0.66
0.74 0.64
0.16
0.42
0.53
0.71
0.84
0.37
0.50
0.70
0.68
0.63
0.60
0.35
0.46
0.62
0.74
0.59
0.30
0.42
0.56
0.52
(continued)
10 Chapter 1:
Introduction
Table 1.4.
[continued}
Ionic radii in angstroms of selected elements for various positive charge states a [from Poole
et al.,
1998, p. 79]
z Element + 1 + 2 + 3 + 4 + 5 + 6
27 Co 0.72
28 Ni 0.69
29 Cu 0.96 0.72
30 Zn 0.88 0.74
39 Y
40 Zr 1.09
41 Nb 1.00
42 Mo 0.93
43 Tc
44 Ru
45 Rh
46 Pd
47 Ag 1.26
48 Cd 1.14
72 Hf
73 Ta
74 W
75 Re
76 Os
77 Ir
78 Pt
79 Au
80 Hg
0.63
Second transition series (4d n)
0.8 a
0.68
0.80
0.89
0.97
Third transition series (5d n)
0.80
1.37
0.85
1.27 1.10
Rare earths (4f n)
57 La 1.39 1.06
58 Ce 1.27 1.07
59 Pr 1.06
60 Nd 1.04
61 Pm 1.06
62 Sm 1.00
63 Eu 0.98
64 Gd 0.62
65 Tb 0.93
66 Dy 0.92
67 Ho 0.91
68 Er 0.89
69 Tm 0.87
70 Yb 0.86
71 Lu 0.85
0.79
0.74
0.70
0.67
0.65
0.78
0.70
0.72
0.88
0.68
0.65
0.94
0.92
0.81
0.69
0.68
0.62
0.62
0.69
aAnion radii are 1.32 for
0 2-,
1.33 for F - 1.84 for
S 2-,
1.91 for Se 2-, and 2.11 for
Ye 2-
(Handbook
of Chemistry and Physics,
p. 187).
E. Glossary
of
Terms 11
Glossary of Terms
A-15 compound An A3B binary compound with a cubic structure, many of
which superconduct.
Abrikosov vortex Cylinder or rod of quantized magnetic flux in a Type II
superconductor, usually referred to simply as vortex.
ACAR Angular correlation of annihilation radiation, measured in positron
annihilation studies.
Acoustic
phonon Phonon of low-frequency branch.
Aligned crystal structure
Cuprate lacking body-centered operation.
Aligned grains
Grains with their magnetizations pointing in more or less the
same direction.
Alloy Two or more metals intimately mixed.
Amp~re's law
V • H- OD/Ot + J.
Angstrom
(A) Unit of length equal to 10 -l~ meters.
Anisotropy Variation of properties along different directions.
Annealing
Heating generally followed by gradual cooling.
Antiferromagnetism
(AFM) Ordered magnetic state in which spins of iden-
tical magnetic atoms are alternately in up and down directions.
Antiparamagnon
Antiferromagnetic fluctuation in Cu02 plane.
Antistokes line
Spectral line arising from a Raman scattered photon with a
frequency greater than that of the incident photon.
Antivortex Vortex aligned opposite to the applied magnetic field.
Anyon Particle with fractional statistics and symmetry under interchange
(0 < 0 < n) between the boson case of 0 = 0 and the fermion case 0 = n,
corresponding to ip(r 2, rl) =
ei~
r2).
ARPES Angular resolved photoemission spectra.
Array of
Josephson junctions
An ordered arrangement of coupled Josephson
junctions.
Atomic
number (of
an atom)
The number of protons in its nucleus.
Auger effect
Radiationless transition whereby a photon generated within an
atom ejects an electron from a higher energy level.
Aurivillius phase
Layered compound
BizOz(Mm_lRmO3m+l)
related to the
cuprates.
Avalanche or cascade process
One involving rapid change or discharge.
Band structure
Arrangement of energy bands in k-space.
BCS Superconductivity theory originated by J. Bardeen, L. N. Cooper, and J. R.
Schrieffer in 1957.
Bean model
Most common critical state model in which the critical current
density can equal either 0 or the value Jc.
Bessel function
Radial solution of Laplace's equation in cylindrical coordi-
nates. The zero-order modified Bessel function
Ko(r/2 )
describes the radial
12 Chapter 1: Introduction
dependence of the magnetic field around a vortex for • >> 1, and
Kl(r/2 )
describes the vortex critical current density.
Binary compound One containing two elements.
BIS,
Bremsstrahlen isochromat spectroscopy
The sample is irradiated with
an electron beam and the emitted ultraviolet photons are measured.
Bloch T s law for electrical resistivity
p -
po T5
for T << |
Bloch's theorem
~F(r -+- R) -- e/kR~lJ(r) for wavefunctions in periodic lattice.
Bloch wavefunctions
~p(r) obey Bloch's theorem and are linear combinations
of atomic states ~b(r- R) localized on atoms at positions
R.
Bogoliubon
Quasi-particle excitation.
Bogoliubov transformation
A transformation involving raising and lowering
operators that is used in solving the BCS Hamiltonian.
Borocarbide
A compound containing the elements boron B and carbon C such
as R_N42B2C,
where ordinarily M is Ni and R is a rare earth.
Boronitride
A compound containing the elements boron B and nitrogen N
such as RMzBzN 3, where ordinarily M is the element Ni and R is a rare earth.
Bose-Einstein condensation
Passage of a system of boson particles such as
Cooper pairs or helium atoms to their lowest energy state where they exhibit
special properties such as superconductivity or superfluidity; also called Bose
condensation.
Boson A particle with integer intrinsic spin obeying Bose-Einstein statistics.
Branch imbalance
A nonequilibrium state with an imbalance between quasi-
particles in two branches of k-space.
Brillouin scattering
Light scattering in which the frequency shift of the
reflected beam arises from an acoustic branch phonon.
Brinouin zone
Unit cell in reciprocal space (k-space).
Buckminsterfullerene
Fancy name for a fullerene.
Bucky
ball Nickname for a fullerene.
Bundle of
flux A group of flux lines (vortices) that undergo motions in unison.
Cascade process
See Avalanche.
CDW Charge density wave.
Celsius (centigrade)
C Related to Fahrenheit through C = 5 (F- 32)/9.
Ceramic
A product made from a clay or a related material.
Chalcogenide
One of the four elements oxygen O, sulfur S, selenium Se, and
tellurium Te in row 6 of the periodic table.
Charge density wave
(CDW) Periodic fluctuation in the density of charge.
Charge imbalance
See Branch imbalance.
Charge transfer organics
Class of organic superconductors.
Chemical shift
A shif~ in the resonant frequency of a spectral line; terminology
common to NMR and M6ssbauer spectroscopies.
Chevrel compound
Generally a ternary compound AjMo6X 8, where X is
sulfur S, selenium Se, or tellurium Te, A can be almost any element, and j has
a value between 1 and 2; many Chevrel compounds superconduct.
E. Glossary of Terms
13
Clean superconductor
One in which the electron mean free path l exceeds the
coherence length ~.
Clogston-Chandrasekhar limit See Paramagnetic limit.
Coercive
field The applied magnetic field needed to restore a magnetized
material to the state of zero magnetization.
Coherence
length ~ Distance ~rL over which Ginsburg-Landau order param-
eter qS(r) can change without an appreciable energy change; radius of core of a
vortex;
size
~Pippard of a Cooper pair.
Collective pinning
Restricted motion of many nearby vortices arising from an
assembly of weak pinning centers.
Composite
Made of separate or disparate elements or parts.
Condensation energy density
Energy released by transforming normal elec-
trons to super-electron state, with the value
B2c/2#o .
Conductivity, electrical
Ratio a =
J/E
of the current density J to the applied
electric field E that causes it to flow.
Conductivity, thermal
Ratio lc =
-U/VT
of the heat current per unit area U
to the temperature gradient V T that causes it to flow.
Conductor
Material of low resistivity, i.e., conducts electricity well.
Cooper pair
Paired electrons that constitute the charge carrier of supercurrent.
Coulomb Mks or SI unit of electrical charge.
Coulomb blockade Blockage of single electron jumps in ultrasmall Josephson
junctions (nanobridges).
Coulomb integral
f~t~(rm)~3(rn)g(rmn)~A(rm)~B(rn)d3r.
Coulomb staircase Steps on current vs voltage plots of ultrasmall Josephson
junctions arising from fluctuations in single-electron tunneling.
Covalent bonding
Chemical bonding scheme in which the involved atoms
share electrons; commonly found in organic molecules.
Creation operator
Quantum mechanical operator that creates a particle.
Critical
current density
Je Highest current density that can flow through a
superconducting material without driving it normal.
Critical current I c Highest current that can flow through a particular super-
conducting wire or tape.
Critical magnetic field B c Highest field sustained by a Type I superconductor.
Type II superconductors have lower Bcl and upper Be2 critical fields with
thermodynamic critical field B c =
(Bc1Bc2 /
In
K) 1"2.
Critical state Magnetic field penetration and associated super current flow
configuration in surface region of superconductor in accordance with a particular
model such as the Bean model (q.v.).
Critical surface Surface in B, J, T (magnetic field, current density, temperature)
coordinate system below which a material is superconducting.
Critical temperature T c Temperature below which a material superconducts.
Cryogenic
Adjective signifying low temperature.
Cryostat
Apparatus for maintaining a low, constant temperature.
14 Chapter 1: Introduction
Cryotron Type of superconducting switch.
Cubic structure All lengths equal, a : b - c; all angles 90 ~
Curie law for susceptibility Z =
C/T,
where C =
n#Z/3k B
is the Curie
constant.
Curie-Weiss law X =
C/(T- |
where | is the Curie-Weiss temperature.
Curie temperature The temperature below which a material becomes ferro-
magnetic.
Current density J Electric current per unit cross-section, A/cm 2.
D'Alembertian Differential Operator
I---]2 = V 2 _ c-2
02/0t2.
Debye approximation Assumption of linear dispersion relation 09 =
ck
in
phonon specific heat determination.
Debye temperature | =
h~
a measure of the temperature above which
all vibrational modes begin to be excited in a solid.
dettaas-van Alphen effect Oscillations in the magnetization exhibited by a
sample during magnetic field scanning.
Delocalization Continual wandering throughout a material of electrons that, for
example, carry electrical current.
Demagnetization Removal of magnetization.
Demagnetization factor N i Parameter characterizing shape dependence of
magnetization in a material.
Density of states (DOS) Number of states or levels per unit energy,
dN/dE,
or
number of states per frequency interval,
dn/do3.
Depairing Uncoupling of bound electron pair.
Depinning Release of vortices from being pinned.
Dewar Container for holding a low-temperature liquid.
Diamond A cubic form of carbon with each atom at the center of a regular
tetrahedron.
Dielectric A material with a dielectric constant e > %.
Dirty superconductor One in which the electron mean free path I is less than
the coherence length ~.
Dispersion relation k-dependence of the frequency og(k).
Distribution function Energy dependence of electron density.
Doping Adding a small amount of one atom to replace another, as in the
superconductor (La0.9Sr0.1
)2CUO4 .
Doppler broadening Spread in energy of monochromatic 7-rays arising from
positron annihilation.
DOS Density of states.
Drude model Description of electrical and thermal conductivity of a metal in
terms of a gas of conduction electrons.
d-wave pairing Cooper pair coupling via d-state wavefunctions.
Eccentricity ~ Measure of the deviation of an ellipsoid from a spherical shape.
EDAX or EDX Energy dispersive analysis by X-rays; provides small-area
atomic composition analysis.
EELS Electron energy loss spectroscopy.
E. Glossary of Terms 15
Effective mass of electron
Experimentally measured mass m* that deviates
from actual mass m e .
Electron microscopy
(EM) Electron beam plays role of light beam in conven-
tional microscope.
Electron paramagnetic resonance
(EPR) Alternate name for electron spin
resonance.
Electron-phonon interaction
Cooper pair coupling mechanism in classical
superconductors.
Electron spin resonance
(ESR) Microwave detection of magnetic moments of
unpaired (electron) spins.
Eliashberg relation Integral of the product of the electron phonon coupling
strength ~(~o), the phonon density of states Dph(O~ ), and co -1 for determining the
electron-phonon coupling constant 2.
Energy band
Set of very closely spaced energy levels.
Energy gap
Separation in energy between two energy bands.
Enthalpy
H = U +
P V.
Entropy
S Thermodynamic measure of disorder.
Epitaxial film A
thin-film single crystal, in the case of the cuprates made with
the copper oxide planes parallel to the surface.
Ettingshausen effect
Establishment of transverse temperature gradient in a
conductor carrying a uniform current and maintained in a constant magnetic field
at a constant temperature.
EXAFS Extended X-ray absorption fine structure.
Exchange integral
~t~(rm)~t~(rn)V(rmn)~n(rn)~B(rm)d3r.
Exciton
Bound electron-hole pair.
Exclusion
of flux Inability of magnetic field lines to enter superconductor
cooled below T c in zero field and then placed in a field.
Expulsion
of flux Exit of magnetic field lines from a material when it is cooled
below T c in an applied magnetic field.
Extinction Decrease in intensity of light passing through a medium.
Extrude To thrust out--for example, in forming a material such as a wire to a
desired cross-section by forcing it through a die.
Face centered cubic
(fee) Crystal structure in which the unit cell is a cube with
atoms at the vertices and in the centers of the faces.
Faraday's law V x E =
-OB/Ot.
Fermi-Dirac statistics
Statistics of indistinguishable particles of half integer
spin S = 1/2, 3/2, ... obeying the Pauli exclusion principle.
Fermi energy
Energy of highest occupied level in k space.
Fermi gas
Collection of noninteracting fermions.
Fermi level
Er Uppermost occupied fermion energy level.
Fermi liquid
Collection of weakly interacting fermions confined, at absolute
zero, to occupy all energy states below the Fermi surface.
Fermion
Particle such as an electron with half integer spin obeying Fermi-
Dirac statistics.