Buschow K.H.J. (Ed.) Concise Encyclopedia of Magnetic and Superconducting Materials

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where N(0) is the normal state density of states at the

Fermi level.

A superconductor has a set of quasiparticle exci-

tations in one-to-one correspondence with those of

the normal metal. The energy of such an excitation is

, corresponding to that of a particle above the

Fermi sea or a hole below. Since excitations are

formed in pairs from the ground state, the minimum

energy required for creating a pair of excitations at

the Fermi surface is the energy gap 2D.

The probability that a state k is occupied at ﬁnite

temperatures is given by the Fermi function

1 þ expðbE

ð11Þ

where b

1

¼ k

T. The gap parameter D decreases

with increasing temperature and goes to zero at the

critical temperature T

. The value of DðTÞ can be

found from the equation

1 ¼ Nð0ÞV

ð1  2f

Þdx

ð12Þ

This equation is commonly referred to as the

BCS gap equation. Near T

, DðTÞ=Dð0ÞC1:74½1

ðT=T



1=2

. The gap goes to zero as T-T

and there

are no solutions for T4T

. The value of T

is given

¼ 1:13_o

exp½1=Nð0ÞVð13Þ

The energy gap at T ¼0 is proportional to T

¼ 2Dð0Þ¼3: 52k

ð14Þ

In this weak coupling approximation there is only

one parameter, D(0) or k

, rather than two, _o

and N(0)V. Weak coupling means that k

is very

small compared with the phonon energies so that the

phonon spectrum is not important. In this limit there

is a law of corresponding states since everything

scales with T

. Strong coupling and effects of cou-

lomb interactions will be discussed later.

With a ground state and set of quasiparticle exci-

tations it is possible to calculate transition probabil-

ities and various response functions without much

more difﬁculty than for normal metals. The main

complications arise from coherence factors that are a

result of the pairing. In normal metals scattering from

ks to k

is independent in ﬁrst approximation from

all other scattering. In a superconductor it is possible

to go from km to k

m either directly or by scattering

from k

k to km and these two contributions must

be added coherently. Depending on the nature of the

scattering, they add constructively or destructively.

The coherence factors can be calculated most easily

by a reformulation of the BCS theory given inde-

pendently by Valatin and by Bogoliubov. They deﬁne

quasiparticle operators for a superconductor as linear

combinations of creation and destruction operators

for normal quasiparticles:



¼ u



 v

kk

ð15aÞ



kk

¼ u



kk

þ v

ð15bÞ

These operators do not conserve particle number and

are designed to operate on wave functions of the type

of Eqn. (9) that are not eigenstates of the number of

pairs N. The ground state C

is the vacuum state for

quasiparticle excitations:

¼ g

kk

¼ 0 ð16Þ

When there is current ﬂow, the paired states

ðk

m; k

kÞ all have precisely the same momentum

¼ _ðk

þ k

Þ so that any pair can be scattered into

another pair. There is thus a long-range order of the

momentum distribution as envisaged by London. In-

dividual electrons can be scattered, but such scatter-

ing does not change the common momentum of the

paired states.

As in the Ginzburg–Landau theory, the canonical

momentum p

can be expressed as the gradient of a

phase j (see Josephson Effects in Weak Links). There

is an arbitrary phase factor in the ground-state wave

function since v

can be replaced by v

exp(ij) with-

out changing the energy. The degenerate wave func-

tion is an eigenstate of phase

ðjÞ¼

ðu

þ v

expðijÞc



kk

ÞC

expðiNjÞC

ð17Þ

The phase and the number of pairs N are conjugate

variables, with N

¼i@=@j. If the number of pairs

is ﬁxed, the phase is undetermined. The phase cor-

responds to that of the Ginzburg–Landau theory,

with p

¼ _rj the momentum of a pair.

It was trying to understand whether phase differ-

ences between two superconductors connected by a

weak link could be observed that led Josephson to

predict that a supercurrent should ﬂow through a

tunnel junction separating two superconductors dif-

fering in phase. The importance of phase as a variable

describing the superconducting state had been

stressed by Anderson. The Josephson effect was ﬁrst

observed by Anderson and Rowell. It has led to many

practical applications.

3. Gor’kov–Eliashberg Formulation

Not long after the ﬁrst publications of the BCS

theory, Gor’kov used the method of thermal Green’s

functions to treat problems in which there is a space

variation of the order parameter D(r) as well as

1150

Superconducting Materials: BCS and Phenomenological Theories

problems in which there is strong coupling so that it

is necessary to include the energy dependence of D.

One of his ﬁrst applications was to show that the

theory reduces to the Ginzburg–Landau theory near

, with D(r) playing the role of the order parameter

CðrÞ. An important difference is that the charge of a

pair, 2e, replaces the electronic charge e.

Eliashberg used the Gor’kov formulation to derive

a gap equation sufﬁciently general to include strong

coupling. The electron–phonon interaction enters the

integral equation in the form a

ðoÞFðoÞ, where

FðoÞdo gives the number of phonons with energies

in the interval _do and aðoÞ is the electron–phonon

coupling constant averaged over all directions in

space. There are two coupled integral equations for

DðoÞ and for the mass renormalization factor Z

ðoÞ

from the electron–phonon interaction, both of which

are in general complex.

For weak coupling, Z

is the same as Z

and is

the factor by which the density of states in energy

is enhanced by the electron–phonon interaction.

Expressed in terms of an interaction parameter l,

¼Z

¼1 þl where

l ¼ 2

ðoÞFðoÞdo=o ð18Þ

The parameter l corresponds to N(0)V

of the BCS

theory, with the phonon energy _o

, deﬁned by

¼ 2

ðoÞFðoÞdo=l ð19Þ

The repulsive Coulomb interaction between elec-

trons can be included by renormalizing the screened

interaction V

to the same energy range as the phon-

ons. The interaction constant m ¼ Nð0Þ/V

S for

scattering between states extending over an energy of

the order of E

from the Fermi surface is replaced by

a renormalized constant



¼ m=½1 þ m lnðE

=_o

Þ ð20Þ

for scattering within an energy _o

of E

. Virtual

scattering to states outside of this range is included in

the renormalized interaction. The expression for T

for weak coupling is then

¼ 1:13_o

exp½1=ðl  m



Þ ð21Þ

Normally, m



varies only slowly with electron density

and is typically in the range 0.10–0.15. It may, how-

ever, change the exponent a in the isotope effect

a

from the value

because m



can vary with

isotopic mass through its dependence on _o

McMillan used the Eliashberg equations to calcu-

late T

for a series of transition metals and found

that the results could be ﬁtted approximately by the

semiempirical expression

1:45



exp 

1:04ð1 þ lÞ

½l  m



ð1 þ 0:62lÞ



ð22Þ

where Y

is the Debye temperature. Values of l

range from about 0.25 for metals with low T

above 1.25 for strongly coupled superconductors with

high T

Metals and compounds with high T

are those with

a reasonably large l and high phonon energies as

deﬁned by Eqn. (19). It is difﬁcult to ﬁnd materials

that satisfy both criteria because a large l is usually

associated with a soft lattice and consequently low

phonon frequencies. Examples of soft metals are

mercury (T

C4.2 K) and lead (T

C7.2 K). A high

density of states N(0) such as is found with transition

metals is helpful, but within a range of alloys it is

often found that l increases inversely with the density

of states. The matrix elements of the electron–phonon

coupling go in the opposite direction and more than

compensate for the decrease in N(0). Materials with

high T

have been discovered by empirical methods

rather than by suggestions from theory.

The Gor’kov–Eliashberg theory has been used to

calculate T

essentially from ﬁrst principles for simple

metals and alloys such as aluminum (weak coupling),

tin (moderate coupling) and lead (strong coupling).

The phonon spectra are derived from neutron scat-

tering data and a pseudopotential method is used

to calculate a

ðoÞ. The good agreement with exper-

imental values leaves little doubt about the basic

validity of the theory.

Of even greater interest is the application of

equations to derive a

ðoÞFðoÞ empirically from the

current–voltage characteristics of a tunnel junction

between a normal metal (N) and the superconductor

(S) separated by a thin insulating layer. At low tem-

peratures (dI/dV)

is portional to the density of

states in the superconductor. The ratio to the con-

ductance measured when both sides are normal is for

a voltage V corresponding to eV ¼ E ¼ _o







ðEÞ

Nð0Þ

ðE

 D

1=2

ð23Þ

where N(0) is the uniform density of states in the

normal metal and the expression on the right is for

weak coupling. The current vanishes when EoD.

For strong coupling, when D(E) is complex,

Schrieffer and co-workers showed that

ðEÞ

Nð0Þ

¼ Re

½E

 DðEÞ



1=2

()

ð24Þ

If the conductance is measured over a broad energy

range, Kramers–Kro

nig relations can be used to de-

termine both the real and imaginary parts of D(E)

and thus the complete Green’s functions required to

1151

Superconducting Materials: BCS and Phenomenological Theories

calculate equilibrium and transport properties. This

procedure has given excellent agreement between

theory and experiment, even when there is strong

coupling.

McMillan devised a computer program to deter-

mine a

ðoÞFðoÞ directly from tunnelling data with

use of the Eliashberg integral equations. This method

has been used for a large number of superconducting

elements and compounds to obtain detailed informa-

tion about a

ðoÞFðoÞ and its variation with energy.

It is necessary to estimate m



from the electron den-

sity; the only other variable in the equations is then

ðoÞFðoÞ. Generally, the peaks of a

ðoÞFðoÞ coin-

cide with the peaks of the phonon density of states

FðoÞ measured by neutron scattering.

4. Basic Features and Extensions of the

Pairing Theory

At low temperatures, the normal state is described as

a weakly interacting gas of quasiparticles and phon-

ons. The quasiparticles correspond to electrons above

or holes below the Fermi surface.

The fully renormalized quasiparticles are in one-to-

one correspondence with those of a gas of non-inter-

acting particles. Because of interactions, the quasi-

particle excitations are not exact eigenstates of the

normal metal but have a ﬁnite lifetime. According to

the Landau Fermi liquid theory, the lifetime varies at

low temperature as o

2

, where o is the quasiparticle

energy relative to that of the Fermi surface. Thus the

approximation treating the quasiparticles as inde-

pendent becomes better and better as the temperature

is decreased.

The wave functions of the superconducting state

are described by linear combinations of low-lying

normal-state conﬁgurations. If the energies of the

normal-state conﬁgurations are derived empirically,

the free energy difference between normal and su-

perconducting states may be derived with far greater

precision than would be possible by a ﬁrst-principles

calculation of either phase by itself.

What is needed is the matrix element for scattering

of a pair of quasiparticles in state ðk

; k

Þ to ðk

; k

with conservation of wave vector:

þ k

-k

þ k

ð25Þ

When the interaction is mediated by the electron–

phonon interaction it may be regarded as occurring

in two steps:

-k

þ q

ð26aÞ

þ q

-k

ð26bÞ

where q

is the wave vector of the phonon. An al-

ternative process is the same as in Eqn. (26) with k

and k

interchanged. Energy is conserved only in the

ﬁnal step. The superconducting phase is formed from

a subset of low-lying normal conﬁgurations between

which the matrix elements are predominantly nega-

tive, corresponding to an attractive interaction.

The quasiparticles in the subset are associated in

pairs each of which has exactly the same momentum

as every other pair. When there is no current ﬂow the

paired states have opposite spin and momentum.

One of the earliest extensions of the original BCS

theory, by Mattis and Bardeen, was to include effects

of a scattering mean free path l. In this case, the mo-

mentum k is not a good quantum number to describe

the quasiparticles. A quasiparticle state cm is paired

with the time-reversed state c



k. The main effect is to

replace the inverse of the Pippard distance, x

1

, with

1

þ l

1

, as in Eqn. (4).

Not long after the microscopic theory was pub-

lished suggestions were made to get an attractive in-

teraction between quasiparticles other than that

mediated by phonons. These are not necessarily al-

ternatives to the phonon mechanism but may sup-

plement it so as to enhance the transition

temperature. Although none has been ﬁrmly estab-

lished, it is likely that something besides phonons is

required to account for the high transition temper-

ature of compounds with copper oxide layers, ﬁrst

discovered by Bednorz and Mu

ller.

Other low-lying excitations that have been sug-

gested are, like phonons, deﬁned by a wave vector.

They include excitons (creation of electron–hole

pairs), acoustic plasmons (in a ground state with

equal numbers of conduction electrons and holes in

overlapping bands) and spin and charge-density wave

excitations (where there is nesting of opposite faces of

the Fermi surface).

Little and, independently, Ginzburg suggested an

exciton mechanism in which there is physical separa-

tion of the excited electron–hole pair and the carriers

that give the current ﬂow. Little suggested conduc-

tion in linear chains surrounded by polarizable side

chains. Ginzburg suggested layer structures with con-

ductive layers alternating with polarizable layers in

which the electron–hole excitations take place.

Both may have found realizations in two classes of

compounds with abnormally high superconducting

transition temperatures, although it is still far from

certain that it is the exciton mechanism that is re-

sponsible. Ginzburg’s suggested geometry is similar

to that of the high-T

copper oxide layer structures.

Little’s is similar to that of organic linear-chain com-

pounds that have been found by Williams and co-

workers to have transition temperatures as high as

13 K. However it has been found that in both systems

the normal metal is not described by a simple Fermi

liquid. More radical departures from the usual BCS

theory than getting enhanced attractive interactions

may be required. In both systems, experiments indi-

cate that in the normal state the reciprocal of the

quasiparticle lifetime varies as o rather than as o

1152

Superconducting Materials: BCS and Phenomenological Theories

The energy gap in planar directions in the high-T

oxides is about 8k

, much larger than the BCS

value of 3.52 k

for weak coupling. Some exten-

sions and changes of the BCS theory are required to

account for the high-T

oxides and organic linear-

chain superconductors. It is uncertain whether or not

these changes are minor or require a major restruc-

turing of the theory.

5. Applications to Other Areas of Physics

The concepts and mathematical methods of the the-

ory of superconductivity have been applied success-

fully to the superﬂuid phase of liquid helium of

isotopic mass three, to the structure of nuclei, to

pulsars and neutron stars and to the elementary par-

ticles of high-energy physics. Liquid

He, like elec-

trons in metals, obeys Fermi–Dirac statistics. The

liquid undergoes a pairing transition to a superﬂuid

phase below 3 mK. The pairing is of parallel spins in

a triplet state (S ¼1) rather than the singlet spin

pairing (S ¼0) that occurs with electrons in super-

conducting metals. Extensions of superconductivity

theory have been applied with great success to ac-

count for the pairing energy of neutrons and protons

in nuclei. The pairing energy gap decreases with in-

creasing atomic number. The biggest superﬂuid ob-

jects in the universe are pulsars and neutron stars in

which there is evidence that both neutrons and pro-

tons undergo pairing transitions.

To deﬁne the state of a superconductor it is nec-

essary to specify the superﬂuid velocity v

in addition

to the other macroscopic variables, such as temper-

ature and pressure. This is associated with what is

called broken-gauge symmetry. Analogous concepts

are now being used widely to better understand the

structure of the particles (elementary excitations) of

high-energy physics.

See also: Amorphous Superconductors; Coherence

Length, Proximity Effect and Fluctuations; Electro-

dynamics of Superconductors: Flux Properties; Elec-

trodynamics of Superconductors: Weakly Coupled;

Superconducting Thin Films: Materials, Preparation

and Properties; Superconducting Thin Films: Multi-

layers

Bibliography

Bardeen J 1973 Electron–phonon interactions and supercon-

ductivity. In: Haken H, Wagner M (eds.) 1973 Cooperative

Phenomena. Springer, Berlin

Bardeen J, Schrieffer J R 1961 Recent developments in super-

conductivity. In: Gorter C J (ed.) 1961 Progress in Low Tem-

perature Physics, Vol. 3. North-Holland Amsterdam

Chilton F (ed.) 1971 Superconductivity: Proc. Int. Conf. Science

of Superconductivity. North-Holland, Amsterdam

de Gennes P G 1966 Superconductivity in Metals and Alloys.

Benjamin, New York

London F 1950 Superﬂuids, Vol. 1. Wiley, London

Parks R D (ed.) 1969 Superconductivity, Vols. 1, 2. Dekker,

New York

Rickayzen G 1965 Theory of Superconductivity. Wiley, London

Shoenberg D 1952 Superconductivity. Cambridge University

Press, Cambridge

Schrieffer J R 1964 Theory of Superconductivity. Benjamin, New

York

Tinkham M 1975 Introduction to Superconductivity. McGraw-

Hill, New York

J. Bardeen

University of Illinois, Urbana, Illinois, USA

Superconducting Materials: Irradi ation

Effects

Irradiating superconducting materials changes their

physical properties, such as critical temperature, T

critical current, J

, and critical magnetic ﬁeld, H

(Brown 1981). Therefore, irradiation effects on su-

perconducting materials are important to the techno-

logical applications and basic understanding of

superconductivity. These effects depend on the mate-

rial and the nature of the radiation itself. The radi-

ation source can be x- or g-rays, electrons, fast or

thermal neutrons, or light or heavy ions. The irradi-

ation effects depend on the type of radiation source,

radiation energy, ﬂux density, and accumulated dos-

age. From a technological point of view, fast neutrons

and g-rays are the principal radiation sources. Both

g-rays and neutrons interact with materials through

electrically charged particles—electrons, light or

heavy recoil ions, or charged reaction products in

transmutations. Ion irradiation can be well controlled

for energy, dose, damage type and density, irradiation

temperature, and damage orientation, etc., to provide

an understanding of the physical processes occurring

in a material. Therefore, the emphasis of this article is

placed on irradiation effects caused by ions.

1. Energy Loss and Defect Formation

Energy loss is the most important factor in ion–solid

interactions, because it is a measure of the capacity

of a particle to deposit energy within a material.

Energy loss is a material property, which is a function

of the mass, atomic number, and energy of the

particle. Figure 1 shows the energy dependence of

electronic and nuclear energy loss of various ions in a

typical high-temperature superconductor (HTS),

namely YBa

. Ions of hydrogen, helium, bro-

mine, tin, and gold are used for illustration.

1153

Superconducting Materials: Irradiation Effects

Energy loss is expressed in units of keV nm

1

and

the ion energy is normalized by the mass number of

the projectile. Subscripts ‘‘n’’ and ‘‘e’’ label the nu-

clear and electronic energy loss, respectively, for a

given ion in the HTS. Figure 1 has three straight lines

that mark the approximate boundary of different

regions of significance. The upper left area of the in-

clined line is a region where nuclear energy loss is

greater than electronic energy loss. In this region, the

energy of the projectile is transferred to the material

via collisions that displace the target atoms, creating

point defects. When the energy of the recoiled atoms

is high enough, these recoiled atoms will subsequently

strike others, creating a collision cascade.

At higher ion energy (across the inclined straight

line in Fig. 1), the energy loss is dominated by the

ionization and excitation of the electrons of the target

material with consequential diluded defect formation,

which has very little effect on the properties of su-

perconductors. However, for heavy ions of GeV en-

ergies, when the electronic energy loss is close to the

maximum, the highly striped multicharged heavy ions

can interact with and set in motion a large number of

target electrons. This dense ionization in a small area

will leave a high density of atomic disorders owing to

a thermal spike (Szenes 1996). At electronic energy

losses higher than 11 keV nm

1

in YBCO (marked as

a horizontal dotted straight line in Fig. 1), local de-

fects with high densities of atom disorders may form

droplet-like and discontinuous columnar defects

along the fast ion track. Above a critical electronic

energy loss between 28 keV nm

1

and 35 keV nm

1

(marked as a horizontal straight line in Fig. 1),

entirely continuous amorphous tracks with diameters

of a few nanometers are produced along the ion

path (Wheeler et al. 1993). Different effects on super-

conductivity will result from these defects. Before

we correlate the defect structure with the super-

conductor properties, we need to understand the

magnetic properties of type II superconductors.

2. Defects and Pinning Centers in Type II

Superconductors

HTSs and all the technologically relevant conven-

tional superconductors are type II superconductors.

In these materials, the magnetic ﬁeld penetrates in

the form of an array of ﬂux lines called supercon-

ducting vortices (Blatter et al. 1994). These vortices

are string-like objects consisting of a central core

(whose radius is approximately the coherence length,

x), where the order parameter, c(r), is depressed,

surrounded by supercurrents extending over a larger

distance. In a defect-free superconductor, the ground

state of the vortex array is a triangular Abrikosov

lattice. As electric currents exert force on the vor-

tices, any transport current density, J, results in a

dissipative vortex motion (i.e., ﬁnite resistance) and

the most relevant property for application is lost.

Figure 1

Defect formation in different energy loss regions and its dependence on incident particle energy. See text for details.

1154

Superconducting Materials: Irradiation Effects

Crystallographic defects may locally depress c(r). If

a core sits on one of these defects, the energy of

the vortex decreases. This position-dependent energy

distorts the lattice and produces vortex pinning, i.e.,

ﬂux line motion is prevented until J surpasses J

Controlling the structure of defects in the material

is thus the key for the optimization of vortex pinning.

The relationship between the elementary pinning

force, f

, of one defect and J

is rather complex, but

several basic rules exist (Ullmaier 1975). First, the

best pinners are those defects that are just large

enough to accommodate a vortex core, of transverse

size Bx. Second, to increase the total pinning force

the density of defects should be large. Finally, defects

elongated along the H direction (columnar defects)

are particularly appropriate, as they are able to con-

ﬁne large portions of a core.

3. Irradiation Effects in Superconductors

The methods of optimizing pinning in conventional

superconductors have improved (Brown 1981). The

inclusion of nonsuperconducting particles (as in Nb–

Ti wires) or the introduction of dislocations by met-

allurgical techniques are well-known procedures.

Amorphous superconductors are less sensitive to ra-

diation damage, owing to the absence of pre-irradi-

ation crystalline order. Irradiation of clean materials

with very low initial J

has been used as a tool to

investigate vortex pinning in samples with a well-

characterized structure of defects.

The very small x of HTSs (B1–2 nm) suggests that

irradiation may be a very appropriate tool for pin-

ning enhancement. The small x and the lack of duc-

tility prevent the use of many metallurgical

techniques to create defects. The controlled genera-

tion of defects by irradiation is a powerful tool in

exploring the novel properties of the vortex matter.

Irradiation of HTSs with light particles, which

produce a random distribution of localized defects

(Civale 1993), reduces T

and increases the normal

state resistivity. The initial dT

/dF for different ions

and energies is proportional to the nuclear energy loss

rate, conﬁrming that damage is only due to atomic

collisions. For higher F the defective regions start to

merge, and T

decreases faster.

Vortex pinning effects of irradiation in HTSs are

best explored in high-quality single crystals, where

weak links are absent and J

before irradiation is low.

The introduction of random defects in HTS crystals

enhances J

, and cascade defects generated by fast

neutrons are very effective. In YBCO, for instance, J

may increase by an order of magnitude (up to

B10

Acm

2

) at 5 K and H of a few tesla, and by

two orders of magnitude (up to B10

Acm

2

)at77K

and low H. Protons of 2–3 MeV produce similar re-

sults. J

increases with F at low doses, reaches a

broad maximum, and eventually drops abruptly at

the F range where T

also starts to collapse. As ex-

pected, optimum J

occurs when the distance between

defects is a few times the value of x. The damage

produced by electrons in the 1–3 Me V energy range,

which consists mostly of point defects (Frenkel pairs),

also produces a small, but still considerable increase

in J

in YBCO and Bi-2212.

Aligned columnar defects (CD) produced by

heavy-ion irradiation are the strongest pinning cen-

ters in HTSs, because of simple geometrical reasons

(Civale 1993). First, they can conﬁne the whole length

of the core, gaining a large pinning energy without

the energy cost of an elastic distortion. Second, their

diameter (5–10 nm) is appropriate to exert a very

large pinning force (Wheeler et al. 1993). The advan-

tage with respect to random defects is dramatic at

high T and H. For instance, for YBCO at 77 K and

5T, J

410

Acm

2

can be obtained. The density of

CD can be expressed in terms of the equivalent

matching ﬁeld, the ﬁeld at which the CD and vortex

densities are the same.

CD generate uniaxial pinning, i.e., the J

increase is

anisotropic, being largest when H is parallel to the

tracks. Uniaxial pinning gives rise to the lock-in effect

(Blatter et al. 1994): in the solid phase, H can be tilted

away from the direction of the tracks by a ﬁnite angle

and the vortices will still remain trapped in the CD.

Irradiation of Bi-2212 and Bi-2223 with B1GeV

protons has been used to produce ﬁssion of the bis-

muth nuclei (Civale 1993). The damage created by the

ﬁssion fragments consists of amorphous tracks sim-

ilar to the aligned CD, but randomly oriented. These

isotropic distributions of CD also produce large in-

creases in J

, with the technological advantage that

the penetration range of the protons is large

(B50 cm), in contrast to the few micrometers of

heavy ions. Similar ﬁssion damage has been intro-

duced in uranium-doped YBCO by irradiation with

thermal neutrons.

See also: High-temperature Superconductors: Thin

Film and Multilayers; Superconducting Materials:

Types of; Superconducting Permanent Magnets:

Principles and Results

Bibliography

Blatter G, Feigel’man M V, Geshkenbein V B, Larkin A I,

Vinokur V M 1994 Vortices in high temperature supercon-

ductors. Rev. Mod. Phys. 66, 1125–388

Brown B S 1981 Radiation effects in superconducting fusion-

magnet materials. J. Nucl. Mater. 97, 1–14

Civale L 1993 In: Jin S (ed.) Processing and Properties of High

Superconductors. World Scientific, Singapore, pp. 299–369

Szenes G 1996 Formation of columnar defects in high-T

su-

perconductors by swift heavy ions. Phys. Rev. B 54, 12458–63

Ullmaier H 1975 Irreversible Properties of Type II Supercon-

ductors. Springer, Berlin

1155

Superconducting Materials: Irradiation Effects

Wheeler R, Kirk M A, Marwick A D, Civale L, Holtzberg F H

1993 Columnar defects in YBCO induced by irradiation with

high energy heavy ions. Appl. Phys. Lett. 63, 1573–5

W. K. Chu and J. Liu

University of Houston, Houston, Texas, USA

L. Civale

Comisio

n Nacional de Energı

a Ato

mica, Bariloche

Argentina

Superconducting Microwave Applications:

Filters

1. Fundamentals of Microwave Absorption of

High-temperature Superconductors

The microwave response of a superconductor can be

understood by two physical phenomena. First, by the

Meissner effect (see any textbook on superconduc-

tivity, e.g. Waldram 1996), which causes any mag-

netic ﬁeld applied to the surface of a superconductor

to decrease exponentially inside the superconductor

on the length scale of the London penetration depth

, which is about 160 nm for the most relevant

high-temperature superconductor (HTS) compound

YBa

(called ‘‘YBCO’’) at T-0. The London

penetration depth increases with temperature accord-

ing to ½l

ð0Þ=l

ðTÞ

2

E12 ðT=T

¼ transi

tion temperature, with T

¼ 92 K for YBCO). As T

approaches T

, l

ðTÞ converges to the normal metal

skin depth. Below about 0:8T

, l

ðTÞ is nearly fre-

quency independent up to several hundred gigahertz.

The second fact is that at ﬁnite temperatures below

, Cooper pairs (corresponding to ‘‘ballistic’’ charge

carriers without dissipation) and quasi-particles (cor-

responding to normal conducting charge carriers

obeying Ohm’s law with a temperature dependent

conductivity s

ðTÞÞ coexist (see, e.g., Klein 1997).

According to this simple physical picture, the elec-

trodynamic response of a superconductor can be de-

scribed by a complex valued conductivity:

s ¼ s

ðTÞis

ðTÞ¼s

ðTÞ

ðTÞ

ð1Þ

The imaginary part of the conductivity corresponds to

the inductive response of the Cooper pairs. The ex-

plicit relationship between s

and l

follows from the

London equations. It should be emphasized that the

rigorous treatment of the electrodynamic response of

a superconductor requires a quantum mechanical

treatment based on BCS theory including some pecu-

liarities of high-temperature superconducting oxides

(Klein 1997, Hein 1999). In the context of this article,

only a basic understanding of the microwave response

being necessary for the understanding of supercon-

ducting ﬁlters can be provided. At microwave fre-

quencies s

holds true resulting in a simple

expression for the surface resistance and reactance of

a superconductor (by a Taylor expansion of Eqn. (1)):

ðTÞl

ðTÞ; X

ðTÞ¼om

ðTÞð2Þ

In contrast to normal metals, the surface resistance

exhibits a quadratic frequency dependence. The ab-

solute values for thin ﬁlms of YBCO at a temperature

of 77 K are shown in Fig. 1. According to this data,

there is a clear advantage by orders of magnitude for

using HTS for the whole range of microwave com-

munications bands. Above 77 K, R

increases strongly

towards the critical temperature due to the strong in-

crease of the London penetration depth. Below 77 K,

decreases gradually towards zero temperature by

less than one order of magnitude.

The surface resistance data discussed so far corre-

spond to the linear regime corresponding to low levels

of microwave magnetic ﬁelds. For higher ﬁeld levels

corresponding to higher values of the stored ﬁeld

energy, nonlinearities occur resulting either in an in-

crease of R

and/or intermodulation distortion (Hein

1999). This is of particular importance, because—

as a result of magnetic ﬁeld displacement—planar

conductors possess a strongly peaked edge current

Figure 1

Measured surface resistance of epitaxially grown YBCO

ﬁlms grown at different laboratories at T ¼ 77 K in

comparison to that of copper.

1156

Superconducting Microwave Applications: Filters

density (enhanced by about one order of magnitude).

This effect can only be avoided by employing ‘‘edge

current free’’ modes in circular disk or ring resona-

tors. Typically, intermodulation distortion as well as

a ﬁeld dependent surface resistance occurs at ﬁeld

levels, which depend strongly on the ﬁlm quality. As

an example, misoriented grains in the strongly aniso-

tropic materials and grain boundaries give rise to

strong nonlinearities.

The nonlinear effects in HTS ﬁlms are strongly

temperature dependent, in particular close to T

(Klein 1997, Hein 1999). Therefore, operation tem-

peratures of 50–65 K are more favourable than 77 K.

As discussed in Sect. 5, temperatures in this range can

be attained by low-power closed-cycle cryocoolers.

2. Technology of High-temperature

Superconducting Thin Film Devices

In most cases, HTS planar microwave devices rely on

epitaxially grown ﬁlms of YBCO grown by vacuum

deposition techniques. Nonvacuum techniques like

spraying or sol–gel (Alford et al. 1997) represent very

challenging developments with the focus on low-cost

production. However, the properties of such ﬁlms are

still behind that of epitaxial ﬁlm. However, for some

applications like coating of large area microwave

cavities for base-station ﬁlters, HTS thick ﬁlms are in

use (Remilliard et al. 2001). For a review about the

growth of YBCO ﬁlms see Wo

rdenweber (1999).

Apart from YBCO, thin ﬁlms with reasonable mi-

crowave properties have been prepared from the

thallium-based compounds Tl

CaCu

105 K) and Tl

E115 K) (Klein

1997, Wo

rdenweber 1999). For the thallium-based

and the more recently discovered mercury-based

compounds with T

values up to 140 K, epitaxial

growth is very difﬁcult because of the volatility of

thallium and mercury, respectively.

HTS ﬁlms with reasonable and qualiﬁed micro-

wave properties can be grown on wafers up to a di-

ameter greater than 4 in; the most common sizes are 2

in and 3 in for microwave applications (Wo

rdenwe-

ber 1999). A very important step was the preparation

of double-sided coatings, which have turned out to be

essential for planar microwave devices, where the

ground plane needs to be superconducting in order to

achieve high quality factors.

Substrates for planar HTS microwave devices need

to fulﬁl the criteria for expitaxial growth plus mod-

erated values of the dielectric constant (e

p25) and a

low level of dielectric losses (tan dp10

5

). The most

commonly used substrate materials are lanthanum

aluminate (LaAlO

, e

E23), magnesium oxide (MgO,

E10) and sapphire (Al

, e

E10) (see Krupka

et al. 1994, Zuccaro et al. 1997).

The fabrication of a planar microwave device is

based on a patterning process. As a ﬁrst step, masks

are prepared according to the layout of the structure.

A conventional photolithographic procedure based

on a photoresist layer followed by chemical or ion

beam etching is applied for patterning of the HTS

wafers. In order to provide electrical contacts to mi-

crostrip connectors and to the housing, the HTS ﬁlms

are often covered with an in situ gold layer prior to

patterning. The feature sizes are typically not smaller

than 10 mm, but due to RF current crowding at the

edges, the performance of planar microwave devices

becomes quite sensitive to the quality of the pattern-

ing process. After process optimization no significant

degradation of the microwave properties remains.

3. Planar HTS Resonators

Typically, the building blocks of HTS ﬁlters are pla-

nar HTS resonators. In general, a microwave reso-

nator is characterized by its resonant frequency f

, its

unloaded quality factor Q

of the selected resonant

mode(s), and its spectrum of spurious modes.

The unloaded quality factor of a resonator is de-

termined by the stored energy W and the dissipated

power P

diss

2pf

diss

ð3Þ

diss

contains loss contribution from the circuit layer

and the unpatterned ground plane, dielectric losses

from the substrate materials in use, and radiation

losses (or metal losses from the housing). The dielec-

tric losses are characterized by the loss tangent tan d

of the substrate material. Neglecting radiation losses,

the unloaded quality factor is given by

¼ k tan d þ

; k ¼

VS

dV þ e

;

G ¼

ð4Þ

with k representing the ﬁlling factor indicating the

normalized fraction of electric ﬁeld energy stored in

the dielectric substrate (S). Typically, k values are

slightly smaller than 1. The quantity G is a geometric

factor (in O) representing the surface integral of the

squared RF magnetic ﬁeld over the conducting parts

(C). In case of planar resonators G is roughly pro-

portional to the substrate thickness. For conventional

metals, the dielectric losses are small in comparison

to metal losses. For HTS resonators, substrate losses

can play a significant role. The integrals in Eqn. (4)

with subscript ‘‘V’’ are volume integrals over the en-

tire volume of the resonator, i.e., substrate plus

housing.

Figure 2 shows typical examples of HTS planar

resonators which have been used as building blocks

of multipole ﬁlter structures. Planar HTS resonators

1157

Superconducting Microwave Applications: Filters

are either formed by segments of quasi-TEM (trans-

verse electromagnetic) transmission lines deﬁning the

resonances to be at frequencies where the length L

equals a multiple of half a guided wavelength (b,c) or

lumped element resonators each composed of a dis-

crete inductor L and capacitor C (see (a) and (Hieng

et al. 2001)) with its resonance at a frequency

f ¼ 1=½2pðLCÞ

1=2

. In many cases hybrids of lumped

elements and microstrip lines are used, e.g., folded

microstrip lines disrupted by a capacitive gap (d)

(Hong et al. 1999). In order to obtain a high value of

Q for microstrip and lumped element structures, the

ground plane needs to be formed by a HTS ﬁlm, e.g.,

in general double-sided ﬁlms are required in order to

achieve ultimate performance. As an alternative, co-

planar resonators have been used (c) with some com-

promise on Q in comparison to microstrip lines of the

same size but with the advantage of single sided

coating. However, until now no high performance

multipole ﬁlters based on coplanar resonators have

been built. For the highest Q and ultimate power

handling capability circular two-dimensional disk

resonators excited in the TM

010

-mode have been

used (e), also requiring double sided coating (Kolesov

et al. 1997).

The structures shown in Fig. 2 are generally not

restricted to superconductors, but only here the

achievable quality factors are comparable or higher

than the ones of bulky cavities or dielectric resonators

at room temperature. Therefore, there is strong po-

tential in high-temperature superconductor technol-

ogy for the miniaturization of high-performance

microwave devices and circuits.

4. Planar Multipole Filter Topologies

The class of microwave ﬁlters which can mostly ben-

eﬁt from HTS technology is characterized by some

common structural features (‘‘topologies’’).

Filters of this class are passive and reciprocal two-

ports, where the incident power is transmitted within

certain frequency intervals (passbands) and reﬂected

within the stopbands. Rejection in the stopbands is

achieved by reﬂection. Unavoidable dissipation causes

a degradation of the ﬁlter performance. For a given

amount of dissipation, the degree of degradation de-

pends on the ﬁlter parameters. In turn, this determines

whether an HTS realization is beneﬁcial or not.

In the most common ﬁlter topology all resonators

are identical, possessing the same resonant frequency

in the uncoupled state. A structure where input and

output ports are coupled via one resonator only,

represent a one-pole ﬁlter with a Lorentzian shaped

transmission characteristic. If losses can be neglected,

the frequency bandwidth (frequency interval where

the transmission factor is higher than a prescribed

value) of a one-pole ﬁlter is controlled by the strength

of the coupling between resonator and ports. In order

to achieve a frequency response closer to the rectan-

gular frequency response of an ideal ﬁlter, a multi-

ple-resonant (multipole) ﬁlter structure needs to be

employed. In the most common ﬁlter topology, mul-

tiple-resonant structures are realized by a set of iden-

tical resonators of unperturbed resonant frequency

, which are mutually coupled. Mutual coupling of

N resonators results in a splitting into N (in general)

different frequencies. In the case that each resonator

supports two orthogonal resonant modes (dual-mode

resonator), N represents the number of operational

modes.

Each coupling path (direct connection between

resonator i and j) is characterized by its coupling co-

efﬁcient k

, which is related to the frequency splitting

of f

into f

and f

for the even and odd mode, re-

spectively, by k

¼ðf

 f

Þ=f

. In the special case of

pure adjacent resonator coupling (k

i;j

¼ 0 for

iaj71), a single chain between input and output

port is formed. This allows one to approximate a

rectangular bandpass response with transition width

df decreasing with increasing order N (see Fig. 3).

Different approximations for the insertion loss re-

sponse are possible. The most popular is the Cheby-

shev response with equal-ripple response of the

(reﬂective) passband insertion loss (Matthaei et al.

1980). The required order (number of resonators or

operational modes) increases with increasing ratio

between the passband width Df and the transition

Figure 2

Typical planar resonators being used as building blocks

for planar HTS ﬁlters: quasi-lumped element (a), (design

by Superconductor Technologies Inc.), microstrip (b),

coplanar waveguide (c), folded microstrip with

integrated capacitors (d), and two-dimensional disk

resonator (e). Omitting the capacitive gap in the folded

microstrip design (d) leads to a ring resonator (if circular

shaped), which also represents a quite commonly used

microstrip resonator design.

1158

Superconducting Microwave Applications: Filters

width df . For a given order N the transition width df

can be further reduced by introducing transmission

zeros in the stopband close to the band-edge. These

transmission zeros cannot be realized if the coupling

topology is restricted to adjacent resonator coupling.

Additional coupling between non-adjacent resona-

tors (‘‘crosscouplings’’) with appropriate coupling

sign and strength is required. Filters with cross-cou-

plings are characterized by the ratio between the

number of transmission zeros at ﬁnite frequencies

and the ﬁlter order and referred to as quasi-elliptic or

elliptic ﬁlters (Fig. 4) (Klauda et al. 2000).

In analogy to spectral-analysis where frequency res-

olution increases with integration time, the skirt steep-

ness (in dB MHz

1

) of a ﬁlter determines the minimum

time duration in which the signal has to be stored in

the ﬁlter structure. This time duration t can be de-

scribed by means of the frequency-dependent group

delay which is the frequency derivative of the trans-

mission phase and corresponds to the delay of a nar-

rowband (bandwidth approaching zero) pulse passing

through the ﬁlter. The group delay time t allows for a

simple relationship between the ﬁeld energy W stored

in the resonators to the incident power P

inc

W ¼ tP

inc

ð5Þ

Up to now, degradations of the ﬁlter response caused

by losses were ignored. The in-band insertion loss L

diss

¼ 10 log

out



¼ 10 log 1 þ



ð6Þ

is increased due to the ﬁnite unloaded quality factor Q

by an amount which is proportional to the group de-

lay. The insertion loss is associated with an unwanted

power reduction of in-band signals as well as with an

unwanted amplitude distortion caused by the frequen-

cy dependence of t. The latter effect leads to a round-

ing of the insertion loss response at frequencies close to

the band edge where the group delay has a maximum.

Therefore, the achievable skirt steepness for a

bandpass ﬁlter depends on the unloaded quality fac-

tor and on the ﬁlter topology (Klauda et al. 2000):

S o 4  10

3

ð7Þ

The dimensionless quantity x has a value of 0.33 for

Chebyshev and 1 for elliptic ﬁlters (the values for

quasi-elliptic ﬁlters are in between) (where S is meas-

ured in dB MHz

1

and f in GHz).

5. Planar High-temperature Supercond ucting

Filters and Subsystems

5.1 Filters

Since the late 1990s, tremendous progress has been

achieved in the development of HTS planar ﬁlters

and their integration in subsystems, mostly for base

stations in mobile communication (Willemsen 2001).

For HTS planar ﬁlters the resonator types shown in

Fig. 2 and others have been used. The main emphasis

was to develop ﬁlters for base stations with steep

skirts coming as close as possible to the Q-require-

ments discussed in the previous section. In most cas-

es, these are (quasi) elliptic ﬁlters with 8–17 poles or

Chebyshev ﬁlters with up to 30 poles.

Figure 5 shows an example of an eight-pole quasi-

elliptic ﬁlter based on folded microstrip resonators

similar to ‘‘c’’ in Fig. 2. In this case of monomode

resonators, coupling coefﬁcients of different signs

were achieved by changing between electric and mag-

netic ﬁeld coupling of the resonators. For a 17-pole

elliptic ﬁlter at 1.8 GHz with 5% relative bandwidth

(resonator Q

¼ 50:000 at 65 K) a steepness of skirts

of 85 dB MHz

1

was demonstrated (Fig. 6). Such a

Figure 3

Typical frequency response of a passband ﬁlter with 2

transmission zeros in the stopbands

(Df ¼ passband width, df ¼ transition width).

Figure 4

Filter topology with resonators, adjacent resonator

coupling, and non-adjacent resonator coupling (cross-

coupling).

1159

Superconducting Microwave Applications: Filters