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

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1. Basic Biomo lecule Labeling and Detection with

Magnetic Par ticles

As for many other biomolecular labels, magnetic

labeling can be easily accomplished using specific lig-

and–receptor interactions, as illustrated generically in

Fig. 1. The illustrated ‘‘sandwich’’ conﬁguration pro-

ceeds as follows:

(i) Receptor molecules specific for the target bio-

molecules are attached to the surface of a solid subst-

rate. Often arrays of different probe spots are used to

simultaneously detect multiple targets.

(ii) When target molecules are present in a sample

solution, they are captured by the surface.

(iii) Magnetic particles coated with a second set of

receptor molecules for the target are introduced,

labeling the previously captured targets.

(iv) The label particles are detected by a magnetic

sensor.

Sandwich assays are commonly performed using

antibody–antigen pairs and complementary DNA

strands, and often use the strongly binding biomo-

lecular combination of streptavidin and biotin to

attach the particle labels to the receptor molecules.

For biosensing applications, the magnetic particles

used as labels should be paramagnetic, i.e., only have

a magnetic moment when a magnetic ﬁeld is applied.

The absence of a permanent magnetic moment pre-

vents agglomeration of the particles in solution which

would render them ineffective as labels. When select-

ing label particles, properties of importance include

the size and size uniformity, the type of magnetic

material and its volume fraction within the particle,

and the particle surface chemistry. For example,

though nanometer-scale particles are matched in size

to biological macromolecules such as DNA, the small

magnetic ﬁelds they generate are difﬁcult to detect.

Second, the surface of the particles must be suitable

for chemical attachment of biomolecules. A variety of

paramagnetic beads with biocompatible polymer or

silica shells are commercially available with diameters

that range from 1 mmto5mm.

Because magnetic particles experience a force in the

presence of a magnetic ﬁeld gradient, they may be

manipulated by magnets. This phenomenon can be

used to apply a controlled force that selectively pulls

off only those labels not bound to the surface by

specific binding (Lee et al. 2000). Such a force dis-

crimination assay increases the sensitivity of detec-

tion by greatly reducing the background signal and

thereby permitting very low signal levels to be de-

tected with conﬁdence.

2. Magnetic Par ticle Detection

The detection of magnetic particles can be accom-

plished by a number of methods, as mentioned above.

Here we will focus on the use of solid-state magnetic

ﬁeld sensors embedded in the substrate, a method

that illustrates many of the advantages of magnetic

labeling for biomolecule detection. For example, such

an approach has the potential to eliminate the re-

quirement for an external detection system, thereby

reducing the size, cost, and power of the sensor

system.

One of the simplest solid-state magnetic ﬁeld sen-

sors to incorporate into a microelectronic substrate is

a giant magnetoresistive (GMR) wire (Baselt et al.

1998). GMR materials are multi-component thin

ﬁlms with alternating magnetic and nonmagnetic lay-

ers (see Giant Magnetoresistance). The relative orien-

tation of the magnetic moments of these layers is

altered in the presence of a magnetic ﬁeld, changing

the electrical resistance of the wire. When a magnetic

particle is present above a GMR sensor, the resist-

ance of the sensor changes; the larger the number of

particles present, the larger the change. The availa-

bility of suitably sensitive GMR sensors was made

possible by recent advances in magnetic materials for

high-density data storage, including magnetic disk

drive read-out sensors and nonvolatile magnetic

memory elements (Prinz 1998).

When paramagnetic microbeads are used, such as

those commercially available, GMR sensors have

been designed that are only sensitive to a planar

component of the dipolar magnetic ﬁeld induced in

the particles, as illustrated in Fig. 2. As discussed by

Miller et al. (2001), when an external magnetic ﬁeld is

applied normal to the plane of the GMR sensor, the

ﬁeld strength at the sensor is given by

B B m=z

ð1Þ

Figure 1

Generic illustration of magnetic labeling of targets

captured onto a solid substrate in a ‘‘sandwich’’

conﬁguration using specific biomolecular ligand–

receptor recognition.

120

DNA Microarrays using Magnetic Labeling and Detection

where m is the magnetic moment of the bead and z is

the distance from the center of the bead to the sen-

sor.Therefore, the resistance change in the sensing

wire is roughly proportional to the number of mag-

netic particles present above it, a signal that can be

easily measured. For maximum signal, the particle

should have as large a magnetization as possible and

be as close to the sensor as possible.

3. Detection with a GMR Sensor Microarray

A well-developed system that uses a GMR microarray

to detect different DNA sequences in a sample is the

bead array counter (BARC) sensor (Baselt et al. 1998,

Edelstein et al. 2000, Miller et al. 2001). In this system,

the test is performed inside a small, quartz ﬂow cell

mounted over the sensor chip. The chip itself is wire

bonded to a printed circuit board housed in a dis-

posable plastic cartridge that contains the required

liquid reagents. The cartridge plugs into an automated

electronic controller and connects to a miniature

pumping system (Tamanaha et al. 2002).

Before assembly, the surface of the sensor chip is

coated with a biocompatible polymer ﬁlm (polyeth-

ylene glycol (PEG)) and an array of single-stranded

DNA capture probes (Fig. 3(a)). Note that the GMR

sensors and associated interconnects must be pro-

tected from the reactive and conductive salt solution

used in the biochemical test, in this case by a ﬁlm of

Figure 2

Schematic of the magnetic ﬁeld induced around a

paramagnetic microbead on top of a magnetic ﬁeld

sensor when an external ﬁeld is applied normal to the

substrate.

Figure 3

(a) An illustration of the BARC biosensor approach. Note that the elements are not to scale; in particular, the beads

and sensors are much larger in proportion to the biomolecules. (b) A micrograph of a BARC chip after a DNA test,

with the eight sensing zones outlined. For each of the probe zones, the graph shows the measured signals from the

individual sensing elements (open circles) and the integrated signal for all 16 elements (colored bar). The assay for

sequence ‘‘B’’ was performed with 10 nM DNA hybridized for 30 min. The total test required about B60 min.

121

DNA Microarrays using Magnetic Labeling and Detection

silicon nitride. The chip is also coated with a thin

gold layer so that the DNA probes and PEG can be

attached to the surface by strong gold–sulfur bonds.

Each sensor zone is covered with strands of DNA

complementary for a different target strand. The

PEG ﬁlm inhibits undesirable, nonspecific, sticking of

target DNA and magnetic particles in the areas out-

side of the DNA probe spots.

To test for the presence of specific target DNA

sequences, the sample is introduced into the ﬂow cell

and allowed to hybridize with the capture probes. In

the illustrated example, the target strands have been

pre-modiﬁed to have biotin molecules attached at one

end. Therefore, the presence of target in the sample

will result in double-stranded, biotin-tagged DNA

bound above a sensor. The captured targets are then

labeled with streptavidin-coated magnetic micro-

beads. Beads just resting on the surface can be re-

moved by a magnetic force, and the remaining beads

detected by the underlying GMR sensors.

The results of an actual experiment performed in

this way are shown in Fig. 3(b) on a chip with eight

sensor zones (Miller et al. 2001). A micrograph shows

the surface of the chip after the test. In this test, each

of the three different DNA sequences and the positive

control (a probe with biotin already on it) were each

applied to two zones. Introducing a sample containing

only sequence ‘‘B’’ results in the complementary zones

and the positive control being covered with beads

(appearing dark). The presence of the beads is accu-

rately detected by the eight GMR sensors in each zone.

4. Summary

Magnetic labeling and detection holds great promise

for sensing biomolecules. As demonstrated by the

BARC system, advances in microelectronics and ma-

terials can be adapted to analytical applications in the

biosciences with great success. Magnetic particles,

magnetic force manipulation, and magnetoelectronic

detection can be combined to create relatively simple,

compact, sensor systems. Such technology promises

to fulﬁll the need for ever faster, more sensitive, and

more portable systems in ﬁelds as diverse as home-

land defense, clinical diagnostics, genomics, pro-

teomics, and forensics.

See also: SQUIDs: Biomedical Applications

Bibliography

Baselt D R, Lee G U, Hansen K M, Chrisey L A, Colton R J

1997 A high-sensitivity micromachined biosensor. Proc.

IEEE 85, 672–80

Baselt D R, Lee G U, Natesan M, Metzger S W, Sheehan P E,

Colton R J 1998 A biosensor based on magnetoresistance

technology. Biosens. Bioelectron. 13, 731–9

Besse P-A, Boero G, Demierre M, Pott V, Popovic R 2002

Detection of a single magnetic microbead using a miniatur-

ized silicon hall sensor. Appl. Phys. Lett. 80, 4199–201

Edelstein R L, Tamanaha C R, Sheehan P E, Miller M M,

Baselt D R, Whitman L J, Colton R J 2000 The BARC

biosensor applied to the detection of biological warfare

agents. Biosens. Bioelectron. 14, 805–13

titz R, Matz H, Trahms L, Koch H, Weitschies W,

Rheinla

nder T, Semmler W, Bunte T 1997 SQUID based

remanence measurements for immunoassays. IEEE Trans.

Appl. Supercond. 7, 3678–81

Kriz C B, Ra

devik K, Kriz D 1996 Magnetic permeability

measurements in bioanalysis and biosensors. Anal. Chem. 68,

1966–70

Lee G U, Metzger S, Natesan M, Yanavich C, Dufreˆ ne Y F

2000 Implementation of force differentiation in the immuno-

assay. Anal. Biochem. 287, 261–71

Miller M M, Sheehan P E, Edelstein R L, Tamanaha C R,

Zhong L, Bounnak S, Whitman L J, Colton R J 2001 A

DNA array sensor utilizing magnetic microbeads and

magnetoelectronic detection. J. Magn. Magn. Mater. 225,

138–44

Pearson J E, Gill A, Vadgama P 2000 Analytical aspects of

biosensors. Ann. Clin. Biochem. 37, 119–45

Prinz G 1998 Magnetoelectronics. Science 282, 1660–3

Richardson J, Hawkins P, Luxton R 2001 The use of coated

paramagnetic particles as a physical label in a magneto-

immunoassay. Biosens. Bioelectron. 16, 989–93

Tamanaha C R, Whitman L J, Colton R J 2002 Hybrid macro–

micro ﬂuidics system for a chip-based biosensor. J. Micro-

mech. Microeng. 12, N7–N17

Vo-Dinh T, Cullum B 2000 Biosensors and biochips: advances

in biological and medical diagnostics. Fresenius J. Anal.

Chem. 366, 540–51

Cy. R. Tamanaha

Geo-Centers, Washington, DC, USA

Lloyd J. Whitman

Naval Research Laboratory, Washington, DC, USA

122

DNA Microarrays using Magnetic Labeling and Detection

Electrodynamics of Superconductors:

Flux Properties

The ‘‘mixed state’’ of type II superconductors occurs

when magnetic ﬂux penetrates the material in the

form of ﬂux lines (vortices) without destroying the

superconducting ground state (Tinkham 1996).

Above the ﬁeld of ﬁrst ﬂux penetration, H

, the re-

pelling ﬂux lines spontaneously order into a triangu-

lar (Abrikosov) lattice with sixfold symmetry. The

state of zero resistivity is destroyed by vortex motion,

which, however, is retained up to a critical current

density, J

, when the ﬂux lines are pinned by crystal-

line defects. Vortex motion can also be driven by

thermal ﬂuctuations. When they are sufﬁciently large,

the vortex lattice melts to a vortex liquid and the

superconducting ground state is destroyed.

1. Flux Line Lattice

Thermodynamically, a ﬂux line is stable above the

lower critical ﬁeld, H

(Abrikosov 1957). The ﬂux

line (or vortex) consists of a ‘‘normal’’ core in which

the density of Cooper pairs decreases to zero over a

length scale x(T), the Ginzburg–Landau (GL) coher-

ence length. Around the core superconducting screen-

ing currents circulate, which generate a local

magnetic ﬁeld of about 2 m

, where m

is the mag-

netic permeability of vacuum. Away from the core

both the ﬁeld and the screening currents decay over a

length scale l(T), the GL penetration depth. The line

energy of a ﬂux line per unit length is given by

e ¼ e

ln k ðF

=4pm

Þ ln k ð1Þ

where F

is the ﬂux quantum, which is also the mag-

netic ﬂux carried by a ﬂux line. The GL parameter k

is the ratio l/x.

Usually, the ﬁrst penetration of ﬂux occurs at a

ﬁeld H

, which is considerably different from H

because the ﬂux line has to enter through the surface

or across the edges of the superconductor. This

means that H

is determined by surface and geomet-

rical barriers which depend on sample shape and ﬁeld

orientation.

Well above H

the ﬂux lines form a triangular

Abrikosov lattice with lattice parameter a

¼(2/

ﬃﬃﬃ

)

1/2

, where a

¼(F

/B) and B is the ﬂux density. Be-

cause of the electromagnetic origin, the range of the

vortex interactions is l, which is generally much larg-

er than a

. Elastic deformations of the Abrikosov

lattice are described by elasticity moduli for uniaxial

compression (c

), for tilt deformations (c

), and for

shear deformations (c

). Both c

and c

strongly

depend on the wavelength of the deformation ﬁelds.

Especially for small wavelengths the effect of the de-

formations is considerably reduced by averaging over

the vortex interaction range l, i.e., the ﬂux line lattice

is very soft on scales smaller than l.

For very anisotropic materials, such as high-tem-

perature superconductors, the ﬂux lines caused by a

magnetic ﬁeld perpendicular to the CuO

planes

break up into pancake vortices located in the super-

conducting layers (Blatter et al. 1994). The tilt mod-

ulus is now reduced by a factor equal to the

anisotropy parameter, G, which can be as large as

. Above a crossover ﬁeld B

¼F

/Gs

(s is the in-

terlayer distance) the tilt deformations become irrel-

evant and a quasi two-dimensional interaction results

in which only compression and shear modes between

pancake vortices in the same layer survive. When the

magnetic ﬁeld is parallel to the CuO

plane it pen-

etrates in the form of Josephson vortices (Bulaevskii

1973), which possess no normal core. For arbitrary

ﬁeld directions crossing lattices of Abrikosov pan-

cake vortex stacks and Josephson vortices may

coexist (Koshelev 1999).

2. Thermodynamics

Usually the mixed state is represented by the mag-

netization curve M(H) which in principle consists of

ﬁve regimes: up to H

in the Meissner state M ¼H,

just above H

it drops exponentially, for H

HoH

it is characterized by Mpln(H

/H), near

it is linearly proportional to H

H, and above

M ¼0. Because of surface and geometrical effects

there may occur deviations from this ideal behavior.

In this picture it was believed that the phase transi-

tion to the superconducting state takes place at the

upper critical ﬁeld, H

(T). It is now generally ac-

cepted that the real thermodynamic phase transition

occurs at the melting line, H

(T). In pure single

crystals of high-temperature superconductors the

melting transition is of ﬁrst order. It manifests itself

by a jump in B, the ﬂux liquid is slightly denser than

the solid (Zeldov et al. 1995). In the presence of

quenched disorder the transition becomes of second

order.

Melting occurs when thermal ﬂuctuations generate

elastic modes in the lattice that are strong enough to

destroy its shear strength (Nelson 1988). In a three-

dimensional lattice both shear and tilt deformations

determine the mean square displacement /u

1/2

According to the Lindemann criterion the lattice melts

when this quantity is a significant fraction of the lattice

parameter, i.e., /u

1/2

¼c

,wherec

is the Linde-

mann number (c

¼0.1–0.3). This criterion gives a

good description of the melting line. For thin ﬁlms in a

perpendicular ﬁeld the tilt modes of the ﬂux lines do

123

not contribute to the thermal disorder and the system

should be considered as a two-dimensional solid con-

sisting of rigid ﬂux bars. Thermal ﬂuctuations in two

dimensions give rise to the formation of edge disloca-

tion pairs, which unbind at a second-order melting line

in accord with the Kosterlitz–Thouless theory for two-

dimensional melting.

In very anisotropic materials the low-temperature

phase consists of a solid of ﬂux lines (stacked pancake

vortices) which at the ﬁrst-order transition line decays

to a plasma of decoupled pancake vortices. In all

situations the state of true superconductivity is

destroyed at the melting line and ohmic behavior is

recovered.

3. Flux Pinning and the Critical Current

A moving ﬂux line lattice (FLL) dissipates energy.

When this motion is driven by a transport current

with density J, a steady state with velocity v is

reached when the driving force, JF

, on each vortex is

compensated by a frictional force Zv. An illustrative

model for the friction coefﬁcient, Z, is that of Bardeen

and Stephen in which the vortex core is treated as a

moving cylinder of normal metal. The ratio between

ﬂux ﬂow resistivity, r

, and normal state resistivity,

, is then simply represented by the normal fraction,

i.e., B/B

, which leads to Z ¼F

1

/2px

A moving FLL should thus be considered as a

normal metal with a (much) smaller resistivity r

Obviously, for applications, the ﬂux ﬂow state should

be suppressed. In practice, most materials can be

made with a high density of inhomogeneities acting

as pinning centers which trap the ﬂux lines up to a

critical force J

, where J

is the critical current

density. Defects deviate from the background mate-

rial by a different local density, elasticity, electron–

phonon coupling, electron mean free path P, etc. The

ﬁrst three properties give rise to a local change in T

whereas the latter leads to a variation in P. Conse-

quently, one may distinguish between dT

pinning

and dP pinning, each with its characteristic temper-

ature and ﬁeld dependence (Kes 1991).

The collective action of the pinning centers breaks

up the FLL in domains of average size L

parallel to

the ﬁeld and transverse size R

perpendicular to the

ﬁeld. Its volume is V

. The critical current

density follows from the statistical average over dis-

tributions of pinning centers with pinning strength

W ¼n

S, where n

is the concentration of pins

and /f

S the squared average of the pinning force of

a single defect. This leads to the well-known collective

pinning relationship (Larkin and Ovchinnikov 1979)

BEðW=V

1=2

ð2Þ

In the original theory the domain sizes were deter-

mined by the balance of pinning energy and the

energy of the elastic deformations in the FLL. How-

ever, when the pinning is strong, the local strains in

the FLL may grow beyond the elastic limit so that

topological defects such as edge dislocations are cre-

ated. These now determine the positional disorder in

the FLL and the domain size V

. Optimal pinning

occurs when the transverse order is totally destroyed,

i.e., R

, and individual ﬂux line segments of

length L

make up the average domain size. In an-

isotropic high-temperature superconductors this

length can be as small as the height, s, of a single

pancake vortex. The transition from elastic to plastic

disorder may be seen as a peak effect in J

. It usually

happens near H

or near the melting line. Interesting

ﬁnite size effects may occur in thin ﬁlms in a perpen-

dicular ﬁeld leading to two-dimensional collective

pinning.

When the pinning is weak, as is the case in amor-

phous thin ﬁlms, L

can be larger than the ﬁlm thick-

ness. Now only shear deformations are relevant,

which makes the analysis conveniently transparent

(Kes and Tsuei 1983).

4. Thermal Effects, Depinning, and Creep

The FLL in the presence of pinning centers may be

seen as an elastic medium in a disorder potential with

each domain sitting in a local potential minimum.

With increasing temperature thermal ﬂuctuations will

give rise to positional ﬂuctuations of the domains and

thermal smearing of the potential roughness. When

the mean square displacement is of the order of the

range, r

, of the disorder potential, this effect (called

thermal depinning) is large enough to cause a con-

siderable decrease of the pinning force and of J

. The

corresponding depinning or irreversibility line in the

temperature–ﬁeld diagram separates a strong pinning

regime at low temperatures from a weak pinning re-

gime at high temperatures.

Thermal ﬂuctuations also make the FLL domains

jump over energy barriers between different potential

minimums of almost equal energy. In the presence of

a driving force this process leads to a unidirectional

creep of the FLL with an average velocity Bexp

(U

T), where U

is the typical height of the

energy barrier and k

is the Boltzmann constant. For

relatively small values, U

Tp10, one thus gets

linear resistivity at small J, called thermally assisted

ﬂux ﬂow (TAFF):

TAFF

¼ r

expðU

TÞð3Þ

Although r

TAFF

may be extremely small, TAFF

does not describe a true superconducting state with

zero resistance at small J, but rather a normal metal

with ohmic resistance. The reason is that U

is sup-

posed to be ﬁnite and independent of J. It has been

shown, however, that for collective creep of an elastic

124

Electrodynamics of Superconductors: Flux Properties

medium in random disorder the domain size, the dis-

tance between available metastable states, and the

corresponding energy barriers should grow when J

decreases in order to maintain the balance between

deformation energy and work done by the driving

force. In fact, U

B(J

/J)

with m of order one, which

leads to true zero resistance for J-0, according to

rBr

exp[c(J

/J)

]. A similar result is obtained in

the vortex glass model where the ﬂux ﬂow is supposed

to be triggered by the motion of topological defects in

the FLL. It shows once more that the mixed state

with strong pinning not only is a very practical phe-

nomenon, but that it is a rich area of research for

solid-state and statistical physics.

See also: Electrodynamics of Superconductors:

Weakly Coupled

Bibliography

Abrikosov A A 1957 On the magnetic properties of supercon-

ductors of the second group. Sov. Phys. JETP 5, 1174

Blatter G, Feigelman V M, Geshkenbein V B, Larkin A I,

Vinokur V M 1994 Vortices in high-temperature super-

conductors. Rev. Mod. Phys. 66, 1125–388

Bulaevskii L N 1973 Magnetic properties of layered super-

conductors with weak interaction. Sov. Phys. JETP 37, 1133

Kes P H 1991 In: Cahn R W, Haasen P, Kramer E J (eds.)

Materials Science and Technology. VCH, New York, Vol. 3A,

Part I

Kes P H, Tsuei C C 1983 Two-dimensional collective ﬂux pin-

ning, defects and structural relaxation in amorphous super-

conducting ﬁlms. Phys. Rev. B 28, 5126–39

Koshelev A E 1999 Crossing lattices, vortex chains, and angular

dependence of melting line in layered superconductors. Phys.

Rev. Lett. 83, 187–90

Larkin A I, Ovchinnikov Yu N 1979 Pinning in type II super-

conductors. J. Low Temp. Phys. 34, 409

Nelson D 1988 Vortex entanglement in high-T

superconduc-

tors. Phys. Rev. Lett. 60, 1973–6

Tinkham M 1996 Introduction to Superconductivity. McGraw-

Hill, New York, 2nd edn

Zeldov E, Majer D, Konczykowski M, Geshkenbein V B,

Vinokur V M, Shtrikman H 1995 Thermodynamic observa-

tion of ﬁrst-order vortex lattice melting in Bi

CaCu

Nature 375, 373–6

P. H. Kes

Leiden University, The Netherlands

Electrodynamics of Superconductors:

Weakly Coupled

Systems consisting of two superconducting samples

connected by a weak electrical contact possess several

unusual properties. First, the net current ﬂowing

across the contact—the Josephson junction (see

Josephson Junctions: Low-T

; Josephson Junctions:

High-T

)—contains a supercurrent I

.IfI

is smaller

than the critical current of a junction, I

, then its ﬂow

does not lead to a voltage drop across the structure.

The supercurrent depends on the phase difference

j ¼w

w

of the superconductor order parameters

exp(iw

) and D

exp(iw

) of the samples (stationary

Josephson effect). This function is exactly 2p periodic

and, in the simplest cases, is sinusoidal:

ðjÞ¼I

sinðjÞð1Þ

The critical current, I

, is a constant determined by

the transport parameters of the contact material and

the shape of the Josephson junction. At I

there

is a voltage drop, V, across the junction, which is

related to j by the fundamental law (nonstationary

Josephson effect)

V ð2Þ

where e is the electron charge and _ is Planck’s

constant.

The physical origin of the stationary Josephson ef-

fect lies in the specific form of electron reﬂection from

a superconducting interface. Electrons with energy e

smaller than the superconducting gap, O, of the elec-

trodes are trapped in a contact material. They cannot

penetrate into an electrode since there is a gap in the

density of states near the Fermi level of a supercon-

ductor. However, if the energy e of an electron is in

the interval 0pepO, its interaction with an electrode

results in the creation, with a probability equal to

one, of an extra pair of correlated electrons (Cooper

pair) in the superconductor and a hole with energy e

in the contact material. The latter moves in the

direction opposite to that of the initial electron (so-

called Andreev reﬂection). This hole, in turn, inter-

acts with another electrode and causes annihilation of

a pair of correlated electrons and the creation of an

electron with energy e, which moves in the same

direction as that of the initial particle. As a result of

this cycle, a pair of correlated electrons transfers from

one superconductor to another leading to a super-

current ﬂow across a junction, i.e., to the stationary

Josephson effect.

The nonstationary Josephson effect is a direct con-

sequence of the Schro

dinger equation. In the station-

ary state the wave function of correlated electrons has

the form c(r, t ) ¼c(r)exp(iEt/_), and the Schro

dinger

equation for the phase of the wave function reduces

to _@w

1,2

/@t ¼E

1,2

, where E

1,2

is the energy of the

superconducting condensate in the samples. Thus

_@j/@t ¼E

E

and this latter difference should be

equal to the difference E

E

of the electrochem-

ical potentials E

and E

across the junction, which

is exactly equal to 2 eV.

125

Electrodynamics of Superconductors: Weakly Coupled

1. Current Components

In the general case, the net current, I, across a Joseph-

son junction has several components in addition to

(j):

I ¼ I

ðjÞþI

ðVÞþI

ðtÞð3Þ

where I

is the normal (quasiparticle) current, I

the

displacement current, and I

the ﬂuctuation current.

1.1 Supercurrent

In accordance with Eqn. (1) a d.c. supercurrent in the

range

I

ðjÞpI

ð4Þ

can ﬂow across the junction as a result of a constant

phase difference j. An external system supporting

this phase difference represents work on the super-

current and, thus, stores potential energy in the junc-

tion given by

Uðj Þ¼E

½1  cosðjÞ þ const; E

ð5Þ

Since the energy is conserved in the form of persistent

current, the Josephson junction can be considered as

having a nonlinear inductance

1

¼ L

1

cosðjÞ; L

2eI

ð6Þ

This means that for weak signals the supercurrent is

equivalent to an inductance L

dependent on j(t) and

on the critical current of the junction.

The most impressive property of a supercurrent is

its ability to oscillate (Josephson oscillations) with a

frequency

E483 MHzmV

1

ð7Þ

if a nonzero d.c. voltage,

V, is ﬁxed across the junc-

tion. At typical voltages (10

6

–10

2

V) the oscillation

frequency, f

, is of the order of 10

–10

Hz.

1.2 Normal Current

The scale of the normal current is given by the nor-

mal conductance of a junction G

. Independent of

transport properties of the contact material at volt-

ages larger than (D

þD

)/e, the I

(V) dependence of

the junction is close to Ohmic at all temperatures:

¼ G

V þ dI ð8Þ

The voltage-independent factor dI in Eqn. (8) is

the so-called ‘‘excess’’ (dIp(D

þD

)) or ‘‘deficit’’

(dIp(D

þD

)) current observed if the contact

material has a metallic type of conductivity (SNS

weak links). In structures with transparent interfaces

between the contact material and superconducting

samples there is excess current on I

(V). Andreev

reﬂection of electrons is responsible for this effect.

Owing to a gap in the density of states, an electron

with energy e in the interval

þ eV  O

pepE

þ eV þ O

ð9Þ

cannot penetrate into the superconductor as an elec-

tron, as in the normal case. Instead, in accordance

with the Andreev process, it creates a pair of corre-

lated electrons and a hole in the contact material.

Thus, in this energy interval, (as in the normal case)

two electrons instead of one are injected into the

electrode leading to an ‘‘excess’’ current.

In structures with small transparent interfaces be-

tween the contact material and superconducting sam-

ples (SINIS double barrier structures) there is a

current ‘‘deficit’’ on I

(V). The probability of And-

reev reﬂection for an electron with energy deﬁned by

Eqn. (9) is proportional to D

, where D51 holds for

the transparency of an interface. Thus, in contrast to

the normal case, there is no electron injection into the

superconductor if the electron energy is in the range

given by Eqn. (9) and, therefore, a current deficit

should be observed on I

(V).

In junctions with tunneling-type conductivity of

the contact region (SIS tunnel Josephson junction),

dI ¼0.

1.3 Displacement Current

For most practical Josephson junctions the displace-

ment current is deﬁned by

¼ CV ¼ C

ð10Þ

where the junction capacitance, C, has the same value

as in the normal state and depends on both the junc-

tion type and its size. The effect of the capacitance on

the processes in the junction is characterized by the

dimensionless parameter

b ¼

C ð11Þ

introduced by McCamber (1968) and Steward (1968).

It compares the normal and displacement currents.

Junctions with b51 have small capacitance or high

damping (I

), and junctions with bb1 have large

capacitance or low damping (I

1.4 Fluctuation Current

The necessity of taking ﬂuctuations (‘‘noise’’) into

account often arises in an analysis of the sensitivity of

126

Electrodynamics of Superconductors: Weakly Coupled

superconducting devices. In most cases this is carried

out by including some ﬂuctuation current, I

(t), in

the whole current across the junction. There are sev-

eral sources of current ﬂuctuation. These are external

noise sources, 1/f noise, thermal noise, and shot noise.

Usually, in experimental setups the intensity of an

external noise is effectively suppressed to a level that

is negligibly small compared to that provided by

other sources.

The cutoff frequency of 1/f noise is relatively low.

In a low-temperature Josephson junction it scales

from 1 Hz to 10 Hz, while in high-temperature junc-

tions it is in the range 100–1000 Hz. This means that

at typical Josephson frequencies oEo

the net effect

of this noise is very small. Nevertheless, in some

practical devices (e.g., see SQUIDs: Biomedical Appli-

cations) the useful signal is contaminated with 1/f

noise and special techniques such as signal or bias

modulation are used for suppression of this noise

component.

Thermal noise and shot noise are two types of

classical ﬂuctuations which exist since there is a di-

ssipative current, I

, across a junction. The relative

intensity of the thermal ﬂuctuations can be charac-

terized by the ratio of the thermal energy, k

T, to the

Josephson energy, E

g ¼

; I

½mAE0:0417½Kð12Þ

Thus, if the critical current of the junction is much

larger than I

the inﬂuence of thermal ﬂuctuations

should be small.

Shot noise is important if the voltage across the

junction is large compared to _o/e and k

T/e and can

be characterized by the spectral density

ðoÞ¼

¼ const ð13Þ

Equations (2) and (3) form the system of the basic

equations for the Josephson junction. The concrete

expressions for the current components in Eqn. (3)

depend on the type of conductivity of the contact

material and the geometry of the junction. The prob-

lem of ﬁnding the relationship between the current

and voltage was solved in solid-state theory (We-

rthamer 1966, Larkin and Ovchinnikov 1966) only

for structures with tunnel-type conductivity of the

contact material. For this reason, even now the ma-

jority of dynamics problems in systems with Joseph-

son junctions are solved in the framework of

phenomenological models, which provide approxi-

mations of the real properties of various junctions.

The resistively shunted junction (RSJ) model intro-

duced by McCamber (1968) and Steward (1968) is the

simplest and the most widely used tool for studying

the processes in junctions.

2. RSJ Model

The RSJ model is based on Eqns. (1) and (8) with

dI ¼0, on Eqn. (10) for the current components in

Eqn. (3)

I ¼ I

sinðjÞþG

V þ C

þ I

ðtÞð14Þ

and on the Josephson relationship (Eqn. (2)).

Quantitatively this model is valid for SIS and SNS

Josephson junctions only in a narrow temperature

interval in the vicinity of the critical temperature of

the superconductor. At lower temperatures G

be-

comes a nonlinear function of voltage and often the

(j) relationship is no longer sinusoidal.

Nevertheless, this model is widely used for describ-

ing the processes in Josephson junctions of any type,

if they are shunted by a conductance G

shunted junction can be considered as a new Joseph-

son contact with linear normal conductance, G

which should replace G

in Eqn. (14). A real form of

the I

(j) relationship can be taken into account by

representing a junction as a junction with

(j) ¼I

sin(j) and effective inductance connected

in series (Zubkov et al. 1981).

The RSJ model is of great importance, since it

provides the possibility of studying the processes in

a variety of devices employing Josephson junctions

without having the exact solutions of solid-state

physics problems for the current components in

Eqn. (3).

There are two experimental tests that are generally

used to verify how close the behavior of real Joseph-

son contacts are to the predictions of the RSJ model.

These are the specific reactions of the structure to

external magnetic and microwave ﬁelds.

3. Josephson Junctions in a Magnetic Field

In an external magnetic ﬁeld, B, the phase difference,

j, across a junction depends on a space coordinate,

x, along the junction and is a solution of the equation

¼ Jðx; tÞð15Þ

where J is the local value of a current (Eqn. (3))

through the junction and l

is the Josephson pene-

tration depth given by

2pm

ðd þ l

þ l

ÞJ

ð16Þ

where d is the distance between the superconducting

samples, F

the magnetic ﬂux quantum, m

the vac-

uum magnetic constant, and l

1,2

the London pene-

tration depths of the superconducting samples.

127

Electrodynamics of Superconductors: Weakly Coupled

Solution of Eqns. (14) and (15) essentially depends

on the relationship between the length of the junc-

tion, L, and l

In a long Josephson junction with Lbl

the mag-

netic ﬁeld penetrates inside the structure in the form

of a Josephson vortex. This is a region with a length

of the order of pl

, ﬁlled by magnetic ﬂux lines, which

are screened by the supercurrent, J, circulating

around:

B ¼

pdl

cosh

1

x  x



;

J ¼

72J

sinh½ðx  x

Þ=l



cosh

½ðx  x

Þ=l



In a short junction with L5l

the phase difference is

a linear function of B:

jðxÞ¼

x þ Z; Z ¼ const ð17Þ

The critical current of a junction

¼ max

sin½jðxÞdx



¼ I

sinðpF=F

ðpF=F



ð18Þ

depends in this case on the magnetic ﬂux, F, inside

the junction and exactly equals zero at F ¼nF

. The

form of I

(F) dependence is similar to the pattern of

Fraunhofer diffraction of waves passing through a

slit and has been termed the ‘‘Fraunhofer pattern.’’

4. Josephson Junctions in a Microwave Field

In his original paper Josephson (1962) also predicted

the appearance of a set of vertical current steps on the

current–voltage characteristics of short Josephson

junctions under the inﬂuence of microwave radiation,

V(t) ¼V

cos(ot).

If the normal current component across the junc-

tion is large compared to the supercurrent, then the

solution of Eqns. (2) and (3) has the simple form

j ¼

Vt þ

sinðotÞþa ð19Þ

Substituting Eqn. (19) into the expression for I

(Eqn. (1)) of the RSJ model gives

¼ I

m¼N

2eV



sin m þ



ot þ a

ð20Þ

where J

(x) are Bessel functions of the ﬁrst kind. It is

clear from Eqn. (20) that synchronization of the nth

harmonic of the external radiation with the frequency

of the Josephson oscillation (no ¼o

) should result in

formation of steps, which look like ‘‘spikes’’ in the

current–voltage characteristic and have amplitude

(2eV

/_o). According to the well-known prop-

erties of Bessel functions, as the signal amplitude in-

creases the nth step ﬁrst increases, reaches a

maximum, and then slowly decreases, oscillating with

a period approximately equal to 2p.

These oscillating steps were ﬁrst observed by

Shapiro (1967) and are referred to as Shapiro steps.

Both effects discussed above (Fraunhofer pattern,

Eqn. (18), and Shapiro steps, Eqn. (20)) are the direct

consequence of the sinusoidal relationship between

supercurrent and phase difference j. It is for this

reason that they are used as a standard experimental

test for the applicability of the RSJ model for the

description of the properties of a Josephson junction

of any type and conﬁguration.

See also: Electrodynamics of Superconductors: Flux

Properties

Bibliography

Barone A, Paterno G 1982 Physics and Application of Josephson

Effects. Wiley, New York

Josephson B D 1962 Possible new effects in superconducting

tunneling. Phys. Lett. 1, 251–4

Kulik I O, Yanson I K 1970 Josephson Effect in Superconduct-

ing Tunnel Structures. Nauka, Moscow (in Russian); 1972,

Keter Press, Jerusalem (in English)

Larkin A I, Ovchinnikov Yu N 1966 Tunnel effect between

superconductors in alternating ﬁeld. Zh. Eksp. Teor. Phys.

(Sov. Phys. JETP) 51, 1535

Likharev K K 1986, 1991 Dynamics of Josephson Junctions and

Circuits. Gordon and Breach, Reading, UK

McCamber D E 1968 Effect of AC impedance on DC volatage-

current characteristics of superconductor weak-link junc-

tions. J. Appl. Phys. 39, 3113

Shapiro S 1967 Josephson currents in superconducting tunnel-

ing: the effect of microwaves and other observations. Phys.

Rev. Lett. 11,80

Solymar L 1972 Superconducting Tunneling and Application.

Chapman and Hall, London

Steward W C 1968 Current-voltage characteristics of Josephson

junctions. Appl. Phys. Lett. 12, 277

Werthamer N R 1966 Nonlinear self-coupling of Josephson

radiation in tunnel junctions. Phys. Rev. 147, 255

Zubkov A A, Kupriyanov M Yu, Cemenov V K 1981 Station-

ary Properties of Josephson Structure with Ohmic conduc-

tivity. Fiz. Nizk. Temp. (Sov. J. Low Temp. Phys.) 7 (11),

1365–71

M. Kupriyanov

Moscow State University, Russia

5f Electron Systems: Magnetic Properties

5f electron systems are characteristic of actinides,

i.e., elements between actinium, atomic number

Z ¼89, and lawrencium, Z ¼103. These elements

128

5f Electron Systems: Magnetic Properties

are characterized by a gradual ﬁlling of the 5f elec-

tronic shell, in analogy to the 4f shell ﬁlling in lan-

thanides. The ﬁlling tendencies for free-ion states

with increasing Z resemble strongly those in lantha-

nides, but as part of a solid the behavior of the 5f

electrons resembles 3d materials more. The reason is

the spatial extent of the 5f wave functions, being

larger compared with the 4f states in lanthanides.

Therefore the magnetism of most 5f electron systems

is more similar to the d-band magnetism than to the

localized 4f magnetism of lanthanides. This is partic-

ularly true for light actinides (up to plutonium),

whereas for the heavier actinides (from americium

onwards) 5f localization has been documented for the

pure elements. More systematic information on mag-

netism of heavy-actinide compounds is missing.

The specific characteristics of the 5f electron mag-

netism are manifest in systems based on elements

up to plutonium. The higher actinides behave in an

analogous way to the corresponding 4f systems. The

main characteristics of the 5f magnetism comprise

strong spin-orbit interaction, leading to large orbital

moments formed even in the case of band-like states,

exchange interactions mediated or assisted by the

hybridization of the 5 f states with the ligand states,

and enormous magnetic anisotropy arising from the

anisotropy of the hybridization (bonding anisotropy).

Another characteristic is a high sensitivity of mag-

netic properties to external variables, such as pressure

and magnetic ﬁeld, and to ﬁne details of composition.

1. General Features of Electronic Structure and

Magnetism of Light-actinide Systems

The fundamental difference between the character of

the 4f electron states in lanthanides (see Localized 4f

and 5f Moments: Magnetism) and 5f states in light

actinides can be attributed to a much larger spatial

extent of the 5f wave functions, and thus a much

stronger interaction with the metallic environment,

compared to the 4f case. The 5f electrons are, as

a rule, delocalized due to their participation in

bonding, which leads to a considerable hybridization

of the 5f states with the valence states of neighboring

atoms (5f ligand hybridization). The delocalization of

the 5f electrons has serious consequences. The most

important one is, that the 5f states form a more or

less narrow 5f band intersected by the Fermi energy

(the bandwidth W

is of the order of several eV)

rather than discrete energy levels. Consequently, the

magnetic moments due to the itinerant 5f electrons

are much smaller than expected for a free ion, and

magnetic moments can disappear in a broad-band

limit leading to weak (Pauli) paramagnetism.

This situation resembles to a certain extent the 3d

transition metals. The strength of magnetic coupling

in cases of existing 5f moments is typically much

larger than for the 4f moments interacting via the

RKKY interaction. The impact on magnetic excita-

tions is even more dramatic, no crystal-ﬁeld excita-

tions could be observed by inelastic neutron

scattering in a vast majority of systems studied so

far. Instead, one observes typically a rather broad

quasielastic response reﬂecting the 5f moment’s in-

stability in analogy to, for example, cerium mixed

valence materials. However, the high density of states

at E

is projected into high g-values of the low tem-

perature specific heat (which are further renormalized

by strong e–e correlations), and into highly anoma-

lous transport properties, resulting from the hybrid-

ization of such ‘‘heavy’’ electron states with the non-f

states carrying the electrical current.

2. Pure Actini de Elements

The small separation of the ions in the pure elements

leads to large overlap of the 5f wave functions of

nearest neighbors, formation of a broad 5f band and

weakly paramagnetic behavior. With increasing 5f

occupation the value of the Pauli-type susceptibility

increases due to the increase of N(E

), the density of

states at the Fermi level, which is corroborated by a

similar increase of the low-temperature electronic

specific heat coefﬁcient g (see Table 1). This tendency

Table 1

Overview of essential properties of elemental actinides, summarizing available data on the g-coefﬁcient of the low-

temperature specific heat, temperature independent susceptibility w

,Ne

el temperature T

and Curie temperature T

Th Pa U Np Pu Am

g (mJ mol

1

2

) 4 6.6 10 14 22 2

(10

8

mol

1

) 0.12 0.34 0.48 0.68 0.64 0.85

Cm Bk Cf Es

(K) 64 34 51(T

)

eff

) 7.55 9.7 9.7 11.3 (?)

129

5f Electron Systems: Magnetic Properties