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

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magnetic ﬁeld. The true pressure distribution shows

the proﬁle as indicated by solid line in Fig. 2. How-

ever, if the pressure in the magnetic ﬁeld region is

measured by a manometer outside the magnetic ﬁeld,

the pressure difference in the manometer indicates

only the pressure drop due to pipe friction loss shown

by the chain line. In such a case, the magnetic pres-

sure also acts on the ﬂuid in the connecting tube

across the magnetic ﬁeld and cancels the magnetic

static pressure in the pipe.

Now, we would like to examine in some detail the

pressure distribution measured by the manometer

using the test tube in the axal magnetic ﬁeld distri-

bution as shown in Fig. 3 where the pressure coef-

ﬁcient C

along the pipe axis is plotted

p  p

ð32Þ

where p

is the pressure at the upstream reference

point. It is clearly shown that a peculiar pressure

drop occurs at the edge of the imposed magnetic ﬁeld.

The following loss coefﬁcients may be deﬁned from

Fig. 3:

z ¼ l

þ l

þ z

out

ð33Þ

where l

denotes the resistance coefﬁcient in uniform

magnetic ﬁeld and l

in the absence of magnetic ﬁeld.

and z

out

are loss coefﬁcients in nonuniform mag-

netic ﬁeld regions at the inlet and outlet. Since the

pressure gradient is supposed to be nearly constant in

the uniform ﬁeld region, the resistance coefﬁcient l

was estimated. The relation between the resistance

coefﬁcient l, which corresponds to l

and the mod-

iﬁed Reynolds number Re* is shown in Fig. 4 where

l ¼



and Re



Z þ DZ

ð34Þ

It is clear that in the laminar ﬂow regime the re-

sistance coefﬁcient l increases with the intensity of

magnetic ﬂux density B. In particular, the coefﬁcient

l for water-based ﬂuid is much larger than that for

Figure 2

Typical experimental result of pressure distribution measured by transducer and manometer.

Figure 3

Measured pressure distribution along the pipe axis

measured by manometer.

220

Ferrohydrodynamics

kerosene-based ﬂuid. On the other hand, l for tur-

bulent ﬂow is not inﬂuenced by the application of

magnetic ﬁeld; it has a slightly lower value compared

to the Blasius formula as a Newtonian ﬂuid.

Since it is observed that the large pressure drop

occurs at the nonuniform ﬁeld region, it is important

to examine the effect of nonuniform magnetic ﬁeld on

the ﬂow resistance. An experimental study was, there-

fore, done to clarify the effect of nonuniform mag-

netic ﬁeld on the ﬂow resistance. The nonuniform

magnetic ﬁeld distribution was generated by adjust-

ing the pole gap of the electromagnet as shown in

Fig. 5. Figure 6 shows the experimental results of the

pressure change Dp

1–2

against mean ﬂow velocity at

several values of current I supplied to the electro-

magnet for both water-based and kerosene-based ﬂu-

ids. Dp

denotes the pressure drop over the pipe

length L:

¼ l

ð35Þ

In this case, subscript 1–2 denotes the change be-

tween the locations z

and z

in Fig. 5. The theoretical

values obtained from Eqn. (26), modiﬁed for the

transverse magnetic ﬁeld, are also shown in Fig. 5.

The experimental values for the kerosene-based ﬂuid

agree well with the predicted ones as indicated in Fig.

5. On the other hand, it is easy to see that the ex-

perimental values for the water-based ﬂuid are much

larger than the predicted ones.

Measurements of the ﬂow resistance were also

made for other pipes with variable cross-sectional

area such as venturi and long oriﬁce tubes (Kami-

yama et al. 1987).

2.3 Unsteady Flow

(a) Oscillatory pipe ﬂow characteristics

The unsteady pipe ﬂow problem is also important in

relation to development in the application to mag-

netic ﬂuid dampers and actuators. As an example of

unsteady pipe ﬂow problems, oscillatory ﬂow in a

straight pipe in a transverse magnetic ﬁeld is con-

sidered here. Theoretical analysis of the oscillatory

ﬂow is made under the assumption that the shape of

aggregated particles in the magnetic ﬂuid is a rigid

prolate spheroid as shown in Fig. 7 (Tsebers 1974).

When the basic equations are simpliﬁed by the same

Figure 4

Effect of magnetic ﬁeld on the relation of resistance

coefﬁcient l versus Reynolds number Re* in a uniform

magnetic ﬁeld.

Figure 5

Distribution of magnetic ﬂux density in the test section.

Figure 6

Pressure difference as a function of mean ﬂow velocity.

221

Ferrohydrodynamics

procedure as for steady ﬂow in Sect. 2.1, the follow-

ing dimensionless equation is obtained:





¼

0



þ 1 þ

2DZ





2

þ 1 þ

2DZ





þ 1 þ

2DZ





2

ð36Þ

together with the boundary conditions



¼ 1: u



¼ 0; r



¼ 0:



¼ 0 ð37Þ

Here, W ¼r

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

or=Z

is Womersley number,

0

¼ p





nkT

lnðx

1

 sinh xÞ;

x  tanh x

x þ tanh x

þ l 3L





sin







x  tanh x

x þ tanh x

þ lL



cos



þl 2L





sin

y þ





x  tanh x

x þ tanh x

þ l 3L





cos







ð38Þ

and

a ¼

4fZ

1  d

ð2  d

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  d

1 þ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  d

 d

is the rotational friction per unit ﬂuid volume and

¼ coth x  x

1

¼ 1 

¼ L



l ¼

1  d

1 þ d

d ¼



p=Zu



¼ ot

ð39Þ

where u

is the amplitude of oscillating mean ﬂow

velocity and Nu is the number of aggregated particles.

When the oscillating ﬂow rate is given, the velocity

distribution is obtained by solving Eqn. (36) using

difference calculus. Also, ﬂuctuating pressure differ-

ence between two points in a pipe is expressed by

Eqn. (40):

0

¼u



ð40Þ

where U* ¼u*/(dp*

/dz*), l* ¼l/r

, and l is the length

between two measuring points in a pipe.

Figure 8 shows the calculated results of the effect

of magnetic ﬁeld on the pressure difference in the pipe

compared with the experimental data obtained by

using the water-based magnetic ﬂuid with 20% mass

concentration of particles (Shimada and Kamiyama

1993). The vertical axis shows the increase rate of the

pressure difference of the ﬁrst mode oscillatory ﬂow

between two pressure transducers installed apart for

160 mm apart with an applied magnetic ﬁeld. The

abscissa is the Womersley number. The Reynolds

number Re is deﬁned as 2r

/n, where n is the kin-

ematic viscosity. I ¼20 A is the current supplied to

the electromagnet which produces a nonuniform

magnetic ﬁeld with H

max

¼120 kAm

1

. The results

show that the ﬂuctuating pressure difference increases

with the magnetic ﬁeld strength especially at low

Womersley numbers. The scatter of the experimental

data is shown with error bands in Fig. 8. Theoretical

analysis can also explain the experimental results well

by assuming that the aggregated particle number Nu

depends on the Womersley number.

(b) Pulsating ﬂow characteristics

As another example of the unsteady pipe ﬂow, we

consider here pulsating pipe ﬂow, that is, the velocity

Figure 7

Model of aggregated particles in a magnetic ﬁeld.

222

Ferrohydrodynamics

of the ﬂuid is composed of steady and oscillatory ﬂow

þu

) in a pulsating magnetic ﬁeld (H

þH

). Eqn.

(36) is solved numerically by assuming that each

time-varying variable is expanded in a series of

number n( ¼1, 2 y n) and then ordering the result-

ing terms with respect to the power of n (Shimada

and Kamiyama 1994). Time-varying velocity proﬁles

in a circular pipe are obtained as a function of the

Womersley number W and the magnetic ﬁeld param-

eter. Generally speaking, it is clear from the numer-

ical calculations that the velocity proﬁles are mainly

inﬂuenced by W in the same way as the ordinary ﬂuid

ﬂow; for smaller W, the velocity proﬁle is parabolic-

like and for larger W, it becomes a proﬁle with a

maximum near the wall.

Figure 9 shows the results calculated for increasing

rate of pressure difference with the application of

pulsating magnetic ﬁeld compared with the experi-

mental results. The results in the case of the oscillatory

ﬂow both in the steady and the pulsating magnetic

ﬁeld are also shown for comparison, where b denotes

the phase difference between ﬂuctuating velocity and

magnetic ﬁeld. It is easy to see that the experimental

data for the pulsating ﬂow are smaller than those for

oscillatory ﬂow in the pulsating magnetic ﬁeld and

this result is similar to the theoretical analysis. How-

ever, the experimental data also show that the differ-

ence in the oscillatory ﬂow characteristics appears

between the cases of the pulsating and steady mag-

netic ﬁeld. The cause seems to be the complicated ef-

fect of the magnetic ﬁeld on the particles aggregation

in the unsteady ﬂow state.

2.4 Gas–Liquid Two-phase Flow

Gas–liquid two-phase pipe ﬂow problems are closely

connected with the development of a new energy con-

version system utilizing magnetic ﬂuid. The use of the

gas–liquid two-phase ﬂow is very effective in increas-

ing the driving force of circulating ﬂuid in this device.

(a) Theoretical analysis

Let us consider one-dimensional two-phase ﬂow in

the vertical pipe. Figure 10 shows schematically the

system used in the theoretical analysis as well as in

the experiment. A temperature sensitive magnetic

Figure 9

Comparison of pressure difference between pulsating

and oscillatory ﬂows.

Figure 8

Increase rate of pressure difference versus Womersley

number.

223

Ferrohydrodynamics

ﬂuid is assumed to be heated passing through the

point of the maximum magnetic ﬁeld strength toward

the downstream side region and boiled to form the

gas–liquid two-phase ﬂow. Under the above condi-

tions, the governing equations of the boiling two-

phase ﬂow based on the thermal nonequilibrium two-

ﬂuid model are derived as follows (Kamiyama and

Ishimoto 1995). The equations of continuity are:

ðr

aÞ¼G

ð1  aÞg ¼G

ð41Þ

where a is the void fraction, v is the velocity, and G is

the phase generation rate. The subscript g and l de-

note gas and liquid phase respectively. G

and G

are

expressed by

¼G

¼ G

ðeÞ

 G

ðcÞ

for T

¼ 0 for T

ð42Þ

where T

¼(T

þT

)/2, T

is the liquid-phase tem-

perature at the inlet section (z ¼l

) and T

is the

evaporation temperature. G

(e)

and G

(c)

are the mean

evaporation and condensation rates (Dobran 1988)

and are given by

ðeÞ

¼ e



ðeÞ

ð1  aÞr

að





for T

¼ 0 for T

ð43Þ

ðcÞ

¼ e



ðcÞ

ð1  aÞr

aðT

 T



for T

¼ 0 for T

ð44Þ

where e

(e)

and e

(c)

are the relaxation parameters for

the evaporation and the condensation and R

is the

gas constant. The variables A

and A

are propor-

tional to the contact area between the gas and liquid

phases. They include the effect of the number density

of the vapor bubbles generation N

[1/(m

1

)] which

is assumed to have normal distribution in the region

of z ¼02z

max

. The combined momentum equation

is:

þ r

ð1  aÞv

g¼

ð1  aÞr

þð1  aÞm

þ G

ðv

ðiÞ

 v

ðiÞ



32fZ

þ Z

ð1  aÞv

ð45Þ

where g is acceleration of gravity and sufﬁx (i) de-

notes the interface between the gas and liquid phases.

To consider the effect of slip and radial expansion

of the bubbles, the equation of motion for the gas

phase is replaced here by the translational motion of

a single bubble (Ishimoto et al. 1994). The equation

of motion for the gas phase is:

¼



 F

ð46Þ

where R

is the mean radius of the bubble deﬁned by

Eqn. (51) below. F

is the drag force, F

is the

virtual mass force considering the expansion of the

bubble, and F

is the Basset term which takes into

account the effect of deviation in a ﬂow pattern from

the steady state.

The energy equations for the gas and liquid phases

expressed as follows:

!()

¼r

g þ G

ðiÞ

þ q

ðiÞ

ð47Þ

ð1  aÞv



¼r

ð1  aÞv

g þ G

ðiÞ

þ q

ðiÞ

ð1  aÞm



þ Q

ð48Þ

Figure 10

Analytical model of two-phase ﬂow in a nonuniform

magnetic ﬁeld.

224

Ferrohydrodynamics

where q

(i)

and q

(i)

are the heat ﬂuxes through the gas–

liquid interface (q

þq

¼0). h

(i)

and h

(i)

are the re-

spective enthalpies as the interface. a

(i)

is the interface

area concentration and Q

is the heat transfer rate

per unit volume. It is assumed that the energy trans-

fer is caused by the heat transfer between an isother-

mal spherical bubble and the surrounding liquid.

With the assumption of a spherical bubble with an

equivalent radius R

, the expression for a

(i)

is obtained

as a

(i)

¼3a/R

(Kataoka 1986).

To close the basic equations in this system, the

following three equations are introduced.

The equation of the ideal gas law:

¼ R

ð49Þ

where R

is the gas constant.

The equation of the pressure balance between the

gas and liquid phases:

 p

ð50Þ

where s denotes the surface tension in the liquid

phase.

The mass conservation equation for the single gas

bubble:

ðr

Þ¼



ð51Þ

where N

is the number density of the generation of

the vapor bubbles.

(b) Experimental study

An experimental study to measure precisely the liquid

phase pressure, the temperature, and the mean ve-

locity was conducted using a test loop tube, the inner

diameter and the total length of which are 8.0 mm

and 3.3 m, respectively. Taking advantage of the high

permeability of an ultrasonic wave in the magnetic

ﬂuid, the ultrasonic imaging technique was applied to

the investigation of a boiling two-phase ﬂow in the

magnetic ﬁeld.

Figure 11 shows a schematic diagram of the ex-

perimental set-up for the boiling two-phase ﬂow of

the magnetic ﬂuid. The ultrasonic wave probe is

closely attached to the ﬂow pipe through the gel pad.

The ultrasonic wave echo visualizes a bubble in the

magnetic ﬂuid. The magnetic ﬂuid is heated by a

YAG laser, with a constant heat ﬂux passing through

the point of the maximum magnetic ﬁeld strength

toward the downstream side.

Figure 12 shows the experimental results of the

pressure distribution in the liquid phase, Dp

com-

pared with the numerical analysis. Dp

is the pressure

difference at the inlet section of the applied magnetic

ﬁeld region and is deﬁned by Dp

¼p

r

g(l

z)p

The measured pressure distribution agrees reasonably

with the numerical results. Also, because of the ap-

pearance of the gas phase and the temperature in-

crease after passing through the heated region, Dp

at the exit position increases with the maximum mag-

netic ﬁeld strength H

max

. These experimental and

theoretical results explain the effectiveness of the two-

phase ﬂow in increasing the driving force of the mag-

netic ﬂuid ﬂow in a nonuniform magnetic ﬁeld.

Figure 11

Experimental set-up for two-phase ﬂow in a magnetic

ﬁeld.

Figure 12

Pressure distribution in liquid phase as a function of z

coordinate for different magnetic ﬁeld strength.

225

Ferrohydrodynamics

3. Flow Around a Blunt Body Coated with a

Magnetic Fl uid

To reduce the drag and enhance the heat transfer by

controlling ﬂow separation in the ﬂow around a blunt

body, the magnetic ﬂuid coating on a solid body is

investigated. A cylinder coating is made by using a

permanent magnet cylinder magnetized transverse to

its axis or an electrically conducting cylinder carrying

the electric current. The magnetic ﬁeld keeps the

magnetic ﬂuid as the coating which has a constant

thickness (its radius ¼a). The shape of this coating

changes under the action of the viscous ﬂow as shown

in Fig. 13.

The numerical calculation of this ﬂow problem

with a transforming the boundary shape was carried

out using a new ﬁnite element method technique and

compared with experimental data (Krakov and

Kamiyama 1995). Figure 14 shows the calculated re-

sults of a drag reduction ratio G as a function of the

initial radius ratio a/R in the case when the viscosity

ratio of the outer ﬂuid (Z

) and the magnetic ﬂuid (Z

)

equals 100. The Reynolds number is deﬁned as

Re ¼r

2R/Z

. The experimental data were ob-

tained by free fall of a cylindrical permanent magnet

coated with the magnetic ﬂuid in silicon oil (Berkov-

sky et al. 1990). It is clear from Fig. 14 that the drag

reduction with the ﬁlm coating is effective in high

viscous ﬂuid ﬂow around the blunt body.

See also: Ferroﬂuids: Applications; Ferroﬂuids:

Magnetic Properties; Ferroﬂuids: Preparation and

Physical Properties

Bibliography

Berkovsky B M, Bashitovi V G, Krakov M S 1990 Flow and

heat transfer under inﬂuence of magnetic ﬂuid coating. J.

Magn. Mater. 85, 190–5

Dobran F 1988 Liquid and gas-phase distributions in a jet with

phase change. Trans. Am. Soc. Mech. Eng. J. Heat Transf.

110, 955–60

Ishimoto J, Okubo M, Kamiyama S 1994 Study of the MHD

power-generation system using two-phase ﬂows of tempera-

ture-sensitive electrically conducting magnetic ﬂuids (Analy-

sis of ﬂow characteristics in a pipe and the power-generation

characteristics). Trans. Jpn. Soc. Mech. Eng. Ser. B 60 (571),

891–8, (in Japanese)

Kamiyama S 1992 Pipe ﬂow problems of magnetic ﬂuids. Jpn.

Soc. Mech. Eng. Int. J., Ser. II 35, 131–7

Kamiyama S, Ishimoto J 1995 Boiling two-phase ﬂows of mag-

netic ﬂuid in a non-uniform magnetic ﬁeld. J. Magn. Magn.

Mater. 149, 125–31

Kamiyama S, Koike K, Iizuka N 1979 On the ﬂow of a ferro-

magnetic ﬂuid in a circular pipe (Report 1, Flow in a uniform

magnetic ﬁeld). Bull. Jpn. Soc. Mech. Eng. 22, 1205–11

Kamiyama S, Koike K, Oyama T 1983 Pipe ﬂow resistance of

magnetic ﬂuids in a nonuniform transverse magnetic ﬁeld. J.

Magn. Magn. Mater. 39, 23–6

Kamiyama S, Htwe J, Koike K, Oyama T 1987 Some pecu-

liarities of pipe ﬂow characteristics of water-based magnetic

ﬂuid. J. Magn. Magn. Mater. 65, 317–20

Kataoka I 1986 Local instant formulation of two-phase ﬂow.

Int. J. Multiphase Flow 12 (5), 745–58

Krakov M S, Kamiyama S 1995 Steady ﬂow past a circular

cylinder coated with a magnetic ﬂuid: Flow structure, drag

reduction and coating deformation. J. Fluid Mech. 295, 1–22

Figure 14

Drag reduction ratio as a function of radii ratio.

Figure 13

Flow past a circular cylinder coated with a magnetic

ﬂuid.

226

Ferrohydrodynamics

Neuringer J L, Rosensweig R E 1964 Ferrohydrodynamics.

Phys. Fluids 7, 1927–37

Rosensweig R E 1985 Ferrohydrodynamics. Cambridge Univer-

sity Press, Cambridge, pp. 100–10

Shaposhinikov I C, Shliomis M I 1975 Hydrodynamics of

magnetic ﬂuids. Magnetohydrodynamics 10, 381–93

Shimada K, Kamiyama S 1993 Oscillatory pipe ﬂow of mag-

netic ﬂuid in a strong magnetic ﬁeld. J. Magn. Magn. Mater.

122, 214–6

Shimada K, Kamiyama S 1994 Pulsating ﬂow of magnetic ﬂu-

ids in a pipe under ﬂuctuating magnetic ﬁeld. Jpn. Soc. Mech.

Eng. Int. J. Ser. B 37, 71–6

Shliomis M I 1967 Hydrodynamics of a liquid with intrinsic

rotation. Sov. Phys. JETP 24, 173–7

Tsebers A O 1974 Flow of dipole ﬂuids in external ﬁelds. Mag-

netohydrodynamics 10, 381–93

S. Kamiyama

Akita Prefectural University

Honjo, Japan

Films and Multilayers: Conventional

Superconducting

Thin-ﬁlm multilayers are artiﬁcially fabricated mate-

rials that consist of thin slabs of various materials

deposited sequentially on a substrate. In general,

when the individual layer thicknesses are comparable

with certain physical length scales for electronic in-

teractions, the electronic transport takes on new

characteristics which could be quite different from

the electronic properties of the constituent layers.

Superconductivity is one property particularly ame-

nable to layered modiﬁcations due to its rather long

coherence length in many materials.

The superconducting materials community has his-

torically paid particular attention to ﬁlms and mul-

tilayers, mostly for uncovering new superconducting

mechanisms, but also in studying superconducting

phenomena in ‘‘tunable’’ model systems. Supercon-

ducting thin ﬁlms are also technologically important

as they are a basis for superconducting electronics.

Besides artiﬁcially structured multilayers, layered

structures also exist naturally in transition-metal di-

chalcogenide superconductors such as NbSe

and

high-T

cuprate superconductors such as YBa

and Bi

CaCu

(see High-temperature Supercon-

ductors: Thin Films and Multilayers).

In this article the focus is on presenting a frame-

work for understanding the physics of superconduc-

tivity in layered materials. The discussion centers on

one of the interesting aspects of the layered supercon-

ductors which is their static anisotropic magnetic

properties and dimensional cross-over. For more de-

tailed reviews of materials systems involved see, for

example, Ruggiero and Beasley (1984), Matijasevic

and Beasley (1987), and Schuller et al. (1990). The

phenomenological framework of Ginzburg–Landau

(GL) that is presented here provides for a simple un-

derstanding of the macroscopic superconducting

properties of multilayers. However, if the layering af-

fects the fundamental interactions that bring about the

superconductive electronic pairing, this picture will be

inadequate, and a microscopic model is needed.

1. Types of Multilayers

Superconducting multilayer structures typically con-

sist of two different layers which alternate in a one-

dimensional layering lattice. One material is the pri-

mary superconductor (S) and the second material can

be an insulator (I), another superconductor (S

), a

normal metal (N), or a ferromagnet (F) (see Fig. 1).

The wavelength of the modulation is usually denoted

by s, with s ¼d

þd

, where d

and d

are the thick-

nesses of the superconducting and mediating layers,

respectively. In reality materials issues are often more

complicated, producing, for example, an interfacial

layer which could be another superconducting phase,

so that there are S/S

/N multilayers rather than sim-

ply S/N.

1.1 Superconductive Coupling

In practice there are two different mechanisms for

Josephson coupling of superconductivity between the

superconducting layers in a multilayer: pair tunnel-

ing and superconducting proximity effect coupling

(extension of Cooper pairs into a normal metal). Pair

Figure 1

Schematic of a multilayer. X represents an insulator, a

second superconductor S

, a normal metal N, or a

ferromagnet F.

227

Films and Multilayers: Conventional Superconducting

tunneling is relevant for the S/I systems and proxim-

ity effect for the others. Both couplings decay expo-

nentially with the separation of the S layers, but the

decay scale is different. For tunneling the range is

typically 1–2 nm, while proximity effect coupling is

typically at 10–1000 nm. Pair tunneling is tempera-

ture-independent, for example, so the details of the

coupling are different in the two cases, but for many

macroscopic properties the actual nature of the cou-

pling is not relevant.

There has been increasing interest in superconduc-

tive coupling via ferromagnetic metallic layers. In the

case of coupling through ferromagnetic materials the

magnetic pair breaking reduces the coupling distance

typically to about 1–10 nm. In a ferromagnet adjacent

to a superconductor, the exponentially decaying su-

perconducting pair function is presumed to oscillate,

as was ﬁrst suggested by Buzdin et al. (1982) and

Radovic et al. (1988). This behavior is akin to the

description of an oscillatory superconducting order

parameter that is generated in the presence of an ex-

change ﬁeld, described by the Fulde–Ferrell–Larkin–

Ovchinikov (FFLO) effect (Fulde and Ferrell 1964,

Larkin and Ovchinikov 1964).

The oscillating order parameter reveals itself in an

oscillating T

as a function of d

and therefore is

manifestly different from other types of couplings.

Jang et al. (1995) and Mu

hge et al. (1996) have shown

such behavior in two different S/F systems, but the

experimental situation is not yet clariﬁed. When the

coupling is negative between the S layers it is pre-

sumed that there would be a p phase shift between the

superconducting layers to give a so-called p-junction.

Subsequent reports by Ryazanov et al. (2001) have

provided strong evidence for such an oscillatory su-

perconducting wave function.

Figure 2 contrasts the different types of mediating

layers and the relevant length scales for the coupling.

The present discussion focuses on macroscopic

thermodynamic properties of the superconducting

Figure 2

Comparison of the superconducting order parameter across various interfaces and their relevant length scales in the

clean and dirty material limits. T

is the transition temperature of the S

superconductor, v

is the Fermi velocity, l is

the mean free path, D is the diffusion constant, and E

is the exchange splitting in the ferromagnet.

228

Films and Multilayers: Conventional Superconducting

multilayer. For a discussion of, for example, elec-

tronic transport perpendicular to the layers see Jose-

phson Junctions: High-T

. A number of properties of

superconducting multilayers have been studied ex-

perimentally, with transition temperature and upper

critical ﬁelds being among the most commonly stud-

ied. When a conducting material is adjacent to the

superconductor the overall T

can be reduced due to

an averaging of the pairing function with the N or F

layer. For the case of the magnetic material in contact

with the superconductor, there is additional proxim-

ity-induced magnetic pair breaking in the S layers.

1.2 Dimensionality of Superconducting Multilayers

Multilayer superconductors can be distinguished ac-

cording to whether the individual superconducting

layers are thick or thin compared with their intrinsic,

i.e., intralayer, temperature-dependent GL coherence

length x(T). If the layer thickness d

4x(T) then the

layers themselves are three-dimensional and exhibit

bulk material properties. In this case the layering only

introduces planar defects to the superconductor

which could be of interest for ﬂux pinning and crit-

ical currents when a registration of the vortex lattice

occurs with the multilayer lattice. If d

ox(T), then

the individual layers are two-dimensional. The be-

havior of the whole system will depend on the

strength of the coupling across the mediating layers.

The dimensionality of a superconducting multilay-

er is not dependent on the detailed nature of the

coupling, but can be described macroscopically by a

coherence length in the direction perpendicular to the

layers, called x

. This perpendicular coherence length

reﬂects the coupling between the S layers. For

(T)bs the system will behave as a bulk anisotrop-

ic superconductor. For systems where the x

(T)ps

one can expect a cross-over from three-dimensional

to two dimensional behavior as a function of tem-

perature when x

(T) reaches Bs. This classiﬁcation

is summarized in Table 1. The theoretical model

described below illustrates this cross-over behavior

quantitatively.

2. Magnetic Properties of Superconducting

Multilayers

The upper critical ﬁeld phase boundary is one of the

macroscopic properties that directly reﬂects on the

size of the superconducting coherence length in the

material and the dimensionality of superconductivity.

Another example, which is not discussed here in de-

tail, are the superconducting ﬂuctuations above T

where the size of the volume for averaging the ﬂuc-

tuations is also x(T).

2.1 Ginzburg–Landau Description of Anisotropic

Superconductors

The GL description of superconductivity follows the

Landau theory for a two-component order parameter

phase transition. The GL free energy of a supercon-

ductor in the presence of a magnetic ﬁeld can be

written as

F ¼F

a7c7

7c7



r









where a ¼a

(TT

) and m* ¼2m, e* ¼2e, and upon

variation with respect to c, the usual GL equation is

obtained:

ac þ b7c7

c þ



r





c ¼ 0 ð1Þ

The GL formulation is often generalized to an an-

isotropic form where the mass m* is replaced by an

effective-mass tensor, such as





ð2Þ

for a uniaxial system as a multilayer where z axis is

the symmetry direction.

Associated with this anisotropic mass tensor are

the anisotropic GL coherence lengths and magnetic

ﬁeld penetration depths,

x;z



27a7m

x;z



1=2

and

x;z

4pe



27c7

x;z



1=2

In a layered superconductor x

(T) reﬂects the effec-

tive coupling across superconducting layers and not

Table 1

Classiﬁcation of superconducting multilayers according

to dimensionality.

Length scale

condition

Dimensionality

of the system

Behavior of

macroscopic

superconducting

properties

3D; weakly

coupled

3D isotropic

, x

os 2D; weakly

coupled

2D single ﬁlm-

, x

4s Quasi-2D;

strongly

coupled

3D anisotropic

229

Films and Multilayers: Conventional Superconducting