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

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See also: Magnets Soft and Hard: Magnetic Do-

mains; Micromagnetics: Finite Element Approach

Bibliography

Aharoni A 1962 Theoretical search for domain nucleation. Rev

Mod. Phys. 34, 227–38

Bauer J, Seeger M, Zern A, Kronmu

ller H 1996 Nanocrystal-

line NdFeB permanent magnets with enhanced remanence. J.

Appl. Phys. 80, 1667–73

Becher M, Seeger M, Bauer J, Goll D, Kronmu

ller H 1998

Magnetic viscosity measurements on nanocrystalline NdFeB

and PrFeB magnets. In: Schultz L, Mu

ller K H (eds.) Proc.

10th Int. Symp. on Magnetic Anisotropy and Coercivity in

Rare-Earth Transition Metal Alloys. Werkstoff-Informations-

gesellschaft, Frankfurt, Germany, pp. 307–16

Becker R, Do

ring W 1939 Ferromagnetismus. Springer, Berlin

Brown W F Jr. 1945 Virtues and weaknesses of the domain

concept. Rev. Mod. Phys. 17, 15–9

Brown W F Jr. 1963 Micromagnetics. Interscience, New York

Coehoorn R, de Mooij D B, Duchateau J P, Buschow K H J

1988 Novel permanent magnetic materials made by rapid

quenching. J. Phys. (Paris) Colloq. 49, 669–70

Fischer R, Kronmu

ller H 1996 Static computational micro-

magnetism of demagnetization processes in nanoscaled per-

manent magnets. Phys. Rev. B 54, 7284–94

Fischer R, Kronmu

ller H 1998 The role of grain boundaries in

nanoscaled high-performance permanent magnets. J. Magn.

Magn. Mater. 184, 166–72

Fischer R, Schreﬂ T, Kronmu

ller H, Fidler J 1995 Phase dis-

tribution and computed magnetic properties of high-reman-

ent composite materials. J. Magn. Magn. Mater. 150, 329–44

Fischer R, Schreﬂ T, Kronmu

ller H, Fidler J 1996 Grain size

dependence of remanence and coercive ﬁeld of isotropic

nanocrystalline composite permanent magnets. J. Magn.

Magn. Mater. 153, 35–49

Friedberg R, Paul D 1975 New theory of coercive force of

ferromagnetic materials. Phys. Rev. Lett. 34, 1234–7

Gaunt P, Mylvaganam C K 1981 Domain wall pinning and

nucleation in SmCo

: I. Thermal activation and coercive

ﬁeld. Phil. Mag. B 44, 569–80

Givord D, Rossignol M F 1996 Coercivity. In: Coey J M D

(ed.) Rare-Earth Iron Permanent Magnets. Clarendon Press,

Oxford, UK Chap. 5

Goll D 2001 Ph.D. thesis, University of Stuttgart

Goll D, Kleinschroth I, Sigle W, Kronmu

ller H 2000 Melt-spun

precipitation-hardened Sm

(Co,Cu,Fe,Zr)

magnets with

abnormal temperature dependence of coercivity. Appl. Phys.

Lett. 76, 1054–6

Goll D, Seeger M, Kronmu

ller H 1998 Magnetic and micro-

structural properties of nanocrystalline exchange coupled

PrFeB permanent magnets. J. Magn. Magn. Mater. 185,

49–60

Hadjipanayis G C 1982 Microstructure and magnetic domain

structure of 2:17 precipitation-hardened rare-earth cobalt

permanent magnets. In: Fidler J (ed.) Proc. 6th Int. Workshop

on Rare Earth–Cobalt Permanent Magnets. Technical Uni-

versity of Vienna, Austria, pp. 609–30

Herzer G 1990 Grain size dependence of coercivity and perme-

ability in nanocrystalline ferromagnets. IEEE Trans. Mag.

26, 1397–402

Hilzinger H R, Kronmu

ller H 1975 Investigation of Bloch wall

pinning of antiphase boundaries in RCo

compounds. Phys.

Lett. 51A, 59–60

Hilzinger H R 1977 The inﬂuence of planar defects on the

coercive ﬁeld of hard magnetic materials. Appl. Phys. 12,

253–60

Kersten M 1942 Grundlagen einer Theorie der ferromagnetisc-

hen Hysterese und Koerzitivkraft. Doctoral thesis, Tech-

nische Hochschule Stuttgart, Germany

Kneller E F, Hawig R 1991 The exchange-spring magnet: a new

material principle for permanent magnets. IEEE Trans. Mag.

27, 3588–600

Kronmu

ller H 1966 Magnetisierungskurve der Ferromagnetika.

In: Seeger A (ed.) Moderne Probleme der Metallphysik.

Springer, Berlin, Vol. 2, pp. 24–156

Kronmu

ller H 1972 Magnetic techniques for the study of dis-

locations in ferromagnetic materials. Int. J. Nondestr. Testing

3, 315–50

Kronmu

ller H 1973 Microstructure and micromagnetism. In:

Graham C D, Rhyne J J (eds.) Proc. 18th AIP Conf. on

Magnetism and Magnetic Materials. American Institute of

Physics, New York, pp. 1006–25

Kronmu

ller H 1981 Theory of coercive ﬁelds in amorphous

ferromagnetic alloys. J. Magn. Magn. Mater. 24, 159–67

Figure 24

Vector plots of the magnetization of a nanocrystalline

assembly of grains with /DS ¼20 nm. (a) Remanent

state. Top: magnetization at the surfaces of the cube;

bottom: magnetization along a cut through the PM.

Strong arrows refer to the bulk of the grains, weak

arrows refer to the magnetization within grain

boundaries. (b) Vector plots of magnetization within the

square of (a) for nonideal grain boundaries with f ¼0.1.

Top: remanent state (H

ext

¼0); bottom: applied opposite

ﬁeld m

ext

¼0.6 T showing the nucleation of reversed

magnetization within the grain boundaries (Fischer and

Kronmu

ller 1998).

Coercivity Mechanisms

Kronmu

ller H 1987 Theory of nucleation ﬁelds. Phys. Stat. Sol.

(b) 144, 385–96

Kronmu

ller H 1990 Micromagnetism and magnetization proc-

esses in modern magnetic materials. In: Hadjipanayis G C,

Prinz G A (eds.) Proc. NATO Advanced Study Institute on

Science and Technology of Nanostructured Magnetic Materi-

als. Kluwer, Dordrecht, The Netherlands, pp. 657–75

Kronmu

ller H 1997 Micromagnetism and the microstructure of

modern magnetic materials. In: Hadjipanayis G C (ed.) Proc.

NATO Advanced Study on Magnetic Hysteresis in Novel

Magnetic Materials. Kluwer, Dordrecht, The Netherlands,

pp. 85–108

Kronmu

ller H, Durst K D, Martinek G 1987 Angular depend-

ence of the coercive ﬁeld in sintered Fe

magnets. J.

Magn. Magn. Mater. 69, 149–57

Kronmu

ller H, Durst K D, Sagawa M 1988 Analysis of the

magnetic hardening mechanism in REFeB permanent mag-

nets. J. Magn. Magn. Mater. 74, 291–302

Kronmu

ller H, Fischer R, Seeger M, Zern A 1996 Micromag-

netism and microstructure of hard magnetic materials. J.

Phys. D 29, 2274–83

Landau L, Lifshitz E 1935 On the theory of the dispersion of

magnetic permeability in ferromagnetic bodies. Physikalische

Zeitschrift Sowjetunion 8, 153–69

Livingston J D, Martin D L 1977 Structure of aged

(Co,CuFe)

Sm magnets. J. Appl. Phys. 48, 1350–4

Luborsky F E, Morelock C R J 1964 Magnetization reversal in

almost perfect whiskers. J. Appl. Phys. 35, 2055–66

Manaf A, Buckley R A, Davies H A, Leonowicz M 1991 En-

hanced magnetic properties in rapidly solidiﬁed NdFeB

based alloys. J. Magn. Magn. Mater. 101, 360–2

Martinek G, Kronmu

ller H 1990 Inﬂuence of grain orientation

of the coercive ﬁeld of FeNdB permanent magnets. J. Magn.

Magn. Mater. 86, 177–83

Neu V, Eckert D, Schultz L 1996 Nanocrystalline two phase

Nd–Fe–B powders with Co and small additions of Zr and Si.

In: Missell F B, Villas-Boas V, Rechenberg H R, Landgraf F

J G (eds.) Proc 9th Int. Symp. on Magnetic Anisotropy and

Coercivity in Rare-Earth Transition Metal Alloys. World Sci-

entific, Singapore, pp. 42–8

Rieger G, Seeger M, Kronmu

ller H 1999 Microstructural pa-

rameters in high-remanent sintered NdFeB magnets. Phys.

Stat. Sol. (a) 171, 583–95

Sagawa M, Hirosawa S 1988 Coercivity and microstructure of

REFeB sintered permanent magnets. J. Phys. (Paris) Colloq.

49, 617–22

Schreﬂ T, Fidler J, Kronmu

ller H 1994 Remanence and co-

ercivity in isotropic nanocrystalline permanent magnets.

Phys. Rev. B 49, 6100–10

Stoner E C, Wohlfarth E P 1948 A mechanism of magnetic

hysteresis in heterogeneous alloys. Phil. Trans. R. Soc. A 240,

599–644

Wilcox M, Williams J, Leonowicz M, Manaf A, Daves H 1994

Relations between phase constitution, determined by Mo

ss-

bauer spectroscopy, and the enhanced magnetic properties

for nanocrystalline FeNdB alloys. In: Manwaring C A F,

Jones D, Williams A, Harris I (eds.) Proc. 8th Int. Symp. on

Magnetic Anisotropy and Coercivity in Rare Earth–Transition

Metal Alloys. University of Birmingham, Birmingham, UK,

pp. 87–94

H. Kronmu

ller

Max-Planck-Institut fu

r Metallforschung, Stuttgart

Germany

Coherence Length, Proximity Effect and

Fluctuations

The phenomenological London theory for supercon-

ductivity was able to explain the Meissner effect in

terms of a rigid wave function under the application

of an external magnetic ﬁeld. Under this assumption,

F. and H. London calculated the value of the London

penetration depth l

ð1Þ

where m is the effective mass of the electron and n

the superﬂuid density, which at T ¼0 is equal to the

electron density in the normal state. Since the ratio

m/n does not vary by more than one order of mag-

nitude among common metals, l

does not vary

much from one superconductor to another: roughly

by a factor of 4 from say lead (39 nm) to the high-T

oxides (about 140 nm in YBa

). No explicit

dependence of l

on T

is predicted.

The main modiﬁcation to the London theory pro-

posed by Ginzburg and Landau (GL theory) was the

introduction of a ﬁnite correlation length x, some-

what unfortunately called the coherence length. This

is the minimum length scale over which the superﬂuid

density may vary with only a moderate increase of the

free-energy density. Ginzburg and Landau showed

that under small applied ﬁelds, the spatial variation

of the superﬂuid density remains negligible and the

London result (Eqn. (1)) is recovered, whatever the

respective values of l and x. The effect of a ﬁnite

coherence length is only apparent in strong applied

ﬁelds and when the GL parameter k ¼ l=x is larger

than 1=

ﬃﬃﬃ

. As shown by Pippard, when x4l non-

local corrections lead to an effective value of l some-

what larger than that given by the London theory.

1. Coherence Length

1.1 Coherence Length in the GL Theory

Ginzburg and Landau introduced the coherence

length in the framework of their treatment of the su-

perconducting phase transition as a second-order

transition, occurring at the critical temperature T

The existence of a coherence length that diverges at

the transition is a general property of all second-

order phase transitions:

xðtÞ¼ax

ð1  tÞ

v

ð2Þ

where a is a numerical coefﬁcient of order unity, x

the zero temperature coherence length, t is the re-

duced temperature (T/T

) and v is the critical expo-

nent for the coherence length. In the mean-ﬁeld

approximation x(t) can be obtained from the GL

Coherence Length, Proximity Effect and Fluctuations

free-energy functional:

¼ F

þ AjDj

jDj

þ CjrDj

ð3Þ

where the coefﬁcients A ¼ Nð0 Þðt  1Þ, B ¼

0:098Nð0Þ=ðk

, C ¼ 0:55 x

Nð0ÞwðaÞ; Nð0Þ is

the normal state density of states and wðaÞ is the

Gorkov impurity function. Here D is the pair poten-

tial, equal to the superconducting energy gap when it

is uniform across the sample.

The pair potential D is related to the order param-

eter C by jDj

¼ _

jCj

=ð4mCÞ. In some cases the

free-energy functional is written in terms of the re-

duced-order parameter c ¼ C=C

, where C

is the

equilibrium order parameter at the temperature un-

der consideration. Sometimes it is convenient to use

the pair amplitude F ¼ D=V, where V is the BCS

interaction parameter (e.g., in a normal metal with

V ¼ 0 in contact with a superconductor, the pair po-

tential is zero but the pair amplitude is ﬁnite, as will

be discussed in Sect. 3). In simpliﬁed terms (FV)

and

ðF=pNð0 ÞÞ

are proportional to n

in the two regions.

In the notation used here

jAj

ð4Þ

Since Ap1  t; v ¼

and l is found in this ap-

proximation to diverge with the same exponent. In

the clean limit, therefore, xðtÞ¼0:74 x

ð1  tÞ

1=2

while in the dirty limit xðtÞ¼0:85 ðx

lÞ

1=2

ð1  tÞ

1=2

where l is the normal state electron mean free path.

The applicability of the mean ﬁeld approximation to

the superconducting phase transition will be returned

to later.

The GL theory is phenomenological and does not

predict what the value of x

should be. However, it

does offer a simple method for measuring xðtÞ

through the expression for the nucleation ﬁeld:

ðtÞ¼

2px

ðtÞ

ð5Þ

where F

¼ 2:07  10

15

is the ﬂux quantum. If

the superconductor is type II, H

as given by Eqn.

(5) is the upper critical ﬁeld. If it is type I, H

is the

supercooling ﬁeld. Coherence lengths in the clean and

dirty limits, obtained from H

measurements, are

given in Tables 1 and 2, respectively. Measurements

of the supercooling ﬁeld are usually affected by the

Saint James–de Gennes surface nucleation phenom-

enon which occurs at a ﬁeld H

¼ 1:69 H

in re-

gions of the sample where the surface is parallel to the

applied ﬁeld (Saint James and de Gennes 1963, De-

utscher 1967). Once the superconducting state has

nucleated in some region, it spreads all over the sam-

ple. Hence H

rather than H

is usually the meas-

ured nucleation ﬁeld and x must be calculated

accordingly.

When H

, a stable superconducting surface

sheath of thickness x exists below the nucleation ﬁeld.

Its thickness has been measured directly in lead and

was found to be about twice the coherence length

(Deutscher 1967).

1.2 Coherence Length in the BCS Theory

The microscopic origin of x became clear only with

the advent of the BCS theory: it is the average radius

of a Cooper pair. The value of x

can be simply es-

timated from the uncertainty principle:

ð6Þ

where v

is the Fermi velocity and 2D is the energy

necessary to break a Cooper pair. In the weak cou-

pling limit of the BCS theory, 2D ¼ 3:5 k

. Since v

does not vary much from one metal to another, the

coherence length is essentially inversely proportional

to the critical temperature (Table 1). As a rule of

thumb, x

ðmmÞCT

1

ðK

1

Þ. Since l

is basically in-

dependent from T

, low-T

superconductors should

be type I and high-T

superconductors should be type

II. This is well borne out experimentally. Supercon-

ductors with a T

lower than that of lead (7.2 K) are

type I, superconductors with a higher T

are type II

(niobium, the A15 compounds, the Chevrel phases

and the high-T

oxides).

The effect of impurities on l and x has been cal-

culated by Gorkov. In the limit where the electron

Table 1

Coherence lengths in the clean limit.

(K) x

(mm) x

(mmK)

Aluminum 1.19 1.20 1.4

Indium 3.40 0.33 1.1

Tin 3.72 0.26 0.97

Gallium 5.90 0.16 0.94

Lead 7.20 0.08 0.58

Niobium 9.25 0.035 0.32

YBa

93 0.0015 0.14

Table 2

Coherence lengths in the dirty limit (practical

superconductors).

(K) x(T ¼0) (mm)

Nb–56 at.% Ti 10 0.0050

Sn 17 0.0040

(Nb, Ta)

Sn 18 0.0035

Coherence Length, Proximity Effect and Fluctuations

mean free path l is much shorter than x

(the so-called

dirty limit):

l ¼ 0:64 l

ðx

=lÞ

1=2

ð7aÞ

x ¼ 0:85ðx

lÞ

1=2

ð7bÞ

As ﬁrst observed by Shubnikov, the introduction

of impurities can turn a type I superconductor into a

type II material. Nonmagnetic impurities do not af-

fect the critical temperature (Anderson theorem).

However, magnetic impurities strongly depress T

;

for a low concentration x of paramagnetic impurities

with total angular momentum J this is given by the

BCS Gorkov theory

p

Nð0Þf

ðg  1Þ

JðJ þ 1Þ

where f is the exchange coupling constant and g is the

gyromagnetic ratio. Eqns. (5) and (7b) show that H

is inversely proportional to the mean free path and,

hence, proportional to the normal state resistivity,

that is, to the concentration of impurities. This result

is in excellent agreement with experiment (Serin 1969)

and is put into practice in the commercial Nb–Ti

wires, which have an upper critical ﬁeld of about

10 T, roughly 30 times larger than that of pure

niobium.

A short coherence length, achieved either in a clean

high-T

superconductor or in a dirty low-T

super-

conductor, is therefore the key to practical applica-

tions of superconductivity in strong applied ﬁelds.

However, in general, x remains much larger than in-

teratomic distances. The high-T

oxides are an ex-

ception to that rule: along the CuO planes, x is about

2 nm in YBa

; in the perpendicular direction it

is only 0.3 nm. Practical superconductors such as

NbTi and Nb

Sn have coherence lengths of the order

of 5 nm.

2. Thermal Fluctu ations of the Superconducti ng

Order Para meter

2.1 Quasi-mean-ﬁeld Regime

Thermodynamic ﬂuctuations of the superconducting

order parameter can be estimated from the GL free-

energy functional. The lowest free energy is achieved

for D ¼ D

; D

¼ðA=BÞ

1=2

below T

and D

¼ 0

above T

. Any other conﬁguration D ¼ D

þ dDðrÞ

can be decomposed into its Fourier components, each

of them being characterized by an amplitude and a

wavelength. The Fourier components with the largest

amplitude have a wavelength of order x. This is be-

cause, for larger wavelengths, the energy cost is high

because of the large volume involved and, for shorter

wavelengths, the gradient term in Eqn. (3) is prohib-

itive. Hence, the volume of a typical ﬂuctuation is x

Its amplitude can be estimated by taking its energy to

be of the order of k

T. Developing F

to second order

in dD, one obtains below T

2A/jdDj

T ð8Þ

Apart from numerical coefﬁcients, the relative am-

plitude of the ﬂuctuations is given by:

/jdDj

DFx

ð9Þ

where DF is the condensation energy per unit volume

at equilibrium, DF ¼ A

=2B ¼ m

=2, where H

the thermodynamic critical ﬁeld.

The relative amplitude of the ﬂuctuations is thus

essentially given by the ratio of the thermal energy

T to the value of the condensation energy per co-

herence volume. Eqn. (9) can be immediately gener-

alized to the case of a general dimensionality d by

replacing x

by x

. The relative amplitude of the ﬂuc-

tuations diverges as

/jdDj

pð1  tÞ

ð4dÞ=2

ð10Þ

where do3 is achieved when one or more of the di-

mensions of the sample is smaller than the coherence

length xðtÞ.

In the preceding analysis the ﬂuctuations have been

treated as independent of each other (quasi-mean-

ﬁeld approximation). Equations (9) and (10) are thus

only valid in the limit where the relative amplitude of

the ﬂuctuations is much smaller than unity. This ap-

proximation is in general excellent for conventional

low-T

superconductors (LTS), but not for the high-

oxides (HTS).

The effects of ﬂuctuations are very small in bulk

LTS (e.g., their effect on the heat capacity is negli-

gible; this is not the case for the high-T

oxides). They

are larger for low dimensionalities (thin ﬁlms, wires,

small grains) because the relative amplitude of the

ﬂuctuations then diverges with a stronger exponent

(see Eqn. (10)). The most easily detected ﬂuctuation

effect is an excess conductivity Ds above T

. In two

dimensions within the quasi-mean-ﬁeld approxima-

tion (Azlamazoff and Larkin 1968)

Ds ¼

16_

ðt  1Þ

1

ð11Þ

where the conductivity is expressed in ð O =squareÞ

1

In a d-dimensional sample, the conductivity diverges

as ðt  1Þ

ð4dÞ=2

, that is, with the same exponent as

that of the relative amplitude of the ﬂuctuations given

by Eqn. (10). Additional terms to the excess conduc-

tivity considered by Maki and Thompson lead to a

stronger divergence.

Coherence Length, Proximity Effect and Fluctuations

The effect of ﬂuctuations on the conductivity is

most easily detected when the background normal

state conductivity is small, that is, in dirty ﬁlms

(Glover 1971) or wires. Another case where the excess

conductivity is easily observed is that of granular su-

perconductors (Deutscher and Dodds 1977).

On the contrary, the effect of ﬂuctuations on the

diamagnetism (which diverges with the same expo-

nent as that of the conductivity) is most easily

observed in clean samples, with a large coherence

length. Fluctuation diamagnetism has been studied in

low-T

intercalated compounds.

2.2 Critical Fluctuation Region

The mean-ﬁeld approximation is valid as long as the

relative amplitude of the ﬂuctuations as given by Eqn.

(10) is much smaller than unity. However, since this

amplitude diverges at T

, there exists a temperature

range near T

where this approximation fails. This

range is called the critical region (for a recent review

see Ginzburg 1988). It is customary to measure it on

the reduced temperature scale e ¼ðT  T

Þ= T

.Ife

is the reduced temperature at which the relative am-

plitude of the ﬂuctuations is of order unity, it follows

from Eqn. (9) that

¼ a

DFð0Þx

ð12Þ

where a is a numerical factor (actually important,

a ¼ 2

11

2

) and DF ð0Þ is the condensation energy at

T ¼ 0. According to the BCS theory:

DFð0Þ¼

Nð0ÞDð0Þ

ð13Þ

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

weak coupling limit, D(0) is proportional to T

and

since as seen before x

1

, it follows that e

. e

is extremely small for LTS, typically e

o10

12

.This

fully justiﬁes the use of the GL mean-ﬁeld approxi-

mation. However, for the HTS e

is much larger

ðe

C10

3

Þ and the critical region can be studied ex-

perimentally (Salamon et al. 1990).

As seen from Eqn. (12), the width of the critical

region is determined by the value of the condensation

energy in a coherence volume at T ¼0, compared

with k

. However, U

¼ DFx

is also the typical

energy scale for core pinning of vortices. Through the

coherence length, there exists therefore an intimate

relationship between the critical temperature, the

width of the critical region and the ability of the su-

perconductor to pin vortices. Notice that as T

goes

up, the upper limit for U

(clean limit) goes down

ðU

1

Þ. In LTS, e

is small and vortices are easily

pinned even close to T

but, in HTS, e

becomes

significant and vortex pinning cannot be achieved

when T is of the order of T

(Deutscher 1990). Full

use of their extremely high upper critical ﬁeld (several

hundred tesla) can only be made at temperatures sig-

nificantly lower than T

. Furthermore, the quasi-

mean-ﬁeld approximation appears to be inadequate

for the description of the vortex state near H

(T)in

the bismuth- and thallium-based oxide superconduc-

tors (Kes et al. 1992).

3. Proximity Effect

When a normal metal (N) is in good electrical contact

with a superconductor (S), Cooper pairs can leak

from S to N. If the electron–electron interaction in N

is zero ðV

¼ 0Þ, and in the absence of magnetic im-

purities and applied magnetic ﬁelds, Cooper pairs in

N are only destroyed by the thermal energy k

Their decay length is then given by (Deutscher and de

Gennes 1969)

¼ K

1

2pk

ð14Þ

where v

is the Fermi velocity in N. When pair

breaking is also induced by magnetic impurities and

applied magnetic ﬁelds, a multipair breaking model

in which the various depairing factors simply add can

be used. The value of K is then expressed in terms of a

depairing variable a (see Yang and Finnemore 1984).

The probability amplitude for ﬁnding a Cooper

pair in N at a distance |x| from the SN interface is

given by

FðxÞ¼jðxÞexpðKjxjÞ ð15Þ

where j(x) is a slowly varying function of x. The

probability amplitude F is deﬁned as the ratio of the

pair potential D to the BCS interaction parameter V.

If V

¼ 0; D

¼ 0, but F is ﬁnite (see later for the

boundary conditions for F at the SN interface).

Eqn. (14) is valid when the mean free path in N, l

is large compared to the decay length as given by

Eqn. (14) (clean limit). In the dirty limit

2

2pk



ð16Þ

where D

is the normal state coefﬁcient of diffusion

in N, D

¼v

/3.

If N is a superconductor at some lower tempera-

ture T

oT; K

1

diverges as T-T

. Properties of

the SN contact near T

can be treated with the GL

formalism. Very close to T

, K

1

pðT  T

1=2

symmetrical with the temperature dependence of the

coherence length in the superconducting state below

.AtT ¼T

, the exponential decay law (Eqn.

(15)) must be replaced by a power law dependence

F(x) ¼F(0) [x

/(x þx

)] (Deutscher and de Gennes

1969). At temperatures slightly above T

, this power

Coherence Length, Proximity Effect and Fluctuations

law persists near the interface. It crosses over to the

exponential decay far from the interface where the

pair amplitude is sufﬁciently small so that the non-

linear term in the GL functional becomes negligible.

At low temperatures, K

1

is a macroscopic length

scale. For instance, in copper at 1 K, K

1

is of the

order of 1 mm. This allows the easy observation of the

proximity effect in superconductivity. This is in con-

trast to other ordering phenomena, such as magnet-

ism, where the basic length scale is much shorter.

In the absence of any significant boundary resist-

ance (which may arise either from imperfections of

the boundary, such as a thin insulating layer, or from

a mismatch of the normal state properties, that is,

different Fermi velocities and electron effective mass-

es), the boundary conditions at the SN interface are

(de Gennes 1964)

F=Nð0Þ continuous ð17aÞ

DðdF=dxÞ continuous ð17bÞ

The pair amplitude is depressed on the S side over

a length x(T) and has a tail on the N side over a

length K

1

(T) (see Fig. 1). For this reason K

1

alternatively expressed as the normal state coherence

length.

The consequences of the proximity effect are

(Deutscher and de Gennes 1969) (a) in zero ﬁeld, a

depressed critical temperature of SN bilayers, an ex-

citation spectrum showing states within the gap if

measured from the S side and gapless superconduc-

tivity if measured from the N side; (b) in strong

applied ﬁelds, a reduction of the upper critical ﬁeld of

thin ﬁlms and suppression of the surface sheath

above H

in bulk superconductors; and (c) in weak

applied ﬁelds, existence of a Meissner effect on the N

side of the contact.

A simpliﬁed discussion of the proximity effect on

the transition temperature and on the upper critical

ﬁeld can be made in terms of a length b deﬁned by



ð18Þ

where the value of F and its derivative are taken at

the interface on the S side. According to Eqn. (18), b

is equal to the distance from the interface to the in-

tercept of the linear extrapolation of F(x) (into N)

with the x axis, which is why it is called the extra-

polation length (Fig. 1).

In the dirty limit, and in the absence of any

boundary resistance, b is given by:

b ¼ðr

ÞK

1

cothðKd

Þð19Þ

where r

and r

are the normal state resistivities and

and d

are the thicknesses of N and S, respectively.

The critical temperature T

cNS

of a bilayer is given by

cNS

¼ T



ðT ¼ 0Þ

ðd

þ bÞ

ð20Þ

This expression is only valid in the GL approxima-

tion, that is, when T

cNS

is not too different from T

In the same approximation, the parallel upper crit-

ical ﬁeld of a bilayer is given by (Orsay Group 1966)



ð0Þ

ð1  t

cNS

Þð21Þ

where b is a function of (b/d

One of the most fundamental manifestations of the

proximity effect is the existence of a Meissner effect

on the N side of SN contacts. This induced Meissner

effect was revealed by the study of the susceptibility

of SN contacts in weak ﬁelds, which showed that a

ﬁeld applied parallel to the interface is screened out at

a distance r

scr

from the interface (Deutscher et al.

1969). This induced diamagnetism is easily identiﬁed

because it is destroyed in ﬁelds that are much smaller

than the characteristic ﬁelds of the S side. The screen-

ing length r

scr

was found to be typically of the order

of 100 nm in the conventional liquid (helium) range.

If N is itself a superconductor at a temperature T

scr

diverges when T

is approached. This effect is

felt well above T

, r

scr

¼ 1 mm in zinc (T

¼0.92 K)

at 1.6 K. If N is a ‘‘truly’’ normal metal, r

scr

diverges

as T-0. In the limit of very small applied ﬁelds and

large N thickness ðd

scr

¼ K

1

fln½K

1

=lð0Þ  0:116gð22Þ

Figure 1

Behavior of the pair amplitude F near an SN interface.

For simplicity, it is assumed that the normal state

density of states is the same in N and in S (otherwise, F

is discontinuous at the interface) and that the boundary

resistance is negligible (which also leads to a

discontinuity of F). A case is illustrated where the

coefﬁcient of diffusion in S is shorter than in N. In this

case, the extrapolation length b is shorter than

1

( ¼x

)

Coherence Length, Proximity Effect and Fluctuations

where l(0) is the value of the local penetration depth

at the SN interface. As soon as the temperature is

sufﬁciently low that ½K

1

=lð0Þ> 1, r

scr

is typically of

the order of K

1

At low temperatures, as r

scr

becomes larger, it is

also more sensitive to the applied ﬁeld:

scr

ðHÞ¼K

1

lnð2H

=HÞð23Þ

where H

is a ﬁeld characteristic of the N side,

¼½F

K=2plð0Þ. Notice that H

-0asT-0.

These relations apply for a semi-inﬁnite N layer.

The main effect of a ﬁnite N thickness is that for

1

and ½K

1

=lð0Þ> 1, a ﬁrst-order transition

is observed at a breakdown ﬁeld H

at which the pair

amplitude in N collapses and r

scr

varies essentially

from d

to zero (Orsay Group 1967). A good ap-

proximation for H

¼ 3:8H

expðKd

Þð24Þ

There are at least two important applications

where the proximity effect is important. One is in

the fabrication of practical Josephson devices. For

LTS electronics, niobium, being a refractory metal, is

a preferred choice of material for stable, cyclable de-

vices. Unfortunately, it does not have a good insu-

lating oxide. The solution is then to use a niobium

ﬁlm with a thin aluminum coating and to oxidize the

surface of the aluminum. The operation of the device

is based on the proximity-induced superconductivity

in the aluminum coating (see SQUIDs: Biomedical

Applictions; SQUIDs: The Instrument; Josephson

Junctions: Low-T

). Another example is in the design

of ultrathin ﬁlamentary wires, composed of very ﬁne

Nb–Ti ﬁlaments in a copper matrix, for ac applica-

tions. In these wires, the distance between the ﬁla-

ments is so small that the copper matrix is effectively

superconducting; to minimize ac losses the proximity

coupling has to be reduced either by separating the

ﬁlaments or by suppressing the proximity coupling by

alloying the copper with magnetic impurities such as

manganese (see Superconducting Wires and Cables:

Materials and Processing).

Special mention should be made of proximity ef-

fects where the S side is a high-T

oxide. In this case

is very small, typically of the order of 1 nm. This

makes the fabrication of high-T

devices quite difﬁ-

cult, since electron tunnelling occurs into a surface

region of thickness x

, whose quality is hard to con-

trol. One solution consists of using a high-T

–low-T

(or normal metal) contact, taking care, however, to

use a high-resistivity normal side. This is essential

because of the high normal state resistivity of the

oxides: as can be seen from Eqn. (19), the extrapo-

lation length b is very small when r

, and if b is

very small the pair amplitude will be destroyed right

at the interface (Fig. 1).

Proximity-coupled weak link structures and tunnel

junctions have been made using thin proximity-cou-

pled layers of silver or gold deposited onto YBa

and other high-T

oxides. Various device

heterostructures have been investigated including

SNS, SNS

and SNIS

junctions, where S

is usually

lead or niobium (Akoh et al. 1990, Gijs et al. 1990).

Properties of these junctions have been calculated

within a BCS framework by Kupriyanov and Likha-

rev (1991). Recently, proximity effects have been

measured using a semiconducting layer of perovskite-

type oxide (e.g., La

1.5

7y

). It appears that

the proximity decay length K

1

can be as large as

65 nm in this system (Tarutani et al. 1991).

The small leakage of Cooper pairs into an insulator

is sufﬁcient to weaken the pair amplitude at the in-

terface of an HTS with an insulator (Deutscher and

Muller 1987). The extrapolation length b at an SI

interface is of the order of x

=a where a is an inter-

atomic distance: since in the oxides x

is only a few

interatomic distances, b is not much larger than x

and as a rule the pair amplitude will be reduced at all

surfaces and interfaces. This explains small critical

currents and granular behavior in polycrystalline ox-

ides; a disordered region at grain boundaries is suf-

ﬁcient to depress the pair amplitude at the surfaces of

the adjacent grains. This depression results in an

anomalous temperature variation of the critical cur-

rent of grain boundary junctions, I

pðT  T

in-

stead of the linear dependence of usual Josephson

junctions (Gross et al. 1990). It also explains why I

scales with the normal state resistance R

of the

junction as R

2

instead of the usual dependence R

1

(Deutscher and Chaudhari 1991). Both effects con-

tribute towards reducing the critical current density

in randomly oriented polycrystalline samples.

See also: Electrodynamics of Superconductors: Flux

Properties; Electrodynamics of Superconductors:

Weakly Coupled

Bibliography

Akoh H, Camerlingo C, Takada S 1990 Anisotropic Josephson

junctions of YBaCuO/Au/Nb ﬁlm sandwiches. Appl. Phys.

Lett. 56 (15), 1487–9

Azlamazoff L G, Larkin A L 1968 Effect of ﬂuctuations on the

properties of a superconductor above the critical tempera-

ture. Sov. Phys. Solid State 10, 875–80

de Gennes P G 1964 Boundary effects in superconductors. Rev.

Mod. Phys. 36, 225–37

Deutscher G 1967 Contribution a l’e

tude expe

rimentale de la

supraconductivite

de surface. J. Phys. Chem. Solids 28,741–82

Deutscher G 1990 The potential for large critical currents in the

high T

oxides. Transport properties of superconductors.

Prog. High Temp. Supercond. 25, 2–10

Deutscher G, Chaudhari P 1991 Scaling behavior of the critical

current of grain boundary junctions. Phys. Rev. B 44, 4664–5

Deutscher G, de Gennes P G 1969 Proximity effects. In: Parks

R D (ed.) 1969 Superconductivity. Dekker, New York, pp.

1005–34

Coherence Length, Proximity Effect and Fluctuations

Deutscher G, Dodds S 1977 Critical ﬁeld anisotropy and ﬂuc-

tuation superconductivity in granular aluminium ﬁlms. Phys.

Rev. B 16, 3936

Deutscher G, Hurault J P, van Dalen P A 1969 Electrodynamic

properties of superconducting contacts. J. Phys. Chem. Solids

30, 509–20

Deutscher G, Muller K A 1987 Origin of extrinsic critical cur-

rents and glass behavior in the high-T

oxides. Phys. Rev.

Lett. 59, 1745–7

Gijs M A M, Sholten D, van Rooy Th, Gerrits A M 1990

YBa

7x

/Ag–Al/Al

/Pb tunnel junctions based on the

superconducting proximity effect. Appl. Phys. Lett. 67, 2600–2

Ginzburg V L 1988 Macroscopic theory for superconductors

with small coherence length. Phys. C 153, 1617–21

Glover R E III 1971 Superconducting ﬂuctuation effects above

the transition temperature. In: Chilton F (ed.) 1971 Super-

conductivity. North-Holland, Amsterdam, pp. 3–31

Gross R, Chaudhari P, Dimos D, Gupta A, Koren G 1990

Thermally activated phase slippage in high-T

grain bound-

ary Josephson junctions. Phys. Rev. Lett. 64 (2), 228–31

Kes P H, van der Beek C J, Maley M P, McHenry M E, Huse D

A, Menken M J V, Menovsky A A 1992 Field suppression of

the phase transition in Bi

CaCu

. Phys. Rev. Lett.

(in press)

Kupriyanov M Yu, Likharev K K 1991 Towards the quanti-

tative theory of the high-T

Josephson junctions. IEEE

Trans. Magn. 27, 2460–3

Orsay Group 1966 Strong ﬁeld effects at the surface of a su-

perconductor. In: Brewer D (ed.) 1966 Quantum Fluids.

North-Holland, Amsterdam, pp. 26–67

Orsay Group 1967 Field dependence of the superconducting

properties induced by the proximity effect: breakdown ﬁelds.

Phys. Kondens. Mater. 6, 307–24

Saint James D, de Gennes P G 1963 Onset of superconductivity

in decreasing ﬁelds. Phys. Lett. 7, 306–9

Salamon M B, Inderhees S E, Rice J P, Ginsberg D M 1990

Heat capacity of untwinned YBa

in magnetic ﬁelds:

dimensional cross-over near T

. Phys. A 168, 283–90

Serin B 1969 Type II superconductors: experiments. In: Parks R

D(ed.)1969Superconductivity. Dekker, New York, pp. 925–76

Tarutani Y, Fukazawa T, Kabasawa U, Tsukamoto A, Hira-

tani M, Takagi K 1991 Superconducting characteristics of

a planar-type HoBaCuO–LaBaCuO–HoBaCuO junction.

Appl. Phys. Lett. 58, 2707–9

Yang H C, Finnemore D K 1984 Pair-breaking mechanisms in

superconductor–normal metal–superconductor junctions.

Phys. Rev. B 30 (3), 1260–5

G. Deutscher

Tel Aviv University, Tel Aviv, Israel

Colossal Magnetoresistance Effects: The

Case of Charge-ordered Pr

0.5

MnO

Manganite Thin Films

Recently, perovskite type manganites such as

1x

MnO

(RE ¼rare-earth and A ¼alkaline-

earth ion) have received renewed interest due to the

fact that they display colossal magnetoresistance

(CMR), a huge decrease in resistance when applying

a magnetic ﬁeld (Chahara et al. 1993, Von Helmolt

et al. 1993, McCormack et al. 1994, Prellier et al.

2001). Since most of the technological applications

require thin ﬁlms, it is essential to understand the

effects of the substrate-induced strains on the prop-

erties of these manganites. These materials are

very sensitive to the strains and even very small

perturbations may result in observable effects on the

properties. These effects have indeed been studied

experimentally on various compounds (Prellier et al.

1999, Biswas et al. 2000). Moreover, among all the

properties of these manganite materials, the phenom-

enon of charge ordering (CO) is probably one of the

most remarkable effects which occurs for certain val-

ues of x and for a particular average A-site cation

radius. It corresponds to an ordering of the charges in

two different Mn sublattices. In fact, the metallic

state becomes unstable, below a certain temperature

) and the material goes to an insulating state.

decreases with increasing ﬁeld and the insulating

CO state can be totally suppressed by the application

of an external magnetic ﬁeld (Rao et al. 2000). As an

example, a 25 T magnetic ﬁeld is required to melt the

CO state in bulk Pr

0.5

MnO

(Tokunaga et al.

1998) leading to huge CMR effect.

The substrate-induced strain effects and the thick-

ness of the ﬁlms inﬂuence the structural and physical

properties of manganite thin ﬁlms. As an example,

0.5

MnO

(PCMO), thin ﬁlms grown on La-

AlO

(LAO) and SrTiO

(STO) using the pulsed laser

deposition (PLD) technique are considered here. The

focus is on the changes in the lattice parameters,

the orientation and the transport measurements of

the ﬁlm. On the basis of these results, a temperature–

ﬁeld phase diagram for CO thin ﬁlms is determined

and compared with the bulk material.

Thin ﬁlms of PCMO were grown in situ using the

PLD technique on [100]-SrTiO

(cubic with

a ¼3.905 A

) and [100]-LaAlO

(pseudocubic with

a ¼3.789 A

) substrates. Detailed optimization of the

growth procedure was completed and described pre-

viously (Prellier et al. 2000, Haghiri-Gosnet et al.

2000). The structural study was carried out by X-ray

diffraction (XRD) using a Seifert XRD 3000P for

the Y–2Y scans and a Philips MRD X’pert for the

in-plane measurements (Cu Ka, l ¼1.5406 A

). The

in-plane parameters were obtained from the (1 0 3)

reﬂection (where C refers to the ideal cubic perovskite

cell). An electron microscope JEOL 2010 was used

for the electron diffraction (ED) study. The resistivity

(r) was measured by a four-probe method with a

Quantum Design PPMS and magnetization (M) was

recorded using a Quantum Design MPMS SQUID

magnetometer as a function of the temperature (T)

and the magnetic ﬁeld (H). The composition of the

ﬁlms was checked by energy-dispersive scattering

analyses. It is homogenous and corresponds exactly

Colossal Magnetoresistance Effects: The Case of Charge-ordered Pr

0.5

MnO

Manganite Thin Films

to the composition of the target (i.e., Pr

0.5

MnO)

in the limit of the accuracy.

The structure of bulk PCMO is orthorhombic

(Pnma) with a ¼5.395 A

, b ¼7.612 A

and c ¼5.403 A

(Jirak et al. 1985). Figure 1 shows a typical Y–2Y

scan recorded for a 750 A

thick ﬁlm deposited on

STO (a) and LAO (b). Note that the same type of

pattern is obtained for all of the ﬁlms. This shows

that the ﬁlms are single phases and highly crystal-

lized. From the XRD pattern, the out-of-plane lattice

parameter is, respectively, calculated to be 3.78 A

LAO and 3.85 A

on STO indicating that the ﬁlm is

under compression on LAO and under tension on

STO as already reported.

Figure 2 shows the ED pattern recorded along a

direction perpendicular to the substrate plane. It is

clear that the orientations of the ﬁlms with respect to

the substrate are different. In fact, the ﬁlm is [010]-

oriented, i.e., with the [010] axis perpendicular to the

substrate plane when grown on STO and [101]-ori-

ented on LAO. This surprising orientation results

from the lattice mismatch. Indeed, the mismatch (s)

between the ﬁlm and the substrate can be evaluated

using the formula s ¼100(a

–a

)/a

(where a

and

, respectively, refer to the lattice parameter of the

substrate and the ﬁlm). The smaller mismatch on

LAO is obtained for a [010]-axis in the plane

LAO

¼0.4%), i.e., the [101]-axis is perpendicular

to the substrate plane (s

LAO

¼0.7% for a [101] axis

in the plane). In contrast, the smaller mismatch on

STO (s

STO

¼2.2%) is found for a [101]-axis in the

plane (s

STO

¼2.5% for a [010]-axis in the plane).

Thus, the [010]-axis is normal to the surface of the

substrate as found experimentally (Prellier et al. 2000,

Haghiri-Gosnet et al. 2000).

The samples were cooled down to 92 K. The dif-

fraction patterns recorded at low temperature in the

strained ﬁlm systematically exhibit a system of extra

reﬂections characteristic of an incommensurate CO

parallel to the a

direction on both substrates (not

shown). The q values of the modulation vector are

close to 0.45 and STO and 0.38 on LAO and do not

vary with the temperature below T

, contrary to

what is observed in the bulk samples. This modula-

tion disappears at T

. Small ﬁlm fragments sepa-

rated from the substrate were also observed. In this

case, a modulation vector close to 0.5 was observed.

This value is similar to that of the bulk samples

(Barnabe

et al. 1998). Since the composition is the

same, this reinforces the assumption that the devia-

tion from the ideal q value originates from the strain

induced by the substrate. This demonstrates that the

[100]

STO

[010]

STO

[010]

LAO

[100]

LAO

020

-101

020

101

(a)

(b)

Figure 2

Room temperature electron diffraction of PCMO on

STO (a) and LAO (b); reproduced from W. Prellier et al.

(2000) Spectacular decrease of the melting magnetic ﬁeld

in the charge-ordered state of Pr

0.5

MnO

ﬁlms

under tensile strain. Appl. Phys. Lett. 77, 1023–5.

20 30 40 50 60

(b)

(a)

STO

020

202

101

040

Intensity (a.u.)

2Θ (deg)

LAO

Figure 1

Room temperature Y–2Y XRD pattern of a 75 nm thick

ﬁlm on (a) STO and (b) LAO. Note the high intensity

and the sharpness of the peaks.

Colossal Magnetoresistance Effects: The Case of Charge-ordered Pr

0.5

MnO

Manganite Thin Films

orthorhombic distortion induced by the CO ordering

is strongly reduced by the presence of the substrate,

preventing the locking of the q value of the CO on

the 1/2 value contrary to the bulk sample of the

same composition, leading to q ¼0.45 for the ﬁlm

grown on STO and q ¼0.38 for the one deposited

on LAO.

The r(T) curves are presented in Fig. 3. Different

magnetic ﬁelds (0 T, 5 T, 6 T, and 7 T) were applied

perpendicular to the substrate. The behavior of both

ﬁlms is very different. Clearly, the ﬁlm grown on STO

(Fig. 3(b)) shows an insulating behavior in the whole

temperature range when the magnetic ﬁeld is below

4 T. A small anomaly in the resistivity (around 240 K)

is associated with the CO. But the most interesting

feature is observed when applying a magnetic ﬁeld of

7 T. The ﬁlm then becomes metallic at low temper-

atures and shows an insulator-to-metal transition

) at 120 K. When a magnetic ﬁeld is applied, the

upon heating occurs at a higher temperature

( ¼140 K) indicating a clear hysteresis. This melting

ﬁeld of 7 T is much lower than the 20 T one which is

necessary to induce the metallic phase in the bulk

sample. On the contrary, this effect is not observed

for [101]-PCMO ﬁlms grown on LAO (Fig. 3(a)) with

ﬁelds up to 7 T.

Figure 4 shows the temperature dependence with

different applied magnetic ﬁelds (0 T, 5 T, and 9 T)

for three ﬁlm thicknesses deposited on STO. In the

absence of an applied ﬁeld, one always observes a

semiconducting behavior with an anomaly around

225 K corresponding to the T

(this value is close to

the one found in the bulk material). On the contrary,

the results are drastically different under application

of external magnetic ﬁelds. A ﬁeld up to 9 T has al-

most no effect on the thinner ﬁlm (15 nm): the ﬁlm

is still semiconducting and the magnetoresistance is

very small (Fig. 4(a)). For the intermediate thickness

(75 nm), while a 5 T magnetic ﬁeld keeps the ﬁlm

semiconducting, a ﬁeld of 9 T renders it metallic with

a T

close to 125 K (see Fig. 4(b)). This feature is a

typical characteristic of the melting of the CO state.

On the thicker ﬁlm (110 nm), the melting magnetic

ﬁeld is reduced to 5 T (Fig. 4(c)). The thickness de-

pendence of the properties suggests that the strains

play an important role in determining the melting

magnetic ﬁeld. When the thickness of the ﬁlm is in-

creasing, there is a reduction of the strain on the ﬁlm

at room temperature. However, the properties of the

ﬁlm do not approach those of the bulk as the subst-

rate-induced strain is reduced. In PCMO single crys-

tal, with a 6 T magnetic ﬁeld, the material remains

insulating (Tomioka et al. 1996), whereas the 110 nm

ﬁlm becomes metallic.

The magnetic ﬁeld dependence of the resistivity

shows a strong decrease on a logarithmic scale at a

critical ﬁeld (H

) indicating the ﬁeld-induced melting

of the CO state (see insets of Fig. 4). This ﬁeld-in-

duced insulator-to-metal transition, which accompa-

nies the collapsing of the CO state takes place below

. The critical ﬁelds in the ﬁeld-increasing scan and

in the ﬁeld-decreasing scan will be represented by H

and H



, respectively. This decrease of the resistivity

versus magnetic ﬁeld can be viewed as CMR of about

ﬁve orders of magnitude at 75 K. A clear hysteresis is

seen at these temperatures as previously reported on

several CO compounds (Kuwahara et al. 1995, Tom-

ioka et al. 1996). This hysteretic region is more pro-

nounced when the temperature is decreasing. The

temperature dependence of the large hysteresis region

is a feature of a ﬁrst-order transition and has been

extensively studied for the composition Nd

0.5

MnO

(Kuwahara et al. 1995). From the resistivity

measurements, a phase diagram for the 110 nm thick

ﬁlm (Fig. 5) and for the 75 nm ﬁlm (inset of Fig. 5)

can be derived. Two remarks should be made: First

as already mentioned, the transition to the metastable

state (i.e., the ﬁeld-induced metallic state) is easier or

requires a lower ﬁeld in case of a thin ﬁlm than in the

bulk (Tomioka et al. 1996) (5 T for a 110 nm thick

ﬁlm and 9 T for a 75 nm thick ﬁlm); second, the shape

of the H–T phase diagram is totally different as

50 100 150 200 250 300

-3

-2

-1

LaAlO

7 T

0 T

T (K)

ρ (Ω cm)

0 50 100 150 200 250 300

-3

-2

-1

SrTiO

5 T

6 T

7 T

0 T

T (K)

ρ (Ω cm)

(a)

(b)

Figure 3

r(T) in different magnetic ﬁelds (0 T, 5 T, 7 T, and 9 T)

applied in the plane of the substrate for different ﬁlms

on (a) LAO and (b) STO.

Colossal Magnetoresistance Effects: The Case of Charge-ordered Pr

0.5

MnO

Manganite Thin Films