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

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nonresonant nuclear spin, electron spin, local charge,

etc.) caused by the atomic movements.

For probe nuclei with spin I ¼1/2 (such as

Si, and

P) o

is due to mutual magnetic dipolar

interactions, while for nuclei with I41/2 (alkali

atoms like

Li,

Na,

85,87

Rb,

B, and

Al) the cou-

pling is dominated by the interaction between the

nuclear quadrupole moment of the probe and electric

ﬁeld gradients at the probe site created by neighbor-

ing charges, e.g., ionic point charges. In either case

is related to the distance r

of the particles by

p1=r

For a thermally activated diffusion of dilute par-

ticles in an ordered lattice, G(t) can be expressed to a

very good approximation by GðtÞD/o

Sexpðt=tÞ

which corresponds to a Debye-like behavior of the

diffusion-induced NSR rate 1/T

Diff

in accordance

with JðoÞ¼/o

St=ð1 þ o

Þ. Here, 1=t ¼ 1=t

expðE=kTÞ is the jump rate of the moving particles

with E being the corresponding activation energy and

S ¼



ioj

denotes the mean-square coupling frequency.

In practice, /o

1=2

is roughly equal to the width

of the respective NMR line at sufﬁciently low tem-

peratures (‘‘rigid line width’’). Moreover, one should

note that the diffusive motion is observable not only

by the NSR rate of the mobile particles but also by

measuring the NSR rate of immobile probe nuclei.

This is caused by the pair character of the coupling

frequency o

which becomes time dependent for both

mobile and immobile probes whenever a hopping

event occurs. Finally, as mentioned in the introduc-

tion, the measured NSR rate 1/T

can be induced by

different relaxation processes. If the processes are in-

dependent of each other, the related rates add up to

the observed rate:

1=T

¼ 1=T

Diff

other

1=T

2. Ionic Diffusion in Glasses by Nuclear Spin

Relaxation

Ionic diffusion in glasses gives rise to a NSR rate that

does not exhibit the classical Debye-like features de-

scribed above. As shown as an example in Fig. 1

(diffusion-induced

F NSR in the glassy ﬂuorine ion

conductor 27ZrF

27HfF

20BaF

3LaF

3AlF



20NaF (ZBLAN20) (Kanert et al. 1996) the Ar-

rhenius representation of 1/T

Diff

exhibits an asym-

metric peak with remarkably different slopes on

either side of the maximum. Further, the frequency

dependence of 1/T

Diff

on the low-temperature side

does not obey the o

2

law proposed by the Debye

relation.

Two theoretical approaches exist to interpret this

unusual behavior. On the basis of the random struc-

ture of the glassy network, the correlation time t (and

correspondingly the activation energy E) is assumed

to be distributed according to an appropriate distri-

bution function g(t) (Svare et al. 1993). Then, G(t)is

formed by a superposition of exponential decays ac-

cording to

GðtÞ¼

gðtÞe

t=t

dlnt ð2Þ

This approach represents the motion of a single

particle in a random landscape. As discussed in detail

by Beckmann (1988), a large variety of functions g(t)

can be used to ﬁt properly the experimental data to

the model.

There is strong evidence, however, that G(t) decays

inherently nonexponentially due both to the Cou-

lomb interaction between the mobile ions and to their

interactions with the glass skeleton. There exist dif-

ferent theoretical approaches dealing with the com-

plex coupled motion of charged particles in a random

matrix—e.g., Monte Carlo simulations (Maass et al.

1995), an analytical jump relaxation model (Funke

1993), a diffusion-controlled relaxation model (Elliott

and Owens 1991), and a phenomenological coupling

model (Ngai 1993)—which all result in a nonexpo-

nential time evolution of G(t). So far, the models do

not take into account the real structure of the glasses,

i.e., a quantitative agreement cannot be expected be-

tween the experimental data and such calculations. In

Figure 1

Arrhenius plot of the diffusion-induced

F NSR rates in

the glassy ﬂuorine ion conductor ZBLAN20 observed in

the rotating frame at three different frequencies (after

Kanert et al. 1996).

Amorphous Materials: Nuclear Spin Relaxation

future, molecular dynamics simulations will offer

more realistic results of ionic diffusion-induced NSR

in vitreous solids.

However, from the different models the following

general features can be extracted: above a lower

cross-over time t

(B10

11

s) the complexity of the

ionic motion enhances the initial ionic jump time t ¼

0,N

exp(V/kT) to the apparent jump time

tðtÞ¼f ðt

; nÞt

1=ð1nÞ

¼ t

exp

ð1  nÞkT



ð3Þ

where V is the microscopic barrier height for the

hopping ion and n denotes the so-called coupling

coefﬁcient where 0pnp1. Instead of n, often the

coefﬁcient b ¼1 n is used in the literature. In con-

sequence, the mean-square displacement /r

(t)S of a

moving ion becomes sublinear in time between t

and

t(T) ð/r

ðtÞSpt

1n

Þ which corresponds to a fre-

quency dispersion of the apparent diffusion coefﬁ-

cient (Funke 1993) and gives rise to the non-Debye

behavior of the NSR rate. For t4t(T) the ion enters

the Gaussian region where /r

ðtÞSpt: Here one

observes the d.c. activation energy E which is related

to the barrier height VbyE¼V/(1n).

The corresponding correlation function G(t) can be

expressed to a good approximation by a stretched

exponential function in the relevant time regime

3

pt/tp10

GðtÞ¼/o

Sexp½ðt=tÞ

1n

ð4Þ

The diffusion-induced NSR rate is given by Eqns.

(1), (1a), and (4), where for arbitrary exponents 1n

the Fourier transformation has to be performed nu-

merically. The following analytical relation, however,

represents a good approximation to the numerical

results (Maass et al. 1995):

1=T

Diff

¼ /o

1 þðo

tÞ

2n

1 þð2o

tÞ

2n

ð5Þ

Applying this relation to Fig. 1 one obtains from

the slopes E ¼1.15 eV and V ¼0.34 eV, i.e., n ¼0.70.

The data conﬁrm also the o

n2

-dependence of 1/T

proposed by Eqn. (5) for o

tb1. Moreover, the tem-

perature dependence of the NSR rate maxima, which

is given by the condition otD0.6, leads to a temper-

ature-dependent correlation time t ¼ 10

16

s. exp

(E/kT) with E ¼1.15 eV equal to V/(1n) in accord

with Eqn. (3).

Approaching the glass-transition temperature T

the shear viscosity Z and, therefore, the related cor-

relation time t

of the glass both decrease rapidly.

The question whether the diffusion-induced NSR rate

is affected by the process in the vicinity of T

is an-

swered by the actual magnitude of the decoupling

coefﬁcient R ¼t

/t. For Rb1 the glass structure is

rigid on the time scale of the ionic hops, i.e., the

observed NSR rate is dominated by the diffusion.

By use of site-selective spin excitation, NSR is able

to detect the ionic diffusion separately in different

environments. Kim et al. (1997) observed the motion

of lithium ions in Li

S B

glasses near 3(BS

)- and

4(BS



)-coordinated boron atoms by measuring the

temperature dependence of the

B NSR rate of dif-

ferent parts of the

B NMR spectrum belonging to

and BS



units, respectively. They obtained two

well-separated NSR rate peaks. From the position of

the maxima they drew the conclusion that the lithium

ions move much faster close to the 3-coordinated

boron than to the 4-coordinated boron. Such site-

selective NSR experiments should also be applied to

highly-resolved magic angle spinning (MAS)-NMR

spectra (Eckert 1992) of glasses, opening new insights

into the ionic motion at different sites of the glassy

network.

With increasing temperature, the correlation time t

reaches the magnitude of the inverse NMR line

width, and the width starts to decrease. Such motio-

nal line-width narrowing offers a simple possibility to

distinguish diffusing from immobile probe nuclei:

while for mobile probes the line width approaches

zero for sufﬁciently high temperatures, the corre-

sponding width of the immobile probes remains ﬁnite

(Kanert et al. 1996).

3. Comparison Between Nuclear Spin Relaxation

and Ionic Conductivity

It is very instructive to compare the results of diffu-

sion-induced NSR with those obtained from complex

ionic conductivity (

s) measurements. Though both

1/T

(o,T)and

s(o,T) measurements are probing the

same physical process, in practice the results can differ

due to the different structure of the underlying corre-

lation functions G(t)andG

(t). While G(t) is a pair

correlation function of the coupling frequencies o

, G

is a particle autocorrelation function of the ionic ve-

locities v

(Ngai 1993). In practice, however, the results

of NSR and ionic conductivity agree fairly well, i.e.,

both the correlation functions can be approximated by

a stretched exponential (Eqn. (4)) with the same cou-

pling coefﬁcient n but different time constants t and t

This is due to the fact that t is the mean waiting

time (correlation time) of an ion between consecutive

hops, while t

denotes the relaxation time of the de-

caying electric ﬁeld under constant dielectric dis-

placement. According to Funke (1993), t and t

are

related by

t ¼

¼ C  t

ð6Þ

where n is the density and q the charge of the ions,

and x

denotes the mean jump distance of a hopping

Amorphous Materials: Nuclear Spin Relaxation

ion. Using appropriate values for the different quan-

tities (see below) one ﬁnds CD5C100, i.e., tbt

qualitative accord with experimental ﬁndings (Kanert

et al. 1996).

For a uniform correlation function G (t/t) for both

NSR and conductivity differing only with regard to

the time constant t (i.e., t

s,N

), the NSR rate 1/

can be linked to the imaginary part of the electric

modulus, M

, by the following general expression

(Funke 1993)

1=T

ðo

tðTÞÞp½M

ðo

Þþ2M

ð2o

Þ=

ð7Þ

This relation turns out to be very useful for compar-

ative NSR and conductivity experiments on ionic

transport in glasses.

Unfortunately, until now only a limited number of

investigations exists on ionic diffusion in inorganic

glasses comparing in detail NSR and conductivity

data. Blache et al. (1998) performed an extended

study on a series of lithium silicophosphate glasses

measuring the temperature dependence of Li

diffu-

sion by NMR and electrical conductivity for different

glass compositions. Part of the results are shown in

Fig. 2 exhibiting the dependence of the activation

energy E and barrier height V on the concentration

c(Li) of the lithium ions for a ﬁxed ratio of the net-

work formers SiO

and P

. The data were obtained

from electrical conductivity,

Li and

P NSR, and

Li tracer diffusion D

(T) using the NMR ﬁeld-

gradient technique (Geil 1998). As expected for the

diffusion of a single species of charged carriers, D

(T)

was found to be in accord with the diffusion constant

(T) calculated from the dc conductivity s

¼ k=ðq

nÞs

 T, i.e., the Haven ratio D

about 1.

The ﬁgure demonstrates that the corresponding

quantities V and E obtained from the different ex-

periments agree well, resulting in a uniform coupling

coefﬁcient n ¼0.67 for cX2 at.%. For cp2 at.% the

data indicate a slight decrease of n as proposed by the

coupling model (Ngai 1993). The observed mono-

tonic decrease of the activation energy E with in-

creasing amount of lithium ions could be interpreted

fairly well by the simpliﬁed Anderson–Stuart relation

(Patel and Martin 1992) E ¼E

þE

c(Li)

1/3

using

¼1.5 eV and E

¼1.52 eV (at.%)

1/3

as best-ﬁt

parameters.

Moreover, the authors measured the dependence of

t and t

on temperature and on the concentration of

lithium ions. Evaluation of the data revealed that the

following relation is approximately valid:

tD10t

¼ 5  10

14

s exp

EðcðLiÞÞ



for c(Li)X5 at.%. This result seems to conﬁrm Eqn.

(6): using the suitable values /nSD0.8 10

3

(810at.% Li), /TS ¼500K, e

¼3, x

¼2.4A

, one

obtains CD10 in accord with the experimental ﬁnd-

ings. It has to be noted, however, that in some glassy

ionic conductors the activation energy E observed by

NSR was found to be somewhat larger than the cor-

responding energy E

measured by dc conductivity,

i.e., n4n

using the relation E ¼V/(1 – n) (Kanert

et al. 1996). Such deviations were proposed also by

Monte Carlo simulations (Maass et al. 1995). At

present, however, a ﬁnal decision whether Eqn. (6) is

correct and E equals E

cannot be made because of a

lack of sufﬁcient experimental data.

See also: Amorphous Materials: Nuclear Magnetic

Resonance, New Techniques; Nuclear Magnetic Res-

onance Spectrometry

Bibliography

Beckmann P A 1988 Spectral densities and nuclear spin relax-

ation in solids. Phys. Rep. 171, 85–128

Blache V, Fo

rster J, Jain H, Kanert O, Ku

chler R, Ngai K L

1998 Ionic motion in lithium silicophosphate glasses by nu-

clear spin relaxation and electrical conductivity. Solid State

Ionics 113–5, 723–31

Dieckho

fer J, Kanert O, Ku

chler R, Volmari A, Jain H 1997

Composition dependence of low-frequency excitations in

lithium silicophosphate glasses by NMR and electrical con-

ductivity. Phys. Rev. B 55, 14836–46

Eckert H 1992 Structural characterization of nanocrystalline

solids and glasses using solid state NMR. Progr. NMR Spec-

trosc. 24, 159–293

Elliott S R, Owens A P 1991 Nuclear spin relaxation in ionically

conducting glasses: application of the diffusion-controlled

relaxation model. Phys. Rev. B 44, 47–59

Figure 2

Dependence of the activation energy E and barrier

height V of lithium diffusion on the amount of lithium in

the glassy lithium ion conductor Li

O SiO

P

observed by different methods (see text). Solid curve is a

ﬁt by the Anderson–Stuart model (after Blache et al.

1998).

Amorphous Materials: Nuclear Spin Relaxation

Funke K 1993 Jump relaxation in solid electrolytes. Prog. Solid

State Chem. 22, 111–95

Geil B 1998 Measurement of translational molecular diffusion

using ultrahigh magnetic ﬁeld gradient NMR. Concepts

Magn. Res. 10, 299–321

Kanert O, Dieckho

fer J, Ku

chler R 1996 Recent progress in the

area of NMR characterization of ionic transport and relax-

ation in glasses. J. Non-Cryst. Solids 203, 252–61

Kim K H, Torgeson D R, Borsa F, Cho J P, Martin S W, Svare

I, Majer G 1997 Evidence of complex ionic motion in

xLi

S (1x)B

glassy fast ionic conductors from

and

B NMR and ionic conductivity measurements. J.

Non-Cryst. Solids. 211, 112–25

Maass P, Meyer M, Bunde A 1995 Non-standard relaxation

behavior in ionically conducting materials. Phys. Rev. B 51,

8164–77

Ngai K L 1993 Difference between nuclear spin relaxation and

ionic conductivity relaxation in superionic glasses. J. Chem.

Phys. 98, 6424–30

Patel H K, Martin S W 1992 Fast ionic conduction in

S B

glasses: compositional contribution to non-

exponentiality in conductivity relaxation in the extreme

low-alkali-metal limit. Phys. Rev. B 45, 10292–300

Schmidt-Rohr K, Spiess H W 1994 Multidimensional Solid

State NMR and Polymers. Academic Press, London

Svare I, Borsa F, Torgeson D R, Martin S W 1993 Correlation

functions for ionic motion from NMR relaxation and elec-

trical conductivity in the glassy fast-ion conductor

(Li

0.56

(SiS

)

0.44

. Phys. Rev. B 48, 9336–44

Tang X P, Busch R, Johnson W L, Wu Y 1998 Slow atomic

motion in Zr–Ti–Cu–Ni–Be metallic glass studied by NMR.

Phys. Rev. Lett. 81, 5358–61

O. Kanert

Universita

t Dortmund, Germany

Amorphous Superconductors

Amorphous superconductors are a class of supercon-

ducting materials that lack long-range, translational

ordering in their atomic arrangement. Since the dis-

covery of superconductivity in amorphous bismuth

and gallium (a-Bi, a-Ga) by Bu

ckel and Hilsch in the

1950s, noncrystalline superconductors have been the

subject of numerous experimental and theoretical

studies. Recently, interest in this ﬁeld has been fo-

cused on amorphous transition-metal (TM) super-

conductors as opposed to the amorphous non-

transition-metal (non-TM) superconductors investi-

gated earlier.

The techniques used to synthesize amorphous su-

perconducting materials include quenching from the

vapor (obtained by evaporation or sputtering), par-

ticle irradiation, ion mixing or ion implantation, and

liquid quenching. The technique of quenching from

the melt is capable of producing large quantities of

metallic glasses in the form of ribbons, ﬂakes or

powder suitable for large-scale engineering applica-

tions. Vapor deposition and particle bombardment

methods are used to prepare thin ﬁlms amenable to

small-scale electronic device applications.

In general, the Cooper pairs in superconductors are

formed from time-reversed electron states, no matter

how complicated these states are in real space as a

result of scattering from various disorders. Hence, the

basic mechanism of superconductivity is not expected

to be different in disordered or crystalline materials.

There are, however, several interesting differences

between amorphous superconducting materials and

their crystalline counterparts; these differences will be

discussed.

1. Systematics of the Transition Temperature

1.1 Nontransition Metals

Amorphous non-TM superconductors usually show

an enhancement in transition temperature T

over

that of corresponding crystalline phases. For exam-

ple, a-Bi thin ﬁlms become superconducting at about

6 K, whereas the crystalline phase, a semimetal,

remains normal down to 0.05 K. The value of T

for a-Ga is found to be 8.4 K, significantly higher

than the T

of the crystalline phases (T

¼1.08 K for

a-Ga and 6 K for b-Ga). A more complete listing of

the T

values of amorphous superconductors and

their crystalline counterparts is given in Table 1. It is

Table 1

The T

of some amorphous (or highly disordered)

superconductors and their crystalline counterparts.

Amorphous or

highly

disordered Crystalline

Metals (K) (K)

Nontransition

metals

Pb 7.16 7.2

6.99 8.1

Ga 8.4 6(b-Ga)

1.08 (a-Ga)

Bi 6.1 o0.05

10 0.026

6 1.18

Transition

metals

Nb 5.7 9.2

Mo 8–9 0.9

Ge 3.6 23

Si 3.9 18

Sn 3 18

Si 1.5 17

Si 7.5 1.4

a Extrapolated values.

Amorphous Superconductors

noted that T

for amorphous non-TM superconduc-

tors tends to increase with valence and with atomic

number for a given valence (Johnson 1981; Bergmann

1976).

1.2 Transition Metals

A systematic study by Collver and Hammond (1973)

on amorphous 4d and 5d TM elements and alloys

thereof indicates that the well-known Matthias rule

for crystalline superconductors (see Superconducting

Materials, Types of ) is significantly altered for the

amorphous state. The variation of T

as a function of

e/a (the average valence electrons–atom ratio) shows

a sharp peak at e/a ¼6.4, corresponding to a half-

ﬁlled d shell (see Fig. 1). More recent work has re-

vealed that T

as a function of e/a is more symmetric

than the original Collver–Hammond curve and sim-

ilar systematics are also observed in some liquid-

quenched TM-metalloid metallic glasses. The T

amorphous TM alloys made from elements widely

separated in the periodic table is always significantly

lower than that of the original Collver–Hammond

alloys composed of nearest 4d or 5d elements. The T

of alloys mixing the 4d and 5d elements shows an

even more intriguing behavior. For example, whereas

the values of T

for a-MoRe alloys are slightly higher

than those for the a-MoRu alloys, lower T

values

have been found for all the a-MoRh alloys in the

same range of e/a. The T

values for a-NbIr, a-NbRh,

a-TaRh and a-TaIr alloys are considerably lower

than those for corresponding amorphous near-neigh-

bor TM alloys and show essentially no variation of

e/a. These data along with others are plotted in

Fig. 1. (Recent T

data on amorphous alloy systems

containing 3d elements and noble metals, for example

a-ZrCu, a-ZrFe and a-ZrNi, are not included in this

plot and should be treated separately.) It is clear from

Fig. 1 that there is no universal curve of T

as a

function of e/a for all amorphous TM alloys. It has

been suggested that the effects of spin ﬂuctuations

and charge transfer may be responsible for this.

At present, there is no satisfactory interpretation

for all the systematics of T

previously described. The

data shown in Fig. 1 do suggest, however, that there

is an upper limit of T

for a given e/a in amorphous

4d and 5d TM alloys as roughly outlined by the

modiﬁed Collver–Hammond curve. The fact that the

peak of this upper-bound curve coincides with half-

ﬁlled d shells suggests that there is an atomic param-

eter characterizing the e/a dependence of T

. This

point can be qualitatively discussed with the aid of

the well-known McMillan formula:

ðy

=1:45Þexp

1:04ð1 þ lÞ

l  m



ð1 þ 0:62lÞ



ð1Þ

where l is the dimensionless electron–phonon cou-

pling parameter, m



is the effective electron–electron

Coulomb pseudopotential and y

is the Debye tem-

perature. The parameter l can be expressed as

l ¼ NðE

Þ/I

S=M/o

S ð2Þ

where N(E

) is the bare electronic density of states

at the Fermi level, /o

S is an average square phonon

frequency and /I

S is the Fermi-surface average of

the electron–phonon interaction strength.

In terms of tight-binding models, it has been sug-

gested that the Collver–Hammond curve essentially

reﬂects the e/a dependence of N(E

) and that the

ratio /I

S//o

S is a constant throughout the 4d

series. A systematic study of the low-temperature

specific heat of some of the more stable Collver–

Hammond alloy ﬁlms is crucial in determining N(E

)

quantitatively as a function of e/a. It should be

pointed out that the empirical and calculated values

of /I

S for the 4d transition metals (Butler 1977) are

insensitive to structural change and their e/a varia-

tion closely mimics the T

upper limit shown in Fig. 1.

2. Strength of the Electron–Phonon Interacti on

The coupling between electrons and phonons is fun-

damentally important to the occurrence of super-

conductivity and is of particular interest in the case

of amorphous superconductors in view of possible

Figure 1

The T

of amorphous TM elements and alloys as a

function of e/a. The original Collver–Hammond curves

for amorphous TM superconductors are shown for ()

4d and (m)5d metals; also shown are (J) the T

values

of various amorphous TM alloys made from elements

widely separated in the periodic table and from mixing

the 4d and 5d metals. For a given e/a, there is an upper

limit of T

as outlined by the triangular peak (– – )

Amorphous Superconductors

enhancement of the coupling strength through struc-

tural disorder as suggested by several theories. The

strength of the electron–phonon interaction can be

estimated from (a) the ratio of Bardeen–Cooper–

Schrieffer (BCS) energy gap to transition tempe-

rature, 2 Dð0Þ=k

, and (b) the parameter l (see

Superconducting Materials: BCS and Phenomeno-

logical Theories). The BCS gap can be measured

directly by superconductive tunnelling experiments or

estimated indirectly from calorimetric measurements.

The parameter l can be determined using either the

McMillan formula for T

or the technique of super-

conductive tunnelling spectroscopy through

l ¼ 2

doo

1

ðoÞFðoÞð3Þ

where a

ðoÞFðoÞ is the Eliashberg electron–phonon

spectral function.

The results accumulated so far clearly lead to three

conclusions.

First, amorphous non-TM superconductors are all

characterized by a very strong electron–phonon cou-

pling (i.e., 2 D(0)/k

B4–5 and l42), as are their

crystalline counterparts.

Second, amorphous TM or TM-based supercon-

ductors are found to be in the BCS weak-coupling

limit. In particular, the electron–phonon coupling

strength of several strong-coupled A15 superconduc-

tors is found to be reduced to the BCS limit in the

amorphous state. These experimental ﬁndings suggest

that structural disorder alone does not necessarily

give rise to a significantly enhanced electron–phonon

interaction.

Third, a detailed analysis of the tunnelling data for

various amorphous non-TM superconductors has re-

vealed that the Eliashberg function is enhanced and

varies linearly with o in the low-frequency region as

predicted by various theories. There is no significant

low-frequency enhancement found in a

F(o) for

amorphous nitrogen-stabilized niobium and molyb-

denum ﬁlms. The function a

F(o), in fact, follows the

phonon density of states F(o) based on a structural

model of dense random packing of hard spheres.

From this ﬁnding, it is concluded that the observed

low-frequency enhancement in a

F(o) for amorphous

non-TM superconductors is essentially a manifesta-

tion of an increase in the electron–phonon interaction

caused by structural disorder. The lack of such

enhancement in amorphous TM superconductors is

attributed to the ‘‘nearly localized’’ nature of d elec-

trons, which are relatively insensitive to structural

disorder.

In a more general perspective, this result can also

be understood by noting the difference between the

TM and non-TM amorphous superconductors in the

effectiveness of the phonon modes in contributing to

the electron–phonon coupling. Unlike amorphous

TM superconductors, the phonon density of states in

non-TM superconductors (crystalline and amor-

phous) is distributed over a range from 0 to approx-

imately 10 meV and is most efﬁcient in achieving a

strong electron–phonon interaction (Tsuei 1981).

3. Critical Fields

Amorphous superconductors are all characterized by

relatively large upper critical ﬁelds H

in view of

their generally low T

values. This is mainly a man-

ifestation of a short electron mean free path (lt1 nm)

leading to a short zero-temperature coherence length

x(0) (see Coherence Length, Proximity Effect and

Fluctuations). The upper critical ﬁeld at 0 K is related

to x(0) by

ð0Þ¼F

=2px

ð0Þð4Þ

where F

¼ h=2e is the ﬂux quantum. Typical values

of m

ð0Þ are about 2–5 T (20–50 kG) for amor-

phous non-TM superconductors and are in the range

10–40 T (100–400 kG) for the noncrystalline TM or

TM-based superconductors. The lower critical ﬁeld

ðTÞ for several metallic glasses has been meas-

ured; m

ð0Þ values are typically of the order of

1–10 mT (10–100 G). Both H

ðTÞ and H

ðTÞ are

found to vary linearly with temperature over a wide

range of temperatures below T

(B0.5T

to T

). Ex-

amples of dH

=dT near T

are about 2TK

1

3TK

1

for amorphous TM superconductors and

about 0.5 T K

1

for amorphous simple metals. The

values of dH

=dT are about 1 mT K

1

From critical-ﬁeld measurements, values for x(0)

and the Ginzburg–Landau parameter k can be esti-

mated; the values are of the order of 5 nm and 60,

respectively. Such large values of k represent a true

indication of the extreme type II, or ‘‘dirty limit,’’

behavior expected for amorphous superconductors.

Detailed studies of the temperature dependence of

ðTÞ for several amorphous TM alloys indicate

that H

is enhanced at low temperatures by the effect

of weak localization as predicated by the FEM theory

(Fukuyama et al. 1984). The amount of the H

en-

hancement depends on the alloy system and can be as

large as 20% (Poon 1985a,b). It should be mentioned

that the quantum correction effects of weak localiza-

tion and interaction also manifest themselves in sup-

pressing T

(Fukuyama et al. 1984) and reducing the

temperature coefﬁcient of the normal-state resistivity.

4. Flux Pinning Phenomena

The pinning of the ﬂux line lattice (FLL) in super-

conductors can be studied by measuring the critical

current density J

(T, H) as a function of temperature

and applied ﬁeld. Results of critical-current measure-

ments on amorphous superconductors show that J

at H ¼0 is in general relatively low (B10

–10

2

)

Amorphous Superconductors

and that it decreases rapidly with increasing ﬁeld.

Such a weak FLL pinning can probably be attributed

to the homogeneity and isotropy, on a scale of x(0),

intrinsic in all amorphous materials. It is exactly this

characteristic property which makes direct use of

amorphous superconductors in high-ﬁeld applica-

tions less attractive. A possible solution to this prob-

lem is to incorporate, by appropriate heat treatments,

some second-phase precipitates in an amorphous

superconducting matrix (Johnson 1981). Homogene-

ous amorphous superconductors, however, are ideal

for studying ﬂux pinning phenomena in terms of

existing theory (see Electrodynamics of Superconduc-

tors: Flux Properties). The fact that amorphous TM

or TM-based superconductors are weak coupled

and in the extreme type II limit simpliﬁes the anal-

ysis of the experimental data considerably, because

theoretical expressions in the dirty limit are highly

developed. Furthermore, pinning behavior in these

materials (in the form of carefully prepared thin

ﬁlms) is expected to be insensitive to complications

arising from pinning by surface irregularities, edge

dislocations and buckling of the ﬂux lines.

A recent fundamental study of ﬂux pinning in

amorphous ﬁlms of Nb

Ge and Nb

Si (Kes and Tsuei

1981) led to the following results.

(a) Owing to the large k and low J

, the FLL in

amorphous thin-ﬁlm superconductors is essentially

two dimensional.

(b) The pinning force density F

( ¼J

B, where B is

ﬂux density) as a function of perpendicular ﬁeld and

temperature is found to be in good agreement with

the theory of collective pinning.

is observed near H

A pinning-mediated renormalization of the shear

modulus of the FLL is probably responsible for this

peak effect.

(d) The F

for samples with relatively large J

shows a domelike behavior characteristic of strong

pinning. By annealing, J

decreases by up to two or-

ders of magnitude and both two-dimensional collec-

tive pinning and the peak effect are recovered,

supporting the view that weak pinning is an essen-

tial requirement to observe the collective behavior. It

has also been demonstrated that the ﬂux pinning can

serve as a sensitive probe to study structural relax-

ation in amorphous metals.

The concept of collective ﬂux pinning along with

the consideration of the thermal ﬂuctuation effects

has been used to understand the ﬂux-creep-related

phenomena in high-T

cuprate superconductors

(Feigel’man and Vinokur 1990). It is interesting to

note that both amorphous superconductors and high-

copper oxides are characterized by short coherence

length and relatively high normal-state resistivity

except that there is a large difference in T

and tem-

perature scale.

5. Potential Applications

Most amorphous non-TM superconductors are struc-

turally stable only at very low temperatures. For ex-

ample, a-Bi ﬁlms are found to crystallize at 20 K and

the amorphous phase of gallium ﬁlms transforms into

a crystalline state (b-Ga, T

¼6.4 K) at about 15 K.

Further annealing at 60 K results in the room-tem-

perature stable phase a-Ga with a T

of 1.08 K. In

sharp contrast, many TM or TM-based supercon-

ducting alloys can resist crystallization at temperatures

up to more than 1000 K. In addition to this phase

stability, these materials are ductile, ﬂexible and pos-

sess extremely high mechanical strength, comparable

with that of single crystals. Potential areas for using

amorphous superconductive materials include high-

ﬁeld magnets, especially for use in an environment of

high-level radiation, and possibly some specialized

vortex memory devices for computer applications.

See also : Thin Films, Multilayers and Devices,

Superconducting

Bibliography

Bergmann G 1976 Amorphous metals and their superconduc-

tivity. Phys. Rep. 27, 159–85

Butler W H 1977 Electron–phonon coupling in the transition

metals. Phys. Rev. B 15, 5267–82

Collver M M, Hammond R H 1973 Superconductivity in

‘‘amorphous’’ transition-metal alloy ﬁlms. Phys. Rev. Lett.

30, 92–5

Feigel’man M V, Vinokur V M 1990 Thermal ﬂuctuations of

vortex lines, pinning, and creep in high-T

superconductors.

Phys. Rev. B 41, 8986–90

Fukuyama H, Ebisawa H, Maekawa S 1984 Bulk supercon-

ductivity in weakly localized regime. J. Phys. Soc. Jpn. 53,

3560–7

Johnson W L 1981 Superconductivity in metallic glasses. In:

Guntherodt H, Beck H (eds.) 1981 Glassy Metals 1. Springer,

Berlin, Chap. 9, pp. 191–223.

Kes P H, Tsuei C C 1981 Collective ﬂux pinning phenomena in

amorphous superconductors. Phys. Rev. Lett. 47, 1930–4

Poon S J 1985a Amorphous and highly disordered metallic su-

perconductors. Physica B 135, 259–66

Poon S J 1985b Localization effects on superconductivity in

homogeneous metallic glasses. Phys. Rev. B 31, 7442–5

Tsuei C C 1981 Amorphous superconductors. In: Foner S,

Schwartz B B (eds.) 1981 Superconductor Materials Science.

Plenum, New York, Chap. 12, pp. 735–56.

C. C. Tsuei

IBM, Yorktown Heights, New York, USA

Amorphous Superconductors

Boltzmann Equation and Scattering

Mechanisms

When an external force is applied to the conduction

electron system of a solid, a drift velocity results.

Random scattering effects tend to restore a dynam-

ical balance between the acceleration caused by the

external forces and restoring forces due to scattering

processes.

In general we expect currents J

to develop as a

response of the system to generalized external forces

. If these responses can be considered to be linear

the relation between the forces X

and the currents J

can be written as

k¼1

ð1Þ

The coefﬁcients L

are called the linear transport

coefﬁcients. Their physical nature depends on which

current is connected to which force. If the forces and

the currents are properly selected so that the internal

entropy production per unit time is

i¼1

ð2Þ

then they are canonically conjugated in the sense of

irreversible thermodynamics. In this case Onsager’s

theorem provides the symmetry relation

ðB

Þ¼L

ð B

Þð3Þ

for these transport coefﬁcients, where B

is an external

magnetic ﬁeld.

In principle two different kinds of force can induce

a current J

. The ﬁrst and the more familiar type is an

electric ﬁeld acting on the electron system. The sec-

ond type of force has its origin in statistics; it is not a

‘‘force’’ in the usual sense but arises from concentra-

tion and temperature gradients. It should be noted

that an external magnetic ﬁeld is inﬂuencing the

transport phenomena but does not contribute to a

current in the sense discussed above.

In order to give a theoretical description of the

transport phenomena (electrical and thermal resistiv-

ity, thermopower, Peltier effect, etc.) the Boltzmann

formalism may be used as one among a number of

different other theoretical concepts.

1. Distribution Function

The Boltzmann formalism is based on the assumption

that a distribution function in phase space ðr

;

kÞ can

be deﬁned. Since we are dealing with a strongly in-

teracting quantum mechanical system it is by no

means clear that such a function can be deﬁned be-

cause of the Heisenberg uncertainity principle. In

many cases, however, one considers the distribution

function to be a function of k

only; the Heisenberg

uncertainity principle does not preclude the use of

this distribution. When an r

-dependence is consider-

ed at all, the position in real space usually occurs

through the dependence of the distribution function

on external position dependent parameters, such as

the r

-dependence of the temperature in the case

of an applied temperature gradient. This expression

makes sense if the electrons can be localized over

macroscopic distances. This can be done by using

wave-packets which contain a small number of wave-

numbers. In this limited sense one can use a distri-

bution function which is a function of both

k and r

(Butcher 1973).

1.1 Bloch Electrons

For an electron interacting with a periodic potential

Vð r

þR

Þ¼Vðr

Þ; R

lattice vector ð4Þ

the stationary eigenstates are given by the solutions

of the Schro

dinger equation



D þ Vð r



ns k

ðr

Þ¼E

ðk

Þc

ns k

ðr

Þð5Þ

where n, s, and k

denote the band index, the spin,

and the wave-vector of the conduction electrons,

respectively.

The eigenstates are usually written as Bloch func-

tions

ns k

ðr

Þ¼

ﬃﬃﬃﬃ

 r

ns k

ðr

Þð5aÞ

with the periodic amplitude

ns k

ðr

þR

Þ¼u

ns k

ðr

Þð5bÞ

where N is the number of the unit cells of the crystal.

The eigenvalues E

(

k ) of Eqn. (5) are functions of

the wave-vector

k . From Kramer’s degeneracy

n;s

ð k

Þ¼E

n;s

ðk

Þð5cÞ

it follows that the electrical (and the thermal) currents

for a thermodynamical equilibrium state are zero.

1.2 Semiclassical Motion of Wave-packets

In order to describe the movement of an electron

through the periodic array of the ions in the lattice,

the electrons are represented by wave-packets. Con-

structing a wave-packet from Bloch states in the

vicinity of a wave vector

k in the energy band ns,we

can consider the time dependence of the shape of this

wave-packet in real space as well as in reciprocal

space corresponding to a classical particle with

position r

and momentum p

The corresponding semiclassical equations describ-

ing the movement of a wave-packet are:

d r

ðk

Þð6aÞ

d k

ð6bÞ

is an external force acting on the electron (for an

electromagnetic ﬁeld this force is the Lorentz force).

The mechanical momentum p

is connected in the

usual way with the wave vector:

p ¼ hk.

1.3 Electrical and Heat Current

Using Eqns. (6a) and (6b) each electron in an energy

band ns gives a trajectory in the 6-dimensional phase

space (

k , r

). We therefore deﬁne (in analogy to clas-

sical statistical mechanics) a distribution function

f(n, s,

k , r

, t) with the following properties

ð2pÞ

 f ðn; s ; k

; r

; tÞ¼

number of electrons

in the energy band ns

within the phase space

volume D

ð7Þ

The factor 1/(2p)

ensures, that the distribution

function f (n, s,

r, t) for a local thermodynamical

equilibrium state (characterized by the local temper-

ature T(

r ) and the local chemical potential mð

r Þ)is

identical with the local Fermi distribution function

f ðn; s; k

; Tð r

Þ; mðr

ÞÞ ¼ f

ns k

 mðr

Tð r

ð8Þ

whereby

ðxÞ¼

þ 1

Using the distribution function we are able to calcu-

late the mean values of all single electron properties.

In the following the abbreviation g is used for all

three quantum numbers n, s, and

k (gn, s,

k )in

order to characterize the Bloch states. Furthermore

we use the notation

? ¼

n;s

Vol

ð2pÞ

1:BZ

ky ð9Þ

where the integral is over the 1st Brillouin zone. The

local electron density can now be written as

nðr

; tÞ¼

Vol

f ðg; r

; tÞð10Þ

with the total number of electrons N

rnðr

; tÞð11Þ

It then follows that the electrical current density, I

may be expressed as

ðr

; tÞ¼

Vol

f ðg; r

; tÞðev

Þð12Þ

whereby e ¼7q

7 is the absolute value of the electric

charge of an electron.

A similar expression may be derived for the ther-

mal heat current density

ðr

; tÞ¼

Vol

f ðg; r

; tÞðE

 mÞ v

ð13Þ

The energy is measured relative to the chemical po-

tential m; this is the correct heat-current for a statis-

tical force caused by a temperature gradient. Then we

obtain the usual relation between the thermopower S

and the Peltier coefﬁcient P: P ¼TS.

2. Boltzmann Equation

For the calculation of the current densities I

and J

the distribution function f is needed (see Eqns. (12)

and (13)). This distribution function is a solution of

the Boltzmann equation in presence of external forces

F. The Boltzmann equation describes the time devel-

opment of the distribution function and can be writ-

ten in the form

@f ðg; r

; tÞ

r

f ðg; r

; tÞ

þ v

r

f ðg; r

; tÞ¼



ð14Þ

The left hand side describes the time development of

the distribution function caused by the semiclassical

equations of motion (see Eqns. (6a) and (6b)). These

terms are called drift terms. The right hand side is the

Boltzmann Equation and Scattering Mechanisms

collision term. It takes into account the changes of the

distribution function due to scattering processes.

2.1 Collision Term

The change of f (g, r

, t) caused by scattering proc-

esses may be subdivided as follows:

Scattering processes from electron state g

to elec-

tron state g result in an increase of f (g, r

, t):

f ðg

Þ½1  f ðgÞP

-g

ð15aÞ

Scattering processes from electron state g to electron

state g

result in a decrease of f(g, r

, t):

f ðgÞ½1  f ðg

ÞP

g-g

ð15bÞ

The factor f gives the probability that the initial state

is occupied whereas the factor (1f ) ensures that the

ﬁnal state is empty. P

g-g

is the quantum mechanical

probability per unit time for the transition from the

state g to the state g

. It is expressed using ﬁrst order

perturbation theory by Fermi’s golden rule

g-g

7/gc7H

int

dðE

 E

Þð16Þ

In this equation H

int

is the interaction Hamiltonian

between the Bloch electrons and the system (rest of

the solid) characterized by quantum states c and c

is the probability that one ﬁnds the system in the

state c. If a thermodynamical description of the sys-

tem is possible, this factor corresponds to the Boltz-

mann factor

bE

ð17Þ

whereby b is proportional to the inverse temperature:

b ¼1/(k

T )(k

is the Boltzmann constant, T is the

temperature). The collision term is now the sum over

all possible scattering processes which contribute to a

change of the distribution function at g:

@f ðgÞ



f f ðg

Þ½1  f ðgÞP

-g

 f ðgÞ½1  f ðg

ÞP

g-g

gð18Þ

2.2 Thermal Equilibrium

The thermal equilibrium distribution function f

characterized by a vanishing collision term:



f f

ðg

Þ½1  f

ðgÞP

-g

 f

ðgÞ½1  f

ðg

ÞP

g-g

g¼0 ð19Þ

Because this equilibrium condition must be fulﬁlled

for any interaction, it follows that each term of this

sum is zero and hence:

ðg

Þ½1  f

ðgÞP

-g

¼ f

ðgÞ½1  f

ðg

ÞP

g-g

ð20Þ

This equation allows the definition of a symmetric

equilibrium transition rate

¼f

ðg

Þ½1  f

ðgÞP

-g

¼f

ðgÞ½1  f

ðg

ÞP

g-g

¼ W

ð21Þ

which will be used in the treatment of the linearized

Boltzmann equation.

The thermal equilibrium distribution function f

the Fermi–Dirac function

ðbðE

 mÞÞ ¼

bðE

mÞ

þ 1

ð22Þ

characterized by a temperature T and a chemical po-

tential m. With this function we obtain from Eqn. (20)

the relation

-g

¼ e

g-g

ð23Þ

which has to be fulﬁlled by the transition probabil-

ities P

g-g

for any scattering process.

3. Linearized Boltzmann Equation

Assuming that there is simultaneously a time-inde-

pendent electric and magnetic ﬁeld applied to the

system of conduction electrons we can make use of

the Lorentz force

¼e

 B

þE



ð24Þ

For this time-independent force we can look for a

time-independent distribution function f (g , r

) which

fulﬁls the Boltzmann equation (Eqn. (14))



ðv

 B

Þr

f ðg; r

Þ

r

f ðg; r

þ v

r

f ðg; r

Þ¼



ð25Þ

As it is shown in Eqn. (18) the collision term depends

also on the distribution function.

3.1 Linearization of the Boltzmann Equation

Due to the external forces the local thermodynamical

equilibrium distribution function f

(g, T( r

), m( r

)) is

disturbed. Assuming this distortion to be small we

can use

f ðg; r

Þ¼f

ðg; Tð r

Þ; mðr

ÞÞ þ f

ðg; r

Þð26Þ

Boltzmann Equation and Scattering Mechanisms