Martienssen W., Warlimont H. (Eds.). Handbook of Condensed Matter and Materials Data

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1042 Part 5 Special Structures

electron–phonon coupling is attributed not to phonon

conﬁnement but to a reduction of the screening of the

electron–ion interaction following electron conﬁnement.

5.3.2.5 Electron Transport Phenomena

The charge transport properties in the direction of free-

carrier motion in a restricted-dimensional system have

important consequences for the magnetotransport ef-

fects. This is treated in Sect. 5.3.4. For disordered

systems, when the Mott variable-range hopping mechan-

ism [3.58] dominates the conductivity, the temperature

scaling law depends on the dimensionality: the 3-D con-

ductivity σ

3-D

varies with temperature following the law

log σ

3-D

∝ T

−4

, whereas the 2-D conductivity varies as

log σ

2-D

∝ T

−3

. The transition from 3-D to 2-D behav-

ior has been demonstrated for ﬁlms of amorphous Ge at

a thickness d of 50 nm, as illustrated in Fig. 5.3-9 [3.54].

A vertical conductance, i. e. perpendicular to the

free-motion direction, is necessary in quantum wells

in many device applications. This requires a ﬁnite bar-

rier in order for one to have a continuum of unbound

states, in addition to one or a few bound states, as

illustrated in Fig. 5.3-10. Practical realizations of such

structures are formed from an n-type low-band-gap well,

usually GaAs, sandwiched between barrier layers of

an intrinsic semiconductor with larger band gap, such

as Al

1−x

Al. The conduction electron states form

a structure of the type shown in Fig. 5.3-10. The choice

of the well width andbarrier height (through the compos-

1000

100

10 100 1000

Slope

d (Å)

slope T

slope

–1/

Fig. 5.3-9 Plot of the experimental power-law dependence

of the conductivity on temperature for ﬁlms of amorphous

Ge of various thicknesses d. The transition from a 3-D

−1/4

) behavior (region marked as T

1/4

slope) to a 2-D

−1/3

) behavior (region marked as T

1/3

slope), as ex-

pected for the variable-range hopping mechanism, is clearly

observed d = 500 Å (vertical bar). (After [3.54])

Fig. 5.3-10 Schematic view of the electron states in aﬁnite-

barrier quantum well. States with energies below the barrier

are bound to the well, and states with energies above the

barrier are extended outside the well and form a continuum

of conduction states

ition x of the barrier) permits one to adjust the number

and energy of bound states, and the n-type doping of the

well controls their population.

An important example of a device based on such

a structure is the quantum-well infrared photodetector

(QWIP) [3.59]. The energy difference E

−E

between

the subbands matches the energy of the photons to be

detected. E

is close to the top of the barrier. Excita-

tion of the intersubband transition by absorption of an

IR photon results in the population of the second sub-

band. If an electric ﬁeld is applied, the excited electron

can tunnel through the small remaining barrier towards

the continuum and move away from the well under the

inﬂuence of the electric ﬁeld. This gives rise to a pho-

tocurrent, which can either constitute the signal of the

detector or feed a visible-light-emitting junction in or-

der to achieve IR-to-visible conversion. To increase the

collection efﬁciency in practical devices, several quan-

tum wells are associated in a periodic stack. For a large

enough bias, once an electron has reached the contin-

uum it is accelerated by the electric ﬁeld and has a high

probability of crossing subsequent wells without being

trapped in a bound state.

In applications such as IR detection, the conduc-

tivity is obtained by populating the continuum. In that

case, the interwell spacing is large enough to prevent di-

rect coupling between neighboring wells, and each well

can be considered as isolated. Such systems are usually

called multiple quantum wells.

A nonzero conductivity in the ground state can be

obtained by reducing the width of the barriers, as ori-

ginally proposed in [3.60]. Then, the subband states of

neighboring wells interact through the evanescent parts

Part 5 3.2

Mesoscopic and Nanostructured Materials 3.2 Electronic Structure and Spectroscopy 1043

of their wave functions in the barrier. The ensemble

of bound states consisting of the subband levels from

all of the coupled quantum wells forms a band in the

conﬁnement direction called a miniband. The periodic

set of coupled quantum wells is called a superlattice.

At very low applied voltages, a band-type conduction

governs the electron transport. However, above a thresh-

old voltage, the conduction mechanism becomes highly

complicated. This results in highly nonlinear behaviors,

with the appearance of negative differential resistance

and oscillations at low temperatures [3.61]. These ef-

fects are attributed to the localization of the electron

states when the voltage drop between neighboring wells

induces an energy difference larger than the coupling

energy between them. At even higher ﬁelds, the voltage

drop spontaneously localizes on one single well [3.61].

A particularly spectacular manifestation of this effect

has been observed in weakly coupled wells at low

temperature, as shown in Fig. 5.3-11 [3.62].

Electrical conduction is also very important for

many devices that exploit the huge area of surface or

interface perunit volume inzero-dimensional nanostruc-

tured materials such as nanoporous materials, granular

materials, nanocomposites, and nanoparticle assem-

blies. Examples of such devices are chemiresistor-type

sensors, solar cells, light-emitting diodes, and energy-

storage cells. From the point of view of electron

–2

–4

–6

–8

012345

Conductance (mS)

Bias voltage (V)

Fig. 5.3-11 Left: differential conductance as a function of the applied voltage for a 49-period superlattice at 20 K. Right:

schematic model to explain the 48 negative peaks in the differential conductance.

(a) Zero bias; (b) ground-state resonant-

tunneling conduction;

excited state takes place;

(d) expansion of the high-ﬁeld region by one additional quantum well. (After [3.62])

transport, these materials can often be modeled as

a dense assembly of metal or semiconductor quantum

dots dispersed in an insulating matrix. As in noncrys-

talline materials [3.63], the elementary mechanism of

conduction is phonon-assisted electron tunneling from

one dot to another, or hopping. Above a threshold tem-

perature, the limiting factor for the hopping probability

is the spatial proximity of two dots. Below this thresh-

old, the weak thermal energy available imposes a strong

selection of sites with a low energy difference, tunneling

to such sites at larger distances becomes dominant, and

the variable-range hopping regime is obtained [3.58].

The temperature dependence of the conductance σ

changes from −log(σ) ∝ T

−α

with 1/2 <α<1for

nearest-neighbor hopping to −log(σ) ∝ T

−1/4

for

variable-range hopping [3.64]. However, in contrast to

noncrystalline semiconductors, the sites in such granu-

lar materials are spatially expanded to the particle size,

and the tunnel gap between neighboring particles is usu-

ally much smaller than this size. As a consequence, the

threshold temperature is very low and a T

law with

α ≈ 1/2 is observed over a wide temperature range.

Another important factor comes into play in

nanoparticle-based conductors compared with 2-D well

systems. The addition of an electron to an initially elec-

trically neutral nanoparticle requires an energy e

/2C,

where C is the capacitance between the nanoparticle

Part 5 3.2

1044 Part 5 Special Structures

–0.1 0.0 0.1 0.2

0.6

0.4

0.2

0.0

–0.4 –0.2 0.0 0.2 0.4

–0.2

Current

Voltage (V)

dI/dV (nS)

Current

Voltage (V)

dI/dV (nS)

78.6 K

44.4 K

Ag–SiO

15.0 K

3.1 K

Au–SiO

10 pA 1 nA

29.3 K

14.3 K

3.9 K

2.9 K

Fig. 5.3-12a,b Temperature dependence of I/V characteristics of ﬁlms of (a) Ag nanoparticles dispersed in SiO

(sizes

3.8±1.0 nm, volume fraction 1.5%, ﬁlm thickness 12 nm) and

(b) Au nanoparticles dispersed in SiO

(sizes 2.9nm,ﬁlm

thickness 9 nm). In both cases the electrode area was 100 × 100 µm

. (After [3.68])

and its surroundings. Typical values of C can be in the

region of 10

−17

F or less, so that the charging energy

approaches or is larger than the thermal energy at room

temperature. This produces the Coulomb blockade ef-

fect: conduction through a single nanoparticle requires

a minimum voltage to overcome the cost in charging

energy. The conduction may even show a stepwise de-

pendence on the bias, called a Coulomb staircase, which

has been observed for a single nanoparticle at room

temperature [3.65]. Extension to three-terminal devices

has led to the realization of mesoscopic devices such

as single-electron transistors, where the charge state of

a conﬁned electron pool is controlled by a gate volt-

age [3.66]. The inﬂuence of the single-electron charging

energy on the conductivity of a mesoscopic material

made of an assembly of dots has not yet been com-

pletely elucidated. It is predicted to reinforce the T

law

with α ≈ 1/2 for the conductivity [3.67]. However,

the most prominent experimental effect is the appear-

ance of remnants of the single-particle blockade effect

when the conducting path involves only a few particles,

which occurs mainly for the perpendicular conductivity

in very thin ﬁlms with a low particle volume fraction

(see Fig. 5.3-12) [3.68, 69]. There are some indications

that a regular ordering also favors Coulomb blockade

effects [3.70].

5.3.3 Electromagnetic Conﬁnement

5.3.3.1 Nanoparticle-Doped Materials

The polarizability α(ω) of a spherical inclusion of vol-

ume V with a dielectric constant ε(ω) at an optical

frequency ω, in a matrix with a dielectric constant

(ω),isgivenby

α = 9ε

ε −ε

ε +2ε

, (3.5)

where the explicit frequency dependence of α, ε,andε

has been dropped in the notation. When the medium is

subjected to a propagatingoptical ﬁeld, the polarizability

of the particle will produce both absorption and elastic

scattering of light. The corresponding cross sections,

and σ

, respectively, are

18πV



(ε



+2ε

)

+ε



(3.6)

Part 5 3.3

Mesoscopic and Nanostructured Materials 3.3 Electromagnetic Conﬁnement 1045

and

6ε

(ε



−ε

)

+ε



(ε



+2ε

)

+ε



, (3.7)

where λ

and λ

are the wavelengths of the light in

vacuum and in the matrix, respectively, and ε = ε



+iε



with ε



and ε



real. The dependence of σ

and σ

the volume shows that for nanoparticles with diameters

much smaller than the optical wavelength, absorption

effects, when present, are preponderant over scattering.

Low-Concentration Systems

For semiconductor particles, the absorption spectrum is

changed only slightly compared with the bulk. The scat-

tering losses, present even in the nonabsorbing region,

have been studied in detail, in view of their waveguid-

ing applications and account has been taken of the size

dependence of the refractive index [3.71]. Some results

are reproduced in Fig. 5.3-13.

In contrast, metals often show negative values of ε



In that case both the absorption and the scattering show

resonances when ε



+2ε

=0, which corresponds to the

dipolar plasmon oscillation mode, and a peak appears

in the absorption spectrum; this is observed especially

clearly with alkali and noble metals (Fig. 5.3-14). For

most metals, the plasmon peak shifts towards longer

wavelengths and becomes sharper for matrices with

a larger refractive index. This is clearly observed for

the alkali metals in Fig. 5.3-14; for a computation of

–1

–2

–3

–4

2.8

0246810

2.6

2.4

2.2

Waveguide losses (dB/cm) Refractive index

Mean particle diameter (nm)

1295 nm

1064 nm

Fig. 5.3-13 Scattering losses of a 5% PbS nanoparticle-

doped silica–titania waveguide as a function of the nano-

particle diameter at wavelengths of 1064 nm and 1295 nm

(After [3.71])

matrix-dependent spectra for various metals, see [3.72].

Higher-multipolar-order plasmon modes exist [3.73] but

cannot be excited optically in the quasi-static regime.

However, they have a nonnegligible inﬂuence on the op-

tical absorption spectra of larger particles, as shown in

Fig. 5.3-15, and can be excited by, for example, elec-

tron beams. From the Mie theory, the frequency of the

Lth-order plasmon mode of a small particle of a Drude-

model metal, with ε(ω) =ε

(1 −ω

/ω

), where ω

the plasma frequency, is given by [3.73]

1+(ε

/ε

)(1+1/L)

(3.8)

A continuous tuning of the positions of the plasmon

absorption peak by alloy formation has been demon-

strated. The dependence of the absorption spectra of

−

Ag alloy nanoparticles on the composition is re-

produced in Fig. 5.3-16 [3.74].

Local-Field Effects

In the quasi-static approximation, the electric ﬁeld in-

side a nanoparticle is changed compared with the ﬁeld

outside by a local-ﬁeld factor f ,givenby

f(ω) =

3ε

(ω)

ε(ω) +2ε

(ω)

(3.9)

This factor is taken into account in (3.5) – (3.8) above,

but it can have an even greater importance in nonlinear

effects, since the second-order and third-order nonlin-

ear optical coefﬁcients, χ

(2)

and χ

(3)

, respectively, are

affected by factors f

and f

, respectively, as com-

pared with the bulk material of the nanoparticle. Hence,

for large f , a nanostructured material can have a larger

optical nonlinearity than its bulk constituents. For typ-

ical semiconductor-doped matrices, ε>ε

and f < 1.

However, particularly strong local-ﬁeld enhancements

are observed for metal nanoparticles in the vicinity of

the plasmon resonance [3.75].

Electromagnetic Concentration Effects

For larger particle concentrations, the interactions be-

tween particles inﬂuence the electromagnetic properties.

For interparticle distances much smaller than the wave-

length, the Maxwell–Garnett model applies and leads to

the Lorentz–Lorenz relation for the effective dielectric

constant ε

eff

of the composite medium, which takes the

form

eff

= ε

ε(ω)(1+2p) +2ε

(1− p)

ε(ω)(1− p) +ε

(2+ p)

(3.10)

where p is the volume fraction. This usually leads to

a redshift of absorption peaks. In particular, the fre-

quency of the dipole (L =1) plasmon mode of a particle

Part 5 3.3

1046 Part 5 Special Structures

200 300 400 500 600 700 800 900

100

200 300 400 500 600 700 800 900

200

150

100

200 300 400 500 600 700 800 900

200

150

100

200 300 400 500 600 700 800 900

200

150

100

200 300 400 500 600 700 800 900

800

600

400

200

200 300 400 500 600 700 800 900

600

400

200

200 300 400 500 600 700 800 900

160

120

200 300 400 500 600 700 800 900

100

200 300 400 500 600 700 800 900

240

180

120

200 300 400 500 600 700 800 900

Li Na Mg K

Ca Cu Pd Ag

CsEu Au Hg

Fig. 5.3-14 Absorption spectra, represented by the dimensionless function Cλ/V where C is the cross section, λ the

wavelength, and V the particle volume, of 10 nm diameter spherical particles of selected metallic elements, as calculated

from their frequency-dependent dielectric constant, in vacuum (dashed lines) and in a matrix with a refractive index equal

to that of water, 1.33 (solid lines). Horizontal scale: λ (nm). (After [3.77])

Fig. 5.3-15 Absorption spectra of spherical particles of various materials, as calculated from their frequency-dependent

dielectric constant, in vacuum for various particle radii. (After [3.72])

Fig. 5.3-16 Effect of formation of a gold–silveralloy on the

surface plasmon absorption: measured UV–VIS absorption

spectra of spherical Au

−

Ag alloy nanoparticles of various

compositions. The gold mole fraction x

varies between 1

and 0.27. The plasmon absorption maximum is blueshifted

with decreasing x

. (After [3.74])

in the Drude model is ω

L=1

√

(1− p)/3ω

instead of

√

3, the value in the low-concentration limit (p ≈0).

This effect has found applications in sensing, in par-

ticular the selective colorimetric detection of nucleotides

based on aggregation of gold nanoparticles [3.76].

The electromagnetic concentration effect is also used,

1.0

0.8

0.6

0.4

0.2

350 400 450 500 550 600 650

Wavelength

(nm)

Absorbance

= 1

= 0.8

= 0.54

= 0.27

Part 5 3.3

Mesoscopic and Nanostructured Materials 3.3 Electromagnetic Conﬁnement 1047

hω (eV)

50 70

hω (meV)

hω (eV)

100

hω (eV)

100

0.5

Silver

Extinction constant

2R (nm):

Copper

Extinction constant

2R (nm):

Aluminium

Extinction constant

2R (nm):

Sodium

Extinction constant

2R (nm):

Potassium

Extinction constant

2R (nm):

MgO

Extinction constant

2R (µm):

Silicon

Extinction constant

2R (nm):

Gold

Extinction constant

2R (nm):

Part 5 3.3

1048 Part 5 Special Structures

in addition to quantum size effects (Sect. 5.3.2), to

tune the absorption band edge of semiconductor-doped

glasses.

5.3.3.2 Periodic Electromagnetic Lattices

Coupled Plasmon Modes

The coupling between the plasmon modes of several

closely packed nanoparticles yields new electromag-

netic eigenmodes and spectroscopic properties. The

electromagnetic eigenmodes of some few-particle sys-

tems, such as Ag and Au pairs and triplets with various

geometries, have been computed, for instance in [3.72,

Sect. 2.3.4], and have been observed directly by e-beam

excitation [3.78]. For the lowest eigenmodes, a large

ﬁeld enhancement is observed in the small gap between

the particles.

Coupled plasmon modes have been observed ex-

perimentally in regularly spaced linear chains of gold

nanoparticles [3.79] and in 2-D hexagonal arrays of sil-

ver nanoparticles [3.80]. The controlled propagation of

plasmon excitations in tailored metal nanoparticle struc-

tures is a new domain of application, sometimes called

“plasmonics” [3.81].

Opals and Photonic-Band-Gap Materials

3-D ordered arrays of nanostructures with periods of the

order of a fraction of the optical wavelength may show

intense Bragg diffraction for speciﬁc wavelengths and

diffraction angles. This phenomenon is the origin of the

iridescent colors (opalescence) of opals. Opals consist of

an fcc-like array of silica nanoparticles with sizes in the

range 150–900 nm [3.82], with a size dispersion below

5% [3.83].

For nanoscale objects with large scattering strengths

and particular geometries, spectral intervals where no

propagating mode is allowed in any direction appear,

thus forming “photonic band gaps”. The idea of ex-

ploiting this phenomenon in optoelectronic devices has

been suggested [3.84,85]. However, up to now, this phe-

nomenon has been realized only in the infrared and

microwave regions. Application in the visible, which

would require 3-D nanostructured materials, is still un-

der active development.

5.3.4 Magnetic Nanostructures

The driving force for the research on magnetic nano-

structures is the possibility of their incorporation into

future generations of information storage devices. This

could lead to a real technological breakthrough, and

a huge economic impact is foreseen. As always, the key

question is whether any potential beneﬁt of such tech-

nology will be worth the production costs. Overall, the

current effort in this research ﬁeld is considerable. The

study of the magnetism of nanoscale objects is charac-

terized by the closeness between fundamental research

and applications. What is more, a number of magnetic

devices are currently in commercial use without a full

understanding of the basic magnetic phenomena under-

lying them. An instructive example is the following: spin

electronic phenomena have been applied rapidly since

the ﬁrst observation of giant magnetoresistance (GMR)

was reported in 1988, and magnetic sensors and read

heads based on this effect have been available since

1994 and 1997, respectively.

Most digital information is stored in the form of

tiny magnetized regions or “bits” within thin magnetic

layers on disks. The size of a magnetic bit deter-

mines the information storage capacity of a magnetic

disk drive. The magnetic storage media used in to-

day’s commercial hard disks consist of homogeneous

polycrystalline magnetic ﬁlms. Owing to continuous

improvements in both the magnetic properties of the

media and the read/write heads, the storage density

of hard disk drives has increased considerably during

recent years, as shown in Fig. 5.3-17. Since 1990, the

storage capacity of disk drives has increased at a 60%

0.1

0.01

–3

1975 1980 1985 1990 1995 2000

Areal density (Gb/in

)

Date of general availability

Inductive

read/write head

21%/ year

Magnetoresistive read head

57%/year

Fig. 5.3-17 Evolution of areal storage density of hard disk

drives (data from IBM, Almaden)

Part 5 3.4

Mesoscopic and Nanostructured Materials 3.4 Magnetic Nanostructures 1049

annual compound growth rate. The changes of the head

technology ﬁrst from thin-ﬁlm heads to magnetoresis-

tive heads and then to read heads based on the GMR

effect [3.86, 87] have enabled enormous increases in

volumetric storage capacity.

The overall gain in areal density has been achieved

by scaling the complete disk drive system. The medium

has to accommodate smaller bit dimensions and smaller

transition widths between written bits. However, the

optimization also includes more sensitive read/write

heads, improving the signal processing and coding, and

improving the head–disk tribology. A good discussion

of these aspects can be found in [3.88].

Progress in this ﬁeld would not be possible without

the maturity of growth techniques such as electron- or

ion-beam lithography, and molecular-beam epitaxy and

the ability to manufacture devices on a nanometric scale

by nanoimprinting lithography. Another essential point

has been the addition of magnetic contrast to near-ﬁeld

microscopy and electron microscopy (magnetic-force

and Lorentz microscopy, respectively). These new tools

permit us to establish a link between magnetism and

structure on the atomic scale.

We have selected here some examples to illustrate

the state of the art in nanometric nanostructures, keeping

as the main idea the impact of the underlying technolog-

ical breakthrough. We focus our attention on two very

active ﬁelds: magnetoresistive nanometric multilayers

for spin electronics, and magnetic dots for ultrahigh-

density storage media.

5.3.4.1 Spin Electronics

Conventional electronic devices are based on charge

transport, and their performance is limited by the speed

of the carriers (electrons) and the dissipation of their

energy. Conventional electronics has ignored the spin

of the electron. Nevertheless, in every wire or de-

vice, approximately 50% of the conducting electrons

are generally spin-up and the remainder spin-down

(where “up” or “down” relate to some locally induced

quantization axis). Spin electronics,orspintronics,is

a new ﬁeld of electronics which relies on the different

transport properties of majority-spin and minority-spin

electrons. Mott postulated the basis of spin electron-

ics in the mid-1930s: he attributed certain anomalies

in the electrical transport behavior of metallic ferro-

magnets to the fact that the spin-up and spin-down

conduction electrons are two independent families of

charge carriers, each with its own distinct transport

properties [3.89].

The Concepts of Spin Accumulation

and Spin Diffusion Length

The other necessary ingredient of this model is that the

two families contribute very differently to the electrical

transport processes. This may be because the number

densities of the carrier types are different, or it may

because they have different mobilities. In either case,

the asymmetry which makes spin-up electrons behave

differently from spin-down ones arises because the fer-

romagnetic exchange ﬁeld causes a splitting between

the spin-up and spin-down conduction bands, leaving

different band structures evident at the Fermi surface.

In nonmagnetic materials, the two spin channels are

equivalent because they have the same density of states

(DOS) at the Fermi energy. In a magnetic material, the

spin polarization is deﬁned as the ratio of the differ-

ence between the populations in the up and down spin

channels to the total number of carriers at the Fermi

level,

SP =

↑

−N

↓

↑

↓

. (3.11)

Clearly, if there is a nonzero spin polarization, the num-

bers of electrons participating in the conduction process

are different for the two spin channels. More subtly, this

implies that the susceptibilities to scattering of the two

spin types are different, and this in turn leads to different

mobilities [3.90].

Let us consider two spin channels of different mo-

bility. When an electric ﬁeld is applied to the metal,

there is a shift in momentum space ∆k of the spin-up

and spin-down Fermi surfaces in accordance with the

equation

F = eE =

∆k

(3.12)

where F is the force on the carrier, E is the electric ﬁeld,

e is the electronic charge, and τ is the electron scattering

time. Since the channels have different mobilities, this

shift is different for the spin-up and spin-down Fermi

surfaces, as illustrated in Fig. 5.3-18.

As a consequence, if a current is passed from such

a spin-asymmetric material, for example cobalt, into

a paramagnetic material such assilver (which has no spin

asymmetry between spin channels), there is a net inﬂux

into the silver of up-spins over down-spins. Thus, a sur-

plus of up-spins appears in the silver, and with it a small

associated magnetic moment per unit volume. This sur-

plus is known as a spin accumulation. Evidently, for

a constant current ﬂow, the spin accumulation cannot in-

crease indeﬁnitely. The spin-up electrons injected across

Part 5 3.4

1050 Part 5 Special Structures

Displaced Fermi spheres

Brillouin zone

Electric

field

∆k

k = 0

Fig. 5.3-18 Schematic illustration of the shift in the Fermi

surface when an electric ﬁeld is applied to a ferromag-

net. The solid circles represent the Fermi spheres of up-

and down-spin electrons in the ﬁeld, and the dashed cir-

cle represents the Fermi sphere in zero external ﬁeld.

(After [3.90])

the cobalt–silver interface are converted into spin-down

electrons by spin-ﬂip process, which we have hitherto

ignored. So now we have a dynamic equilibrium be-

tween an inﬂux of up-spins and their destruction by

spin-ﬂipping. This in turn deﬁnes a characteristic length

scale that describes how far the spin accumulation ex-

tends into the silver. In the present description, we have

assumed that both the spin-up and spin-down electrons

are present in the ferromagnetic material in equal num-

bers but that their mobilities are different. Nevertheless,

we can produce spin accumulation either as a direct

consequence of an asymmetric density of states or as

an indirect consequence via asymmetry in the electron

mobility.

It follows from the above discussion that the spin

accumulation decays exponentially away from the inter-

face, on a length scale called the spin diffusion length

(λ

). This concept is crucial for realizing a link be-

tween spin electronics and magnetic nanostructures. In

this section, we shall not go further into this concept, but

we shall do a rough calculation to see how large λ

and on what parameters it depends. We use the following

expression from [3.90]:

λv

↑↓

(3.13)

where v

is the Fermi velocity, τ

↑↓

is the spin-ﬂip time,

and λ is the mean free path. This relationship high-

lights the critical role of the impurity concentration in

determining the spin diffusion length. If the impurity

level is increased in the nonmagnetic material, λ

drops

because λ is shortened. This relation also allows us to es-

timate values for the spin diffusion length. Taking silver

as an example, λ

can vary between several microns for

very pure silver to the order of 10 nm for silver with 1%

gold impurity. In a realistic material, λ

is of the order

of 10–100 nm. This is the reason why the experimen-

tal observation of spin-dependent transport phenomena

has been possible when researchers have controlled the

growth of nanometric multilayers. It is obvious that the

layer thickness has to be small on the scale of the spin

diffusion length.

The Discovery of Giant Magnetoresistance

The era of spin electronics began with the discovery of

a magnetoresistance effect in a magnetic multilayer con-

sisting of a sequence of thin magnetic layers separated

by equally thin nonmagnetic metallic layers: the electric

current is strongly inﬂuenced by the relative orientation

of the magnetizations of the magnetic layers. This exper-

imental observation was made almost simultaneously by

two research groups: Albert Fert’s group in Orsay [3.86]

and Peter Grünberg’s group in Jülich [3.87,91].

In fact, it was found that the resistance of the over-

all multilayer is low when the magnetizations of all the

magnetic layers are parallel (ferromagnetic coupling),

but it becomes much higher when the magnetizations of

neighboring magnetic layers are antiparallel (antiferro-

magnetic coupling).

Figure 5.3-19 shows the magnetoresistance curves

at 4.2KofFe/Cr (magnetic/nonmagnetic) multilayers

with thicknesses of the individual layers of the order

of 1 nm (from [3.86]). In these multilayers, for certain

thicknesses of the Cr interlayer, the magnetizations of

adjacent Fe layers are oriented antiparallel by an antifer-

romagnetic interlayer exchange coupling [3.87]. When

a magnetic ﬁeld is applied, the resistance of the multi-

layer decreases drastically as the magnetizations of the

two layers progressively align in the direction of the

ﬁeld. The GMR ratio is usually deﬁned as

GMR =

− R

, (3.14)

where R

and R

are the resistances in the parallel

and the antiparallel state, respectively. In the case of

Fe/Cr multilayers, the GMR is 80% at liquid-helium

temperature and 20% at room temperature when the

thickness of the Cr is 9 Å.

The relative change of the resistance can be larger

than 200% [3.92], and that is the reason why the ef-

fect is called “giant” magnetoresistance. GMR effects

Part 5 3.4

Mesoscopic and Nanostructured Materials 3.4 Magnetic Nanostructures 1051

–40 –30 –20 –10 0 10 20 30 40

0.8

0.7

0.6

0.5

R/R (H = 0)

(Fe 30 Å/Cr 18 Å)

(Fe 30 Å/Cr 12 Å)

(Fe 30 Å/Cr 9 Å)

Magnetic field (kG)

≈ 80%

Fig. 5.3-19 Magnetoresistance curves at 4.2K of Fe/Cr

multilayers. (After [3.86])

have been reported in a large number of systems com-

bining ferromagnetic transition metals or alloys with

nonmagnetic metals (for a review, see [3.93]). The most

commonly used combinations of magnetic and non-

magnetic layers are Co/Cu and Fe/Cr, but multilayers

based on permalloy as the magnetic component are also

frequently used.

The second key ingredient was discovered by Parkin

and Mauri [3.94]. These authors discovered that the ori-

entation of the magnetic moments of two neighboring

magnetic layers depends on the thickness of the inter-

vening nonmagnetic layer. In fact, the orientation of

the magnetic moments oscillates between parallel and

antiparallel as a function of the thickness of the nonmag-

netic layer [3.96]. This phenomenon is referred to as an

oscillatory exchange coupling. Figure 5.3-20 shows the

results of the original experiment of Parkin and Mauri.

Oscillations of the GMR as a function of the Cr thick-

ness occur because the magnetoresistive effect is only

measurable for those thicknesses of Cr for which the in-

terlayer coupling aligns the magnetic moments of all the

Fe layers so that they are antiparallel.

GMR effects have been obtained in two geometries.

In the ﬁrst one, the current is applied in the plane of the

layer (hereafter denoted by CIP), as in the experiments

for which results are shown in Figs. 5.3-19 and 5.3-20,

while in the second one, the current ﬂows perpendicu-

lar to the plane of the layers (hereafter denoted by CPP).

The ﬁrst measurements in the CPP conﬁguration were

012345

GMR (%)

Chromium thickness (nm)

Fig. 5.3-20 Dependence of the GMR ratio of an Fe/Cr

multilayer on Cr thickness. (After [3.94])

obtained by sandwiching the multilayer between two

superconducting Nb layers [3.97]. A CPP-GMR con-

ﬁguration has also been obtained in nanowires, namely

multilayers electrodeposited in the pores of a nuclear-

track-etched polycarbonate membrane [3.98], and by

oblique deposition on a prestructured substrate [3.99].

The CPP-GMR effect is deﬁnitely larger than the CIP-

GMR effect and occurs at much larger thicknesses.

The mechanism of the GMR effect is illustrated in

Fig. 5.3-21 for α =(ρ

↓

/ρ

↑

)>1, where ρ

↓

and ρ

↑

are

Parallel configuration

MMNM

Antiparallel configuration

MMNM

–

= 1

–

= R

= (1+R)/2

–

= (R+1)/2

Fig. 5.3-21 Schematic picture of the GMR mechanism.

The electron trajectories between two scatterings are rep-

resented by straight lines and the scatterings by abrupt

changes in the direction. The signs + and − are for spins

= 1/2andS

=−1/2, respectively. The arrows rep-

resent the majority-spin direction in the magnetic layers.

M = magnetic, NM = nonmagnetic. (After [3.95])

Part 5 3.4