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

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conﬁguration results immediately in the noncollinear

magnetic structure similar to the structure found in

the experiment. The deviation of the magnetic mo-

ments does not inﬂuence the symmetry of the mag-

netic structure. The minimum of the total energy is at

an accidental point.

With the SOC being neglected, the curve of the

total energy as a function of the deviation angle y is

symmetrical with respect to the change of the sign of

y (Fig. 5). Therefore, the collinear conﬁguration cor-

responding to y ¼0 is distinguished by symmetry and

is stable. The reason for different symmetry proper-

ties of the total energy in relativistic and nonrelativ-

istic problems is the different types of symmetry

groups which describe the relativistic and nonrelativ-

istic problems.

The phenomenon of the symmetry-predetermined

noncollinearity observed in U

has the same phys-

ical nature as the effect of weak ferromagnetism in

, where the iron moments deviate from their

antiparallel directions and result in a ﬁnite ferromag-

netic component.

3. Calculated Magnetic Properties

Magnetic ordering in itinerant magnets is driven by

the spin density, but the orbital contribution to the

magnetic moment,

¼ m

occ

i;k

7L7f

i;k

S ð16Þ

where f

i,k

is an energy band eigenfunction for the ith

branch, with wave vector k and angular momentum

L, is responsible for such technologically important

effects as intrinsic magnetocrystalline anisotropy en-

ergy (MAE), magnetostriction, the magneto-optical

Kerr effect, and magnetic circular dichroism in itin-

erant magnets. In iron, cobalt, and nickel, the calcu-

lated spin (orbital) contributions to the magnetic

moment are, respectively, 2.13 m

(0.08 m

), 1.52 m

(0.14 m

) and 0.57 m

(0.05 m

) and are parallel (Ebert

et al. 1988, Eriksson et al. 1990). The orbital moment

is carried almost entirely by the 3d electrons, whereas

the spin contribution is the sum of local (3d) and

diffuse (sp) parts aligned antiparallel. However, the

calculated orbital moments are smaller than meas-

ured (Ebert et al. 1988, Eriksson et al. 1990).

Experimental information on ground-state spin

and orbital magnetization densities may be obtained

from elastic neutron scattering measurements (see

Magnetism in Solids: General Introduction; Magnetic

Excitations in Solids). The neutron magnetic form

factor, f(Q), where Q is the momentum transfer

( ¼4p siny/l) is a direct measure of the Fourier

transform of the magnetization. In the dipole ap-

proximation the form factor is given by:

f ðQÞ¼/j

S þ m

S=m ð17Þ

where /j

S ¼4p

(r)j

(r) is a Bessel transform of

the calculated spin density, m is the total and m

the

orbital magnetic moment. The orbital contribution to

the magnetic moment is small for transition metal

systems, but can be large in rare earths and actinides

(Brooks and Kelly 1983). Analysis of the form factors

of actinides has been used to show that they have

large orbital moments. In systems that do not order,

or above the ordering temperature in systems that do

order, the application of a magnetic ﬁeld allows the

induced form factor to be measured. Calculations

have been used to show that the relationship between

spin and orbital moments can be quite different when

the form factor is induced, compared with the or-

dered state due to the breaking of Hund’s rules by the

applied ﬁeld (Hjelm et al. 1993).

3.1 Hyperﬁne Interactions

The interactions between the nuclear and electronic

charge and current densities produce hyperﬁne struc-

ture, for which a comparison of calculated and meas-

ured contributions is therefore an excellent test of

density functional theory. Since the length scale of the

nucleus (r

E10

4

) is small compared with the

atomic length scale of one Bohr radius, (a

¼_

/me

the Coulomb potential, Ze

/r, is large in the region

of the nucleus, and the concomitant large electronic

kinetic energy makes relativistic effects important.

Therefore either the Dirac equation, or some ap-

proximation to the Dirac equation that takes proper

account of the boundary condition for the relativistic

case at small values of radius, should be used in any

theory of hyperﬁne interactions. The interaction be-

tween nuclear and electronic charges and currents is

usually expressed in terms of a multipole expansion:

int

k¼2

k¼0

 T

ð18Þ

since the contribution of multipoles with rank greater

than 2 is negligible.

The monopole (k ¼0) interaction gives rise to the

isomer (or chemical) shift, which is proportional to

the contact charge density at the nuclear radius and is

used to study trends in charge transfer, particularly in

ssbauer experiments (see Mo

ssbauer Spectro-

scopy). For example, the isomer shift at the iron site

in dilute FeX alloys is essentially periodic across the

periodic table, being a minimum (more contact

charge) if X is at the start of a row and maximum

(less contact charge) if X is at the end (Akai et al.

1986). The self-consistently calculated isomer shifts

are in good agreement with measurements, and

broadly follow the argument that charge ﬂows from

atoms at the beginning of a period to those at the

right which are more electronegative. However, de-

tailed calculations show that there is a strong angular

110

Density Functional Theory: Magnetism

momentum dependence in the charge transfer and

that, particularly in the case of transition metals, sd-

hybridization is important.

The dipole (k ¼1) interaction may be separated

into Fermi contact, dipolar, and orbital contribu-

tions. The Fermi contact interaction is between the

electronic spin density at the nucleus and the nuclear

dipole moment which experiences a magnetic ﬁeld:

ð19Þ

where m

(r)m

(r)d

r is the average spin mag-

netic moment density over a sphere with the diameter

of the Compton radius, r

¼Ze

/mc

ðrÞ¼ð1=4pr

Þð@SðrÞ=@rÞ;

where S

1

(r) ¼1 þ((ev

eff

(r))/2mc

) is the relativistic

mass enhancement.

The dipolar interaction is between the electronic

vector spin density and the nuclear moment which

experiences a ﬁeld:

¼ m

SðrÞ3

r  SðrÞ



ð20Þ

Finally, the orbital interaction is between the elec-

tronic current distribution and the nuclear moment

which experiences a ﬁeld:

¼2m

ð21Þ

where L

L(r)S(r)/r

is the weighted orbital mo-

ment density measured in units of _.

The Fermi contact interaction, which is propor-

tional to the spin density at the nucleus, is normally

far larger than the dipolar and orbital contributions,

especially in crystals with cubic symmetry and small

spin–orbit interaction, where the dipolar term is neg-

ligible. The spin density at the nucleus, which arises

entirely from s-states, is composed of the contribu-

tion from the unpaired spins of the valence electrons

and a contribution from core polarization. Detailed

calculations have shown that the contact interaction

is dominated by core polarization. For example, in

iron the calculated hyperﬁne ﬁeld due to core polar-

ization is 235 kG, compared with a valence contribu-

tion of 42 kG (Ebert 1988, Ebert et al. 1988).

The most thoroughly studied itinerant electron

metals are the transition metals iron, cobalt, and

nickel (Ebert et al. 1988). The s-electrons—both core

and valence—are polarized by exchange interactions

with the d-electrons. Since the core spin density is

dominant, comparison with measured hyperﬁne ﬁelds

is a severe test of the calculated exchange interactions

between transition metal d-states and core s-states. In

practice, the calculated and measured hyperﬁne ﬁelds

for cobalt are 194 kG and 215 kG, respectively, where

in addition to the Fermi contact contribution, the

contribution from the orbital currents (moment) of

the d-electrons of 50 kG is included. Although this

result is quite reasonable, the calculated hyperﬁne

ﬁelds for iron and nickel are poor.

In iron, the measured and calculated hyperﬁne

ﬁelds are 339 kG and 253 kG, respectively; and in

nickel the corresponding quantities are 75 kG and

39 kG. Since the calculated dipolar contribution from

the orbital currents is relatively small in iron

(24 kG), the errors are due to the calculated core

polarization. In nickel, the orbital current contribu-

tion is relatively larger (37 kG), but is of the wrong

sign to rectify the errors. Since the calculated mag-

netic moments of the transition metals compare well

with experiment, it is difﬁcult to escape the conclu-

sion that errors in LSDA are connected with its

description of core polarization (Akai et al. 1990).

The hyperﬁne ﬁelds at impurity sites in transition

metal alloys have also been studied in some detail.

When the impurity atom is non-magnetic, the meas-

ured hyperﬁne ﬁeld is due to the induced s-spin den-

sity at the impurity nucleus, which arises from

hybridization with the magnetic transition metal d-

electrons and, since the core of the impurity atom is

unpolarized (there is no impurity d-moment to do

this), core polarization plays no role. The induced

spin density at the impurity nucleus depends both

upon the local occupation of the spin-up and spin-

down states and the magnitude of the spin-up and

spin-down wave functions. When transition metal

d-states hybridize with conduction electrons from

other atoms, the sign of the wave function at the

impurity site can change as the energy in the hybrid-

ized bands changes, passing from bonding to anti-

bonding orbitals. The effect upon the spin density at

the impurity nucleus is a quite remarkable systematic

trend throughout the periods in the periodic table,

illustrated by the measured and calculated hyperﬁne

ﬁelds for impurities in nickel (Akai et al. 1990)

(see Fig. 6).

The hyperﬁne ﬁelds are negative at the beginning

of each period and change sign, reaching a maximum

near the end of each period before changing sign

again at the beginning of the next period. When the

impurities are themselves transition metals the mech-

anism controlling the hyperﬁne ﬁelds is different,

since d–d hybridization is important and an impurity

d-moment is induced which, particularly in the case

of 3d impurities, can be quite large. At the beginning

of a period, the less tightly bound impurity d-bands

lie higher in energy than the nickel 3d-bands, and an

impurity d-moment is induced anti-parallel to the

nickel 3d-moment. However, for impurities towards

the end of the transition metal series, the induced

moment can ﬂip to ferromagnetic alignment. On

the whole, although there are some discrepancies,

both trends and magnitude of the hyperﬁne ﬁelds at

the impurity sites are obtained from LSDA calcu-

lations.

111

Density Functional Theory: Magnetism

The next term in the multipole interaction (k ¼2) is

the interaction between the nuclear electric quadru-

pole moment and the electric ﬁeld gradient produced

by the electron density:

ﬃﬃﬃ

lim

r-0

ðrÞ

ð22Þ

where v

is the quadrupolar electronic charge den-

sity. The accurate calculation of electric ﬁeld gradi-

ents requires accurate treatment of non-spherical

components of the charge density. In particular the

quadrupolar charge density, when weighted by 1/r

,is

directly related to the ﬁeld gradient. The ﬁeld gradi-

ent vanishes for cubic crystals, but it has been found

for hcp transition metals that, due to the radial

weighting, the p-electron contribution to the ﬁeld

gradient becomes dominant, since the ﬁrst radial

node of the p-states lies inside that of the d-states

(Blaha et al. 1988).

3.2 Magnetocrystalline Anisotropy

The magnetic anisotropy energy (MAE) is normally

calculated as the difference in total energy for mo-

ment alignment along an easy and a hard axis. For

iron, cobalt, and nickel, the magnetic anisotropy en-

ergy is of the order of meV, which demands extremely

accurate numerical treatment. The magnitude of the

MAE from self-consistent calculations is too small

for iron and cobalt, and has the wrong sign for nickel

(Daalderop et al. 1990, Jansen 1990). Inclusion of the

orbital polarization correction term improves the re-

sults, except for nickel.

The magnetic anisotropy constants are the coefﬁ-

cients in the expansion of the energy in terms of

spherical harmonics which depend upon the moment

direction. In cubic crystals, the lower order anisotro-

py constants vanish by symmetry; consequently the

anisotropy energy is small. Of the non-cubic symme-

tries where the magnetic anisotropy is larger, the

most important is tetragonal, since this is the sym-

metry of many thin ﬁlms and multilayers. It is pos-

sible to take the crystal symmetry from fcc to bcc via

tetragonal along a Bains path, simply by changing the

c/a ratio. Self-consistent total energy calculations

have shown that the anisotropy energy should

increase by orders of magnitude as the structure is

taken to tetragonal and, since the ﬁnal structure is

also cubic, it has at least one maximum along the

path, which for cobalt is about 600 meV. If the

exchange splitting were larger than the bandwidth

(which is seldom true) and the spin–orbit interaction,

a small perturbation in the magnetic anisotropy

would be proportional to the anisotropy of the or-

bital moment (Bruno 1989). No such approximations

are made in total energy calculations, but experience

has shown that the easy (hard) direction is that in

which the orbital moment is largest (smallest).

3.3 Magnetovolume, Invar Effect, and

Magnetostriction

Magnetostriction is the lattice deformation which

accompanies magnetization (see Magnetoelastic

Figure 6

The calculated hyperﬁne ﬁelds for impurities from the short periods in Ni (Akai et al. 1990).

112

Density Functional Theory: Magnetism

Phenomena). Since magnetostriction is due to the

coupling of magnetic and elastic forces, it is of im-

mense technological importance for the conversion of

energy between magnetic and elastic degrees of free-

dom. The magnetovolume effect is the isotropic (or

volume) component of magnetostriction. Itinerant

magnetic materials may have quite large magneto-

volume effects, since the electrons responsible for the

magnetism are also responsible for chemical bonding.

For example, the Friedel (1969) model of transition

metal bonding attributes to the cohesive energy a

term proportional to f(1f ), where f is the fractional

occupation of the d-band, which is responsible for the

parabolic trend in cohesive energy across a transition

metal series.

More detailed modelling has shown that the

change of this term with volume, contributes an elec-

tronic pressure:

P ¼ð2l þ 1Þ

½g

 f ð1  f Þ=3V ð23Þ

where W

is the width of the d-band, l ¼2 for d-states,

are the fractional occupations of the spin-up and

spin-down bands and g is the fractional magnetic

moment, (f

f



)/2. The magneto-volume effect is

due to the repulsive contribution proportional to g

which is a maximum when f ¼m ¼1/2 and the spin-

up band is ﬁlled.

The Invar effect (see Invar Materials: Phenomena)

arises when high and low volume magnetic states

have about the same energy, such that the latter may

be populated by thermal excitations, thus compen-

sating thermal expansion due to lattice vibrations.

The low volume magnetic state was proposed by

Weiss (1963) to be anti-ferromagnetic, but both the

paramagnetic state and the ferromagnetic state can

be simultaneously stable if the Fermi level of the

paramagnetic state density lies in a dip between two

peaks. Williams et al. (1982) used SDFT to show that

in iron–nickel alloys, the total energy of the low

volume paramagnetic state is a function of compo-

sition, and lies about 10 meV above the total energy

of the high volume ferromagnetic state for Fe

Ni.

Subsequently, the low volume state has been shown

to be disordered when non-collinear spin ordering is

included in the simulation (van Schilfgaarde et al.

1999).

The anisotropic part of the magnetostrictive energy

may be obtained from self-consistent calculations if

the anisotropy energy is calculated as a function of

strain. Calculations for transition metals (James

1999) reveal proportionality between the magnitude

of the magnetostriction energy and the MAE, and

between the MAE and the anisotropy of the orbital

magnetic moment. Although the absolute values of

the calculated magnetostrictive energy for Fe/Co and

Co/Ni alloys tend to be too large by up to an order of

magnitude, the correct trend as a function of cobalt

concentration is reproduced.

3.4 X-ray Magnetic Circular Dichroism

First-principles calculations of optical properties are

usually made in two stages. Firstly, the self-consistent

ground state charge and spin densities are calculated;

then the optical conductivity tensor is calculated for

the Kohn–Sham energies and wave functions corre-

sponding to the ground-state density (Sect. 1.4). The

conductivity tensor, s(o), derived from linear res-

ponse theory, is:

sðoÞ¼

ie

f ðe

Þf ðe

ðkÞ

o  o

ðkÞþi=t

ð24Þ

where v is the volume of the unit cell, f(e

) is the

Fermi function, _o

(k) ¼e

e

, the energy differ-

ence of the Kohn–Sham energies e

, t is the average

lifetime of excited electron states, and P

(k) are the

matrix elements of the momentum operator.

When circularly polarized x-rays are absorbed by a

magnetic material, the absorption depends upon

whether the light is left-or right-polarized. The ab-

sorption is determined by the imaginary part of con-

ductivity tensor, where the initial states are the core

states of the x-ray edge. The source of X-ray magnetic

circular dichroism (XMCD) is that the off-diagonal

elements of the conductivity tensor are non-zero,

which is due to spin–orbit interaction in either initial

or ﬁnal states. Calculations for transition metals

(Ebert 1996) have shown that pronounced XMCD

can occur when there is a large spin–orbit splitting of

the core states and a small moment in the conduction

bands, as for the L

2,3

edges of 5d impurities in mag-

netic 3d transition metal hosts. Alternatively, in pure

transition metals the 2p-core state spin–orbit splitting

is relatively small, but the conduction band moment

is large. In both cases agreement with measurements

is good (for experimental aspect of XMCD, see

Demagnetization: Nuclear and Adiabatic).

3.5 Magneto-optical Kerr Effect

When linearly polarized light is reﬂected from a mag-

netized sample, the reﬂected light becomes elliptically

polarized, with the plane of polarization rotated

(see Faraday and Kerr Rotation: Phenomenological

Theory). The complex Kerr rotation angle, f ¼y þie

where y is the rotation and e the ellipticity, is

given by:

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þð4pi=oÞs

ð25Þ

The sign of the Kerr angle changes when the mag-

netization is reversed; therefore, information stored

in the form of a reversed magnetization may be read

by measuring the Kerr rotation. Since high data den-

sity is desirable, and blue light is easier to focus than

113

Density Functional Theory: Magnetism

red, a large Kerr rotation in the 2–3 eV region is

considered optimal. Since the Kerr rotation is

proportional to the off-diagonal elements of the con-

ductivity tensor, which increase with spin–orbit

interaction, materials with large magnetic anisotropy

are also likely to have a large Kerr rotation for some

energy of incident light (Oppeneer 1998). However,

the fact that the Kerr rotation should be large in the

3 eV region has made Fe/Pt and Co/Pt intermetallics

of particular interest.

The magnitude of the Kerr rotation for transition

metal compounds rarely exceeds 0.51, but for Heusler

alloys, particularly the half-metallic ferromagnets

XMnSb (X ¼Ni, Pd, Pt), it can have higher values

(see Half-metallic Magnetism). Calculations, which

are in good agreement with experiment, for PtMnSb

yield a Kerr rotation of over 11 at 1.6 eV (Ebert

1996). The Kerr rotation for actinide compounds,

where the 5f state spin–orbit interaction is large, can

reach several degrees (Oppeneer et al. 1998). Mag-

netic multilayers possessing perpendicular anisotropy

are excellent candidates for recording materials since,

apart from the fact that they yield a polar Kerr effect,

the magnetic layers may be tailored. Here theoretical

predictions are most useful, and extensive investiga-

tions of, for example, multilayers of the type nA/mB

(A ¼Fe, Co and B ¼Cu, Ag, Au, Pd, and Pt) have

been made (see Magneto-optics: Inter- and Intraband

Transitions, Microscopic Models of ).

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M. S. S. Brooks

European Commission Joint Research Centre

Karlsruhe, Germany

M. Richter

Institut fu

r Festko

rper- und Werkstofforschung

Dresden, Germany

L. M. Sandratskii

Institut fu

r Festko

rperphysik, Technische Universita

Darmstadt, Germany

Dilute Magnetic Semiconductors

This family of materials encompasses standard sem-

iconductors, in which a sizable portion of atoms is

substituted by elements which produce localized

magnetic moments in the semiconductor matrix.

Usually, magnetic moments originate from 3d or 4f

open shells of transition metals (TMs) or rare-earth

metals (lanthanides), respectively, so that typical ex-

amples of diluted magnetic semiconductors (DMS)

are Cd

1x

Se, Ga

1x

As, Pb

1x

Te and, in

a sense, Si:Er. A strong spin-dependent coupling be-

tween the band and localized states accounts for out-

standing properties of DMS. This coupling gives rise

to spin-disorder scattering, giant spin-splitting of the

electronic states, and strong indirect exchange inter-

actions between the magnetic moments, the latter

leading to collective spin-glass, antiferromagnetic or

ferromagnetic spin ordering.

Owing to the possibility of controlling and probing

magnetic properties by the electronic subsystem or

vice versa, DMS have successfully been employed to

address a number of important questions concerning

the nature of various spin effects in various environ-

ments and at various length and time scales. At the

same time, DMS exhibit a strong sensitivity to the

magnetic ﬁeld and temperature as well as constituting

important media for generation of spin currents and

for manipulation of localized or itinerant spins by,

e.g., strain, light, and electrostatic or ferromagnetic

gates. These properties, complementary to both non-

magnetic semiconductors and magnetic metals, allow

applications of DMS as functional materials in spin-

tronics devices.

1. Historical Background and Survey of

Growth Methods

Extensive studies of DMS started in the end of the

1970s, when appropriately puriﬁed manganese was

employed to grow bulk II–VI Mn-based alloys by

various modiﬁcations of the Bridgman method. Com-

pared to magnetic semiconductors, such as europium

chalcogenides (e.g., EuS) and chromium spinels (e.g.,

CdCr

) which were investigated earlier, DMS ex-

hibited smaller defect concentrations and were easier

to dope by shallow impurities. Accordingly, it was

possible to examine their properties by powerful

magneto-optical and magnetotransport techniques.

Since, in contrast to magnetic semiconductors, nei-

ther narrow magnetic bands nor long-range magnetic

ordering affected low-energy excitations, DMS were

named semimagnetic semiconductors. More recently,

studies of DMS have been extended towards mate-

rials containing magnetic elements other than man-

ganese as well as to IV–VI, III–VI, and III–V

compounds. As a consequence, a variety of novel

phenomena have been discovered, including effects

associated with narrow-bands and magnetic phase

transformations, making the borderline between

properties of DMS and magnetic semiconductors

more and more elusive.

Rapid progress of DMS research in the 1990s

stemmed, to a large extent, from the development of

methods of crystal growth far from thermal equilib-

rium, primarily by molecular beam epitaxy (MBE),

but also by laser ablation. These methods have made

it possible to obtain DMS with the content of the

magnetic constituent beyond thermal equilibrium

solubility limits. Similarly, the doping during the

MBE process allows the electrical activity of shallow

impurities to be increased substantially. In the case of

III–V compounds, in which divalent magnetic atoms

supply both spins and holes, the use of the low-tem-

perature MBE (LT MBE) provides thin ﬁlms of, for

example, Ga

1x

As with x up to 0.07 and the hole

concentration in excess of 1 10

3

, in which

ferromagnetic ordering is observed above 100 K. Re-

markably, MBE and processes of nanostructure fab-

rication make it possible to add magnetism to the

physics of semiconductor quantum structures.

2. Magnetic Impurities in Semiconductors

A good starting point for the description of DMS is

the Vonsovskii model, according to which the elec-

tron states can be divided into two categories: (i)

localized magnetic d or f shells and (ii) extended band

states built up of s, p, and sometimes d atomic or-

bitals. The former give rise to the presence of local

magnetic moments and intracenter optical transi-

tions. The latter form bands, which are alike as in the

case of nonmagnetic semiconductor alloys. Indeed,

115

Dilute Magnetic Semiconductors

the lattice constant of DMS obeys the Vegard law,

and the energy gap E

between the valence and the

conduction band depends on x in a manner qualita-

tively similar to nonmagnetic counterparts. Accord-

ing to the Anderson model, the character of magnetic

impurities in solids results from a competition be-

tween (i) hybridization of local and extended states,

which tends to delocalize magnetic electrons, and (ii)

the on-site Coulomb interactions among electrons,

which stabilize magnetic moments.

Figure 1 shows positions of local states derived

from 3d shells of TM impurities in respect to the band

energies of the host II–VI compounds. The levels

labeled donors denote the ionization energy of the

magnetic electrons (TM

2 þ

-TM

3 þ

or d

-d

n1

whereas the acceptors correspond to their afﬁnity en-

ergy (TM

2 þ

-TM

1 þ

or d

-d

n þ1

). The difference

between the two is the on-d-shell Coulomb (Hub-

bard) repulsion energy U in the semiconductor ma-

trix. In addition, the potential introduced by either

neutral or charged TM can bind a band carrier in a

Zhang-Rice-type singlet or hydrogenic-like state, re-

spectively. Such bound states are often experimen-

tally important, particularly in III–V compounds, as

they correspond to lower energies than the competing

d-like states, such as presented in Fig. 1.

In the case of manganese, in which the d shell is

half ﬁlled, the d-like donor state lies deep in the va-

lence band, whereas the acceptor level resides high in

the conduction band, so that UE7 eV according to

photoemission and inverse photoemission studies.

Thus, manganese-based DMS can be classiﬁed as

charge transfer insulators, E

oU. The manganese

ion remains in the 2 þ charge state, which means that

it does not supply any carriers in II–VI materials.

However, it acts as a hydrogenic-like acceptor in the

case of III–V arsenides. According to Hund’s rule the

total spin S ¼5/2 and the total orbital momentum

L ¼0 for the d

shell in the ground state. The lowest

excited state d

5

corresponds to S ¼3/2 and its

optical excitation energy is about 2 eV.

Thus, if there is no interaction between the spins,

their magnetization is described by the Brillouin

function. In the case of other transition metals, the

impurity-induced levels may appear in the gap, and

then compensate shallow impurities, or even act as

resonant dopant, e.g., scandium in CdSe, iron in

HgSe, and copper in HgTe. Transport studies of such

systems have demonstrated that intersite Coulomb

interactions between charged ions lead to the Efros–

Shklovskii gap in the density of the impurity states,

which makes resonant scattering inefﬁcient in semi-

conductors. Furthermore, spin-orbit interaction and

the Jahn–Teller effect control positions and splittings

of the levels in the case of ions with La0. If the

resulting ground state is a magnetically inactive sin-

glet there is no permanent magnetic moment associ-

ated with the ion, as in the case of Fe

2 þ

, whose

magnetization is of the Van Vleck-type at low tem-

peratures.

3. Exchange Inter action Between Band and

Localized Spins

The important aspect of DMS is a strong spin-de-

pendent coupling of the effective mass carrier into the

magnetic electrons. Neglecting nonscalar corrections

that can appear for ions with La0, this interaction

assumes the Kondo form,

¼

Ið r

R

Þ s

ð1Þ

where I

is the short-range exchange energy operator.

When incorporated to the kp scheme, its effect is

described by matrix elements /u

S, where u

are

the Kohn–Luttinger amplitudes of the correspond-

ing band extreme. In the case of carriers at G point

of the Brillouin zone in zinc-blende DMS, the two

relevant matrix elements a ¼/S7I

7SS and b ¼

/X7I

7XS involve s-type and p-types wave functions,

respectively.

There are two mechanisms contributing to the Ko-

ndo coupling: (i) the exchange part of the Coulomb

interaction between the effective mass and localized

electrons and (ii) the spin-dependent hybridization

between the band and local states. Since there is no

hybridization between G

and d-derived (e

and t

)

states in zinc-blende structure, the s-d coupling is de-

termined by the direct exchange. The experimentally

determined values are of the order of aN

E0.25 eV,

where N

is the cation concentration, somewhat

reduced from the value deduced from the energy

difference between S7

states of the free singly

ionized manganese atom 3d

, aN

¼0.39 eV. In

Figure 1

Approximate positions of donor (full points) and

acceptor (open circles) levels derived from d-states of

transition metal impurities in II–VI DMS (courtesy of

Heiman and Dobrowolski, data after Zunger 1986 and

Langer et al. 1988).

116

Dilute Magnetic Semiconductors

contrast, there is a strong hybridization between G

and t

states, which affects their relative position,

and leads to a large magnitude of 7 b N

7E1 eV. If the

relevant effective mass state e

is above the t

level

(e.g., Mn-based DMS), bo0 but otherwise b can be

positive (e.g., Zn

1x

Se).

4. Electronic Properties

4.1 Effects of Giant Spin Splitting

In the virtual-crystal and molecular-ﬁeld approxima-

tions, the effect of the Kondo coupling is described by

Mð r

Þ s

=gm

, where

Mð r

)is magnetization

(averaged over microscopic regions) of the localized

spins, and g is their Lande

factor. Neglecting ther-

modynamic ﬂuctuations of magnetization (the mean-

ﬁeld approximation)

Mð r

) can be replaced by

(T, H), the temperature and magnetic ﬁeld de-

pendent macroscopic magnetization of the localized

spins available experimentally. The resulting contri-

bution to the spin-splitting of s-type electron states is

given by

ðT; HÞ

ð2Þ

The effect is known as the giant Zeeman splitting,

as in moderately high magnetic ﬁelds and low tem-

peratures _o

attains values comparable to the Fermi

energy or to the binding energy of excitons and shal-

low impurities. For effective mass states, whose pe-

riodic part of the Bloch function contains spin

components mixed up by a spin-orbit interaction,

the exchange splitting _o

does not depend only on

the product of M

and the relevant exchange integral,

say b, but usually also on the magnitude and direc-

tion of

k, conﬁnement, and strain. The giant Zeeman

splitting is clearly visible in magneto-optic phenom-

ena as well as in the Shubnikov–de Haas effect, mak-

ing an accurate determination of the exchange

integrals possible, particularly in wide-gap materials,

in which competing Landau and ordinary spin split-

tings are small.

The possibility of tailoring the magnitude of spin

splitting in DMS structures offers a powerful tool to

examine various phenomena. For instance, spin en-

gineering has been explored to control by the mag-

netic ﬁeld the conﬁnement of carriers and photons, to

map atom distributions at interfaces, as well as to

identify the nature of optical transitions and excitonic

states. Furthermore, a subtle inﬂuence of spin split-

ting on the quantum scattering amplitude of inter-

acting electrons with opposite spins has been put into

evidence in DMS in the weakly localized regime.

The redistribution of carriers between spin levels

induced by spin splitting has been shown to drive an

insulator-to-metal transition as well as to generate

universal conductance ﬂuctuations in DMS quantum

wires. Since the spin splitting is greater than the

cyclotron energy, there are no overlapping Landau

levels in modulation-doped heterostructures of DMS

in the quantum Hall regime. This has made it possible

to test a scaling behavior of wave functions at the

center of Landau levels. At the same time, it has been

conﬁrmed that in the presence of a strong spin-orbit

coupling (e.g., in the case of p-type wave functions)

the spin polarization can generate a large extraordi-

nary (anomalous) Hall voltage. Finally, optically and

electrically controlled spin-injection has been observed

in all-semiconductor structures containing DMS.

4.2 Spin-disorder Scattering

Spatial ﬂuctuations of magnetization, disregarded in

the mean-ﬁeld approximation, lead-to-spin disorder

scattering. According to the ﬂuctuation-dissipation

theorem, the corresponding scattering rate in the

paramagnetic phase is proportional to Tw(T), where w

is the magnetic susceptibility of the localized spins.

Except in the vicinity of ferromagnetic phase transi-

tions, a direct contribution of spin-disorder scattering

to momentum relaxation turns out to be small. In

contrast, this scattering mechanism controls the spin

lifetime of effective mass carriers in DMS, as evi-

denced by studies of universal conductance ﬂuctua-

tions, line-width of spin-ﬂip Raman scattering, and

time-resolved magneto-optics. Furthermore, thermo-

dynamic ﬂuctuations contribute to the temperature

dependence of the bandgap and band off-set. In the

case when the total potential introduced by a mag-

netic ion is greater than the width of the carrier band,

the virtual crystal and molecular ﬁeld approxima-

tions break down, a case of the holes in Cd

1x

Various nonperturbative schemes have been devel-

oped to describe nonlinear dependencies of the band

gap on x and of the spin splitting on M observed in

such situations.

4.3 Magnetic Polarons

Bound magnetic polaron (BMP), that is a bubble of

spins ordered ferromagnetically by the exchange in-

teraction with an effective mass carrier in a localized

state, modiﬁes optical, transport, and thermodynam-

ic properties of DMS. BMP is formed inside the Bohr

radius of the occupied impurity state but also around

a trapped exciton, as the polaron formation time is

typically shorter than the exciton lifetime. The BMP

binding energy is proportional to the magnitude of

local magnetization, which is built up by two effects:

the molecular ﬁeld of the localized carrier and ther-

modynamic ﬂuctuations of magnetization. The latter

enlarges width of optical lines, particularly in the case

of quantum dots and wires, in which self-trapped

magnetic polaron is stable even without pre-localiz-

ing potential. In contrast, a free magnetic polaron—a

117

Dilute Magnetic Semiconductors

delocalized carrier accompanied by a traveling cloud

of polarized spins—is expected to exist only in mag-

netically ordered phases. This is because coherent

tunneling of quasi-particles dressed by spin polariza-

tion is hampered, in disordered magnetic systems, by

a small quantum overlap between magnetizations in

neighboring space regions.

5. Exchange Interactions Between Localized Spins

As in most magnetic materials, classical dipole–dipole

interactions between magnetic moments are weaker

than exchange couplings in DMS. Direct d–d or f–f

exchange interactions, known from properties of

magnetic dimmers, are thought to be less important

than indirect exchange channels. The latter involve a

transfer of magnetic information via spin polariza-

tion of bands, which is produced by the exchange

interaction or spin-dependent hybridization of mag-

netic impurity and band states. If magnetic orbitals

are involved in the polarization process, the mecha-

nism is known as superexchange, which is merely an-

tiferromagnetic and dominates, except for p-type

DMS. If fully occupied band states are polarized by

the sp–d exchange interaction, the resulting indirect

d–d coupling is known as the Bloembergen–Rowland

mechanism. In the case of Rudermann–Kittel–Kasuya–

Yosida (RKKY) interaction, the d–d coupling pro-

ceeds via spin polarization of partly ﬁlled bands, that

is by free carriers.

Since in DMS carrier concentrations are usually

much smaller than those of localized spins, the en-

ergetics of the latter can be treated in the continuous

medium approximation, an approach referred to as

the Zener model. Within this model the RKKY in-

teraction is ferromagnetic, and particularly strong in

p-type materials, because of large magnitudes of hole

mass and exchange integral b. Finally, in the case of

systems in which magnetic ions in different charge

states coexist, hopping of an electron between mag-

netic orbitals of neighboring ions in differing charge

states tends to order them ferromagnetically. This

mechanism, dubbed the double exchange, can oper-

ate in, for example, n-type (Zn,Co)O.

In general, the bilinear part of the interaction

Hamiltonian for a pair of spins i and j is described by

a tensor J,

¼2S

ð4Þ

which in the case of the coupling between nearest

neighbor cation sites in the unperturbed zinc-blende

lattice contains four independent components. Thus,

in addition to the scalar Heisenberg-type coupling,

¼2J

, there are nonscalar terms (e.g., Dzialo-

shinskii–Moriya or pseudo-dipole). These terms are

induced by the spin-orbit interaction within the mag-

netic ions or within nonmagnetic atoms mediating the

spin–spin exchange. The nonscalar terms, while small-

er than the scalar ones, control spin-coherence time

and magnetic anisotropy. Typically, J

E1 meV for

nearest-neighbor pairs coupled by the superexchange,

and the interaction strength decays fast with the pair

distance.

Thus, with lowering temperature more and more

distant pairs become magnetically neutral, S

tot

¼0.

Accordingly, the temperature dependence of mag-

netic susceptibility assumes a modiﬁed Curie form,

wðTÞ¼C=T

, where go1 and both C and g depend

on the content of the magnetic constituent x. Simi-

larly, the ﬁeld dependence of magnetization is con-

veniently parameterized by the Brillouin function B

ðT; HÞ¼Sg m

eff

½Sgm

H=k

ðT þ T

Þ ð5Þ

in which two x- and T-dependent empirical param-

eters, x

eff

ox and T

40, describe the presence of

antiferromagnetic interactions.

6. Magnetic Collective Phenomena

In addition to magnetic and neutron techniques, a

variety of optical and transport methods, including

subpicosecond spectroscopy and 1/f noise study of

nanostructures, have successfully been employed to

characterize collective spin phenomena in DMS.

Undoped DMS belong to a rare class of systems,

in which spin-glass freezing is driven by purely

antiferromagnetic interactions, an effect of spin frus-

tration inherent to the randomly occupied f.c.c. sub-

lattice. Typically, in II–VI DMS, the spin-glass

freezing temperature T

increases from 0.1 K for

x ¼0.05 to 20 K at x ¼0.5 according to T

, where

dE6, which reﬂects a short-range character of the

superexchange. For x approaching one, antiferro-

magnetic type III ordering develops, according to

neutron studies. Here, strain imposed by the sub-

strate material—the strain engineering—can serve to

select domain orientations as well as to produce spiral

structures with a tailored period.

Particularly important is the carrier-density con-

trolled ferromagnetism of bulk and modulation-

doped p-type DMS. The observed values of the

Curie temperature T

are below 5 K in Zn

1x

Te:N but in the case of Ga

1x

As, in which man-

ganese supplies both spins and holes, T

reaches

110 K for x ¼0.05. At the same time, strain engi-

neering makes it possible to design the direction of

the easy axis. In the case of p-type quantum struc-

tures, the hole concentration, and thus the magnetic

phase, can be changed isothermally, by light or gate

voltage. Furthermore, observations of an exchange

coupling via nonmagnetic layers have been followed

by demonstrations of giant magnetoresistance

(GMR) and tunneling magnetoresistance (TMR)

effects in ferromagnetic III–V DMS.

118

Dilute Magnetic Semiconductors

7. Concluding Remarks

In addition to being a valuable playground in which

to examine various novel phenomena, DMS have al-

ready been commercialized as Faraday effect-based

optical insulators as well as intracenter light emitters.

Future studies will certainly attempt to explore a

great potential of ferromagnetic semiconductors and

their quantum structures. This requires preparation

of ferromagnetic semiconductors with T

above

room temperature, a feasible task according to some

theoretical predictions. Indeed, promising results on

above 300 K in chalcopirite (Cd,Mn,Ge)P

and

wurzite (Zn,Co)O:Ga have been reported.

Properties of short period ferrimagnetic superlat-

tices containing two different magnetic elements,

such as Mn and Co, but perhaps also different rare

earths or actinides, are to be explored. Other possible

functional materials are hybrid structures of semi-

conductors and ferromagnetic metals, the latter in the

form of overgrowth or inserted layers or inclusions.

Magnetoresistance over 10 000% that has been found

at room temperature in such a granular system con-

stitutes a good example of the many opportunities

ahead.

See also: Giant Magnetoresistance; Spintronics in

Semiconductor Nanostructures

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Institute of Physics, Warsaw, Poland

DNA Microarrays using Magnetic

Labeling and Detection

There are myriad applications for biosensors—from

antiterrorism, to food safety, to point-of-care clinical

diagnostics. The goal is to create systems that mimic

the sensitive and specific sensing abilities that come

effortlessly to naturally occurring organisms. A com-

mon approach to detecting biological molecules is to

attach to the target molecule a chemical label that

produces an externally observable signal. Tradition-

ally, this is accomplished using biomolecular recog-

nition between the target molecule and a specific

receptor (e.g., an antibody) that incorporates a label

such as a radioisotope, enzyme, or ﬂuorescent mol-

ecule. Sensing methods have been developed based

on a range of transduction mechanisms, including

optical, electrical, electrochemical, thermal, and pie-

zoelectrical means, which are covered eloquently

by numerous reviews (see, e.g., Pearson et al. 2000,

Vo-Dinh and Cullum 2000).

Recently, magnetic particles have been developed

as labels for biosensing. Magnetic labels have several

potential advantages over other labels. They are not

subject to degradation over time or photobleaching,

making their properties very stable. In addition,

magnetic ﬁelds are not attenuated by biological ma-

terial, offering the possibility of in vivo sensing. Mag-

netism may also be used to remotely manipulate the

particles. Finally, there are a number of sensitive

magnetic ﬁeld detection methods arising from ad-

vances in microelectronics and magnetic storage

technologies that will enable the design of compact,

low-power sensor systems. Recently described detec-

tion methods include the Maxwell bridge (Kriz et al.

1996), micro-cantilever-based force ampliﬁed biolog-

ical sensor (Baselt et al. 1997), superconducting

quantum interference device (SQUID) (Ko

titz et al.

1997), giant magnetoresistive bead array counter

(Baselt et al. 1998), frequency-dependent magneto-

meter (Richardson et al. 2001), and silicon Hall

sensor (Besse et al. 2002).

119

DNA Microarrays using Magnetic Labeling and Detection