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Magneto-Optic Spectroscopy of Magnetic Thin Films: Influence of Structural Properties on Magnetic Ordering, Exchange, and Coercivity Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der Rheinisch-Westfälischen Technischen Hochschule Aachen genehmigte Habilitationsschrift zur Erlangung der venia legendi vorgelegt von Dr. sc. nat. Paul Fumagalli aus Jona (St. Gallen, Schweiz) Referenten: Prof. Dr. Gernot Güntherodt Prof. Dr. Joachim Schoenes (Technische Universität Braunschweig) Prof. Dr. Richard J. Gambino (State University of New York at Stony Brook) Tag der Habilitation: 7. Mai 1997

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Magneto-Optic Spectroscopy of Magnetic Thin Films:

Influence of Structural Properties on Magnetic Ordering,

Exchange, and Coercivity

Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der

Rheinisch-Westfälischen Technischen Hochschule Aachen genehmigte

Habilitationsschrift zur Erlangung der venia legendi

vorgelegt von

Dr. sc. nat. Paul Fumagalli

aus

Jona (St. Gallen, Schweiz)

Referenten: Prof. Dr. Gernot Güntherodt

Prof. Dr. Joachim Schoenes (Technische Universität Braunschweig)

Prof. Dr. Richard J. Gambino (State University of New York at Stony Brook)

Tag der Habilitation: 7. Mai 1997

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TABLE OF CONTENTS I

TABLE OF CONTENTS

SUMMARY ABSTRACT ............................................................................................ III

1 INTRODUCTION...................................................................................................... 1

2 THEORY OF MAGNETO-OPTICS............................................................................. 4

2.1 General remarks about sign convention in magneto-optics............................................4

2.2 Atomic model of dispersion (Lorentz)...............................................................................8

2.3 Origin of the magnetic-field-induced anisotropy ...........................................................10

2.4 Quantum-mechanical description of interband transitions..........................................11

2.5 Sum rules for interband transitions ................................................................................15

2.6 Quantum-mechanical description of intraband transitions..........................................16

2.7 Relation between the experimental quantities and the off-diagonal conductivity......21

2.8 Optical-enhancement effects ............................................................................................232.8.1 Interface effect .........................................................................................................232.8.2 Interference effect ....................................................................................................242.8.3 Plasma-edge enhancement ......................................................................................24

3 EXPERIMENTAL................................................................................................... 25

3.1 Principles of Kerr-effect measurements..........................................................................25

3.2 Description of a Kerr spectrometer with high resolution..............................................27

3.3 Mathematical evaluation of the beam path ....................................................................30

4 (In, Mn)As HETEROSTRUCTURES...................................................................... 33

4.1 (In, Mn)As/GaAs(100) diluted magnetic semiconductors .............................................334.1.1 Introduction..............................................................................................................334.1.2 Growth conditions of (In, Mn)As/GaAs(100) films................................................334.1.3 Optical and magneto-optic properties of (In, Mn)As/GaAs(100) films .................35

4.2 (In, Mn)As/(Ga, Al)Sb heterostructures .........................................................................374.2.1 General properties of (In, Mn)As/(Ga, Al)Sb heterostructures .............................374.2.2 Optical results (reflectivity)......................................................................................404.2.3 Magneto-optic results...............................................................................................434.2.4 Correlation between reflectivity and Kerr spectra ..................................................484.2.5 Thickness and interlayer dependence of ferromagnetic order...............................50

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II TABLE OF CONTENTS

5 Co(EuS) MACROSCOPIC FERRIMAGNETS ......................................................... 53

5.1 Introduction.......................................................................................................................53

5.2 Low-temperature properties (antiferromagnetic exchange).........................................57

5.3 Temperature dependence of the magneto-optic properties...........................................61

5.4 Room-temperature properties (optical-enhancement effect) ........................................66

6 MnBi-BASED MULTILAYERS.............................................................................. 77

6.1 Introduction.......................................................................................................................77

6.2 General properties of the MnBi system ..........................................................................78

6.3 Preparation of MnBi(Al, Pt) thin films ...........................................................................81

6.4 Structural and magnetic properties of MnBi thin films................................................82

6.5 Structural and magnetic properties of MnBiAl thin films............................................88

6.6 Structural and magnetic properties of MnBiPt thin films ............................................95

6.7 Magneto-optic properties of MnBi(Al, Pt) thin films ....................................................996.7.1 MnBi and MnBiAl thin films ..................................................................................996.7.2 MnBiPt thin films ..................................................................................................101

7 CONCLUSIONS AND OUTLOOK .......................................................................... 105

ACKNOWLEDGMENTS............................................................................................ 107

REFERENCES ......................................................................................................... 109

CURRICULUM VITAE ............................................................................................. 117

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ABSTRACT III

SUMMARY ABSTRACT

In this work, the technique of magneto-optic spectroscopy is thoroughly reviewed and its

suitability for the investigation of the interplay between structural, magnetic, and magneto-optic

properties of thin films is elucidated. This is accomplished by discussing three classes of thin-

film materials and showing how their specific properties are established by applying magneto-

optic spectroscopy in combination with structural characterization.

Because an introductory textbook on the principles of magneto-optics is still lacking, a

comprehensive review of the classical atomic models in combination with their relation to

quantum mechanics is given. Although this has been done at various places in the literature and

in previous reviews, focusing on the magneto-optic properties of specific material classes, there

is no publication where the theory of magneto-optics has been discussed on a level where the

general reader can follow the argumentation easily.

A major incentive of this work is to establish a correct definition of the sign convention in

magneto-optics. Due to historical reasons, there exists a large confusion about the correct

relations between the experimental quantities, as, e. g., Kerr rotation and ellipticity, and the

theoretical quantities, as, e. g., the optical conductivity tensor. The problem arises principally

with the choice of the sign of the time dependence of electromagnetic waves. There is a

historical discrepancy in assigning the angular momentum to circularly polarized eigenstates of

electromagnetic waves in classical optics as opposed to quantum mechanics which is seldom

recognized. Starting from the atomic Lorentz model, the consistency of Faraday’s original sign

convention with the atomic model is shown when using the classical definition of angular

momentum. The sign of the Kerr rotation can then be defined independent from the Faraday

rotation by referring to early work on the magnetic transition metals. Having fixed the two sign

conventions and the time dependence of the electromagnetic waves, there are no more

parameters to be chosen. However, it turns out that there is no publication so far, that is

consistent entirely. People usually get around this problem by adjusting the signs of the

experimental data until it fits previously published data. It is demonstrated that in order to keep

the widely used relations between Kerr rotation and optical-conductivity tensor, the only way to

be consistent with all sign conventions would be to alter the definition of the sign of the off-

diagonal elements of the optical conductivity tensor. In addition, a practical help on how to

perform a correct sign calibration of an experimental Kerr spectrometer is given.

In the experimental section, a high-resolution Kerr spectrometer is introduced and the

technique of polarization modulation as well as the actual data-acquisition procedure discussed.

A comparison with other techniques is briefly addressed.

The final three chapters of this work are dedicated to the discussion of three specific classes

of thin-film materials, III-V diluted-magnetic-semiconductor heterostructures, phase-separated

macroscopic ferrimagnets, and MnBi-based multilayers. These systems are chosen in order to

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IV ABSTRACT

show the capability of magneto-optic spectroscopy in various materials. Moreover, there are

some common features linking all materials.

Starting with the (In, Mn)As heterostructures, the behavior of magnetic impurities having

local magnetic moments is studied. The Mn forms local subnanometer-sized MnAs clusters.

Nevertheless, the (In, Mn)As heterostructures grown at a substrate temperature of 200° C are

homogeneous, p-type, and show ferromagnetic order up to 50 K with strong perpendicular

magnetic anisotropy. With the help of magneto-optic spectroscopy, it is revealed that coercivity

and perpendicular magnetic anisotropy depend on the lattice mismatch between the top (In,

Mn)As layer and the adjacent (Ga, Al)Sb layer. In addition, a previously proposed hypothesis

on magnetic exchange across the (In, Mn)As/(Ga, Al)Sb interface is refuted.

By changing the growth conditions, the MnAs clusters grow until forming a macroscopic

ferromagnetic phase within the semiconducting host material, InAs. The effect of ferromagnetic

clusters of several nanometers in size is studied in the second material, Co(EuS). Here, a

macroscopic, ferromagnetic EuS phase, consisting of approximately 10 nm sized crystalline

precipitate particles, strongly exchange couples to the ferromagnetic host material, Co. With the

help of magneto-optic spectroscopy, the exchange coupling is found to be antiferromagnetic at

low temperatures leading to a profound enhancement of the ordering temperature of the EuS

phase by an order of magnitude, i. e., from 16.5 K to 160 K. The origin of the exchange

coupling is explained by the large surface-to-volume ratio of the phase-separated system

yielding a large interaction interface between Co and Eu while retaining in the EuS phase the

divalent valence of the Eu ion, which is necessary for a high magnetic moment. In addition, this

system exhibits an extraordinary optical-enhancement effect at room temperature due to an

accidental matching of the complex index of refraction of EuS and Co in the ultraviolet part of

the spectrum. This leads to polar Kerr rotations of up to 2° at 4.5 eV.

The last materials to be discussed are MnBi-based multilayers which are deposited in

successive sequences of Bi/Mn bilayers and then annealed to form MnBi. MnBi has, on one

hand, the same NiAs-type structure as the local MnAs clusters in the (In, Mn)As

heterostructures. Therefore, MnBi represents the limiting case of an infinitely large cluster, i. e.,

bulk material. On the other hand, by depositing as Bi/Mn bilayers and by adding metallic

interlayers between subsequent Bi/Mn bilayers an artificial separation of individual Bi/Mn

bilayers is achieved, strongly influencing the structural, magnetic and magneto-optic properties

of the material. In pure MnBi films, an enhancement of the surface roughness correlated with a

strong decrease in coercivity is found when increasing the number of Bi/Mn bilayers, i. e., the

overall film thickness. The coercivity of bare MnBi films is five times larger than that of MnBi

films covered with a protective SiOx layer as reported in previous work. Adding 0.3 nm thin Pt

interlayers reduces drastically the surface roughness of the films although the Pt interdiffuses

completely with the Bi/Mn bilayers. From the dependence of coercivity as a function of Pt-

interlayer thickness, it is concluded that the MnBi grain size is effectively reduced by the Pt.

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ABSTRACT V

Adding a 1.5 nm thick Al interlayer leads to a separation of individual Bi/Mn bilayers even

after annealing. As a result, the film consists of individual, magnetically decoupled MnBi layers

separated by a partially oxidized Al interlayer. As shown by magneto-optic magnetometry, this

leads to a smaller coercivity of the inner MnBi layer as compared to the outer one, similar to the

effect observed when adding a protective SiOx layer. This can be explained by a 2.3% volume

increase of the NiAs-type MnBi unit cell as compared to the as-grown Bi/Mn bilayer inducing

internal strain during MnBi formation. In the case of a rigid Al2O3 interlayer or a protective

SiOx layer, the growth is restricted perpendicular to the film surface and the strain cannot be

relieved. This will lead to a pronounced lateral extension and to a better interconnection of the

MnBi grains reducing the wall energy at the grain boundary. Both effects reduce coercivity as

seen in the experiment. The magneto-optic properties are virtually unchanged by adding Al

interlayers while adding a Pt interlayer strongly reduces the magnetization and hence the

magneto-optic effects. The largest Kerr rotations are found in 90 nm thick MnBi films

consisting of at least three Bi/Mn bilayers indicating that a certain film thickness is needed in

order to achieve complete MnBi formation and a density of bulk material in the innermost

Bi/Mn bilayer.

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VI ABSTRACT

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1 INTRODUCTION 1

1 INTRODUCTION

One hundred and fifty years ago, M. Faraday discovered in 1846 the first magneto-optic

effect [1]. While looking for an interaction between magnetic fields and light, he observed a

rotation of the plane of polarization after passing light through a glass rod within a magnetic

field. It took another thirty years until J. Kerr was able to establish 1876 a similar effect in

reflecting polarized light off a polished surface of an iron mirror [2]. The foundation for a

theoretical understanding of magneto-optic effects was laid in 1896 by Zeeman’s discovery of

the magnetic-field-induced splitting of spectral lines in gases [3]. First quantitative descriptions

of magneto-optic effects were developed after the formulation of J. C. Maxwell’s theory of

electrodynamics [4] and the theory of dispersion by H. A. Lorentz [5] and P. Drude [6]. In the

early fifties, the first resonant magneto-optic effect, the cyclotron resonance, was found [7] and

with the advancement of computers the first band-structure calculations were performed. The

discovery of ferromagnetic semiconductors with large magneto-optic effects in the sixties [8]

boosted magneto-optic spectroscopy to a widely used technique and initiated the idea of an

application in data-storage technology. With the discovery of ferromagnetic amorphous thin

films in the early seventies [9] a suitable magneto-optic media has been established and is

utilized today in the first magneto-optic disk drives which appeared on the market just five

years ago.

Despite the one and a half centuries since the discovery of the Faraday effect, magneto-

optics is a thriving field in physics. With the growing importance of magnetic thin films,

magneto-optic magnetometry has become a standard procedure in determining magnetic

properties with submonolayer resolution of in-situ grown single layers and multilayered

structures. As will be reviewed in this work, by taking advantage of spectroscopic information,

magneto-optics can be an element-specific or layer-specific method. It is therefore perfectly

suited to investigate the interplay between structural, magnetic, and magneto-optic properties in

disordered or phase-separated materials as well as in artificially layered structures.

With the progress of using synchrotron facilities in solid state physics as a versatile source of

electromagnetic radiation, a truly element-specific magneto-optic method is now available

through x-ray magnetic circular dichroism (XMCD) [10-13]. In XMCD core levels (mostly K

and L shell) of electrons are excited and, consequently, element specificity is granted by a large

energy separation of the core levels in different elements. In view of the limited beam time and

the elaborate experimental set-up for XMCD measurements, magneto-optic spectroscopy yields

sufficient information in terms of element specificity in various systems.

As was mentioned before, magneto-optic magnetometry is a very sensitive method in

determining magnetic properties of ultra-thin films. In this context, the acronym SMOKE,

surface magneto-optic Kerr effect, is often used. This is nonsense because the penetration depth

of the light is given by the absorption constant, K, which is in metals of the order of

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2 1 INTRODUCTION

105-106 cm--1. Therefore, the penetration depth amounts to 10-100 nm which is by no means

limited to the surface. What makes people think of a surface-sensitive effect, though, is the

monolayer sensitivity of magneto-optics.

However, during the past few years there has been a new development of a truly surface - or

interface - sensitive magneto-optic technique by using second harmonic generation [14, 15].

The nonlinear magneto-optic Kerr effect (NOLIMOKE) can probe magnetism at the interface

because the broken symmetry yields a nonvanishing second harmonic polarizability [16, 17].

For very intense electric fields, the Kerr rotation at twice the frequency of the incident laser

beam can be determined. It has been predicted theoretically [18] and verified

experimentally [19] that NOLIMOKE exhibits rotations which are one order of magnitudes

larger than the corresponding linear magneto-optic effects. Although this seems to be a very

promising new technique, the experimental set-up is quite involved making this method

accessible only to few people.

Another recent development in magneto-optics is the advance towards higher spatial

resolution by using near-field optical methods [20]. By scanning a nanometer-sized aperture in

close proximity across the surface, i. e., in the near-field regime, a lateral resolution of down to

10 nm has been confirmed [21] which is far beyond the diffraction limit of the laser light used

to illuminate the aperture. By employing polarized light, magnetic contrast can be imaged with

similar resolution providing the possibility of applying magneto-optics on a nanometer scale

[22].

Regarding theoretical calculations of magneto-optic effects, from the beginning of band-

structure calculations in the fifties until recently there has been just a small progress. This is due

to the fact that treating magneto-optics theoretically is very complex as one has to take into

account spin polarization, relativistic effects and spin-orbit coupling. With increasing

computational speed of modern computers and applying the density-functional formalism

[23, 24], an enormous improvement has been achieved within the last five years. Using the

local spin-density approximation (LSDA) [25] in combination with the augmented spherical-

wave (ASW) technique [26] or with the linear muffin-tin orbital (LMTO) method [27, 28], it is

now possible to calculate accurately from first principles the Kerr effect of the ferromagnetic

transition elements Fe, Co Ni and related alloys [29-31] and even of more complex systems like

the Heusler compound PtMnSb [32] and MnBi [33] which we will discuss in section 6. First

successful attempts are made for actinides [34, 35] and rare-earth compounds [36] which

manifest the difficulty of treating the coexistence of localized magnetic moments with itinerant

electronic states.

This review will reveal how magneto-optic spectroscopy can be used as a layer-sensitive and

element-specific method to investigate the interplay between structural, magnetic, and magneto-

optic properties in phase-separated materials as well as in artificially layered structures.

Experimental results of three classes of materials will be surveyed: (i) III-V diluted magnetic

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1 INTRODUCTION 3

semiconductor heterostructures as an example of a system based on GaAs containing localized

magnetic moments, (ii) phase-separated macroscopic ferrimagnets as an example of a granular

material where due to the phase separation a large surface-to-volume ratio is induced leading to

a strong magnetic exchange, and (iii) MnBi, a classical ferromagnetic system prepared by a

multilayer technique with a high technological potential as a magneto-optic data-storage

medium for blue recording.

The paper is organized as follows. In section 0, the theory of magneto-optics will be

discussed with an emphasis on the polar Kerr effect. A discussion on the sign convention and

the assignment of selection rules to the polarization state is included as well as a section on

optical-enhancement effects which are very important for interpreting polar Kerr measurements.

In section 3, a standard experimental set-up for wavelength-dependent measurements will be

described in detail and compared to alternative techniques. Sections 4 through 6 contain the

experimental results and discussion on the three classes of materials mentioned before. Each

chapter includes an introduction about the general properties and technological relevance of the

materials.

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4 2 THEORY OF MAGNETO-OPTICS

2 THEORY OF MAGNETO-OPTICS

2.1 General remarks about sign convention in magneto-optics

Before the theory of magneto-optics is outlined, a few words about the sign convention in

magneto-optics are needed. There has been, and still is, a lot of confusion about the definition

of the correct sign of the Kerr rotation and, consequently, its relation to the conductivity tensor.

Another confusion results from an equivocal assignment of selection rules of dipole transitions

to the circular polarization states of the light. Both problems emanate from the choice of the

sign of the time dependence, e i t± ω , of the electromagnetic waves.

Let us first address the assignment of angular momentum to a photon in a quantum-

mechanical picture. In quantum-mechanics textbooks [37] the choice of the sign of the time

dependence is commonly negative. Therefore, the electric-field vector of an electromagnetic

plane wave is described in a complex notation as

( ) ( ) r r

E t E e i g r tω ω, = ⋅ −0 , (2.1)

where g is the wave vector and ω the angular frequency. It has to be remembered at this point

that one should take the real part of the complex field in order to get the real, i. e., physical,

electric field. The imaginary part has no direct physical meaning. Assuming g along the

positive z axis, circularly polarized light is accordingly defined as

( ) ( ) ( ) r r

E t E e e ex yg r t

±⋅ −= ±ω ω, 1

2 0 i i , (2.2)

where ei is a unit vector. Let us define for the moment ‘+’ as pseudo-right circularly-polarized

(rcp*) and ‘-’ as pseudo-left circularly polarized (lcp*) light. In that case, rcp* light describes a

counterclockwise rotation as a function of time at a fixed location in space when looking into

the beam and a left-handed helix in space at a fixed time. Reducing Eq. (2.2) to Jones vectors

yields

, e e+ −=+

=

12

12

1 1

i i , (2.3)

which is a well-known vector representation of circularly polarized light. However, it has to be

kept in mind that Eq. (2.3) is just a short form of Eq. (2.2) and implicitly contains a spatial as

well as a time dependence.

The standard procedure in quantum mechanics is now to assign a state vector to rcp* and

lcp* light, which will form an orthonormal basis,

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2 THEORY OF MAGNETO-OPTICS 5

R L=+

=

12

12

1 1

i i, . (2.4)

In quantum mechanics, this represents the eigenstates of the photon. Hence, any state ψ can

be expressed by R and L as

ψψ

ψ

ψ ψ ψ ψ=

=

−+

+x

y

x y x yR Li i

2 2 . (2.5)

Acting with the photon spin operator, S, where [37]

S =−

0

0

i

i , (2.6)

on R and L yields the eigenvalues +1 and -1, respectively. By multiplying these eigenvalues

with we get the component of the angular momentum, Lγ = ±, of the photon along the

propagation direction.

On the other hand, the angular momentum, L , of a classical electromagnetic fields is

defined as [37, 38]

( ) ( )[ ] L

cd r r E r t H r t= × ×∫

1

43

π, , . (2.7)

For an infinite plane wave, L has no component in the direction of wave propagation (z axis)

but for a finite plane wave Eq. (2.7) reduces to

( ) ( )[ ] ( ) ( ) ( )[ ] L

cd r E r t A r t

cd r E r t r A r ti i

i

= × + ×∇∫ ∑∫1

4

1

43 3

π π, , , , , (2.8)

where ( ) A r t, is the transverse vector potential. The z component of the second term on the

right can be neglected because only the surface of the beam contributes in contrast to the first

term, which is proportional to the volume. Consequently, by inserting plane waves as defined in

Eq. (2.1) we get [37]

LV E E E E

zneg x y x y=

−−

+

8 2 2

2 2

πωi i

. (2.9)

By comparing with Eq. (2.5), the first term on the right is the component of the rcp* state, R ,

and the second one that of the lcp* state, L . Therefore, we can identify the classical angular

momentum with its quantum mechanical equivalent

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6 2 THEORY OF MAGNETO-OPTICS

( )LV

E Ezneg x y x y= − =

−−

+

+ −8 2 2

2 22 2

πωψ ψ ψ ψ

i i

. (2.10)

Summarizing, rcp* (lcp*) light as defined in Eq. (2.2), corresponds to the quantum mechanical

eigenstate R ( L ) of the photon with an angular momentum of + (-). So, a negative time

dependence of plane waves seems to be the natural choice which is consistent with the

definition of photon spin in quantum mechanics.

If we define the plane waves with a positive time dependence, i. e., rcp (‘+’) and lcp (‘-’)

light as

( ) ( ) ( ) r r

E t E e e ex yt g r

±− ⋅= ±ω ω, 1

2 0 i i , (2.11)

Eq. (2.7) will yield

LV E E E E

zpos x y x y=

+−

8 2 2

2 2

πωi i

, (2.12)

i. e., the classical angular momentum of rcp and lcp light is now - and +, respectively,

corresponding to the quantum mechanical eigenstates L and R of the photon. The confusion

arises from the Jones vectors for rcp and lcp light, which take the same form as for rcp* and

lcp*, respectively (see Eq. (2.3)). However, the identification with the formally identical

quantum mechanical state vectors R and L of Eq. (2.4) is now reversed. This has also some

consequences for the selection rules of dipole transitions. In the case of an electronic interband

transition through photon absorption, the selection rule for the azimuthal quantum number, m,

is ∆m = ±1 for a photon spin of ±. As a result, ∆m = +1 (-1) for lcp (rcp) light in the case of a

positive time dependence of plane waves and vice-versa for a negative time dependence.

It has already been pointed out by Jackson [38] that the whole confusion has an historical

origin. In classical optics, rcp light is defined at a fixed point in space by a clockwise rotation of

the electric field vector as a function of time looking into the source of the beam. This is

equivalent with the definition of Eq. (2.11). As a consequence, the angular momentum of rcp

light is indeed negative and corresponds to a negative spin in quantum mechanics (which is

called ‘negative helicity’ by Jackson [38]). So, we are left with the wrong sign of the angular

momentum as compared to quantum mechanics although the Jones-vector representation is

identical. Unfortunately, literature utilizes both definitions. However, some recent reviews on

magneto-optic properties [39, 40] use a positive time dependence. So, it is helpful to be aware

of the problem and we will use the classical definition of rcp and lcp light according to Eq.

(2.11) throughout this work.

As the last item of this chapter, the sign convention in magneto-optics is discussed. For the

Faraday rotation, θF, M. Faraday himself [1] has given a definition by observing that a glass rod

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2 THEORY OF MAGNETO-OPTICS 7

was rotating the plane of polarization in the direction of the positive current generating the

magnetic field. So, he defined a (Faraday) rotation in this direction as positive. Therefore, glass

has by definition a positive Faraday rotation. The polar Kerr rotation, θK, on the other hand, can

be independently defined. Regarding early measurements on the ferromagnetic transition

elements Fe, Co, and Ni [41], the polar Kerr rotation of Fe (as well as Co and Ni) is defined to

be negative in the visible part of the spectrum, in accordance with recent reviews [39, 40]. The

Faraday rotation of Fe is then predetermined to be positive in that photon energy range. This

definition leads to the fact that a positive polar Kerr rotation is also parallel to the direction of

the positive current producing the magnetic-field. This relationship is visualized in Fig. 2.1.

+θF

positiveellipticity

ηF

E→

B→

ω

E→

positiveellipticity

E→

+θK

ηK

rcp

+

lcp_

Fig. 2.1. Sign convention in magneto-optics.

The experimentalist, who wants to calibrate his magneto-optic set-up and does not have the

possibility to measure a standard Fe film, should proceed as follows. Defining the correct sign

for the Faraday rotation is easy as a piece of glass is available anywhere. By definition its

Faraday rotation is positive. For defining the correct sign of the polar Kerr rotation, a piece of

glass can be useful, too. By placing it in front of a mirror at the location of the sample, it will

produce a double Faraday rotation. This can be interpreted as a pseudo-Kerr rotation and should

yield a positive rotation for a correct definition.

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8 2 THEORY OF MAGNETO-OPTICS

2.2 Atomic model of dispersion (Lorentz)

H. A. Lorentz developed a simple atomic theory at the end of the last century with the aim of

explaining the optical dispersion near absorption lines [5]. His key assumption was that the

electrons are bound to an equilibrium position by Hook’s law leading to a harmonic oscillation.

Adding a friction term, γ r , a periodic electric field, E =

E0

( )e t g zi ω − , with a positive time

dependence (see chapter 0), and the Lorentz force caused by an external static magnetic field,H , the equation of motion for a bound electron is

( ) r r re

mE e r et g z

c z+ + = + ×−γ ω ωω02

0i . (2.13)

Note that the charge, e, is a negative number for electrons, ez is a unit vector in z direction, and

m is the free-electron mass. In Eq. (2.13), the magnetic field is, without loss of generality,

assumed to be along the z axis, i. e., H = (0, 0, H). In Faraday or polar Kerr geometry, the

wave vector g = (0, 0, g) is parallel to

H . Thus, the electric field is in the x-y plane, i. e.,

E0 =

(Ex, Ey, 0). In addition, we have defined the resonance frequency ω0 as

ω02 =

f

m , (2.14)

with f being the force constant, and the cyclotron frequency ωc as

ω c

e H

m c= . (2.15)

Looking at Eq. (2.13), we see immediately that the motion in z direction is purely harmonic.

The x and y components, however, are coupled through the Lorentz force. As the equation of

motion is rotationally invariant along the z axis, it can be solved by transforming into a circular

coordinate system, , , e e e+ − z (as defined in Eq. (2.11)), where

( ) ( ) , e e e e e ex y x y+ −= + = −12

12

i i ,

( ) ( ) , e e e e e ex y= + = −+ −−

+ −12 2

i . (2.16)

Inserting the ansatz

r A E± ± ±= ~

, (2.17)

where ( )E E ei

± ±−

±= 0 e t gzω and r r e± ± ±= 0 , into Eq. (2.13) yields

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2 THEORY OF MAGNETO-OPTICS 9

( )~

~ A E e

e

mE e A E e ec z± ± ± ± ± ± ± ±− + + = + ×0

202

0 0ω γ ω ω ωωi i . (2.18)

Using the identity ( ) ( ) e e e i e e i e i e i e± ±× = ± × = ± ± = ±z x y z x y , the amplitude is

obtained as

( )~A

e

m c± =

− ± +ω ω ωω ω γ02 2 i

. (2.19)

Assuming a concentration, N, of the electrons per unit volume, the total polarization, P± ,

amounts to P N er N e A E E± ± ± ± ± ±= = =~ ~α , where ~α± is the total polarizability. Finally,

the complex dielectric function, ( )~ε ω± , is for rcp (‘+’) and lcp (‘-’) light

~ ~ε πα πω ω ωω ω γ± ±= + = +

− ± +1 4 1

4 12

02 2

N e

m c i . (2.20)

By separating ~ε ε ε± ± ±= −1 2i into real and imaginary part, we get

( )ε π ω ω ωω

ω ω ωω ω γ1

202 2

02 2 2 2 2

14

± = + − ±− ± +

N e

mc

c

,

( )ε π ω γ

ω ω ωω ω γ2

2

02 2 2 2 2

4± =

− ± +N e

mc

. (2.21)

Obviously, the resonance frequency shifts by applying an external magnetic field and the

change depends on the polarization state of the light. The new (magnetic-field-dependent)

resonance frequency, ω0±, satisfies the condition

ω ω ω ω02

02

0 0− ± =± ± c . (2.22)

Using the approximation, ( ) ( ) ( )ω ω ω ω ω ω ω0 0 0 0 0 0 02± ± ± ±− + ≅ − , we obtain

ω ω ω ω0 012± − = ± = ±c L , (2.23)

where ωL is the Larmor frequency. Note, that ωc < 0 for electrons.

The Lorentz model is in accord with a positive Faraday rotation of quartz, θFq , in the region

of normal dispersion (i. e., ∂ ∂ωn > 0 ). From Eq. (2.23) it follows that for electrons

ω0+ < ω0 < ω0-. If ω ω ω0 0 0+ −− << , then the shape of the energy dispersion of the index of

refraction, n(ω) → n±(ω), is to first order unchanged. Therefore, n±(ω) is simply shifted along

the x axis by ωc. Assuming normal absorption, n+(ω) > n-(ω) and hence ( )θFq ∝ −+ −n n > 0

as defined by Faraday.

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10 2 THEORY OF MAGNETO-OPTICS

2.3 Origin of the magnetic-field-induced anisotropy

Within the framework of the Lorentz model, it can be nicely demonstrated that a magnetic-

field-induced shift of the resonance frequency in a circular basis leads to a macroscopic

anisotropy in all optical properties. As an example, let us discuss the magnetic-field-induced

anisotropy of the dielectric tensor, ~ε (neglecting the undisturbed z direction at the moment),

where

~~ ~

~ ~ε =ε ε

ε ε

xx xy

yx yy

. (2.24)

In an isotropic material, ~ε reduces to a scalar complex number. In the presence of a magnetic

field, the dielectric tensor is determined by Eq. (2.20), i. e., in a circular basis, it has a diagonal

form

~~

~ε =±

+

ε

ε

0

0(2.25)

with eigenvalues ε± . Therefore, the electric displacement is given by D E± ± ±= ~ε . In a next

step, ~ε± is transformed into the x-y basis according to Eq. (2.16). Following the standard

procedure of matrix transformation in linear algebra [42], the dielectric tensor, ~ε , in x-y basis

is obtained by

~ ~ε = εT T±−1 , (2.26)

where the rows of the transformation matrix, T, are given by the coefficients of expressing the

old (circular) basis by the new (x-y) one in Eq. (2.16). So, we find

T T-1= =1

2

1 1 1

2

1

1i i

i

i−

, . (2.27)

Inserting this into Eq. (2.26), the dielectric tensor, ~ε , in the x-y system is

( )( )

~~ ~ ~ ~

~ ~ ~ ~

~ ~

~ ~ε = 12

ε ε ε ε

ε ε ε ε

ε ε

ε ε

+ − + −

+ − + −

+ − −

− +

=

i

i

xx xy

yx xx

. (2.28)

It follows at once that ~ε has finite off-diagonal elements, ~ ~ε εyx xy= − . If we take into account

a correct sign convention which leaves the well known relations between the experimental

quantities, θK and ηK, and the off-diagonal elements of the optical conductivity tensor, ~σxy ,

unchanged (see section 2.7, Eq. (2.95)), the dielectric tensor has to be defined as

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2 THEORY OF MAGNETO-OPTICS 11

~

~ ~

~ ~

~

ε =

ε ε

ε ε

ε

xx xy

xy xx

zz

0

0

0 0

, (2.29)

where ~εxy satisfies the relation

~ ~ ~ε ε ε± = xx xy i . (2.30)

Applying Ohm’s law, ~ ~ J E N er= =σ , a similar anisotropy is obtained for the optical-

conductivity tensor, σ , and we get

~

~ ~

~ ~

~

σ =

σ σ

σ σ

σ

xx xy

xy xx

zz

0

0

0 0

, (2.31)

where the connection to the circular basis is given by

~ ~ ~σ σ σ± = xx xy i . (2.32)

This formula is in accordance to recent reviews on magneto-optics [39, 40]. In addition, we

have

~ ~ε πω

σ± ±= −14

i , (2.33)

and furthermore

~ ~ε σ= −14

ω . (2.34)

A description of the optical conductivity tensor, σ , based on symmetry arguments [43, 44],

yields that the diagonal elements, σ ii , depend only on even powers of the magnetic field or

magnetization while the off-diagonal elements, σ ij, depend purely on odd powers. To first

approximation, σ ii is a constant and σ ij is linear in the applied field or magnetization.

2.4 Quantum-mechanical description of interband transitions

Magneto-optic effects are observed in almost any material when applying an external

magnetic field. In nonmagnetic materials, they are due to the Zeeman splitting of the electronic

levels lifting the degeneracy of orbital moment and spin. Being proportional to the thickness of

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12 2 THEORY OF MAGNETO-OPTICS

the material, they are easily observed in a Faraday configuration (transmission). In polar Kerr

geometry (reflection), magneto-optic effects due to Zeeman splitting are very small and are

barely observed because they are no longer proportional to the thickness.

In magnetic materials, however, magneto-optic effects are three orders of magnitude larger

due to strong spin-orbit interactions. This is easily understood from an energetic point of view.

Spin-orbit energies are approximately ≈100-500 meV whereas a Zeeman splitting is only 0.1-

1 meV. For a correct quantum-mechanical description of magneto-optic effects in magnetic

materials, it is therefore essential to include spin-orbit interaction into the Hamiltonian [44-47].

Introducing Pauli’s spin operator, σ = 2s , where

s = ± 1

2 is the electron spin, the spin-

orbit energy, Es.o. is given by

( ) ( )Eg

m cp Vs o. . =

−⋅ × ∇

1

4 2 2σ , (2.35)

where m, g, p , and V are the free electron mass, gyromagnetic ratio, canonical momentum

operator, and central (Coulomb) potential, respectively, and c is the velocity of light. The

classic analogue to this quantum-mechanical operator is the angular momentum, L , that a

magnetic moment (spin), µ , encounters when moving at a velocity,

v , in an electric field,

E ,

( )[ ] L B v Ec= × − ×µ 1 . (2.36)

For an electron, the magnetic moment is given by

µ = e

m cg s

2 (2.37)

and E = - V1

e ∇ is the internal field resulting from the local Coulomb potential, V.

Before considering the interaction with radiation, one has to find the ground state of the

electron wave function in a static magnetic field, H A= ∇ × M , while taking into account the

spin-orbit interaction (Eq. (2.35)). In the absence of a radiation field, the time-independent

Hamiltonian, H0, is given by

( ) ( ) ( )[ ] ( ) ( )H H0 0 2

21

2

1

4= = − +

−× ∇

+ +∑ ∑i

ii M i i i i eff i

i mp

e

cA r

g

mcV r V r V r

σ .

(2.38)

Here we have assumed that the Coulomb interaction between the electrons may be replaced by

an effective one-body operator, Veff, which maintains the lattice periodicity, and that, to a good

approximation, the wave functions for electrons in the solid are products of single-electron

wave functions. Therefore, the system is described as the sum of one-body operators, H0i. The

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2 THEORY OF MAGNETO-OPTICS 13

kinetic momentum operator, πi , is defined in the Heisenberg representation by the equation of

motion,

[ ] r

mi i, H0 = i π . (2.39)

Evaluating this equation yields

( ) ( ) ( )[ ] π σi i M i i ipe

cA r

g

mcV r= − +

−× ∇

1

4 2 . (2.40)

After having switched on the radiation field, described by the (transverse) vector potential,

( ) A r tL i , , the total, time-dependent Hamiltonian, H(t), is written as

( ) ( )H Ht tii

= ∑

( ) ( ) ( )[ ] ( ) ( ) ( )= − +−

× ∇ −

+ +∑ 1

2

1

4 2

2

mp

ec

A rg

mcV r

ec

A r t V r V ri M i i i L i ii

eff i

σ , .

(2.41)

In first order perturbation theory, neglecting second order effects in ( ) A r tL i , , this can be

linearized by introducing the interaction Hamiltonian, Hint(t), and we find

H(t) = H0 + Hint(t) , (2.42)

where

( ) ( )H int ,te

m cA r ti L i

i

= − ⋅∑ π . (2.43)

This describes the interaction of radiation with an electron, which is excited from the ground

state, α , into the final state, β , in the presence of an external magnetic field. Both, ground

and final state, are eigenstates of the time-independent H0i with eigenvalues Eα and Eβ,

respectively. In what follows, we will neglect a direct interaction between the electron spin and

the magnetic field of the electromagnetic radiation as the spins are not able to follow the

radiation field at optical frequencies ω ≥ 1014 s-1 (ω ≥ 0.1 eV) [44]. Furthermore, the influence

of the magnetic field of the light on the orbital motion of the electrons is neglected as well

because it is by a factor of v/c smaller than the effect of the electric field limiting our

considerations to the nonrelativistic case.

By applying Fermi’s golden rule, time-dependent perturbation theory [44] yields the

transition probability per unit time, Wxα β→ , for plane-polarized light (visualizing, along x

direction) from an initial state, α , to the final state, β . As a result, we get

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14 2 THEORY OF MAGNETO-OPTICS

( ) ( )[ ]WE e

mx xα β

βα βαπ

ωβ π α δ ω ω δ ω ω→ =

− + +

20

22

(2.44)

and for rcp (‘+’) and lcp (‘-’) light as defined in Eq. (2.11)

( ) ( )[ ]WE e

m±→

±=

− + +α β

βα βαπ

ωβ π α δ ω ω δ ω ω

40

22

, (2.45)

where ( )ωβα β α= −1h

E E and ( )π π± ±= ± e iex y with ( )π π π± = 1

2 x y i .

In a last step, we have to derive from the transition probabilities the material properties

which are responsible for the magneto-optic activity. According to Section 0, these are the off-

diagonal elements, ~σxy , of the optical conductivity tensor, ~σ . The connection between the two

quantities can be made by considering the (time-averaged) power, P, absorbed by a solid of

volume V,

P J E dV J E dVt

= ⋅ = ⋅∫ ∫ ∗Re Re Re

12 . (2.46)

Neglecting the spatial variation of the electric field and applying Ohm’s law, J E= ~σ , we get

for the absorbed power, Px, of a plane-polarized wave (x direction)

P V Ex xx= 12 1 0

2σ (2.47)

and for P± of rcp (‘+’) and lcp (‘-’) waves

( )P V E xx xy± = ±12 0

21 2σ σ . (2.48)

Solving for the conductivity elements yields

σ102

2xx

xP

V E= (2.49)

and

σ202xy

P P

V E= −+ − . (2.50)

On the other hand, the power absorbed in an energy interval ( )[ ] ω ω ω, + d is

Pd W dunoccocc

ω ω ωα β

βα

= →∑∑::

, (2.51)

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2 THEORY OF MAGNETO-OPTICS 15

where the summation extends only over occupied initial states, α , and unoccupied final

states, β . Combining Eqs. (2.44)-(2.45) and (2.49)-(2.51), the absorptive elements of the

conductivity tensor are derived as a function of the quantum-mechanical matrix elements as

( ) ( )[ ]σ πω

β π α δ ω ω δ ω ωβα βαβα

1

2

2

2

xx xunoccocc

e

m V= − + +∑∑

::

, (2.52)

[ ] ( ) ( )[ ]σ πω

β π α β π α δ ω ω δ ω ωβα βαβα

2

2

2

2 2

4xyunoccocc

e

m V= − − + ++ −∑∑

::

. (2.53)

The dispersive elements are obtained by employing the Kramers-Kronig relations [44] as

( )σ ω β π α

ω ω ωβα βαβα2

2

2

2

2 2

2xx

x

unoccocc

e

m V=

−∑∑

::

, (2.54)

σβ π α β π α

ω ωβαβα1

2

2

2 2

2 22xyunoccocc

em V

= −−−

+ −∑∑ ::

. (2.55)

From Eqs. (2.53) and (2.55) it follows immediately, that the magneto-optic effects are

proportional to the difference in the transition probabilities of rcp and lcp light. This is one of

the most important conclusions in magneto-optics.

2.5 Sum rules for interband transitions

The elements of the conductivity tensor obey specific sum rules and high-frequency limits

which will be summarized here without proof. A derivation of the formulas based on thecommutation relations and the completeness of the set of states, α , for the one-body

Hamiltonian, H0i, is given in [44]. The diagonal elements, ~σ σ σxx xx xx= +1 2i , comply with the

same sum rules as in the case of zero magnetic field, namely

( )σ ω ω π1

0

2

2xxoccd

N e

mV

∫ = (2.56)

and

( )limω

σ ωω→∞

= −2

2

xxoccN e

m V , (2.57)

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16 2 THEORY OF MAGNETO-OPTICS

where Nocc is the number of electrons that are optically excited. The first equation states that the

area under the absorptive part of the diagonal conductivity element is proportional to the

number of electrons that are excited by the electromagnetic wave. Therefore, partially

integrating σ1xx over a spectral feature yields the number of the electrons that are excited in that

energy range. This is very helpful in assigning specific transitions to features in the optical

spectra. The second equation expresses the decrease at high frequencies which can be used as

well to derive the number of electrons that are involved in the optical transitions.

For the off-diagonal conductivity elements, similar relations are computed. In contrast to the

diagonal elements, they are dependent on the external magnetic field and on the expectationvalue of the spin operator, σz . We find

( ) ( )ω σ ω ω π πσ ∂

∂∂∂2

0

3

2

2

3 2

2

2

2

22

1

8xyocc occ

zdN e

m cVH

g N e

m c V

V

x

V

y

∫ = −−

+

( )≅ − ∗π ω σ ωN e

mVocc

c z

2

2 (2.58)

and

( ) ( )limω

σ ωω ω

σ ∂∂

∂∂→∞

= −−

+

1

3

2 2

2

3 2 2

2

2

2

2

1

4xyocc occ

z

N e

m c VH

g N e

m c V

V

x

V

y

( )≅ − ∗N e

m Vocc

c z

2

2ωω σ ω . (2.59)

The term on the right in Eq. (2.58) is the result of Eq. (2.56) times the cyclotron frequency, ωc,

less the expectation value of the electron spin, σ z , times a frequency, ω*, which depends on

the central potential, V.

2.6 Quantum-mechanical description of intraband transitions

In discussing the magneto-optic effects of intraband transitions, i. e., the conduction electrons,

we follow the treatment given in [48] based on the total current and the Boltzmann equation.

Spin-flip transitions will be neglected. In what follows, a spherical Fermi surface of the

conduction band will be assumed, leading to a parabolic wave-vector dependence of the energy,

and the electron wave functions are given in terms of Bloch states,

( ) ( )ψ r

r

r

r

k

k r

kr e u r= i , (2.60)

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2 THEORY OF MAGNETO-OPTICS 17

where ( )u rr

k

obeys the lattice periodicity. Each electron state with wave vector, k , has two spin

expectation values, k ,↑ and

k ,↓ . The electron spins are supposed to be ordered yielding a

macroscopic magnetization,

M . The effect of the spin-orbit interaction causes an electron to

have associated with it a dipole moment, ( ) P k , per unit cell, where

( ) P k A k M= × , (2.61)

which is normal to

M and to k . The spin-orbit interaction is contained in A. This relation can

be understood by rewriting the spin-orbit energy, Es.o., as a dipole energy,

( ) ( ) ( ) ( ) ( )E E Pg

m cV p

g e

m cE k A E k Ms o. . = ⋅ = −

−∇ ⋅ × = −

−⋅ × = ⋅ ×

1

4

1

42 2

2

2 2σ σ . (2.62)

Here, we have used e E V

= − ∇ ,

p k= , and

M ∝ − σ . Introducing the Fermi-Dirac

distribution function of the electrons, ( )f k

, the macroscopic dipole moment, P0 , is given by

( ) ( ) P f k P k d k0

3= ∫ . (2.63)

For a nonmagnetic solid, k ,↑ and

k ,↓ are evenly distributed with opposite dipole moment,

so there is no resultingP0 . In the case of a magnetic solid in equilibrium state,

k ,↑ and

k ,↓ are unbalanced but

k ,↑ ↓ and − ↑ ↓

k , are evenly distributed. Hence, there is still no

resultingP0 . Only if

k ,↑ and

k ,↓ as well as

k ,↑ ↓ and − ↑ ↓

k , are unevenly distributed,

there is a resulting P0 . This will lead to an induced current,

J dP dtpol = 0 and finally to a total

current, J , which is

( ) ( )

J J J e v k f k d kdP

dtcond pol

BZ

= + = +∫ 3

1

0

. .

, (2.64)

where the velocity, v , is defined as

( ) ( ) ( ) ( )[ ]

v k E k P k tk= ∇ − ⋅1 ε . (2.65)

Here, ( )E k is the Bloch energy and ( )ε t is the electric field. The simultaneous presence of

scattering, the electric field, and spin-orbit effects in Eq. (2.64) poses a problem in solving the

equation for the current. This dilemma is avoided by introducing an instantaneous electric field,

( )ε inst t , which we choose to lie along the x axis,

( ) ( )ε ε δinst xt et = 0

. (2.66)

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18 2 THEORY OF MAGNETO-OPTICS

There is no scattering during the application of ( )ε inst t as it occurs in an infinitesimal time.

Therefore, scattering effects are separated from electric-field effects. The effect of ( )ε inst t on

the electron wave vector, k , can be seen from solving the Boltzmann equation,

dk

dtf

df

dtk

⋅∇ = , (2.67)

where we have assumed that f exhibits neither an explicit time nor a spatial dependence.

Inserting the equation of motion, ( )dk dt e t

h= 1 ε inst , yields as a result that

k is shifted from its

initial value k0 by a uniform vector

k k

eex= +0

0ε . (2.68)

As a consequence, the Fermi distribution of the electrons, f, at the time, t = 0+ given by

( )( )f k t f ke

et

x

= += +

0

0 00ε . (2.69)

Here f0 is the equilibrium distribution at t « 0. For t > 0, there are no driving fields present any

more, so we can derive the time evolution of ( )k t by solving the equation of motion in the

relaxation-time approximation,

( )− = − +

× −dk

dt

k k k k

s

0 0

τ

σ

τ . (2.70)

Here σ is twice the electron spin, taken along the z axis (σz = ±1),

k0 is the initial wave vector

before applying ( )ε inst t , τ is the normal scattering lifetime, and τs is the skew scattering lifetime

which accounts for the spin-orbit scattering in ferromagnetic metals [49]. A solution is easily

computed by assuming circular wave-vector states ( ) ( ) ( )k t k t e i e± ±= ±1

2 x y . We get

( )( )− = −

±

± ±dk

dtk t k z

s0

1

τστ

i (2.71)

with the solutions

( )k t k e kz

s

t

± ±

±= +1

1

στ

m i

. (2.72)

Transforming back to the x-y basis, we get the components of ( )k t for t > 0

( ) ( ) ( ) ( ) ( )k t k e t k t k e t k t kx x

t

z y y

t

z z z= + = − =− −

0 0 0∆ Ω ∆ Ωτ τσ σcos , sin , , (2.73)

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2 THEORY OF MAGNETO-OPTICS 19

with ∆ = eε0 and Ω = −τs1 . Inserting ( )

k t and Eq. (2.63) into Eq. (2.64) and integrating

for t > 0, i. e., ( )ε inst t = 0, we find the y component of the total current,

( ) ( ) ( )Je

mk f k d k

d

dtf k P k d ky y y= +∗ ∫ ∫

3 3

( )( ) ( ) ( ) ( ) ( )( )[ ]= ′ + ′ ′ + ′ ′ + ′∗ ∫ ∫e

mk t f k d k

d

dtf k P k P t d ky y y y

∆ ∆0

30

3 . (2.74)

To perform the integration, we have substituted ( ) ( ) k t k t= ′ + ∆ as defined in Eq. (2.73).

Using Eq. (2.61) the integration in k space can be performed. The first term on the right in Eq.

(2.74) is zero as it is odd in ky and the third term vanishes because it is independent of time. In

conclusion, we get

( ) ( ) ( ) ( )J t n ne

me t

d

dt

e P

m ve ty

t

F

t

= − − +

↑ ↓ ∗

−ε τ τ

0

2

sin cosmaxΩ Ω , (2.75)

where we have used that ( ) ( )P A k A m vmax = × = ∗

F z Fσ σ and ( )n f k d k↑↓ ↑↓ ↑↓= ′ ′∫ 03

. The

separation of the integration into spin-up and spin-down contribution is due to the spin

dependence of the trigonometric functions in Eq. (2.73). Taking into account the signconvention of Eq. (2.31), the off-diagonal conductivity, ( )~σ ωxy , is computed by Fourier

transforming ( )~σ εxy yt J= 0 . We finally find

( )( )

( )( )

~ maxσ ω σω

π τ ω

ω τ ω

τ ωxy z

p

F

P

e v= −

+ ++ −

+

+ +

2

2 1 2

1

2 1 241

ΩΩ Ωi

ii

i , (2.76)

with spin polarization, ( )σ z = −↑ ↓n n n , plasma frequency, ω πp

2 24= ∗ne m , and electron

density, n = n n↑ ↓+ . Solving for the absorptive (imaginary) part yields

( )( )

( )( )

σ ω σω

πτ ω

τ ω τ ω

τ ω τ ω

τ ω τ ω2

2 1

2 2 2 2 2 2

1 2 2 2

2 2 2 2 2 242

4 4xy z

p

F

P

e v=

+ − ++

+ +

+ − +

− −

− −

− −

ΩΩ

Ω

Ωmax .

(2.77)

The first term reduces to the classical Drude conductivity if we set Ω = ωc and σz = 1. The

second term is nonclassical and is proportional to the spin-orbit-induced polarization. In the

high-frequency limit, we get

( )σ ω ω ω23 1

xy Drude s o→ ∞ = +− −c c . . . (2.78)

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20 2 THEORY OF MAGNETO-OPTICS

We immediately notice that at finite photon energies the leading term of the intraband

contribution to the off-diagonal conductivity is the spin-orbit term with its characteristic ω-1

fall-off. Therefore, the quantity ω σ2xy is usually plotted in order to separate intraband from

interband contributions in magnetic metals [48].

At last, an interpretation of the skew-scattering frequency, Ω, and the maximum spin-orbit-

induced polarization, Pmax, will be given [50, 51]. The quantum-mechanical analogue to Eq.

(2.70) is in the Heisenberg representation the equation of motion of the momentum operator,

p k= , i. e., the commutation relation with a Hamiltonian, H, including spin-orbit

interaction

[ ]i

dp

dtp= , H , ( ) ( )H = + + ⋅1

22

mp V r l

ξ σ . (2.79)

By evaluating the commutation relation, we get

( ) ( )[ ]i i i

dp

dtV r p

dk

dt= − ∇ − × =ξ σ 2 . (2.80)

Comparing with Eq. (2.70), we can identify the spin-orbit constant with the skew-scatteringfrequency, ξ = Ω .

The maximum spin-orbit-induced polarization, Pmax, is analyzed in first order perturbation

theory by the influence of the spin-orbit interaction Hamiltonian, Hs.o. on the Bloch-waveeigenstates, n,k , of an electron in a crystal lattice. The perturbed states, N k, , are then given

by

N k n k n kn k n ks o

n nn n

, , ,, ,. .= + ′′

− ′′ ≠∑ H

ε ε . (2.81)

The spin-orbit-induced polarization, ( ) P k is now defined by the expectation value of the space

operator, r , with respect to the spin-orbit-perturbed states,

( )

P k e N k r N ke n k r n k n k n ks o

nnn n

= =′ ′

′′ ≠

∑, ,, , , ,. .2

ReH

ω . (2.82)

Here we have used that n k r n k, , = 0 by symmetry reasons [51] and ωnn’ = εn - εn’. Defining

an average transition frequency, ωnn′ , where

1

ω ωnns o

n n

s o

nnn n

n k r n k n k n kn k r n k n k n k

′ ′ ≠ ′′ ≠′ ′ =

′ ′∑ ∑, , , ,, , , ,

. .. .

HH

, (2.83)

and using the completeness relation for Bloch states,

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2 THEORY OF MAGNETO-OPTICS 21

′ ′ = ′ ′ =′ ′ ≠

∑ ∑n n n nn n n

1 , (2.84)

we derive

( )

P k

en k r n k

nns o=

2

ωRe , ,. .H . (2.85)

Inserting ( ) ( )Hs o s o. . . .= ⋅ = ×ξ σ ω σ

l r p and approximating

p kFmax ≅ (Fermi momentum)

and r amax ≅ 0 (Bohr radius), the maximum spin-induced polarization, Pmax, is finally

estimated as

P e k aFs o

nnmax

. .≅′

2 02 ω

ω . (2.86)

In conclusion, this explains why the second term in Eq. (2.77) is called the spin-orbit term. It is

directly proportional to the spin-orbit frequency, ωs.o..

2.7 Relation between the experimental quantities and the off-diagonal conductivity

In this section, the relation between the experimental quantities, θK (polar Kerr rotation) andηK (polar Kerr ellipticity), and the off-diagonal elements, σxy , of the optical conductivity tensor

will be derived. For this purpose, we define a complex polar Kerr rotation

~θ θ ηK K K= − i . (2.87)

A phenomenological explanation of the polar Kerr effect is given in terms of Fresnel’s formulas

at normal incidence. In the presence of a magnetic field or a spontaneous magnetization, it is

assumed that the complex coefficient of reflection, ρ± , is different for rcp (‘+’) and lcp (‘-’)

light. Therefore, we set

~~

~ρ±±

±

= −+

n

n

1

1 , (2.88)

where ( )~n n i k= − is the complex index of refraction. By writing ρ± in a polar coordinate

system, ~ρ ρ φ± ±= ±e i , the polar Kerr rotation, θK, and the polar Kerr ellipticity, ηK are given by

θ φ φK = − −+ −

2 , tan η ρ ρ

ρ ρK = − −+

+ −

+ −

. (2.89)

These two equations can be related to the complex polar Kerr rotation by

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22 2 THEORY OF MAGNETO-OPTICS

~ ~ ~

~ ~θ θ η ρ ρρ ρK K K= − ≅ −

++ −

+ −

i i . (2.90)

This equation is only valid in the limit of small angles θK and ηK. In view of the discovery of

polar Kerr rotations in CeSb [52, 53] reaching almost 90° at low photon energies, we present

the exact relation [54] which is

( ) ( )( ) ( )

( )( ) ( )

sin cos

cos cos

sin

cos cos

~ ~

~ ~2 2

1 2 2

2

1 2 2

θ ηθ η

ηθ η

ρ ρρ ρ

K K

K K

K

K K+−

+= −

++ −

+ −

i i . (2.91)

This reduces to Eq. (2.90) for θK, ηK << 1. Inserting Eq. (2.88) in Eq. (2.90) and using the

approximations ( )~ ~n n+ −+ ≅ 2 ~n and ~ ~ ~n n n+ − ≅ 2 , where ~n is the complex index of refraction

in the absence of magnetic field or magnetization, we get

( )~ ~ ~

~ ~θK

n n

n n= −

−+ −i2 2

22 1 . (2.92)

Using Eq. (2.32) and

( )~ ~ ~n n k± ± ± ± ±= − = = −2 21

4i iε π

ωσ , (2.93)

we finally obtain

( )~

~

~ ~

~

~ ~θσσ

πω

σK

xy

xx

xy

n n n= =

−4

1 2

i . (2.94)

Separation into real and imaginary part yields

θ πω

σ σK

xy xyB A

A B=

++

4 1 2

2 2 , η π

ωσ σ

Kxy xyA B

A B=

−+

4 1 2

2 2 . (2.95)

where the prefactors, A and B, are polynomial functions of n and k,

A n n k n= − −3 23 , B k n k k= − + −3 23 . (2.96)

With these relations it is possible to compute from the experiment the off-diagonal conductivity

elements provided the optical functions have been obtained by a supplementary measurement.

By having used the sign convention of Eq. (2.31), these equations agree with recent reviews on

magneto-optics [39, 40].

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2 THEORY OF MAGNETO-OPTICS 23

2.8 Optical-enhancement effects

Optical-enhancement effects play an important role in magneto-optics. They are intentionally

used to optimize magneto-optic storage materials by adding dielectric and reflecting layers to

the magneto-optic layer [55]. In general, optical-enhancement effects are inherently connected

with Kerr-rotation studies in thin magnetic films and have to be considered in interpreting the

spectra. As is evident from Eqs. (2.94), the Kerr rotation is strongly dependent on the relative

complex index of refraction, ( )~n n i k= − , at the interface to the magneto-optic layer. Only in

the case of a magneto-optic layer facing vacuum, ~n will represent the intrinsic complex index

of refraction of the material. Else, characteristic optical-enhancement effects occur, which will

be discussed in the following sections.

2.8.1 Interface effect

Supposing the magneto-optic layer is not facing vacuum but a transparent (nonabsorbing)

material, like, e. g., a glass substrate, then the complex index of refraction of the magneto-optic

material, ~n , has to be replaced by the relative (complex) index of refraction, ~nrel , at the

substrate/magneto-optic-layer interface [56]. So we have

~~

nn

nrels

= , (2.97)

where ~n is the complex index of refraction of the magneto-optic material and ns is the real

index of refraction of the substrate. Deriving the relations between ~θK and ~σxy , analogous to

section 0, the new prefactors of Eq. (2.95), As and Bs, are

( )An

n n k n nA

nss

ss

= − − ≅133 2 2 , ( )B

nk n k n k

B

nss

ss

= − + − ≅133 2 2 . (2.98)

Inserting into Eq. (2.95) yields

θ πω

σ σθKs

s xy s xy

s ss K

B A

A Bn=

++

≅4 1 2

2 2, η π

ωσ σ

ηKss xy s xy

s ss K

A B

A Bn=

−+

≅4 1 2

2 2 . (2.99)

i. e., the Kerr effect measured through a transparent substrate is enhanced by the value of the

index of refraction, ns, of the substrate. Visualizing, the polar Kerr rotation as measured through

a glass substrate (ns = 1.5) is 50% larger than measured from the film side. This effect is often

ignored in literature and leads to a lot of confusion when polar Kerr rotations are compared

which have been measured by different authors.

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24 2 THEORY OF MAGNETO-OPTICS

2.8.2 Interference effect

A second optical-enhancement effect will happen if the magneto-optic layer itself or an

adjacent (dielectric) layer has a thickness, d, comparable to the wavelength, λ, of the incident

light Then, an enhancement due to multiple internal reflections may occur because the

denominator of Eq. (2.94) will become very small. This is the case if the thickness of the

dielectric layer fulfills an antireflection condition

( ) dm

nm

d

= + ∈λ 1 2

40 12, , , , . (2.100)

where nd is the real index of refraction of the dielectric overlayer. This leads to destructive

interference of the light reflected from front and back surface of the overlayer and, therefore, to

a pronounced structure in ~θK at a certain photon energy. Such interference effects often occur if

an oxide layer is present on top of a magneto-optic material. Because the thickness of native

oxide layers is of the order of 10 nm, the enhancement is usually observed in the far ultraviolet.

This effect is applied in optimizing magneto-optic data-storage materials by using a trilayer or

quadrilayer design [55].

2.8.3 Plasma-edge enhancement

The last mechanism relates to intrinsic properties of the magneto-optic layer itself. In metals,

the free-electron gas can be excited as a whole entity against the positive-charged atomic cores.

This is called a plasma oscillation. It shows up in the reflectivity spectra as a plasma minimum.

Near the plasma minimum, ~n goes to 1 and, therefore, the denominator of (2.94) tends to zero

giving rise to an enhancement of ~θK . Such effects have been found in rare-earth chalcogenides,

as e.g., TmS [50], TmSe [57], NdS [58]. A typical finger print of a plasma-edge enhancement

is a resonant behavior of the prefactors C and D of Eq. (2.95), where

CA

A B=

+1

2 2ω , D

B

A B=

+1

2 2ω . (2.101)

This leads to a pronounced resonance in ~θK while ~σxy , which is the real measure of the

magneto-optic activity, shows no resonance.

In conclusion, an optical enhancement of the polar Kerr rotation can most likely be expected

near a prominent local minimum in the reflectivity. When discussing the results, several

examples of optical-enhancement effects will be encountered.

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3 EXPERIMENTAL 25

3 EXPERIMENTAL

3.1 Principles of Kerr-effect measurements

The basic objective in magneto-optics is to determine the polarization state of an initially

linearly-polarized electromagnetic wave either after having transmitted a transparent material

(Faraday effect) or after being reflected off a specular surface (Kerr effect). In order to resolve

the state of polarization, which is in general elliptical, two quantities have to be measured: The

rotation of the major axis from the incident direction and the ellipticity, i. e., the ratio of the two

principal axes of the polarization ellipse. Concerning polar Kerr-effect measurements, rotations

as small as a few thousandths of a degree have to be resolved, especially in thin magnetic

layers. In magneto-optic spectroscopy, the Kerr effect is measured as a function of photon

energy. Therefore, the high resolution has to be maintained over a wide photon-energy range

despite large changes in e. g., light intensity, monochromator transmission, and sensitivity of

the detector. In the case of magneto-optic magnetometry, Kerr hysteresis loops are determined

at a fixed wavelength. In that case, the Kerr rotation is usually acquired by simply analyzing the

normalized difference, ∆I, of the intensities of two polarization states normal to each other and

at an angle of 45° to the incident plane of polarization. Therefore, we get

( )∆II I

I I

K K

K K

K

KK= −

+=

+

− +

+

+ +

= −+

≅2 1

1 2

2 2

2 2

4 4

4 4

22

22

sin~

cos~

cos~

sin~

cos

cosh

π θ π θ

π θ π θ

π θ

ηθ . (3.1)

The ellipticity, ηK, can be measured by introducing a quarter-wave plate into the beam path

following the polarizer [59]. According to the Senarmont principle, which is explained in

Fig. 3.1, if the retardation axes are aligned with the principal axes of the polarization ellipse, the

retardation of π/2 between the latter will be compensated and the elliptical state transforms into

a linearly polarized one which is rotated by an angle ηK from the major axis. For transverse or

longitudinal Kerr geometry, there are other detection schemes which are not discussed here.

y

x

fs

E→

ηK

θK

x

y

fs

E´→

θK

ηK

Fig. 3.1. Sketch of the Senarmont principle. The fast (f) and slow (s) retardation axesof the λ/4 plate are aligned with the principal axes of the polarization ellipse.

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26 3 EXPERIMENTAL

Fig. 3.2. Relation between the Kerr angles, θK and ηK, and the ellipsometric angles,tan(ψ) = Ey / Ex and ∆ = ∆y - ∆x.

In principle, for an energy-dependent evaluation any ellipsometric detection scheme can be

utilized as e. g., rotating analyzer [60] or photoelastic modulator techniques [61]. The

experimental resolution in the ellipsometric angles, ψ and ∆, is typically not better than

δψ ≈ 0.015° and δ∆ ≈ 0.08°. This can be related to the Kerr effect by the ellipsometric

equations (see Fig. 3.2)

( ) ( ) ( ) ( ) ( ) ( )tan tan cos , sin sin sin2 2 2 2θ ηK K= = −Ψ ∆ Ψ ∆ . (3.2)

Taking into account error propagation, we get an inaccuracy in the Kerr rotation of

δθK ≈ δψ ≈ 0.015° and in the ellipticity of δηK ≈ δ∆ + δψ ≈ 0.095°. This is not sufficient to

investigate thin magnetic layers with monolayer sensitivity.

In order to get to a higher resolution of 0.001°, a lock-in technique has to be applied

implying some sort of modulation. There are several alternatives of modulation. The best one

generally is to modulate the quantity that has to be determined, i. e., the state of polarization. By

using a Faraday modulator, the plane of polarization is wiggled at a low frequency, typically

Ωmod ≅ 80 Hz. This device is easy to be built and consists of a coil with a low-strain glass rod

inside. By applying an ac current, a Faraday rotation proportional to the current-induced

magnetic field is generated. The selection of the glass rod depends on the wavelength. In the

visible and ultraviolet region, quartz glass is a good choice and in the infrared a heavy lead

glass with a high Verdet constant (e. g., a SCHOTT SF57 or SF59) is appropriate.

Another way of modulation is to use a photoelastic modulator. Here, the phase shift between

two orthogonal polarization directions is altered by inducing a uniaxial strain in an otherwise

isotropic glass. Thus, the modulation varies the state of polarization between linear and circular.

Disadvantage as compared to the Faraday modulator is a high dc intensity, besides the 2Ωmod-

component to be measured, on the detector, which reduces the dynamic range. The advantage of

a phase modulation is the high frequency which is in the range of Ωmod ≅ 40-80 kHz, depending

on the geometry of the modulator. Another advantage is the simultaneous accumulation of Kerr

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3 EXPERIMENTAL 27

rotation and ellipticity, which renders this technique, to my knowledge, to the quickest method

but not the most accurate one. Provided the dc intensity, I0, the Ωmod component, IΩ, and the

2Ωmod component, I2Ω, can be determined concurrently, the Kerr rotation and ellipticity are

computed [62] by using

( ) ( )( )( ) ( ) ( ) ( )

I

I

J

JPEM

PEM

Ω

∆0

12

20

2 1

1 2=

−+ −

δ ψψ δ ψ

tan

tan tan sin , (3.3)

( ) ( ) ( )( ) ( ) ( ) ( )

I

I

J

JPEM

PEM

2

0

22

0

4

1 2Ω ∆

∆=

−+ −

δ ψψ δ ψ

tan sin

tan tan sin , (3.4)

where δPEM is the phase-shift amplitude of the photoelastic modulator and

( ) ( )( )θ ηψψK K= − = −

−+

12

1

1∆ , tan

tan

tan . (3.5)

This method requires, similar to the ellipsometric method [61], an accurate calibration of the

phase shift, δPEM, as a function of photon energy, in order to determine the Bessel functions,

Ji(δPEM). Its resolution is furthermore limited by the precision of the sine modulation of the

phase shift.

A last modulation method to be mentioned here, is the modulation of the applied field,

which results in a modulation of the magnetization [63]. The quantity measured is the dynamic

Kerr rotation, i. e., the first derivative of the Kerr rotation with respect to the magnetic field,

∂θ ∂K H . With this technique, resolutions of 10-4-10-6 degrees can be achieved. As a

consequence, one has to measure in a nonsaturated magnetic state which is troublesome

especially in exchange-coupled systems. However, this is an excellent method in magneto-optic

magnetometry, where nonlinearities of Kerr hysteresis loops can be studied very efficiently.

3.2 Description of a Kerr spectrometer with high resolution

In this section, an automated Kerr spectrometer is described with a resolution of 0.002°

across a wide photon-energy range. It employs a lock-in technique and the modulation is

performed with a Faraday modulator, which was outlined in the previous section. The set-up is

drawn in Fig. 3.3. As a light source, a high-pressure Xe lamp (450 W) is used for the visible

and ultraviolet photon-energy range and a halogen tungsten-filament lamp (150 W) for the

infrared. A double monochromator, with a quarter meter focus length and four sets of gratings,

gives a good stray-light reduction in the ultraviolet part of the spectrum. Suitable low-pass

optical filters are added after the monochromator. As a detector, a photomultiplier tube

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28 3 EXPERIMENTAL

(Hamamatsu R955), a Si-diode, and a dry-ice-cooled PbS photoresistor are used for the

ultraviolet and visible, the near infrared and the infrared part of the spectrum, respectively. The

photon-energy range covered that way is 0.6 - 5.3 eV (2000 - 230 nm). The monochromatic

light is linearly polarized by a Glan prism, which is optimized for a low beam displacement.

The focusing of the light is performed by imaging the monochromator exit slit onto the sample

with the help of two spherical mirrors with a focus length of 300 mm, each. This assures that

the shape of the illuminated area on the sample is independent of the wavelength because there

is no chromatic aberration present.

Fig. 3.3. Illustration of a high-resolution Kerr spectrometer.

The sample is mounted in polar Kerr geometry in the 15 mm air gap of an electromagnet which

is capable of making a magnetic field of 2.77 T at room temperature. A linear sample stage

allows for subsequent measurements of up to ten samples. A second pair of pole shoes with a

larger air gap of 30 mm permits to put an optical 4He cryostat into the air gap. Therefore,

temperature-dependent measurements can be made from 4.2 - 300 K in a field of 1.9 T. A high-

temperature optical insert extends the temperature range up to 800 K.

For measuring the polar Kerr rotation, θK, the reflected light is modulated by a Faraday

modulator at Ωmod ≅ 80 Hz and analyzed by a second Glan prism. The modulation amplitude

varies with photon energy but is of the order of a few degrees. The signal output of the detector

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3 EXPERIMENTAL 29

is amplified with the help of a 2-phase lock-in amplifier and the intensity of the

Ωmod component, IΩ, is minimized by adjusting the analyzer. The dependence of IΩ on the

analyzer angle, α, has the typical shape of a ‘V’ curve because it is proportional to

|sin(2[θK - α])| as will be shown in the next section. The polar Kerr ellipticity, ηK, is obtained

by introducing prior to the Faraday modulator a Soleil-Babinet phase shifter set to a phase shift

of π/2. According to the Senarmont principle, ηK is obtained by aligning the retardation axis of

the phase-shifter with the principal axis of the polarization ellipse and tuning the analyzer in

order to minimize again the lock-in amplitude. The polarizing prisms and the Soleil-Babinet

phase shifter are mounted on backlash-free rotation stages with an angular resolution of 0.002°.

The latter is, in addition, fixed on a linear stage to move it into and out of the beam path.

The whole set-up is computer controlled allowing an automated collection of the data. Still,

the monochromator grating, optical filter, modulator glass rod, and detector have to be changed

manually depending on the photon energy. The tuning of the analyzer angle is done in a first

step by a coarse adjustment [54]. Then, the analyzer is rotated within a range of approximately

2° while continuously reading the lock-in amplifier amplitude. In order to avoid drifts due to a

finite relaxation time of the lock-in amplifier, each ‘V’ curve is measured back and forth and

then averaged. The data is thereafter fit to the expression

( ) ( ) ( )[ ] ( )I s s cΩ Θ Θα α α α α α α= − + − − ++ −min min min , (3.6)

where Θ is the Heavyside step function, using a generalized least square procedure [54]. The

fitting parameters are the positive and negative slope, s±, the vertical offset, c, and the angle of

the minimum position, αmin, which relates directly to the Kerr rotation or ellipticity.

To determine θK and ηK, four ‘V’ curves have to be measured. In a first step, a reference,

usually an aluminum mirror with virtually no magneto-optic activity, is measured in order to

determine any spurious rotations of the set-up, as e. g., optical components within the stray field

of the magnet or the Faraday rotation of the cryostat windows. Then a ‘V’ curve for each sample

is measured yielding the Kerr rotation, θ α αKsample reference= −min min . To measure the ellipticity, the

Soleil-Babinet phase shifter (SB) is introduced into the beam path and the retardation axis are

aligned with the incident polarization plane by taking another ‘V’ curve with the Al mirror in

place. This step accounts for any depolarization within the optical beam path. A last ‘V’ curve

is collected for each sample after aligning the retardation axes of the phase shifter with the

principal axis of the polarization ellipse by using the previously measured θK. This yields the

ellipticity, ( )η α αKsample sample= −min minSB . In order to cancel any birefringence of nonmagnetic

origin, as e. g., stress-induced birefringence or natural anisotropy of the crystal, the spectra are

taken in both positive and negative magnetic fields of the same magnitude.

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30 3 EXPERIMENTAL

3.3 Mathematical evaluation of the beam path

In this section, the beam path is mathematically described by using the Jones-matrix

formalism. Each polarization state can be described by a Jones vector (see section 2.1). The

action of any optical element is then defined by a 2×2 matrix, M, and

′ =E EM . (3.7)

The matrices needed to evaluate the magneto-optic set-up are a polarizer, P, where

( ) ( ) ( )( ) ( ) ( )P =

cos sin cos

sin cos sin

2

2

β β ββ β β

, (3.8)

a Faraday modulator, F, where

( )( ) ( )( )( )( ) ( )( )F =

cos sin

sin cos

γ γγ γ

t t

t t , (3.9)

with ( ) ( )γ γt t= 0 sin Ω , and a phase shifter (Soleil-Babinet compensator), C, where

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )C =

+ −− +

e e

e e

i i

i i

∆ ∆

∆ ∆

cos sin sin cos

sin cos sin cos

2 2

2 2

1

1

φ φ φ φφ φ φ φ

. (3.10)

Here, φ is the angle of the fast axis to the x axis and ∆ is the retardation. When applying the

Senarmont principle, we set ∆ = π/2. Finally, the matrix, K describing a sample exhibiting a

polar Kerr effect is

K =−

− −

1

1

ΣΣ

, (3.11)

where

( ) ( )( ) ( )

( )( ) ( )Σ =

+−

+≅ −

sin cos

cos cos

sin

cos cos

2 2

1 2 2

2

1 2 2

θ ηθ η

ηθ η

θ ηK K

K K

K

K KK Ki i . (3.12)

The electric field vector at the detector position can now be derived from

E Eout = P F C K P2 1 . (3.13)

This describes the rather complicated situation of determining ηK. In order to make the

calculation somewhat easier, we start with the electric-field vector, ′′ =E EK P1 , after

reflection from the magneto-optically active sample,

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3 EXPERIMENTAL 31

′′ = =

− −

+ −

E E Ee

e

K K

K K

K

K

K P112 0

2

2

4 4

4 4

cos sin

sin cos

θ π θ π

θ π θ π

η

η

i

i . (3.14)

Here, we have assumed that the polarization of the incident light, ′ =E EP1 , is parallel to the

x axis. Inserting the matrices from above and applying trigonometric sum rules, the electric-field vector,

Eout , incident on the detector, is obtained as

E Eout = ′′P F C2

( ) ( )= − +

− + + − +

− +

12 0 4 4

E eK Ksin sin cos cosφ θ π φ α γ φ θ π φ α γi ∆

( )− − +

− +e K

Ki 2

4η φ θ π φ α γcos sin

( ) ( )( )

( )+ − +

− +

+e K

Ki ∆ 2

4η φ θ π φ α γ

α

αsin cos

cos

sin , (3.15)

where α is the analyzer angle with respect to the x axis, γ is the Faraday-modulation angle, ∆and φ are the phase shift and orientation of the fast axis relative to the x axis of the Soleil-

Babinet compensator, and θK and ηK are the Kerr rotation and ellipticity, respectively. The

detector is merely sensitive to the intensity of the light, so we get after employing again

trigonometric sum rules

2 2 2

0

2

02

02 2

2I

I

E

E EEout out= = ′′

P FC

( ) [ ]( ) ( )[ ]( )= + − − +1 2 2 2cos cos cosη φ θ φ α γK K t

( ) ( ) [ ]( ) ( ) ( )[ ] ( )[ ]( )+ − − − +cos cos sin sin sin sin∆ ∆2 2 2 2η φ θ η φ α γK K K t . (3.16)

Let us first evaluate the situation when measuring the Kerr rotation, θK. In that case, the phase

shifter is removed. Thus we set ∆ = 0, i. e., C = 1, and find

( ) ( )[ ]( )2 2 21 2 2

0

2

02

02 2

2I

I

E

E EE tout out

K K

K Kθ θ

η α θ γ= = ′′ = + − −

P F cos cos . (3.17)

The lock-in amplifier collects only the Ωt component of the Fourier transformed intensity. So

we have to expand Eq. (3.17) in a power series of Bessel functions. Therefore, we first rewrite

the equation as

( ) [ ]( ) ( )( ) [ ]( ) ( )( )[ ]21 2 2 2 2 2

0

I

It tout

K K K

η α θ γ α θ γ= + − + −cos cos cos sin sin . (3.18)

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32 3 EXPERIMENTAL

Using the relations

( )( ) ( ) ( ) ( )cos sin cos2 2 2 2 20 0 0 2 01

γ γ γΩ Ωt J J n tnn

= +=

∑ , (3.19)

( )( ) ( ) ( )( )sin sin sin2 2 2 2 10 2 1 01

γ γΩ Ωt J n tnn

= −−=

∑ , (3.20)

where Jn is the nth Bessel function, the intensity, IΩθK , of the fundamental frequency component

is obtained as

( ) ( ) [ ]( ) ( )I

IJ t

K

K KΩ Ωθ

γ η α θ0

1 02 2 2= −cos sin sin . (3.21)

A two-phase lock-in amplifier set to R-θ mode, i. e., set to determine the amplitude only, will

measure

( ) ( ) [ ]( )I

IJ

K

K KΩθ

γ η α θ0

1 02 2 2= −cos sin , (3.22)

which has as a function of analyzer angle, α, the shape of a ‘V’ curve with a very sharp

minimum position at α θ π= +K l 2 , where l ∈ Z. Thus, regardless of the value of ηK, the Kerr

rotation, θK, is directly related to the analyzer position, α, and minima occur for crossed as well

as for parallel polarizer-analyzer configuration. The crossed position is to be preferred because

then the light intensity is minimal on the detector yielding the largest signal-to-noise ratio.

In a last step, the Kerr ellipticity, ηK, is determined from Eq. (3.16) by inserting the Soleil-

Babinet compensator with φ = θK and ∆ = π/2 (Senarmont principle). This yields

( )[ ]( )21 2

0

I

Itout

K K

α θ η γ= + − − −cos . (3.23)

By separating the modulation angle, γ(t) and expanding in a power series of Bessel functions,

the lock-in amplitude of the first harmonic is

( ) [ ]( )I

IJ

K

K KΩη

γ α θ η0

1 02 2= − −sin , (3.24)

which has again the shape of a ‘V’ curve with a very sharp minimum position now at

α θ η π= + +K K l 2 , where l ∈ Z. Hence, the sum of Kerr ellipticity and rotation is directly

resolved by adjusting the analyzer angle. The advantage of this lock-in technique becomes

immediately clear from Eqs. (3.22) and (3.24) as the determination of the Kerr effect is simply

reduced to the determination of the minimum position of the analyzer angle with respect to the

lock-in amplitude of the first harmonic.

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4 (In, Mn)As HETEROSTRUCTURES 33

4 (In, Mn)As HETEROSTRUCTURES

4.1 (In, Mn)As/GaAs(100) diluted magnetic semiconductors

4.1.1 Introduction

Diluted magnetic semiconductors are suitable for studying magnetic and electronic

properties as well as cooperative effects caused by spin-exchange interaction between

conduction carriers and local magnetic moments. Up to now, primarily II-VI and IV-VI based

diluted magnetic semiconductors have been investigated and interesting effects, like giant

magnetoresistance and large Faraday rotations, have been found [64]. Regarding III-V

materials, there were only a few promising studies on the magnetic scattering of charge carriers

in Mn-doped III-V compounds, as e. g., GaAs, InAs, or InSb. The maximum concentration of

impurities was limited in these materials to approximately 1019 cm-3 [65] due to a low

equilibrium solubility of magnetic ions [66, 67]. Therefore, the incorporation of a large amount

of magnetic ions in order to cause strong exchange interactions while retaining the crystal

structure is prevented. By using the nonequilibrium growth condition of low-temperature

molecular-beam epitaxy, this limitation can be circumvented to a certain extent. The successful

demonstration of epitaxial growth of (In, Mn)As thin films [68-72] provides now the

opportunity to investigate a new class of technologically important III-V diluted magnetic

semiconductors. Taking advantage of III-V semiconductor properties and fabrication

technology, it is possible to control the type and number of carriers over a wide range and, most

importantly, to design heterostructures to change average distance between carriers and local

magnetic moments. To give an example, an optical Kerr rotator based on these heterostructures

has been proposed as follows [73]. The carrier concentration is controlled by applying an

electric field to two p-n junctions separated by a suitable distance. Under bias condition, holes

are depleted from the surface causing the ferromagnetic order to be lost as well as the magneto-

optic properties while under zero-bias condition, a large number of holes is present which

renders the surface region ferromagnetic.

We will review in the following sections the optical and magneto-optic properties of

(In, Mn)As thin films and heterostructures and discuss them in regard of the cooperative effects,

especially long-range magnetic order and carrier-induced ferromagnetism.

4.1.2 Growth conditions of (In, Mn)As/GaAs(100) films

The physical properties of (In, Mn)As films are very sensitive to growth parameters, as

e. g. substrate temperature, Ts, and Mn composition, x. The (In, Mn)As films of thickness

2.5 µm were grown on a 300 nm GaAs buffer layer on a GaAs(100) substrate by molecular-

beam epitaxy (MBE). Details of the sample preparation are given elsewhere [68] Depending on

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34 4 (In, Mn)As HETEROSTRUCTURES

the substrate temperature, the growth can be classified into three regimes, A, B, and C, as

illustrated in Fig. 4.1. Note, that Ts refers to the growth condition for the (In, Mn)As layer. The

GaAs buffer layers were grown at Ts = 480-580° C.

homgen.,

p-type

100

200

300

400

Mn concentration x

InMnAs + MnAs

x ~ 0.03n-type InMnAs

0.20.1

T (°C)s

with ferromagnetic MnAs clusters

growth inhibited

homogeneous, paramagnetic

ferromagn.

InMnAs polycrystalline

inhomogeneousInMnAs

BC

A

Fig. 4.1. Schematic diagram of InMnAs/(In, Ga)As/GaAs(100) film properties inrelation to the two growth parameters, substrate temperature and Mn composition.Lines are meant to be a rough guide. This figure is plotted according to Ref. 74.

Region A with 180 < Ts < 275º C and x ≤ 0.24 consists of a homogeneous paramagnetic n-

type In1-xMnxAs alloy in the zinc-blend structure without a macroscopic ferromagnetic MnAs

second phase as confirmed by TEM measurements [75]. Electron concentration at room

temperature [76] is less than 1019 cm-3 and decreases monotonically with x. The magnetization

follows a Curie-Weiss law with a paramagnetic Curie temperature, Θp, which is less than zero

[68, 74] (Θp ≅ -20 K for 0.08 < x < 0.2). Assuming complete quenching of the orbital angular

moment, it is concluded that Mn is incorporated as Mn2+ ions. X-ray-absorption fine-structure

(XAFS) measurements [77, 78] reveal for the Mn sites a first shell consisting of six As atoms

as in the NiAs structure common to bulk MnAs. However, the second NiAs-type shell of two

Mn atoms is absent or at least strongly disordered and the third shell consists of 12 In atoms as

is typical for the host zinc-blend structure. This suggests that the Mn is imbedded in the host

InAs zinc-blend structure as small centers (r ≤ 4 Å) with NiAs structure.

In region B, where 275 < Ts < 380º C and x > 0.03, we find an inhomogeneous alloy

composed of ferromagnetic MnAs clusters in an (In, Mn)As paramagnetic matrix.

Magnetization measurements show a Curie temperature TC = 310 K for the clusters similar to

bulk MnAs [70].

In a small Mn-concentration range, x < 0.03 and at high substrate temperatures (275 < Ts <

380º C) we find an interesting region (C). Here, a homogeneous (In, Mn)As alloy is formed

again which is now p-type and ferromagnetic with TC = 7.5 K. Hole concentration, p [71],

reaches a maximum at x = 0.004 which is almost two orders of magnitude higher

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4 (In, Mn)As HETEROSTRUCTURES 35

(p ≅ 1020 cm-3) than the electron concentration in n-(In, Mn)As (region A). At higher Mn

concentrations, p decreases by more than a factor of 100. At temperatures below 10 K,

ferromagnetic hysteresis is observed in transport measurements [71]. The coexistence of

remanent magnetization and unsaturated spins as well as a large negative magnetoresistance

effect is explained by the formation of large bound magnetic polarons [72].

4.1.3 Optical and magneto-optic properties of (In, Mn)As/GaAs(100) films

In this section, the magneto-optic properties of representative samples from different regions

of the phase diagram (see Fig. 4.1) are discussed [79]. The polar Kerr rotation, θK, at a

temperature of 5.5 K is plotted in a magnetic field of 3 T as a function of photon energy in

Fig. 4.2(a) for three samples: A homogeneous, paramagnetic n-type In0.88Mn0.12As/GaAs film

(), grown at Ts = 200° C (i. e., in region A in Fig. 4.1), an inhomogeneous (In, Mn)As/GaAs

film with ferromagnetic MnAs clusters (), grown at Ts = 300° C (i. e., in region B in Fig. 4.1),

and a ferromagnetic MnAs/GaAs sample (Σ). In Fig. 4.2(b), the corresponding reflectivity

spectra in the same energy range and at the same temperature are shown.

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1(a)

InMnAs, paramagnetic InMnAs with MnAs clusters MnAs/GaAs

In0.88Mn0.12As/GaAs(100)

T = 5.5 KB = 3 T

Pola

r K

err

Rot

atio

n (d

eg)

0 1 2 30

20

40

60 In0.88Mn0.12As/GaAs(100)

InMnAs, paramagnetic InMnAs with MnAs clusters MnAs/GaAs

(b)

Ref

lect

ivity

(%

) T = 5.5 K

Photon Energy (eV)

Fig. 4.2. The polar Kerr rotation (a) and the reflectivity (b) at 5.5 K as a function ofphoton energy of a homogeneous, paramagnetic n-type (In, Mn)As sample (), aninhomogeneous (In, Mn)As sample with ferromagnetic MnAs clusters (), and aferromagnetic MnAs/GaAs sample (Σ).

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36 4 (In, Mn)As HETEROSTRUCTURES

The homogeneous, paramagnetic n-type (In, Mn)As film (h-InMnAs) exhibits a rather

featureless Kerr spectrum with only a small peak at 1.2 eV. The reflectivity is very low across

the whole photon-energy range. The inhomogeneous (In, Mn)As film with ferromagnetic MnAs

clusters (i-InMnAs) shows a much richer polar Kerr spectrum. Two features can be

distinguished, a positive peak near 1 eV and an ‘S’-shaped structure at 2.5 eV. The reflectivity

of i-InMnAs is more than a factor of two higher than of h-InMnAs and exhibits an increase at

lower energies, typical of metals. The thick MnAs film, which was grown as a reference, has a

much larger value of θK and reveals some structure at 2.15 eV. The reflectivity is even larger

than for i-InMnAs, showing a more pronounced metallic behavior.

The small Kerr signal of h-InMnAs in applied fields of 3 T is to be expected because the

Mn-Mn interaction is antiferromagnetic as indicated by the negative paramagnetic Curie

temperature Θp ≅ -20 K [68]. The increased structure in the Kerr spectrum of i-InMnAs,

however, might be due to two effects. Firstly, because of the presence of MnAs clusters, the Mn

composition in the (In, Mn)As phase is slightly lower than in the h-InMnAs sample which

could lead to more localized magnetic moments, an effect well known in uranium compounds

[80]. This would result in sharper magneto-optic transitions. Secondly, as indicated by

magnetization measurements [70], with increasing dilution of Mn in the (In, Mn)As phase, |Θp|

decreases, indicating a softening of the antiferromagnetic Mn-Mn coupling. Hence, the same

applied field can induce more net magnetization and the Kerr response, being proportional to

the magnetization, increases. The presence of ferromagnetic MnAs clusters, which are

estimated to be ~10% to 25% of the entire volume [74], could in addition give rise to an

exchange coupling between the Mn local moments in the (In, Mn)As phase and the MnAs

clusters.

0 1 2 3

-0.05

0.00

0.05

0.10

B = remanence

B = 3 T

ferromagn. MnAs clustersinhomogeneous with

T = 5.5 K

In0.88Mn0.12As/GaAs(100)

Pola

r K

err

Rot

atio

n (d

eg)

Photon Energy (eV)

Fig. 4.3. The polar Kerr rotation at 5.5 K as a function of photon energy of aninhomogeneous InMnAs bulk sample with ferromagnetic MnAs clusters in a field of 3T and in the remanence state. Note the sign reversal of the peak near 1 eV and of the‘S’-shaped feature at 2.5 eV.

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4 (In, Mn)As HETEROSTRUCTURES 37

In order to clarify this point, we plot in Fig. 4.3 the polar Kerr rotation, θK, of i-InMnAs as

measured in a field of 3 T in combination with θK in the remanence state. Both features, the

peak at 1 eV and the ‘S’-shaped structure reverse sign in the remanence state as compared to 3

T. This sign reversal strongly indicates an antiferromagnetic coupling between the Mn local

moments in the (In, Mn)As phase and the MnAs clusters. Similar effects are found in

macroscopic ferrimagnets which will be discussed in section 5.

4.2 (In, Mn)As/(Ga, Al)Sb heterostructures

4.2.1 General properties of (In, Mn)As/(Ga, Al)Sb heterostructures

Starting from the phase diagram of Fig. 4.1, (In, Mn)As/(Ga, Al)Sb heterostructures were

grown. Such heterostructures are expected to provide additional parameters in controlling the

spatial confinement of both, carriers and magnetic ions, in either the same or different layers. In

addition, dimensional effects on the spin exchange might be observed as the band alignment for

(In, Mn)As/AlSb is likely to be similar to the staggered type II in the InAs/AlSb system [81],

i.e., both conduction and valence-band edges of (In, Mn)As are lower in energy than the

corresponding band edges in AlSb. This infers that electron and hole states are confined

separately in the (In, Mn)As and AlSb layers, respectively.

AlSb (130-150 nm)

GaSb (400-600 nm)

In Al As (0-24 nm)0.5 0.5

T = 200° Cs

T = 480-580° Cs

In Mn As (9-19 nm)0.88 0.12

GaAs buffer (300 nm)

GaAs(100) substrate

Fig. 4.4. Typical layering sequence in (In, Mn)As heterostructures grown by MBE.The GaAs substrate is semi-insulating. A GaAs buffer layer is for smoothing thesubstrate surface while the GaSb layer is used to accommodate the large latticemismatch between GaAs and AlSb which has a relatively good lattice match to(In, Mn)As.

The heterostructures reviewed in this work were all grown at Ts = 200° C, which is a growth

condition where an n-type (In, Mn)As layer is anticipated (zone A in Fig. 4.1). A typical

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38 4 (In, Mn)As HETEROSTRUCTURES

layering sequence is sketched in Fig. 4.4. All nonmagnetic III-V layers were grown at substrate

temperatures Ts = 480-580° C. The semi-insulating GaAs(100) substrate is first covered with

300 nm GaAs for smoothing the substrate surface. In a second step, a GaSb buffer layer of

400-600 nm is added in order to reduce the large lattice mismatch between the AlSb and the

GaAs layers. An AlSb layer is grown next with a thickness of 100-200 nm. Finally, the

In1-xMnxAs layer is deposited at Ts = 200° C with x ≅ 0.12 and with a thickness of 9-19 nm. To

study the influence on the magnetic exchange of a potential barrier between the magnetic

(In, Mn)As and the AlSb layer, an In0.5Al0.5As spacer layer of thickness up to 24 nm was grown

in between.

100

200

300

400

p-type (In, Mn)As + MnAs

x ~ 0.03

p-type (In, Mn)As on (Ga, Al)Sb/GaAs(100)

0.20.1

T (°C)s

with ferromagnetic MnAs clusters

growth inhibited

homogeneous, ferromagnetic

InMnAs polycrystalline

inhomogeneous

Mn concentration x in In Mn As1-x x

Fig. 4.5. Schematic diagram showing the physical properties of (In, Mn)As grown on(Ga, Al)Sb/GaAs(100) in relation to the two growth parameter, substrate temperatureand Mn composition. Lines are meant to be a rough guide.

Surprisingly, such heterostructures exhibit p-type characteristics and show a ferromagnetic

transition at TC ≅ 35 K with strong perpendicular magnetic anisotropy, which has previously

been interpreted as being caused by carrier (hole)-induced ferromagnetism [74, 76, 82]. As a

result, a new phase diagram is derived for the heterostructures as shown in Fig. 4.5. From the

Hall-resistance of In0.88Mn0.12As(dMS)/AlSb(0.1-0.3 µm)/GaSb(0.3-1.0µm) heterostructures it is

concluded that ferromagnetism dominates only in a narrow thickness regime of dMS = 5-20 nm

[82]. Similar results were found in (In, Mn)As/Ga1-yAlySb heterostructures in the entire range

0 ≤ y ≤ 1 [82]. However, in (In, Mn)As/AlSb1-zAsz structures perpendicular ferromagnetism is

significantly weaker and vanishes completely at z ≅ 0.15. At this As concentration, a nearly

lattice-matched condition between the constituent layers is established. Consequently, the

occurrence of perpendicular magnetic anisotropy can be attributed to strain-induced crystal

anisotropy by a lattice mismatch of 0.6%-1.3% between (In, Mn)As and (Ga, Al)Sb layers [82].

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4 (In, Mn)As HETEROSTRUCTURES 39

A lower limit of TC can be estimated from the temperature dependence of the remanent polar

Kerr rotation as plotted in Fig. 4.6 for three (In, Mn)As/Ga1-xAlxSb heterostructures withx = 0, 0.5, and 1. The measurements were obtained at a photon energy of 2.4 eV. Eachsample clearly exhibits a TC larger than 35 K, reaching more than 50 K in the sample with

x = 0.5.

0 10 20 30 40 50

-0.04

-0.02

0.00

In0.88 Mn0.12 As(9 nm)/Ga1-x Alx Sb

x = 0 x = 0.5 x = 1.0

B = remanence

Pol

ar K

err

Rot

atio

n (

deg)

Temperature (K)

Fig. 4.6. Temperature dependence of the remanent polar Kerr rotation of threeIn0.88Mn0.12As/Ga1-x,AlxSb heterostructures with x = 0, 0.5, and 1. The measurementswere taken at a photon energy of 2.4 eV.

Introducing an In0.5Al0.5As spacer layer between the (In, Mn)As and the (Ga, Al)Sb layer

leads to a pronounced decrease in coercivity and hysteresis behavior of the Hall resistance [76].

This observation lead to the idea of carrier-induced ferromagnetism across the (In, Mn)As/AlSb

interface involving two-dimensional hole states [76]. The basic idea is that the holes in the

AlSb layer would establish a Ruderman-Kittel-Kasuya-Yoshida (RKKY)-like exchange

interaction with the Mn ions in the (In, Mn)As layer which would lead to long-range

ferromagnetic order. A spacer layer acting as a potential barrier will suppress the penetration of

the hole wave function into the (In, Mn)As layer and, therefore, reduce the ferromagnetic

exchange between the Mn ions.

In the following sections, we review the optical and magneto-optic properties of

(In, Mn)As/(In, Al)As/(Ga, Al)Sb III-V diluted-magnetic-semiconductor heterostructures and

discuss the dependence of ferromagnetism on structure parameters such as the thickness of the

(In, Mn)As layer and the influence of an (In, Al)As spacer layer on the long-range magnetic

order. Concerning the spacer-layer dependence, we will clarify the hypothesis of a potential

ferromagnetic exchange across the (In, Mn)As/AlSb interface as suggested by previous

experiments [76]. Magneto-optic spectroscopy is shown in this context to be a magnetic-layer-

specific method with which we can focus on the exchange within the ferromagnetic (In, Mn)As

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40 4 (In, Mn)As HETEROSTRUCTURES

top layer [83]. This is in contrast to magneto-transport experiments where carriers (holes)

existing not only in the (In, Mn)As but also in the non-magnetic AlSb layers are contributing to

the transport effects. Therefore, the influence of those carriers which are involved in the

magnetic exchange might be small.

4.2.2 Optical results (reflectivity)

The purpose of this section is to understand qualitatively how (In, Mn)As-layer thickness

and (In, Al)As-spacer-layer thickness influence the reflectivity. This is important for the

comprehension of the magneto-optic properties of the III-V diluted-magnetic-semiconductor

heterostructures as the polar Kerr rotation strongly depends on the optical functions n and k as

discussed in sections 2.7 and 2.8. The individual layers of the heterostructures are mostly

semiconducting. Their thickness is except for the GaAs substrate less than the wavelength of

the incident light. Therefore, the reflectivity spectra are expected to be dominated by structures

relating to the direct energy gaps of individual layers and by interference effects. Table 4.1collects for the heterostructures reviewed in this work the direct energy gap, Eg

dir , at low

temperatures [84], and the thickness of the individual layers. Within the energy range covered

in this work, only AlSb and (In, Al)As are transparent up to ~ 2 eV. Therefore, we can divide

the energy range into two sections.

In0.88Mn0.12As In0.5Al0.5As AlSb GaSb GaAs

direct gap Egdir

at 4.2 K [84]

~ 0.42 eV ~ 1.77 eV 2.32 eV 0.81 eV 1.52 eV

R1350 9 nm 0 nm 136 nm 400 nm 300 nm

R1351 9 nm 0 nm 0 nm 600 nm 300 nm

R1356 9 nm 3.5 nm 140 nm 600 nm 300 nm

R1357 9 nm 7 nm 140 nm 600 nm 300 nm

R1361 9 nm 0 nm 159 nm* 600 nm 300 nm

R1385 19 nm 0 nm 145 nm 600 nm 300 nm

R1388 18.5 nm 24 nm 145 nm 600 nm 300 nm

Table 4.1. Direct energy gap at low temperatures and thickness of the individuallayers of the heterostructures. * Ga0.5Al0.5Sb layer instead of AlSb.

Below 2 eV, spectral features can originate only from electronic transitions in the (In, Mn)As

and the GaSb layers. Interference effects are merely to be expected from the AlSb and the GaSb

layers because the (In, Mn)As and the (In, Al)As layers are too thin. The critical thickness, d,

for a constructive optical-interference effect in reflection is roughly given by the same equation

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4 (In, Mn)As HETEROSTRUCTURES 41

as Eq. (2.100). This equation is valid for a free-standing layer facing air. A phase shift of π at

the front surface and none at the back surface is taken into account. In the heterostructures, the

situation is much more complicated because the layers are either not transparent or they are

facing an absorbing layer instead of air. Yet, Eq. (2.100) is appropriate for a rough estimate.

Taking d = 20 nm and n between 2-4, we get an interference wavelength, λ = 160-320 nm,

which is more than 3 eV in photon energy. Therefore, such thin layers as the (In, Mn)As or the

(In, Al)As layers do not show an interference effect within the energy range covered in this

work. The GaAs layer and substrate will barely contribute to the reflectivity spectra as the

penetration depth of the light is not deep enough. Above 2 eV, no interference effects will be

present any more and electronic transitions may arise from all layers, including AlSb and

(In, Al)As.

0 1 2 330

40

50

60

C’C’’C

B

D’D

A

InMnAs(9nm)/GaSbT = 5.5 K

Ref

lect

ivit

y (

%)

Photon Energy (eV)

Fig. 4.7. Reflectivity spectrum of an In0.88Mn0.12As(9 nm)/GaSb(600 nm) hetero-structure at 5.5 K.

The reflectivity spectrum at a temperature of 5.5 K of the simplest heterostructure,

(In, Mn)As/GaSb with a (In, Mn)As-layer thickness of 9 nm, is shown in Fig. 4.7. Several

features can be distinguished, labeled A through D. By referring to an analysis [85] of the

optical properties of InAs and GaSb, an assignment of some of the structures is attempted. The

most prominent peak A is assigned to an E1 transition (notation according to Ref. 84) in GaSb

at 2.15 eV which appears as a sharp kink at 2 eV in the absorption spectra [85]. We may furtherassign structure B, which is lowest in energy, to the direct band gap Eg

dir = 0.8 eV in GaSb. At

slightly higher photon energies, three peaks are found, C, C’, C’’, with an equal energy spacing

of 0.25 eV whose amplitudes decrease monotonically with increasing photon energy. They

could be explained in terms of interference within the GaSb layer. Using Eq. (2.100), neglecting

absorption (k ≅ 0), and assuming n = 4 for GaSb [84], an energy separation of 0.26 eV of the

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42 4 (In, Mn)As HETEROSTRUCTURES

interference maximums is calculated for a 600 nm thick GaSb layer, in good agreement with the

energy separation of the structures C-C’’. On the other hand, the absorption coefficient, KGaSb, is

larger than 2⋅104 cm-1 above 1 eV yielding a penetration depth of the light of less than 500 nm.

Therefore, the argument that the C-C’’ peaks are due to an interference effect is a rather

tentative conclusion at present. Features D and D’ are found in all heterostructures independent

of the underlying layers. It is tempting to assign them to the (In, Mn)As layer. However, it

should be noted that such features have not been observed in 2.5 µm thick (In, Mn)As films

(see Fig. 4.2(b) and Ref. 79).

The influence of an (In, Al)As-spacer layer on the reflectivity spectra is depicted in Fig. 4.8,

where the reflectivity spectra at a temperature of 5.5 K is plotted for four

(In, Mn)As/(In, Al)As/AlSb heterostructures with varying spacer-layer thickness, dsp. The

structures are labeled A through E. Most features are present in all heterostructures. The main

peak A shifts when the spacer-layer is introduced. In the sample without spacer layer, it is closeto Eg

dir = 2.32 eV of AlSb and shifts then in (In, Mn)As/(In, Al)As/AlSb to a lower value, close

to the Egdir = 1.77 eV of the spacer layer. Therefore, peak A is assigned to Eg

dir in AlSb, and (In,

Al)As, respectively.

0 1 2 30

20

40

60D2

C1’

C1

B1

A1

A4

B4

B2

A2

A3

E4

D4

D1B3

D3

InMnAs(9nm)/AlSb InMnAs(9nm)/InAlAs(3.5nm)/AlSb InMnAs(9nm)/InAlAs(7nm)/AlSb InMnAs(19nm)/InAlAs(24nm)/AlSbT = 5.5 K

Ref

lect

ivity

(%

)

Photon Energy (eV)

Fig. 4.8. Reflectivity spectra of In0.88Mn0.12As/In0.5Al0.5As/AlSb heterostructures withdifferent In0.5Al0.5As-layer thickness dsp = 0 nm (), 3.5 nm (), 7 nm (∆), and24 nm ().

The structures D and D’ shift as well while peak B appears at about the same energy in the

samples with and without spacer layer. Features C and C’ are only present in the samples

without spacer layer. Interference maximums due to the AlSb layer are expected to be separated

by 1.35 eV (n = 3.3, k ≅ 0). The only feature at that energy is peak C’. But a sample with thicker

(In, Mn)As layer (not shown here) does not exhibit a peak at that energy although the AlSb-

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4 (In, Mn)As HETEROSTRUCTURES 43

layer thickness is almost the same [83]. Peak E only appears in samples with thicker (In, Mn)As

layer [83]. This suggests that peak E might be due to a Mn transition as they have a larger

oscillator strength in thicker layers.

4.2.3 Magneto-optic results

In Fig. 4.9, the polar Kerr-rotation spectra of three characteristic heterostructures are plotted

to point out the sensitive dependence of the magneto-optic spectra on heterostructure

parameters such as (In, Mn)As-layer thickness and the layers adjacent to the (In, Mn)As layer.

The measurements are all taken at a temperature of 5.5 K, which is well below the Curie

temperature, TC ≅ 35 K. Each sample is depicted at a high magnetic field of 3 T, where

magnetic saturation is achieved, and in the remanence state.

The spectrum of In0.88Mn0.12As(9 nm)/AlSb(136 nm) (see Fig. 4.9(a)) at 3 T is almost

identical to the spectrum in the remanent state. This indicates that paramagnetic contributions

are negligible and the complete spectrum is related to spin-polarized electronic transitions

which can solely be due to Mn-3d local moments. The Kerr spectra show three distinct features.

At 1.65 eV (peak α), the Kerr rotation, θK, reaches a maximum value of 0.18°. A small

structure β appears at 1 eV. Finally, an 'S'-shaped structure, labeled γ, is observed at 2.5 eV.

Besides these dominant features, some very small peaks (δ, ε, λ) arise at 0.8, 1.3, and 2.15 eV,

respectively, reflecting the rich structure of the reflectivity spectrum (see Fig. 4.8).

The 3 T spectrum of an In0.88Mn0.12As(19 nm)/AlSb(145 nm) sample (Fig. 4.9(b)) shows

only a weak feature α at 1.8 eV. The maximum θK has shifted down to 1.25 eV (peak β) as

compared to the sample with thinner (In, Mn)As layer. However, an 'S'-shaped structure γ is

observed again at 2.55 eV. Furthermore, two small features (δ, λ) can be distinguished at 0.85

and 2.2 eV, respectively. The remanence spectrum is only about 50% of the 3 T preserving all

features.

The saturation and remanence spectra of an In0.88Mn0.12As(9 nm)/GaSb(600 nm)

heterostructure are depicted in Fig. 4.9(c). As in the case of the In0.88Mn0.12As(9 nm)/

AlSb(136 nm) sample (see Fig. 4.9(a)), the remanence spectrum is almost identical to the one at

3 T, yielding a squareness of the Kerr hysteresis loops close to one. However, the spectral

dependence is quite different from Fig. 4.9(a). There is no sharp structure with the exception of

an 'S'-shaped feature (peak γ) at 2.6 eV. The rest of the Kerr spectra is dominated by a broad

peak α, centered at 1.6 eV, with a few small features (β, δ, ε, λ) being superimposed.

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44 4 (In, Mn)As HETEROSTRUCTURES

0.0

0.1

0.2

B = 3 T B = remanence

λ

δ γ

α

β

T = 5.5 K

(b)

In 0.88 Mn 0.12 As (19 nm) / AlSb

Pola

r K

err

Rot

atio

n (d

eg)

0 1 2 3

-0.1

0.0

0.1T = 5.5 K

B = 3 T B = remanence

λ

δ

ε

γα

β

In 0.88 Mn 0.12 As (9 nm) / GaSb

(c)

Photon Energy (eV)

-0.1

0.0

0.1

0.2

γ

λ

α

ε

β

δ

B = 3 T B = remanence

In 0.88 Mn 0.12 As (9 nm) / AlSb

T = 5.5 K

(a)

Fig. 4.9. Polar Kerr-rotation spectra at 5.5 K of three heterostructures as a functionof photon energy in a field of 3 T () and in the remanence state (∆):(a) In0.88Mn0.12As(9 nm)/AlSb(136 nm), (b) In0.88Mn0.12As(19 nm)/AlSb(145 nm), and(c) In0.88Mn0.12As(9 nm)/GaSb(600 nm).

Ferromagnetic order of the heterostructures is nicely demonstrated in polar Kerr hysteresis

loops as plotted in Fig. 4.10. All loops were taken at the photon energy, ω, corresponding to

the maximum θK. The vertical displacement of the hysteresis loops by 0.05-0.07° is due to

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4 (In, Mn)As HETEROSTRUCTURES 45

intrinsic birefringence of the samples, which originates from stress in the layers due to, e. g.,

lattice mismatch. Such birefringence leads to an extra field-independent rotation of the

polarization plane of the light.

-0.2

-0.1

0.0

0.1

0.2

0.3

ω = 1.2 eVT = 5.5 K

(b)

In 0.88 Mn 0.12 As (19 nm) / AlSb

Pola

r K

err

Rot

atio

n (d

eg)

-1.0 -0.5 0.0 0.5 1.0-0.05

0.00

0.05

0.10

ω = 1.5 eVT = 5.5 K

In 0.88 Mn 0.12 As (9 nm) / GaSb

(c)

Magnetic Field (T)

-0.1

0.0

0.1

0.2

ω = 1.65 eVT = 5.5 K

In 0.88 Mn 0.12 As (9 nm) / AlSb

(a)

Fig. 4.10. Polar Kerr hysteresis loops at 5.5 K of three representative heterostructures:(a) In0.88Mn0.12As(9 nm)/AlSb(136 nm) measured at a photon energy of ω = 1.65 eV,(b) In0.88Mn0.12As(19 nm)/AlSb(145 nm) at ω = 1.2 eV, and (c) In0.88Mn0.12As(9 nm)/GaSb(600 nm) at ω = 1.5 eV.

The In0.88Mn0.12As(9 nm)/AlSb(136 nm) heterostructure (Fig. 4.10(a)) exhibits strong

perpendicular magnetic anisotropy, also found in Hall measurements [82], yielding asquareness θ θK

remKsat = 1. The high squareness is also reflected in the similarity of the remanent

spectra with the one at 3 T (see Fig. 4.9(a)). The coercive field, Hc, amounts to 0.055 T, which

is quite small. The coercive field of the (In, Mn)As/AlSb sample with 19 nm (In, Mn)As-layer

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46 4 (In, Mn)As HETEROSTRUCTURES

thickness (Fig. 4.10(b)) and of the (In, Mn)As/GaSb sample (Fig. 4.10(c)) reaches only

Hc ≅ 0.03 T which is half the value of Hc of the (In, Mn)As/AlSb sample with 9 nm thick

(In, Mn)As layer. Such a reduction in coercivity has been observed before in Hall

measurements [82].

-0.1

0.0

0.1

0.2 dsp = 24 nm

dsp = 7 nm dsp = 3.5 nm

λ 2,3

α2

λ 1γ 1,2,3

α1

β 3B = 3 TT = 5.5 K

(a)

In 0.88 Mn 0.12 As / In 0.5 Al 0.5As (dsp) / AlSb

Pola

r K

err

Rot

atio

n (d

eg)

0 1 2 3-0.10

-0.05

0.00

0.05

dsp = 24 nm dsp = 7 nm

α2

α1

λ 1

γ 1

In 0.88 Mn 0.12 As / In 0.5 Al 0.5As (dsp) / AlSb

dsp = 3.5 nm

B = remanenceT = 5.5 K β 3

(b)

Photon Energy (eV)

Fig. 4.11. Polar Kerr-rotation spectra as a function of photon energy at 5.5 K ofIn0.88Mn0.12As/In0.5Al0.5As(dsp)/AlSb heterostructures with In0.5Al0.5As-spacer layerthickness dsp = 3.5 nm (), 7 nm (∆), and 24 nm (). The spectra are measured in (a)a field of 3 T and (b) in the remanence state.

The influence of an In0.5Al0.5As-spacer layer on the Kerr rotation is shown in Fig. 4.11 for a

spacer layer thickness dsp = 3.5, 7, and 24 nm. The spectra are measured in (a) a magnetic field

of 3 T capable of saturating the samples and (b) in the remanent state. At 3 T (Fig. 4.11(a)), θK

reaches a maximum at 1.45 eV in the heterostructures with dsp = 3.5 and 7 nm (peak α). The

sample with the thickest spacer layer dsp = 24 nm does not show a peak at that photon energy

but reaches a maximum θK at 1 eV (peak β). An ’S’-shaped structure (peak γ) centered at

2.55 eV is found in all three samples with spacer layer. Looking at the remanence spectra (Fig.

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4 (In, Mn)As HETEROSTRUCTURES 47

4.11(b)), a Kerr rotation of 25% of the saturation value is obtained for the sample with the

thinnest spacer layer (dsp = 3.5 nm), whereas for the samples with dsp = 7 and 24 nm only 14%

is reached. Because of the small θK values, a remanence rotation can only be separated from

noise in the latter samples at 1.45 and 1 eV, respectively, where θK reaches a maximum value.

-0.2

0.0

0.2

dsp = 3.5 nm

In 0.88 Mn 0.12 As / In 0.5 Al 0.5As (dsp) / AlSb

ω = 1.45 eVT = 5.5 K

(a)

Pola

r K

err

Rot

atio

n (d

eg)

0.2

0.3

0.4

0.5

dsp = 7 nm

ω = 1.45 eVT = 5.5 K

(b)

-1.0 -0.5 0.0 0.5 1.0

0.25

0.30

0.35

0.40 (c)

dsp = 24 nm

ω = 1.2 eVT = 5.5 K

Magnetic Field (T)

Fig. 4.12. Polar Kerr hysteresis loops at 5.5 K of In0.88Mn0.12As/In0.5Al0.5As(dsp)/AlSbheterostructures with an In0.5Al0.5As-spacer layer thickness dsp = 3.5 nm (a), 7 nm (b),and 24 nm (c) at a photon energy ω = 1.45 eV, 1.45 eV, and 1.2 eV, respectively.

The change in the polar Kerr hysteresis loops with spacer-layer thickness, dsp, is

demonstrated in Fig. 4.12. The loops are taken at the photon energy of the maximum θK. The

squareness as well as the coercivity decrease with increasing dsp. While Hc = 0.02 T for

dsp = 3.5 nm, the coercivity decreases to Hc = 0.015 T for dsp = 7 nm and vanishes for

dsp = 24 nm. However, even for the heterostructure with the thickest spacer layer investigated in

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48 4 (In, Mn)As HETEROSTRUCTURES

this work a typical ferromagnetic hysteresis curve is found which is in apparent contradiction to

Hall measurements on these samples [76].

4.2.4 Correlation between reflectivity and Kerr spectra

The very structured reflectivity spectra of the heterostructures have a considerable influence

on the spectral dependence of the Kerr rotation. In order to separate the effects due to the

reflectivity - or, more precisely, due to the optical functions n and k - from intrinsic magneto-

optically active electronic transitions one has to define a quantity which represents the intrinsicmagneto-optic activity. Such a quantity is the off-diagonal element, ~σ σ σxy xy xy= +1 2i , of the

complex optical conductivity tensor as defined in Eq. (2.31), where the magnetization of the

sample and the direction of the light beam are assumed along the z axis and the sample surfaceis in the x-y plane. It can be shown from Eq. (2.53), that ~σxy is to first order proportional to the

joint density of states, Jαβ of the initial (α) and final (β) electronic states [48]. As a conse-quence, ~σxy is proportional to the product of density of states of initial (nα) and final (nβ) state

times the joint spin density, σ joint , where

σ α β α β

α β α βjoint =

+↑ ↑ ↓ ↓

↑ ↑ ↓ ↓

n n n n

n n n n . (4.1)

This leads in general to the well known proportionality between polar Kerr rotation andperpendicular component of the magnetization of the sample. The relation between ~σxy and

the experimentally accessible Kerr effect, is given by Eq. (2.95). As had been discussed insection 2.8 on optical-enhancement effects, the Kerr effect is not simply proportional to ~σxy but

depends in a non-linear way on the relative index of refraction, ~n . If ~n ≅ 1, an enhancement in~θK will occur. An appropriate layer adjacent to the magneto-optic material can also produce an

optical-enhancement effect as shown in Section 2.8.2 and we strongly believe that this is the

situation in these heterostructures.

By comparing the position of the major peak α in the Kerr spectra of the

In0.88Mn0.12As(9 nm)/AlSb (see Fig. 4.9) and of the In0.88Mn0.12(9 nm)/In0.5Al0.5As(dsp)/AlSb

samples (see Fig. 4.11) with the position of the features in the corresponding reflectivity spectra

(see Fig. 4.8), we notice that peak α lies at a photon energy where the reflectivity, R, goes

through a local minimum before reaching its maximum at peak A. A similar correspondence is

found in the heterostructures with 19 nm thick (In, Mn)As layer. Here, the main feature β of the

Kerr spectra (see Fig. 4.11) corresponds again to a local minimum in R (see Fig. 4.8) before R

reaches a local maximum at peak E. The exception is the (In, Mn)As(9 nm)/GaSb sample

which shows only a broad feature A in the reflectivity (see Fig. 4.7) and, accordingly, a broad

peak α in the Kerr rotation (see Fig. 4.9). This is probably due to the absorption of light in the

GaSb layer whose band gap is 0.81 eV.

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4 (In, Mn)As HETEROSTRUCTURES 49

no. ω0i (eV) ωpi (eV) γi (eV)

1 1.0 3 3

2 1.5 2 0.25

3 2.15 3 0.25

4 3.3 8 2

Table 4.2. Parameters of the Lorentzian curves used to calculate the reflectivityspectrum of Fig. 4.13. Curves 2 and 3 represent the relevant transitions while curves 1and 4 serve to approximate the influence of the remaining part of the reflectivityspectrum.

In order to understand this agreement between peak position in the Kerr spectra and local

minimum in reflectivity, the magneto-optic scaling factor, C and D, as defined in Eq. (2.101)

have to be considered. We first approximate a reflectivity spectra which is typical for the

heterostructures by the sum of four Lorentzian curves,

( )~

,εω

ω ω ω γLor ipi

i i

= +− +

12

02 2 i

, (4.2)

using the parameters given in Table 4.2. The calculated reflectivity spectra, R, is plotted in

Fig. 4.13 as a solid line. Let us assume that the off-diagonal elements σ1xy and σ2xy are slowly

varying functions with photon energy and - for simplicity - that they are equal in magnitude.

Then the spectral dependence of θK is given by the difference or sum of C and D depending on

the relative sign of σ1xy and σ2xy.

0 1 2 30

10

20

30

40

50

Ref

lect

ivit

y (

%)

Photon Energy (eV)

0.0

0.5

C + D D - C

Magneto-O

ptic Scaling Factors (a. u.)

Fig. 4.13. Calculated reflectivity spectrum (solid line) using four Lorentzian functionsas described in the text. The dashed and dotted lines are, respectively, the sum anddifference of the two magneto-optic scaling factors C and D which are defined inEq. (2.101).

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50 4 (In, Mn)As HETEROSTRUCTURES

The result is shown in Fig. 4.13 as a dashed line (C + D) and a dotted line (D - C). Obviously,

both combinations of the magneto-optical scaling factors show a sharp peak close to the

minimum of R which proves the general correlation between local minimum in the reflectivity

spectrum and peak position in θK. If σ1xy and σ2xy are themselves varying with photon energy,

the two effects combine to yield a more complicated structure in θK.

The situation is different in the case of the ’S’-shaped feature γ which appears in all Kerr

spectra. It relates to the structures D and D’ in the reflectivity spectra. But although D and D’

are just minor features in R with a peak-to-valley difference of less than 5%, the ’S’-shaped

feature γ is equal in magnitude to the major peaks α and β in θK or even dominates the Kerr

spectrum as in the case of In0.88Mn0.12As/GaSb (see Fig. 4.9). This indicates that the ’S’-shapedfeature is due to an intrinsic magneto-optic structure in ~σxy and not to an optical-enhancement

effect. This conclusion is corroborated by the appearance of an ’S’-shaped feature in an

inhomogeneous bulk (In, Mn)As sample with MnAs clusters (see Fig. 4.2). Magneto-optically

active electronic transitions involving the Mn-3d band are also found in the same photon-

energy range in ferromagnetic perovskite-type La1-x (Ba, Sr)x MnO3 thin films [86].

4.2.5 Thickness and interlayer dependence of ferromagnetic order

Although the energy dependence of the Kerr spectra is influenced by the optical functions n

and k, the magnetic-field dependence of θK is solely due to spin-polarized electronic states

which are contributing to the magnetic moment and give rise to an off-diagonal conductivity,~σxy . A large remanent value of θK is observed, which amounts in some samples (see Fig. 4.9)

to 100% of the saturation value over the entire photon energy range. This indicates that

electronic transitions involving ferromagnetically ordered spins are magneto-optically active

over the complete photon-energy range measured in this work.There are two different types of electronic transitions which contribute to ~σxy , interband and

intraband transitions, displaying different photon-energy dependencies as discussed in sections2.4-2.6. The contribution to ~σxy of intraband transitions, i. e. transitions within the conduction

band for n-type and the valence band for p-type charge carriers is given by Eq. (2.76). From the

high-frequency limits of σ1xy and σ2xy (see Eq. (2.78)) it is concluded that intraband

contribution to the Kerr spectra will be negligible at higher photon energies. By evaluating the

parameters involved in Eqs. (2.76) (see section 2.6), a typical threshold photon energy of 1 eV

can be estimated [48, 50]. Hence, the Kerr spectra of the (In, Mn)As heterostructures are

dominated by interband transitions because spin-polarized electronic transitions are magneto-

optically active over a much wider photon-energy range.

The magnetization is due to localized, spin-polarized Mn-3d electrons. The magnetic

exchange is mediated by holes which are spread in both, the (In, Mn)As and AlSb layers, and of

quasi two-dimensional nature [82]. The hole states participating in the magnetic exchange will

become spin-polarized themselves. Therefore, magneto-optically active interband transitions

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4 (In, Mn)As HETEROSTRUCTURES 51

will involve in one way or another these electronic states. The Mn ions are surrounded by six

As ions similar to the NiAs structure of bulk MnAs [77, 78]. Magneto-optically active

transitions are hence transitions between the Mn-3d and the As-4p band and vice versa. Such

transitions are expected to be very strong because they are dipole-allowed, highly spin-

polarized and their radial overlap integral is large due to a high coordination number and small

Mn-As distance. Furthermore, the transitions are rather broad because the width of the p and d

band is usually more than 1 eV. In conclusion, the Kerr-rotation spectra are due to electronic

interband transitions confined to the (In, Mn)As layer. Magneto-optic spectroscopy therefore

probes the magnetic behavior of exclusively the Mn-3d local moments [83].

With the help of the observed results, we are able to discuss the influence of (In, Mn)As-

layer and (In, Al)As-interlayer thickness on the ferromagnetic exchange. By comparing the

magneto-optic hysteresis curves of (In, Mn)As/AlSb heterostructures with (In, Mn)As-layer

thickness dMS = 9 nm and 19 nm (see Fig. 4.10), we find that the coercive field Hc decreases bya factor of two with increasing dMS. Furthermore, the squareness, θ θK

remKsat decreases from

100% to 65%. A heterostructure with 9 nm thick (In, Mn)As layer grown on a GaSb layer (see

Fig. 4.10(c)), however, displays a squareness of 100% while Hc is reduced to half the value of

the corresponding (In, Mn)As/AlSb heterostructure. This indicates a strong correlation between

perpendicular magnetic anisotropy and strain-induced crystal anisotropy caused by a lattice

mismatch of 0.6% and 1.3% between the (In, Mn)As layer and, respectively, the GaSb and

AlSb layer [82].

Finally, the influence on the ferromagnetic exchange of an In0.5Al0.5As-spacer layer between

(In, Mn)As and AlSb layer is discussed. Introducing the spacer layer leads to a pronounced

decrease in coercivity and hysteresis behavior in the Hall resistance [76] indicating a

disappearance of ferromagnetic order with increasing spacer-layer thickness, dsp. This

observation has previously been explained by carrier-induced ferromagnetism across the

(In, Mn)As/AlSb interface [76] involving two-dimensional hole states specifically in the AlSb

layer. The basic idea is that the holes in the AlSb layer would establish an RKKY-like exchange

interaction with the Mn ions in the (In, Mn)As layer which would lead to long-range

ferromagnetic order. A spacer layer acting as a potential barrier will suppress the penetration of

the hole wave function into the (In, Mn)As layer and therefore reduce the ferromagnetic

exchange between the Mn ions. Nevertheless, the corresponding Kerr hysteresis loops, which

are plotted in Fig. 4.12, clearly demonstrate ferromagnetic order even in a heterostructure with

dsp = 24 nm, where the Hall resistivity gives just a linear dependence on magnetic field [76].

Because the Kerr rotation is only sensitive to the (In, Mn)As layer, as has been justified above,

it is concluded that the In0.5Al0.5As interlayer does not upset ferromagnetic ordering but

suppresses perpendicular magnetic anisotropy [83]. The disappearance of low-field hysteretic

behavior in the Hall resistance for the dMS = 24 nm sample [76] may be explained in terms of

parallel conduction due to two hole channels in both (In, Mn)As and AlSb layers, which

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52 4 (In, Mn)As HETEROSTRUCTURES

sometimes occurs in modulation-doped (Ga, Al)As/GaAs heterostructures with heavy doping.

In summary, we can conclude that the ferromagnetic exchange between the Mn-3d local

moments is carried by the hole states within the (In, Mn)As layer.

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5 Co(EuS) MACROSCOPIC FERRIMAGNETS 53

5 Co(EuS) MACROSCOPIC FERRIMAGNETS

5.1 Introduction

Magnetic thin films are of continuous interest to the scientific community as they have a

high technological potential. They are widely used in mass storage devices, as e.g. in

conventional magnetic hard and floppy disks, magnetic tapes, and - more recently - in magneto-

optic data-storage devices. With the discovery of the giant magnetoresistance (GMR) effect in

magnetic multilayers [87], an application of magnetic thin films as a sensor in disk reading

heads seems likely in near future. The fact that the interface plays a crucial role in producing

GMR effects and oscillatory exchange [88] has prompted for granular or phase-separated

systems where the interface-to-volume ratio is increased as compared to multilayers [89]. A

new class of such phase-separated materials has been recently found and named macroscopic

ferrimagnets [90]. In these materials two macroscopic magnetic phases couple

antiferromagnetically across the phase boundary. They show many of the properties of

ferrimagnets as, e. g., a magnetic compensation point.

The phenomena of phase-separation and its influence on the magnetic and magneto-optic

properties is further pursued in this chapter on macroscopic ferrimagnets. In the (In, Mn)As

diluted magnetic semiconductors discussed in the previous chapter, the manganese provides the

local magnetic moments and forms local MnAs clusters with NiAs-type structure. As a function

of substrate temperature, the MnAs will either exist as subnanometer-sized local clusters within

the semiconducting InAs host lattice or segregate to a macroscopic metallic MnAs phase. This

radically changes the magnetic properties of the material. In the macroscopic ferrimagnets, the

clusters are several nanometers in size. In contrast to the (In, Mn)As heterostructures, the

situation is reversed in that in macroscopic ferrimagnets the clusters are semiconducting and the

matrix is metallic. Furthermore, both phases are magnetic.

Films of thickness 200 nm were prepared by electron-beam co-deposition of Co and EuS at a

rate of 0.3 nm s-1. The Co source was a conventional e-beam hearth. The EuS was purchased

as powder from Ames Laboratory and was evaporated from a tungsten hearth liner. The

substrates were glass, fused SiO2 or silicon. The substrates were cooled to -15° C by passing

helium gas cooled to liquid nitrogen temperature through the substrate support block. Phase

separation is achieved solely by co-evaporating onto slightly cooled substrates. No further heat

treatment like annealing or quenching is needed to accomplish a phase separation. As a result,

the semiconducting compound forms nanocrystalline particles which couple antiferromagneti-

cally to the metallic matrix across the phase boundary. The nature of the magnetic exchange has

been established with the help of magneto-optic spectroscopy [90] which has proven to be an

efficient tool to investigate such complicated systems. We will discuss in the following

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54 5 Co(EuS) MACROSCOPIC FERRIMAGNETS

sections the magnetic and magneto-optic properties at low temperatures and at room

temperature.

Co

EuS crystals (~ 10 nm diameter)

200 nm

Fig. 5.1. Outline of a Co1-x

(EuS)x macroscopic ferrimagnet.

A typical example of a macroscopic ferrimagnet is Co1-x

(EuS)x consisting of a Co

matrix containing precipitate crystalline particles of EuS. A schematic drawing is given in

Fig. 5.1. For Co-rich samples, the EuS phase forms nanocrystalline particles in a metallic Co

matrix. For low Co content, the Co acts a magnetic impurity in a semiconducting matrix. The

latter case had been studied thoroughly when rare-earth chalcogenides were investigated twenty

years ago [91, 92]. The NaCl lattice constant of EuS, aEuS = 5.968 Å, is much larger than

aCo = 3.530 Å of fcc Co. Hence, the volume per formula unit is larger by a factor of five. So,

adding 10% EuS means a volume fraction of 35% EuS. Table 5.1 summarizes the relation for

the samples discussed in this section. The actual EuS concentration has been obtained by

energy-dispersive x-ray analysis (EDX).

xEuS (nominal) 0 0.1 0.2 0.3 0.4 0.5 0.6

xEuS (EDX) 0 0.11 0.23 0.31 0.48 0.59 0.72

vol. EuS (%) 0 35 54 67 76 83 88

Table 5.1. Relation between atomic percentage and volume fraction of EuS in themacroscopic ferrimagnet Co1-x(EuS)x.

The phase-separation of Co and EuS is reflected in the magneto-optic [90] and optical prop-

erties [93] as will be demonstrated later. However, it was first discovered in x-ray-diffraction

measurements [94]. The conductivity, σ, as a function of volume fraction Co, pCo, is shown in

Fig. 5.2. The conductivity increases rapidly with pCo from 2.174 ⋅ 102 Ω-1m-1 at pCo = 0.12 to

1.84 ⋅ 105 Ω-1m-1 at pCo = 0.65. The concentration dependence suggests a percolation

behavior

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5 Co(EuS) MACROSCOPIC FERRIMAGNETS 55

1 10 100102

103

104

105

0.3

x = 0.2

0.5

0.4

0.6 σm = 2.2 ⋅ 10 5 Ω -1m -1

pcrit = 0.12

Co 1-x (EuS) x

σ (pCo) = σm (pCo - pcrit )1.7

σ ( Ω

-1 m

-1

)

pCo - pc (vol. fraction in % )

Fig. 5.2. Conductivity of Co1-x(EuS)x as a function of volume fraction, pCo. The fit isaccording to a percolation model.

indicating that the EuS is present as a separate insulating phase. The conductivity can be fitted

according to a simple percolation model by

( ) ( )σ σp p pCo m Co crit= − 1 7. . (5.1)

Here, σm is roughly the conductivity of the metallic phase, pcrit is the volume fraction at the

percolation threshold, and the exponent 1.7 follows from theory in a three dimensional system.

Although the theory treats percolation in a metal-insulator system while Co-EuS is a metal-

semiconductor system, the conductivity can be fitted up to a volume fraction of 50% Co. The

value σm = 2.2 ⋅ 105 Ω-1m-1 obtained from the fit is about half the value of the conductivity of

amorphous cobalt alloys [95]. This suggests that the Co is amorphous. However, it is also

conceivable that the Co has some sulfur or oxide impurities, reducing the conductivity.

X-ray analysis does not show a crystalline Co peak but a broad feature near 2Θ ≅ 21°

corroborating an amorphous Co matrix. However, Co is difficult to observe when using Cu-Kα

radiation. What is clearly established in the x-ray analysis [94] as shown in Fig. 5.3 is the

crystalline nature of the EuS phase further confirming the phase-separated nature of Co(EuS).

The peak at 2Θ = 29.9° is identified as the EuS (200) peak consistent with a NaCl structure

with a lattice constant of 5.97 Å, which is equal to the value of bulk EuS [96]. The (200) peak

is the strongest, as expected for randomly oriented EuS. Line broadening of the EuS x-ray peaks

indicates a crystallite size of the EuS phase of approximately 10 nm [94]. This analysis is based

on a model of line broadening by the finite size of the scattering volume [97]. The nanometer

size of the EuS precipitate particles is responsible for the interesting magnetic properties of the

compound. Let us estimate the mean free path, λCo, of the electrons in Co by a free electron

model [98],

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56 5 Co(EuS) MACROSCOPIC FERRIMAGNETS

20 30 40

EuS (200)x = 60

50

40

30

20Co 1-x (EuS) x

Inte

nsity

(a.

u.)

2Θ (deg)

Fig. 5.3. X-ray diffraction spectra of Co1-x(EuS)x. Note the crystalline EuS (200)peak at 29.9° which corresponds to a lattice constant of 5.97 Å.

λ τCo Fv= , (5.2)

where vF is the Fermi velocity,

vm eF

p=

3

4

2

2 2

13π ω , (5.3)

τ the relaxation time,

τ π σω

= 4 02p

, (5.4)

and ωp the plasma frequency,

ω πp

N e

m2

24=∗

. (5.5)

Assuming ω pCo = 1.17 ⋅ 1016 s-1 as obtained from optical measurements [99], and a dc

conductivity, σ0 = 1.72 ⋅ 107 Ω-1m-1 for bulk cobalt at room temperature, we get a mean free

electron path, λCo = 12 nm. This is of the same order as the size of the precipitate EuS particles.

Therefore, the EuS particles could act as strong scattering sites. The occurrence of a negative

magnetoresistance effect of δρ/ρ = 2% [94] is consistent with this assumption. Although the

value of 2% is quite low, the absolute change in resistivity, δρ(B = 1 T) = 8 ⋅ 10-7 Ω m is higher

than in granular GMR materials [89, 100]. The reason is the high intrinsic resistivity of

Co(EuS).

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5 Co(EuS) MACROSCOPIC FERRIMAGNETS 57

5.2 Low-temperature properties (antiferromagnetic exchange)

The magneto-optic properties at low temperatures of two compositions of the Co1-x(EuS)x

system, x = 0.2 and 0.3, will be emphasized in this section. Concerning magnetization, they are

representative of Co and EuS dominated compositions, respectively. In the composition

Co0.7(EuS)0.3, the magnetic moment is dominated by the EuS phase up to a compensation point

at about 18 K, then by the Co matrix. In contrast, the magnetization in the Co0.8(EuS)0.2

composition is dominated by the Co matrix at all temperatures up to the Curie point. The

dependence of the Kerr rotation θK on photon energy (eV) is shown in Fig. 5.4 for Co1-x(EuS)x

samples with EuS compositions x between 0.2 and 0.3 [90]. The spectra are measured at a

temperature of 5.5 K in an applied field of 3 T (solid symbols) and in the remanent state (open

symbols) after application of a large magnetic field. We find that the wavelength dependence of

the Co0.9(EuS)0.1 sample (not shown) is very similar to the published Kerr spectra of cobalt

[101].

-0.6

-0.4

-0.2

0.0

0.2

(a)

T = 5.5 K remanence

B = 3 T

Co 0.8 (EuS)

0.2

Pol

ar K

err

Rot

atio

n θ

K (

deg)

0 1 2 3 4 5-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Photon Energy (eV)

(b)

T = 5.5 K

remanence

B = 3 T

Co 0.7 (EuS)

0.3

Fig. 5.4. Polar Kerr rotation spectra at a temperature of 5.5 K of (a) Co0.8(EuS)0.2 and(b) Co0.7(EuS)0.3 in a field of 3 T (solid symbols) and in the remanent state (opensymbols).

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58 5 Co(EuS) MACROSCOPIC FERRIMAGNETS

Note that the remanence-state spectra of Co0.8(EuS)0.2 appears to be very similar to that of the

Co0.7(EuS)0.3 sample but with opposite sign. Also, the remanent state of the Co0.7(EuS)0.3

sample bears a strong resemblance to the Kerr spectrum of EuSe with negative peaks at about

2.2 and 4.2 eV [39, 102].

The different behavior of the remanent and the in-field Kerr spectra is indicative of a

magnetic exchange coupling between two magnetic subsystems. Else, the remanent spectra

should scale with the in-field spectra because the Kerr effect is proportional to the

magnetization. In order to elucidate what the magnetic subsystems could be and what the nature

of the magnetic exchange is, we plot in Fig. 5.5 the polar Kerr spectra of a Co thin film at room

temperature and EuS thin film at 5.5 K [103]. Co exhibits two shallow minima in θK, one at

1.4 eV and one at 3.9 eV while θK of EuS has a pronounced negative peak at 1.8 eV

corresponding to the localized Eu2+ 4f→5d(t2g) transition [63]. Reverting to Fig. 5.4, we notice

that θK of Co0.8(EuS)0.2 at 3 T resembles the Co spectrum and θK of Co0.7(EuS)0.3 looks like the

EuS spectrum. Focusing on the difference between in-field and remanence spectra reveals that

in the case of Co0.7(EuS)0.3 it appears to be a constant positive shift while in Co0.8(EuS)0.2 new

features emerge with opposite sign as θK of Co0.7(EuS)0.3. These findings can be summarized by

assuming an antiferromagnetic exchange coupling between the Co matrix and the EuS

precipitate particles, with Co being the dominating magnetic phase in Co0.8(EuS)0.2 while the

EuS phase dominates in Co0.7(EuS)0.3.

-0.5

0.0T =295 K

B = 3 T(a) Co

Pola

r K

err

Rot

atio

n (d

eg)

0 1 2 3 4 5

-5

0

5

Photon Energy (eV)

(b) EuS T =5.5 K

B = 3 T

Fig. 5.5. Polar Kerr rotation spectra at a temperature well below TC of (a) a Co and(b) a EuS thin film in a field of 3 T.

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5 Co(EuS) MACROSCOPIC FERRIMAGNETS 59

A B

B

θKθK

B

(a) (b)

Fig. 5.6. Hypothetical hysteresis loops of an antiferromagnetically exchange-coupledsystem. (a) Loop of the dominating magnetic subsystem, (b) loop of theantiferromagnetically exchange-coupled subordinate subsystem. Arrows indicate thedirection of the subsystem magnetization. Note the nonlinear behavior of the high-field susceptibility of (b).

Of course, this is just an evidence but not a proof. A proof would be a magnetization

measurement, which is sensitive to the individual magnetic subsystems. Applying this

measurement to the model proposed above would yield hysteresis loops as displayed in Fig. 5.6.

The dominant magnetic subsystem shows a square hysteresis loop. The subordinate magnetic

subsystem, due to the antiferromagnetic exchange coupling, exhibits a nonlinear behavior of the

high-field susceptibility, characteristic of the breaking of the antiferromagnetic coupling.

Naturally, this is only observed in external magnetic fields high enough to break up the

coupling.

Now, we may take advantage of the power of Kerr spectroscopy. Supposing that the two

subsystems are the Co matrix and the EuS precipitate particles, then all we have to do is finding

a spectral range where either the Co or the EuS contribution to the Kerr effect is governing.

From Fig. 5.5, it follows that EuS has a major contribution at the 4f→5d transition energy while

Co is predominant at the lowest photon energies because EuS has hardly any Kerr rotation

below the gap, EgEuS = 1.65 eV. Therefore, polar Kerr hysteresis loops for both compositions are

taken at 1.15 and 2.15 eV, corresponding to an energy much lower than EgEuS and to the position

of the dominating peak in θK (see Fig. 5.4), respectively. The resulting loops are plotted in

Fig. 5.7.

Note the change in sign of the inner loop with photon energy for a given composition and

with composition for a given photon energy. Comparing with the model loops of Fig. 5.6, we

see that Co0.8(EuS)0.2 has a normal loop at 1.15 eV, the photon energy where the Co phase

should be dominating while at 2.15 eV, an nonlinear high-field susceptibility is observed.

Hence, the Co matrix is the governing magnetic subsystem and the EuS phase is

antiferromagnetically exchange coupled to the Co. For the Co0.7(EuS)0.3 sample, the situation is

exactly reversed. However, here the case is not as clear as the Co contribution to θK is not

negligible at 2.15 eV.

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60 5 Co(EuS) MACROSCOPIC FERRIMAGNETS

0.0

0.2

0.4 Co 0.7 (EuS)

0.3

(c)

ω = 1.15 eV0.0

0.2

0.4 Co 0.8 (EuS)

0.2

Pola

r K

err

Rot

atio

n θ

K (

deg)

(a)

ω = 1.15 eV

-5 0 5

-0.4

0.0

0.4

0.8

(d)

T = 5.5 K

ω = 2.15 eV

Magnetic Field (T)-5 0 5

-0.4

0.0

0.4

T = 5.5 K

(b)ω = 2.15 eV

Fig. 5.7. Polar Kerr hysteresis loops at 5.5 K of Co0.8(EuS)0.2 at a photon energy of1.15 eV (a) and 2.15 eV (b) and of Co0.7(EuS)0.3 at a photon energy of 1.15 eV (c) and2.15 eV (d).

To make the antiferromagnetic exchange coupling even more apparent, the polar Kerr loops

are decomposed into the Co and EuS contributions as shown in Fig. 5.8 by assuming that the

Co contribution to θK is of the same size at 1.15 and 2.15 eV. The loops now perfectly agree

with the theoretical prediction. Because of the localized nature of the Eu2+ 4f→5d transition,

0.0

0.2

0.4

0.6 Co 0.7 (EuS)

0.3

(c)

Co contrib.0.0

0.2

0.4

0.6

Pola

r K

err

Rot

atio

n θ

K (

deg)

Co 0.8 (EuS)

0.2

Co contrib.

(a)

-5 0 5-0.8

-0.4

0.0

0.4

T = 5.5 K

(d)

ω = 2.15 eV

EuS contrib.

Magnetic Field (T)-5 0 5

-0.4

0.0

0.4

T = 5.5 Kω = 2.15 eV

EuS contrib.(b)

Fig. 5.8. Decomposition of the polar Kerr hysteresis loop at 2.15 eV of Co0.8(EuS)0.2

in (a) a Co and (b) a EuS contribution and of Co0.7(EuS)0.3 in (c) a Co and (d) a EuScontribution.

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5 Co(EuS) MACROSCOPIC FERRIMAGNETS 61

magneto-optic spectroscopy can be element-specific in this phase-separated system making it

the ideal technique in order to investigate the magnetic properties.

5.3 Temperature dependence of the magneto-optic properties

Having established the antiferromagnetic exchange coupling between the Co matrix and the

EuS precipitate particles, i. e., between two macroscopic phases, questions about the strength

and the origin of the coupling arise. Assuming the crystalline lattice constants for the Co and

EuS phase and the full size of the ordered magnetic moments for the Co (1.72 µB) and EuS

(7 µB) phase, the magnetic compensation point should be at an EuS concentration of x = 0.2.

This is in good agreement with the magnetic compensation observed between x = 0.2 and 0.3.

This means that the major part of the volume of the EuS particles couples to the Co. The

strength of the exchange-coupling field can be estimated from the high-field susceptibility of θK

in Fig. 5.8. Saturation is obtained at fields of ≈10 T. In order to analyze the influence of the

exchange on the TC of the EuS phase, the temperature dependence of the Kerr hysteresis loops

has been studied over a wide energy range [90, 104].

As an example, the temperature dependence of θK of the Co0.7(EuS)0.3 sample is shown in

Fig. 5.9 for a photon energy of 2.15 eV, corresponding to the negative peak in θK [90]. Note

that the inner rectangular loop has a negative sign at 5.5 K and that there is a significant

negative high field susceptibility in θK. This behavior, a negative rectangular inner loop and

negative high field susceptibility, persists up to 20 K. At this temperature, the inner loop has

changed sign to positive but the high field loop is still negative with more curvature at high

fields. The change in sign of the inner loop is consistent with magnetization compensation, so

that at 20 K and above the system is Co dominated. The remanent rotation decreases rapidly

from 22.5 K to 30 K. At 30 K the inner loop has collapsed to about 0.02° but the reverse loop

wings beginning at about 0.6 T indicate that there is still hysteresis at this temperature.

Therefore, the low rotation does not indicate the approach of TC. At 35 K a negative inner loop

reappears. The collapse of θK and the reversal of sign at 30 K is caused by Kerr effect

compensation. At this temperature and photon energy a magneto-optical compensation point is

reached, i. e., the Kerr-rotation contribution of the Co matrix is equal in magnitude and

opposite in sign to that of the EuS phase. Whereas magnetization compensation occurs at the

same temperature independent of the photon energy of the light, Kerr-effect compensation is

extremely wavelength dependent.

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62 5 Co(EuS) MACROSCOPIC FERRIMAGNETS

-0.5

0.0

0.5(a)

Co 0.7 (EuS)

0.3

T = 5.5 K -0.5

0.0

0.5(e)

ω = 2.15 eV

T = 22.5 K

-0.5

0.0

0.5(b)

T = 10 K -0.5

0.0

0.5(f)

T = 25 K

-0.5

0.0

0.5(c)

T = 15 K -0.5

0.0

0.5(g)

T = 30 K

-5 0 5

-0.5

0.0

0.5(d)

T = 20 K

-5 0 5

-0.5

0.0

0.5(h)

Magnetic Field (T)

Pola

r K

err

Rot

atio

n

θK (

deg)

T = 35 K

Fig. 5.9. Temperature dependence of polar Kerr hysteresis loops at 2.15 eV ofCo0.7(EuS)0.3 at (a) 5.5 K, (b) 10 K, (c) 15 K, (d) 20 K, (e) 22.5 K, (f) 25 K, (g) 30 K,and (h) 35 K.

The occurrence of a magnetic and magneto-optic compensation point above TC = 16.5 K of

bulk EuS shows an exchange-induced enhancement of the ordering temperature of the EuS

phase. In order to further investigate the enhancement of the Curie temperature of the EuS

phase, the Kerr spectra in the remanence state have been measured for all samples at

temperatures up to 300 K. As an example, the temperature dependent θK spectra of

Co0.6(EuS)0.4 are plotted in Fig. 5.10 [105]. The characteristic ‘finger print’ of the EuS phase,

the ‘S’-shaped feature between 2 and 3 eV, serves as an indicator of an ordered magnetic

moment in the EuS phase. Between 80 and 90 K, a reversal of the complete spectra is observed,

indicating a magnetic compensation point at 85 K which is more than a factor of five higher

than TC of bulk EuS. The ‘S’-shaped structure persists up to a temperature of 160 K. Therefore,

the Curie temperature of the EuS phase is enhanced through the antiferromagnetic exchange-

coupling to the Co matrix by an order of magnitude as compared to TC of bulk EuS.

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5 Co(EuS) MACROSCOPIC FERRIMAGNETS 63

0 1 2 3

-0.2

0.0

0.2

0.4

160 K

120 K

90 K

7 K

T

80 K

45 KB = remanence

Co0.6(EuS)0.4

Pola

r K

err

Rot

atio

n (d

eg)

Photon Energy (eV)

Fig. 5.10. Temperature dependence of the polar Kerr spectra of Co0.6(EuS)0.4 in theremanence state.

These findings are summarized in Fig. 5.11 where the EuS contribution is extracted from the

θK values at 2.8 eV (EuS dominated photon-energy range) and 1.1 eV (Co dominated part of the

spectrum). The extraction assumes that the optical functions n and k do not change considerably

below 200 K. Then, the Co contribution to θK at 2.8 eV can be estimated from the Kerr spectra

at a temperature, where the ‘S’-shaped structure, and hence the ordering of the magnetic

moments of the EuS phase, has disappeared. This is the case at T0 = 200 K. The Co contributionat 2.8 eV, ( )θK

Co, .2 8 T , then follows at lower temperature from the scaling relation,

( ) ( ) ( ) ( )θ θ θ θKCo

KCo

KCo

KCo

, . , . , . , .2 8 11 2 8 0 11 0T T T T= ⋅ . The EuS contribution as plotted in Fig. 5.11 is then the

difference between θK(T) at 2.8 eV and ( )θKCo

, .2 8 T .

0 50 100 150 200 250

-0.2

0.0

0.2

0.4

TC

Tcomp

ω = 2.8 eV (EuS dominated)

ω = 1.1 eV (Co dominated)

EuS contribution

B = remanence

Co0.6(EuS)0.4Pola

r K

err

Rot

atio

n (d

eg)

Temperature (K)

Fig. 5.11. Extraction of the EuS contribution from the temperature dependent Kerrspectra of Co0.6(EuS)0.4 by using the θK values at 1.1 eV (Co-dominated part of thespectrum) and 2.8 eV (EuS-dominated region). Compensation point and Curietemperature are marked by an arrow.

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64 5 Co(EuS) MACROSCOPIC FERRIMAGNETS

The same relations can be applied to the peak at 2.15 eV [104] yielding the same TC

enhancement to within 10 K. The evaluation at 2.8 eV has a smaller error because the Kerr

rotation is enhanced at that energy due to an optical-enhancement effect which will be

discussed in the next section.

The EuS contribution to the Kerr spectra as a function of temperature and the resulting TC

enhancement have been derived for all EuS concentrations using the method outlined above

[104]. An independent evaluation of the TC can be performed by evaluating the high-field

susceptibility of the polar Kerr hysteresis loops at room temperature [104]. The assumption is

that the high-field susceptibility is solely due to a paramagnetic EuS contribution as the Co

moment is saturated at room temperature. Because of the ‘S’-shaped signature, the polar Kerr

loops at 2.8 eV and 2.15 have the opposite sign of the EuS contribution while the Co

contribution has the same sign. Thus by subtracting both loops, the Co contribution cancels and

we are left with twice the (paramagnetic) EuS contribution, which are straight lines as shown in

Fig. 5.12.

-3 -2 -1 0 1 2 3

-0.05

0.00

0.05 linear extrapolation

Co1-x (EuS)x

x = 0.4 x = 0.5 x = 0.6 T = 295 K

paramagnetic

EuS contribution

x = 0.1 x = 0.2 x = 0.3

Pola

r K

err

Rot

atio

n (d

eg)

Magnetic Field (T)

Fig. 5.12. Difference of the room-temperature polar Kerr hysteresis loops at 2.15 eVand 2.8 eV corresponding to the high-field susceptibility which amounts to twice theparamagnetic Kerr rotation of the EuS phase.

Supposing that the magnetic moments of the EuS phase follow a Curie-Weiß law, the slopeof paramagnetic Kerr rotation of the EuS phase, θK

para H , is written as

( )( )

θ κ χ κ µKpara

paraB

B paraH

N J J g

k T= = +

−1

3

2 2

Θ , (5.6)

where H is the magnetic field, g = 2 is the Landé factor of the EuS 8S7/2 ground state, and κ =

θK(M)/M is the proportionality constant between Kerr rotation and magnetization, M. From this

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5 Co(EuS) MACROSCOPIC FERRIMAGNETS 65

equation, we can calculate the (paramagnetic) Curie temperature, Θpara which is close to TC.

Inserting the definition of the saturation magnetization, Ms,

M N g Js B= µ , (5.7)

and using κ = θK(Ms)/Ms = θKsat /Ms, where θK

sat is obtained from the low-temperature Kerr

spectra, the slope of the Kerr rotation is expressed as

( )( )

θ θ µKpara

Ksat B

B paraB

J g

k T= +

−1

3 Θ . (5.8)

From this equation, Θpara is easily calculated. The result is plotted in Fig. 5.13 in combination

with the TC as derived from the temperature dependence of the remanence Kerr spectra.

0.0 0.2 0.4 0.60

50

100

150 calc. from θ rem

K

Co 1-x(EuS) x calc. from χpara

T c (K

)

EuS concentration x

Fig. 5.13. Curie temperature of the EuS phase as derived from the temperaturedependence of the remanence Kerr spectra () and from the high-field susceptibilityof the Kerr hysteresis loops at room temperature (). Note the high correspondencebetween the two values.

Obviously, the ordering temperature peaks at an EuS composition x = 0.4 close to the

percolation limit. Apparently, at that concentration the exchange between Co and EuS is

maximized. What is the nature of such a strong exchange coupling between macroscopic

phases? From earlier work on rare-earth chalcogenides [92, 103, 106] it is well known, that

adding to EuX (X = O, S, Se, Te) a trivalent ion, as e. g., Gd, Tb or Y, leads to a TC

enhancement of up to 140 K. However, the magnetic exchange is always ferromagnetic and can

be explained in terms of a Ruderman-Kittel-Kasuya-Yoshida (RKKY)-type interaction. The

addition a ferromagnetic transition element, Co, Fe, or Ni, can only be done at low

concentrations without destroying the NaCl structure. Again a TC enhancement up to 190 K is

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66 5 Co(EuS) MACROSCOPIC FERRIMAGNETS

observed in Fe-doped EuO but the exchange is once more ferromagnetic [91]. Adding larger

amounts of Fe leads to phase separation but only the portion of Fe dissolved in the EuO is

responsible for the enhancement [107]. The effect was theoretically described by the formation

of a cluster of EuO spins which are exchange coupled to the impurity transition metal in the

center [108]. The exchange is mediated by an f-d interaction and a single impurity ion can

couple 2000 spins. However, this exchange leads again to a ferromagnetic ordering.

The only antiferromagnetic exchange known between rare-earth ions and transition metals is

in rare-earth-transition-metal alloys [109]. The heavy rare-earth elements, as e. g., Gd, Tb,

exhibit an antiferromagnetic coupling to the magnetic transition metal (Co, Fe) while the light

rare-earths (as e. g., Nd) are ferromagnetically coupled. Alloying with Eu will yield trivalent

Eu3+ which is paramagnetic [110]. However, as divalent Eu2+ is isoelectric to trivalent Gd3+ it is

conceivable that a hypothetical Eu2+-transition-metal alloy would be antiferromagnetically

coupled. We have shown that in Co(EuS) the TC enhancement is largest near the percolation

threshold. There, the intermixing between the two phases is the largest while still having

metallic conductivity in order to guarantee magnetic ordering of the Co phase. Due to the

crystalline nature of the EuS phase, the Eu is forced to be divalent. The small size of the

precipitate EuS particles leads to a very large surface-to-volume ratio yielding close proximity

to the Co phase of the Eu2+ ions at the surface. So in fact, this phase-separated material exactly

creates a hypothetical Eu2+-transition-metal alloy similar to an amorphous Gd-Co alloy. This

would explain the antiferromagnetic exchange coupling between the EuS precipitate particles

and the Co matrix.

5.4 Room-temperature properties (optical-enhancement effect)

In the last section about macroscopic ferrimagnets the room temperature properties will be

discussed [93]. At room temperature, there is no longer a perpendicular magnetic anisotropy in

Co(EuS) as is evident from the polar Kerr hysteresis loops plotted in Fig. 5.14 exhibiting a

typical in-plane shape. There is a small high-field susceptibility, growing with increasing EuS

concentration, x, which had been used to derive the paramagnetic Curie temperature in the

previous section. However, the magnetic ordering is now dominated by the Co phase and at

x ≥ 0.5, long-range magnetic order is lost. Perpendicular magnetic anisotropy can be attained at

room temperature, while retaining phase separation, by adding Tb metal to the Co(EuS) system

[111]. This will also lead to an antiferromagnetic coupling of the EuS phase and the Tb

magnetic moments to Co at low temperatures. Nevertheless, it is not clear to date whether the

coupling persists at room temperature. Some evidence of a room-temperature coupling of the

EuS phase to Co is found in the Kerr spectra [111] but further evidence is needed to proof this

assumption.

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5 Co(EuS) MACROSCOPIC FERRIMAGNETS 67

-3 -2 -1 0 1 2 3-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6Co1-x (EuS)x

x = 0.4 x = 0.5 x = 0.6

T = 295 Kω = 2.15 eV

x = 0.1 x = 0.2 x = 0.3

Pola

r K

err

Rot

atio

n (d

eg)

Magnetic Field (T)

Fig. 5.14. Polar Kerr hysteresis loops of Co1-x(EuS)x at room temperature at a photonenergy of 2.15 eV.

In Fig. 5.15 room-temperature spectra of (a) the polar Kerr rotation, θK, and (b) the Kerr

ellipticity, ηK, of Co1-x(EuS)x are plotted as a function of photon energy. For EuS concentration

x = 0.1 a strong enhancement of θK is found in the ultraviolet region as compared to pure Co

metal (see Fig. 5.5), showing a maximum θK of -1.75° at 4.6 eV, which is four times the

maximum rotation of Co. The ellipticity, ηK, reaches values above 1° in the same energy range.

For higher EuS-doped samples, a sign change occurs in θK and ηK around 4 eV and for x = 0.3 a

maximum value θK = 2° is obtained at 4.4 eV. For x > 0.3, the Kerr response strongly decreases

across the whole energy range.

The enhancement in θK and ηK for x ≤ 0.3 is connected with a broad dip in the reflectivity

above 4 eV as shown in Fig. 5.16. The position of the dip does not to depend on the EuS

concentration for the three samples with the highest Co content. Therefore, the dip cannot be

assigned to a plasma-minimum leading to a plasma-edge enhancement as discussed in section

2.8.3. The plasma frequency, ωp, as defined in Eq. (5.5), is proportional to the square root of the

free-carrier concentration, N. Therefore, the energy position of the dip should scale with thesquare root, NCo , of the metallic Co matrix. Due to the percolation behavior, NCo varies over

several orders of magnitude in these samples (see Fig. 5.2) and hence the dip should shift to

lower photon energies with increasing EuS concentration which is not observed.

At low photon energies, the reflectivity, R, shows an increase with decreasing energy which

is typical for metallic materials. For x > 0.3, no dip in reflectivity and no increase of the

reflectivity towards lower photon energies is observed indicating semiconducting behavior

consistent with transport measurements.

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68 5 Co(EuS) MACROSCOPIC FERRIMAGNETS

-2

-1

0

1

2 x = 0.1 x = 0.2 x = 0.3 x = 0.4 x = 0.5

(a)

Pola

r K

err

Elli

ptic

ity (

deg)

T = 295 K

B = 2.77 T

Co1-x(EuS)x

Pola

r K

err

Rot

atio

n (d

eg)

0 1 2 3 4 5

-1

0

1

Photon Energy (eV)

x = 0.1 x = 0.2 x = 0.3 x = 0.4 x = 0.5

(b)

Co1-x(EuS)x

B = 2.77 T

T = 295 K

Fig. 5.15. Polar Kerr rotation (a) and ellipticity (b) spectra of Co1-x(EuS)x at roomtemperature in an applied field of 2.77 T.

0 1 2 3 4 50

10

20

30

40

50

x = 0.1 x = 0.2 x = 0.3 x = 0.4 x = 0.5

T = 295 K

Co1-x(EuS)x

Ref

lect

ivity

(%

)

Photon Energy (eV)

Fig. 5.16. Reflectivity spectra of Co1-x(EuS)x at room temperature.

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5 Co(EuS) MACROSCOPIC FERRIMAGNETS 69

Kerr effect and reflectivity data clearly reflect the percolation behavior discussed in the

previous sections as large Kerr rotations and ellipticities are only obtained for x ≤ 0.3 and the

decrease in reflectivity above 4 eV vanishes for higher EuS concentrations. In addition, the

reflectivity spectra switch from metallic to semiconducting behavior for x > 0.3 as indicated by

the different energy dependencies at low photon energies. However, the transition from metallic

to semiconducting behavior occurs at lower EuS concentrations (x ≅ 0.4) as obtained from

transport measurements, where the percolation threshold pcrit = 0.12 corresponds to x = 0.6. In

regard of an application as a magneto-optic storage medium, the shot-noise limited figure of

merit, FOM, where

FOM R K K= +θ η2 2 , (5.9)

is displayed in Fig. 5.17 for the three metallic samples with x ≤ 0.3. The FOM reaches a

maximum value of 0.34° in Co0.9(EuS)0.1 at 3.5 eV. At the technologically interesting photon

energy of 3 eV, the FOM is 0.32° which is comparable to Co1-xPtx alloys and multilayers [112].

0 1 2 3 4 50.0

0.1

0.2

0.3

0.4

Figu

re o

f M

erit

(deg

)

Photon Energy (eV)

x = 0.1 x = 0.2 x = 0.3

Co1-x(EuS)x

B = 2.77 T

T = 295 K

Fig. 5.17. Figure of merit of Co1-x(EuS)x at room temperature.

In order to understand the behavior of the Kerr effect, the origin of the enhancement effect

has to be determined. It is well known that optical-enhancement effects are quite often

connected with thin dielectric oxide overlayers on top of the sample leading to interference

effects as described in section 2.8.2. The condition for magneto-optic enhancement by a

dielectric overlayer is a destructive interference (antireflection condition) of the light reflected

from front and back surface of the overlayer according to Eq. (2.100). Considering a CoO

overlayer and setting nd = 2.5, we get a thickness of d = 28 nm for an antireflection condition at

4.5 eV. Although an oxide layer, CoO or Eu3O4, on top of the samples is possible, a thickness

of almost 30 nm seems very unlikely. Furthermore, the thickness of such an extended oxide

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70 5 Co(EuS) MACROSCOPIC FERRIMAGNETS

layer should depend on Co or EuS concentration which would yield a shift with EuS

concentration of the dip in the reflectivity spectra. This is not observed in the experiment.

In order to explain the enhancement effect, one has to consider the phase-separated nature of

these materials. Co is a metal thus highly absorbing while EuS is a semiconductor with a low

absorption up to 4 eV. The penetration depth of light in Co is only 15 nm at 4 eV [99] and

hence the same size as the diameter of the EuS particles. Microscopically, we must therefore

expect a very complicated behavior of the reflectivity. The size of the EuS particles being

smaller than the wavelength of the light, a first approach to model the reflectivity behavior is to

use an effective medium theory and to average the dielectric function, ~ε , of Co and EuS

according to their volume fraction, v. This yields

~ ~ ~ε ε ε= +v vCo Co EuS EuS . (5.10)

The reflectivity, R, at normal incidence is then obtained using the well known relations

~ ~n2 = ε , (5.11)

and

Rn

n=

−+

~

~1

1

2

2 . (5.12)

The result of this calculation is shown in Fig. 5.18 (solid lines). The model describes the

changes at low photon energies from metallic to semiconducting behavior quite well but it fails

to describe the decrease in reflectivity above 4 eV.

0 1 2 3 4 50

20

40

60

80 Co1-x(EuS)x

x = 0.3

x = 0.4

x = 0.2x = 0.1

EuS

Co

Ref

lect

ivit

y (

%)

Photon Energy (eV)

Fig. 5.18. Calculated reflectivity spectra of Co1-x(EuS)x. The dashed lines arepublished data for hexagonal Co [99] and crystalline EuS [113]. The solid lines areobtained by averaging the dielectric functions of Co and EuS according to theirvolume fraction.

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5 Co(EuS) MACROSCOPIC FERRIMAGNETS 71

A better agreement with experiment is obtained by looking at the reflectivity of an infinite

EuS/Co interface. In that case, the complex index of refraction at the interface, ~n , is given by

the ratio of the complex index of refraction of Co and of EuS:

~~

~nn

nEuS

Co

= . (5.13)

The reflectivity, R, can now again be computed by using Eq. (5.12). The solid line in Fig. 5.19

shows the result of this calculation. Obviously, this model can account for the dip above 4 eV

and it also would explain the insensitivity of the position of the dip with EuS concentration.

0 1 2 3 4 50

20

40

60

80

R(ñEuS/ñCo)

EuS

Co

Ref

lect

ivit

y (

%)

Photon Energy (eV)

Fig. 5.19. Calculated reflectivity spectra of for Co1-x(EuS)x. The dashed lines arepublished data for hexagonal Co [99] and crystalline EuS [113]. The solid line is thereflectivity at a EuS/Co interface as described in the text.

Scanning electron microscopy (SEM) indicates a rough surface consisting probably of EuS

particles covered by a few atomic layers of Co. The SEM pictures are plotted in Fig. 5.20 for

Co1-x(EuS)x samples with x = 0.1, 0.2 0.5. At EuS concentrations where the enhancement effect

is observed (Fig. 5.20(a) and (b)), elongated grains are found close to the surface whereas for

semiconducting films a smooth surface is seen (Fig. 5.20(c)). It cannot be distinguished whether

the grains are EuS particles sticking out of the surface or a modulation of a thin Co top layer,

which is formed by EuS crystallites or voids just underneath the surface. In any case, the texture

seen in the SEM pictures is more than an order of magnitude larger than the supposed size of

the EuS crystallites derived from line broadening in the x-ray spectra. It is conceivable that

larger EuS particles form at the film surface, i. e., clusters of 10 nm crystallites.

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72 5 Co(EuS) MACROSCOPIC FERRIMAGNETS

Co (EuS)0.9 0.1(a)

10 µm

10 µm

(b) Co (EuS)0.8 0.2

(c)

10 µm

Co (EuS)0.5 0.5

Fig. 5.20. SEM pictures of Co1-x(EuS)x for x = 0.1, 0.2, and 0.5.

In order to understand why a model assuming an infinite EuS/Co interface as the main

contribution to the reflectivity behavior gives a good agreement with experiment, one could

argue as follows: A continuous, thin Co top layer would not significantly reduce the intensity of

the light before it reaches the EuS/Co phase boundary. Above 4 eV, the reflectivity at the

EuS/Co phase boundary is almost zero (solid line in Fig. 5.19) and most of the light will be

transmitted into the EuS particles. Within the EuS particles, almost no absorption takes place

because of the semiconducting nature of EuS [113] and at the second EuS/Co phase

boundary, the light is again almost completely transmitted. Because the penetration depth of

light in Co is of the same size as the diameter of the EuS particles, the light penetrated that far

will not be able to reach the surface again. Therefore, the reflectivity of the film is dominated

by the reflectivity of the EuS/Co interface.

A better understanding of the optical enhancement would be obtained if one could treat the

propagation of electromagnetic waves in a granular medium, where the small precipitate

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5 Co(EuS) MACROSCOPIC FERRIMAGNETS 73

particles as well as the matrix are absorbing. In principle, this is the same idea as in Mie’s

theory of scattering of small spherical particles in a transparent matrix [114]. However, the

transparency of the matrix is a fundamental requirement in Mie’s theory and an modification to

the case of an absorbing matrix has, to our knowledge, not yet been published. An extension of

Mie’s theory to the case of spherical particles with an absorbing overlayer embedded in a

transparent matrix has been accomplished long ago [115]. A further extension to the case of

multilayered spherical particles [116] and a simple recursion formalism of the same problem,

which is optimized for numerical calculations, has been published just recently [117]. Although

these modifications of Mie’s theory do not exactly describe our problem, we will try to

approximate the Co(EuS) system in terms of this modified Mie theory by a spherical EuS

particle with a thin Co coating in an ambient matrix as outlined in Fig. 5.21. Since the

penetration depth of light into Co is only 15 nm, the main contribution to the reflectivity should

come from a thin surface layer and multiple reflections on EuS precipitate particles are

negligible. What the treatment of a single EuS-Co particle will not be able to account for, is the

high metallic reflectivity of the Co matrix at low photon energies due to the large absorption of

Co. But in the photon-energy range where the optical-enhancement effect occurs, the

characteristic features of the Co/EuS interface should be reproduced. Moreover, the model

describes merely spherical particles and any effect arising from an elongated shape as indicated

in the SEM pictures (see Fig. 5.20) will not be adequately described.

Co

dCo

Con~

λ

rEuS

EuS

EuSn~

ambient

Fig. 5.21. Outline of a spherical EuS particle with Co overlayer in an ambient matrixused for the simulation of the optical behavior of the Co1-x(EuS)x system.

As the formalism of treating multilayered spherical particles in a transparent matrix [117] is

quite complicated, we will only discuss the results of the calculations [118]. Simulations have

been done for various EuS-particle sizes. The optical functions n and k were taken from

literature [99, 113]. It turns out that agreement with the experiment is best for an EuS-particle

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74 5 Co(EuS) MACROSCOPIC FERRIMAGNETS

size of 80 nm (rEuS = 40 nm) which is in between the size of the features observed in the SEM

pictures and the x-ray-derived diameter of the EuS crystallites.

0 1 2 3 4 5 60

10

20

2 nm

5 nm

10 nm

15 nm

dCo = 20 nm

Co(EuS)

Abs

orpt

ion

Cro

ss S

ectio

n (1

0-3

cm2 )

rEuS = 40 nm

Photon Energy (eV)

Fig. 5.22. Total absorption cross section of an EuS particle with Co overlayer in anambient matrix as a function of overlayer thickness.

The total absorption cross section, σabs, of a spherical EuS particle with radius of 40 nm is

plotted in Fig. 5.22 as a function of Co-overlayer thickness between 2 and 20 nm. The total

absorption is a measure of the ability of the particle to dissipate energy from the incoming

electromagnetic radiation. The total rate at which the energy is being absorbed by a single

obstacle, Wabs, integrated over the surface of a large encompassing sphere, is given by [119]

− = + +W W W Wabs in scatt int . (5.14)

Here, Win and Wscatt are the incident and scattered rate, respectively, and Wint is an interference

term describing the interaction between the incident and scattered waves. For a transparentmatrix, Win = 0 and, hence, W W Wabs scatt+ + =int 0 .

In Fig. 5.22, we notice that σabs reaches at photon energies above 4 eV a plateau at each Co-

overlayer thickness indicating strong absorption. This is consistent with the observed low

reflectivity in that energy range (see Fig. 5.16). A pronounced feature shifting from 2.5 to

2.9 eV is related to a local maximum in the absorptive (real) part of the optical conductivity of

crystalline EuS [113] due to the Eu2+ 4f→5d transition. This feature is not observed in the

Co(EuS) system probably because the peak is broadened by the small EuS-particle size.

Because the reflectivity, R, is measured at near-normal incidence, a quantity of our

simulations relating to R is the total backscattering cross section, σbs, as plotted in Fig. 5.23. At

low photon energies, σbs goes to zero because the wavelength of the incident electromagnetic

wave is much larger than the particle size. As a consequence, the particles do not interact any

more with the wave. As mentioned before, this is due the shortcoming of our model that the

metallic reflectivity of the Co matrix is not taken into account because the matrix is simplified

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5 Co(EuS) MACROSCOPIC FERRIMAGNETS 75

by a Co overlayer. Above 4 eV, σbs shows a pronounced decrease with increasing photon

energy. The reduction is the stronger the thicker the Co coating is reaching a maximum value

for dCo = 20 nm. This is again in agreement with the observed behavior of the reflectivity

spectra.

0 1 2 3 4 5 60

2

4

6

2 nm5 nm

10 nm

15 nm

dCo = 20 nmCo(EuS)

rEuS = 40 nmB

acks

catte

ring

Cro

ss S

ectio

n (1

0-3

cm2 )

Photon Energy (eV)

Fig. 5.23. Total backscattering cross section of an EuS particle with Co overlayer inan ambient matrix as a function of overlayer thickness.

In conclusion, these simulations corroborate the interpretation of the pronounced dip in the

reflectivity spectra by an accidental index matching of the complex index of refraction of Co

and EuS above 4 eV.

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76 5 Co(EuS) MACROSCOPIC FERRIMAGNETS

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6 MnBi-BASED MULTILAYERS 77

6 MnBi-BASED MULTILAYERS

6.1 Introduction

The last material to be discussed in this review is MnBi which is one of the most intensively

studied ferromagnetic materials. We started our review with (In, Mn)As diluted magnetic

semiconductors, where manganese provides the local magnetic moments and forms with the

arsenic local MnAs clusters with the same NiAs-type structure as the MnBi compound. As a

function of substrate temperature, the MnAs clusters will either exist as subnanometer-sized

local clusters within the semiconducting InAs crystal lattice or segregate in a macroscopic

metallic MnAs phase. This radically changes the magnetic properties of the material.

The phenomena of phase-separation and its influence on the magnetic and magneto-optic

properties has been further pursued in the next chapter on macroscopic ferrimagnets,

represented by the model system Co(EuS). Here, the EuS clusters are several nanometers in size

and, in contrast to the (In, Mn)As heterostructures, the situation is reversed in that the clusters

are semiconducting and the matrix is metallic. Furthermore, both phases are magnetic. A strong

antiferromagnetic exchange is found between the two macroscopic phases which is related to

the large surface-to-volume ratio of the EuS clusters while their size is still large enough to

allow magnetic ordering.

We conclude our review with the ferromagnetic MnBi system which has the same crystal

structure as the MnAs clusters in the (In, Mn)As heterostructures. It therefore represents the

limit in going from clusters of subnanometer size to an arbitrary large one. MnBi is the heaviest

compound by going along the pnictides series of MnAs, MnSb, MnBi and exhibits the highest

Curie temperature [120, 121]. This is just opposite to the europium chalcogenides, where EuO

has the higher TC than EuS [122, 123]. In MnBi, there is no phase-separation present but a

structural instability leading to a ferromagnetic low-temperature phase and a paramagnetic

high-temperature phase [124] which will be discussed later. Nevertheless, we can perform in

MnBi an artificial separation by adding metallic interlayers between individual MnBi layers.

This artificial separation has again a marked effect on the magnetic and magneto-optic

properties of the material.

The ferromagnetic properties of MnBi have been known since the turn of the century [125]

but an intensive study of the structural and magnetic properties didn’t occur before the early

fifties. In the beginning, MnBi was grown by mixing stoichiometric amounts of Bi and Mn and

reacting the mixture at elevated temperatures of 300-600° C [126]. Enrichment of the MnBi

phase was obtained by grinding to a fine powder and successive magnetic separation, i. e.,

separating magnetic from nonmagnetic powder in a strong magnetic field [127]. Growing MnBi

as thin magnetic films by depositing a layer of Mn and then Bi onto a glass substrate [128]

brought a great improvement in sample preparation. The layering-deposition method was

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78 6 MnBi-BASED MULTILAYERS

further improved by switching the deposition sequence and starting with Bi at low rates in order

to get a template for the MnBi formation [124]. We adapted the same method for the

preparation of our thin-film samples. Recently, MnBi has been obtained by rapid quenching to

room temperature from a melt followed by an annealing step [129]. Up to now, nobody has, to

our knowledge, succeeded in growing MnBi directly by co-evaporation in a ultra-high vacuum.

However, there are first attempts to grow MnBi and related systems using molecular-beam

epitaxy [130] although MnBi formation is still achieved by thermal annealing.

Soon it was realized that MnBi exhibits large magneto-optic effects [131] which could be

used to investigate the magnetic domain structure. At the same time, the idea of

thermomagnetic writing of binary information was brought up [128] in order to construct a

high-density magneto-optic storage device. This objective of using MnBi as a magneto-optic

data-storage material [132] has since been the driving force of further investigations of this

compound [133] and was the reason to start our research. As magneto-optic data-storage

devices, using amorphous rare-earth transition-metal alloys as media, are commercially

available nowadays, the interest focuses on increasing storage density. One way to achieve this

is by reducing the wavelength of the laser light which, in turn, calls for magneto-optic materials

with a high Kerr rotation in the blue spectral range. As we will see, MnBi is a material which

exhibits at room temperature a high Kerr rotation reaching almost 2° at 300-400 nm as

measured through a glass substrate [134].

6.2 General properties of the MnBi system

MnBi crystallizes in a hexagonal NiAs-type structure like the isomorphic compounds MnAs,

MnSb and MnTe. The hexagonal unit cell with two MnBi molecules per unit-cell volume is

outlined in Fig. 6.1.

Mn

Bi

interstitial sites

a

a

c

Fig. 6.1. NiAs-type hexagonal crystal structure of MnBi with two formula units perunit cell.

The Mn ions are located at the 000 and 2100 positions with octahedral coordination. The Bi

ions are at the interstitial sites 13

13

14 and 2

323

34 with bipyramidal coordination [135]. At

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6 MnBi-BASED MULTILAYERS 79

room temperature, the lattice constants are a = 4.29 Å and c = 6.12 Å [136] and the magnetic

moment per Mn ion amounts to 3.84 µB [124]. The hexagonal symmetry of the NiAs structure

induces a uniaxial magnetic anisotropy providing perpendicular magnetic anisotropy when the

compound is grown with a c-axis orientation.

A major drawback of MnBi for usage as a magneto-optic storage material is a structural

phase transition from the ferromagnetic NiAs-type low-temperature phase (LTP) to a high-

temperature phase (HTP) with a distorted NiAs-type structure [137]:

MnBi (LTP) → Mn1.08Bi (HTP) + Bi (liq.) at T = 355° C ,

MnBi (LTP) ← Mn1.08Bi (HTP) + Bi (liq.) at T = 340° C . (6.1)

The HTP is formed by the occupation of about 10 to 15% of the empty interstitial sites by Mn

ions [138] thereby reducing the c-axis lattice constant to c = 6.0 Å and increasing a to

a = 4.38 Å leaving the unit-cell volume almost unchanged [139]. The phase transition to the

HTP occurs before the Curie temperature of MnBi is reached hence destroying ferromagnetic

order [140] as shown in Fig. 6.2. A hypothetical TC of 450° C is extrapolated for the LTP from

the temperature dependence of the magnetization below the phase transition [140]. Neutron

studies indicate that the HTP is paramagnetic with a paramagnetic Curie temperature, Θp, of

167° C. If the HTP is rapidly quenched, a ferromagnetic metastable phase, the quenched high-

temperature phase (QHTP), is obtained with similar structure as the HTP and a TC of 180° C

[141] in good agreement with Θp = 167° C as derived from the HTP. The magnetic moment per

Mn ion in the QHTP is reduced to 1.7 µB as compared to the LTP [136]. At room temperature,

the QHTP transforms slowly back into the LTP by a thermally-activated diffusion process with

a time constant of about two years [142].

Fig. 6.2. Magnetic phase diagram of MnBi.

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80 6 MnBi-BASED MULTILAYERS

A last peculiarity to be mentioned, which is also detrimental for magneto-optic recording, is

the temperature dependence of the coercive field, Hc, as plotted in Fig. 6.3 [143]. Hc increases

with temperature reaching a maximum at 280° C which is very unfavorable for thermomagnetic

writing. The unusual increase in coercivity has been explained by a hybrid domain-wall pinning

model combining a scaling theory of wall bending of anisotropic domain walls with a theory of

thermally-activated unpinning of domain walls [143].

0 100 200 300 400 500 6000.0

0.5

1.0

1.5

2.0 MnBi

Coe

rciv

e Fi

eld

µoH

c (T

)

Temperature (K)

Fig. 6.3. Temperature dependence of the coercive field of MnBi. The figure is takenfrom Ref. [143].

In view of an application of MnBi as a magneto-optic data-storage material, several material

properties have to be optimized. As mentioned before, the extraordinary large Kerr rotation

over a wide photon energy range, a Curie temperature well above room temperature, the high

coercivity, and the perpendicular magnetic anisotropy are all necessary requirements for a

technological application. But, there are mainly three properties which are unfavorable: (i) The

previously mentioned structural phase transition at 340-355° C from ferromagnetic LTP MnBi

to paramagnetic HTP Mn1.08Bi [124], (ii) the temperature dependence of Hc [143], and (iii) a

large grain size of thin films deposited by the standard Bi/Mn-layering technique which

increases noise in magneto-optic recording [144].

Within the pnictide series, MnSb does not exhibit a structural phase instability but it has an

in-plane easy axis of the magnetization [145]. Attempts have been made to substitute part of the

Bi by Sb in order to inhibit the phase transition while retaining perpendicular magnetic

anisotropy. It turns out that only about 4% can be substituted before loosing perpendicular

anisotropy [130].

Another means to change these properties is by adding specific elements to form a ternary

system. A stabilization of the crystal structure can be accomplished if the empty interstitial sites

are filled in with a third element thus preventing Mn ions to occupy the sites. This is realized by

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6 MnBi-BASED MULTILAYERS 81

substituting about 18% of Mn by Ti [146]. The Ti3+ ions prefer a tetrahedral over an octahedral

coordination. Therefore, the bipyramidal interstitial sites will be preferred and the metastable

QHTP will be stabilized. However, the magnetization and Kerr rotation are considerably

reduced. Adding Cu changes the hexagonal NiAs-type to a cubic MnBiCu-type structure, which

does not exhibit a phase transition, while maintaining perpendicular magnetic anisotropy and

ferromagnetic order up to 180° C [147]. Adding Al does not seem to deteriorate the magnetic

and magneto-optic properties [133, 148]. For now, it is not known whether Al occupies the

empty interstitial sites [149] or segregates between the grain boundaries [150].

6.3 Preparation of MnBi(Al, Pt) thin films

We started our investigation by reviewing the standard deposition technique of MnBi films

[124] and by verifying the magnetic and magneto-optic properties [151]. In contrast to previous

work, the MnBi films are not covered with a protective SiOx layer in order to be able to use

surface-analyzing methods like scanning electron microscopy (SEM), scanning force

microscopy (SFM) and - in near future - scanning near-field optical microscopy (SNOM). The

influence of Al and Pt interlayers on structural, magnetic, and magneto-optic properties of

MnBi thin films is reviewed in this chapter. The aluminum as an interlayer materials was

chosen because it does not seem to degrade the magnetic and magneto-optic properties of the

MnBi system as discussed above. Platinum was chosen because it is a noble metal in contrast to

aluminum and therefore inert against oxidation effects.

quartz substrate

12 nm Mn

18 nm Bi

0 - 2 nm Al, Pt

12 nm Mn

18 nm Bi

Fig. 6.4. Typical layering sequence of the MnBi(Al, Pt) thin films. The thickness ofthe Bi and Mn layers is chosen to yield the optimum MnBi stoichiometry afterannealing.

The MnBi thin films are deposited as subsequent Bi/Mn bilayers onto fused SiO2 and glass

substrates at ambient temperatures in a vacuum of 10-6 mbar. Al or Pt is added by placing an

interlayer between subsequent Bi/Mn bilayers. A typical layering sequence is shown in Fig. 1.

The thickness of the Bi and Mn layers is commonly fixed at 18 and 12 nm, respectively, to

obtain after annealing the optimum stoichiometry as derived from energy-dispersive x-ray

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82 6 MnBi-BASED MULTILAYERS

(EDX) analysis. The deposition rate of 0.04 nm/s for Bi is such that a c-axis texture of the Bi

layers is achieved which acts as a template for the MnBi formation. The deposition rates for Al

and Pt are 0.2 nm/s. After deposition, the MnBi films are removed from the chamber and

annealed in a sealed quartz tube for half an hour at 300-310° C [151]. All films containing Al

and Pt layers were annealed for one hour at temperatures between 300 - 380° C yielding films

with c-axis orientation [152].

6.4 Structural and magnetic properties of MnBi thin films

Starting with a single Bi/Mn bilayer, the optimum temperature for the annealing process has

to be found. When annealing for half an hour, there exists only a small temperature window of

about 20° C, where a MnBi compound is successfully crystallized as displayed in Fig. 6.5.

20 30 40 50 60 70 80

sample holder

(d)

(c)

(b)

(a)

Bi(18 nm)/Mn(12 nm)

T = 320° C

T = 300° C

T = 290° C

as grown

* **

Mn3O4

Bi (009)

Bi (006)Bi (003)

MnBi (004)

MnBi (002)

Bi (024)

Bi (012)

Inte

nsity

(a.

u.)

2Θ (deg)

Fig. 6.5. X-ray diffraction spectra of a Bi(18 nm)/Mn(12 nm) sequence as a functionof annealing temperature. (a) as grown, (b) T = 290° C, (c) T = 300° c, and (d)T = 320° C.

For the as-grown films (Fig. 6.5(a)), Bi(00l) reflections are present indicating strong c-axis

orientation of the Bi layer. A few spurious features due to the sample holder are labeled with an

asterisk. Annealing at 290° C (Fig. 6.5(b)), does not form MnBi but Mn3O4. However,

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6 MnBi-BASED MULTILAYERS 83

increasing the annealing temperature by just 10° C to 300° C (Fig. 6.5(c)), yields very strong

MnBi(00l) reflections of the hexagonal NiAs-type structure with c-axis orientation. The Mn3O4

reflections are not present anymore and the Bi(00l) reflections are strongly reduced. Going to

320° C for annealing (Fig. 6.5(d)), destroys the MnBi formation and yields Bi(012) and Bi(024)

reflections indicating randomly oriented Bi. This is probably due to a melting of the Bi layers at

320° C.

20 30 40 50 60 70 80

T = 300° C

T = 310° C

T = 320° C

samlpe holder

as grown

as grown

as grown

annealed

annealed

annealed

(c)

(b)

(a)

Bi (012)

MnBi (004)MnBi (002)

Bi (009)Bi (006)Bi (003)

***

Inte

nsity

(a.

u.)

2Θ (deg)

Bi(18 nm)/Mn(12 nm)

(Bi/Mn)2

(Bi/Mn)3

Fig. 6.6. X-ray diffraction spectra of (a) a single, (b) a double, and (c) a tripleBi(18 nm)/Mn(12 nm) bilayer before and after annealing.

Having determined the optimum annealing conditions, the thickness dependence of the

MnBi formation is studied. In Fig. 6.6 the x-ray diffraction spectra for (a) a single, (b) a double,

and (c) a triple Bi(18 nm)/Mn(12 nm) sequence is displayed before and after annealing. By

increasing film thickness through deposition of additional bilayers, the Bi(012) reflection

becomes stronger relative to the Bi(003) reflection indicating a growing disorientation of the as-

grown Bi layers. Nevertheless, strong MnBi(00l) reflections develop during annealing

independent of thickness. A slight increase in the optimum annealing temperature from 300° C

to 320° C is observed in going from a single to a triple Bi/Mn bilayer.

To further study the influence of film thickness on structural properties, an SEM analysis has

been performed. The SEM micrographs of a single Bi/Mn bilayer in comparison to a (Bi/Mn)2

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84 6 MnBi-BASED MULTILAYERS

double bilayer of twice the thickness is shown in Fig. 6.7 before annealing. By increasing the

film thickness dMnBi, i.e. the number of Bi/Mn bilayers, surface roughness increases

tremendously. The roughness develops during deposition and does not change after annealing.

Fig. 6.7. SEM picture of (a) a Bi(22 nm)/Mn(19 nm) bilayer and (b) a (Bi/Mn)2

double bilayer with a Bi and Mn layer thickness of 18 and 12 nm, respectively. Bothpictures are taken before annealing. The characteristics do not change after annealing.

This is verified by an SFM study [153] of the growth characteristics of Mn onto Bi as

displayed in Fig. 6.8. The comparison between a single Bi layer of 20 nm thickness and a

Bi(20 nm)/Mn(10 nm)/Bi(20 nm) sequence proofs that the roughness is generated during

growth of Mn onto Bi. A Bi layer of 20 nm thickness (Fig. 6.8(a)) has, except for a few large

grains, a homogeneous distribution of Bi islands of about 50 nm lateral size which does not

change considerably when increasing layer thickness to 40 nm (not shown). Growing a 10 nm

thick Mn layer onto Bi induces a large surface roughness. As evidenced in Fig. 6.8(b), large Mn

aggregates of a lateral extension of more than 100 nm are formed identical in size to the clusters

seen in the SEM micrographs. From this point of view, it is actually very astonishing that a

strong c-axis orientation of MnBi is obtained after annealing.

Another thickness dependence is found in the magnetic properties of annealed MnBi films.

The polar Kerr hysteresis loops, measured from the film side, show a strong decrease of the

coercive field, Hc, from 1.25 T in a single Bi/Mn bilayer (dMnBi = 41 nm) to 0.4 T in a (Bi/Mn)3

sequence (dMnBi = 90 nm), as plotted in Fig. 6.9. The reduction in Hc with increasing thickness

can be understood qualitatively as an increase in MnBi grain size on the basis of a simple model

which assumes spherical, single-domain particles [151, 154].

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6 MnBi-BASED MULTILAYERS 85

Fig. 6.8. An SFM analysis of the surface roughness of (a) a single Bi layer of 20 nmthickness and (b) a Bi(20 nm)/Mn(10 nm)/Bi(20 nm) sequence.

From the SEM pictures, we can estimate a grain size of 200-300 nm in thick MnBi films.

The high coercive fields are in contrast to earlier work [132, 143], where for a similar film

thickness values of Hc ≤ 0.5 T were measured. As will be discussed later, this is due to the

altered growth condition when depositing a protective SiOx layer. Besides the decrease of Hc,

an additional change in the characteristics of the magnetization reversal is observed. While a

single Mn/Bi bilayer is reversing smoothly, as revealed in a symmetric ‘S’-shaped slope of the

inner loop (see Fig. 6.9(a)), a thick MnBi film exhibits a sudden onset of magnetization reversal

before leveling off very slowly (see Fig. 6.9(b)).

In order to understand the magnetization reversal from a theoretical point of view, the

hysteresis loops were simulated by a Monte-Carlo method using a micromagnetic model [155]

including dipole interaction, wall energy, Zeeman energy, and an energy barrier describing the

reversal of a single grain. The film is assumed to consist of cells on a square lattice with height,

h, and lateral dimension, L, simulating individual grains [156]. Due to the high magnetic

anisotropy of MnBi, the cells are assumed to be magnetized either parallel or antiparallel to the

surface normal. The energy needed to reverse a cell i with magnetization L2hMsσi (σi = ±1) is

then given by

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86 6 MnBi-BASED MULTILAYERS

-0.5

0.0

0.5 (a)

dMnBi = 41 nm

ω = 1.8 eVT = 295 K

Bi/Mn

-3 -2 -1 0 1 2 3

-0.4

-0.2

0.0(b)

dMnBi = 90 nm

Magnetic Field (T)

ω = 1.8 eVT = 295 K

Pola

r K

err

Rot

atio

n (d

eg)

(Bi/Mn) 3

Fig. 6.9. Polar Kerr-hysteresis loops at a photon energy of 1.8 eV of (a) aBi(22 nm)/Mn(19 nm) bilayer and (b) a (Bi/Mn)3 triple bilayer with a Bi and Mn layerthickness of 18 and 12 nm, respectively. The loops are taken after annealing from thefilm side.

E E E E Ei w dip Z b= + + +

= − + − +∑ ∑≠

12

0 2 23 0

2

4L h S L h M

rH L h Mw i j

js i

j

ijj is i i∆σ ∆σ ∆σσ µ

πσ

µ δ . (6.2)

The first term on the right describes the wall energy, Ew, and the summation is over the fournext neighbors j . A wall energy density, Sw = 4 mJ/m2, is obtained from the simulation which

is smaller than the Bloch-wall energy density, SB = 16 mJ/m2, for MnBi [157]. The second term

is the dipole energy, Edip, which depends on all cells, j, in the lattice. The correct dependence on

relative distance, rij, is taken into account numerically for small values. The external field, H,

goes into the third term which is the Zeeman energy, EZ. The last term, Eb, is an energy barrier,

δi, describing the magnetization reversal of a single cell [158, 159]. By comparing the energy

barrier of coherent rotation, L2hKu, where Ku = 1.16 MJ/m3 [124], with the one for domain-wall

motion through a grain, LhSB, the latter is found to be smaller and, hence, launches

magnetization reversal. Assuming that the energy barrier reaches its maximum value during

magnetization reversal in the middle of a cell, the relevant energy barrier to be taken into

account is ( ) δi B w dip Z= − + +max ,0 12L h S E E E .

Using experimental values for h and L, the model accurately accounts for the change in

coercivity as a function of thickness as demonstrated in Fig. 6.10. The value of h corresponds to

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6 MnBi-BASED MULTILAYERS 87

the known film thickness while L is estimated from the SEM pictures. However, the coercivity

is a factor of two smaller than seen in the experiment indicating that the model is too simple to

predict correct values of Hc.

MnBi

h = 40 nmL = 50 nm

(a)

M/M

s

-1.0 -0.5 0.0 0.5 1.0

h = 90 nmL = 300 nm

MnBi

(b)

Magnetic Field (T)

Fig. 6.10. Hysteresis loops of the simulated MnBi system for two square lattice cellsizes corresponding to a single grain of MnBi with (a) height, h = 40 nm, and lateraldimension, L = 50 nm, and (b) h = 90 nm and L = 300.

With the help of the model, the change in the magnetization reversal can be understood as

well. As is evident from the SEM pictures (see Fig. 6.7), the MnBi grain size is randomly

distributed and therefore, disorder should be incorporated into the model. This would yield a

random distribution of L. As this is very difficult to simulate, due to the complicated evaluation

of the dipole term, Edip, in Eq. (6.2), disorder is added to the model by assuming a random

fluctuation of L only in the Zeeman term, EZ, since this is the leading term being quadratic in L

[155]. The resulting hysteresis loop is plotted in Fig. 6.11 for a Gaussian distribution of L of

width of ∆G = 0.4 and compared to the case without disorder.

From the simulations it can be concluded that in the presence of fluctuations of the grain size

L, the largest grains are the first to reverse their magnetization. As the magnetization is

proportional to the grain volume, the slope of magnetization reversal will be largest in the

beginning of the reversal process, exactly as found in the experimental hysteresis curves (see

Fig. 6.9).

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88 6 MnBi-BASED MULTILAYERS

∆G = 0

h = 90 nmL = 300 nm

MnBi

(a)

M/M

s

-0.5 0.0 0.5

h = 90 nmL = 300 nm

MnBi

∆G = 0.4

(b)

Magnetic Field (T)

Fig. 6.11. Disorder dependence of the hysteresis loops for h = 90 nm and L = 300 nmfor (a) ∆G = 0 (no disorder) and (b) ∆G = 0.4.

6.5 Structural and magnetic properties of MnBiAl thin films

In this section, the influence of an Al interlayer on the structural properties of MnBi thin

films is discussed. As mentioned before, Al as an interlayer material was chosen because it has

been observed, that Al additions do not degrade the magneto-optic properties [133, 148]. The

influence of Al interlayers on the x-ray diffraction spectra of as-grown films is plotted in

Fig. 6.12. A few spurious features due to the sample holder are labeled with an asterisk.

Apparently, the addition of Al does reduce the relative strength of the Bi(012) reflection as

compared to the Bi(003) reflection. This dependence is shown in the inset to the figure.

Because the Bi(012) reflection is strongest in randomly oriented Bi, this implies that an Al

interlayer reduces the disorientation of the Bi layers in thicker MnBi films.

The x-ray diffraction spectra after annealing are shown in Fig. 6.13. Evidently, the addition

of an Al interlayer does not have any effect on the preferred c-axis orientation of the MnBiAl

thin films. Strong MnBi(00l) reflections indicating the hexagonal NiAs-type structure of MnBi

are found for all Al-interlayer thicknesses. Note that there is no evidence for a second phase,

like MnAl or the high-temperature Mn1.08Bi phase, as has been found by other groups [160,

161]. There is, however, a small amount of residual Bi in the Bi/Mn/Al(0.2 nm)/Bi/Mn sample.

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6 MnBi-BASED MULTILAYERS 89

20 30 40 50 60 70 80

sample holder

as grown

Bi/Mn/Al(dAl )/Bi/Mn

dAl = 0.0 nm

dAl = 0.2 nm

dAl = 2.0 nm

* ** Bi (009)

Bi (006)

Bi (012)

Bi (003)

Inte

nsit

y (a

. u.)

2Θ (deg)

0 1 20.0

0.1

0.2

Bi(

012)

/Bi(

003)

interlayer thickness d Al

Fig. 6.12. X-ray diffraction spectra of Bi/Mn/Al(dAl)/Bi/Mn samples with dAl = 0, 0.2,and 2.0 nm before annealing. The inset shows the disorientation of the Bi layer as afunction dAl. The solid line is a guide to the eye.

20 30 40 50 60 70 80

sample holder

annealed

Bi/Mn/Al(dAl )/Bi/Mn

dAl = 0.0 nm

dAl = 0.2 nm

dAl = 2.0 nm

* **

MnBi (004)

Bi (012)

MnBi (002)

Inte

nsit

y (a

. u.)

2Θ (deg)

Fig. 6.13. X-ray diffraction spectra of Bi/Mn/Al(dAl)/Bi/Mn samples with dAl = 0, 0.2,and 2.0 nm after annealing.

As the addition of Al neither reduces c-axis orientation nor changes the MnBi crystal lattice

constant, the question arises what happens to the interlayers at all. In order to elucidate this

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90 6 MnBi-BASED MULTILAYERS

question, Rutherford-backscattering (RBS) experiments were performed using 1.4 MeV He+

ions at a scattering angle of 170° [8]. A comparison between the RBS spectra before (dashed

line) and after (solid lines) annealing of a Bi/Mn/Al(0.2 nm)/Bi/Mn and a

Bi/Mn/Al(2 nm)/Bi/Mn sample is displayed in Fig. 6.14. The surface-backscattering energies of

the components are marked by vertical lines. The offset to lower energies is indicative of the

location of a component inside the film relative to the surface. Typically, heavy elements, like

Bi, have less energy loss than light elements, like Mn or O. The Al interlayer is too thin to be

seen directly. Focusing on the dominant Mn and Bi features, a distinct separation into two

layers, represented by a double peak in the RBS spectra, is recognized in the as-grown state.

This simply represents the layering sequence of the deposition (see Fig. 6.4).

0.4 0.6 0.8 1.0 1.2 1.4

(a)

(c)

(b)

dAl = 2.0 nm, annealed

Energy (MeV)

O Mn Bi

dAl = 2.0 nm, as grown

dAl = 0.2 nm, annealed

Bi/Mn/(dAl )/Bi/Mn

Nor

mal

ized

Yie

ld (

a. u

.)

Fig. 6.14. Rutherford-backscattering spectra before (dotted line) and after (solid lines)annealing of (a & c) a Bi/Mn/Al(2 nm)/Bi/Mn and (b) a Bi/Mn/Al(0.2 nm)/Bi/Mnsequence. The vertical lines mark the surface-backscattering energy of theconstituents.

After annealing, the separation of the two Bi/Mn bilayers is preserved to a great extent in the

Al-doped system with dAl = 2 nm while the sample with thin Al interlayer, dAl = 0.2 nm, reveals

a complete interdiffusion of the two Bi/Mn bilayers. A small amount of oxygen is present

directly at the surface, increasing after annealing by a factor of about 2.5 [162]. It most certainly

forms with Mn a thin Mn3O4 surface layer. This is corroborated by an asymmetry of the Mn

peaks after annealing, implying a higher Mn density at the surface. The asymmetry is more

clearly seen in the double peaks of the Bi/Mn/Pt/Bi/Mn samples as discussed in the next

section. No anisotropy is observed, however, in the Bi/Mn/Al/Bi/Mn samples before annealing.

This could indicate that the Al interlayer getters some of the oxygen during deposition thus

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6 MnBi-BASED MULTILAYERS 91

forming Al2O3, which is a very firm oxide. This would explain why the two Bi/Mn bilayers are

still separated for a thicker Al interlayer after annealing.

Although the MnBi-formation process is not considerably influenced by an Al interlayer, the

effect on the surface roughness is profound. As exhibited in the SEM micrographs of two

Bi/Mn/Al(dAl)/Bi/Mn sequences in Fig. 6.15, even a 0.2 nm thick interlayer smoothes the

surface notably as compared to a (Bi/Mn)2 double bilayer (see Fig. 6.7) while a 2 nm thick

interlayer reduced surface roughness to the value of a single MnBi bilayer.

Fig. 6.15. SEM picture of (a) Bi/Mn/Al(0.2 nm)/Bi/Mn and (b) Bi/Mn/Al(2 nm)/Bi/Mn with a Bi and Mn layer thickness of 18 and 12 nm, respectively. Both picturesare taken before annealing. The characteristics do not change after annealing.

Having found a marked effect on the surface roughness, it is likely that the magnetic

properties are also influenced by an Al interlayer. Therefore, the magnetization of the films is

investigated by the use of polar Kerr hysteresis-loop measurements. In Fig. 6.16, the

dependence of θK on the Al-interlayer thickness of a Bi/Mn/Al(dAl)/Bi/Mn sequence, with Bi

and Mn thickness of 18 and 12 nm, respectively, is plotted. All loops are taken from the film

side, at a photon energy of 2 eV and at room temperature.

For dAl = 0.2 nm, a square loop behavior is found similar to the loops in (Bi/Mn)2 double

bilayers (see Fig. 6.9). When adding a thicker Al interlayer of dAl ≥ 1.4 nm, the Kerr loops

develop a peculiar form. Starting at the highest positive magnetic field, magnetization reversal

takes place in two steps. Shortly after reversing the field, at H ≅ -0.2 T, the Kerr rotation, θK,

shows a sharp decline before increasing again to a positive saturation value at H ≅ -1 T. To

elucidate the origin of the unusual shape, hysteresis loops are measured also through the

substrate. The result for dAl = 2 nm is displayed in Fig. 6.17 in combination with the loop

measured from the film side. Evidently, when looking from the substrate side the loop has the

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92 6 MnBi-BASED MULTILAYERS

usual square form with a negative θK as in MnBi. However, the coercivity is very small,

reaching only 0.4 T. This value coincides with the steep onset of magnetization reversal in the

loop from the film side except that the latter has the opposite direction.

-0.5

0.0

0.5

Bi/Mn/Al(d Al )/Bi/Mn

dAl = 0.2 nmPo

lar

Ker

r R

otat

ion

θK (

deg)

(a)

-0.5

0.0

0.5 dAl = 1.4 nm(b)

-3 -2 -1 0 1 2 3

-0.5

0.0

0.5 dAl = 1.7 nm(c)

Magnetic Field (T)

-3 -2 -1 0 1 2 3

-0.5

0.0

0.5 dAl = 2.0 nm

hω = 2.0 eVT = 295 K

(d)

Fig. 6.16. Dependence of the polar Kerr hysteresis loops of a Bi/Mn/Al(dAl )/Bi/Mnsequence on the Al-interlayer thickness. (a) dAl = 0.2 nm, (b) dAl = 1.4 nm, (c)dAl = 1.7 nm, and dAl = 2.0 nm. All loops are measured from the film side at roomtemperature at a photon energy of 2 eV.

This behavior is explained by assuming that the MnBi film consists of two decoupled

magnetic layers with different values of Hc. The top layer has a large Hc of about 1 T, similar to

a single Bi/Mn bilayer, and the bottom one has a very small Hc of ≅ 0.4 T, as in a thick MnBi

film and opposite sign of θK when measured from the film side.

-3 -2 -1 0 1 2 3

-1.0

-0.5

0.0

0.5

1.0 ω = 2 eVT = 295 K

Bi/Mn/Al(2 nm)/Bi/Mn

film side substrate side

Pola

r K

err

Rot

atio

n (

deg

)

Magnetic Field ( T )

Fig. 6.17. The polar Kerr hysteresis loop of a Bi/Mn/Al(2 nm)/Bi/Mn sequencemeasured from the film side () and through the substrate ().

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6 MnBi-BASED MULTILAYERS 93

-3 -2 -1 0 1 2 3

A B A+B

Ker

r R

otat

ion

(a. u

.)

Magnetic Field (T)

Fig. 6.18. Representation of the polar Kerr hysteresis loop of a system of twodecoupled magnetic layers with different coercive fields and opposite sign of the Kerrrotation.

A superposition of two model loops with the proposed characteristics is plotted in Fig. 6.18.

It correctly describes the experimental findings. An antiferromagnetic coupling between the two

layers can be excluded as in that case the switching should take place before reversing the

external field [162]. Note that the bottom layer now manifests the same value of Hc as reported

in samples with a protective SiOx layer [132, 143]. This accordance will be discussed below.

What seems to be astonishing in our explanation is the change in sign of θK when measuring

from the film side as compared to measuring through the substrate. In sections 2.7 and 2.8, we

have discussed the dependence of the Kerr effect on the relative index of refraction at the

relevant interface, i. e., the interface between a magneto-optic layer and an adjacent layer,

where the light is being reflected. Here, the adjacent layer is transparent glass when measuring

from the substrate side, leading to an interface enhancement in θK (see section 2.8.1). When

measuring from the film side, the adjacent layer is the Al interlayer, which is highly absorbing.

Therefore, the relative index of refraction is the ratio between two complex quantities as in Eq.

(5.13). This can readily introduce a sign reversal in θK as, e. g., observed in the UV region of

the photon spectra in the system Co(EuS) (see Fig. 5.15).

Summarizing, by inserting a reasonably thick Al interlayer, the surface roughness is reduced

to the value of a single Bi/Mn bilayer and the coercivity of the buried Bi/Mn bilayer is largely

reduced by a factor of 3 as compared to the top bilayer or a single bilayer. The coercivity of the

buried bilayer is now similar to MnBi films with a protective SiOx layer. In order to understand

this coercivity reduction and the similarity to adding protective SiOx layer, let us discuss some

consequences of the layering-deposition method [163]. Depositing a Bi/Mn bilayer results in an

average MnxBi1-x composition where the Mn concentration xMn is given by

xd

M

d

M

d

MMnMn Mn

Mn

Mn Mn

Mn

Bi Bi

Bi

= +

ρ ρ ρ1

. (6.3)

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94 6 MnBi-BASED MULTILAYERS

Inserting ρBi = 9.8 g/cm3, ρMn = 7.43 g/cm3, dBi = 18 nm, dMn = 12 nm, MBi = 209 g/mol and

MMn = 55 g/mol, we get xMn = 0.66. This is away from the ideal stoichiometry of xMn = 0.5. As

evidenced by energy-dispersive x-ray analysis, about 10% of the Mn evaporates during

annealing. Furthermore, a small amount of Mn is bound at the surface by oxygen (see

Fig. 6.14). As a result, the complete Bi layer but only part of the Mn layer is engaged in the

MnBi-formation process. We can calculate the average volume, vBi/Mn, per formula unit MnBi

by assuming a Bi and Mn layer thickness corresponding to the ideal stoichiometry. This yields

v v vN

M MBi Mn Bi Mn

A

Bi

Bi

Mn

Mn/ = + = +

1

ρ ρ. (6.4)

where NA is Avogadro’s number. Inserting the values, we calculate vBi/Mn = 47.7 Å3. This has to

be compared with the cell volume per formula unit of NiAs-type MnBi,v a cMnBi MnBi MnBi= 3

42 = 48.8Å3 using aMnBi = 4.29 Å and cMnBi = 6.126 Å [138, 164]. This is

2.3% larger than vBi/Mn . As a consequence, the film has to expand during crystallization

generating internal pressure. This would lead to an increasing surface roughness when enlarging

film thickness. The thicker the films the larger the disorder which is induced by the nucleation

of individual crystallites. However, as has been discussed before, surface roughness develops

already during deposition creating void space (see Fig. 6.8). Therefore, part of the internal

stress induced by the volume increase during crystallization will be relaxed through the void

space.

The change in coercivity behavior of MnBi films can be explained by a volume increase as

well. Depositing a hard interlayer, like Al2O3, or a protective layer, like SiOx, will restrict

expansion of the crystallites perpendicular to the surface, leading to a pronounced lateral

growth of the MnBi crystallites. This will lead to larger lateral dimensions and a better

interconnection of individual grains reducing the wall energy at the grain boundary. Both

effects reduce coercivity which explains the small values observed. This growth model is

visualized in Fig. 6.19.

glass substrate

MnBi

Al

Fig. 6.19. Simple growth model of Bi/Mn/Al/Bi/Mn thin films.

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6 MnBi-BASED MULTILAYERS 95

Direct experimental proof of the model is obtained by transmission electron microscopy

(TEM) of a Bi/Mn/AL(2 nm)/Bi/Mn sequence after annealing. As is seen in Fig. 6.20, the Al

interlayer separates effectively the buried MnBi layer from the top MnBi layer. It is also noticed

that the buried layer is more compact than the top one fully corroborating the proposed growth

model.

Fig. 6.20. TEM micrograph of a Bi/Mn/Al(2 nm)/Bi/Mn sequence with Bi and Mnlayer thickness of 18 and 12 nm, respectively, after annealing.

6.6 Structural and magnetic properties of MnBiPt thin films

The motivation of adding platinum interlayers is that platinum, being a noble metal, will not

form an oxide. It will hence be a minor diffusion barrier to the MnBi formation across the

interlayer than a partially or fully oxidized Al interlayer. As a result, all effects due to interlayer

oxidation should be avoided in using Pt interlayers.

In the x-ray diffraction spectra, which are plotted in Fig. 6.21, a similar behavior as for Al

interlayers is observed. The MnBi formation and the c-axis texture are not influenced by a Pt

interlayer. No evidence of a Mn-Pt phase is found. In order to study the interdiffusion of the Pt

layer into the adjacent Bi/Mn bilayers, RBS spectra have been measured. The results are plotted

in Fig. 6.22. In contrast to Al interlayers (see Fig. 6.14), the double peaks of Mn and Bi are

asymmetric already before annealing indicating surface oxidation of the Mn top layer. This is

not surprising at all as the oxygen will not be chemically bound by the noble metal Pt.

Therefore, all the oxygen is available to form a Mn3O4 surface layer.

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96 6 MnBi-BASED MULTILAYERS

20 30 40 50 60 70 80

Bi (009)

Bi (006)

Bi (012)

Bi (003)

(c)

(b)

(a)

annealed

sample holder

annealed

Bi/Mn/Pt(dPt )/Bi/Mn

as grown

d Pt = 0.3 nm

d Pt = 2.0 nm

* **

MnBi (004)

MnBi (002)

Inte

nsit

y (a

. u.)

2Θ (deg)

Fig. 6.21. X-ray diffraction spectra of (a) a Bi/Mn/Pt/Bi/Mn sequence beforeannealing and (b) a Bi/Mn/Pt(0.3 nm)/Bi/Mn and (c) a Bi/Mn/Pt(2 nm)/Bi/Mn afterannealing.

After annealing, the Pt-doped system reveals a complete interdiffusion of the two Bi/Mn

bilayers in contrast to thick Al interlayers. Again, this can be explained by the noble-metal

properties of Pt. No oxide will be formed in the Pt interlayer and it will hence be a minor

diffusion barrier to the MnBi formation than an partially oxidized Al interlayer.

0.4 0.6 0.8 1.0 1.2 1.4

(a)

(c)

(b)

dPt = 2.0 nm, annealed

Energy (MeV)

O Mn Bi

dPt = 2.0 nm, as grown

dPt = 0.2 nm, annealed

Bi/Mn/Pt(dPt )/Bi/Mn

Nor

mal

ized

Yie

ld (

a. u

.)

Fig. 6.22. Rutherford-backscattering spectra before (dotted line) and after (solid lines)annealing of (a & c) a Bi/Mn/Pt(2 nm)/Bi/Mn and (b) a Bi/Mn/Pt(0.3 nm)/Bi/Mnsequence. The vertical lines mark the surface-backscattering energy of theconstituents.

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6 MnBi-BASED MULTILAYERS 97

Fig. 6.23. SEM picture of Bi/Mn/Pt(0.3 nm)/Bi/Mn with a Bi and Mn layer thicknessof 18 and 12 nm, respectively. The picture is taken before annealing. Thecharacteristics do not change after annealing.

Adding a Pt interlayer between two Bi/Mn bilayers reduces surface roughness even stronger

than in the case of Al as evidenced in the SEM pictures in Fig. 6.23.

As demonstrated in a SFM study [153], the Pt produces a plateau-like structure when

growing on a Bi/Mn bilayer. In the SFM pictures of Bi/Mn/Pt(dPt)/Bi/Mn samples, which are

displayed in Fig. 6.24, large islands are formed which have not been observed in the pure Bi or

Bi/Mn samples (see Fig. 6.8). The islands are growing in size and forming plateau-like

structures when increasing Pt-interlayer thickness from 0.3 nm (Fig. 6.24(a)) to 1 nm

(Fig. 6.24(b)). This explains the strong smoothing of the surface even for very thin Pt

interlayers.

Fig. 6.24. An SFM analysis of the surface roughness of a Bi/Mn/Pt(dPt)/Bi/Mnsequence with (a) dPt = 0.3 nm and (b) dPt = 1.0 nm.

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98 6 MnBi-BASED MULTILAYERS

The influence of a Pt interlayer on the magnetic properties is shown in Fig. 6.25. All polar

Kerr hysteresis loops have been measured from the film side (solid symbols) and through the

substrate (open symbols). Adding a thin Pt interlayer yields a square loop with Hc = 0.4 T

similar to a (Bi/Mn)2 double bilayer. By increasing the Pt-interlayer thickness gradually from

dPt = 0.3 to 1.5 nm, Hc increases progressively to 0.9 T which is almost the value of a single

Bi/Mn bilayer. At dPt = 2 nm, the coercivity is no longer increasing. The same behavior of Hc is

found when measuring through the substrate suggesting a homogeneous film for all Pt-

interlayer thicknesses as derived from the RBS spectra (see Fig. 6.22). The RBS data show that

the Pt interlayer merges completely with the Bi/Mn bilayers. Therefore, an increase in

coercivity with Pt-interlayer thickness is very surprising. One would rather expect that

Hc remains constant at a value of a double Bi/Mn bilayer. This findings indicate that Pt

efficiently reduces the grain size of MnBi. The effect on the magneto-optic properties will be

discussed in the next section.

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5 Bi/Mn/Pt(d Pt )/Bi/Mn

d Pt = 0.4 nm

Pola

r K

err

Rot

atio

n θ

K (

deg)

(a) -0.5

0.0

0.5

d Pt = 1.0 nm

(b)

-3 -2 -1 0 1 2 3

-0.5

0.0

0.5

d Pt = 1.5 nm(c)

Magnetic Field (T)

-3 -2 -1 0 1 2 3

-0.5

0.0

0.5

d Pt = 2.0 nm

hω = 2.0 eVT = 295 K

(d)

Fig. 6.25. Polar Kerr-hysteresis loops at a photon energy of 2 eV of aBi/Mn/Pt(dPt)/Bi/Mn sequence with (a) dPt = 0.4 nm, (b) dPt = 1.0 nm, (c) dPt = 1.5 nm,and (d) dPt = 2.0 nm. The Bi and Mn layer thickness is 18 and 12 nm, respectively.The loops are taken from the film side (solid symbols) and through the substrate (opensymbols) after annealing.

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6 MnBi-BASED MULTILAYERS 99

6.7 Magneto-optic properties of MnBi(Al, Pt) thin films

6.7.1 MnBi and MnBiAl thin films

In the last section we will discuss the magneto-optic properties of Al and Pt doped MnBi

thin films at room temperature. All spectra were measured through the substrate in order to

avoid interference effects of a Mn3O4 surface oxide layer or, especially in thick MnBi films,

enhancement effects due to the surface roughness. In addition, the polar Kerr spectra were

obtained in the remanence state except for the (Bi/Mn)3 sequence. The remanence state is

chosen because in that case no spurious effects of the Faraday rotation of the substrate are

present that could falsify the spectra, especially in the UV part of the photon-energy range. Each

rotation and ellipticity spectra is divided by a value ns = 1.5 equal to the index of refraction of a

quartz or glass substrate according to Eq. (2.99). This is done in order to account for the

interface-enhancement effect when measuring through a transparent substrate (see section

2.8.2).

The polar Kerr rotation, θK, and ellipticity, ηK, at room temperature is plotted as a function

of photon energy in Fig. 6.26 for a (Bi/Mn)3 triple bilayer (), a (Bi/Mn)2 double bilayer (Σ),

and a Bi/Mn/Al(0.3 nm)/Bi/Mn sequence (∇). The nominal overall thickness of the films is 90,

60, and 60.3 nm, respectively. All samples are deposited onto quartz substrates except for the

(Bi/Mn)3 triple bilayer which has been deposited onto glass limiting the spectra to 3.8 eV. For

comparison with literature, the uncorrected, i.e., as-measured, spectra of the (Bi/Mn)3 triple

bilayer is shown as a dashed line as well.

Comparing pure MnBi films with Al doped ones, no change in the spectral features is

observed. The Kerr rotation is slightly larger for a sample with an Al interlayer than for a

corresponding (Bi/Mn)2 bilayer of the same nominal thickness but smaller than for a (Bi/Mn)3

triple bilayer. The increase is not significant as claimed in other work [133, 148]. It rather

represents the natural scattering of the Kerr spectra of various samples.

The largest Kerr effect is found in the thickest sample, i. e., the (Bi/Mn)3 triple bilayer. This

suggests a thickness dependence of the Kerr effect. The penetration depth of the light is equal to

the inverse of the absorption constant, K-1, which is related to the absorption coefficient, k, by

K = 2ωk/c. Using the experimental value at 2 eV of k = 2.6 [165], an absorption constant

K = 5.3 ⋅ 105 cm-1 is calculated. For a typical Bi/Mn bilayer thickness of dBi/Mn = 30 nm in our

films, an exponent KdBi/Mn = 1.59 is obtained corresponding to an extinction of the intensity by

a factor of 5. Therefore, a third bilayer will give neither a contribution to the Kerr effect nor to

the optical functions n and k. In conclusion, the change in θK is not simply due to the increase in

film thickness per se. Nevertheless, adding additional bilayers will impose a larger resistance to

a volume increase of the innermost bilayer during MnBi formation thus leading to a denser film

close to the substrate. Because the magnetization is larger in a dense film, the Kerr effect will

be larger, too. This assumption is corroborated by a slight increase in the magneto-optic

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100 6 MnBi-BASED MULTILAYERS

properties when adding an Al interlayer which acts as a diffusion barrier as discussed before in

section 6.5.

-2.0

-1.5

-1.0

-0.5

0.0Bi/Mn/Al(0.3 nm)/Bi/Mn

Bi/Mn/Bi/Mn

(Bi/Mn) 3

(Bi/Mn) 3 uncorr.(a)

Pola

r K

err

Elli

ptic

ity (

deg)

T = 295 K

MnBiAl(dAl )Po

lar

Ker

r R

otat

ion

(deg

)

0 1 2 3 4 5

-1.0

-0.5

0.0

0.5

1.0

Bi/Mn/Bi/Mn

Bi/Mn/Al(0.3 nm)/Bi/Mn

(Bi/Mn) 3

(Bi/Mn) 3 uncorr.

Photon Energy (eV)

(b)

Fig. 6.26. Polar Kerr rotation (a) and ellipticity (b) spectra of a (Bi/Mn)3 triple bilayer(), a (Bi/Mn)2 double bilayer (Σ), and a Bi/Mn/Al(0.3 nm)/Bi/Mn sequence (∇). Thesolid line is the uncorrected spectra of (Bi/Mn)3, all other spectra are reduced by afactor of 1.5 to account for the interface enhancement. The thickness of the individualBi and Mn layers is 18 and 12 nm, respectively. All measurements were obtainedthrough the substrate at room temperature.

In all samples, two major peaks are observed in θK at 1.9 and 3.4 eV, equal in size and with

an absorptive line shape. The large Kerr effect at photon energies above 3 eV is very

advantageous for blue recording. The maximum substrate-corrected Kerr rotation amounts to -

1.19° at 1.9 eV in the (Bi/Mn)3 triple bilayer. The even larger values of θK at photon energies

larger than 3 eV in that sample are not intrinsic. They are due to a residual Faraday effect of the

substrate because this particular spectrum was not measured in the remanence state. The

corresponding features in the Kerr ellipticity, ηK, exhibit dispersive line shapes. The maximum

value, ηK = -0.9°, is found at 4.5 eV. Our measurements are in good agreement with recently

published spectra [134, 165]. However, these studies were made on SiOx-protected MnBi films.

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6 MnBi-BASED MULTILAYERS 101

There, the peaks positions are also found at 1.9 and 3.4 eV but, unlike our results, the second

peak (θK = 1.5°) is smaller by 15% than the one at lower energy (θK = 1.75°). To compare with

our findings, the rotations have to be corrected for the interface effect. This yields θK = 1.16°

and 1° which is slightly smaller than our highest values.

6.7.2 MnBiPt thin films

The polar Kerr rotation (a) and ellipticity (b) spectra for Pt additions are shown in Fig. 6.27.

As in the case of pure and Al-doped MnBi, two major features are observed in θK at 1.9 and

3.4 eV. However, adding to MnBi a thin Pt interlayer with dPt = 0.3 nm reduces the Kerr

rotation evenly across the photon-energy range. Adding a 1.5 nm thick Pt reduces the Kerr

effect even more. In addition, in both samples the first peak at 1.9 eV is now considerably

smaller in size than the peak at 3.4 eV.

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Bi/Mn/Pt(1.5 nm)/Bi/Mn

Bi/Mn/Pt(0.3 nm)/Bi/Mn

(a)

Pola

r K

err

Elli

ptic

ity (

deg)

T = 295 K

MnBiPt(dPt )

Pola

r K

err

Rot

atio

n (d

eg)

0 1 2 3 4 5

-0.8

-0.6

-0.4

-0.2

0.0

0.2Bi/Mn/Pt(1.5 nm)/Bi/Mn

Bi/Mn/Pt(0.3 nm)/Bi/Mn

Photon Energy (eV)

(b)

Fig. 6.27. Polar Kerr rotation (a) and ellipticity (b) spectra of a Bi/Mn/Pt(dPt)/Bi/Mnsequence with dPt = 0.3 nm () and dPt = 1.5 nm (◊). The thickness of the individual Biand Mn layers is 18 and 12 nm, respectively. All measurements were obtained throughthe substrate at room temperature and reduced by a factor of 1.5 to account for theinterface enhancement.

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102 6 MnBi-BASED MULTILAYERS

In order to evaluate the origin of the strong reduction in the Kerr effect despite the favorable

structural properties of Pt-doped MnBi thin films (see section 6.6), the magnetization, µ, is

measured in a commercial SQUID magnetometer. It is found that µ decreases in the same way

as the Kerr effect. As a function of Pt content, a reduction from µ = 536 kA/m to 304 kA/m for

dPt = 0.3 and 1.5 nm is observed [153]. Normalizing the maximum Kerr rotation, θKmax at 3.4 eV

to the magnetization, we get a ratio θ µKmax = 1.31⋅10-3 and 1.20⋅10-3 deg m/kA, respectively,

which is equal to within 10%. This indicates that the addition of Pt reduces primarily the

magnetization of the MnBi films. As a consequence, θK is decreasing in the same way being in

first order proportional to µ. Of course, this analysis does not take into account any changes in

the optical functions n and k. For a complete analysis, the off-diagonal elements, ~σxy , of the

optical conductivity tensor should be considered as mentioned in section 2.4. As the optical

functions have not been measured for our samples, this discussion cannot be concluded in this

work. However, the strong correlation of the Kerr effect to the magnetization indicates that the

reduction is not due to optical effects and would be found in ~σxy as well. A decrease of the

magnetization could indicate that part of the magnetic moment of Mn is not ordered or

quenched. This could happen if a Mn-Pt phase would be formed. However, theoretical [32] as

well as experimental data [166, 167] show a strong peak in θK between 1 and 1.7 eV but no

feature above 3 eV for MnPt3 and PtMnSb in contrast to our results. The large coercivity in all

Pt-doped MnBi films indicates a small grain size as mentioned before (see section 6.6). But, a

reduction of the magnetization due to the small grain size seems very unlikely. In conclusion,

we do not know at present what the reasons are for the observed magnetization reduction in Pt-

doped MnBi thin films.

The reflectivity spectra have been measured at room temperature relative to an Al reference

mirror. Therefore, the absolute values might have an error of several percent while the relative

features should be accurate. As seen in Fig. 6.28, the reflectivity, R, decreases continuously

with photon energy in all samples. It is the largest for the Al doped film where R declines from

50% at 1.1 eV to 30% at 4 eV. This is in agreement with recently published data [165]. The Pt-

doped films have lower values of R starting at 45% at 1.1 eV and decreasing to 20% at 4 eV. If

we calculate the shot-noise limited figure of merit, FOM, according to Eq. (5.9) a value

FOM = 0.46° and 0.54° is derived at 3 eV for the Al-doped and the (Bi/Mn)3 sample,

respectively. This is considerably more than in Co/Pt multilayers [112] explaining the high

interest in this compound as a potential magneto-optic storage material.

As ab-initio calculations of the Kerr effect have made an immense progress in the last few

years, we would like to compare our data with recent first-principles studies on the Kerr effect

of MnBi [33, 168]. The theoretical studies show that the high intrinsic Kerr effect in MnBi is

due to the coincidence of three favorable reasons: (i) a large magnetic moment of 3.7 µB, (ii) a

large spin-orbit coupling of Bi, and (iii) a large hybridization between the d and p states of Mn

and Bi. This leads to strong, dipole-allowed Mn-3d→Bi-5p as well as Bi-5p → Mn-3d transi-

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6 MnBi-BASED MULTILAYERS 103

1 2 3 4 50

10

20

30

40

50

60

MnBi(Al, Pt)

(Bi/Mn) 3 Bi/Mn/Al(0.3 nm)/Bi/Mn Bi/Mn/Pt(0.3 nm)/Bi/Mn Bi/Mn/Pt(1.5 nm)/Bi/Mn

T = 295 K

Ref

lect

ivity

(%

)

Photon Energy (eV)

Fig. 6.28. Reflectivity spectra of a (Bi/Mn)3 (), a Bi/Mn/Al(0.3 nm)/Bi/Mn (∇), aBi/Mn/Pt(0.3 nm)/Bi/Mn (), and a Bi/Mn/Pt(1.5 nm)/Bi/Mn (◊) sequence. Thethickness of the individual Bi and Mn layers is 18 and 12 nm, respectively. Allmeasurements were obtained through the substrate at room temperature and correctedfor an Al reference mirror.

tions which are magneto-optically active because of (i) and (ii). Although the first peak at

1.9 eV is computed in good agreement with experiment [33] the second peak at 3.4 eV does not

appear in the primary calculations. At present, there are two approaches to account for the high-

energy peak. One is by assuming a change in stoichiometry from MnBi to a Heusler-like Mn2Bi

[33], the other one is by assuming oxygen or Mn filling in the interstitial sites of the MnBi unit

cell [168]. Future measurements have to clarify if any of these mechanisms is responsible for

the peak at 3.4 eV or if the primary calculations need to be modified. A study which is sensitive

to the local coordination of atoms, like EXAFS, would be very helpful.

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104 6 MnBi-BASED MULTILAYERS

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7 CONCLUSIONS AND OUTLOOK 105

7 CONCLUSIONS AND OUTLOOK

An attempt is made in this work of establishing a sound sign convention in magneto-optics

by discussing some general aspects of the theory of magneto-optics. A detailed description of a

high-resolution magneto-optic spectrometer is given in order to support the experimentalist

who wants to start with his own magneto-optic research. The suitability of magneto-optic

spectroscopy in investigating thin magnetic film is demonstrated by reviewing the structural,

magnetic, and magneto-optic properties of three material classes which are of interest on their

own.

In (In, Mn)As heterostructures, a potential for a technological application is given by the

possibility to tailor the carrier density by varying the concentration of the magnetic ions or by

adding a donor material [73]. The substrate temperature has a profound effect on the

homogeneity of the system and the type of conduction carriers. The type and density of the

charge carriers, in turn, determine the magnetic properties of the heterostructures. In epitaxial p-

type (In, Mn)As heterostructures ferromagnetic ordering is confirmed up to 50 K with a strong

perpendicular magnetic anisotropy. The latter is shown in this work to be due to internal stress

induced by the lattice mismatch between the magnetic (In, Mn)As and the adjacent (Ga, Al)Sb

layer. The magnetic ordering is thought to be due to an RKKY-type of exchange interaction

between localized Mn moments. The hypothesis of an exchange interaction across the

(In, Mn)As/(Ga, Al)Sb interface is refuted in this study. With the successful growth of a

ferromagnetic III-V compound on a GaAs substrate a combination of GaAs-based integrated

circuits with magnetic properties seems feasible. However, a major restriction is the low

ordering temperature of 50 K which makes cryogenic techniques indispensable. A main

objective is thus to increase the ordering temperature without loosing the characteristic

properties of these heterostructures.

In the macroscopic ferrimagnetic system, Co(EuS), a strong antiferromagnetic exchange is

established with the help of magneto-optic spectroscopy. The ordering temperature of the EuS

phase is enhanced from 16.5 K of bulk EuS to 160 K. The origin of the exchange coupling is

explained by the large surface-to-volume ratio of a phase-separated system, yielding a large

interaction interface between Co and Eu, while the crystalline precipitate EuS particles retain

the divalent valence of the Eu ion, necessary for a high magnetic moment. At room

temperature, an extraordinary optical-enhancement effect due to an accidental matching of the

complex index of refraction of EuS and Co in the ultraviolet part of the spectrum is observed.

This leads to polar Kerr rotations of up to 2° at 4.5 eV. A magneto-optic figure of merit is

derived which is similar in size as in Co-Pt multilayers and alloys. A technological potential of

this material is restricted by the ordering temperatures of the EuS phase which is below room

temperature. The addition of Tb leads to perpendicular magnetic anisotropy [111]. It enhances

the ordering temperature of the EuS phases even further, presumably up to room temperature.

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106 7 CONCLUSIONS AND OUTLOOK

However, the precipitate particles in the Tb doped system are much smaller than in Co(EuS)

diminishing the optical-enhancement effect. For an application as a magneto-optic storage

material, a small grain size is essential. With particle sizes less than 10 nm, even a granular or

phase-separated material as Co(EuS) could be used. Consequently, it would be advantageous to

be able to control the size of the precipitate particles and to know the influence on the optical

enhancement. In conclusion, the possibility of exchange coupling macroscopic phases opens

new ways of tailoring materials to specific needs.

In MnBi, a thickness dependence of the surface roughness and the magneto-optic properties

is found. Adding metallic interlayers between subsequent Bi/Mn bilayers leads to a distinct

reduction of surface roughness. In the case of Al interlayers, the magneto-optic properties do

not change considerably as compared to MnBi films of the same thickness. Adding Pt, a

decrease in magnetization in conjunction with a decrease of the magneto-optic properties is

observed while the coercivity behavior suggests a reduction of MnBi grain size. Adding a

reasonably thick Al interlayer or a SiOx protective layer will efficiently lower the coercivity.

These findings are explained by a 2.3% volume increase when MnBi is formed from a Bi/Mn

bilayer. Summarizing, MnBi, although being one of most intensively studied magnetic

materials, still bears a lot of unresolved questions. The outstanding magneto-optic properties,

especially in the ultraviolet part of the spectrum, justify further investigations. As shown in this

work, it is possible to influence surface roughness, grain size and coercivity by suitable

additions of metallic interlayers. However, the principal difficulties of this material, like the

structural phase instability and the temperature dependence of the coercivity, have not been

removed so far. An substantial step towards better control of the structural and magnetic

properties of MnBi would be an epitaxial deposition by co-evaporation. This goal has not been

achieved, yet.

One incentive of this work is to demonstrate the suitability of magneto-optic spectroscopy

for the investigation of magnetic thin films. Although magneto-optic effects have been

discovered 150 years ago, the method is still thriving. The extension of magneto-optic

techniques to higher photon energies, using synchrotron radiation, to higher lateral resolution,

with the help of near-field optical microscopy, and to higher time resolution as well a second

harmonic generation, using pulsed lasers, will certainly open a variety of new fields in the

physics of magnetic materials. We can look forward to finding out what for precious secrets

nature still has to reveal, lost in amazement like a child.

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7 CONCLUSIONS AND OUTLOOK 107

ACKNOWLEDGMENTS

This work summarizes the scientific results of the research in my group over the past five

years. Of course, these data would not have been obtained without the help of my students and

without the collaboration of dear friends all over the world. In particular, I thank R. J. Gambino

and H. Munekata for providing me with the Co(EuS) and the (In, Mn)As samples and, in the

first place, for their valuable discussions and friendship since we started our collaboration.

Furthermore, I like to thank my Ph.D. students, U. Rüdiger (who is in charge of making the

MnBi samples), G. Eggers, and A. Rosenberger and my former and present graduate students,

C. Spaeth, H. Berndt, A. Schirmeisen, P. Dworak, T. Roos, and G. Hoffmann for their

numerous scientific contributions to this work. The collaboration with the theoreticians

U. Nowak on magnetization reversal, M. Quinten on the optical properties of Co(EuS),

P. M. Oppeneer and J. Köhler on theoretical aspects of magneto-optic properties of the MnBi

system is gratefully acknowledged. I am obliged to J. Auge for performing SEM and

B. Holländer for performing RBS measurements. Last but not least, I am indebted to

G. Güntherodt who offered me the opportunity to pursue my scientific research at his institute.

Finally, the financial support of part of this work by the German Ministry of Education,

Science, Research and Technology, BMBF, under grant no. FKZ 13N6178/2 is gratefully

acknowledged.

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108 ACKNOWLEDGMENTS

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CURRICULUM VITAE 117

Nationality: Swiss

July 4, 1960 Born in Aarau (Switzerland).

Apr. 1967 - Sep. 1979 Basic education in Zofingen (Switzerland). Maturity type C (mathematics &physics) in September 1979.

Oct. 1979 - Oct. 1984 Study of physics at the ETH Zurich (Swiss Institute of Technology) withemphasis in high-energy physics and nuclear physics.

Nov. 1984 - May 1985 Diploma thesis at the ETH Zurich on x-ray production with atomic andmolecular beams in thin foils performed in the group of Prof. Dr. W. Wölfli.Diploma with distinction on May 7, 1985.

May 1985 - Oct. 1985 Research assistant in the group of Prof. Dr. H. Hofer at the Paul-ScherrerInstitute (formerly: Swiss Institute of Nuclear Research).

Nov. 1985 - Dec. 1989 Ph.D. thesis at the ETH Zurich under supervision of PD. Dr. J. Schoenes in thegroup of Prof. Dr. P. Wachter. Ph.D. thesis entitled "Magneto-optical Kerreffect study on the high field superconductors Eu1-xPbxMo6S8 andEu1-xSnxMo6S8-ySey", DISS. ETH No. 9082, accepted on May 28, 1990.

July 14, 1986 Teaching degree for physics and mathematics at high-school level.

Jan. 1990 - June 1990 Scientific assistant in the group of Prof. Dr. P. Wachter at the ETH-Zurich .

Oct. 1990 - Sep. 1992 Post-doctoral fellow in the group of Dr. T. Penney and Dr. R. J. Gambino at theIBM Thomas J. Watson Research Center in Yorktown Heights, New York.

Nov. 1992 - Oct. 1994 Substitute of a C3 professorship at II. Physikalisches Institut, Lehrstuhl A(Prof. Dr. G. Güntherodt), RWTH Aachen.

Nov. 1994 - Sep. 1996 Senior research assistant at II. Physikalisches Institut, Lehrstuhl A(Prof. Dr. G. Güntherodt), RWTH Aachen.

since Oct. 1996 Professor (C3) at the Institut für Halbleiterphysik und Optik, TU Braunschweig,Germany.

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118 CURRICULUM VITAE