UNIVERSIT DEGLI STUDI

ROMATRE

UNIVERSIT DEGLI STUDI ROMA TRE

Dipartimento di Matematica e FisicaCorso di Laurea in Fisica

Tesi Di Laurea

The Anomalous Hall Eect

Laureando

Arianna Dol

Matricola 461263

Relatore Relatore Esterno

Prof. Roberto Raimondi Prof. Ulrich Eckern

Anno Accademico 2014/2015

To all of you who have always supported me during this journey

Acknowledgements

First of all, I wish to thank Prof. Roberto Raimondi for giving me the opportunity to

write my Bachelor thesis with him and for his helpful critiques and suggestions on this

work.

Second, I wish to thank Prof. Dr. Ulrich Eckern and Sebastian Tlle for their constant

guidance during the entire work, for their availability and patience in answering my

questions and doubts. It really helped me to better understand the topic.

Third, I would like to thank my family, my mother Liliana and my brother Lorenzo, for

supporting me during the entire period of studying and for always encouraging me in

every choices I took. I do not think I could have reached this objective without you.

Special thanks go to my aunt, Maria Civita Pulcini, because it is also thanks to her

help and unconditional support if I was able to keep going forward despite problems

and diculties. I will always remember and cherish all the moments spent with her. I

hope I can be strong as she was.

Finally, I would like to thank all my friends for being such an important part of my

life. Thanks for all the beautiful and joyful moments, for all the laughter and for all

the experiences and adventures. Just thinking about all that we have done, all people

we have met and all places we have discovered, it makes me so nostalgic and willing to

do more.

Thanks also to all my Erasmus friends, who I met during my period abroad in Augsburg.

You really are special and I feel so happy and lucky to have met you and to have shared

such a beautiful experience as Erasmus together with all of you. If I could just go back

in time, I would not change a thing of that whole year spent together and I seriously

could not imagine now an Erasmus without you.

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In addition I want to say hello to my friends of the Mary Jane RT Hockey Club: Miguel,

Emilie, Juliette, Nina, Alfonso, Kubra, Atis and Darja. You were the rst group of

people I met when I arrived in Augsburg in September and since then we have done so

many experiences and traveling. I just want to say thank you for the great memories

and I am looking forward to meeting you once again.

Contents

Contents v

1 Transport equation 4

1.1 The classical Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . 4

1.2 The generalized Boltzmann equation . . . . . . . . . . . . . . . . . . . . 8

1.2.1 SU(2) formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Results 12

2.1 Edelstein eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Anomalous Hall eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Conclusion 29

Bibliography 30

v

Introduction

The Hall eect, discovered in 1879 by E. Hall Ref. [Hal79], states that when a current-

carrying conductor is placed in a magnetic eld, a voltage, transverse to the direction

of the current density, is induced. The appearance of this transverse voltage is a con-

sequence of the Lorentz force, ~F = e(~v ~B

), which acts on the charge carriers and

deects them against one side of the conductor.

For a magnetic eld along z-direction, ~B = Bzez, this results in a Hall resistivity

xy = Ey/jx, which has a linear dependence on the applied magnetic eld Ref. [Hal79].

The discovery of the Hall eect became important for determining the density and po-

larity of charge carriers in nonmagnetic materials.

Two years later, E. Hall found that this eect was much stronger in ferromagnetic con-

ductors and that the dependence of the Hall resistivity on the applied magnetic eld

was dierent from the analyzed case of a nonmagnetic material.

In fact, it is experimentally observed that in ferromagnetic conductors the resistivity

xy was at rst linearly increasing in weak magnetic elds and then, when the resistivity

reaches a certain value, it saturates. This saturation value was found to be nearly Bz

independent.

It was thought that in the expression of the resistivity in ferromagnetic materials must

appear an extra term, which depends on the magnetization, i.e. Mz, of the material.

Therefore this new eect was denoted as the anomalous Hall eect Ref. [NSO+10].

New experiments were carried out for the understanding of this "anomalous" behav-

ior of the resistivity in ferromagnetic conductors, which led to the formulation of an

empirical relation between xy, Bz and Mz

xy = R0Bz +RsMz. (1)

1

2

In Eq. (1), the rst term represents the ordinary Hall resistivity and the coecient R0

mainly depends on the density of charge carriers and the second term is the "anomalous"

Hall resistivity due to the spontaneous magnetization with a proportionality coecient

Rs that is the "anomalous" Hall coecient [NSO+10].

The main mechanism, which gives rise to an anomalous Hall eect, is spin-orbit coupling.

We may distinguish between two dierent types of spin-orbit interactions, extrinsic or

intrinsic, depending on its origin.

The intrinsic one depends on the crystalline potential associated with the band struc-

ture and is largely independent of scattering; the extrinsic one depends on the potential

due to random impurities of the sample.

The two dierent extrinsic spin-orbit coupling mechanisms are: skew-scattering which

takes in consideration the inuence of asymmetric scattering of electrons from impurities

due to the eective spin-orbit interaction of the electron and impurity and side-jumps

in which the electron velocity is deected in opposite directions by the electric elds

experienced when approaching or leaving an impurity [NSO+10] and [RSGV12].

The objective of this thesis is to describe the anomalous Hall conductivity in ferromag-

netic conductors as a consequence of spin-orbit coupling mechanisms. In particular in

this work, we will focus on the Rashba (intrinsic) spin-orbit coupling.

The thesis will be structured as follows:

-Chapter 1: We will introduce the classical Boltzmann equation, with which we are

able to describe charge transport phenomena in metals, considered as a two-dimensional

electron gas. This equation will allow us to evaluate the Drude conductivity.

At this stage spin-orbit coupling eects will not be considered.

Later, we will use a generalized Boltzmann equation within a SU(2) formulation. This

equation takes spin-orbit coupling into consideration and it allows us to study the trans-

port properties of both charge and spin. In the end we will be able to get the expression

of the anomalous Hall conductivity and its dependence on the considered mechanisms.

-Chapter 2: In this chapter we will use the generalized Boltzmann equation to study

explicitly two eects: the Edelstein and the anomalous Hall eect.

The Edelstein eect: we will consider a paramagnetic conductor (no spontaneous mag-

netization) and we will see that through spin-orbit coupling interactions a transverse

3

spin polarization can be induced in the sample by an electric eld.

The anomalous Hall eect: we will consider a ferromagnetic conductor with a sponta-

neous magnetization and we will study how the expressions of the induced spin polar-

ization and of the anomalous Hall conductivity are modied when spin-orbit coupling

together with a magnetization are present. In the latter case we will perform calcula-

tions considering the Elliot-Yafet spin relaxation (extrinsic spin-orbit coupling term).

Chapter 1

Transport equation

In this chapter we want to introduce some important tools for the description of trans-

port phenomena in current-carrying materials when external elds are applied. The

solid is considered as a simple metal given by a two-dimensional electron gas.

The study of these phenomena requires in particular the use of a transport equation,

the Boltzmann equation, which describes in both the classical and semiclassical limit the

transport properties of charge and spin. In the latter case it is extended to a generalized

Boltzmann equation, in which the spin-orbit coupling is taken into account.

The classical Boltzmann equation is useful to study the electrical and thermal conduc-

tion by applying the kinetic theory of gases to a metal, considered as a gas of electrons.

The generalized Boltzmann equation will be subsequently needed to understand and ex-

plain the anomalous Hall eect in ferromagnetic materials, that is a transverse particle

current would be generated in response to an electric eld due to spin-orbit coupling.

1.1 The classical Boltzmann equation

In this section we focus on studying the transport phenomena in metals, considered

as two-dimensional electron gases, ~p = ~p2/2m, to nd the result obtained by Drude,

which establishes a linear dependence of the current density ~j on the external applied

eld ~E

~j = ~E, (1.1)