+ All Categories
Home > Documents > MASTER OF SCIENCE DEGREE

MASTER OF SCIENCE DEGREE

Date post: 02-Jan-2022
Category:
Upload: others
View: 3 times
Download: 0 times
Share this document with a friend
38
Transcript
Page 1: MASTER OF SCIENCE DEGREE

TOPOLOGICAL ANTIFERROMAGNETIC MAGNONS

A Thesis Presented to the Department of Theoretical and Applied Physics,

African University of Science and Technology, Abuja

In partial ful�lment of the requirements for the award.

MASTER OF SCIENCE DEGREE

By

JAMIU ADAM AJIBOLA

Supervised by

PROF. AURELIEN MANCHON

African University of Science and Technology

www.aust.edu.ng

P.M.B 681, Garki, Abuja F.C.T

Nigeria.

July, 2019

Page 2: MASTER OF SCIENCE DEGREE

TOPOLOGICAL ANTIFERROMAGNETIC MAGNONS

By

ADAM JAMIU AJIBOLA

A THESIS APPROVED BY THE DEPARTMENT OF THEORETICAL AND

APPLIED PHYSICS

RECOMMENDED:

..................................................

Supervisor: PROF. AURELIEN MANCHON

..................................................

Head: Department of Theoretical Physics

APPROVED:

............................................................

Chief Academic O�cer (Prof. C. E. Chidume)

..................................................

Date

Page 3: MASTER OF SCIENCE DEGREE

ABSTRACT

We have studied a model for a non-collinear but coplanar antiferromagnetic spin tex-

ture on a two dimensional kagome lattice structure in the presence of Dzyaloshinskii-

Moriya Interaction (DMI). We observed some interesting topological properties in

our system i.e. the presence of non-trivial edge state in the wave function. This

non-trivial edge state, which mainly surfaced in the presence of the DMI, showed

robustness against the external magnetic �eld and thus can be further studied to

see how important transport properties can be computed.

i

Page 4: MASTER OF SCIENCE DEGREE

ACKNOWLEDGMENTS

My acknowledgement and appreciation goes to my amiable supervisor Professor Au-

relien Manchon, the group leader of the Spintronics research group at KAUST, for

his guidance during this research work and the entire personnel in the group who at

some points supported me. I look forward to continuing my PhD research under your

mentorship. My Masters degree was fully sponsored by the Pan-African Materials

Institutes and the African Development Bank, I acknowledge this support. I also

appreciate the entire academic and non-academic personnel, students and colleagues

of the African University of Science and Technology and the Nelson Mandela Insti-

tution for the love and experience shared. To my head of department Prof. Kenfack

Anatole, my ever supporting faculty, Dr. AbdulHakeem Bello and my mentor, Dr.

Collins A. Akosa thank you for your support. All softwares and hardwares used for

this research were licensed to King Abdullah University of Science and Technology.

Finally, much appreciation to my beloved wife, Hassanat and precious daughter Fa-

timah, for their patience, perseverance and understanding, I love you so much. MAY

ALMIGHTY GOD REWARD YOU ALL ACCORDINGLY. Above all, I give glory

to Almighty Allah for His Grace, Favour and Mercy throughout the journey of my

life so far. ALHAMDULILLAH.

The journey indeed has just begun . . .

ii

Page 5: MASTER OF SCIENCE DEGREE

DEDICATION

Glory be to the endower of knowledge, we have no knowledge except what He has

taught us; verily the All-knower, the All-wise [Quran 2:32]. This project is

dedicated to my parents Mr. and Mrs. Jamiu.

iii

Page 6: MASTER OF SCIENCE DEGREE

CONTENTS

Abstract i

Acknowledgments ii

Dedication iii

Table of Contents iv

1 Introduction 1

1.1 THEORETICAL BACKGROUND . . . . . . . . . . . . . . . . . . . 4

1.1.1 Linear spin-wave theory . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Holstein - Primako� Transformation . . . . . . . . . . . . . . 5

1.1.3 Topological Insulators and the Edge States . . . . . . . . . . . 7

2 LITERATURE REVIEW 9

3 METHODOLOGY 12

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 The Kagome Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Without Dzyaloshinskii Moriya Interaction . . . . . . . . . . . 14

3.2.2 With Dzyaloshinskii Moriya Interaction . . . . . . . . . . . . . 16

3.3 Construction of ribbon . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 RESULT AND DISCUSSION 22

iv

Page 7: MASTER OF SCIENCE DEGREE

LIST OF FIGURES

1.1 A wave packet of magnon moving from hot to cold is de�ected by

DMI. [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Condition for Anomalous Hall E�ect . . . . . . . . . . . . . . . . . . 3

1.3 Schematic diagram of edge states of a semi-in�nite plane of a 2D

system for (a) 2D ordinary insulator and (b) topological insulator [10]. 7

2.1 Collective spins excitation. . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 1200-coplanar antiferromagnets spin texture in the kagome lattice . . 13

3.2 Kagome lattice with lattice vectors a1 and a2. Dzaloshinskii-Mariya

vectors are aligned to the lattice plane and are denoted by red dots:

along -z (+z) for a clockwise (anticlockwise) chirality: C-B-A (A-B-

C)[21] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Kagome lattice showing the hopping direction (red and blue lines) . . 19

3.4 Edges of kagome lattice for two di�erent terminations. (a) smooth

termination (b) rough termination. . . . . . . . . . . . . . . . . . . . 21

4.1 Magnon bulk bands energy dispersion. (a) Without any component

of the DMI, J =0.5. (b) With the out-of-plane component of the DMI,

J = 0.5, D⊥ = 0.1 (c) With out-of-plane and in-plane components of

the DMI, J = 0.5, D⊥ = 0.1, D‖ = −0.25. (d) With h = -0.01. . . . . 23

4.2 Edge magnon in kagome lattice for N = 5 nanoribbon widths. The

parameters are J = 0.5, D⊥ = 0.1, D‖ = −0.25, h = −0.01 . . . . . . . 25

4.3 Trivial edge magnon in kagome lattice for (a) J = 0.5, D⊥ = 0.0, D‖ =

0.0, h = −0.0(b)J = 0.5, D⊥ = 0.0, D‖ = 0.0, h = −0.01 . . . . . . . . 26

4.4 Edge magnon in kagome lattice for varied external �eld. The param-

eters are J = 0.5, D⊥ = 0.1, D‖ = −0.25, h = -0.04 . . . . . . . . . . 26

v

Page 8: MASTER OF SCIENCE DEGREE

CHAPTER 1

INTRODUCTION

The last two decades have witnessed tremendous work in the study of collective

excitations of electron's spins called spin waves[1],[2] . Spin waves are disturbance of

the magnetic ordering that travel through the magnetic material. In the quantum

picture, the excitations can be described by magnons. Magnons are chargeless low-

energy collective excitations of localized neighboring spins, with a �xed amount of

energy and play the vital role of an elemental magnetic carrier in numerous insulating

magnets. Due to their intrinsic bosonic nature, magnons can form a macroscopic

coherent state and propagate spin information over several millimeters,[3] much

farther than spin-polarized conduction electrons in metals. With these interesting

and fascinating properties, there are possibilities of utilizing magnons as a carrier to

propagate information in a unit of Bohr magneton in spin-based devices, these have

attracted considerable attention [2],[4],[5],[6] over the last decade.

Antiferromagnetic materials (AFMs) are materials in which electron spin asso-

ciated to individual atom at particular lattice point are ordered antiparallel relative

to each other such that their net magnetization is equal to zero. This phenomenon

occurs below a reference temperature, known as the Neel temperature (TN) , above

this temperature, the material loses its order and behaves just like a paramagnetic

material. Among the properties that make AFMs a promising material for new

functionalities include:- absence of stray �elds, robustness against magnetic �eld

perturbation, ultrafast dynamics [7]. Looking at these properties, in addition to the

fascinating properties of magnons mentioned above, it is only natural to be funda-

mentally interested in understanding the basis of magnon transport in insulating

antiferromagnets.

Unlike the charge-based electronics device, which dissipates heat, spin-based elec-

tronics produces very low dissipation by replacing charge currents with spin currents

(currents magnetic moments). However, in conducting materials, the spin current

is transported by charge carriers that experience dissipation and thereby contribute

to Joule heating. So, to achieve less dissipative materials, one considers magnetic

1

Page 9: MASTER OF SCIENCE DEGREE

insulators, in which magnon is used to transport spin current. Among the promising

physical mechanisms adjunct to spin transport, spin Hall e�ect [8], which is the gen-

eration of a pure spin current transverse to an injected charge current, is particularly

interesting. This mechanism enables the generation of pure spin current which can

be used in spin-based devices. When spin current is carried by magnons, rather than

electrons, magnon Hall e�ect can also emerge and give new functionalities [2].

In 1879, Edwin Hall discovered the Hall E�ect, a phenomenon that occurs when

a current-carrying conductor is placed in a magnetic �eld in a direction perpen-

dicular to that of the �ow of current. Due the Lorentz force exerted by the mag-

netic �eld on the �owing electrons, a transverse electric �eld is created across the

conductor. The quantum-mechanical version of the Hall e�ect i.e., the quantum

Hall e�ect, is observed in a two-dimensional system subjected to low temperatures

and strong magnetic �elds, in which the Hall conductance undergoes quantum Hall

transitions to take on the quantized values. It was established that ferromagnetic

conductors which have nonzero intrinsic magnetization exhibit anomalous Hall e�ect

which is not a consequence of the Lorentz forces but of spin-orbit coupling (SOC)[2]

and therefore has nonzero transverse conductivity without external magnetic �eld.

Anomalous Hall E�ect (AHE) also exists in a "quantized version", called the Quan-

tum Anomalous Hall E�ect (QAHE), where charge edge currents are quantized [9].

In an insulating magnetic material with SOC present, [Fig. (1.2)], novel proper-

ties emerges which include topological states and unconventional magnetic ordering.

The presence of SOC in such a material presents itself as Dzyaloshinskii Moriya

Interaction to nontrivial topological magnetic states. Several interesting works have

proposed the use of collective magnetic excitations as a tool for understanding topo-

logical magnetic materials, [10], [2].

The Dzyaloshinskii Moriya Interaction (DMI) is an antisymmetric interaction

induced by SOC and it is responsible for the AHE observed in ferromagnets. About

six decades ago, the presence of DMI was demonstrated in non-centrosymmetric

systems [11],[12]. This antisymmetric interaction has great impact in the dynamics of

spin system, as it favours noncollinear magnetic con�guration and, as a consequence,

gives rise to topological spin texture. Futhermore, the presence of DMI brings about

non-trivial topologies of magnons which has promising applications.

In this research work, we considered an antiferromagnetic kagome lattice because

of its non-collinear magnetic structure which displays topological features. Topol-

ogy has been a concept used in mathematics for about three centuries . Topological

2

Page 10: MASTER OF SCIENCE DEGREE

properties are the properties of a system that are preserved under continuous defor-

mations. These properties depend on the entire structure of the system rather than

on the local features. The non-trivial topology of a material means that at the edges

- where the continuity of the material is broken - the topological invariant is broken

too. In the case of topological insulators, it implies the presence of edge states that

can transport spin and/or charge currents. Now, the concept has been extended

to magnonic materials, where the non-trivial topology of the magnon states of the

volume implies the existence of magnonic edge states[13]. In the presence of lon-

gitudinal temperature gradient, magnon edge current produces transverse thermal

current. This e�ect was �rst observed experimentally by Onose et al in a pyrochlore

ferromagnet [2], [Fig.(1.1)] and latter in kagome ferromagnet [4], where the DMI

acts, as a vector potential.

Figure 1.1: A wave packet of magnon moving from hot to cold is de�ected by DMI.

[2]

In this thesis, we aim to investigate the role of the topology of the magnon

wave function in 1200 coplanar AFM spin texture (kagome lattice) in the presence

of Heisenberg interaction, DMI and exchange �eld. To track the presence of edge

states which will serve as a basis to compute the spin Seebeck e�ect, a crucial

quantity that is relevant to experiments. Also, this thesis will serve as a basis to

future investigations on the transport properties of the magnonic edge states.

Figure 1.2: Condition for Anomalous Hall E�ect

3

Page 11: MASTER OF SCIENCE DEGREE

1.1 THEORETICAL BACKGROUND

1.1.1 Linear spin-wave theory

The linear spin-wave theory (LSWT) enables us to study the physical properties of

magnetic ordering at non-zero temperature. It was �rst introduced by F. Bloch and

Slater independently and later developed by Holstein - Primako� using the second

quantization bosonic operators [14]. Here, we introduce the linear spin-wave analysis

based on the work of S. Toth and B Lake [15] as well as Mook et al [16], given the

spin Hamiltonian in the form:

H =1

2

∑ij

STi JijSj (1.1)

The Heisenberg exchange Hamiltonian above describes a system of interacting spins

in an insulating magnet and can be used to study the behavior of magnons. As shown

in the equation above, the sum goes over all nearest neighbors' pairs i and j on the

lattice sites. With the spin operator Si and Sj at the lattice sites i and j respectively,

and the magnetic interaction or coupling term between both spins is de�ned by Ji,j.

This coupling constant is positive for the case of an antiferromagnetic ordering.

By treating these quantum mechanical spins as classical vectors, one obtains the

classical ground state, which can be minimized to obtain the ground state energy.

Once this is achieved, the number of spins in the magnetic unit cell and their local

directions are known. Now, we sum the Hamiltonian over the entire M sites of the

magnetic unit cell, referred to by the indices i, j, and a sum over the N unit cells of

the magnetic volume, indexed by l, n.

H =1

2

M∑i,j

N∑l,n

STl,iJijSn,j (1.2)

To investigate the magnon band structure, one needs to express each magnetic mo-

ment from its local frame of reference onto a global frame of reference. So, Using

the Cartesian coordinate system, each spin coordinate system is localized, thus we

rotate each of the spins on to a global coordinate system (x,y, z) by introducing a

rotation matrix. This will also make it easy to transform from the spin Hamiltonian

to the desired bosonic Hamiltonian. The associated rotation matrix is,

Rn = (xn, yn, zn) (1.3)

4

Page 12: MASTER OF SCIENCE DEGREE

The associated transformation matrix for the n spin operator in the j unit cell reads:

Sn,j = RnSn (1.4)

Here, Sn,j is de�ned in the global cartesian system (x, y, z) while Snj is de�ned in

the local coordinate system (xj, yj, zj).

1.1.2 Holstein - Primako� Transformation

To study how magnons behave in our antiferromagnetic kagome system using the

LSWT, we �rst make the magnon operator appear explicitly in our Hamiltonian.

This is usually done using the Holstein-Primako� transformation [14], that is, the

mapping of the spin operators to the bosonic annihilation and creation operators,

Eq.(1.5). The spin operator S can be written explicitly in terms of annihilation and

creation operators as shown below:

Sn,j =

Sn2

(an,j + a†n,j)

−i√

Sn2

(an,j − a†n,j)(S − a†n,jan,j)

(1.5)

With this an,j(a†n,j) represents the bosonic annihilation (creation) operators and

obeys the commutation relation [ai, a†j] = δi,j. Also the operator n = a†n,jan,j counts

the number of bosons (magnons) placed at site j and it is often called the number

operator. The spin operators in the global reference frame can be expressed as

follow:

Sn,j =(√

Sn2

(u∗nan,j + una†n,j) + zn(S − a†n,jan,j)

)(1.6)

Where: un = xj + iyj

Next, we obtain the magnon Hamiltonian in momentum space by taking the

Fourier transform of the magnon operator from the position space to momentum

space using the Bloch state of a single magnon given by:

a(†)n,j =

1√M

∑i,j

e(−)i(rn+tj)a(†)n,k (1.7)

With rn as the position of the nth basis and tj the vector to the jth spin in the

basis. By substituting Eq.(1.7) into the Hamiltonian equation, Eq.(1.2), one gets

the bilinear Hamiltonian as:

H =1

2

∑k

Ψ†kHkΨk (1.8)

5

Page 13: MASTER OF SCIENCE DEGREE

with Ψ†k as the row vector of the bosonic operators given by:

Ψ†k = (a(†)k,1, ..., a

(†)k,N , a−k,1, ..., a−k,N) (1.9)

the non-hermitian Hamiltonian Hk of the linear spin-wave matrix contains the fol-

lowing submatrices:

Hk =

(Ak − C Bk

B†k ATk − C

)(1.10)

elements of the sub-matrices of matrix read [16],

(A)l,n =

√SlSm2

uTnJlmk u∗n

(B)l,n =

√SlSm2

uTnJlmk un

(C)l,n = δl,n∑p

SpzTl J

ln0 zl

This Hamiltonian can be diagonalized by using paraunitary Bogoliubov transforma-

tion which gives the magnetic normal modes bnk with corresponding bands energies

εnk [17], where n here is the band index.

H =1

2

∑k

(b†k b−k

)Ek

(bk

b†−k

)=∑k

∑n

εnk(b†nkbnk + 1

2

)(1.11)

The basic idea of the paraunitary Bogoliubov transformation is to introduce a set of

operators such that the Hamiltonian has only terms proportional to

(bk

b†−k

). These

operators are de�ned below in terms of the coe�cient matrix Tk. With

b†k =(b†1,k..., b

†N,k

)(1.12)

(bk

b†−k

)= Tk

(ψk

ψ†−k

)(1.13)

Ek = T †kHkTk = daig (ε1,k, ...εN,k, ε1,−k, ...εN,−k) (1.14)

This transformation arises from the need to ensure the bosonic commutation relation

is satis�ed, i.e

T †kσ3Tk = σ3

6

Page 14: MASTER OF SCIENCE DEGREE

Tkσ3T†k = σ3

where ,σ3= daig (IN×N ,−IN×N) showing that the Hamiltonian consists of two copies

of the same eigenvectors.

1.1.3 Topological Insulators and the Edge States

Topological insulators (TI) are nonmagnetic materials that have a bulk band gap like

an ordinary insulator but have protected conducting gapless states on their edges or

surfaces. These insulators are in the same category of topological phase as Quantum

Hall systems. In topological insulators, time-reversal symmetry is assumed, so it

does not require a magnetic �eld to produce edge state in a 2D material. Edge

states consist of pairs of states where up-spin and down-spin counter propagate due

to the combination of spin-orbit interactions and time-reversal symmetry [10], [13].

The spin-orbit coupling behaves like a spin-dependent magnetic �eld and gives rise

to spin-dependent quantum Hall e�ect.

Figure 1.3: Schematic diagram of edge states of a semi-in�nite plane of a 2D

system for (a) 2D ordinary insulator and (b) topological insulator [10].

Edge states arise at the interface between two insulators with di�erent topological

orders. These states reveal the topological phase transition between trivial and non-

trivial insulators. The trivial and non-trivial insulators themselves are characterized

by a Z2 index, v, (v = 0→ ordinary insulator , v = 1→ topological insulator ).

Now, because the edge states occur at the topological phase boundary, they are

determined by the topology of the two insulators on each side of the interface. As

such, they are robust against non-magnetic impurities, provided the topology of

the two insulators is una�ected by the impurities. This is because Z2 topological

number cannot change continuously when nonmagnetic impurities are added, the TI

remains invariant [10]. Edge states have been experimentally observed in materials

7

Page 15: MASTER OF SCIENCE DEGREE

like CdTe/HgTe [18]. The topological properties of a system can be determined from

its Hamiltonian. A way to determine these properties of the system is by considering

the wave function. These properties are robust against perturbations as mentioned

above. This indicates that one could obtain currents without dissipation (like in

quantum Hall e�ect or in superconductivity).

8

Page 16: MASTER OF SCIENCE DEGREE

CHAPTER 2

LITERATURE REVIEW

Antiferromagnetic spintronics is an aspect of solid state physics that deals with spin

transport in antiferromagnets. In the past few decades, this �eld of study has wit-

nessed tremendous development [7]. Each electron in a material possesses intrinsic

spin which behaves like a small magnet and has a magnetic moment. Antiferro-

magnets are materials which have their magnetic moments align anti-parallel to the

neighboring moments and this occurs below a certain temperature called the Neel

temperature (TN). Above this temperature, the material loses this alignment due

to thermal agitation. Despite having it net magnetization to be zero, it is di�cult

to harness for practical applications, antiferromagnets possess quite a number of

interesting properties, such as the absence of stray �elds, robust against magnetic

�elds, ultrafast dynamics; these properties and more make it a suitable candidate

for novel applications.

Generally, in a spin system, the total magnetic exchange interaction between

pair of spins have contributions from the symmetric and antisymmetric exchange

interactions. The symmetric exchange, also known as the Heisenberg exchange,

promotes collinear alignment between pairs of spins, while antisymmetric exchange

interactions favor an orthogonal orientation. This exchange is called Dzyaloshinskii-

Moriya interaction and arises in non-centrosymmetric environment in the presence of

spin-orbit coupling [19]. Both symmetric and antisymmetric exchange interactions

could be anisotropic.

In the classical picture of the ground state of a simple insulating ferromagnet

with all spins aligning parallel, a possible excitation occur when one spin is re-

versed. Also the spins can share the reversal as shown in the [Fig.(2.1)] below [20].

The elementary collective excitation of the magnetic system has a wave-like form

which is called magnon. Magnon is a quantum chargeless quasiparticle correspond-

ing to an elementary increase of energy and an elementary reduction of the total

magnetization. Just as phonon is a quantum elastic wave and photon a quantum of

electromagnetic wave, magnon is a quantum spin-wave.

9

Page 17: MASTER OF SCIENCE DEGREE

Figure 2.1: Collective spins excitation.

There have been several experimental and theoretical reports on the AHE caused

by collective spin excitations (magnons), commonly called magnon Hall E�ect (MHE),

in both insulating ferromagnets and antiferromagnets. Experimental result for an

insulating ferromagnets with pyrochlore lattice structure showed transverse heat

current when a longitudinal temperature gradient was applied [2]. This transverse

current is associated to Hall E�ect [21]. Using both the semi-classical approach

and the linear response theory, Matsumoto and Murakami predicted a new form of

topological insulator, inspired from Onose's experiment [1]. They explained that the

MHE is due to non-compensated magnon motion along the boundary of the sample

(magnon edge current) and that this net current is a consequence of the topology of

the system, hence, the existence of topological magnon insulator [9]. The presence

of spin-orbit coupling which manifests as DMI is responsible for nontrivial topology

in kagome ferromagnets [21, 4] and honeycomb lattice [6] systems.

In a 2D like Dirac Hamiltonian, time reversal symmetry de�ned with sublattice

pseudo spins is preserved and in a magnon system, the Dirac points are robust

against magnon-magnon interaction [5] . The Dirac point is the crossing point of

the linear energy dispersion curves. Opening a gap at the Dirac points lead to

magnon edge states as the case of Haldane electronic system [22]. It was reported

that Mn3Sn a hexagonal antiferromagnet that exhibits non-collinear ordering at

the TN , shows large anomalous Hall e�ect [23] .i.e a phenomenon initially associated

with ferromagnets. This empirical evidence has since open a new page for the intense

research in the �eld of antiferromagnetic spintronics.

The AHE observed in ferromagnetic materials �nds its origin from the spin-orbit

10

Page 18: MASTER OF SCIENCE DEGREE

coupling which favors certain scattering direction for up-spins and down-spins, each

moving on opposite edge. For a ferromagnet there is the uneven spin population at

these opposite edges, thereby setting up a transverse electric �eld in the absence of

external magnetic �eld.

11

Page 19: MASTER OF SCIENCE DEGREE

CHAPTER 3

METHODOLOGY

3.1 Introduction

In this section, we describe how magnons propagate on 1200-coplanar antiferromag-

nets spin texture in kagome lattice in the presence of Heisenberg interaction, DMI

and external �eld. The quantum mechanical Heisenberg model Hamiltonian consid-

ered is

H = HH +HDM +Hext (3.1)

where,

HH = −∑i 6=j

J ijSi · Sj (3.2)

HDM = −∑i 6=j

Dij · Si × Sj (3.3)

Hext = −gµB∑i 6=j

H · Si (3.4)

With the Heisenberg interaction J ij = J ji being symmetric and the DMI Dij =

−Dji constrained by symmetry. In this system, we are considering both the in-plane

and out-of-plane components of the DMI given by: Dij = νij (D‖nxij, D‖n

zij, D⊥)

12

Page 20: MASTER OF SCIENCE DEGREE

3.2 The Kagome Lattice

Figure 3.1: 1200-coplanar antiferromagnets spin texture in the kagome lattice

The kagome lattice has three atoms in its unit cell which are arranged at the corners

of an equilateral triangle with the side length equal half of the lattice constant (�gure

3.1). The lattice vectors in the Cartesian coordinate are

a1 = a2(x+

√3y)

a2 = a2(−x+

√3y)

13

Page 21: MASTER OF SCIENCE DEGREE

Figure 3.2: Kagome lattice with lattice vectors a1 and a2. Dzaloshinskii-Mariya

vectors are aligned to the lattice plane and are denoted by red dots: along -z (+z)

for a clockwise (anticlockwise) chirality: C-B-A (A-B-C)[21]

Now we apply the linear spin-wave method discussed in chapter one to solve the

Hamiltonian of our system. In this system, each atom is surrounded by four nearest

neighbors. We started by performing the calculation for the Hamiltonian without

the DMI.

3.2.1 Without Dzyaloshinskii Moriya Interaction

Let H1, H2 and H3 be the Hamiltonian of each of the three atoms with their nearest

neighbors, and HH be the total Hamiltonian of the unit cell.

H1 =1

2J [SA · (SB1 + SB2 + SC1 + SC2)] (3.5)

H2 =1

2J [SB · (SA1 + SA2 + SC1 + SC2)] (3.6)

H3 =1

2J [SC · (SA1 + SA2 + SB1 + SB2)] (3.7)

HH = H1 +H2 +H3 (3.8)

We have to rotate the local coordinate system (xj, yj, zj) of each spin onto the

global coordinate system (x, y, z) to perform the Holstein-Primako�'s transforma-

tion. To do so we de�ned a local rotation matrix for each spin as: Rn = (xn, yn, zn).

14

Page 22: MASTER OF SCIENCE DEGREE

In the x - z plane, we have:-

R =

−1

20 −

√3

2

0 1 0√

32

0 −12

(3.9)

Which gives the transformation as: Sn = RSn,j.In the global reference frame of the Cartesian coordinate, the Holstein Primako�'stransformation for each spin is:-

Sj,n =

√Sn2

(aj,n + a†j,n)

−i√Sn2

(aj,n − a†j,n)

(S − a†j,naj,n)

(3.10)

SA =

(√Sn

2(aj,A + a†j,A),−i

√Sn

2(aj,A − a†j,A), (S − a†j,Aaj,A)

)(3.11)

SB =

(−

1

2

√Sn

2(aC,A + a†B,j) +

√3

2(S − a†B,jaB,j),−i

√Sn

2(aB,j − a†B,j),−

√3

2

√Sn

2(aC,A + a†B,j)−

1

2(S − a†B,jaB,j

)(3.12)

SC =

(−

1

2

√Sn

2(aC,A + a†C,j) +

√3

2(S − a†C,jaC,j),−i

√Sn

2(aC,j − a†C,j),−

√3

2

√Sn

2(aC,A + a†C,j)−

1

2(S − a†C,jaC,j)

)(3.13)

Now inserting the Holstein Primako�'s transformation described in Eq.(3.11) �

(3.13) in the Hamiltonian and ignoring magnon-magnon interaction, i.e. only terms

in the quadratic form are considered, we have;

HH =3

2JZNS2 −

3JS

8

A,B,C∑α6=β

[a†α,ja†β,1 + a†α,ja

†β,2]−

3JS

8

A,B,C∑α6=β

[aα,jaβ,1 + aα,jaβ,2]

+JS

8

A,B,C∑α 6=β

[a†α,jaβ,1 + a†α,jaβ,2] +JS

8

A,B,C∑α6=β

[aα,ja†β,1 + aα,ja

†β,2] +

2JS

8

A,B,C∑α6=β

[a†α,jaα,j ]

Next, we perform the Fourier transformation of the operators to diagonalize the

Hamiltonian. We choose the vectors k to lie in the Brillouin zone associated with

each sub-lattice.

The Fourier transform is of the form:

aα,j =1√L

∑k

ak,αeik·(rα+rj)

15

Page 23: MASTER OF SCIENCE DEGREE

a†α,j =1√L

∑k

a†k,αe−ik·(rα+rj) (3.14)

Where rj is the position of the jth basis and rα the vector of the spin in the α th

basis,

α, β = A,B,C, j = 1, 2...L

δαβ = rβ − rα (3.15)

We now evaluate each term present in the Hamiltonian, i.e.∑j

a†α,ja†β,j =

1

L

∑j

∑k

∑k′

a†k,αe−ik·(rα+rj)a†k′,βe

−ik′·(rβ+rj) (3.16)

=∑k

∑k′

δk′,ka†k′,βe

−i(k′rβ+krα) (3.17)

=∑k

a†k,αa†−k,βe

−ik·(rβ−rα) (3.18)

=∑k

a†k,αa†−k,βe

−ikδα,β (3.19)

By performing similar transformation to other components in the Hamiltonian,(shown in appendix). The Hamiltonian becomes:

HH = −3

2JZNS2 +

3JZS

4

A,B,C∑k,α6=β

[ak,αak,β + a†k,αa†k,β ]γk −

JZS

4

A,B,C∑k,α6=β

[a†k,αak,β + ak,αa†−k,β ]γk − 2JS

A,B,C∑k,α6=β

a†k,βak,α

(3.20)

With γk = 2z

∑δα,β

cos(k·δα,β) and the operators satisfying the commutation relation

[ak, a†k′ ] = δk′,k.

This Hamiltonian has the general form as shown in Eq.(3.20) detailed in appendix

B

HH = −3

2JZNS2 +

s

2

∑k

Ψ†HKHΨk (3.21)

Ψ†k = (a†A,k, a†B,k, a

†C,k, a

†A,−k, a

†B,−k, a

†C,−k) (3.22)

3.2.2 With Dzyaloshinskii Moriya Interaction

By adding the DMI terms derived in the appendix, we obtained the bilinear Hamil-

tonian in Eq.(3.27). Here we let H ′1, H′2 and H ′3 be the Hamiltonian of each of the

three atoms with their nearest neighbors, and HDM be the Hamiltonian contribution

from DMI.

H′1 =1

2[SA × (SB1 + SB2 + SC1 + SC2)] (3.23)

16

Page 24: MASTER OF SCIENCE DEGREE

H′2 =1

2[SB × (SA1 + SA2 + SC1 + SC2)] (3.24)

H′3 =1

2[SC × (SA1 + SA2 + SB1 + SB2)] (3.25)

HDM = Dij · (H′1 +H′2 +H′3) (3.26)

H = H0 +S~2

2

∑k

Ψ†Hkψk (3.27)

We �nally obtain:

Hk =

(Ak − C BkB†k A†−k − C

)(3.28)

And the matrix representation has the form:

Ak =

0 (j − iD‖(cosθ +√

3sinθ))cAB (j + iD‖(cosθ −√

3sinθ))cAC

(j + iD‖(cosθ +√

3sinθ))cAB 0 (j + 2iD‖cosθ)cBC

(j − iD‖(cosθ −√

3sinθ))cAC (j − 2iD‖cosθ)cBC 0

(3.29)

A−k =

0 (j + iD‖(cosθ +√

3sinθ))cAB (j − iD‖(cosθ −√

3sinθ))cAC

(j − iD‖(cosθ +√

3sinθ))cAB 0 (j − 2iD‖cosθ)cBC

(j + iD‖(cosθ +√

3sinθ))cAC (j + 2iD‖cosθ)cBC 0

(3.30)

Bk =

0 (j3 − iD‖(3cosθ −√

3sinθ))cAB (j3 + iD‖(cosθ +√

3sinθ))cAC

(j3 − iD‖(cosθ −√

3sinθ))cAB 0 (j3 − 2√

3iD‖sinθ)cBC

(j3 + iD‖(cosθ +√

3sinθ))cAC (j3 − 2√

3iD‖sinθ)cBC 0

(3.31)

B†k =

0 (j3 + iD‖(3cosθ −√

3sinθ))cAB (j3 − iD‖(cosθ +√

3sinθ))cAC

(j3 + iD‖(cosθ −√

3sinθ))cAB 0 (j3 + 2√

3iD‖sinθ)cBC

(j3 − iD‖(cosθ +√

3sinθ))cAC (j3 + 2√

3iD‖sinθ)cBC 0

(3.32)

C =

j0 0 0

0 j0 0

0 0 j0

(3.33)

with

j0 = −4(√

3D⊥ + J)

j = (√

3D⊥ − J)

j3 = (√

3D⊥ − 3J) (3.34)

δAC = (1

2, 0)

17

Page 25: MASTER OF SCIENCE DEGREE

δAB = (1

2,

√3

2)

δBC = (−1

2,−√

3

2) (3.35)

cαβ = cos(k·δαβ) (3.36)

To diagonalize this Hamiltonian, we introduce the paraunitary Bogoliubov trans-

formation as discussed in section (1.3) in chapter one.

Hk =1

2

∑k

(b†k b−k

)εk

(bk

b†−k

)=∑k

∑n

εnk(b†kb−k +

1

2) (3.37)

With b†k = (b†1,k, ..., b†N,k) (

bk

b†−k

)= T−1

k

(Ψk

Ψ†−k

)εk = T †kHkTk = diag(ε1,k, ..., εN,k, ε1,−k, ..., εn,−k)

This canonical transformation arises from the need to ensure the bosonic commu-

tation relation is satis�ed. This Hamiltonian diagonalization was done numerically

in the Bogoliubov transformation basis.

3.3 Construction of ribbon

In order to track the presence of edge states, we constructed a model for the nanorib-

bon from the bulk Hamiltonian. To build this model, we rewrote the cosine terms

in exponential form. The cosine comes from the assumption that the wave functions

are Bloch states. In a ribbon, the wave functions are Bloch states along the ribbon

direction, but are quantized along the transverse direction. Therefore, one manner

to represent this symmetry lowering is to write the cosine functions into exponen-

tial form, retaining only the exponential factors along the ribbon direction, while

discarding the exponents along the transverse direction. The ribbon was consid-

ered to be in�nite along the x-direction and �nite along the y-direction. The green

lines show the chain direction, the red and blue lines show the hopping along the

transverse direction [Fig.(3.3)]. We as well considered two di�erent types of edges

18

Page 26: MASTER OF SCIENCE DEGREE

truncations [Fig.(3.4)] for several numbers of ribbons, N, ranging from N = 5, N =

30 to N = 80.

Figure 3.3: Kagome lattice showing the hopping direction (red and blue lines)

The expansion of the cosine functions of interests reads

cAB = cos

(−1

2kx +

√3

2ky

)=

1

2

[exp−

12ikxa expi

√3

2kya + exp−−

12ikxa exp−i

√3

2kya]

cBC = cos

(−1

2kx −

√3

2ky

)=

1

2

[exp−

12ikxa exp−i

√3

2kya + exp−−

12ikxa exp−−i

√3

2kya]

(3.38)

which allows us to write the bulk Hamiltonian in the form:

H = Hchain + τ expi√

32kxa +τ † exp−i

√3

2kxa (3.39)

In other words, the bulk Hamiltonian is separated into chains of along x, represented

byHchain, and hopping matrices along y, represented by τ . The expression of Hchain

and τ are given explicitly in the next

a. Ribbon Hamiltonian

19

Page 27: MASTER OF SCIENCE DEGREE

HChain

=1 2

2J

0(J−iS

1)ex

p−

ikxa

42(J

+iS

2)cos(kxa

2)

0(J

3−iS

4)ex

p−

ikxa

42(J

3+iS

3)cos(kxa

2)

(J+iS

1)ex

pikxa

42J

0(J

+2D‖icosθ

)ex

p−

ikxa

4(J

3−iS

4)ex

pikxa

40

(J3−iD‖2√

3sinθ)ex

p−

ikxa

4

2(J−iS

2)cos(kxa

2)

(J−

2D‖icosθ

)ex

pikxa

42J

02(J

3+iS

3)cos(kxa

2)

(J3−iD‖2√

3sinθ)ex

pikxa

40

0(J

3+iS

4)ex

p−

ikxa

42(J

3−iS

3)cos(kxa

2)

2J

0(J

+iS

1)ex

p−

ikxa

42(J−iS

2)cos(kxa

2)

(J3

+iS

4)ex

pikxa

40

(J3

+iD‖2√

3sinθ)ex

p−

ikxa

4(J−iS

1)ex

pikxa

42J

0(J−

2D‖icosθ

)ex

p−

ikxa

4

2(J

3−iS

3)cos(kxa

2)

(J3

+iD‖2√

3sinθ)ex

pikxa

40

2(J

+iS

2)cos(kxa

2)

(J+

2D‖icosθ

)ex

pikxa

42J

0

(3.40)

b.Hoppingmatrix

τ=

1 2

0(J

+iS

1)ex

pikxa

40

0(J

3−iS

4)ex

pikxa

40

00

00

00

0(J−

2D‖icosθ

)ex

p−

ikxa

40

0(J

3+iD‖2√

3sinθ)ex

p−

ikxa

40

0(J

3+iS

4)ex

pikxa

40

0(J

+iS

1)ex

pikxa

40

00

00

00

0(J

3+iD‖2√

3sinθ)ex

p−

ikxa

40

0(J−

2D‖icosθ

)ex

p−

ikxa

40

(3.41)

20

Page 28: MASTER OF SCIENCE DEGREE

Figure 3.4: Edges of kagome lattice for two di�erent terminations. (a) smooth

termination (b) rough termination.

c. The ribbon system

For the Hamiltonian of individual chains, Hchain, and the Hamiltonian of transverse

hopping, τ , we can reconstruct the Hamiltonian of the nanoribbon. For instance,

the Hamiltonian of a nanoribbon with N=5 chains, reads

H5 =

Hbdary τ 0 0 0

τ † Hchain τ 0 0

0 τ † Hchain τ 0

0 0 τ † Hchain τ

0 0 0 τ † Hbdary

(3.42)

21

Page 29: MASTER OF SCIENCE DEGREE

CHAPTER 4

RESULT AND DISCUSSION

Let us �rst study the properties of the bulk antiferromagnet. We observed that when

the DMI components are added to the system, the band's degeneracy is lifted and

a bandgap is seen between the lowest band and the top bands. The band structure

showed particle-hole symmetry, there's one �at band and the dispersion is linear

close to kx = 0.5 Fig. (4.1a). When the out-of-plane DMI is turned on Fig. (4.1b),

the band degeneracy is lifted although no band gap due to touching points on the

bands. The in-plane DMI opens up gap at the touching points between the top bands

and the lowest band Fig. (4.1c). Although, the linear dispersion close to kx = 0.5

is maintained, it seems like a reminiscence of the linear dispersion obtained for a

collinear antiferromagnetic chain without anisotropy. But, the addition of external

�eld to the system, eliminates the Dirac cpoint at kx = 0.5. Thus, the coexistence

of DMI and magnetic gap Fig. (4.1d) suggests that topological edge states might

arise in this system. So, we now move on to the ribbon case in order to observe the

emergence of such states.

22

Page 30: MASTER OF SCIENCE DEGREE

Figure 4.1: Magnon bulk bands energy dispersion. (a) Without any component of

the DMI, J =0.5. (b) With the out-of-plane component of the DMI, J = 0.5, D⊥ =

0.1 (c) With out-of-plane and in-plane components of the DMI, J = 0.5, D⊥ =

0.1, D‖ = −0.25. (d) With h = -0.01.

Let us now turn on the DMI, as depicted in Fig.4.2 for various nanoribbon widths.

Figures 4.2(a), (c) and (e) represent the one-dimensional band structure for non-zero

external �eld and out-of-plane DMI, while keep the in-plane DMI zero. For a narrow

ribbon [N=5, Fig. 4.2(a)], the band structure is gapped. This gap is systematically

quenched upon increasing the width as shown in Figs. 4.2(c) and 4.2(e). This band

structure is comparable to the one computed in Fig. (4.1c).

We now turn on the in-plane DMI. Again, a central gap is observed for the

23

Page 31: MASTER OF SCIENCE DEGREE

narrow nanoribbon [N=5, Fig. 4.2(b)], which is progressively closed upon increasing

the ribbon width [Figs. 4.2(d) and 4.2(f)]. The remarkable feature is observed at

positive energy, around E = 2.5t. There, a bulk gap opens upon increasing the ribbon

width, as already observed in Fig. 4.1(d). However, while Fig. 4.1(d) exhibits a

bulk gap, Fig. 4.2(f) reveals the emergence of single gap state which corresponds to

the topological magnon edge state.

24

Page 32: MASTER OF SCIENCE DEGREE

Figure 4.2: Edge magnon in kagome lattice for N = 5 nanoribbon widths. The

parameters are J = 0.5, D⊥ = 0.1, D‖ = −0.25, h = −0.01

25

Page 33: MASTER OF SCIENCE DEGREE

Figure 4.3: Trivial edge magnon in kagome lattice for (a) J = 0.5, D⊥ = 0.0, D‖ =

0.0, h = −0.0(b)J = 0.5, D⊥ = 0.0, D‖ = 0.0, h = −0.01

Figure 4.4: Edge magnon in kagome lattice for varied external �eld. The parame-

ters are J = 0.5, D⊥ = 0.1, D‖ = −0.25, h = -0.04

In conclusion, we have studied a model for a non-collinear and coplanar antiferro-

magnetic spin texture on a two dimensional kagome lattice. We observed topological

properties in our system i.e. the presence of non-trivial edge states in the energy

spectrum. This edge state showed robustness against the external magnetic �eld

[Fig.(4.4)]. For further study, we hope to use the linear response Kubo formula to

compute the spin Hall and spin thermal Hall conductivities, two important proper-

ties relevant to the experiment.

26

Page 34: MASTER OF SCIENCE DEGREE

APPENDIX

Explicit expression of the magnon Hamiltonian given

in Eq. (3.20).

HH = H1 +H2 +H3

HH = −JS

8

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(3aA1aB1 + 3aB1aaA1 + 3aA1aB2 + 3aB1aA2 + 3aA1aC1 + 3aC1aA1 + 3aA1aC2 + 3aC1aA2)

+(3a†A1a†B1 + 3a†B1a

†A1 + 3a†A1a

†B2 + 3a†B1a

†A2 + 3a†A1a

†C1 + 3a†C1a

†A1 + 3a†A1a

†C2 + 3a†C1a

†A2)

+(3aB1aC1 + 3aC1aB1 + 3aB1aC2 + 3aC1aB2 + 3a†B1a†C1 + 3a†C1a

†B1 + 3a†B1a

†C2 + 3a†C1a

†B2)

+(−a†A1aB1 − a†B1aA1 − a†A1aB2 − a†B1aA2 − a†A1aC1 − a†C1aA1 − a†A1aC2 − a†C1aA2)

+(−aA1a†B1 − aB1a

†A1 − aA1a

†B2 − aB1a

†A2 − aA1a

†C1 − aC1a

†A1 − aA1a

†C2 − aC1a

†A2)

+(a†B1aC1 − a†C1aB1 − a†B1aC2 − a†C1aB2 − aB1a†C1 − aC1a

†B1 − aB1a

†C2 − aC1a

†B2)

−2(−12S + 6aA1aA1 + 2aA2aA2 + 6aB1aB1 + 2aB2aB2 + 6aC1aC1 + 2aC2aC2)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣(4.1)

by taking 2cαβ = γk, and writing HH in matrix form gives:

HH = −3

2JZNS2 +

s

2

∑k

Ψ†HkHΨk (4.2)

HkH = J

−4 −cAB −cAC 0 3cAB 3cAC

−cAB −4 −cBC 3cAB 0 3cAC

−cAC −cBC −4 3cAB 3cBC 0

0 3cAB 3cAC −4 −cAB −cAC3cAB 0 3cAC −cAB −4 −cBC3cAB 3cBC 0 −cAC −cBC −4

(4.3)

The matrix is given as:

HkH =

(AkH − CH BkHB†kH A†−kH − CH

)(4.4)

AkH = J

0 −cAB −cAC−cAB 0 −cBC−cAC −cBC 0

(4.5)

BkH = J

0 3cAB 3cAC

3cAB 0 3cBC

3cAC 3cBC 0

(4.6)

CH = J

−4 0 0

0 −4 0

0 0 −4

(4.7)

27

Page 35: MASTER OF SCIENCE DEGREE

Explicit expression of the magnon Hamiltonian given

in Eq. (3.27).

HDM = Dij · (H′1 +H′2 +H′3) (4.8)

Recall the expressions for SA, SB and SC from Eq. (3.11) - (3.13)

SA×SB =

∣∣∣∣∣∣∣∣x y z√

Sn2

(aA,j + a†A,j) −i√Sn2

(aA,j − a†A,j) (S − a†A,jaA,j)

− 12

√Sn2

(aB,j + a†B,j) +√

32

(S − a†B,jaB,j) −i√Sn2

(aB,j − a†B,j) −√

32

√Sn2

(aB,j + a†B,j)−12

(S − a†B,jaB,j)

∣∣∣∣∣∣∣∣(4.9)

SzASxB − S

xAS

zB =

√3

4S(a†A,ja

†B,j + aA,jaB,j + aA,ja

†B,j + a†A,jaB,j) +

√3

4S(2S − 2a†A,jaA,j − 2a†B,jaB,j)y (4.10)

SyASzB − S

zAS

yB = i

√3

4S(aA,jaB,j − a†A,ja

†B,j + aA,ja

†B,j − a

†A,jaB,j)x (4.11)

SxASyB − S

yAS

xB = −i

S

4(3aA,jaB,j − 3a†A,ja

†B,j − aA,ja

†B,j + a†A,jaB,j)z (4.12)

SA×SC =

∣∣∣∣∣∣∣∣x y z√

Sn2

(aA,j + a†A,j) −i√Sn2

(aA,j − a†A,j) (S − a†A,jaA,j)

− 12

√Sn2

(aC,j + a†C,j)−√

32

(S − a†C,jaC,j) −i√Sn2

(aC,j − a†C,j)√

32

√Sn2

(aC,j + a†C,j)−12

(S − a†C,jaC,j)

∣∣∣∣∣∣∣∣(4.13)

SzASxC − S

xAS

zC =

√3

4S(a†A,ja

†C,j + aA,jaC,j + aA,ja

†C,j + a†A,jaC,j) +

√3

4S(2S − 2a†A,jaA,j − 2a†C,jaC,j)y (4.14)

SyASzC − S

zAS

yC = i

√3

4S(aA,jaC,j − a†A,ja

†C,j + aA,ja

†C,j − a

†A,jaC,j)x (4.15)

SxASyC − S

yAS

xC = i

S

4(3aA,jaC,j − 3a†A,ja

†C,j − aA,ja

†C,j + a†A,jaC,j)z (4.16)

SA×SC =

∣∣∣∣∣∣∣∣x y z

− 12

√Sn2

(aB,j + a†B,j) +√

32

(S − a†B,jaB,j) −i√Sn2

(aB,j − a†B,j) −√

32

√Sn2

(aB,j + a†B,j)−12

(S − a†B,jaB,j)

− 12

√Sn2

(aC,j + a†C,j)−√

32

(S − a†C,jaC,j) −i√Sn2

(aC,j − a†C,j)√

32

√Sn2

(aC,j + a†C,j)−12

(S − a†C,jaC,j)

∣∣∣∣∣∣∣∣(4.17)

SzBSxC − S

xBS

zC =

√3

4S(a†B,ja

†C,j + aB,jaC,j + aB,ja

†C,j + a†B,jaC,j) +

√3

4S(S − a†B,jaB,j − a

†C,jaC,j)y (4.18)

SyBSzC − S

zBS

yC = i

√3

4S(2aB,jaC,j − 2a†B,ja

†C,j)x (4.19)

SxBSyC − S

yBS

xC = i

S

4(2aB,jaC,j − 2a†B,ja

†C,j)z (4.20)

The Fourier transform for each component is of the form:

SySz − SzSy = i

√3

2S

(ak,Aa−k,B − a†k,Aa

†−k,B + ak,Aa

†k,B − a

†k,Aak,B)cAB

+(ak,Aa−k,C − a†k,Aa†−k,C + ak,Aa

†k,C − a

†k,Aak,C)cAC

−(2ak,Ba−k,C − 2a†k,Ba†−k,C)cBC

x (4.21)

SxSy − SySx = iS

2

−(3ak,Aa−k,B − 3a†k,Aa

†−k,B − ak,Aa

†k,B + a†k,Aak,B)cAB

+(3ak,Aa−k,C − 3a†k,Aa†−k,C − ak,Aa

†k,C + a†k,Aak,C)cAC

+(2ak,Ba−k,C − 2a†k,Ba†−k,C)cBC

z (4.22)

SzSx − SxSz =

√3

2S

(ak,Aa−k,B + a†k,Aa

†−k,B + ak,Aa

†k,B + a†k,Aak,B)cAB

+(ak,Aa−k,C + a†k,Aa†−k,C + ak,Aa

†k,C + a†k,Aak,C)cAC

+(ak,Ba−k,C + a†k,Ba†−k,C + ak,Ba

†k,C + a†k,Bak,C)cBC

+(6S − 4a†k,Aak,A − 4a†k,Bak,B)cBC − 4a†k,Cak,C

y (4.23)

28

Page 36: MASTER OF SCIENCE DEGREE

Explicit expression of the DMI given in Eq. (3.27).

Dij = vij(D‖nxij , D‖n

zij , D⊥)

(nxij , nzij) = (sinθx, cosθz) (4.24)

Where θ is the angle between the spin vectors at i and j, depending on the orien-tation of the bond i, j, the signs of the DMI components changes.

HDM = D‖nxij · i√

3

2S

(ak,Aa−k,B − a†k,Aa

†−k,B + ak,Aa

†k,B − a

†k,Aak,B)cAB

+(ak,Aa−k,C − a†k,Aa†−k,C + ak,Aa

†k,C − a

†k,Aak,C)cAC

−(2ak,Ba−k,C − 2a†k,Ba†−k,C)cBC

x

−D‖nzij · iS

2

−(3ak,Aa−k,B − 3a†k,Aa

†−k,B − ak,Aa

†k,B + a†k,Aak,B)cAB

+(3ak,Aa−k,C − 3a†k,Aa†−k,C − ak,Aa

†k,C + a†k,Aak,C)cAC

+(2ak,Ba−k,C − 2a†k,Ba†−k,C)cBC

z

+D⊥ ·√

3

2S

(ak,Aa−k,B + a†k,Aa

†−k,B + ak,Aa

†k,B + a†k,Aak,B)cAB

+(ak,Aa−k,C + a†k,Aa†−k,C + ak,Aa

†k,C + a†k,Aak,C)cAC

+(ak,Ba−k,C + a†k,Ba†−k,C + ak,Ba

†k,C + a†k,Bak,C)cBC

+(6S − 4a†k,Aak,A − 4a†k,Bak,B)cBC − 4a†k,Cak,C

y

AkDM =

0 (√

3D⊥ − iD‖(cosθ +√

3sinθ))cAB (√

3D⊥ + iD‖(cosθ −√

3sinθ))cAC

(√

3D⊥ + iD‖(cosθ +√

3sinθ))cAB 0 (√

3D⊥ + 2iD‖cosθ)cBC

(√

3D⊥ − iD‖(cosθ −√

3sinθ))cAC (√

3D⊥ − 2iD‖cosθ)cBC 0

(4.25)

A−kDM =

0 (√

3D⊥ + iD‖(cosθ +√

3sinθ))cAB (√

3D⊥ − iD‖(cosθ −√

3sinθ))cAC

(√

3D⊥ − iD‖(cosθ +√

3sinθ))cAB 0 (√

3D⊥ − 2iD‖cosθ)cBC

(√

3D⊥ + iD‖(cosθ +√

3sinθ))cAC (√

3D⊥ + 2iD‖cosθ)cBC 0

(4.26)

BkDM =

0 (√

3D⊥ − iD‖(3cosθ −√

3sinθ))cAB (√

3D⊥ + iD‖(cosθ +√

3sinθ))cAC

(√

3D⊥ − iD‖(cosθ −√

3sinθ))cAB 0 (√

3D⊥ − 2√

3iD‖sinθ)cBC

(√

3D⊥ + iD‖(cosθ +√

3sinθ))cAC (√

3D⊥ − 2√

3iD‖sinθ)cBC 0

(4.27)

B†kDM =

0 (√

3D⊥ + iD‖(3cosθ −√

3sinθ))cAB (√

3D⊥ − iD‖(cosθ +√

3sinθ))cAC

(√

3D⊥ + iD‖(cosθ −√

3sinθ))cAB 0 (√

3D⊥ + 2√

3iD‖sinθ)cBC

(√

3D⊥ − iD‖(cosθ +√

3sinθ))cAC (√

3D⊥ + 2√

3iD‖sinθ)cBC 0

(4.28)

CDM =

4√

3D⊥ 0 0

0 4√

3D⊥ 0

0 0 4√

3D⊥

(4.29)

HkDM =

(AkDM − CDM BkDMB†kDM A†−kDM − CDM

)(4.30)

which gives:-HDM =

S

2

∑k

Ψ†HkDMψk (4.31)

Adding Eq.(4.2) and Eq.(4.31) gives the bilinear bulk Hamiltonian in Eq.(3.27),

where H0 = 32JzNS2.

29

Page 37: MASTER OF SCIENCE DEGREE

BIBLIOGRAPHY

[1] Ryo Matsumoto and Shuichi Murakami. Rotational motion of magnons and

the thermal Hall e�ect. Physical Review B - Condensed Matter and Materials

Physics, 84(18):1�9, 2011.

[2] Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, and Y. Tokura. Ob-

servation of the magnon hall e�ect. Science, 329(5989):297�299, 2010.

[3] Kouki Nakata, Se Kwon Kim, Jelena Klinovaja, and Daniel Loss. Magnonic

topological insulators in antiferromagnets. Physical Review B, 96(22):1�14,

2017.

[4] R. Chisnell, J. S. Helton, D. E. Freedman, D. K. Singh, R. I. Bewley, D. G.

Nocera, and Y. S. Lee. Topological Magnon Bands in a Kagome Lattice Ferro-

magnet. Physical Review Letters, 115(14):1�5, 2015.

[5] S. A. Owerre. Magnon Hall e�ect in AB-stacked bilayer honeycomb quantum

magnets. Physical Review B, 94(9):1�10, 2016.

[6] S. A. Owerre. A �rst theoretical realization of honeycomb topological magnon

insulator. Journal of Physics Condensed Matter, 28(38), 2016.

[7] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich. Antiferromagnetic

spintronics. Nature Nanotechnology, 11(3):231�241, 2016.

[8] Jairo Sinova, Sergio O Valenzuela, J Wunderlich, C H Back, and T Jungwirth.

Spin Hall e�ects. Rev. Mod. Phys., 87(4):1213�1260, 2015.

[9] Lifa Zhang, Jie Ren, Jian Sheng Wang, and Baowen Li. Topological magnon

insulator in insulating ferromagnet. Physical Review B - Condensed Matter and

Materials Physics, 87(14):1�8, 2013.

[10] M. Z. Hasan and C. L. Kane. Colloquium: Topological insulators. Reviews of

Modern Physics, 82(4):3045�3067, 2010.

[11] Magnetic Symmetry. 3 It Amounts, According. 5(6):1259�1272, 1957.

[12] Tôru Moriya. Anisotropic Superexchange Interaction and Weak Ferromag-

netism. Phys. Rev., 120(1):91�98, 1960.

30

Page 38: MASTER OF SCIENCE DEGREE

[13] Ryo Matsumoto and Shuichi Murakami. Theoretical prediction of a rotating

magnon wave packet in ferromagnets. Physical Review Letters, 106(19):1�4,

2011.

[14] T. Holstein and H. Primako�. Field dependence of the intrinsic domain mag-

netization of a ferromagnet. Physical Review, 58(12):1098�1113, 1940.

[15] S. Toth and B. Lake. Linear spin wave theory for single-Q incommensurate

magnetic structures. Journal of Physics Condensed Matter, 27(16), 2015.

[16] Alexander Mook, Jürgen Henk, and Ingrid Mertig. Thermal Hall e�ect in

noncollinear coplanar insulating antiferromagnets. Physical Review B, 99(1):1�

8, 2019.

[17] Ryo Matsumoto, Ryuichi Shindou, and Shuichi Murakami. Thermal Hall e�ect

of magnons in magnets with dipolar interaction. Physical Review B - Condensed

Matter and Materials Physics, 89(5):1�12, 2014.

[18] Shuichi Murakami. Two-dimensional topological insulators and their edge

states. Journal of Physics: Conference Series, 302(1), 2011.

[19] I. Kzialoshinskii. The Magnetic Structure of Fluorides of the Transition Metals.

Soviet Journal of Experimental and Theoretical Physics, 6(46):1120, 1958.

[20] Charles Kittel and Joseph F Masi. Introduction to Solid State Physics. 1953.

[21] Alexander Mook, Jürgen Henk, and Ingrid Mertig. Edge states in topologi-

cal magnon insulators. Physical Review B - Condensed Matter and Materials

Physics, 90(2):1�7, 2014.

[22] Ralph M. Kaufmann, Dan Li, and Birgit Wehefritz-Kaufmann. Notes on topo-

logical insulators. 2015.

[23] Satoru Nakatsuji, Naoki Kiyohara, and Tomoya Higo. Large anomalous

Hall e�ect in a non-collinear antiferromagnet at room temperature. Nature,

527(7577):212�215, 2015.

31


Recommended