Time-Evolution of Magnetic Vector Fields through Disorder ... · waves are excited thanks to the...

UNIVERSITA DEGLI STUDI DI MODENA E REGGIO EMILIA

DIPARTIMENTO DI SCIENZEFISICHE, INFORMATICHE E MATEMATICHE

CORSO DI LAUREA MAGISTRALE IN PHYSICS

Tesi di Laurea Magistrale

Time-Evolution of Magnetic VectorFields through Disorder-Averaged

Green Functions in LayeredAntiferromagnets

Supervisors:Prof Carlo Maria Bertoni

Prof Mikhail Titov, Radboud University - NijmegenCandidate:Giacomo Sesti

ANNO ACCADEMICO 2016-17

Contents

1 Introduction 11.1 The Spin Transfer Torque . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Antiferromagnetic Spintronics . . . . . . . . . . . . . . . . . . . . . . 61.3 Outline of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Linear response theory 112.1 The conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 The spin polarisation under an electric field . . . . . . . . . . . . . . 202.3 The Gilbert Damping term . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Dirac AFM on a honeycomb Lattice 273.1 The Landau-Lifshitz-Gilbert equations . . . . . . . . . . . . . . . . . 283.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 The Self Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 The Dressing of the Interaction . . . . . . . . . . . . . . . . . . . . . 373.5 The Spin Orbit Torque and the Conductivity . . . . . . . . . . . . . 413.6 The Spin Transfer Torque . . . . . . . . . . . . . . . . . . . . . . . . . 443.7 The Gilbert Damping contribution . . . . . . . . . . . . . . . . . . . 51

4 AFM Square Lattice 574.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 The Self Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3 The Spin Orbit Torque . . . . . . . . . . . . . . . . . . . . . . . . . . 674.4 The Spin transfer Torque . . . . . . . . . . . . . . . . . . . . . . . . . 694.5 The Gilbert Damping contribution . . . . . . . . . . . . . . . . . . . 71

5 Conclusions 73

Bibliography 79

Appendix 83

III

Chapter 1

Introduction

1.1 The Spin Transfer Torque

Figure 1.1. Pictorial illustration ofthe magnetic trilayer made up of twoferromagnets and a normal metal inbetween. From Ref. [1]

A cornerstone event in the field ofspintronics is the discovery of the ex-istence of spin transfer torques. Theprototypical system in which this ef-fect was initially observed is a mag-netic trilayer, whose sketch is shownin Figure 1.1. In this system, thereare two ferromagnetic(FM) layers: oneof which has a fixed magnetization.Between the two layers there is a non-magnetic metal. A current is injectedinto the magnetic layer with givenmagnetization.In a ferromagnetic material the electrons interact through the exchange interac-tion. Thus, since the exchange favours the spin alignment, in a ferromagnet aspin unbalance is always established. Hence, the population of spin up and spindown are different. This property is reflected in the band structure of the ferro-magnet, as we can observe in Figure 1.2, where the spin majority band and thespin minority band are split by an energy shift, the ”exchange splitting”. As theconductance of a metal is established by the properties of the electronic statesclose to the Fermi surface, the number of majority spin states around the Fermienergy is much bigger than the one of the spin minority states. Thus, a ferromag-net favours the motion of the electrons in the majority spin states, which are thosealigned to its internal magnetization. Therefore, in the passage through the firstferromagnetic layer the current becomes spin-polarized.

1

1 – Introduction

Figure 1.2. (a) Bands structure of a ferromagnet, in particular cobalt, alongone high symmetry direction for majority and minority spin states and theirdensity of states (b). From Ref. [1]

Then, the current propagates into the normal metal, where the spin unbalance ispreserved, until it reaches the second ferromagnet. Once again, in the passagethrough the second ferromagnet, the polarized spins turn in the direction of theFM magnetization due to the difference of states in the two spin bands [2, 3].However, if the magnetization of the second layer is oriented differently from thefirst one, the magnetization in this case can rotate as it experiences a torque thatoriginates from the conservation of the total angular momentum of the system.This happens because the conduction electrons spins are turned in the passagethrough the ferromagnet. This torque is the so called spin transfer torque and itappears as a result of the spin angular momentum transfer from a spin-polarizedcurrent to the magnetic moments.The interaction between the flowing electrons and the local magnetic momentsof ferromagnets is often modelled as a s-d interaction, where the s stands for theelectrons in the conduction band and the d for the electrons responsible for thelocal magnetizations. The interaction makes the s and the d electrons precessaround each other.Recently, it has been discovered that the magnetic moments are turned even bya different kind of spin torque (called spin orbit torque) originated by the spin-orbit interaction of the system. Indeed, in the presence of an external currentflowing through the system, the spin orbit interaction contributes to create a non-equilibrium spin polarization in the system[4, 5, 6]. A full description of the mag-netic dynamic should account for the presence of both the effects, if the spin orbitinteraction is present.

2

1.1 – The Spin Transfer Torque

Depending on the magnitude of the torques, the magnetization dynamics can ei-ther be perturbative and generate spin waves excitations, or can be global andstronger, giving rise to magnetization reversal or rotations. Since the local mag-netic moments time response is slightly less than some ns and the response ofthe conduction electrons is of the order of ps, the magnetic moments motion canbe decoupled from the s electrons one. The magnetization can be considered inquasi-equilibrium during the transport process. The motion of the magnetizationcan then be studied through the Landau-Lifshitz-Gilbert(LLG) equation, whereterms associated to external fields, damping and torques are present (the direc-tions of the forces are shown separately in Figure 1.3).

∂M∂t

= −γM × He f f + αM × ∂M∂t

+ TSOT + TSTT . (1.1)

Figure 1.3. Illustration of thetorques acting over the magneti-zation of the system. From Ref.[7]

The first term on the right side ofEq.(1.1) is the usual Lorentz term dueto the effective magnetic field, thatarises from the contributions of the ex-ternal magnetic field, the diamagneticfield, the anisotropy field and so onHe f f = Hext + Hdem + Han + ..., andγ is the gyromagnetic ratio. The sec-ond contribution is the damping termthat accounts for the dissipation effectsand tends to stabilize the magnetiza-tion along the effective field direction.In its expression is present α, the as-sociated constant, whose dimension isthe opposite of a magnetization. Atthe end, there is the contribution of thetwo spin torques.The most general expression of the spin transfer torque at the microscopic levelis given by:

TSTT = α(E · ∇)M + β∥M × (E · ∇)M∥ + β⊥M × (E · ∇)M⊥,

where M∥ and M∥ are respectively the component of the magnetization paralleland perpendicular to the direction of the spin orbit interaction. In any way, thecomponents of the STT, and similarly those of the SOT, lay in the same directionof the field (for this reason called field-like) and another in the direction of thedamping, both parallel or antiparallel to it [8, 9].

3

1 – Introduction

The most interesting effects arise when the torques are completely antidamping-like. In this case,when the torque is large enough, it is possible to observe a spinswitching. Indeed the magnetization, initially on a energy minimum condition,as a consequence of the spin torques, is pushed further and further from the min-imum state. Consequently, the precession orbit becomes very large and it is sounstable that it finds favourable to reach a new energy minimum state, repre-sented by the flipped state. After the flip, however, the current allows stabiliz-ing the turned magnetization because now the spin torques act to enhance thedamping. This allows to have a well-controlled switching, robust with respectto fluctuations of the driving current parameters. This process is very promis-ing for applications, in particular in the field of magnetic random access mem-ory (MRAM). Indeed many efforts are focused on realizing the switching processwith higher speed, smaller dimension, lower energy consumption and higher ro-bustness. However, the main problem is the fact that the currents for which theswitching occurs are too high.

Figure 1.4. Position of thecenter of the domain wall asa function of time in the adi-abatic case and in the case ofEF > Jsd. From Ref. [11]

Another relevant effect, triggered by spintorques, is the domain wall motion [10, 11]. Ina ferromagnet, the domain wall separates re-gion with a different orientation of the mag-netization. Typical systems in which this ef-fect is very relevant are nanowires, where spintorques are mainly adiabatic, that means thatthe spin polarization lies in the same the di-rection of the local magnetization. The spintorques are unable to move the domain wallsover long distances, because the domain walldeforms slightly as the spin torque pumps en-ergy into the system. Then, initially the veloc-ity is large, but because of the deformation themagnetization it develops a small out-of-planecomponent in the direction of the electron cur-rent. Consequently, the rate of energy damp-ing increases and the domain wall slows down until it stops completely, as it canbe noticed from Figure 1.4.In real systems, a non-adiabatic component of the torque is always present andthus the domain wall velocity tends to remain always finite even below the criti-cal current.The domain motion changes completely when the Fermi energy EF of the con-duction electrons is larger than the exchange energy Jsd between their spins and

4


the local spins of the domain wall, indeed in that case the state of conductionelectrons is spin mixed. The main relevant phenomenon at this scale is that it ispossible to have spin wave generation in the direction of the current; these spinwaves are excited thanks to the spin precession originated by the spin interfer-ence effect. The effect of the spin waves is to distort the shape of the domain wallcreated at the saturation in the adiabatic approximation and then the domain wallcan move beyond the saturation, as it displayed in Figure 1.4.It is clear that, by exploiting the movement of the walls, it is potentially possibleto control the desired magnetization in a certain region.A third effect is the creation of a steady precession, this is possible when a largemagnetic field is applied to the sample [12]. As a consequence of the large field,the energy profile of the free layer is modified and only one local energy mini-mum is present, along the magnetic field direction. Then, since the applied cur-rent creates a spin torque that tends to drift away the magnetization from theminimum, the magnetization will remain in a dynamic state because there is noother stationary equilibrium state. In particular it will precess steadily around thefield direction, as shown in Figure 1.5, because in a cycle of precession, the spintorque cancels out the damping torque on average. The steady-state precessionproduces an oscillation of the resistance and hence a voltage oscillation in the fre-quency range of microwaves. The microwave frequency can be tuned throughmodifications of the magnitude of the magnetic field or of the applied current. Inthis way, spin transfer devices can even be used as tunable sources of microwaveoscillations at the nanoscale.

Figure 1.5. Illustration of the trajectory of a simulated persistent magnetic preces-sion around the magnetic field direction(a) and its characterization in frequencydomain (b) (modified from Ref. [7]

5

1 – Introduction

1.2 Antiferromagnetic Spintronics

Since standard ferromagnetic materials present the defects discussed above, theresearch, in the field of spintronics, moved to different kinds of materials, likedoped semiconductors and antiferromagnets. In particular, the focus of this mas-ter thesis research will be on antiferromagnetic (AFM) systems, which only re-cently have started to be used as active elements in spintronics devices.

It has been suggested that antiferromagnetic materials have a great potentialto reduce device power consumption and device scale, therefore they could begood candidates for future spintronic applications. Other good features of anti-ferromagnets systems are their robustness against perturbations of external mag-netic fields, due to their vanishing internal magnetization, the absence of produc-tion of parasitic demagnetizing fields, the ultrafast dynamics and the generationof large magneto-transport effects.

Also in AFM materials, one of the first type of set-up in which spin transfertorques have been studied are spin valves [13], composed of two AFM layers sep-arated by a metallic spacer.At the beginning, it was investigated the structure of the scattering matrix at theinterface of an AFM. The researchers discovered that the transmitted electronsmaintain their spins orientations and, conversely, the reflected electron emergefrom the system with their spin orientations rotated around the order parameter.The rotation angle turns out to be dependent on the direction of incidence. This isvery different compared with the case of a ferromagnetic scatterer, for which boththe transmitted and reflected electrons are rotated, and in addition rotations areindependent on the direction of incidence. Then the spin polarization of a cur-rent is conserved when it crosses an isolated antiferromagnetic element, thoughthe reflected current becomes spin polarized along the order parameter direction.Such a behaviour was later confirmed by Saidaoui, Manchon and Waintal [14]who, studying an AFM-paramagnet interface through a simple tight-binding model,found that, indeed, the transmitted current remains unpolarised, but there is apolarized spin density in reflection. This result suggests that there is a reflectedspin-mixing conductance at the interfaces between normal metals and antiferro-magnets. The spin-mixing conductance is a parameter that allows to determinethe amount of spin absorbed at the interfaces normal to the magnetization uponreflection or transmission [15][16]. For this reason, it can be used to describe thespin pumping and then the spin transfer torque. In particular Cheng et al. [17]and Takei et al. [18] have computed the reflected mixing conductance and theyconfirmed that it is non vanishing.Even if the transmitted current passing through a layer is not spin-polarized, a

6

1.2 – Antiferromagnetic Spintronics

spin valve made of non-collinear antiferromagnets can create a non trivial spin-current configuration due to multiple reflection processes at each interface. In ad-dition, since the spin density is periodic in each antiferromagnet, it will not decayaway from the interface in each layer and will allow spin transfer torques to actcooperatively throughout the volume of the antiferromagnet elements. Similarresults were obtained in other work based on spin valve system like FeMn/Cu/FeMn[19] and Cr/Au/Cr [20], in which it was found that the injected spin can deeplypenetrate,confirming the non-local effects of the spin torque in antiferromagnets.Since a polarized current can be established, one can observe the creation ofa torque acting on the local magnetic moments of the second antiferromagnet.Given that spin torques can be generated only in the presence of a polarized spindensity, a spin valve configuration made up of a FM, a normal metal and thenthe AFM seems very promising. Indeed, the ferromagnets is very efficient increating a polarized current that can afterwards generate the spin torques in theAFM. Such a configuration was investigated by Haney and MacDonald [21] andby Merodio et al. [22]. In the former work it is obtained that the torque vanishesif the magnetization of the FM is perpendicular to the Neel vector, in particularthey found a sin(2θ) dependence, where θ is the angle between the two vectors.

Considering the direction of the spin torques, it was found, referring again tothe work on the AFM valve system [13], that the in-plane component of the torqueis very large and, at the same time, it is also present a spatially inhomogeneousout-of-plane component. The Neel vector in AFMs plays a role analogous to themagnetization in a FM since it is the relevant order parameter [23]; moreover itwas found that the switching can be realized via the out-of-plane component,unlike in FM systems. Since the spin torques act cooperatively and the anistropyfield is absent, in antiferromagnetic materials the critical current for which theswitching occurs turns out to be much smaller than in ferromagnets.There are other interesting effect that can be observed in AFM systems [24]. Theantidamping-like component of the torque is an efficient generator of oscillationsof the Neel vector in configurations where the polarization of the spin current isperpendicular to the order parameter. Indeed, under this arrangement, it gen-erates parallel torques on the opposite magnetic sublattices. Above a threshold,these torques are able to create a stable precession of Neel vector in the plane per-pendicular to the spin current polarization [25], [26].Differently, if the spin polarization is parallel to the Neel vector, the polarized cur-rent can generate instabilities. As the spin polarization is always antiparallel toone of the magnetic sublattices in an AFM with this configuration, the spin trans-fer torque produced competes with the internal damping on one of the sublatticesand the Neel vector tends to rotate even in this case towards the plane perpen-dicular to the spin polarization. This originates a stable precession, so an AFM

7

1 – Introduction

is a natural spin-torque oscillator, differently from a FM system for which thisphenomenon occurs only at very large external magnetic fields. It turns out thatthe typical oscillating frequency falls into the THz range. This feature of AFMsis very attractive for applications, because, as the precession of the Neel vectorcreates a non zero dynamic magnetization, such spin torque oscillator can pumpa spin current into the neighbouring non-magnetic layer [27] [29] thus inducinga polarized current similarly to FM systems.

1.3 Outline of this work

Given the wide interest and the potential application of AFM in the field of spin-tronics, it is relevant to study the torques in AFM systems starting from a micro-scopic approach. The general structure of the possible terms present inside thetorques at the microscopic level has been suggested by Velkov and Gomonay [30].This study was developed based on a phenomenological approach that uses theOnsager reciprocity relations, which link the process of inducing charge currentsin a time-varying magnetic texture to the effect that charge currents have on themagnetization dynamics. However, it is likely that in a generic system not all thepossible terms are present, or that they are not equally relevant.The purpose of this master thesis work is to analyse the structure of the torques intwo different antiferromagnetic models: a honeycomb lattice and a square lattice.In particular in both these models the lattice is seen as made up of two ferromag-netic interpenetrating sublattices of A and B type, in which the lattice itself canbe partitioned. The conduction is described through a tight-binding model and anearest neighbour hopping, the interaction is modelled a s-d interaction term.Since the spin transfer torque depends on the spin polarization of the conductionelectrons, it has been necessary to derive the expression of this physical quantity.The computation of the spin polarization is performed in accordance to the linearresponse theory. By the fact that the linear response theory is a fundamental toolin this work, we illustrate its derivation throughout Chapter 2. The laws of thetwo model systems are obtained computing the susceptibility tensors, that relatethe spin polarization to the mean fields. In this chapter the spotlight is put inexplaining how these susceptibility tensors are built. After that, we pass to anal-yse in a deep way the characteristics of the models actually studied, and discusstheir laws. In particular, in Chapter 3 we look at the honeycomb lattice, whereasthe analysis of the square one is carried out in Chapter 4. In both Chapter 3 andChapter 4 we will follow this scheme:

• Definition of the model with the corresponding Hamiltonian. A term defin-ing the disorder is included inside the Hamiltonian

8

1.3 – Outline of this work

• Derivation of the Landau-Lifshitz-Gilbert equations for the characteristicfields

• The Hamiltonian referred to the conduction electrons is represented in thebasis of its eigenfunction

• Computation of the Self-Energy and of the Dressed Vertices

• Computation of the SOT, of the conductivity, of the STT and of the GilbertDamping term originated by the current

The last chapter Conclusions is left for the recapitulation and the interpretationof the results. In addition a brief comment over the possible way to improve themethods exploited is included.After the bibliography there is a short Appendix in which it is illustrated the SelfConsistent Born Approximation(SCBA), that has been applied for the computa-tion of the propagators.

9

Chapter 2

Linear response theory

In this Chapter we derive the linear response theory starting from the Hamilto-nian of a many body system. Let suppose to have a collection of particles in thepresence of an external field. The Hamiltonian H can be split into two parts, themany body Hamiltonian H0, that describes the motion and the interaction of theparticles of the system, and the external field contribution, that is time dependentand acts as a source of perturbation to H0:

H = H0 + V(t) (2.1)

The unperturbed Hamiltonian defines a complete set of eigenfunctions with de-fined energies {ψn, En}:

H0|ψn⟩ = En|ψn⟩ (2.2)

In order to find the energy of the perturbed system it is possible to use Schrodingerequation, surely at a given time t the state of the system can always be rewrittenthrough the unperturbed eigenstates |Ψ(t)⟩ = ∑n an(t)|ψn⟩e−iEnt.

i∂|Ψ(t)⟩

∂t= (H0 + V(t))|Ψ(t)⟩. (2.3)

∑n(i ˙an(t)|ψn⟩e−iEnt + an(t)|ψn⟩Ene−iEnt) =∑

n(En + V(t))an(t)|ψn⟩e−iEnt. (2.4)

Let now suppose that initially the system was on a eigenstate of the unperturbedHamiltonian ψm, this corresponds to affirm that am(−∞) = δm,n. The effect of theperturbing Hamiltonian is to drive away the system from its initial state:

ak(t) = −i ∑m

am(t)⟨ψk|V(t)|ψn⟩ei(Ek−Em)t. (2.5)

11

2 – Linear response theory

This is the starting point for perturbation theory, the reason for adopting thistechnique is that we need to substitute the evolved state Ψ(t) with a quantity thatis known. The first step is to start from am(−∞) in such a way one finds a(1)k (t) atfirst order. The second order result for am(t) is obtained starting from the resultof the previous order, and so on and so for. Each order corresponds to higherpower in the perturbing Hamiltonian. The aim of this work is to study linearresponses to external fields, then only the first order terms have to be accounted.Consequently, one can write:

ak(t) = −i t

−∞dt′Vkn(t′)ei(Ek−En)t′ . (2.6)

Hence, at a general time t the state of the system is given by:

|Ψn(t)⟩ = |ψn⟩e−iEnt − i ∑k

t

−∞dt′Vkn(t′)ei(Ek−En)t′ |ψk⟩e−iEkt. (2.7)

For now, we have supposed that we know the initial state of the system. How-ever, this is not actually true because there is a finite probability of finding thesystem in every possible state. In particular, the probability depends on the en-ergy of the state and it is given by the distribution function, in the case of fermionsthis is the Fermi-Dirac distribution that we will call f (En).Our aim is to study the average value of an observable A. Specifically, we wantto link the variations of its average value to the presence of the external probe. Inorder to do it, we need to look at the average value at the linear level at a certaintime t.

δ<A(t)> = ∑n

f (En)⟨Ψn(t)|A|Ψn(t)⟩ − ∑n

f (En)⟨ψn(t)|A|ψn(t)⟩ =

− i ∑n

f (En)⟨ψn|eiEntA∑k|ψk⟩

t

−∞dt′Vkn(t′)ei(Ek−En)t′e−iEkt + h.c.

= −i ∑nk

f (En) t

−∞dt′Vkn(t′)Ankei(En−Ek)(t−t′)+

i ∑nk

f (En) t

−∞dt′Vnk(t′)Akne−i(En−Ek)(t−t′).

(2.8)

Without loss of generality, it is possible to consider the case in which the averagevalue of the observable A is trivial in the equilibrium state of the system. Themagnitude of the observable is then given by the perturbations generated by theexternal field. Reconsidering the final expression wrote in Eq.(2.8), we can noticethat it is possible to exchange the k and the n indices in the last term because the

12

sum is on all the possible states. This rearrangement enables to write the twoterms in forms very similar, differing only for the distribution functions.

<A(t)>=− i ∑nk( f (En)− f (Ek))

t

−∞dt′Vkn(t′)Ankei(En−Ek)(t−t′). (2.9)

In order to understand the meaning of this expression, let us put it in energy orfrequency space.

δ<A(ω)>=− i ∑nk( f (En)− f (Ek))

∞

−∞dt ∞

−∞dt′Θ(t − t′)Vkn(t′)Ankei(En−Ek)(t−t′)e−iωt, (2.10)

where Θ(τ) is the time step function:

Θ(τ) =

0 if τ ≤ 01 if τ > 0

. (2.11)

Through a change of variable t = t − t′, the expression becomes:

δ<A(ω)>=− i ∑nk( f (En)− f (Ek))Ank

∞

−∞dtΘ(t)ei(En−Ek−ω)t

∞

−∞dt′Vkn(t′)e−iωt′ . (2.12)

The second integral is simply the Fourier transform of the matrix element of theperturbing Hamiltonian. The first integral, instead, can be solved introducing asmall parameter ϵ in the definition of the step function:

Θϵ(x) = e−ϵxΘ(x) and Θϵ(x) ϵ→0−−→ Θ(x). (2.13)

Now, using the property of the step function, we can reduce the dominion and,thanks to the small imaginary part, the integral can be calculated: ∞

0dtei(En−Ek−ω+iϵ)t =

iEn − Ek − ω + iϵ

, (2.14)

and one gets:

δ<A(ω)>= ∑nk

f (En)− f (Ek)

En − Ek − ω + iϵVkn(ω)Ank. (2.15)

In principle one should send the quantity ϵ to 0, because it has been introduced asa mathematical tool to perform the calculation. However, it is possible to keep theimaginary value because, as we will see later in Chapter 3, the energy acquires animaginary part. Then, the tiny ϵ can be seen as the imaginary part of the energy.

13


The fraction present in the right-hand-side of Eq.(2.15) can be expressed as thedifference of two integrals:

f (En)− f (Ek)

En − Ek − ω + iϵ=

12πi

dE

f (E)E − Ek − ω + iϵ

1

E − En − iϵ− 1

E − En + iϵ

− 1

2πi

dE

f (E − ω)

En − E + iϵ

1

E − Ek − ω − iϵ− 1

E − Ek − ω + iϵ

.

(2.16)

This relation is easily obtainable as the differences inside the round brackets are,respectively, 2πδ(E − En) and 2πδ(E − Ek − ω) . It is possible to rewrite therelations making the propagators appear:

GR/A =1

E − H ± iϵ= ∑

n

1E − En ± iϵ

|ψn⟩⟨ψn|. (2.17)

Let consider the first term of Eq.(2.15), after the substitution of Eq.(2.16), to showthe procedure that will be followed:

dE f (E)∑

k

1E − Ek − ω + iϵ

⟨ψk|V ∑n

1E − En − iϵ

|ψn⟩⟨ψn|A|ψk⟩

=

dE f (E)∑k⟨ψk|GR(E − ω))VGA(E)A|ψk⟩

=

dE f (E)Tr[GR(E − ω)VGA(E)A],

(2.18)

using the properties of the product of the trace of a product of operators. Byrepeating it for the other three terms we obtain:

δ <A(ω)>=1

2πi

dE Tr

( f (E)(GA(E)− GR(E))AGR(E − ω))

+ f (E − ω)GA(E)A(GA(E − ω)− GR(E − ω))))V(ω).

(2.19)

So, through a rearrangement of all the terms, we get:

δ <A(ω)>=1

2πi

dE Tr

(( f (E)− f (E − ω))GA(E)AGR(E − ω))

+ f (E − ω)GA(E)AGA(E − ω)− f (E)GR(E)AGR(E − ω)))V(ω).

(2.20)

2.1 The conductivity

As a reference example to obtain the laws that govern the linear response, let usconsider the case of the conductivity. The perturbing Hamiltonian depends, in

14

2.1 – The conductivity

such a case, on the vector potential:

V(t) = − j · A|t = − ∑α

jα Aα

t

. (2.21)

The observable A is the current j = −ev, or better its component jα. One obtains,from Eq.(2.20), that the response of the system, that we can call γαβ, is linked tothe vector potential:

jα(ω) = γαβ(ω)Aβ(ω), (2.22)

γαβ(ω) =− 12πi

dE Tr

(( f (E)− f (E − ω))GA(E)jαGR(E − ω)+

f (E − ω)GA(E)jαGA(E − ω)− f (E)GR(E)jαGR(E − ω))jβ.

(2.23)

The interest is to study the variations of the current due to the electric field, so:

jα(t) =

dtσαβ(t − t′)Eβ(t′), by convolution jα(ω) = σαβ(ω)Eβ(ω). (2.24)

The electric field and the vector potential are linked through E(t) = −∂t A(t), soin Fourier space this is equivalent to E(ω) = iωA(ω), then one gets:

σαβ(ω) =γαβ(ω)

iω. (2.25)

Typically the energies of the particles are much bigger than the characteristic fre-quency of the field, therefore it can be performed the limit for small ω. In Eq.(2.24)the terms with two advanced or two retarded propagators seems to be a sourceof problems if one tries to take the limit, because there is no explicit frequencydependence. However, it is possible to show that for small ω the limit can betaken. This is related to the fact that a part of the current has been neglected: theso called diamagnetic current.In order to introduce this additional contribution one needs to reconsider theaverage value of the current in the equilibrium state of the system. Having in-troduced a perturbation originated by the vector potential, the momentum of thesystem has changed to p → p − eA. Consequently the current is given by twoterms:

jtot = j + jA, (2.26)

where the second term is the one originated by the vector potential. The averagevalue of the current on the initial states is:

∑n

f (En)⟨ψn|− e

mp +

e2

mA|ψn⟩ = 0. (2.27)

15


Correctly, the average value of the momentum has been chosen to be trivial sincewe expect that the initial state is at equilibrium. However, the vector potentialgives rise to a non trivial quantity, the diamagnetic current:

<jdiamα > = −e ∑

nf (En) < n| ∂j

∂pα· A|n >

= −e ∑n

dE2πi

1

E − En − iϵ− 1

E − En + iϵ

< n| ∂j

∂pα· A|n >

= − e2πi

Tr

dE(GA(E)− GR(E))∂j

∂pα· A

=e

2πiTr

dE∂

∂pα(GA(E)− GR(E))j · A

− e

2πi∂

∂pαTr

dE(GA(E)− GR(E))j · A

.

(2.28)

The second term at the end of Eq.(2.28) can be transformed back into:

∂

∂pα∑n

f (En) < n|j · A|n >, (2.29)

using once again the relations that link the propagators to the density function.This quantity is surely trivial because the average value of the part of the currentdue to the linear momentum, as previously noticed, is null. Thus, the diamag-netic current is directly given by the other term, which becomes more under-standable after performing the derivatives on the momentum:

<jdiam>= − 12πi

Tr

dEGA(E)jGA(E)(−j · A)− GR(E)jGR(E)(−j · A)

. (2.30)

Clearly this term corresponds to a zero frequency response to the applied vectorpotential. It can be associated to a susceptibility tensor γαβ(0) responsible of thecreation of the diamagnetic current.

γαβ(0) = − 12πi

Tr

dE

GA(E)jαGA(E)− GR(E)jαGR(E)

jβ

. (2.31)

16


It is finally possible to compute the conductivity, in order to do so, one has toperform expansion to the first order in the energy:

σαβ(ω) =γαβ(ω) + γαβ(0)

iω= − 1

2πTr

dE

f (E)− f (E − ω)

ωGA jαGR jβ

+

− 12πω

Tr

dE

f (E)GA jαGA − ω f (E)GA jα∂GA

∂E− ω

∂ f (E)∂E

GA jαGA

− f (E)GR jαGR + ω f (E)GR jα∂GR

∂E− f (E)GA jαGA + f (E)GR jαGR

jβ

.

(2.32)

In the first integral, the difference between the distribution function can be turnedinto the derivative of the distribution function with respect to the energy in theusual way. In the second integral the terms without ω cancel each other, thereforeit is possible to simplify the frequency. In addition, the derivative of a propagatorwith respect to the energy is the square of the propagator itself. Therefore, theconductivity becomes:

σαβ(ω) =− 12π

Tr

dE

∂ f (E)∂E

GA jαGR jβ

− 1

2πTr

dE− f (E)(GA)2 jαGA+

+∂ f (E)

∂EGA jαGA − ∂ f (E)

∂EGA jαGA − f (E)GR jα(GR)2

jβ

.

(2.33)

σαβ(ω) =1

2πTr

dE− ∂ f (E)

∂EGA jαGR + f (E)((GA)2 jαGA + GR jα(GR)2)

jβ

. (2.34)

Eq.(2.34) can be rewritten, via partial integration, in order to put in evidence thetwo separate contributions of the conductivity, as they are expressed in the well-known article by Streda [28].

σαβ(ω) =1

4πTr

dE−∂ f (E)

∂E

2GA jαGR − GA jαGA − GR jαGR

jβ

+ f (E)

GA jαGA jβGA − GA jβGA jαGA − GR jαGR jβGR + GR jβGR jαGR

.

(2.35)

17


This leads to:

σIαβ(ω) =− 1

4πTr

dE

∂ f (E)∂E

GA jα(GR − GA)jβ − (GR − GA)jαGR jβ

.

σI Iαβ(ω) =

14π

Tr

dE f (E)

GA jαGA jβGA − GA jβGA jαGA

− GR jαGR jβGR + GR jβGR jαGR

.

(2.36)

The second term enters only in the transverse current because it is completely an-tisymmetric. In standard conditions, the response of the system is mainly deter-mined by the electronic states at the Fermi energy. The conductivity of the systemis determined by the terms containing the derivatives of the distribution function,because these are typically peaked around the Fermi energy. In particular at zerotemperature the Fermi distribution is simply a step function Θ(E − EF), whereEF is the Fermi energy. So ∂ f (E)/∂E = −δ(E − EF), and then σI

αβ is given by theKubo formula:

σIαβ(ω) =

14π

Tr

GA jα(GR − GA)jβ − (GR − GA)jαGR jβ

. (2.37)

In general the two terms that contain only retarded or advanced propagators donot contribute significantly when the system is far from a gap region, as theyhave, outside this energy range, a regular behaviour. Thus, it is the retarded-advanced term that normally describes the response of the system.To control the validity of our derivation, it is possible to compare Eq.(2.36) withthe expression of the conductivity, at the Fermi energy, obtained in Streda’s arti-cle, in which the longitudinal and transverse conductivity are distinguished.

σStredaxx =− 1

4πTr[jx(GR − GA)jx(GR − GA)]

14π

Tr[jx(GR − GA)jxGA − jx(GR − GA)jxGR] = σIxx.

(2.38)

18


It is evident that the two longitudinal polarisation are indeed equal. Similarly, itis possible to prove that even the expression for the transverse ones are the same:

σStredaxy = − 1

2π

dE f (E)

∂GR

∂Ejx(GR − GA)jy − (GR − GA)jx

∂GA

∂Ejy

=1

4π

dE

f (E)(GR)2 jxGR jy − f (E)GR jx(GR)2 jy +∂ f (E)

∂EGR jxGR jy

+ 2 f (E)GR jx(GA)2 jy − 2∂ f (E)

∂EjxGR jyGA − 2 f (E)GR jx(GA)2 jy

+ f (E)GA jx(GA)2 jy − f (E)(GA)2 jxGA jy +∂ f (E)

∂EGA jxGA jy

= σI

xy + σI Ixy.

(2.39)

In order to understand the last equality, some rearrangements are necessary. Byseparating the terms containing the derivative of the distribution function andby knowing that it is just a delta function peaked at the Fermi energy, we obtainthe expression of σI

xy. The remaining terms, after having being simplified, giveexactly σI I

xy.The expressions of the propagators can be written in Fourier space. To do this,we have to consider the explicit dependence of the quantities inside σI

αβ. Theoperator jα inside Eq.(2.37) enters as jαδ(r0 − r1), because we are evaluating thevalue of the average current density at the point r0. Thus, all the possible productsof retarded and advanced propagators in this equation can be written in the form:

dr

dr′G(r, r′)jα(r′)δ(r0 − r′)G(r′, r)jβ(r) =

drG(r, r0)jα(r0)G(r0, r)jβ(r).(2.40)

So:

σIαβ(r0, ω) =

14π

Tr

drGA(r, r0)jα(r0)(GR(r0, r)− GA(r0, r))jβ(r)

− (GR(r, r0)− GA(r, r0))jα(r0)GR(r0, r)jβ(r)

(2.41)

We rewrite the expression in terms of the Fourier transforms, to realize it we canexploit this relation:

jα(r)G(r, r′) =

jα(p)G(p)eip·(r−r′). (2.42)

It is pretty immediate to prove it proceeding backwards, first of all we have to re-member that the current appearing is just the one proportional to the momentum.

19


So:

− em

pαG(p)eip·(r−r′). (2.43)

The momentum can be expressed through a derivative acting over the exponen-tial:

iem

(∇r)αG(p)eip·(r−r′). (2.44)

The derivative can be brought out from the integral, that has become exactly theFourier transform of the propagator, so:

iem(∇r)αG(r, r′) = jα(r)G(r, r′). (2.45)

Since the former is the representation of the current in coordinate space.

Using the relation Eq.(2.42) inside Eq.(2.41), we obtain:

σIαβ(r0) =

e2

4πTr

dr1

dp(2π)D

dq(2π)D

GA(p)vα(q)(GR(q)− GA(q))vβ(p)

− (GR(p)− GA(p))vα(q)GR(q)vβ(p)

eip·(r−r0)e−iq·(r−r0)

.

(2.46)

The exponentials with r1, integrated over this variable, give a delta function(2π)Dδ(p − q). Performing the integral on q the other exponential simplifies to 1and we get:

σIαβ(r0) =

e2

4πTr

dp(2π)D

GA(p)vα(p)(GR(p)− GA(p))

vβ(p)− (GR(p)− GA(p))vα(p)GR(p)vβ(p)

.

(2.47)

2.2 The spin polarisation under an electric field

It is possible to proceed in the study of the spin polarisation density followingthe same strategy adopted for the conductivity. In this case, the current operatorjαδ(r − r′) has to be replaced with the spin operator σαδ(r − r′).

<sα(ω)>=1

2πi

dE Tr

(( f (E)− f (E − ω))GA(E)σαGR(E − ω))+

f (E − ω)GA(E)σαGA(E − ω)− f (E)GR(E)σαGR(E − ω)))(−j · Aω).

(2.48)

20

2.2 – The spin polarisation under an electric field

For the spin polarisation density there is not a contribution similar to the dia-magnetic term because the spin operator does not depend on the vector poten-tial. However, it turns useful to follow the same procedure as before introducinga null ”diamagnetic”-like term that, if properly subtracted to Eq.(2.48), allows tosimplify it:

∑n

f (En) < n|∂σα

∂pA|n >= 0, (2.49)

and at the same time:

∑n

f (En) < n|∂σα

∂pA|n >=

12πi

Tr

dE f (E)(GA(E)jGA(E)− GR(E)jGR(E))(σα Aω)

. (2.50)

Thus, one gets:

<sα(ω)>=1

2πi

dE Tr

( f (E)− f (E − ω))GA(E)σαGR(E − ω))

+ f (E − ω)GA(E)σαGA(E − ω)− f (E)GR(E)σαGR(E − ω))

− fEGA(E)σαGA(E) + fEGR(E)σαGR(E)(−j · Aω)

.

(2.51)

It is evident that if one introduces the electric field in place of the vector potentialand performs expansions in Eq.(2.51) up to first order in the energy, the expres-sion will get the same form as in Eq.(2.32), under the opportune replacement ofthe current operator with the spin operator. Hence, the development of the cal-culation is as in the previous section. Once again, the result can be split in apart which gives contribute mostly at the Fermi energy and in another that isoff-diagonal and strongly simplifies at low temperatures.

sIα(ω) =− 1

4πTr

dE

∂ f (E)∂E

GAσα(GR − GA)jβ − (GR − GA)σαGR jβ

Eβ,

sI Iα (ω) =

14π

Tr

dE f (E)

GAσαGA jβGA − GAσβGA jαGA − GRσαGR jβGR

+ GRσβGR jαGR

Eβ.

(2.52)

As in the case of the conductivity, the first term can be further simplified in thelow temperatures case, hence one gets:

sIα(ω) =

14π

Tr

GAσα(GR − GA)− (GR − GA)σαGRjβEβ

. (2.53)

21


In this work this will be the only term considered because the interest is put onthe contributions in the energy region close to the Fermi energy.Starting from Eq.(2.53), one can derive the tensors of the spin orbit torque (SOT)and the spin transfer torque (STT). It is known that the latter type of torques ariseonly from derivatives of a Zeeman like term M · σ.Let us suppose that inside the Hamiltonian a M · σ term appears, if the magneti-zation M is sufficiently homogeneous, then we can perform a gradient expansion:

G = (E − H −∇(M · σ))−1 (2.54)

Exploiting Dyson equation up to first order, it is possible to rewrite the propaga-tor as:

G = G + G∇(M · σ)G, (2.55)

where G = (E − H)−1. In a more explicit way, the previous relation correspondsto:

G(r, r1) = G(r, r1) +

dr2G(r, r2) ∑δ

(r2 − r0)δ∇δ(M(r0) · σ)G(r2, r1). (2.56)

Such an expression has to be inserted inside the expression of the spin polarisa-tion density sI Eq.(2.53). In addition, only the retarded-advanced combinationare relevant because the integration over poles cancels the other terms. Then onehas:

sα(r0, ω) =e

2πTr

dr1σαGR(r0, r1)vβ(r1)GA(r1, r0)Eβ

= +e

2πTr

dr1σαGR(r0, r1)vβ(r1)GA(r1, r0)Eβ

+e

2πTr

dr1

dr2σαGR(r0, r2)∑

δ

(r2 − r0)δ

∇δ(M(r0) · σ)GR(r2, r1)vβ(r1)GA(r1, r0)Eβ + h.c.

.

(2.57)

The first term gives the spin orbit torque (SOT), that relates the spin polarisationto the external electric field:

sSOTα = KαβEβ. (2.58)

22

2.2 – The spin polarisation under an electric field

Bringing the propagators of this term in momentum space, as it has been donefor the conductivity in Eq.(2.46), this tensor becomes:

Kαβ =e

2πTr

dp(2π)d σαGR(p)vβGA(p)

. (2.59)

The second term, instead, has to be identified with the Spin Transfer Torque(STT).It can be put in a simpler form by putting again the terms in momentum spacethrough Eq.(2.42):

sSTTα (r0, ω) =

e2π

Tr

dr1

dr2

dp(2π)d

dq(2π)d

dq′

(2π)d σαGR(p)eip(r0−r2) ∑δ

(r2 − r0)δ

∇δ(M(r0) · σ)GR(q)eiq(r2−r1)vβ(q′)GA(q′)eiq′(r1−r0)Eβ

+ h.c.

(2.60)

It is possible to replace (r0 − r2) with the derivative over the momentum p of theexponential:

sSTTα (r0, ω) =i ∑

δ

e2π

Tr

dr1

dr2

dp(2π)d

dq(2π)d

dq′

(2π)d σαGR(p)∂

∂pδeip(r0−r2)


+ h.c.

(2.61)

Then by integrating by parts over the momentum p it becomes:

sSTTα (r0, ω) =− i ∑

δ

e2π

Tr

dr1

dr2

dp(2π)d

dq(2π)d

dq′

(2π)d σα∂GR(p)

∂pδeip(r0−r2)


+ h.c.,

(2.62)


δ

e2π

Tr

dr1

dr2

dp(2π)d

dq(2π)d

dq′

(2π)d σαGR(p)vδGR(p)

∇δ(M(r0) · σ)GR(q)vβ(q′)GA(q′)Eβeir2(q−p)eir1(q′−q)eir0(p−q′)

+ h.c.

(2.63)

23


The integrals over the two space coordinates r1 and r2 give two delta functions,this allows to perform the integrals over q1 and q2. The result is:


γδ

e2π

Tr

dp(2π)d σαGR(p)vδGR(p)σγGR(p)vβ(p)

GA(p)

Eβ∇δMγ(r0) + h.c.

(2.64)

Therefore, as for the SOT, it is possible to define a susceptibility tensor that linksthe spin polarisation density to the spatial variation of the field:

sSTTα (r0, ω) = Rγδ

αβEβ∇δMγ(r0), (2.65)

with:

Rγδαβ = −i

e2π

Tr

dp(2π)d σαGR(p)vδGR(p)σγGR(p)vβ(p)GA(p)

+ h.c. (2.66)

2.3 The Gilbert Damping term

If the Zeeman term in the Hamiltonian is generated by a field evolving in time,then it is possible to compute the response of the system due to these time varia-tions. Once again, one can proceed in the same way to what has been done in theprevious sections. In order to estimate this quantity, we have to look again to thespin polarisation density, that is described by the vertex σαδ(r − r′). The externalfield are the variations of the magnetization σ · δM .

<sα(ω)>=1

2πi

dE Tr

(( f (E)− f (E − ω))GA(E)σαGR(E − ω)) + f (E − ω)

GA(E)σαGA(E − ω)− f (E)GR(E)σαGR(E − ω)))σ · δMω

.

(2.67)

As in case of the diamagnetic current for the vector potential, the variations of themagnetic field are responsible of a static response of the system, given by:

<sα(ω)>st=∑n

f (En) < n|∂σα

∂pA|n >=

12πi

Tr

dE f (E)(GA(E)σGA(E)

− GR(E)σGR(E)) · (σα · δMω)

.

(2.68)

24

2.3 – The Gilbert Damping term

Thus, the dynamic response is obtained by subtracting the previous quantityto the total spin polarisation:

<sα(ω)>dω=

12πi

dE Tr

( f (E)− f (E − ω))GA(E)σαGR(E − ω)) + f (E − ω)

GA(E)σαGA(E − ω)− f (E)GR(E)σαGR(E − ω))− f (E)GA(E)

σαGA(E) + f (E)GR(E)σαGR(E)(σ · δMω)

.

(2.69)

The Gilbert damping is related to the time variations of the field, so it pos-sible to identify a part of the spin polarization that contributes to this quantity:sGD

α (t) = Uαβ∂tMβ(t). By passing in reciprocal space, the previous relation be-comes:

sGDα (ω) = UαβδMβ(ω), (2.70)

so:

Uαβ =1

2πiω

dE Tr

( f (E)− f (E − ω))GA(E)σαGR(E − ω)) + f (E − ω)GA(E)

σαGA(E − ω)− f (E)GR(E)σαGR(E − ω))− f (E)GA(E)

σαGA(E) + f (E)GR(E)σαGR(E)σβ

.

(2.71)

There are two contributes that can be distinguished, nevertheless our interest isonly confined on the term due to the electrons at the Fermi surface. For thisreason, we will consider as the Gilbert Damping term response function is givenby:

Uαβ(ω) =1

4πTr

GAσα(GR − GA)σβ − (GR − GA)σαGRσβ

. (2.72)

By explicitly writing the space integrals and by passing in Fourier space as per-formed in the previous sections, this becomes:

Uαβ =1

4π

dp(2π)dTr

GA(p)σα(GR(p)− GA(p))σβ − (GR(p)− GA(p))σαGR(p)σβ

. (2.73)

Therefore, summarizing the results of the derivations performed, we can rewritethe total spin polarization density response through a unique expression, in whichwe put all the terms together:

<sα(ω)>= KαβEβ + Rγδαβ∂δMγEβ + Uαβ∂tMβ (2.74)

25

Chapter 3

Dirac AFM on a honeycomb Lattice

The first model we address is the Honeycomb AFM system. We have chosen thiskind of lattice since it is known that it allows a very good electronic conductionthanks to the existence of low energy electronic states around the K point of theBrillouin zone, that allows a Dirac-like electronic propagation. For this reason,the creation of multilayer structure based on Dirac layers has become of growinginterest in spintronics. This kind of lattice, until now, has been mainly consideredfor FM structures [31] [32] [33], however its extension in AFM has been proposedand realized [38].

a

Figure 3.1. Pictorial repre-sentation of the honeycomblattice created by the A andB atoms.

In our model we imagine that it is possible todescribe the whole honeycomb lattice as madeup of two kind of atoms, named A and B, thathave differently oriented local magnetic mo-ments. In particular, the localized magneticmoments are chosen as almost oppositely ori-ented to ensure the presence of a small mag-netization. Each atom A has three neighbour-ing atoms B separated by the characteristiclength of the lattice, called a. As is commonlydone, we use the s-d interaction approach: re-membering that the d electrons, placed on theatomic sites, generate the localised magneti-zations (this occurs for both the A and the Batoms, however the orientation of these fieldsis in directions almost opposite on the twosites) whereas the s electrons are responsiblefor the conduction. The propagation is created thanks to a tight-binding interac-tion between the atoms. For simplicity, we consider only a hopping, of constant

27

3 – Dirac AFM on a honeycomb Lattice

t, between next nearest neighbours. Consequently the Hamiltonian will read:

H = −J ∑<i,j>

Si · Sj − Jsd ∑i

c†i Si · σci − t ∑

<i,j>(c†

i cj + c†j ci) + ∑

ic†

i Vici. (3.1)

In Eq.(3.1) the first term is the AFM coupling (J < 0), the second the interactionbetween the travelling electrons and the local magnetizations, the third describesthe hopping and the last is the impurity disorder. In the present research thestudy of the system is developed in order to grasp the general picture, for thisreason fine details of the models are neglected inside the Hamiltonian, like theintroduction of magnetic anisotropy or the Dzyaloshinskii-Moriya(DM) interac-tion. Indeed these quantities, that are very relevant to describe the evolution of areal system, do not enter in the microscopic processes that determine the magni-tude and the shape of the spin torques, which are the aim of this research.In addition, again for the sake of simplicity, between all the possible shape of thedisorder we will adopt a weak Gaussian scalar disorder:

⟨V(r)V(r′)⟩ = (m∗τ)−1δ(r − r′), ⟨V⟩ = 0. (3.2)

In eq.(3.2) m∗ is the effective mass and τ is the scattering time, for brevity we willindicate the quantity (m∗τ)−1 as αD. Since this quantity is rather small, the focuswill be put over the leading terms in αD.

3.1 The Landau-Lifshitz-Gilbert equations

In accordance to the usual way of proceeding, we look for the time evolution ofthe local spin magnetizations. In order to determine it, we apply the Heisenbergequation of motion:

dSi

dt= −i[Si, H]. (3.3)

By performing the calculations, one finds that there is a term that is dependent onthe local spin polarization of the neighbouring sites and an other one dependenton the non-equilibrium spin polarization density of the conduction electrons incorrespondence of the site si =

1Ac†

i σci, where A is the area of the unit cell.

dSi

dt= J ∑

<i,j>Si × Sj + JsdASi × si. (3.4)

The local spin magnetization can be turned into classical fields because they areslowly varying compared to the motion of the travelling electrons. In addition,

28

3.1 – The Landau-Lifshitz-Gilbert equations

δ1

δ2

δ3

ζ1

ζ2ζ3

ζ4

ζ5

ζ6

Figure 3.2. Pictorial representation of the neighbouring atoms and therespective distances

these fields do not vary much from one site to another, and they are very smoothwith respect to the lattice parameter a. Thus, it is possible to take into considera-tion the distance between the sites through an expansion:

MA/B(r0 + δi) = MA/B(r0) + δi · ∇MA/B(r0) +12(δi · ∇)2MA/B(r0), (3.5)

where the δi are the distances of the first neighbours, whose representation isobservable in Fig.(3.2). In particular their values are with respect to an A site (theB sites distances have only opposite abscissae):

δ1 = (−a,0), δ2 =

a2

,

√3

2a

and δ3 =

a2

,−√

32

a

. (3.6)

By applying these simplifications into Eq.(3.4) in both the cases and performingthe summation over all the neighbours, we get the expression of the Landau-Lifshtiz-Gilbert(LLG) equations for the magnetizations of the sublattices:

∂MA

∂t= 3JMA × MB +

34

a2 JMA ×∇2MB + JsdMA × sA,

∂MB

∂t= 3JMB × MA +

34

a2 JMB ×∇2MA + JsdMB × sB.(3.7)

The linear terms are absent in the equations above because these terms all enterwith opposite signs and then they cancel, as we expect for symmetry reason.

29


Interesting enough, the magnetic subsystem can be fully described through a clas-sical free energy functional F [MA(r, t), MA(r, t)] (it is worth to mention that suchan approach is valid even in the case in which more complex magnetic subsys-tems are taken into consideration, like for example when one includes magneticanisotropy or the DM interactions). In the case under consideration, the func-tional is given by:

F = 3J

dra2

4 ∑i

∂i MA · ∂i MB − MA · MB. (3.8)

The local spin magnetization in an AFM are typically re-expressed through themagnetization M = (MA + MB)/2 and the Neel vector N = (MA − MB)/2.These two fields completely characterize an AFM and their time evolution isgiven by the correspondent Landau-Lifshitz-Gilbert equations, that can be ob-tained introducing these two field inside Eq.(4).

∂N∂t

= 3JM × N +3a2

4JN ×∇2M − 3a2

4JM ×∇2N +

Jsd2

M × (sA − sB)

+Jsd2

N × (sA + sB),

∂M∂t

=3a2

4JM ×∇2M − 3a2

4JN ×∇2N +

Jsd2

M × (sA + sB) +Jsd2

N × (sA − sB).

(3.9)

It is relevant to mention that, throughout the derivation, we will exploit thefact that the scalar product between the characteristics fields is trivial N · M = 0.Moreover, we will assume in this model that the two characteristic fields have afixed modulus, N2 = cost and M2 = cost; however the modulus of the magneti-zation is much smaller than the one of the Neel vector as we are considering anantiferromagnetic material. Only changes of the directions of the characteristicfields can be realized.

3.2 The Hamiltonian

The remaining part of the Hamiltonian describes the motion of the s electrons:

H = −∑iσ

c†iσ M i · σciσ − t ∑

<i,j>σ

(c†iσcjσ + c†

jσciσ). (3.10)

In Eq.(3.10), the index σ stands for the spin component correspondent to ↑ or ↓spin in the basis states for the conduction electrons.As previously stated, the low-energy states are those responsible for the conduc-tion. In order to highlight this behaviour, some modifications on the Hamiltonian

30

3.2 – The Hamiltonian

are necessary. The Hamiltonian can be rewritten in terms of the eigenfunctions incorrespondence of the sites, that will be called ΨA and ΨB. By looking for theireigenvalues, we find two set of coupled equations. Since the system is symmetric,the eigenvalues of the two wavefunctions have to be equal:

ϵΨAσ (R) = ∑

σ′MA · sA

σ′ΨAσ′(R)δσσ′ − t ∑

α∑σ′

ΨBσ′(R + δα)δσσ′ ,

ϵΨBσ(R) = ∑

σ′MB · sB

σ′ΨAσ (R)δσσ′ − t ∑

α∑σ′

ΨAσ′(R + δα)δσσ′ .

(3.11)

The siσ′ are the on site spin operators and the δα are the distances with the neigh-

bouring sites that have already been defined in Eq.(3.6), we remark that the onewritten are referred to the A sites and the B sites ones have opposite abscissae.The wavefunction ΨA and ΨB can be expressed through a packet of Block’s wave-function, whose wavevectors k belong to the Brillouin zone. In such a way wehave written our wavefunctions as combinations of free waves multiplied byfunctions that have the same periodicity of the lattice one.Therefore:

kx

ky

K

K ′

K

K ′

K ′

K

Figure 3.3. Location of theK and K’ points in the Bril-louin zone

ΨAσ (R) = ∑

kφA

σk(R)eikR,

ΨBσ(R) = ∑

kφB

σk(R)eikR.(3.12)

Through an analysis of the typical magnitudeof the quantities in the Hamiltonian, the ex-change interaction turns out to give a smallercontribution than the hopping term. Thus,the energy of the electrons is mainly origi-nated by this last term. As only the electronsclose to the K points are able to propagate inthe system, it is possible to find the wave-vectors associated to these points by solving:

∑α

eikδα = 0, (3.13)

and this leads to six possible k vectors:

k1,2 =

0,± 4π

3√

3a

, and k3,4,5,6 = ±2π

3a

1,± 1√

3

.(3.14)

However, there are effectively only two independent vectors because all the oth-ers can be obtained summing or subtracting the unit vectors of the reciprocal

31


lattice: b1,2 = 2π3a

1,±

√3

.Between all the possible equally valid choices, the vectors here kept are:

k =2π

3a

1,

1√3

and k′ =

2π

3a

1,− 1√

3

. (3.15)

The wave-functions can be now rewritten in terms of these two wave-vectors, as:

ΨAσ (R) = φA

σ1(R)eikR + φAσ2(R)eik′R,

ΨBσ(R) = φB

σ1(R)eikR + φBσ2(R)eik′R.

(3.16)

By replacing these expressions inside Eq.(3.11), one obtains a set of four equa-tions for each φ. These equations are decoupled, because φA

1 is linked only to φB1

and not to φB2 and the same is valid for the other terms. As previously stated,

it is relevant to study only the small variations, for this reason it is performed agradient expansion in δα. Let then look at how the equation for φA

1 becomes:

ϵφAσ1(R) = MA · sA

σ φAσ1(R)− t ∑

α

φBσ1(R)eikδα − t ∑

α

eikδα δα ·∇φBσ1(R). (3.17)

The second term is zero because the wave-vector k has been chosen to satisfy thiscondition, then by developing the calculations on the third term one obtains:


σ φAσ1(R) +

3at2

e−2iπ

3 (∂x + i∂y)φBσ1(R). (3.18)

The coefficients in front of the second term at right in Eq.(3.18) determine theFermi velocity of the material; the phase factor, instead, can be included insidethe definition of the wave-function, that is complex in general. This redefinitionis consistent with the equation obtained applying the Hamiltonian over φB

1 , thatis equal to (by developing the calculations in the same manner):

ϵφBσ1(R) = MB · sB

σ φBσ1(R) +

3at2

e2iπ

3 (∂x − i∂y)φAσ1(R). (3.19)

So, the phase factor can be effectively placed inside the definition of φB1 (or of φA

1depending on how we look at the equations). Proceeding in a similar way, theother two equations read:


σ φAσ2(R) +

3at2

e−2iπ

3 (∂x − i∂y)φBσ2(R),

ϵφBσ2(R) = MB · sB

σ φBσ2(R) +

3at2

e2iπ

3 (∂x + i∂y)φAσ2(R).

(3.20)

32

3.2 – The Hamiltonian

Therefore if one combines all the equations together and multiplies by the properphase factors, the system becomes:

ϵ

φA

σ2e−2iπ

3 φBσ2

e−4iπ

3 φBσ1

e−2iπ

3 φAσ1

=

MA ·σ v f (∂x − i∂y)σ0 0 0v f (∂x + i∂y)σ0 MB ·σ 0 0

0 0 MB ·σ v f (∂x − i∂y)σ00 0 v f (∂x + i∂y)σ0 MA ·σ

φAσ2

e−2iπ

3 φBσ2

e−4iπ

3 φBσ1

e−2iπ

3 φAσ1

, (3.21)

where σ0 is the identity matrix in dimension 2.This is the expression of the Hamiltonian in the basis of its eigenvectors, it can beput in a nicer form by replacing the local fields with the characteristic ones andby expressing the partial derivatives as the momentum operator p.

Hel =

(M + N)·σ v f (px − ipy)σ0 0 0

v f (px + ipy)σ0 (M − N)·σ 0 00 0 (M − N)·σ v f (px − ipy)σ00 0 v f (px + ipy)σ0 (M + N)·σ

. (3.22)

The matrix written above is 8×8. We can notice that the Hamiltonian is acting intwo ways; one is on the spin space where the action is described through the σ’s.The other way is between the eigenfunctions of the A and B sites, for which wecan define a new set of Pauli matrices called τ’s and τ0 that permit to distinguishA and B sites.It is evident that the upper and the lower part of the Hamiltonian are well sep-arate, for this reason it is possible to split it into two sub-Hamiltonians. Quitemanifestly, the two sub-Hamiltonians differ only for the fact that the two localmagnetizations are swapped. Despite this difference, any physical law valid forone subsystem is valid even for the other one. Therefore, it is possible to focus onthe study of only one of the two parts, since the contribution of the other one canbe simply obtained by reversing the sign of the Neel vector in the results of thespin polarization density that will be obtained later on. Naturally, the laws of theoverall system are determined by the sum of those of the two sub-Hamiltonians.Let’s then focus on the upper part of the Hamiltonian, written through the τ’sand the σ’s:

H = vFσ0p · τ + N · στ3 + M · στ0, (3.23)

where p is the in plane momentum, vF the Fermi velocity and σ the spin operator.It is possible to treat the magnetization M as a perturbation in an AFM because itis much smaller than the Neel vector. Then, the unperturbed Hamiltonian reads:

33


H0 =

N · σ vF(px − ipy)σ0

vF(px + ipy)σ0 −N · σ

. (3.24)

Figure 3.4. Bandsstructure of the AFMat the edge of theBrillouin zone

As first step, let us consider the eigenvalues ofthis Hamiltonian, we find an energy dispersionsdegenerate in spin:

ϵ1 = −

N2 + p2v2F, ϵ2 =

N2 + p2v2

F. (3.25)

It is possible to notice that both the eigenvalues de-pend only on the modulus of the momentum andof the Neel vector, therefore each energy defines acircular line in momentum space. The Neel vectoropens a gap at the edge of the Brillouin zone, asone can notice in Fig.(3.4), naturally in its absencethe branches have again a Dirac cone dispersion.Our purpose is to compute the spin polarizationdensity sα(r, t) in order to find the torques appliedto N and M. In the mean field approach bothE(r, t) and the two characteristic fields are slow,therefore to obtain the non equilibrium spin po-larization density sα(r, t), it is possible to make agradient expansion that takes into account the nonlocality in an approximate way:

sα =KαβEβ + RγδαβEβ∂δNγ + Rmγδ

αβEβ∂δMγ+

Uαβ∂tNβ + Umαβ∂tMβ + ...(3.26)

The first term is associated to the so called spin-orbit torque (SOT) and describesthe swift of the spins due to the electric field. Instead the second two element arerelated to the spin transfer torque (STT), that takes into consideration the spatiallocal variation of the fields. The last two terms contribute to the so called Gilbertdamping, which has the effect of introducing dissipation effects to stabilise thelocal magnetization.

3.3 The Self Energy

The susceptibility tensors appearing in Eq.(3.26) depend on the propagators ofthe travelling electrons. In order to proceed in the analysis of this model, it is

34

3.3 – The Self Energy

necessary to determine them, therefore one needs to compute the self-energy. Toperform this computation, we will adopt the self-consistent Born approximation(SCBA), whose details are illustrated in the Appendix. Going straight to the point,this theory consists in replacing the propagators with the disorder averaged ones.As already stated, in order to understand the general behaviour, we will supposethat the disorder is a Gaussian spin-independent disorder, whose correlator is< V(r)V(r′) >= αDδ(r′ − r). Since the disorder depends on the effective mass1

of the system, it is relevant to find it through the energy dispersion, in particularwe can write:

1m∗ =

d2ϵ

dp2 =1p

d2

N2 + v2F p2

d2p,

m∗ =ϵ3

N2v2F

.

(3.27)

This expression is appropriate in the region of the gap with N2 > 0, but it issingular for N2 → 0.A more adequate definition [34] could be:

1m∗ =

1p

dϵ

dp=

1p

d

N2 + v2F p2

dp,

m∗ =ϵ

v2F

,

(3.28)

that has a more clear value in the limit for N2 → 0.The SCBA permits to find the ”bare” contributions of the physical quantities ofthe system, like the conductivity, the STT and so on. However, it will be possibleto reconstruct the dressed physical quantities by including the corrections orig-inated by the disorder. Let’s then start looking at the self energy in the SCBA,in particular we analyse the case of the retarded one, keeping in mind that theadvanced and retarded self-energies differ only for having opposite imaginaryparts.

ΣR = 2παD

dp2

(2π)2 GR and GR = [ϵI − Hel − ΣR]−1, (3.29)

Since the magnetization is treated as a perturbation GR = GR +GR(HM +ΣR1 )G

R,where ΣR

1 is the first order contribution of the self-energy that contains the mag-netization and GR = [ϵI − H0 − ΣR

0 ]−1 is the unperturbed propagator. To avoid

1After the derivation of this quantity, we will omit the Fermi velocity, that can be imagined asentering in a redefined momentum; we will restore vF only in the final results.

35


excessive and repetitive calculations, we will illustrate only the derivation of ΣR0 ,

indeed the first order term is a simple extension.

ΣR0 = 2παD

dp2

(2π)2 GR. (3.30)

Given that the retarded propagator depends on the self-energy itself, effectivelyone has to solve a self-consistent equation. A way to realize so is to take a guesson the correct shape of the self energy, in this particular case we suppose that theeffect of the self energy is to modify the magnitude of the components of the Neelvector and the of energy and make them acquiring an imaginary part.

GR =1

ϵI − (N · σ)τz − (p · τ)σ0 − ΣR0=

1ϵRI − (NR · σ)τz − (p · τ)σ0

, (3.31)

where ϵR = ϵ′ + iϵ′′ and NRj = N′

j + iN′′j with j = (x, y, z). The momentum

remains unmodified, because we are integrating over it. Once performed theinversion of the matrix, we have that:

GR =1

ϵR 2 − p2 − NR 2

ϵRI + NR · σ (px − ipy)σ0(px + ipy)σ0 ϵRI − NR · σ

. (3.32)

Inserting Eq.(3.32) inside Eq.(3.30) we obtain:

ΣR0 =

αD

2π

2π

0dφ ∞

0pdp

1ϵR 2 − p2 − NR 2

ϵRI + NR · σ (p cos φ − ip sin φ)σ0

(p cos φ + ip sin φ)σ0 ϵRI − NR · σ

.(3.33)

The integration over the phases is null for the terms containing the sine and thecosine, for the reaming terms, that are phase independent, the integral gives 2π.Therefore, we get:

ΣR0 =− αD

∞

0

pdpp2 − ϵR 2 + NR 2

ϵRI + NR · σ 0

0 ϵRI − NR · σ

=

− αD

2(iπ + Λ(ϵ′, N′)

ϵ′I + N ′ · σ 0

0 ϵ′I − N ′ · σ

,

(3.34)

where Λ(ϵ′, N′) = ln(∆/√

ϵ′ 2 − N ′ 2) is a real function dependent on the ultra-

violet momentum cut-off ∆. The cut-off has to be inserted to prevent the logarith-mic divergence. In the resolution of the integral we disregarded the imaginaryparts of the energy and of the order parameter because they give a lower ordercontribution in the disorder parameter. At this stage, we can insert the expression

36

3.4 – The Dressing of the Interaction

of the self-energy, obtained solving the integral, back in Eq.(3.32) so that we canfind the real and the imaginary part:

ϵ′ + iϵ′′ = ϵ + αDϵ′Λ(ϵ′, N′) +iαDπ

2ϵ′, N′

j + iN′′j = Nj − αDN′

j Λ(ϵ′, N′)− iαDπ

2N′

j . (3.35)

Hence:

ϵR = iϵ′

1 +iαDπ

2

, and NR

j = N′j

1 − iαDπ

2

, (3.36)

where ϵ′ and N′j , that are dependent on ϵ, N, ∆ and αD, are defined through the

transcendental equations ϵ′ = ϵ + αDϵ′Λ(ϵ′, N′) and N′j = Nj + αDN′

j Λ(ϵ′, N′).The renormalization group approach permits to account for the logarithmic di-vergence. This procedure, in a very basic way, consist in exploiting the old valuesof the physical quantities to eliminate the divergences. In particular, the old val-ues are set to be different from the divergent quantities of exactly the true valueof the correspondent physical quantity of the system. In this particular case, thetrue values of the physical quantities are ϵ′, N′

j . By the fact that all the compo-nents of the Neel vector behave in the same way, it is evident that it is actuallythe modulus of the order parameter that gets renormalized.

As a matter of the magnetization, whose relations are obtainable developingthe calculations for the first order, it follows a behaviour similar to the Neel vector.Indeed, what gets renormalized is the modulus of the magnetization. In addition,even in this case the imaginary part turns out to be dependent on the real one.

MRj = M′

j

1 +

iαDπ

2

. (3.37)

For completeness, it is worth to remind that the relations associated to the corre-spondent advanced quantity can be found just taking the complex conjugate. Asadditional remark, we point out that from now on ϵ′, N′

i and M′i will be indicated

without the primed index because they are actually the true (observed) values ofthe physical quantities of the system.

3.4 The Dressing of the Interaction

Let us now pass to consider how the vertices are modified due to the presenceof the disorder, the analysis is done focusing separately on sA and sB, the spinvertices of the conduction electrons.The dressing process consists in considering all the possible way in which we

37


introduce the sources of disorder in the propagation. As we have chosen a Gaus-sian disorder, the only possible interactions are between couple of points; fromthe diagrammatic point of view these interactions are depicted as lines as we cannotice in Fig.3.5. In addition, the interaction lines cannot cross because, as it isexplained in the Appendix, this would give rise to quantities sub-leading in thedisorder parameter. Consequently the contributes, that one needs to analyse toobtain the full vertex in the case of sA, are given from a diagrammatic point ofview by:

Figure 3.5. Diagrammatic expression of the dressed spin vertex sA

As starting point to determine such a quantity, let’s consider the case of havingsimply one interaction line, in such a case the expression of the diagram is simplygiven by:

VAα = αD

d2p(2π)2 GA

σα 00 0

GR. (3.38)

We expect that it is possible to rewrite each Vα in terms of a combination of Paulimatrices, so like:

VAα = Uαβ

σβ 00 0

+Wαβ

0 00 σβ

, (3.39)

where α and β run on (x,y,z). The reason for which we have introduced only thesetwo terms, instead of considering even off-diagonal ones, is related to the fact thatthe coefficients of the off diagonal terms are zero. Indeed, the off diagonal terms,put in Eq.(3.32), make appear phase factors whose integrals over the radial anglevanish.The values of the coefficients of Eq.(3.39) can be determined by substituting theexpression of the vertex inside Eq.(3.38) and then by taking the trace with one of

38

3.4 – The Dressing of the Interaction

Figure 3.6. Diagrammatic expression of the computation done to deter-mine the elements of Uαβ

the elements of the linear combination; this corresponds to close the diagram asit can observed in Fig.(3.6) that is referred to the elements of Uαβ.This computation consists on one side simply in evaluating the traces of productsof Pauli matrices, whereas on the other to solve an integral. By performing all thecomputations we get:

Uαβ =1

2(N2 + ϵ2)

ϵ2 + N2x − N2 NxNy NxNz

NxNy ϵ2 + N2y − N2 NyNz

NxNz NyNz ϵ2 + N2z − N2

. (3.40)

The coefficients of Wαβ are obtained putting the combinations of Pauli ma-trices in the position 2-2, instead of the position 1-1. The path is same followedbefore, however in such a case some computational problems arise. Indeed, ifone performs a careful analysis of the integrals, one can notice that these inte-grals are of smaller order in the disorder parameter αD and tend to go like thelogarithm of the momentum cut-off. Such a dependence was the same observedduring the computation of the self-energy; for this reason it possible to get rid ofthis divergences thanks a re-normalization process; in particular in such a caseby re-normalizing the disorder parameter αD [35] [36]. Through this redefinition,Wαβ becomes of smaller order in αD and negligible, the dressed vertex is com-pletely determined by Uαβ. At this stage, we have included only the dressing ofone interaction line, to count all the dressings at different orders we have to re-iterate the same operation over and over again. Then the general expression forthe dressed vertex is (omitting the bare vertex for now):

Vα + αD

d2p(2π)2 GAVαGR + αD

d2p′

(2π)2 GA

αD

d2p(2π)2 GAVαGR

GR + ...(3.41)

By rewriting every Vα through M and the Pauli Matrices, and performing the

39


integrals, one obtains:

Uαβ

σβ 00 0

+ UαβUβγ

σγ 00 0

+ UαβUβγUγη

ση 00 0

+ ... (3.42)

The products of several U are all multiplied to the same matrix, then one obtainsthat the result is a geometric series in U. This series is surely converging becausethe denominator is bigger than all the other terms. Then if one includes even thebare vertex (whose coefficient is the identity matrix I3) the sum of the elementsof the series will give the fully dressed vertex Γα:

Γα = (I3 − U)−1αβ

σβ 00 0

. (3.43)

This leads to:

Figure 3.7. The diagrammatic expression of the dressed vertices ΓAα and ΓB

α .

ΓAα =

13N2 + ϵ2

2ϵ2 + 4N2x − N2 4NxNy 4NxNz

4NxNy 2ϵ2 + 4N2y − N2 4NyNz

4NxNz 4NyNz 2ϵ2 + 4N2z − N2

αβ

σβ 00 0

.(3.44)

It is possible to perform the same analysis for the sB spin vertex, logically thebehaviour and the results are almost identical to the previous ones. Therefore onegets:

ΓBα =

13N2 + ϵ2

2ϵ2 + 4N2x − N2 4NxNy 4NxNz

4NxNy 2ϵ2 + 4N2y − N2 4NyNz

4NxNz 4NyNz 2ϵ2 + 4N2z − N2

αβ

0 00 σβ

.(3.45)

Another quantity that gets corrections in the dressing process is the velocity, it isworth to point out that the velocity for such a system is:

vx = τ1σ0 =

0 σ0σ0 0

, vy = τ2σ0 =

0 −iσ0

iσ0 0

, (3.46)

40

3.5 – The Spin Orbit Torque and the Conductivity

with the assumption to insert the modulus of v inside the definition of p.The vertex with one dressed interaction line will be in such a case:

αD

d2p(2π)2 GAvαGR. (3.47)

As before, we expect that each velocity vertex, at first order in the dressing pro-cess, can be rewritten as a linear combination of the zero order velocities. To findthe coefficients we take even in this case the trace, closing the loop with anothervelocity vertex. Luckily, in this case the integrals do not give problems of di-vergence and then no further re-normalization is necessary. If we rewrite all thecoefficients in a matrix form once again, we simply find: ϵ2−N2

2(N2+ϵ2)0

0 ϵ2−N2

2(N2+ϵ2)

. (3.48)

Thus, there is no mixing between the x and y components, the correction factor isobtained as the result of a geometric series:

1 +ϵ2 − N2

2(N2 + ϵ2)+

ϵ2 − N2

2(N2 + ϵ2)

2

+

ϵ2 − N2

2(N2 + ϵ2)

3

+ ... =

=1

1 −

ϵ2−N2

2(N2+ϵ2)

= 2(N2 + ϵ2)

3N2 + ϵ2 .(3.49)

The dressed velocity vertex then is:

vα =2(N2 + ϵ2)

3N2 + ϵ2 vα (3.50)

Interesting enough, in the limit of high energy (or small characteristic fields), allthe spin vertices and all the velocity vertices get simply corrected by a factor 2.

3.5 The Spin Orbit Torque and the Conductivity

In order to reconstruct the susceptibility tensor Kαβ, one has to use the Kubo-Streda formula (whose derivation is in the previous section):

Kαβ = dp2

(2π)2 Trsα(GR − GA)vβGA − sαGRvβ(GR − GA)

, (3.51)

41


Figure 3.8. The diagrammatic expression of the dressed velocity vertex vα

where sα stands for the spin vertex either on A or B site.

sAα =

σα 00 0

and sB

α =

0 00 σα

(3.52)

This expression is valid in general, as we work in the SCBA we have to insert thecorresponding propagators and the dressed vertices to reconstruct the dressedphysical quantities. As previously stated, the computations are performed upto first order in the magnetization, this is realized using again Dyson’s equationG = G + GHMG. Then, when one computes the tensor K there are zero and firstorder contributions in the magnetization, as it is shown in Fig.(3.9).

Figure 3.9. Diagrammatic expansion for the computation of the SOT

The zero order term is given by:

K0αβ =

dp2

(2π)2 TrΓαGRvβGA

. (3.53)

As the values of the integrals are established by the poles of the propagators,the combinations in which there are only two retarded propagators GRGR or two

42

3.5 – The Spin Orbit Torque and the Conductivity

advanced ones GAGA are trivial. Indeed, in these cases it is always possible to in-tegrate over the half-plane in which, respectively, the retarded and the advancedpropagator do not have poles. Following the same reasoning, even the first orderterms can be simplified to:

K1αβ =

dp2

(2π)2 TrΓαGRHMGRvβGA + ΓαGRvβGAHMGA

. (3.54)

By developing the calculation for both ΓA and ΓB, one finds that the tensor Kαβ

is trivial in both the cases because the traces are phase dependent. It is usefulto compute the conductivity tensor in this system, so that it is possible to relatethe variations of the Neel vector and of the magnetization not only to the electricfield, but even to the current density J, where J = σE. The conductivity is givenby the Kubo formula (starting from its simplified version for the same reason asabove):

σαβ = dp2

(2π)2 Tr[vαGRvβGA]. (3.55)

Introducing the dressed vertices and performing the diagrammatic expansion inthe magnetization, the zero and first order term read:

2(N2 + ϵ2)

3N2 + ϵ2

dp2

(2π)2 Tr[vαGRvβGA], (3.56)

2(N2 + ϵ2)

3N2 + ϵ2

2 dp2

(2π)2 Tr[vαGRHMGRvβGA + vαGRvβGAHMGA]. (3.57)

By developing the computations above, one gets:

σαβ =

8(ϵ2−N2)αD(3N2+ϵ2)

32N·Mπ(N2−ϵ2)(3N2+ϵ2)2

32N·Mπ(N2−ϵ2)(3N2+ϵ2)2

8(ϵ2−N2)αD(3N2+ϵ2)

. (3.58)

However, the Neel vector and the magnetization are orthogonal in this sys-tem. Consequently, only the longitudinal conductivity turns out to be non trivial.

σαβ =

8(ϵ2−N2)αD(3N2+ϵ2)

0

0 8(ϵ2−N2)αD(3N2+ϵ2)

. (3.59)

43


3.6 The Spin Transfer Torque

In this system, we expect that the Neel vector N and the magnetization M areboth subjected to STT. These STT, as according to Eq.(9), are due to the couplingof N and M themselves with the spin polarization density. In order to find theseSTT, then, we start performing the computation of the si for ΓA and ΓB. Naturally,the torques of the two local magnetizations will have some dependence both onN and on M.The part of the spin polarization related to the variations of N can be computedthrough the tensor Rγδ

αβ, this is given by the formula:

Rγδαβ =− i

d2p(2π)2 Tr

sαGRvγGR

σδ 00 −σδ

GRvβGA

+ i d2p(2π)2 Tr

sαGRvβGA

σδ 00 −σδ

GAvγGA

.

(3.60)

This is the general expression of the susceptibility tensor. We modify Eq.(3.60)inserting the disorder averaged propagators and taking into consideration thedressing of the vertices. Even the susceptibility tensor Rγδ

αβ is computed at zeroand first order in the magnetization, we can visualize the different terms of thediagrammatic expansion in Fig.(3.10). For completeness, their explicit expressionare:

−i d2p(2π)2 Tr

ΓαGRvγGR

σδ 00 −σδ

GRvβGA

+ h.c. (3.61)

− i d2p(2π)2 Tr

ΓαGRHMGRvγGR

σδ 00 −σδ

GRvβGA+ ΓαGRvγ

GRHMGR

σδ 00 −σδ

GRvβGA+ ΓαGRvγGR

σδ 00 −σδ

GRHM


σδ 00 −σδ

GRvβGAHMGA

+ h.c.

(3.62)

44


Figure 3.10. Diagrammatic expansion for the computation of the STT

In a similar way, the part connected to the derivatives of M is given by thetensor Rmγδ

αβ:

Rmγδαβ =− i

d2p(2π)2 Tr

sαGRvγGR

σδ 00 σδ

GRvβGA

+ i d2p(2π)2 Tr

sαGRvβGA

σδ 00 σδ

GAvγGA

.

(3.63)

As for the previous tensor the disorder averaged propagators and the dressedvertices are introduced, moreover it is used the Dyson’s equation in the magneti-zation. The result of the expansions are alike to Eq.(3.61-3.62) with of course thechange of τz in τ0.The general results of the calculations performed to establish the value of thesetensors could be very complex, however, luckily some simplifications are possi-ble. First of all, the terms proportional to the scalar product between the Neelvector and the magnetization can be cancelled. Moreover, during the computa-tion several terms similar to each other appear and it is possible to eliminate themsumming null tensors. These null tensors are originated from the fact that both

45


the Neel vector and the magnetization have a fixed modulus:

∂αN2 = Nx ∂αNx + Ny ∂αNy + Nz ∂αNz = 0. (3.64)

By multiplying for the components of the electric field it is possible to obtain twodifferent relations valid both for the Neel vector and the magnetization:

(E ·∇)N2 = 0, (E ×∇)⊥N2 = 0. (3.65)

(E ·∇)M2 = 0 and (E ×∇)⊥M2 = 0. (3.66)

Each of these expressions, in the same formalism in which we wrote the tensorRγδ

αβ in the equation of the spin polarization density Eq.(3.26), corresponds to threenull tensors, one for each possible direction:

OrtNx =

Nx 0Ny 0Nz 0

0 Nx0 Ny0 Nz

0 00 00 0

0 00 00 0

0 00 00 0

0 00 00 0

, OrtN

y =

0 00 00 0

0 00 00 0

Nx 0Ny 0Nz 0

0 Nx0 Ny0 Nz

0 00 00 0

0 00 00 0

,

OrtNz =

0 00 00 0

0 00 00 0

0 00 00 0

0 00 00 0

Nx 0Ny 0Nz 0

0 Nx0 Ny0 Nz

,

(3.67)

46


CrossNx =

0 Nx0 Ny0 Nz

−Nx 0−Ny 0−Nz 0

0 00 00 0

0 00 00 0

0 00 00 0

0 00 00 0

, CrossN

y =

0 00 00 0

0 00 00 0

0 Nx0 Ny0 Nz


0 00 00 0

0 00 00 0

,

CrossNz =

0 00 00 0

0 00 00 0

0 00 00 0

0 00 00 0

0 Nx0 Ny0 Nz


.

(3.68)

These matrices have a dimension (3x2,3x2), so they are described through fourindices α, β, γ and δ. α and β run, respectively, on the lines and columns of thematrix, whereas γ and δ run on the lines and columns of the sub-matrices.

Figure 3.11. Illustration of the running of the indeces α, β, γ and δ.

Obviously, it is possible to generate analogous tensors with the magnetizationinstead of the Neel vector. It is worth to point out that these null tensors canbe summed only to tensors associated to the derivatives of the respective field.Among all the tensors we are evaluating, the null tensors with the Neel vector

47


can be summed only to Rγδαβ and those with magnetization only to Rmγδ

αβ.Let’s now pass to look at the results of the computation for the vertex ΓA at zeroorder in HM. As first thing, we point out that the simplification we have per-formed are:

RA γδ0 αβ =RA γδ

0 αβ − ∑i

4[(3N2 + ϵ2)(N2 + 3ϵ2)(M × N)i + 8παDϵ(N2 + ϵ2)2Ni]

πα2D(N2 + ϵ2)2(3N2 + ϵ2)2

OrtN γδi αβ

− ∑i

2(N2 − ϵ2)2(M × N)i

α(N2 + ϵ2)2(3N2 + ϵ2)2 CrossN γδi αβ ,

RmA γδ0 αβ =RmA γδ

0 αβ + ∑i

8ϵ(ϵ2 − N2)Ni

πα2D(N2 + ϵ2)(3N2 + ϵ2)2

CrossM γδi αβ .

(3.69)

Inserting the result inside Eq.(3.26) and using that TSTT0 A = MA × sSTT

0 A , we get:

TSTT0 A =− 8N2ϵ

αD(3N2 + ϵ2)2 (E · ∇)N +8ϵ

αD(3N2 + ϵ2)2 M × (N × (E · ∇)N)

− 4(N2 + ϵ2)

αD(3N2 + ϵ2)2 (N + M)× (E ×∇)⊥N +4(N4 + 10N2ϵ2 + 5ϵ4)

πα2D(N2 + ϵ2)2(3N2 + ϵ2)2

(N × M)(N · (E · ∇)M) +4(ϵ2 − N2)

πα2D(3N2 + ϵ2)2

(N + M)× (E · ∇)M

+2(ϵ2 − N2)2

αD(N2 + ϵ2)(3N2 + ϵ2)2 (N + M)× (E ×∇)⊥M +8ϵ(ϵ2 − N2)

πα2D(N2 + ϵ2)(3N2 + ϵ2)2

N × (N × (E ×∇)⊥M)− 2ϵ2(3ϵ2 + 5N2)(N × M)

παD(N2 + ϵ2)2(3N2 + ϵ2)2 (N · (E ×∇)⊥M).

(3.70)

The second tensor is obtained inserting the vertex ΓB, the simplifications per-formed are very similar:

RB γδ0 αβ =RB γδ

0 αβ + ∑i

4[(3N2 + ϵ2)(N2 + 3ϵ2)(M × N)i + 8παDϵ(N2 + ϵ2)2Ni]

πα2D(N2 + ϵ2)2(3N2 + ϵ2)2

OrtNi

+ ∑i

2(N2 − ϵ2)2(M × N)i

αD(N2 + ϵ2)2(3N2 + ϵ2)2 CrossNi ,

RmB γδ0 αβ =RmB γδ

0 αβ − ∑i

8ϵ(ϵ2 − N2)Ni

πα2D(N2 + ϵ2)(3N2 + ϵ2)2

CrossMi .

(3.71)

48


Then by proceeding as before we get:

TSTT0 B =

8N2ϵ

αD(3N2 + ϵ2)2 (E · ∇)N +8ϵ

αD(3N2 + ϵ2)2 M × (N × (E · ∇)N)

+4(N2 + ϵ2)

αD(3N2 + ϵ2)2 (N − M)× (E ×∇)⊥N +4(N4 + 10N2ϵ2 + 5ϵ4)

πα2D(N2 + ϵ2)2(3N2 + ϵ2)2

(N × M)(N · (E · ∇)M)− 4(ϵ2 − N2)

πα2D(3N2 + ϵ2)2

(N − M)× (E · ∇)M

+2(ϵ2 − N2)2

αD(N2 + ϵ2)(3N2 + ϵ2)2 (N − M)× (E ×∇)⊥M − 8ϵ(ϵ2 − N2)

πα2D(N2 + ϵ2)

1(3N2 + ϵ2)2 N × (N × (E ×∇)⊥M) +

2ϵ2(3ϵ2 + 5N2)(N × M)

παD(N2 + ϵ2)2(3N2 + ϵ2)2

(N · (E ×∇)⊥M).

(3.72)

The STT acting over N and M are given, respectively, by the difference andthe sum of those of the two local spin polarizations.

TSTT0 N =

12(TSTT

0 A − TSTT0 B ) = − 8N2ϵ

αD(3N2 + ϵ2)2 (E · ∇)N − 4(N2 + ϵ2)

αD(3N2 + ϵ2)2

N × (E ×∇)⊥N +4(ϵ2 − N2)

πα2D(3N2 + ϵ2)2

N × (E · ∇)M +2(ϵ2 − N2)2

αD(N2 + ϵ2)

1(3N2 + ϵ2)2 M × (E ×∇)⊥M +

8ϵ(ϵ2 − N2)

πα2D(N2 + ϵ2)(3N2 + ϵ2)2

N × (N × (E ×∇)⊥M)− 2ϵ2(5N2 + 3ϵ2)

αD(N2 + ϵ2)2(3N2 + ϵ2)2

(N × M)(N · (E ×∇)⊥M),

(3.73)

TSTT0 M =

12(TSTT

0 A + TSTT0 B ) = +

8ϵ

αD(3N2 + ϵ2)2 M × (N × (E · ∇)N)

− 4(N2 + ϵ2)

αD(3N2 + ϵ2)2 M × (E ×∇)⊥N +4(ϵ2 − N2)

πα2D(3N2 + ϵ2)2

M × (E · ∇)M

+2(ϵ2 − N2)2

αD(N2 + ϵ2)(3N2 + ϵ2)2 N × (E ×∇)⊥M +4(N4 + 10N2ϵ2 + 5ϵ4)

πα2D(N2 + ϵ2)2(3N2 + ϵ2)2

(N × M)(N · (E · ∇)M).

(3.74)

These relations, describing the zero order torques applied to the Neel vector andthe magnetization, have been obtained starting from the upper sub-Hamiltonian.

49


A similar analysis has to be repeated for the other sub-Hamiltonian in order tofind the total torques. When one sums the contribution of both the sub-Hamiltonians,it turns out that only certain term from Eq.(3.73) and Eq.(3.74) remain, specificallyall the terms of the kind (E ×∇)⊥ cancel because they appear with an oppositesign in the lower sub-Hamiltonian. Then the simplified zero order results are:

TSTT0 N = − 16N2ϵ

αD(3N2 + ϵ2)2 (E · ∇)N +8(ϵ2 − N2)

πα2D(3N2 + ϵ2)2

N × (E · ∇)M, (3.75)

TSTT0 M =+

16ϵ

αD(3N2 + ϵ2)2 M × (N × (E · ∇)N) +8(ϵ2 − N2)

πα2D(3N2 + ϵ2)2

M × (E · ∇)M

+8(N4 + 10N2ϵ2 + 5ϵ4)(N × M)

πα2D(N2 + ϵ2)2(3N2 + ϵ2)2

(N · (E · ∇)M).(3.76)

The first order terms, computed through Eq.(3.62) and the analogous expres-sion with the derivatives of the magnetization, can be simplified in the same man-ner of the zero order terms. Moreover, we can directly disregard the (E ×∇)⊥terms because it can be proven that even in this case they vanish. We start con-sidering again the results for the spin vertex ΓA, for this case the only relevantsimplification is:

RA γδ1 αβ =RA γδ

1 αβ − ∑i

2π(N2 + ϵ2)2(3N2 + ϵ2)2 [4(2ϵ4 − 5N2ϵ2 − N4)(M × N)i

− παDϵ(ϵ4 − 14N2ϵ2 − 3N4)Mi]OrtN γδi αβ .

(3.77)

Then, if one computes the tensor, that is done again finding the spin polarizationand then performing the external product with the correspondent local magneti-zation, he would get:

TSTT1 A =−

32ϵ(ϵ2 + 2N2)(ϵ2 − J2sdN2)

π2α3D(N2 + ϵ2)2(3N2 + ϵ2)2

N × (M × (E ·∇)M)

− 2ϵ2

π(N2 + ϵ2)2(5N2 + 3ϵ2)

(3N2 + ϵ2)2 (N × M)(N · (E ·∇)M).

(3.78)

For the vertex ΓB the simplification we have to perform is:

RB γδ1 αβ =RB γδ

1 αβ + ∑i

2π(N2 + ϵ2)2(3N2 + ϵ2)2 [4(2ϵ4 − 5N2ϵ2 − N4)(M × N)i

+ παDϵ(ϵ4 − 14N2ϵ2 − 3N4)Mi]OrtN γδi αβ .

(3.79)

50

3.7 – The Gilbert Damping contribution

and then:

TSTT1 B =−

32ϵ(ϵ2 + 2N2)(ϵ2 − J2sdN2)

π2α3D(N2 + ϵ2)2(3N2 + ϵ2)2

N × (M × (E ·∇)M)

− 2ϵ2

π(N2 + ϵ2)2(5N2 + 3ϵ2)

(3N2 + ϵ2)2 (N × M)(N · (E ·∇)M).

(3.80)

Therefore the full expressions for the STT torques applied to the characteristicfields are:

TSTTN =− 16N2ϵ

αD(3N2 + ϵ2)2 (E · ∇)N +8(ϵ2 − N2)

πα2D(3N2 + ϵ2)2

N × (E · ∇)M

−32ϵ(ϵ2 + 2N2)(ϵ2 − J2

sdN2)

π2α3D(N2 + ϵ2)2(3N2 + ϵ2)2

N × (M × (E ·∇)M),(3.81)

TSTTM =+

16ϵ

αD(3N2 + ϵ2)2 M × (N × (E · ∇)N) +8(ϵ2 − N2)

πα2D(3N2 + ϵ2)2

M × (E · ∇)M +8(N × M)

π2α2D(N2 + ϵ2)(3N2 + ϵ2)

(N · (E · ∇)M).(3.82)

3.7 The Gilbert Damping contribution

The last contributes we consider are the so called Gilbert damping terms, thesecome from the time derivatives of the Neel vector and of the magnetization. Asfor the previous quantities, they can be obtained through susceptibility tensorsUαβ and Umαβ. These tensors are given by:

Uαβ = d2p

(2π)2 Tr

GA sα(GR − GA)

σβ 00 −σβ

− (GR − GA)sαGR

σβ 00 −σβ

,

Umαβ = d2p

(2π)2 Tr

GA sα(GR − GA)

σβ 00 σβ


σβ 00 σβ

.(3.83)

As usual, disorder averaged propagators and dressed vertices are introduced andthe Dyson’s equation in HM is applied, consequently the diagrams that one hasto evaluate are at zero and first order in the perturbation originated by the mag-netization. Their pictorial illustration can be observed in Fig.(3.12). The explicitform of the diagrams, that are shown in the figure, is: d2p

(2π)2 Tr

GAΓα(GR − GA)

σβ 00 −σβ

− (GR − GA)ΓαGR

σβ 00 −σβ

. (3.84)

51


d2p(2π)2 Tr

GAHMGAΓαGRΓβ + GAΓαGRHMGRΓβ

. (3.85)

Figure 3.12. Diagrammatic expansion for the computation of Uαβ. The expansionof Umαβ differs for having in the second vertex τ0 instead of τz.

At zero order, the terms with two retarded propagators GRGR or two ad-vanced ones GAGA do not vanish inside the integral, this is very helpful becauseotherwise logarithmic divergences would arise. The first order term, instead, isas usual.Similarly to what has been done in the computation of the STT, we start findingthe Gilbert damping terms for ΓA and then for ΓB. The expressions of the tensorsobtained with calculation can be, once again, simplified summing null matrices.These null matrices are originated, in this case, from the time-derivatives of N2

and M2 (that are supposed to be zero). In particular for the Neel vector we have:

∂tN2 = Nx ∂tNx + Ny ∂tNy + Nz ∂tNz = 0, (3.86)

and once again this corresponds to three null matrices, one for each direction:

TortNx =

Nx Ny Nz0 0 00 0 0

, TortNy =

0 0 0Nx Ny Nz0 0 0

, TortNz =

0 0 00 0 0

Nx Ny Nz

.(3.87)

These (3x3) matrices can be labelled in term of two coefficients, named α and βthat run on (1,2,3). As before, it is possible to write a set made up of analogousmatrices with the magnetization that can be summed to Umαβ

52


We start by considering the zero order tensors for the vertex ΓA, the simplifica-tions we perform in this case are:

U0 Aαβ = U0 A

αβ − ∑i

4[(M × N)i + παDϵNi]

αD(N2 + ϵ2)TortN

i ,

Um0 Aαβ = Um0 A

αβ − ∑i

4ϵπNi

(3N2 + ϵ2)TortM

i .(3.88)

At this point let us then look at the torque for MA (just for completeness even inthis case TGD

0 A = sGD0 A × MA) :

TGD0 A =− 4πϵN2

3N2 + ϵ2 ∂tN +4πϵ

3N2 + ϵ2 M × (N × ∂tN) +4(ϵ2 − N2)

αD(3N2 + ϵ2)

(N + M)× ∂tM +4πϵ

3N2 + ϵ2 N × (N × ∂tM)− 4(N2 + 3ϵ2)

αD(N2 + ϵ2)(3N2 + ϵ2)

(N × M)N · ∂tM.

(3.89)

Let us repeat the same procedure when we use the vertex ΓB, the simplifica-tions we have to do in this case are:

U0 Bαβ = U0 B

αβ + ∑i

4[(M × N)i + παDϵNi]

αD(N2 + ϵ2)TortN

i ,

Um0 Bαβ = Um0 B

αβ + ∑i

4ϵπNi

(3N2 + ϵ2)TortM

i .(3.90)

Therefore its Gilbert damping term is:

TGD0 B =+

4πϵN2

3N2 + ϵ2 ∂tN +4πϵ

3N2 + ϵ2 M × (N × ∂tN)− 4(ϵ2 − N2)

αD(3N2 + ϵ2)

(N − M)× ∂tM

+4πϵ

3N2 + ϵ2 N × (N × ∂tM)− 4(N2 + 3ϵ2)

αD(N2 + ϵ2)(3N2 + ϵ2)

(N × M)N · ∂tM.

(3.91)

The zero order torques acting over N and M are given, once again, by thedifference and the sum of the previous two:

TGD0 N =

12(TGD

A − TGDB ) = − 4πϵN2

N2 + ϵ2 ∂tN +4(ϵ2 − N2)

αD(3N2 + ϵ2)N × ∂tM, (3.92)

53


TGD0 M =

12(TGD

A + TGDB ) =

4πϵ

3N2 + ϵ2 M × (N × ∂tN) +4(ϵ2 − N2)

αD(3N2 + ϵ2)M × ∂tM

+4πϵ

N2 + ϵ2 N × (N × ∂tM)− 4(N2 + 3ϵ2)(N × M)

αD(N2 + ϵ2)(3N2 + ϵ2)N · ∂tM.

(3.93)

The same scheme as before is applied as regarding the terms at first order in theexpansion of the magnetization. Let us start considering the case of ΓA:

U1 Aαβ = U1 A

αβ − ∑i

16N2(M × N)i + 4παDϵ(3N2 + ϵ2)N i

αD(N2 + ϵ2)(3N2 + ϵ2)TortN

i , (3.94)

In such a way one can obtain the expression of the first order terms:

TGD1 A =

16ϵ(ϵ2 − N2)

πα2D(N2 + ϵ2)(3N2 + ϵ2)

N × (M × ∂tM)

16ϵ2

αD(N2 + ϵ2)(3N2 + ϵ2)(N × M)N · ∂tM.

(3.95)

Similarly, in the case of ΓB one has:

U1 Bαβ = U1 B

αβ + ∑i

16N2(M × N)i + 4παDϵ(3N2 + ϵ2)N i

αD(N2 + ϵ2)(3N2 + ϵ2)TortN

i , (3.96)

Then, its first order torque becomes:

TGD1 B =− 16ϵ(ϵ2 − N2)

πα2D(N2 + ϵ2)(3N2 + ϵ2)

N × (M × ∂tM)

+16ϵ2

αD(N2 + ϵ2)(3N2 + ϵ2)(N × M)N · ∂tM.

(3.97)

The first order terms of the characteristic fields read:

TGD1 N =

16ϵ(ϵ2 − N2)

πα2D(N2 + ϵ2)(3N2 + ϵ2)

N × (M × ∂tM), (3.98)

TGD1 M = +

16ϵ2

αD(N2 + ϵ2)(3N2 + ϵ2)(N × M)N · ∂tM. (3.99)

Therefore, by adding the zero order and the first order contributions, thetorques for the Neel vector and the magnetization are:

TGDN =− 4πϵN2

N2 + ϵ2 ∂tN +4(ϵ2 − N2)

αD(3N2 + ϵ2)N × ∂tM

+16ϵ(ϵ2 − N2)

πα2D(N2 + ϵ2)(3N2 + ϵ2)

N × (M × ∂tM),(3.100)

54


TGDM =

4πϵ

3N2 + ϵ2 M × (N × ∂tN) +4(ϵ2 − N2)

αD(3N2 + ϵ2)M × ∂tM

+4πϵ

N2 + ϵ2 N × (N × ∂tM)− 4(ϵ2 − N2)(N × M)

αD(N2 + ϵ2)(3N2 + ϵ2)N · ∂tM.

(3.101)

Again we have to include the contribution of the lower Hamiltonian, in thiscase, however, it turns out that the Gilbert damping terms are exactly equal there-fore the effect is just to double the previous coefficients:

TGDN =− 8πϵN2

N2 + ϵ2 ∂tN +8(ϵ2 − N2)

αD(3N2 + ϵ2)N × ∂tM

+32ϵ(ϵ2 − N2)

πα2D(N2 + ϵ2)(3N2 + ϵ2)

N × (M × ∂tM),(3.102)

TGDM =

8πϵ

3N2 + ϵ2 M × (N × ∂tN) +8(ϵ2 − N2)

αD(3N2 + ϵ2)M × ∂tM

+8πϵ

N2 + ϵ2 N × (N × ∂tM)− 8(ϵ2 − N2)(N × M)

αD(N2 + ϵ2)(3N2 + ϵ2)N · ∂tM.

(3.103)

55

Chapter 4

AFM Square Lattice

4.1 The model

a

Figure 4.1. Pictorialrepresentation of thesquare lattice created bythe A and B atoms.

The other model we consider is, instead, a squareAFM system. The lattice, represented in Figure4.1, is created, even in this case, by two kind ofatoms, named A and B, that have almost oppositelocal magnetizations. The two types of atoms canbe seen as belonging to two square lattices, whoseunit cell length is equal to

√2 a, that interpene-

trate so that A atoms are exactly at the center ofthe squares created by the B atoms. Consequentlyan A atom has only B neighbours and viceversa, inaddition the interatomic distance will be equal to a.As in the previous model, we adopt the s − d inter-action approach, so the d band electrons give riseto the localized spin polarizations (once again forboth the A and B atoms), whereas the s band elec-trons hop between the sites. Therefore the Hamil-tonian of this system will turn out to be:

H =− J1 ∑<n,m>

SAn∥(r, t) · SB

m∥(r + δm, t) + t ∑<n,m>α

(c†nαcmα + c†

mαcnα)

− Jsd∑nαα′

(SAn (r, t) + SB

n(r + δn, t))c†nασαα′cnα′ + ∑

nc†

nVncn.(4.1)

Only nearest neighbours interactions are taken into consideration because theelectronic orbitals are thought to be well localized. The first term describes the in-teraction between the spin polarization of different atoms and, since the coupling

57

4 – AFM Square Lattice

J1 < 0, an anti-ferromagnetic alignment is favoured. The second term representsthe hopping of the conduction electrons, the third describes the interaction be-tween the local spin polarizations and the conduction electrons.The last term introduces a source of disorder from impurities, we will adopta very simple random Gaussian spin-independent disorder to understand thegeneral behaviour of the system. The disorder correlator is given by: <V(r)V(r′) >= (mτ)−1δ(r′ − r), where τ is the scattering time (again for brevitywe will indicate the product (mτ)−1 with the symbol αD).

It is possible to obtain the evolution law of the localized spin polarizationsthrough the Heisenberg equation:

dSA(r, t)dt

= −∑nn

J1SA(r, t) · SB(r + δnn, t) + JsdSA(r, t)× ∑αα′

c†rασαα′crα′ . (4.2)

Since the dynamic of the localized spins is very slow in comparison with theone of the conduction electrons in most of the magnetic metals, the local spinpolarizations S(r, t), in a small volume, can be transformed into classical magne-tization fields Mk(r, t), where k runs on the number of sub-lattices. The Mk(r, t)fields do not vary much from one lattice site to another, and they are very smoothwith respect to the lattice parameter a. Hence, it is possible to perform a Taylorexpansion to account for the nearest neighbours distances:

MA(r + δnn) = MA(r) + δnn · ∇MA(r) +12(δnn · ∇)2MA(r). (4.3)

Replacing the Taylor expansion inside Eq.(4.2), the linear terms, that depend on∇Mk, drop in a square lattice as the neighbours have exactly opposite coordi-nates. thus, only the zero order and second order terms remain.It is more practical to introduce the characteristic fields, the linear combinationof the sublattices magnetizations Mk(r, t). Specifically, these fields are the Neelvector, that as we know is the AFM order parameter, and the magnetization. TheNeel vector is given by the difference of the Mk(r, t), whereas the magnetizationby the sum:

N(r, t) = 1/2(MA(r, t)− MB(r, t)), (4.4)M(r, t) = 1/2MA(r, t) + MB(r, t)). (4.5)

electrons on the sites.It is necessary to point out that in this model the two characteristic fields are takenwith a fixed modulus, in accordance to our previous way of proceeding, conse-quently some constrains are valid: N2 = cost, M2 = cost and N · M = 0. As

58

4.1 – The model

before, the modulus of the magnetization is assumed to be much smaller than theone of the Neel vector.The evolution equation characteristic field, the so-called Landau-Lifshitz-Gilbertequations, are determined by the difference and the sum of the evolution equa-tions of the sublattices magnetization. they turn out to be:

∂N∂t

=− 4J1M × N + J1a2N ×∇2M − J1a2M ×∇2N + JsdM × (sA − sB)

+ JsdN × (sA + sB),(4.6)

∂M∂t

= J1a2M ×∇2M − J1a2N ×∇2N + JsdM × (sA + sB)− JsdN × (sA − sB),(4.7)

where sA/Bα are the spin polarization density of the conduction electrons in corre-

spondence of the atomic sites.It is possible to express the whole magnetic subsystem through a a classical freeenergy functional, that is dependent only on the characteristic fields:

F = J1

dr−4M2 + 4N2 + a2(∇M)2 − a2(∇N)2

. (4.8)

In order to determine the time-evolution of the fields under the passage of cur-rent, it is necessary to analyse the transport of the conduction electron in the AFMso that it is possible to reconstruct the spin polarization density. This will be doneby looking at the remaining terms of the Hamiltonian:

Hcond = −t ∑<n,m>α

(c†nαcmα + c†

mαcnα)− ∑nαα′

c†nαM i · σαα′cnα′ . (4.9)

By applying this Hamiltonian on the wave function of the conduction electrons,it is possible to obtain the eigenvalues of the system; once again the energies incorrespondence of the A and B sites are supposed to be equal:

MA · σΨAr − t ∑

α

ΨBr+δα

= ϵΨAr ,

MB · σΨBr − t ∑

α

ΨAr−δα

= ϵΨBr .

(4.10)

Using a packet of Block’s wavefunction of different wavevector k, one can de-compose the wave-function as:

Ψir = ∑

kφik(r)eikr . (4.11)

59


By replacing this expression inside Eq.(4.10) and neglecting the δα dependenceinside φik(that is supposed to be small) one gets:

MA · σφAk(r)− t ∑α

eikδα φBk(r) = ϵφAk(r),

MB · σφBk(r)− t ∑α

eikδα φAk(r) = ϵφBk(r).(4.12)

The neighbours in a square lattice are in the positions:

δ1,2 = (±a,0) and δ3,4 = (0,±a) . (4.13)

Then, by performing the summation one gets:

δ1

δ2

δ3

δ4

ζ1

ζ2ζ3

ζ4

Figure 4.2. Pictorial representation of the neighbouring atoms and therespective distances

MA · σφAk(r)− 2t(cos(kxa) + cos(kya))φBk(r) = ϵφAk(r),MB · σφBk(r)− 2t(cos(kxa) + cos(kya))φAk(r) = ϵφBk(r).

(4.14)

So the Hamiltonian in the basis of its eigenstates reads:

Hcond =

MA · σ −2t(cos(kxa) + cos(kya))

−2t(cos(kxa) + cos(kya)) −MB · σ

. (4.15)

Since the magnetization is rather small compared to the Neel vector in anAFM, its interaction with the local spin polarization is treated as a perturbation:

Hcond = H0 + M · στ0. (4.16)

60

4.1 – The model

The unperturbed hamiltonian is then:

H0 =

N · σ −2t(cos(kxa) + cos(kya))

−2t(cos(kxa) + cos(kya)) −N · σ

. (4.17)

By looking for the eigenvalues, one can reconstruct the energy dispersion inthis system, depicted in Figure 4.3. There are two bands, that are well sepa-rated at k = 0 and get close at the edge of the Brillouin zone, where there isa gap created by the order parameter. In our case, we will suppose that theFermi surface lies towards the bottom of the first band, then only the lower bandhas accessible states at the Fermi energy because the parameter t is very large.

Figure 4.3. Bands structureof the AFM in the unit cell.

Thanks to this choice, the Fermi energy corre-sponds to a region of low wavevectors, andthen it is possible to perform an expansion ofthe cosine term. The Hamiltonian now looks:

H0 =

N · σ −2t + p2

2m

−2t + p2

2m −N · σ

. (4.18)

As a consequence, the dispersion of the lowerband is:

ϵ = −

p2

2m− t2

+ N2. (4.19)

Some transformations permit to visualize thisdispersion in a better way: first the fact thatthe parameter t is large allows to expand thesquare root, in addition it turns useful to shiftthe zero of energy at the bottom of the band.In such a way, one obtains a parabolic disper-sion with effective mass m plus a correctioninversely proportional to t, that then is rathersmall:

E =p2

2m− N2

2t. (4.20)

It is relevant to mention that all the results of the computations will be obtainedin the limit of the Fermi energy, the magnitude of the fields much smaller thanthe tight-binding constant t. Thus, all the terms featuring once of these quantitiesdivided by t will be dropped.

61


Our purpose is to compute the spin polarization density sα(r, t) in order to findthe torques to the Neel vector and the magnetization. Recalling that in the meanfield approach E(r, t), M(r, t) and N(r, t) are slow, the non equilibrium spin po-larization density sα(r, t) can be obtained again as:

sα =KαβEβ + RγδαβEβ∂δNγ + Rmγδ

αβEβ∂δMγ+

Uαβ∂tNβ + Umαβ∂tMβ + ...(4.21)

The terms appearing in the equation and their functions are logically the sameas in the previous model. Higher order terms in the expansion will be neglectedbecause they represent fine correction to the overall behaviour of the system.

4.2 The Self Energy

In order to proceed in the analysis of this model, it is necessary to determine thepropagators of the conduction electrons, as they enter in the susceptibility ten-sors of Eq.(4.21). Therefore, one needs to compute the self-energy, that enablesto compute the modifications of the physical quantities of the system introducedby the disorder. To perform this computation we will adopt, even in this model,the self-consistent Born approximation (SCBA) (whose details can be found in theAppendix).We remind that the SCBA permits to find the ”bare” contributions of the physicalquantities of the system, like the conductivity, the STT and so on. However, it ispossible to reconstruct the dressed ones by including the vertex corrections. Letus then start looking at the self energy in the SCBA. Since we have seen, in theprevious model, that the effect of its real part is to produce a redefinition of thephysical quantities of the system that eliminates the logarithmic divergences; wecan avoid looking again at this procedure supposing to have already performedthe renormalization. As a consequence, it is sufficient to focus on the computa-tions of the imaginary part of the self-energy:

ℑ(Σ) = παD

dp2

(2π)2 (GR − GA) and GR/A = [ϵ − Hcond − ΣR/A]−1, (4.22)

where the propagators GR/A depend on ΣR/A themselves. Notably, the advancedand retarded self-energies differ only for the sign of their imaginary part.In order to solve this self consistent equation one needs to choose the correctshape of the self energy; for this system what will happen is that every termacquires an imaginary part (excluding naturally the momentum). Since the mag-netization is treated like a perturbation, we rewrite the propagator in terms of

62


the one independent on the magnetization: G = G + G(HM + ΣR/AM )G, where

ΣR/AM is the part of the self-energy that contains the magnetization and GR/A =

[ϵ − H0 − ΣR/A]−1 is the unperturbed propagator.Hence, by solving the integral (keeping both the zero and first order terms in themagnetization) and comparing the final expression of the self energy with theinitial choice; one is capable of finding the expression of the imaginary parts ofthe physical quantities. It turns out that the imaginary parts of the Neel vectorand of the magnetization are inversely proportional to the half-gap t. As we haveassumed that t is very large, both the characteristic fields remain real. Thus, therenormalized expressions of the physical quantities are:

ϵR/A = ϵ′ ± iπmαD

2, NR/A = N′ MR/A = M′ and tR/A = t′ ± iπmαD

2.(4.23)

For sake of brevity and clearness, the primed quantities, that we remember arethe true values of the physical quantities, will be written unprimed.

Let us now pass to consider the modification to the vertices originated by theinclusion of the interaction, as before the analysis is performed looking initially atthe spin vertices sA and sB then at the velocity. Starting with sA, the contributionthat one needs to analyse is again given by Fig.(4.4).

Figure 4.4. Diagrammatic expression of the dressed vertex sA

In the same manner of the previous method, let us consider the case of havingonly one interaction line, in such a case the diagram is simply given by:

VAα = αD

d2p(2π)2 GA

σα 00 0

GR. (4.24)

We expect that even this vertex can be rewritten via a combination of Pauli ma-trices, so like:

VAα = Uαβ

σβ 00 0

+Wαβ

0 00 σβ

, (4.25)

63


where α and β run on (x,y,z). The off-diagonal terms have not been included be-cause their coefficients are trivial even in this model.The values of the elements of the matrices Uαβ and Wαβ can be determined by re-placing, in each case, the previous expressions inside Eq.(4.24) and then by eval-uating the closed loops:

Figure 4.5. Diagrammatic expression of the computation done to determine theelements of Uαβ. The elements of Wαβ are obtained with Pauli matrices in position2-2 instead of 1-1 as appears in the picture

We recall that the computation consists on one side in evaluating the traces ofproducts between Pauli matrices, whereas on the other to solve an integral. As aresult of the computations, we obtain simply for both the matrices:

Uαβ = Wαβ =14

δαβ. (4.26)

The two expressions are very easy: it is the identity matrix multiplied by a properconstant. We can now pass to consider the effect of the dressing of more interac-tion lines:

Vα + αD

d2p(2π)2 GAVαGR + αD

d2p′

(2π)2 GA

αD

d2p(2π)2 GAVαGR

GR + ...(4.27)

Every Vα can be expressed through the combination of Pauli Matrices, whoseintegrals are well known. Nevertheless, since in this case there are two termsinside the vertex VA

α at each step the number of terms doubles:

Uαβ

σβ 00 0

+Wαβ

σβ 00 0

+ (UαβUβγ +WαβWβγ)

σγ 00 0

+ (UαβWβγ +WαβUβγ)

σγ 00 0

+ ...

(4.28)

Logically, the summations of the terms appearing in the expression can be per-formed only if these terms are multiplied to Pauli matrices in the same positions.

64


It is clear that this operation, even if it is not too complex in this case, it is notso practical. For this reason, we decide to proceed in another way. This differentstrategy could be particularly useful in cases of more complex expressions of Uαβ

and Wαβ matrices. In order to follow this different method, we directly pass toconsider the vertex VB, obtained dressing one interaction line to the spin vertexsB.

VBα = αD

d2p(2π)2 GA

0 00 σα

GR. (4.29)

Again, we expect that it is possible to rewrite the vertex in terms of a combinationof Pauli matrices, so as:

VBα = Uαβ

0 00 σβ

+Wαβ

σβ 00 0

. (4.30)

Since the calculations are very similar to those performed above, not surprisinglywe obtain exactly the same result:

Uαβ = Wαβ =14

δαβ. (4.31)

Thanks to this similarity, we can express the results for the two spin vertices in avery compact way. In order to do so, we have to introduce a new matrix:

Oαβ =

Uαβ Wαβ

Wαβ Uαβ

. (4.32)

Then the first order spin vertices are:

VA

α

VBα

= Oαβ

σβ 00 0

0 00 σβ

. (4.33)

This way of proceeding is more comfortable because, as it can be noticed, thevertex correction at first order can be written through a unique expression. In topof this, even the dressing of further interaction lines can be accounted in a verypractical way. For example the two second order terms inside Eq.(4.28) appear inthe first line of:

OαβOβγ

σγ 00 0

0 00 σγ

. (4.34)

65


It is then clear that the fully dressed vertex for both sA and sB is given by:

ΓA

α

ΓBα

= (δαβ +Oαβ +OαγOγβ +OαγOδβ +Oδβ + ...)

σβ 00 0

0 00 σβ

. (4.35)

By replacing Oαβ with its value and performing the calculations, we obtain:

ΓAα =

32

σα 00 0

+

12

0 00 σα

ΓB

α =12

σα 00 0

+

32

0 00 σα

(4.36)

Figure 4.6. Diagrammatic expression of the dressed vertices ΓAα and ΓB

α .

Another quantity that is typically modified by the dressing of interaction linesis the velocity. In this system the velocity is:

vx =

0 p

m cos(ϕ)σ0pm cos(ϕ)σ0 0

, vy =

0 p

m sin(ϕ)σ0pm sin(ϕ)σ0 0

.(4.37)

In order to determine the changes due to the dressing, it is necessary to considerinitially only one interaction line as usual:

αD

d2p(2π)2 GAvαGR. (4.38)

The velocity vertex at first order can be expressed through a linear combinationof the zero order velocities. However, in this case the coefficients of the linearcombination turn out to be all trivial, therefore the bare velocity is the velocity ofthe system.

66

4.3 – The Spin Orbit Torque

4.3 The Spin Orbit Torque

Since we know the shape of the dressed operators, we can proceed to determinethe response functions of the system. In particular we start considering the spin-orbit torque. The susceptibility tensor Kαβ given by the Kubo-Streda formula:

Kαβ = dp2

(2π)2 TrsαGRvβGA

, (4.39)

where sα are again the spin vertex, either on the A or B sites.

sAα =

σα 00 0

and sB

α =

0 00 σα

(4.40)

The vβ are the components of the velocity and GR/A are again the retarded andadvanced propagators.The study of the SOT, in such a system, can be performed just using symmetryarguments. The SCBA is not necessary for these considerations, because the sym-metries we look at are valid for the full propagator.As first step, it is relevant to study the symmetries of the propagator under theexchange of the components of the Neel vector:

σz 00 σz

G[±p]

σz 00 σz

= G[−Nx,−Ny],

σx 00 σx

G[±p]

σx 00 σx

= G[−Nz,−Ny],

σy 00 σy

G[±p]

σy 00 σy

= G[−Nz,−Nx].

Similar relations can be obtained for the velocities:σz 00 σz

vx/y

σz 00 σz

= vx/y,

σx 00 σx

vx/y

σx 00 σx

= vx/y,

σy 00 σy

vx/y

σy 00 σy

= vx/y.

All these relations permit to test whether the tensor Kαβ is symmetric or an-tisymmetric under the exchange of two components of Neel vector. To performthis control, one flips two components of the Neel vector. The propagators willbe modified in accordance to the first table and one has to introduce the corre-spondent relation inside Eq.(4.39). Operating like this we obtain, in the generalcase: dp2

(2π)2 Trsα

σj 00 σj

GR

σj 00 σj

vβ

σj 00 σj

GA

σj 00 σj

, (4.41)

67


At this point, we can transform the components of the velocity operator basedupon the relations shown in second table and we can use the relations betweenthe Pauli matrices to simplify the remaining ones. Moreover, in the case in whichwe use the relation valid for −p, it is useful to modify the integral on the mo-menta through p → −p, this will give rise to a change of sign coming from thevelocity as everything else is dependent only on the modulus of p . In such a way,we can find relations that are equal to the initial expressions apart from an even-tual sign factor. Since the symmetry relations used are valid both in the case inwhich the momentum is kept fix or it is flipped, one can obtain two different re-lations for the same flip of two component of Neel vector. The relations differ fora sign factor. However, the susceptibility tensor is independent on the directionof the momentum as we are performing an integration over it. The only possibleexplanation is that the tensor Kαβ is all vanishing.The triviality of the tensor Kαβ can be further confirmed by analysing, in a betterway, all the terms inside Eq.(4.39). By the fact that the Hamiltonian depends onthe square modulus of the momentum p, also the advanced and retarded prop-agators have the same dependence; differently the velocity depends directly onp itself, thus its component have an angular dependence. Given that the compo-nents of the velocity introduce a phase term, the integration over the angle φ willalways turn out to be 0, that is equivalent to affirm that Kαβ is trivial.In any way, as a check, the calculations of the tensor have been performed in theSCBA and these confirmed that the tensor Kαβ is effectively null. The fact thatthe SOT is all vanishing in this system is justifiable for the same reason as in theprevious model, indeed even inside this Hamiltonian the spin-orbit interactionsare not present.

It is useful to compute even the conductivity tensor in this system, so that itis possible to relate the variations of the Neel vector not only to the electric fieldbut even to the current density J, where J = σE. The conductivity is given by theKubo formula:

σαβ = dp2

(2π)2 Tr[vαGRvβGA]. (4.42)

It is developed the expansion of the propagator, using Dyson’s equation, to in-clude the effect of the magnetization, then the expanded expression of the con-ductivity that has been used is:

σαβ = dp2

(2π)2 Tr[vαGRvβGA] + dp2

(2π)2 Tr[vαGRMR · στzGRvβGA

+ vαGRvβGAMA · στzGA].(4.43)

68

4.4 – The Spin transfer Torque

The conductivity is directly obtained performing the integration above; no dress-ing operations are done because the vertex corrections on the velocities are nulls.The computation leads to:

σ =

− 2EF

mαD0

0 −2EFmαD

=

−2EFτ 0

0 −2EFτ

, (4.44)

where we reintroduced the scattering time. The conductivity in such a way getsa very simple expression that depends only on the Fermi energy and on the scat-tering time.

4.4 The Spin transfer Torque

The response functions, through which it is possible to determine the spin-polarizations,are again:

Rγδαβ =− i

d2p(2π)2 Tr

sαGRvγGR

σδ 00 −σδ

GRvβGA

+ i d2p(2π)2 Tr

sαGRvβGA

σδ 00 −σδ

GAvγGA

.

(4.45)

Rmγδαβ =− i

d2p(2π)2 Tr

sαGRvγGR

σδ 00 σδ

GRvβGA

+ i d2p(2π)2 Tr

sαGRvβGA

σδ 00 σδ

GAvγGA

.

(4.46)

In the computation, we use the disorder averaged propagator, so we have to re-place the vertices with the dressed ones, that have been previously found. More-over, the calculations are performed up to first order in the magnetization. Atfirst, we look only at the zero order contributions, that are obtained replacing theG’s with the unperturbed G. The computations are performed for both the ver-tices, to determine the evolution of MA and MB.We start looking at the result in the case of ΓA. The dressed vertex is introducedinside Eq.(4.45) and Eq.(4.46), through which we determine the spin polarizationlinked to the STT in accordance to Eq.(4.21). Since we are more interested in theexpression of the torques, than the one of the spin polarization, we look at theresult after having already performed the external product with the local magne-tization:

TSTT0 A = (M + N)× sSTT

0 A = − 1m2πα2

D(M + N)× (E ·∇)M. (4.47)

69


By proceeding in the same manner, we get for ΓB:

TSTT0 B = (M − N)× sSTT

0 B = − 1m2πα2

D(M − N)× (E ·∇)M. (4.48)

The STT for N and M are given, respectively, by the difference and the sum ofthose of the two local spin polarizations, then:

TSTT0 N =

12(TSTT

0 A − TSTT0 B ) = − 1

m2πα2D

N × (E ·∇)M, (4.49)

TSTT0 M =

12(TSTT

0 A + TSTT0 B ) = − 1

m2πα2D

M × (E ·∇)M. (4.50)

The first order terms are given by the first order term in the Dyson’s equationfor the perturbation due to the magnetization; this means that in Eq.(4.45) andEq.(4.46) one of the propagators G has to be sobsituted with: G → GM · στzGwhereas the others have to be replaced with G. In order to understand the effectof this substitution in a better way, we can look at the expression of Rγδ

αβ at firstorder:

− i d2p(2π)2 Tr

ΓαGRMR · στzGRvγGR

σδ 00 −σδ

GRvβGA+ ΓαGRvγGRMR · στzGR

σδ 00 −σδ


σδ 00 −σδ

GRMR · στzGRvβGA+ ΓαGRvγGR

σδ 00 −σδ

GRvβGAMA · στzGA

+ h.c.

(4.51)

It is very clear that Rmγδαβ at first order differs only for being related to variations

of the magnetization, instead of the Neel vector. The first order terms, once per-formed the calculations and the external product with the correspondent localmagnetization, turn out to be:

TSTT1 A = (M + N)× sSTT

1 A = − 1m3π2α3

DN × (M × (E ·∇)M). (4.52)

By proceeding in the same manner, we get for ΓB:

TSTT1 B = (M − N)× sSTT

1 B = +1

m3π2α3D

N × (M × (E ·∇)M). (4.53)

Then the first order terms for the characteristic fields are:

TSTT1 N =

12(TSTT

1 A − TSTT1 B ) = − 1

m3π2α3D

N × (M × (E ·∇)M), (4.54)

70


TSTT1 M =

12(TSTT

1 A + TSTT1 B ) = 0. (4.55)

Therefore the full expressions for the STT torques applied to the characteristicfields at first order in the magnetization are:

TSTTN = − 1

m2πα2D

N × (E ·∇)M − 1m3π2α3

DN × (M × (E ·∇)M), (4.56)

TSTTM = − 1

m2πα2D

M × (E ·∇)M. (4.57)

4.5 The Gilbert Damping contribution

The last contributes we consider are the Gilbert damping terms, determined bythe associated susceptibility tensor Uαβ and Umαβ. These link, respectively, thespin polarization to the time variation of the Neel vector and of the magnetiza-tion: sGD

α = Uαβ∂tNβ + Umαβ∂tMβ. Specifically, the expressions of the tensorsare:

Uαβ = d2p

(2π)2 Tr

GAsα(GR − GA)

σβ 00 −σβ


σβ 00 −σβ

,

Umαβ = d2p

(2π)2 Tr

GAsα(GR − GA)

σβ 00 σβ


σβ 00 σβ

.(4.58)

As before, the disorder averaged propagators are used, then the dressed verticeshave to be introduced in the expression of the susceptibility tensors. Dyson’sequation is applied to account for the effect of perturbation due to the magneti-zation. By the fact that even for this kind of torque only the terms up to first orderin M are relevant, the expansion is stopped at first order. For completeness andclarity, we rewrite the expression of the first order corrections that appear insideUαβ: d2p

(2π)2 Tr

GAMA · στzGAΓαGRΓβ + GAΓαGRMR · στzGRΓβ

. (4.59)

As the susceptibility tensors allow to compute the spin polarization throughEq.(4.21), by performing the external product with the local magnetization oneobtains the expression of the GD term torques. We directly look at the expressionin which both zero order and first order terms are included.The GD term referred to ΓA is:

TGDA =(M + N)× sGD

A =2

αD(M + N)× (E ·∇)M − 2

mπα2D

N × (M × (E ·∇)M). (4.60)

71


Whereas, for ΓB:

TGDB =(M + N)× sGD

B =2

αD(M − N)× (E ·∇)M +

2mπα2

DN × (M × (E ·∇)M). (4.61)

Therefore the GD terms of the characteristics fields are:

TGDN =

12(TGD

A − TGDB ) =

2αD

N × (E ·∇)M − 2mπα2

DN × (M × (E ·∇)M),(4.62)

TGDM =

12(TGD

A + TGDB ) =

2αD

M × (E ·∇)M. (4.63)

72

Chapter 5

Conclusions

The study of the structure of the torques obtained in the two models allows tounderstand what will be the trend of the evolution of the characteristic fields. Forthis reason, we make a summary of the expressions of the torques obtained in thetwo models. First of all, we remind that the SOT is trivial in both the cases andtherefore is of no interest in these models. As a consequence, we focus on theGilbert Damping term and on the STT.However, in the expressions previously written we kept hidden some constants:the area, the coupling constant Jsd and the Fermi velocity. Actually, the last twohave been taken into consideration in a redefined momentum and in redefinedcharacteristic fields. By making explicit these constants and replacing α with itscorrespondent value, we obtain for the first model:

TSTTN =

Jsd

v2F(3J2

sdN2 + ϵ2)2Ae

16τ J2sdϵ2N × (N × (E · ∇)N) +

8Jsd(ϵ2 − J2

sdN2)ϵ2τ2

πv2F

N × (E · ∇)M+

+32J2

sdϵ4τ3(ϵ2 + 2J2sdN2)(ϵ2 − J2

sdN2)

π2v4F(J2

sdN2 + ϵ2)2N × (M × (E ·∇)M)

,

(5.1)

TSTTM =

JsdAev2

F(3J2sdN2 + ϵ2)2

16τ J2

sdϵ2M × (N × (E · ∇)N) +8Jsd(ϵ

2 − J2sdN2)τ2ϵ2

πv2F

M × (E · ∇)M+

+8J3

sdτ2ϵ2(ϵ2 − J2sdN2)

πv2F(J2

sdN2 + ϵ2)N × (N × (M × (E · ∇)M))

,

(5.2)

TGDN =

JsdA(3J2

sdN2 + ϵ2)v2F

8πϵJ2

sdN × (N × ∂tN) +8Jsdϵτ(ϵ2 − J2

sdN2)

v2F

N × ∂t M+

+32J2

sdϵ3τ2(ϵ2 − J2sdN2)

πv4F(J2

sdN2 + ϵ2)N × (M × ∂t M)

,

(5.3)

73

5 – Conclusions

TGDM =

JsdAv2

F(3J2sdN2 + ϵ2)

8πϵJ2

sd M × (N × ∂tN) +8Jsdϵτ(ϵ2 − J2

sdN2)

v2F

M × ∂t M+

+ 8πϵN × (N × ∂t M) +8J3

sdτϵ(ϵ2 − J2sdN2)

v2F(J2

sdN2 + ϵ2)N × (N × (M × ∂t M))

.

(5.4)

A first glance of the results shows, quite clearly, that in antiferromagnetic sys-tems the magnetization and the Neel vector are deeply related one to another.This is the main reason for which we have decided to study a non collinear anti-ferromagnetic system. As a consequence, the dynamic of the system is, generally,much more complex that in a ferromagnetic material for the interplay of the twofields that give rise to not so simple equation of motion.We can notice that all the terms obtained have the correct internal symmetry, in-deed all the terms acting on the Neel vector are odd in the exchange of the twosublattices magnetizations, whereas all the terms acting on the magnetization areeven.Analysing with more care the STT applied to the Neel vector, we notice that itsfirst and its last term are reactive tensors (in the same direction as the externalfield), while the middle term is a dissipative torque. Whereas, in the case of theSTT on the magnetization, only the first one is reactive and the others are dissi-pative.As a further investigation, we can compare the results found for this model withthose proposed by Velkov et al [30]. We observe that there are only two terms inaccordance with those proposed by them. In particular, they are the first termsof the two STT, that are characterized by having the same coefficient (because intheir article N2 is set to 1). The other terms proposed by them are absent, thisis highly probably due to the fact that their coefficients are small in the disorderparameter. However, in our research other terms do appear, that have not beenconsidered in their work. An other aspect we notice, that appears even fromtheir work, is that the terms of the the STT acting over the magnetization can beobtained from the one of the Neel vector exchanging the last characteristic fieldand viceversa (the last terms of both Eq.(5.1) and Eq.(5.2) only apparently not re-spect this symmetry; indeed the corresponding symmetric terms cancel as theyor too small in the order of the magnetization or trivial exploiting the vectorialrelations).

74

Passing to consider the second model, we introduce back once again the miss-ing constants, here we obtain:

TSTTN = −Ae

J2sdτ2

πN × (E ·∇)M +

τ3 J3sd

π2α3 N × (M × (E ·∇)M)

, (5.5)

TSTTM = −Ae

τ2 J2sd

πM × (E ·∇)M

, (5.6)

TGDN = A

2τm∗ J2

sdN × ∂t M −2τ2m∗ J3

sdπ

N × (M × ∂t M)

, (5.7)

TGDM = A

2τm∗ J2

sd M × ∂t M

. (5.8)

Even in this case, the Neel vector and the magnetization are linked one to another,however the relations here are less complex. We can notice that the STT acting onthe order parameter has a first term dissipative, whereas the second is reactive.On the other hand, the STT on the magnetization is purely dissipative, however,as it does not depend on the order parameter, we could expect that this torque israther small. Differently from the first model, we do not actually find any of thetorques proposed by Velkov et al. However, at the same time the torques foundhave a structure that is the same of some of the terms obtained in the first model,confirming the possibility of having this kind of evolution.The results of this model are very simple, this is surely originated by the strongapproximation performed. Thus, we actually think that the results obtained areless interesting with respect to the one of the other model, on one side for theapproximation taken and on the other for the fact that the honeycomb lattices arefar more promising for the realization of AFM systems [38][39].A general feature, that can be observed in the results of both the models, is thatthere is a clear similarity in the structure of the STT and of the GD. Indeed, if onetransforms all the terms in the GD by replacing ∂t with (E ·∇) is able to find aterm exactly equivalent in the characteristic field dependence for almost all theterms (only one term found in the GD in the first model does not satisfy this char-acteristic). On top of this, even the ratio between the GD coefficients and the STTones is basically constant. Since it is evident that this similarity is valid in boththe models, we expect that it is linked to a more general feature of the system.However, this property is largely unclear therefore further investigations in thisdirection are necessary.

75

5 – Conclusions

In this work the calculation of the dressed diagrams has been done in the noncrossing approximation, however this can give rise to mistakes in the correct es-timation of the physical quantities. In order to understand why this is the case, itis necessary to analyse the dressing of the interaction in a deeper way.In general the interaction lines can be originated by single particle scatterings ormultiple particle scatterings. Since in this work it has been adopted the Bornapproximation, scatterings from individual impurity is absent, indeed the cross-section of the single particle is symmetric [37]. The only source of disorder is thepair scatterings at distances of the order of the Fermi wavelength, that even ifrare, produce the skew scattering diagrams.In our work we decided to consider only diagrams in which the impurity lines

Figure 5.1. Additional crossing diagrams in Kαβ originated by the skew scattering.

do not cross, because, as it is proved in the Appendix, the crossing diagrams con-tributes to the physical quantities with terms of one order of magnitude smallerin the scattering time τ. However, a proper quantum-mechanical treatment of thepairs scattering by impurities requires to take into account even these diagrams,that can be observed in the specific case of the response function linked to theSOT in Figure 5.1 (for the other response functions the impurity crossing presentis analogous). These diagrams represent an inherent part of the skew scatteringand, actually, they are indistinguishable from the other part of skew-scattering,that has been used to determine the dressed vertex.If we go back to the results of our enquire, we can notice that there are terms thatdiffer for one order of magnitude in the disorder parameter. Since the crossingdiagrams of the leading quantities produce corrections of the same magnitude ofthe non-leading terms, it is clear that it is necessary to include the crossing di-agrams to have the correct behaviour of the non-leading terms. Therefore, theonly the leading terms in the disorder parameter are fully reliable, the others arewrong as a matter of the coefficients but nevertheless we think that they havethe correct dependence on the characteristic fields (this indeed was, as alreadystated, the aim of this research). Naturally an investigation of the effects of thecrossing diagrams is a very interesting proposal to further develop the study ofthese models.

76

Another way to improve the study would be to consider the Rashba spin-orbit in-teraction, as it is very promising in AFM lattices [38] [39]. We expect that the firstconsequence of its introduction would be the appearance of spin-orbit torques,that, as we have seen, otherwise cannot be generated in these models. Indeed, asthe current induces an electron flow the Rashba field can polarise the travellingelectrons along its direction and normal to the electric field. As the electron enterin the AFM material, they will act on its characteristic fields generating torques.We expect that STT will still be present, even if modified for the appearance ofnew terms in its expression due to the Rashba term. However, the main trigger,we believe, will be the SOT that differs from the STT for its orientation and forbeing independent on the spatial mistracking of the magnetization. It is relevantto notice that SOT are observable even if the current is initially unpolarised, so nospin filter is actually necessary.

77

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81

Appendix

Self Consistent Born Approximation

This Appendix is meant to illustrate the theoretical basis upon which it is builtthe Self Consistent Born Approximation, for reference see [40]. The Born Ap-proximation consists in rewriting the self-energy of the system through only oneself-energy diagram, neglecting all the diagrams in which the impurity line cross.As first thing let’s consider the expression for the non crossing diagram:

ΣNCAR (p, E) = ni

dp′

(2π)3 |Vimp(p − p′)|2GR(p′, E), (A.1)

or in a diagrammatic form:

Figure A.1. Self Energy in the non crossing approximation

This is an implicit expression, because the propagator is dependent on theself-energy itself. In order to extimate the effect of the impurity, it is possible tocompute the non crossing self energy in the case of the free propagator:

G0 R(p, E) = −i

dt θ(t)eiEte−iϵpt, with ϵp =p2

2m. (A.2)

Exploiting that θ(t) = 1/(iω) + πδ(ω), it is possible to insert the previous ex-pression into Eq.(A.1).

Σ0R(p, E) =− ini

dp′

(2π)3 |Vimp(p − p′)|2

dt

dω

1

iω+ πδ(ω)

eit(−ω+E−ϵp). (A.3)

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Bibliography

The first term is real while the second one creates an imaginary part that isproportional to the inverse of τ(p, E), the momentum relaxation time:

1τ(p, E)

= πni

dp′

(2π)2 |Vimp(p −√

2mEp′)|2. (A.4)

The impurity averaged propagator then reads:

GR(p, E) =1

E − ϵp −ReΣ0R(p, E) + i/τ(p, E)

. (A.5)

The main interest is for the case in which the energies and the momenta are large,in particular E ≃ EF >> 1/τ and p ≃ pF >> 1/l (where l = vFτ is the impu-rity mean free path). In addition the potential is assumed to have a range muchshorter than the mean free path. In such a regime the real part of the self-energyhas only the effect of producing a shift in the energy scale, because all the relevantquantities of the system (the density of states, the impurity potential...) are essen-tially constant. A good choice for the potential, that it will be the one adoptedin this research, is to take a Gaussian average potential, in such a case it occursthat the real part of the self-energy is divergent. However this is not a problembecause in such a case it is possible to perform a re-normalization of the energyin order to absorb the divergence.At this stage if one considers the first iterated propagator in Eq.(A.5), this turnsout to be peaked at the Fermi momentum. The iterated propagator can be usedto compute the imaginary part of the self energy at first order in the iteration, ifone performs the computation he can notice that the result remains unchangedto order of 1/E f τ. It is then possible to conclude that in general:

ΣR(p, E) = − iτ

. (A.6)

The self-energy, therefore, can be completely re-expressed through the momen-tum relaxation time and, as one can notice, in the Born approximation the self-energy turns out to be independent on the momentum. The impurity averagedpropagator simply becomes:

< G0R(p, (p′, E) >=

1E − ϵp + i/τ

δp,p′ . (A.7)

The results have been extended in the case of a general propagator, because forp ≃ pF all the propagators are proportional to the free one:

GR(p, t) = dE

dte−iEt 1

E − ϵp + i/τ= G0

R(p, t)e−t/τ . (A.8)

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In order to confirm the validity of the Born approximation it is possible to controlthe value of the other skeleton diagrams, once again in the limit of large energiesand momenta. It is evident that if these diagrams result to be much smaller than1/τ they can be easily neglected.The impurity averaged propagator is large in the region for p ≃ pF, therefore themain contributions to a diagram in which there are internal momentum integra-tion comes from the particles (in a metal the electrons) close to the Fermi surface.Let us initially look at a multiple scattering, for example a fourth order diagram:

Figure A.2. Four impurity scattering diagram.

ni

dp1

(2π)3

dp2

(2π)3

dp3

(2π)3 Vimp(p − p1)Vimp(p1 − p2)Vimp(p2 − p3)

Vimp(p3 − p1)GR(p1, E)GR(p2, E)GR(p3, E).(A.9)

One of the integrals can be turned into a self energy contributions, while theother two are integrations over internal momenta, can be estimated as: dp

(2π)3 GR(p, E) ∼ iN0(EF) ∼ mpF. (A.10)

Then one gets that the four impurity scattering diagram is approximatively givenby:

Figure A.3. Estimation of the four impurity scattering.

By the fact that the potential has a range of the order of the mean free path, theBorn criterion assures p3

FVimp(p = 0) << EF. By performing the ratio between

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Bibliography

this diagram and the single impurity line one, it comes out that the multiple scat-tering term is small if the Born criterion is valid:

Vimp(p = 0)pF

vF

2

<<

EF

vF

2 1p2

F∼ 1. (A.11)

The other class of skeleton diagrams that are present inside the self energy arethose with crossed impurity lines, for example:

Figure A.4. Crossing diagram

n2i

dp1

(2π)3

dp2

(2π)3 |Vimp(p − p1)|2|Vimp(p − p2)|2GR(p1, E)

GR(p1 + p2 − p, E)GR(p2, E).(A.12)

The intermediate momentum integration is not free in the case in which one con-siders the large contributions, for which the energy and the momenta are close tothe Fermi ones. In order to have all the momenta in a thin shell of extension 1/laround pF, one of the angular integrations is restricted to a cone of angle 1/l pF.As a consequence, the crossed diagram will be result to be smaller of such a factorand it is consequently typically neglected.It is interesting to point out that the self consistent Born approximation is a per-turbation theory in the quantity 1/l pF ∝ 1/τ, that is linked to the scattering time.Including crossing diagrams, then corresponds to take into account terms smallerin the scattering time.

Gaussian Approximation

In the SCBA it is possible to neglect multiple scatterings, therefore it is clear thatin such an approximation performing an impurity average corresponds to tie po-tential vertices pairwise together, through impurity correlators, in all possibleways. It is possible to choose the impurity average as a Gaussian average withthe potential mean value equal to zero < V(x) >= 0. It is evident that only av-erages with an even number of vertices will contribute, because otherwise there

86

Bibliography

will always be a spare line.The general average on the impurity potential will be given by the sum on allpossible permutations that couple two vertices together:

<V(x1)V(x2)...V(x2N)>=∑P

<V(xP(1))V(xP(2))> ... <V(xP(2N−1))V(xP(2N))> . (A.13)

In particular it will be adopted a delta correlated potential: < V(x)V(x′) >=(m∗τ)−1δ(x − x′). It is possible, for this choice, to rewrite the self-energy in thecase of the Born approximation. As first thing let us consider the self energy in aGaussian approximation in real space:

ΣR(x, x′, E) =< Vimp(x)Vimp(x′) > GR(x, x′, E). (A.14)

Now let transform the expression passing in momentum space and substitutingthe shape of the correlator:

ΣR(p, p′, E) =(m∗τ)−1

dDx

dDx′eipxe−ip′x′δ(x − x′) dDq(2π)D e−iq(x′−x′)GR(q, E). (A.15)

By exploiting the delta functions one gets:

ΣR(p, p′, E) =(m∗τ)−1

dDx ei(p−p′)x′ dDq

(2π)D GR(q, E) =

(m∗τ)−1 dDq

(2π)D GR(q, E)δp,p′ .(A.16)

Correctly the self-energy does not directly depend on the momentum, howeverthe momentum conservation is assured. Then the self energy can be simply writ-ten, in the SCBA as:

ΣR(E) = (m∗τ)−1 dD p

(2π)D GR(p, E). (A.17)

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