Cornell University

Applications of Magnon Excitations inSpintronic and Optomagnetic Devices

Emeka V. Ikpeazu, Jr.ECE 5370—Nanoscale Device Physics

Professor Sandip TiwariMay 13, 2016

Abstract

In the field of emerging technologies, spintronics and opto-magnetics are two areas that

are gaining ground, particularly with continuing study of magnons. Magnons are the quanta of

spin waves that occur in crystal lattices, similar to acoustic phonons. Spin is the discretized

quantum mechanical analog of angular momentum in classical systems. Spin waves propagate in

magnetic lattices that possess continuous symmetry in the same way that phonons propagate in

nuclear lattices. Magnons arise when materials lose their magnetization properties, i.e. when

they are heated to a temperature above the Curie point—of which more anon—and the material’s

spin states become disordered even when there is no external magnetic field to impel such

behavior. In this paper, I will discuss how the effects of magnons in materials and metamarials

alike can be exploited for use in modern and emerging devices.

Introduction

Magnons are understood at a quantum mechanical level of analysis. For this reason, we

start with a generic Hamiltonian in order to build up to our understanding of spin waves and their

properties. Consider the following Hamiltonian of a Helium atom with an atomic number of

Z=2:

H=p1

2

2 m0+

p22

2 m0− 2e2

4 π ε 0 ( 1|r1|

+ 1|r2|)⏟

Central field

+ e2

4 π ε0|r1−r2|⏟Electroninteraction

The goal is to find an eigenfunction that is dependent on the spin and position observables—for

reasons to be soon explained—that is invariant under particle exchange, one that maintains

exchange symmetry. We than craft a coordinate mapping variableρ (r , ms ) and an eigenfunction

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for whichψ ( ρ1 , ρ2 )=ψ1 ( ρ1) ψ2 ( ρ2 )=ψ1 ( ρ2 ) ψ2 ( ρ1 )Here, we introduce the Pauli exclusion principle.

In regards to electrons, the Pauli exclusion principle says the following:

In the case of electrons, it can be stated as follows: it is impossible for two electrons of a

poly-electron atom to have the same values of the four quantum numbers: n, the principal

quantum number, ℓ, the angular momentum quantum number, mℓ, the magnetic quantum number,

and ms, the spin quantum number.

This means that electrons in the same orbital must have opposite spins. More rigorously put,

electrons must be antisymmetric with respect to the spin observable if they are symmetric with

respect to the position observable, and vice versa. For this reason the wavefunctionψ must be

deconstructed into symmetric and antisymmetric spin and positional components. These states

will take the form of a singlet, with one eigenfunction, and a triplet, with three eigenfunctions.

The singlet—spatially symmetric and antisymmetric in spin—will take the form

φS=ψ S=M S=0={ϕs (r1 , r2 )= 1√2 [φ (r1 ) φ (r 2)+φ (r2 ) φ (r1 ) ]

ζ a ( s1 , s2 )= 1√2

[|↑ ⟩1|↓ ⟩2−|↓ ⟩1|↑ ⟩2 ]The triplet—spatially antisymmetric and

symmetric in spin will take one of three forms:

φT=ψ S=1 ;M S=−1,0,1={ϕa (r1 , r2 )= 1√2 [φ (r1 ) φ (r 2)−φ (r 2) φ ( r1 ) ]

ζ s ( s1 , s2 )={ |↑ ⟩1|↑ ⟩21√2

[|↑ ⟩1|↓ ⟩2−|↓ ⟩1|↑ ⟩2 ]|↓ ⟩1|↓ ⟩2

.

Altogether, taking either the triplet or singlet wavefunction inner product with the central field

Hamiltonian will yield the same energy. The same is true of the electrostatic Coulombic repulsion

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Hamiltonian. This not true, however, of the energy exchange integral which gives us energy splitting as

EeeS =⟨φxs

S (r1 , r2 )|H ee|φxsS (r 1, r2 )⟩=I+J∧Eee

T = ⟨φxsT (r1 ,r2 )|H ee|φxs

T (r1 , r2 )⟩=I−J In this way,

EeeS −Eee

T =2J=Δ E∧E ij=−2∑i< j

J ij S i ∙ S j ,whereSi ∙ S j=14 if spins are symmetric as in the triplet

case andSi ∙ S j=−34 if spins are antisymmetric as in the singlet case.

The conditions under which the exchange energy is at its minimum depends on the sign

ofJ ij. IfJ ij is positive, then the value ofEij is at its minimum whenSi ∙ S j=14 and the spins are

aligned; likewise, ifJ ij is negative, then the value ofEijis at its minimum whenSi ∙ S j=−34 and the

spins are anti-aligned. For a more rigorous understanding, consider the following:

H ee (r1 ,r 2)= e2

4 π ε0 [ 1|r1−r2|

+ 1|R1−R2|

− 1|r1−R1|

− 1|r 2−R2|]

2J=2∫ d3 r1d3 r2 [ φT φS ] H ee (r1 , r2 ) [φT¿ φS

¿ ] → J=⟨ UE ⟩+U ec−⟨ U 1 ⟩−⟨ U2 ⟩ ,where

⟨ U E ⟩ is the average interaction energy between the electrons,

U ec=e2

4 π ε0|R1−R2| is the interaction energy between the two electron clouds,

⟨ U 1 ⟩ is the average Coulombic (electrostatic) energy of the first electron cloud, and

⟨ U 2 ⟩ is the average Coulombic (electrostatic) energy between the second electron cloud.

If the average interaction energy, ⟨ U E ⟩+U ec⏟ is greater than the average Coulombic energy

⟨ U1 ⟩+ ⟨U 2 ⟩⏟, then theJ is positive and the electrons are more correlated. In this way, a positive

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value ofJ favors alignment between the electron spin. Likewise, energy coupling is weaker

whenJ is negative as the electrons are mostly interacting with their respective clouds.

Spin alignment is indicative of correlations between electrons in a material. The anti-

alignment is indicative of the lack of correlation between electrons. Intuitively, one might

attribute the lack of spin alignment to strictly thermodynamic effects, but anti-alignment is

achievable under conditions where such effects are negligible2. Such effects are very powerful

as they produce novel properties in structures. In the case whereJ>0, the subsequent alignment

results in a phenomenon known as ferromagnetism.

Figure 1. Ferromagnetic ordering due to spin alignment

However, in the opposite case, where electrostatic interaction is greater than the energy

exchange interaction, spins will tend to align in opposite directions. This is because the electrons

will tend to be localized to their clouds, decreasing the correlation between neighboring electrons

in the material. The emerging phenomenon here is anti-ferromagnetism. Both spin alignment

and disalignment are examples of ordered magnetism.

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Figure 2. Anti-ferromagnetic ordering due to spin alignment

What has been apparent so far is that electron-electron correlations and electron

localization have become central to a continued discussion of this topic. This contrasts with the

free and/or nearly free electron model that is assumed in linear combination of atomic orbitals

(LCAO) models, semiconductor physics, and transistor fundamentals.

Symmetry

Symmetry, an invariance principle, is fundamental to discussions of not only magnon behavior,

but also phonon, plasmon, and other quasiparticle excitation behavior. This is because these

excitations will be produce by the spontaneous breaking of certain symmetries. An example of

symmetry would be the spherical symmetry of Gauss’s law applied to a hypothetical point

particle of chargeQ wherein

ΦE=∯S

E ⋅ⅆ A=E r (4 π r2 )=Qε0

→ E r=Q

4 π ε0r 2 .Perfect symmetry assumes that the system is

subject to no perturbation. Intuitively, one would think that the thermal agitation, which

produces disorder, would be indicative of broken symmetry but this is in fact false. The increase

in disorder would be an increase in the number of microstatesΩ and thus the entropyS as

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S∝ ln ΩThe increase in microstates leads to symmetry as the amount of randomness increases.

In this way there is symmetry in every direction one looks; it is not organized symmetry in all

directions was would be the case in a Bravais lattice. The noise and disturbance terms

disorganize and envelop modal excitations. The third law of thermodynamics, which states that

the entropy of a perfect crystal at absolute zero is exactly equal to zero, shows that symmetry

breaking occurs atT=0 K . This idea is further evinced in the quantum mechanical principle of

decoherence. In large aggregations of particles, thermodynamic irreversibility forces the

dephasing of particle ensembles. This is also known as heat bath and it deprives the system of

directionality; the system is stripped of coherence. This is the significance of the Curie’s

temperature, above which ordered magnetism more or less ceases.

Figure 3. Rabi oscillations decay faster as temperature increases.

Here a shift from coherence is taking place for temperatures on the

order of K. In the Fourier domain, one can easily see that

decreasing the temperature also decreases the bandwidth.

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For Rabi oscillations, optical oscillations in two-level systems, thermodynamic coupling with the

heat bath will force the atom into a symmetric state. In the frequency domain, this corresponds

to a widening of the bandwidth around a central frequency ofω=0.

As was explained in the introduction, a strong interaction energy increases the coupling.

This interaction energy can take the form of thermodynamic coupling with the environment

which leads to a loss of coherence. This coupling, however, can be intra-atomic or

intramolecular; even in a periodic structure like a crystal symmetry is severely diminished as the

constituent atoms of such a structure become more coupled to one another. This is the case, as

was previously explained, for the exchange energyJ whose being nonzero denotes the breaking

of symmetry and whose being zero denotes the forming of symmetry.

Goldstone’s Theorem

Goldstone’s theorem, named for physicist Jeffrey Goldstone, says that continuous symmetry

breaking in close range interactions produces collective excitations that have no gap, or no

positive valueΔ E above which they are excited. Referring back to the previous section with the

example of a crystal at absolute zero, we see that there is only one unique way for the entropyS

to equal zero. This is to say that the ground state is the only truly unique nondegenerate state of

a system.

Per the Boltzmann distribution wherePi∝e−Ei

kB T , high-energy thermal fluctuations will be

statistically nonexistent, especially under the conditions where symmetry breaking, continuous or

spontaneous, is expected to take place. In this way the only excitations one needs to consider for

this analysis are those lying at the ground state, hence the gaplessness of the excitations. At just

a slight deviation from the ground state—whatever form that may take—the excitations will take

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place due to the interaction and coupling between various particles in the solid. Here, solidity is

an illusion as particles in nanoscale condensed matter systems exhibit strange behaviors that

betray the appearance of their emergent structure.

Goldstone’s theorem is especially important in understanding phase transitions10. Phase

transitions are commonly associated with temperature. In the first paragraph of this section I

even gave the example of such a transition wherein a crystal had no entropy as it was at absolute

zero. However, temperature is not the only variable to consider when discussing phase

transitions and it is in here where the concept of an order parameter enters. An order parameter

—commonly denoted asη—is the operative parameter for a system undergoing a phase

transition, the variable whose change in value is driving the phase transition. For water boiling

on a hot stove the order parameter would be temperature. For a material undergoing

crystallization, the order parameter would be the density of said system. In the context of

ordered magnetism as discussed earlier, the order parameter for ferromagnetism is the

magnetization and the order parameter for anti-ferromagnetism is the staggered magnetization.

The latter parameter is a type of magnetization where spin anti-alignment is favored.

If we consider phonons, for example, what we see are excitations in periodic crystal

lattices. While rigidity is an apparent feature of crystals, this feature is entirely nonexistent on

the atomic and subatomic scales. On this scale, neighboring atoms interact with each other in

such a way that they hold each other to their respective lattice points. The expected

displacement of one or more atoms from respective lattice points stimulate the pulling forces of

its neighbors to its equilibrium position. This process initializes the start of lattice excitations in

the crystal as its constituent atoms are now exhibiting oscillatory behavior. One significant thing

to remember in understanding collective excitations is that they can be approximated

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independent and identically distributed interactions between atoms affected by the breaking of a

certain symmetry. This is important because this phenomenon emerges due to a series of

interactions between its parts. It is a convenient approximation to make as it relieves us of the

burdens of the many-body as applied to quantum mechanics. However, there exist certain

materials where this approximation is not always valid, namely strongly correlated materials. In

the context of magnetic phase transitions which break discrete symmetries, this approximation is

made with Ising model and can be fairly accurate.

Magnons

Magnons are excitations that arise from small perturbations from the ferromagnetic ground state.

These perturbations show up in the form of deviations from the spin alignment. The Ising model

allows us to get treat two neighboring particles as one system whose spin inner product is either

¼ or ¾ depending on whether or not the spins are aligned. However, the Ising model does not

particularly work spin wave excitations. The Ising model only allows for discrete symmetry, a

spin binary wherein there are no slight perturbations. In this model particles are either spin up or

spin down. If we recall from the abstract, magnons arise from the breaking of continuous

symmetry, not discrete symmetry. The breaking of continuous spin symmetries in the system

produces individual sites of magnetization around which a sort of Larmor precession takes place.

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Figure 4. Larmor precession is the precession of a magnetic moment

about a magnetic field. In the case of magnons, each site produces an

effective magnetic field known as the mean field.

The spin waves are thus the result of perturbative precessions around a localized mean

field. The contribution to the magnetic moment comes primarily from spin as the total angular

momentum J ≈ S. In this way we can model the spin halves using the Landau-Lifshitz-Gilbert

(LLG) equation such that

∂ m∂ t

=m×(−γ H eff +α ∂m∂ t )whereγ is the gyromagnetic ratio andα is the Gilbert damping

parameter.

This equation as applied to spin waves is indicative of a oscillatory excitation that

simultaneously decays with time. The frequencies of precession are usually in the GHz range

but some materials see resonances on the order of THz.

Magnons additionally exhibit properties similar to their analogues in sound and light.

The can scatter, reflect, diffract, and propagate. In the section on Goldstone’s theorem, it was

mentioned that the mathematics of the Boltzmann distribution quells any need to consider

thermal fluctuations. This condition is enforced even more by the fact that this ordered

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magnetism is observed at very low temperatures where T is just a few epsilons above absolute

zero. These low temperatures transfer bosonic excitations to a gaseous state of matter. In this

way Bose-Einstein behavior has been observed in magnetic system under the appropriate

conditions3. Low temperature is not always a prerequisite for this observation as Bose-Einstein

condensation has been observed at room temperature in yttrium garnet3. The applications of the

aforementioned phenomenon—among others—will be discussed in a later section.

Spintronic Magnonic Applications

It would be nearly impossible to discuss the physics of magnons and symmetry breaking in

general with engineers without them asking about its practical application. Right now, electrons

form the basis of modern computational systems. That is, charge flow between transistors and in

CMOS (complementary metal-oxide-semiconductor) circuits forms the basis of binary logical

systems and their subsequent hierarchical realizations. Magnons could come in handy in the

realm of data processing as information can be modeled as functions of the phases of spin waves

as opposed to the accumulation and/or depletion of charge; here, Mach-Zender interferometers

would be fundamental to truly harnessing these excitations for data processing6. Such an

application of magnonics would be less computationally taxing on its respective processor as it

opens the door for the possibility of fuzzy logic systems4.

Ohmic losses contribute to a lack of energy efficiency in conventional electronics. These

losses are caused by a phenomenon known as Joule heating, where the passage of electric current

in a conductor releases heat. Magnonic approaches to computing and information processing

bypass this effect altogether as the gaplessness of the excitation implies reversibility. In this way

magnons can provide a more adiabatic medium for information transmission and communication

systems. A recent investigation into magnon spintronics has shown that one can take advantage

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of not only the spin currents carried by electrons, as is the case in spintronics, but also the spin

waves they carry. The exchange interaction in magnetic material systems consists of two

components that dominate on different lengths scales. The long-range component is the dipolar

interaction that can be easily detected my antennas and other microwave devices. Magnonic

interactions arise from short-range spin interactions. Our interest in magnonic behavior is thus

an interest in the nanoscopic intra-atomic interactions in (anti)-ferromagnetic structures.

Recently, various thin-film materials and garnets are being used to understand

magnonics even more. Yttrium-iron-garnets (YIG, Y3Fe5O12) and ferromagnetic Heusler

compounds, which are crystal structures that exhibit metallic bonding, have been used as sources

for spin wave excitations with low Gilbert damping constants4. The classical approach to

magnon spintronics takes advantage of the more macroscopic effects the excitations. This sort of

analysis would be seen in more microwave applications. As was said before, dipolar spin waves

dominate magnonic behavior on length scales of this sort; however, there are two types of

dipolar spin waves. The first type is the backward volume magnetostatic spin wave (BVMSW),

a field whose direction is both in the plane of the film and parallel to the direction of

propagation. The second type is the magnetostatic surface spin wave (MSSW), a field directed

in the film plane but perpendicular to the direction of propagation4. Here, applying an EM field

to a microwave would excite Larmor precession in the material via the existence of an Ørsted

field in the antenna.

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Figure 5. Experimental setup for spin wave geometries and their

propagation roughly 40ns after excitation. (a) BVMSW geometry

is shown with an Ørsted field. (b) Spatially resolved simulation

of BVMSW propagation. (c) MSSW geometry is shown with Ørsted

field. (d) Spatially resolved simulation of BVMSW propagation5.

In magnonic devices electrical conversion of magnonic excitations is optimal to the

function of such devices.

Figure 6. Cyclical conversion of spin currents to electrical currents is shown. Electrical charge

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and spin currents are the substrate of information medium but those are then converted to

magnon currents and then reconverted4.

Magnons could be instrumental in microwave and THz communication systems as magnonic

crystals allow for the successful propagation of magnons over long distances. Crystal structures

of this sort can fabricated to confine magnon excitations for propagation. Magnonic wave guides

will be discussed in the next section.

Figure 7. Dispersion curves for (a) ferromagnetic (FM) and

(b) antiferromagnetic (AFM) couplings in magnonic crystals

(a=20 nm ) under an applied field ofH o=.06 T

μ0. The frequency (GHz) is

a function of the k-vector (nm-1)7.

Optical Magnonic Applications

Ultrafast lasers and optical systems have allowed researchers to delve into a plethora of

interesting phenomena that occur in magnetic materials. Laser-guided control of magnetism is

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possible, as the angular momentum of circularly polarized light, for example, can affect photons

which share that same polarization. The spins of electrons can also be affected in this way.

It has been demonstrated for almost a decade that ultrafast laser pulses can act as

equivalently powerful magnetic fields and induce opto-magnetic excitations. Moreover, such

excitations occur by way of stimulated Raman scattering8. The magneto-optical Faraday effect

enters in here as the magnetization of a material affects the polarization of light. The material

becomes what is known as a Faraday rotator; in this case the material would expectedly have a

tensorial permeabilityμ whereB (ω)=μ H (ω ). The angle of rotationθF is expressed:

θF=V Bd ,where

V is the material’s Verdet constant,

B is the magnetic flux density applied to the material,

andd is the length of propagation in the material.

The inverse effect can happen as well wherein circularly polarized light can affect the

magnetization of a material. We know that in isotropic media, the external electric field can act

as a magnetic field whose strength is proportional to the tensor productE (ω )× E (ω)¿. If we then

consider the case of an external field whose period2 π ΩLO−1 is faster than both the spin relaxation

time and the precession period and its effect on a non-degenerate ground state singlet, we see

that spin orbit coupling begins to take place as the spin-flip process intensifies. This is because

the energy from the field may not be enough to bridge the gap between state|1 ⟩ and state|2 ⟩. The

results of the this spin excitation will be the emission of a photon with less energy.

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Figure 8. Stimulated Raman scattering occurs so that a magnon

of energyℏω2=ℏΩm is released. The intensification of the

spin-flip process means that photon-to-magnon energy conversion

can take place on very short time scales8.

It has only been in the last decade that the inverse Faraday effect has been captured in

magnetically ordered materials9. A 2005 experiment used 100fs pulses of

1.55 eV (ω=2.355 ∙1015rads ∙ s−1 ) photons at a rate of 1 KHz. In this experiment two pump

beams were guided by different paths onto the same spot on a sample of dysprosium orthoferrite

DyFeO3. Manipulation of the phase delay of one pump beam relative to another allowed for the

direct observation of the time evolution of magnetization in the sample. This can be very useful

for understanding the propagation of mageto-optical signals is communication systems.

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Figure 9.

The phase change due to Faraday rotation in the optical magnetic effect is shown9.

Pulse-shaping for the manipulation quantum interference effects has also caught the eye

of many researchers of the past few years, particularly in the field of biomolecular physics. The

use of holmium orthoferrite (HoFeO3) for laser excitation has resulted in magnon oscillations of

roughly 280 GHz. The success of magneto-photonic conversion in the spin-flip process has

allowed for the possibility of controlled energy flow between biological objects. Here, laser

pulses can be used to steer chemical reactions.

Conclusions

More research is need to properly understand magnons and their behavior under certain

conditions. There are several downsides that cause concern, such has how they—are any

excitation that arises from the breaking of symmetry—can be used as a substrate by which

information is encoded. There is also the issue of how they can propagate without severe

attenuation and under what conditions. Currently there is an issue with photonic computing

wherein they require very cold conditions to prevent heat bath from erasing the hard drive.

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The Faraday effect has been shown to be very useful in communication systems and

shows promise for ultrafast propagation of information over long ranges. Powerful laser pulses

can trigger picosecond phase transitions and be used to manipulate the direction of chemical

reactions. What is the most significant, however, is the speed of the photon to magnon energy

conversion that is seen with the intensification of the spin-flip effect. Here, modal excitation is

not necessary.

A recent study11 showed that nanomagnon injections allow for controllable excitations of

up to 20 THz in strongly correlated materials. The main issue as was elucidated in the previous

paragraph is effective control of spin dynamics and excitations. Laser pulses have been used for

spin wave amplitude modulation. The spintronic applications like spin-based transistors, anti-

ferromagnetic storage media with faster switching times, etc. are also on the horizon for

promising research.

Figure 10. Amplitude of spin wave oscillations is periodic in the

phase delay between to pumps11.

What is clear so far is that strongly correlated systems will be the basis for magnonic devices as

the control of such devices must come from a material system whose constituent parts are

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working together. This will allow for the optical and spintronic effects in such materials to be

more widespread and observable in the technologies in which they are applied.

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