1. Cornell University
Applications of Magnon Excitations in
Spintronic and Optomagnetic Devices
Emeka V. Ikpeazu, Jr.
ECE 5370—Nanoscale Device Physics
Professor Sandip Tiwari
May 13, 2016

2. 1
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 =
𝒑̂1
𝟐
2𝑚0
+
𝒑̂2
𝟐
2𝑚0
−
2𝑒2
4𝜋𝜀0
(
1
| 𝒓1|
+
1
| 𝒓2|
)
⏟
Central field
+
𝑒2
4𝜋𝜀0| 𝒓1 − 𝒓2|⏟
Electron interaction
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 variable 𝜌( 𝐫, 𝑚 𝑠) and an eigenfunction
for which

3. 2
𝜓( 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 wavefunction 𝜓 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
𝜑 𝑆 = 𝜓 𝑆=𝑀 𝑆=0 =
{
𝜙 𝑠( 𝐫1, 𝐫2) =
1
√2
[ 𝜑( 𝐫𝟏) 𝜑( 𝐫𝟐)+ 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:
𝜑 𝑇 = 𝜓 𝑆=1;𝑀𝑆=−1,0,1 =
{
𝜙 𝑎( 𝐫1, 𝐫2) =
1
√2
[ 𝜑( 𝐫𝟏) 𝜑( 𝐫𝟐)− 𝜑( 𝐫𝟐) 𝜑( 𝐫𝟏)]
𝜁 𝑠( 𝐬1, 𝐬2) =
{
|↑⟩1|↑⟩2
1
√2
[|↑⟩1|↓⟩2 − |↓⟩1 |↑⟩2]
|↓⟩1|↓⟩2
.
Altogether, taking either the triplet or singlet wavefunction inner product with the centralfield
Hamiltonian will yield the same energy. The same is true of the electrostatic Coulombic repulsion
Hamiltonian. This not true, however, of the energy exchange integral which gives us energy splitting as

4. 3
𝐸𝑒𝑒
𝑆
= ⟨𝜑 𝑥𝑠
𝑆 ( 𝐫1, 𝐫2)|ℋ̂ 𝑒𝑒|𝜑 𝑥𝑠
𝑆 ( 𝐫1, 𝐫2)⟩ = 𝐼 + 𝐽 and 𝐸𝑒𝑒
𝑇
= ⟨𝜑 𝑥𝑠
𝑇 ( 𝐫1, 𝐫2)|ℋ̂ 𝑒𝑒|𝜑 𝑥𝑠
𝑇 ( 𝐫1, 𝐫2)⟩ = 𝐼 − 𝐽
In this way,
𝐸𝑒𝑒
𝑆
− 𝐸𝑒𝑒
𝑇
= 2𝐽 = Δ𝐸 and 𝐸𝑖𝑗 = −2∑ 𝐽𝑖𝑗 Si ∙ 𝑆j
𝑖<𝑗
,
where Si ∙ Sj =
1
4
if spins are symmetric as in the triplet case and Si ∙ Sj = −
3
4
if spins are
antisymmetric as in the singlet case.
The conditions under which the exchange energy is at its minimum depends on the sign
of 𝐽𝑖𝑗. If 𝐽𝑖𝑗 is positive, then the value of 𝐸𝑖𝑗 is at its minimum when Si ∙ Sj =
1
4
and the spins are
aligned; likewise, if 𝐽𝑖𝑗 is negative, then the value of 𝐸𝑖𝑗is at its minimum when Si ∙ Sj = −
3
4
and
the spins are anti-aligned. For a more rigorous understanding, consider the following:
ℋ̂ 𝑒𝑒( 𝐫1, 𝐫2) =
𝑒2
4𝜋𝜀0
[
1
| 𝐫1 − 𝐫2|
+
1
| 𝐑1 − 𝐑2|
−
1
| 𝐫1 − 𝐑1|
−
1
| 𝐫2 − 𝐑2|
]
2𝐽 = 2 ∫ 𝑑3
𝑟1 𝑑3
𝑟2[ 𝜑 𝑇 𝜑𝑆 ]ℋ̂ 𝑒𝑒( 𝐫1, 𝐫2)[ 𝜑 𝑇
∗
𝜑𝑆
∗] → 𝐽 = ⟨ 𝑈 𝐸⟩ + 𝑈ec − ⟨ 𝑈1⟩ − ⟨ 𝑈2⟩,
where
⟨ 𝑈 𝐸⟩ is the average interaction energy between the electrons,
𝑈ec =
𝑒2
4𝜋 𝜀0| 𝐑1−𝐑2|
is the interaction energy between the two electron clouds,
⟨ 𝑈1⟩ is the average Coulombic (electrostatic) energy of the first electron cloud, and
⟨ 𝑈2⟩ is the average Coulombic (electrostatic) energy between the second electron cloud.
If the average interaction energy, ⟨ 𝑈 𝐸⟩ + 𝑈ec⏟ is greater than the average Coulombic
energy ⟨ 𝑈1⟩ + ⟨ 𝑈2⟩⏟ , then the 𝐽 is positive and the electrons are more correlated. In this way, a

5. 4
positive value of 𝐽 favors alignment between the electron spin. Likewise, energy coupling is
weaker when 𝐽 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 where 𝐽 > 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.

6. 5
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 charge 𝑄 wherein
Φ 𝐸 = ∯ 𝐄 ⋅ 𝑑𝐀
𝑆
= 𝐸𝑟(4𝜋𝑟2) =
𝑄
𝜀0
→ 𝐸𝑟 =
𝑄
4𝜋𝜀0 𝑟2
.
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

7. 6
microstates Ω and thus the entropy 𝑆 as
𝑆 ∝ ln Ω
The 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
at 𝑇 = 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.

8. 7
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 of 𝜔 = 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 energy 𝐽 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 value Δ𝐸 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 entropy 𝑆
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 where 𝒫𝑖 ∝ 𝑒
−
𝐸 𝑖
𝑘 𝐵 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

9. 8
excitations will take 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 as 𝜂—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

10. 9
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.

11. 10
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 × (−γ𝐇 𝐞𝐟𝐟 + 𝛼
𝜕𝐦
∂𝑡
)
where 𝛾 is the gyromagnetic ratio and 𝛼 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

12. 11
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

13. 12
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.

14. 13
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

15. 14
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 of 𝐻 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

16. 15
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 permeability 𝝁̅ where 𝑩( 𝜔) = 𝝁̅𝑯( 𝜔). The angle of rotation 𝜃F is expressed:
𝜃F = 𝒱𝐵𝑑,
where
𝒱 is the material’s Verdet constant,
𝐵 is the magnetic flux density applied to the material,
and 𝑑 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 product 𝑬( 𝜔)× 𝑬( 𝜔)∗
. If we
then consider the case of an external field whose period 2𝜋ΩLO
−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.

17. 16
Figure 8. Stimulated Raman scattering occurs so that a magnon
of energy ℏ𝜔2 = ℏΩ 𝑚 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 ∙ 1015
rads ∙ 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.

18. 17
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.

19. 18
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

20. 19
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.

21. 20
References
1. Schlosshauer, Maximilian (2005). "Decoherence, the measurement problem, and
interpretations of quantum mechanics". Reviews of Modern Physics 76 (4): 1267–1305.
arXiv:quant-ph/0312059
2. "Louis Néel - Nobel Lecture: Magnetism and the Local Molecular Field". Nobelprize.org.
Nobel Media AB 2014. Web. 10 May 2016.
http://www.nobelprize.org/nobel_prizes/physics/laureates/1970/neel-lecture.html
3. Kruglyak V.V, Demokritov S.O, Grundler D Magnonics J. Phys. D Appl. Phys. 43
264001 (2010), doi:10.1088/0022-3727/43/26/264001
4. Chumak, Andrii V., V.I. Vasyuchka, A.A. Serga, and B. Hillebrands. "Magnon
Spintronics." Nature. Nature Physics, 21 Apr. 2015. Web. 12 May 2016.
<http://www.nature.com/nphys/journal/v11/n6/full/nphys3347.html>.
5. Schneider, T., Serga, A. A., Neumann, T., Hillebrands, B. & Kostylev, M. P. “Phase
reciprocity of spin-wave excitation by a microstrip antenna”. Phys. Rev. B 77, 214411
(2008).
6. Jamali, M., Kwon, J. H., Seo, S. M., Lee, K. J. & Yang, H. “Spin wave nonreciprocity for
logic device applications”. Sci. Rep. 3, 3160 (2013).
7. Di, K. et al. “Enhancement of spin-wave nonreciprocity in magnonic crystals via
synthetic antiferromagnetic coupling”. Sci. Rep. 5, 10153; doi: 10.1038/srep10153
(2015).
8. A. V. Kimel, A. Kirilyuk, and T. Rasing, Laser Photonics Rev. 1, 275 (2007).
9. A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pisarev, A. M. Balbashov, and Th.
Rasing Nature 435, 655 (2005).

22. 21
10. Nanoscale Device Physics, by Sandip Tiwari.
11. Bossini, D. et al. Macrospin dynamics in antiferromagnets triggered by sub-20
femtosecond injection of nanomagnons. Nat. Commun. 7:10645 doi:
10.1038/ncomms10645 (2016).