# Magnonics in collinear magnetic insulating systems

B. Flebus  Department of Physics, Boston College, 140 Commonwealth Avenue
Chestnut Hill, MA 02467

###### Abstract

In the last decades, collinear magnetic insulating systems have emerged as
promising energy-saving information carriers. Their elementary collective spin
excitations, i.e., magnons, can propagate for long distances bypassing the
Joule heating effects that arise from electron scattering in metal-based
devices. This tutorial article provides an introduction to theoretical and
experimental advances in the study of magnonics in collinear magnetic
insulating systems. We start by outlining the quantum theory of spin waves in
ferromagnetic and antiferromagnetic systems and we discuss their quantum
statistics. We review the phenomenology of spin and heat transport of the
coupled coherent and incoherent spin dynamics and the interplay between
magnetic excitations and lattice degrees of freedom. Finally, we introduce the
reader to the key ingredients of two experimental probes of magnetization
dynamics, spin transport and NV-center relaxometry setups, and discuss
experimental findings relevant to the outlined theory.

††preprint: AIP/123-QED

##  I Introduction

The possibility of using the electron’s spin as a new degree of freedom for
transmitting information has received significant attention in the last
decades [Spin, ,Spin1, ]. Of particular interest for novel device applications
are magnetic insulating systems [insulator, ,magnonics, ]: solids in which
electrons are “frozen” in their atomic positions while a pure spin current,
i.e., a flow of spin angular momentum, can be transmitted through a wave-like
collective motion of the electrons’ spins. Transmitting signals via spin flows
in insulators has the potential to circumvent the heating and energy loss that
is generated when the electrons collide, e.g., with impurities, as they move
within a crystal lattice. The discovery of the spin Hall effect (SHE) and the
inverse spin Hall effect (ISHE) [SHEref, ; SHEref1, ; SHEref2, ; SHEref3, ;
SHEref4, ; SH1, ; SH2, ], which convert a charge current into a spin current
and vice versa, has further advanced the field by allowing electrical
injection and detection of spin currents in heterostructures comprised of
magnetically ordered insulators interfaced with a normal metal. More recently,
Uchida and co-authors showed that a pure spin flow can be generated in a
magnetic insulator not only via electrical injection, but by heat gradients as
well via the spin Seebeck effect (SSE) [SSE, ], suggesting the possibility of
converting waste heat into spin signals and, thus, opening up intriguing
prospects for “greener” information technology [reviewCal, ].
Experimental observations of electrically- and thermally-generated spin
transport have confirmed the potential of magnetic insulating systems as long-
range information carriers [spindiff1, ; spindiff2, ; spindiff3, ;
AFdifflength, ].

There is a plethora of magnetic insulating systems, which display ground-state
spin configurations ranging from collinear to non-collinear depending on the
intrinsic spin-spin interactions. Non-collinear configurations stem from spin
couplings that can endow the system with a non-trivial topological band
structure and engender exotic spin phases and textures. However, for the
purpose of spin current transmission, these systems are not ideal platforms
due to the lack of spin conservation; thus, the magnetic materials currently
deployed in spin and heat transport setups provide spins that are mainly
collinear.

In this tutorial we aim at introducing the readers to the key properties of
collective spin excitations in collinear ferromagnetic and antiferromagnetic
insulating systems. In section I, we outline the fundamental ingredients of
the theory of spin waves and we review the coupled equations describing the
coherent (long-wavelength) magnetization dynamics and the spin and heat
transport driven by the thermal magnon cloud. Finally, we briefly discuss the
coupling between magnetic and lattice degrees of freedom.

In section II, we introduce spin transport and NV relaxometry setups as probes
of magnetization dynamics and we discuss recent experimental results making
use of the theoretical tools outlined in section I.

##  II  Collinear magnetic insulators

In this section, we introduce the fundamental principles of spin-wave theory
taking as examples monoatomic ferromagnetic and two-sublattice
antiferromagnetic collinear insulators. We discuss the interaction between
their coherent and incoherent spin dynamics, and their coupled spin and heat
transport properties. Finally, we review the coupling between magnetic and
lattice degrees of freedom.

###  II.1 $U$(1)-symmetric ferromagnetic systems

We consider a $U$(1)-symmetric ferromagnetic insulating system. At each $i$th
site of the monoatomic lattice, we can define a spin
$\mathbf{S}_{i}=S\mathbf{n}_{i}$, where $S$ is the classical spin and the unit
vector $\mathbf{n}_{i}$ the spin orientation. At a temperature $T$ far below
the magnetic ordering (Curie) temperature $T_{c}$, we can map spin
fluctuations around the ground state onto second-quantized operators via the
Holstein Primakoff (HP) transformation [Holstein, ]. For a ground state with
spins uniformly oriented along the $-z$-direction, the HP transformation
reads, in the macrospin limit, as

$\displaystyle\hat{S}_{iz}=\hat{a}_{i}^{\dagger}\hat{a}_{i}-S,\;\;\;\;\;\;\;\;\hat{S}_{i-}=\frac{\hat{S}_{ix}+i\hat{S}_{iy}}{2}\simeq\sqrt{2S}\hat{a}_{i}\,,$
(1)

where $\hat{a}_{i}^{\dagger}$ ($\hat{a}_{i}$) is the canonical bosonic second-
quantized operator that creates (annihilates) a magnon carrying spin angular
momentum $\hbar$. In the limit of small excitation amplitude, i.e., for $T\ll
T_{c}$, the magnon-magnon interaction corrections are small
[magnoninteraction, ]. Thus, any term higher than quadratic in the magnon
creation and annihilation operators can be omitted from the magnetic
Hamiltonian $\mathcal{H}_{\text{m}}$, i.e.,

$\displaystyle\mathcal{H}_{\text{m}}=\sum_{\mathbf{k}}\hbar\omega_{\mathbf{k}}\hat{a}^{\dagger}_{\mathbf{k}}\hat{a}_{\mathbf{k}}+E_{0}\,.$
(2)

Here $E_{0}$ is the ground state classical energy, $\omega_{\mathbf{k}}$ the
spin-wave dispersion, and

$\displaystyle\hat{a}_{\mathbf{k}}=\frac{1}{\sqrt{N}}\sum_{i}e^{i\mathbf{k}\cdot\mathbf{r}_{i}}\hat{a}_{i},\;\;\;\;\;\;\;\hat{a}^{\dagger}_{\mathbf{k}}=\frac{1}{\sqrt{N}}\sum_{i}e^{-i\mathbf{k}\cdot\mathbf{r}_{i}}\hat{a}^{\dagger}_{i}\,,$
(3)

where $N$ is the number of lattice sites.

The Hamiltonian $\mathcal{H}_{\text{m}}$ possesses $U$(1) symmetry, i.e.,
$[\hat{S}_{iz},\mathcal{H}_{\text{m}}]=0$, i.e., it is invariant for rotations
in spin space around the $z$-axis. Owing to this symmetry, Noether’s theorem
guarantees that the $z$-component of the spin,
$S_{iz}=\langle\hat{S}_{iz}\rangle$, and, consequently, the total magnon
number,
$\tilde{N}=\sum_{\mathbf{k}}\langle\hat{a}_{\mathbf{k}}^{\dagger}\hat{a}_{\mathbf{k}}\rangle$,
are conserved quantities, where $\langle...\rangle$ stands for the equilibrium
(thermal) average. Such conservation law is, however, an approximation: in
reality, magnon-magnon and magnon-lattice interactions that invalidate spin
conservation are present in any insulating system. Even in the long-wavelength
and low-temperature limit, one must include a dimensionless Gilbert damping
parameter $\alpha$, with $\alpha\ll 1$, in the Landau-Lifshitz-Gilbert
equation [LLG1, ,LLG2, ] in order to account for the experimentally observed
broadening of the ferromagnetic resonance (FMR), i.e.,

$\displaystyle\dot{\mathbf{n}}(\mathbf{r})=-\gamma\mathbf{n}(\mathbf{r})\times\mathbf{H}(\mathbf{r})-\alpha\mathbf{n}(\mathbf{r})\times\dot{\mathbf{n}}(\mathbf{r})\,.$
(4)

Here, $\gamma$(>0) is the gyromagnetic ratio, $\mathbf{H}(\mathbf{r})$ is the
effective (Landau-Lifshitz) field acting on the order parameter
$\mathbf{n}(\mathbf{r})$, with $\mathbf{n}(\mathbf{r})=\mathbf{n}_{i}$ in the
continuum limit. Equation (4) can be solved in the linear regime, i.e., by
neglecting terms quadratic in spin fluctuations, i.e., $n_{i}n_{j}$ for
$i,j=x,y$. By performing a HP and Fourier transformation (1), one can find a
direct correspondence between the linearized classical dynamics (4) and the
second-quantized Hamiltonian via the Heisenberg relation

$\displaystyle\dot{\hat{a}}_{\mathbf{k}}=\frac{i}{\hbar}\left[\mathcal{H}_{\text{m}},\hat{a}_{\mathbf{k}}\right]\,.$
(5)

According to Eqs. (4) and (5), however, non-Hermitian terms violating magnon
number conservation will appear in the Hamiltonian $\mathcal{H}_{\text{m}}$
[FlebusNH, ]. Until recently, such terms have been neglected in the quantum
theory of spin waves; a discussion of such non-Hermicity will be addressed
elsewhere.

Here we focus on magnetic systems with low damping, for which spin
nonconserving terms can be (approximately) neglected. Namely, we discuss the
limit in which the energy scales of dissipative processes are very small
compared to the exchange interactions that control the thermalization of the
magnon distribution function. It has been extensively debated whereas the
elementary quanta of spin waves, i.e., magnons, can be described as quasi-
particles whose density is (approximately) conserved in collinear magnetic
systems with low damping. The assumption of magnon density (quasi-)
conservation carries important consequences. From a statistical standpoint, it
implies the existence of a well-defined magnon chemical potential, and thus of
magnon Bose-Einstein condensation, achievable even at relatively high
temperatures [BEC, ]. Magnon density conservation has also been predicted to
underlie a spin superfluid phase that shares many similarities with
supercurrent of electric charge in superconductors and the mass superflow in
helium [SpinSup, ; SpinSup1, ; Yuan, ].

Recent experiments have shown that the injection of spin current into a
collinear magnet can drive the magnon gas into a quasi-equilibrium state
described by a Bose–Einstein statistics with non-zero chemical potential
[Cornelissen2016, ; chemDem, ; Du2017, ], suggesting that the inclusion of a
chemical potential is necessary to properly capture the experimental features
of incoherent magnon transport. Thus, we will describe thermal magnons as a
(quasi-equilibrium) thermalized Bose-Einstein ensemble with a well-defined
chemical potential $\mu$, i.e.,

$\displaystyle
n_{\text{BE}}\left(\frac{\hbar\omega_{\mathbf{k}}-\mu}{k_{B}T}\right)\,,$ (6)

where $n_{\text{BE}}(x)=\left(e^{x}-1\right)^{-1}$ is the Bose-Einstein
distribution function and $k_{B}$ the Boltzmann constant. The equation
governing the diffusive dynamics of thermal magnons density $\tilde{n}$ can be
written as

$\displaystyle\dot{\tilde{n}}(\mathbf{r})+\bm{\nabla}\cdot\mathbf{j}_{s}(\mathbf{r})=-g_{n\mu}\mu(\mathbf{r})\,.$
(7)

Here, $g_{n\mu}$ parametrizes the spin relaxation rate and
$\mathbf{j}_{s}(\mathbf{r})=-\sigma\bm{\nabla}\mu(\mathbf{r})\,$ the spin
current, where $\sigma$ is the magnon spin conductivity. Thermal and long-
wavelength magnons interact, e.g., via exchange coupling or single-ion
anisotropies. The form of such interaction between coherent (4) and incoherent
(7) spin dynamics can be derived phenomenologically, using the symmetry and
reciprocity principles discussed in Ref. [Flebus2016, ]. For a
$U$(1)-symmetric system, it suffices to supplement Eqs. (4) and (7) with the
following terms [Flebus2016, ]

$\displaystyle\hbar\dot{\mathbf{n}}(\mathbf{r})=$
$\displaystyle-\eta\mathbf{n}(\mathbf{r})\times\left[\hbar\dot{\mathbf{n}}(\mathbf{r})-\mu(\mathbf{r})\mathbf{z}\times\mathbf{n}(\mathbf{r})\right]\,,$
(8) $\displaystyle\dot{\tilde{n}}(\mathbf{r})=$
$\displaystyle-\eta\tilde{s}\mathbf{z}\cdot\mathbf{n}(\mathbf{r})\times\left[\dot{\mathbf{n}}(\mathbf{r})-\frac{\mu(\mathbf{r})}{\hbar}\mathbf{n}(\mathbf{r})\times\mathbf{z}\right]\,,$
(9)

where $\eta\ll 1$ is a phenomenological (dimensionless) coefficient
parametrizing the strength of interactions between the order parameter
$\mathbf{n}$ and the thermal magnon cloud. Here, we have introduced the
reduced spin density $\tilde{s}=S/V-\tilde{n}$, with $V$ being the system
volume. Equations (8) and (9) show that the magnon chemical potential can be
tuned by driving the coherent spin dynamics. The later, in turn, can pump spin
angular momentum into the incoherent thermal magnon cloud [Flebus2016, ].
Using this mechanism, Du and coauthors measured the magnon chemical potential
via single spin magnetometry based on nitrogen-vacancies [Du2017, ], as we
will describe in detail later.

###  II.2 $U$(1)-symmetric antiferromagnetic systems

Due to their lack of a net magnetic moment, antiferromagnets can not be
controlled via an external magnetic field with the same ease as ferromagnetic
systems. However, the absence of production of stray fields, which limit the
packing density of ferromagnetic elements, and their fast spin dynamics
($\sim$THz) make them desirable for the development of spintronic devices
[reviewAF, ,reviewAF1, ]. To illustrate the fundamental properties of
antiferromagnetic magnons, here we focus on a $U$(1)-symmetric
antiferromagnetic insulator with a ground-state staggered magnetic order. The
neighbouring spins are aligned in opposite directions: this magnetic
arrangement can be conveniently described in terms of a unit cell with two
magnetic sublattices, $A$ and $B$. For each $i$th unit cell, we can define the
spin $\mathbf{S}_{A(B)_{i}}=S\mathbf{n}_{A(B)i}$, where the unit vector
$\mathbf{n}_{A(B)i}$ represents the spin orientation of the magnetic
sublattice $A(B)$. We consider sublattices with the same magnitude of
magnetization; if the sublattices have different magnetizations, i.e.,
$|\mathbf{S}_{A}|\neq|\mathbf{S}_{B}|$, the system is ferrimagnetic and in
several respects behaves like a ferromagnet, e.g., yttrium iron garnet (YIG)
[YIG, ].

We set the orientations of the order parameters as
$\mathbf{n}_{A(B)i}\parallel\pm\hat{\mathbf{z}}$. This assumption is valid
only below the critical field at which a spin-flop transition occurs, i.e.,
when the two-sublattice spins rotate suddenly to a direction perpendicular to
the easy-magnetization direction, as shown in Fig. 1(a). Below the spin-flop
transition and far below the magnetic ordering (Néel) temperature $T_{N}$, the
HP transformation reads, in the macrospin limit, as

$\displaystyle\hat{S}_{Aiz}$
$\displaystyle=S-\hat{a}_{i}^{\dagger}\hat{a}_{i},\;\;\;\;\;\;\;\;\hat{S}_{A+}=\sqrt{2S}\hat{a}_{i}\,,$
$\displaystyle\hat{S}_{Biz}$
$\displaystyle=\hat{b}_{i}^{\dagger}\hat{b}_{i}-S,\;\;\;\;\;\;\;\;\hat{S}_{B+}=\sqrt{2S}\hat{b}_{i}^{\dagger}\,,$
(10)

where $\hat{a}_{i}^{\dagger}$ ($\hat{a}_{i}$) and $\hat{b}^{\dagger}$
($\hat{b}$) are the creation (annihilation) operators for spin deviations on
the sublattice $A$ and $B$, respectively. The sublattice spins are coupled by,
e.g., nearest-neighbor exchange interactions; thus, terms of the form
$\hat{a}_{\mathbf{k}}\hat{b}_{-\mathbf{k}}$
($\hat{a}_{\mathbf{k}}^{\dagger}\hat{b}_{-\mathbf{k}}^{\dagger}$) appear in
the Hamiltonian upon HP and Fourier transformation. The magnetic Hamiltonian
can be brought in a diagonal form, i.e.,

$\displaystyle\mathcal{H}_{\text{m}}=\sum_{\mathbf{k}}\left[\hbar\omega_{\mathbf{k},\alpha}\hat{\alpha}^{\dagger}_{\mathbf{k}}\hat{\alpha}_{\mathbf{k}}+\hbar\omega_{\mathbf{k},\beta}\hat{\beta}^{\dagger}_{\mathbf{k}}\hat{\beta}_{\mathbf{k}}\right]\,,$
(11)

via a Bogoliubov-de-Gennes transformation [BdG, ], i.e.,

$\displaystyle\hat{\alpha}_{\mathbf{k}}=u_{\mathbf{k}}\hat{a}_{\mathbf{k}}-v_{\mathbf{k}}\hat{b}^{\dagger}_{-\mathbf{k}}\,,\;\;\;\;\;\;\;\;\;\;\;\hat{\beta}_{\mathbf{k}}=u_{\mathbf{k}}\hat{b}_{\mathbf{k}}-v_{\mathbf{k}}\hat{a}^{\dagger}_{-\mathbf{k}}\,,$
(12)

where $\hat{\alpha}_{\mathbf{k}}^{\dagger}$
($\hat{\beta}_{\mathbf{k}}^{\dagger}$) is the creation operator for a magnon
mode with dispersion $\omega_{\mathbf{k},\alpha(\beta)}$. Setting
$u^{2}_{\mathbf{k}}-v^{2}_{\mathbf{k}}=1$ guarantees the Bogoliubov-de-Gennes
operators to obey the canonical commutation relations. By rewriting the
$z$-component of the spin operator (in units of $\hbar$) in terms of the
Bogoliubov quasi-particles (12) as [Rezende, ]

$\displaystyle\hat{S}_{z}=\sum_{i}\left(S_{Aiz}+S_{Biz}\right)=\sum_{\mathbf{k}}\hbar\left(-\hat{\alpha}^{\dagger}_{\mathbf{k}}\hat{\alpha}_{\mathbf{k}}+\hat{\beta}^{\dagger}_{\mathbf{k}}\hat{\beta}_{\mathbf{k}}\right)\,,$
(13)

we can identify bosonic operator $\hat{\alpha}_{\mathbf{k}}^{\dagger}$
$(\hat{\beta}_{\mathbf{k}}^{\dagger})$ as the operator creating a magnon with
spin angular momentum $\mp\hbar$. $U$(1) symmetry implies the conservation of
the $z$-component of the total spin $S_{z}=\langle\hat{S}_{z}\rangle$ (13).
Generally, the magnon chemical potential accounts for how much the free energy
of the system in thermal equilibrium will change by adding a quanta of spin
angular momentum. In an antiferromagnetic system, net spin injection is
achieved by creating an imbalance between the population of the two magnon
species (13); thus, there is a single chemical potential associated with both
magnonic species [FlebusCP, ]. Assuming (approximate) spin conservation, we
can assign a chemical potential $\mp\mu$ to the Bose-Einstein distribution
function $n_{\text{BE},\alpha(\beta)}$ of the antiferromagnetic eigenmode
$\alpha$ ($\beta$) [FlebusCP, ], i.e.,

$\displaystyle
n_{\text{BE},\alpha(\beta)}\left(\frac{\hbar\omega_{\alpha(\beta)k}\pm\mu}{k_{B}T}\right)\,.$
(14)

Figure 1: (a) Spin configurations in collinear antiferromagnets subjected to a
magnetic field $H$ oriented along the magnet symmetry axis. Below the spin-
flop transition, the order parameters $\mathbf{n}_{A}$ and $\mathbf{n}_{B}$
are antiparallel. Increasing the magnetic field induces a spin-flop
transition, leading to a canted spin structure. (b) Magnon eigenmodes,
labelled as $\alpha$ and $\beta$. The antiferromagnetic mode $\alpha$
($\beta$), which carries $\pm\hbar$ spin angular momentum, can be visualized
as a combination of both up and down spins, precessing counterclockwise
(clockwise) at the frequency $\omega_{\alpha(\beta)}$. The two
antiferromagnetic modes are associated with an equal and opposite chemical
potential $\mu$.

Thermal fluctuations engender a finite thermal magnon density of both
antiferromagnetic modes, i.e., $\tilde{n}_{\alpha}$ and $\tilde{n}_{\beta}$;
since the contributions of the two modes to the spin current have opposite
signs, it is convenient to rewrite the spin diffusion equation, in the
continuum limit, in terms of the net density of thermal magnons carrying
angular momentum $\hbar$, i.e.,
$\tilde{n}=\tilde{n}_{\beta}-\tilde{n}_{\alpha}$, as

$\displaystyle\dot{\tilde{n}}(\mathbf{r})+\bm{\nabla}\cdot\mathbf{j}_{s}(\mathbf{r})=-g_{\mu}\mu(\mathbf{r})\,,$
(15)

which is formally analogous to Eq. (7). Their classical coupled dynamics can
be modeled by the phenomenological Landau-Lifshitz equation [kittelAF, ]

$\displaystyle\dot{\mathbf{n}}_{A(B)}=-\gamma\times\left[\mathbf{H}_{A(B)}-H_{c}\mathbf{n}_{B(A)}\right]\,,$
(16)

where $\mathbf{H}_{A(B)}$ is the effective (Landau-Lifshitz) field acting on
the order parameter $\mathbf{n}_{A(B)}$ and $H_{c}$ is the, e.g., exchange,
field coupling the spin dynamics of the two sublattices.

The linearized dynamics of the long-wavelength magnetic order parameters (16)
can be diagonalized in terms of two normal modes $\mathbf{n}_{\alpha}$ and
$\mathbf{n}_{\beta}$ [kittelAF, ]. In the mode $\alpha$ both up and down spins
precess clockwise with frequency $\omega_{\alpha\mathbf{k}}$, while in the
$\beta$ mode the spins undergo a counterclockwise precession with frequency
$\omega_{\beta\mathbf{k}}$, as depicted in Fig. 1(b). The corresponding
Landau-Lifshitz-Gilbert equation reads as [kittelAF, ,RezendeAFF, ]

$\displaystyle\dot{\mathbf{n}}_{\alpha(\beta)}=-\gamma\mathbf{n}_{\alpha(\beta)}\times\mathbf{H}-\alpha_{\alpha,\beta}\mathbf{n}_{\alpha(\beta)}\times\dot{\mathbf{n}}_{\alpha(\beta)}\,,$
(17)

where we have included a mode-selective Gilbert damping parameter
$\alpha_{\alpha(\beta)}$ [GilbertAF, ]. In terms of the normal modes, the
interplay between coherent and incoherent spin dynamics can be accounted for
by supplementing Eqs. (15) and (17) with [FlebusCP, ]

$\displaystyle\hbar\dot{\mathbf{n}}_{\alpha}=$
$\displaystyle-\eta_{\alpha}\mathbf{n}_{\alpha}\times(\hbar\dot{\mathbf{n}}_{\alpha}-\mu\mathbf{z}\times\mathbf{n}_{\alpha})\,,$
(18) $\displaystyle\hbar\dot{\mathbf{n}}_{\beta}=$
$\displaystyle-\eta_{\beta}\mathbf{n}_{\beta}\times(\hbar\dot{\mathbf{n}}_{\beta}-\mu\mathbf{z}\times\mathbf{n}_{\beta})\,,$
(19) $\displaystyle\dot{\tilde{n}}=$
$\displaystyle-\eta_{\alpha}\tilde{s}_{\alpha}\mathbf{z}\cdot\mathbf{n}_{\alpha}\times(\dot{\mathbf{n}}_{\alpha}-\frac{\mu}{\hbar}\mathbf{n}_{\alpha}\times\mathbf{z})$
$\displaystyle-\eta_{\beta}\tilde{s}_{\beta}\mathbf{z}\cdot\mathbf{n}_{\beta}\times(\dot{\mathbf{n}}_{\beta}-\frac{\mu}{\hbar}\mathbf{n}_{\beta}\times\mathbf{z})\,,$
(20)

with $\tilde{s}_{\alpha(\beta)}=s-\tilde{n}_{\alpha(\beta)}$. Here,
$\eta_{\alpha(\beta)}\ll 1$ is a phenomenological (dimensionless) coefficient
parametrizing the strength of interactions between the order parameter
$\mathbf{n}_{\alpha(\beta)}$ and the thermal magnon cloud.

###  II.3 Coupled spin and heat transport

Spin and heat transport in collinear magnetic insulators have been extensively
investigated both theoretically and experimentally in a variety of setups.
Here, for simplicity, we present the theory of coupled spin and heat transport
by considering a normal metal$|$magnetic insulator$|$normal metal
heterostructure in a 1D geometry, shown in Fig. 2. The metallic reservoirs act
as thermal baths, set at two different temperatures, i.e., $T_{l}$ and
$T_{r}$. The temperature bias generates a linear temperature gradient $\nabla
T$ across the sample. The normal metals have strong spin–orbit coupling: if a
charge current flows through them, a spin accumulation is generated at the
metal$|$magnetic insulator interface and vice versa.

The bulk spin, $\mathbf{j}_{s}$, and heat, $\mathbf{j}_{q}$, currents carried
by magnons in the magnetic insulating system can be written as

$\displaystyle\begin{pmatrix}\mathbf{j}_{s}\\\
\mathbf{j}_{q}\end{pmatrix}=-\begin{pmatrix}\bm{\sigma}&\bm{\zeta}\\\
\bm{\rho}&\bm{\kappa}\end{pmatrix}\begin{pmatrix}\bm{\nabla}\mu\\\
\bm{\nabla}T\end{pmatrix}\,,$ (21)

where the tensors $\bm{\sigma}$, $\bm{\kappa}$ , $\bm{\zeta}$, and
$\bm{\rho}$($=T\bm{\zeta}$ by the Onsager-Kelvin relation [Onsager, , Callen,
]) are, respectively, the spin and (magnetic) heat conductivities, and the
spin Seebeck and Peltier coefficients. The hydrodynamics equations for spin
and heat transport can be easily constructed within the Boltzmann transport
theory [Boltzmann, ], i.e.,

$\displaystyle\dot{\tilde{n}}+\bm{\nabla}\cdot\mathbf{j}_{s}=-g_{n\mu}\mu-
g_{nT}\left(T-T_{p}\right)\,,$
$\displaystyle\dot{u}+\bm{\nabla}\cdot\mathbf{j}_{q}=-g_{u\mu}\mu-
g_{uT}\left(T-T_{p}\right)\,.$ (22)

Here, $T$ ($T_{p}$) is the magnon (phonon) temperature, $u$ the energy density
of the thermal cloud, while $g_{n(u)\mu}$ and $g_{n(u)T}$ parametrize,
respectively, the relaxation of magnons via inelastic magnon-magnon and
magnon-phonon interactions and the magnon thermalization to the phonon
temperature.

Figure 2:  Metal|magnetic insulator|metal heterostructure. Via the spin Hall
effect, a charge current $I$ sent through the metal along the $y$-direction
generates a spin accumulation $\bm{\mu_{l}}=\mu_{l}\hat{\mathbf{z}}$ at the
metal$|$magnetic insulator interface, which injects a $z$-polarized spin
current $j_{s}$ into the magnetic insulator. Due to the Joule heating
associated with the charge current, a temperature gradient $\nabla T$ is set
across the sample. The injected spin current and the temperature gradient
trigger the dynamics of the magnetic order parameter $\mathbf{n}$ and of the
thermal magnon cloud density $\tilde{n}$. The coherent and incoherent spin
dynamics at the magnetic insulator$|$metal interface pump a spin current
$j_{s}$ at $x=L$, leading to a spin accumulation
$\bm{\mu}_{r}=\mu_{r}\hat{\mathbf{z}}$ that can be converted into a measurable
voltage in the metal via the inverse spin Hall effect.

The relations (21) and (22), complemented by Eqs. (4,8,9) and by Eqs.
(17,18,19,20) for, respectively, a ferromagnetic and an antiferromagnetic
system, describe the coupled bulk spin and heat transport of the coherent and
incoherent spin dynamics.

The transport equations must be determined consistently with the boundary
conditions for spin and heat transport at the interfaces. For a ferromagnet,
we take the order parameter to be aligned along the $z$-direction, i.e.,
$\mathbf{n}\parallel-\hat{\mathbf{z}}$. For a two-sublattice antiferromagnet,
it is convenient to introduce the Néel unit vector, i.e.,
$\mathbf{n}=\left(\mathbf{n}_{A}-\mathbf{n}_{B}\right)/2$. In this geometry,
the relevant nonequilibrium spin accumulation at the metal$|$magnetic
insulator interface at $x=0(L)$ is polarized along $z$-direction [spinpumping,
,spinpumping1, ], i.e., $\bm{\mu}_{l(r)}=\mu_{l(r)}\hat{\mathbf{z}}$. The
boundary conditions read as

$\displaystyle j_{s}|_{x=0(L)}=$ $\displaystyle
G_{l(r)}\left(\mu_{l(r)}-\mu\right)|_{x=0(L)}+S_{l(r)}\left(T_{l(r)}-T\right)|_{x=0(L)}\,,$
$\displaystyle j_{q}|_{x=0(L)}=$ $\displaystyle
K_{l(r)}\left(T_{l(r)}-T\right)|_{x=0(L)}+\Pi_{l(r)}\left(\mu_{l(r)}-\mu\right)|_{x=0(L)}\,.$
(23)

Here, the coefficients $G_{l(r)}$, $K_{l(r)}$, $S_{l(r)}$, and
$\Pi_{l(r)}$(=$TS_{l(r)}$ by Onsager reciprocity) are the magnon spin and
thermal conductances and spin Seebeck and Peltier coefficients, respectively,
at the metal$|$magnetic insulator interface at $x=0(L)$. Furthermore, the
coherent magnetization dynamics pumps a spin current into an adjacent
conductors [SpinSup1, ,spinpumping, ; spinpumping1, ; spinpumping2, ;
spinpumpingAF2, ], i.e.,

$\displaystyle
j_{s}|_{x=0(L)}=-g_{l(r)}\hat{\mathbf{z}}\cdot\left(\mathbf{n}\times\dot{\mathbf{n}}\right)|_{x=0(L)}\,,$
(24)

where $g_{l(r)}$ is the spin pumping efficiency of the metal$|$magnetic
insulator interface at $x=0(L)$, related to the magnon spin conductance
$G_{l(r)}$ by Onsager reciprocity [spinpumping1, , spinpumping2, ]. The bulk
and interfacial transport coefficients appearing, respectively, in Eqs. (21)
and (23) can be obtained via algebraic manipulation and integration of the
magnon Boltzmann equation over momentum. Their estimate depends on the details
of the transport regime under investigation; for a detailed discussion of some
examples, we refer the reader to Refs. [Cornelissen2016, ] and [Flebus2016, ].

It is worth remarking that the outlined phenomenology can be easily extended
to a setup comprised of an insulating non-magnetic substrate$|$magnetic
insulator$|$normal metal, commonly used, e.g., in longitudinal SSE
measurements. At the substrate|magnetic insulator interface, the spin flow is
blocked as there are no spin carriers in the substrate. Nevertheless, heat
still can be transmitted via inelastic spin-preserving scattering processes
between magnons and phonons. The corresponding boundary conditions at the
substrate$|$insulator interface at $x=0$ read as

$\displaystyle j_{s}|_{x=0}=0\,,\;\;\;\;\;\;\;\;\;\;j_{q}|_{x=0}=$
$\displaystyle K_{l}\left(T_{l}-T\right)|_{x=0}\,.$ (25)

###  II.4 Magnon-phonon coupling

Coupling between collective magnetic and elastic excitations, i.e., magnons
and phonons, is ubiquitous in magnetic insulating systems. It relies on
relativistic effects such as dipole-dipole interactions and spin-orbit
coupling, as well as on the dependence of the spin-spin exchange interactions
on the phonon coordinates, and it provides a pathway for thermalization and
dissipation of the magnetization dynamics. While the exact form of the phonon-
driven magnon thermalization and dissipation rates can be computed from
microscopic models [phononmagnonlifetime, ], often they are accounted for by
introducing phenomenological parameters, see, e.g., Eq. (22).

However, magnon thermalization and dissipation are not the only relevant
mechanisms emerging from the magnetoelastic coupling. When the magnetoelastic
coupling term that is quadratic in magnon and phonon operators does not
vanish, magnons and phonons can hybridize to form quasiparticles that are an
admixture thereof, dubbed as magnon-polarons [mp1, , mp2, ]. The Hamiltonian
of a coupled magnon-phonon system reads generally as

$\displaystyle\mathcal{H}=\mathcal{H}_{\text{m}}+\mathcal{H}_{\text{ph}}+\mathcal{H}_{\text{mec}}\,,$
(26)

where $\mathcal{H}_{\text{m}(\text{ph})}$ is the magnon (phonon) quadratic
Hamiltonian and $\mathcal{H}_{\text{mec}}$ the long-wavelength magnetoelastic
coupling Hamiltonian, whose explicit form depends on the symmetries of the
underlying lattice. The Hamiltonian (26) can be brought into diagonal form via
a Bogoliubov-de-Gennes transformation [BdG, ], which maps magnon and phonon
operators into composite quasi-particles, i.e., magnon-polarons.

Figure 3: (a) The dispersions of uncoupled magnons and transverse acoustic
(TA) phonons are degenerate at certain wavevectors. (b) In the presence of
magnon-phonon coupling, the system can be diagonalized in terms of magnon-
polarons, i.e., hybridized magnon-phonon quasi-particles. The magnetoelastic
coupling lifts the degeneracy between the magnonic and phononic bands and the
crossing points become anticrossing points. At the anticrossing points, the
mixing between magnons and phonons is maximized, while at other wave-vectors
the magnon-polaron bands retain a phonon- or magnon-like character. (c)
Longitudinal spin Seebeck setup comprised of a magnetic insulating system
(YIG) sandwiched between a gadolinium gallium garnet (GGG) substrate and a
platinum (Pt) film. A temperature gradient set across the sample generates a
spin current that can be detected as a voltage in the Pt film [MP1, ]. (d) At
the magnetic field $H_{\text{TA}}$, the magnon dispersion shifts upwards such
that the TA phonon branch becomes tangential and the effects of the
magnetoelastic coupling are maximized, i.e., the magnon and phonon modes
become strongly coupled over a relatively large volume in momentum space. The
voltage measured in a spin Seebeck setup displays a peak at the field
$H_{\text{TA}}$: the magnon-polaron spin transport properties are enhanced
with respect to their purely magnonic counterpart [MP1, ].

In the absence of coupling, transversal acoustic (TA) and longitudinal
acoustic (LA) phonon bands are degenerate with the spin-wave dispersion at
certain energies, as depicted in Fig. 3(a). When the coupling is sufficiently
strong, the mutual interaction between the magnons and phonons lifts the
degeneracy, leading to an anticrossing around which the magnon-polaron bands
deviate from the purely phononic and magnonic ones, as shown in Fig. 3(b).

Figure 4: (a) Non-local spin transport setup and experimental data [spindiff1,
]. Two Pt strips, separated by a distance $d$, are deposited on top of a
magnetic insulating film (YIG) grown on a GGG substrate. One metallic strip
(injector) injects spin current into the magnetic insulating system. The spin
current propagates through the insulator and generates a voltage $V$ in the
other metallic strip (detector). (b,c) By tuning the orientation of the in-
plane magnetic field $\mathbf{B}$, the magnetic insulator order parameter
$\mathbf{n}$ can be oriented at a generic angle $\alpha$ within the sample
plane, i.e., $\mathbf{n}\parallel\cos\alpha$. A charge current in the injector
leads to a spin accumulation $\bm{\mu}_{l}=\mu_{l}\hat{\mathbf{z}}$, with
$\mu_{l}\propto I$, at the metal$|$magnetic insulator interface; the effective
spin accumulation that injects non-equilibrium magnons in the insulator is
parallel to the order parameter, i.e.,
$\bm{\mu}_{\text{eff}}=\mu_{l}\cos\alpha$. Similarly, a spin accumulation
parallel to $\mathbf{n}$ is generated at the detector. The charge current $I$
yields as well a temperature gradient $\nabla T\propto I^{2}$ that injects
non-equilibrium magnons with no dependence on the angle $\alpha$. (b) The
first harmonic signal, which measures electrically-generated spin waves, is a
product of the effects at the injector and detector, leading to a
$\cos^{2}\alpha$ dependence. (c) The measured second harmonic signal, which
measures thermally-generated spin waves, displays a $\cos\alpha$ dependence.
(d) The measured first harmonic signal is fitted to the model in Eq. (28).

Generally, strong hybridization occurs only in proximity of these
anticrossings points and it is not likely to affect relevantly spin and heat
transport coefficients (22), whose evaluation requires integration of the
Boltzmann transport equation over the full reciprocal space. However, it was
recently found that, at critical magnetic field values, the magnon and phonon
dispersions tangentially touch each other and the magnon–phonon hybridization
effects are maximized [MP1, , MP2, ]. In correspondence of these fields, spin
transport properties can be enhanced or suppressed, depending on the ratio
between the phonon and the magnon relaxation times. If the phonon relaxation
time is longer (shorter) than the magnon relaxation time, magnon-polarons will
have a longer (shorter) lifetime than the pure magnetic excitations and thus
spin transport properties will be enhanced (suppressed). These features have
been recently observed in the field dependence of thermally generated current
in YIG in a longitudinal SSE transport setup, shown in Figs. 3(c) and (d).
Similar effects have been investigated as well in antiferromagnetic systems
[MPAF, ; MPAF1, ; MPAF2, ].

##  III Probing spin dynamics

In this section we review two experimental probes of magnetization dynamics.
We introduce the reader to spin transport setups, by focusing on the
measurements of the YIG spin diffusion length [spindiff1, ] and of thermally-
generated spin currents in the antiferromagnetic insulator $\text{MnFe}_{2}$
[AFdifflength, ]. Secondly, we discuss a new minimally-invasive magnon sensing
technique, i.e., NV-center relaxometry, and review the recent measurements of
the magnon chemical potential in YIG [Du2017, ] and of the spin diffusion
length in antiferromagnetic $\alpha$-$\text{Fe}_{2}\text{O}_{3}$ [WangAF, ].

###  III.1 Spin transport

A non-local transport set-up is commonly used as a probe of spin transport
properties of magnetic insulating systems. This setup is comprised of two
metallic strips, operating, respectively, as spin current injector and
detector. The metals are deposited on top of a magnetic insulating sample
grown on a substrate, as shown in Fig. 4(a). A charge current $I$ modulated at
frequency $\omega$ is sent through the injector. The SHE converts the charge
current into a spin accumulation $\mu_{l}\propto I$ at the metal$|$magnetic
insulator interface, while Joule heating generates a thermal gradient $\nabla
T\propto I^{2}$.

Spin and thermal injection trigger a spin current that might propagate to the
magnetic insulating$|$detector interface. In the detector, owing to the ISHE,
the impinging spin current is converted into a charge current, which generates
a voltage $V$ under open-circuit conditions. Using lock-in amplifiers, one can
separate higher order contributions in the voltage by measuring higher
harmonics:

$\displaystyle V=R_{1}I+R_{2}I^{2}+...\;\,,$ (27)

where $R_{i}$ is the $i$th harmonic response of the non-local resistance
$R=V/I$ [exptransp1, ]. The first, $R_{1}$, and second, $R_{2}$, harmonic
signals are a measure of, respectively, the electrically- and thermally-
generated spin currents. An external magnetic field $\mathbf{B}$, oriented at
an angle $\alpha$ within the sample plane, sets the equilibrium direction of
the magnetic order parameter $\mathbf{n}$. The effective component of the spin
accumulation, $\mu_{\text{eff}}$, that exerts a spin-transfer torque on the
order parameter is parallel to the magnetic order, i.e.,
$\mu_{\text{eff}}=\mu_{l}\cos\alpha$. Similarly, the spin current injected via
spin pumping into the detector is modulated by a factor $\cos\alpha$. Thus,
the electrically-generated signal scales as $R_{1}\propto\cos^{2}\alpha$,
while for thermal injection one has $R_{2}\propto\cos\alpha$, as shown in
Figs. 4(b) and (c).

Numerous experiments have been performed using a non-local transport setup. In
this tutorial, we focus on the first experimental report of room-temperature
long-range spin transport in a magnetic insulating system. Cornelissen and
coauthors [spindiff1, ] investigated spin transport in YIG using the setup
depicted in Fig. 4(a). In the data analysis, the spin and heat transport in
the Pt$|$YIG$|$Pt heterostructure is modelled using the phenomenology
presented in Sec. IIC. The magnon-phonon energy relaxation length
$\ell_{u}=\sqrt{\kappa/g_{uT}}$, associated with spin-preserving relaxation of
magnon distribution towards the phonon temperature, is assumed to be much
shorter than the magnon relaxation (i.e., spin diffusion) length
$\ell_{s}\equiv\sqrt{\sigma/g_{n\mu}}$. This can be intuitively understood as
spin conserving phonon-magnon interactions can stem from Heisenberg exchange
coupling, whose strength is much larger than the spin-orbit interactions
responsible for spin non-conserving processes. In this limit, by plugging Eq.
(21) into Eq. (22), one obtains, in a steady-state, the spin diffusion
equation

$\displaystyle\nabla^{2}\mu=\frac{\mu}{\ell_{s}}\,.$ (28)

Fitting Eq. (28) to the experimental signal, shown in Fig. 4(d), leads to the
estimate $\ell_{s}\sim 10\;\mu$m, which has been corroborated by further
experiments [spindiff2, ,spindiff3, ]. It worth noting that, few years later,
the magnon-phonon energy relaxation length $\ell_{u}$ in YIG was measured to
be about 250 nm [magnonrelax, , magnonrelax1, ], i.e., much shorter than the
spin diffusion length, confirming the assumption that underlies the analysis
of the experimental data in Ref. [spindiff1, ].

An alternative technique for probing thermally-generated spin currents is
offered by a longitudinal spin Seebeck setup, comprised of a magnetic
insulator sandwiched between a substrate and a metallic strip. A heat gradient
applied across the sample generates a spin current in the magnetic insulator,
which is then converted into a measurable voltage in the metal via ISHE. This
setup, shown in Fig. 3(c), was used to detect the signatures of magnon-phonon
hybridization in YIG [MP1, ], and, more recently, as a probe of thermally-
driven spin transport in the easy-axis antiferromagnetic insulator
$\text{MnFe}_{2}$ [SHEAF, ].

Figure 5: Temperature dependence of the spin Seebeck voltage response at
various magnetic fields [SHEAF, ].

Wu and coauthors revealed a dependence of the spin Seebeck voltage on
temperature that is not observable in ferromagnetic system. The experimental
signal, shown in Fig. 5, displays peaks at temperatures much lower than the
magnetic ordering temperature. The position of the peaks varies with the
intensity of the applied magnetic field. While a detailed model and analysis
of the data can be found in Ref. [Rezende, ], here we can develop an intuitive
understanding by using the concepts outlined in Sec. IIB. $\text{MnFe}_{2}$
can be modeled as a two-sublattice uniaxial antiferromagnet, with two normal
magnon modes that carry opposite spin angular momentum (13), and, thus,
generate spin currents propagating in opposite directions. The spin current
density operator $\hat{\mathbf{j}}_{s}$ reads as

$\displaystyle\hat{\mathbf{j}}_{s}=\frac{\hbar}{V}\left[-\mathbf{v}_{\mathbf{k},\alpha}\hat{\alpha}^{\dagger}_{\mathbf{k}}\hat{\alpha}_{\mathbf{k}}+\mathbf{v}_{\mathbf{k},\beta}\hat{\beta}^{\dagger}_{\mathbf{k}}\hat{\beta}_{\mathbf{k}}\right]\,,$
(29)

where
$\mathbf{v}_{\mathbf{k},\alpha(\beta)}=\partial_{\mathbf{k}}\omega_{\mathbf{k},\alpha(\beta)}$
is the group velocity of the $\alpha$ ($\beta$) magnon mode. For an uniaxial
antiferromagnet subjected to an external magnetic field $B$ oriented along its
symmetry axis, the dispersion can be written as

$\displaystyle\omega_{\mathbf{k},\alpha(\beta)}=\omega_{\textbf{k}}\pm\gamma
B\,,$ (30)

where $\omega_{\mathbf{k}}$ is the zero-field spin-wave dispersion, whose
explicit form depends on the specific microscopic model. In the absence of a
magnetic field, the two magnon modes are degenerate; thus, they have equal
occupation numbers in thermal equilibrium. Since they also have equal group
velocities, the spin current (30) vanishes. When a magnetic field is applied,
the magnon dispersions split according to Eq. (30). At a given temperature,
the magnon mode $\beta$ has a higher occupation density than the magnon mode
$\alpha$. The difference between the occupation density of the $\alpha$ and
$\beta$ modes is proportional to the splitting $2\gamma B$ between their
dispersions. For temperatures lower than the bottom of the dispersion curve of
the $\alpha$ mode, i.e., $T<\omega_{\mathbf{k}=0,\alpha}$, increasing the
temperature increases the thermal population of the $\beta$ mode and thus the
net spin current (29). At higher temperatures, the thermal population of the
$\beta$ mode start increasing, leading to a reduction of the net current (30).

Figure 6: NV relaxometry setup and experimental data [Du2017, ]. (a) A
stripline excites long-wavelength magnons in a YIG film deposited on a GGG
substrate. A NV center lies at a height $d$ above the sample. The magnetic
noise generated by magnons affects the NV-center relaxation rates, which are
probed by a laser. (b) Dependence on the external magnetic field of the FMR
(black line) and of the NV-center relaxation frequencies for the transitions
$0\leftrightarrow+1$ (blue line) and $0\leftrightarrow-1$ (red line). The
orange shaded area corresponds to the magnon population at room temperature.
(c) In the absence of external drives, the magnon chemical potential vanishes
due to dissipative interactions with the phonon bath. When a drive excites
long-wavelength magnons, the thermal magnon density and, thus, the magnon
chemical potential increase. (d) The increase of the magnon chemical potential
is proportional to the power of the driving field $B_{AC}$, until it saturates
at the ferromagnetic resonance value.

Thus, the resulting temperature dependence of the spin Seebeck voltage will
display a peak, as observed in the experimental data in Fig. 5. The position
and the height of the peak depend on the applied magnetic field. With
increasing fields, the dispersion of the $\alpha$ mode moves further up in
energy. The thermal population of the $\beta$ mode can increase further with
increasing temperatures without populating the $\alpha$ mode, which leads to
an enhanced signal at larger temperatures.

###  III.2 NV-center relaxometry

NV centers in diamond are point defects where one carbon atom in the diamond’s
crystal lattice is replaced by a nitrogen atom (N) and an adjacent lattice
site is left empty [NV, ]. Due to their exceptional sensitivity to magnetic
fields, they have recently emerged as a minimally-invasive probe of magnetic
systems, which provides a frequency resolution not achievable by other
techniques [NVreview, ]. For the purpose of this tutorial, it suffices to
model a NV center as a three-level spin system ($|\mathbf{S}|=1$) subjected to
an external field $\mathbf{B}(\mathbf{r}_{\text{nv}})$. The external field
accounts for the magnetic noise generated by an adjacent system and for a
static field applied to split the degeneracy of the $m_{s}=\pm 1$ NV spin
states. The NV spin $\mathbf{S}$, set at the position
$\mathbf{r}_{\text{nv}}$, is oriented along its anisotropy axis
$\hat{\mathbf{n}}_{\text{nv}}$, with
$\hat{\mathbf{n}}_{\text{nv}}\cdot\hat{\mathbf{z}}=\cos\theta$. The
corresponding Hamiltonian can be written as

$\displaystyle\mathcal{H}_{\text{nv}}=DS^{2}_{z}+\gamma_{e}\mathbf{S}\cdot\mathbf{B}_{\text{nv}}(\mathbf{r}_{\text{nv}})\,,$
(31)

where $D=2.87$ GHz is the ground state zero-field splitting between the
$m_{s}=0$ and the degenerate $m_{s}=\pm 1$ states, and
$\gamma_{e}=28\;\text{GHz}\cdot\text{T}^{-1}$ is the gyromagnetic ratio of the
electronic spin. Here, we have introduced
$\mathbf{B}_{\text{nv}}(\mathbf{r}_{\text{nv}})=\mathcal{R}_{x}(\theta)\mathbf{B}(\mathbf{r}_{\text{nv}})$,
where $\mathcal{R}_{x}(\theta)$ is a rotation matrix that allows us to easily
distinguish between the longitudinal and transverse components of the field
with respect to the NV anisotropy axis.

NV magnetometry and relaxometry are used as probes of, respectively, static
and dynamical properties of magnetic systems [NVreview, ]. The quantity
measured in a magnetometry experiment is the projection of the magnetic field
onto the NV anisotropy axis, $B_{\text{nv},z}$, to which the NV electron spin
resonance splitting is first-order sensitive. The full vector field can be
reconstructed from the field component $B_{\text{nv},z}$ and compared with the
one generated by a given magnetic texture [NVreview, ].

In a relaxometry setup, the relevant quantities are the field components
transverse to the NV anisotropy axis, $B_{\text{nv},x}$ and $B_{\text{nv},y}$.
Up to leading order in perturbation theory, the Zeeman coupling between the NV
spin and the transverse field components induces NV transitions between the
spin states $m_{s}=0\leftrightarrow\pm 1$ at the resonance frequency
$\pm\omega$. For a NV-center spin set at a height $d$ above a $U$(1)- and
translationally-symmetric magnetic system, the relaxation rate at frequency
$\omega$ can be written as [FlebusQI, ]

$\displaystyle\Gamma(\omega)=$ $\displaystyle
f(\theta)\int^{\infty}_{0}dk\;k^{3}e^{-2kd}\left[C_{xx}(k,\omega)+C_{zz}(k,\omega)\right]\,,$
(32)

with $f(\theta)=(\gamma\gamma_{e})^{2}(5-\cos 2\theta)/16\pi$. Here,
$C_{\alpha\beta}(\mathbf{k},\omega)$ is the Fourier transform of the spin-spin
correlator
$C_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};t)=\langle\\{\hat{s}_{\alpha}(\mathbf{r}^{\prime},t),\hat{s}_{\beta}(\mathbf{r},0)\\}\rangle$
of the magnetic system, with $\alpha,\beta=x,y,z$. Equation (32) shows that
the NV relaxation rate is a measure of the magnetic noise transverse,
$C_{xx}$, and longitudinal, $C_{zz}$, to the equilibrium orientation of the
order parameter $\mathbf{n}\parallel\hat{\mathbf{z}}$ of the magnetic system.
Invoking the HP transformations (1) and (10) for, respectively, a
ferromagnetic (FM) and an antiferromagnetic (AFM) system, it is easy to see
that the transverse spin-spin correlator accounts for one-magnon processes,
i.e., the creation (or annihilation) of a magnon at frequency $\omega$. Thus,
it is proportional to the thermal magnon distribution function at frequency
$\omega$, i.e.,

$\displaystyle C^{\text{FM}}_{xx}(k,\omega)\propto$
$\displaystyle\;n_{\text{BE}}\left(\frac{\hbar\omega-\mu}{k_{B}T}\right)\,,$
(33) $\displaystyle C^{{}^{\text{AFM}}}_{xx}(k,\omega)\propto$
$\displaystyle\;n_{\text{BE}}\left(\frac{\hbar\omega\pm\mu}{k_{B}T}\right)\,.$
(34)

Equations (32), (33) and (34) show that the NV relaxation rate can provide a
direct measurement of the magnon chemical potential, which has proven to be
hardous to perfom in spin transport setups [Cornelissen2016, ]. The first
measurement of the magnon chemical potential via NV-center relaxometry was
performed by Du and coauthors [Du2017, ]. In their room-temperature setup,
sketched in Fig. 6(a), a NV center is set above a YIG film. A stripline drives
the ferromagnetic resonance of the YIG film, i.e., it excites the coherent
spin dynamics obeying Eqs. (4), (8) and (9). The coherent spin dynamics, in
turn, pumps spin angular momentum into the thermal magnon cloud. This
mechanism increases thermal magnon density and, consequently, the magnon
chemical potential (9), as depicted in Fig. 6(b).

Figure 6(c) shows the dependency of the NV-center relaxation frequencies and
of the YIG ferromagnetic resonance frequency on the static external field. The
transverse noise corresponding to single-magnon processes can be probed by a
NV-center relaxation rate at frequencies for which the magnon thermal
population is finite. The increase in the magnon chemical potential was found
to be directly proportional to the driving power, and, as predicted by Bose-
Einstein statistics, to saturate at the ferromagnetic resonance frequency, as
shown in Fig. 6(c). A further analysis of the experimental data [Du2017, ]
lead to the extrapolation of the parameter $\eta$ (8).

The chemical potential of a $U$(1)-symmetric antiferromagnetic system might be
probed in a similar fashion [FlebusCP, ]. The two antiferromagnetic normal
modes undergo precessions with opposite handedness, as shown in Fig. 1(b);
thus, they can be selectively excited by an ac field with matching
polarization. Resonantly driving the coherent spin dynamics of the
antiferromagnetic modes $\alpha$ and $\beta$ increases their thermal magnon
populations, as dictated by Eqs. (20). The increase in the thermal population
of $\alpha$ ($\beta)$ leads to a increase of a negative (positive) chemical
potential, which can be measured according to Eqs. (32) and (34).

Figure 7: The interaction between the NV-center spin and a nearby magnetic
system, here depicted as gas of magnons with spin $\hbar$ and frequency
$\omega_{k}$ (with $\omega_{k=0}=\Delta$), leads to a NV-center transition
rate $\Gamma(\omega)$ with emission of energy $\omega$. When $\omega>\Delta$,
the latter can result in the creation of a magnon at frequency
$\omega_{k}=\omega$ or in a magnon scattering with energy gain $\hbar\omega$.
These events are accounted for, respectively, by the transverse, i.e.,
$C_{xx}(k,\omega)$, and by the longitudinal, i.e., $C_{zz}(k,\omega)$, spin-
spin correlation function. When $\omega>\Delta$, the relaxation rate is
typically dominated by one-magnon processes, i.e., $C_{xx}(k,\omega)\gg
C_{zz}(k,\omega)$. Conversely, for $\omega<\Delta$, one-magnon events are
suppressed, i.e., $C_{xx}(k,\omega)\rightarrow 0$ [FlebusQI, ].

In order to probe the transverse noise, the NV transition frequency $\omega$
must be larger than the gap $\Delta$ of the spin-wave dispersion. At lower
frequencies, there are no available magnon states and, thus, Eqs. (33) and
(34) vanish. For $\omega>\Delta$, the transverse noise dominates over the
longitudinal noise [Du2017, ,FlebusQI, ], while for $\omega<\Delta$ the
longitudinal noise represents the leading contribution to the relaxation rate
(32), as sketched in Fig. 7. By invoking the HP transformations (1) and (10),
for, respectively, a ferromagnetic and an antiferromagnetic system, the
longitudinal spin-spin correlator $C_{zz}(k,\omega)$ can be expressed in terms
of two-magnon scattering processes, i.e., a magnon with frequency
$\omega_{1}+\omega$ scatters into a magnon state with frequency $\omega_{1}$,
or vice versa, emitting (or absorbing) magnetic noise at frequency $\omega$.
The details of how two-magnon scattering processes occur depend on the spin
transport properties of the system. For diffusive magnon transport in the
absence of a heat gradient (7, 15), the imaginary part of the spin
susceptibility can be found as [FlebusQI, ]

$\displaystyle\chi^{{}^{\prime\prime}}_{zz}(k,\omega)=\frac{\chi\hbar^{2}}{D}\frac{\omega
k^{2}}{\left(k^{2}+1/\ell^{2}_{s}\right)^{2}+(\omega/D)^{2}}\,,$ (35)

where $\chi$ is the static uniform longitudinal susceptibility and
$D=\sigma/\chi$ the spin-wave diffusion coefficient. In thermal equilibrium,
the fluctuation-dissipation theorem dictates [Kubo, ]

$\displaystyle
C_{zz}(k,\omega)=\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)\chi^{{}^{\prime\prime}}_{zz}\left(k,\omega\right)\,.$
(36)

Wang and coauthors [Dupreparation, ] recently measured the relaxation rate
(32) of a YIG film at frequencies below the magnetic gap. Using Eqs. (35) and
(36) they extracted a spin diffusion length of $\ell_{s}\sim 1.5$ $\mu$m,
which was further corroborated by a non-local spin transport measurement
performed on the same sample. An analogous method can be used to probe non-
invasively the spin transport properties of collinear antiferromagnets. Wang
and coauthors [WangAF, ] probed the time-dependent fluctuations of the
longitudinal spin density of $\alpha$-$\text{Fe}_{2}\text{O}_{3}$. They
estimated the spin diffusion length $\ell_{s}$ to be $3$ $\mu$m at 200 K,
which is in agreement with the values reported by non-local spin transport
experiments [AFdifflength, ].

##  IV Summary and perspectives

In this tutorial, we have outlined the fundamental properties of collective
spin excitations, i.e., magnons, in collinear magnetic insulating systems. We
have reviewed their statistical and transport properties and discussed the key
ingredients of the coupling between magnetic and lattice degrees of freedom.
Making use of this theoretical framework, we have introduced the reader to two
magnon sensing techniques, i.e., spin transport setups and NV-center
relaxometry.

Here we have restricted our discussion to simple monoatomic ferromagnetic and
two-sublattice antiferromagnetic systems. However, there is a variety of
collinear insulators with different crystalline structures and spin
interactions that continue to be discovered and that can be addressed with the
methods presented here.

Finally, while non-collinear magnetic systems do not support long-range
diffusive spin transport, their spin non-conserving interactions might endow
the spin-wave bands with a nontrivial topological structure. The emergence of
topologically-protected long-range propagating magnon modes and their non-
Hermitian topology represent an emergent promising platform to investigate
spin transport and novel magnetic phenomena.

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