# Magnonics vs. Ferronics

Gerrit E.W. Bauer  Ping Tang  Ryo Iguchi  Ken-ichi Uchida  WPI-AIMR, Tohoku
University, 2-1-1 Katahira, 980-8577 Sendai, Japan  IMR, Tohoku University,
2-1-1 Katahira, 980-8577 Sendai, Japan  Center for Spintronics Research
Network, Tohoku University, Sendai 980-8577, Japan  Zernike Institute for
Advanced Materials, University of Groningen, 9747 AG Groningen, Netherlands
National Institute for Materials Science, Tsukuba 305-0047, Japan

###### Abstract

Magnons are the elementary excitations of the magnetic order that carry spin,
momentum, and energy. Here we compare the magnon with the ferron, i.e. the
elementary excitation of the electric dipolar order, that transports
polarization and heat in ferroelectrics.

††journal: Journal of LaTeX Templates

##  1 Introduction

Much of condensed matter physics addresses weak excitations of materials and
devices from their ground states. More often than not the complications caused
by electron correlations can be captured by the concept of approximately non-
interacting quasi-particles with well-defined dispersion relations. Quasi-
particles are not eigenstates and eventually decay on characteristic time
scales that depend on material and environment. In conventional electric
insulators, the lowest energy excitations are lattice waves with associated
bosonic quasi-particles called phonons. The elementary excitations of magnetic
order are spin waves and their bosonic quasi-particles are the “magnons” [1].
In magnetic insulators, magnons and phonons coexist in the same phase space
and can form hybrid quasi-particles that may be called “magnon polarons”.
Yttrium iron garnet (YIG) is the material of choice to study magnons, phonons,
and magnon polarons because of their longevity in YIG bulk and thin-film
single crystals. For more than half a century, Professor Sergio Rezende and
his team seminally contributed to our understanding of magnons, phonons, and
their hybrids by experimental and theoretical research, see e.g. [2, 3].
Recently we extended Prof. Rezende’s Boltzmann theory of the spin Seebeck
effect [4] in magnetic insulators by introducing “ferrons”, i.e. the
excitation of the electric dipolar order in ferroelectrics (FEs), and explored
their ability to transport heat and polarization, including the associated
Seebeck and Peltier effects [5, 6]. Here we expound the analogies and
differences between magnets and FEs in their ground states [9] as well as in
their dynamic and transport properties, thereby comparing the magnonics of
ferromagnetic insulators with the “ferronics” of ferroelectric insulators.

##  2 Magnetism vs. ferroelectricity

###  2.1 Dipoles

A magnetic dipole $\mathbf{M}$ is called “Ampèrian” [7] because it generates
magnetic fields by the intrinsic or orbital angular momentum of charged
particles, implying broken time reversal symmetry. The interference with an
external magnetic field $\mathbf{H}$ causes the Zeeman interaction

$E_{M}=-\mathbf{M}\cdot\mu_{0}\mathbf{H}$ (1)

with vacuum permeability $\mu_{0}$ that leads to the Landau-Lifshitz equation
of motion.

$\mathbf{\dot{M}}=-\gamma\mathbf{M}\times\mathbf{H,}$ (2)

where $\gamma=g_{e}\mu_{B}/\hbar$ is the modulus of the gyromagnetic ratio,
$g_{e}$ is the electron g-factor and $\hbar=h/\left(2\pi\right)$ Planck’s
reduced constant.

Two local spins $\mathbf{S}_{1}$ and $\mathbf{S}_{2}$ with magnetic moments
$\mathbf{M}_{i}=-\gamma\mathbf{S}_{i}$ can couple by the exchange interaction

$E_{x}=-J\mathbf{S}_{1}\cdot\mathbf{S}_{2},$ (3)

where the exchange integral $J$ vanishes exponentially as a function of
distance between the local moments on an interatomic length scale. The
magneto- and electro-dipolar interactions have the same angle and distance
dependence, but different prefactors [10]. The interaction energy of two
parallel magnetic moments with $\mathbf{M}_{i}=\mu_{B}$ at a distance $r$ is

$F_{M}=\frac{\mu_{0}\mu_{B}^{2}}{4\pi}\frac{1}{r^{3}}\sim\frac{\alpha^{2}}{4\pi}\left(\frac{a_{B}}{r}\right)^{3},$
(4)

where $\alpha=1/137$ is the fine structure constant, $\mu_{B}$ the Bohr
magneton, and $a_{B}$ the Bohr radius.

An electric dipole $\mathbf{P}$ is called “Gilbertian” [8], i.e. a directional
difference of positive and negative charges. $\mathbf{P}$ breaks inversion
symmetry, but it is time-reversal invariant. A dipole interacts with an
electric field $\mathbf{E}$

$E_{P}=-\mathbf{P}\cdot\mathbf{E.}$ (5)

A point electric dipole is not affected by a constant electric field, so
$\mathbf{\dot{P}}=\mathbf{0.}$

There is no such thing as an exchange interaction between electric dipoles, so
they interact only electrostatically. The coupling of two parallel electric
dipoles with $P\sim ea_{B}\sim 10^{-29}\operatorname{C}\operatorname{m}$
scales like

$F_{P}=\frac{P^{2}}{4\pi\epsilon_{r}\epsilon_{0}}\frac{1}{r^{3}}\sim\frac{1}{4\pi\epsilon_{r}}\left(\frac{a_{B}}{r}\right)^{3},$
(6)

where $\epsilon_{r}$ is the relative dielectric constant of the medium that
embeds the dipoles. Therefore

$\frac{F_{P}}{F_{M}}=\frac{1}{\alpha^{2}\epsilon_{r}}\gg 1.$ (7)

###  2.2 Ground states

Magnets - The magnetic order in electric insulators can be described by the
Heisenberg model of local spins coupled by the exchange interaction $J$
between nearest neighbors (n.n.)

$H=-\frac{J}{2}\sum_{ij}^{n.n.}\mathbf{S}_{i}\cdot\mathbf{S}_{j}.$ (8)

For a positive $J$ the ground state is ferromagnetic with a coarse-grained
continuous field $\mathbf{m}M_{0},$ where
$M_{0}=\left|\gamma\mathbf{S}/\Omega\right|$ is the magnetization and $\Omega$
is the volume occupied by a single spin. The direction and position dependent
unit vector $\mathbf{m}$ of the ground state minimizes the magnetic energy
that includes the exchange energy and the Zeeman interaction with applied and
effective magnetic fields.

FEs - Ferroelectricity is the ordered state of a large number of microscopic
electric dipoles with an associated permanent electric polarization. Roughly
two types of FEs can be distinguished, viz. “displacive” and “order-disorder”.
In the former, the order emerges during a soft-phonon structural phase
transition that breaks inversion symmetry of the unit cell, while in the
latter stable molecular dipoles order during the phase transition. We can
write the unit cell dipole as $\mathbf{P}_{0}=Q\boldsymbol{\delta}$, where $Q$
is the ionic charge and $\boldsymbol{\delta}$ a displacement vector. The
polarization density $p_{0}=P_{0}/\Omega$, where $\Omega$ is the unit cell
volume.

FEs have a large dielectric constant $\epsilon_{r}\sim 10^{3},$ so according
to Eq. (7) $F_{P}/F_{M}\sim 20$. Since the magnetostatic interaction governed
by $F_{M}$ are typically in the GHz regime, the electric dipolar interactions
are still much smaller than ambient thermal energies. Since there is no
exchange interaction that order the dipoles, ferroelectricity appears to be a
consequence, but not the origin of the phase transition to the symmetry-broken
ground state. However, the dipolar interactions affect the polarization
direction and texture including domain formation in larger structures stronger
than in ferromagnets.

###  2.3 Excitations

Magnonics and ferronics address the small-amplitude dynamics of the dipolar
order at temperatures below the phase transition.

Magnets - In magnets, the low energy excitations are spin waves, i.e. plane-
wave-like modulated precessions around the equilibrium magnetization with a
time and position dependent phase. The exchange energy contribution
$\omega_{k}^{\left(\mathrm{x}\right)}\sim Jk^{2}$ vanishes at small wave
numbers $k.$ In this limit the magnetodipolar interaction with
$\omega_{k}^{\left(\mathrm{dip}\right)}-\omega_{0}^{\left(\mathrm{dip}\right)}\sim
k$ dominates. Spin and lattice waves coexist in the same regions of reciprocal
space. When their coupling by the magnetic anisotropy and magnetoelasticity is
larger than the level splitting and broadening, they form hybrid states or
magnon polarons that transport magnetization with the sound velocity [14]. The
interaction between spins and lattice is relativistic and in general weak, so
in most magnets magnon polarons form, if at all, only in relatively small
volumes of phase space.

FEs - The electric dipoles in FEs are defined by the coordinates of the
charged ions that have mass and are subject to electric fields and elastic
forces. In contrast to magnets, the polarization dynamics for $k\rightarrow 0$
is governed by transverse optical phonons in the THz regime. The dipolar
contribution to the restoring force constant can be estimated for a film
geometry as

$C_{P}=\frac{Q^{2}}{\epsilon_{r}\epsilon_{0}\Omega}=\mathcal{O}\left(10^{2}\,\mathrm{J/m^{2}}\right),$
(9)

which is much smaller than typical elastic ones. To a good approximation the
elementary excitations of the FE order therefore appear to be mechanical, i.e.
electrically polarized phonons rather than oscillating massless dipoles, which
are solutions of a lattice dynamics problem.

##  3 Transport

Transport of magnetic order in the form of spin currents is crucial in
spintronics and spin caloritronics [15]. Surprisingly, the transport of the FE
order has attracted little attention, see bibliography in [5]. Here we review
our formulation of polarization transport in FEs that is inspired by the
models for spin transport in magnetic insulators [4, 11, 14, 16].

###  3.1 Linear response

Magnets - Magnons in magnetic insulators carry energy and spin currents when
subject to temperature and magnetic field gradients $\partial T$ and $\partial
H$. In the linear response regime of a homogeneous and isotropic magnet, the
extended “Ohm’s Law” reads

$\left(\begin{array}[c]{c}-j_{m}\\\
j_{q}\end{array}\right)=\sigma_{m}\left(\begin{array}[c]{cc}1&S_{m}\\\
\Pi_{m}&\kappa/\sigma_{m}\end{array}\right)\left(\begin{array}[c]{c}\partial
H\\\ -\partial T\end{array}\right),$ (10)

where $j_{m}$ $(-j_{m})$ is the magnetization (magnon) current density defined
by the conservation law $\partial j_{m}=-\dot{M}$, $j_{q}$ is the heat current
density, $\sigma_{m}$ is a magnetization or magnon conductivity, $\kappa$ the
thermal conductivity, while $\Pi_{m}$ and $S_{m}=\Pi_{m}T$ are the spin
Peltier and Seebeck coefficients, respectively. The signs are chosen such that
the transport parameters are positive for a simple ferromagnet.

FEs - Similar equations govern electric polarization transport, but $H$ is
replaced by an electric field $E$:

$\left(\begin{array}[c]{c}-j_{p}\\\
j_{q}\end{array}\right)=\sigma_{p}\left(\begin{array}[c]{cc}1&S_{p}\\\
\Pi_{p}&\kappa/\sigma_{p}\end{array}\right)\left(\begin{array}[c]{c}\partial
E\\\ -\partial T\end{array}\right),$ (11)

and $j_{p}$ ($-j_{p})$ is the polarization (ferron) current density defined by
the conservation law $\partial j_{p}=-\dot{P}$, $\sigma_{p}$ is a polarization
or ferron conductivity, $\Pi_{p}$ and $S_{p}=\Pi_{p}T$ are the polarization
Peltier and Seebeck coefficients, respectively. The signs are chosen such that
the transport parameters are positive for simple FEs. The polarization current
should be distinguished from the displacement current $\dot{P},$ which is not
a transport property.

###  3.2 Diffusion theory

Magnets - In magnetic insulators, the diffusion picture of transport has
successfully made contact with experiments with only a small number of
adjustable parameters. At equilibrium with field $H_{0}$ and temperature
$T_{0}$ the magnons are distributed according to the Bose-Einstein function
with zero chemical potential (or Planck’s function)

$f_{BE}^{\left(0\right)}\left(\varepsilon\right)=\left[\exp\left(\frac{\varepsilon+\hbar\gamma
H_{0}}{k_{B}T_{0}}\right)-1\right]^{-1}.$ (12)

The crucial assumption is approximate local equilibration of the magnon
spectral distribution to the form [18]

$f_{BE}\left(x,\varepsilon\right)=\left[\exp\left(\frac{\varepsilon+\hbar\gamma
H\left(x\right)-\mu_{m}\left(x\right)}{k_{B}T_{m}\left(x\right)}\right)-1\right]^{-1}$
(13)

in terms of a slowly varying magnetic field $H\left(x\right)=H_{0}+\triangle
H\left(x\right),$ non-equilibrium magnon chemical potential or magnon
accumulation $\mu_{m}\left(x\right)$ and temperature
$T_{m}\left(x\right)=T_{0}+\triangle T_{m}\left(x\right)$. In magnetic
insulators, this assumption can be justified by efficient magnon-conserving
magnon-phonon and magnon-magnon scattering, while magnon-non-conserving
damping processes are weak [11]. The magnon number density is then reduced to

$-m\left(x\right)=\int
f_{BE}\left(x,\varepsilon\right)\rho_{m}\left(\varepsilon\right)\frac{d\varepsilon}{2\pi},$
(14)

where $\rho_{m}\left(\varepsilon\right)$ is the magnon energy density of
states. In linear response and otherwise homogeneous systems

$\displaystyle f_{BE}\left(x,\varepsilon\right)$
$\displaystyle=f_{BE}^{\left(0\right)}\left[1-f_{BE}^{\left(0\right)}\exp\left(-\frac{\varepsilon+\hbar\gamma
H_{0}}{k_{B}T_{0}}\right)\right.$
$\displaystyle\left.\left(\frac{\hbar\gamma\triangle
H-\mu_{m}}{k_{B}T_{0}}+\frac{\left(\varepsilon+\hbar\gamma
H_{0}\right)\triangle T}{k_{B}T_{0}^{2}}\right)\right]$ (15)

and

$m\left(x\right)-m_{0}=\chi_{m}\left(\triangle
H-\frac{\mu_{m}}{\hbar\gamma}\right)+\chi_{T}\triangle T_{m},$ (16)

where $m_{0}$ is the thermal equilibrium value. The susceptibilities

$\displaystyle k_{B}T_{0}\chi_{m}$
$\displaystyle=-\int\left(f_{BE}^{\left(0\right)}\right)^{2}e^{-\frac{\varepsilon+\hbar\gamma
H_{0}}{k_{B}T_{0}}}\rho_{m}\frac{d\varepsilon}{2\pi}$ (17) $\displaystyle
k_{B}T_{0}^{2}\chi_{T}$
$\displaystyle=-\int\left(f_{BE}^{\left(0\right)}\right)^{2}e^{-\frac{\varepsilon+\hbar\gamma
H_{0}}{k_{B}T_{0}}}\left(\varepsilon+\hbar\gamma
H_{0}\right)\rho_{m}\frac{d\varepsilon}{2\pi}$ (18)

parameterize the response to _constant_ field and temperature changes and are
crucial parameters for magnetocaloric effects. The finite lifetime
$\tau_{m}\left(\tau_{q}\right)$ of the magnon (energy) density modifies the
conservation relations

$\displaystyle-\partial j_{m}$
$\displaystyle=\dot{m}+\frac{m-m_{0}}{\tau_{m}}$ (19) $\displaystyle-\partial
j_{q}$ $\displaystyle=\dot{q}+\frac{q-q_{0}}{\tau_{q}}.$ (20)

where the subscript $0$ indicates the equilibrated values. Efficient magnon-
conserving magnon-phonon interactions render a short $\tau_{q}$ and the
lattice has a large heat capacity. Then $T=T_{0}$, i.e. the magnon temperature
equals the lattice temperature everywhere. This approximation should be
reconsidered in the low temperature regime [17]. Combining Eqs. (10,14-19) and
assuming $\partial^{2}H=0$ we find for the steady state

$\partial^{2}\mu_{m}=\frac{\mu_{m}}{\ell_{m}^{2}}.$ (21)

with magnon diffusion length $\ell_{m}=\sqrt{\sigma_{m}\tau_{m}/\chi_{m}}.$
The transport parameter $\sigma_{m}$ indicates the realm of spin caloritronics
[15]. We note that the divergence of a spin accumulation is equivalent to
magnetic charges and stray magnetic fields that most studies neglect.

FEs - By the same arguments, we arrive at the polarization diffusion equation
for the chemical potential or polarization accumulation $\mu_{p}$

$\partial^{2}\mu_{p}=\frac{\mu_{p}}{\ell_{p}^{2}}.$ (22)

with $\ell_{p}=\sqrt{\sigma_{p}\tau_{p}/\chi_{p}}$. Equation (22) relies on
assumptions that are well-tested for magnetic insulators at room temperature,
but not for ferroelectrics. The existence of an accumulation $\mu_{p}$ as a
driving force for transport implies that the relaxation time $\tau_{p}$ is
much larger than the scattering life time $\tau_{r}$ that limits the
conductivity $\sigma_{p}.$ In contrast to a magnon accumulation, a
polarization accumulation is not protected in the non-relativistic limit,
however, so the assumption $\tau_{r}\gg\tau_{p}$ should be better justified.
An example of a scattering process that limits transport but conserves
polarization is intermode phonon back scattering by defects.$\tau_{p}$ should
be measurable by the electrocaloric response to small amplitude pulsed or AC
electric fields along the equilibrium polarization. Polarization accumulations
betray their presence with gradients that generate space electric charges and
observable electric fields.

##  4 Microscopic theory

Here we discuss theories that address the material and device-dependent
parameters discussed above. In magnonics, the results compare well with the
parameters fitted to experiments. No experiments are available for
ferroelectrics, however.

###  4.1 Excitations

Magnets - The Heisenberg model is the starting point of most calculations. For
large spins such as $S=5/2$ for half-filled 3d-shells, atomistic simulations
give an appropriate picture for the spatiotemporal dynamics at finite
temperatures [12]. In the long wave-length limit, the Landau-Lifshitz Gilbert
(LLG) equation

$\mathbf{\dot{m}=}-\gamma\mathbf{m}\times\mathbf{H}_{\mathrm{eff}}+\alpha_{G}\mathbf{m}\times\mathbf{\dot{m}}$
(23)

is well established, in which $\mathbf{H}_{\mathrm{eff}}$ is an effective
magnetic field and $\alpha_{G}$ the Gilbert damping constant.
$\mathbf{H}_{\mathrm{eff}}$ contains a stochastic term with a correlation
function that in equilibrium obeys the fluctuation-dissipation theorem in
terms of the Gilbert damping and temperature, and can be used to compute the
spin-spin correlation and response functions [12]. The linearized LLG
equation, equivalent to the Heisenberg equation to lowest order in the
Holstein-Primakoff expansion [1], is appropriate for weakly or thermally
excited magnon gases. Its solutions are the magnon dispersion relations,
amplitudes, and group velocities, which are the starting point for transport
theories such as the Boltzmann formalism. The magnetization dynamics consists
of small transverse fluctuations $\mathbf{m}_{\bot}$ that reduce the net
magnetization magnetization $M$ relative to the ground state $M_{0}$

$-m_{z}=\frac{\mathbf{m}_{\bot}^{2}}{2}.$ (24)

The (quantum) thermal average $-\overline{m_{z}}$ may be interpreted as the
local number of magnons and
$-m=\overline{\mathbf{m}_{\bot}^{2}}M_{0}/\left(2g\mu_{B}\right)$ in Eq. (14).

Magnetoelastic interactions mix the magnons with phonons to create hybrid
magnon polarons in reciprocal space that are beyond the scope of
micromagnetics. An appropriate model for small wave numbers is an ensemble of
coupled harmonic oscillators with weak level repulsions at the crossing points
of the magnon and phonon dispersions [14].

FEs - We focus on the symmetry-broken phase at temperatures well below the
phase transition with a finite macroscopic polarization per unit cell

$\mathbf{P}_{0}=\sum_{s}^{\mathrm{unit\,cell}}Q_{s}\mathbf{r}_{s}^{\left(0\right)}.$
(25)

$\mathbf{r}_{s}^{\left(0\right)}$ is the location of the $s$-th ion in a unit
cell with net charge $Q_{s}$ and $\sum_{s}Q_{s}=0.$ We do not address here the
complications caused by surface charges. Finite temperatures or external
excitations induce lattice vibrations that affect the polarization. The
elementary excitation of the crystal is a phonon with wave number $\mathbf{k}$
in a band $\sigma,$ polarization $\mathbf{e}_{\mathbf{k}\sigma}$ and frequency
$\omega_{\mathbf{k}\sigma}$ that modulates the position $\mathbf{r}_{ls}$ of
an ion $s$ with mass $M_{s}$ in the $3N$ unit cells with index
$l=\left\\{1,\cdots,3N\right\\}:$

${\boldsymbol{u}}_{\mathbf{k}\sigma}(l,s)=\frac{1}{\sqrt{NM_{s}}}\mathbf{e}_{\mathbf{k}\sigma}(s)e^{i\mathbf{k\cdot
r}_{l}-i\omega_{\mathbf{k}\sigma}t}.$ (26)

The local dipole then fluctuates according to

$\triangle\mathbf{P}_{l}=\sum_{\mathbf{k}\sigma
s}a_{\mathbf{k}\sigma}Q_{s}{\boldsymbol{u}}_{\mathbf{k}\sigma}(l,s)$ (27)

where $a_{\mathbf{k}\sigma}$ is the phonon amplitude. The polarization
$\mathbf{e}$ has components along and normal to the equilibrium polarization.
The dipolar dynamics consists of a rotation by an angle $\theta$ normal to and
a deformation in the direction $z$ of the dipolar order, which modifies the
projection

$\left(\triangle P_{l}\right)_{z}=\left(P_{0}+\triangle
P_{l}^{\parallel}\right)\cos\theta.$ (28)

Since harmonic oscillators do not change the average position of the ions and
therefore the polarization, we have to take into account non-linearities. In
magnets, the high energy cost of changing the modulus of the magnetization
leads to the transverse dynamics described by the LLG Eq. (23). In “order-
disorder” FEs such as KNO3 or NaNO2 we have a similar situation, because the
nitrate and nitrite molecular units are characterized by stable permanent
dipoles. The low-frequency phonons are then polarized along the minimum energy
path that switches the polarization without deforming the strongly bound
molecular units. Small transverse fluctuations
$\triangle\mathbf{P}_{l}^{\perp}$ reduce the polarization projection and
thereby the macroscopic polarization by

$\left(\triangle
P_{l}\right)_{z}=-\frac{\left|\triangle\mathbf{P}_{l}^{\perp}\right|^{2}}{2P_{0}}+\triangle
P_{l}^{\parallel}+\mathcal{O}[(\triangle\mathbf{P}_{l}^{\perp})^{2},\triangle
P_{l}^{\parallel}].$ (29)

Here the longitudinal fluctuations $\triangle P_{l}^{\parallel}$ are
corrections that are disregarded in the Landau-Lifshitz-Gilbert dynamics of
the magnetization (24). The transverse fluctuations or _ferrons_ reduce the
polarization, just as the magnons reduce the magnetization. This “ferron”
approximation should be accurate for the order-disorder type as argued above,
but it is as yet untested for displacive FE’s.

The averaged polarization is reduced by

$\overline{\triangle p}=\frac{1}{V}\sum_{\mathbf{k}\sigma}\triangle
p_{\mathbf{k}\sigma}=-\frac{1}{2P_{0}V}\sum_{\mathbf{k}\sigma}\left|a_{\mathbf{k}\sigma}\right|^{2}\left|\mathbf{F}_{\mathbf{k}\sigma}\right|^{2}$
(30)

where

$\mathbf{F}_{\mathbf{k}\sigma}=\sum_{s}\frac{Q_{s}}{\sqrt{M_{s}}}\mathbf{e}_{\mathbf{k}\sigma}^{\perp}(s)$
(31)

After quantizing the oscillators, we arrive at the thermally averaged
polarization per unit cell

$\overline{\triangle
p}=-\frac{\hbar^{2}}{4P_{0}V}\sum_{\mathbf{k}\sigma}\frac{\left|\mathbf{F}_{\mathbf{k}\sigma}\right|^{2}}{\varepsilon_{\mathbf{k}\sigma}}f_{BE}^{\left(0\right)}\left(\varepsilon_{\mathbf{k}\sigma}\right)\
$ (32)

and the susceptibilities that govern the electrocaloric and pyroelectric
susceptibilities $\chi_{p}=\partial\overline{\triangle p}$/$\partial E$ and
$\chi_{T}=\partial\overline{\triangle p}$/$\partial T.$ $-\overline{\triangle
p}/p_{0}$ may be interpreted as an effective number of ferron excitations. In
contrast to the pure magnon case, but similar to that for the magnon polaron,
each quasiparticle excitation contributes with a state-dependent weight to the
reduction of the polarization.

These properties can be computed by conventional lattice dynamics codes for
realistic models. We may capture the essential physics by a one-dimensional
harmonic oscillator model of a diatomic chain of atoms with equal masses $M,$
opposite charges $\pm Q$, and force constants for longitudinal
$\left(C_{L}\right)$ and transverse $\left(C_{T}\right)$ motions. The FE phase
transition shifts the ions in each unit cell to generate a permanent electric
dipole $\mathbf{P}_{0}=Q\boldsymbol{\delta}$ with shift vector
$\boldsymbol{\delta}$. The polarization $\mathbf{p}_{0}=\mathbf{P}_{0}/a^{3}$
can point in any direction, but we focus here on a dipolar order along or
normal to the chain. The chain has three phonon branches, one longitudinal and
two transverse modes, in a Brillouin zone with boundaries
$\left|k\right|\leq\pi/a$. The FE transition doubles the size of the unit
cell, folding the bands at $\left|k\right|\leq\pi/\left(2a\right)$ into
acoustic and optical ones. We simplify this already primitive model even
further by assuming that all transverse force constants are the same. The
longitudinal phonons are not ferroelectrically active but contribute to the
heat conductance. We may entirely disregard the high-frequency longitudinal
optical phonon mode. In [6] we plot the phonon bands for the two main
polarization directions. Remarkably, the polarization of the acoustic mode can
be switched off completely by rotating the direction of the FE order from in-
chain to perpendicular to the chain. This should generate drastic effects on
the caloric and caloritronic properties.

###  4.2 Transport

The transport coefficients can be computed straightforwardly with the above
models for diffuse and ballistic transport. Rezende et al. [4] and Cornelissen
et al. [11] formulated a Boltzmann equation in the relaxation time
approximation, where the latter focusses on the role of the magnon chemical
potential. Flebus et al.’s [14] spin Seebeck effect theory for magnon polarons
can be easily adapted to handle ferrons. Meier and Loss’ [16] scattering
theory for magnon transport in ballistic spin chains inspired our ferron
transport formulation [6]

####  4.2.1 Boltzmann equation

Magnets - The starting point of the Boltzmann formalism is the non-equilibrium
distribution function in real and reciprocal space
$f_{\sigma}\left(\mathbf{k},\mathbf{r}\right)$ that in equilibrium reduces to
$f_{BE}^{\left(0\right)}\left(\varepsilon_{\mathbf{k}\sigma}+\hbar\gamma
H_{0}\right).$ In the steady state, constant relaxation time $\tau_{r},$ and
one spatial dimension $x,$ the distribution is modified by a field or
temperature gradient as

$\displaystyle\triangle f_{\sigma}\left(k,x\right)$
$\displaystyle=\tau_{r}\frac{\partial\varepsilon_{k\sigma}}{\hbar\partial
k}\frac{\partial f_{\sigma}\left(k,x\right)}{\partial x}$
$\displaystyle=-\tau_{r}\frac{\partial\varepsilon_{k\sigma}}{\hbar\partial
k}\left(\frac{\hbar\gamma\partial
H(x)-\partial\mu(x)}{k_{B}T}-\frac{\varepsilon_{k\sigma}+\hbar\gamma
H_{0}}{\hbar k_{B}T^{2}}\partial T\right)$
$\displaystyle\times\left(f_{BE}^{\left(0\right)}\right)^{2}e^{\frac{\varepsilon_{k\sigma}+\hbar\gamma
H_{0}}{k_{B}T}},$ (33)

where we used the expansion Eq. (15). Here we assume that the relaxation time
$\tau_{r}$ that includes all scattering processes is shorter than $\tau_{m}$
in the spin diffusion equation.

The magnon spin current in the magnon-polaron system for constant gradients
then, for example, reads

$-j_{m}=\sum_{k\sigma}\triangle m_{k\sigma}\triangle
f_{\sigma}\left(k\right)=\sigma_{m}\left(\partial
H-\frac{\partial\mu}{\hbar\gamma}\right)-\sigma_{m}S_{m}\partial T,$ (34)

which leads to the conductivity

$\sigma_{m}=\frac{\tau_{r}}{k_{B}T}\sum_{k\sigma}\triangle
m_{k\sigma}\left(\frac{\partial\varepsilon_{k\sigma}}{\hbar\partial
k}f_{BE}^{\left(0\right)}\right)^{2}e^{\frac{\varepsilon_{k\sigma}+\hbar\gamma
H_{0}}{k_{B}T}},$ (35)

where $\triangle m_{k\sigma}$ is the magnetic component in the magnon-polaron
wave with index $k\sigma,$ which follows from diagonalizing the magnetoelastic
Hamiltonian [14].

FEs - The transport coefficients in ferroelectrics can be computed under the
ferron approximation analogously ($E_{0}=0$)

$\sigma_{p}=\frac{\tau_{r}}{k_{B}T}\sum_{k\sigma}\triangle
p_{k\sigma}\left(\frac{\partial\varepsilon_{k\sigma}}{\hbar\partial
k}f_{BE}^{\left(0\right)}\right)^{2}e^{\frac{\varepsilon_{k\sigma}}{k_{B}T}}.$
(36)

where $\triangle p_{k\sigma}$ is the polarization (change) of a phonon in
state $k\sigma$ introduced in Eqs. (30,31). The transport coefficients in the
1D model with polarization parallel to the gradients can be obtained
analytically in the high and low temperature limits [5].

####  4.2.2 Ballistic transport

Magnets - The Landauer-Büttiker scattering theory of transport is designed to
treat transport that is limited by geometry, for example, by point contacts
that are connected adiabatically to large thermodynamic reservoirs. Eq. (11)
then becomes

$\left(\begin{array}[c]{c}-J_{m}\\\
J_{q}\end{array}\right)=G_{m}\left(\begin{array}[c]{cc}1&S_{m}\\\
\Pi_{m}&K/G_{m}\end{array}\right)\left(\begin{array}[c]{c}\triangle H\\\
-\triangle T\end{array}\right),$ (37)

where the driving forces are now the field and temperature differences, $K$
and $G_{m}$ are heat and polarization conductances, respectively, and on the
l.h.s. we have currents rather than current densities. The transport
coefficients are governed by the matrix of transmission probabilities
$\mathbf{T}$ of magnons that propagate from the left to the right, which in
the absence of scattering becomes a unit matrix in the space of propagating
states. The magnon conductance of the spin chain in the low temperature limit
[16]

$G_{m}=\hbar\gamma^{2}f_{BE}^{\left(0\right)}\left(\varepsilon_{m}\right).$
(38)

$H_{\mathrm{eff}}$ in $\varepsilon_{m}=\hbar\gamma H_{\mathrm{eff}}$
represents applied or magnetic anisotropy fields that fix the direction of the
magnetic order. $G_{m}\sim e^{-\varepsilon_{m}/\left(k_{B}T\right)}$ vanishes
exponentially when the temperature falls below the magnon gap.

FEs - In ferroelectrics

$\left(\begin{array}[c]{c}-J_{p}\\\
J_{q}\end{array}\right)=G_{p}\left(\begin{array}[c]{cc}1&S_{p}\\\
\Pi_{p}&K/G_{p}\end{array}\right)\left(\begin{array}[c]{c}\triangle E\\\
-\triangle T\end{array}\right),$ (39)

where, e.g., the polarization conductance for the dipolar chain [6]

$G_{p}=\frac{1}{k_{B}T}\sum_{\sigma}\int\left(\xi_{\mathbf{p}}f_{BE}^{\left(0\right)}\right)^{2}e^{\varepsilon/(k_{B}T)}\frac{{d\varepsilon}}{{h}}.$
(40)

$G_{p}$ may be compared with the conductivity $\sigma_{p}$ of a diffuse wire
(36). The latter differs by being proportional to the relaxation time and the
increased importance of the group velocities in the integral.

For perpendicular FE order only the high-frequency optical branch contributes
with

$\xi_{\mathbf{p}}=-\frac{\left(\hbar Q\right)^{2}}{MP_{0}\varepsilon}\text{
for }\mathbf{p}\perp\mathbf{\hat{x},}$ (41)

while for parallel polarization with $\varepsilon\geq 0$

$\xi_{\mathbf{p}}=-\frac{\varepsilon P_{0}}{4C\delta^{2}}\text{ for
}\mathbf{p}\|\mathbf{\hat{x}.}$ (42)

We see that the ferron conductance dramatically differs from the magnon
conductance by its strong dependence on the direction of the dipolar order.
When $\mathbf{p}\perp\mathbf{\hat{x}}$, $G_{p}$ vanishes exponentially with
temperature because only the high-frequency optical phonon branch contributes.
For $\mathbf{p}\|\mathbf{\hat{x},}$ however, polarization transport is gapless
and the ferron conductance scales linearly with low temperatures $G_{p}\sim T\
$[6].

##  5 Devices

The theory can be tested by experiments with concrete devices fabricated from
different materials and flexible configurations. Magnonics has a great
advantage: magnon currents can be measured when injected into heavy metal
contacts by means of a transverse electromotive force induced by the inverse
spin Hall effect [19, 20, 21].

We are not aware of a “polarization Hall effect” that could detect a ferron
current. Nevertheless, ferrons do cause observable effects. Here we introduce
two simple devices that are tailored to find evidence for a polarization
current in the diffuse and ballistic regimes.

###  5.1 Planar capacitors

A capacitor is a slab of a dielectric insulator between metal contacts. Taking
advantage of the spin Hall effects, films with one or two Pt contacts on YIG
have been investigated extensively. Electric charges and capacitances appear
to play only a minor role in magnonics, however.

Planar FE capacitors are common devices because the high dielectric constant
of FE’s ensures high capacitances and the switchability of the FE order allows
their use in FE random access memory (FeRAM) cells. Metallic contacts screen
the FE surface charges in equilibrium and residual electric fields are small
[22]. A global temperature change modulates the equilibrium polarization and
generates thermovoltages, while applied voltage steps heat or cool the FE
spacer, governed by susceptibilities $\chi_{p}$ and $\chi_{T}$.

A temperature gradient generates a directional heat and associated
polarization current, but because $\partial E=0$ in these devices, an applied
voltage difference does not. We expect that a polarization current injected
into a metal contact decays on a very short length scale without obvious
observable signatures. However, opaque contacts accumulate a polarization on
the length scale of the diffusion length $\ell_{p}$. In an open-circuit
configuration, a thermally excited polarization $p\left(x\right)$ generates a
voltage over the contacts

$\triangle V=-\int_{0}^{L}\frac{\triangle
p\left(x\right)}{\epsilon}dx=-\frac{\chi_{E}}{\epsilon
P}\int_{0}^{L}\mu\left(x\right)dx,$ (43)

where $L$ is the thickness of the FE barrier. In a capacitor with one contact
interface opaque and another one transparent and in the limit $\ell_{p}\gg L$
becomes $\triangle V_{\mathrm{\max}}=-\chi_{E}LS\triangle
T_{\mathrm{ext}}/\left(2\epsilon_{0}\epsilon_{r}\right)$. $\triangle V$ can be
used by inserting the capacitor into an electric circuit, analogous to
pyroelectric devices, but operating on temperature differences rather than
global temperature changes. The effects can be switched off at temperatures
below the optical phonon band by rotating the polarization, as discussed
above.

Returning to magnonics: a temperature gradient over a capacitor with a
electrically insulating magnetic spacer generates a magnon accumulation and
stray magnetic fields, which to the best of our knowledge have not yet been
reported, however.

###  5.2 Lateral structures

More flexible than the planar capacitors are lateral structures on and of thin
films. Propagating magnons can be injected and detected by narrow microwave
striplines [23], while Pt contacts on YIG films allow the study of diffuse
magnon transport by injecting them by the spin Hall effect or Ohmic heating
and detection in another contact by the emf generated by the inverse spin Hall
effect from spin pumping [24].

We envisage analogous experiments in gated thin FE films. Their polarization
can be perpendicular to the plane, either spontaneously or induced by gates.
The ferroelectricity in few-monolayers of van der Waals materials such as
bilayer boron nitride in an especially interesting phenomenon [25]. In
contrast to the planar capacitor, it should be easy to generate electric field
gradients and electrically controlled polarization currents. Thin films may be
nanostructured such that large islands are connected by point contacts or
strips over which the electric field and temperature drop selectively, see
Figure 1. The scattering theory of transport is applicable and may be
approximated by the single wire model introduced above. Additional local
electrostatic gates should allow rotation of the FE order in the constriction.
These structures are open to local probes that can measure, for instance, the
significant magnetic stray fields predicted for the electric dipolar current
through the constriction [6].

Figure 1: Lateral polarization valve on a thin film FE. The two large FE pads
are thermodynamic reservoirs assumed to be at local thermal equilibrium, but
at variable electric fields and temperatures, controlled by metallic gates
with voltages $\Delta V_{1}$ and $\Delta V_{2}$ as well as temperatures
(indicated by the color). Here a voltage $\Delta V_{G}$ over side gates
controls the FE polarization direction angle with the sample plane.

##  6 Summary and outlook

Magnonics has developed into one of the most active fields in magnetism and
spintronics, while the field of “ferronics” does not yet exist. Our models are
primitive, some approximations are untested, and essential parameters are not
known. Experimental results are necessary to guide better theoretical models
and understanding. We hope that our initial theoretical steps will motivate
experimentalists with frontier spirit to test our predictions and on the way
discover completely new physics and applicable new functionalities. Realistic
lattice dynamic calculations of the polarization amplitudes in ferroelectric
materials can assess the accuracy of the ferron approximation. The
polarization relaxation time and diffusion lengths are key parameters that
have to be established with better accuracy. We have only scratched the
surface of the field, not addressing fascinating extensions of ferronics that
include antiferroelectrics, ferroelectric textures, polar metals, and
multiferroics.

## Acknowledgements

We thank K. Yamamoto, J. Heremans, and B. van Wees for helpful discussions.
G.B. acknowledges support by JSPS KAKENHI Grant No. 19H00645. R. I. and K. U.
are supported by JSPS KAKENHI Grant No. 20H02609, JST CREST “Creation of
Innovative Core Technologies for Nano-Enabled Thermal Management” Grant No.
JPMJCR17I1, and the Canon Foundation.

## References

  * [1] S.M. Rezende, Fundamentals of Magnonics (Springer Nature, Heidelberg, 2020) ISBN 978-3-030-41317-0 .
  * [2] S.M. Rezende and F. R. Morgenthaler, Magnetoelastic Waves in Pulsed Magnetic Fields, Appl. Phys. Lett. 11, 24 (1967), DOI: 10.1063/1.1754945 .
  * [3] S. M. Rezende, D. S. Maior, O. Alves Santos, and J. Holanda, Theory for phonon pumping by magnonic spin currents, Phys. Rev. B 103, 144430 (2021), DOI: 10.1103/PhysRevB.103.144430 .
  * [4] S. M. Rezende, R. L. Rodríguez-Suárez, R. O. Cunha, A. R. Rodrigues, F. L. A. Machado, G. A. Fonseca Guerra, J. C. Lopez Ortiz, and A. Azevedo, Magnon spin-current theory for the longitudinal spin-Seebeck effect, Phys. Rev. B 89, 014416 (2014), DOI: 10.1103/PhysRevB.89.014416 .
  * [5] G.E.W. Bauer, R. Iguchi, and K. Uchida, Theory of Transport in Ferroelectric Capacitors, Phys. Rev. Lett. 126, 187603 (1-4) (2021), DOI: 10.1103/PhysRevLett.126.187603
  * [6] P. Tang, R. Iguchi, K. Uchida and G.E.W. Bauer, The Ferroelectric Point Contact, arXiv:2105.14791 .
  * [7] A.-M. Ampère, La détermination de la formule qui represénte l’action mutuelle de deux portions infiniment petites de conducteur Voltaïques, Ann. Chem. Phys. 20, 398 (1922).
  * [8] W. Gilbert, De Magnete (1700),
  * [9] N.A. Spaldin, Topics Appl. Phys. 105, 175 (2007).
  * [10] P. Chandra and P. B. Littlewood, Topics Appl. Physics 105, 69 (2007).
  * [11] L.J. Cornelissen, K.J.H. Peters, G.E.W. Bauer, R.A. Duine, and B.J. van Wees, Phys. Rev. B 94. 014412 (2016).
  * [12] J. Barker and G.E.W. Bauer, Semiquantum thermodynamics of complex ferrimagnets, Phys. Rev. B 100, 140401(R) (2019), DOI: 10.1103/PhysRevB.100.140401.
  * [13] N.A. Spaldin, A beginner’s guide to the modern theory of polarization, J. Sol. St. Chem. 195, 2 (2012), DOI: 10.1016/j.jssc.2012.05.010 .
  * [14] B. Flebus, K. Shen, T. Kikkawa, K. Uchida, Z. Qiu, E. Saitoh, R.A. Duine, and G. E. W. Bauer, Phys. Rev. B 95, 144420 (2017).
  * [15] G.E.W. Bauer, E. Saitoh, and B.J. van Wees, Spin Caloritronics, Nat. Mat. 11, 391 (2011).
  * [16] F. Meier and D. Loss, Magnetization transport and quantized spin conductance, Phys. Rev. Lett. 90, 167204 (2003).
  * [17] M. Elyasi and G. E. W. Bauer, Cryogenic spin Seebeck effect, Phys. Rev. B 103, 054436 (2021), DOI: 10.1103/PhysRevB.101.054402.
  * [18] R. A. Duine, A. Brataas, S. A. Bender, and Y. Tserkovnyak, In: Universal Themes of Bose-Einstein Condensation, edited by N. Proukakis, D, Snoke, and P. Littlewood (Cambridge University Press, 2017), arXiv:1505.01329v1.
  * [19] A. Azevedo, L.H. Vilela Leão, R. L. Rodriguez-Suarez, A. B. Oliveira, and S. M. Rezende, dc effect in ferromagnetic resonance: Evidence of the spin-pumping effect?, J. Appl. Phys. 97, 10C715 (2005), DOI: 10.1063/1.1855251.
  * [20] E. Saitoh, M. Ueda, and H. Miyajima, Conversion of spin current into charge current at room temperature: Inverse spin-Hall effect, Appl. Phys. Lett. 88, 182509 (2006), DOI: 10.1063/1.2199473 .
  * [21] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Spin Seebeck insulator, Nat. Mat. 9, 894 (2010), DOI: 10.1038/nmat2856.
  * [22] R. R. Mehta, B. D. Silverman, and J. T. Jacobs, J. Appl. Phys. 44, 3379 (1973).
  * [23] H. Wang, J. Chen, T. Yu, C. Liu, C. Guo, H. Jia, S. Liu, K. Shen, T. Liu, J. Zhang, M. A. Cabero Z, Q. Song, S. Tu, M. Wu, X. Han, K. Xia, D. Yu, G. E. W. Bauer and H. Yu, Nonreciprocal coherent coupling of nanomagnets by exchange spin waves, Nano Research, s12274-020-3251-5 (2020), DOI: 10.1007/s12274-020-3251-5.
  * [24] L.J. Cornelissen, J. Liu, R.A. Duine, J. Ben Youssef, B.J. van Wees, Long-distance transport of magnon spin information in a magnetic insulator at room temperature, Nat. Phys. 11, 1022 (2015), DOI: 10.1038/nphys3465 .
  * [25] Z. Zheng, Q. Ma, Z. Bi, S. de la Barrera, M.-H. Liu, N. Mao, Y. Zhang, N. Kiper, K. Watanabe, T. Taniguchi, J. Kong, W. A. Tisdale, R. Ashoori, N. Gedik, L. Fu, S.-Y. Xu & P. Jarillo-Herrero, Unconventional ferroelectricity in moiré heterostructures, Nature 588, 71 (2020), DOI: 10.1038/s41586-020-2970-9.