# Nonlinear spin dynamics of ferromagnetic ring in the vortex state and its
application for spin-transfer nano-oscillator

Vera Uzunova  Institute of Theoretical Physics, Faculty of Physics, University
of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland  Institute of Physics of
the National Academy of Sciences of Ukraine, 46 Nauky Ave., 03039 Kyiv,
Ukraine  Boris A. Ivanov<EMAIL_ADDRESS>Institute of Magnetism,
National Academy of Sciences of Ukraine, 03142 Kiev, Ukraine  Institute for
Molecules and Materials, Radboud University Nijmegen, Heyendaalseweg 135 6525
AJ Nijmegen The Netherlands

###### Abstract

We study a nonlinear spin dynamics of a ferromagnetic ring in a vortex state
induced by the spin-polarized current. We also suggest to use the
ferromagnetic ring as a free layer of a coreless vortex spin-transfer nano-
oscillator. The calculated working frequency is about several GHz, that is
much higher than the gyromode frequency of the disk-based vortex oscillator.
The response of the vortex-state ring to the spin-polarized current has
hysteretic behavior with the reasonable values of the thresholds current
densities: ignition threshold is about $10^{8}\text{A}\text{cm}^{-2}$, and
elimination current to maintain the oscillations has much lower values about
$10^{6}\text{A}\text{cm}^{-2}$. The output signal can be extracted by the help
of the inverse spin Hall effect or by the giant magnetoresistance. The output
electromotive force averaged over all sample vanishes, and we suggest to use a
ferromagnetic ring or disk in a vortex state as a GMR analyzer. For an inverse
spin Hall analyser we advise to use two heavy metals with different signs of
Spin-Hall angle. The ring-based STNO is supposed to increase the areas of
practical application of the STNOs.

###### pacs:

75.47.-m., 75.30.Ds, 75.40.Gb., 85.75.-d

##  I Introduction

Spin-polarized current flowing into a magnetic material creates a large torque
acting on the magnetization due to the direct transfer of spin angular
momentum, that opens up opportunities to manipulate mesoscopic magnetic
elements Slonczewski . Discovery of a spin transfer mechanism laid the
foundation of a new generation of spintronic devices Kiselev ; BA7 ; Ralph ;
Bader , including auto-oscillators Ruiz-Calaforra ; ISHE1 ; ISHE2 ;
Houssameddine , magnetic random access memory Vaysset  and logic gates
Murugesh ; BA5 .

One of the key spintronic elements applicable in wireless communications
technology are spin-transfer nano-oscillators (STNOs), GMR2 . These devices
are using the spin transfer torque to generate microwave signals by
transforming energy from a dc electric current into high-frequency magnetic
oscillations. Traditionally designed STNOs are fabricated as sandwich
structures of magnetic and nonmagnetic nanolayers in forms of nanocontacts
Consoloa ; Soucaille ; Houshang  or nanopillars, Tarequzzaman ; Zeng ;
GuslienkoAranda . The main structural elements of the STNO are a free layer,
i. e. a thin magnetic layer allowing rotation of the magnetization, a
polarizer and an analyzer. Polarizer, the thicker magnetic layer with
relatively fixed magnetization, gives a spin polarisation to direct current
passing through the device. Above a certain critical current density its spin-
transfer torque acting on a thinner free magnetic layer can locally overcome
the intrinsic damping and excites a steady-state spin auto-oscillations on one
or more spin-waves modes of the system. In the analyser these spin
oscillations are converted to a microwave power by the magnetoresistive effect
or by the Inverce Spin Hall effect.

Advantages of the STNOs are small size, broad range of working temperatures
and easy integration into standard semiconductor technology. A characteristic
feature of all STNOs is nonlinear dynamics of the free layer magnetization.
Typical current densities for creating a steady-state spin dynamics are
$10^{7}-10^{8}\text{A}\text{cm}^{-2}$ in nanocontacts and
$10^{6}-10^{7}\text{A}\text{cm}^{-2}$ in nanopillars Zeng1 .

A promising alternative to the single-domain oscillators are vortex STNOs
based on ferromagnetic nano-disks in a so-called vortex state. The vortex
configuration has a closed magnetic flux and creates demagnetising fields only
in a small region of the vortex core. Therefore magnetic vortex can realize
the ground state of magnetic nanoparticles. The vortex STNOs are characterized
by a uniquely narrow line width, require lower current densities and dissipate
less energy comparing to single-domain oscillators Zvezdin2009 ; Zaspel_07 .
They are useful for applications in arrays of oscillators, memory systems on
vortex states, and others. The primary excitable mode in the vortex-state
particle is the gyrotropic mode (gyromode) that corresponds to the slow
precessional motion of the vortex around the disc center. The frequency of the
gyromode is usually lower then 1 GHz. This slow dynamics of the vortex core is
used in the standard design of the vortex-state STNO. Operating frequencies of
the vortex oscillators are low compared to single-domain oscillators, where
frequencies of $10$-$15$ GHz can be achieved.

The generated frequency can be doubled by using of a dual-free-layer STNO
because of superposition of two opposite current-driven precessional motions
of vortices in two free layers, Prokopenko . One more way to increase the
working frequency of the vortex-state STNOs is to use ferrimagnets near the
point of compensation of the angular momentum. They have gyrotropic
frequencies much higher than those of ferromagnets Tserkovnyak ;
ZaspelGalkinaIvanov19  that allows the operating frequency up to $20$ GHz.
Limitations of this design are related to the larger critical size of the
vortex state ferrimagnetic disk with the thicknesses less than 10 nm, which is
needed to spin-torque applications. Therefore, the problem how to rise the
working frequency of the vortex-state STNO is quite challenging.

In this paper we investigate nonlinear spin dynamics of a ferromagnetic ring
in a vortex state under an action of spin-polarized current and discuss the
usage of the ferromagnetic ring as the base of a vortex-state STNO.
Ferromagnetic nanorings are widely investigated due to their unique optical
and electromagnetic properties, BA3 ; biomedical applications of vortex-state
magnetic rings BA1 ; BA2  are one of the current interests. The dynamics of
magnetization in the vortex-state ring differs significantly from that for the
vortex-state disk, ZaspelJMM . The frequency of the lowest magnon mode for the
ring is much higher then frequency of the gyromode of vortex-state magnetic
disc, that opens up new opportunities to rise the working frequency of the
STNO. The ring-based vortex oscillator is expected to have some significant
advantages comparing to the case of oscillators based on vortex-state discs,
providing high values of the working frequencies, and smaller size.

The adequate theoretical description of the non-linear spin dynamics for the
vortex-state ring is much more complex then for the particles with uniform (or
quasi-uniform) magnetization, which can be analysed within the macrospin
approximation. In our work we used the collective variables approach, which is
valid for highly-nonlinear regimes of oscillations for the wide variety of
magnetic solitons, see e.g. Oleg . We found the inertial behavior of the spin
dynamics of the vortex-state ferromagnetic ring, formally common to one for
antiferromagnetic STNO, with corresponding analogy of oscillators based on
superconducting Josephson junction SCR . One of the important consequences is
the lower value of the critical current, the elimination critical current,
which is needed to support the generation once it has been started. Thus the
ring-based STNO can have suitable geometric dimensions and threshold current
densities and therefore is a good candidate for practical applications.

The article is organized as the following. In Section II basic equations
describing magnetization dynamics in the system are given. The ground state of
the ferromagnetic ring and spin dynamics above this ground state are
considered in Sections III and IV, respectively. In Section V we obtain
expressions for critical currents and working frequency. Finally, in Section
VI we discus ways of extracting a useful ac electric output signal for the
case of interest when the output electrical power averaged over the contact is
zero.

##  II  Ferromagnetic ring under the action of the spin-polarized current.

The problem under consideration is to investigate magnetization dynamics
induced by the spin-polarized current in a thin ferromagnetic ring. Spatial
dimensions of the ring are: thickness $L$, inner and outer radii $R_{i}$ and
$R_{o}$ respectively; ring is considered thin, $L\ll R_{i,o}$. Ground state of
the ring is the vortex state, in which magnetization $\mathbf{M}$ aligns
circularly around the ring axis. It is discussed in section III. In the
Cartesian reference frame with the $z$-axis along the ring axis the components
of the ground-state magnetization are
$M_{s}(-\sin\theta_{0}\sin\chi,\sin\theta_{0}\cos\chi,\cos\theta_{0})$, where
$\chi$ is the azimuthal coordinate in the plane of the ring and $\theta_{0}$
is the polar angle of the ground-state magnetization. The schematic picture of
the vortex-state ring and its magnetization are shown in Fig. 1.

The spin-polarized current flows into the ring and its spin-polarization
$\mathbf{p}$ is along $z$. The current flow is assumed to be uniform within
the sample. Spin-polarized current creates a torque $\mathbf{T}$, which tends
to rotate the in-plane component of $\mathbf{M}$ in the plane of the ring.

Figure 1: Ground state and dynamics of magnetization $\mathbf{M}$ of the
vortex-state ring under the action of spin-transfer torque: (a) standard in-
plane vortex with the polar angle $\theta_{0}=\pi/2$. (b) cone-vortex state,
which realizes the ground state of the free layer when the external magnetic
field $\mathbf{H}$ is applied along the $z$-axis, the angle
$\theta_{0}\neq\pi/2$. The spin-polarized current with polarization
$\mathbf{p}$ flows into the free layer. Vector $\boldsymbol{\omega}$ shows
schematically the direction of rotation of magnetic moments in different
points of the ring.

In addition we are considering an option of applying a weak (with the value
less than saturated) magnetic field $\mathbf{H}$ along $z$-axis: $h=H/4\pi
M_{s}$, where $M_{s}$ is the saturation magnetization of the ferromagnetic
material of the ring. If the external magnetic field is zero, $h=0$,
magnetization $\mathbf{M}$ lies entirely in the plane of the ring and the
polar angle $\theta_{0}=\pi/2$ , Fig. 1(a). The case of nonzero $\mathbf{H}$
is shown in Fig. 1(b). The small external field, $h<1$, leads to appearance of
a small out-of-plane component of magnetization equal to $M_{s}h$. Then the
polar angle $\theta_{0}$ in the ground state is determined by the external
field as $\cos\theta_{0}=h$. We show below that small external field does not
change the nature of the excitations but may give an additional way of the
device control.

The dynamics of magnetization $\mathbf{M}$ is described by the Landau-Lifshitz
equation

$\displaystyle\frac{\partial\mathbf{M}}{\partial
t}=\gamma\left[\mathbf{M}\times\frac{\delta
W}{\delta\mathbf{M}}\right]+\frac{\alpha}{M_{s}}\left[\mathbf{M}\times\frac{\partial\mathbf{M}}{\partial
t}\right]+\mathbf{T}.$ (1)

Here $W\equiv W[\mathbf{M}]$ is the ferromagnet’s energy written as a
functional of the magnetization $\mathbf{M}$, the vector $-\delta
W/\delta\mathbf{M}$ has the sense of the effective field,
$\gamma=g\mu_{B}/\hbar$ is the gyromagnetic ratio, $g$ is Lande factor,
$\mu_{B}$ is the Bohr magneton, $\hbar$ is the Planck’s constant, $\alpha$ is
the dimensionless parameter of the Gilbert damping. The particular form of the
energy functional $W$ will be presented in the next section. The spin-transfer
torque is given by $\mathbf{T}=\sigma
JM_{s}\left[\mathbf{m}\times[\mathbf{m}\times\mathbf{p}]\right]$, where
$\mathbf{m}=\mathbf{M}/M_{s}$ is the unit magnetization vector of the free
layer, $\mathbf{p}$ is the spin-polarization of the current. The value of
$\sigma$ is given by the formula

$\displaystyle\sigma=\frac{\epsilon g\mu_{B}}{2eM_{s}LS},$ (2)

where $e$ is the modulus of the electron charge, $\epsilon$ is dimensionless
spin-polarization efficiency, and $S$ is the area of the current-carrying
region that in our case coincides with the area of the ring face.

Considering the spin dynamics of the ring it is convenient to parameterize the
vector of normalized magnetization $\mathbf{m}$ by two angles: azimuthal angle
$\varphi$ and polar angle $\theta$. Then the Cartesian components of
$\mathbf{m}$ are $(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$.
The energy functional $W$ can be also presented as a functional of these
angles $W=W[\theta,\varphi]$. Then the dynamic equations take the form

$\displaystyle\frac{M_{s}}{\gamma}\sin\theta\frac{\partial\varphi}{\partial
t}=\frac{\delta W}{\delta\theta}+\frac{\alpha
M_{s}}{\gamma}\frac{\partial\theta}{\partial t},$ (3)
$\displaystyle-\frac{M_{s}}{\gamma}\sin\theta\frac{\partial\theta}{\partial
t}=\frac{\delta W}{\delta\varphi}+\frac{\alpha
M_{s}}{\gamma}\sin^{2}\theta\frac{\partial\varphi}{\partial t}-\frac{\sigma
JM_{s}\sin^{2}\theta}{\gamma}.$ (4)

These equations can be presented as the Lagrange equations for a non-
conservative system

$\displaystyle\frac{\partial\mathcal{L}}{\partial\theta}-\frac{\partial}{\partial
t}\frac{\partial\mathcal{L}}{\partial(d\theta/dt)}=\frac{\partial
Q}{\partial(d\theta/dt)},$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\varphi}-\frac{\partial}{\partial
t}\frac{\partial\mathcal{L}}{\partial(d\varphi/dt)}=\frac{\partial
Q}{\partial(d\varphi/dt)},$ (5)

with a dissipative function $Q$ and a Lagrangian density

$\displaystyle{\mathcal{L}}=\frac{M_{s}L}{\gamma}\int\left[\cos\theta\frac{\partial\varphi}{\partial
t}-W[\theta,\varphi]\right]d^{2}\mathbf{r}.$ (6)

For a thin magnetic particle, the problem is essentially two-dimensional and
the integration is performed over two-dimensional vector $\mathbf{r}$, that
lies in the plane of the ring with an origin at its center. The integration
over $z$, where $z$-axis is perpendicular to the ring plane, simply results in
factor $L$.

The dissipative function of the ferromagnet is
$Q=Q_{\text{G}}+Q_{\text{STT}}$, where

$\displaystyle Q_{\text{G}}=\frac{\alpha
M_{s}L}{2\gamma}\int\left[\left(\frac{\partial\theta}{\partial
t}\right)^{2}+\sin^{2}\theta\left(\frac{\partial\varphi}{\partial
t}\right)^{2}\right]d^{2}\mathbf{r},$ (7) $\displaystyle
Q_{\text{STT}}=-\frac{\sigma
JM_{s}L}{\gamma}\int\sin^{2}\theta\left(\frac{\partial\varphi}{\partial
t}\right)d^{2}\mathbf{r}.$ (8)

Here $Q_{\text{G}}$ is the Gilbert dissipative function, leading to the
friction force linear in velocity, as in Eqs. (3, 4), whereas the second term
$Q_{\text{STT}}$ describes the contribution of the spin-transfer torque. The
value $Q_{\text{G}}>0$ has a definite sign, while the sign of $Q_{\text{STT}}$
depends on the relative orientation of the spin-current polarization $\bm{p}$
and the magnetization $\bm{M}$. It can play the role of positive and negative
friction, which determines the possibility of "anti-damping" effects.

The full time derivative of the energy can be presented as
${dW}/{dt}=-2Q_{\text{G}}-Q_{\text{STT}}$. At a certain value of current $J$,
these two terms can be compensated on average, giving $dW/dt=0$. It means that
a regime of a steady-state oscillations can exist in the system. Our goal is
to investigate possible oscillations of the system and to find conditions
necessary for this regime to appear. In the following two sections we consider
in details the ground state of the ring and its excited states.

##  III Vortex state of the ferromagnetic ring

Let us briefly discuss the ground state of the ring in the vortex state. The
energy of soft ferromagnet like permalloy contains the isotropic exchange
interaction, the magnetostatic interaction and the interaction with the
external magnetic field:

$\displaystyle
W[\mathbf{M}]=\int\left(\frac{A}{2M_{s}^{2}}\sum_{i}{\partial_{i}\bm{M}}^{2}-\frac{1}{2}\bm{M}\cdot\bm{H}_{m}-\bm{M}\cdot\bm{H}\right)d\bm{r}.$
(9)

Here $A$ is the exchange constant and ${\bm{H}}=4\pi M_{s}h\bm{e}_{z}$ is the
external magnetic field, normal to the plane of the ring, $\bm{e}_{z}$ is the
unit vector along $z$-axis. The field $\bm{H}_{m}$ is determined by the
magnetostatic equations $\textmd{div}(\bm{H}_{m}+4\pi\bm{M})=0$ and
$\textmd{rot}\bm{H}_{m}=0$ with the standard boundary conditions: the
continuity of the normal component of $(\bm{H}_{m}+4\pi\bm{M})$ and the
tangential component of $\bm{H}_{m}$ on the border of the sample. These
equations can be rewritten as $\textmd{div}\bm{H}_{m}=4\pi\rho_{M}$, where
$\rho_{M}$ is the ‘‘density of magnetic charges’’, which include volume
charges $-\textmd{div}\bf{M}$ and surface charges, which equal to
$-(\bm{M}\cdot\bm{n})$, where $\bm{n}$ is the unit vector normal to the
border.

If the distribution of magnetization within the particle is known, what is the
main assumption of the collective coordinate approach used below, the
magnetostatic field $\bm{H}_{m}$ can be presented as a sum of the volume
contribution $\boldsymbol{H}_{\textmd{m}}^{\textmd{vol}}$ and surface
contributions of the lateral surfaces ${\bm{H}}_{\textmd{m}}^{\textmd{edge}}$
and of the ring faces $\boldsymbol{H}_{\textmd{m}}^{\textmd{face}}$. This
representation was used for description of both vortex ground state and magnon
modes on the vortex background, and is proved to give a good agreement with
the experimental data for vortex-state discs and rings, see Zaspel_05 ;
ZaspelJMM .

For the given form of the function $\bm{M}(\mathbf{r})$, the magnetostatic
energy can be present through the known $\bm{H}_{m}$ as the sum of
corresponding volume and surface contributions. For the case of interest, the
thin magnetic ring, the main term arising from the surface charges of the two
ring faces is
${\bm{H}}_{\textmd{m}}^{\textmd{face}}=-4\pi(\bm{M}\cdot\bm{e}_{z})$ that
leads to the corresponding energy density of the form of
$2\pi(\bm{M}\cdot\bm{e}_{z})^{2}$.

The energy functional $W[\mathbf{M}]$ in terms of angles $\theta$ and
$\varphi$ is

$\displaystyle
W[\theta,\varphi]=L\int\left\\{\frac{A}{2}\left[(\nabla\theta)^{2}+\sin^{2}\theta(\nabla\varphi)^{2}\right]+2\pi
M_{s}^{2}\cos^{2}\theta-
HM_{s}\cos\theta\right\\}d^{2}\mathbf{r}+W_{vol}+W_{edge}.$ (10)

First term in Eq. (10) represents the exchange energy with exchange constant
$A$. The second term is the approximate contribution to the magnetostatic
energy from upper and lower faces of the ring; this term can be treated as an
effective easy-plane anisotropy, see e. g., Zaspel05 . This term is the
largest of the magnetostatic terms in the case of a thin particle. The third
term in the integrand is the Zeeman energy $-(\mathbf{M}\cdot\mathbf{H})=-4\pi
hM_{s}^{2}\cos\theta$ of the interaction with external magnetic field (if
there is any) applied along the $z$-axis. Terms $W_{vol}$ and $W_{edge}$
denote the parts of the magnetostatic energy caused by volume and edge
‘‘magnetic charge densities’’, respectively. At the moment we do not specify
the form of these contributions to the magnetostatic energy.

The ground state magnetization distribution in the ferromagnetic ring is the
coreless magnetic vortex. Introducing planar coordinates $(r,\chi)$ in the
plane of the ring and $z$-axis along the axis of the ring, we can write it as
$\varphi=\chi+\pi/2$ and $\theta=\theta_{0}(r)$, see Fig. 1. The function
$\theta_{0}(r)$ is a result of minimization of the energy functional Eq. (10).
Here it is taken into account that volume and edge "magnetic charge densities"
for both vortex-type configurations, in-plane and cone, do not contribute in
the energy. So, the ground state distribution of $\theta_{0}(r)$ is described
by the solution of ordinary differential equation

$\displaystyle
l_{0}^{2}\left(\frac{d^{2}\theta_{0}}{dr^{2}}+\frac{1}{r}\frac{d\theta_{0}}{dr}-\frac{1}{r^{2}}\sin\theta_{0}\cos\theta_{0}\right)+\sin\theta_{0}\cos\theta_{0}-h\sin\theta_{0}=0,$
(11)

where $l_{0}^{2}=A/4\pi M_{s}^{2}$ is the exchange length, and $h=H/4\pi
M_{s}$ is the normalized external magnetic field, see for details Guslienko ;
Otani .

This equation describes vortex solutions for the out-of-plane projection of
magnetization $M_{\perp}$; it is applicable to both disk-shaped and ring-
shaped particles. For a zero external field, $h=0$, the ground state
corresponds to the in-plane vortex shown in Fig. 1(a), whereas for non-zero
external field smaller then saturation value, $h<1$, the cone vortex state
shown in Fig. 1(b) is realized, Wysin_02 . Eq. (11) has a singularity at
$r\rightarrow 0$ that leads to appearance of vortex core (small region with
out-of plane magnetization) for the ferromagnetic disk. However, for the ring
the singular region of ferromagnet is just ‘‘cutted off’’ and the dynamics of
excitations substantially changes. Size of the vortex core is about
$l_{0}/\sqrt{1-h^{2}}$ and in what follows we assume that the field is not
very close to saturation such that the core size do not exceed the inner
radius of the ring. Therefore, for estimates in the main approximation in
small parameters $l_{0}/R_{o}$ and $l_{0}/R_{i}$ asymptotic of Eq. (11) can be
used: the out-of-plane component of magnetization far from the origin is
$M_{\perp}=M_{s}\cos\theta_{0}=M_{s}h(1+l_{0}^{2}/r^{2})$, implying
$\theta_{0}(r)=\pi/2$ in case $h=0$. Note, that the magnetic field applied
along $z$ axis reduces the out-of-plane component of the magnetization and, as
a consequence, suppresses the effect of the spin-polarized current.

##  IV Radially-symmetric nonlinear dynamics

Excitation spectra of soft-magnetic nanoparticles are of interest both from a
fundamental point of view and in context of their applications in spintronic
devices. If such a particle is used as a working layer of STNO, the lower part
of its spectrum determines the operating frequencies of the device. The linear
dynamics of magnetic particles in the vortex state is well studied. Excitation
modes in linear approximation are denoted by two integers: the number of
radial nodes in the out-of-plane component of the dynamical magnetization,
radial mode number $n\geq 1$, and the azimuthal mode number $m$, which
determines the angular dependence of this component. The spectrum of the
vortex-state disk includes the gyrotropic mode ($n=1,\ m=1$) with low (sub-
gigahertz) frequency, corresponding to the slow vortex core precession around
the disc center, and a system of higher modes.

Contrary, in the spectrum of the vortex-state ring there is no low-frequency
gyroscopic mode, since the vortex core is absent for the ring-shaped sample,
ZaspelJMM . At the same time, the higher mods for discs and rings are similar.

In the ring-shaped magnetic particle the mode with the lowest frequency is the
breathing mode with the azimuthal number $m=0$. It is characterized by the
deviation of magnetization from the ground state of the form
$\varphi=\chi+\pi/2+\psi(\mathbf{r},t)$, $\theta=\theta(\mathbf{r},t)$, with
the small deviation $\psi$. The spin waves are described by the set of dynamic
Landau-Lifshitz equations. There, Eqs. (3,4), the ferromagnetic energy
$W(\theta,\varphi)$ contains, among other terms, the nonlocal contribution to
the magnetostatic energy caused by the volume and edge magnetic charges, which
greatly complicates the analytical description. However, in the linear
approximation this problem can be solved for small values of parameter
$L/R_{o,i}$, see e.g., Zaspel05 ; Zaspel_05 ; Sheka2004 .

The challenge of our problem is that the spin dynamics excited by the spin-
polarized current is essentially nonlinear and approaches suitable for small
excitations do not work. For this reason, we use the collective variable
approach, derived in works ZaspelIvanov09 ; Galkin  to describe the nonlinear
analogue of the breathing mode as well as the total in-plane rotation of the
magnetization vector at each point of the ring. The last one is the case of
our spesial interest, as this regime can potentially provide the generation of
microwave signal.

Within the collective variable approach applied to the case under
consideration the Lagrangian of the ferromagnet can be written in the form

$\displaystyle\mathcal{L}=\frac{L}{\gamma}\int
M_{\perp}\frac{d\varphi}{dt}d^{2}\mathbf{r}-W[\mathbf{M}],$ (12)

where it is taken into account that magnetization $\mathbf{M}$ depends only on
the in-plane coordinates. We assume that $\varphi$ is coordinate-independent
variable, and its time derivative ${d\varphi}/{dt}$ can be taken out of the
integral, see for details ZaspelIvanov09 ; Galkin . With this assumption the
breathing mode can be described as excitations with radial symmetry:
$\varphi=\chi+\pi/2+\psi(t)$, $\theta=\theta({r},t)$. Here $\psi(t)$ is the
averaged value of $\psi(\mathbf{r},t)$, the deviation from the ground state
configuration. Then the Lagrangian describing the radially-symmetric dynamics
of the vortex-state ring can be written through collective variables as

$\displaystyle\mathcal{L}=\frac{1}{\gamma}\mu\frac{d\psi}{dt}-W(\mu,\psi),$
(13)

where $\psi(t)$ has the meaning of generalized coordinate corresponding to the
azimuthal rotation of the magnetization. The quantity ${\mu}/{\gamma}$ has the
sense of generalized momentum conjugated to $\psi(t)$; it is expressed through
the total out-of-plane magnetic moment of the ring, ${\mu}$, namely

$\displaystyle\frac{\mu}{\gamma}=\frac{L}{\gamma}\int
M_{\perp}d^{2}\mathbf{r}=\frac{LM_{s}}{\gamma}\int\cos\theta d^{2}\mathbf{r}.$
(14)

In accordance with the collective coordinate approach, the Hamilton function
$W(\mu,\psi)$ represents the ferromagnet’s energy functional $W[\mathbf{M}]$
minimized over the distribution of magnetization $\mathbf{M}$ at fixed values
of generalized variables $\mu$ and $\psi$. This dependencies of $W(\mu,\psi)$
need to be found in the main approximation, considering the ratios
$l_{0}/R_{i,o}$ to be small parameters.

The dependence of the energy $W$ on the generalized momentum ${\mu}/{\gamma}$
can be understood by considering how the external magnetic field applied along
the axis of the ring increases the out-of-plane component of the magnetization
$M_{\perp}$ and transforms the in-plane magnetic vortex into the cone-state
vortex. The minimum of the energy $W(\mu,\psi)$ at a given value of $\mu$ can
be obtained by the method of indefinite Lagrange multipliers, see Galkin . As
a result, the extremals of the energy functional correspond to the solutions
of the Eq. (11), where the role of magnetic field $h$ is played by the
Lagrangian multiplier $h(\mu)$. Thus, apart from the vortex core region we can
assume that $\cos\theta=h$ . The relation between $\mu$ and $h$ is described
by Eq. (14) and in the first order approximation has the form $\mu=M_{s}LSh$,
where $S=\pi(R_{0}^{2}-R_{i}^{2})$ is the area of the ring face.

To obtain the explicit form of the $\mu$-dependence of the energy, we multiply
Eq. (11) for the cone state by $r^{2}d\theta/dr$, integrate it over $r$, and
substitute in the energy functional Eq. (10), see Voronov . Finally,

$\displaystyle W(\mu)=4\pi
M_{s}h\mu+4\pi^{2}M_{s}^{2}l_{0}^{2}L(1-h^{2})\ln\left(\frac{R_{o}}{R_{i}}\sqrt{\pi\Lambda(h)}\right)-SL(2\pi
M_{s}^{2}h^{2}),$ (15)

where the function $\Lambda(h)\approx 5.27$ as $h\rightarrow 0$ and linearly
decreases to zero at $h\rightarrow 1$, see ZaspelIvanov09 .

The $\psi$-dependence of the ferromagnet energy $W$ is determined by nonlocal
effect of edge $(\mathbf{r}\cdot\mathbf{M})/r=M_{s}\sin\theta\sin\psi$ and
volume $(\nabla\cdot\mathbf{M})=[(1/r)d(r\sin\theta)/dr]\cos(\varphi-\chi)$
magnetostatic charges, that can be estimated by asymptotic evaluation of the
magnetostatic integral, ZaspelIvanov09 ; Zaspel_05 . In the leading
(logarithmic) order we can write

$\displaystyle W(\psi)=2\pi
M_{s}^{2}L^{2}\left[\left.R_{o}\sin^{2}\theta_{0}\right|_{r=R_{o}}\ln\left(\frac{\eta
R_{o}}{L}\right)+\left.R_{i}\sin^{2}\theta_{0}\right|_{r=R_{i}}\ln\left(\frac{\eta
R_{i}}{L}\right)\right]\sin^{2}\psi.$ (16)

This expression includes the energy of the outer, $r=R_{o}$, and inner,
$r=R_{i}$, edges of the ring. The contribution of the volume charge can be
taken into account by a phenomenological coefficient $\eta$, it has values
between $4$ and $6$, see. Zaspel05 . For the ring with $R_{o,i}\gg L$ the
value of $\sin\theta_{0}$ on the outer and inner edges can be replaced by
their asymptotic value $\sqrt{1-h^{2}}$.

Variables $\mu$ and $\psi$ can be considered as conjugated coordinate and
momentum in accordance with Lagrangian Eqs. (12,13) and the corresponding
equations for them have the Hamiltonian form. Taking into account the explicit
dependence of $W$ on $\psi$, equations for collective variables $\mu$ and
$\psi$ can be written as

$\displaystyle-\frac{1}{\gamma}\frac{\partial\mu}{\partial
t}=\left.\frac{\partial W}{\partial\psi}\right|_{\mu=const}=$ (17)
$\displaystyle=4\pi(1-h^{2})M_{s}^{2}L^{2}\left[R_{o}\ln\left(\frac{\eta
R_{o}}{L}\right)+R_{i}\ln\left(\frac{\eta
R_{i}}{L}\right)\right]\sin\psi\cos\psi,$
$\displaystyle\frac{1}{\gamma}\frac{\partial\psi}{\partial
t}=\left.\frac{\partial W}{\partial\mu}\right|_{\psi=const}.$ (18)

To go further, note that the in-plane deviation of magnetization from the
ground state weakly changes the ferromagnet energy, which makes possible
nonlinear radial oscillations of the system. On the contrary, the deviation of
the out-of-plane component $M_{\perp}$ strongly changes the vortex energy.
This fact allows to separate in the main approximation the energy
contributions from the azimuthal and out-of-plane deviations of magnetization.
Since $\mu$ is a more rigid variable that does not contain any small
parameter, it is sufficient to take into account $\mu$ in the linear
approximation even for description of highly nonlinear dynamics with unlimited
change of $\psi$. So, in the linear approximation on $\mu-\mu_{0}$, the
derivative $\partial W/\partial\mu$ can be replaced by
$(\mu-\mu_{0})(d^{2}W(\mu)/d\mu^{2})|_{\mu=\mu_{0}}$, where $\mu_{0}$ the is
equilibrium value of the out-of-plane component of the magnetic moment in the
vortex core.

Then the dynamic equations take the form of the Hamilton equations for
coordinate $\psi$ and conjugated momentum $p=(\mu-\mu_{0})/\gamma$. Within
this approximation, we can significantly simplify the Hamilton function of our
problem

$\displaystyle
H=\frac{p^{2}}{2m}+W(\psi),\;\;\frac{1}{m}=\gamma^{2}\left.\frac{d^{2}W(\mu)}{d\mu^{2}}\right|_{\mu=\mu_{0}},$
(19)

where the effective mass $m$ does not depend on $\psi$ and $p$, and the
potential energy $W(\psi)$ does not depend on $p$. The form of the Hamiltonian
is the same as for a massive particle subject to potential energy $W(\psi)$.
To the lowest order the effective mass can be evaluated using
$dW(\mu)/d\mu=h(\mu)$ and the first order approximation relation between $h$
and $\mu$,

$\displaystyle\frac{1}{m}\approx\frac{4\gamma^{2}}{L(R_{0}^{2}-R_{i}^{2})}.$
(20)

The second Hamilton Eq. (18) takes the form $\mu=\mu_{0}+m\gamma(d\psi/dt)$.
Taking the time derivative of it and eliminating $d\mu/dt$ from Eq. (17) we
come to the second-order differential equation for coordinate $\psi(t)$ in the
form

$\displaystyle\frac{d^{2}\psi}{dt^{2}}+\omega_{0}^{2}(1-h^{2})\sin\psi\cos\psi=0,$
(21)

where

$\displaystyle\omega_{0}=\omega_{m}\sqrt{R_{o}\ln\left(\eta\frac{R_{o}}{L}\right)+R_{i}\ln\left(\eta\frac{R_{i}}{L}\right)}\sqrt{\frac{L}{\pi(R_{0}^{2}-R_{i}^{2})}},$
(22)

and $\omega_{m}=4\pi\gamma M_{s}$ is a characteristic frequency of the
ferromagnetic material (of the order of 30 GHz for permalloy). It can be seen
that the frequency of $\omega_{0}$ contains the small parameters $L/R_{o,i}$,
and is much lower than $\omega_{m}$. Note, that in the linear approximation,
the frequency is $\sqrt{1-h^{2}}\omega_{0}$ decreases not only with decrease
of relative thickness of the ring, but also with the application of the
external field $h$.

Eq. (21) has an integral of motion $E=K+U$ that can be considered as a sum of
kinetic and potential parts,

$\displaystyle
E=\frac{1}{2\omega_{m}}\left(\frac{d\psi}{dt}\right)^{2}+(1-h^{2})\frac{\omega_{0}^{2}}{2\omega_{m}}\sin^{2}\psi.$
(23)

The first term can be treated as a kinetic energy and the second term as a
periodic potential of the restoring force of a magnetostatic nature. Thus,
within the above collective coordinate approach, the typical ‘‘inertial’’
features, common to ones that are known for usual mechanics of massive
particles, are found here for in-plane magnetization dynamics in the vortex-
state ferromagnetic ring. Before going further, it worth discussing these
features.

The inertial dynamics is widely accepted for antiferromagnets, whereas the
spin dynamics of ferromagnets is commonly believed to be gyroscopic. In
particular, the equation similar to Eq. (21) was obtained for
antiferromagnetic STNO and was claimed as the clear manifestation of inertial
character of spin dynamics SCR . For antiferromagnets, the inertial dynamics
appears naturally within the sigma-model approach; it is based on the presence
of the uniform exchange interaction between antiferromagnetic sublattices. It
is valid up to extremely high frequencies of the order of the exchange
frequency values, exceeding usual magnon frequencies for antiferromagnets.
Inertial dynamics for some models of ferromagnets is known for a long time for
some particular cases, see e.g., Mikeska1978 ; Kulagin ; it was also employed
for description of STNO based on easy-plane magnets in uniform state
inertia2022 . The models allowing such simplification are characterized by the
presence of effective anisotropy with significantly different constants: hard
for one angular variable and weak for the other one. In fact, it is the case
of our problem. Then the inertial approach is valid up to frequency
corresponding to "hard" anisotropy field. In our case, this upper limit
frequency is $\omega_{m}$; it is much lower than the exchange frequency, but
much higher than the value of interest, $\omega_{0}$. Thus, the inertial
approximation is valid for our problem.

The inertial approximation leads to the simple and transparent physical
picture of the nonlinear motion of magnetization. Using the first integral
(23), it is possible to write down solutions of Eq. (21) in quadratures and to
present them explicitly through the elliptic integrals. However, it is useful
to carry out a qualitative analysis using the phase plane, the phase diagram
is shown in Fig. 2.

Stationary points of the function $E$ define singular points of the phase
plane, $d\psi/dt=0$ and $\sin 2\psi=0$, and the sign of $\cos 2\psi$
determines types of these points. There are two types of singular points:
centers, $\psi=0,\pm\pi$ and $d\psi/dt=0$, around which small oscillations can
occur with the frequency $\sqrt{1-h^{2}}\omega_{0}$; and saddle points,
$\psi=\pm\pi/2$ and $d\psi/dt=0$. Saddle points are characterized by the
constant $E=\omega_{0}^{2}/2\omega_{m}$, and separatrices follow the equation

$\displaystyle\frac{d\psi}{dt}=\pm\omega_{0}\sqrt{1-h^{2}}\cos\psi.$ (24)

Figure 2: Phase plane representation of radially-symmetric oscillations
without dissipation. Here we used the energy units of
$2\omega_{m}/\omega_{0}^{2}(1-h^{2})$, and time units of
$\omega_{0}\sqrt{1-h^{2}}$. Black circles show the singular points of the
center type and black rectangles show saddle points. Separatrix trajectories
connect the saddle points. Closed phase trajectories below the separatrix
correspond to the finite motion, whereas phase trajectories above the
separatrix correspond to infinite dynamics with non-limited growing of $\psi$.

This phase diagram shows that there are two regimes of oscillations: the
regime with the finite oscillations of $\psi$ around the equilibrium position
and the regime with unlimited increase in $\psi$ with time. These two regimes
are separated by separatrices passing through the saddle points. As we will
show below, these two regimes are essentially different when a ferromagnetic
ring is used as an active element of the nanooscillator. In the finite regime
the time average of $d\psi/dt$ equals to zero and the spin current cannot
excite this type of oscillations. We are interested in the case when the
initial vortex ground state of the ring loses its stability under the action
of a spin-polarized current and the spin system goes into the regime of
infinite circular oscillations. It takes place when the current overcome some
critical values that we are about to estimate in the next section.

##  V Non-conservative dynamics and critical currents

The non-conservative effects of the spin-polarized current and damping are
taking into account by use of the general form of the dissipative function
$Q=Q_{\text{G}}+Q_{\text{STT}}$ with Eqs. (7,8) and explicit form of the
radially-symmetric excitation, $\theta=\theta_{0}$ and
$\varphi=\chi+\pi/2+\psi(t)$. Substituting it into the Lagrangian density, Eq.
(6), and using the Lagrange equation for non-conservative system, Eqs. (II),
we obtain the wave equation for $\psi$ in the Newtonian form

$\displaystyle\frac{d^{2}\psi}{dt^{2}}+\omega^{2}_{0}(1-h^{2})\sin\psi\cos\psi+\omega_{m}(1-h^{2})\left(\alpha\frac{d\psi}{dt}-\sigma
J\right)=0.$ (25)

This equation describes a wide variety of physically interesting systems. In
particular, the same equation describes the planar dynamics of the Neel vector
for biaxial antiferromagnet SCR . Note as well a simple mechanical analogy:
after the substitution $\psi=\phi/2$, Eq. (25) coincides with the equation of
a physical pendulum, a massive suspended particle moving in the gravitational
field with accounting for viscous friction force (the term with $\alpha$ here)
and subject to some eddy force, which is perpendicular to the radius vector of
the material point that mimicries the action of spin current, see e.g., Kent ;
Ivanov2020 . Probably the most interesting analogy is that the above equation,
written for the variable $\phi=2\psi$, also coincides with the equation for
the phase $\phi$ of the superconducting order parameter of the point Josephson
contact, see for details Josephson1 ; Josephson2 .

From the Eq. (25) it follows the law of the energy evolution

$\displaystyle\frac{dE}{dt}=\frac{M_{s}LS}{\gamma}(1-h^{2})\left[\sigma
J-\alpha\frac{d\psi}{dt}\right]\frac{d\psi}{dt},$ (26)

which can be rewritten in the form $d\tilde{E}/dt=-\alpha(d\psi/dt)^{2}$,
where $\tilde{E}=K+\tilde{U}$ and

$\displaystyle\tilde{U}=(1-h^{2})\frac{\omega_{0}^{2}}{2\omega_{m}}\sin^{2}\psi-\sigma
J(1-h^{2})\psi.$ (27)

Function $\tilde{U}$ includes the contribution of the spin-polarized current,
which turns it into a ‘‘tilted washboard’’. So the eddy force in Eq. (25),
that is proportional to $\sigma J$, can be formally presented as a part in the
‘‘potential’’ $\tilde{U}$. Then its non-potential nature manifests itself in
the fact, that function $\tilde{U}$ is changing after a full rotation of
magnetization, i.e., after the angle $\psi$ changes by $2\pi$.

First note, that the system becomes absolutely unstable when local minima and
maxima of potential $\tilde{U}(\psi)$ disappear, i.e., $d\tilde{U}/d\psi$
never equal to zero. Then the magnetization vector rotates through the full
angle for any initial conditions. To achieve this regime, the current $J$
should overcome a critical value that is natural to call the ignition
threshold

$\displaystyle J_{cr1}=\frac{\omega_{0}^{2}}{2\omega_{m}\sigma}.$ (28)

If the current value is less than $J_{cr1}$, the behavior is more complicated,
but its full qualitative analysis can be done through the phase plane method.
To illustrate it, some characteristic phase trajectories obtained from the
numerical solution of the Eq. (25) for various initial conditions and a fixed
current value $J$ are shown in Figs. 3,4.

For non-conservative system, the maxima of the effective potential still are
saddle points, whereas the centers for the weak enough dissipation transform
to focuses, which, in principle, could be either stable or unstable (spiral
sinks or spiral sources for the phase trajectories, respectively).

The steady-state motion and antidamping features are possible only for the
infinite motion of $\psi$, i.e., for the full rotation of the magnetization.
Thus, the minima of the effective potential corresponds to the stable focuses,
and the phase trajectories approach them with spiralling as it shown on the
figures 3 and 4 below. Obviously, for zero current it is the fate of the
separatrix trajectories, which coming out from the saddle points, and it is
clear that the same behavior will be present at small enough values of the
current. This behavior is shown in Fig. 3. All trajectories, regardless of the
initial conditions, are twisted around one of the focuses, and all
oscillations in the system will decay in time.

Figure 3: Phase plane representation of the wave equation with the spin-
polarized current $J$ and damping for the case $J<J_{cr2}$. All trajectories
are tend to the focuses. We also show some of the trajectories, passing as
close as possible to the separatrix trajectories from above and below.

For the oscillations to become steady-state the change of energy of the system
over a period of rotation has to be zero. This behavior appears starting from
some current value, such that the phase trajectories coming out from one
saddle point come into the adjacent saddle point. This condition is held for
all pairs of the neighboring saddle points, and the separatrix has the same
form as for the system without dissipation and pumping, see Fig. 2 above.

The corresponding critical value of the current can be found from the
condition of compensation of the damping and spin pumping on this separatrix
trajectory. This brings us to a concept of the second critical current, the
so-called elimination threshold, $J_{cr2}<J_{cr1}$.

Here and below we consider small damping regime, $\alpha\ll 1$, to obtain the
analytical formulae for the characteristics of oscillations. This condition
allows us to estimate the integrals over trajectories with use of non-
perturbed equations found through Eq. (23), see the previous Section. All
quantities are obtained in the linear approximation over the small parameter
$\alpha$.

The value $J_{cr2}$ is found by substituting the equation for the separatrix
trajectory, Eq. (24) into Eq. (26) and integrating it over time. Finally,

$\displaystyle J_{cr2}=\sqrt{1-h^{2}}\frac{2\alpha\omega_{0}}{\pi\sigma}.$
(29)

When the current value is above $J_{cr2}$, still some of the trajectories are
finite, tending to the focuses on the phase plane. They correspond to the
damped oscillations. But the trajectory coming out from one saddle point is
going above the neighboring saddle point, and later on, these trajectories are
going higher and higher, see Fig. 4. On the other hand, for the trajectories
in the region of the phase plane with high enough ‘‘velocity’’ the dissipation
is more efficient, and in the average they are going ‘‘downstairs’’. Such a
behavior is typical to the system with limit circle: these two sets of
trajectories can be fitted to each other if and only if we suppose that all of
them are tending, from both above and below, to the some periodic trajectory,
limit cycle, which corresponds to the steady-state oscillations. The center
line of the limit cycle is horizontal showing that in this regime the action
of spin current and damping are compensated at average.

Figure 4: Phase plane representation of the wave equation with the spin-
polarized $J$ current and damping for the case ignition. All trajectories
below the separatrix are finite. All trajectories above the separatrix are
approaching the limit cycle. The limit cycle (red, color online) is shown
between the two horizontal dashed lines.

It is worth to stress here the difference of two regimes of oscillations. The
ignition current is strong enough to remove the potential barriers and the
system unavoidably begin to roll down the "washboard". The elimination current
gives the system enough energy to inertially overcome the potential barriers
of potential $U$ like a pendulum uses its kinetic energy to maintain the
oscillations.

We are interested in the regime of the steady-state oscillations,
characterized by the full rotation, that can be used to produce a useful
microwave signal. To trigger this regime, the ignition current higher than
$J_{cr1}$ need to be applied for a short time $\Delta t$. The required time
can be estimated from the angle needed for a complete turn rotation as
$(d\Psi/dt)\Delta t=2\pi$, that gives $\Delta t=2\pi\alpha/(\sigma J_{cr1})$.

The method of the phase plane is a good tool for understanding the dynamical
behavior of the system and it allows to find the conditions for different
oscillation regimes. The magnetization dynamics can be easily imagined, if we
notice, that the axes on the phase diagram have the following meaning. The
axis $\Psi$ is the instantaneous value of deviation of magnetization from the
ground state and $d\Psi/dt$ is its instantaneous velocity.

Magnetization distribution is shown in Fig. 1 for two different ground states.
The dynamics of the breathing mode is the simultaneous oscillations of the
magnetization vector at each point of the ring around the ground state
position. For the linear breathing mode the maximal deviation from the ground
state $\Psi$ is small, for nonlinear breathing mode it is up to the angle of
$2\pi$. These oscillations are in the plane of the ring, and the direction of
the initial deviation is symbolically shown by circled arrow denoted with
$\boldsymbol{\omega}$. Spiral or closed phase trajectories on the phase
diagram correspond to periodic or damped oscillations of the breathing mode
with the maximal deviation smaller then $2\pi$. The wavy phase trajectories
reaching the limit cycle, the regime of the steady-state oscillations,
correspond to the permanent full in-plane rotation over the angle $2\pi$ of
the magnetization vector at each point of the ring in the direction of
$\boldsymbol{\omega}$.

The starting points of the phase trajectories in Fig. 3 and 4 play the role of
initial conditions, but, in fact, we do not need to specify them to achieve
one of the oscillation regimes. Since the phase trajectories never intersect,
it is enough to get into one of the points of the phase plane leading to the
trajectory of the corresponding type. In practice the initial velocity, i.e
the initial value of $d\Psi/dt$, can be easily controlled by the magnitude of
the ignition current, applied to trigger the oscillations. The ignition
current above $J_{cr1}$ is enough to reach the regime of steady-state
oscillations.

Let us estimate the frequency of the steady-state oscillation, corresponding
to the limit cycle on the phase diagram. It will play the role of the
generation frequency $\omega$ of the STNO based on the ring-shaped vortex-
state free layer. In the approximation $\alpha\ll 1$ the generation frequency
is found by integrating of Eq. (23) over the period, that gives

$\displaystyle\omega=\frac{\pi}{2}\frac{\omega_{0}\sqrt{1-h^{2}}}{kK(k)},$
(30)

where $K(k)$ is the elliptic integral of the first kind. To find the value of
argument $k$, corresponding to the limit cycle, we use the condition that at
the limit cycle the integral over the period of $dE/dt$ is zero. The
integrating of Eq. (26) gives the equation $\sigma J\pi
k=2\alpha\sqrt{1-h^{2}}\omega_{0}E(k)$, where $E(k)$ is the elliptic integral
of the second kind. These two equations give the dependence of $\omega$ on the
current in the implicit form, this dependence is shown in Fig. 5.  Expressions
for elliptic integrals we are using are the following

$\displaystyle
K(k)=\int_{0}^{\pi/2}\frac{d\beta}{\sqrt{1-k^{2}\sin^{2}\beta}},$ (31)
$\displaystyle E(k)=\int_{0}^{\pi/2}{d\beta}{\sqrt{1-k^{2}\sin^{2}\beta}}.$
(32)

Figure 5: The generation frequency $\omega$ via the value of the spin-
polarized current $J$ (red line, color online). Black dashed line shows the
linear asimptotic $\omega_{lin}$. The frequency units are chosen as
$\tilde{\omega}=\omega_{0}\sqrt{1-h^{2}}\pi/2$.

It turns out, that at current values 2-3 times higher than the threshold value
$J_{cr2}$, the frequency grows almost linearly with the current,
$\omega_{lin}=\sigma J/\alpha$. For small current values, $\omega$ is more
complicated function of the current; it vanishes at $J\to J_{cr2}$. Despite
the fact that the dependence $\omega(J)$ present in Fig. 5 is derived in the
small-$\alpha$ approximation, $\omega_{lin}$ coincides with the linear
frequency expected at the extremely high current values. The reason is, that
at current values much larger than $J_{cr1}$ the role of the potential $U$ is
negligible, and the limit cycle is presented by almost strait horizontal line,
which corresponds to the constant solution of Eq. (25), namely
$d\psi/dt=\omega_{lin}$ .

To evaluate practical benefits of the ring-based STNO we make estimates for
the critical current values. We use the following parameters of the system:
$R_{0}=200$ nm, $R_{i}=150$ nm, $L=5$ nm, $h=0$, $\eta=5$, $\epsilon=1$. In
typical magnetic materials which are used in spintronic devices (we use values
for permalloy), the Gilbert damping is $\alpha\sim 0.01$, $M_{s}=8\cdot
10^{5}\text{A}\text{m}^{-1}$, characteristic frequency $\omega_{m}\approx 30$
GHz. For these parameters we obtain $\omega_{0}=11.4$ GHz,
$J_{cr1}=0.055\text{A}$, $J_{cr2}=0.0017\text{A}$. Dividing by the current
carrying area $S=5.5\cdot 10^{-10}\text{cm}^{2}$ we obtain the critical
current densities $j_{cr1}=10^{8}\text{A}\text{cm}^{-2}$. The value of
$j_{cr1}$ is high but reasonable Demidov , especially having in mind that it
should be applied for a short time only to trigger the oscillations. The
estimated time of its application is less than $0.1$ ns, which is short
enough. After the ignition the applied current can be reduced to smaller
densities above the elimination threshold $j_{cr2}=3.1\cdot
10^{6}\text{A}\text{cm}^{-2}$, which is sufficiently low. This makes the ring-
based STNO promising for the practical implementation. Let’s consider the
configuration of this device in more detail.

##  VI Extracting an electric output signal

In the previous sections we made theoretical description of the self-
sustaining nonlinear dynamics of magnetization in the ferromagnetic ring. We
suggest to use the ring as a free layer of the STNO. For its practical
application as a microwave generator it is necessary to convert the energy of
the oscillations into an alternating microwave electrical signal. There are
two known phenomena allowing to extract electrical current from the
magnetization oscillations: giant magnetoresistance (GMR)Katti ; GMR  and the
inverse spin Hall effect (ISHE)SH1 ; SH2 . In what follows we are considering
both possibilities.

Geometries of the STNO for these two cases shown in Figs. 6,7. The basic
design is a sandwiched structure with three main elements: polarizer, free
layer, and analyser. The direct current $J$ is injected in the device along
its axis. Spin-polarization of electrons in the current, $\mathbf{p}$, is
provided by a thick ferromagnetic polarizer of magnetization $\mathbf{M}_{p}$,
which assumed to be fixed. The vector $\mathbf{p}\|\mathbf{M}_{p}$ and
directed perpendicular to the plane of the free layer. Polarizer and the free
layer are separated by nonmagnetic spacer. As it is described above, the spin-
polarized current flowing through the free layer leads to the nonlinear
oscillations of its magnetization $\mathbf{M}$. The Analyzer, in turn,
converts the oscillations into a useful signal.

In contrast to mono-domain oscillators, in the vortex-state free layer the
oscillations of the total magnetic moment are negligibly small. The deviation
of the magnetization from the ground state during the oscillations lies in the
plane of the free layer, and its value averaged over the layer area equals to
zero. For this reason, the question of the signal extraction is nontrivial. We
suggest a solution of this problem by using special non-uniform analyzers.

Figure 6: Nanopillar geometry and the scheme of extracting the electric output
signal by GMR. The analizer is a ferromagnetic ring (optionally a disk) in the
vortex state. The gray arrow indicates the direction of the current flowing
into the nanopiler (opposite to the direction of electron movement). Vector
$\mathbf{p}$ shows the direction of spin polarization.

First, we discuss the possibility to obtain an alternating electrical signal
at the STNO output by the GMR effect. In the spin valve scheme based on GMR
the analyzer consist of a ferromagnetic layer with the fixed magnetization
$\mathbf{M}_{a}$ and a thin nonmagnetic interlayer separating it from the free
layer of magnetization $\mathbf{M}$. The magnetization of analyser are usually
fixed by usage of a hard ferromagnet of large thickness or by an additional
antiferromagnetic layer. When the spin-polarized current passes through this
system of layers, the scattering of electrons depends on the mutual
orientation of the magnetizations of the layers. The GMR contribution to the
resistance of the spin valve is locally (at each point of the layer area)
proportional to the scalar product of these magnetizations
$(\mathbf{M}\cdot\mathbf{M}_{a})$.

In the case under consideration the ground state of the free layer is the
vortex state, and projection of its magnetization on a fixed direction
averaged over the layer area is zero. Moreover, the same is valid for the
highly-nonlinear oscillational regime providing the generation, see Fig. 1.
Thus the use of a traditional homogeneous spin valve with
$\mathbf{M}_{a}=const$ is impractical.

Let consider an analyzer in a form of a thick magnetic ring (optionally a
disk) with the saturation magnetization $M_{as}$ in a vortex state, which is
stable enough even being in contact with free layer and under the action of
the current. In order to stabilize the vortex ground state of the analyzer, it
has to be made thick enough. Optionally, the analyzer in the vortex state can
be made from a chiral magnet, where the vortex state is stabilized by the
Dzyaloshinskii-Moriya interaction (DMI).

The magnetization of this vortex analyzer can be written as
$\mathbf{M}_{a}/M_{as}=-\mathbf{e}_{x}\sin(\chi+\varphi_{0})+\mathbf{e}_{y}\cos(\chi+\varphi_{0})$,
where $\mathbf{e}_{x}$ and $\mathbf{e}_{y}$ are orthogonal unit vectors in the
plane of the layer, and $\mathbf{e}_{z}$ is the unit vector along the device
axis. Here the constant $\varphi_{0}=0$ is for the ‘‘standard’’ vortices (Fig.
1(a)) with closed magnetic flux, and $\varphi_{0}=\pm\pi/2$ for so-called
radial vortices, stabilized by interfacial DMI. For the nonlinear regime of
the oscillations the scalar product
$(\mathbf{M}\cdot\mathbf{M}_{a})=M_{s}M_{sa}\sin\theta_{0}\cos(\psi(t)+\varphi_{0})$.
So the GMR contribution to the electrical resistance $\Delta R(t)$ of this
vortex spin valve scheme is proportional to
$\sin\theta_{0}\cos[\psi(t)-\varphi_{0}]$. The factor $\sin\theta_{0}$ is
determined by the external magnetic field, and the best efficiency of the
vortex spin valve is achieved for zero external field, when
$\theta_{0}=\pi/2$.

Figure 7: Devise geometry and the scheme of extracting the electric output
signal by ISHE. The nonhomogeneous analyzer contains two parts made of heavy
metals with different signs of spin Hall angle. The gray arrow indicates the
direction of the current flowing into the nanopillar (opposite to the
direction of electron’ motion). Vector $\mathbf{p}$ shows the direction of
spin polarization.

We also consider an alternative possibility to extract a useful electric
signal by the inverse spin Hall effect. Magnetization precession that appears
in the ferromagnetic ring-shaped free layer can acts as a spin pump to an
adjacent nonmagnetic metal. The direction of the spin current flow is
perpendicular to the interface between the magnetic free layer and the
nonmagnetic analyzer layer, which in our case coincides with $z-$axis. A spin
current generated via the spin pumping mechanism, i.e. the direction of
polarization of electrons in the current,
$\boldsymbol{j}_{sp}\varpropto\mathbf{m}\times\partial\mathbf{m}/\partial t$.
It can be presented as a sum of in-plane and out-of-plane components,
$\boldsymbol{j}_{sp}=\boldsymbol{j}_{sp}^{in}+\boldsymbol{j}_{sp}^{out}$. The
component $\boldsymbol{j}_{sp}^{in}$ lies in the plane of the layers and
$\boldsymbol{j}_{sp}^{out}$ is along $\mathbf{e}_{z}$. The spin pumping
generated due to the radially-symmetric mode is

$\displaystyle\boldsymbol{j}_{sp}\varpropto\frac{1}{2}\sin
2\theta_{0}\left[\mathbf{e}_{x}\sin(\chi+\psi)-\mathbf{e}_{y}\cos(\chi+\psi)\right]\frac{\partial\psi}{\partial
t}+\mathbf{e}_{z}\sin^{2}\theta_{0}\frac{\partial\psi}{\partial t},$ (33)

where $\psi=\psi(t)$ determines the spin precession.

If the external magnetic field is absent the ground state of the ferromagnetic
ring is the in-plane vortex with $\theta_{0}=\pi/2$ and only nonzero component
of the spin pumping is $\boldsymbol{j}_{sp}^{out}$, which, as we show below,
does not lead to the generation of an electrical signal. However, in the
presence of the magnetic field, i.e., for the cone state vortex, the situation
is absolutely different.

This spin current injected into nonmagnetic metal conductor attached to the
free layer can be converted into a charge current $\mathbf{j}_{c}$ via the
inverse spin Hall effect

$\displaystyle\mathbf{j}_{c}\varpropto\theta_{SH}\boldsymbol{\mathbf{e}}_{z}\times\mathbf{j}_{sp},$
(34)

where $\theta_{SH}$ is the spin Hall angle. It gives rise to an alternating
electromotive force, the source of a useful microwave signal. In the
considered geometry in-plane component of the spin current
$\boldsymbol{j}_{sp}^{in}$ is responsible for the occurrence of the electric
current,

$\displaystyle\mathbf{j}_{c}\varpropto\theta_{SH}\sin
2\theta_{0}\left[\mathbf{e}_{x}\cos(\chi+\psi)+\mathbf{e}_{y}\sin(\chi+\psi)\right]\frac{\partial\psi}{\partial
t}.$ (35)

Due to of the factor $\sin 2\theta_{0}$, the signal is nonzero only when
external magnetic field is applied and the free layer is in the cone vortex
state with $\theta_{0}<\pi/2$. The direction of the electric current
$\mathbf{j}_{c}$ lies in the plane of the layers perpendicular to
$\boldsymbol{j}_{sp}^{in}$. The maximum value of the $\sin 2\theta_{0}$ and
hence the amplitude of the electrical signal is reached at $\theta_{0}=\pi/4$,
that corresponds to the magnetic field $H\approx 2\sqrt{2}\pi M_{s}$.

To extract the output signal by the inverse spin Hall effect the analyzer
layer has to be made of the nonmagnetic metallic conductor. The signal that
can be taken is proportional to the value of the current density Eq. (35)
averaged over the interface between the free layer and the analyzer layer, i.e
$\int_{R_{i}}^{R_{o}}rdr\int_{0}^{2\pi}d\chi\;\mathbf{j}_{c}$. For the
homogeneous material of the analyzer this integral is equal to zero because of
the alternating periodic functions $\sin(\chi+\psi)$ and $\cos(\chi+\psi)$. To
solve this technical problem, it is advisable to fabricate different areas of
the analyzer made of two heavy metals with different signs of spin Hall angle
$\theta_{SH}$, e.g., platinum and tantalum, as has been realized in ISHE3 .
Then the electrical signals from different areas are summed up. For example,
we can select areas according to the sign of $\sin\chi$, then averaged in-
plane component of the spin current is along $\mathbf{e}_{x}$ and the
electrical current flows in the direction $\mathbf{e}_{y}$.

In summary, the proposed ring-based STNO is compatible with both known
mechanisms of microwave signal extraction: the GMR effect and ISHE.

##  VII Conclusions

In the paper we investigated the nonlinear spin dynamics of the ferromagnetic
ring in the vortex state under the action of the spin-polarized current and
suggested the basic design of the ring-based vortex-state STNO. Proposed
nanooscillator is characterised by the working frequencies up to $10$ GHz,
that makes it advantageous over the disc-based vortex oscillators operating in
the sub-GHz range.

Indeed, the working frequency of STNO is determined by the first excited spin-
wave mode. For the magnetic disk it is the gyromode, the collective mode,
manifesting itself as a homogeneous precession of the vortex core. The
frequency of the gyromode is very low, it is of the order of $1$ GHz. The high
generation frequency of the ring-based STNO is achieved due to the fact that
the ring-shape particle does not have the mode of core precession, because it
does not have a vortex core region. The first mode exited by the spin-torque
in the ring is the nonlinear breathing mode, that has higher frequency. It is
the radially-symmetric in-plane oscillations of magnetization with small out-
of-plane component. The regime of steady-state oscillations characterised by
the full radially-symmetric rotation of the magnetization vector provides the
generation of useful microwave signal. The frequency estimates for the
permalloy ring of spatial dimensions usual for the oscillators give $11.4$
GHz.

Also it is worth noting, that due to the vortex state of the proposed STNO the
in-plane total magnetic moment of the ferromagnet remains zero during the
oscillations. Thus the working oscillator does not create stray fields and it
is possible to pack the oscillators more densely, for example, for creating
arrays of oscillators. In the contrary, oscillator based on a homogeneous
state or on a quasi-homogeneous state have nonzero magnetic moment in the
ground state of the system. The rotation of this magnetic moment provides the
generation, and such oscillators create disadvantageous stray fields.

We also found that the nonlinear spin oscillations are described by the simple
and universal equation, Eq. (25). This equation appears for many physical
problems; probably, the most important system is a point Josephson contact.
This formal analogy allows usage of serious groundwork already developed for
application of Josephson systems in electronics Josephson1 ; Josephson2 .
Despite the fact that the spin dynamics of considered device is described by
the same equation as in cases of Josephson contact and antiferromagnetic
oscillator, the ring-based oscillator has different operational properties.
Indeed, the useful alternating electrical signal obtained by the ring-based
STNO has different behavior with increasing of the spin-polarized current,
and, as a result, different mechanisms of signal extraction are needed. In the
antiferromagnetic oscillator the alternating part of the signal is related to
inhomogeneity of precession of the Neel vector and decreases with the current.
Due to the vortex state of the ring-based STNO the alternating signal can be
obtained for the homogeneous precession of magnetization, and therefore there
is no effect of signal decreasing. This fact simplifies the basic design of
the analyzer for the ring-based STNO. Nevertheless, the signal extraction
technology in the case has to be nontrivial, since the oscillation of
magnetization averaged over the area of the ring is zero, which will result in
a zero average electromotive force in output of standard schemes. We suggested
to solve this problem by usage of special nonhomogeneous analyzers described
in Sec. VI. We showed that the ring-based STNO can be used both in the spin
valve circuit and in combination with the inverse spin Hall effect.

We expect the proposed device to have the following advantages. It can be made
of convenient size: ring radii are of the order of hundreds of nanometers, and
ring thickness is thin enough for spin transfer effects, namely
$1-5\text{nm}$. The system allows the easy adjustments of the main parameters,
like working frequency, by application of a weak enough out-of-plane magnetic
field. It is also worth noting the linear growth of the generation frequency
with the current. The threshold current has usual values for the vortex STNOs:
spin-transfer torque overcome the damping at current density about
$10^{6}\text{A}\text{cm}^{-2}$, whereas the ignition current is of the order
of $10^{8}\text{A}\text{cm}^{-2}$. In addition, it is possible to slightly
reduce the elimination current by the magnetic field.

Aside from the continuous auto-oscillations discussed in the paper, note that
the system also allows some impulsive regimes, which can be used, for example,
for creation of artificial neurons for neuromorphic computing, see
Refs.inertia2022 ; Khymyn-neuron ; Khymyn-logic . All of the above makes the
proposed ring-based STNO interesting for practical applications.

##  VIII Acknowledgment

V. Uzunova thank K. Byczuk for support and helpful discussions. The work of
B.A. Ivanov was partly supported by National Scientific Foundation of Ukraine
under Grant No. 2020.02/0261.

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# Optical Text Recognition in Nepali and Bengali: A Transformer-based Approach

S M Rakib Hasan  Department of Computer Science and Engineering
BRAC University
Dhaka, Bangladesh
<EMAIL_ADDRESS>Aakar Dhakal  Department of Computer Science and
Engineering
BRAC University
Dhaka, Bangladesh
<EMAIL_ADDRESS>Md Humaion Kabir Mehedi  Department of Computer
Science and Engineering
BRAC University
Dhaka, Bangladesh
<EMAIL_ADDRESS>Annajiat Alim Rasel  Department of
Computer Science and Engineering
BRAC University
Dhaka, Bangladesh
<EMAIL_ADDRESS>
###### Abstract

Efforts on the research and development of OCR systems for Low-Resource
Languages are relatively new. Low-resource languages have little training data
available for training Machine Translation systems or other systems. Even
though a vast amount of text has been digitized and made available on the
internet the text is still in PDF and Image format, which are not instantly
accessible. This paper discusses text recognition for two scripts: Bengali and
Nepali; there are about 300 and 40 million Bengali and Nepali speakers
respectively. In this study, using encoder-decoder transformers, a model was
developed, and its efficacy was assessed using a collection of optical text
images, both handwritten and printed. The results signify that the suggested
technique corresponds with current approaches and achieves high precision in
recognizing text in Bengali and Nepali. This study can pave the way for the
advanced and accessible study of linguistics in South East Asia.

###### Index Terms:

Low-Resource Languages, OCR, Transformers, Machine Translation, Digitized text

††publicationid: pubid: 979-8-3503-0544-9/23/$31.00 ©2023 IEEE

##  I Introduction

With the advent of ever-changing digital media and technology in recent years,
there has been a sharp rise in the demand for automated text detection.
Optical Character Recognition(OCR) is a technology that enables the
recognition of printed or handwritten text characters from scanned images. It
improves access to information, preserves rare texts and enables the
development of language technologies such as voice assistants and machine
translation systems. Despite numerous works on text recognition in various
languages, Bengali and Nepali text recognition has not received enough
attention due to its resource deficiency or being morphologically complex.
Bengali and Nepali are two extensively spoken languages in South Asia, and
understanding them is essential for computer translation, document
digitisation, and language processing. In this paper, an image-based method is
suggested for identifying Bengali and Nepali written text. A model is
developed using Microsoft TrOCR, an encoder-decoder based transformer
architecture as the base model and its efficacy was assessed using a
collection of text images. The outcomes demonstrate that the suggested
technique achieves high precision in deciphering typed text in Bengali and
Nepali. This study could aid in the advancement of linguistic technology in
South Asia and be useful in a number of industries, including automation,
administration, and education. This paper discusses the previous research,
some unique characteristics of Bengali and Nepali text, segmentation and
feature extraction methods followed by the experimental results and
conclusion.

##  II Previous Works

There have been many OCR models for low-resource languages like Bengali or
Nepali and some remarkable research exists in this field. An end-to-end word
identification system for handwritten Bengali words from pictures is
introduced in the paper [1]. Deep convolutional neural networks (CNNs) are
used by authors as feature extractors, followed by RNNs and a completely
connected layer that produces the end prediction. The Connectionist Temporal
Classification (CTC) loss function is used to teach the algorithm. The
efficacy of four distinct baseline models—Xception, NASNet, MobileNet, and
DenseNet—as feature extractors are examined in this article. The writers come
to the conclusion that for Bengali handwritten OCR, deeper systems with
residuals work better. The BanglaWriting dataset, a reputable Bengali dataset,
is used to assess the suggested technique. With a word recognition accuracy of
90.3%, the authors describe encouraging findings.

The study [2] presents an optical character recognition (OCR) framework for
printed Bengali and English text. The system was developed utilizing a
solitary, 128-unit hidden bidirectional long short-term memory (BLSTM)
architecture with connectionist temporal classification (CTC). The proposed
methodology consists of a dual-phase process. In the initial stage, the
vertical strip-based projection profile valleys of the document components
were regarded as the primary suppositions for the line end/beginning. The
aforementioned hypothesis underwent further development in the subsequent
stage, wherein depictions of objects and additional artefacts were excluded.
The text lines are subjected to normalization to achieve a height of 48 pixels
before being presented to the BLSTM-CTC-based OCR system for training and
validation. Performance evaluation employs test data that serves as a
reference standard. During the testing phase, the CTC functions as the
classifier and produces the most probable classifications for a given input
sequence as the final outcome. The OCR system’s accuracy at the character
level was 99.32%, while its accuracy at the word level was 96.65%.

The difficulties of optical character recognition (OCR) for Bangla text are
discussed in the work[3], along with a useful character segmentation method.
Using vertical and curved scanning, the segmentation approach uses line, word,
and character segmentation. The segmented character extraction procedure
utilizing the flood-fill method is also covered in the article. The suggested
method had 99% character segmentation accuracy, 99.8% line segmentation
accuracy, and 99.5% word segmentation accuracy. In the publication, character
segmentation accuracy experimental findings are presented. For five separate
pictures, 99% accuracy was attained. The suggested method can enhance the
accuracy of OCR for printed Bangla text.

A low-cost convolutional neural network design for Bengali handwritten
character identification is suggested by the writers in the study [4]. For the
training phase, the researchers used openly accessible standard datasets, such
as the CMATERdb Bengali handwritten character dataset, and created various
dataset forms in accordance with prior research. BengaliNet, the suggested
design, outperformed earlier work by a significant margin, achieving a total
accuracy of 96-99% for various databases of Bengali characters. The study aids
in the creation of an instrument that can automatically and effectively
recognize Bengali handwriting symbols.

The paper[5] suggests a customized version of a cutting-edge deep learning
methodology to identify handwritten Bengali symbols. The suggested approach
makes use of transfer learning on the ImageNet dataset along with a pre-
trained Resnet-50 deep convolutional neural network model. By altering the
input image sizes, the technique also changes the one-cycle strategy to
guarantee quicker training. The 84-class BanglaLekha-Isolated dataset is used
to assess the suggested methodology. The findings demonstrate that the
suggested approach outperforms other contemporary methods, achieving 97.12%
accuracy in just 47 epochs. The architecture and regularizations, including
batch normalization, dropout, learning rate schedule, and optimizers, used to
boost speed are also covered in the article. The authors come to the
conclusion that ResNet-50 is successful at classifying Bengali handwritten
characters using their suggested technique.

A novel feature set for handwritten Bangla alphabet detection is described in
the paper[6]. The feature collection comprises 132 features, including quad-
tree-based longest run features, distance-based features, modified shadow
features, octant and centroid features, and features based on centroid and
distance. The features are calculated from binary pictures of Bangla
alphabetic letters with a dimension of 64x64 pixels. Four longest-run features
are calculated for every sub-image at every point of the quad-tree structure.
The novelty of the present study is the partitioning of any character sequence
using the CG-based quad-tree structure. From the 75.05% seen in the authors’
earlier work to 85.40% on 50 character classes with an MLP-based classifier on
the same dataset, the identification performance using this feature set rises
significantly.

The authors of the study [7] established a novel benchmark for 60 low-resource
languages that possess low-resource scripts. This was accomplished through the
utilization of authentic and synthetic data that was augmented with noise. The
researchers conducted a benchmarking study to identify the most common errors
in Optical Character Recognition (OCR) models that are utilized for both
academic research and commercial applications. This study utilized state-of-
the-art OCR mistakes to evaluate the impact of fine-tuning machine translation
(MT) models with OCR-ed data in comparison to pre-trained MT models and MT
models fine-tuned with initial data. The study aimed to contrast the MT models
and determine the effect of SOTA OCR mistakes on them. The principal finding
is that the utilization of back translation in optical character recognition-
processed monolingual data results in an improvement in machine translation.
Overall, the majority of current optical character recognition (OCR) models
exhibit a high level of recognition accuracy, which allows for the training of
machine translation (MT) models. However, there are certain scripts, such as
Perso Arabic, that present challenges to OCR models. Additionally, while this
augmentation process is generally robust to various types of errors, it may be
susceptible to substitutions in particular. This work will enable future
research endeavours focused on data augmentation for machine translation
utilizing OCR documents.

The study[14] introduces a novel Optical Character Recognition (OCR) system
designed for scripted Bangla characters. The system has been constructed
utilizing a Convolutional Neural Network (CNN) for the purpose of character
recognition. The system also integrates various techniques such as
binarization and noise reduction to enhance the precision of the results. The
results of the performance evaluation indicate that the proposed system
exhibits superior accuracy in comparison to existing methods with respect to
line detection, word segmentation, and character recognition. The study
involved conducting an experiment on the recognition of a maximum number of
distinct characters, which was 202. Additionally, the number of training and
test images utilized in the experiment exceeded those used in prior research.
A precision rate of 99.33% was attained. The system exhibits robust
performance in diverse conditions, including images that are affected by noise
or blurriness. The duration of computation may pose a challenge contingent
upon the capacity of the system in which it is executed, and the system’s
efficacy may be compromised in instances where the image is excessively
rotated.

##  III Characteristics of Bengali and Nepali Texts

By nature, Bengali, Nepali and other oriental languages have quite irregular
and unique forms than Latinized languages. These features make the feature
extraction and mapping of these languages difficult. However, in its written
form, the Bengali language has a total of 50 letters in its alphabet, 11 being
vowels and 39 consonants and the Nepali language has the same number of vowels
with 33 consonants. Both these languages have some similarities in their
written form which makes their interpretation harder, described as follows:

###  III-A Dependant Letters

There are vowels and consonants joined with the consonants to produce the
sounds in the words. So, the regular shape of the letters gets changed very
frequently. These can be in many different shapes(each different for short and
long sounds), which can get very confusing for the machine to interpret. These
also change the normal distancing between letters which makes it more
difficult to segment the letters. Fig-1 and Fig-2 depict these patterns for
both languages

Figure 1: Dependant letters in Bengali Text.

Figure 2: Dependant letters in Nepali Text.

###  III-B Overlapping Characters

Often the letters of these alphabets do not remain bounded to a certain region
and enter the region of its next character. It happens because the dependent
letters join and get into the next letters. It is shown in figure-3 and
figure-4:

Figure 3: Overlapping Character in a Bengali word.

Figure 4: Overlapping Character in a Nepali word.

###  III-C Word Separation

There is a line on the letters of each word which keeps the word together.
This line, known as the “Matra” line can determine the words separately.
However, there are letters which skip the line above them, making the feature
extraction difficult. Also, above the matra line, there may be parts of a
letter and below there may be vowels. The figures below show this phenomenon.

Figure 5: Separation of words in Bengali texts.

Figure 6: Separation of words in Nepali texts.

###  III-D Confusing Shapes

Both languages have some letters that are often mistaken to be the same. For a
machine, it becomes tough to differentiate between these letters having very
low data to train on. The following figures show these letters in both
languages.

Figure 7: Some confusing letters in Bengali.

Figure 8: Some confusing letters in Nepali.

##  IV Methodology

Apart from common OCR techniques in low-resourced languages that use CNNs or
RNNs, a transformer-based method is proposed. The Transformer models, in
contrast to the features retrieved by the CNN-like network, do not contain
picture-specific inductive biases but do the image processing as a series of
patches, making it simpler for the model to pay attention to either the entire
image or the independent patches. In this study, the transformer model
TrOCR[9] is used as the base model and extended to extract features from
Bengali and Nepali images and map these features to texts. TrOCR uses typical
encoder-decoder methods. It can be divided into two distinct parts

###  IV-A Feature Extraction Encoder

An image transformer is used in this part which takes some input images and
extracts their features as outputs. Here, the VisionEncoderDecoder model with
the pre-trained Vision Transformer (ViT)[10] is used as the encoder. The
google/vit-base-patch16-384 from huggingface was used which is pre-trained on
14 million images and fine-tuned on one million images. The model receives
images as a series of linearly embedded, fixed-size patches with a 16x16
resolution. In order to employ a sequence for classification tasks, a [CLS]
token is also added to the beginning of the sequence. Before feeding the
sequence to the layers of the Transformer encoder, one may additionally
include absolute position embeddings. An instance of utilizing a dataset of
labelled images involves training a conventional classifier through the
stacking of a linear layer onto the pre-existing encoder. Through pre-
training, the model acquires an internal representation of images that can be
leveraged to extract features that are advantageous for subsequent tasks. The
final hidden state of the [CLS] token is often regarded as a comprehensive
representation of an entire image and is therefore commonly augmented with a
linear layer. Figure 9 depicts the architectural design of the model. After
the feature extraction, the language modelling is done by the decoder.

Figure 9: The ViT architecture[8].

###  IV-B Language Modelling

A text transformer is used to map the extracted features to texts. In this
instance, the xlm-roberta-base decoder has been chosen. This is a multilingual
text transformer model that has been trained on 2.5 terabytes of data from 100
languages. There are not many language decoder models specifically for Bengali
and Nepali. Therefore, this model is selected that can perform the required
tasks in both languages. This is a multilingual rendition of RoBERTa. It was
pretrained with the objective of Masked language modelling (MLM). Using a
sentence as input, the model conceals 15% of the words at random before
running the entire sentence through the model and attempting to predict the
hidden words. This differs from conventional recurrent neural networks (RNNs),
which typically see the words one after the other, and autoregressive models
such as GPT, which conceal the future tokens internally. It enables the model
to learn a representation of the sentence that is bidirectional. Thus, the
model learns an internal representation of 100 languages that can then be used
to derive features useful for downstream tasks[11].

Figure 10: Proposed workflow

##  V Experimental Setup

###  V-A Datasets

For the Bengali language, the open-source BanglaWriting[12] dataset is used
for fine-tuning the transformer model. It includes the single-page handwriting
of 260 people of various ages and personalities This data set includes a total
of 21,234 words and 32,784 characters. Additionally, this dataset comprises
5,470 unique Bangla words. The dataset also contains 261 comprehensible
overwriting and 450 handwritten strikes and errors, in addition to the usual
words. The word labels are generated manually. For the Nepali language, a
manually prepared dataset is taken which includes 50 images from different
people. It contains around 7 thousand words with about 10 thousand characters.
It also has many unique words and the word labels are manually generated. Both
language datasets have been used separately for training and validation.

###  V-B Data Preprocessing

The selected dataset contained images of varying sizes. Each image was resized
to 384x384 pixels for input to the encoder, ViT. The images had already been
denoised, so the denoising step was omitted. Since the encoder performs
effectively with regular images, the data was not converted to grayscale. The
images were enhanced through random flipping and random rotation (-5 to 5
degrees).

###  V-C Training

The model was trained on the dataset for 2000 epochs for each language using a
batch size of 4 images. Training took place on a machine equipped with a
3.6GHz CPU, 16 GB RAM, and an RTX 3070 GPU. Optimization was performed using
the AdamW algorithm, a stochastic gradient descent technique that incorporates
adaptive prediction of first-order and second-order moments, along with a
weight decay method according to the techniques[13]. Visualization of training
and validation loss was accomplished using the WandB and DataCollator
libraries.

###  V-D Testing

The performance of the proposed trained Optical Character Recognition (OCR)
model was assessed by analyzing a set of 10 images obtained from various
sources. The chosen images were curated to exemplify a broad spectrum of font
styles, sizes, and orientations that are frequently encountered in practical
situations. The initial step in the testing procedure involved image
preprocessing, which entailed resizing the images to a uniform size of 512x512
pixels and converting them to grayscale. Subsequently, the preprocessed images
were inputted into the OCR model, and the resulting recognized text was
documented. In order to assess the precision of the optical character
recognition (OCR) model, the subsequent metrics were computed:

  * •

The metric of character-level accuracy evaluates the proportion of accurately
identified characters within the completely recognized text. The computation
involves the division of the count of accurately identified characters by the
overall count of characters present in the reference text.

  * •

The metric of word-level accuracy evaluates the proportion of accurately
identified words within the complete recognized text. The computation involves
the division of the count of accurately identified words by the overall count
of words present in the reference text.

The ground truth text for each of the 10 images was manually annotated and
compared with the recognized text generated by the OCR model to calculate the
metrics.

##  VI Results and Evaluation

The text recognition performance of the suggested model in Bengali and Nepali
languages was encouraging. During training, the model achieved low Word Error
Rates (WER) of 0.10 and 0.14 for Bengali and Nepali, respectively, and
Character Error Rates (CER) of 0.04 and 0.10 for Bengali and Nepali,
respectively. The Levenshtein distance between the recognized and ground truth
texts was used to determine the CER and WER values.

Figure 11: CER and WER for Bengali on Training Dataset Figure 12: CER and WER
for Nepali on Training Dataset

The CER and WER values were plotted over 2000 epochs to show the model’s
performance. The data demonstrated that the model, for both Bengali and Nepali
languages, reached a consistently low CER and WER after roughly 1500 epochs.

The CER and WER values were marginally higher than those attained during
training, though, when the model was tested on the test set. Bengali had an
average CER and WER of 0.07 and 0.12, while Nepali had an average CER and WER
of 0.11 and 0.15.

The modest variation in the CER and WER values between training and testing
may be explained by the test set’s greater diversity and changing image
quality when compared to the training set. The test set may have included
images with more intricate backgrounds or uncommon font styles that were not
present in the training set, despite the fact that the model was trained on a
varied selection of images with different font styles, sizes, and
orientations.

##  VII Conclusion

In this research, a model for Bengali and Nepali text recognition was
proposed. Images containing handwritten and printed text in both languages
served as training data for the model. According to the experimental findings,
the proposed model exhibited a low Character Error Rate (CER) and Word Error
Rate (WER) during training and performed effectively on the test set. During
training, the model achieved CERs of 0.04 and 0.09, as well as WERs of 0.10
and 0.14, for Bengali and Nepali, respectively. When evaluated on the test
set, the model’s average CER and WER for Bengali and Nepali were 0.07 and
0.12, and 0.11 and 0.15, respectively. These results demonstrate the model’s
strong performance in accurately recognizing text in Bengali and Nepali. Due
to its low CER and WER values during training and its effectiveness on the
test set, the model holds promise for practical applications such as document
digitization and text extraction. The suggested OCR methodology underscores
its potential for practical uses like text extraction and document
digitization. The model’s capability to accurately identify text from diverse
images in Bengali and Nepali languages illustrates its competence in handling
complex and varied input.

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