# Ultrahigh Frequency and Multi-channel Output in Skyrmion Based Nano-
oscillator

Abhishek Sharma  Saumya Gupta  Debasis Das  Ashwin. A. Tulapurkar  Bhaskaran
Muralidharan Abhishek Sharma and Saumya Gupta are co-first authors. Abhishek
Sharma is with the Department of Electrical Engineering, IIT Ropar, Rupnagar,
Punjab 140001, India (e-mail: abhishek@iitrpr.ac.in). Saumya Gupta, Bhaskaran
Muralidharan, and Ashwin. A. Tulapurkar are with the Department of Electrical
Engineering, IIT Bombay, Mumbai 400076, India (e-mail:
<EMAIL_ADDRESS>bm@ee.iitb.ac.in;ashwin.tulapurkar@iitb.ac.in).The
author BM acknowledges the support by the Science and Engineering Research
Board (SERB), Government of India, Grant No. CRG/2021/003102 and Grant No.
MTR/2021/000388, and the Ministry of Human Resource Development (MHRD),
Government of India, Grant No. STARS/APR2019/NS/226/FS under the STARS scheme.
Debasis Das is with the Department of Electrical and Computer Engineering,
National University of Singapore, Singapore 117583 (e-mail: eledd@nus.edu.sg).

###### Abstract

Spintronic nano-oscillators can generate tunable microwave signals that find a
wide range of applications in the field of telecommunication to modern
neuromorphic computing systems. Among other spintronic devices, a magnetic
skyrmion is a promising candidate for the next generation of low-power devices
due to its small size and topological stability. In this work, we propose a
multi-channel oscillator design based on the synthetic anti-ferromagnetic
(SAF) skyrmion pair. The mitigation of the skyrmion Hall effect in SAF and the
associated decimation of the Magnus force endows the proposed oscillator with
an ultra-high frequency of 41GHz and a multi-channel frequency output driven
by the same current. The ultrahigh operational frequency represents an
$\sim$342 times improvement compared to the monolayer single skyrmion
oscillator featuring a constant uniaxial anisotropy profile. Using
micromagnetic simulations, we demonstrate the effectiveness of our proposed
multi-channel oscillator design by introducing multi-channel nanotracks along
with multiple skyrmions for enhanced frequency operation. The ultrahigh
operational frequency and multi-channel output are attributed to three key
factors: The oscillator design accounting for a finite spin-flip length of the
spacer (such as Ru) material, tangential velocity proportionality on input
spin current along with weak dependence on the radius of rotation of the
skyrmion-pair, skyrmion interlocking in the channel enabled by the multi-
channel high Ku rings and skyrmion-skyrmion repulsion, therefore resulting
ultrahigh frequency and multi-channel outputs.

Bilayer, gigahertz frequency, nano oscillators, skyrmions, spintronics,
Synthetic antiferromagnet, Spin Transfer Torque

##  1 Introduction

The spin-transfer torque (STT)[1], along with magnetoresistance (MR)[2], has
emerged as a technologically relevant phenomenon that caters to various
applications such as spin transfer torque-magnetic random access memory (STT-
MRAM)[3], spin-transfer torque nano-oscillators (STNOs)[4], racetrack
memories[5], to name a few. The STNOs are a class of non-linear inductorless
oscillators in which the STT excites uniform magnetization dynamics, and non-
collinear MR translates the precession of magnetization to the microwave
signal[6]. Among various spintronic devices, magnetic skyrmion show a huge
potential for next-generation memory and computing devices, due to its small
size [7], topological protection[8], low depinning current density[9],
electrical manipulation and detection[10]. Thus, making them suitable
candidate for many applications such as nano-oscillator[11], race track
memories[12], logic gates [13, 14], neuromorphic computing[15, 16, 17] and
transistors[18]. The observation of magnetic skyrmions in a bulk non-
centrosymmetric ferromagnet (FMs)[19, 20, 21] and thin films[22, 23] has paved
a way for many advanced electronic applications. The magnetic skyrmions are
stable vortex-like spin textures with a non-trivial topology. The magnetic
skyrmions are stabilized by a mutual competition between Heisenberg exchange
interaction favouring collinear alignment and Dzyaloshinskii-Moriya
interaction (DMI) interaction favouring perpendicular alignment of spins [24].
Bulk non-centrosymmetric magnets with bulk DMI support the Bloch-type magnetic
skyrmions, whereas in thin films, Neel-type hedgehog skyrmions are stabilized
by the interfacial DMI acting on the FM layer attached to a heavy metal (HM)
layer [25, 26]. It has been shown that a thin FM layer with a perpendicular
magnetic anisotropy (PMA) grown on a heavy-metal layer, can stabilize Neel-
type Skyrmion[22, 23]. Robustness against the thermal fluctuation [27, 28],
makes the Neel-type of skyrmion, one of the promising candidates for future
spintronic applications. Thus, in this work, we focus on the Neel-type
skyrmion to design the spintronic oscillator.
In particular, the skyrmion-based oscillators[11, 13, 29, 30, 31] have been
explored theoretically for microwave generation featuring a small line-width
and dynamical frequency range (MHz $\sim$ few GHz) reporting manipulation of
PMA, DMI, polarization angle, or design level variations such as introducing a
groove in the ferromagnet layer to enhance operational frequencies. Jin et.
al. reported an operation frequency of 15.63GHz by moving a skyrmion in an
annular groove[31]. Our proposed skyrmion STNO is feasible for introducing
multiple skyrmions together to improve the oscillation frequency, having both
a smaller size and topological protected dynamics compared to regular
STNOs[11, 32, 33]. The main obstacle faced by the skyrmion is the Magnus
force[34] which adds a transverse component to the velocity, leading to a path
deviation governed by the injected spin current, a phenomenon known as the
skyrmion Hall effect[8]. This may lead to skyrmion annihilation at the edge of
the FM layer for large spin currents, thus limiting the gyration frequency of
a skyrmion-based oscillator[11, 33]. To resolve this issue of the unwanted
Magnus force, we utilize the bilayer system, where two FM layers are coupled
by an antiferromagnetic (AFM)[35, 36, 37] exchange coupling[38]. In such a
system, sufficient coupling constant ensures the cancellation of the Magnus
force acting on the skyrmions in the individual FM layers[39].
Several other variations of skyrmion-based oscillators utilizing the AFM or
bilayer geometry[40] as reported by Shen et. al. depicted an AFM system with
skyrmion having a frequency of 25GHz by varying the nanodisk radius, damping
constant, and applied current[40].
In this work, we explore a skyrmion-based oscillator design using an AFM
exchange-coupled bilayer system. Our work underscores the significance of our
design in addressing a limitation often overlooked in prior studies involving
bilayer skyrmion geometries, namely, the substantial spin-flip length
associated with Ru[41]. Consequently, our work yields a notably improved
upper-limit operational frequency compared to earlier reported works.

Using micromagnetic simulation, we show that our proposed bilayer system can
generate a broad range of frequencies compared to its single-layer FM
counterpart. We also show that the frequency range can be enhanced by
designing a multi-channel circular racetrack, which acts as a multi-frequency
generating oscillator using a single device. Using the repulsion
characteristics between skyrmions with the same topology, we simulate the
device by nucleating multiple skyrmions in the racetracks and demonstrate that
the generated frequency range can be increased further. The rest of the paper
is organized as follows. In Sec.2, we describe our proposed bilayer model for
the skyrmion oscillator. Sec.3 describes the mathematical framework used for
the simulation. In Sec. 4, we discuss the results obtained from the
simulation, and finally, Sec. 5 concludes the paper.

##  2 Device schematic and working principle

Figure 1: Schematic diagram of (a) regular skyrmion oscillator and (b) bilayer
skyrmion based oscillator.

A regular skyrmion-based oscillator[11], consists of two circular FM layers
known as the reference layer (RL) and free layer (FL) separated by an
insulator layer (spacer) as shown in Fig. 1(a). The FL is assumed to have an
uniaxial perpendicular magnetic anisotropy (PMA), which is grown on a heavy
metal (HM) layer. This induces DMI at the interface of the HM/FM structure due
to the broken inversion symmetry. A skyrmion is nucleated in the FL, where the
competition between the DMI and Heisenberg exchange interaction in the FL
stabilizes magnetic skyrmion [24]. On the other hand, the magnetization of the
RL is fixed for this structure with in-plane vortex-like spin polarization as
shown in Fig. 1(a). The vortex-like RL magnetization is denoted by
$\mathbf{m}_{\text{RL}}=(cos\phi,sin\phi,0)$, where
$\phi=tan^{-1}(\frac{y}{x})+\Psi$ and $(x,y)$ are the spatial coordinates of
the RL with the origin being at the center of nano-disk(dot) [42, 43] and the
polarization angle $\Psi$ is $180^{o}$. During the injected current flow,
electrons passing through the RL get spin-polarized according to
$\mathbf{m}_{\text{RL}}$. The spin-polarized electrons upon reaching the FL,
move the skyrmion in a circular path as shown in Fig. 1(a). The vortex-like
spin-polarized current exerts a tangential force on the skyrmion (see Eq. 14),
which causes the motion of the skyrmion in the tangential direction. On the
other hand, the Magnus force proportional to the skyrmion velocity (see Eq. 9)
acts along the radially outward direction. This force is balanced by the
repulsion force on the skyrmion acting towards the center of the nano-disk due
to the canted spins at the edge of the nano-disk[44]. This causes the skyrmion
to move in a uniform circular path. The uniform circular motion of the
skyrmion can be translated to the electrical signal by the MR effect of a
sensing magnetic tunnel junction (MTJ) on top of the FL[27]. we also propose
an extended schematic design Fig.2 for reading via Magnetoresistance in light
of recent experimental demonstration[45, 46], the proposed device utilizes
multiple synchronized MTJ’s corresponding to each skyrmion in a particular
channel Fig.7 which can be phase-locked to deliver an enhanced additive output
power from each MTJ compared to a single MTJ to read the nanodisk. The
frequency of the skyrmion oscillator increases with current density (J)
attributed to the rise in skyrmion tangential velocity. The intensified Magnus
force as an effect of increased skyrmion velocity can surpass edge repulsion
beyond a critical current density value leading to skyrmion annihilation at
the edge of the nano-disk[33]. This limits the gyration frequency of the
skyrmion oscillator as well as any possibility of multichannel operation.

Figure 2: Schematic diagram of Multi-channel skyrmion-based bilayer nano
oscillator with multiple MTJ to detect the corresponding skyrmions present in
channel F1(see Fig.7)

To alleviate this issue, we explore a bilayer device for the skyrmion-based
oscillator as shown in Fig. 1(b). It has a similar device design to the
regular skyrmion-based oscillator, except for the free layer. In the bilayer
device, a single free ferromagnet layer is replaced by two free ferromagnet
layers known as the bottom free layer (BFL) and the top free layer (TFL) which
are separated by another spacer layer (spacer-2). The BFL and TFL have the
same uniaxial PMA with magnetization pointing along $+\hat{z}$(red) and
$-\hat{z}$(blue) directions, respectively. These layers are anti-
ferromagnetically coupled (AFC) via spacer-2 consisting of a non-magnetic
metal such as Ru through Ruderman-Kittel-Kasuya-Yosida(RKKY) type interlayer
coupling[47]. Additionally, the Ru also provides DMI for stabilizing the
skyrmion in the SAF multilayer. The experimental feasibility of such
structures in nucleating stable skyrmions is demonstrated by Juge et. al[37].
We have made the spacer-2 layer such that it touches both the BFL and TFL for
different disc radii i.e., a lesser disc surface for TFL compared to the BFL
to limit or minimize the spin current flow in the TFL. The design
modification(etched design) ensures the application of higher current density
in the BFL, and the spacer-2 ensures reduced spin polarization within the SAF
multilayer, also accounting for the Ru’s spin-flip length of 14nm attributing
to higher frequency generation. if the spin current translates into the TFL
the magnetization dynamics would be completely different, and the skyrmion in
the TFL and BFL may no longer be in unison, undermining the objective of the
bilayer structure.
We have made contact inside the spacer-2 layer touch the BFL (Fig 1(b)) to
avoid or minimize the spin current flow in the TFL [41]. Without the
integration of this design modification, the bilinear surface exchange
coefficient reduces beyond the spin-flip length of 14nm for the Ru material
leading to a spin memory loss. Under high current densities, the skyrmion in
the TFL and BFL will no longer be in unison, undermining the objective of a
bilayer structure.

This type of contact can be made via selectively etching out the section of
TFL along with a significant part of the spacer-2 layers. This technique is
usually employed in heterostructure-based device fabrication and is well
within current fabrication capabilities[48].

Figure 3: Monolayer skyrmion oscillator. (a) The two-dimensional color plot of
the $m_{z}$ ($z$-component magnetization) on the $x-y$ plane of the FL. (b)
The frequency of oscillation (c) The rotation radius of skyrmion on nano-dot
(d)skyrmion diameter with current density.

Along with the monolayer and bilayer skyrmion oscillators with constant PMA we
have simulated the bilayer oscillators with other uniaxial anisotropy profiles
such as a high ku selective disc and multichannel ring, as described in sec IV
can be fabricated using He+ ion irradiation technique[49] or VCMA methods
which can create desired PMA profile via inducing voltage on the electrode
gate through an insulator[50].

The bilayer skyrmion-based oscillator works similarly to the regular skyrmion
oscillator except for the absence of Magnus force. The TFL and BFL skyrmions
form an AFC skyrmion pair with equal and opposite skyrmion number[39, 35]. The
Magnus force on the skyrmion pair is zero due to the vanishing gyro-vector(Eq.
17). The vertically injected vortex-like spin-polarized current in the BFL
exerts a tangential force on the BFL-skyrmion and due to the AFC, both the
skyrmions in BFL and TFL move in unison along a uniform circular path as shown
in Fig. 1(b). The spin current only exerts a tangential force(Eq. 3) on the
skyrmion pair, which enables a large spin current ejection and high frequency
of oscillation. The absence of Magnus force allows the skyrmion pair to move
in a designated track opening up the possibility of multi-channel frequency of
operation. The uniform circular motion of the skyrmion-pair can be translated
to the electrical signal by the MR effect of a sensing magnetic tunnel
junction (MTJ)[51, 52, 46, 45] on top of the TFL[27] in the path of TFL-
skyrmion.

##  3 Mathematical description and simulation details

The Hamiltonian of the layer $l$ is given by:

$\displaystyle H_{l}$ $\displaystyle=$ $\displaystyle-
J_{ex}\displaystyle\sum_{<i,j>}\textbf{m}_{i}^{l}\cdot\textbf{m}_{j}^{l}+\displaystyle\sum_{<i,j>}\textbf{D}_{i,j}\cdot(\textbf{m}_{i}^{l}\times\textbf{m}_{j}^{l})$
(1)
$\displaystyle+K_{u}\displaystyle\sum_{i}[1-(\textbf{m}_{i}^{l}\cdot\textbf{z})^{2}]+H_{DD}$

where, $l$ represent the layer index ($l$=TFL, BFL),
$\textbf{m}_{i}=\textbf{M}/M_{s}$ is the normalized local magnetic moment at
lattice site $i$, $<i,j>$ runs over neighboring sites and $M_{s}$ is the
saturation magnetization. The first term represents the FM exchange
interaction ($J_{ex}>0$). The second term represents the DMI interaction with
D being DMI vector favoring canted magnetic moments. The third represents
uniaxial PMA with $K_{u}$ being the anisotropy constant and the last term
represents the dipole-dipole interaction. Apart from $H_{\text{TFL}}$ and
$H_{\text{BFL}}$, there is also another energy term that represents the
interlayer exchange coupling between TFL and BFL, given by:

$H_{\text{interlayer}}=A_{\text{inter}}\displaystyle\sum_{i}\textbf{m}_{i}^{\text{TFL}}\cdot\textbf{m}_{i}^{\text{BFL}}$
(2)

where, $A_{\text{inter}}<0$ represents the interlayer AFM interaction. The
total Hamiltonian for the bilayer system is
$H_{T}=H_{\text{TFL}}+H_{\text{BFL}}+H_{\text{interlayer}}$. In the continuous
limit of the magnetization, the total energy can be written as:

$\displaystyle E$ $\displaystyle=$ $\displaystyle\int
dV\big{[}\big{.}A(\nabla\textbf{m})^{2}+\varepsilon_{DMI}+K_{u}\left[1-(\textbf{m}\cdot\textbf{z})^{2}\right]$
(3)
$\displaystyle-\frac{\mu_{0}}{2}\textbf{m}\cdot\textbf{H}_{d}+\varepsilon_{interlayer}\big{.}\big{]}$

where, A is exchange stiffness constant, $H_{d}$ is demagnetization field,
$\mu_{0}$ is the free space permeability, $\epsilon_{DMI}$ and
$\epsilon_{interlayer}$ are interfacial form of DMI and interlayer AFM
coupling energy densities, respectively and are given by:

$\varepsilon_{interlayer}=\sigma\left[1-\textbf{m}^{TFL}\cdot\textbf{m}^{BFL}\right]$
(4) $\varepsilon_{DMI}=D\left(m_{z}\frac{\partial m_{x}}{\partial
x}-m_{x}\frac{\partial m_{z}}{\partial x}+m_{z}\frac{\partial m_{y}}{\partial
y}-m_{y}\frac{\partial m_{z}}{\partial y}\right)$ (5)

where, $\sigma=\frac{A_{inter}}{t_{sp2}}$, $A_{inter}$ being inter-layer
exchange coupling and $t_{sp2}$ being thickness of spacer-2 region and $D$ is
interfacial DMI constant.
In the presence of spin current perpendicular to the plane (CPP) of the BFL,
the dynamics of skyrmion for layer $l$ of the oscillator is obtained by
solving the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation described as
follows:

$\displaystyle\frac{d\textbf{m}^{l}}{dt}$ $\displaystyle=$
$\displaystyle-|\gamma|\textbf{m}^{l}\times\textbf{H}^{l}_{eff}+\alpha\left(\textbf{m}^{l}\times\frac{d\textbf{m}^{l}}{dt}\right)$
$\displaystyle+|\gamma|\beta_{j}\epsilon\left(\textbf{m}^{l}\times\textbf{m}_{RL}\times\textbf{m}^{l}\right)-|\gamma|\beta_{j}\epsilon^{\prime}\left(\textbf{m}^{l}\times\textbf{m}_{RL}\right)$

where,$\gamma=2.211\times 10^{5}~{}\text{m/(A.s)}$ is the gyromagnetic ratio,
$\textbf{m}_{RL}$ is the magnetization of RL, $\beta_{j}=\hbar
J/(\mu_{0}et_{z}M_{s})$,
$\epsilon=\frac{P\Lambda^{2}}{\left(\Lambda^{2}+1\right)+\left(\Lambda^{2}-1\right)\cdot\left(\mathbf{m}\cdot\mathbf{m}_{RL}\right)}$,
and $\epsilon^{\prime}$ is the secondary spin transfer term. Here $e$,$\hbar$,
$P$, $t$, and $J$ are electronic charge, reduced Planck’s constant,
polarization, thickness of the BFL, and current density, respectively. The
effective magnetic field $\textbf{H}_{eff}$ is given by:

$\textbf{H}_{eff}=-\frac{1}{\mu_{0}M_{s}}\frac{\delta E}{\delta\textbf{m}}$
(7)

The motion of the center of the skyrmion in the presence of spin current is
described by the Thiele equation:

$\textbf{G}\times\textbf{v}-\alpha\mu_{0}M_{s}t_{z}\textbf{v}.\textbf{d}/\gamma+F_{STT}+F=0$
(8)

where the first term represents the Magnus force with the gyro-vector G and
velocity of skyrmion v. The gyromagnetic coupling vector $\textbf{G}=(4\pi
Q\mu_{0}M_{S}t_{z}/\gamma)\hat{z}$ depends on the skyrmion number
$Q=-\frac{1}{4\pi}\int
dxdy[\textbf{m}\cdot(\partial_{x}\textbf{m}\times\partial_{y}\textbf{m})]$,
which is equal to $\pm 1$ for an isolated FM skyrmion. The second term
represents the dissipative (viscous) force with dissipative tensor
$\textbf{d}=\begin{pmatrix}d&0\\\ 0&d\end{pmatrix}$, where
$d=\frac{1}{4\pi}\int
dxdy(\partial_{x}\textbf{m}\cdot\partial_{x}\textbf{m})$. The third term
represents the force due to spin transfer torque and the last term is the
force due to anisotropy gradient (if any) or/and due to boundary-induced
force.
The driving force due to spin transfer torque and velocity of skyrmion can be
decomposed in two parts on the plane of nanodisk
$\textbf{F}_{STT}=F_{r}\hat{r}+F_{t}\hat{t}$ and
$\textbf{v}=v_{r}\hat{r}+v_{t}\hat{t}$. The radial force ($F_{r}$) and
tangential force ($F_{t}$) and are given by
$F_{r/t}=-\mu_{0}\beta_{j}M_{s}t_{z}\int
dxdy[(\textbf{m}\times\textbf{m}_{p})\cdot\partial_{r/t}\textbf{m}]$. In the
steady state motion of skyrmion on the nano-disk (See Fig. 1(a)) have
$v_{r}=0$ and constant $v_{t}$ for the given current density. The equation 8
can be rewritten along the $\hat{r}$ & $\hat{t}$ respectively, as:

$\displaystyle v_{t}G+F_{r}+F=0$ (9)
$\displaystyle\alpha\mu_{0}M_{s}t_{z}v_{t}d/\gamma+F_{t}=0$ (10)

Equation 9 implies that the Magnus force ($v_{t}$G) is balanced by boundary
force or/and anisotropy gradient ($F$). Whereas, the equation 10 gives the
steady state motion of skyrmion and can be simplified to give $v_{t}$. For
ease of calculation the $F_{t}$ during the steady state motion
($v_{t}=$constant) can be evaluated when the skyrmion is along the x-axis on
nano-disk such that $\textbf{m}_{p}=-\hat{x}$, $\hat{t}=\hat{-y}$ and
$\hat{r}=\hat{x}$.

$F_{t}=-\mu_{0}\beta_{j}M_{s}t_{z}\int
dxdy[(\textbf{m}\times\textbf{m}_{p})\cdot\partial_{y}\textbf{m}]$ (11)

The profile of skyrmion is given by:

$\textbf{m}=\hat{x}sin\big{(}\theta(r^{\prime})\big{)}cos\psi+\hat{y}sin\big{(}\theta(r^{\prime})\big{)}sin\psi+\hat{z}cos\big{(}\theta(r^{\prime})\big{)}$
(12)

where, $\theta(r^{\prime})$ azimuthal angle is function of $r^{\prime}$
distance measured from the center of the skyrmion such that
$\frac{d\theta}{dr^{\prime}}\neq 0$ only at $r^{\prime}=R_{S}$. The
$\theta(r^{\prime})$ changes from $0$ to $\pi$ and $\psi$ changes from $0$ to
$2\pi$. The $F_{t}$ can be simplified the Eq. 11 in the polar coordinate
system as:

$F_{t}=-\mu_{0}\beta_{j}M_{s}t_{z}\int
r^{\prime}dr^{\prime}d\psi\Big{[}sin^{2}\psi\frac{d\theta}{dr^{\prime}}+\frac{cos^{2}\psi
sin2\theta}{2r^{\prime}}\Big{]}$ (13)

Assuming $\frac{d\theta}{dr^{\prime}}\neq 0$ only at $r^{\prime}=R_{S}$, the
Eq. 13 can be further simplified as:

$F_{t}\approx-\mu_{0}\pi^{2}\beta_{j}M_{s}t_{z}R_{S}$ (14)

Using the Eq. 10 & 14 the $v_{t}$ is given by:

$v_{t}\approx\frac{\gamma\beta_{j}\pi^{2}R_{S}}{\alpha d}$ (15)

In the uniform circular motion of skyrmion on the nano-disk the frequency of
rotation can be estimated as:

$f^{S}\approx\frac{\gamma\beta_{j}\pi^{2}R_{S}}{2\pi R_{R}\alpha d}$ (16)

where $R_{R}$ is the radius of rotation of skyrmion in the nano-dot.
Thiele equation in the bilayer skyrmion oscillator for an AFC skyrmion pair
such that both skyrmions move in unison with the same velocity can be written
as:

$\displaystyle\textbf{G}_{net}\times\textbf{v}-\alpha^{BFL}\mu_{0}M_{s}t_{z}\textbf{v}.\textbf{d}^{BFL}/\gamma$
$\displaystyle-\alpha^{TFL}\mu_{0}M_{s}t_{z}\textbf{v}.\textbf{d}^{TFL}/\gamma+\textbf{F}^{BFL}_{STT}+\textbf{F}$
$\displaystyle=$ $\displaystyle M^{SP}a$ (17)

where, $\textbf{G}_{net}=\textbf{G}^{TFL}+\textbf{G}^{BFL}\approx 0$ is the
vanishing gyro-vector for the AFC skyrmion pair as the total skyrmion number
for the top($Q_{TFL}$=1) and bottom free layer($Q_{BFL}$ = -1) is ($Q_{TFL}$
\+ $Q_{BFL}$= 0) which means suppressing the SkHE and canceling the Magnus
force, $\alpha^{BFL(TFL)}$ and $\textbf{d}^{BFL(TFL)}$ are damping factor and
dissipative tensor, respectively of the BFL(TFL). The saturation magnetization
and thickness of the BFL and TFL are the same. The vortex spin current is
injected only in the BFL which is represented by the $F^{BFL}_{STT}$. $M^{SP}$
is the mass and $a$ is the acceleration of the skyrmion pair. Assuming that
the skyrmion pair is AFC both TFL and BFL skyrmions move together with the
same tangential velocity under the influence of $F^{BFL}_{STT}$, in the steady
state motion Eq. 17 can be decomposed along the $\hat{r}$ & $\hat{t}$
respectively, as:

$\displaystyle v_{t}G_{net}+F_{r}+F=M^{SP}a$ (18)
$\displaystyle\alpha^{BFL}\mu_{0}M_{s}t_{z}v_{t}d^{BFL}/\gamma$
$\displaystyle+\alpha^{TFL}\mu_{0}M_{s}t_{z}v_{t}d^{TFL}/\gamma+F_{t}^{BFL}$
$\displaystyle=$ $\displaystyle 0$ (19)

STT only acts on the BFL and $F_{t}^{BFL}$ is given by the Eq. 14. Eq. 3 and
can be simplified to give the steady state $v_{t}$ of the skyrmion pair
frequency of oscillation ($f^{SP}$) as:

$\displaystyle
v_{t}\approx\frac{\gamma\beta_{j}\pi^{2}R_{S}}{\alpha^{BFL}d^{BFL}+\alpha^{TFL}d^{TFL}}$
(20) $\displaystyle f^{SP}\approx\frac{\gamma\beta_{j}\pi^{2}R_{S}}{2\pi
R_{R}(\alpha^{BFL}d^{BFL}+\alpha^{TFL}d^{TFL})}$ (21)

We have simulated the proposed device using Object Oriented MicroMagnetic
Framework (OOMMF) public code[53] by incorporating the module for the DMI[54].
For the regular single free-layer skyrmion oscillator, the diameter is
considered to be 200 nm and the thickness is 0.5 nm. For the bilayer skyrmion
oscillator, the diameter for the BFL (TFL) is considered 220 (200) nm and the
thickness is 0.5 nm. For both devices, we discretize the free layer into the
cell size of 2$\times$2$\times$0.5 $\mathrm{nm^{3}}$. Material parameters are
adopted for the Co/Pt system [10], which includes exchange stiffness
$A=15~{}\text{pJ/m}$, Gilbert damping coefficient $\alpha=0.3$,saturation
magnetization $M_{s}=580~{}\text{kA/m}$, the interfacial DMI constant
$D=3~{}\mathrm{mJ/m^{2}}$,polarization $P=0.4$, anisotropy constant
$K_{u}=0.8~{}\mathrm{MJ/m^{3}}$, and bilinear surface exchange coefficient
$\sigma=-0.25~{}\mathrm{mJ/m^{2}}$. For simplicity, we assume $\Lambda$=1, and
the secondary spin transfer term $\epsilon^{\prime}$=0.

Figure 4: Single skyrmion(initial:15nm) trajectory in a Bilayer disk with
constant Ku profile under the different applied current density of (a)J = -25
$\mathrm{MA/cm^{2}}$(Orange), -50$\mathrm{MA/cm^{2}}$(Blue),
-100$\mathrm{MA/cm^{2}}$(Black) for 100ns (b) J = 25
$\mathrm{MA/cm^{2}}$(Orange), 50 $\mathrm{MA/cm^{2}}$(Blue),100
$\mathrm{MA/cm^{2}}$ (Black) for 100 ns

Figure 5: Bilayer-skyrmion based oscillator. The magnetization of (a.1) the
TFL and (a.2) the BFL. The color-coded anisotropy profile of (a.3) the TFL and
(a.4) the BFL. The red and blue color represent the magnetization/anisotropy
along +$\hat{z}$ and -$\hat{z}$ direction respectively. (b) The frequency of
oscillation for both simulation and theoretical obtained from (Eq. 16)along
with the bilinear surface exchange coefficient versus frequency in the bilayer
skyrmion oscillator at J$=100~{}\mathrm{MA/cm^{2}}$ on the bottom right
corner.(c) left y-axis: skyrmion diameter and right y-axis: rotation radius of
skyrmion on nano-dot with current density.(d)The Analytical (obtained from
(Eq. 15) ) and simulated velocity of skyrmion at different current densities
and their corresponding radius of rotation for the bilayer skyrmion nano-
oscillator

##  4 Results and discussion

We reported simulations on both monolayer and bilayer oscillators to
investigate the impact of Magnus force on skyrmions. The device assumes that
the initial magnetization of the FL without the skyrmion is oriented towards
$+z$-direction. Previously reported result[11] shows that a skyrmion at the
center of the nanodisk remains immobile even after injecting the spin current.
Thus for the monolayer oscillator (Fig. 1), we first nucleate a skyrmion on
the FL at a distance of 25 nm from the center of the nanodisk as shown in Fig.
3(a). Then, we apply the current density (J) with the spin polarization
according to $\textbf{m}_{\text{RL}}$ into the FL. We start with a value of
J=2 $\mathrm{MA/cm^{2}}$, which initially generates a force on the skyrmion
along radial ($F_{r}$) as well as tangential ($F_{t}$) direction, and
eventually the skyrmion moves towards the edge of the nanodisk. Due to the
tilted magnetization at the edge of the nanodisk, it exerts a repulsive force
along the radially inward direction. When the skyrmion reaches near the edge,
this repulsive force ($F_{b}$) counterbalances $F_{r}$, resulting in a stable
circular motion of the skyrmion on the nanodisk as shown in Fig. 3(a)[55] with
a frequency of $\sim$45 MHz (Fig. 3(b)). With the increase of $J$, the
frequency of this oscillation also increases as shown in Fig. 3(b). This is
justified, as the velocity of the skyrmion increases with the injected current
density[10, 33], which eventually increases the frequency, and Eq. 16 also
supports this claim. With the increase of $J$, the tangential velocity also
increases (Eq. 15), which further increases the Magnus force acting on the
skyrmion. This pushes the skyrmion toward the nanodisk edge. This is evident
from Fig. 3(c), which shows the radius of the circular path
($\mathrm{R_{rot}}$) traversed by the skyrmion increases with $J$. As the
skyrmion moves toward the edge of the nanodisk, the edge repulsion force
becomes more dominant, which acts against the radial force felt by the
skyrmion due to the injected current. This shrinks the size of the skyrmion as
we can see from Fig. 3(d) that shows the decrease of the skyrmion diameter
with an increasing $J$. We have obtained the highest frequency of 120 MHz at
$J$=7 $\mathrm{MA/cm^{2}}$. Further increment of $J$ increases the velocity,
which generates a large Magnus force that pushes the skyrmion to the edge of
the nanodisk, and eventually annihilates the skyrmion.
Similarly, for the case of a bilayer skyrmion oscillator having a constant Ku
profile in both layers the skyrmion moves radially outward in the anti-
clockwise direction (Fig.4a)(clockwise direction(Fig.4b)) for the different
applied current densities in 100ns simulation time depicting skyrmion motion
to be unaffected by the current magnitude. Now, we design the bilayer skyrmion
oscillator with different uniaxial anisotropy profiles for enhanced frequency
operations.

We show in Fig. 5(a.1 & 2)[55] magnetization of the TFL and BFL of the bilayer
skyrmion oscillator. We show in fig. 5(a.3 & 4) the anisotropy profile of the
TFL and BFL. We have used a high anisotropy disc in the TFL of diameter 120 nm
(see Fig.4 a(3)) to avoid the slow falling of the skyrmion pair.Slow falling
refers to the gradual movement of the skyrmion towards the center of the nano-
disk until equilibrium in the absence of current. This behavior is
predominantly due to the edge repulsion forces, with no driving force from the
applied current. The skyrmion gets trapped at the center[11], making it
difficult to restart oscillation even after injecting the spin current.
To prevent this, a high anisotropy disc is added to the TFL. Fig.6 illustrates
the skyrmion’s x-position and snapshots at various time instants, showing its
slow fall between 20 ns and 850 ns without applied current density. The
vortex-like spin-polarized current is injected in the BFL, which moves the AFC
skyrmion pair.

Figure 6: The plot shows the x-position of the skyrmion’s center over time,
illustrating its oscillatory motion before 20 ns with current and its slow
falling towards the nano-disk center from 20 ns to 850 ns without current.
Snapshots of the skyrmion’s position at different time instants are included.
The simulation uses a current density of J=25 $\mathrm{MA/cm^{2}}$ and runs
for 850ns.

We show in Fig. 5(b) the theoretical(Analytical) curves of the frequency of
oscillation for the different current densities in the bilayer skyrmion
oscillator derived from (Eq. 16) with a solid orange curve is in very good
agreement with the simulation curve shown with a solid blue curve.The absence
of Magnus force ($G=0$) in the AFC skyrmion pair allows the large injection of
current density that makes the bilayer skyrmion oscillator a suitable
candidate for generating high-frequency oscillations. We also observe the AFC
skyrmion-pair diameter also grows at large vortex spin current densities as
shown in Fig. 5(c). The skyrmion pair in the bilayer skyrmion oscillator can
not enter in the high anisotropy region ring, which results in the expansion
of the skyrmion center towards the edge of the nano-disk. This increases the
skyrmion rotation radius as shown in Fig. 5(c). This design works for a
bilinear surface exchange coefficient values of above -0.1090 mJ$/m^{2}$ (Fig.
5(d))

The absence of the Magnus force in the bilayer skyrmion oscillator paves a
pathway for a multi-channel frequency oscillator excited by the same current
density. The tangential velocity (Eq. 20) of the skyrmion-pair in the bilayer
skyrmion oscillator depends on the current density. As shown in Fig.5(d) both
the analytical and simulation curves are in accord. Also, we observe a very
minute change of 4-5nm in the radius of rotation of the skyrmion for the
different current densities. thus, velocity is weakly dependent on the radius
of rotation.The skyrmion pairs at different radial distances from the center
of the nano-disk under the same current density have the same tangential
velocities and different angular velocities. This can be used to make a multi-
channel oscillator driven by the same current source. We present the design of
a 3-channel oscillator shown in Fig. 7[55] out of a large landscape of designs
for such multi-channel oscillators. The physical dimensions of the bilayer
nano-disk for the multi-channel oscillator are shown in Fig. 7(a) such that
the skyrmion-pairs in two different channels do not reflect each other. The
centripetal force by the effective mass of the AFM skyrmion in the bilayer
device with channel-f3 gyrates the skyrmion in the absence of Magnus force. In
channels f1 and f2, skyrmions are bound by the repulsive forces of the high Ku
nanorings, ensuring balance and stability in the formed spaces by PMA
patterning[49] between the high Ku nanorings.The high anisotropy energy
barrier region in the TFL separates the skyrmion pairs of different channels.
It also provides robustness to the motion of the skyrmion pairs against any
perturbation in their respective channels. We show in Fig. 7(b) the frequency
of oscillation for the channel-1, channel-2, and channel-3 of the oscillator.
The Channel-1 has the highest frequency ($\approx$11GHz) due to a smaller
radius of rotation(Eq. 21).

Figure 7: Multi-channel bilayer-skyrmion based oscillator. The magnetization
of (a.1) the TFL and (a.2) the BFL. The color-coded anisotropy profile of
(a.3) the TFL and (a.4) the BFL. (b) The frequency of oscillation.

Figure 8: Multi-channel bilayer-skyrmion based oscillator with multi-skyrmion
(a) The magnetization and anisotropy profile of the TFL in channel-1.(b)The
frequency of oscillation of the different channels in channel-1.(c)The
magnetization and anisotropy profile of the TFL in channel 2.(d) The frequency
of oscillation of the different channels in channel 2. (e)The magnetization
and anisotropy profile of the TFL in channel-3 (f) The frequency of
oscillation of the different channels in channel-3.

The frequency of the multi-channel bilayer skyrmion oscillator can be
increased further by nucleating multiple skyrmions in the channel as now the
presence of the next skyrmion in the skyrmion lattice shadows the current one,
resulting in a shorter time period and higher frequency generation compared to
using a single skyrmion in the modified design. The frequency of the multi-
channel bilayer skyrmion oscillator can be increased further by nucleating
multiple skyrmion in the channel. We show in Fig. 8(a)[55] the magnetization
profile of the TFL of the multi-channel skyrmion oscillator with 4-skyrmion
pairs in channel-f1. The maximum number of skyrmion pairs that can be
generated in channel-1 is given by $N_{s}\approx\frac{2\pi R_{R}}{4R_{S}}$. We
have 4-skyrmion pairs in the channel-1 (Fig. 8(a)) which increases the
frequency of oscillation of the channel-1 by a factor of 3.4 as shown in the
Fig. 8(b). The number of skyrmion pairs in a particular channel should be
maximum such that different skyrmion pairs are interlocked due to skyrmion-
skyrmion repulsion. The constricted high Ku rings that form the nano-tracks
motivate the skyrmion interlocking secured by skyrmion-skyrmion repulsion.
This leads to single-frequency output and robustness against deflection of
skyrmion motion due to scattering.
The frequency of channel-2 can also be increased by using multi-skyrmions in
channel-2. We show in Fig. 8(c)[55] multi-skyrmions in channel-2 with a
maximum of 12 skyrmion pairs. In our design of the multi-channel skyrmion
oscillator, the radius of rotation of channel-2 skyrmion-pair is three times
that of channel-1 (see Fig. 8(b)). It allows the maximum generation of 12
skyrmion pairs in channel 2. The frequency of oscillation of channel 2 is
shown in Fig. 8(d). It can be noted that the maximum frequency ($\approx$41
GHz) of channel-2 with multi-skyrmion pairs is increased to that of
channel-1($\approx$37 GHz) with multi-skyrmion pairs (Fig. 8(d) & Fig. 8(b)).
We show in Fig. 8(e)[55] multi-skyrmions in channel-3. The radius of rotation
of channel-3 skyrmion-pair is five times that of channel-1 (see Fig. 8(b)). In
channel 3, we can generate a maximum of 20 skyrmion pairs. The multi-skyrmion
increases the frequency of channel-3 as shown in Fig. 8(f). It has to be noted
that the maximum frequency ($\approx$41 GHz) is the same as that of channel 2.
In all three profiles with multi-skyrmion, the spacing between the neighboring
skyrmions remains nearly constant, with only a small difference of
approximately 2.2 nm. This small variation contributes to the nearly identical
high frequencies observed in these profiles.

##  5 Conclusion

The paper presents a multichannel SAF skyrmion-based STT nano-oscillator
design capable of generating ultrahigh microwave frequencies in the broad
range of 0-41GHz. Such enhanced frequency range is achieved by eliminating the
Magnus force using a bilayer device as well as design modification considering
the Ru spin-flip length by incorporating a reduced spacer region that
minimizes the spin current flow in the TFL. We further demonstrate the
tunability of the frequencies based on the number of skyrmions interlocked in
the multichannel high anisotropy rings for the same current density variation.
We conclude that the multi-channel bilayer oscillator with 1,12,1 skyrmions in
their respective channels has an enhanced frequency range(0-41GHz) as compared
to various other designs such as the monolayer skyrmion
oscillator(max$\sim$120MHz), A bilayer skyrmion oscillator with a high Ku
disc(max$\sim$3.8GHz) and a multi-channel bilayer oscillator with one skyrmion
per channel (max$\sim$10GHz) are demonstrated in the paper. Finally, we
conclude that this work provides a generic method to generate multi-channel
frequency output along with ultra-high frequency in a controllable manner
which may open new pathways to utilize skyrmion-based nano-oscillators in
various applications.

## References

  * [1] John C Slonczewski  “Current-driven excitation of magnetic multilayers”  In _Journal of Magnetism and Magnetic Materials_ 159.1-2 Elsevier, 1996, pp. L1–L7
  * [2] WH Butler, X-G Zhang, TC Schulthess and JM MacLaren  “Spin-dependent tunneling conductance of Fe— MgO— Fe sandwiches”  In _Physical Review B_ 63.5 APS, 2001, pp. 054416
  * [3] M Gajek et al.  “Spin torque switching of 20 nm magnetic tunnel junctions with perpendicular anisotropy”  In _Applied Physics Letters_ 100.13 AIP Publishing, 2012
  * [4] Alina M Deac et al.  “Bias-driven high-power microwave emission from MgO-based tunnel magnetoresistance devices”  In _Nature Physics_ 4.10 Nature Publishing Group UK London, 2008, pp. 803–809
  * [5] Stuart SP Parkin, Masamitsu Hayashi and Luc Thomas  “Magnetic domain-wall racetrack memory”  In _Science_ 320.5873 American Association for the Advancement of Science, 2008, pp. 190–194
  * [6] Abhishek Sharma, Ashwin A Tulapurkar and Bhaskaran Muralidharan  “Resonant spin-transfer-torque nano-oscillators”  In _Physical Review Applied_ 8.6 APS, 2017, pp. 064014
  * [7] XS Wang, HY Yuan and XR Wang  “A theory on skyrmion size”  In _Communications Physics_ 1.1 Nature Publishing Group, 2018, pp. 1–7
  * [8] Karin Everschor-Sitte and Matthias Sitte  “Real-space Berry phases: Skyrmion soccer (invited)”  In _Journal of Applied Physics_ 115.17, 2014, pp. 172602
  * [9] Niklas Romming et al.  “Writing and Deleting Single Magnetic Skyrmions”  In _Science_ 341.6146 American Association for the Advancement of Science, 2013, pp. 636–639
  * [10] J Sampaio et al.  “Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructures”  In _Nature Nanotechnology_ 8.11, 2013, pp. 839–844
  * [11] F Garcia-Sanchez et al.  “A skyrmion-based spin-torque nano-oscillator”  In _New Journal of Physics_ 18.7 IOP Publishing, 2016, pp. 075011
  * [12] R Tomasello et al.  “A strategy for the design of skyrmion racetrack memories”  In _Scientific Reports_ 4.1, 2014, pp. 6784
  * [13] Xichao Zhang, Motohiko Ezawa and Yan Zhou  “Magnetic skyrmion logic gates: conversion, duplication and merging of skyrmions”  In _Scientific reports_ 5.1 Nature Publishing Group, 2015, pp. 1–8
  * [14] Shijiang Luo et al.  “Reconfigurable skyrmion logic gates”  In _Nano letters_ 18.2 ACS Publications, 2018, pp. 1180–1184
  * [15] Yangqi Huang et al.  “Magnetic skyrmion-based synaptic devices”  In _Nanotechnology_ 28.8 IOP Publishing, 2017, pp. 08LT02
  * [16] Tinish Bhattacharya et al.  “Low-power (1t1n) skyrmionic synapses for spiking neuromorphic systems”  In _IEEE Access_ 7 IEEE, 2019, pp. 5034–5044
  * [17] Debasis Das, Yunuo Cen, Jianze Wang and Xuanyao Fong  “Bilayer-skyrmion-based design of neuron and synapse for spiking neural network”  In _Physical Review Applied_ 19.2 APS, 2023, pp. 024063
  * [18] Xichao Zhang et al.  “Magnetic skyrmion transistor: skyrmion motion in a voltage-gated nanotrack”  In _Scientific Reports_ 5.1, 2015, pp. 11369
  * [19] S. Mühlbauer et al.  “Skyrmion Lattice in a Chiral Magnet”  In _Science_ 323.5916 American Association for the Advancement of Science, 2009, pp. 915–919
  * [20] F. Jonietz et al.  “Spin Transfer Torques in MnSi at Ultralow Current Densities”  In _Science_ 330.6011 American Association for the Advancement of Science, 2010, pp. 1648–1651
  * [21] X Z Yu et al.  “Real-space observation of a two-dimensional skyrmion crystal”  In _Nature_ 465.7300, 2010, pp. 901–904
  * [22] Stefan Heinze et al.  “Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions”  In _Nature Physics_ 7.9, 2011, pp. 713–718
  * [23] A Hrabec et al.  “Current-induced skyrmion generation and dynamics in symmetric bilayers”  In _Nature Communications_ 8.1, 2017, pp. 15765
  * [24] Albert Fert, Vincent Cros and Joao Sampaio  “Skyrmions on the track”  In _Nature nanotechnology_ 8.3 Nature Publishing Group, 2013, pp. 152–156
  * [25] Naoto Nagaosa and Yoshinori Tokura  “Topological properties and dynamics of magnetic skyrmions”  In _Nature Nanotechnology_ 8.12, 2013, pp. 899–911
  * [26] Albert Fert, Nicolas Reyren and Vincent Cros  “Magnetic skyrmions: advances in physics and potential applications”  In _Nature Reviews Materials_ 2.7, 2017, pp. 17031
  * [27] Wang Kang et al.  “Skyrmion-electronics: An overview and outlook”  In _Proceedings of the IEEE_ 104.10 IEEE, 2016, pp. 2040–2061
  * [28] Kang Wang et al.  “Single skyrmion true random number generator using local dynamics and interaction between skyrmions”  In _Nature communications_ 13.1 Nature Publishing Group, 2022, pp. 1–8
  * [29] JH Guo et al.  “A ferromagnetic skyrmion-based nano-oscillator with modified perpendicular magnetic anisotropy”  In _Physics Letters A_ 392 Elsevier, 2021, pp. 127157
  * [30] JH Guo et al.  “A ferromagnetic skyrmion-based nano-oscillator with modified profile of Dzyaloshinskii-Moriya interaction”  In _Journal of Magnetism and Magnetic Materials_ 496 Elsevier, 2020, pp. 165912
  * [31] Chendong Jin et al.  “High-frequency spin transfer nano-oscillator based on the motion of skyrmions in an annular groove”  In _New Journal of Physics_ 22.3 IOP Publishing, 2020, pp. 033001
  * [32] Senfu Zhang et al.  “Current-induced magnetic skyrmions oscillator”  In _New Journal of Physics_ 17.2 IOP Publishing, 2015, pp. 023061
  * [33] Debasis Das, Bhaskaran Muralidharan and Ashwin Tulapurkar  “Skyrmion based spin-torque nano-oscillator”  In _Journal of Magnetism and Magnetic Materials_ 491, 2019, pp. 165608
  * [34] Michael Stone  “Magnus force on skyrmions in ferromagnets and quantum Hall systems”  In _Physical Review B_ 53.24 APS, 1996, pp. 16573
  * [35] Xichao Zhang, Motohiko Ezawa and Yan Zhou  “Thermally stable magnetic skyrmions in multilayer synthetic antiferromagnetic racetracks”  In _Physical Review B_ 94.6 APS, 2016, pp. 064406
  * [36] William Legrand et al.  “Room-temperature stabilization of antiferromagnetic skyrmions in synthetic antiferromagnets”  In _Nature materials_ 19.1 Nature Publishing Group UK London, 2020, pp. 34–42
  * [37] Roméo Juge et al.  “Skyrmions in synthetic antiferromagnets and their nucleation via electrical current and ultra-fast laser illumination”  In _Nature Communications_ 13.1 Nature Publishing Group UK London, 2022, pp. 4807
  * [38] Joseph Barker and Oleg A Tretiakov  “Static and dynamical properties of antiferromagnetic skyrmions in the presence of applied current and temperature”  In _Physical review letters_ 116.14 APS, 2016, pp. 147203
  * [39] Xichao Zhang, Yan Zhou and Motohiko Ezawa  “Magnetic bilayer-skyrmions without skyrmion Hall effect”  In _Nature communications_ 7.1 Nature Publishing Group UK London, 2016, pp. 10293
  * [40] Laichuan Shen et al.  “Spin torque nano-oscillators based on antiferromagnetic skyrmions”  In _Applied Physics Letters_ 114.4 AIP Publishing, 2019
  * [41] K. Eid et al.  “Current-perpendicular-to-plane-magnetoresistance properties of Ru and Co/Ru interfaces”  In _Journal of Applied Physics_ 91.10, 2002, pp. 8102–8104
  * [42] C Phatak, AK Petford-Long and O Heinonen  “Direct observation of unconventional topological spin structure in coupled magnetic discs”  In _Physical review letters_ 108.6 APS, 2012, pp. 067205
  * [43] Sebastian Wintz et al.  “Topology and origin of effective spin meron pairs in ferromagnetic multilayer elements”  In _Physical review letters_ 110.17 APS, 2013, pp. 177201
  * [44] Xichao Zhang et al.  “Skyrmion-skyrmion and skyrmion-edge repulsions in skyrmion-based racetrack memory”  In _Scientific reports_ 5.1 Nature Publishing Group UK London, 2015, pp. 7643
  * [45] Sai Li et al.  “Experimental demonstration of skyrmionic magnetic tunnel junction at room temperature”  In _Science Bulletin_ 67.7 Elsevier, 2022, pp. 691–699
  * [46] NE Penthorn et al.  “Experimental observation of single skyrmion signatures in a magnetic tunnel junction”  In _Physical Review Letters_ 122.25 APS, 2019, pp. 257201
  * [47] See Hun Yang, Kwang Su Ryu and Stuart Parkin  “Domain-wall velocities of up to 750 m s-1 driven by exchange-coupling torque in synthetic antiferromagnets”  In _Nature Nanotechnology_ 10.3, 2015, pp. 221–226
  * [48] D. Banerjee et al.  “Electrical spin injection using GaCrN in a GaN based spin light emitting diode”  In _Applied Physics Letters_ 103.24, 2013, pp. 242408
  * [49] Roméo Juge et al.  “Helium ions put magnetic skyrmions on the track”  In _Nano Letters_ 21.7 ACS Publications, 2021, pp. 2989–2996
  * [50] Haiyan Xia et al.  “Skyrmion motion driven by the gradient of voltage-controlled magnetic anisotropy”  In _Journal of Magnetism and Magnetic Materials_ 458 Elsevier, 2018, pp. 57–61
  * [51] Christian Hanneken et al.  “Electrical detection of magnetic skyrmions by tunnelling non-collinear magnetoresistance”  In _Nature nanotechnology_ 10.12 Nature Publishing Group, 2015, pp. 1039–1042
  * [52] Xueying Zhang et al.  “Skyrmions in magnetic tunnel junctions”  In _ACS applied materials & interfaces_ 10.19 ACS Publications, 2018, pp. 16887–16892
  * [53] Michael Joseph Donahue and Donald Gene Porter  “OOMMF User’s Guide, Version 2.0a0” http://math.nist.gov/oommf/, 2019
  * [54] Stanislas Rohart and Andre Thiaville  “DMExchange6Ngbr: OOMMF Oxs Extension Modules” https://math.nist.gov/oommf/contrib/oxsext, 2012
  * [55] S. Gupta  “Ultrahigh Frequency and Multi-channel Output in Skyrmion Based Nano-oscillator video files” Accessed on 2023-11-27, https://t.ly/cw_Fn, Nov. 27,2023
  * [56] Youhua Feng et al.  “A skyrmion-based spin-torque nano-oscillator with enhanced edge”  In _Journal of Magnetism and Magnetic Materials_ 491, 2019, pp. 165610
  * [57] William Legrand et al.  “Room-temperature stabilization of antiferromagnetic skyrmions in synthetic antiferromagnets”  In _Nature Materials_ , 2019
  * [58] Gong Chen  “Skyrmion hall effect”  In _Nature Physics_ 13.2 Nature Publishing Group, 2017, pp. 112–113
  * [59] Wanjun Jiang et al.  “Direct observation of the skyrmion Hall effect”  In _Nature Physics_ 13.2 Nature Publishing Group, 2017, pp. 162–169


# Efficient Fine-Tuning of BERT Models on the Edge ††thanks: * Equal
contribution.

Danilo Vucetic*, Mohammadreza Tayaranian*, Maryam Ziaeefard,
James J. Clark, Brett H. Meyer and Warren J. Gross  Department of Electrical
and Computer Engineering,
McGill University
Montreal, Canada
{danilo.vucetic<EMAIL_ADDRESS>
{maryam.ziaeefard, james.clark1, brett.meyer<EMAIL_ADDRESS>
###### Abstract

Resource-constrained devices are increasingly the deployment targets of
machine learning applications. Static models, however, do not always suffice
for dynamic environments. On-device training of models allows for quick
adaptability to new scenarios. With the increasing size of deep neural
networks, as noted with the likes of BERT and other natural language
processing models, comes increased resource requirements, namely memory,
computation, energy, and time. Furthermore, training is far more resource
intensive than inference. Resource-constrained on-device learning is thus
doubly difficult, especially with large BERT-like models. By reducing the
memory usage of fine-tuning, pre-trained BERT models can become efficient
enough to fine-tune on resource-constrained devices. We propose Freeze And
Reconfigure (FAR), a memory-efficient training regime for BERT-like models
that reduces the memory usage of activation maps during fine-tuning by
avoiding unnecessary parameter updates. FAR reduces fine-tuning time on the
DistilBERT model and CoLA dataset by 30$\%$, and time spent on memory
operations by 47$\%$. More broadly, reductions in metric performance on the
GLUE and SQuAD datasets are around $1\%$ on average.

###### Index Terms:

Transformers, BERT, DistilBERT, NLP, Language Models, Efficient Transfer
Learning, Efficient Fine-Tuning, Memory Efficiency, Time Efficiency, Edge
Machine Learning

††publicationid: pubid: 978-1-7281-0397-6/22/$31.00 ©2022 IEEE

##  I Introduction

Large language models employing attention-based architectures have
revolutionized Natural Language Processing (NLP) with the likes of BERT and
GPT-3, achieving numerous state-of-the-art scores on well-studied tasks [1,
2]. As language models move from the cloud to resource-constrained devices
(i.e., edge devices), their deployment is made difficult by constraints in
available resources such as memory, computation, energy, and time [3, 4, 5].
Solely considering inference for edge learning is insufficient as the
operating environment may be dynamic, and input data may change over time [5].
Learning on edge devices allows models to continually adapt to their
environments and to the requirements of their users, without revealing data to
the cloud [4, 6, 5]. However, training requires about three times more memory
operations than inference111During inference only the parameters and data are
retrieved from memory (2 operations). During training, the parameters must be
retrieved at least twice and stored once after being updated, activations are
stored and retrieved, and data is retrieved (6 operations). and more resource
utilization generally. Tools like model compression are not enough to bridge
the gap (i.e., a model 1/3 the size with similar performance is hard to
achieve), and compressed models may not be training-efficient [7]. The problem
specifically lies with the greater memory requirements of training compared to
inference, since: 1) accesses to main memory are two orders of magnitude more
energy intensive than computation; and 2) large models can not fit in cache,
meaning that higher latency accesses to main memory are required [7, 8, 9].

Enabling on-device learning for large language models necessitates two
conditions: 1) using a compressed model to reduce baseline memory
requirements, and 2) training in a resource-efficient manner. To achieve the
latter condition we propose Freeze And Reconfigure (FAR), a memory-efficient
fine-tuning regime for BERT-like models. We focus first and foremost on
reducing memory usage during the backward pass by reducing the largest
contributors to said usage: activations and gradients. We introduce a
methodology that identifies which parameters to freeze and which to continue
training. Freeze here refers to disabling parameter updates during back-
propagation. We also propose a method of dynamic architecture reconfiguration
such that frozen and non-frozen parameters are grouped separately.
Reconfiguration reduces poorly-structured memory accesses which, for example,
encumber unstructured pruning methods. The contributions of this paper are
summarized as follows:

  * •

We introduce a novel learning metric that tracks the size of weight updates
over contiguous groups of parameters during fine-tuning using the $L_{1}$
norm. According to the metric, the best-learning subsets of nodes are
selected.

  * •

We introduce a novel parameter freezing scheme that exploits architectural
reconfiguration, grouping frozen and nonfrozen parameters, reducing training-
time resource consumption.

FAR achieves sizeable reductions in the resource consumption of fine-tuning.
Our results show that when fine-tuning DistilBERT on CoLA with 60$\%$ of the
model parameters frozen, fine-tuning time is reduced by 30$\%$ and memory
access time is reduced by 47$\%$, while metric performance is maintained.
Similar results are achieved for other GLUE and SQuAD tasks. Evidently, FAR
makes training more efficient while maintaining metric performance.

##  II Background and Related Work

###  II-A Transformers and BERT Models: Training and Efficiency

Transformers and BERT-like models make use of Attention and Feed-Forward
Network (FFN) sublayers to extract linguistic knowledge and achieve state-of-
the-art scores[10]. BERT in particular uses the Transformer Encoder
architecture to model languages by pre-training on large datasets of
relatively simple language tasks. To deploy BERT-like models to a new
downstream task, the model must be fine-tuned [1]. This allows the use of a
single set of weights, the pre-trained BERT weights, as the starting point for
various, faster converging, fine-tuning tasks, avoiding the large overhead of
pre-training the full BERT model222As reported in [1], 4 cloud tensor
processing units running for 4 days are required to pre-train BERT’s 110
million parameters.. Various methods have been proposed to compress BERT
including DistilBERT [11], MobileBERT [12], and TinyBERT [13]. A detailed
discussion of compressing large language models is found in [14], including
analysis of the major bottleneck of Transformer-based models: the FFN
sublayers, which contribute more parameters, runtime memory consumption, and
inference latency than all other sublayers. Model compression is useful for
enabling on-device learning, but greater efficiency during training is
desirable to further enable the realization of large models on edge devices.

###  II-B Efficient Learning

On-device learning has been detailed in [5], including methods of data
compression and theoretical approaches to efficiency. When considering on-
device learning, a fundamental trade-off must be kept in mind to train in a
reasonable amount of time: either the training-time resource requirements of a
model must be decreased, or the hardware resources must be increased. The
former approach is usually studied in terms of model compression such as in
[11, 12, 13]. Others take the problem as reducing the real requirements of
training without altering model architecture. One such approach, [15], trains
only the best-learning weights of a feed-forward neural network, while keeping
the others at their initialization values. Trained weights are selected by
tracking the largest accumulated weight updates over some initial training
steps. The authors also further reduce memory requirements by regenerating
frozen weights from a pseudorandom number generator, thereby trading memory
for extra computation. This approach does not work for BERT-like models as the
weights of the pre-trained model are not randomly generated. Additionally, the
unstructured selection of weights leads to inefficient memory accesses, an
issue which we address with structured selection in FAR. A similar approach,
BitFit [16], freezes all BERT weights during fine-tuning and only applies
updates to bias terms. The authors report near-baseline performance despite
the large drop in trained parameters. While BitFit works on uncompressed
(i.e., highly overparameterized) BERT models, on the more compressed variants
like DistilBERT, it is expected to fail due to the smaller network size and
lower capacity. We address this issue by fine-tuning a greater proportion of
the parameters.

##  III Freeze And Reconfigure

On-device learning of large language models requires compressed models and
efficient training procedures. The reduction in memory usage provided by these
requirements has knock-on effects: higher energy efficiency, reduced
computation, and reduced memory access times. In the case of memory access
time, reducing the number of parameters decreases the amount of information
transferred across a system.

FAR is specifically designed for use with compressed language models,
specifically DistilBERT, whose capacity and adaptability are diminished from
their uncompressed counterparts [17]. This is accomplished by fine-tuning a
subset of linear layer weights (called FFNs in DistilBERT) as well as their
biases, in contradistinction to BitFit, which trains only the biases of BERT.

In FAR, the sets of fine-tuned weights in the linear layers are selected in a
structured manner to avoid sparse memory accesses which would delay memory
operations [18]. Each node in an FFN layer is considered a single group of
parameters. These nodes are classified as learner and nonlearner nodes
depending on how well a node performs during fine-tuning. The parameters of
the nonlearner nodes are frozen while learner nodes are fine-tuned. Frozen
nodes no longer require gradient calculations during the backward pass,
storage of activations, or memory accesses during the backward pass.
Reconfiguration of the FFN layers is completed during fine-tuning by
separating learner and nonlearner nodes into distinct, newly created, FFN
sublayers.

Each Transformer block in DistilBERT is made up of a multi-head attention
layer which feeds an FFN. The FFN consists of two dense layers which contain
3072 and 768 nodes respectively. The weights of the FFNs make up more than
$66\%$ of the parameters of the whole network. Thus, freezing a subset of
these weights has a considerable effect on the overall number of updated
parameters of the model. Past work has shown that freezing up to 80$\%$ of
feed-forward neural network parameters results in minimal changes in accuracy
for smaller models and simpler tasks [15].

###  III-A Selection of Learner Nodes

The set of learner nodes is decided upon after an initial set of fine-tuning
iterations during a process we call priming (cf. Figure 1, green nodes in left
figure are learner nodes). During priming, a copy of the pre-trained FFN
weights is stored and the network is fine-tuned for some percentage, $p$, of
the total number of optimization steps. The priming percentage, $p$, is kept
small so as to avoid the additional burden of fully fine-tuning a large model
for a greater proportion of the optimization steps. It is expected, however,
that more priming should result in higher metric performance. After fine-
tuning the model during priming, the initial FFN weight vectors,
$\boldsymbol{w}^{e,i}_{n}$, are subtracted from the fine-tuned FFN weight
vectors, $\boldsymbol{\phi}^{e,i}_{n}$, for each Transformer Encoder, $e$, FFN
sublayer, $i$, and node, $n$. We employ the $\text{L}_{1}$ norm to compute the
total amount learned by each node in the FFN. This is in distinction to [15],
where each weight is chosen individually based on the difference in its
initial and trained value. Usage of the $\text{L}_{1}$ norm allows for the
weights of the FFN node to be grouped so that architectural modifications can
be made later for greater memory savings. The learning metric is computed as:

$m^{e,i}_{n}=\left\|\boldsymbol{\phi}^{e,i}_{n}-\boldsymbol{w}^{e,i}_{n}\right\|_{1}$
(1)

Finally, to compute the set of learner nodes, a retention percentage, $r$, is
defined as the ratio of learner nodes to the total number of nodes. After
sorting the learning metric for each FFN, the $r\%$ of nodes with the largest
metric are classified as learners; the rest as nonlearners. Note that
decreasing the node retention results in more nonlearner nodes, which in turn
reduces memory utilization during fine-tuning. Using higher retention values
is expected to have a positive impact on metric performance as it increases
the model’s ability to adapt to the down-stream task. In addition, the priming
and retention hyperparameters are defined as percentages to allow them to
scale to larger datasets and varying models. For example, a more compressed
model may require higher node retention to avoid substantial losses in metric
performance.

###  III-B Dynamic FFN Reconfiguration

Figure 1: Overview of priming steps and the dynamic reconfiguration of nodes
in FFN. In the left, the learner (green) and nonlearner (red) nodes are chosen
after the priming steps. In the right, the nodes are reconfigured to disable
gradient computation for red weights. The output of the FFN is permuted after
reconfiguration.

After assigning the learner and nonlearner nodes, the FFNs of each Transformer
block are reconfigured to separate the nonlearner nodes from learner nodes. As
shown in Figure 1, FFN nodes are reconfigured into two parallel sub-modules.
Such reconfiguration ensures that the memory of learner nodes is grouped
together, allowing only the fine-tuned parameters to be retrieved during fine-
tuning. Finally, the reconfigured FFNs need to have their output order
restored such that their outputs are coherent with the parameters of the
model, so a permutation is applied to the output vectors. The combination of
reconfiguration and permutation trades memory accesses for additional time
spent on computation and data movement, which is far less costly [8].

Using nodes as parameter groupings and reconfiguring the model this way allows
us to avoid memory access issues that arise in methods like unstructured
pruning. Since individual weights are never solely accessed, we avoid sparse
memory accesses. Reconfiguration is also beneficial for practical
implementation since in PyTorch, automatic gradient computation can not be
disabled for singular parameters, but instead only for whole layers or
sublayers. Thus, unlike unstructured pruning, no special hardware or libraries
are needed to implement FAR.

##  IV Experiments

TABLE I: Effectiveness of priming and retention with 5 averaged runs on DistilBERT for each dataset | MNLI | CoLA
---|---|---
$p$ $r$ | 10$\%$ | 25$\%$ | 40$\%$ | 10$\%$ | 25$\%$ | 40$\%$
1$\%$ | 81.8 | 81.9 | 82.1 | 52.8 | 53.1 | 53.3
5$\%$ | 81.8 | 82.1 | 81.9 | 51.9 | 52.7 | 53.7
10$\%$ | 81.9 | 82.0 | 82.2 | 53.4 | 52.5 | 54.0
TABLE II: Metric performance of DistilBERT with and without FAR on GLUE and SQuAD 2.0 - $1\%$ priming and $10\%$ and $40\%$ retention. Each FAR run is accompanied with its percentage drop compared to the baseline. The last column shows the average score drop percentage across all tasks. $\dagger$ Shows our implementation. | CoLA | MNLI | QNLI | QQP | SST2 | SQuAD 2.0 |
---|---|---|---|---|---|---|---
| Matthews Corr. | Acc. | Acc. | Acc. | Acc. | EM | F1 | Avg
Baseline $\dagger$ | 53.6 | 82.1 | 89.1 | 90.2 | 91.1 | 65.0 | 67.9 |
| 52.8 | 81.9 | 88.2 | 90.0 | 91.1 | 63.3 | 66.4 |
FAR10 | -1.58% | -0.24% | -1.01% | -0.22% | 0.00% | -2.59% | -2.27% | -1.13%
| 53.3 | 82.1 | 88.5 | 90.2 | 91.3 | 63.6 | 66.6 |
FAR40 | -0.55% | 0.00% | -0.67% | 0.00% | 0.22% | -2.14% | -2.01% | -0.74%

We test FAR using the compressed BERT-based model DistilBERT [11]. DistilBERT
is a pre-trained language model and can be fine-tuned to downstream tasks.
DistilBERT has six Transformer encoder layers, each with two FFN sublayers
(i.e., $e\in\left\\{1,\cdots,6\right\\}$ and $i\in\left\\{1,2\right\\}$ in
Equation 1). We fine-tune the model using a learning rate of $2e^{-5}$, with a
linear learning rate schedule, batch size of 16, using at most 5 epochs.

Experiments are run on NVIDIA Tesla V100 Volta GPU accelerators to gather
metric results for all datasets. Wherever timing is reported, these results
have been gathered on the NVIDIA Jetson Xavier NX edge device. Memory access
time is computed using the NVIDIA’s profiling tool, nvprof. Memory access time
is the summation of the execution time for memory-related API calls, namely
cudaStreamSynchronize, cudaMemcpyAsync and cudaMemsetAsync.

FAR is tested against conventional and efficient fine-tuning approaches using
standard NLP datasets from the GLUE benchmark and SQuAD 2.0 [19, 20]. All the
GLUE tasks are included in the experiments except for RTE, MRPC, STS-B because
of their small dataset sizes and WNLI because of its confounding results [21].
This leaves MNLI, QQP, QNLI, SST-2, and CoLA. CoLA (9594 training examples)
and MNLI (433k training examples) are used to find effective priming and
retention percentages in grid searches (Cf. Table I). MNLI and CoLA are
believed to represent small and large datasets, respectively [19]. Each
reported value is the average of 5 fine-tuning attempts with random seeds to
ensure consistency.

In order to test the hypothesis that higher priming and higher retention
percentages produce better metric performance, we complete a grid-search style
test on CoLA and MNLI. The results are listed in Table I. The table shows that
with higher retention percentage and higher priming percentage, fine-tuning
produces a higher metric score, with the highest score occurring with $10\%$
priming and $40\%$ retention. Moreover, the difference between $1\%$ and
$10\%$ priming is small enough to warrant the use of $1\%$ priming for FAR.
This also means that the training overhead is made minimal since extra priming
does not necessarily correlate to better performance. Henceforth,
$\text{FAR}_{x}$ will represent FAR using $1\%$ priming and $x\%$ node
retention. $\text{FAR}_{10}$ and $\text{FAR}_{40}$ are used to demonstrate two
trade-off points between metric performance and resource utilization. The
former has lower resource utilization while also producing lower metric
performance and vice-versa for the latter. The results of this comparison are
listed in Table II alongside with the baseline implementation of DistilBERT
using the Hugging Face python library[22], without freezing any weights during
fine-tuning. We observe higher metric scores for $\text{FAR}_{40}$ compared to
$\text{FAR}_{10}$ and a lower drop in performance compared to the baseline at
just $0.74\%$ compared to $1.13\%$ respectively. The largest drops in
performance occur on SQuAD 2.0, which is the most complex dataset tested.

Figure 2: Overall time and total memory access time for fine-tuning DistilBERT
on CoLA using FAR with various retention percentages on the NVIDIA Jetson
Xavier NX. Priming percentage is set to 1% across all runs. Values for FAR10,
FAR40 and the baseline (FAR100) are indicated with green, red and blue dots,
respectively.

Figure 2 shows the effect of changing the retention percentage on training
time and memory access time. A similar trend is observed in both figures,
showing that using setups with lower retention values reduces both fine-tuning
and memory access time. These results indicate that FAR reduces the resource
consumption of fine-tuning, and as a result also reduces training time.
$\text{FAR}_{10}$ and $\text{FAR}_{40}$ are also indicated with green and red
dots, respectively. In terms of overall fine-tuning time, $\text{FAR}_{10}$
and $\text{FAR}_{40}$ each respectively show $30\%$ and $17\%$ reductions
compared to the baseline. For memory access time, $47\%$ and $36\%$ are the
time reductions associated to $\text{FAR}_{10}$ and $\text{FAR}_{40}$.

TABLE III: Comparison of FAR10 with random node selection and BitFit | CoLA | MNLI | SQuAD 2.0
---|---|---|---
Mathews Corr. | Acc. | EM | F1
FAR10 | 52.8 | 81.9 | 63.3 | 66.4
| Random
---
Selection10
52.0 | 81.8 | 62.8 | 65.7
BitFit | 51.7 | 81.5 | 49.8 | 49.8

Finally, we carry out an ablation study to measure the effectiveness of our
method against related works and to show that FAR is in fact necessary for
efficient fine-tuning. First, to show the effect of using priming, we perform
an experiment in which the learner and nonlearner nodes are randomly selected.
Table III shows that for the same retention value of $10\%$, FAR, which uses
priming to select nodes, performs better than random node selection, with the
gap being wider in the more complex SQuAD task. Comparing FAR with BitFit,
which freezes all the weights of all dense layers across the network, we see a
drop in performance of less than 2$\%$ for CoLA and MNLI. However, BitFit
performs poorly on SQuAD 2.0, proving its failure to cope with complex tasks
on compressed models like DistilBERT. The memory access and training times of
fine-tuning on DistilBERT using BitFit on CoLA are 237 and 384 seconds.
Clearly, BitFit spends more time accessing memory, but around the same time
fine-tuning as FAR10 (Figure 2).

##  V Conclusions and Future Work

In this paper we propose FAR, a novel method to reduce fine-tuning memory
consumption by exploiting the overparameterization of large language models.
We show that for a compressed BERT-based model, it is not enough to simply
freeze large sections of the model during fine-tuning, but that freezing must
be selective and structured. Despite large decreases in the number of fine-
tuned parameters, metric performance remains near baseline levels, while fine-
tuning time and memory usage are significantly reduced. Future work involves
investigations into new domains such as Transformers on NLP and vision tasks.
We will also apply FAR to other backbone models such as MobileBERT to
demonstrate the generalizability of the method.

## Acknowledgment

We thank Huawei Canada for their sponsorship of this work and Compute Canada
for providing computational resources.

## References

  * [1] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “BERT: Pre-training of deep bidirectional transformers for language understanding,” in _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_. Minneapolis, Minnesota: Association for Computational Linguistics, Jun. 2019, pp. 4171–4186. [Online]. Available: https://aclanthology.org/N19-1423
  * [2] T. Brown, B. Mann, N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, S. Agarwal, A. Herbert-Voss, G. Krueger, T. Henighan, R. Child, A. Ramesh, D. Ziegler, J. Wu, C. Winter, C. Hesse, M. Chen, E. Sigler, M. Litwin, S. Gray, B. Chess, J. Clark, C. Berner, S. McCandlish, A. Radford, I. Sutskever, and D. Amodei, “Language models are few-shot learners,” in _Advances in Neural Information Processing Systems_ , H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, Eds., vol. 33. Curran Associates, Inc., 2020, pp. 1877–1901. [Online]. Available: https://proceedings.neurips.cc
  * [3] E. Strubell, A. Ganesh, and A. McCallum, “Energy and policy considerations for deep learning in nlp,” in _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , 2019, pp. 3645–3650. [Online]. Available: https://aclanthology.org/P19-1355.pdf
  * [4] R. Sharma, S. Biookaghazadeh, B. Li, and M. Zhao, “Are existing knowledge transfer techniques effective for deep learning with edge devices?” in _2018 IEEE International Conference on Edge Computing (EDGE)_ , 2018, pp. 42–49.
  * [5] S. Dhar, J. Guo, J. J. Liu, S. Tripathi, U. Kurup, and M. Shah, “A survey of on-device machine learning: An algorithms and learning theory perspective,” _ACM Trans. Internet Things_ , vol. 2, no. 3, Jul. 2021. [Online]. Available: https://doi.org/10.1145/3450494
  * [6] M. Satyanarayanan, “The emergence of edge computing,” _Computer_ , vol. 50, no. 1, pp. 30–39, 2017.
  * [7] H. Cai, C. Gan, L. Zhu, and S. Han, “Tinytl: Reduce memory, not parameters for efficient on-device learning,” in _Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual_ , 2020. [Online]. Available: https://proceedings.neurips.cc
  * [8] S. Han, X. Liu, H. Mao, J. Pu, A. Pedram, M. A. Horowitz, and W. J. Dally, “Eie: Efficient inference engine on compressed deep neural network,” in _Proceedings of the 43rd International Symposium on Computer Architecture_ , ser. ISCA ’16. IEEE Press, 2016, p. 243–254. [Online]. Available: https://doi.org/10.1109/ISCA.2016.30
  * [9] J. L. Hennessy and D. A. Patterson, _Computer architecture: a quantitative approach_ , 5th ed. Elsevier, 2012, pp. B–3.
  * [10] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” in _Advances in neural information processing systems_ , 2017, pp. 5998–6008.
  * [11] V. Sanh, L. Debut, J. Chaumond, and T. Wolf, “Distilbert, a distilled version of BERT: smaller, faster, cheaper and lighter,” 2019. [Online]. Available: http://arxiv.org/abs/1910.01108 [unpublished]
  * [12] Z. Sun, H. Yu, X. Song, R. Liu, Y. Yang, and D. Zhou, “MobileBERT: a compact task-agnostic BERT for resource-limited devices,” in _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_. Online: Association for Computational Linguistics, Jul. 2020, pp. 2158–2170. [Online]. Available: https://www.aclweb.org/anthology/2020.acl-main.195
  * [13] X. Jiao, Y. Yin, L. Shang, X. Jiang, X. Chen, L. Li, F. Wang, and Q. Liu, “TinyBERT: Distilling BERT for natural language understanding,” in _Findings of the Association for Computational Linguistics: EMNLP 2020_. Online: Association for Computational Linguistics, Nov. 2020, pp. 4163–4174. [Online]. Available: https://aclanthology.org/2020.findings-emnlp.372
  * [14] P. Ganesh, Y. Chen, X. Lou, M. A. Khan, Y. Yang, D. Chen, M. Winslett, H. Sajjad, and P. Nakov, “Compressing large-scale transformer-based models: A case study on bert,” _Transactions of the Association for Computational Linguistics_ , vol. 9, pp. 1061–1080, 2021.
  * [15] M. Golub, G. Lemieux, and M. Lis, “Full deep neural network training on A pruned weight budget,” in _Proceedings of Machine Learning and Systems 2019, MLSys 2019, Stanford, CA, USA, March 31 - April 2, 2019_ , A. Talwalkar, V. Smith, and M. Zaharia, Eds. mlsys.org, 2019. [Online]. Available: https://proceedings.mlsys.org/book/260.pdf
  * [16] E. B. Zaken, S. Ravfogel, and Y. Goldberg, “Bitfit: Simple parameter-efficient fine-tuning for transformer-based masked language-models,” _CoRR_ , vol. abs/2106.10199, 2021. [Online]. Available: https://arxiv.org/abs/2106.10199 [unpublished]
  * [17] S. I. Mirzadeh, M. Farajtabar, A. Li, N. Levine, A. Matsukawa, and H. Ghasemzadeh, “Improved knowledge distillation via teacher assistant,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 34, no. 04, 2020, pp. 5191–5198.
  * [18] W. Wen, C. Wu, Y. Wang, Y. Chen, and H. Li, “Learning structured sparsity in deep neural networks,” _Advances in neural information processing systems_ , vol. 29, pp. 2074–2082, 2016.
  * [19] A. Wang, A. Singh, J. Michael, F. Hill, O. Levy, and S. Bowman, “Glue: A multi-task benchmark and analysis platform for natural language understanding,” in _Proceedings of the 2018 EMNLP Workshop BlackboxNLP: Analyzing and Interpreting Neural Networks for NLP_ , 2018, pp. 353–355.
  * [20] P. Rajpurkar, R. Jia, and P. Liang, “Know what you don’t know: Unanswerable questions for squad,” in _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , 2018, pp. 784–789.
  * [21] M. A. Gordon, K. Duh, and N. Andrews, “Compressing bert: Studying the effects of weight pruning on transfer learning,” _ACL 2020_ , p. 143, 2020.
  * [22] T. Wolf, L. Debut, V. Sanh, J. Chaumond, C. Delangue, A. Moi, P. Cistac, T. Rault, R. Louf, M. Funtowicz, J. Davison, S. Shleifer, P. von Platen, C. Ma, Y. Jernite, J. Plu, C. Xu, T. Le Scao, S. Gugger, M. Drame, Q. Lhoest, and A. Rush, “Transformers: State-of-the-art natural language processing,” in _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_. Online: Association for Computational Linguistics, Oct. 2020, pp. 38–45. [Online]. Available: https://aclanthology.org/2020.emnlp-demos.6