nonlinear dynamics of spin transfer nano-oscillators

PRAMANA c© Indian Academy of Sciences Vol. 84, No. 3— journal of March 2015

physics pp. 473–485

Nonlinear dynamics of spin transfer nano-oscillators

B SUBASH1, V K CHANDRASEKAR2 and M LAKSHMANAN1,∗1Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University,Tiruchirapalli 620 024, India2Centre for Nonlinear Science & Engineering, School of Electrical and Electronics Engineering,SASTRA University, Thanjavur 613 401, India∗Corresponding author. E-mail: [email protected]

DOI: 10.1007/s12043-014-0922-3; ePublication: 18 February 2015

Abstract. The evolution equation of a ferromagnetic spin system described by Heisenbergnearest-neighbour interaction is given by Landau–Lifshitz–Gilbert (LLG) equation, which is afascinating nonlinear dynamical system. For a nanomagnetic trilayer structure (spin valve or pil-lar) an additional torque term due to spin-polarized current has been suggested by Slonczewski,which gives rise to a rich variety of dynamics in the free layer. Under appropriate conditions thespin-polarized current gives a time-varying resistance to the magnetic structure thereby inducingmagnetization oscillations of frequency which lies in the microwave region. Such a device iscalled a spin transfer nanooscillator (STNO). However, this interesting nanoscale level source ofmicrowaves lacks efficiency due to its low emitting power typically of the order of nWs. To over-come this difficulty, one has to consider the collective dynamics of synchronized arrays/networksof STNOs as suggested by Fert and coworkers so that the power can be enhanced N2 times that of asingle STNO. We show that this goal can be achieved by applying a common microwave magneticfield to an array of STNOs. In order to make the system technically more feasible to practical levelintegration with CMOS circuits, we establish suitable electrical connections between the oscilla-tors. Although the electrical connection makes the system more complex, the applied microwavemagnetic field drives the system to synchronization in large regions of parameter space.

Keywords. Magnetic properties of nanostructures; macrospin simulation; synchronization.

PACS Nos 75.75.−c; 85.75.−d; 05.45.Xt

1. Introduction

Ferromagnetism of a bulk material is defined as the phenomenon of long-range order-ing in the magnetic moments of atoms [1]. Quantum mechanically the long-rangeordering is explained by the nearest-neighbour exchange interaction which arises in atwo-body problem. There are several other interactions which can influence the magneticmoment of atoms, such as applied external magnetic field, magnetocrystalline anisotropy,demagnetization field [2], etc. Using the effective field due to the above interactions,

Pramana – J. Phys., Vol. 84, No. 3, March 2015 473

B Subash, V K Chandrasekar and M Lakshmanan

Landau and Lifshitz in 1935 [3] derived the dynamical equation for the magnetizationwhich is now called the Landau–Lifshitz (LL) equation (to which a damping term canalways be added). Another form of LL equation, namely the Landau–Lifshitz–Gilbert(LLG) equation, derived by Gilbert in 1954 [4] using a Lagrangian approach also hasan explicit form for damping. Note that both the equations are equivalent and derivablefrom each other, respectively. It is important to mention that the field of applied mag-netism is dominated by LLG equation in explaining many of the experimentally observedphenomena [1,2,5–7].

Due to the advent of nanostructures in magnetic materials several nonlinear phenom-ena have been identified, including ultrafast switching, microwave oscillations, dropletsolitons [2–12], etc. Apart from these, new physical phenomenon such as spin transfertorque has also been introduced due to the spin/magnetic moment-dependent transportin ferromagnetic metals. Spin transfer torque is the force exerted by spin-polarized elec-trons to the magnetization of a thin film layer when injected perpendicular to it [8,9,11].Under certain circumstances this torque can balance the inherent damping in the system,thereby leading to the exciting possibility of producing spin waves whose frequency liesin the microwave range. This nanostructure device, capable of producing magnetizationoscillations, is termed as spin transfer torque nano-oscillator or simply spin transfer nano-oscillator (STNO). However, the above nanoscale level microwave source lacks efficiencyon two counts: (1) low output power (∼ nW), (2) high signal-to-noise ratio. Both theissues can be handled by phase locking a large array of oscillators [13–21]. Phase lockingand synchronizing several STNOs is a typical nonlinear problem to enhance the powerand to reduce the signal-to-noise ratio of the oscillators.

In this paper, we summarize the recent efforts on the synchronization of an array ofSTNOs by applying a common oscillating external magnetic field. The STNOs consid-ered in this study are electrically uncoupled similar to the case discussed in [14] withadditional oscillating external magnetic field. Further, to make the mechanism more fea-sible for technical implementation in electronic circuits we have established electricalconnections between them and discussed their effects on the synchronization dynamics.

This paper is organized as follows. In §2 we introduce the dynamical equation for abulk ferromagnetic material, namely LLG equation by considering a lattice of spins. Wehave also explained the modification of LLG equation due to spin transfer torque whicharises as a result of spin-polarized electric current for a trilayer nanostructure in §2. In §3,we have briefly discussed the nonlinear dynamics of the system considered by carryingout a macromagnetic simulation of the underlying dynamical equation. In §4 we indicatethe already proposed mechanisms for synchronizing STNOs and also present the resultsin synchronizing a large array of it by using a common oscillating external magnetic field.In §5, we have given the latest improvements in synchronization dynamics of STNOs byestablishing electrical connections for technical feasibility. In the last section we havegiven the concluding remarks.

2. LLG equation

To begin with, in this section, for a bulk ferromagnetic material, we deduce the phe-nomenological LLG equation, by considering the case of a lattice of spins having nearest-neighbour interaction [3,7].

474 Pramana – J. Phys., Vol. 84, No. 3, March 2015


2.1 Dynamics of lattice of spins and continuum case

For a lattice of spins �Mi , i = 1, 2, . . . N , representing a ferromagnetic material, forsimplicity, considering an one-dimensional lattice of N spins with nearest-neighbourinteractions, applied magnetic field, onsite anisotropy, demagnetizing field, etc., thedynamics of the ith spin can be defined [7] as

d �Mi

dt= �Mi × �Heff, i = 1, 2, ..., N, (1)

where

�Heff = ( �Mi+1 + �Mi−1 + AMxi �nx + BM

y

i �ny + CMzi �nz + �H(t) + · · · )

−λ{ �Mi+1+ �Mi−1+AMxi �nx+BM

y

i �ny +CMzi �nz+ �H(t)+· · · }× �Mi, (2)

�Mi = (Mxi ,M

y

i ,Mzi ),

�M2i = 1. (3)

Here A,B,C are anisotropy parameters and �nx, �ny, �nz are unit vectors along the x, y

and z directions, respectively. In the long-wavelength and low-temperature limit (in onespatial dimension), i.e., in the continuum limit, one can write

�M�i (t) = �M(x, t),

�M�i±�1 = �M(x, t) ± a∂ �M∂x

+ a2

2

∂2 �M∂x2

+ higher orders, (4)

(a: lattice constant) so that the LLG equation takes the form of a vector nonlinear partialdifferential equation (as a → 0),

∂ �M(x, t)

∂t= �M×

[{∂2 �M∂x2

+AMx �nx +BMy �ny +CMz�nz + �H(t) + · · ·}]

+λ

[{∂2 �M∂x2

+AMx �nx+BMy �ny+CMz�nz+ �H(t)+· · ·}× �M(�x, t)

],

= �M × �Heff, (5)

�M(�x, t) = (Mx(x, t),My(x, t),Mz(x, t)), �M2 = 1. (6)

In fact, the three spatial dimensional version of eq. (5) was deduced from phenome-nological grounds for bulk magnetic materials by Landau and Lifshitz originally in1935 [3].

2.2 Hamiltonian structure of the LLG equation in the absence of damping

The dynamical equations for the lattice of spins (1) in one dimension (as well as in higherdimensions) results in a Hamiltonian structure in the absence of damping.

Defining the nearest-neighbour spin Hamiltonian

Hs = −∑{i,j}

�Mi · �Mi+1+A(Mxi )2+B(M

y

i )2+C(Mzi )

2+μ( �H(t)· �Mi)+· · · (7)



and the spin Poisson bracket between any two functions of spin A and B as

{A,B} =3∑

α,β,γ=1

N∑i=1

εαβγ

∂A∂Mα

i

∂B∂M

β

i

Mγ

i , (8)

one can obtain the evolution eq. (1) for λ = 0 from (d �Mi/dt) = { �Mi,Hs}. Here εαβγ

is the Levi–Civita tensor. Similarly, for the continuum case, one can define the spinHamiltonian density

Hs = 1

2[(∇ �M)2+A(Mx)2+B(My)2+C(Mz)2+( �H · �M)+Hdemag+· · · .] (9)

along with the Poisson bracket relation{Mα(x, t),Mβ(x ′, t ′)

} = εαβγ Mγ δ(x − x ′, t − t ′), (10)

and deduce the spin field evolution eq. (5) for λ = 0.

2.3 Spin transfer torque – LLGS equation

It is well known that the storage memory of all electronic devices are based on the mag-netoresistive phenomenon of ferromagnetic materials. Numerous experiments have beenconducted to improve the magnetoresistance of ferromagnetic materials. In 1988, thegroups of Fert and Grünberg independently found enhanced magnetoresistance calledgiant magnetoresistance (GMR) in ferromagnetic thin film trilayer structures [6,14]. Theyfound that the magnetoresistance of the trilayer structure consisting of two ferromagneticthin films separated by the conducting non-magnetic thin film has an improved value ofthe order of 50% compared to the existing magnetoresistance of single-layer ferromag-netic thin film. Later, in 1996 Slonczewski [8] and Berger [9] independently proposeda quantum mechanical treatment to an extended GMR trilayer structure consisting of athick fixed layer and a thin free layer separated by a conducting nonmagnetic spacer.They predicted that when an electric current is injected into the thick layer, it gets spinpolarized in the direction of the magnetization (mp) of the fixed layer. This spin-polarizedcurrent exerts a torque on the free layer, which is termed as spin transfer torque (STT), andthis exerts a rich variety of dynamics in the free layer. The expression for this torque hasbeen given by Slonczewski and the redefined LLG equation with the STT is read as [7,15]

∂ �m∂t

= −γ �m × �Heff + λ �m × ∂ �m∂t

− γ a �m × ( �m × mp), �m = �M(x, t), (11)

where

�Heff = �Hexchange + �Hanisotropy + �Hdemag + �Happlied. (12)

Equation (11) is called the LLGS equation. The effective field ( �Heff) consists of field�Hexchange due to exchange interaction of the magnetic moments, �Hanisotropy due to anistropy

of the material, �Hdemag due to crystal dimension of the material and �Happlied due to theexternal applied magnetic field. Here γ is the gyromagnetic ratio, a is the spin currentdensity defined as a = hη/2m0V e, where η = 0.35 is the spin polarization ratio, m0 isthe saturation magnetization and V is the volume of the free layer.



We assume homogeneous magnetization in the ferromagnetic free layer. So�Hexchange = 0. We also assume an easy plane anisotropy, �Hanisotropy = (κmx, 0, 0),

where κ is the strength of the anisotropy. For a nanopillar geometry, we assume�Hdemag = −4πm0(0, 0,mz) and �Happlied = (H(t), 0, 0) so that the external magnetic field

is applied along the x-direction. We choose the parameters corresponding to permalloythin film [16]. In this case the resultant dynamical equation for the spin �m(t) is designatedas the spin transfer nano-oscillator.

2.4 LLGS equation in stereographic representation

Under a stereographic projection, eq. (11) can be equivalently rewritten using a complexscalar function [17]

ω(t) = mx + imy

1 + mz

,

mx = ω + ω

1 + |ω|2 ; my = −i(ω − ω)

1 + |ω|2 ; mz = 1 − |ω|21 + |ω|2 . (13)

For example, with Hexchange = 0, eq. (11) can be rewritten as

(1 − iλ)dω

dt= γ

(a − iH

2

)(1 − ω2) − 1

2iγ κ

ω + ω∗

1 + ωω∗ (1 − ω2)

−iγ 4πm01 − ωω∗

1 + ωω∗ ω. (14)

The stereographically projected LLGS equation has several advantages over the conven-tional Cartesian/polar coordinate representation:

(1) It has the natural structure to preserve the normalization of the magnetization vector(| �m|2 = 1).

(2) There are no trigonometric quantities involved in the equation, and hence it improvesthe numerical simulation time and also helps to eliminate errors in calculating thesequantities.

In the analysis, we shall make use of both the representations for confirmation of the results.

3. Spin transfer torque-induced switching, bifurcations and chaos

In a typical ferromagnetic material, the asymptotic orientation of the magnetic momentis due to the phenomenological damping theoretically introduced by Landau and Lifshitzand Gilbert. In trilayer extended GMR structure, the STT due to spin-polarized currentacts opposite to the damping created by the effective field. This action of STT leads toa rich variety of dynamics in the free layer magnetization, including switching, bifurca-tions and chaos without altering the applied magnetic field. In the following analysis,as illustration, we choose the parameters typically as that of a permalloy thin film. Sowe have fixed the Gilbert damping parameter as λ = 0.02, saturation magnetization as4πm0 = 8.4 kOe and the gyromagnetic ratio as γ = 0.01767 Oe−1 ns−1, respectively.



We assume the spin polarization vector �mp = (1, 0, 0) and that the applied externalmagnetic field consists of both DC and AC components. Therefore, in this case, wehave

�Heff = Hi + κmxi − 4πm0mzk, H(t) = hDC + hAC cos( t). (15)

3.1 Switching

Switching of magnetization of a ferromagnetic thin film used typically in magnetic mem-ory devices is generally achieved by varying the applied external magnetic field [15,16].Alternatively, switching can be achieved and the switching time can be enhanced by usingthe STT. By using this mechanism it has been proved that the density of storage elementscan be improved tremendously due to its nanostructure dimension. Macromagnetic sim-ulation of the device considered is carried out by numerically integrating eq. (11) or (14)using variable step size Runge–Kutta method. In figure 1 we have shown the switchingdynamics of the magnetization in the free layer [15]. It can be observed that the directionof magnetization gets reversed at the point �m = (1, 0, 0).

3.2 Self-sustained oscillations

Any system capable of converting the energy supplied to it into oscillations is termed as anoscillator. Self-sustained oscillation is a unique nonlinear behaviour of a typical nonlin-ear dynamical system. In a trilayer extended GMR structure, on the one hand the injecteddirect current exerts STT (acting opposite to damping) which is used to switch the mag-netization of the free layer. On the other hand, in certain circumstances the produced STTbalances the inherent damping due to effective field and makes the magnetization to oscil-late in time [18]. This typical device capable of converting the injected direct current intomagnetization oscillation is termed as spin torque/transfer nano-oscillator (STNO/STO).It is important to note that here the produced oscillations have the same frequency as thatof linear waves in the ferromagnetic material, namely magnons, and hence the frequencyof oscillation is in the wide range of hundreds of KHz to few tens of GHz [5,10,14]. Due

mxmy

-1-0.5

0 0.5

1

mz

-1-0.5

0 0.5

1-1-0.5

0 0.5

1

mz

Figure 1. Trajectory of magnetization vector �m, obtained by macromagnetic simula-tion of eq. (11) for isotropic sphere of permalloy thin film with anisotropy κ = 45 Oe,damping coefficient λ = 0.007 and spin current density a = 5.6 Oe.



3 4 5 60

20

40

60

80

100

Frequency (GHz)

Pow

er (

a.u)

a=208 Oea=228 Oe

a=198 Oe

a=178 Oe a=158 Oe

Figure 2. Power spectral density of the x-component (mx ) of the magnetization vec-tor �m for increasing values of spin current density (a). Frequency of the magnetizationoscillation is decreasing with increase in a, showing the typical in-plane oscillationbehaviour of STNO.

to this, STNO becomes a vital candidate for nanoscale level source of microwaves in thetelecommunication industry. Apart from technological application, STNO is one of thesmallest ever known self-sustained oscillator.

Two types of oscillations [11] are exhibited by STNO depending upon the currentinjected into the fixed layer, namely

(1) In-plane oscillation – the magnetization vector ( �m) oscillates symmetricallyaround an axis which lies in the plane of the effective field ( �Heff) or easy planethereby giving rise to low-frequency oscillations. Further, in the earlier studies, itis characterized by the phenomenon of decrements in frequency of oscillations byincreasing the spin current density (see figure 2). Typical in-plane magnetizationis shown in figure 3.

(2) Out-of-plane oscillation (as shown in figure 4) – the magnetization vector ( �m)oscillates symmetrically around an axis which lies out of the easy plane exhibitinghigh-frequency oscillations. The typical behaviour of out-of-plane oscillations isthat the frequency increases with increase in the spin current density (a) as shownin figure 5.

mxmy

-1-0.5

0 0.5

1

mz

-1-0.5

0 0.5

1-1-0.5

0 0.5

1

mz

Figure 3. In-plane oscillation of the magnetization vector �m.



-1 -0.5 0 0.5 1 -1-0.5 0 0.5 1

-1

-0.5

0

0.5

1

mz

mx

my

mz

-1

0

1

-1

0

1

-0.5

0mz

mxmy

mz

(a) (b)

Figure 4. Out-of-plane oscillation of the magnetization vector �m. (a) �m is plotted forthe spin current density a = 658 Oe, (b) magnetization vector �m is plotted for threevalues of spin current density a = 338, 458 and 658 Oe.

3.3 Chaos in the presence of oscillating magnetic field

Although the equation of motion, eq. (11), is a coupled nonlinear evolution equation (forthe components of the spins mx, my, mz), it is to be noted that its dimension gets reducedby one with the constraint | �m|2 = 1. In other words, the dynamics of the magnetizationvector �m is constrained on the surface of a unit sphere, and so chaos is precluded. Byapplying an oscillating external magnetic field/current, the effective dimension of theunderlying system increases and hence a rich variety of nonlinear dynamics includingbifurcations and chaos can be expected [16]. In order to identify the chaotic behaviourof the system, we have numerically calculated the maximal Lyapunov exponent in theparameter space (hDC vs. a) as shown in figure 6. It clearly shows the onset of bifurcationsand chaos.

4. Synchronization in the presence of oscillating magnetic field

Although a single STNO is a favourable source of microwave power over the conven-tional device in various aspects such as wide band of frequency tunability, low bias

5 6 7 80

20

40

60

80

100

120

140

Frequency (GHz)

Pow

er (

a.u.

)

a=458 Oea=358 Oe

a=338 Oe

a=298 Oea=658 Oe

Figure 5. Power spectral density of the x-component (mx ) of magnetization vector�m for increasing values of spin current density (a). Frequency of the magnetizationoscillation is increasing with increase in a, showing the typical out-of-plane oscillationbehaviour of STNO.



150

200

250

300

350

400

100 150 200 250

a (O

e)

hDC (Oe)

Λ

-5

-4

-3

-2

-1

0

1

2

3

Figure 6. Maximal Lyapunov exponent (�) is plotted in the region of static magneticfield (hDC) vs. spin current density (a). Violet colour regimes are periodic regimeswhere � is negative. On the other hand, the red/yellow regimes are the chaoticregimes where � is positive.

voltage, resistance to external radiation, easy integration with CMOS circuits throughMRAMs, it lacks efficiency due to its low output power which is in the order of nanoWatt[19,20]. Several researchers have suggested that synchronization of a large array ofSTNOs coupled through various mechanisms can improve the power output of the device,viz.,

(1) Mutual phase lock – STNOs are in close proximity – electrically uncoupled [14].(2) Dipolar coupling – electrically uncoupled [21].(3) Stimulated microwave current – electrically coupled [22,23].(4) Injection locking to external microwave current [20,24].

Spacer

Free layer

Fixer layer

+

........ ........

Sp Sp

H app

Y

X

Z

Figure 7. Schematic representation of an array of N electrically uncoupled oscillatorsmagnetically coupled through a medium of oscillating magnetic field. Here ‘+’ signrepresents the microwave power combiner [14,26].



150

200

250

300

350

400

100 150 200 250

a (O

e)

hDC (Oe)

σ

0

0.2

0.4

0.6

0.8

1

Figure 8. Synchronization regimes (violet/blue) for an array of N = 100 oscillatorsmagnetically coupled through a medium of oscillating magnetic field. Red/yellowregimes correspond to phase-desynchronized regimes characterized by the values oftime-averaged standard deviation of the x-component of the magnetization vector. Thevalues of standard deviation are encoded in the colour palette.

(5) Common noise induced synchronization – electrically connected STNOs [25](6) Common oscillating magnetic field-induced synchronization – electrically uncou-

pled [26,27].

In this study, we have considered the collective dynamics of N = 100 electricallyuncoupled oscillators placed under a common oscillating magnetic field (15). As theoscillators are electrically uncoupled and influenced magnetically through a commonmedium (see figure 7), eq. (11) remains the same for all the N oscillators [26]. To identifythe regions of synchronization we calculate the time-averaged standard deviation (σ ) ofthe x-component of the magnetization vector (mx

i ) [28]

σ =⟨√

1

N

∑i

(mxi − μ)2

⟩t

, μ = 1

N

∑i

mxi , (16)

where μ is the average time series of the magnetization of N STNOs. Note that if all theoscillators are completely synchronized, σ takes a value zero, otherwise it takes a valueother than zero. To illustrate the synchronization dynamics in this case we have calculatedthe value of σ and plotted it in the parameter space hDC vs. a as shown in figure 8. Bycomparing figures 6 and 8, it is evident that the oscillating magnetic field synchronizesthe periodic/limit cycle oscillations of the STNOs in different regions.

5. Serially coupled STNOs – enhanced synchronization

From a technological point of view, it is hard to keep all the STNOs electrically uncou-pled because it requires separate DC power source for each STNO. To make the devicemore feasible we serially couple all the STNOs which are under an oscillating external



Rl

RN(t)

R1(t)

IDC

ISTNO

Figure 9. Equivalent electronic circuit for N coupled STNOs of resistance Ri(t).

microwave magnetic medium. By such a serial electrical coupling the microwave outputof the first oscillator is injected into the fixed layer of the second layer and so on. Themodified LLGS equation for the ith STNO for a system of N coupled oscillators is givenas [28,29]

d �mi

dt= −γ �mi × �H eff

i + α �mi × d �mi

dt

−γβ(t) �mi × ( �mi × Mp), i = 1, 2, ...N. (17)

The effective field for the ith STNO is given by

H effi = �Happ + κim

xi i − 4πM0m

zi k, (18)

J (t) = RlIDC

Rl + R0i − �Ri

∑Ni=1 cos θi(t)

, (19)

-0.9

-0.6

-0.3

0

0.3

0.6

0.9

100 100.2 100.4 100.6 100.8 101

mx1-

500

t (ns)

(a)

-0.9

-0.6

-0.3

0

0.3

0.6

0.9

100 100.5 101 101.5 102

mx1-

500

t (ns)

(b)

Figure 10. Time series of x-component of the magnetization vector �m for N = 500STNOs are plotted in (a) the absence of external oscillating component of the magneticfield and (b) the presence of external oscillating component of the magnetic field.


which contains an external applied magnetic field ( �Happ), along x-axis and all the remain-ing terms are as defined in §2.3. The density of the spin current β(t) = hηJ (t)/2M0V e,where η = 0.35 is the spin polarization ratio, and the shared microwave current J (t)

is derived by using a current divider relation between the resistance of N serially cou-pled STNOs, each having a resistance Ri(t) = R0

i + �Ri cos[θi(t)], R0i = (R

pi +

Rapi )/2,�Ri = (R

api − R

pi )/2 and load resistance Rl . Then one can show [28,29] that


where Rp and Rap are the resistances of the STNOs when free and fixed layers are in par-allel and antiparallel magnetization states, respectively. The equivalent electronic circuitfor the model is given in figure 9. We find that the oscillating medium synchronizes thephase desynchronized magnetization oscillations as shown in figure 10.

Further, it is essential to find the regions of synchronization in the frequency determin-ing parameter space, namely hDC vs. a, where we can tune the synchronized oscillationswithout compromising the output power. This has been successfully carried out. Thedetails will be published elsewhere [28].

6. Conclusion

To conclude, we have briefly reviewed the method for obtaining the model equation,namely LLG equation, to describe the spin dynamics of ferromagnetic materials, byconsidering the Heisenberg nearest-neighbour interaction and using spin Poisson bracketrelation for a chain of magnetic spins. The continuum limit was then discussed. Thenwe have pointed out how the introduction of Slonczewski’s STT term into the LLG equa-tion for an extended GMR trilayer structure can be made. Macrospin simulation of thetrilayer structure has been done by numerically integrating the underlying LLGS equa-tion. A rich variety of free layer magnetization dynamics has been discussed, includingtorque-induced switching to chaos. One important dynamical state of magnetization dueto STT is the self-oscillation which makes the device a vital candidate for nanoscale levelmicrowave power generation. To enhance the power output, several possible mechanismshave been discussed. In particular, synchronization due to a common external oscillatingmagnetic field is discussed. Another possible mechanism of establishing electrical con-nections between the magnetically coupled oscillators is introduced to make the devicemore feasible for practical level of integration with electronic circuits. The approach lookspromising for actual device applications.

Acknowledgements

The work forms part of a Department of Science and Technology (DST), Government ofIndia, IRHPA project and is also supported by a DST Ramanna Fellowship of ML. He hasalso been financially supported by a DAE Raja Ramanna Fellowship.

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nonlinear dynamics of spin transfer nano-oscillators

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