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Metastable state in a shape-anisotropic single-domain nanomagnet

subjected to spin transfer torque

Kuntal Roy1,∗ Supriyo Bandyopadhyay1, and Jayasimha Atulasimha21Dept. of Electrical and Computer Engg., Virginia Commonwealth University, Richmond, VA 23284, USA2Dept. of Mechanical and Nuclear Engg., Virginia Commonwealth University, Richmond, VA 23284, USA

(Dated: June 8, 2011)

The magnetization of an unperturbed shape-anisotropic single-domain nanomagnet has two stableorientations along the easy axis. Here, we show analytically that when the nanomagnet is perturbedwith a spin-polarized current delivering a spin-transfer torque, a third metastable orientation canemerge, which can prevent switching between the two stable orientations. Above a threshold cur-rent, this state no longer appears and switching is guaranteed. This has immense technologicalconsequences for spin-transfer-torque based magnetic memory and logic systems.

Spin-angular-momentum-transfer or simply spin-transfer-torque (STT) mechanism is an electric current-induced magnetization switching mechanism that canrotate the magnetization vector of a shape-anisotropicnanomagnet through ∼180 by exerting a torque on itdue to the passage of a spin-polarized current [1, 2].STT-induced switching has been verified in nanomagnetswith ∼100 nm size [3] and has been demonstrated innumerous experiments involving both spin-valves [4]and magnetic tunnel junctions (MTJs) [5]. The latteris becoming the staple of nonvolatile magnetic randomaccess memory (STT-RAM) [6, 7].

STT-RAM technology demands fast switching with-out failures. In this Letter, we point out that switchingfailure can occur owing to a hitherto unsuspected mech-anism; namely, the magnetization vector getting trappedin a metastable state (orientation) during rotation. Oncetrapped, it cannot be dislodged by anything except ther-mal fluctuations, which appears to be too weak at roomtemperature [see Fig. S4 in supplementary material]. Wederive the existence of the metastable state analyticallyand show that it arises for certain ranges of switchingcurrent. There is also a critical switching current beyondwhich it does not appear.

Consider a single-domain nanomagnet shaped like anelliptical cylinder with elliptical cross section in the y-z plane (see Fig. 1). The major (easy) and the minor(in-plane hard) axes of the ellipse are aligned along thez-direction and y-direction, respectively. Let θ(t) be theangle subtended by the magnetization vector with the+z-axis at any instant of time t and φ(t) be the anglebetween the +x-axis and the projection of the magneti-zation vector on the x-y plane. Thus, θ(t) is the polarangle and φ(t) is the azimuthal angle.

At any instant of time t, the energy of the unper-turbed nanomagnet is the uniaxial shape anisotropy en-ergy which can be expressed as [8]:

E(t) = B(t)sin2θ(t) + C, (1)

FIG. 1. A nanomagnet shaped like an elliptical cylinder. Thenanomagnet cross-sections are on the y-z plane. The magneti-zation direction can be rotated with a spin polarized current.

where

B(t) = B(φ(t)) =µ0

2M2

sΩ[

Nd−xxcos2φ(t)

+Nd−yysin2φ(t) −Nd−zz

]

(2)

C =µ0

2M2

sΩNd−zz. (3)

Here Ms is the saturation magnetization, Nd−xx, Nd−yy

and Nd−zz are the x-, y- and z-components of demagne-tization factor [9], and Ω is the nanomagnet’s volume.

The magnetization M(t) of the nanomagnet has a con-stant magnitude at any given temperature but a variableorientation, so that we can represent it by the vector ofunit norm nm(t) = M(t)/|M| = er where er is the unitvector in the radial direction in spherical coordinate sys-tem represented by (r,θ,φ). The other two unit vectorsin the spherical coordinate system are denoted by eθ andeφ for θ and φ rotations, respectively. The coordinates(θ,φ) completely describe the motion of M(t) [2].

The torque acting on the magnetization within unitvolume due to shape anisotropy is [8]

TE(t) = −er × [2B(t)sinθ(t)cosθ(t)eθ

−B0e(t) sinθ(t)eφ]

= −2B(t)sinθ(t)cosθ(t)eφ − B0e(t) sinθ(t)eθ,

(4)

where

B0e(t) = B0e(φ(t)) =µ0

2M2

sΩ(Nd−xx−Nd−yy)sin(2φ(t)).

(5)

2

Passage of a constant spin-polarized current I perpen-dicular to the plane of the nanomagnet generates a spin-transfer-torque that is given by [10, 11]

TSTT(t) = −snm(t)× [asnm(t) + bsns(t)

+ csnm(t)× ns(t)], (6)

where s = (~/2e)ηI is the spin angular momentumdeposition per unit time and η is the degree of spin-polarization in the current I. The coefficients bs(V ) andcs(V ) are voltage-dependent dimensionless terms thatarise when the nanomagnet is coupled with an insulatinglayer as in an MTJ [12] and the spin-polarized currenttunnels through this layer, while as(V ) is somewhat ir-relevant in this context since the term involving as(V )vanishes in the expression for torque.The unit vector ns is in the direction of the spin polar-

ization of the switching current I and lies in the y-z plane.For simplicity, we will assume it to be time-invariant. Ifthe initial orientation of the magnetization is along the-z-axis and we wish to switch it to the other stable ori-entation along the +z-axis, then the spin polarizationvector ns must be along the positive z-axis. In that case,the spin-transfer torque generated by I is given by

TSTT(t) = s [bs(V ) sinθ(t) eφ − cs(V ) sinθ(t) eθ] . (7)

We will use constant values of bs(V ) and cs(V ) forsimplicity [13]. Furthermore, we will assume bs(V ) =0.3 cs(V ) and cs(V ) = 1 to be in approximate agreementwith the experimental results presented in Refs. [11, 14].The magnetization dynamics of the single-domain

nanomagnet under the action of various torques is de-scribed by the Landau-Lifshitz-Gilbert (LLG) equationas

dnm(t)

dt+ α

(

nm(t)×dnm(t)

dt

)

=γ

MV

(TE(t) +TSTT(t)) (8)

where α is the dimensionless phenomenological Gilbertdamping constant, γ = 2µBµ0/~ is the gyromag-netic ratio for electrons and is equal to 2.21 × 105

(rad.m).(A.s)−1, where µB is the Bohr magneton, andMV = µ0MsΩ. In the spherical coordinate system, withconstant magnitude of magnetization

dnm(t)

dt=

dθ(t)

dteθ + sinθ(t)

dφ(t)

dteφ. (9)

Accordingly,

dnm(t)

dt+ α

(

nm(t)×dnm(t)

dt

)

= [θ′(t)− αsinθ(t)φ′(t)] eθ+[sinθ(t)φ′(t) + αθ′(t)] eφ

where ()′ denotes d()/dt. Equating the eθ and eφ com-ponents in both sides of the Equation (8), we get

(

1 + α2)

θ′(t) = −γ

MV

[s (cs(V )− α bs(V ))

+B0e(t)sinθ(t) + 2αB(t)sinθ(t)cosθ(t)] (10)

(

1 + α2)

φ′(t) =γ

MV

[s (α cs(V ) + bs(V )) + αB0e(t)

− 2B(t)cosθ(t)] (sinθ 6= 0). (11)

Note that when the magnetization vector is alignedalong the easy axis (i.e. θ = 0, 180), the torque dueto shape anisotropy, TE(t) and the torque due to spin-transfer-torque, TSTT(t) both vanish (see Equations (4)and (7)), which makes dθ(t)/dt as well as dφ(t)/dt equalto zero. Hence the two mutually anti-parallel orientationsalong the easy axis become “stable”. If the magnetiza-tion is in either of these states, no amount of switchingcurrent I can budge it (because the spin transfer torquevanishes), which is why these two orientations are alsoreferred to as “stagnation points”. Fortunately, thermalfluctuations can dislodge the magnetization from a stag-nation point and enable switching. We will now showthat there can be a third set of values (θ3, φ3) for θ(t)and φ(t) for which both dθ(t)/dt and dφ(t)/dt will vanish.Unlike in the case of the other two stable states whereboth shape-anisotropy torque and spin-transfer-torqueindividually vanish, here neither vanishes, but they areequal and opposite so that they cancel to make the nettorque zero. If the magnetization ends up in this orien-tation, then it will be stuck and not rotate further unlesswe change the switching current I to change the spin-transfer torque. Since changing I can dislodge the mag-netization from this state, it is not a stagnation pointunlike θ = 0, 180. Hence, we call it a metastable state.

We determine the values of (θ3, φ3) as follows. FromEquations (10) and (11), by making both dθ(t)/dt anddφ(t)/dt equal to zero, we get

2αB(φ3)cosθ3 = αs bs(V )− s cs(V )−B0e(φ3) (12)

2B(φ3)cosθ3 = s bs(V ) + αs cs(V ) + αB0e(φ3). (13)

From the above two equations, we get B0e(φ3) =−scs(V ), which translates to (see Equation (5))

µ0

2M2

sΩ(Nd−xx −Nd−yy)sin(2φ3) = −scs(V ). (14)

If we put B0e(φ3) = −scs(V ) in Equation (12) or inEquation (13), we get 2B(φ3)cosθ3 = sbs(V ), or

µ0M2

sΩ[

Nd−xxcos2φ3 +Nd−yysin

2φ3 −Nd−zz

]

cosθ3

= sbs(V ). (15)

3

FIG. 2. Switching delays for a range of switching current700 µA – 40 mA producing spin transfer torque (cs(V ) = 1,bs(V )=0.3). No switching occurs for the following switch-ing current ranges: 2-2.05 mA, 2.33-2.49 mA, 2.83–3.26 mA,3.69–5.09 mA, 5.6–10.24 mA, 11.41–24.51 mA. The switchingfailure is either due to the metastable state or the magneti-zation vector ending up in a state where it stops rotating andbegins to oscillate around a mean orientation. The latter isa different state not discussed here. For this calculation, theswitching current is varied in steps of 10 µA.

Accordingly, we can determine the values of (θ3, φ3) as

φ3 =1

2sin−1

(

−(~/2e)ηIcs(V )

(µ0/2)M2sΩ(Nd−xx −Nd−yy)

)

(16)

θ3 = cos−1

(

(~/2e)ηIbs(V )[

µ0M2

sΩ]

−1

Nd−xxcos2φ3 +Nd−yysin2φ3 −Nd−zz

)

.

(17)

Note that φ3 depends on cs(V ) while θ3 depends on bothcs(V ) and bs(V ). Neither depends on the Gilbert damp-ing factor α.If bs(V )=0, then θ3 = 90, which means that the mag-

netization will be stuck somewhere in the x-y plane per-pendicular to the easy axis, if it lands in the metastablestate. This plane is defined by the in-plane and out-of-plane hard axes. Note that when cs(V ) is negative, φ3

is in the range 0 ≤ φ3 ≤ 90, but when cs(V ) is pos-itive, 90 ≤ φ3 ≤ 180. Therefore, in the former case,the magnetization vector will be stuck above the plane ofthe magnet, whereas in the latter case, it will be stuckbelow the magnet’s plane. The quantity bs(V ) cannot benegative. Note also that when I = 0 so that there is nospin-transfer-torque, θ3 = 90 and φ3 is either 0 or 90.Consequently, the state is either along the in-plane hardaxis (y-axis) or the out-of-plane hard axis (x-axis). Sincethese are obviously metastable states in an unperturbedshape-anisotropic nanomagnet, we call the state (θ3, φ3)a “metastable” state.For numerical simulations, we consider a nanomagnet

of elliptical cross-section made of CoFeB alloy which hassaturation magnetization Ms = 8 × 105 A/m [15] and aGilbert damping factor α = 0.01. We assume the lengthsof major axis (a), minor axis (b), and thickness (l) tobe 150 nm, 100 nm, and 2 nm, respectively. These di-mensions (a, b, and l) ensure that the nanomagnet willconsist of a single ferromagnetic domain [16, 17]. The

FIG. 3. Switching delays for the same range of switchingcurrent 700 µA – 40 mA producing spin transfer torque(cs(V ) = 1, bs(V )=0). Within the range of current consid-ered here, switching always occurs, although switching maynot happen outside this range of current. The switching cur-rent is varied in steps of 10 µA.

combination of the parameters a, b, l, and Ms makesthe in-plane shape anisotropy energy barrier height ∼32kT at room temperature. The spin polarization of theswitching current is always assumed to be 80%.We assume that the magnetization is initially along

the -z-axis, which is a stagnation point. Hence, at 0 K,no switching can occur. However, at a finite tempera-ture, thermal fluctuations will dislodge the magnetizationfrom the stagnation point and enable switching. Our pastsimulations showed that at room temperature, the ther-mal fluctuations will deflect the magnetization vector by∼4.5 from the easy axis when averaged over time [18],so that we will assume the initial value of the polar an-gle to be θinit = 175.5. We choose the initial azimuthalangle φinit as +90 [18]. Similar assumptions are madeby others [13]. We then solve Equations (10) and (11) si-multaneously to find θ(t) and φ(t) as a function of time.Once θ(t) reaches 4.5, regardless of the value of φ(t),we consider the switching to have completed. The timetaken for this to happen is the switching delay.Figs. 2 and 3 show the switching delays versus switch-

ing current for two different values of bs(V ). Theswitching delay is ‘infinity’ in some current ranges whenbs(V ) 6= 0 because switching failed. Most of the failureshappen because the magnetization vector gets stuck inthe metastable state. However, beyond the current of24.51 mA, switching always occurs within a finite time,meaning that the magnetization never ends up in themetastable state.The important question is why switching fails only for

certain ranges of the current I, i.e. why does the mag-netization vector land in the metastable state for certainvalues of I and not others? The answer is that start-ing from some initial condition (θinit, φinit), the anglesθ(t) and φ(t) must reach the values θ3 and φ3 at thesame instant of time t. This may not happen for anyarbitrary I. Hence, only certain ranges of I will spawnthe metastable state. It is also clear from Equation (16)that above a certain value of |I|, there will be no realsolutions for φ3 since the argument of the arcsin func-

4

(a) (b)

FIG. 4. Temporal dynamics of the polar angle θ(t) and theazimuthal angle φ(t) when the switching current is 24.51 mA(cs(V )=1, bs(V )=0.3). (a) Dynamics of θ(t). (b) Dynamics ofφ(t). The magnetization gets stuck in a metastable state withθ3 = 82.42 and φ3 = 335.87. Simulations in the presence ofthermal fluctuations have shown that the latter cannot untrapthe magnetization from this state at room temperature.

tion will exceed unity. This value will be |I|threshold =[

eµ0M2

sΩ (Nd−xx −Nd−yy)]

/ [~ηcs(V )]. By maintainingthe magnitude of the switching current above |I|threshold,we can ensure that the magnetization vector will neverget stuck in the metastable state. For the nanomag-net considered, |I|threshold = 32.7 mA, but switching be-comes guaranteed at even lower current of 24.52 mA sincein the range [24.52 mA, 32.7 mA], the coupled θ and φ-dynamics expressed by Equations (10) and (11) do notallow θ(t) and φ(t) to reach θ3 and φ3 simultaneously.In order to understand the physical origin of the

metastable state, consider the fact that the effective mag-netic field associated with the total torqueM(t)×Heff (t)can be deduced from Equations (4) and (7) as

Heff (t) = (−2B(t)cosθ(t) + sbs(V )) sinθ(t) eθ

+(B0e(t) + scs(V )) sinθ(t) eφ, (18)

where we have neglected any component of Heff (t) thatis collinear with M(t) since it will not produce a torque.From Equations (16) and (17), we immediately see thatHeff (t) vanishes when θ(t) = θ3 and φ(t) = φ3. Hencethere is no torque acting on the magnetization vectorif it reaches the state θ(t) = θ3 and φ(t) = φ3 at thesame instant of time t. Thereafter, it cannot rotate anyfurther since the torque has vanished. Consequently, themagnetization will be stuck in this state and it will be ametastable state.Figure 4 shows the dynamics of the polar angle θ(t)

and the azimuthal angle φ(t) for a switching currentof 24.51 mA. We observe that the magnetization getsstuck in a metastable state with θ3 = 82.42 and φ3 =335.87. These are also the values of the angles predictedby Equations (16) and (17), thereby confirming that themetastable state indeed has the origin described here.We also notice that slight oscillations precede settlinginto the metastable state. This is due to coupled θ- andφ-dynamics governing the rotation of the magnetizationvector. The metastable state appears in the switchingcurrent range 11.41 mA to 24.51 mA since within this

range, θ(t) and φ(t) can reach θ3 and φ3 simultaneouslystarting with the initial conditions (θinit, φinit). If theinitial conditions are changed, the range can change aswell.

In conclusion, we have shown the existence of ametastable magnetization state in a shape anisotropicnanomagnet carrying a spin-polarized current. It ap-pears due to cancellation of the spin-transfer torque andthe torque due to shape anisotropy. If the magnetiza-tion gets trapped in this state, switching will fail unlessthermal fluctuations dislodge the magnetization from thisstate. In order to avoid that with certainty, the switchingcurrent should be maintained above a certain thresholdvalue.

∗ royk@vcu.edu[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1

(1996).[2] J. Z. Sun, Phys. Rev. B 62, 570 (2000).[3] J. Z. Sun, IBM J. Res. Dev. 50, 81 (2006).[4] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,

and D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).[5] G. D. Fuchs, J. A. Katine, S. I. Kiselev, D. Mauri, K. S.

Wooley, D. C. Ralph, and R. A. Buhrman, Phys. Rev.Lett. 96, 186603 (2006).

[6] S. S. P. Parkin, K. P. Roche, M. G. Samant, P. M. Rice,R. B. Beyers, R. E. Scheuerlein, E. J. O’sullivan, S. L.Brown, J. Bucchigano, and D. W. Abraham, J. Appl.Phys. 85, 5828 (1999).

[7] C. Chappert, A. Fert, and F. N. V. Dau, Nature Mater.6, 813 (2007).

[8] K. Roy, S. Bandyopadhyay, and J. Atulasimha,arXiv:1103.0352, Phys. Rev. B (2011).

[9] S. Chikazumi, Physics of Magnetism (Wiley New York,1964).

[10] S. Salahuddin, D. Datta, and S. Datta, Arxiv preprintarXiv:0811.3472 (2008).

[11] H. Kubota, A. Fukushima, K. Yakushiji, T. Naga-hama, S. Yuasa, K. Ando, H. Maehara, Y. Nagamine,K. Tsunekawa, and D. D. Djayaprawira, Nature Phys.4, 37 (2008).

[12] I. Theodonis, N. Kioussis, A. Kalitsov, M. Chshiev, andW. H. Butler, Phys. Rev. Lett. 97, 237205 (2006).

[13] D. E. Nikonov, G. I. Bourianoff, G. Rowlands, and I. N.Krivorotov, J. Appl. Phys. 107, 113910 (2010).

[14] J. C. Sankey, Y. T. Cui, J. Z. Sun, J. C. Slonczewski,R. A. Buhrman, and D. C. Ralph, Nature Phys. 4, 67(2008).

[15] J. Fidler and T. Schrefl, J. Phys. D: Appl. Phys. 33, R135(2000).

[16] M. Beleggia, M. D. Graef, Y. T. Millev, D. A. Goode,and G. Rowlands, J Phys. D: Appl. Phys. 38, 3333(2005).

[17] R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E.Welland, and D. M. Tricker, Phys. Rev. Lett. 83, 1042(1999).

[18] K. Roy, S. Bandyopadhyay, J. Atulasimha, andA. Ghosh, unpublished.

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Supplementary Information

Metastable state in a shape-anisotropic single-domain nanomagnet

subjected to spin transfer torque

Kuntal Roy1, Supriyo Bandyopadhyay1, and Jayasimha Atulasimha2

Email: royk, sbandy, jatulasimha@vcu.edu

1Dept. of Electrical and Computer Engg., 2Dept. of Mechanical and Nuclear Engg.

Virginia Commonwealth University, Richmond, VA 23284, USA

June 8, 2011

We provide here a few more plots of magnetization dynamics with corresponding descriptions.

We point out that there is a minimum current needed to switch magnetization with spin transfer

torque [Ref. 2 in the main Letter]. The current ranges that we have considered here are above this

minimum current.

2 Roy, K. et. al., XXX (2011)

Figure S1: The trajectory traced out by the tip of the magnetization vector in three-dimensionalspace. The nanomagnet is made of CoFeB and has the geometric and material parameters mentionedin the main Letter. It is subjected to spin-transfer-torque switching current of 24.51 mA (cs(V )=1,bs(V )=0.3). The magnetization vector eventually gets stuck at the metastable state and fails torotate further. Note the precessional motion before settling into the metastable state. This motionis responsible for the oscillations in θ(t) and φ(t) observed in Fig. 4 of the main Letter.

3 Roy, K. et. al., XXX (2011)

Figure S2: Switching delay versus switching current in the range 670 µA – 40 mA. Any lowerswitching current will increase the delay to more than 5 ns and is hence avoided. We used theparameters cs(V ) = −1, bs(V )=0.3 in this plot. The opposite signs of cs(V ) and bs(V ) correspondto parallel-to-antiparallel switching in MTJs. The initial state of the magnetization vector wasφinit = 90 and θinit = 175.5. Unlike when cs = +1 (antiparallel-to-parallel switching), theswitching delay always remains finite showing that the magnetization vector never gets stuck at themetastable state within this range of switching current. Changing φinit did not make any difference.Obviously this happens since θ(t) and φ(t) cannot reach the values θ3 and φ3 simultaneously withinthis current range.

4 Roy, K. et. al., XXX (2011)

Figure S3: Switching delay versus switching current in the range 700 µA – 40 mA. We used theparameters cs(V ) = 1 and bs(V )=0.3 in this plot. The initial state of the magnetization vectorwas φinit = 0 and θinit = 175.5. No switching takes place for the following ranges of switchingcurrents: 2.49-5.18 mA, 7.1-31.08 mA. The switching dynamics was simulated for every 10 µA ofswitching current in this interval. Obviously switching fails in these ranges since θ(t) and φ(t) reachthe values θ3 and φ3 simultaneously, so that the magnetization vector ends up in the metastablestate. Note that the ranges of current where switching fails are sensitive to initial conditions sincethese ranges are different from those in Figure 2 of main paper, where the initial condition wasφinit = 90 and θinit = 175.5.

5 Roy, K. et. al., XXX (2011)

(a)

(b)

Figure S4: Time evolution of θ(t) and φ(t) when the switching current is 24.51 mA (cs(V ) = 1,bs(V ) = 0.3). The magnetization vector gets stuck in the metastable state and cannot be unstuck bythermal fluctuations at room temperature. This plot was obtained by the solution of the stochasticLandau-Lifshitz-Gilbert equation in the presence of a random thermal torque to simulate the effectof thermal fluctuations. Occasional small perturbations around the metastable state show up inboth the dynamics of θ(t) and φ(t) due to thermal fluctuations; however, such small fluctuationsare not sufficient to dislodge the magnetization axis from the metastable state. A large number ofsimulations (10,000) and simulation over a long time duration (500 ns) were carried out to confirmthis. (a) Dynamics of θ(t). (b) Dynamics of φ(t).

6 Roy, K. et. al., XXX (2011)

(a)

(b)

Figure S5: Time evolution of θ(t) and φ(t) when the switching current is 14.9 mA (cs(V ) = 1,bs(V ) = 0.3). The oscillations in θ(t) and φ(t) asymptotically decay to θ3 and φ3, whereupon thesystem gets stuck at the metastable state. The oscillation amplitude in φ(t) starts out as verylarge (between nearly 0 and 360) and then abruptly diminishes at around 2.5 ns. Such trends areobserved for switching currents exceeding 11.41 mA. (a) Dynamics of θ(t). (b) Dynamics of φ(t).In this case, θ3 = 85.94 and φ3 = 346.51.