micromagnetic study of high-power spin–torque oscillator with perpendicular magnetization in...

Micromagnetic study of high-power spin–torque oscillatorwith perpendicular magnetization in half-metallic Heusleralloy spin valve nanopillar under external magnetic fields

H.B. Huang a,b,n, X.Q. Ma a, C.P. Zhao a, Z.H. Liu a, L.Q. Chen b

a Department of Physics, University of Science and Technology Beijing, Beijing 100083, Chinab Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA

a r t i c l e i n f o

Article history:Received 2 December 2013Received in revised form7 March 2014Accepted 17 March 2014

Keywords:Micromagnetic simulationSpin–torque oscillatorHeusler alloySpin-valve nanopillarMagnetization precession

a b s t r a c t

We investigated the high-power spin–torque oscillator in a half-metallic Heusler alloy Co2MnSi spin-valve nanopillars with perpendicular magnetization under external magnetic field using micromagneticsimulations. Our simulations show that the narrow optimum current of magnetization precession in theHeusler-based spin valve is broadened by introducing the surface anisotropy. The linear decrease offrequency with the out-of-plane magnetic field is obtained in our simulation. Additionally, the in-planemagnetic field dependence of frequency shows a parabolic curve which is explained by the magnetiza-tion trajectory tilting. Furthermore, we also discussed the decrease of output power using the excitationof non-uniform magnetization precession in the in-plane magnetic fields.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Since the initial predictions that a spin-polarized current couldexert a torque named spin transfer torque (STT) [1,2] in themagnetic nanopillar, much progress has been made in understand-ing the effect and developing the new functional magnetic devices.The magnetization precession could be produced to get the highfrequency microwave when STT input equals the magnetic dampingtorque. The high frequency precession has been investigated inmultilayers with both spin valve nanopillar [3] and point contactgeometries [4]. Coupled with the giant magnetoresistance effect(GMR), this stable precession produces a voltage response thatmakes these devices high-frequency oscillators, called spin–torqueoscillators (STO). It is difficult to increase the output power andmaintain narrow linewidth for any useful applications. There weremany attempts to increase the output power, e.g., a magnetic tunneljunction (MTJ) based STO [5,6]. Compared with MTJ–STO, theadvantages of using a full metal STO are their narrow linewidthand low resistance enabling good impedance matching.

Recently, the high polarization materials like Heusler alloys areproposed to potentially increase the GMR signal [7,8] and hencethe power output from the oscillator. In the experiments, a high rfoutput power of 1.1 nW was achieved in Co2MnSi (CMS) alloy [9]

and a large emission amplitudes exceeding 150 nV/Hz and narrowlinewidth below 10 MHz were obtained [10]. In our previouspaper, we demonstrated that the output power can be improvedsignificantly due to the out-of-plane (OPP) precession by usingmicromagnetic simulations [11]. However, the narrowing opti-mum current region for the rf oscillation due to the high spinpolarization constant of Heusler alloy is the barrier to get thestable OPP oscillation which is demonstrated by Seki et al. usingmacrospin model [12]. Since the balance between STT and thedamping torque is the key to stabilize the precessional oscillation,the magnitude of the effective field is not sufficiently large tocompete with the large STT originating from the high spinpolarization constant in Heusler-based spin valve nanopillars.Furthermore, the surface and interfacial anisotropies are non-negligible and often dominate the bulk contributions in theultrathin films. It was demonstrated that the interfacial perpendi-cular anisotropy decreases the critical current of spin transfermagnetization switching in MgO-based magnetic tunnel junctionnanopillars with perpendicular full-Heusler alloy electrodes [13].Thus the surface anisotropy which is characterized by the thin filmroughness also plays a vital role in the STT induced magnetizationdynamics [14,15].

This paper is structured as follows: in Section 2, we first describeour method based on micromagnetic simulations by introducing thesurface anisotropy. In Section 3.1, we present the phase diagrams ofthe frequency as a function of out-of-plane magnetic fields. In Section3.2, we demonstrate that the in-plane magnetic field dependence of

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Journal of Magnetism and Magnetic Materials

http://dx.doi.org/10.1016/j.jmmm.2014.03.0500304-8853/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author.E-mail address: [email protected] (H.B. Huang).

Please cite this article as: H.B. Huang, et al., Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j.jmmm.2014.03.050i

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frequency is a parabolic curve. We discuss the decrease of outputpower using the excitation of non-uniform magnetization precessionin the in-plane magnetic fields. Finally, we summarize the high outputpower of spin–torque oscillator by introducing the surface anisotropy.

2. Model description

We studied a half Heusler-based spin valve nanopillar withCMS (16 nm)/Ag (4 nm)/CMS (2 nm) and the elliptical cross sectionarea of 100�76 nm2 by using micromagnetic simulations. Fig. 1shows a Cartesian coordinate system where the x-axis and y-axisare respectively the long-axis and short-axis of the ellipse. The topCMS layer is the free layer whose magnetization dynamics istriggered by a spin-polarized current. The bottom CMS layer is thereference layer with the initial magnetization P along the positivez axis. The initial magnetization vector M of the free layer is alongthe negative z axis. A middle Ag layer is to avoid the exchangecoupling between the two CMS layers. In our simulation, a positivecurrent is defined as a current flowing from the free layer to thereference layer, and the current will be polarized perpendicularlyby the magnetization of the reference layer. After the electrons gothrough the reference layer, the spin transfer torque will act on thefree layer magnetization. At a small current density, the dampingtorque will push M back towards the low energy configurationalong the negative z axis. With the increase in current density, astable magnetization precession will be produced as STT inputequals to the magnetic damping torque. At the end, the magne-tization will switch to another stable configuration along thepositive z axis if the current is large enough to overcome theenergy barrier.

The dynamics of the magnetization M are described by using ageneralized Landau–Lifshitz–Gilbert–Slonczewski (LLGS) equation [1],

dMdt

¼ �γ0M�Hef f �

αγ0

MsM� ðM�Hef f Þ

� 2μBJ

ð1þα2ÞedMs3gðM;PÞM� ðM� PÞ

þ 2μBαJ

ð1þα2ÞedMs2gðM;PÞðM� PÞ ð1Þ

where the effective field Heff could be expressed by

Hef f ¼ � 1μ0

δEδM

ð2Þ

γ0 ¼γ/(1þα2), γ is the electron gyromagnetic ratio, and α is thedimensionless damping parameter. The last two terms on the righthand side of Eq. (1) describe STT which tends to drag the magnetiza-tion away from its initial state and drives the magnetization preces-sion. μB, J, d, e, Ms, are the Bohr magneton, the current density, thethickness of the free layer, the electron charge and the saturationmagnetization, respectively. Here, we only consider the first term oftransverse torque and ignore the second term of smaller field-liketorque since it is very small in metallic alloys [20,21]. The scalarfunction g(M,P) is given by [1]

gðM;PÞ ¼ ½�4þð1þηÞ3ð3þMUP=M2s Þ=4η3=2��1 ð3Þ

where η is the spin polarization factor, the angle betweenM and P is θ.M�P/M2

s ¼cos θ. The total effective fields include magnetocrystallineanisotropy, surface anisotropy, demagnetization, external magnetic,and exchange fields, namely,

Htotal ¼HanisþHsurf þHdemþHextþHexch ð4Þ

Different with the previous model, we introduce the surface aniso-tropy field [16,17],

Hsurf ¼ Ksm2z=d ð5Þ

where Ks is the surface anisotropy constant. The expressions of otherenergies could be found in our previous publications [18–21]. Hsurf willsignificantly increase the effective field to overcome the large STTinput and produce the high power precessional oscillation.

The dynamics of magnetization is investigated by numericallysolving the time-dependent LLGS equation using the Gauss–Seidelprojection method [22] with a constant time step Δt¼0.0268858 ps [23]. We adopt the following magnetic parameters:saturation magnetization Ms¼8.0�105 A/m [24] smaller than theexperimental value for thinner thickness; exchange constantA¼2.0�10�11 J/m [25]; magnetocrystalline anisotropy constantK1¼3.0�103 J/m3 [26] and surface anisotropy constant Ks¼�1.3�106 J/m3 [16,17]. Other parameters are Gilbert dampingparameter α¼0.008 [26,27], spin polarization factor η¼0.56 [28]and electron gyromagnetic ratio γ¼2.3245�105 m/(A s). Addition-ally, the easy axis is along [1 0 0], [0 1 0] and [0 0 1] if the surfaceanisotropy is not included in the effective field. After introducingthe surface anisotropy, the easy axis is along [0 0 1].

Fig. 1. Schematic illustration of Co2MnSi (CMS)/Ag/CMS spin valve nanopillar (left). Directions of damping and spin transfer torque vectors in the free layer (right).

H.B. Huang et al. / Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎2






3. Results

3.1. Out-of-plane external magnetic field dependence

Fig. 2(a) shows the phase diagram of the normalized resistanceas a function of current and out-of-plane external magnetic fieldalong the z axis. The blue and red colors correspond to the high(antiparallel, AP) and low (parallel, P) normalized resistance,respectively. The definition of normalized resistance can beexpressed as R¼2r/(RAP�RP)�(RAPþRP)/(RAP�RP), where r, RAPand RP are the real resistance, resistances of AP and P, respectively.At zero magnetic field, the blue area is larger than the red,resulting from the asymmetric spin transfer torque of Slonczews-ki's model, i.e., the relationship of STT vs. θ is not symmetricalabout 901, and STT reaches its maximum value at the angle about1451. It causes the asymmetry of the area dependence for differentdirections of current [29]. Furthermore, the positive z axis mag-netic field favors the parallel configuration while the negative zaxis magnetic field impedes the parallel configuration. If we applythe positive z axis magnetic field, the blue area increases gradually.While the negative z axis magnetic field is applied, the blue area isdecreasing gradually. Therefore, the optimum current range forspin transfer precession can be increased or decreased dependingon the direction of external magnetic fields. As shown in Fig. 2(b),the corresponding magnetic domains at different current densities

of 0.4�107 A/cm2, 1.0�107 A/cm2 and the external magneticfields of �0:3Ms

z , 0 and 0:3Msz show the out-of-plane spatial

magnetization precession. The numbers 1, 2, 3, 4, 5, and 6 corre-spond to the numbers of Fig. 2(a). The colors represent themagnitude of the magnetization components /mzS (�1.0 to 1.0)while the arrows indicate the in-plane magnetization. The z axismagnetic field just changes the magnitude of effective field, and itdoes not change the direction of effective field. Therefore, the colorof domain distributions changes with the external magnetic fields.

Fig. 3 shows the phase diagram of the frequency as a functionof current and z axis external magnetic field. The pink and yellowcorrespond to the highest and lowest frequencies respectively.With the increase of the positive z axis magnetic field, thefrequency increases gradually (blue shift). When the negative zaxis magnetic field is applied, the frequency decreases gradually(red shift). The phenomena could be explained by the change ofthe effective field by the external magnetic field. The precessionfrequency can be estimated by [1,30–32]

f ¼ γ

2π4πMs/mzS ð6Þ

From Eq. (6), the frequency is proportional to /mzS. As shown inFig. 1, the initial effective field is along the negative z axis. Thenegative magnetic field will keep the magnetization stay in theinitial antiparallel configuration, and /mzS increases with the

Fig. 2. (a) Phase diagram showing the normalized resistance as a function of applied current and out-of-plane external magnetic field along the z axis. (b) Snapshot ofmagnetic domains at different current and magnetic fields. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 3. (a) Phase diagram showing the frequency as a function of applied current and external magnetic field along the z axis. (b) Power spectral density (PSD) as a function ofcurrent densities and the constant external magnetic fields. For clarity, spectra are shifted vertically with different constants. (For interpretation of the references to color inthis figure, the reader is referred to the web version of this article.)

H.B. Huang et al. / Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3






external magnetic field. The positive magnetic field will induce themagnetization to switch to the parallel configuration, and /mzSdecreases with the external magnetic field. Therefore, the blue orred shifts can be obtained with the external magnetic field alongnegative or positive z axis, which are consistent with the experi-ment and macrospin modeling in the metallic spin valve [3,30,32].Fig. 3(b) shows the corresponding power spectrum densities (PSD)at different current densities of 0.4�107 A/cm2, 1.0�107 A/cm2

and the z axis external magnetic fields of �0:3Msz , 0 and 0:3Ms

z .

For clarity, spectra are shifted vertically with different constants.The numbers 1, 2, 3, 4, 5, and 6 correspond to the numbers of Fig. 3(a). For the points 1, 2, and 3, they are under the same currentdensity. The maximum of the output power depends on theexternal magnetic field, which will be discussed in the following.

As shown in Fig. 4, PSD as a function of current density andmagnetic fields indicate that the maximum output power appearsat a specific current density. The colors represent the magnitude ofoutput power. Comparing with the output power without the

Fig. 4. (a and b) Phase diagrams of PSD showing the frequency as a function of applied current at z axis external magnetic field. The colors represent the magnitude of outputpower. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. (a and c) Phase diagrams showing the frequency as a function of applied current and external magnetic fields along the x and y axes. (b and d) Snapshot of magneticdomains at different current densities and magnetic fields. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)







magnetic field [33], the maximum power can be gained at smallercurrent density if a positive z axis magnetic field is applied. Due tothe smaller current density, the frequency also decreases at themaximum output power. For example, the power output max-imum is achieved at the specific current density of 4.0�106 A/cm2

at the frequency 21 GHz under 0:3Msz . For the negative z axis

magnetic fields, the maximum power can be gained at a largercurrent density. Due to larger current density, the frequencyalso increases at the point of maximum power. For example, themaximum power can be obtained at the current density of9.5�106 A/cm2 at frequency 43 GHz under �0:3Ms

z . Since theout-of-plane external magnetic field suppresses the magnetizationoscillation, the magnitudes of output power decrease from 12.0 to4.0 after applying the z axis magnetic fields of �0:3Ms

z and0:3Ms

z. One of the interesting features of relationships betweenthe frequency and output power vs. magnetic fields is that thenegative or positive z axis magnetic field has different influence onthe frequency but the same influence on the output power.Furthermore, the multiple local maximums of PSD result fromthe non-uniform magnetization distribution (vortex domain) inthe external magnetic field as shown in Fig. 2(b). The vortexdomain leads to the decrease in output power, and the vortexdomain is moving driven by both the current and the magneticfield. As we discussed it previously, a positive z magnetic field willinduce the magnetization to switch to the parallel configuration.Under a positive magnetic field, the stable magnetization preces-sion is obtained at a smaller current density. Therefore, the widerange of PSD at the lower current density is obtained under apositive magnetic field.

3.2. In-plane external magnetic field dependence

Fig. 5(a) and (c) shows the phase diagrams of frequency as afunction of current and in-plane external magnetic fields along thex and y axes. The pink and yellow correspond to the highest andlowest frequency respectively. With the increase of the magneticfields along x or y axis, the frequency decreases in a parabolic way(red shift). The reason for the parabolic dependence is due to thefact that the in-plane magnetic fields pull the magnetization backinto the plane, thus decrease the magnetization component /mzSwhich is proportional to the frequency according to Eq. (6). Ourmicromagnetic simulation results of frequency vs. in-plane exter-nal magnetic fields are consistent with both of experiment [30]and macrospin model [31,32] in the metallic nanopillar.

As shown in Fig. 5(b) and (d), the corresponding magneticdomains at different current densities of 0.4�107 A/cm2 and1.0�107 A/cm2 and the external magnetic fields of �0:3Ms

x,0:3Ms

x, �0:3Msy and 0:3Ms

y are labeled with the numbers from1 to 8 which correspond to the numbers in Fig. 5(a) and (c). Thecolors represent the magnitude of the magnetization components/mzS (�1.0 to 1.0) while the arrows indicate the in-planemagnetization. Since the in-plane magnetic fields make theeffective field tilt into the plane, the spatial magnetizations ofthe domains will force to the field directions. The non-uniformmagnetization precession leads to the decrease of output powerunder the influence of the in-plane magnetic fields. Therefore,we observe that the output power under the in-plane magneticfields is smaller than the power under the out-of-plane magneticfields.

Fig. 6. (a–d) Phase diagrams of PSD showing the frequency as a function of applied current at x and y axes external magnetic fields. The colors represent the magnitude ofoutput power. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

H.B. Huang et al. / Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 5






As shown in Fig. 6, PSD as a function of current density underdifferent in-plane magnetic fields indicate that the maximum outputpower appears at the specific current density. The colors representthe magnitude of output power. The frequency is also increasingwith the current, but the magnitudes of output power are signifi-cantly decreased compared with the power of our previous simula-tion without the magnetic field [33]. This phenomenon can beexplained by the non-uniform magnetization precession induced bythe in-plane magnetic fields.

4. Conclusions

In summary, we investigated the high-power STT with perpen-dicular polarization in a half-metallic Heusler alloy CMS spin-valvenanopillars using micromagnetic simulations. We obtained thehigh output power spin transfer precession at a wide range ofcurrent density by introducing the surface anisotropy under zeroexternal magnetic field and with a high spin polarization constant.The phase diagrams of the frequency as a function of currentindicate the linear increase, however the linear decrease with theout-of-plane magnetic field can be obtained. Especially, the in-plane magnetic field dependence of frequency shows a paraboliccurve which is explained by the magnetization trajectory tilting.Furthermore, we also discussed the decrease of output powerusing the excitation of non-uniform magnetization precession inthe in-plane magnetic fields. The influence of external magneticfield may give the guidance for designing the Heusler-based spin–torque oscillator in improving the output power and adjusting thefrequency in future.

Acknowledgments

This work was sponsored by the US National Science Founda-tion under the grant number DMR-1006541 (Huang and Chen),and the National Natural Science Foundation of China (11174030).The computer simulations were carried out on the LION andCyberstar clusters at the Pennsylvania State University.

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