Scholarly article on topic 'Oscillation characteristics of zero-field spin transfer oscillators with field-like torque'

Oscillation characteristics of zero-field spin transfer oscillators with field-like torque Academic research paper on "Physical sciences"

CC BY
0
0
Share paper
Academic journal
AIP Advances
OECD Field of science
Keywords
{""}

Academic research paper on topic "Oscillation characteristics of zero-field spin transfer oscillators with field-like torque"

Oscillation characteristics of zero-field spin transfer oscillators with field-like torque

Yuan-Yuan Guo, Hai-Bin Xue, and Zhe-Jie Liu

Citation: AIP Advances 5, 057114 (2015); doi: 10.1063/1.4920941 View online: http://dx.doi.Org/10.1063/1.4920941

View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/5/5?ver=pdfcov Published by the AIP Publishing

Articles you may be interested in

Large amplitude oscillation of magnetization in spin-torque oscillator stabilized by field-like torque J. Appl. Phys. 117, 17C504 (2015); 10.1063/1.4907696

Current-induced switching of magnetic tunnel junctions: Effects of field-like spin-transfer torque, pinned-layer

magnetization orientation, and temperature

Appl. Phys. Lett. 104, 022413 (2014); 10.1063/1.4862182

Intrinsic frequency doubling in a magnetic tunnel junction-based spin torque oscillator J. Appl. Phys. 110, 076102 (2011); 10.1063/1.3647759

Direct experimental measurement of phase-amplitude coupling in spin torque oscillators Appl. Phys. Lett. 97, 032502 (2010); 10.1063/1.3467043

Measurement of nonlinear frequency shift coefficient in spin-torque oscillators based on MgO tunnel junctions Appl. Phys. Lett. 95, 022507 (2009); 10.1063/1.3176939

(■) CrossMark

VHi «-dick for updates

Oscillation characteristics of zero-field spin transfer oscillators with field-like torque

Yuan-Yuan Guo,1,2 Hai-Bin Xue,1,2,a and Zhe-Jie Liu3,b

1Key Laboratory of Advanced Transducer and Intelligent Control system, Ministry of Education, Taiyuan University of Technology, Taiyuan 030024, China 2Department of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, China

3Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576

(Received 12 February 2015; accepted 29 April 2015; published online 6 May 2015)

We theoretically investigate the influence of the field-like spin torque term on the oscillation characteristics of spin transfer oscillators, which are based on MgO magnetic tunnel junctions (MTJs) consisting of a perpendicular magnetized free layer and an in-plane magnetized pinned layer. It is demonstrated that the field-like torque has a strong impact on the steady-state precession current region and the oscillation frequency. In particular, the steady-state precession can occur at zero applied magnetic field when the ratio between the field-like torque and the spin transfer torque takes up a negative value. In addition, the dependence of the oscillation properties on the junction sizes has also been analyzed. The results indicate that this compact structure of spin transfer oscillator without the applied magnetic field is practicable under certain conditions, and it may be a promising configuration for the new generation of on-chip oscillators. © 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/L4920941]

I. INTRODUCTION

Since the initial theoretical predictions1,2 and subsequent experimental observations3-6 of the magnetization dynamics driven by spin transfer torque (STT), there has aroused tremendous and continuous research interest.7-12 The phenomenon of the spin transfer can be used to switch the magnetization from one static configuration to another one4,5 or produce a steady precession of magneti-zation.6-8 In particular, this steady-state precession can generate an AC voltage oscillation with the same frequency, that known as spin-torque oscillators (STOs). The STOs have high frequency microwave, large tuning range via applied field or current,13 high modulation rates,14 small footprints, and the same compatibility with the standard complementary metal-oxide-semiconductor process. Thus, it can be used as the frequency-tunable transmitters and receivers for wireless communication pur-poses,15 nanoscale dynamic magnetic field sensors16 and recording heads of high-density hard disk drives.17,18 However, the necessity of an applied magnetic field has severally limited the STOs' potential applications in microwave generation and other fields. Recently, various solutions in the zero magnetic field case have been suggested to produce the steady-state precession, namely, STO with a perpendicularly magnetized pinned layer,19,20 magnetic vortex oscillators,21,22 a tilted magnetization of the fixed layer respect to the film plane,23 and a synthetic ferromagnetic free layer.24

Recently, the discovery of interfacial perpendicular anisotropy (IPA) between the ferromagnetic electrodes and the tunnel barrier of MTJs25-27 enabled the realization of a STO with a perpendicularly magnetized free layer (PMF) and an in-plane magnetized pinned layer, which may be referred to

aElectronic address: xuehaibin@tyut.edu.cn bElectronic address: pandanlzj@hotmail.com

2158-3226/2015/5(5)/057114/7

5, 057114-1

© Author(s) 2015

FIG. 1. Device layout: the system consists of an in-plane magnetized pinned layer (PL) and a perpendicularly magnetized free layer (FL). The positive current indicates that electrons flow from the free layer to the pinned layer. The field is applied perpendicular to the film planes.

as PMF-STO. This structure was able to exhibit a reduced threshold current, a high power and a high Q factor.28,29 It has been theoretically demonstrated that this configuration is capable of generating microwave signal in zero applied magnetic field by taking into account the field-like torque.30 However, the dependence of oscillation behavior on the field-like torque still remains unclear.

In this work, we have studied the spintronic device architecture with a perpendicularly magnetized free layer (PMF) and an in-plane magnetized pinned layer. We report on the oscillation characteristic of spin torque oscillators as a function of field-like torque when the ratio between the field-like torque and the spin transfer torque ¡3 takes up a negative value without the applied magnetic field. The critical current for onset of the STO precession stays almost constant with increasing the absolute value of 3, while the frequency is observed to be increased. In particular, the working current window has a non-monotonic dependence on the value of 3, with a maximum range about (-0.19, -0.29). In addition, the dependence of the oscillation behavior on the junction sizes has also been analyzed; it was found that the critical current Jc and the corresponding oscillation frequency decrease with increasing junction surface area.

II. MODEL DESCRIPTION

The schematic structure of the PMF-STO is illustrated in Fig. 1. The units vectors m and P denote the magnetization directions of the free and pinned layers, respectively. The z-axis is normal to the film plane, and the x-axis is parallel to P . The current I flows uniformly along the z-axis, where the positive current corresponds to electrons flow from the free layer to the pinned layer and the external applied field Happi is applied along the z-axis. In the framework of the macrospin model, the time evolution of the free-layer magnetization m is found by solving the Landau - Lifshitz - Gilbert -Slonczewski (LLGS) equation which also includes the field-like torque,

d m _^ _^ _^ d m _^ _^ _^ _^ _^

-= -y m x Heff + a m x-+ yP(I) m x ( m x p) - yBP{I) m x p (1)

Here y is the gyromagnetic ratio, a is the Gilbert damping parameter, and H eff is the effective magnetic field, which the applied field Happi, demagnetizing field Hdemag, crystalline uniaxial anisotropy Hanis and shape anisotropy Hsha with Happl = (0, 0, Happl), Hanis = (0, 0, Hk±mz), Hdemag = (0,0, -Hdmz), and Hsha = (Hkmx, 0,0). Because the pinned layer is taken to be in the film plane, an asymmetric angle dependence of the spin torque is essential for dynamics to be induced within the single-domain model. The coefficient P(I) is defined as31-33

Pd) = -1 ^ ^ (2)

2eMsSd[1 + A(m ■ p)]

where S and d are the cross-sectional area and the thickness of the free layer, respectively. Ms is the saturation magnetization of the FL, I is the current, e is the elementary charge. Two dimensionless parameters n and A determine the magnitude of the spin polarization of the injected current and the dependence of the spin torque strength on the relative angle of the magnetizations, respectively. The dimensionless parameter 33 represents the ratio of the field-like torque against the spin torque.

Equation (1) can be expressed as the following set of differential equations in Cartesian coordinates:

r^m1 = -[myHa + mymz(Hk± - Hd)] - a[-Hkmxmy2 - Hkmxmz2 + Hamxmz + (Hk± - Hd)mxmz2]

+(1 - a/3)P(I)(-my2 - m2) (3)

dmy 2 2

T~dT = -[Hk mx mz - Hamx - (Hk ± - Hd)m x mz ] - a[Ham y mz + (Hk± - Hd)m y mz + Hk mx my]

+(1 - a/3)P(I)mxmy - (a + ¡3)P(I)mz (4)

rdmmz = Hkmxmy - a[Hkmx2mz - Ha(1 - mz2) - (Hk± - Hd)mz(1 - mz2)]

+(1 - a/3)P(I)mxmz + (a + 3)P(I)my (5)

with r = (1 + a2)/y.

In order to determine whether the STO is oscillating, a so-called current-field phase diagram is constructed.34-37 The diagram shows the value of mxrms = J1 x £ (mxi - mx a „)2,37 which can be

Vn i=1

used to distinguish between the static states and steady precessional regimes, as a function of applied current and magnetic field after the STO considered reaches a steady state. Here, mxi is the value of mx at i-th time-step, mx,a v is the average value of mx for a "waiting time" t using n time steps of length At.38 It is important to note that mx,rms = 0 corresponds to the static states regime; while mx,rms > 0 the steady precessional regime.36

III. RESULTS AND DISCUSSIONS

We initially focus on a free layer with a radius of 60nm and thickness of 2nm. The other parameters of the STO are chosen as: 4nMs = 18.2 kOe, Hk± = 18.6 kOe, n = 0.54, A = n2, Y = 17.32 MHz / Oe and a = 0.005. The anisotropy fields Hd and Hk can be expressed as a function of demagnetizing factors Nx, Ny, Nz, which are associated with the shape of the nanomagnet with Hd = Ms x (Nz - Ny) and Hk = Ms x (Nx - Ny).Forthe finite-sized free layer considered here, thecompo-nents of N are calculated in the following simulations as Nx = Ny = 0.0256 and Nz = 0.9488, and this leads to Hd = 16.8 kOe, Hk = 0 kOe.

Figure 2 displays a comparison between the current-field phase diagrams for different values of 3, i.e. ¡3 = 0, 0.1, - 0.1, - 0.2, respectively. From different values of 33, it can be seen that precessional states (colored triangular regions) can only be realized at positive current polarities for both positive and negative applied fields. This is a direct result of the spin transfer torque asymmetry parameter A. But the oscillation is independent of the sign of Happl because the system has inversion symmetry about the xy-plane. The critical current for the onset of the STO precession increases with the increasing applied field. For the case of 3 = 0, this configuration required a finite field to obtain precession, as shown in Fig. 2(a). For positive 33, as shown in Fig. 2(b), we observe a larger applied field is required for steady-state precession. On the other hand, for negative 33, stable self-oscillation can be realized for negative 33 at zero applied magnetic field, as shown in Figs. 2(c) and 2(d). This result coheres with Ref. 30. The results shown in Fig. 2 indicate that the field-like torque plays a key role in this STO. When taking into account the field-like torque, it breaks the energy balance. For 33 > 0, the energy supplied by the spin torque is enhanced by the field-like torque compared to that for 3 = 0. For 33 < 0, the energy supplied by the spin is suppressed by the field-like torque. The energy supply

FIG. 2. Current-field phase diagrams obtained by evaluating mx, rms over the length of the time trace for (a) p = 0, (b) P = 0.1,(c) p = -0.1,(d) p = -0.2.

FIG. 3. (a) The phase diagram of the value of mx,„„s as a function of the applied current and negative ß at zero applied magnetic field, and (b) the oscillation frequencies of STO on the applied current for various negative value of ß = -0.1, -0.2, -0.5.

from the spin torque balances with the energy dissipation due to the damping above the film-plane, and therefore, the stable self-oscillation of the magnetization can be realized.

From Figs. 2(c) and 2(d), we can observe that this device enters a state of self-oscillation at zero applied magnetic field for ^ = -0.1, -0.2. Then, we extend the jS range from 0 to -0.5, the phase diagram of the value of mx, rms as a function of the applied current and negative jS at zero applied magnetic field is shown in Fig. 3(a). The critical current for onset of the STO precession stays almost working current window has a non-monotonic dependence on the value of jS, with a maximum around € (-0.19, -0.29). Figure 3(b) shows the oscillation frequencies for various values of jS. The oscillation frequency for ^ = -0.1 is low compared to that for ^ = -0.5 because the energy supplied by the spin torque decreases as the absolute value of jS increase. When the current above the critical current the oscillation frequency monotonically decreases with increasing current magnitude for = -0.1, whereas for ^ = -0.3 or ^ = -0.5 the oscillation frequency decreases initially with

Frequency (GHz)

FIG. 4. (a) Power spectral density with the increase of current, and (b) Magnetization trajectories under different current.

Radius (nm) Radius (nm)

FIG. 5. Comparison between the current-field phase diagram for different radii of the free layer (a) R = 40 and (b) 80 nm, and (c) critical current Jc and (d)oscillation frequency as functions of radius of the free layer.

the increasing of the current, before increasing. This is because the field-like torque gives rise to an effective field 33Hs p along the x-axis, and increases with an increasing absolute value of 33 as well as applied current. If the precession amplitude is made larger by increasing the field-like torque, the net magnetization of the free layer becomes smaller, thereby leading to a larger total magnetic field and a higher oscillation frequency. These behaviors are exactly what were predicted in the theory described in Ref. 39. Figure 4(a) shows the power spectral density with the increase of current at zero applied magnetic field for 33 = -0.1. For clarity, all the spectra are shifted vertically by different constants. Both the amplitude and peak frequency varied significantly with the current. The magnetization trajectories for several values of current are given in Fig. 4(b). At the lowest current, the free layer undergoes roughly circular precession about the axis perpendicular to the film plane and expands as the current increases (compare trajectories at 1.2 and 1.4 mA). At large currents the oscillation axis moves towards the xy-plane (trajectories at 2.2 and 3.7mA) and for values of I > 6 mA the AP-state is obtained.

The output characteristic, such as frequency and critical current, can be tuned by varying the surface area of MTJs. Figures 5(a) and 5(b) display a comparison between the current-field phase diagrams for the two different radii of the free layer R = 40 and 80 nm. It can be seen that the effect of the junction sizes has a non-trivial influence on the dynamic properties of the STOs. Figures 5(c) and 5(d) illustrate the critical current Jc and oscillation frequency at Jc variations for different radii of the free layer. It can be observed that the critical current Jc and the corresponding oscillation frequency decrease with increasing junction surface area. This is due to the increase in demagnetizing factor Nz for an enlarged junction size, thereby leading to a change in the effective magnetic field.

IV. CONCLUSION

In summary, we have calculated the current-field phase diagrams for a spin-transfer-driven oscillator consisting of an in-plane polarizer layer and an out-of-plane free layer using a macrospin model. By introducing the field-like torque, it is shown that a steady-state precession in the absence of an applied magnetic field can be achieved. Moreover, the field-like torque offers an additional degree of freedom for tuning the frequency response of the device. In particular, the junction size plays an important role in determining the oscillation properties of the STOs. The results also indicate that a higher current density is needed with smaller lateral dimensions.

ACKNOWLEDGMENTS

This work was supported by the Nations Science Foundation of China (Nos. 11204203 and 61274089), and International Technology Collaboration Program of Shanxi Province (No. 201481029-2).

1 J. C. Slonczewski, "Current-driven excitation of magnetic multilayers," J. Magn. Magn. Mater. 159, L1 (1996).

2 L. Berger, "Emission of spin waves by a magnetic multilayer traversed by a current," Phys. Rev. B 54, 9353 (1996).

3 M. Tsoi, A. G. Jansen, J. Bass, W. -C. Chiang, M. Seck, V. T. Tsoi, and P. Wyder, "Excitation of a magnetic multilayer by an electric current," Phys. Rev. Lett. 80, 4281 (1998).

4 E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, "Current-induced switching of domains in magnetic multilayer devices," Science 285, 867 (1999).

5 J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, "Current-driven magnetization reversal and spin wave excitations in Co/Cu/Co pillar," Phys. Rev. Lett. 84, 3149 (2000).

6 M. Tsoi, A. G. M. Jansen, J. Bass, W. -C. Chiang, V. Tsoi, and P. Wyder, "Generation and detection of phase-coherent current-driven magnons in magnetic multilayers," Nature (London) 406, 46 (2000).

7 S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, "Microwave oscillations of a nanomagnet driven by a spin-polarized current," Nature (London) 425, 380 (2003).

8 W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, andT. J. Silva, "Direct-Current induced dynamics in Co90Fe10/Ni80Fe20 point contacts," Phys. Rev. Lett. 92, 027201 (2004).

9 O. Boulle, V. Cros, J. Grollier, L. G. Pereira, C. Deranlot, F. Petroff, G. Faini, J. Barna, and A. Fert, "Shaped angular dependence of the spin-transfer torque and microwave generation without magnetic field," Nature Phys. 3, 492 (2007).

10 V. S. Pribiag, I. N. Krivorotov, G. D. Fuchs, P. M. Braganca, O. Ozatay, J. C. Sankey, D. C. Ralph, and R. A. Buhrman, "Magnetic vortex oscillator driven by d. c. spin-polarized current," Nature Phys. 3, 498 (2007).

11 J. H. Chang, H. H. Chen, C. R. Chang, and Y. W. Liu, "Phase locking of dynamical modes in a nanomagnetic oscillator with a circularly spin-polarized current," Phys. Rev. B 84, 054457 (2011).

12 M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti, G. Gubbiotti, F. B. Mancoff, M. A. Yar, and J. Akerman, "Direct observation of a propagating spin wave induced by spin-transfer torque," Nature Nanotech. 6, 635 (2011).

13 S. Bonetti, P. Muduli, F. Mancoff, and J. Akernan, "Spin torque oscillator frequency versus magnetic field angle: The prospect of operation beyond 65 GHz," Appl. Phys. Lett. 94, 102507 (2009).

14 M. R. Pufall, W. H. Rippard, S. Kaka, T. J. Silva, and S. E. Russek, "Frequency modulation of spin-transfer oscillators," Appl. Phys. Lett. 86, 082506 (2005).

15 H. S. Choi, S. Y. Kang, S. J. Cho, I. Y. Oh, M. Shin, H. Park, C. Jang, B. C. Min, S. Y. Park, and C. S. Park, "Spin nano-oscillator-based wireless communication," Sci. Rep. 4, 5486 (2014).

16 P. M. Braganca, B. A. Gurney, B. A. Wilson, J. A. Katine, S. Maat, and J. R. Childress, "Nanoscale magnetic field detection using a spin torque oscillator," Nanotechnology 21, 235202 (2010).

17 J. G. Zhu, X. Zhu, and Y. Tang, "Microwave assisted magnetic recording," IEEE Trans. Magn. 44, 125 (2008).

18 K. Kudo, T. Nagasawa, K. Mizushima, H. Suto, and R. Sato, "Numerical Simulation on temporal response of spin-torque oscillator to magnetic pulses," Appl. Phys. Express 3, 043002 (2010).

19 D. Houssameddine, U. Ebels, B Delaet, B. Rodmacq, I. Rirastrau, F. Ponthenier, M. Brunet, C. Thirion, J. -P. Michel, L. Perjbeanu-Buda, M. -C. Cyrille, O. Redon, and B. Dieny, "Spin-torque oscillator using a perpendicular polarizer and a planar free layer," Nature Mater. 6, 447 (2007).

20 H. Zhang, Z. W. Hou, J. W. Zhang, Z. Z. Zhang, and Y. W. Liu, "Precession frequency and fast switching dependence on the in-plane and out-of-plane dual spin-torque polarizers," Appl. Phys. Lett. 100, 142409 (2012).

21 A. Dussaux, B. Georges, J. Grollier, V. Cros, A. V. Khvalkovskiy, A. Fukushima, M. Konoto, H. Kubota, K. Yakushiji, S. Yuasa, K. A. Zvezdin, K. Ando, and A. Fert, "Large microwave generation from current-driven magnetic vortex oscillators in magnetic tunnel junctions," Nature Comm. 1, 8 (2010).

22 A. Dussaux, E. Grimaldi, B. Rache Salles, A. S. Jenkins, A. V. Khvalkovskiy, P. Bortolotti, J. Grollier, H. Kubota, A. Fukushima, K Yakushiji, S. Yuasa, V. Cros, and A. Fert, "Largr amplitude spin torque vortex oscillations at zero external field using a perpendicular spin polarizer," Appl. Phys, Lett. 105, 022404 (2014).

23 Y. zhou, H. Zhang, Y. W. Liu, and J. Akerman, "Macrospin and micromagnetic studies of tilted polarizer spin-torque nano-oscillators," J. Appl. Phys. 112, 063903 (2012).

24 C. Klein, C. Petitjean, and X. Waintal, "Interplay between nonequilibrium and equilibrium spin torque using synthetic ferrimagnets," Phys. Rev. Lett. 108, 086601 (2012).

25 S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayalawa, F. Matsukura, and H. Ohno, "A perpendicular-anisotropy CoFeB-MgO magnetic tunnel junction," Nature Mater. 9, 721 (2010).

26 H. Kubota, S. Ishibashi, T. Saruya, T. Nozaki, A. Fukushima, K. Yakushiji, K. Ando, Y. Suzuki, and S. Yuasa, "Enhancement of perpendicular magnetic anisotropy in FeB free layers using a thin MgO cap layer," J. Appl. Phys. 111, 07C723 (2012).

27 Z. Zeng, P. K. Amiri, I. N. Krivorotov, H. Zhao, G. Finocchio, J. P. Wang, J. A. Katine, Y. Huai, J. Langer, K. Galatsis, K. L. Wang, and H. Jiang, "High-power coherent microwave emission from magnetic tunnel junction nano-oscillators with perpendicular anisotropy," ACS Nano. 6, 6115 (2012).

28 H. Kubota, K. Yakushiji, A. Fukushima, S. Tamaru, M. Konoto, T. Nozaki, S. Ishibashi, T. Saruya, S. Yakata, T. Taniguchi, H. Arai, and H. Imamura, "Spin-torque oscillator based on magnetic tunnel junction with a perpendicularly magnetized free layer and in-plane magnetized polarizer," Appl. Phys. Express 6, 103003 (2013).

29 Z. Zeng, G. Finocchio, B. Zhang, P. K. Amiri, J. A. Katine, I. N. Krivorotov, Y. Huai, J. Langer, B. Azzerboni, K. L. Wang, and H. Jiang, "Ultralow-current-density and bias-field-free spin-transfer nano-oscillator," Sci. Rep. 3, 1426 (2013).

30 T. Taniguchi, S. Tsunegi, H. Kubota, and H. Imamura, "Self-oscillation in spin torque oscillator stabilized by field-like torque," Appl. Phys. Lett. 104, 152411 (2014).

31 J. C. Slonczewski, "Current, torques, and polarization factors in magnetic tunnel junctions," Phys. Rev. B 71,024411 (2005).

32 J. C. Slonczewski and J. Z. Sun, "Theory of voltage-driven currrent and torque in magnetic tunnel junctions," J. Magn. Magn. Mater. 310, 169 (2007).

33 T. Taniguchi, H. Arai, S. Tsunegi, S. Tamaru, H. Kubota, and H. Imamura, "Critical field of spin torque oscillator with perpendicularly magnetized free layer," Appl. Phys. Express 6, 123003 (2013).

34 S. E. Russek, S. Kaka, W. H. Rippard, M. R. Pufall, and T. J. Silva, "Finite-temperature modeling of nanoscale spin-transfer oscillators," Phys. Rev. B 71, 104425 (2005).

35 O. Boulle, V. Cros, J. Grollier, L. G. Pereira, C. Deranlot, and F. Petroff, "Microwave excitations associated with a wavy angular dependence of the spin transfer torque: Model and experiments," Phys. Rev. B 77, 174403 (2008).

36 P. Bala, M. Gmitraz, and J. Barnas, "Current-induced dynamics in noncollinear dual spin values," Phys. Rev. B 80, 174404 (2009).

37 C. Fowley, V. Sluka, K. Bernert, J. Lindner, J. Fassbender, W. H. Rippard, M. R. Pufall, S. E. Russek, and A. M. Deac, "Zero-field spin-transfer oscillators combining in-plane and out-of-plane magnetized layers," Appl. Phys. Express 7,043001 (2014).

38 J. Xiao and A. Zangwill, "Macrospin models of spin transfer dynamics," Phys. Rev. B 72, 014446 (2005).

39 S. Tamaru, H. Kubota, K. Yakushiji, T. Nozaki, M. Konoto, A. Fukushima, H. Imamura, T. Taniguchi, H. Arai, T. Yamagi, and S. Yuasa, "Bias field angle dependence of the self-oscillation of spin torque oscillators having a perpendicularly magnetized free layer and in-plane magnetized reference layer," Appl. Phys. Express 7, 063005 (2014).