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The spin-torque nano-oscillator (STNO), which is a novel type of nano-sized microwave oscillator driven by direct current, is considered as a promising candidate for future radio frequency (RF) transceivers owing to its scalability, nanoscale size and high frequency tunability. However, the potential application of STNO is limited because its stable oscillation requires an external magnetic field. In this work, the influences of the field-like torque and applied current intensity on the stable oscillation of STNO with a perpendicularly magnetized free layer are studied theoretically based on the macrospin model (also known as the single-spin or single-domain model) and the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation in the absence of magnetic field. It is demonstrated numerically that a stable oscillation of STNO can be observed when the ratio between the field-like torque and the spin torque is a negative value and larger than a certain value that depends on the damping coefficient and the current intensity, whose physical mechanism can be understood by the energy balance equation. Moreover, the frequency of stable oscillation of STNO can be modulated by the ratio between the field-like torque and the spin torque and also by the current intensity. Particularly, the larger the absolute value of the ratio between the field-like torque and the spin torque and the smaller the applied current intensity (above the critical current intensity), the more conducive it is to suppressing the formation of second and third oscillation frequencies, thereby enhancing the STNO’s “single-frequency” feature. Our findings provide a theoretical scheme for realizing a frequency tunable zero-field STNO, which may be useful for designing future RF transceivers.
[1] Slonczewski J C 1996 J. Magn. Magn. Mater. 159 L1
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Google Scholar
[3] Julliere M 1975 Phys. Lett. A 54 225
Google Scholar
[4] Miyazaki T, Tezuka N 1995 J. Magn. Magn. Mater. 139 L231
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Google Scholar
[6] Kiselev S I, Sankey J C, Krivorotov I N, Emley N C, Schoelkopf R J, Buhrman R A, Ralph D C 2003 Nature 425 380
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Fang B, Zeng Z M 2014 Chin. Sci. Bull. 59 1804
Google Scholar
[13] Taniguchi T, Arai H, Tsunegi S, Tamaru S, Kubota H, Imamura H 2013 Appl. Phys. Express 6 123003
Google Scholar
[14] Awad A A, Houshang A, Zahedinejad M, Khymyn R, Åkerman J 2020 Appl. Phys. Lett. 116 232401
Google Scholar
[15] Arun R, Gopal R, Chandrasekar V K, Lakshmanan M 2022 J. Phys. Condens. Matter 34 125803
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[16] Taniguchi T, Tsunegi S, Kubota H, Imamura H 2014 Appl. Phys. Lett. 104 152411
Google Scholar
[17] Taniguchi T, Arai H, Kubota H, Imamura H 2014 IEEE Trans. Magn. 50 1400404
Google Scholar
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Google Scholar
[20] Arai H, Matsumoto R, Yuasa S, Imamura H 2015 Appl. Phys. Express 8 083005
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[21] Arai H, Matsumoto R, Yuasa S, Imamura H 2016 J. Phys. Soc. Jpn. 85 063802
Google Scholar
[22] Guo Y Y, Zhao F F, Xue H B, Liu Z J 2016 Chin. Phys. Lett. 33 037501
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[41] Zeng L, Liu Y, Chen H H, Zhou Y, Zhang D M, Zhang Y G, Zhao W S 2020 Nanotechnology 31 375205
Google Scholar
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图 1 由三个圆形薄膜组成的自旋转移力矩纳米振荡器的示意图. 该结构上面为厚度d和截面半径r分别为2 nm和60 nm的自由层, 且其磁矩垂直于膜面; 中间为1 nm厚度非磁性材料形成的隧道势垒层; 下面为磁矩平行于膜面的极化层. 在笛卡尔坐标系中, x-y平面和z轴分别平行和垂直于膜面, 而在球坐标系中
$\varphi $ 和θ分别为自由层磁矩的方位角和极角Figure 1. Schematic diagram for the considered spin-transfer torque nano-oscillator consisting of the trilayer circular thin films. Here, the top, middle and bottom layers are the 2 nm-thick perpendicular magnetized free layer with a radius of 60 nm, the 1 nm-thick tunnel barrier layer formed by non-magnetic material and the in-plane polarizer pinned layer, respectively. Moreover, in the Cartesian coordinate system, the x-y plane and z-axes of are parallel and perpendicular to the free layer, respectively; while the
$\varphi $ and θ in the spherical coordinate system are the azimuth and polar angles of the magnetization of the free layer, respectively.
图 2 自由层磁矩
$x$ 轴分量${m_x}$ 的均方根${m_{x{\text{-rms}}}}$ 在不同阻尼系数下随β和I变化的相图 (a)$\alpha = 0.003$ ; (b)$\alpha = 0.005$ ; (c)$\alpha = 0.007$ ; (d)$\alpha = 0.009$ Figure 2. Phase diagrams of the root-mean-square value of
${m_{x{\text{-rms}}}}$ of the${m_x}$ component of the free layer as a function of β and I with different Gilbert damping constants: (a)$\alpha = 0.003$ ; (b)$\alpha = 0.005$ ; (c)$\alpha = 0.007$ ; (d)$\alpha = 0.009$ 
图 3 当电流为1.5 mA时, 在一个周期内, 自旋转移力矩提供的平均能量
${\overline W_{\text{s}}}$ , 阻尼消耗的平均能量${\overline W_\alpha }$ , 以及其平均能量之差${\overline W_{\text{s}}} - {\overline W_\alpha }$ 随自由层磁矩z分量的变化 (a) β = 0; (b) β = –0.2; (c) β = 0.2Figure 3. In a precession period, the average work done by the spin-transfer torque
${\overline W_{\text{s}}}$ , the average energy dissipation due to the damping${\overline W_\alpha }$ , and the energy difference between${\overline W_{\text{s}}}$ and${\overline W_\alpha }$ as a function of the z component of the free layer magnetization at given current intensity I = 1.5 Ma: (a) β = 0; (b) β = –0.2; (c) β = 0.2.
图 4 (a) 自旋转移力矩纳米振荡器的振幅和频率对β的依赖关系; (b), (c) 自由层磁矩的x分量和z分量在不同类场矩β情形下的稳态振荡情况(I = 1.5 mA)
Figure 4. (a) Dependences of the amplitude and frequency of spin-transfer torque nano-oscillator on the β; (b), (c) the x and the z components of the free layer magnetization as a function of time for the different ratios between the spin-tranfer torque and the field-like torque β (I = 1.5 mA).
图 5 α = 0.005时, 自旋转移力矩纳米振荡器稳定自激振荡的频率(a)和振幅(b)随β和I变化的相图
Figure 5. Phase diagrams of the frequency (a) and amplitude (b) of stable oscillation of spin-transfer torque nano-oscillator as a function of the β and the I at α = 0.005.
图 6 (a)自旋转移力矩纳米振荡器的振幅和频率对施加电流强度I的依赖关系; (b), (c) 自由层磁矩的x分量和z分量在不同电流强度情形下的稳态振荡情况(β = –0.2)
Figure 6. (a) Dependences of the amplitude and frequency of spin-transfer torque nano-oscillator on the applied current intensity I; (b), (c) the x and the z components of the free layer magnetization as a function of time for the different values of current intensity (β = –0.2).
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[1] Slonczewski J C 1996 J. Magn. Magn. Mater. 159 L1
Google Scholar
[2] Berger L 1996 Phys. Rev. B 54 9353
Google Scholar
[3] Julliere M 1975 Phys. Lett. A 54 225
Google Scholar
[4] Miyazaki T, Tezuka N 1995 J. Magn. Magn. Mater. 139 L231
Google Scholar
[5] Moodera J S, Kinder L R, Wong T M, Meservey R 1995 Phys. Rev. Lett. 74 3273
Google Scholar
[6] Kiselev S I, Sankey J C, Krivorotov I N, Emley N C, Schoelkopf R J, Buhrman R A, Ralph D C 2003 Nature 425 380
Google Scholar
[7] Houssameddine D, Ebels U, Delaet B, Rodmacq B, Firastrau I, Ponthenier F, Brunet M, Thirion C, Michel J P, Prejbeanu-Buda L, Cyrille M C, Redon O, Dieny B 2007 Nat. Mater. 6 447
Google Scholar
[8] Georges B, Grollier J, Cros V, Fert A 2008 Appl. Phys. Lett. 92 232504
Google Scholar
[9] Ranjbar M, Drrenfeld P, Haidar M, Iacocca E, Balinskiy M, Le T Q, Fazlali M, Houshang A, Awad A A, Dumas R K, Kerman J 2014 IEEE Magn. Lett. 5 3000504
Google Scholar
[10] Kubota H, Yakushiji K, Fukushima A, Tamaru S, Konoto M, Nozaki T, Ishibashi S, Saruya T, Yuasa S, Taniguchi T, Arai H, Imamura H 2013 Appl. Phys. Express 6 103003
Google Scholar
[11] Litvinenko A, Iurchuk V, Sethi P, Louis S, Tyberkevych V, Li J, Jenkins A, Ferreira R, Dieny B, Slavin A, Ebels U 2020 Nano Lett. 20 6104
Google Scholar
[12] 方彬, 曾中明 2014 科学通报 59 1804
Google Scholar
Fang B, Zeng Z M 2014 Chin. Sci. Bull. 59 1804
Google Scholar
[13] Taniguchi T, Arai H, Tsunegi S, Tamaru S, Kubota H, Imamura H 2013 Appl. Phys. Express 6 123003
Google Scholar
[14] Awad A A, Houshang A, Zahedinejad M, Khymyn R, Åkerman J 2020 Appl. Phys. Lett. 116 232401
Google Scholar
[15] Arun R, Gopal R, Chandrasekar V K, Lakshmanan M 2022 J. Phys. Condens. Matter 34 125803
Google Scholar
[16] Taniguchi T, Tsunegi S, Kubota H, Imamura H 2014 Appl. Phys. Lett. 104 152411
Google Scholar
[17] Taniguchi T, Arai H, Kubota H, Imamura H 2014 IEEE Trans. Magn. 50 1400404
Google Scholar
[18] Guo Y Y, Xue H B, Liu Z J 2015 AIP Adv. 5 057114
Google Scholar
[19] Arun R, Gopal R, Chandrasekar V K, Lakshmanan M 2022 J. Appl. Phys. 132 094301
Google Scholar
[20] Arai H, Matsumoto R, Yuasa S, Imamura H 2015 Appl. Phys. Express 8 083005
Google Scholar
[21] Arai H, Matsumoto R, Yuasa S, Imamura H 2016 J. Phys. Soc. Jpn. 85 063802
Google Scholar
[22] Guo Y Y, Zhao F F, Xue H B, Liu Z J 2016 Chin. Phys. Lett. 33 037501
Google Scholar
[23] Zhao C P, Ma X Q, Huang H B, Liu Z H, Jafri H M, Wang J J, Wang X Y, Chen L Q 2017 Appl. Phys. Lett. 111 082406
Google Scholar
[24] 郭圆圆, 蒿建龙, 薛海斌, 刘喆颉 2015 64 198502
Google Scholar
Guo Y Y, Hao J L, Xue H B, Liu Z J 2015 Acta Phys. Sin. 64 198502
Google Scholar
[25] Kowalska E, Kákay A, Fowley C, Sluka V, Lindner J, Fassbender J, Deac A M 2019 J. Appl. Phys. 125 083902
Google Scholar
[26] Shirokura T, Hai P N 2020 J. Appl. Phys. 127 103904
Google Scholar
[27] Zhang W, Zhang Y, Jiang B, Fang B, Zhong H, Li H, Zeng Z M, Yan S S, Han G, Liu G, Yu S, Kang S 2021 Appl. Phys. Lett. 118 012405
Google Scholar
[28] Slonczewski J C 1989 Phys. Rev. B 39 6995
Google Scholar
[29] Slonczewski J C 2005 Phys. Rev. B 71 024411
Google Scholar
[30] Fowley C, Sluka V, Berent K, Lindner J, Fassbender J, Rippard W H, Pufall M R, Russek S E, Deac A M 2014 Appl. Phys. Express 7 043001
Google Scholar
[31] Taniguchi T, Kubota H 2016 Phys. Rev. B 93 174401
Google Scholar
[32] Taniguchi T, Ito T, Tsunegi S, Kubota H, Utsumi Y 2017 Phys. Rev. B 96 024406
Google Scholar
[33] Taniguchi T, Isogami S, Shiokawa Y, Ishitani Y, Komura E, Sasaki T, Mitani S, Hayashi M 2022 Phys. Rev. B 106 104431
Google Scholar
[34] Mancilla-Almonacid D, Arias R E, Escobar R A, Altbir D, Allende S 2018 J. Appl. Phys. 124 162102
Google Scholar
[35] Talatchian P, Romera M, Araujo F A, Bortolotti P, Cros V, Vodenicarevic D, Locatelli N, Querlioz D, Grollier J 2020 Phys. Rev. Appl. 13 024073
Google Scholar
[36] Zeng L, Xu X J, Chen H H, Zhou Y, Zhang D M, Wang Y J, Zhang Y G, Zhao W S 2020 Appl. Phys. Lett. 117 082404
Google Scholar
[37] Kaka S, Pufall M R, Rippard W H, Silva T J, Russek S E, Katine J A 2005 Nature 437 389
Google Scholar
[38] Mancoff F B, Rizzo N D, Engel B N, Tehrani S 2005 Nature 437 393
Google Scholar
[39] Singh H, Konishi K, Bhuktare S, Bose A, Miwa S, Fukushima A, Yakushiji K, Yuasa S, Kubota H, Suzuki Y, Tulapurkar A A 2017 Phys. Rev. Appl. 8 064011
Google Scholar
[40] Singh H, Bose A, Bhuktare S, Fukushima A, Yakushiji K, Yuasa S, Kubota H, Tulapurkar A A 2018 Phys. Rev. Appl. 10 024001
Google Scholar
[41] Zeng L, Liu Y, Chen H H, Zhou Y, Zhang D M, Zhang Y G, Zhao W S 2020 Nanotechnology 31 375205
Google Scholar
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