Frequency Modulation and Voltage Locking of the Voltage Controlled Spin Oscillators (VCSOs)
Lang Zeng Hao-Hsuan Chen Deming Zhang Tianqi Gao Mingzhi Long Youguang Zhang Weisheng Zhao
cond-mat.mes-hall





























[]zenglang@buaa.edu.cn



Fert Beijing Institute, Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, China, 100191
School of Microelectronics, Beihang University, Beijing, China, 100191
Hefei Innovation Research Institute, Beihang University, Hefei, Anhui, China, 230013











Lang Zeng and Hao-Hsuan Chen contributed equally to this work.

Fert Beijing Institute, Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, China, 100191
School of Microelectronics, Beihang University, Beijing, China, 100191
Hefei Innovation Research Institute, Beihang University, Hefei, Anhui, China, 230013






Fert Beijing Institute, Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, China, 100191
School of Electronic and Information Engineering, Beihang University, Beijing, China, 100191






Fert Beijing Institute, Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, China, 100191
School of Microelectronics, Beihang University, Beijing, China, 100191






Fert Beijing Institute, Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, China, 100191
School of Electronic and Information Engineering, Beihang University, Beijing, China, 100191






School of Electronic and Information Engineering, Beihang University, Beijing, China, 100191


[]weisheng.zhao@buaa.edu.cn



Fert Beijing Institute, Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, China, 100191
School of Microelectronics, Beihang University, Beijing, China, 100191











The oscillating frequency of typical Spin Torque Nano Oscillators (STNOs) can be modulated by injected DC current or bias magnetic field. And phase locking of STNOs to an external Radio Frequency (RF) signal can be imposed by AC current or RF bias magnetic field. However, in this study, we have proposed a Voltage Controlled Spin Oscillators (VCSOs) by introducing Voltage Controlled Magnetic Anisotropy (VCMA) effect. The oscillating frequency of VCSOs can be modulated by VCMA voltage as well as injected DC current. Furthermore, we have shown a novel locking mechanism caused by AC VCMA voltage. Both the frequency modulation and voltage locking mechanism are analyzed theoretically by Nonlinear Auto-oscillator theory and verified by numerical simulation. At last, we proposed that by utilizing negative capacitance material to enhance VCMA effect, the locking range for voltage locking can be expanded thus may lead to easy mutual synchronization of multiple VCSOs.








Frequency Modulation and Voltage Locking of the Voltage Controlled Spin Oscillators (VCSOs)
    Weisheng Zhao
    December 30, 2023
===========================================================================================





§ INTRODUCTION


Spin Torque Nano Oscillators (STNOs) are very promising devices which can be used to replace traditional Voltage Controlled Oscillator (VCO) in semiconductor chips <cit.>. Comparing with VCO, STNO has advantages of small footprint size (∼ 100 nm) and wide tunable frequency range (∼GHz). However, the STNO is suffering from low emitted power (∼μ W) and high phase and amplitude noise<cit.>. It is both theoretically predicted and experimentally demonstrated that for N synchronized STNOs, the emitted power can be increased as N^2 while the frequency line width can be decreased as N^-2 <cit.>. The synchronization of multiple STNOs can be performed through electrical connection <cit.>, spin wave propagation <cit.> as well as dipolar effect <cit.>. Among all these synchronization mechanisms, the most convenient one to be controlled and implemented in semiconductor chip is electrical connection. Since the low emitted power of single STNO, the mutual synchronization by electrical connection is very difficult to experimentally achieve <cit.>. The only example for such success is mutual synchronization of electrically connected two vortex STNOs because of large output power of vortex STNOs <cit.>. The oscillating frequency of vortex STNOs is only several hundreds of MHz which is far below frequency band used in modern communication system <cit.>. The electrical synchronization is closely related to injection locking phenomenon in STNO <cit.>. If the locking range can be expanded, it is postulated that the electrical synchronization can be made easily.

The stable rotation of magnetic moment in the free layer of STNOs is sustained by injected DC current which will induce Spin Transfer Torque (STT) effect. This implies that STNO is basically a current controlling device. Thus the way for electrical mutual synchronization of multiple STNOs is through injecting AC current <cit.>. However, the Voltage Controlled Oscillator composed by semiconductor devices using in chips are voltage controlling devices <cit.>. If the voltage can be introduced into the controlling of STNOs as an additional degree of freedom, not only the controlling of STNOs can be more versatile, but also novel locking and synchronization by applying AC voltage can be imaged.

The emitted power of STNOs is related to their magnetic configurations of free layer and reference layer. There are four kinds of different configurations <cit.>: magnetic directions of both layers are in-plane (case 1), magnetic directions of both layers are perpendicular (out-of-plane) (case 2), magnetic direction of free layer is in-plane and that of reference layer is perpendicular (case 3) and vice versa (case 4). Among all of these four configurations, the case 1 and case 2 can not provide large magnetoresistance change thus lead to small emitted power while case 3 need another in-plane magnetization layer for signal readout. The configuration of case 4 can provide intrinsically large magnetoresistance change which is beneficial for power emission <cit.>. Experimentally the high emitted power of STNO with case 4 configuration is demonstrated. The μ W emission power is almost the highest one reported in the literature for single STNO device excluding vortex type <cit.>. The oscillating frequency of the STNO with in-plane reference layer and perpendicular free layer can be controlled by injected DC current, bias magnetic field as well as Perpendicular Magnetic Anisotropy (PMA) field of the free layer. The PMA field of the free layer can be modulated by Voltage Controlled Magnetic Anisotropy (VCMA) effect with an applying voltage <cit.>. This inspires us how to make the STNO device to be voltage controlled.

In this work, we proposed Voltage Controlled Spin Oscillators (VCSOs) which has the voltage as an additional degree of freedom by introducing VCMA effect for the controlling of oscillating. The frequency modulation and the locking phenomenon by both injected current and applied voltage are theoretically analyzed with Nonlinear Auto-Oscillator theory <cit.>. In contrast with current locking mechanism, the voltage locking mechanism can be explained by the tilted oscillation axis. The locking range of the voltage locking is comparable to that of the current locking. However, we furthermore proposed that a negative capacitance layer can be added in the stack which can enhance the VCMA effect by tens of times <cit.>. This will make the locking range of the voltage locking expand correspondingly and it implies an easy and novel way for electrical mutual synchronization of multiple VCSOs.



§ FREQUENCY MODULATION OF PROPOSED VCSO



 §.§ Device Structure of Proposed VCSO


The device we proposed is illustrated in Fig. <ref>. The proposed device has three terminals: T_1, T_2 and T_3 (Ground). The terminal T_2 connects to a thin CoFeB layer which holds perpendicular magnetic direction because of interfacial PMA effect and serves as free layer. The terminal T_3 connects to a thick CoFeB layer which holds in-plane magnetic direction and serves as reference layer. The MgO insulating layer between free layer and reference layer is about 0.8 nm thick to keep the RA product of Magnetic Tunneling Junction (MTJ) small. The MgO layer between terminal T_1 and free CoFeB layer is about 1.2 nm thick to make the resistance between terminal T_2 and T_1 large. Applying a voltage between terminal T_2 and T_1, the current can be negligible since large resistance. However, the VCMA effect will take place and the applying voltage will modify the PMA field of the free layer.

As shown in Fig. <ref>, the perpendicular direction is defined as z axis in our study while the in-plane direction is x and y axis. When there is injected DC current between terminal T_2 and T_3, the spin torque exerted on the magnetic moment of the free layer will make it rotate around the z axis. From previous analytical solution, it can seen that the oscillating frequency of STNO is related by the external magnetic field, the PMA field and the injected DC current. Since the applying VCMA voltage can modify the PMA field of the free layer, the oscillating frequency can be tuned by the applying VCMA voltage. This makes our proposed device is voltage controllable and we named it Voltage Controlled Spin Oscillator (VCSO).





 §.§ Theoretical Analysis on Frequency Modulation


The analytical solution for the frequency modulation of STNO with perpendicular free layer and in-plane reference layer is already derived by energy balance theory and verified with experimental results. However in energy balance theory the energy change due to spin torque term and damping term is averaged over one rotation period, thus some information about the oscillating dynamics is lost. Here in this study, the phase locking phenomenon will be explored. We expanded Nonlinear Auto-oscillator theory developed by J. V. Kim, V. Tiberkevich and A. Slavin for our study <cit.>. The Nonlinear Auto-oscillator theory will provide unified and powerful framework for the analysis of VCSO.

The LLG equation which describes the magnetic dynamics in the free CoFeB layer is as follows

    ∂ m/∂ t = -γ(m× H_eff)+α(m×∂ m/∂ t)-σ J m×(m× e_p)

where m is the magnetic direction in the free layer, H_eff is the effective field of the free layer, α is the Gilbert damping constant, γ is the gyromagnetic ratio, σ is the spin torque efficient, J is the injected current density and e_p is the vector describing the magnetic direction in the reference layer.

Eq. <ref> can be rearranged as

    ∂ m/∂ t =    -γ/1+α^2(m× H_eff)-αγ/1+α^2(m×(m× H_eff))
       -σ J/1+α^2[m×(m× e_p)-α(m× e_p)]

Since Gilbert damping constant is very small, Eq. <ref> can be approximated by

    ∂ m/∂ t =    -γ(m× H_eff)-αγ(m×(m× H_eff))
       -σ J (m×(m× e_p))


In our VCSOs, the magnetic direction of the reference layer is in-plane and it can be assumed in the x axis.

    e_p =  ([ P_x;   0;   0;     ])

If the applied VCMA voltage is zero, then the effective field H_eff can be written as

    H_eff =  ([                      0;                      0; H_appl+(H_k-4π M_s)m_z;                        ])

Both of the external magnetic field H_appl and the effective PMA field (H_k-4π M_s)m_z are along z axis.

The LLG equation Eq. <ref> is expanded to be
 

    ([ ṁ_̇ẋ; ṁ_̇ẏ; ṁ_̇ż;      ])
    =  -γ([ m_x; m_y; m_z;     ])×([                      0;                      0; H_appl+(H_k-4π M_s)m_z;                        ])
       -αγ([ m_x; m_y; m_z;     ])×[([ m_x; m_y; m_z;     ])×([                      0;                      0; H_appl+(H_k-4π M_s)m_z;                        ])]
       -σ J ([ m_x; m_y; m_z;     ])×[([ m_x; m_y; m_z;     ])×([ P_x;   0;   0;     ])]



A new variable a defined by

    a = m_x-i m_y/√(2(1+m_z))

is introduced. With this definition, m_x, m_y and m_z can be written as

    m_x = (a+a^*)√(1-|a|^2)
       m_y = i(a-a^*)√(1-|a|^2)
       m_z = 1-2|a|^2

With this transformation, the precessional term -γ(m× H_eff) is derived as

    .∂ a/∂ t|_Precession   = -iγ[[ (H_appl+H_k-4π M_s)a; -2(H_k-4π M_s)a^2a^*;                      ]]
       = -iγ[[ (H_appl+H_k-4π M_s)a;     -2(H_k-4π M_s)pa;                      ]]

where p=|a|^2 is the auto-oscillation power.

Similarly, the damping term -αγ(m×(m× H_eff)) is written as

    .∂ a/∂ t|_Damp   = αγ[[        -(H_appl+H_k-4π M_s)a; (H_appl+3(H_k-4π M_s))a^2a^*;        -2(H_k-4π M_s)a^3a^*2;                              ]]
       = αγ[[    -(H_appl+H_k-4π M_s)a; (H_appl+3(H_k-4π M_s))pa;       -2(H_k-4π M_s)p^2a;                          ]]

As we can see, both of the precessional term and damping term can be written as polynomial expansion of a and p. However, the STT term -σ J (m×(m× e_p)) is transformed to

    .∂ a/∂ t|_STT = σ J /4√(1-aa^*)[[        2;   -3aa^*;    -3a^2; 2a^2a^*2;  2a^3a^*;          ]]

Since there is √(1-aa^*) in the denominator, direct following analysis in the framework of Nonlinear auto-oscillator is not possible.

The spin torque strength σ is defined by

    σ    = γħη/2e(1+λ m_x)M_s Sd
       =γσ^'/(1+λ m_x)

where σ^' is

    σ^' = ħη/2eM_s Sd

λ is a small value, and for not very large oscillation angle m_x≪ 1, thus the spin torque coefficient can be in Taylor expansion form as

    σ = γσ^'(1-λ m_x+λ^2 m_x^2-λ^3 m_x^3+O(λ^4 m_x^4))

Also for not very large angle oscillation, p≪ 1. Thus the denominator √(1-aa^*) can be Taylor expanded also

    √(1-aa^*) = 1-aa^*/2-(aa^*)^2/8-(aa^*)^3/16+O((aa^*)^4)


With these two Taylor expansions, the STT term can be written to the order of p^2

    .∂ a/∂ t|_STT = γσ^'λ J/2(1-(3-3λ^2)p+(2-12λ^2)p^2)a

and to the order of p^3

    .∂ a/∂ t|_STT = γσ^'λ J/2(1-(3-3λ^2)p+(2-12λ^2)p^2+15λ^2p^3)a

Now all of the terms in LLG Equation Eq. <ref> is the function of p and a. Keep in mind that in Eq. <ref> and <ref>, only conservative terms around the oscillation orbit are written. For the derivation of oscillation frequency, only conservative terms are reqiured. However, for the exploration of phase locking phenomenon, non-conservative terms in Eq. <ref>. So we can apply similar analysis for the frequency modulation and locking range in the framework of Nonlinear Auto-oscillator theory. It is worth noting that the calculation for Eq. <ref> and <ref> is tedious and complicated, we invented a new method call Virtual Perpendicular System for the  fast theoretical derivation. The details are presented in  Appendix. <ref>.

After the above calculation, the oscillation dynamics of the proposed VCSO can be described by the general nonlinear oscillator model

    da/dt+iω(p)a+Γ_+(p)a-Γ_-(p)a=0

where a is the the complex amplitude of the auto-oscillation, and p is the power of the auto-oscillation and ϕ = arg(a) is the phase of the auto-oscillation. ω(p) is the oscillating frequency of the auto-oscillator, and the detail expression for it is written as

    ω(p) = ω_0+Np

where N is the coefficient of the nonlinear frequency shift for the normally magnetized film in the free CoFeB layer. N can be written as

    N = -2ω_k

In the above two equations, ω_0 and ω_k is

    ω_0 = γ(H_appl+H_k-4π M_s)
       ω_k = γ(H_k-4π M_s)


The Γ_+(p) in Eq. <ref> is the damping rate for the natural energy dissipation of the auto-oscillation and it comes from the damping torque. The exact form of Γ_+(p) is

    Γ_+(p) = Γ_G(1+Qp+Q^'p^2)

where

    Γ_G = αω_0
       Q = -(1+2ω_k/ω_0)
       Q^' = 2ω_k/ω_0

For physically-relevant range of complex amplitudes a, the damping term Γ_+(p) is positive thus Γ_+(p) describes the reduction of the spin wave amplitude a with time.

Contrary to damping term Γ_+(p), the STT term Γ_-(p) comes from the STT torque of the injected current and it describes the effective action of the external energy source. To calculate Γ_-(p), as shown in the above Eq. <ref> and <ref>, extensions to the Nonlinear Auto-oscillator theory should be made. After Taylor expansion, the expression for Γ_-(p) to the order of p^3 is

    Γ_-(p) =    γσ^'λ J/2(1-(3-3λ^2)p
       +(2-12λ^2+10λ^4)p^2+(15λ^2-50λ^4)p^3)


The condition for the threshold injected current of the appearance of an auto-oscillation having a nonzero power is

    Γ_+(0) = Γ_-(0)

From Eq. <ref>, the threshold current for the onset of auto-oscillation is

    I_th   = 2Γ_G/γσ^'λ
       = 4α e M_sSd/ħηλ(H_appl+H_k-4π M_s)

This threshold current I_th agrees exactly with the results obtained from the energy balance theory.

For the stability of the stationary solution of Eq. <ref>, three more new variables can be defined as

    G_+ = dΓ_+(p)/dp
       G_- = dΓ_-(p)/dp
       Γ_p = (G_+-G_-)p_0

where the derivatives are calculated at the stationary point p=p_0. Γ_p is the damping rate for small power deviations δ p at p_0. For the stationary solution p=p_0 is stable, it is required that

    Γ_p > 0

If both of the damping term Γ_+(p) and the STT term Γ_-(p) is expanded to the order of p, the condition for stationary stable oscillation Eq. <ref> can be written

    Q > -(3-3λ^2)

Substituting Q with Eq. <ref> and <ref>, the condition for stationary stable oscillation is got

    H_appl > 0

The requirement for an external magnetic field in the z axis to sustain a stable oscillation which is the main conclusion of Ref. 33 is derived by us using Nonlinear Auto-oscillator theory.

For the stable stationary oscillation condition Eq. <ref>, if the damping term Γ_+(p) and the STT term Γ_-(p) is expanded to the order of p^2, then we can get

    1+Qp+Q^'p^2   =1-(3-3λ^2)p+(2-12λ^2)p^2
    
    Q+Q^'p   =-(3-3λ^2)+(2-12λ^2)p
    
    p   =Q+3-3λ^2/2-12λ^2-Q^'

For stable stationary oscillation, p>0 should be fulfilled and after calculation it is got that

    H_appl > 3λ^2/2-3λ^2(H_k-4π M_s)

This result means that there exists a minimum external magnetic field in the z axis to make the stable stationary oscillation possible. It is also worth noting that this is the main conclusion of Ref. 32 derived by energy balance theory.

For the calculation of oscillation frequency for any specific inject DC current, first the stationary oscillation power p will be calculated from

    Γ_+(p) = Γ_-(p)

Then Eq. <ref> will be used to calculate the oscillation frequency. It should be noted that for Γ_-(p) expanded with the order of p^2 and p^3, there will be two or three roots for the solution of Eq. <ref>. However, since

    p   =   aa^*
       =   1-m_z/2

So 0<p<1/2. There will be only one root in this range. This implies our oscillation is single mode.

The analytical solution for the oscillation frequency with varied injected DC current is shown in Fig. <ref>. We expanded the STT term Γ_-(p) to the order of p, p^2 and p^3, and compared the results with that got by energy balance theory. Fig. <ref>(a) shows the oscillation frequency for injected current in the range of 0-45 mA, and Fig. <ref>(b) shows the oscillation frequency for injected current in the range of 1.2-2.0 mA. It is can be observed that there are large deviations when only expanded to the order of p and p^2. However, when expanded to the order of p^3, the oscillation frequency calculated from Eq. <ref> is very close to the oscillation frequency derived from energy balance theory, especially when inject current is small.



Besides oscillation power p_0, there are two important variables defined as

    Γ_p = (G_+-G_-)p_0

and

    ν = N/G_+-G_-

ν is related to the influence of the nonlinear frequency shift N in comparison with the nonlinear damping coefficients G_+ and G_-. For conventional oscillators ν≪ 1, the oscillation power p has negligible effect on the oscillation frequency ω. In contrary, for typical spin torque oscillators ν≫ 1, thus spin torque oscillators is strong nonlinear oscillator.

The p_0, Γ_p and ν calculated from Eq. <ref> and extracted from macro simulation are plotted in Fig. <ref>. It is also observed that expansion to the order of p^3 can give us very accurate solutions agreeing well with macro simulation results.



When there applies VCMA voltage, the effective field can be written as

    H_eff =  ([                                      0;                                      0; H_appl+(H_k-4π M_s)(1+V_VCMA/V_ref)m_z;                                        ])

where V_VCMA is the applied VCMA voltage, and V_ref which means an applied VCMA voltage of -V_ref will fully compensate the intrinsic PMA field of the free CoFeB layer. The oscillation frequency with varied applied VCMA voltage for injected DC current of 1.2, 1.6 and 2.0 mA respectively is shown in Fig. <ref>. The analytical results are calculated with the order of p^3. The analytical results agree very well with that of macro simulation results.



The variables of p_0, Γ_p and ν calculated analytically and extracted from macro simulation are plotted in Fig. <ref>. The expansion to the order of p^3 can give us very accurate results for these three variables.



In this section, we derived frequency modulation by injected DC current and applied VCMA voltage under the framework of the Nonlinear Auto-oscillator theory. We got the same results as the analytical solution derived by energy balance theory, and repeated several main conclusions in the literature. Also our theoretical results agree very well with the macro simulation results. For the macro simulation, all of the parameters used are list in Table. <ref> extracted from experimental work <cit.>.





§ CURRENT AND VOLTAGE LOCKING OF PROPOSED VCSO


In the previous section, the Nonlinear Auto-oscillator theory is employed to analyze frequency modulation by injected DC current and applied VCMA voltage. Same results also can be got by energy balance theory. In this section, we will use Nonlinear Auto-oscillator theory to analyze phase locking phenomenon caused by RF current as well as RF voltage. Such theoretical derivations cannot be performed by energy balance theory.



 §.§ Theretical Analysis on Current Locking of VCSO


For phase locking caused by injected RF current, the injected current can be described

    J=J_DC(1+ϵcos(ϕ_e))

where ϕ_e is the phase of the external injected RF current, J_DC is the amplitude of the injected DC current and ϵ is the amplitude ratio of the external injected RF current and DC current. Eq. <ref> should not be substituted into Eq. <ref> or <ref> since only conservative terms are kept thus no phase locking will be observed. Instead Eq. <ref> should be substituted into Eq. <ref>, and keep all of the coupling factors that allow for resonant with external RF current. After some tedious algebra calculation, we got

    .∂ a/∂ t|_STT =    γσ^'λ J_DC/2[(1-(3-3λ^2)p+(2-12λ^2)p^2)a
       +1/2λϵcos(ϕ_e)(1+2λ^2acos(ϕ)√(1-p))(2-3p)]

where ϕ is the phase of the spin torque oscillator. Actually, the complex amplitude of the auto-oscillation a can be written as

    a = √(p)e^iϕ

Thus we can write equations as follows

    a^*(da/dt)=1/2dp/dt+ipdϕ/dt

Multiple a^* in the left of Eq. <ref>, we can get

    a^*(da/dt)_STT=   Γ_-(p)p+γσ^' J_DC/4ϵcos(ϕ_e)*
       (√(p)(cos(ϕ)-isin(ϕ))
       +2λ^2pcos(ϕ)√(1-p))(2-3p)


For the study of phase locking phenomenon with external RF current, the phase difference ψ is introduced

    ψ = ϕ_e +ϕ

where the phase of the spin torque oscillator can be written as

    ϕ = -ω t+ϕ_0

Noting since there is plus sign in the definition of phase difference, thus there should be minus sign in the definition of phase of the spin torque oscillator. Substituting ψ into Eq. <ref>, we can get

    a^*(da/dt)_STT=   Γ_-(p)p+γσ^' J_DC/4ϵ*
       (1/2√(p)(cos(ψ)-isin(ψ))
       +λ^2pcos(ψ)√(1-p))(2-3p)

Rearranged Eq. <ref> with oscillation power p and oscillation phase ϕ

    dp/dt=   -2(Γ_+(p)-Γ_-(p))p+γσ^' J/2ϵ*
       (1/2√(p)cos(ψ)+λ^2pcos(ψ)√(1-p))(2-3p)
    dϕ/dt=   -(ω_0+Np)-γσ^' Jϵ1/8√(p)sin(ψ)(2-3p)

Eq. <ref> is derived by STT term expanded to the order of p^2. If the STT term just is expanded to the order of p, a simpler equation can be got

    dp/dt=   -2(Γ_+(p)-Γ_-(p))p+γσ^' J/2ϵ√(p)cos(ψ)
    dϕ/dt=   -(ω_0+Np)-γσ^' Jϵ1/4√(p)sin(ψ)


With an injected RF current, the stationary oscillation power will have small deviation from its free-running oscillation power p_0

    p_s = p_0 +δ p_s

where p_s is the stationary forced power which is the stationary power under the impact of the injected RF current, and δ p_s is the forced power deviations due to injected RF current. Here we will use Eq. <ref> corresponding to the STT term expanded to the order of p  to derive phase locking range and phase difference, and give the phase locking range and phase difference when the STT term is expanded to the order of p^2 directly. Substituting p_s back into Eq. <ref>, and use stationary stable oscillation conditions of p_0

    0=-2(Γ_+(p)-Γ_-(p))(p_0+δ p_s)+γσ^' J/2ϵ√((p_0+δ p_s))cos(ψ)

Eq. <ref> can be further simplified to be

    0=-2Γ_p(p_0)δ p_s+γσ^' J/2ϵ√(p_0)cos(ψ)

Thus the small power deviation δ p_s can be computed

    δ p_s/p_0 = γσ^' J/4Γ_p(p_0)√(p_0)ϵcos(ψ)

With this relation, Eq. <ref> is transformed to

    dϕ/dt   =-(ω_0+Np)-γσ^' Jϵ1/4√(p)sin(ψ)
    dψ/dt   =-ω+ω_e-Nδ p_s-γσ^' Jϵ1/4√(p_0)sin(ψ)
       =δω_e-νγσ^' J/4√(p_0)ϵcos(ψ)-γσ^' Jϵ1/4√(p_0)sin(ψ)

where ω is the free-running auto-oscillation, and ω_e is the oscillation frequency of the external RF current. δω_e = ω_e-ω is the detuning which measures the frequency difference between the free-running spin torque oscillator frequency ω and the external RF current frequency ω_e. Eq. <ref> is rearranged as

    dψ/dt=δω_e+ΔΩsin(ψ+ψ_0)

where

    ΔΩ = γσ^' Jϵ/4√(p_0)√(1+ν^2)


For stable phase locking, dψ/dt should be zero. Thus ΔΩ is the phase locking range and ψ_0 is the phase difference between forced spin torque oscillator and injected RF current

    ψ_0 = -arctan(ν)


If the STT term is expanded to the order of p^2, the locking range ΔΩ is

    ΔΩ = γσ^' Jϵ/8√(p_0)√((2-3p_0)^2+[ν(1/2+λ^2√(p_0(1-p_0)))(2-3p_0)]^2)

and the phase difference is

    ψ_0 = -arctan(ν(1/2+λ^2√(p_0(1-p_0)))(2-3p_0))


The phase locking phenomenon for spin torque oscillator with external injected RF current is simulated with macro-magnetic model, and the results are shown in Fig. <ref>. The DC current is 1.6 mA and the current amplitude ratio ϵ between RF current and DC current varies from 0.01 to 0.05. It is observed that the larger ϵ is, the larger the locking range is. The locking range for different injected DC current is extracted and plotted in Fig. <ref>.





For comparing the macro simulation results and the analytical results, we plotted locking range extracted from macro simulation and computed with Eq. <ref> and <ref> for varied DC current of 1.2 to 2.0 mA and current amplitude ratio ϵ of 0.01 to 0.05 in Fig. <ref>. It is can be seen that Eq. <ref> over estimates the locking range while Eq. <ref> under estimates the locking range. We can get similar observation from Fig. <ref> where expansion to the order of p under estimates the oscillation frequency and expansion to the order of p^2 over estimates the oscillation frequency, and expansion to the order of p^3 can give accurate estimations. Also we can observe that when ϵ = 0.01, expansion to the order of p can give a closer estimation for the locking range comparing with expansion to the order of p^2 while when ϵ = 0.05, expansion to the order of p^2 can give very accurate results. For ϵ = 0.05, the estimated locking range by Eq. <ref> deviates more significantly when injected DC current is larger. This can be explained that the oscillation power p increases with large injected DC current. So higher order expansion is needed for larger DC current.





 §.§ Theretical Analysis on Voltage Locking of VCSO

The phase locking phenomenon of the spin torque oscillator is not novel. There are plenty reports in the literature about injection locking. However, since the applied VCMA voltage can tune the oscillation frequency of the VCSOs, a question will naturally be raised: will a RF VCMA voltage lead to phase locking? However, we followed the steps shown in Sec. <ref> and surprisingly we find there is no coupling factors for resonant with external RF VCMA voltage. It implies RF VCMA voltage will not lock the phase of VCSO. But as plotted in Fig. <ref>, macro simulation results show that for injected DC current of 1.6 mA and no DC VCMA voltage, there does exist locking ranges when the frequency of the external RF voltages sweeps. Also it is seen that the locking range increases when the voltage amplitude ratio ϵ between RF VCMA voltage and reference VCMA voltage V_ref increases from 0.01 to 0.05. The locking range with varied injected DC currents for voltage amplitude ϵ from 0.01 to 0.05 are extracted and plotted in Fig. <ref>. It is worth noting that there are similar experimental results in the literature which reports a comparable voltage induced torque with spin transfer torque <cit.>.




The 3D plot for the trajectory of the oscillating magnetic moment of VCSO is shown in Fig. <ref>. Carefully observed, it can be seen that the oscillation orbit is slightly tilted away from the blue equal energy line. This means that the oscillation axis is not fully along the z axis, but is a little tilted away from the z axis. And this cant oscillation axis is the origin for voltage locking in VCSO.



Let us assume that the oscillation axis is tilted away from the z axis with angle θ, and the magnetization direction in the energy minimum can be written as

    m_θ = m_xsin(θ)+m_zcos(θ)

Thus the effective field H_eff is got by

    H_eff =  ([ H_applsin(θ)+(H_k-4π M_s)sin(θ)m_θ;                                  0; H_applcos(θ)+(H_k-4π M_s)cos(θ)m_θ;                                    ])

The precessional term -γ(m× H_eff) is derived as
 

    .∂ a/∂ t|_Precession =     -iγ[[ H_appl+(H_k-4π M_s)acos(θ);     -2(H_k-4π M_s)pacos(θ);                            ]]cos(θ)
       -iγa^2+3p-2/4√(1-p)H_applsin(θ)
       -iγa^2+3p-2/4(a+a^*)(H_k-4π M_s)sin(θ)^2
       -iγa(5a+11a^*-6pa)/4√(1-p)(H_k-4π M_s)sin(θ)cos(θ)
       +iγ(1+5p^2)/2√(1-p)(H_k-4π M_s)sin(θ)cos(θ)
 


We considering this is no DC VCMA voltage and only RF VCMA voltage, the PMA field H_appl reads

    H_appl=(H_k-4π M_s)(1+ϵcos(ϕ_e))

where ϵ is the voltage amplitude ratio of RF VCMA voltage and reference VCMA voltage V_ref, and ϕ_e is the phase of external RF VCMA voltage. Substituting H_appl back into Eq. <ref>, considering θ≪ 1 and doing rearrangement we can get

    .∂ a/∂ t|_Precession =     -iγ[[ (H_appl+H_k-4π M_s)a;     -2(H_k-4π M_s)pa;                      ]]
       -iγ5a^2+11p-6pa^2-10p^2-2/4*
       (H_k-4π M_s)ϵcos(ϕ_e)sin(θ)

Keep in mind that in Eq. <ref> only conservative term and resonating term with RF VCMA voltage are written.

Multiple a^* in the left of Eq. <ref>
 

    a^*(da/dt)_Precession =     -iω(p)p
       -iγ5pa+11pa^*-6p^2a-10p^2a^*-2a^*/4*
       (H_k-4π M_s)ϵcos(ϕ_e)sin(θ)
    
    =     -iω(p)p
       -iγ((cos(ϕ)+isin(ϕ))(5p-6p^2)/4.
       .+(cos(ϕ)-isin(ϕ))(11p-10p^2-2)/4)*
       (H_k-4π M_s)ϵcos(ϕ_e)sin(θ)√(p)
 

Further transform into oscillation power p and oscillation phase ϕ if expanded to the order of p^2

    dp/dt=   -2(Γ_+(p)-Γ_-(p))p+(H_k-4π M_s)ϵγ*
       -6p+4p^2+2/4sin(ψ)√(p)sin(θ)
    dϕ/dt=   -(ω_0+Np)-(H_k-4π M_s)ϵγ*
       16p-16p^2-2/8√(p)cos(ψ)sin(θ)

and expanded to the order of p

    dp/dt=   -2(Γ_+(p)-Γ_-(p))p+
       (H_k-4π M_s)ϵγ√(p)/2sin(ψ)sin(θ)
    dϕ/dt=   -(ω_0+Np)+
       (H_k-4π M_s)ϵγ1/4√(p)cos(ψ)sin(θ)


The equation for the small power deviation δ p_s caused by forced external RF VCMA voltage is

    0=-2Γ_p(p_0)δ p_s+(H_k-4π M_s)ϵγ√(p_0)/2sin(ψ)sin(θ)

and δ p_s can be calculated as

    δ p_s/p_0 = (H_k-4π M_s)ϵγsin(ψ)sin(θ)/4Γ_p(p_0)√(p_0)


Using the same technology as the locking range derivation for injected RF current, we can get

    dϕ/dt=   -(ω_0+Np)+
       (H_k-4π M_s)ϵγ1/4√(p_0)cos(ψ)sin(θ)
    dψ/dt   =-ω+ω_e-Nδ p_s+
       (H_k-4π M_s)ϵγ1/4√(p_0)cos(ψ)sin(θ)
       =δω_e-ν(H_k-4π M_s)ϵγsin(ψ)sin(θ)/4√(p_0)
       +(H_k-4π M_s)ϵγ1/4√(p_0)cos(ψ)sin(θ)

Finally we can wirte

    dψ/dt=δω_e+ΔΩsin(ψ+ψ_0)

where the locking range for voltage locking is written for expansion to the order of p

    ΔΩ = (H_k-4π M_s)ϵγsin(θ)1/4√(p_0)√(1+ν^2)

and the phase difference is written

    ψ_0 = π-arctan(ν)

For the expansion to the order of p^2, we got

    ΔΩ =    (H_k-4π M_s)ϵγsin(θ)1/8√(p_0)*
       √((16p-16p^2-2)^2+ν^2(-6p+4p^2+2)^2)


The voltage locking range for varied injected DC current and no DC VCMA voltage are shown in Fig. <ref> for voltage amplitude ratio ϵ of 0.01 and 0.05 respectively. In Fig. <ref>, the macro-magnetic simulation results are compared with Eq. <ref> for expansion to the order of p and Eq. <ref> for expansion to the order of p^2. Similar to the observation for current locking in Fig. <ref>, Eq. <ref> can give a good estimation for ϵ = 0.01 and Eq. <ref> can give a good estimation for ϵ = 0.05. This results are solid proof for the correctness of our theoretical derivation.



The case for non-zero DC VCMA voltage is also explored. Fig. <ref> shows macro-magnetic simulations for DC current of 1.6 mA and DC VCMA voltage of 0.5 V when varied voltage amplitude ϵ when the frequency of the external RF VCMA voltage sweeps. And the voltage locking range for varied DC VCMA voltage extracted from macro-magnetic simulations are shown in Fig. <ref>. The comparison between macro-magnetic simulation results and analytical results is plotted in Fig. <ref>. When ϵ = 0.01 and DC VCMA voltage is smaller than 0.4 V, Eq. <ref> for expansion to the order of p can give very good agreement. When ϵ = 0.05 and DC VCMA voltage is smaller than 0.2 V, Eq. <ref> for expansion to the order of p^2 can give very good agreement. However, when DC VCMA voltage is larger than 0.2 V, both of Eq. <ref> and <ref> over estimate the voltage locking range. When DC VCMA voltage is larger than 0.2 V, the oscillation power p will be small from Fig. <ref> and <ref> and ν drops dramatically. Thus maybe expansion to higher order of p is required when DC VCMA voltage is large than 0.2 V.









 §.§ Expanding Locking Range by Enhanced VCMA effect


Although STNO is very promising device to replace VCO based on semiconductor transistors, low emitted power and high phase noise hinders its practical use in modern communication system. However, both of the figure of merit for emitted power and phase noise can be dramatically improved by mutual synchronization of multiple STNOs. There are several synchronization mechanisms proposed in the literature: spin wave, magnetic field, dipolar effect as well as electrical connection. Among all of these means, the synchronization through electrical connection is most favorable for implemented on chip. As shown in Ref. 17 only in vortex STNO the mutual synchronization through electrical connection is experimentally demonstrated. This is because of high emitted power of vortex STNO can overcome the random walking phase error induced by thermal fluctuation.

The phase locking phenomenon caused by injected RF current and applied RF voltage is closely related to the mutual synchronization through electrical connection. The wider the locking range is, the easier the mutual synchronization is. So it will have significant impact on mutual synchronization by electrical connection if the locking range can be expanded. From the previous sections about the theoretical and numerical results of locking range for current and voltage locking, we can see that the locking range for current and voltage locking are comparable. However, there are two methods which can significantly expand the locking range of voltage locking.

1) From Eq. <ref> and <ref>, the locking range for current locking ΔΩ is directly proportional to sin(θ). From Fig. <ref>, we can see that θ is very small and the oscillation axis is only a little bit tilted away from z axis. As pointed out in Ref. 35 and 37, if the field like torque of the spin polarized current taken into account, the oscillation axis will be tilted away in a large angle from the z axis. This will lead to a tremendously expanded locking range to facilitate easy mutual synchronization by electrical connection. However, the details analysis for mutual synchronization by voltage coupling should be carefully taken and is beyond the scope of this work. It will be our future work.

2) In our previous study, we proposed that negative capacitance ferroelectric material can be used to amplify the VCMA effect for the first time <cit.>. As shown in Fig.<ref> (a), the structure we proposed to amplify VCMA effect just requires an additional ferroelectric layer. The principle for why the capacitance of the ferroelectric layer can be negative is proposed in Ref. 47, and it is very hot topic in the semiconductor device community to achieve steep slope transistors <cit.>. Recently there are two back to back papers published in Nature to report direct observation of negative capacitance phenomenon <cit.>. The simplified capacitance model for the proposed structure is shown in Fig. <ref> (b). As shown, the internal voltage V_x which is the effective voltage applied on MgO insulating layer to induce VCMA effect can be calculated as when V_1 = 0

    V_x = V_2 C_FE/C_MgO+C_FE

V_x will be much larger than the external applied voltage V_2 if C_FE+C_MgO≈ 0. The amplification ratio of the VCMA effect is defined as

    V_x/V_2=C_tot/C_MgO = C_FE/C_MgO+C_FE

In our previous study, we have proved that by this novel method, the VCMA effect can be amplified to tens or even hundreds of times. And based on the amplification of the VCMA effect, we have proposed memory devices as well as all spin logic device with features of ultra fast, ultra low power consumption and ultra low working voltage <cit.>. According to Eq. <ref> and <ref>, the locking range for current locking ΔΩ is directly proportional to ϵ. ϵ is proportional to VCMA coefficient thus giant VCMA effect will lead to very large locking range. More details analysis on this kind of method is beyond the scope of this work and will be our next work.





§ CONCLUSION

In this work, we proposed a Voltage Controlled Spin Oscillator (VCSO). The VCMA effect is employed to play as an additional degree of freedom to tune oscillation frequency of the VCSO. We also proposed that applying a RF voltage can lead to novel phase locking phenomenon as well as current locking in our VCSO. Both of the frequency modulation and current and voltage locking are theoretically investigated in the framework of Nonlinear Auto-oscillation theory. The analytical results agree very well with the numerical simulation results. In the end, two kinds of method which can expand the locking range for voltage locking are proposed. Especially we proposed that by integrating an additional ferroelectric layer, the VCMA effect can be amplified thus lead to largely expanded locking range as well as possible easy mutual synchronization in uniform precession STNOs by voltage coupling through electrical connection.


The authors wish to acknowledge the support from the National Natural Science Foundation of China under Grants 61471015, 61504006 and 51602013, International mobility project under Grant B16001, National Key Technology Program of China under Grant 2017ZX01032101, National Postdoctoral Program for Innovation Talents under Grant BX20180028 and China Postdoctoral Science Foundation funded project under Grant 2018M641153.




§ VIRTUAL PERPENDICULAR SYSTEM FOR FAST THEORETICAL DERIVATION


As shown by Eq. <ref> and <ref>, the calculation of the STT term expressed with a involved Taylor expansion and merging of  items with the same order. This process is tedious, cumbersome and easy to make mistakes. Here we provide a novel method which can get the conservative terms of the STT term in a very convenient way.

Let us start from a STNO which has both of the free layer and reference layer in the perpendicular direction. The STT terms of such configuration can be calculated as

    .∂ a/∂ t|_STT = σ J(1-p)a

where σ is the spin torque strength in this STNO

    σ    = γħη/2eM_s Sd

From Eq. <ref>, we can see that in such a STNO there is only conservative term but no resonant term. It is can be concluded that injection locking by RF current is absent in such system.

By using the LLG equation, the work done by the spin torque W is given by

    W =    γ M_s σ[e_p· H_eff-(m· e_p)(m· H_eff)
       -α e_p·(m× H_eff)]


W for the STNO with both of the free layer and reference layer in the perpendicular direction (we call it perpendicular system) is very easy to calculate

    W_PS = γ M_s ħη J/2eM_sSd(H_appl+(H_k-4π M_s)cos(θ))sin(θ)^2

where a constant tilted angle θ for a steady precession around the z-axis is assuming.

While in Ref. 35, W for the STNO with the same configuration as our proposed VCSO (we can call it Virtual Perpendicular System, and the reason will be explained in the following) is given by <cit.>

    W_VPS =    γ M_s ħη J/2eλ M_sSd(1/√(1-λ^2sin(θ)^2)-1)
       [H_appl+(H_k-4π M_s)cos(θ)]cos(θ)

where a constant tilted angle θ for a steady precession around the z-axis is assuming.

Actually, calculation of the work done by spin torque or damping term is the typical operation in energy balance theory. Since W_PS or W_VPS is integration over one period of the oscillation orbit, it should be the same as calculation of the conservative terms like Eq. <ref>, <ref> or Eq. <ref>. Thus we postulate the following equation should be established

    .∂ a/∂ t|_VPS = W_VPS/W_PS·.∂ a/∂ t|_PS

Then we can get

    .∂ a/∂ t|_VPS = (1-p)(1/√(1-λ^2sin(θ)^2)-1)cos(θ)/λsin(θ)^2

Substituting p=(1+cos(θ))/2

    .∂ a/∂ t|_VPS =    (1-p)(1/√(1-λ^2(1-(1-2p)(1-2p)))-1)*
       1-2p/λ(1-(1-2p)(1-2p))

Taylor expansion of Eq. <ref> to the order of p^2, Eq. <ref> can be easily got while Taylor expansion of Eq. <ref> to the order of p^3, Eq. <ref> can be easily got. Any we can calculate STT term to any order of p as we want.

Although VCSO is not perpendicular system which should have perpendicular free layer and reference layer, we successfully make a connection between the work done by spin torque and the STT term. Thus we can calculate the STT term from that of the perpendicular system. In other words, the VCSO can be treated as a Virtual Perpendicular System. It is worth noting that such connection is general relation, thus our VPS method can be applied to other STNOs with different configurations. It is a general fast theoretical derivation method.



























































































