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Nonlinear spin-torque microwave resonance near the loss of spin state stability
Solid State Communications 248 (2016) 140–143
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Fast-track communication
Nonlinear spin-torque microwave resonance near the loss of spin state stability
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⁎
A.F. Popkova,b, N.E. Kulaginc, G.D. Demina,b, a b c
Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Moscow, Russia National Research University of Electronic Technology, 124498, Zelenograd, Moscow Russia Moscow Aviation Institute (National Research University), 125993 Moscow, Russia
A R T I C L E I N F O
A BS T RAC T
Keywords: C. Magnetic tunnel junction C. Spin-transfer torque D. Spin diode effect D. Microwave sensitivity D. Microwave hysteresis
The paper analyzes microwave resonant response of the spin-torque diode. The considered spin-torque diode is a magnetic tunnel junction with a nano-pillar structure. The magnetization of the free layer has a tilt caused by an action of the inclined magnetic field in the plane of the structure. Taking into account the effect of spin torque transfer we define stability regions of stationary states of magnetization in the free layer as a function of the azimuth angle of the magnetic field and bias DC current. Microwave volt-watt sensitivity of the spin diode for the obtained stationary states is calculated. It is shown that non-linear frequency shift of the resonance line width limits maximum sensitivity of the spin diode near the critical bias current corresponding to the point of the transition to the self-oscillating mode. Besides that, overlapping of frequency branches occurs in the resonant response as the critical point approach, which is different from the foldover effect in the nonlinear ferromagnetic resonance induced by an alternating magnetic field.
1. Introduction Magnetic tunnel junction (MTJ), which is promising to create a new generation of non-volatile memory, demonstrates a spin diode rectification effect of the microwave signal [1–10]. Due to the giant magnetoresistance and spin torque transfer effects spin diode, in principle, allows to reach a thermodynamic limit of the volt-watt sensitivity. The effect of the DC voltage occurrence at the MTJ under microwave current has been analyzed for different mechanisms of spin torque effects, bias current values, linear and nonlinear oscillation modes (see. e.g. [5,9,10] and cit. lit.). It was shown that volt-watt sensitivity of the spin-torque diode increases with the increase of a DC bias by more than an order of value for a weak deviation from the mutually perpendicular geometry of magnetization in magnetic layers of the MTJ. The increase may exceed ultimate sensitivity of the semiconductor Schottky diode [5,10]. In Ref. [5] the increase of the microwave sensitivity was associated with the drop of the resonance line width to zero, when the DC bias approaches critical value. The effects of instability have been observed in the vicinity of the transition to the oscillatory regime, mechanisms of which are not entirely clear. Some questions of microwave sensitivity limits for different modes of spin self-oscillation in the supercritical region were considered in [9]. Another mechanism of the sensitivity behavior in the region of spin
⁎
self-oscillation associated with the injection locking effects was investigated in [10]. The microwave sensitivity limitations of the spin-torque diode before the transition to a self-oscillations mode under the DC bias for different tilted states of its magnetization we analyze below. We have shown that microwave spin diode sensitivity is limited by non-linear shift of the resonance line width near the critical bias current, when the line width of forced oscillations tends to zero in linear approximation. At the same time there is a hysteresis effect near the resonance frequency, which is significantly different from the non-linear effect of overlapping of frequency branches in the ferromagnetic resonance induced by an alternating magnetic field. 2. Basic equations Let us consider the spin diode with a magnetization tilt in the free layer and fixed magnetization in the reference layer having cylindrical nano-pillar symmetry as shown in Fig. 1. Similar structure was considered in [6] within the framework of the linear theory and without bias current. We assume that nano-pillar diameter D of the spin diode is comparable to the exchange length in its magnetic layers and therefore spin dynamics is described in the macrospin approximation. In this case we can neglect the inhomogeneous modes of spin
Corresponding author at: Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Moscow Russia. E-mail address: [email protected] (G.D. Demin).
http://dx.doi.org/10.1016/j.ssc.2016.10.004 Received 15 July 2016; Received in revised form 7 August 2016; Accepted 3 October 2016 Available online 04 October 2016 0038-1098/ © 2016 Elsevier Ltd. All rights reserved.
Solid State Communications 248 (2016) 140–143
A.F. Popkov et al.
diode resistances for parallel and anti-parallel magnetization states correspondingly. The dissipated power P0 on the resistance R is
P0 = R S 2Jp2
β12 cos2 (ωt ) [1 + ρm (t ) ep ]
, (4)
On the other side, the input power is related to the dissipated power 4Z 0 R by the well-known matching ratio P0 = Pin 2 . Taking into account (Z 0 + R )
this expression, the microwave sensitivity of the spin diode becomes
<ΔV > =A ε= Pin
β1 cos (ωt ) [1 + ρm (t ) ep] β12 cos 2 (ωt )
,
[1 + ρm (t ) ep]
where A =
(5)
1 4R Z 0 . Jp S (R + Z 0 )2
We take the following parameters of the spin diode structure. d = 2 nm , D = 140 nm , S = π 70 × 70 nm2 = 1.5⋅10 4 nm2 , Ms = 10 6 A/m, SR = SdV / dI = 10 Ω μm2 , R = 700 Ω ,ΔR / R = 130%, η||0 = 0.63, η⊥0 = 0.3, keff = −4πNz = −12 , d = 2πd2 / D = 0.27, ρ = (η||0 )2 = 0.4 ,α = 0.01, kb = 4π (Ny − Nx ) = 0 . Chosen resistive parameters are close to those of the spin-torque diode structure investigated in [10], but for different anisotropy parameters. For these parameters the frequency is normalized by [ω] = 3.2 GHz , current density is normalized by Jp = 6.4⋅1011 А/m2 , sensitivity is normalized by the factor A = 26 mV/mW . The magnitude of the external magnetic field is regarded as fixed and equal to H = 3.98⋅10 4 A/m , or in normalized units h = 0.5. Magnetic field direction in the first magnetic layer is fixed in the layer plane, and the azimuth angle ϕ1 is varied. The direction of the magnetization in the second layer is fixed in the layer plane by ϕ2 = ϕp = 0 .
Fig. 1. (Color online) Considered nano-pillar magnetic tunneling diode with the tilt of the magnetization in the free magnetic layer and fixed magnetization in the reference layer.
oscillations. In general case one should take into account spin time variation in both magnetic layers. However, if the thickness of the upper and lower layers is substantially different, i.e. d1 < < d2 , and the exchange bias field in the reference layer is sufficiently large, magnetization oscillations in the reference layer are negligible near the spintorque resonance for the free layer. In this case it is possible to adopt the approximation in which the magnetization of the second layer is fixed i.e. m2 (t ) ≈ ep = const , where the vector ep is directed along the pinning field in the second layer. Dynamical system is simplified and is described by the Landau-Lifshitz-Gilbert -Slonczewski equation for the free layer
3. Stability regions of magnetization states in the anglecurrent plane in the absence of microwave irradiation
ṁ = −m × heff + α m × ṁ − η β (t ) m × [m × e p] − η⊥ β (t )[m × e p], (1)
Based on the fact that the microwave sensitivity maximum is reached near the critical lines of loss of stationary states stability, it is necessary to choose the optimal angle of inclination of the magnetic field in accordance with the “angle - current” diagram of the transition to the self-oscillating mode. In this connection we first perform linear dynamic analysis of the system in the absence of an external alternating current, i.e. for β1 = 0 . Qualitative analysis of the dynamic system described by Eq. (1) is carried out by classification of the stationary equilibrium points types after considering characteristic equation for the linearized system near stationary points. Fig. 2 presents the result of the bifurcation analysis of the dynamical system considered for the selected account parameters. It shows stability regions and critical lines on the “angle-current” diagram. It is seen that there are regions in which macro spin equilibrium points are absolutely absent (region II), and only macrospin self-oscillations are possible. The line width of spin-torque resonance tends to zero near the critical lines of transition from the static to the self-oscillating mode (between regions I and II, see e.g. Ref. [5]), and therefore the sensitivity of the microwave excitation should have a maximum. Note that there is no region of the spin precession in the vicinity of angles close to 90° , where microwave sensitivity at zero DC bias reaches a maximum in the frame of linear theory. On the other hand, microwave response will decrease due to the fall of the magnetoresistance amplitude1 to zero when approaching a state of magnetization of layers to the collinear geometry. In this connection the optimum angle of inclination corresponding to the maximum sensitivity is close to either left border of the upper region of instability or right edge of the bottom region. We
M
where heff = keff (m⋅ e n) e n + h − κ e p , β (t ) = β0 + β1 cos(ωt ), m = M , s H h = M is the external magnetic field, normalized by the saturation s J Ms , β = J is the spin-polarized current, normalized by magnetization p 2edM2 Jp = ℏ s , where d is the thickness of the free magnetic layer, e is the electron charge, ℏ is the Planck constant, η||, η⊥ are the spin-polarized prefactors. The time is normalized by t0 = (γ Ms )−1, where γ is the gyromagnetic ratio and the frequency is normalized by [ω] = γ Ms . The spin-polarized prefactors that determine spin-torque components transferred by the current depend on the relative magnetization directions in the layers of the structure according to the relations
η|| =
η||0 1 + ρm⋅ep
, η⊥ =
η⊥0 1 + ρm⋅ep
,
(2)
(η||0 )2 ,
keff , kb are the parameters of in-plane and basic where ρ = anisotropy, κ ~2πd2 / D is the coefficient of the magneto-dipole interaction. We assume β = β0 + β1 cos ωt , where β0 is a DC bias current and β1 is the AC current amplitude, which is proportional to the square root of the input signal power Pin according to the relation β1 =
2 2Pin Z 0 , (R + Z 0 ) SJp
where S is the cross-sectional area of the spin-torque diode, R is the diode resistance, Z 0 = 50 Om is the impedance of the line. < ΔV > The sensitivity of the spin diode equals to ε = P , where <ΔV > is in the time-averaged voltage on the spin diode, andPin is the incident microwave power. In the frame of the proposed model the average voltage is determined by the formula
<ΔV (t ) > = Jp R S where R −1 =
β1 cos(ωt ) [1 + ρm (t ) ep]
−1 + R −1 R↑↑ ↑↓
2
, ρ=
R↑↓ − R↑↑ R↑↓ + R↑↑
, (3)
1 It is easy to show that the microwave sensitivity ε for zero DC bias of spin diode having the tilt of the magnetization in the free layer by an angle ϕ in the linear theory is proportional to sin2 ϕ .
, R↑↑ and R↑↓ are the spin-torque 141
Solid State Communications 248 (2016) 140–143
A.F. Popkov et al.
Fig. 4. (Color online) Frequency hysteresis of microwave sensitivity at the critical current Ic = 1.18 mA ( β0с = 0.123) for ϕ1 = 120° . The inset shows full frequency depen-
Fig. 2. (Color online) Spin states stability regions for the dynamical system described by the Eq. (1) on the “angle-current” plane: I is the stability region of the inclined spin states, II is the region of spin precession (self –oscillation state), III and IV are regions of additional equilibrium tilted states.
dence of the sensitivity, demonstrating overlapping and self-intersection of stable and unstable resonance branches, responsible for the frequency hysteresis.
⎡ ∂Δω ⎤−1/3 δϕ → С1/3 ⎢⎣ ⎥ . ∂(δϕ)2 ⎦ Monotonic increase of the sensitivity with the growth of bias current continues until the width of resonance peak becomes comparable to the non-linear shift in the line width. Besides that, the bias current increase near the critical point leads to the unusual hysteresis effect of frequency dependence of microwave sensitivity. Fig. 4 shows the nonlinear change in the shape of the resonance curve with increasing oscillation amplitude, which gives rise to the ambiguity of the frequency response (amplitude hysteresis). Attention is drawn to the occurrence of self-intersection of the lower frequency branches of the left resonance peak with increasing bias current, as shown in the inset to Fig. 4. Below the self-intersection point resonance branches become unstable, which is the reason why negative resonance peak is abruptly transferred to the right neighboring peak. This feature distinguishes the nonlinear spin-torque resonance from the non-linear ferromagnetic resonance demonstrating foldover effect under the high alternating magnetic field [11–13]. In the latter case there is only overlapping and not intersection of resonance branches accompanying by the sign change of the resonance peak. A further sensitivity limiting factor at the critical current point is the growth of an influence of thermal fluctuations [5], which is not taken into account here. Thermal fluctuations may lead to spontaneous transitions from one to another stable branch of resonant oscillations. Away from the critical line the sensitivity falls sharply and, as already noted, “angle-current” diagram has tilt angle range, where there is no transition from the tilted states to the precession. In this range giant sensitivity cannot be obtained by changing the bias current. The variation of the angle of inclination changes critical current and maximum sensitivity. Table 1 shows calculated sensitivity at bias currents up to the critical values for different tilting angles of the
consider the effect of giant increasing sensitivity for the angle ϕ1 = 120°. 4. Microwave sensitivity near the critical point Numerical analysis of the periodic solutions shows that under DC bias there is sharp increase in the microwave sensitivity at the approach to the critical point of the spin state stability. Fig. 3 shows frequency dependence of the sensitivity for three increasing values of current I = 0 , I = 0.912 mA , I = 1.152 mA at ϕ1 = 120°. Resonance frequency changes slightly near the value fres ~9.2 GHz with the current increase. Calculations show that microwave sensitivity increases by more than an order of magnitude when the bias current approaches the critical line. As was noted above the sensitivity growth in this case is caused by line width of the spin resonance approaching zero Δω (J , ϕ1) → 0 . An unlimited increase in sensitivity is confined by the effect of nonlinear shift of the resonance line width with the amplitude growth. Nonlinear ∂ω0 2 effects of the resonance frequency shift ω0 (J , δϕ) = ω0 + 2 (δϕ ) and ∂(δϕ)
the line width Δω (J , δϕ) = Δω0 (J ) +
∂Δω ∂(δϕ)2
(δϕ)2 with the oscillation am-
plitude δϕ are associated with nonlinear offset of anisotropy and magnetostatic fields averaged during spin precession. If we consider that maximum amplitude of the spin resonance is inversely proportional to the C line width, i.e. δϕ~ Δω , then amplitude of oscillation at the point of vanishing of the line width in the linear approximation Δω0 (J ) → 0 remains finite due to the effect of the nonlinear shift, so that
Table 1 Extreme (peak) microwave spin diode sensitivity, calculated for different angles of the magnetizations tilt. For angles of 30, 120 and 150° maximum bias currents correspond to the critical values. I, mA
ε (0°) mV/ mW
−4.8
0 0 0 0 0 0 0
−2.03 0.0 +0.71 +1.18 +4.8
Fig. 3. (Color online) Frequency dependence of the microwave sensitivity for the tilted by ϕ1 = 120° magnetic field for different bias currents: 1 - I = 0 ( β0 = 0.0 ), 2 -
I = 0.912 mA ( β0 = 0.095), 3 - I = 1.152 mA ( β0 = 0.12 ). AC amplitude in calculations is
β1 = 0.000156 , which corresponds to the input microwave power Pin ~0.003 μW .
142
ε (30°) mV/ mW
+6550 −52
ε (60°) mV/ mW
ε (90°) mV/mW
−52
−19,5
−104
−104
−26
−101,4
ε (120°) mV/mW
−65 +19750
ε (150°) mV/mW
−22,7 +1820
ε (180°) mV/mW
0 0 0 0 0 0 0
Solid State Communications 248 (2016) 140–143
A.F. Popkov et al.
additional layer of an antiferromagnet can play the role of the external magnetic field in the free layer [14]. In the latter case it is possible to make highly sensitive microwave detectors without an external magnetic field, which may have prospects in the microwave holography and other fields [15–17].
magnetization. It is seen that in the range of angles where there are no transition points to the self-oscillation mode high sensitivity cannot be achieved. For the collinear geometry the sensitivity drops to zero. 5. Conclusion
Acknowledgement
Thus, the microwave sensitivity of the spin diode is analyzed on the basis of macrospin model in dependence on the azimuth angle of the free layer magnetization and the bias current. We have shown that maximum sensitivity for the fixed magnetization in the reference layer is not achieved at 90° of the magnetization tilt in the free layer, but only when the angular inclination and bias current are close to the critical lines on the “angle – current” diagram, which correspond to the boundaries of stability. This is due to the fact that spin-polarized bias current for inter-perpendicular geometry cannot cause the loss of stability of the stationary states when magnetic damping is compensated by spin torque pumping. On the other hand, magnetoresistance modulation by the alternating current tends to zero in the collinear magnetization geometry, so sensitivity also goes to zero. There is a finite inclination angle range, where the transition from stable spin state to the self-oscillatory regime responsible for the dramatic increase in sensitivity does not exist for any bias currents. The effects of overlapping and mutual intersection of frequency branches of forced microwave oscillations occur when critical lines of the spin states stability loss are approached and hysteresis phenomena arise at frequency scanning. These effects differ significantly from nonlinear ferromagnetic resonance phenomena under alternating magnetic field for the case of Lorentz line shape of the resonance peak at low pumping (foldover effect). To conclude we should say that self-oscillations arising at the supercritical bias current can lead to the complex phenomena, which can cause instability in the observed sensitivity of the spin diode, which requires special consideration. The second aspect of the problem, which also requires additional analysis is the impact of the second layer oscillation. Account of its impact can significantly skew the results obtained due to the occurrence of nonlinear interference of the selfoscillations in both layers. This is especially important for comparable thicknesses of both layers. It should be noted that the exchange pinning field created by an
The authors are grateful to K.A. Zvezdin, P.A. Skirdkov and A.V. Lobachev for the fruitful discussion. The work was supported by the Russian Science Foundation, project number 16-19-00181. References [1] A.A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D.D. Djayaprawira, N. Watanabe, S. Yuasa, Nature 438 (2005) 339. [2] J.C. Sankey, P.M. Braganca, A.G.F. Garcia, I.N. Krivorotov, R.A. Buhrman, D.C. Ralph, Phys. Rev. Lett. 96 (2006) 227601. [3] J.N. Kupferschmidt, S. Adam, P.W. Brouwer, Phys. Rev. B 74 (2006) 134416. [4] X. Cheng, J.A. Katine, G. Rowlands, I.N. Krivorotov, Appl. Phys. Lett. 103 (2013) 082402. [5] S. Miwa, S. Ishibashi, H. Tomita, T. Nozaki, E. Tamura, K. Ando, N. Mizuochi, T. Saruya, H. Kubota, K. Yakushij, T. Taniguchi, H. Imamura, A. Fukushima, S. Yuasa, Y. Suzuki, Nat. Mater. 13 (2014) 50. [6] C. Wang, Y.-T. Cui, J.Z. Sun, J.A. Katine, R.A. Buhrman, D.C. Ralph, J. Appl. Phys. 106 (2009) 053905. [7] T. Zeng, Y. Zhou, K.W. Lin, P.T. Lai, P.W.T. Pong, IEEE Trans. Magn. 51 (2015) 1401204. [8] J. Zhu, J.A. Katine, G.E. Rowlands, Y.-J. Chen, Z. Duan, J.G. Alzate, P. Upadhyaya, J. Langer, P.K. Amiri, K.L. Wang, I.N. Krivorotov, Phys. Rev. Lett. 108 (2012) 197203. [9] O.V. Prokopenko, I.N. Krivorotov, E. Bankowski, T. Meitzler, S. Jaroch, V.S. Tiberkevich, A.N. Slavin, J. Appl. Phys. 111 (2012) 123904. [10] B. Fang, M. Carpentieri, X. Hao, H. Jiang, J.A. Katine, I.N. Krivorotov, B. Ocker, J. Langer, K.L. Wang, B. Zhang, B. Azzerboni, P.K. Amiri, G. Finocchio, Z. Zeng, Nat. Commun. 7 (2016) 11259. [11] P.W. Anderson, H. Suhl, Phys. Rev. 100 (1955) 1788. [12] H. Suhl, J. Appl. Phys. 31 (1960) 935. [13] Y.S. Gui, A. Wirthmann, N. Mecking, C.M. Hu, Phys. Rev. B 80 (2009) 060402. [14] A.E. Berkowitz, K. Takano, J. Magn. Magn. Mater. 200 (1999) 552. [15] L. Fu, Y.S. Gui, L.H. Bai, H. Guo, H. Abou-Rachid, C.-M. Hu, J. Appl. Phys. 117 (2015) 213902. [16] B.M. Yao, L. Fu, X.S. Chen, W. Lu, L.H. Bai, Y.S. Gui, C.-M. Hu, Appl. Phys. Lett. 104 (2014) 062408. [17] S. Hemour, Y. Zhao, C.H.P. Lorenz, D. Houssameddine, Y. Gui, C.-M. Hu, K. Wu, IEEE Trans. Microw. Theory Tech. 62 (2014) 965.
143
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Fast-track communication
Nonlinear spin-torque microwave resonance near the loss of spin state stability
crossmark
⁎
A.F. Popkova,b, N.E. Kulaginc, G.D. Demina,b, a b c
Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Moscow, Russia National Research University of Electronic Technology, 124498, Zelenograd, Moscow Russia Moscow Aviation Institute (National Research University), 125993 Moscow, Russia
A R T I C L E I N F O
A BS T RAC T
Keywords: C. Magnetic tunnel junction C. Spin-transfer torque D. Spin diode effect D. Microwave sensitivity D. Microwave hysteresis
The paper analyzes microwave resonant response of the spin-torque diode. The considered spin-torque diode is a magnetic tunnel junction with a nano-pillar structure. The magnetization of the free layer has a tilt caused by an action of the inclined magnetic field in the plane of the structure. Taking into account the effect of spin torque transfer we define stability regions of stationary states of magnetization in the free layer as a function of the azimuth angle of the magnetic field and bias DC current. Microwave volt-watt sensitivity of the spin diode for the obtained stationary states is calculated. It is shown that non-linear frequency shift of the resonance line width limits maximum sensitivity of the spin diode near the critical bias current corresponding to the point of the transition to the self-oscillating mode. Besides that, overlapping of frequency branches occurs in the resonant response as the critical point approach, which is different from the foldover effect in the nonlinear ferromagnetic resonance induced by an alternating magnetic field.
1. Introduction Magnetic tunnel junction (MTJ), which is promising to create a new generation of non-volatile memory, demonstrates a spin diode rectification effect of the microwave signal [1–10]. Due to the giant magnetoresistance and spin torque transfer effects spin diode, in principle, allows to reach a thermodynamic limit of the volt-watt sensitivity. The effect of the DC voltage occurrence at the MTJ under microwave current has been analyzed for different mechanisms of spin torque effects, bias current values, linear and nonlinear oscillation modes (see. e.g. [5,9,10] and cit. lit.). It was shown that volt-watt sensitivity of the spin-torque diode increases with the increase of a DC bias by more than an order of value for a weak deviation from the mutually perpendicular geometry of magnetization in magnetic layers of the MTJ. The increase may exceed ultimate sensitivity of the semiconductor Schottky diode [5,10]. In Ref. [5] the increase of the microwave sensitivity was associated with the drop of the resonance line width to zero, when the DC bias approaches critical value. The effects of instability have been observed in the vicinity of the transition to the oscillatory regime, mechanisms of which are not entirely clear. Some questions of microwave sensitivity limits for different modes of spin self-oscillation in the supercritical region were considered in [9]. Another mechanism of the sensitivity behavior in the region of spin
⁎
self-oscillation associated with the injection locking effects was investigated in [10]. The microwave sensitivity limitations of the spin-torque diode before the transition to a self-oscillations mode under the DC bias for different tilted states of its magnetization we analyze below. We have shown that microwave spin diode sensitivity is limited by non-linear shift of the resonance line width near the critical bias current, when the line width of forced oscillations tends to zero in linear approximation. At the same time there is a hysteresis effect near the resonance frequency, which is significantly different from the non-linear effect of overlapping of frequency branches in the ferromagnetic resonance induced by an alternating magnetic field. 2. Basic equations Let us consider the spin diode with a magnetization tilt in the free layer and fixed magnetization in the reference layer having cylindrical nano-pillar symmetry as shown in Fig. 1. Similar structure was considered in [6] within the framework of the linear theory and without bias current. We assume that nano-pillar diameter D of the spin diode is comparable to the exchange length in its magnetic layers and therefore spin dynamics is described in the macrospin approximation. In this case we can neglect the inhomogeneous modes of spin
Corresponding author at: Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Moscow Russia. E-mail address: [email protected] (G.D. Demin).
http://dx.doi.org/10.1016/j.ssc.2016.10.004 Received 15 July 2016; Received in revised form 7 August 2016; Accepted 3 October 2016 Available online 04 October 2016 0038-1098/ © 2016 Elsevier Ltd. All rights reserved.
Solid State Communications 248 (2016) 140–143
A.F. Popkov et al.
diode resistances for parallel and anti-parallel magnetization states correspondingly. The dissipated power P0 on the resistance R is
P0 = R S 2Jp2
β12 cos2 (ωt ) [1 + ρm (t ) ep ]
, (4)
On the other side, the input power is related to the dissipated power 4Z 0 R by the well-known matching ratio P0 = Pin 2 . Taking into account (Z 0 + R )
this expression, the microwave sensitivity of the spin diode becomes
<ΔV > =A ε= Pin
β1 cos (ωt ) [1 + ρm (t ) ep] β12 cos 2 (ωt )
,
[1 + ρm (t ) ep]
where A =
(5)
1 4R Z 0 . Jp S (R + Z 0 )2
We take the following parameters of the spin diode structure. d = 2 nm , D = 140 nm , S = π 70 × 70 nm2 = 1.5⋅10 4 nm2 , Ms = 10 6 A/m, SR = SdV / dI = 10 Ω μm2 , R = 700 Ω ,ΔR / R = 130%, η||0 = 0.63, η⊥0 = 0.3, keff = −4πNz = −12 , d = 2πd2 / D = 0.27, ρ = (η||0 )2 = 0.4 ,α = 0.01, kb = 4π (Ny − Nx ) = 0 . Chosen resistive parameters are close to those of the spin-torque diode structure investigated in [10], but for different anisotropy parameters. For these parameters the frequency is normalized by [ω] = 3.2 GHz , current density is normalized by Jp = 6.4⋅1011 А/m2 , sensitivity is normalized by the factor A = 26 mV/mW . The magnitude of the external magnetic field is regarded as fixed and equal to H = 3.98⋅10 4 A/m , or in normalized units h = 0.5. Magnetic field direction in the first magnetic layer is fixed in the layer plane, and the azimuth angle ϕ1 is varied. The direction of the magnetization in the second layer is fixed in the layer plane by ϕ2 = ϕp = 0 .
Fig. 1. (Color online) Considered nano-pillar magnetic tunneling diode with the tilt of the magnetization in the free magnetic layer and fixed magnetization in the reference layer.
oscillations. In general case one should take into account spin time variation in both magnetic layers. However, if the thickness of the upper and lower layers is substantially different, i.e. d1 < < d2 , and the exchange bias field in the reference layer is sufficiently large, magnetization oscillations in the reference layer are negligible near the spintorque resonance for the free layer. In this case it is possible to adopt the approximation in which the magnetization of the second layer is fixed i.e. m2 (t ) ≈ ep = const , where the vector ep is directed along the pinning field in the second layer. Dynamical system is simplified and is described by the Landau-Lifshitz-Gilbert -Slonczewski equation for the free layer
3. Stability regions of magnetization states in the anglecurrent plane in the absence of microwave irradiation
ṁ = −m × heff + α m × ṁ − η β (t ) m × [m × e p] − η⊥ β (t )[m × e p], (1)
Based on the fact that the microwave sensitivity maximum is reached near the critical lines of loss of stationary states stability, it is necessary to choose the optimal angle of inclination of the magnetic field in accordance with the “angle - current” diagram of the transition to the self-oscillating mode. In this connection we first perform linear dynamic analysis of the system in the absence of an external alternating current, i.e. for β1 = 0 . Qualitative analysis of the dynamic system described by Eq. (1) is carried out by classification of the stationary equilibrium points types after considering characteristic equation for the linearized system near stationary points. Fig. 2 presents the result of the bifurcation analysis of the dynamical system considered for the selected account parameters. It shows stability regions and critical lines on the “angle-current” diagram. It is seen that there are regions in which macro spin equilibrium points are absolutely absent (region II), and only macrospin self-oscillations are possible. The line width of spin-torque resonance tends to zero near the critical lines of transition from the static to the self-oscillating mode (between regions I and II, see e.g. Ref. [5]), and therefore the sensitivity of the microwave excitation should have a maximum. Note that there is no region of the spin precession in the vicinity of angles close to 90° , where microwave sensitivity at zero DC bias reaches a maximum in the frame of linear theory. On the other hand, microwave response will decrease due to the fall of the magnetoresistance amplitude1 to zero when approaching a state of magnetization of layers to the collinear geometry. In this connection the optimum angle of inclination corresponding to the maximum sensitivity is close to either left border of the upper region of instability or right edge of the bottom region. We
M
where heff = keff (m⋅ e n) e n + h − κ e p , β (t ) = β0 + β1 cos(ωt ), m = M , s H h = M is the external magnetic field, normalized by the saturation s J Ms , β = J is the spin-polarized current, normalized by magnetization p 2edM2 Jp = ℏ s , where d is the thickness of the free magnetic layer, e is the electron charge, ℏ is the Planck constant, η||, η⊥ are the spin-polarized prefactors. The time is normalized by t0 = (γ Ms )−1, where γ is the gyromagnetic ratio and the frequency is normalized by [ω] = γ Ms . The spin-polarized prefactors that determine spin-torque components transferred by the current depend on the relative magnetization directions in the layers of the structure according to the relations
η|| =
η||0 1 + ρm⋅ep
, η⊥ =
η⊥0 1 + ρm⋅ep
,
(2)
(η||0 )2 ,
keff , kb are the parameters of in-plane and basic where ρ = anisotropy, κ ~2πd2 / D is the coefficient of the magneto-dipole interaction. We assume β = β0 + β1 cos ωt , where β0 is a DC bias current and β1 is the AC current amplitude, which is proportional to the square root of the input signal power Pin according to the relation β1 =
2 2Pin Z 0 , (R + Z 0 ) SJp
where S is the cross-sectional area of the spin-torque diode, R is the diode resistance, Z 0 = 50 Om is the impedance of the line. < ΔV > The sensitivity of the spin diode equals to ε = P , where <ΔV > is in the time-averaged voltage on the spin diode, andPin is the incident microwave power. In the frame of the proposed model the average voltage is determined by the formula
<ΔV (t ) > = Jp R S where R −1 =
β1 cos(ωt ) [1 + ρm (t ) ep]
−1 + R −1 R↑↑ ↑↓
2
, ρ=
R↑↓ − R↑↑ R↑↓ + R↑↑
, (3)
1 It is easy to show that the microwave sensitivity ε for zero DC bias of spin diode having the tilt of the magnetization in the free layer by an angle ϕ in the linear theory is proportional to sin2 ϕ .
, R↑↑ and R↑↓ are the spin-torque 141
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A.F. Popkov et al.
Fig. 4. (Color online) Frequency hysteresis of microwave sensitivity at the critical current Ic = 1.18 mA ( β0с = 0.123) for ϕ1 = 120° . The inset shows full frequency depen-
Fig. 2. (Color online) Spin states stability regions for the dynamical system described by the Eq. (1) on the “angle-current” plane: I is the stability region of the inclined spin states, II is the region of spin precession (self –oscillation state), III and IV are regions of additional equilibrium tilted states.
dence of the sensitivity, demonstrating overlapping and self-intersection of stable and unstable resonance branches, responsible for the frequency hysteresis.
⎡ ∂Δω ⎤−1/3 δϕ → С1/3 ⎢⎣ ⎥ . ∂(δϕ)2 ⎦ Monotonic increase of the sensitivity with the growth of bias current continues until the width of resonance peak becomes comparable to the non-linear shift in the line width. Besides that, the bias current increase near the critical point leads to the unusual hysteresis effect of frequency dependence of microwave sensitivity. Fig. 4 shows the nonlinear change in the shape of the resonance curve with increasing oscillation amplitude, which gives rise to the ambiguity of the frequency response (amplitude hysteresis). Attention is drawn to the occurrence of self-intersection of the lower frequency branches of the left resonance peak with increasing bias current, as shown in the inset to Fig. 4. Below the self-intersection point resonance branches become unstable, which is the reason why negative resonance peak is abruptly transferred to the right neighboring peak. This feature distinguishes the nonlinear spin-torque resonance from the non-linear ferromagnetic resonance demonstrating foldover effect under the high alternating magnetic field [11–13]. In the latter case there is only overlapping and not intersection of resonance branches accompanying by the sign change of the resonance peak. A further sensitivity limiting factor at the critical current point is the growth of an influence of thermal fluctuations [5], which is not taken into account here. Thermal fluctuations may lead to spontaneous transitions from one to another stable branch of resonant oscillations. Away from the critical line the sensitivity falls sharply and, as already noted, “angle-current” diagram has tilt angle range, where there is no transition from the tilted states to the precession. In this range giant sensitivity cannot be obtained by changing the bias current. The variation of the angle of inclination changes critical current and maximum sensitivity. Table 1 shows calculated sensitivity at bias currents up to the critical values for different tilting angles of the
consider the effect of giant increasing sensitivity for the angle ϕ1 = 120°. 4. Microwave sensitivity near the critical point Numerical analysis of the periodic solutions shows that under DC bias there is sharp increase in the microwave sensitivity at the approach to the critical point of the spin state stability. Fig. 3 shows frequency dependence of the sensitivity for three increasing values of current I = 0 , I = 0.912 mA , I = 1.152 mA at ϕ1 = 120°. Resonance frequency changes slightly near the value fres ~9.2 GHz with the current increase. Calculations show that microwave sensitivity increases by more than an order of magnitude when the bias current approaches the critical line. As was noted above the sensitivity growth in this case is caused by line width of the spin resonance approaching zero Δω (J , ϕ1) → 0 . An unlimited increase in sensitivity is confined by the effect of nonlinear shift of the resonance line width with the amplitude growth. Nonlinear ∂ω0 2 effects of the resonance frequency shift ω0 (J , δϕ) = ω0 + 2 (δϕ ) and ∂(δϕ)
the line width Δω (J , δϕ) = Δω0 (J ) +
∂Δω ∂(δϕ)2
(δϕ)2 with the oscillation am-
plitude δϕ are associated with nonlinear offset of anisotropy and magnetostatic fields averaged during spin precession. If we consider that maximum amplitude of the spin resonance is inversely proportional to the C line width, i.e. δϕ~ Δω , then amplitude of oscillation at the point of vanishing of the line width in the linear approximation Δω0 (J ) → 0 remains finite due to the effect of the nonlinear shift, so that
Table 1 Extreme (peak) microwave spin diode sensitivity, calculated for different angles of the magnetizations tilt. For angles of 30, 120 and 150° maximum bias currents correspond to the critical values. I, mA
ε (0°) mV/ mW
−4.8
0 0 0 0 0 0 0
−2.03 0.0 +0.71 +1.18 +4.8
Fig. 3. (Color online) Frequency dependence of the microwave sensitivity for the tilted by ϕ1 = 120° magnetic field for different bias currents: 1 - I = 0 ( β0 = 0.0 ), 2 -
I = 0.912 mA ( β0 = 0.095), 3 - I = 1.152 mA ( β0 = 0.12 ). AC amplitude in calculations is
β1 = 0.000156 , which corresponds to the input microwave power Pin ~0.003 μW .
142
ε (30°) mV/ mW
+6550 −52
ε (60°) mV/ mW
ε (90°) mV/mW
−52
−19,5
−104
−104
−26
−101,4
ε (120°) mV/mW
−65 +19750
ε (150°) mV/mW
−22,7 +1820
ε (180°) mV/mW
0 0 0 0 0 0 0
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additional layer of an antiferromagnet can play the role of the external magnetic field in the free layer [14]. In the latter case it is possible to make highly sensitive microwave detectors without an external magnetic field, which may have prospects in the microwave holography and other fields [15–17].
magnetization. It is seen that in the range of angles where there are no transition points to the self-oscillation mode high sensitivity cannot be achieved. For the collinear geometry the sensitivity drops to zero. 5. Conclusion
Acknowledgement
Thus, the microwave sensitivity of the spin diode is analyzed on the basis of macrospin model in dependence on the azimuth angle of the free layer magnetization and the bias current. We have shown that maximum sensitivity for the fixed magnetization in the reference layer is not achieved at 90° of the magnetization tilt in the free layer, but only when the angular inclination and bias current are close to the critical lines on the “angle – current” diagram, which correspond to the boundaries of stability. This is due to the fact that spin-polarized bias current for inter-perpendicular geometry cannot cause the loss of stability of the stationary states when magnetic damping is compensated by spin torque pumping. On the other hand, magnetoresistance modulation by the alternating current tends to zero in the collinear magnetization geometry, so sensitivity also goes to zero. There is a finite inclination angle range, where the transition from stable spin state to the self-oscillatory regime responsible for the dramatic increase in sensitivity does not exist for any bias currents. The effects of overlapping and mutual intersection of frequency branches of forced microwave oscillations occur when critical lines of the spin states stability loss are approached and hysteresis phenomena arise at frequency scanning. These effects differ significantly from nonlinear ferromagnetic resonance phenomena under alternating magnetic field for the case of Lorentz line shape of the resonance peak at low pumping (foldover effect). To conclude we should say that self-oscillations arising at the supercritical bias current can lead to the complex phenomena, which can cause instability in the observed sensitivity of the spin diode, which requires special consideration. The second aspect of the problem, which also requires additional analysis is the impact of the second layer oscillation. Account of its impact can significantly skew the results obtained due to the occurrence of nonlinear interference of the selfoscillations in both layers. This is especially important for comparable thicknesses of both layers. It should be noted that the exchange pinning field created by an
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