Abstract
Modulation of the energy landscape by external perturbations governs various thermallyactivated phenomena, described by the Arrhenius law. Thermal fluctuation of nanoscale magnetic tunnel junctions with spintransfer torque (STT) shows promise for unconventional computing, whereas its rigorous representation, based on the NéelArrhenius law, has been controversial. In particular, the exponents for thermallyactivated switching rate therein, have been inaccessible with conventional thermallystable nanomagnets with decadelong retention time. Here we approach the NéelArrhenius law with STT utilising superparamagnetic tunnel junctions that have high sensitivity to external perturbations and determine the exponents through several independent measurements including homodynedetected ferromagnetic resonance, nanosecond STT switching, and random telegraph noise. Furthermore, we show that the results are comprehensively described by a concept of local bifurcation observed in various physical systems. The findings demonstrate the capability of superparamagnetic tunnel junction as a useful tester for statistical physics as well as sophisticated engineering of probabilistic computing hardware with a rigorous mathematical foundation.
Introduction
A dynamical system is classified by the stability of its potential landscape, especially by the local bifurcation of the dynamical equation. Under finite stochasticity, the Arrhenius law, a general principle for thermallyactivated events, describes a variety of dynamical phenomena ranging from chemical reactions to physical processes. According to the Arrhenius law, the relaxation time τ in staying at a certain state is given by τ = τ_{0} expΔ with the thermal stability factor Δ ≡ E_{0}/k_{B}T, where τ_{0} is the intrinsic time constant of each system, k_{B} the Boltzmann constant, T the absolute temperature, and E_{0} an intrinsic energy barrier for switching to different states without perturbation. Under the perturbation by normalised external input x, E_{0} is replaced by an effective energy barrier E = \({E}_{0}{\left(1x\right)}^{{n}_{x}}\), where the switching exponent \({n}_{x}\) is determined by the effective energy landscape with x. In general, when the dynamical equation dΘ/dt = f(Θ,x) is given, where t is the time and Θ is the state variable (e.g. particle position, spin quantisation direction, amount of unreacted substance, etc.), the types of local bifurcation of f(Θ,x) determines \({n}_{x}\); for example, \({n}_{x}\) = 2 for the pitchfork bifurcation and \({n}_{x}\) = 3/2 for the saddlenode bifurcation^{1}. This suggests that the switching exponent serves as a unique lens for the local structure, especially the stability of the energy landscape under the perturbations in the relevant system.
In magnetic materials, which have served as a model system to study the physics of thermallyactivated phenomena, the basis of the Arrhenius law was built by Néel^{2} and Brown^{3}, known as the Néel–Arrhenius law. For singledomain uniaxial magnets with a magnetic field H applied along the easy axis, E under magnetic field can be simply derived as E = E_{0}(1 – H/H_{K}^{eff})^{2} by the Stoner–Wohlfarth model, where H_{K}^{eff} is the effective magnetic anisotropy field^{4}. However, theoretical studies pointed out that the value of exponent n_{H}, 2 in the above equation, should vary when one considers some realistic factors such as misalignment of magnetic field^{5,6} and higherorder terms of anisotropy^{7}; in other words, the local bifurcation varies with them.
The magnetisation of nanomagnets can also be controlled by spintransfer torque (STT) under current application through the angular momentum transfer^{8,9,10,11,12,13}. The STTinduced magnetisation switching in thermallystable magnetic tunnel junctions (MTJ) is a key ingredient for nonvolatile magnetoresistive random access memory^{14,15,16}. Moreover, recent studies have demonstrated an unconventional paradigm of computing, e.g. neuromorphic computing with population coding^{17}, and probabilistic computing^{18}, which utilises a combinatorial effect of STT and thermal fluctuation in superparamagnetic tunnel junctions (sMTJs). By further combining the effect of external magnetic fields, additional tunabilities of the sMTJs for probabilistic computing have been shown^{19}. In this regard, understanding how the effective energy of sMTJs is characterised under STT, as well as magnetic field, is of significant interest not only from fundamental but also from technological aspects. The Néel–Arrhenius law under STT has been a longstanding question partly because the STT itself does not modulate the energy landscape due to its nonconservative nature, preventing one from defining its effective potential energy. Despite the difficulty, the expectation value of event time of the magnetisation switching, i.e. the Néel relaxation time τ, under field H and current I is phenomenologically expressed in a form:
where τ_{0} is ~1 ns in magnetic systems^{3}, and I_{C0} an intrinsic critical current. Regarding the exponent n_{I} for the factor of current, different values, 1 (refs. ^{20,21}) or 2 (refs. ^{22,23,24}), have been theoretically derived, where the former was obtained by considering a fictitious temperature, whereas the latter was obtained by analysing the stochastic process based on the Fokker–Planck equation. Experimentally, it has been practically inaccessible as far as one examines conventional thermallystable MTJs and consequently, their decadelong unperturbed retention property has been extrapolated from limited data obtained in a reasonable time while assuming a certain number for n_{H} or n_{I}^{25,26,27,28,29,30,31,32,33}. For applications with superparamagnetic tunnel junctions that actively utilise thermal fluctuation under STT, such uncertainty makes sophisticated engineering impractical as a rigorous description of modulation of the effective energy landscape is indispensable.
Here we experimentally study the Néel–Arrhenius law of a nanomagnet under STT utilising superparamagnetic tunnel junctions that allow direct determination of the event time under fields and currents^{34,35,36,37,38,39}. Through measurements of homodynedetected ferromagnetic resonance (FMR) under current biases, highspeed STT switching with various pulse widths, and random telegraph noise (RTN) under various fields and currents, values of n_{H} and n_{I} are uniquely determined for given conditions. Furthermore, we show that by considering the local bifurcations under magnetic field and STT, which have not been considered for magnetic systems, the obtained results can be comprehended with the effects of the torques of the field and current without the difficulties to define the effective potential of STT.
Sample preparation and strategy of following experiments
As shown in Fig. 1a, a stack structure, Ta(5)/Pt(5)/[Co(0.3)/Pt(0.4)]_{7}/Co(0.3)/Ru(0.45)/[Co(0.3)/Pt(0.4)]_{2}/Co(0.3)/Ta(0.3)/CoFeB/MgO(1.0)/CoFeB(t_{CoFeB} = 1.88)/Ta(5)/Ru(5) (numbers in parenthesis are nominal thickness in nm), is deposited by dc/rf magnetron sputtering on a sapphire substrate. The stack possesses essentially the same structure as what was utilised in the demonstration of probabilistic computing^{18}. Resistance (R)area (A) product, RA, of the MgO tunnel barrier is 5.5 Ωμm^{2}. The stack is patterned into MTJ devices, followed by annealing at 300 °C. The MTJ device we will mainly focus on hereafter (device A) has a diameter D of 34 nm (results for device B with t_{CoFeB} = 1.82 nm, RA = 8.1 Ωμm^{2}, and D = 28 nm will be also shown later). In this size range, the magnetisation can be represented by a single vector without significant effects of spatial inhomogeneity for the present stack structure^{16,27,40}. Both CoFeB layers have a perpendicular easy axis. Figure 1b shows the junction resistance R as a function of the perpendicular magnetic field H_{z}. Gradual variation of mean R with H_{z} and scattering of data points at the transition region reflects a superparamagnetic nature of the MTJ whose switching time is shorter than the measurement time of R (~ 1 s).
To determine n_{H} and n_{I} in actual MTJs, we take into account the following two effects: electricfield modulation of magnetic anisotropy^{41,42,43,44} and uncompensated stray field H_{S} from the reference layer. Consequently, Δ, the argument of the exponential in Eq. (1), is rewritten as
with \( {{\it\Delta}}_{0}\equiv {E}_{0}/{k}_{{{{{{\rm{B}}}}}}}T\), \(h({H}_{z},V)\equiv \left({H}_{z}{H}_{{{{{{\rm{S}}}}}}}\right)/{H}_{K}^{{{{{{\rm{eff}}}}}}}(V)\), and \({v}_{{{{{{\rm{P}}}}}}\left({{{{{\rm{AP}}}}}}\right)}(V)\equiv V\!/\!{V}_{{{{{{\rm{C}}}}}}0,{{{{{\rm{P}}}}}}({{{{{\rm{AP}}}}}})}\). V_{C0,P(AP)} denotes the intrinsic critical voltage for STT switching from parallel, P, to antiparallel, AP, states (AP to P states). Because the electricfield effect on anisotropy is governed by the applied voltage V, we use an expression based on voltage input rather than current input. Equation (2) contains so many unknown variables (E_{0}, H_{S}, H_{K}^{eff}(V), V_{C0,P(AP)}, n_{H} and n_{I}) that one cannot directly determine the exponents by only measuring RTN. Thus, in the following, we first separately determine H_{K}^{eff}(V) from a homodynedetected FMR and the next V_{C0} from the STT switching probability. After that, we determine H_{S} and the exponents n_{H} and n_{I} from RTN measurement^{34,35,36,37,38} as a function of V.
Electricfield effect on anisotropy field
Firstly, we determine H_{K}^{eff}(V) from homodynedetected FMR under dc bias voltage^{45,46}. Figure 2a shows the circuit configuration for the measurement. Homodynedetected voltage spectra are measured while sweeping H_{z} at various frequencies and H_{K}^{eff} is determined from the peak position (see Methods and Supplementary Information for details). We perform this measurement at various dc biases and obtain H_{K}^{eff} vs. V, as shown in Fig. 2b. H_{K}^{eff} changes nonlinearly with V. We fit a quadratic equation to the obtained dependence and determine the coefficients for constant, linear, and quadratic terms to be μ_{0}H_{K}^{eff}(0) = 77.0 ± 0.5 mT, μ_{0}dH_{K}^{eff}/dV = −57.8 ± 1.6 mT V^{−1}, and μ_{0}d^{2}H_{K}^{eff}/dV^{2} = −49.9 ± 7.5 mT V^{−2} (μ_{0} is the permeability of vacuum). Note that the constant term represents the magnetic anisotropy field at zero bias whereas the linear and quadratic terms mainly originate from the electricfield modulation of anisotropy and an effect of Joule heating, respectively. The determined coefficients will be used in the analysis of RTN later.
Intrinsic critical voltage for STT switching
Secondly, we determine V_{C0} for the STT switching. Because the fluctuation timescale of the studied system is on the order of milliseconds, we need to measure the junction state right after the switching pulse application when they are still nonvolatile. To this end, we use a circuit configuration shown in Fig. 3a. This configuration is similar to that we used in our previous work to study the switching error rate^{47}; a voltage waveform composed of initialisation, write, and read pulses [Fig. 3b] is applied to the MTJ by an arbitrary waveform generator, and the transmitted signal at the read pulse is monitored by a highspeed oscilloscope to identify the final state of magnetisation configuration. The typical transmitted signals for P and AP states are shown in Fig. 3c. A clear difference is observed in the amplitude of the transmitted signal for different configurations due to the tunnel magnetoresistance. Write and read pulses are separated by 30 ns, which is much shorter than the shortest relaxation time shown later (~0.3 ms), ensuring a sufficiently low readerror rate (unintentional switching probability before the read pulse) <10^{−4} (see Methods). The waveform is applied 200 times repeatedly and switching probability is evaluated.
Figure 3d shows the write pulse voltage dependence of the switching probability with different write pulse durations t_{pulse}. The switching voltage V_{C}, defined as the voltage at 50% probability, is plotted as a function of the inverse of t_{pulse} in Fig. 3e. In the precessional regime (typically t_{pulse} ≲ several nanoseconds) where the switching/nonswitching is determined by an amount of transferred angular momenta, V_{C} is known to linearly depend on t_{pulse}^{−1} and the intercept yields V_{C0}^{20,48}. From a linear fitting, V_{C0,P} and V_{C0,AP} are obtained as 313 ± 45 mV and −247 ± 42 mV, respectively.
Random telegraph noise measurement to determine the switching exponents
With the results above, we are now ready to determine the switching exponents, n_{H} and n_{I}, from RTN measurement under various V and H_{z}. Figures 4a, b show the circuit configurations for the measurement with V ~ 0 and V ≥ 25 mV, respectively. For V ~ 0, we apply a small direct current I = 200 nA and monitor R by an oscilloscope connected in parallel to the MTJ and probe the temporal magnetisation configuration. For V ≥ 25 mV, we monitor the divided voltage at reference resistor R_{r} serially connected to the MTJ using an oscilloscope. Note that R_{r} (= 470 Ω) is set to be much smaller than R to prevent a change in the electricfield effect on the magnetic anisotropy between P and AP states. Figure 4c shows typical results of RTN with various H_{z} and the definition of the magnetisation switching event time t. Figure 4d shows the distribution of the number of unique t for μ_{0}H_{z} = −30.5 mT. As expected, the exponential distribution is confirmed, i.e., the number of events ∝ τ^{−1}exp(t/τ), indicating that the fluctuation is characterised by a Poisson process. From the fitting, expectation values of the event time for P and AP states, i.e., the relaxation time τ_{P} and τ_{AP}, are obtained as a function of H_{z} as shown in Fig. 4e. Subsequently, Δ_{P(AP)} can be determined from the relation τ = τ_{0}expΔ_{P(AP)}.
We measure τ_{P(AP)} and Δ_{P(AP)} for various H_{z} and V. Figure 5a shows the obtained Δ_{P} and Δ_{AP} as a function of H_{z} for various V. Δ_{P(AP)} increases (decreases) with increasing H_{z} for each V, as expected from the energy landscape modulation by H_{z}. Also, the mean Δ gradually decreases with increasing V, which is also consistent with the trend of H_{K}^{eff} shown in Fig. 2b. To derive n_{H} and n_{I}, we then take the natural logarithm of the ratio between Δ_{P} and Δ_{AP}, which can be expressed from Eq. (2) as
ln(Δ_{P}/Δ_{AP}) vs. H_{z} for each V is plotted in Fig. 5b. At a small perturbation limit, i.e., h, v_{P}, v_{AP} ≪ 1, Eq. (3) is reduced to ln(Δ_{P}/Δ_{AP}) = 2n_{H}h + n_{I}(v_{AP} − v_{P}); thus, the slope and intercept of the linear fit to the data shown in Fig. 5b give n_{H} and n_{I}, respectively. One can see that the results are well fitted by the linear function, validating the employed model.
We analyse Fig. 5 with Eq. (3) and obtain n_{H} and n_{I} as a function of V for device A as shown in Fig. 6a. One can see that both n_{H} and n_{I} show similar values at each V and gradually decreases to about 1.5 with decreasing V. We perform the same procedure for the device B, whose properties are determined as μ_{0}H_{K}^{eff}(0) = 129.0 ± 0.7 mT, μ_{0}dH_{K}^{eff}/dV = −61.7 ± 2.3 mT V^{−1}, μ_{0}d^{2}H_{K}^{eff}/dV^{2} = −58 ± 13 mT V^{−2}, V_{C0,P} = 672 ± 4 mV and V_{C0,AP} = −541 ± 2 mV. The obtained n_{H} and n_{I} are shown in Fig. 6b. At V = 0, the two devices show the same value for n_{H} within experimental inaccuracy. Also, both n_{H} and n_{I} of device B show similar values with each other as in device A. However, in contrast to device A, they do not show meaningful variations at around 2 with V.
Discussion
As shown above, we have found that n_{H} and n_{I} show virtually the same value with each other for both devices A and B. Also, they are almost constant at around 2.0 for device B whereas change from 2.0 to 1.5 with V for device A. The main difference between devices A and B is t_{CoFeB}, which manifests in a difference in μ_{0}H_{K}^{eff}(0) [77.0 ± 0.5 mT for device A and 129.0 ± 0.7 mT for device B]. In the following, we will discuss the mechanism that can account for the obtained results in the context of the energy landscape and its bifurcation.
In systems with uniaxial anisotropy where the magnetic field is applied along the easy axis where the macrospin approximation holds, Brown derived n_{H} = 2 in a high barrier region, E_{0} ≳ k_{B}T, using Kramers’ analysis on the Fokker–Planck equation that is equivalent to the Landau–Lifshitz–Gilbert (LLG) equation with the Langevin term^{3}. Taking into consideration the secondorder anisotropy, where the magnetic anisotropy energy density is given by \({{{{{\mathcal{E}}}}}}\) = K_{1}^{eff}sin^{2}θ + K_{2}sin^{4}θ, n_{H} was pointed out to vary with K_{2}/K_{1}^{eff} ^{7}, where K_{1}^{eff}, K_{2} and θ are the first and secondorder effective anisotropy fields and polar angle of magnetisation vector, respectively. In the CoFeB/MgO system, positive voltage, which decreases electron density at the interface, was found to increase K_{1}^{eff}, while keeping μ_{0}H_{K2} (≡μ_{0}K_{2}/4M_{S}) constant at around 45 mT^{46,49,50}. Accordingly, as shown in the upper axes of Fig. 6a, b, in the present cases, K_{2}/K_{1}^{eff} is calculated to be around 0.22 for device B whereas it increases up to 0.45 for device A. The numerical calculation, assuming material parameters of magnetic recording media (Δ_{0} ≳ 60), shows that n_{H} decreases from 2.0 to 1.5 in the range of K_{2}/K_{1}^{eff} from 0 to 0.25^{7}. In general, a dynamical system with pitchfork bifurcation leads to the switching exponent of n_{x} = 2, while saddlenode bifurcation results in n_{x} = 3/2. Note that the aforementioned magnetic energy density \({{{{{\mathcal{E}}}}}}\) gives the LLG equation dθ/dt = f(θ,H_{z}) = −αγμ_{0}[(2K_{1}^{eff}/M_{S})cosθ − (4K_{2}/M_{S})cos^{3}θ − H_{z}]sinθ. We show f(θ,H_{z}) takes two types of local bifurcations: pitchfork bifurcation appears at K_{2}/K_{1}^{eff} < 0.25, while saddlenode bifurcation at K_{2}/K_{1}^{eff} > 0.25 as shown in Fig. 6c, d, respectively [see Supplementary Information in detail]. Thus, the experimentally observed transition of the switching exponents is attributed to the transition of the bifurcation of the potential landscape through the modulation of K_{2}/K_{1}^{eff}. However, the experiment shows the transition of n_{H} at K_{2}/K_{1}^{eff} ≈ 0.45, which is larger than that expected by the macrospin model (K_{2}/K_{1}^{eff} = 0.25). This deviation implies that the local bifurcation of the magnetic potential and the resultant n_{H} in the real MTJ device is more insensitive to higherorder anisotropy field K_{2} than the macrospin limit, for example, due to the micromagnetic effects.
Regarding n_{I}, some theoretical studies derived 1 by considering a fictitious temperature in LLG equation with the Langevin term^{20,21}, whereas others derived 2 from an analysis of the Fokker–Planck equation^{22,23,24}. Matsumoto et al. pointed out that n_{I} rapidly decreases from 2 to 1.4 with increasing K_{2}/K_{1}^{eff} from 0 to ~0.25^{51}. Experimentally, some assumed 1^{25,27,28,29,30,36,39} whereas others assumed 2^{26,31,32,33}, and importantly no studies access the number. The present experimental results support the scenario of Matsumoto et al., but, similarly to n_{H}, the reduction of n_{I} is more moderate than the theoretical prediction. This fact indicates that the mechanism for n_{H} could be also applicable for the case with STT perturbation as well. Another important implication of our results is that, despite the nonconservative nature of STT, the pseudo energy landscape under STT can be investigated through the switching exponents. LLG equation with STT τ_{STT} can be represented as dθ/dt = f(θ,x) = {−αγμ_{0}[(2K_{1}^{eff}/M_{S})cosθ − (4K_{2}/M_{S})cos^{3}θ] + τ_{STT}}sinθ. Since n_{H} and n_{I} show virtually the same value for all K_{2}/K_{1}^{eff} conditions, meaning that f(θ,τ_{STT}) takes the same local bifurcation type as that for f(θ,H), our experiment reveals that in MTJ devices with perpendicular easy axis, the magnetic field and the STT effectively similarly modulate the energy landscape.
In summary, this work has experimentally revealed the hithertoinaccessible representation of thermallyactivated switching rate under field and STT, using a relevant material system for applications. The obtained results could allow for sophisticated engineering of nonvolatile memory and unconventional computing hardware. Through the switching exponents, we have also accessed the local bifurcation of energy landscape under STT, and have found that, despite the qualitative difference between magnetic field and STT, their effect on the energy landscape is equivalent in the case of perpendicular MTJ. This work has also demonstrated that superparamagnetic tunnel junctions and analysis of their local bifurcation can serve as a versatile tool to investigate unexplored physics relating to thermallyactivated phenomena in general with various configurations and external perturbations.
Methods
Sample preparation
Stacks with Ta(5)/Pt(5)/[Co(0.3)/Pt(0.4)]_{7}/Co(0.3)/Ru(0.45)/[Co(0.3)/Pt(0.4)]_{2}/Co(0.3)/Ta(0.3)/CoFeB(1.0)/MgO(1.0)/CoFeB(t_{CoFeB})/Ta(5)/Ru(5) (numbers in parenthesis are thickness in nm) were deposited by dc/rf magnetron sputtering on a sapphire substrate. The nominal CoFeB thicknesses t_{CoFeB} = 1.88 nm (device A) and 1.82 nm (device B). After the deposition, the stacks were processed into MTJs by a hardmask process with electronbeam lithography, followed by annealing at 300 °C under a perpendicular magnetic field of 0.4 T for 1 h. The resistance (R)area (A) product (RA) was determined from the physical size determined from scanning electron microscopy observation and measured resistance for large devices with diameter D > 45 nm. The resistancearea product of device A (device B) is 5.5 Ωμm^{2} (8.1 Ωμm^{2}), and the tunnel magnetoresistance ratio is 73% (74%). The nominal thickness of the MgO is 1.0 nm for both devices, and the difference of the RA corresponds to the ~7% variation of the actual thickness due to the process variations between the two runs. D of devices A and B are determined from their resistance and RA to be D = 34 and 28 nm, respectively.
Homodynedetected ferromagnetic resonance (FMR)
With the circuit shown in Fig. 2a, homodynedetected voltageH_{z} spectra were measured at various frequencies. As shown in previous papers, the spectra were well fitted by the Lorentz function and peak position was determined by the fitting^{45,46}. From resonance frequency f_{r} vs. H_{z}, the effective anisotropy field H_{K}^{eff} was determined while assuming a constant secondorder anisotropy field μ_{0}H_{K2} = 45 mT^{46,49,50} (μ_{0} is the permeability of vacuum). The measurement was performed at various dc biases at AP configuration and obtained H_{K}^{eff} vs. V, as shown in Fig. 2b. A quadratic equation was fitted to the obtained dependence and the coefficients for constant, linear and quadratic terms were determined. Note that, in the error of H_{K}^{eff}, we have included the effect of H_{K}^{eff} difference in P and AP configurations due to the different device resistances and resultant Joule heatings in these configurations under the identical bias voltage.
Switching probability measurement
With the circuit shown in Fig. 3a, the switching probability was measured as functions of write pulse voltage amplitude and duration to determine intrinsic critical voltage V_{C0}. A voltage waveform composed of initialisation (0.45 V/300 ns), write (amplitude V_{write}/duration t_{pulse}), and read (V_{read} = 0.15 V/75 ns) pulses as shown in Fig. 3b, was generated by an arbitrary waveform generator (AWG). Both the interval of initialisation/write and write/read pulses were 30 ns, which is much shorter than the shortest relaxation time measured here (~0.3 ms), ensuring a sufficiently low readerror rate due to unintentional switching probability before the read pulse, exp(−30 ns/0.3 ms) \(\le\) 10^{−4}. Singleshottransmitted voltage for write pulse was monitored to determine the magnetisation configuration; the transmitted voltage is ~2Z_{0}V_{read}/(R + Z_{0}), where Z_{0} is characteristic impedance 50 Ω, and due to the tunnel magnetoresistance, the transmitted voltage changes with magnetisation configuration. The typical transmitted signal for P and AP states is shown in Fig. 3c. Transmitted signals for 5 ns (between 15 and 20 ns in Fig. 3c) were averaged. Its averaged value 〈V〉 and standard error 〈(V〈V〉)^{2}〉^{0.5}/N^{ 0.5} for P and AP states were 2.44 ± 0.03 mV and 1.40 ± 0.03 mV, respectively (N is averaged points; 20 Gbit/s × 5 ns duration = 100 points), ensuring low readerror rate due to misassignment of the magnetisation configuration^{47}. As shown in Fig. 3d, switching probability as a function of the voltage amplitude V at MTJ with write pulse duration t_{pulse} from 1 to 5 ns was measured. The probability of the switching was determined from 200 times measurement. The switching measurement was conducted under H_{z} of the stray field H_{S} which was determined from the random telegraph noise measurement. Note that the anomaly of the switching probability at V ~ −900 mV can be attributed to a change of the magnetic easy axis through the electricfield effect on the magnetic anisotropy, which is reported in previous works^{52}. In addition, the slope of the switching probability at P_{sw} ~ 0.5 for P to AP switching increases with decreasing the pulse duration, which is opposite to the thermallystable MTJs. The decreases of the effective field and the thermal stability factor through the electricfield effect on magnetic anisotropy reasonably explain the behaviour.
Random telegraph noise (RTN)
With the circuit shown in Fig. 4a, the RTN signal of the MTJs for V ~ 0 was measured. Small direct current I = 200 nA was applied and R was monitored by an oscilloscope connected in parallel to the MTJ to probe the temporal magnetisation configuration. The voltage applied to MTJ here was up to 2.5 mV (5 mV) for device A (device B), which is small enough to prevent major voltage/currentinduced effects in MTJs focused here. If one utilises the same circuit, the applied voltage for P and AP states varies by a factor of about 1.7 due to the tunnel magnetoresistance effect. Therefore, to prevent variation of effective anisotropy field for P and AP states, the circuit shown in Fig. 4b was utilised for V ≥ 25 mV. Direct voltage V to the MTJ was applied and divided voltage at reference resistor R_{r} connected in serial to the MTJ was monitored using the oscilloscope. For measuring RTN on device A (device B), with setting R_{r} = 0.47 kΩ (1 kΩ) much smaller than R, variation of applied voltages between P and AP states was prevented.
Attempt frequency
In determining Δ with the random telegraph noise measurement, Néel–Arrhenius raw τ_{P(AP)} = τ_{0}expΔ_{P(AP)} with attempt frequency τ_{0} of 1 ns was assumed. This assumption is widely adopted because τ is an exponential function of Δ and the value of τ_{0} does not affect the estimated Δ, and τ_{0} ranges between 0.1 and 10 ns. According to Brown’s calculation with Kramer’s method on the Fokker–Planck equation^{3}, the attempt frequency τ_{0} of the magnetic materials with uniaxial perpendicular magnetic anisotropy is [2αγμ_{0}H_{K}^{eff}(1 − h^{2})(1 + h)]^{−1}(π/Δ_{0})^{0.5} under large barrier approximation Δ_{0}\(\gg\)1, where α, γ, and μ_{0} are damping constant 0.006, gyromagnetic ratio, and permeability of vacuum, respectively. In our devices, the Brown’s attempt frequency above is derived to be 1.1 and 2.4 ns for device A and device B, respectively. Thus, the switching time τ vs. thermal stability factor Δ of device A should be well described by the Néel–Arrhenius law with τ_{0} = 1 ns.
Data availability
The data that support the plots within this paper have been deposited in Zenodo at https://zenodo.org/record/6767828^{53}.
References
Strogatz, S. H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Studies in Nonlinearity) (CRC Press, 2001).
Néel, L. Theorie du trainage magnetique des ferromagnetiques en grains fins avec application aux terres cuites. Ann. Geophys. 5, 99–136 (1949).
Brown, W. F. Thermal fluctuations of a singledomain particle. Phys. Rev. 130, 1677–1686 (1963).
Stoner, E. C. & Wohlfarth, E. P. A mechanism of magnetic hysteresis in heterogeneous alloys. Philos. Trans. R. Soc. A 240, 599–642 (1948).
Victora, R. H. Predicted time dependence of the switching field for magnetic materials. Phys. Rev. Lett. 63, 457–460 (1989).
Tannous, C. & Gieraltowski, J. The StonerWohlfarth model of ferromagnetism. Eur. J. Phys. 29, 475–487 (2008).
Kitakami, O., Shimatsu, T., Okamoto, S., Shimada, Y. & Aoi, H. Sharrock relation for perpendicular recording media with higherorder magnetic anisotropy terms. Jpn. J. Appl. Phys. 43, L115–L117 (2004).
Slonczewski, J. C. Currentdriven excitation of magnetic multilayers. J. Magn. Magn. Mater. 159, L1–L7 (1996).
Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54, 9353–9358 (1996).
Tsoi, M. et al. Excitation of a magnetic multilayer by an electric current. Phys. Rev. Lett. 80, 4281–4284 (1998).
Myers, E. B., Ralph, D. C., Katine, J. A., Louie, R. N. & Buhrman, R. A. Currentinduced switching of domains in magnetic multilayer devices. Science 285, 867–870 (1999).
Katine, J. A., Albert, F. J., Buhrman, R. A., Myers, E. B. & Ralph, D. C. Currentdriven magnetization reversal and spinwave excitations in Co/Cu/Co pillars. Phys. Rev. Lett. 84, 3149–3152 (2000).
Brataas, A., Kent, A. D. & Ohno, H. Currentinduced torques in magnetic materials. Nat. Mater. 11, 372–381 (2012).
Kent, A. D. & Worledge, D. C. A new spin on magnetic memories. Nat. Nanotechnol. 10, 187–191 (2015).
Apalkov, D., Dieny, B. & Slaughter, J. M. Magnetoresistive random access memory. Proc. IEEE 104, 1796–1830 (2016).
Jinnai, B., Watanabe, K., Fukami, S. & Ohno, H. Scaling magnetic tunnel junction down to singledigit nanometersChallenges and prospects. Appl. Phys. Lett. 116, 160501 (2020).
Mizrahi, A. et al. Neurallike computing with populations of superparamagnetic basis functions. Nat. Commun. 9, 1533 (2018).
Borders, W. A. et al. Integer factorization using stochastic magnetic tunnel junctions. Nature 573, 390–393 (2019).
Lv, Y., Bloom, R. P. & Wang, J. Experimental demonstration of probabilistic spin logic by magnetic tunnel junctions. IEEE Magn. Lett. 10, 4510905 (2019).
Koch, R. H., Katine, J. A. & Sun, J. Z. Timeresolved reversal of spintransfer switching in a nanomagnet. Phys. Rev. Lett. 92, 088302 (2004).
Li, Z. & Zhang, S. Thermally assisted magnetization reversal in the presence of a spintransfer torque. Phys. Rev. B 69, 134416 (2004).
Taniguchi, T. & Imamura, H. Thermally assisted spin transfer torque switching in synthetic free layers. Phys. Rev. B 83, 054432 (2011).
Butler, W. H. et al. Switching distributions for perpendicular spintorque devices within the macrospin approximation. IEEE Trans. Magn. 48, 4684–4700 (2012).
Pinna, D., Mitra, A., Stein, D. L. & Kent, A. D. Thermally assisted spintransfer torque magnetization reversal in uniaxial nanomagnets. Appl. Phys. Lett. 101, 262401 (2012).
Fuchs, G. D. et al. Adjustable spin torque in magnetic tunnel junctions with two fixed layers. Appl. Phys. Lett. 86, 152509 (2005).
Sato, H. et al. Perpendicularanisotropy CoFeBMgO magnetic tunnel junctions with a MgO/CoFeB/Ta/CoFeB/MgO recording structure. Appl. Phys. Lett. 101, 022414 (2012).
Thomas, L. et al. Perpendicular spin transfer torque magnetic random access memories with high spin torque efficiency and thermal stability for embedded applications (invited). J. Appl. Phys. 115, 172615 (2014).
Zhao, H. et al. Low writing energy and sub nanosecond spin torque transfer switching of inplane magnetic tunnel junction for spin torque transfer random access memory. J. Appl. Phys. 109, 07c720 (2011).
Heindl, R., Rippard, W. H., Russek, S. E., Pufall, M. R. & Kos, A. B. Validity of the thermal activation model for spintransfer torque switching in magnetic tunnel junctions. J. Appl. Phys. 109, 073910 (2011).
Sukegawa, H. et al. Spintransfer switching in fullHeusler Co2FeAlbased magnetic tunnel junctions. Appl. Phys. Lett. 100, 182403 (2012).
Nakayama, M. et al. Spin transfer switching in TbCoFe/CoFeB/MgO/CoFeB/TbCoFe magnetic tunnel junctions with perpendicular magnetic anisotropy. J. Appl. Phys. 103, 07A710 (2008).
Van Beek, S. et al. Thermal stability analysis and modelling of advanced perpendicular magnetic tunnel junctions. AIP Adv. 8, 055909 (2018).
Lourembam, J., Chen, B. J., Huang, A. H., Allauddin, S. & Ter Lim, S. A noncollinear double MgO based perpendicular magnetic tunnel junction. Appl. Phys. Lett. 113, 022403 (2018).
Urazhdin, S., Birge, N. O., Pratt, W. P. Jr. & Bass, J. Currentdriven magnetic excitations in permalloybased multilayer nanopillars. Phys. Rev. Lett. 91, 146803 (2003).
Wegrowe, J. E. Magnetization reversal and twolevel fluctuations by spin injection in a ferromagnetic metallic layer. Phys. Rev. B 68, 214414 (2003).
Rippard, W., Heindl, R., Pufall, M., Russek, S. & Kos, A. Thermal relaxation rates of magnetic nanoparticles in the presence of magnetic fields and spintransfer effects. Phys. Rev. B 84, 064439 (2011).
Chiba, D., Ono, T., Matsukura, F. & Ohno, H. Electric field control of thermal stability and magnetization switching in (Ga,Mn) As. Appl. Phys. Lett. 103, 142418 (2013).
Enobio, E. C. I., Bersweiler, M., Sato, H., Fukami, S. & Ohno, H. Evaluation of energy barrier of CoFeB/MgO magnetic tunnel junctions with perpendicular easy axis using retention time measurement. Jpn. J. Appl. Phys. 57, 04FN08 (2018).
Zink, B. R., Lv, Y. & Wang, J. P. Independent control of antiparallel and parallelstate thermal stability factors in magnetic tunnel junctions for telegraphic signals with two degrees of tunability. IEEE Trans. Electron Devices 66, 5353–5359 (2019).
Sun, J. Z. et al. Spintorque switching efficiency in CoFeBMgO based tunnel junctions. Phys. Rev. B 88, 104426 (2013).
Chiba, D. et al. Magnetization vector manipulation by electric fields. Nature 455, 515–518 (2008).
Maruyama, T. et al. Large voltageinduced magnetic anisotropy change in a few atomic layers of iron. Nat. Nanotechnol. 4, 158–161 (2009).
Endo, M., Kanai, S., Ikeda, S., Matsukura, F. & Ohno, H. Electricfield effects on thickness dependent magnetic anisotropy of sputtered MgO/Co40Fe40B20/Ta structures. Appl. Phys. Lett. 96, 212503 (2010).
Kanai, S., Matsukura, F. & Ohno, H. Electricfieldinduced magnetization switching in CoFeB/MgO magnetic tunnel junctions. Jpn. J. Appl. Phys. 56, 0802A0803 (2017).
Tulapurkar, A. A. et al. Spintorque diode effect in magnetic tunnel junctions. Nature 438, 339–342 (2005).
Kanai, S., Gajek, M., Worledge, D. C., Matsukura, F. & Ohno, H. Electric fieldinduced ferromagnetic resonance in a CoFeB/MgO magnetic tunnel junction under dc bias voltages. Appl. Phys. Lett. 105, 242409 (2014).
Saino, T. et al. Writeerror rate of nanoscale magnetic tunnel junctions in the precessional regime. Appl. Phys. Lett. 115, 142406 (2019).
Bedau, D. et al. Spintransfer pulse switching: From the dynamic to the thermally activated regime. Appl. Phys. Lett. 97, 262502 (2010).
Okada, A. et al. Electricfield effects on magnetic anisotropy and damping constant in Ta/CoFeB/MgO investigated by ferromagnetic resonance. Appl. Phys. Lett. 105, 052415 (2014).
Okada, A., Kanai, S., Fukami, S., Sato, H. & Ohno, H. Electricfield effect on the easy cone angle of the easycone state in CoFeB/MgO investigated by ferromagnetic resonance. Appl. Phys. Lett. 112, 172402 (2018).
Matsumoto, R., Arai, H., Yuasa, S. & Imamura, H. Efficiency of spintransfertorque switching and thermalstability factor in a spinvalve nanopillar with first and secondorder uniaxial magnetic anisotropies. Phys. Rev. Appl. 7, 044005 (2017).
Ohshima, N. et al. Currentinduced magnetization switching in a nanoscale CoFeBMgO magnetic tunnel junction under inplane magnetic field. AIP Adv. 7, 055927 (2017).
Funatsu, T. et al. Dataset: “local bifurcation with spintransfer torque in superparamagnetic tunnel junctions.” Zenodo. https://zenodo.org/record/6767828 (2022).
Acknowledgements
The authors thank O. Kitakami, M. Stiles, W. A. Borders and Y. Utsumi for fruitful discussion and B. Jinnai, H. Sato, K. Watanabe, J. Igarashi, M. Shinozaki, T. Saino, Z. Wang, T. Hirata, H. Iwanuma, I. Morita, R. Ono, S. Musya and C. Igarashi for technical supports. This work was partly supported by the ImPACT Programme of CSTI (S.F. and H.O.), JSTOPERA JPMJOP1611 (S.F. and H.O.), JSTCREST JPMJCR19K3 (S.F.), JSTPRESTO JPMJPR21B2 (S.K.), JSPS CoretoCore Programme (S.K., S.F. and H.O.), JSPS Kakenhi 19H05622 (S.K., J.I. and S.F.), Shimadzu Research Foundation (S.K.), and RIEC Cooperative Research Projects (S.K., S.F. and H.O.).
Author information
Authors and Affiliations
Contributions
S.F. and H.O. planned the study. S.K. and T.F. designed the experiment and analysis. T.F. prepared samples, performed measurements and analysed the data under the support of S.K. S.K. formulated the bifurcation of the dynamics of nanomagnet. S.K. and S.F. wrote the manuscript with input from H.O. and J.I. All authors discussed the results.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Communications thanks JianPing Wang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Funatsu, T., Kanai, S., Ieda, J. et al. Local bifurcation with spintransfer torque in superparamagnetic tunnel junctions. Nat Commun 13, 4079 (2022). https://doi.org/10.1038/s41467022317881
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467022317881
This article is cited by

Local bifurcation with spintransfer torque in superparamagnetic tunnel junctions
Nature Communications (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.