Spin-torque nano-oscillators (STNO) are studied in terms of the Landau–Lifshitz–Gilbert (LLG) equation. The effect on the limit of detectivity of an STNO concerning externally applied magnetic fields is studied with micromagnetic models by placing adjacent magnetic flux concentrators (MFCs) at different distances from the nanopillar to analyze the effect on the induced auto-oscillations and magnetization dynamics. Perpendicular STNO structures allow for different detectivities with respect to externally applied magnetic fields depending on the distance from the MFCs to the nanopillar. The optimal design of an STNO combined with MFCs is proposed to improve the limit of detectivity, where the STNO consists of two out-of-plane (OP) ferromagnetic (FM) layers separated by a MgO insulating nonmagnetic (NM) thin film, and the MFCs positioned in the vicinity of the STNO are made of permalloy. The time evolution of the free-layer magnetization is governed by the Landau–Lifshitz–Gilbert (LLG) equation. The auto-oscillations induced within the free-layer averaged magnetization are provoked by externally applied magnetic fields. In addition, the DC current-driven auto-oscillations in the STNO structure are studied as a function of the externally applied magnetic field strength, with and without MFCs. The suppression of the DC current-driven auto-oscillations is observed due to the damping effect generated by the MFCs positioned at varying distances with respect to the STNO. By placing MFCs adjacent to the STNO, the lowest detectable magnetic field strength is enhanced from 10 (μT) to 10 (nT). Therefore, it is concluded that MFCs improve the sensitivity of STNO to externally applied magnetic fields thanks to the damped magnetization dynamics. The results presented in this work could inspire the optimal design of STNO and MFC-based ultra-low magnetic field sensors based on nanoscale oscillators and spintronic diodes.
The STNO consists of a double-layer FM structure separated by an NM thin-film spacer. The behavior of the free layer is modeled in Mumax3,1 where the magnetic material consists of permalloy (Ni80Fe20).2 The STNO works based on the principle of the Slonczweski spin-transfer torque (STT) according to the Landau–Lifshitz formalism, which is the working principle of several physical devices, for example, STT-MRAM. As a result, it is possible to proportionally modulate the resonant frequency for different magnetic materials by varying the current density and free-layer orientation, for example, in digital frequency shift-key modulation techniques. In this simulation, a particular DC current-driven steady-state auto-oscillation of the in-plane magnetization is estimated for a given current density and free-layer orientation.3 When positioning the MFCs at different distances from the pillar, it is possible to modulate the frequency of the auto-oscillations of the in-plane averaged magnetization, which reaches an equilibrium steady-state oscillation after ∼1 (ns) for the simulation parameters in Table I. Along the STNO z-axis, the magnetization converges to a constant non-zero value, whereas it oscillates in-plane between the maximum absolute value of the averaged in-plane magnetization, as depicted in Fig. 1(a). For a given current density (Jz) flowing along the z-axis, the DC current-driven STNO auto-oscillations take place along the x- and y-component of the free-layer magnetization, respectively, and it persists for tens of nanoseconds with a constant amplitude. Figure 2(a) shows the time evolution of the magnetization for a perpendicular STNO with a periodic steady-state auto-oscillation. Figure 2(b) characterizes the dynamics of the averaged magnetization ⟨mx,y,z⟩ with a constant timestep. It is possible to observe the transition point from an unstable to a stable behavior of the magnetization along the z-component, where the DC current-driven auto-oscillations are in-plane and of significant amplitude. The direction of the current (Jz) is perpendicular to the horizontally oriented layers. The initial fixed-layer magnetization along the z-axis is suppressed to a decreased constant value, while the magnetization oscillates in-plane. The MFCs surrounding the pillar induce an overall reduction in power emission, as shown in Fig. 1(b).
|Cells size||5 (nm) × 5 (nm) × 4.5 (nm)1,5,15,16|
|Pillar dimension||Φ = 80 (nm)5|
|γLL||1.76 × 1011 (rad/Ts)1,15|
|Ms||800 × 103 (A/m)5,15,16|
|Jc||5 × 1010 (A/m2)5|
|Aex||10 × 10−12 (J/m)1,5,15,16|
|mx,y,z||(0.6, 0.6, 0.15)|
|Cells size||5 (nm) × 5 (nm) × 4.5 (nm)1,5,15,16|
|Pillar dimension||Φ = 80 (nm)5|
|γLL||1.76 × 1011 (rad/Ts)1,15|
|Ms||800 × 103 (A/m)5,15,16|
|Jc||5 × 1010 (A/m2)5|
|Aex||10 × 10−12 (J/m)1,5,15,16|
|mx,y,z||(0.6, 0.6, 0.15)|
STNO devices have a parallel, perpendicular, or mixed free-layer magnetization.4–6 The spectral analysis for DC current-driven STNO devices allows the detection of modifications in steady-state oscillations. In addition, synchronization phase-locking techniques performed on intercoupled arrays of spin-torque oscillators are used to modulate the power spectrum for practical applications.7–9 A desirable feature for spin-torque oscillators is a sustained precession over a prolonged period combined with a consistent amplitude of the averaged magnetization to distinguish the precession occurring along each axis and consequently increase the power emission. A classical STNO device is composed of a relatively thick fixed layer regulating the polarization, and a thin free layer allowing for sustained oscillations. In Fig. 2(a), excluding the initial nonequilibrium transient behavior, the steady-state auto-oscillation stabilizes from 1–2 (ns). In STNOs, the precession of the averaged magnetization induced by STT phenomena increases the threshold of the critical current density, which is equivalent to experimental currents in the order of (mA). In addition, a relatively low resistance-area product is preferable to overcome the barrier breakdown voltage. For application purposes, tunneling magnetoresistance (TMR) effects transduce the precession of the magnetization into detectable microwave voltages.10
II. THEORY OF STNOs
STNO devices consist of either out-of-plane (OP) or in-plane FM layers separated by an NM spacer, for example, copper (Cu) for spin valves, or an insulating layer of magnesium oxide (MgO) for magnetic tunnel junctions (MTJs).11 The free-layer magnetization dynamic follows the Landau–Lifshitz–Gilbert (LLG) equation with the STT term:5
where M = m Ms is the magnetization vector, m the unitary magnetization, Ms the saturation magnetization, and Heff the effective field comprising the external, anisotropy, demagnetizing, exchange, and dipolar contributions. The third and fourth terms of Eq. (1) are described by the Slonczewski spin-transfer and field-like spin torques, respectively.5,12 The critical current density is described by:
where Jc is the critical current density, e is the elementary charge, ħ the reduced Planck constant, g(θ) the efficiency factor for the Slonczewski spin-transfer torque contribution, where P is the polarization, θ the perpendicular angle, Ms is the saturation magnetization, Hk the magnetic anisotropy field, and d the free-layer thickness. Hence, it is possible to estimate the critical current density for different configurations.3,5,13 The precession frequency can be estimated by the following formula described as:
where Ms is the value of the saturation magnetization, d is the free-layer thickness, and P is the free-layer polarization referring to permalloy as material. Since the LLG equation is nonlinear, micromagnetic solvers are implemented to find an approximate solution. The software used for this purpose is Mumax3 for accelerated GPU on Google Colab.1 The interaction between the free layer and the NM spacer is modeled considering the effect at the interface of the material when simulating the free-layer dynamics.5,14 To obtain a reliable solution, different micromagnetic solvers are used to verify the robustness of the DC current-driven steady-state auto-oscillations for similar simulation parameters and initial conditions. The Oersted field contribution estimated with the finite element method (FEM) is also taken into consideration to correct the final results of the micromagnetic model. Because of the high critical current density involved, there is a significant difference in the bias magnetic field between the boundaries and the center of the STNO pillar. Figures S2 and S3 show that, after an initial stabilization period, the steady-state auto-oscillations persist beyond 10 (ns).
III. MICROMAGNETIC MODEL
The micromagnetic model verifies that, for a given combination of damping, magnetization, cell size, and material properties listed in Table I, a steady-state auto-oscillation occurs for a prolonged period. The cell size needs to respect the constraints of the exchange length of the soft magnetic material utilized in the calculations. The micromagnetic simulation shown in Figs. 2(a) and 2(b) illustrates the evolution of the averaged magnetization along the x-, y-, and z-axis, whereas Fig. S2 represents a longer time interval. By modulating the parameters described in Eq. (2), it is therefore, possible to model the correction of the critical current density concerning the DC current-driven auto-oscillations of the in-plane averaged magnetization according to the parameters reported in Table I, as represented in Fig. S1. For a particular solution of the given current density, it is therefore, possible to compare the results with and without MFCs. Multiple magnetized layers can be stacked onto each other to induce a stronger STT effect. For the micromagnetic simulation analyzed, the following parameters are considered:
The cell size is consistent within the minimum exchange length for the magnetic material based on the magnetostatic and anisotropic energy density. Since the current density along the z-axis is also responsible for a relevant static contribution, an in-plane distribution of the Oersted field is estimated as depicted in Fig. 8. The current density is supposed constant along the pillar without spatial distribution, whereas the magnetic induction has a spatial distribution depending on the intensity of the current density.16 The simulation is performed at a constant room temperature. The pillar diameter is Φ = 80 (nm), and the dimensions are experimentally suitable for nanofabrication.17
The initial condition [t = 0 (s)] expressed in Fig. 2(b) is represented by the red cross and the steady-state condition [t = 10 (s)] by the red symbol. Along the z-axis, a rapid stabilization of the magnetization is observed. Once the averaged magnetization stabilizes along the z-axis within a constant value, the precession, and steady-state in-plane auto-oscillation occurs for a prolonged period of at least 200 (ns), as shown in Fig. S2. Using equivalent simulation parameters, the DC current-driven STNO is simulated with another micromagnetic solver (OOMMF) to verify the consistency and robustness of the model, as depicted in Figs. S1, S3, and S4. Excluding the fast initial stabilization of the magnetization, the steady-state oscillation along the x- and the y-axis is of similar magnitude.
IV. POWER SPECTRUM AND SENSITIVITY
The power emitted by the STNO is estimated and plotted for different current densities.10,18 The TMR is derived from theoretical values as:
where the polarization P1 = P2 = 0.35 is supposed to be equal for both the top and bottom electrodes. Therefore, the tunneling magnetoresistance is around ∼27.9%. The matched root-mean-square (RMS) microwave power PSTT is defined by the following formula, where a factor of 8 is considered from the peak-to-peak to RMS conversion, and a factor of 4 is due to the power splitting in the matched circuit:3
If R ≈ Rap and ΔRmax = Rap − Rp, the TMR value can be rewritten as TMR = (Rap − Rp)/Rp. Therefore, Rap = 1.387 × Rp and ΔRmax = 0.387 × Rp, where the current is derived from the current density with respect to the area of the STNO pillar: I = JzA = 0.251 (mA). Therefore, PSTT = 2.12 × 10−10 Rp (W). In case the STNO resistance is estimated to be in the vicinity of R ≈ 450 (Ω),10 herein ΔR ≈ 175 (Ω), and PSTT = 95.4 (nW), in accordance with the expectations. The value is in the range of the expectations, where PSTT ≈ 0.1 (μW).10
Figure 3 represents the estimated power emission for different current densities. The total spectrum is represented in Fig. S5. The narrow and well-defined peak of the power emission has an average Q-factor of ∼77. An increase in the current density corresponds to a power emission peak shift of the DC current-driven STNO. The steady-state auto-oscillations of the STNO nanopillar occur between 3 (GHz) and 3.5 (GHz) for this configuration, with a limit critical current density defined by Eq. (2).3,19,20
Figure 4 represents the effect on the steady-state auto-oscillations of the DC current-driven STNO when different externally applied in-plane magnetic fields are applied. The variability of the steady-state auto-oscillations provoked by the different intensities of the externally applied magnetic field is correlated with the limit of detectivity (LOD) of the theoretical device. For the STNO considered, the LOD is between 1 (µT) and 1 (nT), as shown in Figs. 5 and 12. The frequency shift of the DC current-driven auto-oscillations has not the same directionality in the spectrum and varies depending on the magnitude of the externally applied magnetic field.
As represented in Fig. 5, in the range from 10 (µT) to 1 (µT), the difference in the peak frequency is in the order of ∼100 (MHz), whereas between 1 (µT) and 100 (nT), it is limited to a (kHz) range. The limit of detectivity for steady-state auto-oscillations below 100 (Hz) is in the range of nT for the specific STNO simulated, although it can be improved both through the shape and material optimization of the MFC, for example, using an antiferromagnetic (AFM) or ferrimagnetic material, and exploit physical phenomena, such as RKKY interaction, antiferromagnetic and exchange coupling. The limit of detectivity for this structure is around 1 (µT). A detailed representation of the detectivity is shown in Fig. 12. For low-magnitude magnetic fields, there are no significant modifications of the steady-state auto-oscillations. The oscillation of the magnetization is provoked by the externally applied magnetic field, while the DC current flows perpendicularly through the STNO thanks to quantum tunneling at the insulating layer, and it can be detected using a bias-tee to extract the AC component with the spectrum analyzer.21–23 In the case of spin Hall current or other configurations, a substrate is inserted below the oscillator.3
V. EFFECT OF MAGNETIC FLUX CONCENTRATORS ON THE SENSITIVITY
Magnetic flux concentrators (MFCs) are modeled by rectangular shapes surrounding the circular pillar, where the thickness of MFCs is set equal to the STNO pillar. The material filling the boundaries is air, and the MFCs are made of soft magnetic materials. The distance between the pillar and the MFC is equal to 5 (nm), and the pillar diameter is 80 (nm). The rectangular MFC has a dimension of 80 (nm) × 80 (nm), as shown in Fig. 6. Experimentally, it is possible to have nanometer resolution for the thickness using magnetron sputtering techniques. Optimizing the geometry of MFCs with different soft magnetic material structures has the potential to take advantage of several physical phenomena, such as room-temperature stable skyrmions,24 solitons,25 and chaotic behaviors.26
By positioning the MFCs at a decreasing distance, it is possible to observe a reduction in the power emission, as well as sidelobes of the main frequency, as depicted in Fig. 7 and Fig. S6. When MFCs are not present, the peak power emission frequency is narrower with fewer harmonics. The MFCs are responsible for the damping effect of the steady-state auto-oscillations and reduce the total power emission.
The distance from the pillar to the MFCs, where the suppression of the DC current-driven auto-oscillations occurs, ranges from 74 (nm) to 82 (nm), as shown in Fig. S6. For 80 (nm), there is a suppression of the power emission attributed to the symmetrical structure with respect to the diameter of the STNO pillar. Since the scale of the dimension is in the order of tens of nanometers, and the distance of the MFC is well-defined within the pillar diameter, it is theoretically possible to nanofabricate DC current-driven STNO on thin-film magnetic materials using e-beam lithography fabrication techniques. Since the MFCs are responsible for a reduction of the STNO power emission, relevant applications can be related to spintronics diodes, on-chip devices with spin–orbit torque rectifiers, and sensing platforms, for example, working on the principle of synchronization through phase-locking techniques, as shown in Fig. S8.2,9,21,27–35
The Oersted field static contributions are studied for different relative permeabilities and calculated with COMSOL® Multiphysics Module AC/DC,36 as shown in Figs. 8(a) and 8(b) and Fig. S7. The relative permeability of permalloy decreases to a unitary value for high frequencies. The current density corresponds to the critical one used in the micromagnetic simulations [J = 5 × 1010 (A/m2)], and the thickness of 5 (nm) is the same as the cell dimension utilized in the Mumax3 simulation. The material filling the theoretical gap between the MFCs, and the pillar is air. An increase in the relative permeability of the permalloy material causes a significant increase in the static fields surrounding the nanopillar. The intensity of the effect reaches a magnitude in the order of 7 (mT) for μr = 5.18,37–41 The Oersted field is confined to the edges of the pillar and is negligible at the center. This static field also affects the edges of the MFCs and the nanopillar. By reducing the relative permeability further, the effect of the Oersted field stretches outside the pillar. Figures 8(c)–8(f) represents the eigenfrequencies of the DC current-driven auto-oscillations calculated with the COMSOL® Micromagnetics module in the range of ∼3 (GHz).36,42 Equivalent to the simulation performed either in Mumax3 and OOMMF, the theoretical STNO nanopillar has a steady-state auto-oscillation around 3.11 (GHz), corresponding to a breathing and rotation modes, as shown in Figs. 8(c) and 8(d), respectively. In addition, the simulation converges to another eigenfrequency around 3.135 (GHz), equivalent to its breathing and rotation modes, as represented in Figs. 8(e) and 8(f), respectively. Since there are no other eigenfrequencies in the frequency range from 2 (GHz) to 4 (GHz), this means that the eigenfrequencies represent the steady-state auto-oscillation when the device is resonating and, therefore, working as a spin-torque oscillator. The parameters used in the simulation in Figs. 2(c)–2(f) are the ones listed in Table I converted to proper units, where the STT contribution is considered for the linearized LLG equation embedded in the solver. Differently than the simulation performed in Mumax3 or OOMMF, the cell dimension in this simulation is automatically generated during the meshing process, which is responsible for a small deviation around ∼10–30 (MHz) with respect to the steady-state auto-oscillation depicted in Fig. 3. For this purpose, a coarse mesh is selected to avoid violating the physical minimum exchange length. In addition, it is possible to observe that one eigenfrequency provokes relevant changes of ⟨mx⟩ in the whole region of the pillar, as shown in Figs. 8(e) and 8(f), whereas the other one occurs at the boundary interface of the pillar, according to the results depicted in Fig. S1. This means that the theoretical device is indeed working as a spin-torque oscillator for the chosen critical current density [Jz = 5 × 1010 (A/m2)], and steady-state auto-oscillation frequency is equivalent to the solution obtained with the Mumax3 and OOMMF micromagnetic solvers corroborating the previous results, as illustrated in Fig. 3.
As shown in Fig. 9, it is possible to observe a shift in the steady-state auto-oscillations in DC current-driven STNO for different temperatures. The increase in the temperature corresponds to a reduction in power emission. Above a certain threshold in the vicinity of the Curie temperature, the DC current-driven auto-oscillations start vanishing and becoming uncorrelated with the temperature. The difference in the frequency for the peak power emission is around 12 (MHz) every 100 (K) step with no significant difference between 300 (K) and 400 (K). For the temperature simulation, the solver is the standard one, and the spatial resolution complies with the stochastic field and the cell size.43 The time resolution is enough to resolve the fastest dynamic in the system, converging to a steady-state solution. The threshold also causes a realistic loss of the magnetization of the STNO device. Figure 9 shows the effect on the DC current-driven STNO in the order of tens of (kHz) for temperatures between 200 (K) and 300 (K). For a suitable description of the behavior close to the Curie temperature, an evaluation with atomistic micromagnetic models is required. In addition to the static fields, the observation of temperature-dependent steady-state auto-oscillations implies an experimental calibration of the proposed nanodevice. Since the dynamic Gilbert damping constant α(T, t) depends both on the temperature (T) and thickness (t) of the permalloy thin film, a correction factor needs to be taken into consideration for significant low temperatures below 150 (K). Since the device is supposed to operate at temperatures ranging from 250 (K) to 350 (K), the effect is not critical. In addition, at higher temperatures, the device will also experience demagnetization phenomena and loss of performance.
Figure 10 represents the hysteresis loop for an STNO structure without MFCs. Furthermore, Figs. 11(a)–11(d) shows the effect of MFCs on the hysteresis loop. The magnetization switches with a magnetic field along the y-direction being less than 30 (Oe). By positioning the MFCs nearby the STNO pillar, the coercivity increases, and the behavior turns into a double-shifted hysteresis loop with unconventional effects, as shown in Fig. 11(c). When the MFC is placed in contact with the STNO pillar, the coercivity decreases, and the behavior changes from fast switching to superparamagnetic. One disadvantage of making use of MFCs made of soft magnetic materials in the vicinity of the nanopillars is the increase of the coercivity in the hysteresis loop, which requires higher magnetic fields to switch the device, as represented in Figs. 10 and 11.
By introducing MFCs in contact with the nanopillar, it is possible to observe a reduction in the LOD concerning an externally applied magnetic field by two orders of magnitude according to the Mumax3 simulation, as shown in Fig. 12. The enhancement of the local magnetic field amplified by the MFC structure made of soft magnetic materials causes faster damping of the DC current-driven auto-oscillations. Consequently, this induces higher frequency contributions of the auto-oscillations due to the introduction of instabilities in the steady-state solution. Hence, DC current-driven auto-oscillations of the magnetization are more sensitive to external field contributions. In Fig. 12, it is theoretically feasible to detect externally applied magnetic fields as low as (nT) units. The functionality of MFCs depends on the relative permeability of the magnetic material, as well as its geometry and atomic structure. Spin-torque oscillators work on the principle of nonlinear magnetization processes and spin-dependent transport through magnetic heterostructures. For sufficiently high currents, the magnetic damping and the spin torque can compensate themselves leading either to instabilities, or self-sustained oscillations.44 So far, the physical limit reported in the literature for spin oscillations is within the sub-Hz regime, and it is achievable with nitrogen-vacancy (NV) centers in diamonds.45–49
For a spin-torque nano-oscillator (STNO), it is possible to observe an effect of the magnetization dynamics provoked by the adjacent placing of magnetic flux concentrators (MFCs) surrounding the structure of the nanopillar. The damping of the DC current-driven auto-oscillations induced by the MFCs introduces high-frequency modes in the spectrum. Hence, the phenomenon can be implemented as a sensing platform to detect lower externally applied magnetic fields with a far better resolution. Furthermore, STNO is a promising alternative for spintronic diodes. Phase-locking combined with arrays of STNO is a feasible alternative for magnetic sensors. Thanks to the optimization of the material and geometry of external MFCs, it is possible to control the DC current-driven STNO as a sensing platform, oscillator, or rectifier.2,7–9,13,33,50–58
The supplementary material includes a simulation of an STNO pillar in Mumax3 with a longer period to validate the stability of the steady-state response up to at least 50 (ns), as shown in Fig. S2, as well as a simulation in OOMMF with the visualization of the planar magnetization during the steady-state oscillation represented in Figs. S1, S3, and S4.15 Figure S5 shows the full single-sided spectrum of a DC-driven STNO with different current densities. Figure S6 depicts the power emission of an STNO with MFCs positioned at different distances. Figure S7 shows the limit of static fields for a higher relative permeability in the material. Figure S8 shows the mutual synchronization of 2 and 4 STNO pillars. The video in the supplementary material reproduces the visualization of the in-plane magnetization obtained with the OOMMF micromagnetic solver from ∼0.5 (ns) to 40 (ns). It is possible to distinguish the steady-state auto-oscillations.
This study was financially supported by the Minnesota Partnership for Biotechnology and Medical Genomics under Award Number ML2020. Chap 64. Art I, Section IV. The authors acknowledge the Robert F. Hartmann Chair Professorship, MN Drive Neuromodulation Fellowship, and Minnesota Supercomputing Institute (MSI) at the University of Minnesota for providing resources that contributed to the research results reported in this publication.
Conflict of Interest
The authors have no conflicts to disclose.
Denis Tonini: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Kai Wu: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal). Renata Saha: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal). Jian-Ping Wang: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.