RF signal detector and energy harvester based on a spin-torque diode with perpendicular magnetic anisotropy

With the advances of the Internet of Things (IOT) and radio frequency identification (RFID) technologies and wide application of micro- and nano-scale wireless devices that require independent power supplies, the problems of efficient RF signal detection and energy harvesting from ambient sources of radiation became critically important.^1–5 Both these problems can be solved with the help of spintronic diodes (SDs) based on magnetic tunnel junctions (MTJs).^6–17 In such a diode, the input RF current I(t) = I_RF cos(ωt) applied to an MTJ excites variations of the junction’s resistance R(t) with the angular frequency ω = 2πf of the incident RF signal. As a result, the voltage across the MTJ has a DC component $UDC=I(t)R(t)$ (here the angular brackets denote averaging over the period T = 2π/ω = 1/f of the RF current, $x=T−1∫0Txdt$⁠).^6–8,17

There are several possible regimes of operation of an SD.^7,8,17 Among them, the best-known regime is the regime of the forced in-plane (IP) magnetization precession about the in-plane “easy axis” in the free layer (FL) of the SD. This regime is, sometimes, called a “resonance” regime, because the DC voltage produced by an SD has a maximum magnitude when the driving external RF signal has the frequency equal to the frequency of the ferromagnetic resonance of the SD’s FL.^6–12,17

Another regime, which exists in the case when an SD is biased by a perpendicular bias magnetic field insufficient for the full saturation of the SD FL, is the regime of the forced out-of-plane (OOP) precession, and it was, first, described theoretically in Ref. 14. In the OOP-regime, the equilibrium direction of the FL magnetization lies out-of-plane of the SD FL, and the incident RF signal (if it exceeds a certain amplitude threshold) excites in the SD FL a large-angle OOP magnetization precession about the perpendicular direction of the bias magnetic field.^7,8,14 In this regime, the magnitude of the SD output DC voltage U_DC is negligibly small in the case when the incident RF current amplitude I_RF is below a certain threshold I_th, but increases abruptly as soon as I_RF exceeds a certain threshold value I_th. Above the threshold, the angle of the OOP magnetization precession is only slowly increasing with the increase in I_RF, and the resultant DC voltage produced by the SD virtually does not depend on the magnitude of the driving RF signal. This DC voltage also depends on the driving frequency in a non-resonance way—it increases with the increase in the driving frequency up to a certain magnitude and for higher frequencies vanishes abruptly.^7,8,14 Thus, the SD in the OOP regime works as a non-resonant threshold detector of RF signals having a sufficiently low frequency.^7,8,14

We believe that the OOP regime of the SD operation was observed for the first time in Ref. 13 and is responsible for the extremely large diode volt-watt sensitivity observed in the experiments.¹³ It was also proposed in Refs. 7, 8, and 14 that the OOP regime of the SD operation could be very useful for broadband RF energy harvesting.

Recent experiments,¹⁵ indeed, demonstrated that the efficient broadband RF energy harvesting is possible using an SD working in the OOP regime, and an important practical achievement of Ref. 15 was the demonstration that DC energy harvesting in the OOP regime can be experimentally realized without any bias magnetic field. To move the equilibrium orientation of the FL magnetization out of the plane of the SD FL, the authors of Ref. 15 used an FL with perpendicular magnetic anisotropy (PMA).

The use of PMA in the forced RF dynamics of an SD creates a new situation, which was not previously considered theoretically, and in our current work, we consider both analytically and numerically the OOP regime of the SD operation in the case when the out-of-plane equilibrium orientation of the static magnetization of the SD FL was created by the combination of the FL demagnetization field and the FL PMA. The goal is to elucidate the forced RF magnetization dynamics, in this case, to be able to optimize the operation of the broadband RF energy harvesters based on SDs operation in the OOP regime.

We consider an SD formed by an MTJ nano-pillar having elliptical-shape a × b FL of the thickness l (a/2 and b/2 are the ellipse semi-axes, Fig. 1). We assume that the magnetization of the FL M = mM_s is spatially uniform, and depends on time t only (i.e., we use the macrospin approximation¹⁸), m ≡ m(t) is the unit vector, and M_s is the saturation magnetization of the FL. The magnetization of the lowest pinned layer is assumed to be completely fixed, and its direction is defined by the unit vector $p=x^$⁠, where $x^$ is the unit vector of the x-axis. The FL of the SD (see Fig. 1) has a PMA of the first and second order,¹⁵ characterized by the anisotropy constants K₁ and K₂, respectively, and no in-plane anisotropy. There is no bias magnetic field applied to the structure.

Then, the effective magnetic field B_eff acting on the FL magnetization M is formed by the demagnetization field¹⁹ $Bd=−μ0MsN⋅m$ and the PMA field¹⁵ $BPMA=z^B1+B21−mz2mz$⁠, B_eff = B_d + B_PMA. Here, N = diag{N_x, N_y, N_z} is the diagonal self-demagnetization tensor having the elements N_x, N_y, and N_z (their sum is equal to 1), $z^$ is the unit vector of z-axis, $mz=m⋅z^$⁠, μ₀ is the vacuum permeability, and B₁ = 2K₁/M_s and B₂ = 4K₂/M_s are the fields of the first- and second-order PMA, respectively.

The magnetization dynamics in the FL is governed by the Landau–Lifshitz–Gilbert–Slonczewski equation,^7,8,20

where γ ≈ 2π · 28 GHz/T is the modulus of the gyromagnetic ratio, α is the Gilbert damping constant, σ = σ_⊥/(1 + η² cos β) is the current–torque proportionality coefficient, σ_⊥ = (γℏ/2e)η/(M_sV), ℏ is the reduced Planck constant, e is the modulus of the electron charge, η is the spin-polarization of current, V = πabl/4 is the volume of the FL having thickness l and elliptical cross section a × b, and β = arccos(m · p) is the angle between the magnetizations of the free and pinned layers.

There are three possible equilibrium magnetization states in the considered SD (see Fig. 1): the OOP state (⁠$m=±z^$⁠, m_z = ±1), the IP state (m lies in the x–y plane, m_z = 0), and the cone state (CS) that corresponds to the case 0 < |m_z| < 1. Among these three possible equilibrium magnetization states, the CS is the most interesting for broadband RF signal detection and RF energy harvesting, as in this equilibrium state an external RF signal most easily excites in the SD FL the magnetization precession with a large precession angle. In the absence of the bias magnetic field, the SD dynamics is symmetrical with respect to 180° rotation around x axis; therefore, we shall consider below only the case when the equilibrium magnetization direction lies in the upper half sphere, m_z > 0.

The value of m_z that corresponds to the equilibrium CS of magnetization, m_z,CS, can be found from (1) assuming dm/dt = 0, I(t) = 0: $mz,CS=1−r$⁠, where r = [μ₀M_s(3N_z − 1)/2 − B₁]/B₂ is the dimensionless ratio describing the state of the considered system. The equilibrium magnetization angle θ_CS (m_z = cos θ) that corresponds to the CS is $θCS=arccos(1−r)$⁠.

It is clear that the equilibrium CS of magnetization is possible only when 0 < r < 1, i.e., when the second-order PMA field B₂ is stronger than the positively definite effective first-order field [μ₀M_s(3N_z − 1)/2 − B₁] > 0. Alternatively, this condition can be written as a constraint on allowed values of N_z: 1 + 2B₁/μ₀M_s < 3N_z < 1 + 2(B₁ + B₂)/μ₀M_s. When r → 0, the CS transforms to the OOP state (m_z → 1), while at r → 1 the CS turns into the IP state (m_z → 0).

Using spherical polar coordinates for the unit vector m, $m=x^sin⁡θ⁡cos⁡ϕ+y^sin⁡θ⁡sin⁡ϕ+z^cos⁡θ$ (where $y^$ is the unit vector of y-axis), one can derive from (1) equations for the polar θ ≡ θ(t) and azimuthal ϕ ≡ ϕ(t) magnetization angles,

Here, ω_θ ≡ ω_θ(θ) = (ω₂ sin²θ + ω₁ − ω_MN_z)cos θ, ω₁ = γB₁, ω₂ = γB₂, and ω_M = γμ₀M_s. Assuming that both the Gilbert damping constant α and the magnitude of the input RF current I_RF are rather small, we can substantially simplify (2) by neglecting terms proportional to α² and αI_RF. Note, however, that this approximation should not be used for the case of large-power input signals (I_RF ≫ I_th).¹⁶ 

To estimate the average influence of an input RF current on the magnetization dynamics, we assume that in the OOP-regime the magnetization precesses around some equilibrium inclined axis (corresponding to the equilibrium CS with the polar angle θ_CS) along an approximately circular orbit (see Fig. 1). First, we let θ ≈ const, ϕ ≈ ωt + ψ in the CS, where ψ is the phase shift between the magnetization oscillations and the driving current. Second, we average the simplified equations for θ and ϕ over the period of oscillation T = 2π/ω of the driving current and obtain the following equations for the slow variables θ and ψ:

Here, we used the relation N_x + N_y = 1 − N_z and introduced the frequency of the OOP precession $ωp≡ωp(θ)=ωθ+ωM⁡cos⁡θ(1−Nz)/2=ω2sin2⁡θ−rcos⁡θ$ and the dimensionless functions $u≡u(aη)=1−(qη−1)2/aη2/qη$⁠, $v≡v(aη)=1+(qη−1)2/aη2/qη$ of parameter a_η = η² sinθ, $qη=1−aη2$⁠. In a typical experimental situation (η ≤ 0.7), the values of u and v are close to 1 for all the angles 0 ≤ θ ≤ π/2.

The OOP-regime of magnetization precession corresponds to the following stationary solution of (3): θ = θ_s = const, ψ = ψ_s = const. In this case, one can find from (3) the stationary value of the phase shift ψ_s [sin ψ_s = (2/u)(ω − ω_p)sin θ_s/σ_⊥I_RF, cos ψ_s = −2(α/$v$⁠)ω_p tan θ_s/σ_⊥I_RF] and then obtain the characteristic equation for the stationary polar precession angle θ_s,

Equation (4) has solutions only for RF currents I_RF that are larger than a certain threshold I_th. For small damping (α ≪ 1), the first term in (4) is much larger than the second one unless ω_p ≈ ω. Then, we can assume that at the threshold I_RF = I_th, the OOP eigen-frequency ω_p(θ) coincides with the driving frequency ω, which determines the threshold precession angle θ_th: ω_p(θ_th) = ω. Using this angle in (4), one can obtain the following expression for the threshold current:

The second expression for I_th in (5) was obtained by replacing θ_th ≈ θ_CS, which is valid at sufficiently low frequencies. Note that the threshold I_th vanishes in the limit ω → 0.

To analyze the stability of the magnetization precession in the OOP-regime around the CS of magnetization, we consider small deviations δθ and δψ of angles θ and ψ from their stationary values θ_s and ψ_s, respectively. Using the standard technique of the stability analysis for linearized equations with δθ, δψ, we found the following two conditions of stability: (i) dθ/dI_RF > 0 and (ii) $6⁡sin2⁡θ>4+r−4(2−r)2−3r2$⁠. The condition (i) determines the stable branch of solutions for θ, i.e., the branch for which the angle θ increases with current magnitude I_RF. The condition (ii) is satisfied for θ = θ_th for any 0 < r < 1 and, thus, is always satisfied on the increasing branch dθ/dI_RF > 0.

The output DC voltage generated by an SD in the regime of stationary OOP magnetization precession can be evaluated as $UDC=I(t)R(t)=IRFR⊥cos(ωt)/[1+τ⁡cos⁡β(t)]$⁠, where $R(t)=R⊥/1+τ⁡cos⁡β(t)$ is the MTJ resistance,^11,15 R_⊥ is the junction resistance for β = π/2, τ = TMR/(2 + TMR), TMR is the tunneling magnetoresistance ratio of the MTJ, cos β(t) = m(t) · p = sin θ_s cos(ωt + ψ_s). Calculating analytically$cos(ωt)/[1+τ⁡sinθs⁡cos(ωt+ψs)]$⁠, and using the previously given expression for cos ψ_s and assumption ω ≈ ω_p(θ_s), the output DC voltage can be written in the following form:

where $w$ ≡ $w$(a_τ) = (1 − q_τ)/a_τq_τ is the dimensionless function of parameter a_τ = τ sin θ_s,$qτ=1−aτ2$⁠.

As one can see, close to the threshold [I_RF ≳ I_th(ω)], the output DC voltage U_DC of an SD virtually does not depend on the input RF current magnitude I_RF and linearly increases with the frequency ω.

For an SD with an average resistance R₀, the energy harvesting efficiency ζ can be defined as a ratio between the detector’s output DC power $PDC=UDC2/R0$ and the power of the input RF signal $PRF=IRF2R0/2$⁠,

For I_RF ≥ I_th, the maximum value of ζ (achieved at the threshold) depends on the TMR ratio, $ζmax≈2(1−qτ)2/aτ2$⁠, and can reach ζ_max ≈ 40% for the TMR of 600% experimentally achieved in Ref. 21. However, with a decrease in TMR, the maximum value of ζ_max is substantially reduced; for instance, for TMR = 1, one can obtain only ζ_max ≈ 6%. Note, also, that in real experiments, the measured values of ζ may be substantially smaller than ζ_max value due to the impedance mismatch²² between the input transmission line with the impedance Z_TL and the SD with the impedance Z_SD connected to that line. To account for this effect, one should use in (7) the effective input power delivered into the SD, P_eff = P_RF(1 − |Γ|²), instead of the incident power P_RF, where Γ = (Z_SD − Z_TL)/(Z_SD + Z_TL) is the complex reflection coefficient.²² 

To compare the results of the developed analytical theory to the experimental results, and the results of numerical simulations, we consider the case of a SD based on a MTJ with the following parameters:¹⁵ the FL of the thickness l = 1.65 nm has an elliptical cross section of a × b = 150 × 50 nm²; normalized saturation magnetization of the FL is μ₀M_s = 1194 mT; the first order PMA field is B₁ = 1172 mT; the second order PMA field is B₂ = 64 mT; the Gilbert damping constant is α = 0.02; and the spin-polarization efficiency of the current is η = 0.6. For simplicity, we assume N_z = 1; thus, N_x = N_y = 0. Then, the CS ratio is r = (μ₀M_s − B₁)/B₂ = (ω_M − ω₁)/ω₂ = 0.344, and the equilibrium CS polar angle is $θCS=arccos1−r≈36°$⁠. In addition, using the experimentally found values of the MTJ resistance in parallel (R_P = 640 Ω) and antiparallel (R_AP = 1236 Ω) states and tunneling magnetoresistance ratio TMR = (R_AP − R_P)/R_P = 0.93, one can calculate the resistance of the SD in the perpendicular magnetic state (β = π/2) R_⊥ = 2R_APR_P/(R_AP + R_P) = 843 Ω.

To check the validity of the developed analytical theory, we performed macrospin simulations¹⁸ based on the numerical solution of (1) and, then, numerically calculated the output DC voltage using the general expression $UDC=I(t)R(t)$⁠. The results obtained from the developed analytical theory (green solid lines), from our simulations (crosses), and from the experimental results from Ref. 15 (circles) are presented in Fig. 2. As one can see from the analytical expressions, the response of the SD to an input RF power $PRF=IRF2R0/2$ is non-zero and relatively weakly changes with P_RF for P_RF exceeding the frequency-dependent power threshold $Pth(ω)=Ith2(ω)R0/2$⁠. Indeed, the analytical and numerical RF-power dependence of the output DC voltage U_DC has a step-like shape [see Fig. 2(a)]: U_DC ≈ 0 for P_RF < P_th and U_DC ≈ const. for P_RF ≥ P_th. The non-resonant response of the considered SD to the variation of the RF signal frequency f can be clearly seen in Fig. 2(b). The output DC voltage of the SD obtained in both analytical theory and numerical simulations increases linearly with f for f < f_th, and vanishes (U_DC ≈ 0) when the RF signal frequency exceeds a certain threshold f ≥ f_th.

The existence of the threshold frequency f_th follows from Eq. (5): with an increase in the signal frequency f, the threshold power P_th required for the proper SD operation also increases, and at the point where this threshold P_th exceeds the input power P_RF, the magnetization oscillations in the device FL vanish. Thus, as expected, in the OOP-regime, the SD works as a broadband non-resonant threshold RF detector.

It should be noted that while the analytically and numerically calculated dependences of the output DC voltage U_DC on the input signal power P_RF and frequency f, presented in Fig. 2, are in reasonable agreement, these dependencies demonstrate only qualitative resemblance with the experimental results from Ref. 15. We believe, that this discrepancy between the experiment¹⁵ and theory could be explained by an influence of the in-plane anisotropy in the FL of the SD used in the experiment,¹⁵ which was not taken into account in our theoretical model, and by the possible excitation in the experiment of some transitional regimes of the magnetization precession at rather large values of P_RF and f. Both these effects require an additional theoretical and experimental study. At the same time, we note that the presented theory, nonetheless, allowed us to approximately evaluate the experimentally obtained U_DC in the interval of variation of both P_RF and the signal frequency f. In addition, it is important to note that the efficiency ζ of the RF/DC energy conversion (or RF energy harvesting) for an SD with chosen typical parameters used in our numerical simulations reaches 5.7% at the threshold, which well agrees with the analytical estimation from (7). This number is rather encouraging for the possible use of SD having perpendicular anisotropy of the FL in practical RF energy harvesting. If we take into account the above discussed effect of impedance mismatch, which reduces the efficiency ζ to 1.5%, even this last lower number for the energy harvesting efficiency could be sufficient for many practical applications.

It should be noted that the analytical theory developed in this work could be very important for the optimization of working characteristics of SD-based RF energy harvesters. In particular, it follows from Eqs. (5) and (6) that such important SD performance parameters as RF power threshold P_th and the maximum operational frequency f_th strongly depend on the CS ratio r = (μ₀M_s − B₁)/B₂ (written here for the case N_z = 1). In experiments, the effective PMA field B₂ can be controlled by the variation of the FL thickness,^21,23 which allows one to vary the ratio r in a rather wide range. As one can see from Fig. 3 and Eq. (5), with the increase in the ratio r, the power threshold P_th of energy harvesting strongly increases, while the maximum operational frequency f_th linearly decreases. Thus, it is preferable to work at low values of the ratio r, which correspond to very small thicknesses of the SD FL that are difficult to achieve experimentally.^21,23 Thus, a compromise is necessary, and Eqs. (5) and (6) allow one to find that at the experimentally reachable values of r ≈ 0.2–0.4, it is possible to achieve reasonably small values of the threshold power P_th ≲ 20 nW while keeping the maximum operational frequency around 200 MHz–250 MHz (see Fig. 3).

In conclusion, we have shown theoretically that a spintronic diode (SD) having a first and second order perpendicular magnetic anisotropy of the free layer can be used as an efficient RF signal detector and energy harvester operating in the absence of a bias magnetic field. The device generates a finite output DC voltage, when its input RF power P_RF exceeds a certain frequency-dependent threshold P_th, while at P_RF > P_th, the voltage weakly depends on power. Such a regime of diode operation is possible at sufficiently low frequencies below the threshold frequency dependent on the RF signal power and CS ratio r, which has an optimal range of values r ≈ 0.2–0.4. Finally, it was demonstrated that the energy harvesting efficiency for the harvester could exceed 5% (1.5% with an account of the impedance mismatch effect) that is sufficient for some RF energy harvesting applications.

This work was supported, in part, by the U.S. National Science Foundation (Grant No. EFMA-1641989), the U.S. Air Force Office of Scientific Research under MURI Grant No. FA9550-19-1-0307, and the Oakland University Foundation. This work was also supported, in part, by Grant No. 19BF052-01 from the Ministry of Education and Science of Ukraine, the NATO SPS.MYP Grant No. G5792, and Grant No. 1F from the National Academy of Sciences of Ukraine.