Terahertz Antiferromagnetic Spin Hall Nano-oscillator
We consider the current-induced dynamics of insulating antiferromagnets in a spin Hall geometry. Sufficiently large in-plane currents perpendicular to the Néel order trigger spontaneous oscillations at frequencies between the acoustic and the optical eigenmodes. The direction of the driving current determines the chirality of the excitation. When the current exceeds a threshold, the combined effect of spin pumping and current-induced torques introduces a dynamic feedback that sustains steady-state oscillations with amplitudes controllable via the applied current. The ac voltage output is calculated numerically as a function of the dc current input for different feedback strengths. Our findings open a route towards terahertz antiferromagnetic spin-torque oscillators.
pacs:76.50.+g, 72.25.Mk, 75.78.-n, 75.50.Ee
Introduction.—The discovery of spin-transfer torques (STTs) ref:STT1 (); ref:STT2 () initiated an intense search for current-induced phenomena in magnetic materials because a STT can compensate the magnetic damping and induce spontaneous magnetization dynamics. When such a compensation occurs, the magnetization either switches to another direction ref:SHswitch_Ralph1 (); ref:SHswitch_Ralph2 () or evolves into a steady-state oscillation ref:Tsoi (); ref:Chien (); ref:STNO_Ralph (); ref:STNO_Silva (); ref:STNO_Zeng (). While the former improves writing operations in magnetic memory devices, the latter enables sustainable ac signal generation from dc inputs, known as spin-torque oscillators ref:STOreview (); ref:chapter38 (). In ferromagnets, currents or magnetic fields can tune the output frequency in the range from the megahertz to the gigahertz regime.
Spin-torque oscillators can potentially be operated at much higher terahertz frequencies when antiferromagnets (AFs) replace ferromagnets. Two facts make this possible: (1) the eigenfrequencies of typical AFs fall into the terahertz range ref:AFMR () and (2) a STT can trigger spontaneous excitations of an AF in a similar way as ferromagnets ref:SPAF (); ref:Matthew (); ref:Gomonay (). While most AFs are insulators preventing the STTs to be operative by passing a current through the sample, the spin Hall effect (SHE) ref:SHE () is an alternative that generates STTs even when electrons do not flow through the magnet ref:SHNO_Saitoh (); ref:SHNO_Ralph (). The latter phenomenon provides an avenue towards low-dissipation spin Hall nano-oscillators (SHNOs) ref:SHNO_Urazhdin (); ref:SHNO_simulation ().
However, to realize AF-based SHNOs, current-induced excitations should not grow indefinitely. Instead, they should evolve into steady-state oscillations with substantial output power note (). Although an AF does not suffer magnetic switching even when a STT overcomes the damping, its Néel vector will directly evolve into a right-angle precession around the direction of the spin accumulation ref:Gomonay (). Since the amplitude of such a dynamical motion is not continuously tunable via the applied current, it does not meet the requirements of a SHNO.
Steady-state oscillations are realizable in ferromagnets for the following reasons. According to the original form of the STT ref:STT1 (), its angle dependence and that of the Gilbert damping differ ref:JXangle (). Therefore, as the amplitude of a spontaneous motion is growing, the two competing mechanisms will balance each other at a unique angle—that is where a steady-state oscillation takes place. Nevertheless, this feature is not active when the SHE operates the STT. In the latter scenario, one needs to introduce alternative mechanisms to prevent a spontaneous excitation from growing into magnetic switching. For example, the spatially localized mode ref:Bullet () produces a nonlinearity that can sustain its auto-oscillations ref:SHNO_Urazhdin (). To remain within spatially uniform excitations, the dipolar interaction is often required ref:Rezende (). However, the dipolar interaction is negligible in AFs where the magnetization is vanishingly small.
In this Letter, we exploit a recently proposed feedback mechanism ref:Feedback () to realize a terahertz SHNO in an AF/heavy-metal heterostructure. The feedback originates from the combined effect of the SHE and its reverse process, which connects spin pumping with the spin backflow ref:spinbattery (); ref:backflow (). It is entirely independent of the dipolar interaction. First, we determine the threshold of spontaneous excitations by solving the Néel order dynamics in the linear response regime and relate the threshold to a current density. Then, we numerically explore the nonlinear Néel order dynamics beyond the threshold by including the feedback effect ref:Feedback () and show that the feedback is indispensable to maintain uniform auto-oscillations. Finally, we demonstrate that in contrast to previous studies ref:Gomonay (), our proposed SHNO creates a substantial ac voltage output with its amplitude continuously tunable via the applied dc current.
Dynamics.—We assume that the AF has a single crystal structure, and describe the sublattice magnetizations by two unit vectors and . We introduce the Néel vector , and the small magnetization ; they satisfy and . In the exchange limit, , thus and . The Cartesian coordinates are chosen such that the hard axis is along , and the in-plane easy-axis along . We scale everything in (positive) angular frequency, where the hard axis anisotropy is described by , the easy in-plane anisotropy , and the Heisenberg exchange interaction . In the macrospin description, the free energy is , which defines two thermodynamic forces and ref:phenom (). The coupled equations of motion are
Here, is the vector of spin accumulation; its magnitude (in frequency units) represents the STT strength.
To derive the current-induced excitations, we decompose the Néel vector as , assuming . Restricting to linear order in , we can eliminate in Eqs. (1a) and (1b), and obtain the eigenfrequencies as , where the () sign corresponds to the optical (acoustic) mode supp (). In Fig. 1, we plot the two eigenfrequencies as a function of the STT strength with parameters taken from NiO ref:NiO (). As increases, the real parts and approach each other until they become degenerate at . By contrast, the imaginary parts and remain degenerate and unaffected for . But when , () reduces (grows) rapidly, indicating that the damping is diminished (enhanced) by the STT. At the threshold ref:Gomonay (); supp ()
vanishes, which marks the onset of spontaneous excitation of the optical mode and the breakdown of the linear response approximation. The uniaxial symmetry enforces that also vanishes for so that the auto-oscillation can be triggered by a reversed current as well. Hereafter, we restrict our discussions to positive unless otherwise stated.
In the absence of the hard-axis anisotropy (), the threshold (3) is linear in , so the antidamping effect occurs when the STT is turned on. However, in the general case where , the antidamping effect appears only when as shown by the curve in Fig. 1, whereas a driving STT in the regime modifies the patterns of the eigenmodes as illustrated in Fig. 1 (also see Ref. ref:Gomonay ()). Specifically, an increasing STT drags the long axes of the elliptical precessions away from their original orientations until they are 45 away from the hard-axis. In spite of this change, and always exhibit opposite chiralities, i.e., as seen from the -direction, () rotates counterclockwise (clockwise). However, at the degenerate point , the chirality of () in the optical (acoustic) mode reverses. Consequently, when , both and , hence the Néel vector , all acquire the same chirality. At the threshold , the excited optical mode is right-handed. If changes sign, the excited mode is still the optical mode, but its chirality becomes left-handed. These observations suggest that the direction of the current determines the chirality of the excitation.
Critical current.—Consider a setup consisting of an insulating AF deposited on a heavy-element normal metal (HM) with spin-orbit coupling, as shown schematically in Fig. 2. We assume a current density is applied along the -direction; it is perpendicular to the Néel vector of the AF. The SHE in the HM generates antidamping STTs to drive the Néel vector dynamics, which in turn pumps spin current back into the HM. The pumped spin current converts into a charge voltage due to the inverse SHE ref:ISHE (), which is detected by two voltmeters. Let be the thickness of the AF, the thickness of the HM, and assume that the HM has spin diffusion length , lattice constant , and conductivity . By solving the spin diffusion equation in the presence of the SHE ref:Feedback (); ref:JX () under boundary conditions involving both spin pumping and STTs ref:SPAF (); supp (), we relate the threshold STT Eq. (3) to a critical current density
where is the spin Hall angle, is the electron charge, and is the areal density of the transverse mixing conductance ref:SPAF (). From Eq. (4), we see that the critical current density can be lowered by reducing (increasing) the thickness of the AF (HM ). For example, consider a NiO(1)/Pt(25) (numbers in nm) bilayer. At room temperature, we use Pt material parameters from Ref. ref:Ptdata (), and use per for perfect interfaces ref:SPAF (). Since nm in NiO, Eq. (4) gives A/cm. For uniaxial AFs such as MnF, the hard-axis anisotropy is absent (), thus will be appreciably smaller, so will .
In real AF/HM heterostructures, the critical current density could be higher than the above estimation since the surface roughness can diminish the transverse mixing conductance . Nevertheless, a large spin Hall angle by using e.g. topological insulators ref:TI () can reduce the critical current. In addition, even though domain formation can happen, the Néel vector survives a spatial average over all domains since it is bi-axial (for comparison, the magnetization vector of a ferromagnet is uniaxial). In this sense, the domain formation amounts to a reduction of the volume density of the Néel vector. While these imperfections renormalize the material parameters, they do not qualitatively invalidate the essential physics.
Feedback.—The linear response only allows us to solve the eigenmodes as those depicted in Fig. 1 and to predict the threshold of auto-oscillation excitations. Beyond the threshold, however, the assumption is invalid, and we need to consider nonlinear responses. But in our calculations so far, both the Gilbert damping and the antidamping STT are linear in , so is the total effective damping. This behavior implies that the amplitude of a uniform excitation will grow exponentially with time since . In ferromagnets, this means that the magnetization switches to the opposite direction without any steady-state motion at intermediate configurations. Here in a collinear AF, the terminal status of the Néel vector is a right-angle precession around as shown in the upper-right panel of Fig. 3, whereas oscillations at cone angles are unstable.
However, the above analysis is incomplete as it ignores a crucial feedback effect ref:Feedback (). The pumped spin current from a precessing Néel vector into the HM experiences a backflow that introduces a spin battery effect ref:spinbattery (). In HMs, however, the spin pumping and the spin backflow are also connected via the combined effect of the SHE and its inverse process, which feeds the Néel vector dynamics back into itself. In ferromagnets, such a feedback mechanism manifests as a nonlinear damping effect in the magnetization dynamics ref:Feedback (). Following the same spirit, we can derive a similar feedback-induced damping effect for AFs in the exchange limit where . To this end, recall that the pumped spin current into the HM converts into an electric field due to the inverse SHE. According to Ohm’s law, where is the spin accumulation in the HM. As we fixed the current density through external circuits, a change of the electric field necessarily leads to a change of the spin accumulation . Subsequently, the change of diffuses and generates an additional spin current, which will finally deliver the influence of spin pumping back into the Néel vector through STTs. By closing such a feedback loop supp (), we obtain a feedback torque that should be added to Eq. (2a) as
where the feedback coefficient is
For the NiO(1)/Pt(25) bilayer structure considered earlier, . When manipulating the material parameters, has a maximum at . While the feedback effect seems to be a higher order effect as is proportional to , it can be significantly enhanced by searching for materials with large . For example, it was shown recently that a topological insulator can potentially exhibit an extraordinarily large even greater than unity ref:TI ().
The feedback-induced nonlinear damping is a critical ingredient because it dramatically modifies the dynamical behavior of a SHNO using AFs. We demonstrate its effect by performing a numerical simulation with the result shown in Fig. 3. For a given set of , we first run the simulation for a sufficiently long time so that the oscillation no longer grows. Then, we take a time average of the cone angle over several periods. If we ignore the feedback, , the AF either experiences no oscillation or proliferates into the right-angle precession (). In the latter case, the orbits of and are circles and overlap each other completely (but their phases are still different by ). In the presence of the feedback, , a finite window of stable oscillations at opens up; the larger the , the wider the window. In this novel phase, the terminal angle increases with increasing STT strength (hence ). But at a sufficiently large , the oscillator inevitably jumps into the proliferation phase, which marks a phase boundary separating stable oscillations from the proliferated configuration. However, this phase boundary terminates at extremely large , after which transitions between the stabilized and proliferated oscillations are continuous.
Output.—A salient feature of the novel stable oscillation phase is that the applied dc current density controls the output power and that the output power is substantial: both features are indispensable for a SHNO. To quantify this fact, we explore multiple ac outputs as functions of for different feedback strengths. Considering again the NiO(1)/Pt(25) structure, we run simulations for A/cm and . First, we plot the frequency output in Fig. 4(a). In the stable oscillation phase, the actual frequency output lies between the acoustic and the optical modes. Second, we study the ac voltage output from the inverse SHE and the spin pumping. For a fixed , the total electric field includes a time varying part supp ()
that reflects our desired contribution. By eliminating the dc component of , we compute its ac components in the effective value numerically. As shown in Figs. 4(b) and (c), the () component is appreciably large (essentially zero) in the stable oscillation phase, whereas it vanishes (becomes nonzero) in the proliferation phase. This contrasting property also enables a practical way to observe the phase transition. As illustrated in Fig. 2, the field is measured by two voltmeters. Since the maxima of both and reach 10 V/cm, the actual measured voltages from a nanometer-sized sample can be as large as V.
Acknowledgements.We are grateful to H. V. Gomonay, J. Xiao, M. W. Daniels, and J.-G. Zhu for insightful discussions. Work at Carnegie Mellon University is supported by the Department of Energy, Basic Energy Sciences, Grant No. DE-SC0012509.
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