Introduction

Exceptional points (EPs) are singularities in the parameter space of a system corresponding to the coalescence of two or more eigenvalues and the associated eigenvectors1,2,3,4,5,6. They are a peculiar feature of nonconservative (open) systems that have both loss and gain and they emerge when these two effects compensate. From the fundamental point of view, EPs play an important role in the area of non-Hermitian quantum theory based on \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric Hamiltonians (with simultaneous parity-time invariance)7. In this context, they occur at phase transitions between broken-unbroken \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetry. While initially EPs were regarded as a mathematical-physics concept, in the last decade there has been a growing interest in EPs from the experimental point of view in such areas as atomic spectra measurements8, microwave cavity experiments9,10,11,12,13, chaotic optical microcavities14 or optomechanical systems15. Interestingly, exceptional points arise also in classical systems, such as coupled electric oscillators16,17, optical systems18, classical spin dynamics19,20, and general dissipative classical systems21.

Application-wise, the strong sensitiveness of eigenvalues to perturbations near EPs has been used to devise new types of sensors with unprecedented sensitivity11,22,23. This was demonstrated in highly sensitive optical nanoparticle detection24,25, in laser gyroscopes26, in optically pumped semiconductor rings for temperature detection13, and in coupled microcantilevers for ultrasensitive mass sensing27, however, none has been adapted to CMOS compatible systems.

In the realm of magnetism, a field with profound implications for modern information technology, there has been growing attention on EPs and the effects of non-hermiticity in the last couple of years. Different approaches have been used exploiting the coupling of magnons to other distinct quantum systems, such as phonons28 or explicitly photons29,30,31,32, where great advance could be achieved in the field of cavity-magnonics33. In spintronics, the interest on EPs started with theoretical works on classical spin dynamics19,20,34,35,36,37,38 and lately also includes spin wave physics39,40. Recent theoretical studies identify EPs as signatures of dynamical phase transitions relating linear and nonlinear spin dynamics in their proximity38, or find nontrivial non-hermitian topological phases in large arrays of coupled spintronic oscillators41,42, or evidence complex bifurcations and bistability in a nonlinear coupled spintronic system43 – aspects that are yet to be explored experimentally. It is to be noted that all works predominantly focus on local coupling mechanisms (except Ref. 43). Only recently, also based on a local coupling, the emergence of an EP could be demonstrated experimentally in magnonic \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetry devices44. However, the approach stays passive, i.e. the magnetic dissipation (damping) in two coupled magnetic layers is fixed and the system control (through the coupling) was realized through the choice of the thickness of the separating layer between the ferromagnetic thin films. The experimental study of EPs in coupled discrete nano-devices (with gain) has been so far overlooked and an experimental control of the coupled dynamics on a nonlocal circuit level is completely missing.

In this study, we demonstrate the presence of EPs in a system of two coupled spin-torque nano-oscillators (STNOs) (see Fig. 1) and how it can be used to control their oscillating state. STNOs are typical, CMOS-compatible, spintronic nanoscale devices in which the magnetization dynamics in a thin layer can be converted into electrical microwave signals45. They have both loss, associated with the damping of magnetization oscillations, and gain, provided by the transfer of spin angular momentum (through the spin torque effect) from a spin polarized current injected into the device (see Methods and Supplementary Information for further details). In this respect, coupled STNOs are archetypal non-hermitian systems to evidence the relevance of EPs in spintronics. We use STNOs based on the spin torque gyrotropic dynamics of a magnetic vortex core45 (later on labelled as STVOs), since these show the best rf performance46. However, our results are applicable to all coupled STNO systems. We experimentally demonstrate that, by tuning the dc currents injected into the two STNOs, the position of an EP can be finely controlled leading to the phenomenon of amplitude death47. This describes the vanishing of the oscillation amplitudes of the coupled STNOs, despite the increase of the spin torque (gain) in one oscillator. As shown later, such amplitude death occurs for a certain specific range of injected dc current values.

Fig. 1: Schematic view of the spintronic EP system made of coupled spintronic nano oscillators.
figure 1

The coupling, which is designed to be symmetric, is obtained by feeding strip-line antennas above each oscillator with the microwave current Irf generated by the spin torque dynamics occurring in the other oscillator. This generates a magnetic rf field, Hrf, that hence generates the coupling78. The coupling is nonlocal and can be performed over long distances.

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The presented results intend to create an important connection between the non-hermitian physics of EPs and spintronics, an area of research that has crucial technological implications for data storage and processing48,49, sensor technology48,50, wide-band high-frequency communications51,52,53,54,55,56, and, more recently, bio-inspired networks for neuromorphic computing beyond CMOS57,58. Up to now, coupled STNOs have been studied mainly with respect to the phenomenon of mutual synchronization59,60,61,62,63 and non-hermitian aspects are neglected or are about to be theoretically discovered41,42. Our observation provides the important experimental evidence to exploit EPs in coupled spintronics nano-devices.

Results & discussion

Theoretical modelling of EPs in spintronics

We first theoretically study the regime of small amplitude oscillations of the two coupled STVOs around their rest positions. From the linearized equations governing these small oscillations, the condition for an EP to emerge can be determined, leading to a formula that connects, at the EP, the relevant parameters of the coupled oscillators: frequencies, gain/loss parameters, and coupling coefficient. The position of an EP in the parameter space can hence be controlled in order to determine the interval of injected current values in which amplitude death occurs. The gain mechanism, counteracting the natural dissipation and enabling self-sustained oscillations in each STVO, is provided by the spin-transfer torque that is proportional to the injected dc current. Self-oscillations set in when the injected current I is larger than a critical (threshold) current Ic, which corresponds to the exact compensation of gain and loss61.

Gyrations of the vortex core around the symmetry axis of each oscillator are modeled by the Thiele-like theory for which the overall state of the oscillators is given by the in-plane displacements (ρ1, ρ2) of the vortex cores from the center of the corresponding devices (for details see Supplementary Information). The coupling between the two STVOs, which is assumed to be symmetric, is obtained by feeding strip-line antennas above each oscillator with the microwave voltage generated by the other, that in turn gives rise to a rf magnetic field (see Fig. 1). For vortex core displacements sufficiently small compared to the device radius, it is reasonable to assume a linear coupling47,64 between the STVOs, reflecting the relevant range of the performed experiments. Importantly, the coupling has both dissipative and conservative terms that are described by the coefficients kd and kc, respectively, and relate to the complex impedance of the electrical circuit.

The linearized Thiele equation governing the regime of small oscillations of the vortex cores around the rest position ρ1,2 = 0, written in terms of the complex state variables zl = xl + iyl (with l = 1, 2) associated to the l-th vortex core x- and y-axis position, reads

$$\begin{array}{r}\frac{d}{dt}\left[\begin{array}{c}{z}_{1}\\ {z}_{2}\end{array}\right]=iA\cdot \left[\begin{array}{c}{z}_{1}\\ {z}_{2}\end{array}\right],\end{array}$$
(1)

with

$$\begin{array}{r}A=\left[\begin{array}{cc}{\omega }_{1}-i{\beta }_{1}&k\\ k&{\omega }_{2}-i{\beta }_{2}\end{array}\right],\end{array}$$
(2)

where k = kc − ikd, ωl are the angular frequencies of vortex free oscillations, and βl are the loss/gain parameters. These latter parameters are given by βl = ClIl − dlωl, where dl are the damping constants and Cl are parameters determining the efficiency of the spin transfer effect, i.e. effectively βl > 0 ( < 0) corresponds to gain (loss).

The matrix A in eq. (2) is non-hermitian and has indeed the typical form for systems exhibiting EPs1,4. In order to study the natural frequencies of the system (1), we assume a dependence of z1,2 on the time of the type eiνt. The natural frequencies ν1,2 are then obtained as eigenvalues of the matrix A and they are given by the following formula:

$$\begin{array}{r}{\nu }_{1,2}=\bar{\omega }-i\bar{\beta }\pm \sqrt{{k}^{2}+{\left(\tilde{\omega }-i\tilde{\beta }\right)}^{2}},\end{array}$$
(3)

where

$$\bar{\beta } = (\,{\beta }_{1}+{\beta }_{2})/2,\quad \bar{\omega }= ({\omega }_{1}+{\omega }_{2})/2,\\ \tilde{\beta }= (\,{\beta }_{1}-{\beta }_{2})/2,\quad \tilde{\omega }= ({\omega }_{1}-{\omega }_{2})/2.$$

Stability of solutions is given by the following condition:

$$\begin{array}{r}\,{{\mbox{Im}}}\,(\nu )\ge 0\,\,\iff \,\,\,{{\mbox{Re}}}\,(i\nu )\le 0.\end{array}$$
(4)

By definition, EPs emerge when two natural frequencies coalesce along with the corresponding eigenvectors. This occurs when the square root term in eq. (3) is zero, leading to the following condition on the parameters to obtain an EP:

$$\begin{array}{r}{k}_{c}-i{k}_{d}=\pm \left[\frac{(\,{\beta }_{1}-\,{\beta }_{2})}{2}+i\frac{(\,{\omega }_{1}-\,{\omega }_{2})}{2}\right],\end{array}$$
(5)

where we have explicitly expressed \(k,\tilde{\omega }\) and \(\tilde{\beta }\).

We consider here the case that will be also presented in the experimental section: the current I1 is fixed to a value \({I}_{1}^{*}\) and the second current I2 is swept from values below to values above the threshold current Ic,2. By using the condition (5), the values of the parameters can be adjusted to have an EP at the desired value of the current I2. The effect of the EP on the eigenvalues of the matrix A is illustrated in Fig. 2. The black curves in Fig. 2a are the real and imaginary parts of the STVOs’ eigenvalues when there is no coupling (kc = kd = 0). When coupling is taken into account, an EP exists at I2,EP = 7.1 mA (green star), and it has the effect of attracting the two eigenvalues to one point in the complex plane. According to eq. (3), if, at the EP, \(\bar{\beta }=(\,{\beta }_{1}+{\beta }_{2})/2\) is negative (this occurs when β2 < 0, ∣β2∣ > β1 > 0), then both eigenvalues iν have negative real parts and the system is lossy. This is also visible in Fig. 2b where the eigenvalues are plotted in the complex plane (Re(iν), Im(iν)): In the proximity of the EP, both eigenvalues are in the plane Re(iν) < 0. This implies that the rest position of the two oscillators is stable, leading to the disappearance of both STVOs’ oscillations in the nonlinear interpretation. This phenomenon is called amplitude death and, as we have illustrated above, it can be controlled by the appropriate placing of the EP in the parameter space. Since the condition for the onset of an EP is very sensitive to perturbations, it might happen in experiments that the EP is not reached in a strict sense. Nevertheless, if the parameters are such that the condition (5) is nearly verified, the amplitude death phenomenon is expected to be reliably observed as well.

Fig. 2: Eigenvalues iν1,2 from eq. (3) when the EP is placed at \({I}_{1}^{*}={I}_{2,{{{{{{{\rm{EP}}}}}}}}}=7.1\) mA (green star).
figure 2

a Real and imaginary part of the eigenvalues as a function of the dc current I2. b Eigenvalues in the complex plane (Re(iν), Im(iν)). Black lines in (a) as well as black line and symbol in (b) refer to eigenvalues computed in the uncoupled case. Dashed gray lines depict the stability criterion (4): for Re(iν) < 0, the rest position is stable and no auto-oscillations in the nonlinear sense occur. The system parameters can be found in the Supplementary Information. The coupling constant is kEP = 9.76 − 25.13i.

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It is important to note that, when the rest position is stable in terms of the linearized model, we find that this stability is also exhibited by the rest position in the full nonlinear equations (see Supplementary Information). An important consequence is that, for predicting the phenomenon of amplitude death, the linear theory is strictly appropriate. On the other hand, when the real part of the eigenvalue iν becomes positive – this happens when Re(iν) crosses zero in Fig. 2 – the rest state becomes unstable. The regime that sets in after instability has an amplitude determined by the nonlinear saturation term in the Thiele equation and an approximate frequency of Im(iν) at the aforementioned crossing. This phenomenon is referred to as a supercritical Andronov-Hopf bifurcation65. For values of parameters which correspond to a Hopf bifurcation point and no other bifurcations take place, the linear analysis can be used to estimate the frequency of the self-oscillating regimes by considering the imaginary parts of the natural frequency iν at the Hopf bifurcation. In the assessment of the experimental results, this concept is applied in order to identify the appropriate parameter values describing the amplitude death region as a function of the injected currents.

Experimental emergence of EPs

After having theoretically established the condition for the existence of an EP, we describe the experimental results that demonstrate the emergence of EPs and the correlated amplitude death regions in our coupled STVO system. All measurements have been conducted at room temperature. In the performed experiments, the current injected into the STVO 1 is kept constant to \({I}_{1}^{*}\), while sweeping the current I2 injected into STVO 2. Note that the onset for self-sustained oscillations in the uncoupled case is Ic,1 ≈ 6.95 mA and Ic,2 ≈ 8 mA for STVO 1 and 2, respectively. The frequency evolution of the uncoupled STVOs with the applied current is similar (see Supplementary Information).

In Fig. 3a, we display the frequency spectra measured at \({I}_{1}^{*}=8\) mA while I2 is changed. For these conditions, no amplitude death is observed, however, a square-root-like frequency branching for I2 ≥ 8 mA is present. We ascribe this phenomenon to the presence of an EP in the linearized model. Therefore, we use formula (5) collocating the EP at \(({I}_{1,{{{{{{{\rm{EP}}}}}}}}}^{*};{I}_{2,{{{{{{{\rm{EP}}}}}}}}})=(8;8)\) mA. Based on this identification, the linear coupling constant can be determined and the theory parameters adjusted in order to compute the eigenvalues iν1,2 as a function of the current I2 (Fig. 3b). From the general point of view, the eigenvalues iν1,2 give information about the linear dynamics around the rest position. Their imaginary parts show a branching similar to the experimentally observed oscillation frequencies (Fig. 3a). Indeed, the theoretical linear approach provides a good access to the analysis of the intrinsically nonlinear regime of the experimental self-oscillations. In the Supplementary Information, we show the consistency of the linear model with numerical computations of the mutually coupled nonlinear dynamics based on the Thiele equations. Note that in a strict sense, the eigenvalues in the coupled system cannot be directly assigned to the single STVOs, but the modes must be regarded collectively, referring to the system matrix A. However, throughout the article we label the experimental signals corresponding to the STVO in which the oscillation mode is mainly localized, as corroborated by the nonlinear simulations and the theoretical analysis of the system eigenvectors under relatively small coupling (see Supplementary Information). In Fig. 3b, the real part confirms that the rest-position is unstable over the entire range I2, i.e. Re(iν) > 0, and hence self-oscillations are stabilized. The decrease of Re(iν) in the proximity of the EP is consistent with the experimental linewidth broadening in the range I2 ∈ [7.2; 8] mA in Fig. 3a, where noise-induced fluctuations become important due to the vicinity of the stability axis (Re(iν) = 0).

Fig. 3: Measured frequency spectra of the coupled system vs. current I2 for \({I}_{1}^{*}=8\) mA.
figure 3

a The labeling corresponds to the STVO in which the oscillation mode is mainly localized. b Corresponding theoretically determined evolution of eigenvalues exhibiting an EP (green star). System parameters in Supplementary Information. The resulting coupling constant: kEP = 10.68 − 25.13i.

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From the prediction of our model, we expect that the decreasing of the gain effect (through the adjustment of the spin transfer torque in our case) together with the attraction of the eigenvalues around the EP will make the amplitude death phenomenon observable. To confirm this behavior, in Fig. 4a–d, we perform measurements of the coupled system for smaller \({I}_{1}^{*}\) for which the eigenvalue real part can explicitely become negative due to the EP and hence, amplitude death occurs in this regime. With respect to the critical currents of the uncoupled STVOs, STVO 1 is undercritical in Fig. 4a and overcritical in Fig. 4b–d. The overall range of oscillation death evolves with \({I}_{1}^{*}\) (Fig. 4b–d), whereas rather the smaller value I2 defining the amplitude death interval is affected than the larger one which remains quasi-constant. Increasing \({I}_{1}^{*}\) tends to stabilize the oscillation of STVO 1 and in consequence, counteracts the occurrence of the amplitude death. This leads to a decrease of the current range in which no oscillation is detected (see Fig. 4). Furthermore, for \({I}_{1}^{*} \, < \,7.8\) mA and I2 > 8 mA, the oscillations from STVO 1 show a lower output power together with a larger linewidth than it would be expected for self-sustained oscillations. When the current \({I}_{1}^{*}\) is close to 8 mA, in the vicinity of the EP, thermal noise can induce stochastic transitions between the oscillatory regime and the rest state corresponding to amplitude death (clearly visible in Fig. 4d). For currents \({I}_{1}^{*}\, \gtrsim \, 8\) mA (see Fig. 3a for \({I}_{1}^{*}=8\) mA), oscillation death is no more occurring, however, the linewidth of the oscillation is clearly enhanced in a small range I2 ∈ [7; 8] mA. This range however decreases with increasing currents \({I}_{1}^{*}\). At even larger currents \({I}_{1}^{*}\, \gtrsim \, 9\) mA (see Supplementary Information), the two STVOs tend to mutually synchronize, a phenomenon that is commonly known for STNOs59,60,61,62,63 and which refers to the strongly nonlinear characteristics of the oscillator, far from the Hopf bifurcation point.

Fig. 4: Amplitude death and stochastic stability in the vicinity of the exceptional point.
figure 4

Measured frequency spectra of the coupled system vs. current I2 in STVO 2 for different currents \({I}_{1}^{*}\) in STVO 1 (a–d). e–h Real and imaginary part of iν1 (red) and iν2 (blue) fixing current I1 and changing current I2. Black lines refer to the same quantity evaluated when the coupling constant is set to 0. Experimental and theoretical graphs in the same column correspond to the same parameters.

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The experimentally observed amplitude death is very well reproduced by our modelling of the coupled STVOs. In Fig. 4e–h, we present the corresponding real and imaginary parts of the eigenvalues iν1,2 as a function of the current I2. Except for the value of the coupling constant, which in principle depends on the electric interface between the STVOs as well as on their dynamical state, the modelling parameters are the same as those used in Fig. 3b. We find that by only rotating the before determined coupling constant \({k}_{EP}^{*}\to {k}_{EP}^{*}\,{e}^{i{\phi }_{k}}\) in the complex plane (kc, kd) without changing its modulus, the amplitude death phenomena can be completely described. The rotation angle for the two cases where the amplitude death is evident at \({I}_{1}^{*}=7.1\) and 7.5 mA (Fig. 4b, c) is ϕk = 40 and 45∘, respectively. In Fig. 4f–g, the amplitude death current ranges can be recognized by looking at where the condition Re(iν1,2) ≤ 0 is satisfied. Then, at the upper current value I2 of the amplitude death regime, the real part of one eigenvalue crosses the real axis and the corresponding mode becomes unstable. This situation corresponds to a Hopf bifurcation which brings the system to self-oscillations. Such consideration permits to rigorously justify the presence of the upper branch in the measured spectra. The discussed Hopf bifurcation point does not significantly change its position while the square root like upper branch of Re(iν) at lower currents I2 implies a strong dependence of the amplitude death range’s lower boundary on the fixed current \({I}_{1}^{*}\), as also found experimentally. For larger current I2, in the case \({I}_{1}^{*}=7.5\) mA, also the real part of the other eigenvalue (blue curve in Re(iν)) becomes positive, but it stays close to the real axis. In the experiments, which are subject to thermal fluctuations, this manifests as the described linewidth broadening of STVO 1’s oscillation at relative smaller power. Similar situation occurs for \({I}_{1}^{*}=6.9\) mA and \({I}_{1}^{*}=7.1\) mA. In both cases the value of the rotation angle is set to ϕk = 40∘. The main difference with the \({I}_{1}^{*}=7.5\) mA case is that only the real part of one eigenvalue crosses the real axis. The other stays close to it. Similar to before, thermal fluctuations shall permit oscillations, however exhibiting a large linewidth in the experiments. For \({I}_{1}^{*}=7.8\) mA, the measured spectra are similar as for \({I}_{1}^{*}=8\) mA and hence we set ϕk = 0∘. The oscillations’ death for this case is experimentally observed (see Fig. 4d), but is not described by the linear theory. Note that nonlinearity might become more important in this regime. However, the stochasticity of the transitions between oscillation regime and rest state suggests that also thermal fluctuations play in this case a dominant role in determining the stability of the oscillators.

Indeed, the main characteristics of the coupled system can be accessed by the developed linearized theory. The study of the eigenvalues as a function of the current permits to unravel the key features of the coupled STVO system’s frequency response.

In conclusion, we exploit the non-hermiticity of two coupled spintronic nano-oscillators and demonstrate the emergence of EPs in this spintronic system, which is in fact promising candidate for multiple potential applications45. The existence of an EP drastically influences the eigenvalue characteristics leading to various complex phenomena, such as oscillation death or stochastic oscillation stability. We develop a theoretical modelling and show that the main experimental features at this stage can be well reproduced by linearized coupled spintronic equations.

Outlook

One of the interesting specificities of the spintronic nano-oscillators is their strong nonlinearity which makes them a promising candidate for various applications and leads to a tremendous manifold of physical phenomena unified in these nanoscale devices. The emergence of an EP in a nanoscale nonlinear system is to our opinion of fundamental interest. Beyond the already mentioned implications for the development of novel types of spintronic sensors operating at exceptional points11,22, these systems are anticipated to unravel fascinating physics. This includes phenomena such as chaos, complex bifurcations43, or the emergence of topological operations around the EP15,66. Complex dynamics and as well the demonstrated occurrence of stochastic stability might furthermore complement the field of hardware-based neuromorphic computing that recently gained attention in the context of spintronics67, for instance as stochastic spiking neurons. Non-hermiticity in this respect adds an additional complex response of the system to input signals68,69,70, implying abrupt phase transitions which are also inherent in neural networks71. Characteristics of non-hermiticity have been found in the description of the brain, for instance in EEG measurements72,73, or the inhibitory and excitatory balance in neocortical neurons74, similar to nonconservative elements of gain and loss in our STNO system. We emphasize that higher dimensionally coupled systems have been realized with STNOs63,75,76 which are anticipated to facilitate the emergence of higher order exceptional points13,36,40 or other complex dynamics41,42. All these different aspects are still to be explored and potentially lead to intriguing findings in nonlinear non-Hermitian systems on the nanoscale.

Methods

More extensive information along with additional data and discussion regarding theory, simulations, and experiment can be found in the supplementary information file.

Device fabrication

The studied STVO devices are magnetic tunnel junctions containing a pinned layer made of a conventional synthetic antiferromagnetic stack (SAF), a MgO tunnel barrier and a NiFe-free layer in a magnetic vortex configuration (blue, green and yellow layers in Fig. 1, resp.). The magnetoresistive ratio related to the tunnel magnetoresistance effect (TMR) lies around 110% at room temperature and the area resistance product is RA ≈ 2 Ωμm2. In detail, the SAF is composed of IrMn(60)/Co70Fe30(2.6)/Ru(0.85)/Co40Fe40B20(2.6) and the total layer stack is Ta(5)/CuN(50)/Ta(5)/Ru(5)/SAF/MgO(1)/Co40Fe40B20(2)/Ta(0.2)/Ni80Fe20(7)/Ta(10)/CuN(30)/Ru(5), with the nanometer layer thickness in brackets. The growth of the amorphous NiFe-free layer is decoupled from the lower CoFeB layer by a 0.2 nm Ta-layer. This structure permits to exploit the high tunnel magnetoresistance (TMR) ratio of the crystalline CoFeB-junction and the magnetically softer NiFe for the vortex dynamics. The layers are deposited on high resistivity SiO2 substrates by ultrahigh vacuum magnetron sputtering and subsequently annealed for 2 h at T = 330 ∘C at an applied magnetic field of 1 T along the SAF’s easy axis. The patterning of the circular tunnel junctions is conducted using e-beam lithography and Ar ion etching. They have an actual diameter of 2R = 370 nm and the microwave field line of 1 μm × 300 nm is lithographied 300 nm above the nanopillar.