Temperature-dependent collective magnetization reversal in a network of ferromagnetic nanowires Open Access

Many modern technologies and sciences rely on steady advances in nanofabrication techniques. Today, it is possible to design and create a wide range of different nanostructures. Nanofabrication has also propelled the development of metamaterials, which are artificial materials with properties not usually found in nature. Towards this end, artificial spin ice (ASI) networks are nanofabricated magnetic materials, in which the magnetic material is modulated in space. The main driver for the exploration of ASI is the discovery of lattice frustration,¹ emergence of magnetic monopoles,² as well as exotic phases,³ complex magnetic ordering,⁴ and collective behavior.⁵ Owing to those properties, ASI are of interest to both fundamental research and applied sciences on equal footing.⁶ For instance, ASI networks can provide a pathway for new concepts in storage media⁷ and neuromorphic computing.⁸ 

In this context, the high-frequency dynamics and the magnetoresistance (MR) behavior in those structures are of particular interest.^9–17 The GHz-frequency resonance and spin-wave properties can effectively be modulated in ASI offering new functionalities for microwave applications such as filters or magnonic position transducers; exploiting the complex magnetoresistance behavior in ASI, on the other hand, could potentially be used in information technologies such as computing and storage. The first MR studies in ASI indicated that anisotropic magnetoresistance (AMR) in combination with domain wall reversal in the nanowires, are responsible for the observed field dependence:^18,19 the resistance of each link depends on the angle between the magnetization and the local electric current density. More recent works shed light on the question of how the local magnetization configuration in the vertex regions affects angular- and field-dependent magnetoresistance measurements.^11,15 Le et al. proposed a combination of micromagnetic simulations and the phenomenology of AMR to explain the experimental findings.¹⁵ Those results also suggest that the prominent, sharp features observed in the MR data occur due to sudden changes in the magnetization configuration when the field is swept and that the angular-dependent alignment of the vertex region that determines the occurence of magnetic monopoles in the ASI structure strongly affects the resistance. These processes can be thought of as “avalanches,” where magnetic domain walls are nucleated and quickly move through an interconnection as they trigger the magnetization in a neighboring nanowire to flip. Dynamic studies utilizing spin-torque ferromagnetic resonance measurements in the GHz regime support these conclusions.¹¹ At finite temperatures, thermal fluctuations result in the nucleation and propagation of magnetic domain walls or, in other words, create pairs of separated monopoles creating a so-called “Dirac string” of reversed spins. Recent proposals envision that magnetic monopoles, which are a point of frustrated interactions between the nanowire legs in ASI, could offer interesting perspectives for bio-inspired computing.⁸ A crucial aspect here is the targeted control of the vertex region,^15,20 so that the switching process can be manipulated.

Here, we study the temperature and current dependence of the switching fields in a connected square spin-ice lattice made of a Ni₈₀Fe₂₀/Pt bilayer measured by magnetoresistance. We find that the magnitude of the switching field increases when the temperature is lowered and as the temperature is increased the magnitude of the collective magnetization reversal field decreases. The experimental observations are confirmed by micromagnetic simulations that give further insights in the magnetization alignment and reversal process. Furthermore, we identify the impact of local heating effects due to the probing electric current used for the magnetotransport measurements. Our results not only have direct consequences for the development of magnetoresistive devices based on complex magnetic nanostructures with potential applications in bio-inspired computing, but also indicate that driving auto-oscillations^21–24 in this kind of structures is accompanied with heat generation that may alter the switching behavior.²⁵ 

The samples were fabricated in the following way: the connected square spin-ice lattice is patterned using electron beam lithography (Raith eLine). The total area of the connected network is approximately 75 × 10 μm². Subsequently, 15 nm Ni₈₀Fe₂₀ (permalloy, Py) and 5 nm Pt were deposited using magnetron sputtering without breaking vacuum between depositions, followed by lift-off. Please note that the Pt layer acts as a capping layer to avoid oxidation of the Py layer. While there may be some contributions from spin-Hall spin torque, we estimate that this effect is negligible and does not alter the switching process in the used range of electric currents, using the expression of the switching current density produced by spin-Hall effect^26–29 j = α2eM_St(H_k + M_S/2)/(θ_SHℏ), with α ≈ 0.01 the damping constant, 4πM_S = 1000 G the saturation magnetization of Py, t = 15 nm the Py thickness, H_k ≈ 0 Oe the magnetocrystalline anisotropy, and θ_SH ≈ 0.068 the spin-Hall angle.³⁰ We also note that general temperature-dependent trends of the switching field are independent of the applied field angle (not shown) ruling out spin-Hall driven magnetization reversal. A shortened coplanar waveguide (CPW) is patterned using optical lithography and electron beam evaporation of Ti (3 nm)/Au (120 nm). Figure 1(a) shows a sketch of the experimental configuration; the lattice constant of the spin ice is a = 872 nm, the hole width b = 620 nm. A scanning electron microscopy image of the sample is shown in Fig. 1(b).

The temperature-dependent resistance of the ASI is measured in a Physical Property Measurement System (PPMS) cryostat for different current values. For this purpose, the sample holder is rotated so that the relative angle θ between the applied magnetic field and the electric current can be adjusted. Figure 1(a) shows the geometry of the sample with the direction of the electric current and applied magnetic field. Note that the direction of the (global) current I is defined here as parallel to the DC leads. As it has been shown in Refs. 15 and 31, the current distribution on a microscopic level can be modeled using an effective circuit model, in which each vertex corresponds to a four-terminal node. The magnetic field is swept down from +1500 Oe to −1500 Oe and back up to +1500 Oe. During this field sweep, the MR signal is recorded using a Keithley current source (6221) and nanovoltmeter (2182A).

Figure 2 shows the AMR curves of the ASI with a current of 6 mA, recorded at temperatures of 8, 20 and 50 K. The magnetic field is set at an angle θ = 35°. During the sweep with increasing fields, we observe that the resistance increases to a maximum at zero field and, as the magnetic field changes its polarity, a sudden drop in resistance is observed. At a field of ∼ +220 Oe, the resistance increases abruptly and then follows again the main resistance trace. As the magnetic field is further increased, the resistance decreases, following a symmetrical slope as in the opposite field value. The sweep with decreasing fields shows a symmetric behavior under field reversal. This general observation is in agreement with earlier reports,^11,15 where the sharp features in the magnetoresistance were explained by a collective magnetization reversal. In particular, it was shown that the sharp features in the magnetoresistance are strongly affected by the vertex region.¹⁵ 

As is obvious from Fig. 2, the switching field (defined as the field at which a sharp step-like feature in the magnetoresistance ΔR is observed) changes as a function of the base temperature. Figure 2(b) shows the low-field regime, where the collective magnetization reversal is observed. To illustrate this more clearly, we plot the switching field as a function of the probing current for the three different base temperatures at a fixed angle of θ = 35° in Fig. 3. The angular-dependent magnetoresistance behavior is reported elsewhere.¹¹ We first focus on the switching behavior at low electric currents below 1 mA. As is obvious from the experimental observations, the magnitude of the switching field is the highest for the lowest temperature (8 K, blue curve) and is the lowest for highest temperature tested (50 K, green curve). The change in the switching field for low probing currents compared to high currents is approximately 40 Oe. This suggests that thermal fluctuations of the magnetization at higher temperatures assist the collective magnetization reversal that is detected by the magnetoresistance behavior, and that magnetotransport is an effective tool to probe the temperature-dependent reversal effect in the network.

In the following, we focus on how the switching behavior is altered as a function of the applied current. The results are shown in Fig. 3. For the highest temperature of 50 K, the switching field magnitude is independent of the probing current in the investigated range of currents (green curve in Fig. 3). However, the behavior changes for the lower temperature values of 20 K and 8 K, orange and blue curves in Fig. 3, respectively. The switching field at base temperatures of 20 K and 8 K decreases as the electric current is increased, with an asymptotic value corresponding to the 50 K switching field. Interestingly, in the intermediate current range between 0.5 and 5 mA, the 20 K curves lies between the 8 K and 50 K data. This behavior indicates that the connected network of nanowires is locally heated, resulting in an increase of thermal energy in the system that supports the collective magnetization reversal. Our results show that only at a sufficiently low temperature (in the investigated network <50 K), this local resistive heating effect changes the switching behavior. This means that, at a temperature between 20 K and 50 K, the effect of local heating is small compared to thermal heating.

To further confirm and interpret the experimental observations we conduct micromagnetic simulations using mumax3.³² The following standard simulation parameters for Py are used: Gilbert damping parameter α = 0.01, exchange constant A = 1.3 · 10⁶ erg/cm. To qualitatively model the change in temperature, the saturation magnetization 4πM_S is varied to 800, 1000 and 1200 G. The large variations in M_S allow to better qualitatively observe any effects on the switching fields. The simulation space is divided in 872 × 872 × 1 cells with dimensions 3 × 3 × 15 nm³, resulting in a total of three unit cells. Two-dimensional periodic boundary conditions are set to simulate an infinitely long lattice. The magnetic field direction is set to 35° in agreement with experiments. The magnetic field is then swept from −1500 Oe to +1500 Oe in 301 steps. We time-evolve the magnetization for 1 ns at each field step to determine the most stable magnetic configuration.

In order to obtain the magnetoresistance from the simulated magnetic distribution, we assume that the global current I flows from left to right; i.e., the local current j flows parallel to the two incoming legs, it converges at the vertex while it rotates 45°, and then it splits into the two outgoing legs; see Fig. 4(b). At each cell, the square of the dot product of the local current direction j with the local magnetization m is proportional to the cell resistance r_i. The total resistance R can be obtained by the sum of the resistances r_i of each cell,^15,25,31 R = ∑r_i ∝ ∑(j · m)².

Figure 4(a) shows the results of the simulated magnetoresistance traces of the lattice with three different values of M_S. The switching field in the lattice with the highest saturation magnetization is the highest, and the switching field decreases as the saturation magnetization decreases. This can be seen in the evolution of the magnetization curves in Fig. 4(a) from the blue curve (corresponding to 4πM_S = 1200 G), to the orange curve (4πM_S = 1000 G), and finally to the green curve (4πM_S = 800 G). The value of the saturation magnetization is expected to decrease with temperature according to Bloch’s Law, and hence our micromagnetic simulations show that the (temperature-dependent) decrease in M_S is at least partially responsible of the decrease in the switching field, in agreement with our experimental observations. A more quantitative evaluation of the contribution of the reduction of magnetization would require an evaluation of the energy in each situation taking into account thermal activation in our micromagnetic model, which is beyond the scope of this work.

Having the spatial distribution of the magnetization as well as the current, we can determine which lattice regions contribute the most to the magnetoresistance and produce the characteristic features in the magnetoresistance traces. In Fig. 4(b) we show the spatial distribution of the product (j · m)² at the switching field, which is characterized by a minimum in resistance as shown in the experiments (Fig. 2), as well as in the simulations [Fig. 4(a)]. We can see that in the vertex region and its vicinity, the dot product (j · m)² ∝ R is minimum. On the other hand an increased value of (j · m)² ∝ R is found in the legs. This indicates that it is the vertex region contributing the most to the characteristic minimum in the magnetoresistance curve confirming previous reports on AMR in connected ASI lattices.^15,31

While it would be interesting to increase the current density so that a spin-torque assisted magnetization reversal is triggered, we did not observe this effect here. Another outstanding question is how local pinning sites change the reversal process and if the electric current in the magnetotransport measurement affects this magnetization pinning. To answer this question the temperature- and current dependent magnetoresistance measurements would have to be correlated with temperature-dependent magnetometry.

In summary, we presented the temperature dependence of the magnetization reversal behavior in a connected square artificial spin-ice system and demonstrated that a magnetotransport approach is an effective tool to probe this behavior. We find a decreased switching field at an elevated base temperature in the cryostat, and an increased switching field when the base temperature is lowered. This can be associated with a global temperature that alters the switching behavior in the network. Using current-dependent measurements, we could identify the effect of resistive heating in the network. In particular, we showed that the local temperature increase due to resistive heating only affects the collective magnetization reversal below a base temperature of 50 K.

Our experimental observations are consistent with micromagnetic modeling, which suggests that the additional thermal energy at elevated temperatures results in a decrease of the saturation magnetization ultimately assisting the magnetic-field-driven reversal process. However, within the tested range of applied currents, we could not identify any spin-torque assisted switching behavior or signatures for the onset of self-oscillations in the connected network of ferromagnetic nanowires, which is presumably due to too low current densities.

We thank Dr. Joseph Sklenar for valuable discussions. This work was supported by the U.S. Department of Energy, Office of Science, Materials Science and Engineering Division. Lithography was carried out at the Center for Nanoscale Materials, an Office of Science user facility, which is supported by DOE, Office of Science, Basic Energy Science under Contract No. DE-AC02-06CH11357. M.B.J. acknowledges support by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0020308.

The data that support the findings of this study are available from the corresponding author upon reasonable request.