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Short-term memory capacity analysis of Lu3Fe4Co0.5Si0.5O12-based

The date of: 2023-04-03
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Short-term memory capacity analysis of Lu3Fe4Co0.5Si0.5O12-based spin cluster glass towards reservoir computing



Reservoir computing is a brain heuristic computing paradigm that can complete training at a high speed. The learning performance of a reservoir computing system relies on its nonlinearity and short-term memory ability. As physical implementation, spintronic reservoir computing has attracted considerable attention because of its low power consumption and small size. However, few studies have focused on developing the short-term memory ability of the material itself in spintronics reservoir computing. Among various magnetic materials, spin glass is known to exhibit slow magnetic relaxation that has the potential to offer the short-term memory capability. In this research, we have quantitatively investigated the short-term memory capability of spin cluster glass based on the prevalent benchmark. The results reveal that the magnetization relaxation of Co, Si-substituted Lu3Fe5O12 with spin glass behavior can provide higher short-term memory capacity than ferrimagnetic material without substitution. Therefore, materials with spin glass behavior can be considered as potential candidates for constructing next-generation spintronic reservoir computing with better performance.


With the society transitioning into the high-speed informatization era, artificial intelligence technology has become crucial as an efficient method of data processing. However, the separation of memory and computation of traditional computers restricts the embedded interaction between artificial intelligence systems and modern devices and significantly increases the power consumption of computing1. In recent years, to overcome this bottleneck, researchers have developed neuromorphic devices with high integration and low power consumption by simulating the function of neurons and synapses2,3,4,5. Among the existing neuromorphic devices, a framework called reservoir computing has been favored by researchers6,7,8. The basic principle is based on mapping data to a high-dimensional space using large reservoir units and learning its characteristics9. For hardware devices, a large number of reservoir units in the reservoir computing software can be easily simulated by time multiplexing a single unit10. In the training process, only the output layer data must be trained using simple linear regression11. In essence, a reservoir computing system is mainly required to have two characteristics: short-term memory ability and nonlinearity12. Short-term memory refers to a small amount of external input information that remains a transient active state in the memory system for a short time 13. The short-term memory ability can cause the historical information entering the reservoir to gradually decay and disappear with time. As another key characteristic, nonlinearity can be easily realized in various physical devices. Therefore, reservoir computing is simple in terms of training, fast convergence, and easy physical implementation.

So far, physical reservoir computing has been developed based on various systems14,15,16, including electronic systems17,18, mechanical systems19,20, optical systems21,22, spintronics devices23,24. Among them, magnetic-tunnel-junction based spintronic devices are considered to be a promising choice for physical reservoir computing, owing to their non-volatility, read/write endurance, high-speed operation, low power consumption and low size25,26,27,28. Besides, spin wave29,30, skyrmion31, antiferromagnetic systems32 are also used for constructing spintronics reservoir computing systems with different physical principles. Nontheless, whether it is based on magnetic tunnel junction or other spintronics systems, the present spintronic reservoir computing realizes the required short-term memory ability through the relaxation process originates from certain physical phenomena27,28 or the delay module in the system29,30. The nonlinearity can be simply provided by the magnetic materials and the spin dynamics28,29,30. Interestingly, when the short-term memory and nonlinearity are simultaneously provided from multiple components, the abilities of the entire system intensify more than those of a single component28,29,33. Therefore, in an ideal case, it is expected that all components in spintronic reservoir computing can provide short-term memory ability and nonlinearity. However, in the existing studies, the magnetic characteristics of ferromagnets used in spintronic devices mainly contribute to the nonlinearity28,29. Among various magnetic materials, spin glasses are disordered magnets, where spin-freezing state occurs below the spin-freezing temperature (Tg), and exhibit characteristic memory and aging effects related to the slow magnetic relaxation. From the perspective of the mean-field theory, this characteristic property of spin glass is explained by the multi-valley potential, which is caused by the randomness of incoming atoms and the frustration between spin interactions34,35. Although a number of spin glasses have been reported to show memory effects, particularly pertaining to memorizing the thermal and/or magnetic field history36,37,38, there are no quantitative studies discussing their connection with short-term memory ability in physical reservoir computing.

To address this research gap, we quantitatively tested the short-term memory capacity of Lu3Fe4Co0.5Si0.5O12 (LFCS), which has demonstrated spin cluster glass behaviors of memory effect with high Tg39, through the short-term memory task commonly used by reservoir computing. It is well-known that there is an upper limit on the amount of new information for the memory system to remember in a short time 40. This limit is the short-term memory capacity, also known as memory span. For a reservoir computing system, the memory span can be quantitatively evaluated by two different methods. The first is benchmarking the system on short-term memory task with quantitatively defined short-term memory capacity as an index41. The second method is using information processing capacity to comprehensively evaluate the nonlinear and linear capacities of the binary input-driven reservoir computing system42,43. Because we mainly focus on the short-term memory capacity of spin glass, the first method is chose in this work. Numerical simulation and experimental results showed that the short-term memory capacity of LFCS magnetization is higher than that of typical ferrimagenetic material of Lu3Fe5O12 (LuIG). Specifically, when the input binary sequence is mapped from {0, 1} to applied magnetic field {0, 100 Oe} and the working temperature is 150 K immediately below Tg, the magnetization of spin glass has an short-term memory capacity around 2. In contrast, LuIG does not have an effective short-term memory capability. We also found that the short-term memory ability in the experimental results is attenuated compared with the ideal simulation that can be attributed to the relaxation introduced by noise and the measurement system. In addition, the relationship between the amplitude of external write magnetic field for LFCS and the short-term memory ability is explored. It was revealed that the short-term memory ability of LFCS is suitable for utilization under a field strength of approximately 100 Oe. Because the aging memory behaviors of LFCS are common in the spin glass family 37,44,45, the conclusions of this work can be similarly extended to typical spin glass materials. These results contribute a quantitative connection between the spin glass behavior and the short-term memory capacity of reservoir computing. Hence, this work is expected to offer a potentail material candidate for improving the spintronics reservoir computing.

Material and methods

Spin cluster glass of Lu3Fe4Co0.5Si0.5O12

The sample preparation and spin cluster glass behaviors of LFCS have already been reported35,36,46. The general chemical formula of rare-earth iron garnet is (R3) [Fe2] a[Fe3]dO12 (where R denotes the rare-earth elements). Octahedral a-sites and tetragonal d-sites are occupied by 16 and 24 Fe3+ ions, respectively, whereas the dodecahedral c-sites are occupied by 24 R3+ ions. Among various rare-earth iron garnets, magnetic moments in Fe3+ in octahedral and tetragonal sites are dominant in the magnetization of lutetium iron garnet (Lu3Fe5O12, LuIG) because the 4f. level is filled in Lu3+. The magnetic moments of Fe3+ in octahedral and tetragonal sites are antiferromagnetically aligned with each other; hence, the net moment corresponds to that for one Fe3+ per formula unit, resulting in ferrimagnetism. Among various magnetic materials, LuIG and YIG (Y3Fe5O12) show exceptionally low damping constants for spin wave propagation, which means long lifetime and free path. The small magnetic loss is necessary for spin wave application including reservoir computing. Although typical magnetic materials such as permalloy, CoFeB, and Heusler alloys show the damping constants of 10–3, LuIG and YIG show one digit or more smaller values as 10–5–10–447. Both LuIG and YIG are garnet-structured magnetic insulator and composed of the spectroscopic S-state of all the ions that carry the magnetic moment, which result in very small spin–orbit interaction and magnetic loss. Therefore, LuIG and YIG are commonly used candidates for spin wave devices. Because spin glass behaviors have been reported in Co and Si-doped LuIG in addition to the characteristics of spin wave propagation 39, we selected LuIG in this work. According to previous studies39, Co2+ and Si4+ prefer to substitute the octahedral and tetrahedral sites, respectively. The incorporation of anisotropic Co2+ and nonmagnetic Si4+ induces randomness and frustration of magnetic interactions. Thus, similar spin cluster glass behavior as that previously reported in spinel-type ferrimagnets, was observed48,49. LuIG and LFCS films were deposited on Y3Al5O12 (111) substrates through pulsed laser deposition using an ArF excimer laser. The source targets were prepared using a conventional solid-state reaction. Powders of Lu2O3 (99.9%), α-Fe2O3 (99.99%), CoO (99.9%), and SiO2 (99.9%) were mixed stoichiometrically and sintered at 1400 °C for 6 h. During the film growth, the substrate temperature and ambient oxygen pressure were maintained at 750 °C and 1 Pa, respectively. The as-grown films were post-annealed in air at 800 °C for 2 h to improve their crystallinity.

Figure 1a shows the temperature dependence of magnetization in zero-field-cooling (ZFC) and field-cooling (FC) procedures. The irreversibility given by the divergence between ZFC and FC magnetization is a characteristic of spin glass indicating spin freezing state at temperature lower than Tg. As illustrated at inset of Fig. 1a, randomly spin frozen state and ferrimagnetic state are realized at low temperature under ZFC and FC protocols, respectively. Prior to the measurements, the sample was heated up to 400 K for demagnetization to ensure uniformity of the initial state. Thereafter, it was cooled from 400 to 10 K in the absence of field, and the magnetization was recorded based on the ZFC procedure upon heating to 400 K in the weak fields (that is, 50 or 200 Oe). Subsequently, the magnetization was recorded upon cooling to 10 K based on the FC procedure. The magnetization in the ZFC shows a steep decrease at low temperatures that is a characteristic of spin glass. The ZFC curves exhibit cusps at blocking temperatures that are defined as the spin-freezing temperature Tg. The Tg shifts systematically from 220 to 150 K as the applied magnetic field increases from 50 to 200 Oe that is a typical characteristic of spin cluster glass. Generally, the time-dependent magnetic relaxation can be fitted based on the stretched exponential function39:



where M0 and M1 are constants which can reflect the glassy component of magnetization and initial remnant magnetization. β is the stretching exponent. τ represents the relaxation time constant that reflects the duration of the relaxation behavior. A is the amplitude of external magnetic field, which can influence the τ and β. Because the initialization of the curve fit affects the specific value of T, we compared the order of τ at different temperatures under the normalization condition. As shown in Fig. 1 (b), among the measured temperature conditions of LFCS, τ reaches the maximum value at 150 K with an external magnetic field of 100 Oe. With an increase in temperature over 190 K which is the Tg under applied magnetic field of 100 Oe39, τ gradually decreases. At 300 K, τ decreases by two orders of magnitude compared with τ at 150 K. This can be attributed to the ferrimagnetic behavior of LFCS at room temperature39, which reveals that magnetization is independent of time. If the temperature drops to 80 K and 10 K, τ is significantly reduced by nearly an order of magnitude, owing to the deeper frozen state of the system at lower temperatures. Figure 1 (c) displayed the experimental magnetic relaxation data with the fitting curve, which is more intuitive to reveal the effect of τ on spin glass relaxation. The index used to assess the goodness of fit is the root mean square error. The small RMSE given in Fig. 1c shows the accuracy of our fitting calculation.

In contrast to traditional reservoir computing implemented by software, the physical reservoir computing system with time-multiplexing approach depicted in Fig. 2b relies on a unique technique called masking to realize the function of the input matrix Win51,52. However, in the following short-term memory tasks, it is not necessary to preprocess the input by masking. Moreover, to reduce energy consumption, a physical reservoir computing system generally has only one physical unit. Therefore, it is necessary to sample the continuous response of the physical system and treat each sampling point as a virtual node33. It is noteworthy that f and W are determined by the inherent characteristics of the spin cluster glass as a physical reservoir computing system, as shown in Fig. 2c.

Figure 3 shows the experimental schematic of realizing the above training process in spin cluster glass material. Firstly, the input training sequence u→(t) and input weight matrix Win are generated by computer. Then, the masked input sequence Winu→(t) is converted into a series of applied magnetic field signal and the magnetic relaxation is recorded as x→(t) by the physical property measurement system with vibrating sample magnetometer (PPMS-VSM). After collecting all the x→(t) containing at all time step, we can obtain X. Finally, according to the initial determined target data matrix Y and obtained X, Wout can be calculated offline in the computer.

Results and discussion

Simulation evaluation for short-term memory capability of spin glass relaxation

In order to numerically test the short-term memory ability of the relaxation term of the spin glass, we first randomly generate binary sequence with 200 codes. Each code contains 100 points to simulate the discrete sampling physical signal, which indicates that the virtual node number of the spin glass system is 100. Thereafter, we determine the time of each positive magnetization relaxation and negative magnetization relaxation according to the binary code. According to Eq. (1), we can then obtain the relaxation sequence of spin glass for the corresponding binary input sequence. Notably, as the premise of simulation, we need to carry out the magnetization relaxation measurement experiment described in Section 'Spin cluster glass of Lu3Fe4Co0.5Si0.5O12' to determine the relaxation parameters under specific temperature scheme. Since M0 and M1 can be normalized during training, accurate τ and β are more important. Because the magnetization relaxation characteristic of spin glass is the strongest near Tg, we extracted τ and β at a constant temperature of Tg using Eq. (1). The obtained [τ,β] values for the FC and ZFC relaxations under an external applied magnetic field of A=100 Oe are [2741 s, 0.60] and [3848 s, 0.53], respectively39. In contrast, the short-term memory benchmark was also tested on tanh function. The tanh function is representative of the threshold activation function in reservoir computing56 as well as other artificial neural networks57,58,59, which is used to simulate the excitation and inhibition of human neurons. In this work, tanh can be used to numerically simulate the magnetodynamics of LuIG.

The time-domain responses of three systems to random binary sequences are shown in Fig. 4a. It can be seen that when the input signal is switched between 0 and 1, the output of the tanh function reaches the steady state instantaneously without any relaxation transition. Different from the tanh without relaxation behavior, spin glass in the form of ZFC and FC form shows a relatively long-time relaxation duration. Generally, the shorter the relaxation time, the weaker is the short-term memory capability29. Moreover, in Fig. 4b, the relationship between the input and output of the spin glass forms a loop, whereas that of the tanh function forms a straight line. The loop relationship between input and output is the embodiment of the system memory effect60,61,62. Therefore, from Fig. 4a and b, we can qualitatively understand the short-term memory capability comparison of the three systems. Specifically, the tanh function has no short-term memory capability. It should be noted that when the relaxation time τ is shortened, the relaxation of spin glass will weaken and approach the output by tanh function. Thereby, one can expect that the short-term memory capacity of spin glass will decrease when the temperature is significantly higher / lower than Tg.

For further quantitative analysis, Fig. 4c and d show the short-term memory benchmark test results of the three systems. Consistent with the qualitative analysis, the short-term memory capability of spin glass in both the ZFC and FC relaxation states is better than that of the tanh function, which is 0. In Fig. 4d, rSTM(d)2 of the spin glass under both ZFC and FC condition first decreases with the delay distance, and thereafter fluctuates near 0. Although there is no significant difference in the attenuation rate of rSTM(d)2, the rSTM(d)2 of ZFC state is higher than that of FC state at d=1, which contributes to the short-term memory difference between the two conditions. In essence, the different short-term memory capacities under ZFC and FC condition can be attributed to different magnetization relaxation processes caused by different external magnetic field conditions. Specifically, under the condition of FC, the magnetization of the spin glass is subject to additional constraints compared with that of ZFC, which also weakens the response to the binary input, resulting in a lower short-term memory capacity. From the simultaneous relaxation performance, the relaxation amplitude of spin glass in FC is less than that in ZFC63.

Experimental testing for short-term memory capability of spin glass relaxation

When A=100 Oe, the short-term memory benchmark was experimentally tested on LFCS and LuIG. Figure 5a and b show the short-term memory capacities of the two materials at different temperatures and cooling conditions. First, it can be seen that for the LFCS of ZFC or FC relaxation, its short-term memory capacity is the best at 150 K. This can be explained by the effect of the change in thermal energy on the height of the potential barrier resulting from the multivalley potential34. At a low temperature of 10 K, the potential barrier is high, and the spin state is completely frozen, with almost no change in the magnetic field. In contrast, at temperatures around 300 K, the potential barrier reduces, the metastable state is lost, and the spin-glass-like characteristic is lost. Therefore, the short-term memory capacity is expected to be small at 10 K and 300 K. However, at around 150 K, the temperature is slightly less than the spin-freezing temperature in the magnetic field64, therefore the short-term memory capacity is expected to be larger because the magnetic field causes a slow magnetic relaxation that slowly exceeds the potential barrier of the magnetic state. Similar to the simulation results in Fig. 5 (a), the short-term memory capacity of FC relaxation at 150 K is lower than that of ZFC. Moreover, from Fig. 5 (c), one can expect the difference in short-term memory capacity under the two conditions is mainly contributed by rSTM(d=1)2. Therefore, the short-term memory capability of LFCS under ZFC relaxation is better than that of FC relaxation that is consistent with the simulation results. Notably, a delay distance of 1 means a time scale of 1 s, which is the minimum measurement interval of PPMS-VSM. For the traditional spintronics RC, its short-term memory can only maintain microsecond level29. In contrast, the LFCS can maintain the short-term memory ability at the second level. It implies that even for the high-speed signal in a shorter time scale, the spin cluster glass can still be expected to provide considerable short-term memory capacity.

Furthermore, for LuIG, the short-term memory capability should be close to the tanh function in theory because of the extremely short relaxation time of ferrimagnetic materials65. In Fig. 5b, one can show that the short-term memory capacity fluctuates sharply under the tested conditions. In Fig. 5d, for almost every d, the error of rSTM(d)2 is large compared to its average value. Hence, it can be considered that the short-term memory capacity of LuIG is extremely low and the measurement result indicates the error fluctuation caused by the environment and equipment disturbance.

Although the aforementioned results preliminarily prove that LFCS has a short-term memory capacity around 2 at 150 K, there is a noticeable gap between the experimental results and simulation. To further analyze the source of this gap, the ideal input and the input magnetic field applied by PPMS-VSM under the same binary test sequence are depicted in Fig. 6a and b. In the ideal case, the binary sequence should be instantaneously switched between 0 and 1, without transition values. In the actual case, the externally applied physical signal cannot be changed instantaneously, which will produce a transition state, such as the scattered points in Fig. 6b. The positions of these scattered points are irregular, and the intervals are not uniform; thus, random relaxation independent of input information is introduced. In addition, it can be observed that in the magnetization curves measured by PPMS-VSM (Fig. 6c), regardless of FC or ZFC relaxation, the magnetization of spin glass is markedly disturbed by noise during the relaxation process. This implies that noise also introduces random relaxation independent of the input information. The directions of input relaxation and noise relaxation are shown in Fig. 6d. In the ideal condition of the simulation, the input–output curve with short-term memory capability should form a rectangle (Fig. 4b). However, the random relaxation caused by the input and noise obviously distorted the input–output relationship curve. Comparing the durations of the two types of random relaxation, when A=100 Oe, one can expect that the duration of the input relaxation is approximately 3–4 s, which is significantly shorter than the duration (100 s) of a binary code. However, noise relaxation is always present. In addition, in a previous study28, the short-term memory capacity of physical reservoir computing was increased by approximately 1 after the system noise was suppressed. Therefore, it can be perceived that the main reason behind the difference in the short-term memory capability of spin glass between the experimental and simulation results, is the random relaxation introduced by noise. For the input relaxation, it is difficult to completely eliminate its negative influence due to the inevitable transition process of the physical system. As a feasible method, we can reduce the effect of input relaxation by increasing the duration of each input code to decrease the proportion of the transition process in the whole signal input process. For reducing the impact of noise, filters and time-domain averaging techniques are effective methods which has been widely used in other reservoir computing systems


Based on the short-term memory benchmark commonly used in physical reservoir computing, we quantitatively evaluated the short-term memory ability of magnetization in LFCS, both numerically and experimentally. Under ideal conditions, the short-term memory capability of LFCS is approximately 2 and comparable to some physical reservoir computing systems with a single node28,33. Although in the experiment, we found that the additional irregular relaxation caused by noise and the measurement system degenerate the short-term memory capacity of LFCS, its short-term memory capacity is still larger than that of ferrimagnetic LuIG without spin glass behavior.

It is worth noting that the above conclusions are not intended to encourage the development of a physical reservoir computing system that depends only on the relaxation behavior of spin cluster glass materials. Rather, the conclusion of this study should be understood as providing an improved basic material candidate for spintronic reservoir computing based on magnetodynamic relaxation. When combining spin glass with existing spintronic reservoir computing systems, some additional physical considerations need to be taken into account according to the specific system. For example, if combing the spin glass with spin wave equipment, it should be careful to consider the change of damping constant after replacing materials, which affects the shape designing of the spin cluster glass. Because the nonlinearity and short-term memory capability of the spintronic reservoir computing result from the interaction of various components in the system28,29, choosing a material that can provide short-term memory capability will undoubtedly be more conducive to the enhancement of spintronic reservoir computing performance. However, an optimal short-term memory capacity of LFCS is obtained near 150 K. Therefore, it is a challenging and an important task in future research to realize materials exhibiting spin glass behavior at higher temperatures.

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