Large Exchange Bias Triggered by Transition Zone of Spin Glass

Exchange bias has increasingly practical significance in magnetoresistive and spintronic devices. However, the underlying mechanism of exchange bias in bulk compounds with the structural single‐phase and inhomogeneous magnetic phases is still elusive. Herein, based on experimental and simulation results, two important parameters are studied, i.e., the antiferromagnetic (AFM) volume fraction and the ferromagnetic (FM)/AFM interface area, which essentially determine the (spontaneous) exchange bias of Mn‐rich Ni44Co6Mn44‐xSn6+x (x = 0 ∼ 6) magnetic shape memory alloys. The substitution Sn for Mn changes magnetic ground state following the sequence of superparamagnetic/AFM → dilute spin glass/AFM → transition zone → cluster spin glass/AFM → FM, accompanying the growth of FM cluster and the weakening of AFM interactions. The results reveal that the magnetic ground state for exchange bias is optimized at transition zone between dilute spin glass/strong AFM and cluster spin glass/weak AFM, in which the optimal AFM volume fraction and FM/AFM interface area are achieved by tuning magnetic fields. A giant exchange bias field of 702.7 mT and a spontaneous exchange bias field of 318.7 mT are demonstrated. The work contributes to in‐depth understanding of (spontaneous) exchange bias in magnetically inhomogeneous compounds.


Introduction
Interfacial functionalities of ferromagnet and antiferromagnet are critical constitutes in magnetoresistive and spintronic DOI: 10.1002/apxr.202200043 devices. [1,2] Exchange bias (EB), as one of the most important fundamental magnetic properties, has attracted long-standing attention. This kind of field-induced unidirectional anisotropy, a pinning effect of antiferromagnetic (AFM) on ferromagnetic (FM) at interface, is usually presented by a shift of the hysteresis loop along the field axis. It has been widely investigated in various systems, such as nanostructures, [3] multilayers, [4] thin films, [5] 2D van der Waals layered structures, [6] and bulk compounds with inhomogeneous magnetic phases. [7] Since its discovery in Co/CoO fine particles by Meiklejohn and Bean in 1956, [8] the physical mechanism of EB has long been a subject of discussion. Many interpretations have been made in systems with the well-defined FM/AFM interfaces, such as uncompensated interfacial spins, [9,10] a domain-state model, [11] and the roughness of FM/AFM interfaces. [12] On the other hand, the understandings on mechanism and characteristic of bulk compounds with inhomogeneous magnetic phase are still limited.
Recently, diverse EB behaviours and large bias fields have been reported in off-stoichiometric Mn-rich Ni-Mn-Z (Z = Sb, Sn, In, Ga) magnetic shape memory alloys (MSMAs), in which magnetic ground states exhibit their significant influence on EB. [13][14][15][16] It is well known that Mn-Mn distance of Mn rich Ni-Mn-Z MSMAs plays an important role for determining the type and intensity of magnetic interaction. [17] Generally, the FM and AFM interactions of magnetic ground state result from the Mn-Mn exchange interaction within the regular Mn sublattice and the Mn-Mn exchange interaction between the regular Mn sublattice and Z sublattice, respectively. Thus, the variation of Ni-Mn-Z composition can generate complicated magnetic ground states, such as FM/AFM, [16,[18][19][20][21][22][23][24] FM/spin glass (SG), [25,26] super spin glass/AFM, [27][28][29] cluster SG (CSG)/AFM, [30] canonical spin glass/AFM, [31,32] SG/AFM, [33] FM/super ferromagnetic, [34] dilute spin glass (DSG)/AFM, [13] and reentrant spin glass/AFM, [14,15] as summarized in Table S1 (Supporting Information). The complexity in magnetic ground state makes the comparison of EB behaviours difficult. One little enlightenment can be given by the comparison of bias field (H EB ) in Table S1 (Supporting Information), which shows that the FM phases exhibit small H EB while SG states interacting with AFM give much larger H EB . The unidirectional anisotropy of EB is usually induced by cooling from a high temperature to the desired temperature under a constant field (H FC ) and then modified by a maximum measuring field (H Max ) in the subsequent isothermal magnetization. [13] Interestingly, some SG states can isothermally produce unidirectional magnetic anisotropy by applying a large enough H Max after zero field cooling (H FC = 0 T), leading to spontaneous exchange bias (SEB). [27] It was reported that the upper limit for the bias field of SEB (H SEB ) in MSMAs is 0.3 T. [17] However, the effective way to improve the (spontaneous) exchange bias by tuning magnetic ground states is still far from established. There are very limited comprehensive researches on (spontaneous) exchange bias of various magnetic ground states in the same system of MSMAs. Thus, it is important to explore the relationship of the (spontaneous) exchange bias mechanism with their corresponding magnetic ground states.
In this study, Ni 44 Co 6 Mn 44-x Sn 6+x (x = 0, 1, 2, 3, 4, 5, and 6) MSMAs were designed to achieve a series of magnetic ground states. The substitution of Sn for Mn can decrease the Mn-Mn AFM exchange interaction between the regular Mn sublattice and Sn sublattice, which thus adjusts the magnetic ground state. The phase diagram of this alloy system was established, in which the state changes in the sequence of superparamagnetic (SPM)/AFM → DSG/AFM → transition zone (TZ) → CSG/AFM → FM, indicating the growth of FM cluster and the weakening of AFM interactions. It was found that H EB versus H FC (H SEB versus H Max ) curves display a peak shape with larger H EB (H SEB ) values at TZ range (2 ≤ x ≤ 4), being different from those of other compositions. The giant bias fields H EB = 702.7 mT and H SEB = 318.7 mT were obtained at x = 3 (H FC = H Max = 1 T) and x = 4 (H FC = 0 T, H Max = 5 T), respectively. Experimental and theoretical investigations reveal that (spontaneous) exchange bias is essentially determined by the AFM volume fraction and the FM/AFM interface area, which can be optimized through tuning the FM cluster size of magnetic ground state (composition x) and the value of magnetic fields (H FC , H Max ). This work helps to comprehensively understand the physical mechanism of (spontaneous) exchange bias, and design giant bias field in bulk compounds with inhomogeneous magnetic phases.

Results and Discussion
To investigate the evolution of phase transition and magnetic ground state as a function of content x, a compositiontemperature phase diagram of Ni 44 Co 6 Mn 44-x Sn 6+x (x = 0, 1, 2, 3, 4, 5, 6) MSMAs is established in Figure 1. The experimental results of x = 4.5 from our previous study were also added in this phase diagram. [35] The martensitic transition temperatures (M ts ) in the phase diagram were obtained by DSC measurements in Figure S1 (Supporting Information) and the Curie temperatures (T c ) were determined by M-T curves in Figure S2a (Supporting Information). With substituting Mn by Sn, M ts dramatically decreases and the martensitic transition disappears when the content is close to x = 6. The decrease of M ts indicates that the stability of martensite is greatly reduced by doping Sn, which is due to the fact that the dopant of defects can suppress the long-range strain ordering of martensitic transition. [36] T c of austenite appears at x = 4.5 and it slightly changes with the increase of Sn content.
It has been reported that the AFM interaction is enhanced by the martensitic transition in Mn-rich Ni-Mn-Z MSMAs. [37] And the sharp drop of magnetization at M ts in M-T curves ( Figure S2a, Supporting Information) proves that the AFM interaction becomes stronger in the martensite of the Ni-Co-Mn-Sn system. To explore low-temperature magnetic ground states of martensite, the M-T curves with zero field cooling (ZFC)/field cooling (FC) histories and AC susceptibility have been performed as shown in Figure S2a and S3 (Supporting Information). The ZFC and FC curves deviate from each other below M ts and the ZFC curves exhibit a peak at T P , indicating the existence of inhomogeneous magnetic state in the martensite of Ni 44 Co 6 Mn 44-x Sn 6+x (x = 0 ∼ 5) alloys. It has been reported that superparamagnetic (SPM) domains coexist with the surrounding AFM matrix in martensite upon cooling from M ts . [27,38] Previous investigations demonstrate that the magnetization of FC curve increases monotonically with decreasing temperature, as the SPM/AFM state persists upon further cooling to approaching 0 K, while it tends to saturate or decrease with decreasing temperature, as the SPM embedded in AFM matrix transforms into SG on further cooling. [30] Accordingly, the M-T curves ( Figure S2a, Supporting Information) reveals that the SPM behavior persists even approaching to 0 K for x = 0 and the SPM is collectively frozen into SG for other compositions x = 1 ∼ 5. The corresponding SG transition temperature (T g ) increases with increasing x. The parameters (Table S2, Supporting Information) obtained by fitting AC susceptibility curves verify that the low temperature SG can be divided into three ranges: DSG for x = 1, CSG for x = 5, and a wide transition zone (TZ) from DSG to CSG for x = 2 ∼ 4. By substituting the Sn for Mn, the AFM Mn-Mn interaction between the regular Mn sublattice and Z sublattice are weakened. Consequently, the FM strength enhances and the size of FM clusters increases. The magnetic frozen state of martensite changes from DSG with small cluster size into CSG with large cluster size through the intermediate regime of TZ. Note that SG has been widely investigated in particles, [39] ribbons, [40] assemblies, [41] and intercalation compounds, [42] in which such a wide range of TZ has not been observed. In summary, as displayed in the phase diagram of Figure 1 for Ni 44 Co 6 Mn 44-x Sn 6+x MSMAs, the SPM/AFM state of martensite freezes into three different low temperature  Figure 2b. H EB and H C initially increase before x = 3 and then decrease, resulting in a peak. Meanwhile, the composition at this peak exhibits an enhanced M E . These results reveal that a significant improvement on bias field and magnetization appears in the TZ, as shown in Figure 2b.
To further verify the EB behaviours, H EB as a function of H FC for all the samples is shown in Figure 2c. It is interesting to note that the behaviours of H EB versus H FC curves can be classified into two types. For the first type, the H EB monotonically increases with increasing H FC when 0 ≤ x < 2. For the second one, a H EB peak is formed at a critical cooling field (H Cri ) when x ≥ 2. Meanwhile, the H EB peak value and its corresponding H Cri dramatically reduces as x increases from 3 to 5. Obviously, these different EB behaviours stem from the different magnetic ground states. The compositions at TZ exhibits larger H EB than other compositions.
A maximum H EB = 702.7 mT is obtained, which, to the best of our knowledge, is the largest value reported in bulk Ni-Mn-Z MS-MAs.
In addition, M-H loops for SEB measurement were also recorded at 3 K with different H Max after cooling from 400 K under H FC = 0 T. The selected M-H loops (x = 0, 2, 4) and the H SEB versus H Max curves (x = 0, 3, 4, 5) are shown in Figure 2d,e, respectively. The asymmetric M-H curves demonstrates the appearance of SEB when x > 2 ( Figure 2d). The H SEB versus H Max curves exhibit the monotonic increase for x = 3 but show a peak shape for x = 4 and 5 (Figure 2e). Such a peak curve occurs in the terminal of TZ nearby the CSG side. This is distinguished with the H EB versus H FC curves in Figure 2c, of which the peak shape exists in the whole range of TZ and CSG. A maximum value of H SEB = 318.7 mT is obtained for x = 4 under H Max = 5 T, which surpasses the largest H SEB (296.4 mT) in MSMAs reported so far. [28] Apparently, Figure 2 reveals that there exists close relationship between EB (SEB) and the corresponding magnetic ground states. In particular, the TZ between DSG/strong AFM and CSG/weak AFM can achieve the large EB.
Under different temperature and magnetic field histories, the interaction between the FM clusters and the AFM matrix greatly varies, leading to the different magnetic domain structures and magnetic states. However, direct experimental observation of magnetic domain structures related to EB behaviours has long been challenging. To understand the observed EB behaviours (Figure 2), the domain structures with the changes of composition x and magnetic field were investigated by simulations based on micromagnetism theory and Monte Carlo algorithm. Figure 3 shows the low-temperature domain structures to simulate the magnetic states of Ni 44 Co 6 Mn 44-x Sn 6+x (x = 0, 1, 2, 3, 4, 5) To unravel the relationship between EB behaviours (Figure 2b) and domain structures (Figure 3), some parameters of domain structure should be further quantified. The changes on FM and AFM domain structures with the variation of composition x are difficult to be defined manually. Machine learning (ML) as a promising analyzing tool in materials science is a powerful method for rapid statistical analysis in condensed-matter research. [43] The main idea of machine learning is to feed ML models with large amount input data and corresponding labels, allowing ML models to learn the mapping from input data and labels. Traditionally, the segmentation tasks are supervised learning, which requires a vast amount of data with the corresponding label. However, due to the complex configuration of the mixed phase domain, for example, the spin configurations of the inhomogeneous magnetic phase, it is difficult to mark a big enough dataset manually. A learning strategy called transfer learning was used to tackle the problem of the shortage of labeled training data. [44] It largely relieves the data shortage and accelerates the data collection processes because it is much easier to acquire the spin configurations of FM domains and those of AFM with corresponding labels. Thus, a model to distinguish different magnetic phases based on a convolutional neural network can be well trained, which is convenient for counting parameters of domain structures.
It is well known that the EB of inhomogeneous magnetic ground states is caused by the unidirectional anisotropy generated by the FM/AFM interfaces and their interfacial interactions. Thus, the area of FM/AFM interfaces (S inter ) is an important parameter associated with EB. The deep learning method was used to count the S inter , and the S inter versus content x curve (denoted by squares) is plotted in Figure 4a. It exhibits a basically upward trend with a fluctuation of a small peak occurring at TZ. This S inter versus content x curve is different from the H EB versus content x relationship (Figure 2b), indicating that the S inter is not the only parameter to determine EB. It is reported that AFM components www.advancedsciencenews.com www.advphysicsres.com  also play important role on the magnitude of EB, [45][46][47] as they can provide robust pinning effect on interfacial spins. Therefore, the volume of AFM matrix (V AFM ) was chosen as another important parameter associated with EB. The composition dependence of V AFM (denoted by circles) is shown in Figure 4a. The V AFM descends from x = 0 to x = 5. This decreasing tendency is consistent with experimental results in Figure S2b (Supporting Information), where the magnetization of martensite at 6 K considerably increases with increase x. To simultaneously consider the effects of V AFM and S inter on EB, the frequently-used way, i.e., the product of V AFM and S inter was adopted. The V AFM ·S inter as a function of content x is shown in Figure 4b. The shape of it is almost coincide with the H EB versus content x curve in Figure 2b, which soundly demonstrate that H EB of Ni 44 Co 6 Mn 44-x Sn 6+x (x = 0, 1, 2, 3, 4, 5) MSMAs is essentially determined by the product of V AFM and S inter . Note that the V AFM and S inter with their corresponding standard deviations (error bars) are the average values of the simulations repeated by 6 times to reduce the randomness of the domain structure of each composition, and the average V AFM and S inter are used to calculate the product results in Figure 4b. Figure 4 demonstrates that the TZ with optimal H EB has maximum V AFM ·S inter , which is attributed to the intermediate V AFM and S inter . Either large V AFM & small S inter or small V AFM & large S inter lead to small V AFM ·S inter , which is unfavorable for achieving giant H EB . V AFM and S inter can be tuned by the magnetic ground state (composition x), which show opposite change tendency with composition as shown in Figure 4a.
As mentioned above, the H EB versus H FC curves in Figure 2c display two different kinds of characteristics for the composition range of x = 0 ∼ 2 and x = 2 ∼ 5. Such an interesting phenomenon accompanying magnetic ground state evolution are also related to the changes of V AFM and S inter with H FC . For x = 0 ∼ 2, the system stays in the ground states of (SPM, DSG)/strong AFM, and the H EB versus H FC curve of these two states show monotonically increasing relationship (Figure 2c). Strong AFM matrix leads to the strong AFM pinning effect on interfacial spins and the high barrier energy for the growth of FM clusters. Thus, S inter slowly increases and V AFM slightly reduces with the application of H FC . The monotonically increasing relationship of H EB versus H FC (Figure 2c) demonstrates that the H EB for (SPM, DSG)/AFM is dominated by S inter . This mechanism can also be used to explain the similar phenomenon reported in the compensated state with small FM cluster and strong AFM matrix of Mn 2.4 Pt 0.6 Ga alloy. [47] For x = 2 ∼ 5, the system stays in the ground states of ZT or CSG/AFM. The H EB versus H FC curve of these two states show a peak when H FC is close to the critical cooling field H Cri (Figure 2c). Because the strength of AFM interactions of them are medium or weak, pinning effect on interfacial spins and the barrier energy for the growth of FM clusters both decrease. When applying small H FC (<H Cri ), large FM clusters easily grow up and get closer with each other, which results in the increase of S inter and the decrease of V AFM . The enhancement of H EB with H FC suggests that the H EB is mainly determined by S inter for H FC < H Cri . When applying large H FC (> H Cri ), the long-range FM interactions can be established, which accompanies the merging of large FM clusters. [13] As a consequence, both S inter and V AFM decrease, resulting in the decrease of H EB . Obviously, the formation of peak in H EB versus H FC curve (Figure 2c) for TZ and CSG/AFM is alternately dominated by S inter and V AFM ·S inter .
In addition, Figure 2e shows that the SEB occurs in the TZ and CSG/AFM, where the large FM clusters are imbedded in the AFM matrix. The frozen state of large FM clusters in the TZ terminal can be broken by isothermally applying large enough H Max at low temperatures after ZFC. Simultaneously, the superferromagnetic exchange is established between them, leading to the SEB. [27] As shown in Figure 2d, the optimal composition for achieving the largest H SEB = 318.7 mT under H Max = 5 T is located at TZ with x = 4, which possesses the intermediate V AFM and S inter . This demonstrates that the SEB of Ni 44 Co 6 Mn 44-x Sn 6+x (x > 2) MSMAs is also determined by both V AFM and S inter .

Conclusion
In summary, EB and SEB behaviours of various magnetic ground states have been investigated in Ni 44 Co 6 Mn 44-x Sn 6+x (x = 0, 1, 2, 3, 4, 5, 6) MSMAs. Experiments and calculations demonstrate that (spontaneous) exchange bias is essentially determined by V AFM and S inter ., which can be tuned by the magnetic ground state (composition x) and the external magnetic fields (H FC , H Max ). The substitution Sn for Mn gradually changes the magnetic ground state following the sequence: SPM/AFM → DSG/AFM → TZ → CSG/AFM → FM, which is accompanied by the growth of FM cluster and the weakening of AFM interactions. Our work reveals that the composition located at TZ between DSG/AFM and CSG/AFM has the optimal FM cluster size and intermediate AFM strength, resulting in the peak value of V AFM ·S inter and giant bias fields. The largest values of H EB and H SEB are 702.7 and 318.7 mT, respectively. This study provides new insight to understand the mechanism and characteristic of (spontaneous) exchange bias in bulk compounds with inhomogeneous magnetic phase.

Experimental Section
Materials and Methods: Polycrystalline Ni 44 Co 6 Mn 44-x Sn 6+x (x = 0, 1, 2, 3, 4, 5, 6) MSMAs were prepared by arc melting stoichiometric amounts of the high-purity (99.99%) Ni, Co, Mn, and Sn in an argon atmosphere. Ingots were turned upside down and re-melted for four times. Approximately 3% extra manganese was added in advance to compensate the weight loss during fabrication. As-cast ingots were subsequently annealed at 1173 K for 24 h in evacuated quartz tubes and then quenched in room temperature water. The actual compositions were verified by a scanning electron microscopy (ESEM-JSM 7000F, JEOL) equipped with the energy-dispersive X-ray spectroscopy analyzer, as summarized in Table S3 (Supporting Information). Differential scanning calorimetry (DSC-Q2000, TA instruments) was used to determine the martensitic transition temperature. The magnetic properties, including magnetic hysteresis (M-H) loops, magnetization temperature (M-T) curves, and alternating current (AC) susceptibility, were measured using a superconducting quantum interference device (SQUID) magnetometer (Quantum Design, MPMS-XL).
Simulation Method: Based on micromagnetism theory and Monte Carlo algorithm, [48,49] the domain structures of Ni 44 Co 6 Mn 44-x Sn 6+x (x = 0, 1, 2, 3, 4, 5) MSMAs were simulated with the change of composition x and magnetic field. A 2D crystal structure with 30 × 30 = 900 cells was employed. The Mn-Mn exchange interaction within the regular Mn sublattice was set to be ferromagnetic while the Mn-Mn exchange interaction between the regular Mn sublattice and Z sublattice was set to be antiferromagnetic. For every site, a magnetic moment with a constant value was assigned and it could be rotated freely. Furthermore, the position and interaction of the atomic magnetic moments evolved freely with composition x and magnetic field. In this way, domain structures were obtained without losing the generality. The empirical parameters of micromagnetism, such as exchange coefficient and anisotropy coefficient, were chosen to make the domain structures more readable but do not change the physical essence of the process. After setting the physical model, the Monte Carlo method was applied to search for the state with the lowest total energy. To improve the repeatability of these results, the codes were listed in the supporting information.
The typical simulation process of Monte Carlo method is the follows: 1) an initial moment arrangement was set; 2) a moment was randomly chosen in the moment cluster; 3) a trial moment-flip was executed for the chosen moment; 4) the local Hamiltonian of the chosen moment was calculated for the state before and after moment-flip. The energies of the state before and after the moment-flip are E 1 and E 2 respectively, the energy difference between the two states is ΔE = E 2 -E 1 . Then a probability P was calculated through the Metropolis algorithm: P = exp(−∆E/kT), where k is the Boltzmann constant. A random number R that is distributed uniformly in the interval [0, 1] was generated and compared with P. If R < P, the moment-flip was approved. Otherwise, the original state remained unchanged; 5) Each moment state for the chosen moment was decided by making a comparison between their energies. One Monte Carlo step (1 MCS) means scanning up to the total cell number of times for the process from (2) to (5). Finally, Monte Carlo steps were repeated until reaching an equilibrium state.
Machine Learning: Single-phase spin configurations were first used to train a classification model and then deploy such a model to the segmentation task. A fully connected neural network was built, which took the angle of four neighbor spins as the input data and had two nodes in the output layer, corresponding to the FM phase and the AFM phase, respectively, as shown in Figure S4a (Supporting Information). The simple idea was to train the weights of the fully connected neural network to classify the magnetic phase of the area surrounded by the input spins. For example, if the four inputs shared nearly the same direction, it was classified into FM phase, shown as the red block in Figure S4b (Supporting Information). If the inputs showed AFM properties, the neural network was classified into the AFM area.
To be more specific, all the neighbor spins in the simulated FM phase spin configurations were selected, and labeled as FM phase. In the same process, the AFM neighbor spins and their labels were obtained. Then, these data were used to train the neural network. The input layer of the fully connected network had four nodes, which were denoted as X 1 , X 2 , X 3 , X 4 . The input layer was connected to a five-node hidden layer through the "Relu" activation function. Then, the hidden layer was connected to the second five-node hidden layer through the "Relu" activation function. The second hidden layer was connected to a two-node hidden layer through "sigmod" activation function. The hidden layer was connected to the output layer with two nodes through the "softmax" activation function. The two nodes of the output layer were denoted as y 1 , y 2 , and the values of the two output nodes represented the probability of FM and AFM phase. When the training data was FM phase, the corresponding output node label was y 1 = 1, y 2 = 0; when the training data was AFM phase, the corresponding output node label is y 1 = 0, y 2 = 1. The training was stopped when the training accuracy rate reached more than 99%. The segmentation result was shown in Figure S5 (Supporting Information). The color bar shows the value of y 1 at each area. If y 1 > 0.5, the area was FM, otherwise the area was AFM area.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.