# Concomitant interfacial spin fractal transformation and exchange bias in a magnetic shape memory alloy

L. Shen, Y. Gong, A. Alshemi, E. Campillo, R. Cubitt, N.-J. Steinke, E.

Blackburn

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L. Shen, Y. Gong, A. Alshemi, E. Campillo, R. Cubitt, et al. Concomitant interfacial spin fractal transformation and exchange bias in a magnetic shape memory alloy. Applied Physics Letters, 2023, 123 (13), pp.132402. 10.1063/5.0166232. hal-05330187

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# Concomitant interfacial spin fractal transformation and exchange bias in a magnetic shape memory alloy

L. Shen ; Y. Gong ; A. Alshemi ; E. Campillo ; R. Cubitt ; N.-J. Steinke ; E. Blackburn

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Appl. Phys. Lett. 123, 132402 (2023)

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# Concomitant interfacial spin fractal transformation and exchange bias in a magnetic shape memory alloy

Cite as: Appl. Phys. Lett. 123, 132402 (2023); doi: 10.1063/5.0166232

Submitted: 4 July 2023 · Accepted: 6 September 2023 ·

Published Online: 25 September 2023

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L. Shen,<fnref n="1" />,ª) Y. Gong,<fnref n="2" />) A. Alshemi,<fnref n="1" />) E. Campillo,<fnref n="1" />) R. Cubitt,<fnref n="3" />) N.-J. Steinke,<fnref n="3" />) and E. Blackburn<fnref n="1" />,ª) ID

## AFFILIATIONS

## ABSTRACT

Small-angle neutron scattering is used to study the exchange bias effect in Mn₅₀Ni₄₀Sn₁₀. The martensitic transformation is found to be responsible for the nanometer-scale spin clusters (SCs) inside the low-temperature ferromagnetic (FM) martensite phase. The magnetic field dependences of the SCs and FM domains exhibit an asymmetry that characterizes the exchange bias. We find that the surface geometries of the two magnetic phases are strongly susceptible to magnetization changes. While the FM domains and SCs are dominated by diffuse and fractal surfaces around the coercive fields, this configuration is reversed in the saturation regime, where the FM and SC surfaces become fractal and diffuse, respectively. These geometry changes are driven by the spin reorientation inside the heterogeneous FM/SC domain walls, where exchange anisotropy arises. Our work reveals the microscopic mechanism underlying the exchange bias effect in magnetic shape memory alloys.

© 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0166232

Exchange bias describes the asymmetry of the left and right coercive fields: |μ₀H₁| ≠ |μ₀H₂|, which leads to a shift of the magnetization loop along the magnetic field axis. This effect is the cornerstone of a variety of applications, e.g., spintronics.<fnref n="1" /> As a result, understanding its microscopic mechanism is of particular importance in applied physics. In theory, it is widely accepted that exchange bias relies on the exchange anisotropy at heterogeneous interfaces, e.g., those between ferromagnetic (FM) and antiferromagnetic (AFM) phases.<fnref n="2" /> Typically, interfaces with exchange anisotropy are introduced by phase separation.<fnref n="3" />⁻¹⁰ However, the relationship between the microscopic interfacial spin geometry and macroscopic exchange bias effect has not been investigated.

The mathematical establishment of self-similarity in random structures is an important milestone in the 20th century.¹¹ It resulted in the concept of fractal geometry, which is now being broadly employed to understand the scale invariance in complex systems.¹² In general, there exist two types of fractals in a d-dimensional space—surface and volume fractals. One parameter for characterizing, and more

importantly, distinguishing their geometries is fractal dimension. Unlike d, which must be an integer number, both surface (D_s) and volume (D_v) fractal dimensions are non-integers. Moreover, they must fulfill the following relations:

$$d - 1 < D_s < d \quad \text{and} \quad 1 < D_v < d. \qquad (1)$$

D_s measures the roughness of a surface. Its lower limit (d - 1) corresponds to a perfectly smooth surface, where the geometry is trivial. On the other hand, if a surface is so wrinkled that it fills the entire d-dimensional space, D_s takes the upper limit (d). Similarly, the volume fractal dimension D_v is a descriptor of volume roughness. Typically, objects with D_s ≠ (d - 1) or D_v ≠ d are geometrically nontrivial.

Spin fractals have a critical role in stabilizing an array of exotic phenomena, which promise applications in information technology. These phenomena include exchange bias,¹³,¹⁴ colossal magnetoresistance,¹⁵,¹⁶ and new topological states of matter.¹⁷,¹⁸ However, magnetic objects undergo complex changes in an external magnetic field,¹⁹ and it remains far from clear if these changes also involve their fractal

<footnote n="1">¹Division of Synchrotron Radiation Research, Department of Physics, Lund University, SE-22100 Lund, Sweden</footnote>

<footnote n="2">²MIIT Key Laboratory of Advanced Metallic and Intermetallic Materials Technology, Nanjing University of Science and Technology, Nanjing 210094, China</footnote>

<footnote n="3">³Institut Laue-Langevin, CS 20156, 38042 Grenoble Cedex 9, France</footnote>

<footnote>ª) Authors to whom correspondence should be addressed: lingjia.shen@sljus.lu.se and elizabeth.blackburn@sljus.lu.se</footnote>
geometry. This is an important question, as many of the aforementioned phenomena, including the exchange bias effect studied here, are pro- moted by magnetic field. In order to fully understand their driving mechanism, the spin fractal geometry in the field-induced state, together with its coupling to these phenomena, must be thoroughly evaluated.

Small-angle neutron or x-ray scattering is well suited for the study of fractal objects.⁹,¹⁶,²⁰⁻²⁴ A two-component unified Guinier/ power-law approach has been widely used²² to approximate the rela- tionship between the momentum transfer (q) and scattering cross sec- tion (dΣ/dΩ) arising from polydispersive fractals, i.e., systems where the fractal size distribution is spatially nonuniform,

$$ \frac{d\Sigma}{d\Omega}(q) = \sum_i \left\langle G_i \exp \left[ -\frac{(qR_i)^2}{3} \right] + P_i \exp \left[ -\frac{(qR_0)^2}{3} \right] \left\{ \left[ \frac{\text{erf}(qR_i/\sqrt{6})}{q} \right]^3 \right\}^{n_i} \right\rangle, \quad (2) $$

where erf() is the error function. For a given type of fractals labeled by the subscript i, its Guinier and power-law coefficients Gᵢ and Pᵢ can be expressed as

$$ G_i \propto N_i \rho_i^2 V_i^2 \quad \text{and} \quad P_i \propto N_i \rho_i^2 S_i, \quad (3) $$

where Nᵢ, ρᵢ, Sᵢ, and Vᵢ are the total number of fractal objects probed by neutrons or x rays, scattering contrast at the surface, average surface area, and volume of a single fractal object, respectively. The radius of gyration Rᵢ characterizes the size of the type-i fractals, while the power-law exponent nᵢ measures the fractal geometry. In three dimensions (d = 3), the power-law exponent is between 1.0 and 3.0 for volume fractals and between 3.0 and 4.0 for surface fractals. These limits are imposed by Eq. (1).²⁰ Accordingly, a power-law term with an exponent of 3.0 corresponds to surface fractals with maximum roughness, while that with an exponent of 4.0 comes from perfectly smooth surfaces (Dₛ = 2.0), where the Porod formula, dΣ/dΩ(q) ∝ q⁻⁴, is recovered. These descriptions for fractals are valid if the scattering contrast ρᵢ is discontinuous across the surface. However, there exist cases where ρᵢ changes continuously over a finite length scale, i.e., diffuse surfaces. This will lead to a power-law exponent greater than 4.0.²¹,²² There is also a cutoff parameter R₀ in Eq. (2); it defines the length scale below which the scale invariance for fractals no longer holds.

In this Letter, we study the exchange bias seen in the phase-segregated magnetic shape memory alloy Mn₅₀Ni₄₀Sn₁₀, using small-angle neutron scattering (SANS).²⁵ Our major discovery is a magnetization-induced, reversible spin fractal geometry transformation inside the heterogeneous domain walls separating the ferromagnetic (FM) domains and local spin clusters. Importantly, this transformation shares the same asymmetry that describes the large exchange bias effect in this system. Based on these observations, we propose this as the microscopic mechanism underlying the exchange bias in materials of this class.

A polycrystalline Mn₅₀Ni₄₀Sn₁₀ ingot was prepared by arc-melting.⁷ The sample quality was checked by x-ray powder diffraction using a STOE STADI MP diffractometer (λ = 1.5406 Å, see the supplementary material). A small piece (~0.5 × 0.5 × 0.5 mm³) was cut for the magnetization measurements in a Physical Property Measurement System (Quantum Design). A much larger piece (~7 × 5 × 1 mm³) was cut from the same batch and then polished for

the SANS investigation on the D33 instrument at the Institut Laue-Langevin (ILL).²⁶ A cryomagnet with field parallel to the incident neutron beam was used.²⁷ The SANS data reduction procedure in this scattering geometry is described in the supplemental material.

Mn₅₀Ni₄₀Sn₁₀ is a Mn-excess variant of the stoichiometric Heusler alloy Ni₅₀Ni₂₅Sn₂₅.⁷ It belongs to a wider category of Ni-Mn-X-based (X = Sn, In, and Sb) shape memory alloys where the magnetic properties are closely linked to the underlying atomic configuration.²⁸,²⁹ On cooling, their atomic lattice undergoes a symmetry reduction, known as martensitic transformation; the low- and high-temperature phases are called martensite and austenite, respectively. Deep inside the martensite phase, exchange bias is commonly observed.⁶⁻⁹

The zero-field-cooled (ZFC) and field-cooled (FC) magnetization vs temperature curves are shown in Fig. 1(a). They develop several distinct kinks when the temperature is varied and are in broad agreement with previous studies.⁷,⁸ At high temperatures, Mn₅₀Ni₄₀Sn₁₀ is a paramagnetic austenite. A magnetization plateau starts to form at T<sub>C</sub> = 277(2) K, marking the onset of FM order. The magnetization drops sharply at the martensitic transformation temperature T<sub>MA</sub>, which is 201(2) K on heating and 191(2) K on cooling. The magnetic properties in the martensitic phase are complicated. On the one hand, the magnetization starts to increase at T<sub>C</sub><sup>M</sup> = 167(2) K, and there is a sizeable remnant magnetization in the isothermal magnetization data [Fig. 1(b)]. These results support the existence of bulk FM domains. On the other hand, the ZFC and FC curves diverge below T<sub>C</sub><sup>M</sup>, pointing to a glass-like state.⁸

A kink is visible at T<sub>f</sub> = 120(2) K in the ZFC magnetization branch [Fig. 1(a)]. This is due to the freezing of the glass-like state.⁸ In Fig. 1(b), we show the full magnetization loop at 10 K after implementing a +5.0 T FC protocol. A strong asymmetry with respect to the zero-field line is observed. The strength of this exchange bias effect can be characterized by the exchange bias field, μ₀H<sub>E</sub> = μ₀H₁/2 + μ₀H₂/2.

<figure><img src="image_12.png" /><figcaption>FIG. 1. Exchange bias. (a) ZFC (heating) and FC (cooling) M curves at 5 mT. Red lines mark the characteristic temperatures described in the main text. (b) Magnetization (M) loop. Red lines mark the coercive fields. (c) Top: spin cluster (SC) and surrounding ferromagnetic (FM) domain spaced by a heterogeneous domain wall (HDW). Bottom: the HDW is populated by nonmagnetic (empty circles) and magnetic (filled circles) defects.</figcaption></figure>
In our sample of Mn₅₀Ni₄₀Sn₁₀, µ₀H_E is about 0.09 T, in good agreement with the previous work.⁷

We have studied the entire magnetization loop [Fig. 1(b)] using SANS, after preparing an initial state by the same +5.0 T FC protocol for the magnetization measurements. Previously, a power-law-based model has been used to analyze the spin fractals.⁹,²³,²⁴ This model is an approximation of Eq. (2) in the intermediate q range.²² Our SANS measurements cover an extensive q window spanning two decades (Fig. 2), allowing us to evaluate not only the surface geometries of the scattering objects but also their volume. In other words, they require the full description of Eq. (2). Phase separation has been reported in Mn₅₀Ni₄₀Sn₁₀⁸,³⁰ and other Ni–Mn–X-based alloys.⁹,³¹ With these considerations, the SANS cross sections in absolute units, $\frac{d\Sigma}{d\Omega}(q)$, are fitted to the following three-phase model:

$$ \frac{d\Sigma}{d\Omega}(q) = \frac{A_L}{q^2 + \xi^{-2}} + \left(\frac{d\Sigma}{d\Omega}\right)_f(q). \quad (4) $$

In this equation, the Lorentzian term comes from atomic defects in the sample with a characteristic length scale $\xi \simeq 2.0$ nm (see the supplementary material). The $(\frac{d\Sigma}{d\Omega})_f$ term describes the Guinier/power-law fractals labeled by $i=1, 2$ in Eq. (2). These fractals will be referred to as type-1 and type-2 fractals.

Equation (4) fails to fit the data at and above $T_{MA}$ (not shown), supporting the idea that the physics reported below are specific to the low-temperature martensite phase. In our fitting, $R_0$ is fixed at 0.6 nm—the cubic unit cell size of Mn₅₀Ni₄₀Sn₁₀.⁷ As shown in Fig. 2 for the selective datasets, this model can reproduce well the experimental observations below $T_{MA}$. It is also worth pointing out that the three terms in this model, which are separately plotted in Fig. 2, dominate the SANS cross section in different q regions. This reduces the possibility of overfitting.

We discuss the nature of the type-1 fractals. An initial fitting revealed that their $R_1$ is beyond the upper detection limit (~250 nm, estimated by $2\pi/q$). In the final fitting, we have fixed the Guinier coefficient $G_1$ to zero and $R_1$ to 1000 nm. This means that only the domain walls can be resolved for these fractals. Regarding the choice of $R_1$, any arbitrary number above 250 nm (the upper detection limit) does not

alter the results reported below. These objects can come from either the FM domains, which are on the order of a few micrometers³⁰ or the crystalline grains in Mn₅₀Ni₄₀Sn₁₀. To further confirm their nature, we study the temperature dependence. These measurements were conducted at zero field and on heating. The power-law coefficient $P_1$ and exponent $n_1$ are plotted in Figs. 3(a) and 3(b) (filled solids). Both parameters are temperature independent below $T_f$. Between $T_f$ and $T_C^M$, $P_1$ increases by almost three orders of magnitude, while $n_1$ decreases from about 4.1 to 2.7. The change in $n_1$ suggests the suppression of diffuse surfaces ($n_1 > 4.0$) and emergence of volume fractals ($n_1 < 3.0$). The resulting fractal geometry dramatically increases the surface area. It is responsible for the increase in $P_1$ based on Eq. (3). These fractal changes occur well below the martensitic transformation temperature $T_{MA}$, and they are intimately coupled to the FM transition in the martensite phase. We conclude that the type-1 fractals must be predominantly magnetic, i.e., they mainly come from the FM domains. Above $T_C^M$, $P_1/n_1$ remains almost unchanged in a narrow temperature window and then start to decrease/increase when approaching $T_{MA}$. The latter is due to the development of austenitic FM domains [Fig. 1(a)].

We describe the type-2 fractals, the temperature dependence of which is shown in Fig. 3 (empty solids). No clear response to $T_f$ or $T_C^M$ can be established. While approaching $T_{MA}$, $n_2$ increases from about 3.5 to above 4.0, indicating a change in fractal geometry. Notably, both $P_2$ and $G_2$ decrease sharply at high temperatures. At 198 or 3 K below $T_{MA}$, $P_2$ can no longer be resolved, while the value of $G_2$ is 3.2(7). These data strongly indicate the suppression of type-2 fractals by the martensitic transformation. Accordingly, we assign them to the local clusters generated by this structural transition, with a characteristic length scale of 10–20 nm [Figs. 3(d) and 4(f)]. Nanometer-scale clusters have been observed in the martensitic phase of Mn₅₀Ni₄₀Sn₁₀ by transmission electron microscopy.³⁰ Their magnetic properties, which are crucial for understanding the exchange bias, will be discussed later.

With the nature of the type-1 and type-2 fractals assigned, we study their magnetic field dependences at 2.5 K. The results are shown in Fig. 4. While the left and right coercive fields cannot be

<figure><img src="image_13.png" /><figcaption>FIG. 2. Modelling the SANS cross section $\frac{d\Sigma}{d\Omega}(q)$ using Eq. (4). [(a) and (b)] and [(c) and (d)] are datasets collected after the ZFC and +5.0 T FC protocols, respectively.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>FIG. 3. Temperature induced fractal changes at zero magnetic field. (a) $P_{1,2}$, (b) $n_{1,2}$, (c) $G_2$, and (d) $R_2$. They are defined in Eq. (2). Red lines mark $T_f$, $T_C^M$, and $T_{MA}$ (on heating, main text).</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>FIG. 4. Magnetic field induced fractal changes at 2.5 K. (a) and (b) $P_1$ and $n_1$ describing the type-1 fractals. (c)–(f) $P_2$, $n_2$, $G_2$, and $R_2$ describing the type-2 fractals. The field ramping directions are marked by $\pm$ symbols.</figcaption></figure>

unambiguously separated, an asymmetry with respect to the zero-field line is present for all of these parameters. This asymmetry follows that of the magnetization loop [Fig. 1(b)], indicating that the exchange bias effect is strongly correlated with these two types of spin fractals.

In the magnetization process, the power-law coefficient $P_1$ increases by more than 3 orders of magnitude and then levels off when saturation is realized [Fig. 4(a)]. According to Eq. (3), the power-law coefficient is proportional to the average surface area of the fractals. The behavior of $P_1$ does not fit the descriptions of conventional FM domain walls because their population should be suppressed in the saturation regime, where a single-domain state is formed.¹⁹ Neither is it consistent with the FM domain percolation observed in other phase-segregated materials.³² This peculiarity can be understood by the changes in the exponent $n_1$. As shown in Fig. 4(b), $n_1$ is around 4.75 at the coercive fields. This value is larger than 4.0, meaning that the $P_1$ term is dominated by diffuse surfaces in the low-magnetization region.²¹,²² These surfaces are FM domain walls, as has been reported in other magnetic alloys.²³ Notably, $n_1$ decreases significantly as the magnetization increases. In the saturation regime, it approaches 3.0, meaning that the FM domain walls become spin fractals and possess the maximum roughness permitted in three dimensions. This magnetization-induced development of fractal geometry can lead to a

dramatic increase in the surface area, which is responsible for the increase in $P_1$ [Eq. (3)].

What is the driving mechanism for the geometry changes in the FM domain walls? We turn to study the type-2 fractals. As shown in Figs. 4(c)–4(f), this phase also strongly responds to the external magnetic field. One plausible scenario is a change in the atomic configuration. However, such changes are always accompanied by a metamagnetic transition due to the magnetoelastic coupling. In $Mn_{50}Ni_{40}Sn_{10}$, a metamagnetic transition occurs in a narrow temperature window below $T_{MA}$ and at very high fields.³³ No metamagnetic transition can be observed at 2.5 K [Fig. 1(b)]. As a result, the response of type-2 fractals to the external magnetic field must be related to the spins.

In sharp contrast to the type-1 counterparts, the surfaces of these objects are fractal at the coercive fields with $n_2 \simeq 3.5$ and become diffuse in the saturation regime with $n_2 \simeq 4.4$ [Fig. 4(d)]. This removal of fractal complexity, which greatly suppresses the surface area of the type-2 fractals, explains the decrease in $P_2$ in the magnetization process [Eq. (3); Fig. 4(c)]. Interestingly, the Guinier coefficient $G_2$ and radius of gyration $R_2$ are found to anti-correlate with magnetization [Figs. 4(e) and 4(f)]. In theory, $G_2$ is proportional to the square of average volume of a single fractal [Eq. (3)]. Based on this, the anticorrelation can be explained by the partial disintegration of local spin clusters.

The above-mentioned results allow us to draw a microscopic scenario describing the magnetization-induced interfacial geometric transformation. This is schematically displayed in Fig. 1(c). There exist heterogeneous domain walls (HDWs) with a finite thickness between the type-1 bulk FM domains and local spin clusters inside the type-2 fractals. The HDWs are populated by magnetic and nonmagnetic defects. The magnetic defects form the pathways connecting the type-1 and type-2 fractals. Around the coercive fields, the magnetic defects share the same configuration as the spins inside the spin clusters. This means that, first of all, the FM domains only “see” the spin clusters via the diffuse HDWs, and second, that the surface geometry of the spin clusters is fractal. These features account for the $n_1 > 4.0$ and $n_2 < 4.0$ observed. In the saturation regime, the magnetic defects are re-oriented to be ferromagnetically aligned with the FM domains, reversing the characteristics of $n_1$ and $n_2$.

The magnetization-induced interfacial fractal transformation shares the same asymmetry with the exchange bias. On the theoretical front, it is well known that heterogeneous interfaces are critical for generating an exchange bias effect.² For example, some models predict that exchange bias occurs at two-dimensional interfaces, but the exact influence of interfacial geometry on enhancing or suppressing this effect is strongly debated.³,⁴ Moreover, a so-called “domain state” model based on nonmagnetic defect induced fractal magnetic domains can reproduce a wide range of features associated with exchange bias, although in this model the HDW configuration was not studied.¹³,¹⁴ Neither of the theories above has accounted for a magnetization tunable interfacial spin fractal geometry. Most importantly, our work reveals that, in the saturation regime, the heterogeneous interfaces on the FM side possess the *maximum* roughness allowed in three dimensions, while those on the spin cluster side are diffuse. These observations unambiguously support that this type of fractal/diffuse interfacial configurations, which maximize the surface area for exchange anisotropy, can accommodate a large exchange bias effect [Fig. 1(b)].
In summary, we have used SANS to study the exchange bias in Mn₅₀Ni₄₀Sn₁₀. We find that this effect strongly correlates with the fractal geometry of the HDWs separating the bulk FM domains and local spin clusters. The latter results from the martensitic transformation, which is a common feature in magnetic shape memory alloys.²⁸,²⁹ As a result, the scenario proposed in Fig. 1(c) should also be applicable to understanding the exchange bias effect in a wide range of Ni-Mn-X-based (X = Sn, In, Sb) Heusler alloys.

See the supplementary material for details regarding x-ray powder diffraction characterization on the Mn₅₀Ni₄₀Sn₁₀ sample studied, SANS data reduction process for generating the one-dimensional scattering cross section data (e.g., see Fig. 2), and atomic defects represented by the Lorentzian term in Eq. (4).

L.S., A.A., E.C., and E.B. acknowledge the financial support from the Swedish Research Council under Project No. 201804704. The authors gratefully acknowledge the ILL for the allocated beam time.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Lingjia Shen:** Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).
**Yuanyuan Gong:** Data curation (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal); Validation (equal); Writing – review & editing (equal).
**Ahmed Alshemi:** Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Writing – review & editing (equal).
**Emma Campillo:** Investigation (equal); Methodology (equal); Resources (equal); Visualization (equal); Writing – review & editing (equal).
**Robert Cubitt:** Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Writing – review & editing (equal).
**Nina-Juliane Steinke:** Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Writing – review & editing (equal).
**Elizabeth Blackburn:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (lead); Methodology (lead); Resources (lead); Software (equal); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are openly available at Ref. 26.

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