Magnetic states of Ni-Mn-Sn based shape memory alloy: a combined muon spin relaxation and neutron diffraction study

TL;DR

This study investigates the complex magnetic behavior of Ni-Mn-Sn shape memory alloy using muon spin relaxation and neutron diffraction, revealing multiple magnetic transitions and a likely spin-glass ground state.

Contribution

It combines muSR and neutron diffraction to elucidate the magnetic states and transitions in Ni-Mn-Sn alloy, highlighting the presence of a disordered spin-glass phase.

Findings

Bulk ferromagnetic order below 320 K

Collapse to paramagnetic state below 290 K

Re-entrant magnetic order below 260 K

Abstract

The fascinating multiple magnetic states observed in the Ni-Mn-Sn based metamagnetic shape memory alloy are addressed through a combined muon spin relaxation (muSR) and neutron powder diffraction studies. The material used in the present investigation is an off-stoichiometric alloy of nominal composition, Ni[2.04]Mn[1.4]Sn[0.56]. This prototypical alloy, similar to other members in the Ni-Mn-Sn series, orders ferromagnetically below T[CA] (= 320 K), and undergoes martensitic type structural transition at T[MS] (= 290 K), which is associated with the sudden loss of magnetization. The sample regains its magnetization below another magnetic transition at T[CM] = 260 K. Eventually, the composition shows a step-like anomaly at T[B] = 120 K, which is found to coincide with the blocking temperature of exchange bias effect observed in the alloy. In our study, the initial asymmetry A_[10] ) of the $\mu$SR data falls rapidly below T[CA], indicating the onset of bulk magnetic order. A[10] regains its full asymmetry value below T[MS] suggesting the collapse of the ferromagnetic order into a fully disordered paramagnetic state. Below the second magnetic transition at T[CM], asymmetry drops again, confirming the re-entrance of a long range ordered state. Interestingly, A[10] increases sluggishly below T[B], indicating that the system attains a disordered/glassy magnetic phase below T[B], which is responsible for the exchange bias and frequency dispersion in the ac susceptibility data as previously reported. The neutron powder diffraction data do not show any magnetic superlattice reflections, ruling out the possibility of a long range antiferromagnetic state at low temperatures. The ground state is likely to be comprised of a concentrated metallic spin-glass in the backdrop of an ordered ferromagnetic state.

Tables

Table 1. Table 2: Crystallographic parameter of the sample at 8 K ( P ​ m ​ m ​ a 𝑃 𝑚 𝑚 𝑎 Pmma ) as obtained from the refinements.

Atom	Site	X	Y	Z	$B_{iso}$
Ni	4$h$	0.0000	0.249(6)	0.5000	1.229(7)
Ni	4$k$	0.2500	0.2484(4)	0.091(5)	0.544(4)
Mn	2$a$	0.0000	0.0000	0.0000	1.616(6)
Mn	2$f$	0.2500	0.5000	0.574(7)	1.500(5)
Mn	2$b$	0.0000	0.5000	0.0000	0.990(4)
Mn	2$e$	0.2500	0.0000	0.562(6)	0.990(4)
Sn	2$b$	0.0000	0.5000	0.0000	0.990(4)
Sn	2$e$	0.2500	0.0000	0.562(6)	0.990(4)

Equations

$$\text{centering}{G}_z(t)=A_{10}\exp (-{\lambda }_{1}t)+A_{20}\exp (-{\lambda }_{2}t)+b_g\text{@add@centering}$$

$$\text{centering}{G}_z(t)=A_{10}\exp [(-\lambda t{)}^{\beta }]+b_g\text{@add@centering}$$

Full text

I Introduction

Advanced functional materials play an ever increasing role in the modern technological developments, which encompass areas such as energy harvesting, computation, communication as well as to combat environmental pollution. Broadly, they can be classified into five groups depending upon their functionality; Adaptive, Magnetic, Electric, Optical, and Energy and Environmental materials. It is needless to mention that a proper investigation of their physical properties is important in understandings the essential physics associated with their functionality. For example, the study of CuO-based high $T_{C}$ superconductors has enriched our understanding of electronic properties of correlated metal oxides. It is to be noted that many of these functional materials are actually multifunctional, i.e., they show two or more functional properties. Ni2.04Mn1.4Sn0.56, the title composition of this work, is one such material having functionality as adaptive, magnetic, electric as well as energy and environmental material.

Ni2.04Mn1.4Sn0.56 belongs to a class of materials known as metamagnetic shape memory alloys (MSMAs). They show large magnetic field induced strain, magneto-resistance, magneto-caloric, baro-caloric, and exchange bias effects. kainuma1 ; krenke ; kainuma2 ; manosa ; chatterjee ; barocaloric The general compositions of these alloys can be expressed as Ni2Mn1+pZ1-p (Z= In, Sn, Sb and $p<$ 1), and the observed functionality arises from their bi-ferroic nature with the simultaneous presence (as well as their mutual interplay) of ferromagnetism and ferroelasticity. The alloys are characterized by first-order martensitic type structural phase transition occurring at a temperature $T_{MS}$ and a second order paramagnetic (PM) to ferromagnetic (FM) transition at the Curie point ($T_{C}$). For the practical realization of magnetic shape memory and other magneto-functional properties, one should have $T_{C}>T_{MS}$, i.e., the sample should be in a magnetically ordered state when martensitic phase transition (MPT) takes place.

The presence of two critical temperatures (namely $T_{C}$ and $T_{MS}$) makes the system to have a rather exotic phase diagram. There are few important aspects associated with these alloys, which remain elusive till date. Firstly, what happens to the FM state of the alloy (we are considering $T_{C}>T_{MS}$) right below the MPT? The high temperature structural phase is called austenite with cubic lattice symmetry, while the low temperature phase (below $T_{MS}$) is called martensite with a tetragonal/orthorhombic/modulated structure. It is found that magnetization drops sharply below $T_{MS}$, indicating the loss of ordered FM moment. Some diffuse peaks were observed in the Neutron Powder Diffraction (NPD) data of Ni-Mn-Sn alloys, which were assigned to the existence of incipient antiferromagnetic (AFM) coupling brown . As evident from the chemical compositions, a fraction of Z atoms is replaced by Mn (we call it Mn*′), and theoretical calculations indicate the existence of AFM correlation between regular Mn and Mn′* atoms. montecarlo ; xps ; tang ; vvs-th ; hbx-th ; cmli-th ; pirolkar On the other hand, 57Co-Rh Mössbauser spectroscopy on a Ni-Mn-Sn alloy (with small amount of enriched 57Fe doped at the Mn site) indicated a PM state below $T_{MS}$ of the sample. umetsu NPD studies, performed on various Ni-Mn-Z alloys, fail to identify well defined magnetic Bragg peaks associated with an ordered AFM state. brown ; brown-In Neutron polarization study indicates the existence of FM correlations, which vanishes below $T_{MS}$ with the concomitant occurrence of Mn-Mn*′* AFM correlations. aksoy Therefore, the nature of the magnetic state just below $T_{MS}$ remains uncertain; it may be incipient AFM in the backdrop of a PM/FM phase, a long range ordered AFM state with weak moment or an FM state with reduced Mn moment.

The second unresolved point is associated with the magnetic ground state of these alloys. The off-stoichiometric Ni-Mn-Z alloys show a step-like feature well below $T_{MS}$ in the zero-field-cooled magnetization data and exchange bias (EB) effect was observed below $T_{B}$. Our group previously reported that field-cooling from just above $T_{B}$ is sufficient to observe EB, and $T_{B}$ actually signifies a spin freezing temperature. souvik2 Subsequently, there have been numerous reports on the glassy magnetic state of the Ni-Mn-Z alloys. cong ; wang ; umetsu1 ; sangam Nevertheless, ambiguities remain on the nature of the glassy state, and the ground state has been described as re-entrant spin glass, cluster glass or super spin glass by various authors (cited in the previous line). EB generally requires two different magnetic phases (say, FM $+$ AFM or spin-glass $+$ FM) to be present. Presence of AFM clusters below $T_{MS}$ is highly possible, since there is a strong evidence for incipient antiferromagnetism. It remains unclear whether the ground state is characterized by a (i) mixture of AFM and FM phase fractions along with interfacial glassyness, (ii) coexisting spin-glass (SG) and FM phases, or (iii) stand alone SG phase.

In the present work, we have addressed these aspects using muon spin resonance/rotation ($\mu$SR) as well as NPD techniques on an MSMA of nominal composition, Ni2.04Mn1.4Sn0.56. One can notice that a small amount of excess Ni is doped at the expense of Sn, which is required to elevate $T_{MS}$. While ferromagnetic Curie point of the high temperature austenite is around $T_{CA}$ = 320 K, the structural transition takes place around $T_{MS}$ = 290 K. The reason for choosing this composition lies with the fact that $T_{MS}$ is very close to $T_{CA}$, and the FM state becomes unstable below the MPT (see fig. 1). The sample regains its magnetization below a second transition at $T_{CM}$ = 260 K in the martensitic phase. The step like anomaly is seen below $T_{B}$ = 120 K, and considerable EB is observed at low temperature. sabya The sample shows the signature of field-cooled-field-stop memory (FCFS) sabya1 as shown in the lower panel of fig. 1. The observed FCFS in this bulk sample indicates the presence of glassy magnetic ground state.

II Experimental Details

The polycrystalline sample of Ni2.04Mn1.4Sn0.56 for the present study was prepared by argon arc melting the constituent elements. sabya The temperature ($T$) dependent dc magnetization ($M$) measurements were performed using a commercial Quantum Design SQUID magnetometer (MPMS 3). The $\mu$SR measurements were performed at ISIS facility, Rutherford Appleton Laboratory, UK using EMU (for zero magnetic field) and HIFI (for longitudinal magnetic field) spectrometers. The sample was mounted on a silver sample holder to minimize the background and measurements were performed at different temperatures. The neutron powder diffraction was carried out at the WISH time of flight diffractometer at the ISIS facility between 8 and 363 K. The powdered sample was inserted in a cylindrical vanadium container of 6 mm diameter. A standard Helium closed cycle refrigerator was used to cool the sample down to 8 K. Nuclear and magnetic structure refinements were performed by the Rietveld method using the FULLPROF program. fp The diffraction peaks were described by a pseudo-Voigt profile function.

III Results

III.1 Muon Spin Relaxation

$\mu$SR is an accomplished local probe technique to study the magnetism of a material. blundell Spin polarized positive muons ($\mu^{+}$) are implanted into the sample. The implanted muons decay into positrons, which are emitted preferentially in the direction of the muon spin. The spin of the implanted $\mu^{+}$ precesses around the effective magnetic field vector at the site of implantation. Random and fluctuating fields within the sample can depolarize the muons. In the actual zero field (ZF) or longitudinal field (LF) experiment, the emitted positrons are counted parallel and anti-parallel to the initial muon spin direction. The difference between the number of positrons in the forward and backward directions is generally measured as a function of time ($t$), and it is called the asymmetry function, $G_{z}(t)$. Since, $G_{z}(t)$ is a measure of muon depolarization, one can get significant information on the magnetic state of the material out of it. ryan ; dalmas

For a typical magnetic material, the relaxing part of the asymmetry often obeys an exponential law, $G_{z}=A_{0}\exp{(-\lambda t)}$, where $A_{0}$ is the initial asymmetry, and $\lambda$ is the (spin-lattice) relaxation rate. For the present Ni-Mn-Sn alloy, the simple exponential law is found to be inadequate to describe the data at all temperatures. The simplest approach is to have a bimodial distribution with two relaxation rates, $\lambda_{1}$ and $\lambda_{2}$, which results, manganite1

[TABLE]

Here $b_{g}$ denotes the time independent background of asymmetry. Similar two exponent model has been used for diverse magnetic systems successfully, which include perovskite manganites, manganite1 ; kawasaki ; heffner cobaltates, guo , as well as Hesuler based intermetallic alloys. hiroi Muons, whose nearest atom on the Z sublattice is either Mn*′* or Sn, are subject to different internal fields. The regression converges better over full $T$ range, if we put the constrain $A_{20}=0.4A_{10}$, and the fittings presented in the subsequent parts have been performed considering the above constrain. The value of $b_{g}$ for a particular fixed magnetic field was estimated from $T$ = 30 K data and it is kept constant at all $T$ for a particular value of $H$. For the ZF case, the value of $b_{g}$ was kept constant at 0.0094(4), while for the LF data it is fixed at the value 0.1973(4).

Considering glassy ground state in the studied alloy, we have additionally used a stretched exponential function to fit the data below 150 K. ryan ; stretch1 ; nav2o5 ; mncosi ; pbfenbo

[TABLE]

Here $\beta$ is called the shape parameter and $b_{g}$ is the background. Similar to the double exponential fitting, we have kept $b_{g}$ fixed for all $T$ values for a particular $H$.

Fig. 2 (a) shows the time domain $\mu$SR data recorded at different temperatures in ZF condition while cooling. At $T$ = 340 K, the asymmetry shows an exponential like decay, which is expected for a PM state. On lowering the temperature below the FM Curie point ($T_{CA}$ = 320 K), $G_{z}(t)$ shows a fast relaxation component, which coexists with the slow relaxing part. This is clearly due to the onset of FM transition in the system. On further lowering $T$ below $T_{MS}$ (= 290 K), the fast relaxation component appears to get diminished. This corresponds to the rapid fall of $M$ below $T_{MS}$ in the magnetization data [see fig. 1 (a)]. The fast relaxation reappears when the sample is cooled below the second magnetic ordering point $T_{CM}$. Most interestingly, the damping gradually gets reduced, when the sample is cooled below the blocking temperature, $T_{B}$ = 120 K. We also recorded the $\mu$SR spectra in presence of 45 kOe of applied longitudinal field, as depicted in fig. 2(b).

In order to elucidate the magnetic state of the sample at different $T$’s, we have fitted the time domain data with the double exponential function stated in eqn. 1. The solid lines in figs. 2(a) and (b) represent the fit to the data. The values of $A_{10}$ and ($\lambda_{1}$, $\lambda_{2}$), obtained by fitting the ZF data, are plotted in figs. 3(a) and (b) respectively.

The $T$ variation of $A_{10}$ [see fig.3 (a)] provides noteworthy information on the magnetic state of the studied alloy. In the PM state (above 310 K), the initial asymmetry is found to be almost $T$ independent. On cooling, $A_{10}$ falls sharply with the onset of ferromagnetism at $T_{CA}$, and it attains its lowest value just below 300 K. The structural transition is beautifully echoed in that data, as $A_{10}$ again rises sharply below $T_{MS}$ and attains a value ($=$ 0.11) slightly lower than the value in the high-$T$ PM state ($=$ 0.13). Below 260 K, $A_{10}$ shows another sharp fall, which can easily be assigned to the second magnetic transition occurring at $T_{CM}$. The most remarkable observation is the sluggish rise (as opposed to the sharp change at magnetic Curie points) of $A_{10}$ with decreasing $T$ below $T_{B}$. This is an indication for the loss of magnetic order, and it nicely fits with the conjecture of a glassy magnetic ground state of the system. It is evident that even at the lowest temperature, the value of $A_{10}$ ($=$ 0.03) is much smaller than the value observed in the high-$T$ PM state. The $T$ variation of $\lambda_{1}$ and $\lambda_{2}$ are plotted in fig. 3 (b). Both the parameters show well defined peaks at the long range magnetic ordering temperatures, which corroborates well with the nature of the $A_{10}(T)$ curve. It is worthwhile to mention that across the transition from PM to a magnetically ordered phase, one would expect the relaxing part of asymmetry to decrease by $\frac{1}{3}$ for a powder/polycrystalline sample in ZF measurement. The residual $\frac{2}{3}$ component in the time domain data will correspond to muon’s damped oscillations due to the internal magnetic field. ryan ; dalmas In the present case, the frequency of oscillation may be too high too be observed in the spectrometer at ISIS due to the large internal field from Mn.

We have fitted the LF relaxation curves recorded under 45 kOe, and the $T$ variation of resulting parameter $A_{10}$ is depicted in fig. 4(a). For the LF data, we recorded relaxation during both heating and cooling. The variation of LF $A_{10}(T)$ traces similar nature as that of ZF one. However, it provides few pieces of additional information, which are not obvious in the ZF data. Firstly, the sharp rise in $A_{10}$ due to MPT gets shifted to a lower temperature under LF. It is well known that the magnetic field favors austenite and it reduces $T_{MS}$, which gets well reflected in our $\mu$SR data. Secondly, clear thermal hysteresis is seen in $A_{10}(T)$ around the martensitic transition occurring at $T_{MS}$. The hysteresis is present in the plot of $\lambda_{1}$ and $\lambda_{2}$ as well [fig. 4 (b)], and it can be accounted by the first order nature of the structural transition. A second thermal hysteresis is present just below $T_{CM}$, which can be traced back to the similar thermal hysteresis observed in the bulk magnetization data (see fig. 1). Such hysteresis may be linked with the first order nature of the magnetic transition at $T_{CM}$.

Apart from the double exponential fitting (eqn. 1), we have used the stretched exponential function (as described by eqn. 2) to fit the time domain data. A stretched exponential form of muon depolarization is generally expected both above and below the spin freezing temperature ($T_{g}$). For a simple PM to SG transition, $\beta$ attains a value close to unity at a temperature well above $T_{g}$. On cooling towards $T_{g}$, $\beta$ decreases. mncosi It has been found that for so-called concentrated canonical spin glasses (where there is a distribution of the frequency of fluctuation of the local magnetic field), $\beta$ attains a value of $\frac{1}{3}$ at $T_{g}$. stretch1

We find that both the ZF and LF relaxation data can also be fitted well with a stretched exponential function as described in eqn. 2 below about 150 K. In fact the quality of the fittings in the temperature range 30-150 K is found to be better (as evident from the lower values of $\chi^{2}$ of the fits) in case of stretched exponential as compared to two exponential function. However, stretched exponential fitting turns poorer above 150 K, and it fails to converge with physically meaningful values of the fitting parameters. In figs. 5 (a), (b), and (c), we have shown the $T$ variations of the parameters $A_{0}$, $\beta$and $\lambda$ respectively only below 150, which were obtained by fitting the ZF and LF relaxation data recorded while the sample is being cooled. The initial asymmetry ($A_{0}$) shows a rise below 150 K, similar to the behavior of $A_{10}$ obtained from the two exponential model. The exponent $\beta$ decreases as we approach $T_{B}$ from high temperature, and shows a minimum at 120 K [fig. 3 (c)]. The values $\beta$ at $T_{B}$ are found to be $\beta$(ZF)${}_{T_{B}}$ = 0.33(4) and $\beta$(LF)${}_{T_{B}}$ = 0.30(6). These values are fairly close to the $\beta$ = $\frac{1}{3}$ law (particularly the ZF one) for concentrated metallic SG. Below $T_{B}$, it rises again and attains a value of 0.62 at 30 K. On the other hand, $\lambda$ shows a decreasing trend on cooling along with peak like feature at $T_{B}$.

III.2 Neutron powder diffraction measurements

Figs. 6 (a), (b) and (c) show the high resolution NPD data measured at different temperatures. The sample was first heated to 363 K (which is well above $T_{MS}$ and $T_{CA}$) and diffraction data were recorded while cooling from 363 to 8 K within the closed cycle refrigerator. The diffraction pattern at 363 K can be well indexed by the cubic L21 structure with space group $Fm\bar{3}m$ as expected for a pure austenitic phase [see fig. 6 (a)]. At 363 K, the sample is in the PM state, and a good refinement is obtained by considering only the nuclear contribution of the cubic austenite phase with L21 geometry. The refined cubic lattice parameter is found to be $a_{c}$ = 5.991(1) Å. On cooling below $T_{MS}$, the cubic peaks start to disappear, along with the appearance of martensitic peaks [see the NPD data at 275 K in fig. 5 (c)].

At 8 K, the data can be described by a single orthorhombic phase as shown in fig. 6 (b). We do not observe any well resolved magnetic superlattice reflection, which matches well with the previous report. brown This rules out the possibility of an ordered AFM state below $T_{MS}$. In case of related Ni-Co-Mn-Ga based Hesuler alloys, distinct AFM state was observed below the MPT. orlandi A stable refinement is achieved assuming the nuclear phase coming from $Pmma$, and one magnetic phase with propagation vector $k$ = (0, 0, 0). Here we have assumed that the ordered moments arise from the Mn atoms residing at 2$a$ and 2$f$ positions only, and neglected any contribution from Ni. We have fitted our data with several possible options of collinear magnetic structure with $k$ = (0, 0, 0), and the best fit is obtained when the moments are aligned along orthorhombic $c$ axis. Mn atoms, situated at 2$a$ and 2$f$ sites, have ordered moment 2.76 $\mu_{B}$ and 2.30 $\mu_{B}$ respectively (see Table 3).

In fig 6. (c), we have plotted NPD data for an intermediate temperature of 275 K, where both $Fm\bar{3}m$ (austenite) and $Pmma$ (martensite) phases coexist, and we have considered both the phases to refine the diffraction data. As evident from our $\mu$SR data, the asymmetry rises below $T_{MS}$, indicating a PM martensitic phase. However, the residual austenite fraction may still be present in the sample having an ordered FM state. Our effort to fit 275 K data considering only the nuclear contributions coming from cubic and orthorhombic phases do not provide a good convergence. A better fit is obtained, when the FM contribution from the cubic phase is taken into consideration. Fig 6 (c) shows the experimental data as well as the refinements. The ratio of the volume fraction of the cubic and orthorhombic phases is found to be $\frac{2}{3}:\frac{1}{3}$. This indicates that below $T_{MS}$, the major phase fraction is martensite, although a sizable austenite phase is still present. The cubic and orthorhombic lattice parameters are found to be $a_{c}$ = 5.991(7) Å and $a_{o}$ = 8.613(4) Å , $b_{o}$ = 5.675(6) Å , $c_{o}$ = 4.360(5) Å respectively. The ordered Mn moments are found to be 1.78 and 1.13 $\mu_{B}$ at 4$a$ and 4$b$ sites respectively (see Table 1). These moment values match quite well with the previous report. brown

Interestingly, a fraction of high-$T$ austenite continues to exist over a wide temperature range well below $T_{MS}$. Eventually, the reflections due to austenite disappears when the sample is cooled below 200 K. In order to determine the $T$ variation of phase fraction, we have performed structural refinements of the NPD data at different temperatures between 8 K and 363 K. Considering the coexistence of the cubic and the orthorhombic phases, the data were refined using two phases. Fig. 6 (d) shows how the fraction of orthorhombic and cubic phases changes with $T$. As expected, the cubic fraction diminishes rapidly on cooling and disappears below 200 K. The orthorhombic phase fraction, on the other hand, increases monotonically and almost saturates below 150 K. It is to be noted that the thermal hysteresis in the magnetization and $\mu$SR data disappears below 150 K. Therefore, 150 K can be assigned as the culminating point of MPT, below which the system attains a stable martensite fraction.

IV Discussions

The complex magnetic phases of the studied alloy get reflected in the $\mu$SR data and in association with the NPD result, it clarifies significantly the prevailing doubts on the magnetic states of such Ni-Mn-Z based MSMAs. We observe that the values of $A_{10}$ obtained from the $\mu$SR data (both ZF and LF) show a sharp rise on cooling below $T_{MS}$, which continues till the second magnetic transition at $T_{CM}$ is attained. The bulk magnetic measurements [as depicted in figs. 1(a) and (b)] indicate a rapid fall of $M$ below $T_{MS}$, which can be due to the development of a (i) long range ordered AFM state, (ii) long range ordered FM state albeit with highly reduced Mn moment, (iii) a state with short range AFM correlations, (iv) ordered FM clusters in the backdrop of a PM state, or (v) a pure PM state.

This sharp rise in $A_{10}$ below $T_{MS}$ summarily rejects cases (i) and (ii), as long range order should not be accompanied with increasing asymmetry. Therefore, we are left with options (iii), (iv) and (v). If we look at the variation of $A_{10}(T)$, the asymmetry does not fully attain its austenite PM state value just below $T_{MS}$. Therefore, the scenario of a pure PM state can be excluded. The magnetic state just below $T_{MS}$ can be either due to the presence of AFM phase fraction, or associated with the residual FM austenite phase which remained untransformed even below $T_{MS}$. In our NPD data, we find clear signature of this cubic austenite down to 200 K. Therefore the mismatch of $A_{10}$ at 264 K ($\approx T_{CM}$) and 350 K in the ZF $\mu$SR data is likely to be associated with this cubic FM fraction. The variation of $A_{10}(T)$ in the LF $\mu$SR data is somewhat similar, although the signature of $T_{MS}$ has shifted slightly to lower temperature. This is due to the fact that an external magnetic field prefers the ferromagnetically ordered austenite. koyama ; planes Logically, the most probable scenario is case (iv), where the transformed martensite is PM (the major phase), residing along with the untransformed FM austenite (the minor phase).

The thermal hysteresis around the martensitic transition is expected, and it is present in the $\mu$SR data too. Interestingly, both $M(T)$ and $A_{10}(T)$ show another thermal hysteresis between 150 and 225 K. There are several reports on two-step martensitic transition, where a second inter-martensite transformation occurs. fan ; mcc In case of one such Ni-Mn-In based alloy, the inter-martensite transition was found to occur just below $T_{CM}$, and it was assigned to a transformation from 10M modulated structure to 14M. huang We carefully looked at the NPD data in this temperature range, however, no anomaly was detected in the form of peak splitting or appearance of additional reflections. The magnetic transition at around $T_{CM}$ is certainly a first order one, however, it may be an iso-structural one where the lattice symmetry remains unaltered.

The most important observation in the present work is the signature of $T_{B}$ in the $\mu$SR data. As evident [see Figs. 3(a) and 4 (a)], the initial asymmetry shows a rise below $T_{B}$ indicating the loss of magnetic order in the system. Notably, this rise is present irrespective of the fitting function used (two-exponential or stretched exponential). If we look at the variation of $\beta$ [see fig. 5 (b)], it shows a minimum at $T_{B}$ = 120 K with the value of $\beta$ close to 0.33. In addition, $\lambda$ shows a weak peak [see fig. 5 (b)] around $T_{B}$. Considering the glassy magnetic state observed in the family of Ni-Mn-Sn alloys below $T_{B}$, we can assign $T_{B}$ to be the spin freezing temperature of the presently studied sample.

It is now pertinent to discuss the nature and origin of SG ground state. From our NPD data, we observe a single phase orthorhombic martensite at the base temperature. Therefore, the spin-freezing is not related to the presence of minority cubic phase in the system. In Ni-Mn-Z based MSMAs, the sign and strength of magnetic interaction depend strongly on the Mn-Mn bond distance. Below $T_{MS}$, the intersite Mn-Mn*′* distance decreases paving the path for enhanced AFM correlation. montecarlo ; vvs-th ; pirolkar ; parijat In addition, chemical and lattice disorders play an important role in determining the magnetic ground state of these materials. vvs-th The AFM correlations between Mn-Mn*′* (particularly below $T_{MS}$), the Mn-Mn FM correlations and the presence of disorder eventually lead to spin freezing below $T_{B}$. The Mn*′*atoms are substituted randomly in the Sn site, which can give rise to random occurrence of FM and AFM bonds. From our analysis of the NPD data it is evident that the AFM correlations is short range in nature, i.e., it does not give rise to a long range ordered AFM state. Nevertheless, the ground state does show long range FM order.

In general for a PM to SG transition, the value of $\beta$ in the stretched exponential fitting assumes a constant value below the spin freezing temperature due to the residual fluctuation of the frozen state. nav2o5 In contrary to the usual observation, the value of $\beta$ increases below $T_{B}$ and attains a value of 0.62 at 30 K. The main reason for such anomalous behavior of $\beta$ lies with the reentrant character of the SG state, where spin freezing takes place on top of a long range FM state. Interestingly, very similar $T$ dependence of $\beta$ was reported in case of Pb(Fe1/2Nb1/2)O3, where $\beta$ shows a minimum at $T_{g}$ = 20 K and approaches to unity on further cooling. pbfenbo . The SG state in this compound is reentrant type which develops in the backdrop of an AFM state. It has been argued that with the critical slowing down of the SG fluctuations at $T_{g}$, muons starts to sense the non-critical fluctuations of the long range ordered state below $T_{g}$ leading to the increase in $\beta$. By analogy, the rise in $\beta$ below $T_{B}$ in the studied Ni-Mn-Sn alloy can be accounted by the presence of long range ordered FM phase with the SG state side by side. This is corroborated by the NPD data, where FM order is indeed present at 8 K. The weak nature of the peak in $\lambda(T)$ data (ideal spin glass should show a sharp peak at spin freezing temperature) may be due to the presence of this FM ordered state alongside the SG phase.

In conclusion, the present work successfully discerns few ambiguities related to the magnetic phase diagram of Ni-Mn-Sn alloy system. Based on our study on a particular Ni-Mn-Sn alloy, it is evident that the alloy assumes a PM state just below the martensitic transition. The $\mu$SR result identifies two long range magnetic ordering temperatures $T_{CA}$, and $T_{CM}$ and they are found to be ferromagnetic in nature. The work categorically justifies the view that the magnetic anomaly at $T_{CM}$ in the martensitic state indeed corresponds to the onset of a long range ordered state. Most importantly, the work identifies that the system transform into a partially disordered magnetic phase below the exchange bias blocking temperature, which can be characterized by the coexistence of ordered FM and frozen spin glass state. The remarkable phenomena of exchange bias observed in Ni-Mn-Z alloys is due to the coupling between the interfacial spins of SG and FM phases.

V Acknowledgment

The work is supported by the India-RAL collaborative project (SR/NM/Z-07/2015). J. Sannigrahi wishes to acknowledge EU’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement (No. 665593) awarded to the STFC, UK.