Introduction

In the past few decades, Heusler alloys have received considerable attention due to a variety of novel physical properties like magneto-optical effect1, magnetocaloric effect2, unconventional superconductivity3, half metallicity4, magnetic shape memory effect5, barocaloric effects6, magnetoresistance7, anomalous Hall Effect8 and large exchange bias9 exhibited by them. Ternary Heusler alloys are represented by the formula X2YZ where X and Y are transition metals and Z is usually sp element from III-VI groups of the periodic table. These systems generally crystallize in L21 structure with Fm-3m space group1,2,3,4,5. Among several compositions, Fe2Y′Al (Y′: 3d transition metals such as V, Cr and Ti) have been extensively investigated in recent years because considerable changes have been observed near the Fermi level in their electronic band structure10. For example, Fe2VAl show semiconducting behaviour in spite of metallic constituents and is nonmagnetic down to 2 K, inspite of the presence of Fe11. Also, Fe2TiAl exhibits optical and electronic properties similar to metals but has a low magnetic moment10,12. On the other hand, it has been reported that this observed non-metallic character can be due to magnetic disorder. Interestingly, the ratio of Y′/Fe can be tuned to study the role of anti-site/chemical disorder on magnetic as well as transport properties. R. Saha et al.13 investigated the magnetic properties of Fe2V1-xCrxAl (x = 0 to 1) and found that increment in Cr content promotes site disorder and alters the magnetic properties of Fe2VAl.

Irrespective of elements present in Heusler alloys, there are significant chances of formation of the disordered structure as they show greater sensitivity towards the environment as well as elemental substitution. One such studied alloy is Fe2CrAl. From first principles calculations it has been predicted to be a half metallic ferromagnet (HMF) system while, experimental investigations deviate from the theoretical results14. It has been reported that this alloy crystallizes in B2 type structure and exhibits semiconducting behaviour. It undergoes ferromagnetic transition with TC ~ 234 K15. Also, through band structure calculations, it is predicted that Mn2CrAl is a HMF with 100% spin polarization16. However, there are no experimental reports available in the literature till now. Hence, it will be interesting to explore the effect of Mn substitution at Fe-site in tuning the structural as well as magnetic properties of the Fe2CrAl alloy.

In this manuscript, we investigate the structural, magnetic and thermodynamic properties of a series of Heusler alloys Fe2-xMnxCrAl (x = 0, 0.25, 0.5, 0.75 and 1). Investigations are carried out through high resolution synchrotron X-ray diffraction, temperature and field dependent magnetization, time dependent magnetization, magnetic memory effect, DC field dependent AC susceptibility along with heat capacity studies. Our studies reveal that: (i) increment of Mn at Fe site in Fe2CrAl destabilizes the L21 structure (ii) the parent compound Fe2CrAl undergoes paramagnetic (PM) to ferromagnetic (FM) phase transition near TC ~ 202 K, followed by a cluster glass (CG)-like state near Tf2 ~ 3.9 K (iii) with increment in Mn concentration, TC is suppressed to 120, 48 and 27 K for Fe1.75Mn0.25CrAl, Fe1.5Mn0.5CrAl and Fe1.25Mn0.75CrAl respectively while it is observed that Fe1.75Mn0.25CrAl and Fe1.5Mn0.5CrAl exhibit double CG-like transitions near Tf1 ~ 22 K, Tf2 ~ 4.2 K and Tf1 ~ 30.4 K, Tf2 ~ 9.5 K respectively (iv) only one CG-like transition is observed near Tf2 ~ 11.5 K for Fe1.25Mn0.75CrAl (v) long ranged magnetic ordering is suppressed in FeMnCrAl and this alloy undergoes CG-like transitions at Tf* ~ 22 K, Tf1 ~ 16.6 K and Tf2 ~ 11 K (vi) in the parent compound Griffiths phase-like singularities is noted and this is shifted down in temperature with an increase in Mn concentration. In this series of alloys, significant suppression in value of TC can be due to the presence of AFM coupling between Fe and Mn. Also, observation of more than one CG phase can be due to enhancement in magnetic anisotropy in the system, which dissociates the infinite clusters into smaller ones. In the present case, these different sized clusters freeze at a different temperature depending upon their anisotropic fields i.e. smaller and highly anisotropic clusters freeze near Tf2 whereas bigger and less anisotropic clusters freeze near Tf1 and Tf*. Additionally, GP-like behaviour is noted in high temperature regime and such feature is observed due to the formation of short range ordered clusters which arise due to quenched/anti-site disorder between Fe and Al.

Results

Structural analysis

The Le-Bail refinements of room temperature XRD patterns of the Fe2-xMnxCrAl (0 ≤ x ≤ 1) alloys are shown in the Fig. 1. All the reflections present in the XRD patterns could be identified using a cubic structure (Fm-3m space group). This confirms that all the samples are in single phase without any impurity. The refined lattice parameters and cell volume for all compositions are listed in Table 1. It can be noted from the Table 1 that the lattice parameter and hence the unit cell volume increases with increasing concentration of Mn. A comparison of the x-ray diffraction patterns of the parent and doped alloys (inset (b) of Fig. 1) reveals a gradual shift of diffraction lines to the lower angle side, thereby establishing that the dopant goes to the respective site. It has been suggested in literature that from the intensity ratio of (200) and (220) reflections i.e. I200/I220 one can determine the degree of site ordering of Fe and Al atoms13. Here, the intensity of (200) is low (as compared to the most intense peak) to be seen clearly. Hence we have plotted it for different alloys (in inset of Fig. 1(b)) along with variation in its Mn concentration. In Fe2-xMnxCrAl, replacing Fe with Mn reduces the intensity of (200) reflection and hence the I200/I220 ratio decreases with Mn, which indicates enhancement in site disorder as shown in inset of Fig. 1(c). This disorder destabilizes the L21 structure and can also alter the physical properties of the Heusler alloys13,17,18,19,20. Therefore, in order to understand the effect of chemical/anti-site disorder on magnetic and thermodynamic properties of this series of alloys, magnetic and heat capacity studies have been carried out.

Figure 1
figure 1

Le-bail fit of the room temperature indexed XRD patterns of (a) Fe2CrAl (b) Fe1.75Mn0.25CrAl (c) Fe1.5Mn0.5CrAl (d) Fe1.25Mn0.75CrAl and (e) FeMnCrAl. Inset of (b) shows the shifting of peak with increasing Mn substitution along with decrease in intensity of (200) peak for selected compositions of Fe2CrAl, Fe1.5Mn0.5CrAl and FeMnCrAl. Inset of (c) depicts the change in intensity ratio of I200/I220 reflection lines with Mn substitution.

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Table 1 Lattice parameters of Fe2-xMnxCrAl (0 ≤ x ≤ 1) obtained from Le-bail fit of XRD patterns.
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Temperature and magnetic field dependent DC magnetization studies

Temperature response of zero-field cooled (ZFC) and field-cooled (FC) magnetization of the series Fe2-xMnxCrAl (0 ≤ x ≤ 1) at 100 Oe are represented in Fig. 2(a–e). It is observed that the parent compound Fe2CrAl undergoes magnetic phase transition near 202 K, which is confirmed through d(M(T))/dT vs. T plot (as shown in Fig. 2(f)). This magnetic transition can be attributed as PM to FM phase transition which is in accordance with the literature15. Interestingly, a weak bifurcation between ZFC and FC curves is also noted in low temperature regime. It is found that increment of Mn concentration in the parent compound suppresses this magnetic transition temperature to 120, 48 and 27 K for Fe1.75Mn0.25CrAl, Fe1.5Mn0.5CrAl and Fe1.25Mn0.75CrAl respectively. Also, in contrast to the parent compound, a large bifurcation between ZFC and FC magnetization curves is observed in low temperature region of Fe1.75Mn0.25CrAl and Fe1.5Mn0.5CrAl. In all these alloys, one can see that the ZFC magnetization curves increase down to a particular temperature but show a downturn at low temperatures, signifying a possible presence of another magnetic phase. This downturn along with bifurcation in ZFC and FC curves is generally attributed to the presence of glassy magnetic phase or due to short range magnetic correlation or owing to the presence of superparamagnetic (SPM) phases and can also originate due to magneto-crystalline anisotropy and/or anti-site disorder21. Additionally, in high temperature regime of Fe1.5Mn0.5CrAl, i.e. in the PM phase a weak feature (deviation of ZFC curve) is noted which is also present in Fe1.25Mn0.75CrAl and FeMnCrAl as illustrated in insets of Fig. 2(c–e) respectively. This high temperature magnetic phase will be discussed further in section “Evidence of the presence of the Griffiths phase in the high temperature region”. The isothermal magnetization curves as a function of the magnetic field at 2 K and 300 K for all the alloys are shown in Fig. 3(a,b)). At 2 K, it is observed that the nature of the curves remains unchanged across the series and all the curves show insignificant hysteresis. It is also observed that there is a significant reduction in magnetization value obtained at 50 kOe for FeMnCrAl as compared with Fe2CrAl. However, at 300 K, it is found that Fe2CrAl and Fe1.75Mn0.25CrAl exhibit non-linear behaviour rather than expected linear behaviour of a pure PM phase, indicating the presence of some short range magnetic correlations above TC.

Figure 2
figure 2

Temperature response of the DC magnetization under ZFC (straight line) and FC (open circle) conditions at 100 Oe in the temperature range 2–250 K for (a) Fe2CrAl. (b) Fe1.75Mn0.25CrAl (c) Fe1.5Mn0.5CrAl; (d) Fe1.25Mn0.75CrAl; (e) FeMnCrAl. Inset of (a) shows the magnified view of ZFC and FC curves in the temperature range 2–80 K, while, insets of (c–e) shows magnified view of the respective ZFC curves (f) dM/dT vs T plot for Fe2-xMnxCrAl (x ≤ 0 < 1) alloys in the temperature range 18–250 K.

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Figure 3
figure 3

Isothermal magnetization as a function of magnetic field at (a) 2 K and (b) 300 K of Fe2-xMnxCrAl (x ≤ 0 ≤ 1) alloys (c) Magnetic field dependent magnetization plots of all alloys at 2 K with fit (red solid line) using Eq. 2.

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Role of magnetic anisotropy in low temperature regime

In order to investigate whether magneto-crystalline anisotropy affects the nature of ZFC and FC curves below ordering temperature, we have analysed the magnetic field dependent magnetization response, using random anisotropy theory (RAT) developed by Chudnosky et al.22. In this theory, the magnetic ground state of an alloy can be determined by the strength of random anisotropy field, leading to the prediction of different magnetic ground states. The strength of H is measured with respect to Hs; Hs = Hr4/Hex3 where Hex and Hr are the exchange and anisotropic fields respectively. In the case of H < Hs, there is an existence of correlated spin glass phase which has a large susceptibility and is similar to Ahrony and Pyte phase23. Presence of random anisotropy causes variation of the direction of magnetization of locally correlated regimes. In this region, magnetization is expressed as

$$M={M}_{0}[1-1/15{({H}_{{\rm{s}}}/H)}^{1/2}]$$
(1)

where M0 is the saturation magnetization. In case of high field (H > Hex), there is virtual alignment of spins with applied field along with slight tipping angle due to anisotropy. Once the applied magnetic energy overcomes the exchange field energy, M(H) curves saturates. In this region, M is expressed as

$$M={M}_{0}(1-1/15({H}_{{\rm{r}}}/{(H+{H}_{{\rm{ex}}})}^{2})$$
(2)

The M-H data obtained at 2 K for all alloys for 5 ≤ H ≤ 50 kOe are analyzed using Eq. 2 and are shown in Fig. 3(c). The fit parameters extracted from the fit are given in the Table 2. It can be seen that both Hr and Hex increases with Mn concentration. This behaviour reflects an increment in strength of magnetic anisotropy and similar trend has been observed in system exhibiting glassy magnetic phase13,24. In other words, at low temperatures, the strength of anisotropy is large which can result in freezing of correlated spins in glassy state. Hence, it can be said that magnetic anisotropy plays a significant role in the low temperature regime in this series.

Table 2 Parameters obtained from RAT fitting using Eq. 2 of M-H data at 2 K.
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Magnetic phase in low temperature regime

Time dependent magnetization study

To shed some light on the magnetic phase in low temperature regime of this series of alloys isothermal remanent magnetization (IRM) measurements has been performed at different temperatures (Fig. 4) using the following protocol: the alloy is cooled in zero field from room temperature to the measurement temperature (2 and 10 K). After that 100 Oe field is applied for 20 minutes. After that the field is switched off and magnetization as a function of time is recorded. The obtained curves are fitted using equation

$$M(t)={M}_{0}-S\,\mathrm{ln}(1+t/{t}_{0})$$
(3)

where M0 is the magnitude of magnetization at t = 0 and S is the magnetic viscosity21,25. The obtained fitting parameters are tabulated in Table 3. For Fe2CrAl, logarithmic relaxation behaviour is observed at 2 K indicating the presence of a large number of intermediate anisotropic metastable states, while this type of behaviour is absent at 10 K (inset of Fig. 4(a)). Across the series, similar type of relaxation behaviour is observed at 2 K. It is seen that at 10 K, all Mn doped alloys show relaxation behaviour. It is also observed that magnetic viscosity decreases with increase in temperature possibly due to a variation in the size of the cluster of spins. Usually, for long ranged ordered systems magnetization remains unchanged with time. Our observations indicate the absence of long range ordering and distribution of energy barrier due to the presence of clusters of spins. This indicates a possible presence of a glassy phase in low temperature regime of this series.

Figure 4
figure 4

(a–e) Magnetization as a function of time at different temperature for all the alloys. Inset of (a) represents the relaxation behaviour of Fe2CrAl at 10 K. Solid red line depicts the logarithmic fit using Eq. 3. Note: Left axis represents the scale for data obtained at 2 K and right axis represents the scale for the data obtained at 10 K.

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Table 3 Parameters obtained from fitting of time dependent magnetization data with Eq. 3.
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Heat capacity study

The presence of glassy phase in low temperature regime is also reflected in the detailed analysis of heat capacity measurements. Left panel of Fig. 5 shows the T2 response of C/T for all the alloys. In the low temperature regime the curves are fitted with the following equation

$$C=\gamma T+{\rm{\beta }}{T}^{3}$$
(4)

where γ and β are the fitting constants. The obtained fitting parameters are listed in Table 4. It can be inferred from the table that the obtained values of γ for all alloys are quite large. It is also determined that there is increment in the value of γ with increasing Mn concentration. Table 4 also contains the value of Debye temperature (θD) obtained using θD = (12π4nR/5β)1/3 where R is ideal gas constant, n is the number of atoms per formula unit. The values of γ and β are in accordance with those obtained in case of magnetic glassy systems26,27,28,29,30,31. Also, it can be noted from right panel of Fig. 5 that C deviates from the typical Debye T3 behaviour at low temperature when temperature response of C/T3 is plotted. At low temperature, such upturn indicates the presence of glassy phase and can also be related to the presence of TLS (two level systems) excitation which is in contrast to expected Debye behaviour seen in crystal26. The above observations signify the presence of glassy phase in the low temperature regime of Fe2-xMnxCrAl (0 ≤ x ≤ 1).

Figure 5
figure 5

Left panel: C/T vs. T2 plot of all the alloys and fitted (solid red line) with Eq. 4. Right panel: C/T3 vs. T plot of all the alloys.

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Table 4 Parameters obtained from fitting of C/T Vs. T2 using Eq. 4.
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ZFC and FC memory effect

In order to discern the origin of low temperature glassy phase in this series of alloys we have performed both FC and ZFC memory effect using stop and wait protocol32,33,34. To investigate the FC memory effect, we have employed the following protocol: the alloy is cooled in the presence of 100 Oe from 300 K to 2 K but cooling is interrupted at different halting temperatures (TH) (for Fe2CrAl, TH = 15, 12 and 8 K; for Fe1.75Mn0.25CrAl, TH = 30 and 15 K; for Fe1.5Mn0.5CrAl, TH = 10 and 5 K; for Fe1.25Mn0.75CrAl, TH = 8 and 4 K and FeMnCrAl TH ~ 8 K) for 2 hours in each case. During the halt, the magnetic field is turned off and after the wait time, it is switched on and cooling is resumed. It is observed that this cooling procedure produces a step-like feature in the obtained FC curve for each sample. This obtained FC magnetization curve is represented as FC-tw in the left panel of Fig. 6. After reaching the 2 K, the alloy is again heated back in presence of 100 Oe to 300 K. Interestingly, the curve obtained after warming also exhibit kinks near the halt temperature. This curve is illustrated as FC curve (in left panel of Fig. 6). It can be said that the alloy remembers its thermal history or magnetic state achieved during FC-tw. Again the alloy is cooled to 2 K without any halt and the warming curve is taken immediately after cooling. The curve is featureless and behaves as normal FC curve which is treated as FC-reference in Fig. 6 (left panel). Both SPM and glassy magnetic systems exhibit FC memory effect, but there is a significant difference between the behaviour of FC curves using the above protocol. In the case of SPM, the FC curves increase continuously with decreasing temperature whereas for glassy magnetic systems such behavior is absent32,33. As observed from the figure, there is no increment in magnetization in FC curves with decreasing temperature; implying absence of SPM behaviour in the low temperature region of these alloys. Additionally, according to refs32,33,34 it is reported that ZFC memory effect is exhibited by glassy magnetic system and is not observed in systems exhibiting SPM behaviour. Hence, we have performed ZFC memory effect using the following protocol: the alloy is cooled in the presence of zero field, and halts are made at different temperatures (TH) for different alloys (for Fe2CrAl, TH = 7 K, for Fe1.75Mn0.25CrAl, TH = 5 K, for Fe1.5Mn0.5CrAl, TH = 5 K, for Fe1.25Mn0.75CrAl, TH = 6 K and for FeMnCrAl, TH = 4 K). Then, 100 Oe is applied and data is obtained during the warming cycle (ZFC-tw). The obtained curve is shown in the right panel of Fig. 6. Again the alloy is cooled to 2 K in absence of magnetic field and then 100 Oe is applied and data is taken during warming cycle without any halt. This curve is treated as ZFC-ref. Here, we have observed that there is a difference between ZFCref and ZFCtw i.e. presence of memory dips at wait temperature (as shown in the inset of the right panel of Fig. 6). The observation of ZFC memory effect indicates the presence of a glassy magnetic phase in the low temperature region of this series of alloys. Based on the above observations, it can be said that in the low temperature region the origin of the glassy magnetic phase is due to the presence of interacting spins clusters rather than blocking of spins.

Figure 6
figure 6

Temperature dependent memory effect of all the alloys at different halt temperatures (TH); Left panel: Under FC condition. Right panel: Under ZFC condition. Insets of right panel: Temperature response of ZFCtw-ZFCref.

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AC susceptibility study

In order to understand the nature of the glassy magnetic phase, we have performed frequency dependent AC susceptibility measurements in the absence of any superimposed DC fields. Figure 7 depicts the temperature response of real (χ′) and imaginary part (χ′′) of AC susceptibility for this series, measured at different frequencies (13–931 Hz) for AC drive field of 1 Oe. In the case of Fe2CrAl, it is observed that χ′ exhibits frequency dependent broad feature in low temperature regime, whereas two frequency dependent peaks near Tf1 ~ 12.5 K and Tf2 ~ 3.9 K are clearly visible in χ′′. However, Tf1 is not clearly visible at 13 and 131 Hz. It can be due to the presence of some interacting clusters in this region which display frequency dependent behaviour at higher applied frequencies. In all the Mn doped alloys, χ′ show frequency dependent broad features, while, χ′′ exhibit well defined peaks. In Fe1.75Mn0.25CrAl, in χ′′ two frequency dependent peaks are observed near Tf1 ~ 22 and Tf2 ~ 4.2 K, while, for Fe1.5Mn0.5CrAl, three peaks are noted at Tf1 ~ 30.4, Tf2 ~ 9.5 K and TC ~55 K. However, in Fe1.25Mn0.75CrAl, two frequency dependent peaks are noted at Tf2~11.5 K and TC ~ 27 K. But these peaks near TC in both compounds does not show any shift with temperature with frequency as compared to Tf1 and Tf2. Interestingly, for FeMnCrAl three frequency dependent peaks in χ′′ are observed at Tf* ~ 22 K, Tf1 ~ 16.6 K and Tf2 ~ 11 K. Hence, it can be said that with increment in Mn concentration, Tf1 and Tf2 peaks shift towards high temperature region except in Fe1.25Mn0.75CrAl, where only Tf2 is observed. Presence of Tf* in FeMnCrAl signifies the absence of long range ordering in the whole temperature range of this alloy.

Figure 7
figure 7

Temperature response of in-phase (left panel) and out-of-phase (right panel) components of AC susceptibility of all the alloys measured at different frequencies.

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In all these alloys, it is observed that Tf*, Tf1, and Tf2 shift towards higher temperature as the magnitude of measuring frequency increases. Such a frequency dependent shift can be due to either glass-like freezing or SPM blocking21. However, the presence of SPM-like behaviour in these systems has been ruled out in the earlier sections. Thus, in order to characterize the nature of glassy transition exhibited by Tf*, Tf1, and Tf2 in all alloys we have determined Mydosh parameter which is often used to distinguish/compare various magnetic glassy systems. This parameter reflects the response of magnetic clusters toward frequencies and it depends on the type of interaction between clusters. As the interaction between magnetic clusters is generally weak, it results in strong sensitivity towards frequencies. In contrast for normal magnetic systems, the interactions between the magnetic ions are strong and such frequency dependence is not observed. Mydosh parameter (δTf) is expressed as21

$${\rm{\delta }}{T}_{f}=\Delta {T}_{f}/{T}_{f}\,\log \,f$$
(5)

where Tf is the peak temperature corresponding to frequency (f). The obtained Mydosh parameters are listed in Table 5. For Fe2CrAl, it is found that the value of δTf for Tf2 is similar to that reported for other CG systems21,35,36,37,38,39, whereas it is not calculated for Tf1. For the Mn-doped alloys, the values of δTf obtained for the respective freezing temperatures indicate that they also belong to the CG category. We have further investigated the behaviour of magnetic clusters, by analyzing the temperature dependence of relaxation time (τ) using standard critical slowing down model, which is given by dynamic scaling theory21,40

$${\rm{\tau }}={{\rm{\tau }}}^{\ast }{(({T}_{{\rm{f}}}-{T}_{{\rm{g}}})/({T}_{{\rm{g}}}))}^{-zv}$$
(6)

where τ* is the microscopic flipping time, Tg is the true spin glass (SG) transition temperature, z is the dynamic critical exponent and v is the critical exponent of the correlation length. Also, the temperature maxima is fitted with Vogel-Fulcher (V-F) law (which takes into account the interaction between spins) of the form21,41

$${\rm{\tau }}={{\rm{\tau }}}_{0}\exp \,(({E}_{{\rm{a}}})/{k}_{{\rm{B}}}({T}_{f}-{T}_{0}))$$
(7)

where T0 is the VF temperature, which represents the strength of interactions between clusters. The fittings using Eqs (6) and (7) is shown in Fig. 8. The values of parameters obtained after fitting are summarized in Table 6. In the case of Fe2CrAl, we have obtained non-zero value of T0 and our data agrees well with V-F law which suggests a finite interaction among spins and thus leading to formation of clusters42. It can also be inferred that the magnitude of τ* and τ0 obtained in case of Tf2 is quite higher than that obtained in case of SG systems [10−10–10−13 s]. Such large values have been observed in various CG systems21,25. For all the Mn-doped alloys, for each freezing temperature, the value of Tg is found large than T0 which is in accordance with the trend found in CG systems. Similarly, the obtained values of zν for all the freezing temperature also support the preceding statement43,44,45. Thus, it can be said that Tf*, Tf1, and Tf2 correspond to CG-type freezing in the respective alloys. It also suggests that this series of alloys exhibit slow spin dynamics due to cluster formation. This type of existence of more than one glassy transition in a system is not unusual; it has been reported in various systems43,44.

Table 5 Mydosh parameters obtained from Eq. 5.
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Figure 8
figure 8

(a) V-F law fit of relaxation time (τ) as function of reduced temperature 1/(Tf-T0) using Eq. 7 (b) Critical law fit of relaxation time (τ) as function of reduced temperature (Tf/Tg −1) using Eq. 6 for Tf*, Tf1 and Tf2 for respective alloys.

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Table 6 Parameters obtained from critical slowing down model and V-F law using Eqs 6 and 7.
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In order to see the effect of anisotropic behaviour in cluster glass region, we have also performed AC susceptibility measurements in the presence of superimposed DC fields (0–1500 Oe) for all alloys by fixing the AC excitation field at 1 Oe and 531 Hz. Field response of AC susceptibility is plotted in Fig. 9. These measurements will help to discern the highly anisotropic from less anisotropic cluster region. In the presence of large DC fields, the less anisotropic regions get aligned and their responses to small AC fields are suppressed46. As it can be seen from Fig. 9, in case of Fe2CrAl, Tf2 transition shifts towards lower temperature with increasing HDC, thereby indicating the highly anisotropic nature of CG transition. A different behaviour is observed in the case of Tf1. In the case of Tf1, the peak gets transformed into a broad shoulder when HDC is applied and gets suppressed under the application of 750 Oe. Similar shifting of Tf2 peak towards lower temperature and transformation of Tf1 peak into broad shoulder along with suppression in the presence of 500 and 300 Oe is observed in case of Fe1.75Mn0.25CrAl and Fe1.5Mn0.5CrAl respectively. For Fe1.25Mn0.75CrAl, it is observed that Tf2 shifts towards lower temperature. However, in case of FeMnCrAl, it is found that Tf* and Tf1 combines into single broad shoulder like feature which suppresses under 300 Oe. This type of behaviour exhibited by Tf1 in all alloys (except Fe1.25Mn0.75CrAl) can be due to presence of less anisotropic cluster regions near Tf1 which gets aligned in the direction of applied HDC and their response to AC field gets suppressed. It is also noted that on the application of the DC field, Tf2 moves towards lower temperature. Hence, we have focussed on the response of Tf2 towards applied HDC. In the case of glassy magnetic systems, typically two irreversible lines are observed in H-T phase diagram: de Almeida-Thouless line (A-T line) and Gabay-Toulose (G-T line) line. In case of Heisenberg spin systems, we can observe both the lines, in the strong anisotropy regime we can expect that the line follows A-T character while in weak anisotropic region G-T behaviour is followed. On other hand, only A-T line is usually observed for Ising spin glass systems47,48. However, we can expect a quantitative difference in A-T line in the mean field and non mean field model. According to non-mean field theory, in low field region the variation of Tf with HDC follows49

$${T}_{{\rm{f}}}(H)={T}_{{\rm{f}}}(0)(1-A^{\prime} {H}^{Q})$$
(8)

where A′ is the anisotropic parameter, Tf (0) is the value of freezing temperature in the absence of magnetic field and Q is the exponent. In the mean field model, Q has a value of 0.66 in strong anisotropic regime whereas it has a value of 2 in the weak anisotropic regime. The obtained parameters from fitting the response of Tf2 on the application of HDC (with Eq. 8) are mentioned in Table 7. The obtained values of Q corresponding to each alloy reflect that Tf2 does not follow the mean field theory; rather a non-mean field type exponent has been obtained similar to other SG systems which do not belong to same universality class50,51. Also, from the table, it can be inferred that all alloys neither follows A-T line nor G-T line but it can be said that as Mn concentration increases, the system transits from weak irreversibility to strong irreversibility regime (Fig. 10). Therefore, it can be said that Mn concentration increases the role of anisotropy in the system. Also, we would like to mention that, in case of Fe2CrAl it is observed that χ′′ near TC shows frequency dependent behaviour i.e. there is a decrement in magnitude with increment in frequency (not shown) but no shifting of peak temperature with frequency is noted. This type of frequency dependence can be due to irreversible domain wall movement or pinning effect52. Similar type of frequency dependent behaviour is also observed in Fe1.75Mn0.25CrAl, Fe1.5Mn0.5CrAl and Fe1.25Mn0.75CrAl. Hence, it implies that magnetic anisotropy plays an important role and affects the nature of magnetic phase for the various concentration of Mn. This is also responsible for the absence of long range magnetic ordering in the FeMnCrAl compound.

Figure 9
figure 9

Left panel: Temperature response of out-of phase component of AC susceptibility at different HDC. Right panel: Tf vs. H plot obtained from out-of-phase component of AC susceptibility and fitted (solid red line) with Eq. 8.

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Table 7 Parameters obtained from fitting of field response of freezing temperature using Eq. 8.
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Figure 10
figure 10

Phase diagram to illustrate the transition of the system from weak irreversibility regime to strong irreversibility regime.

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Evidence of the presence of the Griffiths phase in the high temperature region

As mentioned in section “Temperature and magnetic field dependent DC magnetization studies”, in PM phase of Fe1.5Mn0.5CrAl, a weak feature is noted. In order to see whether the similar feature is present in Fe2CrAl and Fe1.75Mn0.25CrAl, the temperature response of magnetization is measured upto 400 K at 100 Oe (not shown). It is observed that there is a weak bifurcation between ZFC and FC curves in high temperature regime, which indicates to the presence of short range correlations among spins. Additionally, similar to Fe1.5Mn0.5CrAl, some anomaly is noted around 364 and 300 K for Fe2CrAl and Fe1.75Mn0.25CrAl respectively. To shed some light on the observed anomaly in this series of alloy, we have plotted temperature response of inverse DC susceptibility (χ−1DC) (left panel of Fig. 11). For all these alloys, a strong downward deviation from Curie Weiss (CW) law is observed. For Fe2CrAl it is observed near 364 K and is shifted to 300, 206, 180 and 214 K for Fe1.75Mn0.25CrAl, Fe1.5Mn0.5CrAl, Fe1.25Mn0.75CrAl and FeMnCrAl respectively. It has been reported in the literature that the observation of a downward deviation of χ−1DC curve well above TC is a signature of the presence of the GP regime53,54. Hence the observation of similar feature gives an indication of the possibility of the presence of GP regime in this series of alloys. Under the application of the magnetic field, it is noticed that the downturn softens (shown in Fig. 11(k,l) corresponding to Fe2CrAl and FeMnCrAl, respectively). Similar suppression of the downturn has also been reported in the GP regime of other magnetic systems55,56,57. In general, Griffith’s singularity can be characterized using the exponent (λ) which can be obtained from the expression

$${{\rm{\chi }}}^{-1}\propto {(T-{{T}_{{\rm{C}}}}^{{\rm{R}}})}^{(1-\lambda )}$$
(9)

where 0 ≤ λ < 1 and TCR is the critical T where χ diverges. This relation is the modified form of CW law where λ signifies the deviation from CW behaviour. This value has a finite value between 0 and 1 above TC but is approximately 0 in PM region. We have plotted log10 χ−1 vs log10 (T/TCR − 1) (right panel of Fig. 11) and the value of λ is obtained from the slopes, fitted with Eq. 9. In PM and GP region, the exponent is defined as λP and λG respectively. The obtained values are mentioned in Table 8. For all these alloys, we have observed larger values of λG (with the value lying between 0 and 1) which signifies the strong deviation from CW behaviour in high temperature region. It is also observed that as Mn concentration is increased, the GP becomes stronger. This behaviour can be attributed to the formation of short range clusters. Such features are in accordance to the theoretical studies, where it is argued that coexistence of competing magnetic phases stabilizes as well as enhances the possibility of clusters to be present above TC, resulting in an inhomogeneous PM phase58. Interestingly, AC susceptibility also shows an anomaly in this high temperature region. However, no shift in maxima with frequency in AC susceptibility is observed in all alloys but it suppresses under the application of superimposed HDC = 100 Oe (not shown), which is in contrast to that observed for the clusters in the low temperature regime. These observations rule out the possibility of formation of glassy magnetic state or SPM-like behaviour in this temperature region. In low temperature regime, anisotropy is dominant and plays a crucial role in the formation of CG phase but the origin of clusters in this temperature regime is different from those in low temperature regime. Here, they are formed as a result of anti-site disorder between Fe and Al59,60,61 and anisotropy has negligible effect in this regime. It is interesting to note that Fe2CrAl also shows the presence of this phase inspite of its stoichiometric composition. As we have noted from Fig. 1(b) and inset of (c) that the intensity ratio of I200/I220 is low (2.7%), which indicates the presence of substantial amount of anti-site disorder between Fe/Mn and Al. This anti-site disorder is responsible for the observation of GP in the parent compound. It is also observed that the disorder increases with Mn concentration (as discussed in section “Structural analysis”). A similar trend is noted in the value of λG, which reflects the role of anti-site disorder in the formation of GP in these alloys. It results in the formation of the highly inhomogeneous phase where different values of exchange coupling are allocated randomly at different lattice sites. It may lead to coexistence of short range FM and AFM correlations. Such anti-site disorder and competition between AFM-FM interactions are responsible for evolution of GP in these alloys.

Figure 11
figure 11

Left panel: (a,c,e,g,i) represent Inverse DC susceptibility as function of temperature plot at 100 Oe. Solid red line represents the fitting using CW law. Right panel: (b,d,f,h,j) represent the log-log plot of χ−1DC vs. T/TCR −1. Solid red lines are straight line fitting in GP and PM region. (k,l) depicts Inverse DC susceptibility as function of temperature plot at different applied fields for Fe2CrAl and FeMnCrAl respectively.

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Table 8 Parameters obtained from straight line fitting of log-log plot of χDC−1 as function of (T/\({T}_{C}^{R})\) − 1 in PM and GP region.
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Discussions

The Heusler alloy Fe2CrAl crystallizes in L21 structure. The Mn substituted Fe2CrAl alloys have similar XRD spectra but with increasing Mn concentration, significant changes in the intensity of (200) reflection line and increment in lattice parameter are noted. These observed changes can be due to atomic site disorder between Fe and Al arising out of substitution of Mn in crystal lattice, which also plays a significant role in altering the magnetic correlations in the system. The observed drastic decrease in value of TC with increment in Mn concentration is due to development of AFM coupling between Fe and Mn. Presence of more than one CG phase transition is due to the fact that magnetic anisotropy plays a significant role and it increases with the increment in Mn concentration. At low temperature, this anisotropy grows larger such that it weakens the coupling that holds together infinite clusters. Hence, these clusters dissociate into smaller clusters. It is noted that less anisotropic and bigger clusters freeze at higher temperature (near Tf* and Tf1); whereas those with higher anisotropic field freezes at lower temperature (near Tf2). Along with anisotropy, site disorder also plays a significant role in altering the magnetic properties of this series of alloys. GP-like features are observed high temperature regime due to the presence of anti-site disorder, which increases with Mn concentration.

Phase diagram of Fe2-xMnxCrAl (0 ≤ x ≤ 1)

Figure 12 depicts the T-x phase diagram of Fe2-xMnxCrAl. The phase boundaries in the diagram are estimated from the DC magnetization and AC susceptibility studies. Phase (CG-I) is the CG phase it is composed of small sized and highly anisotropic clusters. At higher temperature, there is another CG phase (Phase CG-II) consisting of less anisotropic and bigger sized clusters. However, this phase is absent in Fe2CrAl and Fe1.25Mn0.75CrAl. Phase (CG-III) (present only in FeMnCrAl) also has similar character as phase CG-II. Phase FM represents the FM phase of this series. This phase gets suppressed with increment in Mn due to development of AFM coupling between Fe and Mn and it vanishes in FeMnCrAl. The intermediate phase (IM) is the region lying between the FM and GP phase region. In this phase, as observed from magnetization and AC susceptibility results, there is an abrupt decrease in magnetization value as the temperature is increased (after TC). Such abrupt drop has been noted in ferromagnetic systems, when phase transition takes place from ordered (FM) to disordered phase. In high temperature region, GP (Phase GP) is observed, which gets suppressed towards lower temperature with increment in Mn content. Phase PM is the PM phase of these alloys.

Figure 12
figure 12

Temperature-concentration phase diagram of Fe2-xMnxCrAl (0 ≤ x ≤ 1).

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Summary

In summary, it is concluded that Fe2CrAl is a ferromagnetic Heusler alloy crystallizing in L21 structure with TC ~ 202 K and exhibits CG state in low temperature regime. With increase in Mn concentration, TC is significantly suppressed towards lower temperature. However, it is noted that below TC, Fe1.75Mn0.25CrAl and Fe1.5Mn0.5CrAl exhibit double CG-like transitions. Interestingly for Fe1.25Mn0.75CrAl; a single CG-like transition is noted below TC. For FeMnCrAl, no long ranged magnetic ordering is observed and this alloy undergoes three CG-like transitions. In this series of alloys, observation of more than one CG state has been attributed to increment in magnetic anisotropy in the system. It weakens the coupling between infinite clusters and dissociates them into small clusters. These clusters freeze at different temperatures depending upon their anisotropic fields. In present case, we have found that bigger and less anisotropic clusters are near Tf* and Tf1 whereas smaller and highly anisotropic near Tf2. Additionally, at high temperatures, GP phase is observed in Fe2CrAl which is shifted towards lower temperature with increasing Mn concentration and this feature arises as a result of anti-site disorder between Fe and Al.

Methods

The Heusler alloys Fe2-xMnxCrAl (x = 0, 0.25, 0.5, 0.75 and 1) are prepared by arc melting the stoichiometric ratio of the high purity of the constituent elements (>99.9%) in an atmosphere of argon. The ingots are re-melted several times to ensure the homogeneity of the alloys. The weight loss after the final melting for each alloy is less than 1%. The resultant ingots are sealed in evacuated quartz tubes and subjected to 900 °C for 1 week, followed by water quenching. The elemental analysis is carried using energy dispersive x-ray analyzer attached with FESEM (Nova Nano SEM-450, JFEI U.S.A). The average atomic stoichiometry of each alloy is in accordance with the expected values. X-ray diffraction (XRD) measurements are performed at room temperature at P02 beamline in the Petra III Synchrotron radiation, Hamburg, Germany using a wavelength of 0.20712 Å. Temperature (T) and magnetic field (H) dependent magnetization (M) measurements are carried using Magnetic property measurement system (MPMS) from Quantum Design, U.S.A. Heat capacity (C) is measured using the heat capacity option of Physical property measurement system (PPMS) from Quantum design, U.S.A.