A study of cluster spin-glass behaviour at the critical composition Mn0.73Fe0.27 NiGe

Journal Pre-proofs A study of cluster spin-glass behaviour at the critical composition Mn0.73Fe0.27 NiGe Pallab Bag, K. Somesh, R. Nath PII: DOI: Reference:

S0304-8853(19)31361-7 https://doi.org/10.1016/j.jmmm.2019.165977 MAGMA 165977

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Journal of Magnetism and Magnetic Materials

Received Date: Accepted Date:

16 April 2019 10 October 2019

Please cite this article as: P. Bag, K. Somesh, R. Nath, A study of cluster spin-glass behaviour at the critical composition Mn0.73Fe0.27 NiGe, Journal of Magnetism and Magnetic Materials (2019), doi: https://doi.org/10.1016/

j.jmmm.2019.165977

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© 2019 Published by Elsevier B.V.

A study of cluster spin-glass behaviour at the critical composition Mn0.73 Fe0.27 NiGe Pallab Bag,1, 2 K. Somesh,1 and R. Nath1, ∗ 1

School of Physics, Indian Institute of Science Education and Research Thiruvananthapuram, Kerala-695551, India 2 Department of Physics, National Dong Hwa University, Hualien 97401, Taiwan (Dated: September 26, 2019) The solid solution Mn1−x Fex NiGe with x ∼ 0.27 has been investigated via powder x-ray diffraction, dc magnetization, ac susceptibility, heat capacity, and magnetic relaxation measurements. The alloy Mn0.73 Fe0.27 NiGe crystallizes in a Ni2 In-type hexagonal structure at room temperature. A coexistence of hexagonal and orthorhombic phases with a phase fraction of nearly 88:12 was found below 80 K. The dc susceptibility measurements reveal a paramagnetic to ferromagnetic transition at around Tt ≃ 80 K and a thermal hysteresis observed during field-cooled-cooling and fieldcooled-warming suggests the first order nature of the magnetic transition. The splitting between zero-field-cooled and field-cooled dc susceptibilities and the appearance of a frequency dependent peak in ac susceptibility indicate the onset of a spin-glass transition at Tg ≃ 58 K. The relative shift in freezing temperature (δTf ) is calculated to be ∼ 0.03 and ∼ 0.09, respectively from the real and imaginary parts of ac susceptibility indicating the formation of cluster spin-glass state. The analysis of frequency dependent Tf using power law yields a characteristic time for a single spin flip τ ∗ ≃ 1.5 × 10−10 s and critical exponent zν ′ ≃ 5.5. Similarly, the analysis using the Vogel-Fulcher law results a characteristic time for a single spin flip τ0 ≃ 1.2 × 10−8 s, Vogel-Fulcher temperature T0 ≃ 54.8 K, and an activation energy Ea /kB ≃ 72.8 K. The magnitude of τ ∗ and τ0 together with a non-zero value of T0 add further evidence for the formation of cluster spin-glass. The magnetic relaxation and memory effect measurements also demonstrate the low temperature cluster spin-glass behaviour. The reason for the cluster spin-glass behaviour could be the difference in local environment of Mn atoms in the coexisting Mn-rich antiferromagnetic and Fe-rich ferromagnetic phases. Furthermore, Mn0.73 Fe0.27 NiGe shows the asymmetric response of magnetic relaxation with a change in temperature, below Tf , which can be explained by the hierarchical model. The low temperature heat capacity gives a large electronic coefficient γ ≃ 25 mJ/mol K2 and the Debye temperature θD ≃ 300 K. This value of θD is in close agreement with that calculated from the temperature variation of unit cell volume. PACS numbers: 75.47.Np, 75.50.Lk, 75.10.Nr

I.

INTRODUCTION

In the past few years, equiatomic ternary alloys of the general formula M M ′ X (M , M ′ = transition metals and X = Si, Ge, and Sn) have been extensively studied primarily because of their numerous physical properties.1–4 These materials often have coupling between structural and magnetic degrees of freedom leading to various magneto-responsive effects. Manifestation of such effects are expected to facilitate the multifunctional applications of the studied materials. The most common magneto-responsive effects exhibited by such systems are magnetocaloric effect (MCE) which can be projected as an alternative for gas-compression refrigerators,1,5–7 magnetoresistance,2,8 ferromagnetic shape memory effect,9,10 exchange bias effect,11 etc. In particular, the alloy MnNiGe is widely pursued because of its potential to produce the desired paramagnetic (PM) to ferromagnetic (FM)-type magnetostructural transition (MST) by tuning the chemical compositions. It shows the onset of a martensite transition (MT) from Ni2 In-type hexagonal structure with space group P 63 /mmc (high temperature) to TiNiSi-type orthorhombic structure with space group P nma (low temperature) at Tt ≃ 470 K.12 The low temperature marten-

site phase shows the onset of a spiral antiferromagnetic (AFM) transition at TNM ≃ 350 K which is formed by six nearest Mn atoms surrounding the Ni atom. The magnetic moments of 2.8 µB are only carried by the Mn atoms.1,13–15 The Curie temperature (TCA ) of the high temperature austenite phase is around 205 K which is much lower than TNM .1 Therefore, it is expected that by chemical substitution, one can reduce the MT below its magnetic transition and establish a magneto-structural coupling which is of interest to the physics community. In order to unveil its functional behaviour, Liu et al substituted Fe at the Ni and Mn sites and reported their complete phase diagrams.1 In Mn1−x Fex NiGe, with a small doping (x = 0.08), Tt is lowered to meet TNM and then introduces ferromagnetism at a relatively high temperature of 350 K. The Fe substitution not only converts AFM martensite into FM one but also drives the FM austenite phase into a spin-glass (SG) like state. The MT eventually vanishes and the austenitic-phase zone with x ≥ 0.26 enters into a spin-glass-like state at the freezing temperature below about 70 K. Overall, the magnetic phase diagram of Mn1−x Fex NiGe shows the martensite AFM, martensite FM, and austenite SG states for x < 0.08, 0.08 < x < 0.26, and x ≥ 0.26, respectively.1 The MST associated with the PM to FM transition was observed over a wide temperature range

from 350 K to 100 K by changing x from 0.08 to 0.26 and a large MCE has also been reported in this window.16–18 However, the critical doping concentration at which the austenite shows the SG-like phase is found to vary depending on the sample preparation conditions.1,11 These alloys also show remarkable magneto-transport properties such as a negative magneto-resistance and a clear virgin line effect at 5 K for x = 0.20.2 Though there are several studies reported in this series, majority of the works are focused in the FM region (x ≤ 0.26) and a little is known about the nature and origin of the SG state for x > 0.26. In this work, we report a detail investigation of the structural, static, and dynamical properties of Mn1−x Fex NiGe alloy at the critical doping concentration x = 0.27. It is found that the system undergoes a hysteretic ferromagnetic transition below Tt ≃ 80 K which is of first order type, below which it exhibits a glassy behavior. The analysis of ac susceptibility data using conventional dynamical models pinpoint the formation of cluster SG at low temperatures. We also demonstrate the magnetic relaxation and magnetic memory effect using different measurement protocols and explained our observations in the framework of hierarchical model.

II.

EXPERIMENTAL DETAILS

Polycrystalline sample of Mn0.73 Fe0.27 NiGe was prepared by arc melting the elements in stoichiometry ratio. The ingot was remelted several times to get better homogeneity and then annealed continuously for five days at 800 0 C. Phase purity of the sample was confirmed from the powder x-ray diffraction (XRD) measurements performed as a function of temperature T (15 K≤ T ≤ 300 K). For low-T measurements, a lowT attachment (Oxford Phenix) was used. The dc and ac magnetization (M ) measurements were performed using the vibrating sample magnetometer (VSM) and ACMS attachments to the physical property measurement system (PPMS, Quantum Design). Heat capacity (Cp ) was measured using the heat capacity option in the PPMS.

III.

RESULTS AND DISCUSSION A.

X-ray Diffraction

In order to check the phase purity and the structural transition, the temperature dependent powder XRD was measured during cooling from 300 K to 15 K of the polycrystalline Mn0.73 Fe0.27 NiGe. Rietveld refinement of the XRD patterns were carried out using the FullProf package.19 Figure 1(a) and 1(b) present the representative XRD pattern collected at T = 300 K and 15 K along with the Rietveld refinement. All the Bragg peaks in the XRD pattern at T = 300 K could be refined by considering the Ni2 In-type hexagonal (space group: P 63 /mmc)

structure suggesting that the sample under investigation is phase pure.13,14 Below 80 K, few extra peaks were observed in the XRD pattern which correspond to low temperature orthorhombic phase. These peaks are highlighted in the inset of Fig. 1(b) by the downward arrows. A two phase analysis below 80 K reveals the hexagonal to orthorhombic phase fraction of 88:12. The obtained lattice constants (a and c for the hexagonal and a, b, and c for the orthorhombic phases) and the unit cell volume (Vcell ) as a function of T are plotted in Fig. 1(c) and (d), respectively for both the phases. As expected, Vcell of hex the high temperature hexagonal phase (Vcell ) is found to decrease with temperature, suggesting a contraction hex in the unit cell. The temperature dependent Vcell was 20 fitted by the equation Vcell (T ) = γU (T )/K0 + V0 ,

(1)

where V0 is the cell volume at T = 0 K, K0 is the bulk modulus, and γ is the Gr¨ uneisen parameter. The internal energy U (T ) is related to the Debye temperature (θD ) as ( U (T ) = 9pkB T

T θD

)3 ∫

θD /T 0

x3 dx, ex − 1

(2)

where, p is the number of atoms in the specimen and hex kB is the Boltzmann constant. The best fit of Vcell vs T using the Eq. (1) is shown in Fig. 1(c), which yields θD ≃ 340 K. B.

DC Magnetization

The upper panel of Fig. 2 displays the temperature dependent dc susceptibility [χ(T ) ≡ M/H] of Mn0.73 Fe0.27 NiGe measured in different applied magnetic fields (H) for three protocols: zero-field-cooled (ZFC), field-cooled-cooling (FCC), and field-cooledwarming (FCW). As the temperature is lowered, χ(T ) increases rapidly, shows a broad peak at around 90 K, and then the FCC and FCW data saturate at very low temperatures. Such a behaviour is reminiscent of a low temperature FM transition. For H = 1 T, a clear thermal hysteresis is visible between the FCC and FCW data around 80 K, characterizing the magnetic transition at Tt ≃ 80 K as first order type. Interestingly, the temperature range of the hysteresis gets reduced and the peak at 90 K gets suppressed as we increase the applied field. A significant bifurcation between the ZFC and FCW data was observed below Tt and the difference between them (∆χ) in the low temperature region gets reduced with increasing H. Such an irreversibility and reduction in ∆χ with increasing H are characteristic features of a frozen SG state below Tt , as also anticipated for x > 0.26 from the reported T − x phase diagram.1 A small kink in χ(T ) has also been reported earlier for x = 0.25 − 0.30 in the low field values.1 Similar peaking behaviour in χ(T ) is also reported for Mn3 Ga0.45 Sn0.55 .21 The first order magnetic transition at Tt ≃ 80 K is indeed consistent with our

(c)

(a)

6.0

I

Intensity (arb. units)

obs

T = 300 K

5.9

a

I

orth

cal

a

hex

Bragg peaks I

4.09

- I

obs

cal

Å

Lattice constants ( )

4.08 3.76

b

orth

3.72

3.68

7.2

c

5.32

c

hex

2

44

48

(d)

156.8

degree

156.0

orth

V

cell

hex

V

cell

76.8

Eq. (1)

20

3

40

Å)

36

5.28

(

T = 15 K

7.0

40

60 2

degree

80

160

ce ll

Intensity (arb. units)

(b)

V

Intensity (arb. units)

orth

76.0

240

T (K)

FIG. 1. Powder XRD patterns of Mn0.73 Fe0.27 NiGe measured at (a) T = 300 K and (b) 15 K along with the Rietveld refinements. The observed and calculated patterns are presented by the symbols and solid lines, respectively. The vertical ticks indicate the Bragg positions. The difference between the observed and calculated intensities is shown by a solid line at the bottom. Inset of (b): Magnified pattern at 15 K. The downward arrows mark the orthorhombic phase. (c) Lattice constants and (d) unit cell volume (Vcell ) as a function of T for both the phases. The solid line in (d) represents the fit using Eq. (1) to hex Vcell (T ) of the hexagonal phase.

XRD analysis where we have observed the co-existence of both the orthorhombic martensite and hexagonal austenite phases. This also implies the existence of competing AFM and FM interactions below Tt and the occurrence of low temperature SG transition is a consequence of that. The inset of Fig. 2 displays the 1/χ vs T for H = 0.1 T. A fit of the high-T data (T ≥ 270 K) by the CurieWeiss (CW) law, [χ = C/(T − θCW ), where C and θCW are the Curie constant and CW temperature, respectively] results the paramagnetic effective moment µeff ≃ 4.4 µB /f.u. and θCW ≃ 176 K. A positive and large value of CW temperature suggests a dominant FM interaction in the compound. To gain further insight, we have also measured the magnetization isotherms [M (H)] at T = 2 K, 100 K, and 300 K which are shown in the lower panel of Fig. 2. The measurement at each temperature was performed after the zero-field-cooling from 300 K. At T = 300 K, M (H) is nearly a straight line, as expected in the PM austen-

ite regime. With lowering temperature, at T = 100 K, it forms a pronounce curvature and the measurements during increasing and decreasing magnetic field do not show any hysteresis. On the other hand, the isotherm at T = 2 K shows the virgin curve (first increasing field cycle) lying outside the envelope curve (second increasing field cycle) while cycling H. Such a magnetic field irreversibility further supports the complex interplay of AFM and FM interactions at low temperatures. The lack of complete saturation even at H = 9 T also reflects the coexistence of a fraction of the AFM phase along with the majority FM phase.11,18 It is to be noted that the parent compound is antiferromagnetic and Fe doing induces ferromagnetism in the system. Therefore, the partially Fe doped samples are expected to have both FM and AFM interactions in it. It also shows a small hysteresis with a coercive field of ∼ 1 kOe, typical for ferromagnets.

80

H =1T

80

0.20

1/

64

CW fit

40

20

0 0

100

200

300

T (K)

C

p

0.04

32

240

/

T (K) 160

p

0.8

0.04

2 K T =

0.03

100

K

16

/f.u.)

0

20

40

60

80

100

T2 (K)

0.4 300 K

0

0.0

0

50

100

M (

B

fit

p

80

C /T

C T

0

0.05

48

2

8T

(J/mol K )

3

5T

60

(J/mol K)

-1

(cm /mol)

ZF C

3T

1/

(cm /mol)

0.08

C C F

0.12

W C F

0.16

H = 0.1 T

-0.4

200

250

300

FIG. 3. Temperature dependent heat capacity Cp (T ) measured in zero field. Inset: Cp /T vs T 2 in the low temperatures regime and the solid line is the fit, as discussed in the text.

-0.8 -6

150

T (K)

-4

-2

0

2

4

6

H (T)

FIG. 2. Upper panel: Temperature dependent dc susceptibility χ(T ) of Mn0.73 Fe0.27 NiGe in different applied fields (H) for zero-field-cooled (ZFC), field-cooled-cooling (FCC), and field-cooled-warming (FCW) conditions. Inset: Inverse susceptibility (1/χ) as a function of temperature for H = 0.1 T in FCW condition and the solid line represents the CW fit for T ≥ 270 K. Lower panel: Magnetic isotherms M (H) measured at different temperatures.

C.

tronic contribution and β represents the lattice contribution. The best fit of the data for T < 10 K yields γ ≃ 25 mJ/mol K2 and β ≃ 0.23 mJ/mol K4 . This value of γ is found to be quite large. It is worth mentioning that a large value of γ is reported for several SG and weak FM systems, the origin of which is not yet understood.23,24 The Debye temperature (θD ) is estimated from the value ( 4 )1/3 of β by using the standard expression θD = 12π5βmR . The obtained θD ≃ 300 K is found to be close to the value hex estimated from the Vcell vs T analysis.

Heat Capacity

Temperature dependent Cp (T ) measured in the absence of magnetic field is presented in Fig. 3. We did not see any anomaly associated with the MT which could be due to a very small change in magnetic entropy involved at Tt and/or the ordering temperature is so high that the large phonon contribution masks the associated peak. In certain cases, inhomogeneity in the polycrystalline sample leads to a distribution of the transition temperature. This results in either the disappearance of the peak or a broad peak in the Cp (T ) data. The value of Cp at T = 300 K is about 80 J/mol-K which is close to the expected classical Dulong-Petit value CV = 3mR = 74.8 J/mol K, where R is the gas constant and m is the number of atoms per formula unit.22 In the inset of Fig. 3, Cp /T vs T 2 is plotted in the low temperature regime. It shows almost a linear behavior which was fitted by the equation, Cp (T ) = γT + βT 3 . Here, γ is the Sommerfeld coefficient which represents the elec-

D.

AC Susceptibility

In order to further study the low temperature glassy behavior, the frequency (ν) dependent ac susceptibility was measured in the ZFC condition at a small ac field. The real (χ′ ) and imaginary (χ′′ ) parts of the ac susceptibility as a function of temperature are plotted in Fig. 4(a) and (b), respectively. The χ′ (T ) exhibits a broad peak at around 60.5 K similar to the dc χ(T ) which is found to be frequency dependent and the height of the peak decreases with increasing ν. Similar to χ′ (T ), χ′′ (T ) also exhibits a pronounced maximum at ∼ 42.5 K which shows frequency dependency. In addition, another frequency independent broad hump is exhibited by χ′′ (T ) at around 80 K which is close to the first order PM-FM transition, observed in dc χ(T ). Such a frequency independent behaviour is reported for first-order MT in Ni2 Mn1.36 Sn0.64 and Mn2 Ni1.6 Sn0.4 .25,26 The shifting of peak position at

(a)

-2

50 Hz

(a)

(b) data

100 Hz

fit

10 kHz

fit

( )

1 kHz

-8 ln ( )

log

-4

data

3

62

f

'

(K)

0.8

-10

64

fit

T

(cm /mol)

10

1.2

-6

data

-1.4 60 2

0.4

3

-1.2

log

10

4

-1.0

(T / T f

g

- 1)

0.12

0.16

1/(T -T ) f

0

log

10

0.4

(b) 50 Hz 100 Hz 1 kHz

FIG. 5. (a) Plot of log10 (τ ) vs log10 (Tf /Tg − 1) obtained from the χ′ (T ) data. The solid line represents the fit using Eq. (4). (b) ln τ vs 1/(Tf − T0 ) extracted from the χ′ (T ) data. The solid line is the fit using Eq. (6).

0.2

data

(δTf = 0.086),23 and Mn3 Ga0.45 Sn0.55 (δTf = 0.0797).21 This classifies our system as a cluster SG type. In SG systems, the standard critical slowing down (power law) model can be used to describe the frequency dependent Tf where the characteristic relaxation time (τ = 1/2πν) is expected to diverge at a critical temperature. The form of the behaviour is given by27,30 ( )−zν ′ Tf − Tg ∗ τ =τ , (4) Tg

52

T

f

48

(K)

fit

''

3

(cm /mol)

10 kHz

44

2

3

4

log

10

0.0 40

80

T (K)

′

′′

FIG. 4. (a) Real (χ ) and (b) imaginary (χ ) part of the ac susceptibility measured at different frequencies (ν). The arrows point to the shifting of the freezing temperature. Insets: Tf vs log10 ν obtained from the respective data and the solid lines are the fits using Eq. (3).

∼ 60.5 K in χ′ (T ) and ∼ 43.5 K in χ′′ (T ) correspond to spin freezing temperature Tf , which is a common characteristic feature of a glass transition. The relative shift of freezing temperature Tf with ν, known as the Mydosh parameter (δTf ), is often used to classify different SG systems which is defined as27–29 δTf =

∆Tf . Tf ∆(log10 ν)

(3)

In the insets of Fig. 4(a) and (b), Tf is plotted as a function of log10 (ν), obtained from the χ′ (T ) and χ′′ data, respectively. Both the data sets exhibit a desired linear behavior and from the fit, δTf is estimated to be ∼ 0.03 and ∼ 0.09, respectively. These values are almost one order of magnitude larger than the ones reported for most canonical SG systems [e.g. AuMn (δTf = 0.0045)28 and CuMn (δTf = 0.005)27 ] and less than that of superparamagnets (e.g. α-[Ho2 O3 (B2 O3 )], δTf ≃ 0.28).27 In fact, these values are comparable to the ones reported for cluster SG systems e.g. Cr0.5 Fe0.5 Ga (δTf = 0.017),20 PrRhSn3

where τ ∗ is the relaxation time of a single spin flip, Tg is the freezing temperature as ν approaches zero, z is the critical exponent, and ν ′ is the critical′ exponent of the correlation length ζ = (Tf /Tg − 1)−ν . Here onwards, for the analysis of the dynamic behaviour we have only used the χ′ (T ) data. Here, we have taken Tg ≃ 58 K which is the y-intercept of the linear fit of Tf vs log10 ν plot shown in the inset of Fig. 4(a). In Fig. 5(a), we have plotted log10 (τ ) vs log10 (Tf /Tg − 1) which shows a linear behaviour and a straight line fit results τ ∗ = (1.5± 0.5) × 10−10 s and zν ′ = 5.5 ± 0.2. The dynamic scaling implying the divergence of the relaxation time indicates a phase transition from PM to SG in Mn0.73 Fe0.27 NiGe. The obtained value of τ ∗ falls in-between the range of values expected for conventional SG (10−10 to 10−13 s) and cluster-SG (10−7 to 10−10 s) systems. This suggests that the spin dynamics in the present system occurs in a slow manner, probably due to interacting spin clusters instead of individual spins.23,31,32 For SG systems, the value of zν ′ typically lies between ∼ 4 and ∼ 1220,32,33 . Indeed, our obtained value of zν ′ falls within the range of typical SG systems.20,33–35 To understand the interaction among the magnetic entities, we first undertake the analysis using the Arrhenius law which describes the relaxation of non-interacting spins.31 It can be written as ( ) Ea τ = τ0 exp , (5) kB Tf

where T0 is the VF temperature. For the purpose of fitting, Eq. (6) is rewritten as lnτ = lnτ0 + kB (TEf a−T0 ) . Figure 5(b) presents the variation of lnτ with 1/(Tf − T0 ) over the measured frequency range. To make the fit physically meaningful, the lnτ vs 1/(Tf − T0 ) was plotted by fixing T0 ≃ 54.8 K, obtained from the best fit of the data by Eq. (6). From the slope and intercept of the straight line fit, the parameters are obtained to be Ea /kB = (72.8 ± 1.5) K and τ0 = (1.2 ± 0.6) × 10−8 s. A finite value of T0 and the agreement of our data with the VF law imply a finite interaction among the spins in the alloy. The characteristic relaxation time constant obtained here is again higher than for conventional SGs and indeed falls in the expected range for the typical clusterSG. Further, the VF model also gives information about the coupling strength between the interacting entities of the cluster. For instance, T0 ≪ Ea /kB represents a weak coupling and T0 ≫ Ea /kB represents that the coupling is a strong one.38 For our system, T0 is about 0.75Ea /kB , which is in the intermediate regime indicating a finite interaction among the dynamic entities.35,39 From the above assessment, it is clear that both the laws fit equally well to our Tf vs ν data in the measured frequency regime. The value of T0 obtained from the fit of Eq. (6) is found to be slightly smaller than Tg obtained from the fit using Eq. (4). A similar trend is also reported for other cluster-SG systems.33 On the other hand, the characteristic time constant τ ∗ obtained using the power law is almost two order of magnitude smaller than τ0 obtained using the VF law. Such a difference in characteristic time constant is also reported in many cluster SG systems e.g. Fe2 O3 ,35 Ni doped La1.85 Sr0.15 CuO4 ,33 Cr0.5 Fe0.5 Ga,20 etc. Altogether, our experimental results classify Mn0.73 Fe0.27 NiGe as a cluster SG system. Our observations are quite similar to that reported for Mn3 Ga0.45 Sn0.55 C where Ga-rich and Sn-rich phases coexist. The difference in local structures of Mn atoms that find themselves in Ga-rich or Sn-rich environment is found to be the main reason for the formation of cluster glass.21 In the present study, the parent compound MnNiGe has AFM ground state with AFM Mn-Mn interaction. Upon the substitution of Fe at the Mn site FM is evolved. Due to random substitution, there may remain some Mn-rich phases, which will act as AFM islands.

T

44 0.

1.02 41 30

s,

03 33

=

5 K

53 0. s,

K 10

43 0.

(

3 0.4

, 1 s 270

20 K

1.01

(

) /

K 15

s,

=0)

45 32

Mt Mt

where τ0 has the same physical meaning as τ ∗ and Ea /kB is the activation energy of the relaxation barriers. Although our lnτ varies linearly with 1/Tf (not shown here) over the measured frequency regime, the obtained values of Ea /kB = (4953 ± 102) K and τ0 ≃ 7.7 × 10−39 s are completely unphysical. It means that the dynamics of the system is governed a cooperative character due to inter-cluster interactions.36 In order to include the interaction among the spins and to estimate the activation energy, we employed the phenomenological Vogel-Fulcher (VF) law27,37 ( ) Ea τ = τ0 exp , (6) kB (Tf − T0 )

Eq. (7)

1.00 0

40

80

120

t (min)

FIG. 6. Magnetic relaxation measured under ZFC condition at different temperatures (T = 5 K, 10 K, 15 K, and 20 K) for H = 200 Oe, as discussed in the text. The solid lines are the fits using Eq. (7).

The co-existence of these AFM islands and the Fe-rich FM phase may give rise to conflicting magnetic interactions leading to magnetic frustration and hence cluster glass like states. However, to understand the actual reason for the cluster formation, high resolution experiments like neutron scattering experiments are desirable.

E.

Non-equilibrium Dynamics 1.

Magnetic Relaxation

Magnetic relaxation measurement is also performed as a function of time (t) to explore the metastable behavior in the glassy state. In Fig. 6, magnetization normalized to its value at t = 0 i.e. [M (t)/M (t = 0)] is plotted as a function of t measured in ZFC condition at different temperatures, below Tf . For this measurement, the sample was cooled in zero field from 300 K (PM state) to the measurement temperature and then after a waiting time of tw = 60 s, a magnetic field of 200 Oe was applied. The M was then measured as a function of t for t = 2 hours. It shows a continuous growth of magnetization with time and does not saturate even after 2 hours. This indicates that it takes long time to align the excited spins for an applied field H = 200 Oe along the field direction, as expected in a glassy state. The t dependence of the magnetization curves can be

H

2.

int FCC mem

M

FCW

300

ref

M

3

(G cm /mol)

FCC

M

290

T

int 2

= 5 K

T

int 1

= 12 K

280 0

20

T

Magnetic Memory Effect

M

= 0.02 T

40 (K)

FIG. 7. Magnetization as a function of temperature in FCC condition for H = 200 Oe, as described in the text. The int measurement was interrupted at T int 1 = 12 K and T 2 = 5 K for 2 hours each, indicated by vertical upward arrows.

modeled by the stretched exponential function20,32 [ ( ) ] β t M (t) = M0 − Mg exp − , τ

(7)

where M0 is the intrinsic magnetization, Mg is related to the glassy component of the magnetization, τ is the relaxation time constant, and β is the exponent. This function is typically used to fit the magnetic relaxation data to understand the SG systems.40 For SG systems, β usually ranges from 0 to 1. In this function, when β = 0, M (t) is constant i.e. no relaxation at all while β = 1 implies that the system relaxes with a single time constant, typically observed in ferromagnets. β = 1 also reflects that there exists a uniform energy barrier but when β value is between 0 and 1, it reflects a distribution of energy barriers. The best fit of the data at each temperature is presented by solid lines and the obtained parameters are also mentioned in Fig. 6. From the fit, the value of β is found to vary from 0.43 to 0.53 which are within the range (0 to 1) of different glassy systems reported earlier.27,32,34,41 Moreover, the reduced value of β suggests that the activation takes place against the multiple anisotropic barriers. The values of τ obtained for our system are also comparable with other reports on SG systems.20,42,43

In order to study the memory effect, the dc magnetization was measured under FCC condition and the results are depicted in Fig. 7. For this measurement, we cooled the sample from well above the first order PM-FM transition to 2 K at a constant ramp rate (1 K/min) in an applied field of 200 Oe. The cooling process was interrupted int at T int 1 = 12 K and T 2 = 5 K for a duration tw = 2 hours each. At each interruption temperature, the magnetic field was switched off for the time tw . After the time tw , the same magnetic field was applied once again and the FCC process was continued. The measured M during this process is denoted as M int FCC . It shows step-like features at 12 K and 5 K. After reaching 2 K, the sample was heated continuously in the same field under the same constant ramp rate and M (T ) was recorded upto 50 K which is referred to as M mem FCW . It also exhibits a change of slope at T int = 12 K and T int 1 2 = 5 K. This indicates that the system shows the features at each interruption point in M int FCC , as an attempt to follow the history of the magnetization, revealing the magnetic memory effect. A FCC curve (M ref FCC ) in the same applied field without any interruption is also measured as a reference. To understand the memory effect in further detail, we performed the relaxation measurements for both negative and positive T -cycles. The relaxation behaviour was recorded for the negative T -cycle in both ZFC and FCC conditions and the results are presented in Fig. 8(a) and 8(b), respectively. In the ZFC process, the sample was first cooled from the PM phase to the desired temperature in zero field. At the measurement temperature T = 10 K (below Tf ), a field of 200 Oe was applied and M (t) was recorded for t1 = 70 minutes. During the time t1 , M (t) is found to increase exponentially with t. After that, the sample was cooled down to 5 K in the same applied field and at 5 K M (t) was recorded for t2 = 66 minutes which is found to be almost constant with t. Finally, the sample was warmed back to 10 K in the same applied field and at 10 K M (t) was recorded for t3 = 65 minutes which is also found to vary exponentially with t. In the FCC process, at first, the sample was cooled down to 10 K in H = 200 Oe. At 10 K, the field was switched off and M (t) was measured for t1 = 80 minutes which is found to decay exponentially with t. Then, the sample was cooled down to 5 K in zero field and M (t) was recorded for t2 = 75 minutes which is found to be constant with t. By keeping the zero field, the sample was warmed back to 10 K and M (t) was measured for t3 = 65 minutes which again decays exponentially with t. The M (t) data collected at 10 K during t1 and t3 for ZFC and FCC conditions are put together and are shown in the insets of Fig. 8(a) and 8(b), respectively. We found that the combined curve follows a continuous exponential growth and decay behavior for the ZFC and FCC processes, respectively. It implies that the state of the sample at a particular temperature below Tf is recovered

[a]

2

216

3

220

10 K

5 K

220

t

1

t

2

3

192

M

t

1

3

M

216

t

216 0

70

140

180

t (min)

[d]

168

M

t

1

204 3

192

160 0

70

5 K

140

t (min)

t

t

2

5 K

180

t

3

M

M

1

10 K

3

3

(G. cm /mol)

(G cm /mol)

164

t

164

(G cm /mol)

[b]

168

3

204

t

3

(G. cm /mol)

3

5 K

3

M (G cm /mol)

10 K t

t

1

from that observed during t1 . Thus, positive T -cycling revives the magnetic relaxation process in both ZFC and FCC processes and erases the memory effect.

[c]

5 K

t

(G cm /mol)

10 K

160 10 K

5 K

t

t

1

0

2

100 t (min)

168

10 K t

3

200

0

100

200

t (min)

FIG. 8. Magnetic relaxation measurements in the negative T -cycle in an applied field of H = 200 Oe for (a) ZFC and (b) FCC methods. Insets: M (t) data at 10 K for negative ZFC and FCC T -cycles, respectively along with the fit by Eq. (7). For the positive T -cycle, ZFC and FCC data are shown in (c) and (d), respectively.

back to the same state even after temperature cycling, which represents the memory effect. A stretched exponential function [Eq. (7)] was used to fit these curves which gives β ≃ 0.5. This type of behavior is often observed in the SG systems and is an indication that the memory effect is quite strong in this alloy, below Tf . Similar to the negative T -cycle, the relaxation measurements were also performed for the positive T -cycle in both ZFC and FCC conditions and are shown in Fig. 8(c) and 8(d), respectively. In the ZFC process, at first, the sample was cooled from the PM state to 5 K in zero field. At 5 K, M (t) was measured for t1 = 65 minutes after switching on a magnetic field of H = 200 Oe which shows an exponential increase with t. After that, the sample was heated upto 10 K in the same field and M (t) was measured for t2 = 66 minutes which also shows a exponential increase with t. Finally, the sample was cooled back to 5 K and M (t) was measured for t3 = 64 minutes which is found to be almost constant with t. In the FCC process, the sample was cooled down to 5 K in H = 200 Oe. At 5 K, the same sequence was repeated (as for the ZFC process) after switching off the magnetic field. As one can see in Fig. 8(d), the overall M (t) data follow the similar behaviour as for the ZFC results in Fig. 8(c) but in the opposite direction. It is also evident that there is no continuity in the M (t) data measured during t1 and t3 at 5 K, suggesting that the nature of the magnetic relaxation during t3 is different

The memory effect in SG systems can be explained either by the droplet model or the hierarchical model.44,45 At a particular temperature, there exists a multi-valley spin structure on the free-energy landscape in the hierarchical model, whereas in the droplet model only one spin configuration is favoured. Experimentally, one can distinguish the two models by measuring magnetic relaxation with respect to the temperature cycling, below Tf . In the droplet model, the original spin configuration is restored while in the hierarchical model, the original spin configuration is lost after a positive temperature cycle. Thus, no memory effect observed in the positive T -cycle indicates an asymmetric behavior in magnetic relaxation for both ZFC and FCC processes. This supports the hierarchical model of the relaxation rather than the droplet model. Overall, our magnetic relaxation and magnetic memory effect experiments support the cluster SG behavior of the alloy under investigation.

IV.

CONCLUSIONS

In conclusion, we present a detailed study of structural and magnetic properties of Mn0.73 Fe0.27 NiGe. Temperature dependent XRD measurement show the coexistence of both hexagonal and orthorhombic phases with a phase fraction of 88:12, below 80 K. The temperature dependent dc susceptibility reveals a first order PM to FM transition at Tt ≃ 80 K followed by a SG transition at low temperatures. The low temperature SG behaviour is possibly arising due to the competing FM and AFM interactions coming from the coexisting Fe-rich and Mnrich phases, respectively. A magnetic field irreversibility observed in the magnetic isotherm further supports the complex interplay of AFM and FM interactions at low temperatures. The SG transition is again justified by the ac susceptibility measurements. The relative shift in Tf and the parameters obtained using dynamical scaling laws from the ac susceptibility data pinpoint the cluster SG behaviour of the system. The average activation energy of the energy barriers is calculated to be Ea /kB ≃ 72.8 K. A clean demonstration of the magnetic relaxation and magnetic memory effects is presented, which confirms that the glassy magnetic state is associated with cooperative spin freezing. In the positive T -cycle, a small heating reinitializes the relaxation process making the magnetization unable to restore its initial value. Such an asymmetric response of magnetic relaxation with respect to positive temperature change can be described by the hierarchical model. The Debye temperature obtained from the Cp (T ) data is found to be consistent with that obtained from the Vcell (T ) analysis.

V.

ACKNOWLEDGEMENT

We would like to acknowledge BRNS, India for financial support bearing sanction No.37(3)/14/26/2017BRNS. PB was supported by the Indian Institute of Science Education and Research Thiruvananthapuram postdoctoral programme.

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