Experimental and Theoretical Investigations of Fe-Doped Hexagonal MnNiGe.

We report a comprehensive investigation of MnNi0.7Fe0.3Ge Heusler alloy to explore its magnetic, caloric, and electrical transport properties. The alloy undergoes a ferromagnetic transition across T C ∼ 212 K and a weak-antiferromagnetic transition across T t ∼ 180 K followed by a spin-glass transition below T f ∼ 51.85 K. A second-order phase transition across T C with mixed short and long-range magnetic interactions is confirmed through the critical exponent study and universal scaling of magnetic entropy and magnetoresistance. A weak first-order phase transition is evident across T t from magnetization and specific heat data. The frequency dependent cusp in χAC(T) along with the absence of a clear magnetic transition in specific heat C(T) and resistivity ρ(T) establish the spin glass behavior below T f. Mixed ferromagnetic and antiferromagnetic interactions with dominant ferromagnetic coupling, as revealed by density functional calculations, are experimentally evident from the large positive Weiss temperature, magnetic saturation, and negative magnetic-entropy and magnetoresistance.

Introduction

The physics of phase transitions is important to understand the properties of compounds. Exploration of the critical phenomena across the transition temperature unveils the nature of the phase transition. A second-order phase transition (SOPT) from paramagnetic (PM) to ferromagnetic (FM) state is characterized by a continuous variation of the spontaneous magnetization, an order parameter. Nevertheless, the magnetic properties of the compounds are governed by the exchange interactions among the spins. FM compounds with a large change in magnetic entropy and adiabatic temperature Tad (change in temperature of the system under adiabatic condition without exchange of heat) across the transition temperature near room temperature are potential candidates for technological applications such as magnetic refrigeration. In recent times, Heusler compounds have become materials of topical interest for their multifunctional and peculiar properties such as topological insulators,[1,2] Weyl semimetals,[3] spin-gapless semiconductors,[4] shape memory effect,[5,6] half-metals,[7−9] exchange bias, large magnetoresistance, and magnetocaloric effect.[10,11]

Heusler alloys with 1:1:1 stoichiometry have been receiving a great deal of attention due to their tunable magnetic properties with substitution, magnetic field, and hydrostatic pressure. Substitution/disorder driven suppression of a first-order magneto-structural transition in 1:1:1 stoichiometric Heusler alloy MnNiGe was reported.[12,13] It undergoes a structural transition at Tt = 470 K from high-temperature Ni2In-type hexagonal austenite to a TiNiSi-type orthorhombic martensite structure.[14,15] In addition, it is reported to order antiferromagnetically at = 346 K in the martensitic state, followed by a ferromagnetic phase below = 205 K. The magnetic properties of MnNiGe have been reported to be substitution and site specific. A gradual replacement of Fe at Mn site, Mn1–FeNiGe, has suppressed Tt down to 84 K at x = 0.26. The alloys with x > 0.26 crystallize in a Ni2In-type hexagonal structure with a glassy phase at low temperatures.[12] On the other hand, in MnNi1–FeGe where Ni is gradually replaced by Fe, the systems remain in the ferromagnetic austenite phase for x > 0.3.[12] MnNi0.7Fe0.3Ge is reported to undergo a structural transition at Tt = 189 K (below which a small thermal hysteresis was noticed) and austenite ferromagnetic transition at = 211 K.[12] Recently, MnNi0.8Fe0.2Ge has been investigated for its successive magnetic transitions with a field-induced conversion of the low-temperature magnetic state to the FM state.[16] Nevertheless, the universality class of the high-T SOPT and transport behavior of MnNi0.7Fe0.3Ge are unclear.

In the present study, we report on the universality class and critical magnetic behavior of MnNi0.7Fe0.3Ge using the combined results of magnetization and specific heat, combined with ab initio calculations. The study has been focused on understanding the phenomena across and Tt. Our results reveal that the alloy crystallizes in a Ni2In-type hexagonal structure at room temperature and undergoes a second-order phase transition at TC ∼ 212.5 K. A narrow thermal hysteresis below 180 K disappears in a field of ∼5 kOe above which the system behaves ferromagnetically down to 2 K. Mixed (short- and long-range) interactions are suggested by the critical exponents. The Sommerfeld coefficient of electronic specific heat (γel = 15.9 ± 0.5 J·mol–1.K–2) and electrical resistivity confirm the metallic character. The self-consistency of the critical exponents, extracted using magnetization, is established through the analysis of the magnetocaloric and the magnetoresistance methods. A second-order phase transition across TC ∼ 212 K is confirmed through universal scaling of the magnetic-entropy and magnetoresistance data. A cluster-glass type behavior with weakly coupled magnetic clusters is reported below 50 K. The dual transitions can be carefully manifested/tuned to achieve a table-like magneto-caloric effect for magnetic refrigeration. The effect of the magnetic field on the electrical resistivity across the magnetic transition temperature can be utilized in magneto-resistive applications.

Methods

A polycrystalline MnNi0.7Fe0.3Ge has been prepared by the arc-melting method. The constituent elements Mn, Ni, Fe, and Ge (of purity better than 99.999%) were taken in a stoichiometric ratio and were loaded into a copper hearth. The elements were melted under a continuous supply of argon gas. The ingot was melted several times by flipping each time. X-ray diffraction pattern at room temperature is collected on a powder specimen using PANalytic X’Pert Pro X-ray diffractometer with Cu–Kα radiation. Energy-dispersive X-ray (EDAX) spectroscopy measurements (not shown here) were carried out using JSM-7600F. The atomic percentages of Mn, Ni, Fe, and Ge are in good agreement with the originally taken stoichiometric ratio within the experimental error. Magnetization was measured with the help of a commercial superconducting quantum interference device-vibrating sample magnetometer (SQUID-VSM) under zero-field cooling (ZFC), field-cooled cooling (FCC), and field-cooled warming (FCW) conditions. Under ZFC conditions, magnetization was recorded during warming (under the ambiance of required set field) after the sample was cooled to 2 K from 400 K in zero-field. Under FCC conditions, magnetization was recorded while the sample was cooled in a finite magnetic field. Consequently, under FCW conditions, magnetization was recorded under warming, without switching off magnetic field. Isothermal magnetization versus field curves were measured, under ZFC condition, by ramping the magnetic field. Electrical resistivity was measured using a standard dc-four probe method using a 9 T-Physical Property Measurement System (PPMS). The specific heat was measured using relaxation calorimetry with the help of commercial 14 T PPMS.

The electronic structure was calculated in the Quantum ESPRESSO software[17] using the scalar-relativistic potentials in a local density approximation of Perdew–Zunger-type, included in the standard QE library. Wave functions were decomposed into plane waves, and interactions between ions and valence electrons were taken into account of method of attached plane waves (PAW). To model the concentration of Fe closest to the experiments, we constructed a supercell with 4 f.u. of MnNiGe with 1 Ni ion substituted by Fe that finally resulted in the MnNi0.75Fe0.25Ge composition. For the sufficient convergence in our first-principles calculations, the energy cutoff, i.e., energetic limit, 60 Ry, was taken. A k-mesh of 8 × 8 × 8 k-points was used for the tetrahedron method integration in a reciprocal space.

Results

Figure a shows the room-temperature X-ray diffraction pattern of MnNi0.7Fe0.3Ge, along with Rietveld refinement using the FullProf suite.[18] The alloy crystallizes in a Ni2In-type hexagonal structure with the P63/mmc space group. The lattice parameters are a = b = 4.102 Å and c = 5.368 Å. The obtained structure is drawn using visualization for electronic and structure analysis (VESTA),[19] as shown in Figure b. Mn occupies the 2a (0, 0, 0) position and Ni/Fe share the 2d (1/3, 2/3, 3/4) positions, while Ge occupies the 2c (1/3, 2/3, 1/4) position. The shortest Mn–Mn distance along the c-axis is 2.6839(1) Å.

Figure 1

(a) Refined X-ray diffraction pattern of MnNi0.7Fe0.3Ge using the P63/mmc space group. The global χ2 = 2.31 with Bragg factor = 14.6 and RF-factor = 15.2. (b) Crystal structure with lattice parameters a = b = 4.102 Å and c = 5.368 Å. Ni atom shares 30% of its occupancy with Fe. Mn–Mn nearest distance along the c-axis is 2.684 Å. (c) Temperature dependence of magnetization under 100 Oe in ZFC, FCC, and FCW processes. ZFC and FCW curves are separated from each other below Tt = 180 K. Inset: A narrow thermal hysteresis is noticed between FCC and FCW in the temperature range 75–215 K. (d) Curie–Weiss fit of the inverse susceptibility. The Weiss temperature and the effective magnetic moment are found to be (233.64 ± 0.11) K and μeff = (4.981 ± 0.003) μB/f.u., respectively.

The temperature dependence of the magnetization M(T) curve under the influence of 100 Oe is shown in Figure c. M(T) rises sharply below 250 K before it takes a down turn around 180 K along with a separation between the ZFC and FCW magnetization curves. In addition, a narrow thermal hysteresis between FCC and FCW is observed in the temperature range 75 to 215 K, as shown in the inset of Figure c. Further, M(T) exhibits a down turn below Tf, noted as a freezing temperature. In 100 Oe, the inverse susceptibility χ–1(T) is fit, as shown in Figure d, to the Curie–Weiss law using eq where C is the Curie–Weiss constant from which the effective magnetic moment is calculated as (kB is the Boltzmann constant and NA is the Avogadro number). Thus, the obtained effective magnetic moment is μeff = 4.981 ± 0.003 μB/f.u. A positive and large Weiss temperature θW = (233.64 ± 0.11) K indicates the prevailing ferromagnetic exchange correlations above the transition temperature. M(T) in 1 kOe is shown in Figure a. The bifurcation between ZFC and FCW is found to decrease with increasing H. However, the kink at Tt and the thermal hysteresis, observed below 180 K in 100 Oe, shift toward low-temperature with increasing H up to 3 kOe. In H ≥ 5 kOe, the kink at Tt is smeared out with simultaneous vanishing of thermal hysteresis. Such a phenomenon resembles the field-induced weak-AFM to FM transition. Shown in Figure b are M(T) curves, measured in constant magnetic fields ranging from 100 Oe to 50 kOe.

Figure 2

(a) M(T) measured in 1 kOe in ZFC, FCC, and FCW protocols. Reduced bifurcation of ZFC and FCW curves is noticed when compared to 100 Oe. (b) M(T) measured under a few representative magnetic fields. (c) Isothermal magnetization versus field at 2 K. M increases sharply with H with a saturation magnetization of Ms ∼ 2.6 μB/f.u. at a saturation field of Hs ∼ 6.6 kOe. (d) Isothermal magnetization versus field curves at a few representative temperatures. In the paramagnetic region (at 300 K), M develops linearly with H.

Figure c shows an isothermal magnetization versus field M(H) up to 70 kOe, recorded at 2 K. M(H) is measured in five quadrants (0 → 70 kOe → 0 kOe → −70 kOe → 0 kOe → 70 kOe). Remnant magnetization MR is zero, indicative of soft ferromagnetic behavior which is supported by zero coercive field Hc. The saturation magnetization Ms is about 2.6 μB/f.u. Figure d shows the isothermal M(H) curves, measured at a few selected temperatures. As the temperature is increased, the linearity from the high-field region is extended to low-fields. In the paramagnetic state (at 300 K), M linearly increases with H.

Discussion

Critical Behavior across TC

In order to understand the magnetism of MnNi0.7Fe0.3Ge i.e. whether localized or itinerant, Rhodes–Wohlfarth (RW) ratio qc/qs[20,21] is calculated, where qc is the number of magnetic carriers per atom and qs is the saturation magnetic moment. qs = 2.6 μB/f.u. for the present case. qc = 4.08 μB/f.u. is estimated from the effective magnetic moment as μeff = . The RW ratio is obtained as qc/qs > 1, indicating an itinerant magnetic behavior of the alloy. Further, with an aim of realizing the universality class and the type of magnetic interactions in MnNi0.7Fe0.3Ge, a critical study has been carried out with the help of isothermal magnetization curves. A set of M(H) curves were measured in the temperature range 201–209 by 1 K difference, 210–220 by 0.5 K difference, and 221–230 K by 1 K difference, as shown in Figure a. In regard to the SOPT from the PM to the FM state, the spontaneous magnetization Ms below the critical transition temperature TC, the inverse susceptibility χ–1 above TC, and the isothermal magnetization at TC follow power-laws given by eqs , 3, and 4, respectively[22,23]where t = 1 – T/TC is the reduced temperature and the critical exponents associated with Ms, χ0, and TC are β, γ, and δ, respectively, while Ms0, χ00 ,and D are the critical amplitudes. Arrott plots[24] with mean-field theory exponents (β = 0.5 and γ = 1.0), i.e., M2 versus H/M, are shown in Figure b. In these isotherms, the downward concave curvature of the Arrott plot in high fields hints at SOPT, following Banerjee’s criterion.[25] To estimate the correct critical exponents, we have used the modified Arrott plot method (MAP) using Arrott and Noakes magnetic equation of state, eq (26)where c1 and c2 are constants. Different universal magnetic behaviors such as 3D-Heisenberg (β = 0.365, γ = 1.386), 3D-Ising model (β = 0.325, γ = 1.24),[27] and tricritical mean field theory (β = 0.25, γ = 1.0)[28] were tested by plottng M1/β versus (H/M)1/γ curves (not shown here).

Figure 3

(a) Isothermal magnetization versus curves of MnNi0.7Fe0.3Ge at a few selected temperatures in the temperature range 201–230 K. In particular, M(H) curves are measured with 0.5 K difference in the vicinity of critical transition. (b) Arrott plot M2 versus H/M with mean-field theory exponents (β = 0.5 and γ = 1). The positive slope of the high-field curves indicate the second order phase transition. (c) Modified Arrott plot M1/β versus (H/M)1/γ with parallel set of lines with critical exponents β = 0.395 and γ = 1.381. Critical transition isotherm passing through origin is shown in red and the linear fit is shown in dashed lines. (d) M(H) at 212.5 K (≡ TC), a power-law fit (shown in solid line) using eq yields an exponent δ = 4.176 ± 0.011.

Various trials were made to obtain the correct critical exponents by taking initial values of β = 0.365 and γ = 1.386. M(H) curves were subjected to the demagnetization correction Heff = Happlied – NDM. Every time, the newly obtained exponents are validated for the sufficient condition using eq and checked that the modified isotherm of transition temperature passes through the origin (M1/β = 0, (H/M)1/γ = 0). After a rigorous exercise, a set of parallel isotherms were obtained, satisfying eq with β = 0.395 and γ = 1.381, for which M1/β versus (H/M)1/γ curves are shown in Figure c. Ms and are extracted from the intercepts of the M1/β and (H/M)1/γ axis, respectively. Thus, obtained Ms(T) and , shown in Figure a,b, are fit using the respective eqs and 3. The obtained critical exponents and transition temperatures through the MAP method are TC = (212.51 ± 0.14) K and β = 0.315 ± 0.076; TC = (212.57 ± 0.10) K, γ = 1.327 ± 0.045. Further, more accurate exponents are obtained through the Kouvel–Fisher (KF) method. The plots of Ms[1/(dMs/dT)] and χ–1[1/(dχ–1/dT)] as a function of temperature are shown in Figure c,d. The inverse slopes of the linearly fit curves give the exponents β = 0.409 ± 0.002 and γ = 1.293 ± 0.019 with critical temperatures TC = (212.45 ± 0.23) K and TC = (212.63 ± 0.17) K, respectively. Using Widom’s relation,[29,30] δ = 1 + γ/β, the estimated δ values through MAP and KF methods are δ = 5.212 ± 1.162 and 4.161 ± 0.062, respectively, which are in good agreement with δ (= 4.176 ± 0.011) directly obtained through the critical isotherm fit using eq , as shown in Figure d. The critical exponents do not straight away indicate a single universality class but closely resemble 3D-Heisenberg and 3D-Ising models. The exchange interaction J(r), where r is the distance of interaction, depends on the spatial dimensionality d and the length of interaction σ through a relation J(r) ∼ r–(σ+. σ can be calculated from γ using eq (31)where Δσ = (σ – d/2) and G(d/2) = 3 – (d2/16). For the present compound, with d = 3, σ ∼ 1.75 is obtained and exchange interaction varies as J(r) ∼ r–4.75 which falls in between the ranges for Mean-field model (r–4.5, σ ≤ 3/2) and 3D Heisenberg model (r–5, σ ≥ 2). This indicates the mixed exchange interactions of long-range and short-range MnNi0.7Fe0.3Ge. Nevertheless, the closeness of the critical exponents to other models 3D-Ising or 3D-XY point out anisotropic exchange interactions. On the other hand, Pinninti et al. reported a single magnetic transition and 3D-Heisenberg universality with short-range magnetic interactions in MnCo0.7Fe0.3Ge[32] and enhancement of transition temperature with Fe substitution in Mn0.7Fe0.3Co0.7Fe0.3Ge.[33]

Figure 4

(a) Temperature dependence of spontaneous magnetization Ms. A critical exponent fit of the data using eq produces β = 0.315 ± 0.076 and TC = (212.51 ± 0.14) K. (b) Inverse susceptibility 1/χ0(T). γ = 1.327 ± 0.10 and TC = (212.57 ± 0.045) K are obtained by fitting the data using eq . (c) Kouvel–Fisher plot of Ms(T) yielding the exponent β = 0.409 ± 0.002 and critical temperature TC = (212.45 ± 0.23) K. (d) Kouvel-Fisher fit of 1/χ0(T) with γ = 1.293 ± 0.019 and TC = (212.63 ± 0.17) K.

Spin-Glass Behavior below 52 K

Real (χ′) and imaginary parts (χ″) of AC-susceptibility χAC are shown in parts a and b, respectively, of Figure . The measurements were carried out in an AC-drive field of HAC = 5 Oe and HDC = 0 Oe under the effect of a few selected frequencies ν = 1, 47, 97, 197, 297, 397, and 497 Hz. The temperature variation of dχ′/dT (not shown here) exhibits a dip around 213 K, followed by a peak around 180 K, which are in good agreement with TC and Tt. The dip around 213 K is found to be frequency independent, indicating the long-range magnetic order. However, a dispersion in χ′(T) is visible below Tt. In ν = 1 Hz, dχ′/dT exhibits a peak around the freezing temperature Tf ∼ 51.85 K which shifts toward high temperatures with increasing frequency, indicating short-range correlations among the spins. In order to understand these short-range correlations, Mydosh parameter which represents the relative shift of a freezing temperature is estimated as ϕ = ΔTf/TfΔ[ log10ν] where ΔTf = and Δ[ log10ν] = log10ν2 – log10ν1, with ν1 = 1 Hz and ν2 = 497 Hz. For the present compound ϕ ∼ 0.12, which is larger than ϕ reported for cluster spin-glasses and matches with ϕ reported for superparamagnetic systems (ϕ ∈ [0.10, 0.13]).[34] Further, relaxation time is obtained with the help of relation between Tf and ν for the dynamical slowing down of spin fluctuations[34,35] above the glass transition temperature of a spin glass, i.e., τ = , where τ = 1/ν, τ0 is the single spin-flip relaxation time, and zν′ is an exponent. Figure c shows a linear fit of log10τ = log10τ0 – zν′log10(Tf/Tg – 1). The currently obtained value of zν′ (= 3.67 ± 0.05) is close to that reported for spin-glass systems.[34] However, τ0 ∼ 2.67× 10–5 s is 2 orders of magnitude higher than that reported for cluster spin-glass systems (10–7–10–10 s).[36]Figure d shows a linear fit of Tf = T0 – (Ea/kB)[1/ ln(ν/ν0)] which is a Vogel–Fulcher law, a modified Arrhenius relation,[34,37] where Ea is the energy barrier arising from the anisotropy and volume of the particle (in case of nanosystems), T0 is the characteristic temperature, and kB is the Boltzmann constant (8.617 × 10–5 eV·K–1). The Vogel–Fulcher fit results are T0 = (46.45 ± 0.36) K and Ea/kB = (92.71 ± 1.82) K. Non-zero T0(38]δTTh = 0.1 suggests the cluster-glass nature (δTTh ∈ [0.05, 0.5]). Although τ0 is relatively large, it can be deduced from the results of AC susceptibility that MnNi0.7Fe0.3Ge behaves as a cluster spin-glass with weak coupling among the clusters, below 50 K.

Figure 5

(a, b) and measured in labeled frequencies under HAC = 5 Oe and HDC = 0 Oe. The curves show a peak at 213 K along with a hump around 180 K. Below about 52 K, dispersion in with a shift in Tf toward high-T. (c) Fit of critical dynamical slowing in relation to Tf(ν) which gives a relaxation time τ0 ∼ 2.7× 10–5 s. (d) Fit of Vogel–Fulcher law which yields a characteristic temperature T0 = (46.45 ± 0.36) K and Ea/kB = (92.71 ± 1.82) K.

Density Functional Calculations

To theoretically calculate the electronic structure and magnetic properties of the experimental MnNi0.7Fe0.3Ge composition, a supercell comprising 4 f.u. was taken with 1 Ni ion substituted by Fe; this resulted in the very close composition MnNi0.75Fe0.25Ge. For this setup, the calculations were carried out for ferromagnetic and different antiferromagnetic configurations of the Mn and Fe magnetic moments. The ferromagnetic solution was found to be the most stable with the total energy of −3235.1737 Ry per supercell. Other initial AFM configurations converged to the ferrimagnetic solution with 0.13 μB/f.u. and have a total energy 2.5 mRy (34 meV) higher.

This calculated FM ordering of the Mn and Fe magnetic moments has a total magnetic moment of 2.8 μB/f.u., including 2.7 and 2.9 μB per each of the two Mn ions, 0.8 μB/Fe, 0.1 μB/Ni, and −0.2 μB/Ge. The calculated FM total magnetic moment 2.8 μB/f.u. is very close to the saturation magnetic moment 2.6 μB/f.u. obtained from the experimental measurements reported above. In ref (39), we reported the calculated exchange interaction parameters for MnNiGe with the strong AFM nearest neighbor Mn ions JNN = 705 K coupling between the Mn ions, much larger than the FM coupling JNN = −302 K and AFM JNNN = 67 K. In MnNi0.7Fe0.3Ge, due to the presence of Fe in the Ni positions, the FM coupling JNN = −881 K between the Mn ions becomes large and dominates in the Mn subsystem over the AFM coupling JNN = 299 K and JNNN = 45 K. The moderate magnetic moment of the Fe ion causes much smaller values of the exchange coupling with the highest value between Fe and the nearest Mn ion as FM JNN = −95 K. Thus, the FM and smaller AFM couplings among the Mn ions are determining the magnetic properties of MnNi0.7Fe0.3Ge.

In Figure , the first-principles electronic structure for MnNi1–FeGe (x = 0.25) is shown corresponding to the ferromagnetic arrangement of the Mn and Fe magnetic moments. The partial densities are plotted for the total and Fe-3d states on the highest panel, then the Mn and Ni-3d states, and finally, the Ge-4p and Ge-4s states. In the calculated total and partial densities of states (DOS), the strongly spin-polarized Mn-3d contribute from −5 to +5 eV with strong peaks near −2.8, −1.1, and +1.0 eV. Notice the two different types of the Mn ions (plotted as Mn and Mn1) due to the presence of Fe. The Ni and Ge ions also subdivided into two types; this difference in DOS and magnetic moments is negligible. The Ni states are almost nonmagnetic with deviating densities of states mostly below the Fermi energy (EF) in both spin projections with the peaks from −5.0 to −0.5 eV below EF. The selected MnNi0.75Fe0.25Ge supercell is very close to the experimental composition. In order to check exact electronic concentration for Fe with x = 0.30, we added the rigid band approximation line in Figure , and it came very close to the Fermi energy and accounts for the 0.05 difference in x.

Figure 6

Calculated total and partial densities of states for MnNi1–FeGe (x = 0.25) given for two spin projections (↑ and ↓). The solid orange line close to the Fermi energy (EF = 0 eV) corresponds to the rigid band shift for x = 0.30.

Specific Heat and Resistivity

Specific heat C(T), measured in the presence of a few selected magnetic fields, is shown in Figure a. It exhibits two successive peaks at TC and Tt which are in agreement with the magnetization data (see Table ). The peak at TC is broad and indicates a second-order phase transition from PM to FM, while a relatively sharp peak at Tt gives a hint of first-order phase transition. In addition, TC gradually shifts toward high-T in applied magnetic fields, whereas Tt is found to be independent of H up to 5 kOe. Nevertheless, in higher fields (typically H > 5 kOe) C(Tt) gets smeared out with a gradual shift of Tt toward high-T, implying a field-induced transition from a weak-AFM to FM state. This observation is in good agreement with a tiny/narrowed thermal hysteresis between FCC and FCW curves across Tt. Except for a strong FOPT, a tiny thermal hysteresis (weak-FOPT) cannot be traced out by heating and cooling curves of specific heat using commercial PPMS.[40] Zero-field C versus T is shown in the inset of Figure a. The data below 10 K is fit to the equation C = γelT + βT3 to estimate the Sommerfeld parameter for the electronic contribution of specific heat (γel) and the Debye temperature (θD). Thus, obtained γel and (where p is the number of atoms in a formula unit) are (3.86 ± 0.1) mJ mol–1·K–2 and (499 ± 5) K, respectively. γel indicates the metallic nature of the alloy and the density of states at the Fermi level are found to be about 1.64 states/eV/f.u. The absence of a clear transition at Tf in C(T) along with a frequency dependent cusp in χAC(T) indicates the glassy behavior.

Figure 7

(a) Specific heat as a function of temperature from 160 to 240 K, measured in a few selected magnetic fields. TC and Tt are shown by arrows. Inset: Zero-field data; a fit of C(T) = γelT + βT3 below 10 K with γel = (3.86 ± 0.1) mJ mol–1·K–2 and θD = (499 ± 5) K and the specific heat data from 2 to 70 K. (b) Change in magnetic entropy under the effect of labeled magnetic fields. It is negative and exhibits two dips, respectively, at TC and Tt. (c) Magnetic field dependence of is fit to a power-law using eq , yielding an exponent n = 0.559 ± 0.012. (d) RCP versus magnetic field. A power-law fit using eq yields an exponent δ = 4.852 ± 0.465.

Table 1

List of Transition Temperatures Inferred from the Temperature Dependence of Magnetization, Specific Heat, and Resistivity Data

temp	magnetization	specific heat	resistivity
TC (K)	212	210	210
Tt (K)	180	178.5	180
Tf (K)	51.85	no feature	no feature

Figure d shows the temperature variation of magnetic entropy ΔSmag (referred to MCE hereafter) which is estimated using eq . Conventionally, MCE is negative for a ferromagnet. For MnNi0.7Fe0.3Ge, MCE shows two dips, respectively, TC and Tt. Negative MCE around Tt suggests the dominant ferromagnetic interactions in the alloy. An absolute MCE value at TC, in an applied of 70 kOe, is found to be 0.5 J·kg–1·K–1.Here, RCP is the relative cooling power defined as RCP = . Though MCE and RCP values of MnNi0.7Fe0.3Ge are not comparable to that of some of the prominent MCE materials,[41,42] magnetically distinct dual transitions (one at high-T) with a temperature difference of about 32 K can be tuned (by hydrostatic pressure or substitution) to achieve a table-like ΔSmag(T). Parts c and d of Figure show the field dependence of and RCP, which are fit to the power-law dependences given in eqs and 8b, respectively. The critical exponents, obtained from MCE plots using the relations given by eqs and 8d,[43,44] are β = 0.331 ± 0.086, γ = 1.186 ± 0.322, and δ = 4.852 ± 0.465.

Figure a shows the temperature dependence of resistivity ρ(T), measured in a few representative magnetic fields. The metallic character of the samples is evident from the positive slope of ρ(T). As shown in the inset of Figure a, dρ/dT exhibits two peaks respectively at TC and Tt which are in agreement with that of obtained from magnetization and specific heat (see Table ). Magnetoresistance, MR = [ρ(H) – ρ(0)]/ρ(0), is shown in Figure b. Under an applied field of 90 kOe, absolute MR is found to be about 6.0%. It is negative arising from the suppression of magnetic fluctuations by the application of field. The dip temperature Td increases with application of magnetic fields, as expected for a ferromagnet. In addition, there is a minor kink at Tt = 180 K. Parts c and d of Figure show the magnetic field variation of and , respectively. A similar method, described to extract the exponents from MCE using eqs and 8b, is followed. Thus, extracted critical exponents, β = 0.392 ± 0.130, γ = 1.779 ± 0.672, and δ = 5.545 ± 1.239, are in good agreement with those obtained MAP, KF, and magnetic entropy methods. For the sake of completeness, we have obtained an exponent α from resistivity. Considering the magnetic contributions, it has been theoretically proposed[45−52] and experimentally verified[53−56] that the temperature derivative of resistivity (dρ/dT) and specific heat (C) exhibit similar temperature dependent behaviors in the critical region. Figure a shows zero-field (1/ρ(TC)(dρ/dT) versus the t curve. A critical exponent α is extracted using a power-law given by eq (27)where A and B are constants, α is a critical exponent, and t is the reduced temperature. For the present alloy, the critical fit yields α = 0.55 ± 0.05. Positive α indicates the Ising type universality class, though it is larger about four times. α along with β and γ satisfies the universal scaling equation α + 2β + γ = 2.[27] Experimentally obtained exponents α, β, and γ using modified Arrott plots, Kouvel Fisher, magnetocaloric, and magnetoresistance methods, listed in Table , are in good agreement within the error bars, establishing the self-consistency of the critical exponents.

Figure 8

(a) Resistivity as a function of temperature, measured in labeled magnetic fields. It decreases with reduction in the temperature, revealing the metallic nature of MnNi0.7Fe0.3Ge. Inset shows the derivative dρ/dT which depicts two peaks corresponding to TC and Tt. (b) Percentage of magnetoresistance. It is negative, indicating the suppression of magnetic fluctuations in external magnetic fields. In 90 kOe, MR is about 6.0%. (c) Maximum of absolute MR (taken from (b)) versus change in the magnetic field, ΔH. A power-law fit to the data yields an exponent n = 0.72 ± 0.01. (d) Power-law fit of RCPMR versus ΔH, yielding an exponent δ = 5.545 ± 1.239. The significance of the power-law fits is discussed in the text.

Figure 9

(a) Reduced temperature dependence of in zero field. The data above critical temperature is fit (red solid line) to the eq , which yields a critical exponent α = 0.55 ± 0.05. (b) Magnetic field dependence of MR%. The curves are fit (red solid line) to the power-law MR ∝ −Hm. A quadratic field dependence in the paramagnetic state suggests the suppression of spin fluctuations by applying external fields. As T approaches TC, the magnitude of MR increases. Below TC, a reduction in MR% with linear dependence on H is noticed. (c, d) Rescaled magnetic entropy and magnetoresistance against a reduced temperature θ. The data scales well by falling onto a single universal curve, irrespective of the strength of H, indicating a SOPT. However, the deviation from the universal scaling around Tt hints at a first-order phase transition.

Table 2

Critical Exponents of MnNi0.7Fe0.3Ge

method	TC (K)	β	γ	δ	α	ref
modified Arrott Plot	212.51 ± 0.14	0.315 ± 0.076	1.327 ± 0.045	5.212 ± 1.162		this work
Kouvel-Fisher	212.45 ± 0.23	0.406 ± 0.002	1.293 ± 0.019	4.161 ± 0.062a		this work
critical isotherm	212.5			4.176 ± 0.011a		this work
magnetoresistance	∼213	0.392 ± 0.130	1.779 ± 0.672	5.545 ± 1.239	0.55 ± 0.05b	this work
magnetocaloric	∼213	0.331 ± 0.086	1.186 ± 0.322	4.852 ± 0.465		this work
mean-field theory		0.5	1.0	3.0	0.0	(27)
3D Heisenberg model		0.365	1.386	4.80	–0.0115	(27)
3D-Ising model		0.325	1.241	4.82	0.110	(27)
3D-XY model		0.346	1.316	4.81		(27)
tricritical model		0.25	1.0	5.0		(27)

δ is obtained using Widom’s relation δ = 1 + γ/β.

α is extracted from the resistivity and specific heat using eq .

Magnetic field variation of MR, measured at a few representative temperatures, is shown in Figure b. It is negative, as expected for a ferromagnetic metal, and isotropic against positive and negative applied fields. The absolute magnitude of MR increases as T → TC. MR is fit to a power-law dependence MR ∼ −H.[57] A near quadratic dependence (−H1.8) in the paramagnetic region (at 300 K) which indicates the suppression of spin fluctuations in the presence of magnetic fields.[58] Near the Curie temperature MR varies as – H0.74. While further reducing the temperature below 20 K, MR is observed to follow near-linear dependence with m = 0.92. The isotropic (independent of magnetic field’s direction) negative MR was attributed to the effect of spin alignment toward the interference contribution.[59] Later, Agrinskaya et al.[60] have attributed the observed unusual negative linear MR in narrow-band gap two-dimensional (2D) quantum well structures of GaAs-AlGaAs to the magnetic exchange interactions between localized and delocalized (thermally activated charge carriers) spins. For the present case, negative linear MR can be attributed to arise from the scattering of conduction electrons (s and p) from the itinerant d-electrons of constituent magnetic atoms.

In Figure c,d, the rescaled magnetic entropy and magnetoresistance are plotted against a reduced temperature θwhere Tp is the temperature at which absolute ΔSmag and MR are maximum and Tcold and Thot are the hot and cold temperatures taken from the full width at half-maximum of ΔSmag and |MR|. Overall, the data scale well by falling onto a single universal curve, irrespective of the strength of the applied magnetic field, inferring a second-order nature of the phase transition.[44,61] However, the curves are observed to deviate from the universal scaling below the structural transition Tt. Such a deviation from the scaling behavior is attributed to the first order phase transition below Tt.

Summary

MnNi0.7Fe0.3Ge alloy has been studied using the combined results of magnetization, specific heat, and resistivity along with electronic structure calculations. The alloy shows a ferromagnetic transition (TC ∼ 212 K) and an antiferromagnetic-like transition at (Tt ∼ 180 K) followed by a spin-glass transition (Tg ∼ 51.85 K). Across TC, (i) a well-scaled normalized magnetoresistance and magnetic-entropy, (ii) frequency independent hump in χAC(T), (iii) the critical exponents and J(r) ∼ r–4.75 suggest mixed (short and long-range) magnetic interactions with unclassified universality class of a second-order phase transition. Across Tt, the deviation from the scaling, a narrow thermal hysteresis between FCC and FCW, and relatively sharp transition in C(T) and 1/ρ(dρ/dT) indicate a weak first-order phase transition. Across Tg, the absence of a clear magnetic transition in C(T) and the analysis of a frequency-dependent cusp in χAC(T) reveal cluster spin-glass behavior. Density functional calculations reveal mixed ferro- and antiferromagnetic interactions with the dominating ferromagnetic coupling, which is experimentally evident from the large positive Weiss temperature, magnetic saturation, and overall negative magnetic-entropy and magnetoresistance. The magnetically distinct dual transitions of MnNi0.7Fe0.3Ge can be tuned to achieve large table-like MCE.