Local strains, calorimetry, and magnetoresistance in adaptive martensite transition in multiple nanostrips of Ni39+x Mn50Sn11-x (x ⩽ 2) alloys.

Ni39+x Mn50Sn11-x (x = 0.5, 1.0, 1.5 and 2) alloys comprise multiple martensite nanostrips of nanocrystallites when cast in small discs, for example, ∼15 mm diameter and 8 mm width. A single martensite phase with a L10 tetragonal crystal structure at room temperature can be formed at a critical Sn content of 9.0 at.% (x = 2), whereas an austenite cubic L21 phase turns up at smaller x ⩽ 1.5. The decrease in the Sn content from x = 2 to 0.5 also results in a gradual increase in the crystallite size from 11 to 17 nm. Scanning electron microscopy images reveal arrays of regularly displaced multiple martensite strips (x ≽ 1.5) with an average thickness of 20 nm. As forced oscillators, these strips carry over the local strains, magnetic dipoles, and thermions simultaneously in a martensite-austenite (or reverse) phase transition. A net residual enthalpy change ΔHM↔A = -0.12 J g-1 arises in the process that lacks reversibility between the cooling and heating cycles. A large magnetoresistance of (-)26% at 10 T is observed together with a large entropy change of 11.8 mJ g-1 K-1, nearly twice the value ever reported in such alloys, in the isothermal magnetization at 311 K. The ΔHM↔A irreversibility accounts for a thermal hysteresis in the electrical resistivity. Strain induced in the martensite strips leads them to have a higher electrical resistivity than that of the higher-temperature austenite phase. A model considering time-dependent enthalpy relaxation explains the irreversibility features.

Introduction

The martensite transition in ferromagnetic Ni–Mn–Sn Heusler alloys adapts unique properties of the shape-memory effect, inverse magnetocaloric effect, and magnetoresistance (MR) [1-11]. Functional properties crop up when structural transition coincides with a magnetic transition in the first-order diffusionless martensite transition, which is mediated through a thermoelastic strain arising in a shear-like atomic displacement. A fine compositional detuning from a stoichiometric Ni2MnSn Heusler alloy results in a phase transition from a cubic (austenite, A) to a tetragonal/orthorhombic (martensite, M) structure near room temperature. Although it is yet unclear why non-stoichiometry is required for the martensite transition, a theoretical prediction [6] explains the features by excess Mn(3d54s2) over Ni(3d84s2) as follows. The cubic phase is destabilized with a tetragonal distortion when the local electronic structure changes by hybridization between 3d orbitals of the excess Mn atoms at the Sn and Ni sites. Despite extensive studies of magnetism and the magnetocaloric effect in Ni50Mn50−Sn (5 ⩽ x ⩽ 25) [1–3, 8–11], there are very few reports on electrical resistivity in alloys with 13 ⩽ x ⩽ 16 [4, 5, 7]. A ferromagnetic reordering occurs in a narrow x = 13–15 range, resulting in a ferromagnetic to paramagnetic transition in the martensite phase at the Curie point, , before a reverse ferromagnetic reordering takes place at the martensite to austenite transition temperature, TA.

A large difference between magnetizations in the martensite and austenite phases in such Ni–Mn–Sn alloys results in a large inverse magnetocaloric effect and magnetic shape memory. The transition carries over a large thermoelastic strain and a small irreversibility in spin reordering. A concurrent magnetic transition near room temperature is required for applications. In Ni50Mn50–Sn, the martensite transition falls either well below or above room temperature [4, 5, 7, 12, 13]. Also a Ni-deficient Ni50−Mn39+Sn11 (5 ⩽ x ⩽ 7) alloy exhibits a large gap (30 K at x = 5, or 80 K at x = 7) between the two transitions and the transitions occur at temperatures TM ∼ 190 K at x = 7 or ∼260 K at x = 5 [14], which are well below room temperature. Another Ni43−Mn47+Sn10 (x ⩽ 5) series, which were studied as Ni-deficient alloys, has a TM value well below room temperature [15]. In this system, the gap between TM and the Curie point (austenite) increased from 20 K at x = 0 to as much as 170 K at x = 5.

In the Ni50Mn50−Sn (5 ⩽ x ⩽ 25) phase diagram [3], the valence–electron concentration e/a (8.0–8.2 per atom) controls the structural and magnetic transitions that approach each other very closely at e/a ∼ 8.0. In this paper, we study multiple nanostrips (of crystallites) in Ni39+Mn50Sn11− alloys (x = 0.5, 1.0, 1.5 and 2) with e/a varying sharply from 7.87 to 7.96, which is smaller than that (8.0) proposed in the phase diagram. As x approaches 2.0 (e/a → 7.96), the two transitions concur with a large entropy change, nearly twice the value ever reported in such alloys, in alliance to a large MR at ∼311 K. A local strain that the nanostrips restore and carry over imparts a magnetocaloric peak to the A → M transition. A thermal hysteresis in electrical resistivity and an enthalpy loss in the reverse transition arise due to a lack of reversibility in spin reordering.

Experimental details

The nanostrips of Ni39+Mn50Sn11− (0.5 ⩽ x ⩽ 2) were grown by cooling a melt in the shape of a disc (15 mm diameter and 8 mm width) in a mould of copper in a tungsten inert-gas arc-melting furnace. A master alloy made by melting and casting a stoichiometric mixture of pure metals in small discs was flipped and remelted three, four times ensuring a homogeneous mixing. The final compositions were measured using inductively coupled plasma optical emission spectroscopy (JY-ULTIMA spectrophotometer, France) and energy dispersive x-ray analysis performed with a Jeol JSM-5800 scanning electron microscope. Rectangular bars were freshly sliced from middle parts of the alloy discs by electrodischarge machining for x-ray diffraction (XRD) measurements. The measurements were carried out with a high-resolution diffractometer (X'Pert PRO PANalytical) operated at 40 kV and 40 mA, using filtered 0.15410 nm CuK radiation. Nanostrips were observed in a field emission scanning electron microscope (FESEM, Zeiss SUPRA-40) at an accelerated voltage of 10 kV. Vickers microhardness was measured using a microhardness tester (UHL VMHT, Germany) with a load of 200 gf, at three different locations (including the disc center) with a 4 mm gap on the cross-section of the alloy disc sliced through the diameter. The heat outputs in the M ↔ A phase transitions were studied by heating and cooling a specimen at 10 K min−1 in a differential scanning calorimeter (DSC Q100, TA Instruments). Electrical resistivity (ρ) was studied over the 10–300 K range using a standard four-probe method (Lakeshore Hall effect measurement instrument) by passing 100 mA of dc current through a sample of 10 × 1 × 1 mm3 size. Magnetic field (B) dependent isothermal magnetization and ρ were studied using a superconducting quantum interface device and a physical properties measurement system (Quantum design), respectively, with B varied up to 10 T.

Results and discussion

Local strains and microstructure in nanostrips

The XRD patterns shown in figures 1(a)–(d) reveal distinct changes in the number and/or relative intensities of the peaks when varying the Sn content as x = 0.5, 1.0, 1.5, or 2.0 in Ni39+Mn50Sn11− alloys marked as alloys 1, 2, 3, or 4, respectively. A simple pattern of only six peaks arises in a pure tetragonal L10 martensite phase in alloy-4, with lattice parameters a = 0.7808 nm, c = 0.6954 nm and density d = 7.797 g cm−3 (z = 8 formula units). As many as fifteen XRD peaks occur in a L21 cubic crystal structure with the superlattice peaks (all h, k, l are odd) in alloy-1, with a = 0.6039 nm and d = 7.615 g cm−3 (z = 4). The superlattice peaks no longer persist when the Ni content is raised, i.e. decreasing the Sn content to ∼10 at.% (x = 1), in a strain-free fcc-L21 lattice having an enhanced d = 7.696 g cm−3 (a = 0.6034 nm). With Ni increasing further, the L10 phase (a = 0.7884 nm and c = 0.6841 nm) co-precipitates with a simple fcc L21 phase (a = 0.5980 nm) at an intermediate Sn content of 9.5 at.%. A compressive strain that develops in an intergranular structure in both phases enhances the densities by 0.2–2.0% over the values in the single-phase alloys 1 and 4.

Figure 1.

XRD patterns (a)–(d) reflecting the transformation from (a) a cubic austenite L21 to (d) a tetragonal martensite L10 phase in Ni39+Mn50Sn11− (x ⩽ 2.0) crystallites.

When reinforced by the martensite phase (∼29 vol% as per the XRD data of figure 1(c)) an austenite phase compresses greatly in the lattice volume (V) by 3 versus 0.28% for the martensite phase (filler) because of a counterpart effect. This is a result of a surface-reinforced densification when one phase bounds the other phase with a rigid interface. A larger atomic radius of Sn (0.1405 nm) over Ni (0.1240 nm) results in a volume increase upon the Sn → Ni substitution in a base Ni41Mn50Sn9 alloy. Phenomenologically, the substitution takes place primarily in the austenite phase. In table 1, consistently, a large τ value of 3.5% (estimated from inhomogeneous broadening in the XRD peaks [16]) arises in a hybrid composite of two phases with 9.5 at.% of Sn. It reduces the average crystallite size to D = 12 nm in comparison to a single austenite phase with D = 17 nm. Other structural parameters computed from XRD patterns in the different alloys are included in table 1.

Table 1.

Structural parameters in martensite and austenite Ni39+Mn50Sn11−(x ⩽ 2.0) phases.

			Lattice parameters (nm)		D	τ	V	d	Vsl
x	e/a	Phase	a	c	(nm)	(%)	(nm3)	(g cm−3)	(cm3 g−1)
0.5	7.87	L21(A)	0.6039	–	17	2.5	0.2202	7.615	0.1315
1.0	7.90	L21(A)	0.6034	–	13	0.9	0.2197	7.696	0.1300
1.5	7.93	L21(A)	0.5980	–	12	3.5	0.2138	7.766	0.1290
		Ll0(M)	0.7808	0.6841	–	–	0.4240	7.812	0.1280
2.0	7.96	Ll0(M)	0.7884	0.6954	11	1.7	0.4252	7.797	0.1285

The values are accurate to the last reported digits, except Vsl within ±0.05% error.

The FESEM images in figures 2(a) and (b) show martensite nanostrips in Ni39+Mn50Sn11− alloys (x ≽ 1.5) grown in parallel arrays. As shown in figure 2(a), the strips in an XRD-pure martensite phase are separated by a distance of SssM ∼ 0.4 μm and have a thickness of SstM ∼ 0.70 ± 0.05 μm. These strips are grown preferentially in the (222) planes as judged from the most intense (222) peak in figure 1(d), viz. perpendicular to the [111] direction. While cooling a molten alloy, a liquid alloy phase splits up and solidifies in the multiple strips as a result of a directional cooling in thin rods (∼15 mm or smaller diameter). The specific lattice volume (Vsl) decreases by 2.3% (table 1) during the A → M transition that imposes a compressive shear stress leading to a local splitting of the martensite phase into discrete and deep strips. To balance the compositional gradient, a primarily nondiffusional A → M transition sets up with a restoring force. As a result, the strips persist when the alloy is cooled from a dynamic equilibrium at a critical temperature T0 with a decreased Gibbs free energy ΔGA→M = ΔVslΔGV, where ΔGV is the Gibbs free-energy (per unit volume) of the transformation. A self-accommodating microstructural behavior of the resulting alloy can arise in slipping and/or twinning in the local structure so as the strain-energy relieves in a pure martensite structure (figure 1(d)). A small (0.5 at.%) Sn inclusion above a pure martensite phase thus creates a compressive stress with compressed by ∼29%, or Vsl by 2.3%, in a martensite–austenite alloy composite (x = 1.5). The contraction of the strips with reduced by 50% increases the density of strips. An elastic energy that the strips store during the A → M transition creates a compressive stress with in situ generated shear forces, which sets up a dynamic interfacial motion between the strips. A frictional force opposite to the interfacial motion, which boosts up when the drops adiabatically, compels the strips to split up into finer strips (SstM ∼ 0.1 μm) displaced between the parent strips (∼ 0.5 μm), as observed in figure 2(b). At x < 1.5, the strips disappear as soon as the martensite phase converts to the austenite phase near room temperature (figures 2(c) and (d)).

Figure 2.

FESEM images showing that martensite Ni39+Mn50Sn11− strips (a) x = 2.0 and (b) x = 1.5 disappear on conversion to an austenite phase at (c) x = 1.0 and (d) x = 0.5.

The FESEM images shown in figures 3(a)–(d) reveal that the Ni39+Mn50Sn11− (2.0 ≽ x ≽ 1.5) anisotropic bars consist of smaller substrips (4–16 in different regions). The region A of figure 3(a) (x = 2.0) enlarged in figure 3(c) displays the substrips (thin laminates) displaced in regular arrays, which represent a nanotwinned structure (a mirror-like stacking fault of the close-packed atomic layers) in the martensite L10 phase. In figures 3(b) and (d), less distinct laminates appear when two phases breed a net residual strain in an intergranular structure (x = 1.5). The nanotwins arise to overcome the lattice mismatch with entities in the intimate contacts. Figure 3(e) shows the model substrips of a rectangular bar as they are observed in the FESEM images in figure 3(c), with an average thickness of 20 nm. A geometrical theory of martensite by Wechsler et al [17] predicts a periodic twinning of a tetragonal lattice, expressed through the fraction of the twin lamellar widths w1 and w2 as w1/w2 =(am – aa)/(aa– cm), where am, cm, and aa are the lattice parameters with the subscripts m and a denoting martensite and austenite. The values am = 0.7884, cm = 0.6954 nm, and aa = 0.6039 nm calculated from XRD patterns imply w1/w2 ≈ 2, illustrating that periodically arranged nanotwin lamellae, with w1:w2 = 2:1, can interbridge martensite strips as modeled in figure 3(f). The bright (∼20 nm) and dark (10 nm) strips have thicknesses in the same ratio, w1:w2 = 2:1.

Figure 3.

FESEM images showing bundles of multiple martensite strips (4–16 thin laminates in different regions) in Ni39+Mn50Sn11−at (a) x = 2.0 and (b) x = 1.5, with enlarged parts (c) A and (d) B. (e) A model pattern of laminates (f) of nanotwinned atomic layers.

Vickers microhardness

To explore whether the composition and microstructure control the mechanical hardness in Ni39+Mn50Sn11− (x ⩽ 2.0), we studied Vickers microhardness (HV) at three different points on a vertical cross-section sliced from a disc, viz. close to (i) the upper (U) and (ii) lower (L) surfaces by 1.0 mm from the lateral faces and (iii) the midpoint (C). As shown in figure 4, the HV value has an inverse relation with τ value, an atypical effect of the local strain on the alloy hardening. In Hooke's law, the hardness is directly proportional to the local strain, i.e. the strain increases linearly against the stress within the elastic limit. A deviation arises here because the martensite strips that are pinned down at the nanotwins govern the average value. As shown in table 2, the average HV value 2.584 GPa (x = 2.0) is smaller compared to the values for x ⩽ 1.0 because of a significantly weak interfacial motion and weak pinning between effectively thick strips. The HV value falls down by 13% when the pinning weakens further in the same kinds of strips in a mixed M–A phase (x = 1.5). A similar elastic softening is shown in ferroelastic domains that are pinned down weakly in the cubic → tetragonal phase transition in SrTiO3, KMnF3, or KMn0.997Ca0.003F3 at temperatures below 200 K [18,19]. In the Ni39+Mn50Sn11− (x ⩽ 2.0) series, a single austenite phase (x = 1.0) that grows at the expense of the interface with a residual martensite phase acquires a larger hardness HV = 3.321 GPa, but a smaller residual strain τ = 0.9%. Further, when a superlattice structure builds up with a local strain, the HV drops marginally in the alloy-1 (x = 0.5).

Figure 4.

Vickers microhardness (HV) showing martensite (x = 2.0) and austenite (x = 1.0) phase softening at the expense of strains τ in Ni39+Mn50Sn11− crystallites. The data were measured on upper (U), central (C) and lower (L) cross-sections from an alloy disc. ∗Contains superlattices.

Table 2.

Vickers microhardness (HV) and gradient (∂HV/∂y) in Ni39+Mn50Sn11−(x ⩽ 2.0) alloys.

	Hv (GPa)			∂Hv/∂y (GPa mm−1)
x	U	C	L	U→C	C←L
0.5	2.570	3.175	2.635	0.200	0.180
1.0	3.050	3.755	3.155	0.235	0.200
1.5	2.025	2.605	2.135	0.1 90	0.155
2.0	2.545	2.570	2.635	0.010	0.020

The data are measured on upper (U), central (C) and lower (L) cross-sections of three discs. The values are correct with ±0.2% error.

The HV values compared in table 2 for different regions of four Ni39+Mn50Sn11– (x = 0.5, 1.0, 1.5 and 2) alloys are the averaged values measured on three discs prepared under the same conditions. They were reproducible within a standard deviation of ±0.2%, illustrating the veracity of the hardness variation across the disc. The outer region, which had cooled faster in a bulk structure, is a softer austenite phase. During cutting from a disc, a compressive stress in the lower surface would breed surface hardening only. The hardness propagates with local strain that relieves when the martensite phase splits up and self-accommodates as illustrated with the FESEM image of figure 2(a).

Calorimetric signals in magnetostructural transitions

Now let us analyze how the martensite strips and local strains affect the caloric signals jointly in the magnetic and structural transitions. The heat output was measured during heating followed by cooling the Ni39+Mn50Sn11− alloy (x ⩽ 2) in the temperature range 200–400 K. The results so obtained are compared in figure 5 for three alloys (x = 1.0, 1.5, and 2.0). Upon heating, an exothermic peak (x ≽ 1.5) appears at temperature Ap in the M → A transition, with Ap > TA, where TA = (As + Af)/2 is a midpoint of the austenite start As and austenite finish Af temperatures. A modified transition peak occurs at Mp > TM on cooling, where TM is a midpoint of the martensite start Ms and finish Mf temperatures. In other words, both endothermic and exothermic peaks are asymmetric, and show a hysteresis ΔT = Ap − Mp. At x ∼ 2.0, the TM = 310.5 K and TA = 322.5 K are raised considerably above room temperature with a strong caloric signal of an enthalpy change ΔHM←A = 3.325 J g−1, or entropy ΔSM←A = 10.655 mJ kg−1 K−1, and full-width at half maximum βM←A = 29.7 K. The local strains (τ = 1.7%) relieved in regular martensite strips have a higher ΔHM←A value compared to samples with x = 1 or 1.5 (table 3). Reheating gives a smaller ΔHM→A = 3.205 J g−1, or ΔSM→A = 9.800 mJ g−1 K−1, in the process that lacks reversibility in the enthalpy, which is relieved slowly when the paramagnetic martensite state reverts back to the ferromagnetic austenite state. A thermal variation that sets up in the electrical conduction is modified in a spin disorder → reorder transition. In comparison to Ni-rich alloys Ni49Mn38Sn13 (8.35 J g−1) and Ni50Mn35Sn15 (5.1 J g−1) [3, 20], these ΔHM←A values are smaller, but correspond to a larger βM←A ∼ 29.7 K and a higher transition temperature (TM = 310.5 K), 2.2 times the temperature in Ni45Co5Mn40Sn10 of an intermediate Ni content (bulk L10 phase; TM = 130 K) [21].

Figure 5.

DSC thermograms for the M ↔ A transition in cooling and heating of Ni39+Mn50Sn11− at 10 K min−1 in argon; (a) x = 1.0, (b) x = 1.5, and (c) x = 2.0. Martensite strips (x = 2.0) exhibit prominant peaks while only a weak signal is observed near in the austenite (x = 1.0, see the inset).

Table 3.

M↔A caloric transition parameters in Ni39+Mn50Sn11−(x ⩽ 2.0) alloys.

x	RT phase	Ap (K)	MP (K)	ΔT (K)	ΔH (J g−1)		ΔS (mJ g−1 K−1)		ΔG (mJ g−1)
					M←A	M→A	M←A	M→A	M←A	M→A
1.0	L21	225.0	212.0	13.0	0.130	−0.155	0.615	−0.695	3.640	−4.310
1.5	L21+L10	276.2	265.1	11.1	0.435	−0.340	1.645	−1.225	8.155	−6.325
2.0	L21	327.1	312.0	15.1	3.325	−3.205	10.655	−9.800	73.110	−70.490

RT: room temperature. The Ap, Mp, and ΔT are accurate within ±0.5 K error, while other entities within ±0.1% of the reported values.

As shown by the hatched areas in the DSC peaks in figure 6(a), the martensite strips (x = 2.0) in cooling and heating exhibit not only a large irreversibility ΔHM←A–ΔHM→A = 0.120 J g−1 (ΔGM→A = ΔHM→AΔT/2T≅ 70.490 mJ g−1, with +Af)), but also a thermal hysteresis (ΔT = 15.1 K). Possible contributions to these effects arise from (i) well-displaced twinned martensite strips (figure 3(f)) that bear a strong magnetic spin-pinning barrier, (ii) residual stress on a granular structure, (iii) spin relaxation, (iv) magnetoelastic coupling between spins and lattice ordering, (v) thermal conductivity induced in dynamic spins and phonons, and (vi) different heat capacities in the two states (). Further, the strips divide the sample into domains by spins pin-down at the boundaries. A local spin–lattice reconfiguration that sets up in a transition thus adds a large frictional energy (or shear and compressive stresses) according to a large ΔHM←A = 3.325 J g−1 found in this alloy. Low electrical resistivity and stress in the austenite state ease a spin-lattice heat transfer in an exchange-coupled relaxation that drives in situ spin disorder (paramagnetic martensite) ← order (ferromagnetic austenite) transition by absorbing a large ΔHM←A over a large span βM←A with ∼ 0.090 mJ g−1 K−1.

Figure 6.

DSC signals with irreversibile ΔH in M ↔ A transition for x = 2 (a) and 1.5 (c). (b) A model C-diagram in Ni39+Mn50Sn11− (x = 2.0). An IMT peak appears with a weak signal in a premartensite transition (marked by the vertical arrows in (c) in a mixed phase (x = 1.5). FMO: ferromagnetic ordering, SDO: spin disordering; ISRO: irreversible spin-reordering.

A model C-diagram in figure 6(b) explains a positive value assuming a spin–lattice relaxation relieves heat before a heat uptake begins in the M ← A transition from a point Ae (cooling). The heat dissipates faster in the conductive A-state. After the transition terminates at Mf, the C rises up further due to the spins disorder in a paramagnetic phase Me. In reheating from point Me, a value evolves, i.e. expectedly lower than 0.090 mJ g−1 K−1 observed in the transition in cooling, following part of the heat relieved in a spin-lattice relaxation. As indicated in figure 6(b), a C value of 0.270 mJ g−1 K−1 estimated at Mf () from the DSC thermograms (figure 6(a)) is raised to 0.311 mJ g−1 K−1 upon heating to As. For a fixed ratio κe/γ (κe = electronic thermal conductivity and γ = electrical conductivity) in the Wiedemann–Franz law in a metal, a diminished κe value in the martensite strips accounts for the energy loss in a spin–lattice heat transfer, describing a lower. A lower magnetization (σ) thus increases in the A-phase by 58% in a low magnetic field B = 5 mT, or 12% in B = 5 T.

Qualitatively, κe enhanced in a mixed M/A phase Ni39+Mn50Sn11− (x = 1.5) yields smaller =0.010 mJ g−1 K−1, ΔHM←A – ΔHM→A = 0.095 J g−1, ΔGM→A = 6.325 mJ g−1), and ΔT = 11.1 K (figure 6(c)). The residual martensite that transforms into austenite near 383 K (or the reverse transition near 367 K in the cooling cycle) leads to a superimposition of the DSC signal over a background due to the local structural changes. A similar phase transition between the micromodulated phase and a bcc austenite is known in Ni2MnGa near 230 K, but with a smaller ΔT ≈ 7 K [22]. For x ≽ 1.5, and TM (or TA) coincide in a single caloric signal, while a distinct appears near 270 K (figure 5(a)) at x = 1.0. An intermartensite transition (IMT) emerges in a distinct signal at lower temperature to the martensite transition peak, at 288 K for 9.5 at.% Sn (x = 1.5), or 315 K for 9.0 at.% Sn (x = 2.0). Any internal stress that builds up during the local atomic redistribution is relieved in this process [23]. A characteristically faster ΔH change accounts for a stronger IMT signal in the M ← A transition (cooling).

The martensite strips Ni39+Mn50Sn11−, 1.5 ⩽ x ⩽ 2.0, assist both the irreversibility and magnetocaloric effect. As shown in figures 7(a) and (b), upon heating from 309 to 312 K the σ value in the x = 2.0 alloy that exhibits a large irreversibility is enhanced by as much as 7.1 emu g−1 in B = 5 T. This increase corresponds to a large magnetic entropy change ΔSm = 11.8 mJ g−1 K−1, or refrigerant capacity RC = 133.1 mJ g−1, near TA ∼312 K in an inverse magnetocaloric effect in the standard relation A larger TA of 322.5 K is observed in DSC revealing that magnetic field shifted the transition to lower temperatures. The RC was calculated from the area under the ΔSm versus T curve (inset in figure 7(a)) in a field of 5 T. Only a residual ΔSm ∼3.5 mJ g−1 K−1 (RC ∼105.3 mJ g−1) lasts due to the reduction in the irreversibility in a mixed M/A phase (x = 1.5) near TA ∼254 K, viz. the irreversibility helps the magnetocaloric effect. A minor austenite phase also suppresses ΔSm to ∼1.0 mJ g−1 K−1 (ΔB = 9 T) in the Ni50Mn34Sn16 alloy (TM = 190 K) [13]. Pure martensite strips are thus required to enhance ΔSm.

Figure 7.

Magnetization isotherms with a σ jump in the M → A transition in Ni39+Mn50Sn11−; (a) x = 2.0 and (b) x = 1.5, with a larger ΔSm in martensite strips (x = 2.0) in the insets.

Forced oscillations and damping in multiple martensite strips

Let us consider that a compressive stress induced by thermal agitation in the M ← A transition compels substrips in a group to oscillate about their average positions. As small oscillators they carry over the local strains, magnetic dipoles, and thermions along effective force F exerted on the strips. A strip thus obeys the following equation of motion in a single degree of freedom: with where F0 is the oscillation amplitude, ω is angular frequency, m is the mass of a strip, h is the effective damping coefficient, α is the dynamic restoring coefficient in a strip, and y is displacement from the mean position; h arises from the internal friction between the strips Ni39+Mn50Sn11−. Qualitatively, a larger value hM←A arises in the M←A transition because the oscillators transfer the strain, spin, and thermion from a low-strain point (Ms) to a higher value (Mf). A model diagram in figures 8(a)–(d) according to the observed ΔHM↔A (DSC) indicates that the initial hM←A decays faster with time t than hM→A on the heating. Lesser interfacial friction in the strips decreases the damping (hM→A < hM←A) so that the separation of strips no longer ceases rapidly. Here, m is constant and α is a weak function of time.

Figure 8.

Model damping of collective oscillations of multiple martensite strips under a compressive force during (a, b) cooling and (c, d) heating via an M ↔ A transition in Ni39+Mn50Sn11− (x ⩽ 2.0). Damping coefficient hM→A modified during heating inducts irreversibility in the process parameters.

A phase difference ϕ between the force and resultant motion in an oscillator of amplitude R gives on substituting in equation (2), with ξ = h/hc, where hc is the critical h value at which transmissibility Γ of spins and thermions approaches zero, and ωn is the natural frequency of the strips. One can express Here, Rd is a dimensionless response factor, ∼1 at very low ω values. It traces a peak near ωn and approaches zero as ω → ∞. The force that transmits through the strips in carrying over the strain, spin, and thermion can be expressed in a differential equation The forces h dy/dt and αy are phase-shifted by 90°, and thus the FT magnitude is The ratio implies Equation (7) results in a weak peak in figure 9(A) with Γ ⩽ 5 at resonance ω/ωn = 1 for ξ ≽ 0.1 in an M ← A transition. This peak is intensified for a smaller ξ = 0.01, which is possible if strains and thermions flow collectively and rapidly with spins from a disordered configuration (lower κe) before the Mf sets in. A moderate ξ value on smaller h in a reverse transition gives a medium Γ peak. A maximum ψ determined in a wide transition over 1 ⩽ ω/ωn ⩽ 10 transmits a maximum energy (figure 9(B)).

Figure 9.

Variations of (A) transmissibility and (B) phase angle against ω/ωn when the damping parameter ξ is varied from 0.01 to 0.5.

As shown in figure 10(A), a net force in the oscillators can describe an ellipse . It dissipates the following amount of heat energy per unit time: Analogous to equation (3), ω and ξ in a collective oscillation of the strips of period μ describe Integrating equation (9) between μ and μ + 2πω−1 yields or where is a constant and χ is a correlation factor. As shown in figure 10(B), a normalized ∂H/∂t with z = 1 has a peak against ω/ωn in an M ← A transition. As ω/ωn → 1, the ∂H/∂t reaches a maximum at a nodal point, i.e., the TM or TA point. The peak intensity rises progressively with the parameter ξ used in figure 9. As a result, considering an average ξ value can corroborate a weaker heat flow as it is observed when cooling the sample.

Figure 10.

(A) Phase diagram of a damped oscillator and (B) the rate of the enthalpy change ∂H/∂t versus normalized frequency ω/ωn from equation (12).

Magnetoresistance and entropy in magnetostructural transition

To illustrate the effects of the local structural and magnetic changes on the M ↔ A transitions in terms of electrical transport in Ni39+Mn50Sn11− (x ⩽ 2.0), we studied ρ and MR in different conditions as follows. In figure 11, ρ displays a thermal hysteresis in cooling and heating, with a more resistive structure on the heating damping heat flow with moderate ξ value. For x ∼ 2.0, the ρ value rises sharply in the M ←A transition at TM ∼309 K (table 4) due to enhanced local strains and spin disordering, which promotes electron scattering. A large change ΔρM←A = 275 μΩ cm along with κe = LT/ρ is changed by −1.06 = 1.43 W m−1 K−1 using the Lorentz number L = 2.45 × 10−8 W Ω−1 K−2 in the Wiedemann–Franz law. On heating, the transition results in a release of thermoelastic energy in a large ρ–T hysteresis (ρT)h = 17.5 mΩ cm K, or ΔT = TA-TM∼15 K. This sizeable irreversibility is important for tailoring MR and other properties. At x ∼1.5, impeded strips turn up in the M ← A transition with a large residual strain (3.5%) resulting in a lowered ΔρM←A = 62 μΩ cm, or =0.42 W m−1 K−1. (ρT)h is markedly lowered to 1.85 mΩ cm K (ΔT ∼ 9.5 K) in a transient fashion in the reverse transition. Relatively smaller strains stored on low irreversibility in the transition cycles result in average (ρT)h = 5.5 mΩ cm K at x = 1.0, or 3.6 mΩ cm K at x = 0.5. The slope of the ρ–T curve changes at the TCA ∼ 265 K (x = 0.5), or 270 K for x ∼ 1.

Figure 11.

Thermal hysteresis in the M ↔ A transition in electrical resistivity ρ of Ni39+Mn50Sn11−: (a) x = 0.5, (b) x = 1.0, (c) x = 1.5, and (d) x = 2.0. The martensite strips (x = 2.0) exhibit a large hysteresis loss (ρT)h plotted in the inset. The given ρ-scale applies to the sample x = 2.0; it is to be divided by 2.55, 2.28, or 0.97 for the x = 0.5, 1.0 or 1.5 samples.

Table 4.

Martensite transition temperatures measured from DSC thermograms and ρ–T hystereses in Ni39+Mn50Sn11−(x ⩽ 2.0) alloys.

x	Method	Ms (K)	Mf, (K)	As (K)	Af (K)	TM (K)	TA (K)	ΔT (K)
0.5	Resistivity	163	79	102	172	121.0	137.0	15.0
1.0	DSC	222	–	–	242	–	–	13.0
	Resistivity	220	178	1 66	240	189.0	203.0	14.0
1.5	DSC	288	235	237	300	261.5	268.5	11.l
	Resistivity	287	233	238	301	260.0	269.5	9.5
2.0	DSC	334	287	297	348	310.5	322.5	15.1
	Resistivity	337	283	295	350	309.0	324.0	15.0

The temperatures are accurate within ±0.5 K error in general.

Figure 12 shows the field dependence of ρ values at temperatures between 100 and 320 K in Ni39+Mn50Sn11− of low (x ∼ 2) and high (x ∼ 1.5) strains. The data were measured by warming a zero-field-cooled sample from 100 K. A large MR = −26% arises near TA ∼ 320 K (figure 12(A)) in B = 10 T for a low strain of 1.7%. The MR decreases to (−)17% near TA ∼270 K (figure 12(B)) for the higher strain of 3.5% in the other sample. While negative MR arises when B suppresses the spin-disorder scattering of electrons, a significantly large MR lasts if the spin reordering lacks reversibility in a strain-free system. Such large MR is useful for room-temperature applications. A still larger value (−)50% is known in an alloy Ni50Mn36Sn14, but at a low temperature of 150 K and in large B = 18 T [4].

Figure 12.

Magnetic field dependence or resistivity ρ in Ni39+Mn50Sn11−with (A) x = 2.0 and (B) x = 1.5. A lack of fully reversible spin reordering correlates with a large MR = (−)26% in the M → A transition (x = 2.0) at 320 K and ΔB = 10 T.

In the model enthalpy–temperature diagrams of figures 13(a) and (b), thermions propagate easily via martensite strips giving rise to a net change of enthalpy ΔH, or entropy ΔS, in a magnetostructural transition in Ni39+Mn50Sn11− (x ⩽ 2). This change contains three major parts of ΔHchem from a chemical change, ΔHelast from an elastic strain change, and irreversible change ΔHirr in the transition. The Gibbs free energy that leads to change the M ↔ A states yields ΔHchem, while ΔHelast arises in restoring the lattice volume. A nonzero ΔHirr value, requisite of a thermal hysteresis of the transition, describes the work done by the frictional forces opposing the interfacial motions in the strips in spin disordering or reordering. In this approximation, the final ΔH value that an M ← A transition absorbs (cooling), or a reverse transition releases (heating), can be expressed as

Figure 13.

Schematic of hystereses in (a) enthalpy and (b) entropy in the M ↔ A transitions in Ni39+Mn50Sn11−(x ⩽ 2.0) martensite strips.

If dissipates rapidly, as observed in Cu–Al–X (X = Zn, Ni) and Fe–Pt alloys [24], a reversibility ΔHM←A > ΔHM→A could develop in Ni39+Mn50Sn11− (x ⩽ 2) martensite strips due to a time lag in the relaxation process over heating and cooling via the transition. It can be described with an empirical relation, , where tsp is the spin-relaxation time and tst is the structural relaxation time with an exponent N. Setting N ∼1 and tsp ≪ tst in an ergodic transition accounts for the reversibility. A fractional ΔHM→A ∼ 0.964ΔHM←A at x = 2.0, or 0.775ΔHM←A at x = 1.5, observed by DSC in Ni39+Mn50Sn11− is reproduced well with tsp/tst = 0.01 and N = 0.72, viz. a nonergodic transition, tsp < tst and 0 25]. Thus, in the absence of full relaxation over the experimental timescale and exhibit a hysteresis in an entropy–temperature diagram, or a temperature lag ΔT in an enthalpy–temperature diagram. A controlled spin relaxation which determines is required in high-performance spintronics [26, 27].

Conclusions

The Ni39+Mn50Sn11− alloys where Ni is partially substituted by nonmagnetic Sn (x = 0.5, 1.0, 1.5, and 2.0) illustrate that an inverse magnetocaloric effect concurs with a large MR in the multiple martensite alloy nanostrips that transact the carriers in the M ↔ A transition. For an optimal Sn content of 9.0 at.% (x = 2.0), the transition occurs well above room temperature, at TM ∼ 310.5 K, from a cubic L21 austenite phase (x ⩽ 1.0) to a tetragonal L10 martensite phase. A local strain, which builds up in the M ← A transition in cooling, propagates undisrupted through the anisotropic nanostrips that displace and transfer the energy as a forced oscillator. In a nanotwinning process, a martensite strip splits into a regular pattern of 4–16 substrips (∼20 nm average thickness). The spins that are pinned down at the boundaries divide the substrips into the single magnetic domains. These spins also cause surface alloy hardening with effectively large shear stress, compressive stress, and frictional energy on account of a coupled spin-lattice system. A major part of the heat intake ΔHM←A = 3.325 J g−1, such as 3.205 J g−1, is relieved in the transition that reverts back (upon reheating) with reversibility of the heat carriers. A large MR = −26% is observed and it carries over a large entropy change ΔSm ∼ 11.8 mJ g−1 K−1, nearly twice the value reported for such alloys [3, 20, 27]; this large MR is useful for room-temperature applications. The observed features degrade sharply on austenite phase precipitates (x ⩽ 1.5). A model enthalpy–temperature diagram with a time-dependent spin–lattice–thermion relaxation in a nonergodic nature of the M ↔ A transition explains how a local strain is generated with the spins pin-down at the domain boundaries thereby enhancing the ΔHM↔A and ΔSm values.