Small core–shell Mn_0.5Bi_0.5−Bi (⩽3 at%) magnets, the anisotropic growth of crystallite nanoplates, interface-bridging, and tailored magnetic properties

In view of our microscopic observations, wet-milling an alloy Mn_0.5+xBi_0.5−x−Bi, x ⩽ 0.05, with glycine is refined and dispersed via the glycine (step-1) into a fluid of hexane (liquid) in small Mn_0.5+xBi_0.5−x and Bi refined crystallites of refreshed surfaces, as shown in figure 1(a), in a model hierarchical structure. Glycine, a hydrocarbon NH₂–CH₂–COOH, binds over fresh metal surfaces as a 'surface passivation layer', which keeps them separated in such a way that they do not adjoin easily in the continued milling process employed in this experiment. Wihtout the use of glycine, refined alloy species will deform, expand and ultimately collapse into one another, resulting in a rather 'coarser structure' in thermochemical reactions where local heat is produced during the milling process. Soft ball-milling, however, is well controlled, allowing the Mn_0.5Bi_0.5−Bi sample to be finely refined into a nanostructure in the significant LTP-phase, where its magnetic properties may be tailored [20–22]. An inbuilt surface-layer divides the local heat over the refined alloy particles at an average temperature, which is not sufficient to disintegrate the refined Mn_0.5Bi_0.5 → Mn_0.5Bi_0.5−z + zBi alloy further in a peritectic reaction during continuous milling in hexane (a heat sink) in a closed container. This protects the highly coercive sample of surface-stabilized single magnetic domains. Further, in the proposed model, as shown in in figure 1(b), part of the excess Bi used here blends over the refined Mn_0.5+xBi_0.5−x to form small core–shell crystallites. On further milling (step 2), some of the Bi separates over the hot Mn_0.5+xBi_0.5−x facets in tiny droplets, which wet and mediate the alloy, growing slowly via wet facets, as x → 0, in a specific core–shell structure (see figure 1(c)) in local solid–liquid surface reactions.

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Subsequent heating at T_h = 573 K, just above T_m = 544 K of bulk bismuth, yields a refined sample of Mn_0.5+xBi_0.5−x−Bi, which grows preferentially (as shown in the XRD patterns described below) when free Bi slowly diffuses into the 'microscopic through channels' of the mobile vacancies (resulting from the milling process) via reactive wet-facets, in a lattice of small crystallites of Mn_0.5+xBi_0.5−x, x → 0. Presumably, a reverse reaction, Mn_0.5Bi_0.5 → Mn_0.5Bi_0.5−z + zBi, z ⩽ 0.02, occurs only beyond a critical growth on heating over a critical limit of t_p = 72 h at 573 K, in such a way as to maintain its equilibrium over the residual Bi in the hybrid composite. Evidently, a refined sample maintains a larger degree of Bi solubility in the nanostructure. Further, a Mn-rich Mn_0.55Bi_0.45 alloy results in the separation of some Mn during extended milling, but this is readily reincorporated on heating at T_h = 563 K [8, 25], due to the faster diffusivity of the smaller atoms. To obtain a single MnBi-LTP phase alloy is thus limited by the peritectic reaction occurring between Mn and Bi at around 535 K in which part of the Bi precipitates out (due to its lower T_m) in a supersaturated phase [12, 21]. Usefully, some of the free Bi binds to the Mn_0.5+xBi_0.5−x facets in such a way that its μ_ℓ (≅−0.27 μ_B per atom) is set up opposite to μ_s in the core of the robust exchange-coupled core–shell magnet with modified magnetic properties. These can be formed in different model shapes, as shown in figure 1(d). Here, the excess Bi fulfils multiple roles in terms of its blending with Mn_0.5+xBi_0.5−x−Bi in small core–shell crystallites: (i) a grain refiner, (ii) a surfactant or a lubricant, (iii) a surface modifier, (iv) a reactive catalyst, and (v) an erosion resistance, in addition to readily allowing the tuning of lattice strain, twined boundaries, vacancies, and in turn the magnetic properties of the hybrid Mn_0.5+xBi_0.5−x−Bi, x ⩽ 0.05, structure, as shown below.

As anticipated, the master alloy of Mn_0.5+xBi_0.5−x−Bi, x ⩽ 0.05, contains XRD peaks indicative of free Bi, dispersed in a major Mn_0.5+xBi_0.5−x phase of small crystallites, in a Bi reinforced hybrid nanocomposite. When wet milled in powder form with glycine in a fluid medium (hexane), the refined Bi blends with the crystallites (alloy) and is finely dispersed into the phase boundaries in an intergranular phase due to thermally induced mechanochemical reactions. The refined powder thus contains a multiple structure of twins and defects (Bi vacancies, Mn/Bi interstitials, antisite disorders, etc), displaced in different orientations at the nanometer scale in a texture [40, 41]. This makes it a nontrivial precursor to the growth of tiny crystals in anisotropic shapes via thermal annealing. As a result, the XRD patterns in figures 2(a)–(c) reveal that a milled sample, heated at 573 K, slowly diffuses the Bi in a stable Mn_0.5+xBi_0.5−x, which ultimately results in a major phase (x → 0) of small core–shell crystallites over a t_p = 72 h period. Thus, the sample has preferentially grown in multiple facets (see modules given in the inset), exhibiting enhanced (101) and (110) XRD peaks at the expense of the (102) peak. The lattice expansion, when Bi occupies regular sites in the crystallite, results in a shift in the XRD peaks over a larger interplanar spacing d_hkl in the hexagonal Mn_0.5+xBi_0.5−x crystal structure (P63/mmc space group [45]). Figure 3 shows Retvield refined XRD peaks of the critically annealed alloy, t_p = 96 h, owing to the presence of only a 2%–3% residual phase of free Bi (crystalline). A longer period of heating under these conditions results in a greater degree of Bi separation. No separate Mn peaks are visible since it has dissolved into a Bi-rich Mn_0.5Bi_0.5−Bi alloy, with lattice parameters a = 0.4289 nm and c = 0.6118 nm, a crystal density of ρ = 8.994 g cm⁻³), a lattice volume V_c = 0.0975 nm³, and D = 35 nm. Larger values of a = 0.4290 nm and c = 0.6130 nm (ρ = 8.971 g cm⁻³) prevail in bulk Mn_0.5Bi_0.5 [8, 45]. Small core–shell Mn_0.5Bi_0.5−Bi crystallites adopt a larger ρ-value as a consequence of an inbuilt compressive surface layer, and free Bi occupying its vacancies, as described below.

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In the model reaction, free Bi at 573 K in the mixed Mn_0.5+xBi_0.5−x−Bi alloy diffuses via deep interfaces, travels and fills the vacancies in the Mn_0.5+xBi_0.5−x, x → 0, in a pool, resulting in the tailored XRD peaks of the annealed samples. Thus, the most intense (101) XRD peak (normalized intensity I₁₀₁ = 1.0) in figure 4 has slowly grown at the expense of a major (012)* peak of the free Bi, whereby it first shifts over a larger diffraction angle 2θ (smaller d₁₀₁), t_p ⩽ 24 h, before reverting to lower 2θ-values on further heating, t_p ⩽ 72 h. With reference to the second (102) and third (110) most intense peaks, with intensities of I₁₀₂ = 0.45 and I₁₁₀ = 0.35 for the starting sample, I₁₁₀ has increased to 0.41, while I₁₀₂ has reduced to 0.34, based on heating over a period of 96 h. Larger d₁₀₁ and d^* ₀₁₂ values exist due to the presence of significant Bi vacancies, reported to be as much as 8% for a rapidly cooled Mn_0.5Bi_0.5 alloy [12]. The vacancies appear to be filled in a recurring lattice during the course of heating the sample, where a solid–liquid pool tunnels hot Bi atoms diffuse in this example of vacancy-induced shear bands of 'microscopic through channels'. As a result, it acquires an anisotropic growth of small crystallites. Any excess Bi used here not only diffuses via reactive facets and fills-up Bi sites in a growing crystal lattice in an Mn_0.5Bi_0.5−Bi pool, but also prevents the Mn from precipitating over by raising its solubility in a hybrid nanostructure at this temperature. As a result, after 96 h heating, only a small trace of the (012)* Bi peak remains, compressed to d*₀₁₂ = 0.3275 nm, along with d₁₀₁ = 0.3170 nm, due to the compressive effect of the small core–shell Mn_0.5Bi_0.5−Bi crystallites. Presumably, this is the thin inbuilt Bi surface layer, tightly bound over the Mn_0.5Bi_0.5 facets so as to compress the lattice in a conjoint core–shell structure. Other XRD peaks (102), (110)*, and (110) in figure 4 have reordered congruently at the perspective 2θ-values. The (110)* Bi-peak gradually loses its intensity as the free Bi migrates to Mn_0.5Bi_0.5, growing in an anisotropic (101)/(110) crystal texture.

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The microscopic model proposed in figure 4 (bottom) describes a possible pathway for the free Bi whereby local melting causes it to diffuse, travel, and drive a surface reaction in a pool of Mn_0.5+xBi_0.5−x−Bi (on heating at 573 K) in order to fill the Bi sites in the Mn_0.5+xBi_0.5−x, x → 0, lattice. Conversely, as a supersaturated sample of Mn_0.5Bi_0.5−Bi obtains a critical D ⩾ 35 nm size (table 1) due to a longer period of heating, it precipitates out the excess Bi in a reverse reaction, forming an Mn-rich Mn_0.5Bi_0.5−z + zBi, z ⩽ 0.05, alloy of small core–shell crystallites. A high yield LTP-MnBi alloy, made using different methods, thus always ends with a residual 2%–5% Bi [19–29]. An excess of Mn(z), as per a smaller atomic radius of 0.130 nm as compared with Bi at 0.182 nm [46], results in a 0.23% lower V_c in terms of its saturated value at t_p ∼ 72 h. As shown in the analysis of the XRD peak broadening in the Williamson–Hall plot [47], the core–shell exhibits a lattice strain of γ = 0.17–0.22%, which barely decreases even in bigger crystallites, D = 38 nm (27 nm at t_p ∼ 0), grown at 573 K over t_p → 96 h (table 1). This may be ascribed to the shell storing the strain in the small core–shell crystallites. Figure 5 depicts (a) a crystal Mn_0.5Bi_0.5 unit cell of (b) Mn and Bi atoms ordered in terms of magnetic moments of (c) valence electrons, with (d) model facets as observed in its XRD pattern. In this case, the x-ray beam imposes a sample on its (101) facet at an angle, as shown in figure 5(e), so as to exhibit XRD with its most intense (101) peak, though it also shares the increased anisotropic XRD intensities of conjoint facets (002), (102), (110), or (201) proportionally to their cross-sections, as can be observed here for a sample of a (101) crystal texture.

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The HRTEM images shown below, studied along with the SAED patterns, clearly display how the master alloy Mn_0.5+xBi_0.5−x–Bi, x ⩽ 0.05, has grown in multiple facets of small core–shell crystallites after heating at 573 K, and that the free Bi has slowly diffused therein via a solid–liquid interface in small pools at a length scale of depletion of a few nanometers. Multiple twins displaced in different orientations have previously been reported in shape memory alloys of nanostructures [40, 41]. A given alloy thus creates a pattern of multiple martensite/austenite phases. A recurring lattice strain (a vital variant), in Bi diffused into free sites, dictates nontrivial growth of the unified structure of the texture, as described above in terms of the XRD patterns of the annealed samples. For example, well-defined core–shell crystallites are shown in HRTEM images at different scales in figures 6(a)–(c) for the wet-milled alloy in glycine formulated in this work. Uniquely, these are grown in a specific shape of fullerene C₆₀ like balls of multiple facets (primarily pentagons, hexagons, octagons, or dodecagons), with widths in cross-section of W = 40 − 100, bounding a thin peripheral surface layer of θ ≅ 1.9 nm thickness. Their corners appear to be bent inwards, as though tightly reinforcing the surface-to-volume ratio in a rigid core–shell structure, with a fairly sharp size distribution, in that almost 80% of the sample exhibits an average width of W = 60 nm. A typical SAED pattern in figure 6(d) reveals that the crystallite has grown systematically in (101) regular lattice arrays at a regular separation of d₁₀₁ = 0.3215 nm. A few twins (Ts) of white spots are observed at off lattice points (weaker intensities) indicating the presence of a few vacancies and/or interstitials in the regular lattice of a single Mn_0.5+xBi_0.5−x, x ⩽ 0.05, nanocrystal. The arrow marks how lattice arrays twisting on a twin boundary are inbuilt at the facets in a core–shell structure.

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Figure 6(e) displays an HRTEM image of three consecutive edges, A, B, and C, of a cross-section (a polygon) of a Mn_0.5+xBi_0.5−x−Bi core–shell crystallite, showing how it varies in terms of (i) atomic stacking at edges, (ii) internal angles, and (iii) surface layers on its different facets. A single surface strip (whitish) is marked on facet A, where δ = 1.90 nm thickness, while 3 to 4 substrips are split up in a specific pattern on facet B, where δ = 1.95 nm, as per the topotactic structures and stacking energies of the surface atoms. Moreover, edges A and B of the internal angle where φ = 140° are duly stretched over the subsequent B and C edges of φ = 145° in a markedly truncated dodecagon, as given in figure 6(f). Finally, a thin polygon, primarily grown with (101) facets, reveals (101) lattice arrays, where d₁₀₁ = 0.3215 nm, in figure 6(e), in the form of a fullerene-like shape, as shown in close-up in the inset. A Bi-diffusion reaction, proceeding undisrupted at 573 K, results in a regular Mn_0.5+xBi_0.5−x, x → 0, growing slowly (without deforming into a twin) in a presumed 'microscopic through channel' over its reacting facets, as illustrated in the model in figure 6(g), in a (101) crystal texture.

A further closer view of the lattice images of a core–shell crystallite, e.g. PQ in figure 6(e), reproduced in figure 7, produces futher insights into its internal structure, surface topology, and peripheral surface boundaries. A distinct boundary layer (whitish), where δ = 0.65 nm, is marked on facet A, which looks to be rather darker on facet B, where δ = 0.61 nm, possibly due to a (002) stacking of Bi atoms (a rhombohedral structure) in this module, i.e. stretched over a bulk d₀₀₂ = 0.5930 nm Bi-value [45]. Presumably, this interlocks the (101) lattice arrays of a single crystallite enclosed inside it like a cage, leading to slight compression near the surfaces, so that d₁₀₁ = 0.3205 nm in figures 7(a) and (b), over the inner zone D in figure 7(c), d₁₀₁ = 0.3215 nm, due to the tightly binding surface layer (like a belt) of a closely exchange-coupled core–shell structure. In a counterpart effect, characteristically diffuse d₁₀₀ = 0.4765 nm Bi-arrays are marked on the (101) arrays on the BC edges in figure 7(d). Furthermore, the (101) lattice arrays are more converged, so that d₁₀₁ = 0.3168 nm, near surface P in sample PQ in figure 7(e), containing multiple (102) lattice arrays, where d₁₀₂ = 0. 2362 nm, and a topotactic Bi-surface layer (d₁₀₂ = 0.2670 nm), as marked in boundary Q in figure 7(f). Moreover, on heating at 573 K, one core–shell Mn_0.5+xBi_0.5−x, x → 0, grows in a 3D-prism on a basic building block of a 2D-polygon of atomic arrays. This appears with a hexagonal-shaped cross section (20–80 nm long and W = 20–60 nm) in the HRTEM images as viewed at different scales, shown in figures 8(a)–(c), for a sample heated for 48 h at 573 K.

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The above HRTEM images show an example of a surface strained sample that exhibits SAED spots in rings, as shown in figure 8(d), from its (101), (002), (102) and (110) crystallographic planes, with d_hkl (radii) of 0.3170, 0.3050, 0.2445, and 0.2165 nm, respectively. Presumably, the d₀₁₂ Bi-surface layer shares the bright ring (0.3295 nm radius) near the centre of the SAED pattern in a core–shell sample. Moreover, (101) lattice images (d₁₀₁ = 0.3170 nm) obtained from the rectangular facet of a polygonal prism, as shown in figure 8(e), superpose over broad interference fringes (of 0.65 nm and 1.05 nm separations in regions-1, 2) due to electron beams interfering in reflections from successively inbuilt surface layers in the sample. Similar interference fringes had been observed in C-sp² coated spinel oxides [11, 44] and gold [48] core–shell nanostructures. Figure 8(f) compiles the different modules of anisotropic shapes of core–shell crystallites derived from building blocks of a 2D-pattern of atomic arrays in a 3D-hexagonal lattice. In view of the observed lattice pattern of atoms, a model core–shell crystallite in figure 8(g) contains three microscopic regimes (i) an average atomic density in a deep core below its surfaces, (ii) a compressed atomic density near interfaces, and (iii) a stretched surface layer in a hybrid structure. All three factors collectively tailor the magnetic properties in a core–shell structure, as discussed below.

An annealed alloy Mn_0.5+xBi_0.5−x−Bi, grown in a (101) texture, and comprising small core-shell crystallites filled with Bi sites (x → 0) , exhibits markedly improved M_s, M_r, H_c, and K₁ values in a wider M-H hysteresis loop at room temperature. For example, figures 9(a)–(e) compares M-H loops, measured over H ⩽ 50 kOe at temperatures of 300, 325, and 350 K , for samples annealed at 573 K for 0, 24, 48, 72, and 96 h, respectively. Magnetization (M) does not saturate below a specific level of H fields, in lieu of a higher H_a = 2K₁/M_s > 50 kOe in these samples of randomly oriented single domains. In this case, the M_s is stipulated by extrapolating M over high-field susceptibility (χ_m), in accordance with the well-known law relating to saturation [49]:

$$\begin{equation}M = {M_s}\left[ {1 - \frac{4}{{15}}\frac{{K_1^2}}{{M_s^2\,{H^2}}}} \right] + {\chi _m}H.\end{equation} \tag{ 1 }$$

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Here, H → ∞, a nonlinear χ_m(H) extends over H-induced rotations of unsaturated spins in superparamagnetic (SPM), paramagnetic (PM), or antiferromagnetic (AFM) species along a net induced M-value. As given in table 2, a low value of M_s = 20.5 emu g⁻¹ obtained at 300 K in the master alloy has increased to 72.5 emu g⁻¹ for a sample annealed for 96 h at 573 K. A marginally lower 70.0 emu g⁻¹ is measured at 325 K, and 67.4 emu g⁻¹ is recorded at 350 K, with a small thermal coefficient, β = (∂M_s/∂T)M_s ⁻¹ ≅ −0.14% K⁻¹, in terms of heat induced spin relaxations. These are better M_s values in comparison to other reported high H_c values (compared in table 3) achieved using refined alloys of small crystallites [9, 26,27,28–31], which usually lose M_s on canted (or SPM) surface spins.

As plotted in figure 10(a), M_s occurs in two microscopic steps, 1 and 2, as the Mn_0.5+xBi_0.5−x−Bi alloy is heated over t_p at 573 K. In region 1, a peak 1 evolves at t_p ∼ 24 h, superposing over a linearly growing M_s in region 2 into a denser alloy, where ρ_c = 8.994 g cm⁻³. In addition, K₁ = 0.9 × 10⁷erg cm⁻³ has peaked rapidly at this point in figure 10(b), stiffly growing over a given t_p of a critical 6.3 × 10⁷erg cm⁻³ value at t_p ∼ 96 h. H_c is enhanced to 9.850 kOe over 5.010 kOe (at t_p ∼ 0) in a broad peak 2 in figure 10(c) for the sample of critical single domains (size D_c = D ≅ 33 nm) at a critical t_p ∼ 48 h. It falls to 4.345 kOe in mutidomains at later t_p. The H_c is progressively enhanced in peak 2, measured at higher 325 K and 350 K temperatures, with a positive thermal coefficient, (∂H_c/∂T) H_c ⁻¹ ≅ 0.93% K⁻¹. An H_c value as high as 14.435 kOe is thus obtained at 350 K in critical single domains with K₁ = 5.2 × 10⁷erg cm⁻³. Here, H_c falls on either side of D_c ; where D < D_c, this is due to the growth of SPM species (region 1), and where D > D_c, this is due to the growth of multidomains (region 2) in a wide distribution of D-values. Other details of magnetic parameters obtained at three temperatures for the annealed alloys of model D-values of the core-shell crystallites are given in table 2.

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Next, we review the mechanism by which of M_s is promoted as Bi diffuses in a Bi deficient Mn_0.5+xBi_0.5–x, x ⩽ 0.05, sample (having been annealed at 573 K), filling the available Bi vacancies, and replacing some Mn at antisites till x → 0. In contrast, some Bi sites are exchanged with Mn atoms in an Mn doped off-stoichiometric alloy of an antisite lattice disorder, which not hampers only net M_s, but also other functional magnetic properties. Thus, a case where two successive Mn atoms either displace via a vacancy, or an antisite is filled by a Mn atom, would result in an AFM scheme of spins (a so-called a 'spin-blister') in place of an FM order, as would usually be found in pure Mn or its many alloys [7, 35]. As a result, the net M_s decreases, as described via the spin model shown in figure 10(d), due to the shorter Mn–Mn distance, as is clearly reflected in the (101) XRD peak in the sample annealed for 24 h at 573 K in figure 4(b). The value recovers as blisters dissolve in the course of further annealing. A significant value, μ_ℓ > 0, shares the p-d hybridization occurring between Bi and Mn atoms in an exchange-coupled magnet. Further, Mn-d⁵ and Bi-p³ spins, when ordered in opposition at alternate sites, form a ferrimagnetic (FIM) structure with a strong Mn-Bi-Mn superexchange interaction of p-d electrons, as described in figure 5(c). For example, N(sp³), part of the same VA group as Bi, shares a significant μ_s when bonded to C-sp² [50, 51]. A derivative C₃NH_0.135 exhibits the wide M-H loop, with M_s = 35.2 emu g⁻¹ (at 1.8 K) and T_C = 96 K, of a semiconductor magnet [51]. The Mn_0.5+xBi_0.5–x, x → 0, alloy is therefore shown to hold a net spin-magnetic moment per formula unit:

$$\begin{align}{\mu _s}\left( t \right) &= \frac{1}{2}{\left\{ {\mu _{s1}^2 + \mu _{s2}^2 \pm 2{\mu _{s1}}{\mu _{s2}}\mathrm{cos}\, \phi } \right\}^{1/2}} \nonumber\\ &\cong \frac{1}{2}\left\{ {{\mu _{s1}} - {\mu _{s2}}} \right\}\,,\,\,as\,\phi \to 180^\circ ,\\ &\ge\!{1.03}\,{\mu _B}\,{\text{ }}\left( {{\text{FIM spin - lattice}}} \right){\text{,}}\nonumber \end{align} \tag{ 2 }$$

where μ_s1 ≡ g₁{S₁(S₁ + 1)}^½ ≅ 5.92 μ_B, μ_s2 ≡ g₂{S₂(S₂ + 1)}^½ ≅ 3.87 μ_B, g₁ ≅ g₂ ∼ 2.0 denotes the Lande g-factor, and ϕ denotes the angle between the S₁ = 5/2 and S₂ = 3/2 spins of Mn and Bi atoms, respectively. As ϕ → 180°, it tunes a minimum μ_s(t) ≅ 1.03 μ_B, i.e. still nearly twice the value of nickel (M_s = 54.4 emu g⁻¹). Generally, μ_s2 is quenched over a large atomic Bi space, leaving behind a net μ_s(t) ≅ 3.42 μ_B (72.5 emu g⁻¹), as observed here. A huge μ_s(t) ≅ 4.90 μ_B (or 103.7 emu g⁻¹) would incur per formula unit in μ_s1 and μ_s2 set up in parallel. The variants ϕ and μ_s2 can this be seen to control a wide range of M_s values, usually 22 to 81 emu g⁻¹ at 295 K [6–10, 14, 26,27,28–31] in the Mn_0.5+xBi_0.5–x, x → 0, sample, as per its hierarchical structure of spins for a texture of small crystallites. Accordingly, there is scope for the potential scaling-up of M_S (and other magnetic properties) on induced μ_s2 in surface-modified Mn_0.5Bi_0.5−x (x ⩽ 0.05), with N-sp³ and C-sp² spins of light atoms.

In a marked effect of Bi → Mn substitution, the T_C point is gradually enhanced in thermomagnetic curves (measured at a bias field, H = 500 Oe) in figures 11(A) and (B) relating to Mn_0.5+xBi_0.5−x−Bi alloys annealed for t_p ⩽ 72 h at 573 K. As anticipated, this confers a composition-dependent T_C, as a result of the enhanced Mn-Bi-Mn exchange interaction [14, 33], which is maximized in terms of stoichiometry as x → 0. As large a value as T_C = 641.5 K (x ∼ 0) is obtained above 600.5 K in the initial (x ∼ 0.05) after t_p ∼ 72 h. Here, free Mn, if it precipitates at all over a supersaturated alloy, combines with any mobile Bi in the vicinity, forming a stable Mn_0.5Bi_0.5−Bi phase of small core–shell crystallites. A derivative of these curves (figure 11(C)) displays a fairly sharp peak (ΔT_C = 8–20 K width) of a precisely determined value. Here, a wider signal, ΔT_C ≅ 20 K, extends in Bi vacancies and/or antisite Mn order in a wide band of density of states over its average value. An enhanced T_C = 641.5 K, over a bulk Mn_0.5Bi_0.5 value of 628 K [6–8], confers small core–shell crystallites which promote a coupled magnetic and structural LTP → HTP transition. A satellite T_C1 peak, displaced in HTP, converts to a PM state at 645.5 K (figure 11(C)) in a sample heated for up to 96 h, where some Bi segregates into phase boundaries. As expected, well below T_C, the surface magnetism decays in two overlapping micromagnetic regimes, 1 and 2, over two distinct types of exchange-coupled spins in the small core–shell crystallites.

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At near stoichiometry, Mn_0.5+xBi_0.5−x, x ⩽ 0.05, the enhanced FM interaction, in relation to free Bi, diffused to (x → 0) and replaces Mn placed on the antisites in a core–shell crystallite, describing a sharply enhanced T_C(x) in a model DFT relation:

$$\begin{equation}{T_C}\left( x \right)\,\, \equiv \,\,{T_C}\left( 0 \right)\,{\Omega _{12}}\left( x \right)\,{\left[ {\frac{{1 - x}}{{1 + x}}} \right]^p},\end{equation} \tag{ 3 }$$

with

$$\begin{equation}{\Omega _{12}}\left( x \right)\,\, = \,\,\frac{{\,{\Omega _{12}}\left( 0 \right)\,}}{{{{\left( {1 + 2x} \right)}^q}}},\end{equation} \tag{ 4 }$$

where Ω₁₂(x) is a module describing an exchange interaction between the S₁ and S₂ sites of spins, and 'p' is an exponent which modulates them in a recurring variant of x-value. As given in table 2, a normalized Ω₁₂(0) = 1.0 with a model q = 1/3 reproduces an observed T_C = 600.3 K using x = 0.051 (in the initial alloy) and a model p = 1/3 on T_C(0) ≡ 641.5 K, within a deviation ⩽0.4 K. The same model parameters satisfactorily reproduce an intermediate T_C ≅ 629.7 K at x = 0.014, as observed in the 24 h annealed alloy, or T_C ≅ 638.1 K at x = 0.004 in an alloy annealed for 48 h at 573 K. This model duly accounts for and describes an antisite effect of Mn-d⁵ spins on T_C in this alloy of small core–shell crystallites. In figure 11(D), the T_C varies in two maxima in two microscopic regions, 1 and 2, of two overlapping parabolic paths over t_p, as any free Bi in the vicinity diffuses into a stable Mn_0.5+xBi_0.5−x phase. The antisite Mn present in region 1 reorders via vacancies in region 2. A decrease in T_C after a critical 541.5 K value confers a supersaturated alloy, releasing Bi in any further annealing. In general, a module of the magnetic T_C signal in equation (3), which increases the heat, ΔQ = ΔST_C, on disordered spins in an FM → PM transition (with a change of entropy ΔS), can be further extended by formulating its impacts on coupled lattice-expansion, magnetocalory and magnetostriction over the transition, and their associated applications.