Magnetic disorder in TbAl₂ nanoparticles

Measurements of the specific heat c_p(T) on TbAl₂ for the 300 h sample compared to the bulk (unmilled) alloy are shown in figure 2. The λ-anomaly observed for the bulk sample at the ferromagnetic ⇔ paramagnetic transition (T_C = 105 K) is practically wiped out in the nanosized sample. The broadening of the ${c}_{p}({T}_{C})$ transition has been explained as a consequence of the combination of finite-size and surface effects, whose influence increases when decreasing the particle size [19, 20]. However, a careful analysis of the dc-magnetization data reveals that the FM transition does not fully disappear, and the T_C of these nanometric alloys is strongly size dependent and decreases from 103 K for the bulk alloy down to 98 K for the 14 nm (300 h) alloy, as the particle volume is reduced. This reduction of T_C has been explained by a finite-size scaling law [13]. Thus, specific heat measurements seem to be more sensitive to changes in disorder and particle size. Moreover, when comparing the specific heat of this sample with the unmilled one, besides the disappearance of the FM contribution, the c_p(T) shows an enhancement in two different temperature ranges: above 120 K and below 20 K. We aim to better quantify these enhancements by plotting, in the inset of figure 2, the difference curve between the milled and the unmilled samples and marking with arrows the regions in which this difference reaches a positive value. The origin of these two contributions, both at low and high temperatures, has been explained with a model including a modification in the phonon spectrum of atoms at the core and at the surface. Such a modification is not specific of RX₂ nanoalloys but appears on simple nanometric metals as well [21].

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On the other hand, the 300 h milled sample was already reported to show SG behaviour, with a freezing temperature T_f = 45 K [12]. For canonical SG systems, a contribution to the c_p(T), in the shape of a broad maximum, is expected around 1.3–1.5 T_f with a tail, due to short-range magnetic correlations, that may extend up to ≈ 4T_f [22, 23]. Thus, considering the T_f value for the 300 h milled TbAl₂ alloy, a weak SG contribution would be expected around 60 or 70 K. As this contribution is not directly inferred from the curve presented in figure 2, a way to reveal it might be to subtract the electronic as well as the lattice contributions, from both the core and surface atoms in the particle [21]. This is traditionally carried out by choosing a non-magnetic isostructural material (LaAl₂), but if it was to be employed it would be extremely difficult to set in the exactly same nanostructure, affecting the validity of the subtraction.

In this context, we suggest another way round to manifest the feeble magnetic contribution. It is well-known that the application of magnetic field in an ordinary ferromagnet would tend to broaden and shift the λ-anomaly towards higher temperature. Then, assuming that the magnetic field will only influence the contribution with magnetic origin, the differences with regard to the curve measured at zero field will yield information on the magnetic state. These results are presented in figure 3 for applied magnetic fields H = 1, 5 and 9 T. For any applied magnetic field, a maximum around 70 K is observed which is affected by the magnitude of H. Hence this is fingerprint of the magnetic contribution in 300 h nano-TbAl₂. Incidentally the peak temperatures are close to those expected from the above commented freezing temperature.

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The disclosure of a strong disordered magnetic state in c_p(T) agrees well with the behaviour reported for the low-field irreversibility temperature dependence of the saturation magnetization (M_S) of bulk and 300 h milled TbAl₂ alloys [13]. In addition, the analysis of high-field thermal (low-T) demagnetisation of the nanoalloys also provides supportive magnetic information relating the propagation of spin waves whose dispersion relation for ferromagnets is ${\hslash }\omega =\Delta {q}^{2}$, in which Δ is the spin wave stiffness. These experimental data are presented in figure 4 for the two alloys. First we concentrate on the low-T M(T) variation. Visually there is a faster thermal reduction of the magnetisation (Tb³⁺ magnetic moment) in nanometric alloys respect to the bulk. Given that the particles comprise nanometric single crystals it is possible to use the Bloch's law, which considers the temperature dependence of the magnetisation following, if we drop higher order T-dependent terms [26]:

$$\begin{eqnarray}&&M(T)=M(0)(1-{B}_{\alpha }{T}^{\alpha })\end{eqnarray} \tag{ 1 }$$

where M(0) is the saturation magnetisation at 0 K, and ${B}_{\alpha }$ is a coefficient depending on the spin wave stiffness. In conventional ferromagnets, α = 3/2, and hence should be employed for the bulk alloy. However, it has been proposed that there could be modifications in the exponent for nanostructures and α = 2 for MNPs have provided good results [27, 28]. The present fits have been performed within a temperature range up to 0.2 T_C as it is usually considered in conventional magnets [26]. The result gives for the bulk sample ${B}_{3/2}$=12·10⁻⁵ ${{\rm{K}}}^{-3/2}$, which is of order to that reported for GdAl₂ (32·10⁻⁵ ${{\rm{K}}}^{-3/2}$) [29]. In the nanometric alloys, B₂ = 11 · 10⁻⁵ K⁻² and B₂ = 16 · 10⁻⁵ K⁻² for 150 and 300 h, respectively. There is a certain tendency to increase the value as a consequence of a faster demagnetisation process. The tendency to find a B increase when the particle size is reduced has also been studied in other nanometric (ferrite) compounds [27, 30].

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The MNP size-reduction can also be tackled by establishing a correlation between the values of the saturation magnetization and the particle diameter. Using previous magnetisation data [12], it is possible to plot the M_S as a function of ${D}^{-1}$, as shown in the inset of figure 4. There is a linear behavior which can be connected to a reduction of the ${\mathrm{Tb}}^{3+}$ magnetic moment can be explained by the existence of a spin disorder layer at the surface of the particles. In this sense, and as was also applied in other nanoparticle systems [30–32], the width of this surface layer can be estimated from the plot of M_S versus the inverse of the mean diameter of the particles (1/D), assuming a core-shell structure in which the non-magnetic shell has a constant thickness (t). The behaviour of this relationship is well accounted by the expression ${M}_{S}={M}_{\mathrm{bulk}}(1-6t/D)$ [30], being t = 0.8 nm in the case of the TbAl₂ alloys. This value is close to that reported for MnFe₂O₄ nanoparticles (t = 0.7 nm) [31].

Before analyzing the neutron scattering results on the nanosized TbAl₂ alloys, it is convenient to investigate the reference bulk alloy (0 h milled sample). Thus, figure 5 shows the Rietveld refinement (a = 7.7877(6) Å, Bragg error R_B = 3.72) of the neutron diffraction pattern obtained for the bulk TbAl₂ alloy at 10 K, well below the ferromagnetic transition temperature T_C = 105 K [12]. Another diffraction pattern, measured at 150 K, is also plotted in the same figure, in order to notice the differences with the paramagnetic state. At 10 K, the increase of the intensity of several peaks is associated with the magnetic contribution. As expected for a ferromagnetic collinear structure, the magnetic peaks are superimposed to those corresponding to the crystallographic structure. The Rietveld refinement of the neutron diffraction data is consistent with a collinear FM structure with a $\vec{k}=(0,0,0)$ propagation vector, and having the magnetic moments aligned along the c-axis, as shown in the inset of figure 4. The value obtained for the ${\mathrm{Tb}}^{3+}$ ordered magnetic moment from this refinement at 10 K is $8.2{\mu }_{B}$, agreeing with previously reported data [25].

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In the case of the nanosized alloys, we have measured several patterns for the 150 and 300 h milled samples. The Rietveld refinements performed on the 10 K data corresponding to the both milled TbAl₂ alloys are presented in figure 6. The diffraction pattern obtained at 150 K is also shown as a paramagnetic reference. In comparison with the bulk alloy, for 150 h, we can notice a broadening of the peaks and a decrease of their intensity, both effects associated with the particle size reduction. The magnetic core structure is not altered with a ferromagnetic collinear arrangement.

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The results on the 300 h milled alloy follow the same trend (see figure 6), exhibiting a larger broadening of the peaks due to a further reduction of the particle size down to 14 nm. Here, the average ${\mathrm{Tb}}^{3+}$ magnetic moment at 10 K strongly reduces, down to 5.7 ${\mu }_{B}$. This is a microscopic confirmation of the reduction of M_S with decreasing size. From the analysis of all the different neutron diffraction patterns (not shown), we have obtained the temperature dependence of the ${\mathrm{Tb}}^{3+}$ magnetic moment for the bulk sample, the 150 and 300 h milled TbAl₂ alloys; these results are depicted in figure 7. The values obtained at the lowest measured temperature (10 K) are 6.6 ${\mu }_{B}$ for the 150 h sample and 5.7 ${\mu }_{B}$ for the 300 h sample, and they are reduced respect to the value obtained for the bulk alloy. In all cases, the magnetic moment decreases when increasing the temperature, but the reduction is faster for the milled alloys than for the bulk one, in agreement with the above results gathered for the Bloch demagnetisation.

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In order to gain complementary information on the magnetic behaviour of this nanoparticle system, we have performed small angle neutron scattering (SANS) in the sample with the smallest particle size: the TbAl₂ 300 h milled alloy. SANS experiments can supply information on the magnetic microstructure with a resolution of length scales in the nanometer regime [17, 33, 34].

Figure 8 shows the total SANS integrated intensity as a function of the momentum transfer (q), in a standard log-log representation, for different temperatures. SANS total intensity comprises both the nuclear and magnetic contributions (${I}_{\mathrm{tot}}={I}_{\mathrm{nucl}}+{I}_{\mathrm{mag}}$). In the left inset, the intensity variation with the magnetic field is presented, clearly affecting the central q-range. Under an applied magnetic field (H = 8 T), we expect an alignment of the ${\mathrm{Tb}}^{3+}$ magnetic moments along the direction of the magnetic field and, thus, a decrease on the magnetic SANS signal. To extract more information we use the thermal variation of the intensity. At low temperatures (5 K) a definite hump centered around q = 0.04 Å⁻¹ is observed. This contribution decreases for higher temperatures (40 K, 60 K), becoming negligible for T > 100 K. For $q\;\lesssim$ 0.02 Å ⁻¹, the signal remains practically constant. The variation of the signal can be better inspected by the subtraction of the paramagnetic data (T = 150 K) to the lower T patterns, thus extracting the magnetic contribution to the intensity. The result is shown in the right inset of figure 8 where there is peak that decreases with increasing temperature. The peak position is associated to a periodicity of 16 nm [34]. The MNPs are concentrated and closely packed meaning that this is a reasonable value for the particle. The integrated intensity of this peak decreases with increasing temperature (at T = 60 K, the value is 18% respect the intensity at T = 5 K). Respect the correlation length ξ extracted from the peak width, the values are practically constant with temperature giving ${\xi }_{5\;{\rm{K}}}=40.1(4)\mathrm{nm}$; ${\xi }_{40\;{\rm{K}}}=35.8(3)\mathrm{nm}$; ${\xi }_{60\;{\rm{K}}}=36.6(4)\mathrm{nm}.$ This is pointing to magnetic correlations among particles which become slightly more important when the system is at the lowest temperature.

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