IOP PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 24 (2012) 346001 (10pp)
doi:10.1088/0953-8984/24/34/346001
Spin-glass freezing in a Ni–vermiculite intercalation compound 2 , S A Khainakov3 , J Rodr´ıguez Fern´ ¨ C Marcos1 , A Arguelles andez4 and J A Blanco2 1
Departamento de Geolog´ıa e Instituto de Organomet´alica Enrique Moles, Universidad de Oviedo, E-33005 Oviedo, Spain 2 Departamento de F´ısica, Facultad de Ciencias, Universidad de Oviedo, E-33007 Oviedo, Spain 3 Departamento de Qu´ımica Org´anica e Inorg´anica, Universidad de Oviedo, E-33006 Oviedo, Spain 4 CITIMAC, Facultad de Ciencias, E-39005 Santander, Spain E-mail:
[email protected]
Received 23 May 2012, in final form 4 July 2012 Published 7 August 2012 Online at stacks.iop.org/JPhysCM/24/346001 Abstract We report on the magnetic properties of a Ni2+ –vermiculite intercalation compound from Santa Olalla, Huelva (Spain). This modified vermiculite was studied by means of DC and AC magnetic measurements. The existence of two maxima in magnetic susceptibility below 10 K was interpreted in terms of the Cole–Cole formalism as being due to spin-glass freezing in this material. The temperature, frequency and external magnetic field dependences of these anomalies located at temperatures around 2–3 K and 8–10 K in the imaginary part of the magnetic susceptibility, χ 00 , seem to suggest the existence of spin-relaxation phenomena between the magnetic moments of the Ni2+ ions. A dynamic study of the relaxation processes associated with these phenomena considering the Cole–Cole formalism allows us to interpret the anomaly found at 2–3 K according to a law of activated dynamics, obtaining values for the critical exponent, ψν < 1, characteristic of a d = 2 spin-glass-like system, while the maximum observed in χ 00 at 8–10 K can be described by means of a law of standard dynamics with a value of the exponent z of around 5, representative of a d = 3 spin-glass-like system. (Some figures may appear in colour only in the online journal)
1. Introduction
addition, owing to its anisotropic crystal structure, vermiculite is used to examine interesting physical properties such as mixed-cation effects and two-dimensional magnetism [12, 13]. In particular, it is well known that the existence of frustration and disorder is a key feature for understanding the mechanisms of spin-glass (SG), crystallographic disorder or a geometrically frustrated lattice being the principal reasons usually preventing the magnetic moments of a magnetic system from being long-range ordered. The subject of SG has been challenging both experimentalists and theoreticians for more than forty years [14]. The dimensionality of magnetic interactions plays a central role in controlling the critical dynamics of SG systems: the lowest critical dimensionality of short-range Ising SG is between two and three. In the case of dimensionality three, equilibrium of ordered phase is reached at a finite temperature, whereas for two-dimensional systems, this situation is only achieved at zero temperature.
Clay minerals are not only a core problem in geology and soil science, but are also remarkably important as model systems in physics, chemistry and biological science [1–4]. Vermiculite clay is a hydrous phyllosilicate consisting of millions of disordered negatively-charged parallel silicate platelets held together by positively-charged counter ions located in the interlayer region. Disordered materials play a central role in our daily life. In particular, vermiculite is an interesting material suitable for the sorption of heavy metals [5] and organic pollutants that could be very harmful to the environment [6, 7]. These possibilities, together with its low-gas permeability [8] and useful mechanical properties [9], make vermiculite a suitable material for producing clay-based nano-composites [10] that may be employed in many other industrial applications [11]. In 0953-8984/12/346001+10$33.00
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Figure 1. Schematic representation of the unit cells of VO and VNi vermiculites. Right: coordinated tetrahedral and octahedral polyhedrons and interlayer space (T, tetrahedral layer; O, octahedral layer). Left: schematic view of the interlayer space in VO vermiculite showing the interstitial water molecules of types 1 and 2. In VNi, the interstitial water molecules of type 2 are located in the centres of pseudo-hexagonal cavities.
structures using the DIFFaX+ software [22]. The structure of Santa Olalla vermiculite modelled by means of that software [22] corresponds to that of a semiordered layered crystalline material showing the existence of a large density E of defects due to random ± b3 translations along the crystalline [010] direction. A joint analysis combining transmission and reflection x-ray data allowed us to determine both the atomic positions and occupancies of the exchangeable cations and water molecules in the interlayer space (see tables 1–3 of Arg¨uelles et al [20]). Within the experimental uncertainty the structures of the original Santa Olalla vermiculite and the Ni–vermiculite intercalation compounds are almost identical. The 3D structure is described using a disordered stack of two types of 2D building block, which is made up of one talc-type layer and one interlayer space containing hydrated Mg2+ cations (see figure 1). Electron micrograph (TEM and HRTEM) and AFM images show similar morphologies and local structures for both investigated vermiculites, with no indication of the presence of nano-precipitates or amorphous phases [23, 24]. Thus these results confirm the high homogeneity of both samples. The conclusion about the distribution of cations among the different structural positions suggests that the nickel has not only replaced the interlayer Mg2+ cations but has also entered the octahedral layer in the nickel-intercalated vermiculite from Santa Olalla. Additional information on the final details on the location of Ni2+ cations in the nickel-intercalated vermiculite from Santa Olalla can be found in Marcos et al [25]. We have focused our interest on studying Santa Olalla (Huelva, Spain) vermiculite using an exchange synthesis process. In this paper we have undertaken an extensive study of the magnetic properties of Ni2+ –VIC using DC magnetization and AC magnetic susceptibility measured on a superconducting quantum interference device (SQUID). This material behaves like a quasi-2D spin-glass system
Vermiculite intercalation compounds (VICs) are well characterized expanding layered silicates whose basic building block is composed of two sheets of SiO− 4 tetrahedra coupled symmetrically to another sheet of MgO4− octahedra in a tetrahedral–octahedral–tetrahedral 6 layer lattice. Isomorphic substitution within the sheets leads to a net negative charge located on the surfaces of the tetrahedral and octahedral layers. In natural vermiculites, hydrated Mg2+ cations located at the surfaces balance the negative charge of the SiO− 4 anions. As a result, a three-dimensional vermiculite structure is formed by a sequence of tetrahedral–octahedral–tetrahedral layers along the z-axis which are joined to each other by the interlayer hydrated Mg2+ cations. These interlayer Mg2+ cations can be easily exchanged for different magnetic ions forming magnetic VICs [15]. VICs have three kinds of hydration states defined by the number of water layers in the interlamellar space: zero-, one- and two-water-layer hydration states. Magnetic VICs are frequently found in the two-water-layer hydration states [16]. The standard structural formula for the half unit cell can be expressed by X4 Y3 Zi O10 (OH)2 ·n(H2 O) (Si4+ ,
where X = Al3+ ) is the cation of the tetrahedral layer, Y = (basically Mg2+ , Al3+ , Fe2+ , Fe3+ ) is the cation of the octahedral layer, Z = (Mg2+ in natural vermiculite) is the interlayer cation and n and i refer to the numbers of water molecules and interlayer cations, respectively. Interest in these vermiculites has been recently rekindled by the observation that the crystal structure and hydration states evolve with pressure and temperature [17, 18]. The complex crystal structure of Santa Olalla vermiculite has been successfully refined [19–21] from x-ray powder diffraction data by a method based on a recursive description of faulted 2
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The c-axis stacking structure for VO and Ni2+ –VIC with two-water-layered hydration states (2-WLHS) was examined by x-ray diffraction, monitoring (00l) Bragg reflections. A vertical INEL XRD RG3000 Debye–Scherrer goniometer ˚ cobalt tube at 40 kV and 35 operating with a λ = 1.7885 A mA was used. The diffractometer was equipped with a curved position sensitive detector (INEL CPS 120) used to collect the x-ray diffraction patterns in transmission geometry in order to minimize the strong texture effects resulting from the layered nature of vermiculite. The DC magnetization and AC magnetic susceptibility of the VNi sample were measured using SQUID and PPMS magnetometers (Quantum Design). The samples were cooled from 305 K down to 1.9 K at H = 0 Oe. After applying an external magnetic H field at 1.9 K, the zero-field cooled magnetization (MZFC) was measured with increasing T from 1.9 K up to 50 K. Samples were heated up to 100 K and then annealed at this temperature for 10 min. Subsequently, the field-cooled magnetization (MFC) was measured with decreasing T from 50 K down to 1.9 K in the presence of the same magnetic H field. The DC magnetization was measured between 1.9 and 305 K under an applied magnetic field of 1 kOe. AC magnetic susceptibility (χ = χ 0 + iχ 00 ) from 50 K down to 1.9 K was measured under different DC magnetic fields (H = 0, 20, 50, 100, 200, 500, 1000, 2000 Oe), and at several frequencies (ν = 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500 Hz) and amplitudes (h = 1–2 Oe) for the AC magnetic field.
in which competing ferromagnetic and anti-ferromagnetic interplanar exchange interactions are responsible for the complex magnetic behaviour found. The paper is organized as follows. Section 2 is devoted to the experimental details. The results and discussion of the DC and AC magnetic susceptibilities are presented in section 3. Finally, conclusions are drawn in section 4.
2. Experimental details The Mg–vermiculite from Santa Olalla (Huelva, Spain), herein denoted as VO, was used as the starting material for this study. The material was powdered in a Retsch 25SM-1 ultracentrifuge mill in order to obtain a mean particle diameter of ≤80 µm. A vermiculite intercalation compound containing Ni2+ instead of Mg2+ as the interlayer cation, henceforth VNi, was prepared by ion exchange. The ion exchange reaction for the preparation of the inter2+ calation compounds is Mg2+ x –vermiculite + x[Ni(H2 O)6 ] 2+ 2+ → Nix –vermiculite + x[Mg(H2 O)6 ] if one assumes no change in iron cation charge. A suspension (VNi) was prepared by adding 0.67 g of VO to 50 ml of a 1 M aqueous solution of Ni2+ acetate. This concentration ensured a suitable amount of Ni2+ cations for producing the exchange process, according to the theoretical cation exchange capacity of VO. The pH of the suspension after preparation was 1.79 for VNi. Mechanical stirring was carried out at 60 ◦ C for 12 days and the solid was centrifugally separated, washed with deionized water until the chloride ions were removed and then air dried. As the result of the ion exchange mechanism in VO, the hydrated Mg2+ cations that were located in the interlayer space were substituted by Ni2+ cations. Chemical analyses were performed by means of energy dispersive spectroscopy, by using a CAMEBAX-MBXSX-50 electron microprobe (with an acceleration voltage of 15 kV and a beam current of 15 nA). The water content was determined by means of thermogravimetric analysis using a Mettler Toledo TGA/SDTA851 instrument, in an alumina crucible under a dynamic nitrogen atmosphere of 50 ml min−1 at a heating rate of 10 ◦ C min−1 . The cation content was obtained using the MINPET program (MINPET Geological Software, 146 DV Chateau Masson-Angers, Quebec, Canada). The calculations to determine the cation contents were based on 12 anions (10 structural oxide ions and 2 OH− ions). Because electron microprobe techniques do not permit one to distinguish between Fe2+ and Fe3+ , the Fe2+ contents for VO and VNi were estimated from the semi-empirical formula proposed by Laird and Albee [26]. The distribution of cations among the different structural positions was carried out according to F¨oster [27], and the averaged structural formulae obtained are for VO
3. Results and discussion Figure 2 shows the x-ray diffraction patterns for VO and VNi collected at 300 K. All the diffraction peaks can be indexed with single-phase monoclinic C2/c (space group No. 15), thus no other phases were detected in the x-ray powder diffraction patterns. The interplanar distance, d002 , for ˚ the reflection (002) was determined to be 14.33 ± 0.07 A for VNi, which is in good agreement with values reported ˚ and 14.330 ± 0.007 A) ˚ [12, previously (d = 14.36 ± 0.01 A 16]. The high intensities observed for the first diffraction peak (2θ ∼ 8◦ ) and other peaks located at higher 2θ are related to (00l) reflections. Due to the platelet-like shape of vermiculite grains, typical of most clays, a [00l] fibre texture is expected with an enhancement of the x-ray signal from the basal plane. Moreover, the indexing of the x-ray diffraction patterns reveals the presence of some extremely asymmetric broadened peaks corresponding to the semiordered stacks, where the reciprocal space cannot be described by a set of hkl reciprocal points (with h, k, l integer), but rather by modulated reciprocal rods hk with continuous variation with intensity. All these features are reported in Arg¨uelles et al [20] and Marcos et al [25]. Figure 3 shows the temperature dependence of the reciprocal magnetic susceptibility, χ −1 , for VNi. The experimental variations of χ −1 do not behave linearly with temperature. We therefore used a modified Curie–Weiss law with a magnetic susceptibility term, χ0 , independent of
(Si2.83 Al1.17 )(Mg2.46 Al0.30 Fe0.22 Ti0.02 )O10 (OH)2 (Mg0.38 Ca0.03 Na0.02 )·3.82H2 O and for VNi (Si2.71 Al1.29 )(Mg2.20 Al0.25 Fe0.15 Ni0.39 )O10 (OH)2 (Ni0.44 )·2.61H2 O. 3
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Figure 2. Observed (points), calculated (solid line) and differences in (bottom) room temperature x-ray diffraction profiles for Santa Olalla vermiculite (VO) and Ni2+ –VIC (VNi). The vertical marks correspond to the positions of the allowed Bragg reflections associated with the crystal structure. The observed–calculated difference pattern is depicted at the bottom of the figure (see text for more details). The solid lines were calculated using structural information taken from Arg¨uelles et al [19, 20] and Marcos et al [25] (see text).
Figure 4. Temperature dependence of MZFC (red solid points) and MFC (blue solid points) for VNi under several selected magnetic fields (H = 10, 20, 50, 100, 200, 1000, 2000 Oe). The arrows point to the irreversibility temperature, Tirr (see text).
the interval 50 K ≤ T ≤ 305 K leads to values for the effective paramagnetic moment of µeff = 3.1(2) µB and the paramagnetic Curie temperature of θ = 20.4(2) K for VNi, respectively. The value of µeff corresponding to the expected theoretical free-ion value is µeff (Ni2+ ) = 3.2 µB . The value of χ0 ∼ 6 × 10−6 emu g−1 Oe−1 is similar to that found in other vermiculite compounds [15], and includes the diamagnetic contribution of the closed-shell electrons and the possible contribution from Fe2+ and Fe3+ ion impurities. Figure 4 shows the temperature dependence of the DC magnetic susceptibility measured for different magnetic fields. Under low fields (H = 10 and 20 Oe), the magnetic susceptibility exhibits a sharp increase below 15 K and an irreversibility between ZFC and FC susceptibility data below 10 K which are typical of a weak ferromagnetic system and suggest a superparamagnetic or spin-glass behaviour.
Figure 3. Temperature dependence of the reciprocal magnetic susceptibility (points) for VNi measured under a magnetic field of 1 kOe. The inset shows the linear high-temperature behaviour, where the solid line corresponds to a fit to a Curie–Weiss modified law (see text).
temperature [28], χ = χ0 +
C , T −θ
(1)
where C is the Curie–Weiss constant and θ is the paramagnetic Curie temperature. Least squares fitting of the data in 4
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Figure 5. Magnetic field dependence (H2/3 -scale) of the irreversibility temperature Tirr (points). The solid line corresponds to a linear fit (see text for more details).
The aforementioned irreversibility persists even under a magnetic field of 2 kOe. An increase in magnetic field strength tends to smooth out the DC magnetic anomalies and reduces the magnitude of the susceptibility. On the other hand, superparamagnetic and spin-glass systems both have similar characteristics, such as [29] a peak observed at a certain temperature (the so-called blocking temperature for superparamagnets and freezing temperature for spin-glasses) in ZFC magnetization versus temperature and a bifurcation in the ZFC and FC magnetization versus temperature curves below a characteristic temperature, Tirr . In our case, Tirr depends linearly on H2/3 (a linear dependence of Tirr with H2/3 is called the de Almeida–Thouless (AT) line [30]). The existence of the AT line constitutes strong evidence for the existence of spin-glass freezing (see figure 5). We shall discuss this issue further when reporting on the AC susceptibility. The real (χ 0 ) and imaginary (χ 00 ) magnetic susceptibilities are presented in figures 6 and 7 for selected frequencies ν (ω = 2π ν) and applied magnetic fields. Both χ 0 and χ 00 exhibit pronounced anomalies whose amplitude and peak positions depend on the frequency of the applied AC magnetic field and the strength of the DC magnetic field. It is found that χ 0 is characterized by the existence of two clear maxima centred around frequency-dependent freezing temperatures of Tf = 2.2 and 9.7 K for 0.1 Hz, the magnitude at the maxima increasing (decreasing) with increasing frequency (magnetic field); the frequency dependence of χ 0 is extremely strong for both peaks, their position shifting to higher temperatures with increasing frequency, which is a characteristic feature of an SG ordering. A criterion that is often used to compare the frequency dependence of Tf for different SG materials is that of estimating the relative shift in the freezing temperature per decade of frequency, 1Tf /Tf 1(log10 w). For VNi, we obtained values of 0.13 and 0.021 for the maxima located at 2.2 K and 9.7 K, respectively. These values are similar to those expected for a superparamagnet (0.1–0.3) and an insulating SG system (0.005–0.05), respectively [31]. The maxima of χ 0 for ν > 10 Hz are shifted below the lowest measured temperature, 2 K. In order to estimate the relaxation times, τ , we used the phenomenological approach of Cole
Figure 6. Temperature dependence of the real χ 0 (a) and imaginary χ 00 (b) magnetic susceptibilities for VNi measured under an AC magnetic field of 1 Oe. Data were taken at frequencies from 0.1 to 1000 Hz. The solid lines are guides for the eye. The variations have been shifted vertically for clarity by 0.001 (0.0005) emu g−1 for χ 0 (χ 00 ).
and Cole [32], which involves modelling the dynamics at a given temperature onto a symmetric distribution of relaxation times when represented on a logarithmic timescale [13]. The Cole–Cole equation is written as χ (ω) = χs +
χT − χs , 1 − (iωτc )1−α
(2)
where χT and χs are the isothermal and adiabatic susceptibilities, respectively, and τc is the mean relaxation time. The parameter α determines the distribution width. Equation (2) can be decomposed into χ 0 (ω) = χs + " ×
χT − χs 2
sinh [(1 − α) ln (ωτc )] 1− cosh [(1 − α) ln (ωτc )] + sin
# απ 2
, (3a)
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Figure 8. Argand representation (χ 00 versus χ 0 ) corresponding to the observed maxima at Tf1 (a) and Tf2 (b) in the AC magnetic susceptibilities for selected temperatures for VNi. The symbols correspond to the frequencies in figure 7, while the solid lines are fits to equation (4).
This allows us to adjust the parameters χT , χs and α, and use equations (3a) and (3b) to determine the value of τc as a function of temperature. Figure 8 shows some illustrative examples of the Cole–Cole representation. Note the shift of data points for the first peak of χ 0 at Tf1 = 2.2 K from a nearly isothermal susceptibility at 2 K via a maximum in χ 00 around T = 2.6 K to a non-vanishing susceptibility at 3.2 K, owing to the overlapping with the second maximum of χ 0 found at Tf2 = 9.7 K. This feature indicates a mean relaxation time, τc , that evolves from τc < 10−6 s to τc ∼ 10−1 s, moving quickly through our time window covering five decades within a few K. For the selected temperatures in figure 8, the results of these fittings are shown in figures 9 and 10 as a function of frequency. The refined values of the isothermal magnetic susceptibility χT for the first and second anomalies are (1.52 ± 0.04) × 10−3 and (0.92 ± 0.03) × 10−3 emu g−1 at Tf1 = 2.2 K and Tf2 = 9.7 K respectively for 0.1 Hz decrease around these anomalies according to the variations depicted in figure 6, while the adiabatic magnetic susceptibility χs is almost constant and equal to (4 ± 1) × 10−4 emu g−1 . In contrast, the value of α (the magnitude of α determines the width of the distribution of the relaxation time) changes from α = 0.50 ± 0.02 for the first anomaly at Tf1 , to α = 0.76±0.03 for the second one at Tf2 , which is characterized by a wider distribution. This feature indicates that the relaxation processes of the two anomalies are different (see below). It is clear from all these analyses that the Cole–Cole representation provides an adequate description of the experimental data of VNi.
Figure 7. Dependence of the real χ 0 and imaginary χ 00 magnetic susceptibilities taken at 1000 Hz for VNi measured under selected DC applied magnetic fields (H = 0, 20, 50, 100, 200, 500, 1000, 2000 Oe). The variations have been shifted vertically for clarity by 0.001 (0.0005) emu g−1 for χ 0 (χ 00 ).
χT − χs χ (ω) = 2
"
cos
00
απ 2
#
cosh [(1 − α) ln (ωτc )] + sin
απ 2
. (3b)
However, in order to extract the parameters χT , χs , α and τc directly from the experimental data, it is more appropriate to fit the locus of χ in the complex plane (known as Argand’s plot) first, χT − χs χ 00 χ 0 = − 2 tan s +
π 2
1 (1 − α)
(χ 0 − χs ) (χT − χ 0 ) +
(χT − χs )2 . 4tan2 π2 (1 − α) (4) 6
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Figure 9. Frequency dependences of χ 0 and χ 00 for a selection of temperatures around Tf1 . The solid lines are fits to equations (3a) and (3b) for VNi.
Figure 11. Temperature dependence of the mean relaxation time, τ , associated with Tf1 = 2.2 K (a) and Tf2 = 9.7 K (b) derived from the Cole–Cole representation for VNi. The solid line is the fit to (a) τc ∼ exp(b/T (1+ψν) ) and (b) τc ∝ (T − T0 )−zν (see text for more details).
We shall turn in what follows to the temperature dependence of the mean relaxation time, τc , obtained from the fitting to the imaginary magnetic susceptibility using equations (3a) and (3b). Figures 9 and 10 show that the dynamics of VNi slows down by five decades in time within the range of temperature investigated. The solid line in figure 11 for Tf2 = 2.2 K represents a fit to Ln(τc /τ0 ) = (b/T (1+ψν) ), where b is a measure that controls the fluctuations of the energy barriers and 1+ψν is associated with the low-temperature divergence of the correlation length. The fitting given in figure 11 extends over the temperature range of 2–3.5 K for the first peak of χ 0 , yielding values of ψν ∼ 0.54 ± 0.05, τ0 ∼ (4.2 ± 0.9) × 10−12 s, b ∼ 16.5 ± 0.4 K. Note that the finite value of ψν excludes the Arrhenius law (ψν = 0). Below 2 K, the expression given above is inadequate and may be indicating the existence of magnetic correlations, which depart from a zero critical temperature. A fitting to power laws of the form τc = ((Tf − T0 )/Tf )−zν , T0 being the characteristic temperature of the interactions of the spin-glass, fails entirely, yielding unphysical or unacceptably large values of zν for this first peak around Tf1 = 2.2 K. Even forcing the T0 values to be close to Tf leads to unsatisfactory
Figure 10. Frequency dependences of χ 0 and χ 00 for a selection of temperatures around Tf2 . The solid lines are fits to equations (3a) and (3b) for VNi.
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time of relaxation, τ0 , for individual spins, we considered the value obtained by means of adjustment to a law of the type described by the expression τc = τ0 exp(b/T (1+ψν) ) for activated dynamics, with τ0 ∼ (4.2 ± 0.9) × 10−12 s. Many different-scaling attempts were carried out considering several values of the exponent γ , with values around those found by means of simulations obtained using the Monte Carlo method. These calculations lead to values for γ of 4–5 for an Ising-type ±J system and of ∼7 for an Ising-type system with a Gaussian distribution of spin orientations. However, even the best result, obtained for a value of γ = 4.5, presents quite bad convergence, especially for temperatures below 4 K. Given that the expression χ 00 T −p assumes a very strong dependence of the intensity of the maximum of χ 00 on the frequency when including the critical exponent γ , and the experimental fact that this dependence is not experimentally observed, a second scaling law was employed by means of the expression [33] χ 00 /T ψν = F[−Ln(ωτ0 )T ψν+1 ].
This scaling law, which presents a lesser dependence on the temperature than that implied by the above expression, was successfully applied to describe the behaviour of a 2D SG. In this case, the critical exponent γ is not needed. Only two free parameters are necessary: the time of relaxation τ0 and the critical exponent ψν. Applying this last expression to the experimental data of VNi in the proximity of Tf1 leads to a quite good convergence for temperatures T < 4 K, as is shown in figure 13. The values obtained from the fitting to the experimental data lead to ψν = 0.24 ± 0.04 and τ0 = (2.1 ± 0.8) × 10−12 s, in good agreement with those obtained previously, when applying the law of activated dynamics for adjusting the average time of relaxation. The dynamic behaviour of the VNi sample around Tf1 can thus be described relatively well with the scaling laws, corresponding to a d = 2 SG with a logarithmic divergence of the relaxation times when T approaches 0 K. The application of the latter scaling law also suggests that the system is in a non-equilibrium state, at least in the range of temperatures investigated, which can be described according to the theory of critical phenomena for this universality class. The present analysis therefore suggests that VNi behaves cooperatively as an SG. In addition, a scaling procedure applied to χ 00 around the maximum observed at Tf2 was also carried out. In this case, bearing in mind the result of the previous case for the evolution of ln τc with temperature, and after many attempts, we decided to use the mathematical dependence introduced by Geschwind et al [34] for systems whose behaviour can be described by the standard theory of dynamic scaling, according to the expression
Figure 12. The distribution of relaxation times of VNi for a selection of temperatures around Tf1 = 2.2 K and Tf2 = 9.7 K. The shaded area represents the range of relaxation times covered by the experimental measurements.
fits. However, for the second peak observed in χ 0 around 8–10 K (Tf2 = 9.7 K), the latter power law leads to reliable results: τ0 = (2.1 ± 0.1) × 10−8 s, zν = 5.1 ± 0.2 and T0 = 8.4 ± 0.2 K. Note that T0 is slightly smaller than Tf2 and the value of zν is similar to that found in the literature for SG. This result is in good agreement with what is expected for a d = 3 SG system [24]. However, the value of τ0 is large compared to the typical value of τ0 ∼ 10−12 s (such as in Tf1 ) for canonical SG systems, which suggests a slow spin dynamics in VNi around Tf2 probably due to the presence of strongly interacting clusters rather than to individual spins. The distribution of relaxation times associated with the Cole–Cole equation can be calculated from the knowledge of the average relaxation time, τc , and the width parameter, α. Figure 12 shows a number of representative distributions. The most striking fact is that the shape of g(Lnτc ) is symmetrically centred around τc , expanding as much as 10 decades in time. With the aim of confirming the results obtained from the Cole–Cole formalism with respect to the dynamic behaviour of VNi, we carried out an analysis of the activated dynamic scaling of χ 00 around the maxima observed at Tf1 and Tf2 on the basis of the data in figure 6, following the procedure explained by Dekker et al [14]. For the first peak, we used the scale function χ 00 T −p = F[−Ln(ωτ0 )T ψν ],
(6)
χ 00 /ωβ/ζ ν = F[(TT0 )/ω1/zν ],
(7)
where β is the critical exponent associated with the order parameter. The value of T0 used was obtained from the fitting of the average relaxation times with temperature according to τc ∝ (T − T0 )−zν , while the value of β = 0.5 was taken from mean field theory. The refined values of these parameters
(5)
where the scaling parameter p = −1 − γ + ψν, γ is a critical exponent in the usual notation and F is a scaling function. It is assumed that the relaxation spectrum has a slowly varying dependence with respect to ω. As the initial 8
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the spin dynamics of the nickel ions located in different local environments in the intralayer and interlayer spaces of the structure. However, we cannot rule out other possibilities such as the existence of a structural phase transition between 2 and 10 K responsible for the change in the dynamics. This rather unusual situation was evidenced in NaV6 O11 [35], where 2D Ising-like critical scaling and 3D Ising-like scaling coexist. Additional experiments will be needed in order to elucidate the true nature of the spin-glass-like phases.
4. Conclusions AC magnetic susceptibility measurements carried out in Ni2+ –VICs have revealed the existence of two maxima in both the real and the imaginary susceptibility. The evolution with temperature, frequency and external magnetic field of these anomalies located at temperatures of around 2–3 K and 8–10 K for Tf1 and Tf2 respectively indicate the existence of cooperative phenomena between the magnetic moments of the Ni2+ ions, showing a complex magnetic response. A dynamic study of the relaxation processes associated with this phenomenon, using the Cole–Cole formalism, has allowed us to describe the behaviour of the anomaly found at Tf1 = 2.2 K according to a law of activated dynamics, obtaining values of the critical exponent, ψν < 1, compatible with the theoretical values corresponding to a d = 2 spin-glass-like system. However, the same study applied to the anomaly observed at the temperature Tf2 = 9.7 K leads to a good description of its behaviour by means of a law of standard dynamics with a value of the exponent zν of around 5, which is compatible with a d = 3 spin-glass-like system. These results have also been confirmed through an analysis by means of scaling laws, applied to the imaginary part of the AC magnetic susceptibility.
Figure 13. Dynamic scaling plots of the χ 00 data for temperatures around Tf = 2.2 and 9.7 K and frequencies gathered in the figure using (a) dynamic scaling laws and (b) standard scaling laws (see text for more details).
(see figure 11) are 0.63 for β, relatively close to the mean field value, and 8.8 K for T0 . The latter value is slightly larger than that obtained according to standard dynamics when using a representation of ln τc versus T. However, the value of the critical exponent zν = 5.1 ± 1.0 is in fairly good agreement with that found from standard dynamics. It should be pointed out that the convergence worsens for T < T0 . This circumstance could be related to the effect of overlapping with the maximum observed at Tf1 . These results confirm the idea of the existence of critical behaviour characteristic of a d = 3 SG, already implied from the analysis of the reported average relaxation times. In the present work, the Ni–vermiculite intercalation compound from Santa Olalla shows two anomalies (instead of only one) in χ 0 that shift their positions with the frequency. Although both anomalies are very close in temperature to each other, making the analysis difficult, the application of the Cole–Cole formalism allows us to describe the experimental observations in χ 0 according to different dynamics by using scaling procedures. However, an open question in the Ni–vermiculite intercalation compound from Santa Olalla is why two different dimensional dynamics coexist and what the qualitative picture of the spin-glass magnetic ordering involved is. Both Mg–vermiculite and Ni–vermiculite intercalation compounds are chemically homogeneous, as shown in sections 2 and 3, so the most plausible explanation is that each anomaly could be related to
Acknowledgments Financial support from ERDF and under the Spanish MICINN research projects MAT2011-27573-C04-02 and MAT2008-06542-C04 is acknowledged.
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