Size dependence study of the ordering temperature in the Fast Monte Carlo method
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(2) Page 2 of 12. those of bulk systems. Magnetic systems with a variety of geometries such as clusters (Rana et al. 2010), spheres (Pal and Bahadur 2010), cylinders (Roshchin et al. 2009), wires (Peng et al. 2000), rings (Singh et al. 2009), and nanotubes (Albrecht et al. 2010), have been experimentally studied with the aim to determine the effect of shape on their magnetic properties. A deep understanding of these properties contributes to the improvement in the implementation of technological applications related to magnetic memory devices (Gapin et al. 2006), high-resolution magnetic field sensors (Chapman et al. 1998), spintronics (Parkin et al. 2008), cancer detection and treatment (Pankhurst et al. 2009), therapeutic hyper-thermia (Dennis et al. 2009), among others. Monte Carlo (MC) simulations are widely used as a theoretical approach for studying magnetic properties in several types of systems (Landau and Binder 2000). By using this approach, thermal averages of some properties can be obtained when a suitable Hamiltonian is known. Since dipolar energy is a long-range interaction, it becomes decisive to define the behavior of a magnetic system and its dynamical evolution (Mejı́a-López et al. 2010). For large enough systems, at least in uniform or quasi-uniform states, the dipolar energy gives a contribution proportional to the volume of the system. Since the volume of a system increases with the number of spins; the higher this number, the stronger the effect of the dipolar term on the system configuration. Whenever MC simulations are carried out on systems whose size lies within the experimentally studied range of lenghts (*10–500 nm), every spin can interact, at the same time, at least with other 108–109 spins approximately. This number of interactions must be counted for each spin in each MC step, doing the calculation very time consuming. Since the computation time of the dipolar term increases as the square of the number of particles, the study of most of the real systems becomes not reachable with the current computational facilities within reasonable amounts of time. In 2002, d’Albuquerque e Castro et al. (2002) proposed the scaling technique, a method to study the magnetic behavior of nanosized systems. According to this scaling technique (also called sometimes scaling approach in order to avoid any confusion with finite size scaling theory), a system with competitive interactions given by exchange and dipolar energies can be replaced by an equivalent scaled system that is smaller than the original one and behaves magnetically in a. 123. J Nanopart Res (2013) 15:1437. similar way. In fact, they proved that the small scaled system preserves the same magnetic states of the real one if the exchange constant is also scaled in such a way that the competition occurring at the real sample is preserved in the scaled one (d’Albuquerque e Castro et al. 2002). A remarkable advantage can be obtained through this approach: it allows to reduce the computation time by several orders of magnitude and consequently to explore in detail the magnetic configurations of nanostructured samples wherever dipolar interaction plays a relevant role. In order to study the dynamic and thermal properties of magnetic nanostructures, in 2005, Mejı́a-López et al. proposed to combine the scaling technique with MC simulations, and applied it to study the exchange bias phenomenology of ferromagnetic nanodots on antiferromagnetic substrates (Mejı́a-López et al. 2005). Later, in 2006, Vargas et al. proved that the scaling technique could be used along with MC simulations in order to study magnetic properties (Vargas et al. 2006). This approach, combining scaling technique with MC simulations, is currently known as Fast Monte Carlo (FMC) and has been extensively used to describe the magnetic properties of several types of systems such as: (i) nanowires (Bahiana et al. 2006; Allende et al. 2009a, b), (ii) magnetic nanodots (Mejı́a-López et al. 2010), (iii) nanotubes (Landeros et al. 2007, 2009; Allende et al. 2008), (iv) nanorings (Zhang and Haas 2010; Zhang et al. 2008), and (v) elliptical nanoparticles (Zhang et al. 2008). Worthy of attention is the very good agreement that has been found between some results obtained through this method and experimental findings (Mejı́a-López et al. 2010). Most of this work has been developed at low temperature, where the magnetic states are supposed to be very similar to the ground state. Clearly, this particular aspect projects venues for potential research studies that could provide interesting results when considering high temperature since many experiments are carried out at room temperature (RT). Additionally, the temperatures corresponding to order–disorder magnetic phase transitions (Tc) for several systems are around or beyond RT, and there is great interest to know the size dependence of the ordering temperature Tc for magnetic nanostructures (see Lang et al. 2006; Sun et al. 2000 and references therein). In order to use the FMC method within these temperatures, it is necessary to elucidate a suitable scaling relation for the ordering temperature..
(3) J Nanopart Res (2013) 15:1437. Page 3 of 12. Thus, the purposes of this study are (i) to present the way which ordering temperature should scale in, to get true values of Tc when using FMC method for the study of magnetic systems, and, (ii) to focus our attention on the size dependence of the magnetic properties of cylindrical nanostructures and compare our results with other reports. In the following sections, we first describe the computation details, then we start the discussion on our results presenting a brief validation of the framework, we discuss some results related to the items (i) and (ii) of the previous paragraph, and finally present comparisons and conclusions.. Computation details For this study, we have chosen cylindrical-shaped Fe samples (growth along [110] direction) in which magnetic moments occupy the sites of a bcc cubic lattice. In order to test size dependences, several cylinders, with height H = 20 nm and diameters D = 30, 35, 40, 45, 50, 52, 55, 57, 60, 63, 65, 67, 70, 80, 90 and 100 nm, have been studied. In these systems, magnetic structure is mainly determined by the competition between the exchange interaction among neighboring atoms and the long-range dipolar interaction (Mejı́a-López et al. 2010). Therefore, the FMC approach can be used to study their magnetic properties, and make comparisons with experimental and theoretical results. For completeness of this study, we present here a brief summary of the details needed to understand the computation process. The scaling technique consists in replacing the original system by an equivalent sample with the same density (q = N/V, N magnetic moments in a volume V) and geometric shape of the original one, but smaller dimensions. According to this method, the dimensions of the system should be scaled by the following rules (d’Albuquerque e Castro et al. 2002): D0 ¼ xg D;. H 0 ¼ xg H;. ð1Þ. where x is the scaling factor, g is the scaling exponent, and D and H represent the true diameter and height, respectively. Throughout the present report scaled quantities are represented by using the apostrophe sign.. When the size of a system is reduced, the number of magnetic moments decreases. Therefore, the dipolar interaction over a given magnetic moment diminishes; while the exchange interaction—that depends only on its nearest neighbors—does not change. The proposal of d’Albuquerque et al. in order to keep the correct balance between these energy terms, was to decrease the magnitude of the exchange energy (d’Albuquerque e Castro et al. 2002). According to the scaling technique, this can be done by scaling the exchange constant J by a factor x \ 1, so as to reduce its strength J 0 ¼ xJ:. ð2Þ. Therefore, by implementing the scaling scheme with several different x values applied to cylinders with sizes D–H, several different scaled samples with sizes D0 –H0 can be obtained. These systems emulate the magnetic configuration of the real sample with size D–H independently on the x value, as has been proved in (d’Albuquerque e Castro et al. 2002; Vargas et al. 2006). So far, in order to relate temperature and size, there are at least two different proposals in the scaling technique framework. One, based on the mean field theory (Mejı́a-López et al. 2010) [where the ordering temperature of a system is proportional to J (Kittel 1996)], implies that temperature should scale linearly with the scaling factor x as T 0 ¼ xT:. ð3Þ. The second proposal is based on the definition of a blocking temperature; and it relates to the fact that, in order to keep the thermal activation process invariant under the scaling transformation, the energy barriers should be invariant (Bahiana et al. 2006; Allende et al. 2009b). In this case, it must be taken into account that, although the energy landscape is complex due to dipolar interactions, in the vicinity of each local minimum transitions can be considered as ruled by simple energy barriers of the form KeVe (Bahiana et al. 2006; Allende et al. 2009b). Here, Ke is an effective anisotropy constant that takes into account different energy contributions and Ve is an effective volume. Thermally activated transitions lead to the definition of a blocking temperature TBµ KeVe. Therefore, according to this idea, temperature would scale as the volume does (Bahiana et al. 2006; Allende et al. 2009b). 123.
(4) Page 4 of 12. J Nanopart Res (2013) 15:1437. T 0 ¼ x3g T:. ð4Þ. In this study, the temperature is taking into account through MC simulations, which were carried out by using the Metropolis algorithm on a classical Heisenberg spin model with local dynamics and single spin–flip methods, namely, the new orientation of the magnetic moment was chosen randomly with a probability p ¼ min½1; expðDE=ðkB TÞÞ; where DE is the change in energy due to the reorientation of the spin, and kB is the Boltzmann constant (Binder 1997). Total energy Etot for a single nanocylinder with N magnetic moments in a given configuration fli g is given by Etot ¼. 1X ^j Þ: ðEij J^ li l 2 i6¼j. ð5Þ. The exchange constant is J (in units of energy [meV]) for nearest neighbors moments, and J = 0 ^k Þ represents the unit vector along the otherwise. l direction of the magnetic moment lk ; and Eij is the dipolar energy: Eij ¼. li lj 3ðlj n^ij Þðlj n^ij Þ rij3. ;. ð6Þ. ^ij is where rij is the distance between li and lj ; and n the unit vector along the direction that connects the ^ij ¼ rij =rij : two magnetic moments: n Since the sample’s material is iron, we have used jli j ¼ lFe ¼ 2:2lB ; the lattice parameter a0 = 2.86 Å (Kittel 1996), and J = JFe = 43 meV, which, for simulations under periodic boundary conditions, reproduces the ordering temperature for the bulk case [TC = 1,043 K (Kittel 1996)]. In order to get accurate results in the simulations, temperature steps were as small as 1 K in an adequate range around Tc. According to previous reports, the value of the scaling exponent has been set to g = 0.57 for the x values considered here (Mejı́a-López et al. 2010). The scaling factor x is taken as variable in our study with values between 1.16 9 10-3 and 5.81 9 10-3, corresponding to J0 = xJ = 0.05 and 0.25 meV, respectively. At this point, it is worthwhile mentioning that, for every cylindrical sample with diameter D and height H, in the scaling process, the x values are chosen in such a way that the resulting scaled systems preserve the geometrical shape.. 123. Fig. 1 Low temperature equilibration processes. These panelsc show energy as a function on time (expressed in MC steps) for the different system sizes when a scaled temperature 0.01 is used. Insets show the snapshots for a selected plane (perpendicular to the axis of the cylinder) indicating the low temperature magnetic configurations obtained at the end of the equilibration process for each cylinder. In the simulations, 2 9 104 Monte Carlo steps per spin (MCS) were used for equilibration purposes (this number was selected after computing and analyzing the nonlinear relaxation functions for all the system sizes), and 104 more MCS per spin, and at least 10 different random seeds, were used to compute thermal and configurational averages.. Results and discussion Figure 1 shows the low temperature equilibration processes for some selected system sizes. Three different configurations have been selected as starting point of each equilibrium curve: ferromagnetic, random, and vortex. These runs have been carried out at low temperature and using a large enough time (*105 MCS) in order to determine the ground state of each system. Snapshots for a selected plane (perpendicular to the axis of the cylinder) are enclosed as insets in Fig. 1. These images show the lower energy configuration obtained for the spins at the last MCS. As expected, results indicate that the low temperature magnetic ground state of the samples depends on the cylinder size, confirming that the dipolar energy plays a crucial role in the magnetic ground states and the magnetization processes, in agreement with experimental and theoretical studies reported previously (Lebib et al. 2001; d’Albuquerque e Castro et al. 2002). Moreover, from the mentioned snapshots, it could be observed that there is a boundary value, located between 45 and 50 nm diameter, that separates two different magnetic states at low temperature. For system’s sizes below about 50 nm, a collinear ferromagnetic (FM) state is obtained; whereas, for diameters greater than and equal to 50 nm, the low temperature magnetic state corresponds to a vortex with a FM core. In order to test these low temperature magnetic states in an independent way, we have implemented a second computation route. Starting from a random configuration (corresponding to the disordered state at.
(5) J Nanopart Res (2013) 15:1437. Page 5 of 12. 123.
(6) Page 6 of 12. J Nanopart Res (2013) 15:1437. high temperature) we implement a run on temperature from high to low T allowing the relaxation process up to the minimum value of T (this process is followed for every x value and the several different random seeds employed). For each of the temperature values, we register the magnetic configuration of the system. The obtained results were in perfect agreement with the method described previously. Moreover, the findings described are independent of the scaling parameter x, and are in complete agreement with experimental and theoretical studies reported previously (Mejı́a-López et al. 2010), what allows the validation of the method and gives confidence to our results. Authors of that work report, for instance, differences in the reversal modes for the magnetization processes depending on size, which present good agreement with our findings, as will be discussed below.. To analyze the order–disorder process, we have considered the thermal behavior of the samples. Figure 2 shows the magnetic properties as functions of the scaled temperature for two selected cylinders with D = 40 and 80 nm, when a scaling factor of x = 2.1 9 10-3 (J0 = 0.09 meV) is used. Figure 2a, b shows the absolute value of the total magnetization per spin as functions of the scaled temperature T0 . The result for the samples scaled from the 40-nm diameter cylinder (Fig. 2a) corresponds to the wellknown behavior exhibiting a FM–paramagnetic transition for a certain ordering temperature. Insets show views (a selected plane perpendicular to the cylinder axis) for the paramagnetic state obtained at high temperature, and a low temperature FM state. The magnetization behavior for samples scaled from the cylinder D = 80 nm (Fig. 2b) is remarkably. (a). (b). (c). (d). (e). (f). D = 40 nm. Fig. 2 Thermal average properties as functions of the scaled temperature for two selected 20 nm height cylinders with D = 40 and 80 nm when a scaling factor x = 2.1 9 10-3 (J0 = 0.09 meV) is employed: a, b magnetization per spin. Insets. 123. D = 80 nm. show snapshots of a selected plane (perpendicular to the axis of the cylinder) for two magnetic states at low and high temperature. c, d Magnetic susceptibility per spin. e, f specific heat per spin.
(7) J Nanopart Res (2013) 15:1437. different; in those cases a vortex state is obtained. As it is known, in the ideal case of a perfect vortex state a null magnetization would be expected in the low temperature regime. At high temperature, the system is characterized by a disordered paramagnetic state, giving a net magnetization close to zero as well. Thus, in vortices cases, the expected magnetization as a function of temperature would be approximately flat through the whole range of temperature. However, as the system approximates the transition temperature, thermal fluctuations also increase; in consequence magnetization exhibits a small but well-defined peak (Fig. 2b) with the maximum located at a certain temperature that marks the order–disorder transition (i.e., vortex–paramagnetic transition). Additionally, since the system does not reach a perfect vortex state due to the existence of a FM core, magnetization at low temperature has non-zero values (Roshchin et al. 2009). This magnetic behavior, just discussed here, is the one obtained for all the samples with diameter equal to and greater than 50 nm. Figure 2c, d shows magnetic susceptibility as functions of the scaled temperature, for 40 and 80 nm diameter samples, respectively. Figure 2e, f shows the respective specific heat data. These properties have been computed as fluctuations of the magnetization and internal energy of the system, respectively (Landau and Binder 2000). As expected, these figures show well-defined peaks indicating the transition temperatures. Here, it is worthwhile noting that the peaks corresponding to the smaller size present bigger width and rounded shape, contrarily to the sharp and tight peak exhibited by the big sizes. These differences can be understood by taking into account that the response of the system (i.e., the variations of the magnetic state) as T increases occurs in a slower manner for the FM configurations own of the small systems due to the stronger contribution of the exchange interaction obtained for ordered collinear spins in comparison with the tilted spins of the vortex configuration. In other words, due to the tilt among neighboring spins in the vortex configurations, the system responds easier than in the FM state to changes in T and the application of an external magnetic field. In short, for nanostructures smaller than 50 nm in diameter, the magnetic transition takes place from FM to paramagnetic states. In contrast, for the 50-nm diameter cylinder and for the greater diameters,. Page 7 of 12. different magnetic states are obtained, as will be discussed below. Taking into account that thermal fluctuations of the energy are much smaller than those of the magnetization (Landau and Binder 2000), for the scaled ordering temperatures we have used the values obtained from the positions of the maxima of the specific heat data. These values were obtained by fitting the energy data with a polynomial function (order 4) which gives the smallest Chi square in the vicinity of Tc and then taking derivatives of this function to locate the change in its concavity, which led to the identification of the ordering temperature for each case. For illustrative purposes, we have selected a typical sample, the 65-nm diameter cylinder. The scaled ordering temperature for the sample scaled with x = 2.1 9 10-3 was obtained as T0 c = 1.51 K. Now, it should be recalled that, for any material, the ordering temperature of nanostructures must be lower than the one for the bulk case (Lang et al. 2006), i.e., due to the reduction in the magnetic bond density, ordering temperature for a nanosized system cannot be larger than its corresponding bulk-Tc. Then, taking the values T0 c = 1.51 K, x = 2.1 9 10-3 and Tc B Tc bulk-Fe = 1,043 K (Kittel 1996), in the scaling relation for ordering temperature Tc0 ¼ xb Tc ;. ð7Þ. which is a generalized form of Eqs. 3 and 4, the value 1.07 is obtained as an upper limit for the scaling exponent b, i.e., the relationship b B 1.07 holds. We note that the value b = 1 in Eq. 3 lies within this upper limit. Contrarily, the exponent b = 3g = 1.71 that appears in Eq. 4 (obtained with g = 0.57, which is the value used in this study) is well beyond this limit. In accordance, and perhaps as the main inference of this study, this last result suggests that the scaling relation for temperature based on the blocking temperature assumption, Eq. 4, cannot be the correct choice in order to extend FMC simulations with temperature studies. Figure 3 illustrates the behavior of specific heat as a function of the scaled temperature T0 for the cylinder with D = 65 nm, and for different values of the scaling factor in the range of 1.6 9 10-3 \ x \ 5.8 9 10-3. This figure exhibits the well-known behavior for an order–disorder phase transition. 123.
(8) Page 8 of 12. Fig. 3 Specific heat as a function of the scaled temperature for the cylinder H = 20 nm, D = 65 nm, and different scaling factors in the range of 1.6 9 10-3 \ x \ 5.8 9 10-3. occurring at a certain T0 c (that can be computed from the position of the peaks as previously described). It can be observed that the scaled ordering temperature shifts toward lower values as x decreases. This progressive decrease in the T0 c value is in complete agreement with the direct relationship between the exchange constant and the ordering temperature Tcµ J. From the figures cv(T0 ) for the full range of sizes, we get the corresponding scaled temperature as function of the scaled exchange interaction to draw the Fig. 4. This figure shows the behavior of T0 c as a function of J0 (= xJ) for some selected diameters; linear fits of the form T0 c = AJ0 , and error bars, have also been included. It is worthwhile highlighting that a linear behavior is obtained for every case. Since the results obtained through the scaling technique should not depend on the x value, as reported in d’Albuquerque e Castro et al. (2002), it could be assumed that the linear behavior obtained here holds for big x values as well. Then, we can extrapolate the behaviors presented in Fig. 4 to the case J0 = J (x = 1), scilicet, toward the actual value of the exchange coupling in order to get the results back up to normal size, and now the size dependence of the true ordering temperature can be drawn. Figure 5 shows the true ordering temperature as a function of the diameter of the cylinders. It is worthwhile noting that the size dependence of the ordering temperature computed from this atomistic approach for systems including dipolar interactions, which are presented in Fig. 5, are consistent with the. 123. J Nanopart Res (2013) 15:1437. Fig. 4 Scaled ordering temperature T0 c as a function of the scaled coupling constant J0 = xJ for some selected sizes (expressed in nm). Solid lines correspond to linear fits of the form T0 c = AJ0 . Error bars in the scaled ordering temperatures are explicitly shown. expected behavior; i.e., Tc should decrease with decreasing size due to the decrease in the magnetic bond density since the energy cost to carry out the transition is lower, and thus the ordering temperature should also be smaller. Additionally, as D increases Tc(D) shows an incremental tendency. It is also evident the strong dependence of Tc on diameter. Although Tc increases with size as expected; the behavior is not a monotonously increasing function of size. Three different regions (marked as I, II, and III) are observed instead. Our simulations indicate the existence of two metastable states whose energy depends on size, which is in complete agreement with well-known experimental and theoretical results (see Refs. Shinjo et al. 2000; Martı́n et al. 2003; Choe et al. 2004; Mejı́a-López et al. 2010 and references therein). For cylinders with sizes in region I of Fig. 5, there are differences between scaled energies corresponding 0 0 0 to FM and vortex states (DEFV ¼ EFM Evortex :) For example, these energies for the 40-nm diameter 0 ¼ cylinder and J0 = 0.15 lead to the value DEFV 0:01328 meV/spin. Therefore, energy barriers between FM and vortex configurations are big enough to have the system in a FM configuration. The thermaldriven magnetic phase transition takes place from a pure FM to a paramagnetic state, and Tc increases with size..
(9) J Nanopart Res (2013) 15:1437. Page 9 of 12. Fig. 5 True ordering temperature (x = 1, J0 = J) as a function of the diameter of the cylinders. Regions I, II, and III correspond to different thermal-driven magnetic transition modes, as discussed in the text. Error bars are computed by considering the uncertainties coming from the linear fits in Fig. 4. Light shadows are included as a guide to the eye. Inset shows a comparison among our findings (dots) and the theoretical Lagng’s model using a = 10.3 and D0 = 1.4898 nm (black solid line) (see text). As the diameter increases beyond 50 nm (region II), energy for the two configurations becomes very similar in magnitude, and the energy barrier between FM and vortex states decreases around one order of magnitude. For instance, for the 60-nm diameter 0 nanocylinder we have DEFV ¼ 0:00605 meV/spin. Thus, for cylinders belonging to this region II, whenever temperature increases, the system alternately occupy any of these two metastable configurations. Then, for these sizes, the magnetic phase transitions take place from the mentioned metastable configuration (mean configuration) to the high temperature paramagnetic phase, thus, as size increases and while the energy barrier is sufficiently small compared with the thermal energy, the ordering temperature will not depend on size; giving place to the observed plateau (II). As diameter increases beyond 60 nm, for systems belonging to the region III, magnetic configurations are dominated by vortex states (example for D = 80 nm, 0 we have DEFV ¼ 0:01272 mev/spin). Therefore, for these sizes, the magnetic phase transitions occur between the vortex and the paramagnetic state, and Tc rises monotonously with size as expected at least for nanostructured systems. For big sizes (above about 100 nm) magnetic domains can take place in the sample, therefore, magnetic states are not longer of the clean type of vortex or FM states occurring in nanostructures, and the FMC approach loses its. validity. Of course, it is expected to obtain a limiting value for the ordering temperatures while increasing size of the system: the one corresponding to the bulk case Tc-bulk. This interesting question, i.e., determining a nano-bulk boundary, is matter of current research. The phenomenology reported in Fig. 5 and described above is in agreement with previous experimental and theoretical studies on this type of systems (Mejı́a-López et al. 2010). There, authors analyze hysteresis loops of Fe nanocylinders of the sizes considered here, study the magnetization reversal processes and compared with experimental results. In order to understand the kind of comparisons we present here, it is important to have in mind that both of these kind of processes, i.e., magnetization reversals (discussed in Mejı́a-López et al. 2010) and magnetic phase transitions (considered here), are governed by the same kind of energy barriers. Authors of that work also report three different kind of behaviors (in this case for the magnetization reversal processes), which are consistent with our results presented in Fig. 5 for ordering temperatures. Their results can be summarized as: (i) for small diameters a coherent rotation takes place; (ii) for the biggest sizes inversion process occurs mediated by a vortex state; and (iii) for intermediate values of diameter the magnetization reversal implies a mixed behavior; namely, sometimes a coherent rotation, sometimes. 123.
(10) Page 10 of 12. J Nanopart Res (2013) 15:1437. through a vortex. These stages are related to magnetic states and energy barriers among them. Likeness of behaviors and boundaries for such behaviors is remarkable (see Fig. 5 and Mejı́a-López et al. 2010). Regarding a comparison with experimental results we want to stress that to the best of our knowledge, besides it is rather difficult to obtain a diameter dependence of the ordering temperature due to several factors such as shape inhomogeneities, size distribution, and Fe surface oxidation, no experimental results on Fe nanocylinders have been published so far. Despite of that, our results are in good qualitative agreement with some others reported for nanostructures. Magnitudes of the ordering temperatures obtained here are consistent with available simulation results on Fe-based low-dimensional nanostructures reported by other models proposed to understand the mechanisms lying on the effect of the breaking of exchange bonds upon the Tc(D) function for nanoparticles (Lang et al. 2006). In that work, an energyequilibrium criterion between the spin–spin exchange interactions, the thermal vibration energy of atoms at the transition temperature and a size-dependent Debye temperature function were considered to develop a model to fit experimental results to obtain both Tc(D) and TN(D) of FM and anti-FM small systems. Such model yielded the following expression for nanoparticles (Lang et al. 2010): Tc ðDÞ=Tc ð1Þ ¼ exp ½ða 1Þ=ðD=D0 1Þ;. ð8Þ. where a is a measure of the root-mean-square thermal average amplitude of surface atom vibration relative to the core and D0 denotes a critical size at which all atoms of the nanostructure are located on its surface. Remarkable agreement is obtained whenever this model (Eq. 8) is used with adequate parameter values. In this case, with the idea of considering our cylindrical-shaped Fe nanostructures, we have taken into account that, according to the discussion of Lang et al. (2006), the a parameter appears associated to the vibrational entropy, that have a free-surface-dependent effect, and thus a becomes the only adjustable parameter. Since free-surface strongly depends on shape, it is expected to have appreciable variation of a for different kind of particles. In consequence, in this study of cylinders, we have proceeded to kept unaltered the value D0 = 1.4898 nm which stands for the Fe nanoparticles (using the zero dimension d = 0 and the Fe nearest-neighbor bulk distance).. 123. With this in mind we obtain a = 10.3 as an adequate value for describing the Tc(D) dependence within the Lang‘s model. This behavior is represented through the black solid line in the inset of Fig. 5, where a notable concordance is obtained. Small differences in the intermediate zone are attributable to the fact that the Lang’s model does not describe the three different regions obtained here. The good agreement obtained here among our results and those reported in the literature (Mejı́aLópez et al. 2010; Lang et al. 2006) indicates that the FMC approach along with the temperature scaling scheme employed here can be considered as a reliable and useful method for obtaining magnetic properties of nanostructured systems at finite temperature.. Conclusions Traditionally the FMC simulation framework has been used to study magnetic properties in nanostructures, leaving the scaling for temperature an open issue to be explored from different perspectives. The linear scaling of the ordering temperature of nanostructures and the diameter dependence of the magnetic properties of cylindrical systems in the nanometric range of sizes are two of the main results that can be concluded from this study. (a). It was found that there is an upper limit for the exponent b of a scaling relationship of the form: T0 c = xb Tc. This upper limit rules out a scaling relation proportional to x3g that has been proposed by some authors (Bahiana et al. 2006; Allende et al. 2009b), and suggests that, in the framework of FMC calculations, ordering temperature should scale linearly with the scaling factor x, in accordance with the mean field theory-based proposal. (b) The magnetic state of Fe nanocylinders strongly depends on their diameter. For cylinders whose diameter is less than about 45 nm, the ground magnetic state corresponds to FM configurations; whereas for bigger diameters vortex states are found. Moreover, results on size dependence of the ordering temperature indicate that there are three different order–disorder magnetic transition modes. For nanostructures smaller than around 50 nm in diameter, the process takes place directly.
(11) J Nanopart Res (2013) 15:1437. from the low temperature FM phase to the paramagnetic one. For those samples with diameter beyond 60 nm, the transitions occur from a low temperature vortex state toward the high temperature paramagnetic phase. In contrast, for those cylinders with intermediate diameters, due to the small value of the energy barriers between FM and vortex states, the process involves the existence of two metastable states: FM and vortex, which are occupied alternately by the system as temperature increases. This phenomenology is in agreement with previous reports (Mejı́a-López et al. 2010); and the ordering temperatures reported here are consistent with experimental and simulation results (Lang et al. 2010). Results of the method might be of interest since it allows to expand the FMC approach used so far to study magnetic properties at low temperature regime. In other words, by applying this temperature scaling scheme within the framework of FMC simulations it will be now possible to cover a wider kind of phenomenologies occurring at higher temperatures, and study new magnetic configurations and new devices based on nanostructured magnetic materials. It is worthwhile mentioning that results presented here might bring the attention of experimental researchers, who could consider the ordering temperatures for Fe nanostructures a potential field of research. Acknowledgments We acknowledge support from several projects: Chile-Colombia scientific cooperation program CONICYT-COLCIENCIAS under contracts 2008-157 & 279-2009; FONDECYT-Chile Grant No. 1100365; ‘‘Fondo de Innovación para la competitividad-MINECON-Chile’’ Grant ICM P10061-F; ‘‘Financiamiento Basal para Centros Cientı́ficos y Tecnológicos de Excelencia-Chile’’, under project FB0807; Universidad de San Buenaventura-Colombia Project 060102201201; the SUI-CODI ‘‘Magnetismo de nanoestructuras’’ and CODI ‘‘Sostenibilidad-GES-2012’’ projects at Universidad de AntioquiaColombia. J.M.L. thanks Universidad de Antioquia for a stay as Invited Professor during 2012–2013. We want to thank Prof. D. Altbir for a critical reading of the manuscript and J. Restrepo for helpful comments.. 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