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Magnetic features of a mixed ferro-ferrimagnetic ternary alloy cylindrical nanowire
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Magnetic features of a mixed ferro-ferrimagnetic ternary alloy cylindrical nanowire
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Article history: Received 2 July 2017 Received in revised form 21 August 2017 Accepted 29 August 2017 Available online xxxx Communicated by M. Wu Keywords: Ternary alloy Magnetic cylindrical nanowires Monte Carlo simulation
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In the present study, we have investigated the finite temperature magnetic phase transition properties of a mixed ferro-ferrimagnetic ternary alloy cylindrical nanowire of the type A B p C 1− p by Monte Carlo simulation technique. The nanowire system consists of two interpenetrating sublattices, one of which contains type- A magnetic components with spin-3/2 and the other one is randomly occupied type-B and type-C magnetic components with spin-1 and spin-5/2, respectively. We have examined the effect of exchange interaction ratio, R, and the concentration value of type-B magnetic ions, p, on the transition temperature of the system. It has been found that one can modify the transition temperature and saturation magnetization of the system by varying concentration value and interaction ratio. Moreover, our numerical results show that the ternary alloy nanowire system exhibits compensation behavior for particular values of the system parameters. © 2017 Elsevier B.V. All rights reserved.
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1. Introduction
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Magnetic nanowires are one-dimensional nanostructures with peculiar properties such as large proportion of surface atoms and high aspect ratio (ratio of length to diameter) [1,2]. These unusual properties of nanowires make them attractive for a variety potential technological applications in the fields of high density magnetic recording media [3,4] and biomedical applications [5,6]. Theoretically, magnetic nanowires become excellent structures for understanding magnetism in nanoworld. Up to now, magnetic features of nanowires have been successfully investigated with several methods for instance effective field theory (EFT) [7–14], Monte Carlo (MC) simulations [15–23] and micromagnetic simulations [24]. Most of the theoretical studies in literature have been focused on phase transition properties and hysteresis characteristics of nanowires with core/shell structure. Prussian blue analogs [25], which are a class of moleculebased magnets, are in active investigations in the last decades because of their important magnetic features such as photoinduced magnetic pole inversion [26], inverted magnetic hysteresis loop [27], existence of one or two compensation temperatures [28,29] and possible technological applications ranging from hydrogen storage [30] to electrode materials for batteries [31]. Moreover, in recent years, nano-sized magnets based on Prus-
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E-mail address: [email protected]. http://dx.doi.org/10.1016/j.physleta.2017.08.060 0375-9601/© 2017 Elsevier B.V. All rights reserved.
sian blue analogs such as nanoparticles with core/shell structure [30,32,33], nanowire arrays [34,35] and nanotubes [36] have been synthesized with various size-dependent magnetic features. Particularly, the multi-metal Prussian blue analogs have gained remarkable interest with the pioneering works of Okhoshi and his co-workers [37–39]. They have designed ternary metal Prussian blue compounds of the type (NiIIp MnII1− p )1.5 CrIII (CN)6 , which includes both ferromagnetic ( J Ni–Cr > 0) and antiferromagnetic ( J Mn–Cr < 0) superexchange interactions [37–39]. It has been demonstrated that it is possible to modify the magnetic features of ternary metal Prussian blue compounds such as the saturation magnetization, Weiss temperature and coercive field by changing the active concentration of the magnetic ions in the compound [37]. On the theoretical side, magnetic properties of mixed ferromagnetic–ferrimagnetic ternary alloy Prussian blue analogs of the type A B p C 1− p have been widely investigated in literature by various methods for instance mean field theory (MFT) [40–47], EFT [48–50], Bethe lattice approximation [51], Green’s function technique [52], MC simulation method [53–60]. In the above mentioned theoretical investigations, the ternary alloy has been modeled such that type- A magnetic ions are located in one of the sublattice while type-B and type-C magnetic components are distributed randomly over the other sublattice with probabilities p and 1 − p, respectively. Also, the model system contains both ferromagnetic ( J A B > 0) and antiferromagnetic ( J AC < 0) exchange interactions between the nearest neighbor magnetic components in accordance with the real ternary metal Prussian blue analogs
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[26]. The antiferromagnetic to ferromagnetic exchange interaction ratio R = | J AC |/ J A B plays an important role in the magnetic properties of the ternary alloys since this parameter determines the dominant type of exchange interaction in the system. Moreover, there exists a special interaction ratio, R C , for which the critical temperature of the system becomes independent of the concentration value p [55]. Despite the extensive literature on magnetic properties of ternary alloy Prussian blue analogs, as far as we know there is no theoretical study on magnetic nanostructures composed of ternary alloys. Thus, in the present study we have investigated the magnetic features of a mixed ferro-ferrimagnetic ternary alloy nanowire of the type A B p C 1− p . For this purpose, we have performed detailed MC simulations based on Metropolis algorithm. The spins of the magnetic components are selected as S A = 3/2, S B = 1 and S C = 5/2 for type- A, type-B and type-C magnetic components, respectively. We have examined the magnetic phase diagrams of the system in ( R − k B T C / J A B ) and ( p − k B T C / J A B ) planes and compared the magnetic features of the system with those of its bulk counterparts. Besides, for certain values of the concentration value and exchange interaction ratio, the ternary alloy nanowire system is shown to exhibit compensation behavior. The paper is organized as follows. The details of the theoretical model and our MC simulations are given in section 2. Our numerical results regarding the magnetic properties of ternary alloy nanowires are presented in section 3. Finally, our conclusions are summarized in section 4.
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2. Formulation
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We consider a ternary alloy cylindrical nanowire of radius r = 10 and length L = 200 on a simple cubic lattice by imposing free boundary conditions in xy plane and periodic boundary conditions in z-direction (see Fig. 1). The considered ternary alloy nanowire of the type A B p C 1− p includes two interpenetrating sublattices. Type-B magnetic components with spin-1 and type-C magnetic components with spin-5/2 are distributed randomly throughout the one of the sublattice with probabilities p and 1 − p, respectively and they connect to type- A magnetic components with spin-3/2 which are located on the other sublattice. Therefore, the Hamiltonian corresponding to the model ternary alloy nanowire system can be written as
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N A A M A = [m A ] = Si , NA
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respectively. If site j is occupied by type-α atom (α = B or C ), δ j α takes the value of unity and otherwise it is zero. i , j denotes the sum over the nearest neighbors and we assume that the spin–spin exchange interaction between type- A and type-B magnetic ions is ferromagnetic ( J A B > 0) whereas type- A and type-C magnetic ions interact via antiferromagnetic spin coupling ( J AC < 0). In our numerical calculations, J A B is set as unity and the other interested quantities are normalized with J A B . In order to study the magnetic behavior of the ternary alloy nanowire, we use the MC simulation technique based on single spin-flip Metropolis algorithm [61,62]. We perform 20 independent sample realizations. In each computer experiment, type-B and type-C components are distributed randomly through the lattice and their positions do not change during the experiment. We start the simulation at low temperature region with ferromagnetic arrangement of the spins and heat the system slowly with temperature step of k B T / J A B = 5 × 10−2 where k B and T are Boltzmann constant and absolute temperature, respectively. For every temperature step, the first configuration is selected as the final configuration of the previous temperature step. Then, the following configurations are created by selecting the sites sequentially from the lattice and making singlespin-flip attempts. The generated new configuration is accepted or rejected according to Metropolis algorithm. We have performed 105 MC steps per site at each temperature step. Thermal averages of interested quantities are computed after discarding the first 4 × 104 steps in order to obtain the equilibrium state. We have calculated the thermal average and configurational average of sublattice magnetizations (m A , m B , mC ) per spin:
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N B 1 B M B = [m B ] = Si , NB
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and total magnetization, m T :
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where N A , N B , N C and N T = N A + N B + N C are the total number of magnetic components A, B, C and overall of the system, respectively. Also, ... corresponds to thermal averages and [...] denotes the configurational averages. In order to determine the transition temperature of the system, we use the thermal variation of the susceptibility, which is defined by
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3. Results and discussion
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Fig. 1. (Color online.) Schematic representation of the ternary alloy cylindrical nanowire of the type A B p C 1− p with length L and radius r for p = 0.5. Cross section of the nanowire is shown on the right hand side.
The magnetic phase diagrams of the ternary alloy nanowire system in ( R − k B T C / J A B ) plane is presented for selected values of the concentration value p in Fig. 2. Here, R = | J AC |/ J A B specifies the
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Fig. 2. (Color online.) Magnetic phase diagrams in ( R − k B T C / J A B ) plane of the ternary alloy nanowire system for several values of the concentration value p.
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Fig. 3. (Color online.) Variation of the transition temperature of the ternary alloy nanowire with active concentration value of type-B magnetic components, p in a range of 0.41 ≤ R ≤ 0.49.
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dominant sort of exchange interaction (i.e., ferromagnetic, antiferromagnetic or both of them) in the system. It is obvious that p = 1 case corresponds to a mixed spin-3/2 and spin-1 ferromagnetic nanowire with no J AC exchange interaction in the system and hence the transition temperature of the system does not change with R for this concentration value. On the other hand, the system is a ferrimagnetic mixed spin-3/2 and spin-5/2 nanowire for p = 0. In this case, starting from R = 1, the transition temperature of the system decreases with a decrease in R and eventually reduces to zero at R = 0. Besides, if one considers the special case of R = 0, the system consists of type- A and type-B magnetic ions with a mutual ferromagnetic interaction for p = 1. As p gets smaller, type-B magnetic components replace with type-C components (which behave as non-magnetic atoms for R = 0) and the transition temperature of the system decreases. Thus, for all the p values other than p = 1, we have a randomly site-diluted Ising nanowire and on the basis of the percolation theory, there should be no magnetic ordering below a critical concentration value of p c [63]. Since it is beyond the scope of our work to determine the precise value of p c , we have not perform any detailed simulations for this purpose. However, according to Fig. 2, p c should be smaller than p = 0.25. As one can readily see from Fig. 2 that all the phase separation curves belonging to different p values intersect at a special point, R c , which means that the transition temperature of the system does not depend on the concentration value of type-B magnetic components at this point. In order to determine the precise value of R c , we have examined the transition temperature of the system as a function of p in a narrow range of 0.41 ≤ R ≤ 0.49 with an increment of 0.01 in R value in Fig. 3. It can be clearly seen that the special point of the ternary alloy cylindrical nanowire system is R c = 0.47. For this special interaction ratio, a balance occurs between the contribution of the first and second terms of the Hamiltonian (Eq. (1)) such that replacing type-B magnetic components with type-C components (or vice versa) has no effect on the temperature where transition from the ordered to disordered phase occurs. For R > R c transition temperature of the system decreases with increasing p whereas it increases with p for R < R c . We should emphasize that the shape of the phase boundary curves obtained in ( R − k B T C / J A B ) and ( p − k B T C / J A B ) for the ternary alloy nanowire system is in accordance with two-dimensional [53, 60] and three-dimensional [55] mixed-spin ternary alloy bulk systems.
It is worthwhile to compare the special point of the ternary alloy nanowire system with the R c values previously reported in literature. Remarkably, special point of the mixed ferro-ferrimagnetic ternary alloy nanowire is the same with its three dimensional bulk counterpart [55]. Combining the results of ref. [55] with this study, one can infer that reducing the dimension of the system in xand y-directions, does not change the critical interaction ratio. This finding is reasonable since it has been demonstrated that R c value does not vary with the number of nearest neighbors [40] and also the lattice stoichiometry [55] within the framework of MFT and MC simulations, respectively. In a ternary alloy nanowire, applying free boundary conditions also implies decreasing the total number of coordination numbers as compared to the bulk system with periodic boundary conditions in all directions and thus R c is not influenced by the reduced dimension of the system. We should notice that according to our simulation results, the critical interaction ratio is likewise R c = 0.47 for smaller nanowire radii value for instance, r = 6. The proportion of surface atoms with respect to the total number of atoms increases with a reduction in radius and thus the average number of coordination numbers decreases. Nevertheless, the critical interaction ratio does not change with nanowire radius. We have shown the thermal variations of the total magnetization and susceptibility profiles of the ternary alloy nanowire for varying p below (R = 0.2), above (R = 1) and at the critical exchange interaction ratio R c in Fig. 4. When all the spin–spin exchange interaction terms are equal to each other (R = 1), a second order phase transition from ordered to disordered phase occurs at the transition temperature which depends on the p value. The susceptibility curves in Fig. 4(f) exhibit a maximum at the transition temperature of the system. Also, the saturation magnetization of the system strongly depends on p. For example for p = 1, saturation magnetization is maximum (M s = 1.25) whereas it takes the minimum value of M s = 0.0625 for p = 0.25. Therefore, according to our observations it is possible to modify the saturation magnetization of the ternary alloy nanowire by varying the concentration of type-B magnetic components. At the critical exchange interaction ratio, as demonstrated in magnetization and susceptibility curves in Fig. 4(c)–(d), the transition to disordered phase occurs at the same temperature for all the p values under consideration. If one concentrates on an interaction ratio which is below R c (R = 0.2), the magnetization curves exhibit different shapes depending on the concentration value. For instance,
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Fig. 4. Thermal dependence of the total magnetization, M T , and the total susceptibility, χ T , of the ternary alloy nanowire (a)–(b) for R = 0.2, (c)–(d) for R = 0.47 and (e)–(f) for R = 1. The curves are displayed for varying values of concentration of type-B atoms: p = 0, 0.25, 0.5, 0.75 and 1.
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for p = 0 and p = 1, the total magnetization of the system increases to its saturation value above the transition temperature, however it has single maximum in the case of p = 0.5. Moreover, the total magnetization of the system takes a minimum value below the transition temperature p = 0.25. This indicates that the ternary alloy nanowire system can exhibit compensation behavior for particular values of concentration and exchange interaction ratio values.
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The existence of compensation point can be seen explicitly from Fig. 5 where we have illustrated the temperature dependence of the total magnetization of the ternary alloy nanowire for p = 0.25 and selected values of the interaction parameter: R = 0.1, 0.2, 0.3, 0.4, 0.5. According to Fig. 5, for all the R values except for R = 0.5, the total magnetization of the system vanishes at the compensation temperature. Above the compensation point, the total magnetization firstly increases and then decreases
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with raising temperature. Finally, the system becomes disordered at the transition temperature of the system. The compensation temperature shifts to higher temperature region with increasing antiferromagnetic exchange interaction strength between type- A and type-C magnetic components and it disappears for R = 0.5. In the present study, we do not examine the range of p and R at which the compensation behavior occurs for the ternary alloy nanowire and thus a detailed investigation of this point is left for a future work. The compensation effect can also be examined from the thermal variation of positive magnetization because of the A and B sublattices (M + = 1/2( M A + pM B )) and absolute value of the negative magnetization because of the C sublattice (M − = 1/2(1 − p ) M C ) which is shown in Fig. 6 for (a) R = 0.1, (b) R = 0.5 with a concentration value of p = 0.25. The absolute value of the positive and negative magnetization cross at the compensation point at which these magnetization components have equal magnitude and thus the total magnetization of the system is zero for R = 0.1. It is obvious that the existence of compensation phenomenon is due to the different temperature dependence of positive and negative
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Fig. 5. (Color online.) Total magnetization, M T , of the ternary alloy nanowire as a function of reduced temperature for p = 0.25 and selected values of the interaction parameter: R = 0.1, 0.2, 0.3, 0.4, 0.5.
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magnetization components. However, for a higher exchange interaction ratio of R = 0.5, the magnetization curves do not intersect as seen from Fig. 6 (b). It should be underlined that ternary alloy Prussian blue compounds of the type A B p C 1− p are shown to exhibit compensation points according to both experimental [38] and theoretical investigations [39,41,53–55,59].
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4. Conclusions
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To summarize, we have performed detailed MC simulations in order to study the finite temperature magnetic phase transition properties of a single ternary alloy cylindrical nanowire of the type A B p C 1− p . The model ternary alloy nanowire system contains both ferromagnetic ( J A B > 0) and antiferromagnetic ( J AC < 0) exchange spin couplings and R = | J AC |/ J A B specifies the dominant type of interaction in the system. The phase separation curves of the system is presented in ( R − k B T C / J A B ) and ( p − k B T C / J A B ) spaces. We have determined the critical value, R c , where the transition temperature does not depend on the concentration value, p. According to our simulation results, R c value of the quasi-one-dimensional ternary alloy nanowire is the same with its three dimensional bulk counterpart [55]. Magnetization profiles of the system indicate that it is possible to tune the saturation magnetization by changing the concentration value. Also, the ternary alloy nanowire system is shown to display compensation temperatures depending on the values of p and R. It is worth mentioning that, from the point of the theoretical model, the ternary alloy nanowire system can also be modeled by Heisenberg type Hamiltonian with single-ion anisotropy term or/and Zeeman energy. Besides, the present model of the ternary alloy nanowire assumes that the outer part of the nanowire is just as the same with its inner part. On the other side, a more complicated model, for instance core/shell nanowire system with a ternary alloy core coupling ferromagnetically with an antiferromagnetic shell may be considered and many interesting phenomenon such as exchange bias effect may be investigated.
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Fig. 6. (Color online.) Total magnetization, M T , positive magnetization due to the type- A and type-B magnetic components, M + , and absolute value of the negative magnetization due to the type-C magnetic components, M − , of the ternary alloy nanowire as a function of reduced temperature for p = 0.25 in the case of (a) R = 0.1, (b) R = 0.5.
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Acknowledgements
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The numerical calculations reported in this paper were performed at TÜBI˙ TAK ULAKBI˙ M (Turkish agency), High Performance and Grid Computing Center (TRUBA Resources).
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Magnetic features of a mixed ferro-ferrimagnetic ternary alloy cylindrical nanowire
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Article history: Received 2 July 2017 Received in revised form 21 August 2017 Accepted 29 August 2017 Available online xxxx Communicated by M. Wu Keywords: Ternary alloy Magnetic cylindrical nanowires Monte Carlo simulation
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In the present study, we have investigated the finite temperature magnetic phase transition properties of a mixed ferro-ferrimagnetic ternary alloy cylindrical nanowire of the type A B p C 1− p by Monte Carlo simulation technique. The nanowire system consists of two interpenetrating sublattices, one of which contains type- A magnetic components with spin-3/2 and the other one is randomly occupied type-B and type-C magnetic components with spin-1 and spin-5/2, respectively. We have examined the effect of exchange interaction ratio, R, and the concentration value of type-B magnetic ions, p, on the transition temperature of the system. It has been found that one can modify the transition temperature and saturation magnetization of the system by varying concentration value and interaction ratio. Moreover, our numerical results show that the ternary alloy nanowire system exhibits compensation behavior for particular values of the system parameters. © 2017 Elsevier B.V. All rights reserved.
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1. Introduction
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Magnetic nanowires are one-dimensional nanostructures with peculiar properties such as large proportion of surface atoms and high aspect ratio (ratio of length to diameter) [1,2]. These unusual properties of nanowires make them attractive for a variety potential technological applications in the fields of high density magnetic recording media [3,4] and biomedical applications [5,6]. Theoretically, magnetic nanowires become excellent structures for understanding magnetism in nanoworld. Up to now, magnetic features of nanowires have been successfully investigated with several methods for instance effective field theory (EFT) [7–14], Monte Carlo (MC) simulations [15–23] and micromagnetic simulations [24]. Most of the theoretical studies in literature have been focused on phase transition properties and hysteresis characteristics of nanowires with core/shell structure. Prussian blue analogs [25], which are a class of moleculebased magnets, are in active investigations in the last decades because of their important magnetic features such as photoinduced magnetic pole inversion [26], inverted magnetic hysteresis loop [27], existence of one or two compensation temperatures [28,29] and possible technological applications ranging from hydrogen storage [30] to electrode materials for batteries [31]. Moreover, in recent years, nano-sized magnets based on Prus-
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E-mail address: [email protected]. http://dx.doi.org/10.1016/j.physleta.2017.08.060 0375-9601/© 2017 Elsevier B.V. All rights reserved.
sian blue analogs such as nanoparticles with core/shell structure [30,32,33], nanowire arrays [34,35] and nanotubes [36] have been synthesized with various size-dependent magnetic features. Particularly, the multi-metal Prussian blue analogs have gained remarkable interest with the pioneering works of Okhoshi and his co-workers [37–39]. They have designed ternary metal Prussian blue compounds of the type (NiIIp MnII1− p )1.5 CrIII (CN)6 , which includes both ferromagnetic ( J Ni–Cr > 0) and antiferromagnetic ( J Mn–Cr < 0) superexchange interactions [37–39]. It has been demonstrated that it is possible to modify the magnetic features of ternary metal Prussian blue compounds such as the saturation magnetization, Weiss temperature and coercive field by changing the active concentration of the magnetic ions in the compound [37]. On the theoretical side, magnetic properties of mixed ferromagnetic–ferrimagnetic ternary alloy Prussian blue analogs of the type A B p C 1− p have been widely investigated in literature by various methods for instance mean field theory (MFT) [40–47], EFT [48–50], Bethe lattice approximation [51], Green’s function technique [52], MC simulation method [53–60]. In the above mentioned theoretical investigations, the ternary alloy has been modeled such that type- A magnetic ions are located in one of the sublattice while type-B and type-C magnetic components are distributed randomly over the other sublattice with probabilities p and 1 − p, respectively. Also, the model system contains both ferromagnetic ( J A B > 0) and antiferromagnetic ( J AC < 0) exchange interactions between the nearest neighbor magnetic components in accordance with the real ternary metal Prussian blue analogs
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[26]. The antiferromagnetic to ferromagnetic exchange interaction ratio R = | J AC |/ J A B plays an important role in the magnetic properties of the ternary alloys since this parameter determines the dominant type of exchange interaction in the system. Moreover, there exists a special interaction ratio, R C , for which the critical temperature of the system becomes independent of the concentration value p [55]. Despite the extensive literature on magnetic properties of ternary alloy Prussian blue analogs, as far as we know there is no theoretical study on magnetic nanostructures composed of ternary alloys. Thus, in the present study we have investigated the magnetic features of a mixed ferro-ferrimagnetic ternary alloy nanowire of the type A B p C 1− p . For this purpose, we have performed detailed MC simulations based on Metropolis algorithm. The spins of the magnetic components are selected as S A = 3/2, S B = 1 and S C = 5/2 for type- A, type-B and type-C magnetic components, respectively. We have examined the magnetic phase diagrams of the system in ( R − k B T C / J A B ) and ( p − k B T C / J A B ) planes and compared the magnetic features of the system with those of its bulk counterparts. Besides, for certain values of the concentration value and exchange interaction ratio, the ternary alloy nanowire system is shown to exhibit compensation behavior. The paper is organized as follows. The details of the theoretical model and our MC simulations are given in section 2. Our numerical results regarding the magnetic properties of ternary alloy nanowires are presented in section 3. Finally, our conclusions are summarized in section 4.
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2. Formulation
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We consider a ternary alloy cylindrical nanowire of radius r = 10 and length L = 200 on a simple cubic lattice by imposing free boundary conditions in xy plane and periodic boundary conditions in z-direction (see Fig. 1). The considered ternary alloy nanowire of the type A B p C 1− p includes two interpenetrating sublattices. Type-B magnetic components with spin-1 and type-C magnetic components with spin-5/2 are distributed randomly throughout the one of the sublattice with probabilities p and 1 − p, respectively and they connect to type- A magnetic components with spin-3/2 which are located on the other sublattice. Therefore, the Hamiltonian corresponding to the model ternary alloy nanowire system can be written as
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N A A M A = [m A ] = Si , NA
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respectively. If site j is occupied by type-α atom (α = B or C ), δ j α takes the value of unity and otherwise it is zero. i , j denotes the sum over the nearest neighbors and we assume that the spin–spin exchange interaction between type- A and type-B magnetic ions is ferromagnetic ( J A B > 0) whereas type- A and type-C magnetic ions interact via antiferromagnetic spin coupling ( J AC < 0). In our numerical calculations, J A B is set as unity and the other interested quantities are normalized with J A B . In order to study the magnetic behavior of the ternary alloy nanowire, we use the MC simulation technique based on single spin-flip Metropolis algorithm [61,62]. We perform 20 independent sample realizations. In each computer experiment, type-B and type-C components are distributed randomly through the lattice and their positions do not change during the experiment. We start the simulation at low temperature region with ferromagnetic arrangement of the spins and heat the system slowly with temperature step of k B T / J A B = 5 × 10−2 where k B and T are Boltzmann constant and absolute temperature, respectively. For every temperature step, the first configuration is selected as the final configuration of the previous temperature step. Then, the following configurations are created by selecting the sites sequentially from the lattice and making singlespin-flip attempts. The generated new configuration is accepted or rejected according to Metropolis algorithm. We have performed 105 MC steps per site at each temperature step. Thermal averages of interested quantities are computed after discarding the first 4 × 104 steps in order to obtain the equilibrium state. We have calculated the thermal average and configurational average of sublattice magnetizations (m A , m B , mC ) per spin:
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and total magnetization, m T :
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where N A , N B , N C and N T = N A + N B + N C are the total number of magnetic components A, B, C and overall of the system, respectively. Also, ... corresponds to thermal averages and [...] denotes the configurational averages. In order to determine the transition temperature of the system, we use the thermal variation of the susceptibility, which is defined by
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3. Results and discussion
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Fig. 1. (Color online.) Schematic representation of the ternary alloy cylindrical nanowire of the type A B p C 1− p with length L and radius r for p = 0.5. Cross section of the nanowire is shown on the right hand side.
The magnetic phase diagrams of the ternary alloy nanowire system in ( R − k B T C / J A B ) plane is presented for selected values of the concentration value p in Fig. 2. Here, R = | J AC |/ J A B specifies the
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Fig. 2. (Color online.) Magnetic phase diagrams in ( R − k B T C / J A B ) plane of the ternary alloy nanowire system for several values of the concentration value p.
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Fig. 3. (Color online.) Variation of the transition temperature of the ternary alloy nanowire with active concentration value of type-B magnetic components, p in a range of 0.41 ≤ R ≤ 0.49.
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dominant sort of exchange interaction (i.e., ferromagnetic, antiferromagnetic or both of them) in the system. It is obvious that p = 1 case corresponds to a mixed spin-3/2 and spin-1 ferromagnetic nanowire with no J AC exchange interaction in the system and hence the transition temperature of the system does not change with R for this concentration value. On the other hand, the system is a ferrimagnetic mixed spin-3/2 and spin-5/2 nanowire for p = 0. In this case, starting from R = 1, the transition temperature of the system decreases with a decrease in R and eventually reduces to zero at R = 0. Besides, if one considers the special case of R = 0, the system consists of type- A and type-B magnetic ions with a mutual ferromagnetic interaction for p = 1. As p gets smaller, type-B magnetic components replace with type-C components (which behave as non-magnetic atoms for R = 0) and the transition temperature of the system decreases. Thus, for all the p values other than p = 1, we have a randomly site-diluted Ising nanowire and on the basis of the percolation theory, there should be no magnetic ordering below a critical concentration value of p c [63]. Since it is beyond the scope of our work to determine the precise value of p c , we have not perform any detailed simulations for this purpose. However, according to Fig. 2, p c should be smaller than p = 0.25. As one can readily see from Fig. 2 that all the phase separation curves belonging to different p values intersect at a special point, R c , which means that the transition temperature of the system does not depend on the concentration value of type-B magnetic components at this point. In order to determine the precise value of R c , we have examined the transition temperature of the system as a function of p in a narrow range of 0.41 ≤ R ≤ 0.49 with an increment of 0.01 in R value in Fig. 3. It can be clearly seen that the special point of the ternary alloy cylindrical nanowire system is R c = 0.47. For this special interaction ratio, a balance occurs between the contribution of the first and second terms of the Hamiltonian (Eq. (1)) such that replacing type-B magnetic components with type-C components (or vice versa) has no effect on the temperature where transition from the ordered to disordered phase occurs. For R > R c transition temperature of the system decreases with increasing p whereas it increases with p for R < R c . We should emphasize that the shape of the phase boundary curves obtained in ( R − k B T C / J A B ) and ( p − k B T C / J A B ) for the ternary alloy nanowire system is in accordance with two-dimensional [53, 60] and three-dimensional [55] mixed-spin ternary alloy bulk systems.
It is worthwhile to compare the special point of the ternary alloy nanowire system with the R c values previously reported in literature. Remarkably, special point of the mixed ferro-ferrimagnetic ternary alloy nanowire is the same with its three dimensional bulk counterpart [55]. Combining the results of ref. [55] with this study, one can infer that reducing the dimension of the system in xand y-directions, does not change the critical interaction ratio. This finding is reasonable since it has been demonstrated that R c value does not vary with the number of nearest neighbors [40] and also the lattice stoichiometry [55] within the framework of MFT and MC simulations, respectively. In a ternary alloy nanowire, applying free boundary conditions also implies decreasing the total number of coordination numbers as compared to the bulk system with periodic boundary conditions in all directions and thus R c is not influenced by the reduced dimension of the system. We should notice that according to our simulation results, the critical interaction ratio is likewise R c = 0.47 for smaller nanowire radii value for instance, r = 6. The proportion of surface atoms with respect to the total number of atoms increases with a reduction in radius and thus the average number of coordination numbers decreases. Nevertheless, the critical interaction ratio does not change with nanowire radius. We have shown the thermal variations of the total magnetization and susceptibility profiles of the ternary alloy nanowire for varying p below (R = 0.2), above (R = 1) and at the critical exchange interaction ratio R c in Fig. 4. When all the spin–spin exchange interaction terms are equal to each other (R = 1), a second order phase transition from ordered to disordered phase occurs at the transition temperature which depends on the p value. The susceptibility curves in Fig. 4(f) exhibit a maximum at the transition temperature of the system. Also, the saturation magnetization of the system strongly depends on p. For example for p = 1, saturation magnetization is maximum (M s = 1.25) whereas it takes the minimum value of M s = 0.0625 for p = 0.25. Therefore, according to our observations it is possible to modify the saturation magnetization of the ternary alloy nanowire by varying the concentration of type-B magnetic components. At the critical exchange interaction ratio, as demonstrated in magnetization and susceptibility curves in Fig. 4(c)–(d), the transition to disordered phase occurs at the same temperature for all the p values under consideration. If one concentrates on an interaction ratio which is below R c (R = 0.2), the magnetization curves exhibit different shapes depending on the concentration value. For instance,
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Fig. 4. Thermal dependence of the total magnetization, M T , and the total susceptibility, χ T , of the ternary alloy nanowire (a)–(b) for R = 0.2, (c)–(d) for R = 0.47 and (e)–(f) for R = 1. The curves are displayed for varying values of concentration of type-B atoms: p = 0, 0.25, 0.5, 0.75 and 1.
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for p = 0 and p = 1, the total magnetization of the system increases to its saturation value above the transition temperature, however it has single maximum in the case of p = 0.5. Moreover, the total magnetization of the system takes a minimum value below the transition temperature p = 0.25. This indicates that the ternary alloy nanowire system can exhibit compensation behavior for particular values of concentration and exchange interaction ratio values.
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The existence of compensation point can be seen explicitly from Fig. 5 where we have illustrated the temperature dependence of the total magnetization of the ternary alloy nanowire for p = 0.25 and selected values of the interaction parameter: R = 0.1, 0.2, 0.3, 0.4, 0.5. According to Fig. 5, for all the R values except for R = 0.5, the total magnetization of the system vanishes at the compensation temperature. Above the compensation point, the total magnetization firstly increases and then decreases
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with raising temperature. Finally, the system becomes disordered at the transition temperature of the system. The compensation temperature shifts to higher temperature region with increasing antiferromagnetic exchange interaction strength between type- A and type-C magnetic components and it disappears for R = 0.5. In the present study, we do not examine the range of p and R at which the compensation behavior occurs for the ternary alloy nanowire and thus a detailed investigation of this point is left for a future work. The compensation effect can also be examined from the thermal variation of positive magnetization because of the A and B sublattices (M + = 1/2( M A + pM B )) and absolute value of the negative magnetization because of the C sublattice (M − = 1/2(1 − p ) M C ) which is shown in Fig. 6 for (a) R = 0.1, (b) R = 0.5 with a concentration value of p = 0.25. The absolute value of the positive and negative magnetization cross at the compensation point at which these magnetization components have equal magnitude and thus the total magnetization of the system is zero for R = 0.1. It is obvious that the existence of compensation phenomenon is due to the different temperature dependence of positive and negative
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Fig. 5. (Color online.) Total magnetization, M T , of the ternary alloy nanowire as a function of reduced temperature for p = 0.25 and selected values of the interaction parameter: R = 0.1, 0.2, 0.3, 0.4, 0.5.
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magnetization components. However, for a higher exchange interaction ratio of R = 0.5, the magnetization curves do not intersect as seen from Fig. 6 (b). It should be underlined that ternary alloy Prussian blue compounds of the type A B p C 1− p are shown to exhibit compensation points according to both experimental [38] and theoretical investigations [39,41,53–55,59].
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4. Conclusions
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To summarize, we have performed detailed MC simulations in order to study the finite temperature magnetic phase transition properties of a single ternary alloy cylindrical nanowire of the type A B p C 1− p . The model ternary alloy nanowire system contains both ferromagnetic ( J A B > 0) and antiferromagnetic ( J AC < 0) exchange spin couplings and R = | J AC |/ J A B specifies the dominant type of interaction in the system. The phase separation curves of the system is presented in ( R − k B T C / J A B ) and ( p − k B T C / J A B ) spaces. We have determined the critical value, R c , where the transition temperature does not depend on the concentration value, p. According to our simulation results, R c value of the quasi-one-dimensional ternary alloy nanowire is the same with its three dimensional bulk counterpart [55]. Magnetization profiles of the system indicate that it is possible to tune the saturation magnetization by changing the concentration value. Also, the ternary alloy nanowire system is shown to display compensation temperatures depending on the values of p and R. It is worth mentioning that, from the point of the theoretical model, the ternary alloy nanowire system can also be modeled by Heisenberg type Hamiltonian with single-ion anisotropy term or/and Zeeman energy. Besides, the present model of the ternary alloy nanowire assumes that the outer part of the nanowire is just as the same with its inner part. On the other side, a more complicated model, for instance core/shell nanowire system with a ternary alloy core coupling ferromagnetically with an antiferromagnetic shell may be considered and many interesting phenomenon such as exchange bias effect may be investigated.
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Fig. 6. (Color online.) Total magnetization, M T , positive magnetization due to the type- A and type-B magnetic components, M + , and absolute value of the negative magnetization due to the type-C magnetic components, M − , of the ternary alloy nanowire as a function of reduced temperature for p = 0.25 in the case of (a) R = 0.1, (b) R = 0.5.
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Acknowledgements
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The numerical calculations reported in this paper were performed at TÜBI˙ TAK ULAKBI˙ M (Turkish agency), High Performance and Grid Computing Center (TRUBA Resources).
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