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Hysteresis properties of a quenched disordered binary alloy cylindrical nanowire: A Monte Carlo simulation study
Journal of Alloys and Compounds 720 (2017) 388e394
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Hysteresis properties of a quenched disordered binary alloy cylindrical nanowire: A Monte Carlo simulation study Z.D. Vatansever _ Department of Physics, Dokuz Eylül University, Tr-35160, Izmir, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 March 2017 Received in revised form 21 May 2017 Accepted 24 May 2017 Available online 27 May 2017
We have elucidated the hysteresis features of a quenched disordered binary alloy cylindrical nanowire of the type Ap B1p by means of Monte Carlo simulation technique. The nanowire system is composed of two types of magnetic components, A with spin-1/2 and B with spin-1, which are distributed randomly on the sites of the nanowire. The dependence of the remanence magnetization ðMr Þ and coercivity field ðHc Þ values on the active concentration of type-A magnetic components, p, and the strength of the spinspin coupling between type-A and -B components have been investigated in a wide range of temperature values. Our Monte Carlo simulation findings suggest that, it is possible to enhance the magnetic properties (i.e., coercivity, remanence as well as hysteresis loops) of the system by changing the concentration of the magnetic components, and also the exchange coupling strength between unlike atoms. Finally, we compare our numerical findings with recent experimental results, and it is found that there exists a qualitatively agreement between them. © 2017 Elsevier B.V. All rights reserved.
Keywords: Binary alloy systems Magnetic nanowire Hysteresis Monte Carlo simulation
1. Introduction In the last several decades, there has been a remarkable interest on nanostructured magnetic materials not only due to their wide range of technological applications but also because of their outstanding magnetic features, for instance superparamagnetism [1,2], giant magnetoresistance [3,4] and exchange bias effect [5,6]. Among these, one-dimensional nanostructured magnetic nanowires play an important role in nanoscience owing to their unique properties such as large proportion of surface atoms and high length to diameter ratio [7]. Besides, ferromagnetic nanowires provide promising application areas ranging from high density magnetic recording media [8,9] to permanent magnets [10,11]. From the experimental point of view, much attention has been paid to the binary alloy magnetic nanowire arrays fabricated in nanoporous anodic aluminum oxide matrix by electrochemical deposition techniques. The alloy composition and geometrical properties such as nanowire length and diameter can be tuned experimentally making it possible to control the magnetic properties of nanowires [12e14]. In a recent study, Ramazani and coworkers have investigated the effects of geometrical parameters on the magnetic features of FeNi nanowire arrays [15]. Their
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hysteresis loop measurements show that with an increment of nanowire length and diameter, the coercivity and squareness values decrease. Moreover, several experimental studies have been reported the effects of concentration value, p, on the magnetic features of binary alloy nanowire arrays of the type Ap B1p [16e19]. For instance, in Ref. [19], it has been observed that changing Ni content of Cop Ni1p nanowire arrays enables to control magnetic properties like coercive field and the remanent magnetization. On the other side, it is a fact that the experimental results have stimulated theoretical studies regarding the hysteresis characteristics of magnetic nanowires. Mostly, magnetic nanowires with core-shell morphology have been investigated with several methods such as effective field theory (EFT) [20e25] and Monte Carlo (MC) simulations [26e32]. Magnetic properties of cylindrical spin-1/2 core-shell Ising nanowire have been investigated within the framework of EFT [20] and MC simulations [26]. For both methods it has been found that for appropriate values of system parameters, triple hysteresis loops occur in the presence of antiferromagnetic interaction between shell and the core parts of the system. In another interesting work, magnetic and thermodynamic properties of mixed-spin Ising nanowire with spin-1 core and spin3/2 shell has been studied in the framework of MC simulation [29,30]. The authors have stated that the single-ion anisotropy, exchange coupling strength between the core and the shell and the temperature have considerable influences on the coercivity and
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remanence of the system. The effects of random crystal field on the hysteresis features of a spin-1 ferromagnetic cylindrical nanowire has been investigated by Zaim and co-workers in Ref. [31] by using MC technique. Single, double or para hysteresis loops are observed depending on the parameters of the random crystal field and temperature. Hysteresis properties of magnetic nanowires in the presence of random magnetic field [22] and transverse field [23] have been examined within the framework of EFT. Besides, more realistic core/shell nanowires are simulated by micromagnetic simulations. For instance in Ref. [33], exchange bias properties of core/shell coaxial Co/CoO freestanding nanowires have been studied in the presence of cooling and external magnetic fields in the perpendicular nanowire axis. In our previous work, we have elucidated the magnetic phase transition properties of a quenched disordered binary alloy cylindrical nanowire system by means of MC simulations [32]. The phase diagrams of the system are obtained in several planes. However, to the best of our knowledge, there is no theoretical report on hysteresis properties of a binary alloy magnetic nanowire. Therefore, in the present study, we intend to investigate magnetic features of a single ferromagnetic quenched disordered binary alloy cylindrical nanowire system of the type Ap B1p where p represents the active concentration of type-A atoms. The nanowire contains two types of magnetic components A with spin-1/2 and B with spin-1 and they are distributed randomly on the lattice of the nanowire. With MC simulation technique based on Metropolis algorithm, we determine the dependence of magnetic properties such as, coercivity,
389
remanence and hysteresis loops, on the concentration value of the type-A magnetic components and temperature for several values of the exchange coupling between type-A and type-B magnetic components. The paper is organized as follows. In section 2, the details of the theoretical model and our MC simulations are given. The results of our numerical simulations are presented in section 3. Finally, our conclusions are summarized in section 4.
2. Formulation We implement quenched disordered binary alloy cylindrical nanowire of the type Ap B1p with total radius r, and length L. The magnetic components A and B are randomly located on the magnetic nanowire with the concentration p and 1p, respectively. Thus, there exists three types of exchange interactions in the nanowire: JAA , JBB and JAB ¼ JBA between AA, BB and AB atoms, respectively. The Hamiltonian of the studied system is defined as follows:
b b b b H total ¼ H ex þ H anisotropy þ H Zeeman ;
(1)
b ex ; H b b here H anisotropy and H Zeeman terms denote the energy contributions to the system coming from spin-spin interaction between nearest-neighbor spins, single-ion anisotropy and Zeeman terms, respectively. They are given as follows:
Fig. 1. Hysteresis loops of the binary alloy cylindrical nanowire for several values of the nanowire radius, r¼2,4,6,8,10 with p¼0.5 and JAB =JAA ¼ 1:0 at a) kB T=JAA ¼ 1:0, b) kB T=JAA ¼ 2:0, c) kB T=JAA ¼ 3:0 and d)kB T=JAA ¼ 4:0.
390
b ex ¼ H
Z.D. Vatansever / Journal of Alloys and Compounds 720 (2017) 388e394
X 〈i;j〉
JAA diA djA si sj þ JBB diB djB Si Sj þ JAB diA djB si Sj
þdiB djA Si sj ; b H anisotropy ¼ D
X
diB S2i ;
i
b Zeeman ¼ H H
X ðdiA si þ diB Si Þ:
(2)
i
For the sake of simplicity, we select all of the spin-spin couplings such that they are ferromagnetic, i.e., JAA,JBB and JAB > 0. D refers to the single-ion anisotropy term, which acts only on the type-B atoms in the system, and H is the external magnetic field. The s and S are Ising spin variables, and they can take values s¼±1 and S ¼ ±1; 0 for the magnetic components-A and B of system, respectively. In our computer experiments, the symbols dia ¼ 1 (a¼A or B) if site i is occupied by type-a atom and 0 otherwise. JAA and JBB are fixed to unity in the simulations and the other system parameters are normalized with JAA . In order to gain a better understanding of the hysteresis characters of the disordered binary alloy cylindrical nanowire, we realize Monte Carlo simulation method based on the single-spin flip Metropolis algorithm [34,35]. We deal with a quenched disordered binary alloy nanowire which means that the positions of the atoms are frozen during a simulation. The nanowire system is located on a simple cubic lattice and it has a length of L¼200 and a
radius of r. In order to simulate such a finite nanosystem, we use the free boundary conditions in xy plane and periodic boundary conditions in z direction. The computer experiment starts at a high temperature value using random initial condition, which corresponds a paramagnetic phase, and then the system is slowly cooled down with a reduced temperature step of kB DT=JAA ¼ 0:1. At each temperature step, the magnetization is calculated over a number of 105 MC steps after 4104 MC steps are used for thermalization process. Once the specific temperature configuration has been obtained, this configuration has been used as the initial configuration for decreasing field branch of the hysteresis which is obtained by scanning the magnetic field from H to H with a reduced magnetic field step of DH=JAA ¼ 0:05. When the magnetic field reaches the H value, decreasing field branch is completed and the obtained configuration corresponding to the H value has been used as the initial configuration for the increasing field branch of the hysteresis. Similarly, by scanning the magnetic field from H to H, increasing field branch is obtained. At each magnetic field step, the physical quantities are averaged over a number of 104 MC steps after 5103 MC steps are used for thermalization process. 3. Results and discussion In this section, we present our MC simulation results regarding the hysteresis properties of the binary alloy nanowire system. We should notice that p¼1 and p¼0 cases correspond to clean nanowire systems composing of type-A atoms and of type-B atoms, respectively. For both p values, the system does not contain any
Fig. 2. a)-c) Coercivity and b)-d) remanence variations of the cylindrical nanowire system as a function of p at kB T=JAA ¼ 2:0 (first row) and kB T=JAA ¼ 3:0 (second row) for several values of the exchange interaction strength between type-A and type-B atoms: JAB =JAA ¼ 0:01; 0:5; 1:0 and 2.0.
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Fig. 3. Temperature dependence of the coercivity and remanence for exchange coupling strengths of a)-b) JAB =JAA ¼ 0:5 and c)-d) for JAB =JAA ¼ 2:0 for varying concentration of typeA atoms: p¼0,0.2,0.4,0.6,0.8 and 1.
disorder effects originating from different type of magnetic components and therefore the magnitude of the exchange interaction term, JAB =JAA , has no effect on the magnetic properties of the system. Also, for simplicity, we investigate the magnetic characteristics of the system in the absence of single-ion anisotropy (D¼0). Firstly, we investigate the magnetic behavior of the binary alloy nanowire system by changing the magnitude of the nanowire radius. The hysteresis curves for several values of the nanowire radius, r¼2,4,6,8,10, with a concentration value of p¼0.5 and exchange interaction strength of JAB =JAA ¼ 1:0 at different reduced temperature values are shown in Fig. 1 (a)-(d). It can be clearly seen that the thickness of the nanowire has an important effect on the variation of the total magnetization with the external magnetic field. The binary alloy nanowire system is in the ferromagnetic phase at kB T=JAA ¼ 1:0 and kB T=JAA ¼ 2:0 for all the nanowire radius values under consideration. Since the exchange energy contribution to the total energy increases as the nanowire becomes thicker, much more magnetic field is required to reverse the spins of the system and hence, the coercive field takes bigger values with increasing r for a fixed value of reduced temperature. At kB T=JAA ¼ 3:0, the nanowire becomes paramagnetic for r¼2 because the proportion of the surface atoms increases as r gets smaller (Fig. 1 (c)). With a further increment in temperature (Fig. 1 (d)), the nanowire is paramagnetic for all r values. According to these simulation results, in order to reduce surface effects, we deal with a binary alloy cylindrical nanowire of radius r¼10 throughout this study.
We have illustrated the coercive field, Hc, and remanent magnetization, Mr, of the binary alloy nanowire for varying values of p with JAB =JAA ¼ 0:01; 0:5; 1:0 and 2.0 at a reduced temperature value of kB T=JAA ¼ 2:0 in Fig. 2(a) and (b), respectively. This figure explicitly reveals that the number of type-A magnetic components and the spin-spin exchange interaction strength between unlike magnetic components, JAB =JAA , have a significant impact on the coercivity and remanence treatments. At first glance, it is obvious that the coercivity takes larger values as JAB =JAA increases for a fixed value of p. When all the spin-spin coupling terms are equal to each other, which corresponds to JAB =JAA ¼ 1:0, the coercivity displays an almost linearly increment with p due to the enhanced number of spin-1/2 atoms in the system. If the exchange interaction term JAB =JAA takes lower values such as, JAB =JAA ¼ 0:01 and JAB =JAA ¼ 0:5, the coercivity exhibits an initial decrease with increasing p by reaching a minimum p value and then it increases with the concentration value of type-A magnetic components. Besides, starting from p¼0, the coercive field shows an increasing behavior until a maximum value for the case of strong interaction strength of JAB =JAA ¼ 2:0. Then, a reduction in coercivity is observed with a further increment in p. The dependence of the variation coercivity with p on the strength of the exchange interaction term, JAB =JAA , can be understood as follows. If one starts with a pure spin-1 nanowire corresponding to p¼0 state, the number of JAB exchange interaction terms begins to dominate by gradually adding type-A atoms in the nanowire [32]. This means that for small p values, exchange energy, which favors the parallel alignment of the spins, and therefore the
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Fig. 4. Hysteresis loops of the binary alloy cylindrical nanowire at kB T=JAA ¼ 2:0 for a) JAB =JAA ¼ 0:01, b) JAB =JAA ¼ 0:5, c) JAB =JAA ¼ 1:0, d) JAB =JAA ¼ 2:0. The curves are depicted for varying values of active concentration value of type-A magnetic components: p¼0,0.2,0.4,0.6,0.8 and 1.
magnitude of the reverse magnetic field required to destroy the magnetization increases (decreases) with p for strong (weak) ferromagnetic interaction between type-A and type-B magnetic components. Conversely, in the case of large p values, JAA bonds are effective in the system and thus the coercivity decreases (increases) with the number of type-A atoms for strong (weak) JAB =JAA values. It can be seen from Fig. 2(b) that the dependence of remanent magnetization on p is almost the same with that of coercivity behavior. In accordance with the expectations, remanence takes larger values for stronger spin-spin interaction strengths between unlike atoms. Additionally, in the presence of a weak spin-spin interaction strength of, JAB =JAA ¼ 0:01, the coercivity and remanence reduces to zero for p¼0.4 and p¼0.5 which means that the system is in the paramagnetic phase. From the above discussions we can infer that the magnetic properties like coercivity and remanence of a binary alloy nanowire of the type Ap B1p can be controlled by varying the concentration value of type-A magnetic atoms and/or the exchange interaction strength JAB =JAA . It is worthwhile to underline that the numerical results of this study are in accordance with the experimental findings of references which states the controllability of magnetic properties of binary alloy nanowires with varying composition [16e19]. In these experimental studies, the only parameter that has been changed is the concentration value, p while the other parameters remains the same. We should mention that it would be possible to tune JAA , JAB , JBB by changing the type of magnetic components from the experimental point of view. For instance, the exchange interaction strengths in a CoNi [19] binary alloy nanowire should be different than those of FeNi [15].
The concentration value p of the type-A magnetic components strongly affects the transition temperature of the binary alloy nanowire [32] and therefore the variation of Hc and Mr with p should be influenced by the temperature of the system. We show the dependence of Hc and Mr on p for the same parameters as Fig. 2(a) and (b) but for a higher temperature value of kB T=JAA ¼ 3:0 in Fig. 2(c) and (d), respectively. It is clear that the enhanced temperature of the system does not alter the overall dependence of Hc and Mr on p for all the JAB =JAA values under consideration. However, due to the increased temperature, Hc and Mr values shift to lower temperature region and the number of states which become paramagnetic increases. This can be seen in the case weak spin-spin interaction strengths JAB =JAA ¼ 0:01 and JAB =JAA ¼ 0:5 for which the coercivity and remanence become zero for p¼0.2,0.4,0.5 and p¼0.6. In order to gain further insight about the magnetic properties of the binary alloy nanowire, in Fig. 3(a)-(d) we show the temperature dependencies of the coercivity and remanence in the cases of JAB =JAA ¼ 0:5 and JAB =JAA ¼ 2:0 for different values of p. As one can readily see, Hc and Mr depend sensitively on the number of type-A atoms in the nanowire system for a fixed JAB =JAA and kB T=JAA in accordance with our discussions of Fig. 2. In addition, because of the increasing thermal fluctuations which destroy the ferromagnetic order in the system, Hc and Mr decrease with raising temperature and they reduce to zero above the transition temperature of the system for all concentration values. Comparing the coercivity values of two different exchange interaction magnitudes at fixed temperature, one can observe that in the case of JAB =JAA ¼ 2:0 coercivity takes larger values for all p. From the experimental side, in Refs. [12,17,36] temperature dependence of coercivity in binary
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Fig. 5. Hysteresis curves of the binary alloy cylindrical nanowire for a) JAB =JAA ¼ 0:01, b) JAB =JAA ¼ 0:5, c) JAB =JAA ¼ 1:0, d) JAB =JAA ¼ 2:0 with a concentration value of p¼0.5. The curves are shown for several values of the reduced temperature.
alloy nanowire arrays containing Ni and Fe atoms has been investigated. It is interesting to note that, similar to our numerical findings, it has been observed that the coercivity displays a decreasing trend with increasing temperature. As a further investigation, in Fig. 4 (a)-(d) the hysteresis curves of the binary alloy nanowire corresponding to the system parameters of Fig. 2 are presented. If the JAB =JAA term remains constant, the hysteresis curves change with p in agreement with our findings in Fig. 2. For a fixed value of p, the hysteresis loops become wider with increasing JAB =JAA value due to the enhanced spin-spin interaction in the system except for p¼0 and p¼1 since these are concentration values with no JAB bonds in the system. Thus, it is possible to obtain a hard or soft ferromagnetic binary alloy cylindrical nanowire by changing the magnitude of JAB =JAA at a fixed temperature value and p. Moreover, depending on the magnitude of the relative exchange interaction between unlike magnetic components, one can also modify the magnetic phase of the system with changing p. One can observe this from the hysteresis curves of Fig. 4 (a) where the binary alloy nanowire becomes paramagnetic for p¼0.4 and p¼0.5. Finally, the temperature dependence of the hysteresis loops of the binary alloy nanowire with p¼0.5 is presented in Fig. 5 for spinspin interaction strengths of a) JAB =JAA ¼ 0:01, b) JAB =JAA ¼ 0:5, c) JAB =JAA ¼ 1:0 and d) JAB =JAA ¼ 2:0. As one can explicitly observe, with raising temperature, the hysteresis loops become narrower and disappear above the transition of the system for a constant value of JAB =JAA . Also, the hysteresis loop areas increase with increasing spin-spin coupling strength for a constant value of
temperature indicating the controllability of the magnetic features of the considered system. Since the transition temperature of the cylindrical binary alloy nanowire raises as the ferromagnetic exchange interaction term between unlike atoms gets stronger [32], the paramagnetic behavior is observed at higher temperature values for stronger JAB =JAA strengths. As a final comment, it is worthwhile to discuss the assumptions made for the nanowire system simulated in this work. Firstly, we simulate an infinite nanowire system with periodic boundary conditions along the nanowire axis. For such an infinitely long nanowire, the demagnetizing factor in the longitudinal axis is zero. On the other side, a real nanowire system has a finite length which means that the demagnetizing field, (associated with the shape anisotropy), should play an important role in the hysteresis characteristics. Moreover, we deal with a single nanowire throughout the work whereas in the above mentioned experimental studies, binary alloy nanowire arrays are considered for which the magnetostatic interactions between individual nanowires should be taken into account for the determination of the magnetic properties [37]. Lastly, we assume that the properties of the surface of the nanowire are identical with those of the inner part except for the reduced number of nearest neighbours at the surface. However, the outer part of the nanowire may be very different than the inner part due to, for instance, altered exchange interactions or disorder effects. In this sense, many different hysteresis properties, such as exchange bias phenomenon [33,38], may be observed by considering a binary alloy ferromagnetic core and an antiferromagnetic shell with a ferromagnetic interface coupling.
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4. Conclusions To conclude, we have studied the magnetic properties of a single ferromagnetic binary alloy cylindrical nanowire of the type Ap B1p by means of MC simulation with Metropolis algorithm. Firstly, the coercivity and remanence are presented as a function of p for various values of JAB =JAA at kB T=JAA ¼ 2:0. Also, the thermal dependence of coercivity and remanence are shown in the case of varying values of p and JAB =JAA for a broad range of temperature. Our numerical results demonstrate that the magnetic properties of the system can be tuned by altering the active concentration value of type-A magnetic components and the exchange coupling between unlike magnetic components. We remark that the controllability of magnetic features of binary alloy nanowires is also observed experimentally in hysteresis loop measurements. Moreover, the hysteresis curves of the system are displayed for different values of p at fixed temperature. It has been observed that when the spin-spin interaction term remains constant, the magnetic phase of the system can be altered by changing p. Finally, we have considered the hysteresis loops of the system in the case of several values of temperature for p¼0.5. The hysteresis curves become narrower with raising temperature and paramagnetic phase is seen above the transition temperature of the system. As a final conclusion, from the theoretical perspective, we note that the results obtained in the present study may be improved by using a little bit complicate model, such as Heisenberg model including single-ion anisotropy term. From a magnetism point of view, it could be also interesting to focus on the thermal and magnetic phase transition properties of the same nanowire system, but in the form of ternary alloy ABp C1p by means of MC simulation method. Acknowledgements The numerical calculations reported in this paper were per_ _ (Turkish agency), High Performance formed at TÜBITAK ULAKBIM and Grid Computing Center (TRUBA Resources). References [1] Q. Chen, Z. John Zhang, Appl. Phys. Lett. 73 (1998) 3156. [2] M. Mikhaylova, D.K. Kim, N. Bobrysheva, M. Osmolowsky, V. Semenov, T. Tsakalakos, M. Muhammed, Langmuir 20 (2004) 2472e2477. [3] K. Liu, K. Nagodawithana, P.C. Searson, C.L. Chien, Phys. Rev. B 51 (1995) 7381.
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Hysteresis properties of a quenched disordered binary alloy cylindrical nanowire: A Monte Carlo simulation study Z.D. Vatansever _ Department of Physics, Dokuz Eylül University, Tr-35160, Izmir, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 March 2017 Received in revised form 21 May 2017 Accepted 24 May 2017 Available online 27 May 2017
We have elucidated the hysteresis features of a quenched disordered binary alloy cylindrical nanowire of the type Ap B1p by means of Monte Carlo simulation technique. The nanowire system is composed of two types of magnetic components, A with spin-1/2 and B with spin-1, which are distributed randomly on the sites of the nanowire. The dependence of the remanence magnetization ðMr Þ and coercivity field ðHc Þ values on the active concentration of type-A magnetic components, p, and the strength of the spinspin coupling between type-A and -B components have been investigated in a wide range of temperature values. Our Monte Carlo simulation findings suggest that, it is possible to enhance the magnetic properties (i.e., coercivity, remanence as well as hysteresis loops) of the system by changing the concentration of the magnetic components, and also the exchange coupling strength between unlike atoms. Finally, we compare our numerical findings with recent experimental results, and it is found that there exists a qualitatively agreement between them. © 2017 Elsevier B.V. All rights reserved.
Keywords: Binary alloy systems Magnetic nanowire Hysteresis Monte Carlo simulation
1. Introduction In the last several decades, there has been a remarkable interest on nanostructured magnetic materials not only due to their wide range of technological applications but also because of their outstanding magnetic features, for instance superparamagnetism [1,2], giant magnetoresistance [3,4] and exchange bias effect [5,6]. Among these, one-dimensional nanostructured magnetic nanowires play an important role in nanoscience owing to their unique properties such as large proportion of surface atoms and high length to diameter ratio [7]. Besides, ferromagnetic nanowires provide promising application areas ranging from high density magnetic recording media [8,9] to permanent magnets [10,11]. From the experimental point of view, much attention has been paid to the binary alloy magnetic nanowire arrays fabricated in nanoporous anodic aluminum oxide matrix by electrochemical deposition techniques. The alloy composition and geometrical properties such as nanowire length and diameter can be tuned experimentally making it possible to control the magnetic properties of nanowires [12e14]. In a recent study, Ramazani and coworkers have investigated the effects of geometrical parameters on the magnetic features of FeNi nanowire arrays [15]. Their
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hysteresis loop measurements show that with an increment of nanowire length and diameter, the coercivity and squareness values decrease. Moreover, several experimental studies have been reported the effects of concentration value, p, on the magnetic features of binary alloy nanowire arrays of the type Ap B1p [16e19]. For instance, in Ref. [19], it has been observed that changing Ni content of Cop Ni1p nanowire arrays enables to control magnetic properties like coercive field and the remanent magnetization. On the other side, it is a fact that the experimental results have stimulated theoretical studies regarding the hysteresis characteristics of magnetic nanowires. Mostly, magnetic nanowires with core-shell morphology have been investigated with several methods such as effective field theory (EFT) [20e25] and Monte Carlo (MC) simulations [26e32]. Magnetic properties of cylindrical spin-1/2 core-shell Ising nanowire have been investigated within the framework of EFT [20] and MC simulations [26]. For both methods it has been found that for appropriate values of system parameters, triple hysteresis loops occur in the presence of antiferromagnetic interaction between shell and the core parts of the system. In another interesting work, magnetic and thermodynamic properties of mixed-spin Ising nanowire with spin-1 core and spin3/2 shell has been studied in the framework of MC simulation [29,30]. The authors have stated that the single-ion anisotropy, exchange coupling strength between the core and the shell and the temperature have considerable influences on the coercivity and
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remanence of the system. The effects of random crystal field on the hysteresis features of a spin-1 ferromagnetic cylindrical nanowire has been investigated by Zaim and co-workers in Ref. [31] by using MC technique. Single, double or para hysteresis loops are observed depending on the parameters of the random crystal field and temperature. Hysteresis properties of magnetic nanowires in the presence of random magnetic field [22] and transverse field [23] have been examined within the framework of EFT. Besides, more realistic core/shell nanowires are simulated by micromagnetic simulations. For instance in Ref. [33], exchange bias properties of core/shell coaxial Co/CoO freestanding nanowires have been studied in the presence of cooling and external magnetic fields in the perpendicular nanowire axis. In our previous work, we have elucidated the magnetic phase transition properties of a quenched disordered binary alloy cylindrical nanowire system by means of MC simulations [32]. The phase diagrams of the system are obtained in several planes. However, to the best of our knowledge, there is no theoretical report on hysteresis properties of a binary alloy magnetic nanowire. Therefore, in the present study, we intend to investigate magnetic features of a single ferromagnetic quenched disordered binary alloy cylindrical nanowire system of the type Ap B1p where p represents the active concentration of type-A atoms. The nanowire contains two types of magnetic components A with spin-1/2 and B with spin-1 and they are distributed randomly on the lattice of the nanowire. With MC simulation technique based on Metropolis algorithm, we determine the dependence of magnetic properties such as, coercivity,
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remanence and hysteresis loops, on the concentration value of the type-A magnetic components and temperature for several values of the exchange coupling between type-A and type-B magnetic components. The paper is organized as follows. In section 2, the details of the theoretical model and our MC simulations are given. The results of our numerical simulations are presented in section 3. Finally, our conclusions are summarized in section 4.
2. Formulation We implement quenched disordered binary alloy cylindrical nanowire of the type Ap B1p with total radius r, and length L. The magnetic components A and B are randomly located on the magnetic nanowire with the concentration p and 1p, respectively. Thus, there exists three types of exchange interactions in the nanowire: JAA , JBB and JAB ¼ JBA between AA, BB and AB atoms, respectively. The Hamiltonian of the studied system is defined as follows:
b b b b H total ¼ H ex þ H anisotropy þ H Zeeman ;
(1)
b ex ; H b b here H anisotropy and H Zeeman terms denote the energy contributions to the system coming from spin-spin interaction between nearest-neighbor spins, single-ion anisotropy and Zeeman terms, respectively. They are given as follows:
Fig. 1. Hysteresis loops of the binary alloy cylindrical nanowire for several values of the nanowire radius, r¼2,4,6,8,10 with p¼0.5 and JAB =JAA ¼ 1:0 at a) kB T=JAA ¼ 1:0, b) kB T=JAA ¼ 2:0, c) kB T=JAA ¼ 3:0 and d)kB T=JAA ¼ 4:0.
390
b ex ¼ H
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X 〈i;j〉
JAA diA djA si sj þ JBB diB djB Si Sj þ JAB diA djB si Sj
þdiB djA Si sj ; b H anisotropy ¼ D
X
diB S2i ;
i
b Zeeman ¼ H H
X ðdiA si þ diB Si Þ:
(2)
i
For the sake of simplicity, we select all of the spin-spin couplings such that they are ferromagnetic, i.e., JAA,JBB and JAB > 0. D refers to the single-ion anisotropy term, which acts only on the type-B atoms in the system, and H is the external magnetic field. The s and S are Ising spin variables, and they can take values s¼±1 and S ¼ ±1; 0 for the magnetic components-A and B of system, respectively. In our computer experiments, the symbols dia ¼ 1 (a¼A or B) if site i is occupied by type-a atom and 0 otherwise. JAA and JBB are fixed to unity in the simulations and the other system parameters are normalized with JAA . In order to gain a better understanding of the hysteresis characters of the disordered binary alloy cylindrical nanowire, we realize Monte Carlo simulation method based on the single-spin flip Metropolis algorithm [34,35]. We deal with a quenched disordered binary alloy nanowire which means that the positions of the atoms are frozen during a simulation. The nanowire system is located on a simple cubic lattice and it has a length of L¼200 and a
radius of r. In order to simulate such a finite nanosystem, we use the free boundary conditions in xy plane and periodic boundary conditions in z direction. The computer experiment starts at a high temperature value using random initial condition, which corresponds a paramagnetic phase, and then the system is slowly cooled down with a reduced temperature step of kB DT=JAA ¼ 0:1. At each temperature step, the magnetization is calculated over a number of 105 MC steps after 4104 MC steps are used for thermalization process. Once the specific temperature configuration has been obtained, this configuration has been used as the initial configuration for decreasing field branch of the hysteresis which is obtained by scanning the magnetic field from H to H with a reduced magnetic field step of DH=JAA ¼ 0:05. When the magnetic field reaches the H value, decreasing field branch is completed and the obtained configuration corresponding to the H value has been used as the initial configuration for the increasing field branch of the hysteresis. Similarly, by scanning the magnetic field from H to H, increasing field branch is obtained. At each magnetic field step, the physical quantities are averaged over a number of 104 MC steps after 5103 MC steps are used for thermalization process. 3. Results and discussion In this section, we present our MC simulation results regarding the hysteresis properties of the binary alloy nanowire system. We should notice that p¼1 and p¼0 cases correspond to clean nanowire systems composing of type-A atoms and of type-B atoms, respectively. For both p values, the system does not contain any
Fig. 2. a)-c) Coercivity and b)-d) remanence variations of the cylindrical nanowire system as a function of p at kB T=JAA ¼ 2:0 (first row) and kB T=JAA ¼ 3:0 (second row) for several values of the exchange interaction strength between type-A and type-B atoms: JAB =JAA ¼ 0:01; 0:5; 1:0 and 2.0.
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Fig. 3. Temperature dependence of the coercivity and remanence for exchange coupling strengths of a)-b) JAB =JAA ¼ 0:5 and c)-d) for JAB =JAA ¼ 2:0 for varying concentration of typeA atoms: p¼0,0.2,0.4,0.6,0.8 and 1.
disorder effects originating from different type of magnetic components and therefore the magnitude of the exchange interaction term, JAB =JAA , has no effect on the magnetic properties of the system. Also, for simplicity, we investigate the magnetic characteristics of the system in the absence of single-ion anisotropy (D¼0). Firstly, we investigate the magnetic behavior of the binary alloy nanowire system by changing the magnitude of the nanowire radius. The hysteresis curves for several values of the nanowire radius, r¼2,4,6,8,10, with a concentration value of p¼0.5 and exchange interaction strength of JAB =JAA ¼ 1:0 at different reduced temperature values are shown in Fig. 1 (a)-(d). It can be clearly seen that the thickness of the nanowire has an important effect on the variation of the total magnetization with the external magnetic field. The binary alloy nanowire system is in the ferromagnetic phase at kB T=JAA ¼ 1:0 and kB T=JAA ¼ 2:0 for all the nanowire radius values under consideration. Since the exchange energy contribution to the total energy increases as the nanowire becomes thicker, much more magnetic field is required to reverse the spins of the system and hence, the coercive field takes bigger values with increasing r for a fixed value of reduced temperature. At kB T=JAA ¼ 3:0, the nanowire becomes paramagnetic for r¼2 because the proportion of the surface atoms increases as r gets smaller (Fig. 1 (c)). With a further increment in temperature (Fig. 1 (d)), the nanowire is paramagnetic for all r values. According to these simulation results, in order to reduce surface effects, we deal with a binary alloy cylindrical nanowire of radius r¼10 throughout this study.
We have illustrated the coercive field, Hc, and remanent magnetization, Mr, of the binary alloy nanowire for varying values of p with JAB =JAA ¼ 0:01; 0:5; 1:0 and 2.0 at a reduced temperature value of kB T=JAA ¼ 2:0 in Fig. 2(a) and (b), respectively. This figure explicitly reveals that the number of type-A magnetic components and the spin-spin exchange interaction strength between unlike magnetic components, JAB =JAA , have a significant impact on the coercivity and remanence treatments. At first glance, it is obvious that the coercivity takes larger values as JAB =JAA increases for a fixed value of p. When all the spin-spin coupling terms are equal to each other, which corresponds to JAB =JAA ¼ 1:0, the coercivity displays an almost linearly increment with p due to the enhanced number of spin-1/2 atoms in the system. If the exchange interaction term JAB =JAA takes lower values such as, JAB =JAA ¼ 0:01 and JAB =JAA ¼ 0:5, the coercivity exhibits an initial decrease with increasing p by reaching a minimum p value and then it increases with the concentration value of type-A magnetic components. Besides, starting from p¼0, the coercive field shows an increasing behavior until a maximum value for the case of strong interaction strength of JAB =JAA ¼ 2:0. Then, a reduction in coercivity is observed with a further increment in p. The dependence of the variation coercivity with p on the strength of the exchange interaction term, JAB =JAA , can be understood as follows. If one starts with a pure spin-1 nanowire corresponding to p¼0 state, the number of JAB exchange interaction terms begins to dominate by gradually adding type-A atoms in the nanowire [32]. This means that for small p values, exchange energy, which favors the parallel alignment of the spins, and therefore the
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Fig. 4. Hysteresis loops of the binary alloy cylindrical nanowire at kB T=JAA ¼ 2:0 for a) JAB =JAA ¼ 0:01, b) JAB =JAA ¼ 0:5, c) JAB =JAA ¼ 1:0, d) JAB =JAA ¼ 2:0. The curves are depicted for varying values of active concentration value of type-A magnetic components: p¼0,0.2,0.4,0.6,0.8 and 1.
magnitude of the reverse magnetic field required to destroy the magnetization increases (decreases) with p for strong (weak) ferromagnetic interaction between type-A and type-B magnetic components. Conversely, in the case of large p values, JAA bonds are effective in the system and thus the coercivity decreases (increases) with the number of type-A atoms for strong (weak) JAB =JAA values. It can be seen from Fig. 2(b) that the dependence of remanent magnetization on p is almost the same with that of coercivity behavior. In accordance with the expectations, remanence takes larger values for stronger spin-spin interaction strengths between unlike atoms. Additionally, in the presence of a weak spin-spin interaction strength of, JAB =JAA ¼ 0:01, the coercivity and remanence reduces to zero for p¼0.4 and p¼0.5 which means that the system is in the paramagnetic phase. From the above discussions we can infer that the magnetic properties like coercivity and remanence of a binary alloy nanowire of the type Ap B1p can be controlled by varying the concentration value of type-A magnetic atoms and/or the exchange interaction strength JAB =JAA . It is worthwhile to underline that the numerical results of this study are in accordance with the experimental findings of references which states the controllability of magnetic properties of binary alloy nanowires with varying composition [16e19]. In these experimental studies, the only parameter that has been changed is the concentration value, p while the other parameters remains the same. We should mention that it would be possible to tune JAA , JAB , JBB by changing the type of magnetic components from the experimental point of view. For instance, the exchange interaction strengths in a CoNi [19] binary alloy nanowire should be different than those of FeNi [15].
The concentration value p of the type-A magnetic components strongly affects the transition temperature of the binary alloy nanowire [32] and therefore the variation of Hc and Mr with p should be influenced by the temperature of the system. We show the dependence of Hc and Mr on p for the same parameters as Fig. 2(a) and (b) but for a higher temperature value of kB T=JAA ¼ 3:0 in Fig. 2(c) and (d), respectively. It is clear that the enhanced temperature of the system does not alter the overall dependence of Hc and Mr on p for all the JAB =JAA values under consideration. However, due to the increased temperature, Hc and Mr values shift to lower temperature region and the number of states which become paramagnetic increases. This can be seen in the case weak spin-spin interaction strengths JAB =JAA ¼ 0:01 and JAB =JAA ¼ 0:5 for which the coercivity and remanence become zero for p¼0.2,0.4,0.5 and p¼0.6. In order to gain further insight about the magnetic properties of the binary alloy nanowire, in Fig. 3(a)-(d) we show the temperature dependencies of the coercivity and remanence in the cases of JAB =JAA ¼ 0:5 and JAB =JAA ¼ 2:0 for different values of p. As one can readily see, Hc and Mr depend sensitively on the number of type-A atoms in the nanowire system for a fixed JAB =JAA and kB T=JAA in accordance with our discussions of Fig. 2. In addition, because of the increasing thermal fluctuations which destroy the ferromagnetic order in the system, Hc and Mr decrease with raising temperature and they reduce to zero above the transition temperature of the system for all concentration values. Comparing the coercivity values of two different exchange interaction magnitudes at fixed temperature, one can observe that in the case of JAB =JAA ¼ 2:0 coercivity takes larger values for all p. From the experimental side, in Refs. [12,17,36] temperature dependence of coercivity in binary
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Fig. 5. Hysteresis curves of the binary alloy cylindrical nanowire for a) JAB =JAA ¼ 0:01, b) JAB =JAA ¼ 0:5, c) JAB =JAA ¼ 1:0, d) JAB =JAA ¼ 2:0 with a concentration value of p¼0.5. The curves are shown for several values of the reduced temperature.
alloy nanowire arrays containing Ni and Fe atoms has been investigated. It is interesting to note that, similar to our numerical findings, it has been observed that the coercivity displays a decreasing trend with increasing temperature. As a further investigation, in Fig. 4 (a)-(d) the hysteresis curves of the binary alloy nanowire corresponding to the system parameters of Fig. 2 are presented. If the JAB =JAA term remains constant, the hysteresis curves change with p in agreement with our findings in Fig. 2. For a fixed value of p, the hysteresis loops become wider with increasing JAB =JAA value due to the enhanced spin-spin interaction in the system except for p¼0 and p¼1 since these are concentration values with no JAB bonds in the system. Thus, it is possible to obtain a hard or soft ferromagnetic binary alloy cylindrical nanowire by changing the magnitude of JAB =JAA at a fixed temperature value and p. Moreover, depending on the magnitude of the relative exchange interaction between unlike magnetic components, one can also modify the magnetic phase of the system with changing p. One can observe this from the hysteresis curves of Fig. 4 (a) where the binary alloy nanowire becomes paramagnetic for p¼0.4 and p¼0.5. Finally, the temperature dependence of the hysteresis loops of the binary alloy nanowire with p¼0.5 is presented in Fig. 5 for spinspin interaction strengths of a) JAB =JAA ¼ 0:01, b) JAB =JAA ¼ 0:5, c) JAB =JAA ¼ 1:0 and d) JAB =JAA ¼ 2:0. As one can explicitly observe, with raising temperature, the hysteresis loops become narrower and disappear above the transition of the system for a constant value of JAB =JAA . Also, the hysteresis loop areas increase with increasing spin-spin coupling strength for a constant value of
temperature indicating the controllability of the magnetic features of the considered system. Since the transition temperature of the cylindrical binary alloy nanowire raises as the ferromagnetic exchange interaction term between unlike atoms gets stronger [32], the paramagnetic behavior is observed at higher temperature values for stronger JAB =JAA strengths. As a final comment, it is worthwhile to discuss the assumptions made for the nanowire system simulated in this work. Firstly, we simulate an infinite nanowire system with periodic boundary conditions along the nanowire axis. For such an infinitely long nanowire, the demagnetizing factor in the longitudinal axis is zero. On the other side, a real nanowire system has a finite length which means that the demagnetizing field, (associated with the shape anisotropy), should play an important role in the hysteresis characteristics. Moreover, we deal with a single nanowire throughout the work whereas in the above mentioned experimental studies, binary alloy nanowire arrays are considered for which the magnetostatic interactions between individual nanowires should be taken into account for the determination of the magnetic properties [37]. Lastly, we assume that the properties of the surface of the nanowire are identical with those of the inner part except for the reduced number of nearest neighbours at the surface. However, the outer part of the nanowire may be very different than the inner part due to, for instance, altered exchange interactions or disorder effects. In this sense, many different hysteresis properties, such as exchange bias phenomenon [33,38], may be observed by considering a binary alloy ferromagnetic core and an antiferromagnetic shell with a ferromagnetic interface coupling.
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4. Conclusions To conclude, we have studied the magnetic properties of a single ferromagnetic binary alloy cylindrical nanowire of the type Ap B1p by means of MC simulation with Metropolis algorithm. Firstly, the coercivity and remanence are presented as a function of p for various values of JAB =JAA at kB T=JAA ¼ 2:0. Also, the thermal dependence of coercivity and remanence are shown in the case of varying values of p and JAB =JAA for a broad range of temperature. Our numerical results demonstrate that the magnetic properties of the system can be tuned by altering the active concentration value of type-A magnetic components and the exchange coupling between unlike magnetic components. We remark that the controllability of magnetic features of binary alloy nanowires is also observed experimentally in hysteresis loop measurements. Moreover, the hysteresis curves of the system are displayed for different values of p at fixed temperature. It has been observed that when the spin-spin interaction term remains constant, the magnetic phase of the system can be altered by changing p. Finally, we have considered the hysteresis loops of the system in the case of several values of temperature for p¼0.5. The hysteresis curves become narrower with raising temperature and paramagnetic phase is seen above the transition temperature of the system. As a final conclusion, from the theoretical perspective, we note that the results obtained in the present study may be improved by using a little bit complicate model, such as Heisenberg model including single-ion anisotropy term. From a magnetism point of view, it could be also interesting to focus on the thermal and magnetic phase transition properties of the same nanowire system, but in the form of ternary alloy ABp C1p by means of MC simulation method. Acknowledgements The numerical calculations reported in this paper were per_ _ (Turkish agency), High Performance formed at TÜBITAK ULAKBIM and Grid Computing Center (TRUBA Resources). References [1] Q. Chen, Z. John Zhang, Appl. Phys. Lett. 73 (1998) 3156. [2] M. Mikhaylova, D.K. Kim, N. Bobrysheva, M. Osmolowsky, V. Semenov, T. Tsakalakos, M. Muhammed, Langmuir 20 (2004) 2472e2477. [3] K. Liu, K. Nagodawithana, P.C. Searson, C.L. Chien, Phys. Rev. B 51 (1995) 7381.
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