The phase diagrams and the magnetic properties of a ternary mixed ferrimagnetic nanowire

Accepted Manuscript The phase diagrams and the magnetic properties of a ternary mixed ferrimagnetic nanowire B. Boughazi, M. Boughrara, M. Kerouad PII: DOI: Reference:

S0304-8853(16)33070-0 http://dx.doi.org/10.1016/j.jmmm.2017.02.059 MAGMA 62520

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Journal of Magnetism and Magnetic Materials

Please cite this article as: B. Boughazi, M. Boughrara, M. Kerouad, The phase diagrams and the magnetic properties of a ternary mixed ferrimagnetic nanowire, Journal of Magnetism and Magnetic Materials (2017), doi: http:// dx.doi.org/10.1016/j.jmmm.2017.02.059

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The phase diagrams and the magnetic properties of a ternary mixed ferrimagnetic nanowire ∗

B. Boughazi, M. Boughrara, and M. Kerouad

Laboratoire Physique des Matériaux et Modélisation des Systèmes (LP2MS), Unité Assocée au CNRST-URAC: 08, Faculty of Sciences, University Moulay Ismail, B.P. 11201, Zitoune, Meknes, Morocco.

Abstract

1 A hexagonal nanowire consisting of a ferromagnetic spin core, spin1 inter core/shell 2 3 and spin outer shell coupled with ferrimagnetic interlayer couplings has been studied by 2 the use of Monte Carlo simulation based on the heat bath algorithm. The exchange interactions and the uniaxial anisotropy eects on the magnetic properties of the system have been discussed. We found that, the system exhibits the compensation phenomenon, the rst and the second order transitions. Moreover, we have obtained that, the existence of the tricritical point depend on the surface shell interaction.

Keywords:

∗

Monte Carlo simulation, Mixed Nanowire, Magnetic properties.

e-mail: boughrara− [email protected]

1

2

1

Introduction Recently, magnetic nanoparticles have been the subject of a large number of experi-

mental and theoretical studies, because of their great potential for technological [1, 2] and biomedical [35] applications such as information storage devices and drug delivery in cancer thermotherapy. Magnetic properties of nanowires [6, 7], namely saturation magnetization, curie and compensation temperatures are greatly dierent from their bulk counterparts, which are strongly inuenced by surface and nite-size eects. Mixed spin Ising systems, which consist of two inter-penetrating inequivalent sublattices, have attracted a great deal of attention in the last two decades. They have been proposed as possible models to describe a certain type of ferrimagnetic systems such as molecularbased magnetic materials [8]. Moreover, the importance of these systems is mainly related to the potential technological applications in the area of thermomagnetic recording [9]. Several techniques have been used to study these systems such as mean eld theory [10], eective eld theory [11] and Monte Carlo simulation [12] Phase diagrams of a cylindrical nanowire with diluted surface have been studied by the use of Monte Carlo Simulation [13]. Depending on the surface parameters, Some characteristic phenomena are found in the phase diagram, such as the rst and second phase transitions. 1 Boughrara et al. have investigated the magnetic properties of a mixed ferrimagnetic spin- and 2 spin-1 Ising nanowire by using Monte Carlo Simulation and eective eld theory [14]. They studied the eects of the uniaxail anisotropy, the shell coupling, the interface coupling and the concentration of the magnetic atoms at the surface shells on both critical and compensation temperatures. It was found that the system presents very rich critical behavior, which include the rst and the second order phase transitions, the tricritical and the critical end points. In an other work, the magnetic properties of a ferrimagnetic nanowire on a hexagonal lattice with 3 a spin- core surrounded by a spin-1 shell layer with antiferromagnetic interface coupling have 2 been investigated by the use of Monte Carlo simulation [15]. It was shown that, depending on the crystal eld and the exchange interactions, the system can exhibit the rst and the second order phase transitions, the isolated critical and compensation points. E. Kantar et al. have 1 3 proposed a ternary Ising spin ( ,1, ) model to investigate the magnetic properties of a two 2 2 dimensional nanoparticles with core-shell structure within the framework of the eective eld theory with correlations [16]. In particular, behaviors of the core and the shell magnetizations, susceptibilities and internal energies as well as the total magnetization are studied. They have also investigated the free energy of the system to conrm the stability of the solutions. The

3 phase diagrams and the compensation behavoir are discussed in detail. They have found that the system exhibits a tricritical point, special critical points and reentrant phenomenon. It was also found Q, R, P, S and W type compensation behaviors, which depend strongly on the interaction parameters. Moreover, one or two compensation temperatures have been observed. In this work, we are interested in investigating the phase diagrams (critical and compen1 sation behaviors) of a ferrimagnetic nanowire with a spin- and spin-1 core surrounded by a 2 3 spin- shell layer. In our analysis, we use the Monte Carlo Simulation (MCS) according to 2 the heat bath algorithm [17]. The outline of this paper is as follows: In section 2, we give the model and review the basic points of the Monte Carlo Simulation. In section 3, we present the results and discussions, while section 4 is devoted to a brief conclusion.

2

Model and formalism

1 spin, 2 the shell core surrounding the centric wire is occupied by S = 1 spins and the surface shell 3 is occupied by m = spins. Each spin is connected to the nearest-neighbor spins with an 2 exchange interaction. A cross section of the nanowire is depicted in Fig. 1. Therefore, the 1 3 magnetic nanowire is modeled by ternary Ising spins ( , 1, ) system with the following 2 2 Hamiltonian: X X X X X X H = −JC σi σj − J1 Sk Sl − J2 σi Sk − J3 Sk mp − JS mp mr − D1 S2i A hexagonal nanowire consisting of a centric wire which is occupied by σ =

hiji

− D2

X

hkli

hiki

m2j

hkpi

hpri

hii

(1)

hji

where the exchange interactions are dened as:

3 nearest-neighbors at the surface shell. 2 1 •JC : is the exchange interaction between two spin- nearest-neighbors at the centric wire. 2 •J1 : is the exchange interaction between two spin-1 nearest-neighbors at the interface shell. 1 •J2 : is the exchange interaction between spin- and spin-1 nearest-neighbors at the centric 2 wire and at the interface shell. 3 •J3 : is the exchange interaction between spin-1 and spin- nearest-neighbors at the interface 2 shell and at the surface shell. •JS : is the exchange interaction between two spin-

D1 and D2 are the single-ion anisotropies that come from the interface and the surface shell sublattices respectively.

4 Our system consists of two shells, namely one shell of the surface and another shell of 3 the core which surround the centric wire (Fig. 1), the surface shell contains NS spins - , and 2 1 the shell core contains NI spins -1 and the centric wire contains NC spins - . The total number 2 of spins in the nanowire is NTotal = NC + NI + NS . NC = 1 × L, NI = 6 × L, NS = 12 × L, L denotes the wire length's. We use the Monte Carlo Simulation and we ip the spins once a time according to the heat bath algorithm [17]. 4 × 104 Monte Carlo steps were used to obtain each data point in the system, after discarding the rst 2 × 104 steps. The magnetization M of a given conguration is dened by the sum over all the spin values of the lattice sites. There are three longitudinal magnetizations. The rst is of the centric wire (MC ), the second is of the core shell (MI ) and the third is of the surface shell (MS ). By employing the Monte Carlo Simulation, the magnetizations per site of the sublattices are given by: NC 1 X MC = < σi > NC i=1

(2)

NI 1 X MI = < Si > NI i=1

(3)

NS 1 X MS = < mi > NS i=1

(4)

The total magnetization per site is giving by:

MT =< Mi >=

1 NTotal

(NC MC + NI MI + NS MS )

(5)

where Mi is the total magnetisation for each conguration. The total susceptibility is dened as

χT = βNTotal (< Mi >2 − < M2i >)

(6)

1 KB T The internal energy per site is given by: with β =

E=

NTotal

(7)

5 At a compensation temperature (TK ), the total magnetization vanishes below the critical temperature TC and it is due to the fact that the core and the shell magnetizations cancel each other, whereas at TC , they vanishe. Then, the compensation temperature can be determined by solving the following equation:

NC MC (TK ) + NI MI (TK ) + NS MS (TK ) = 0

(8)

The second phase transition is a continuous phase where the order parameter rises continuously from zero below the critical temperature (Tc) while the rst order transition is a discontinuous phase which exhibits a discontinuity of the order parameter at the critical point. The tricritical point is a point where the second and rst order lines meet. The critical temperatures are determined from the maxima of the susceptibility curves and the rst-order phase transitions are obtained by locating the discontinuities of the magnetization and the internal energy curves.

3

Results and discussions In this section, we examine the phase diagrams and the magnetic properties of the

system for some selected values of the Hamiltonian parameters. To investigate the size eect of the system, we have plotted the total susceptibilJ1 JS J2 ity as a function of the temperature for R1 = = 0.1, RS = = 0.1, R2 = = −0.1, JC JC JC D1 D2 J3 R3 = = −0.1, = 0.0, = −3.0 and for dierent lengths of the nanowire (L = 80, 100, JC JC JC 200, 300, 400 and 500) (Fig. 2). We can see, from this gure, that the value of the critical temperature is independent of the size of the system when L is greater than 300. T Fig. 3 represents the phase diagrams of the nanowire in the ( ,−R2 ) plane with JC D1 D2 = = −3.0 and R1 = −R3 = 1.0 and for dierent value of RS . In these phase diagrams, JC JC the black circles, open and black squares lines represent the second and the rst order phase transition and the compensation temperature lines, respectively; the tricritical points, which are points at which the second order lines is connected to the rst ones, are denoted by satrs. As it is seen from the gure, depending on the value of RS , we can have three types of phase diagrams. The rst one is obtained for RS < 0.56, it is presented in Fig. 3(a), for RS = 0.2 and 0.5. It is observed that the system exhibits two types of transition lines, the rst one is a rst order transition line, it starts from R2 = 0.0 and increases with decreasing R2 to terminate

6 at a tricritical point. The second one is a second order transition line, separating ordered and disordered phases, it extends from the tricritical point and increases with decreasing R2 . The coordinates of the tricritical point shift to higher values of T and R2 , when increasing RS (T = 0.7, R2 = −0.43 for RS = 0.2 and T = 0.74, R2 = −0.16 for RS = 0.5). The second phase diagram type is obtained for 0.56 ≤ RS < 1.2 and is plotted in g. 3(b), for RS = 1.0. The phase diagram exhibits a rst order transition line with constant TC in the low temperature region, separating two ordered phases and a second order transition line, separating the ordered phase from the disordered one. The third type obtained for RS ≥ 1.2 is plotted in g. 3(c), as it is seen, the phase diagram contains almost second order line separating ordered and disordered phases. We mention also that, the critical temperature increases almost linearly as

R2 decreases. Concerning the compensation phenomenon, it is found that, for RS < 0.45 the system exhibits a compensation phenomenon below a threshold value of R2 , which depends on RS . The compensation temperature remains constant as R2 decreases. In g. 4, we investigate the thermal behavior of the sublattice magnetizations, susceptibility and the internal energy in order to conrm and understand the results obtained in g. T 3. We can observe that, the system undergoes a rst order phase transition at = 0.06, beJC cause, as it is seen from the gure, at this value, the magnetization MS and the energy's curves undergo a discontinuity. This is due to the fact that there is a competition between JS and D2 ; at low temperature, the strenght of JS overcomes the strength of D2 so the spin state becomes 3 1 3 m = . Thus, this phase may be called as the ferrimagnetic ( ,-1, ) phase. Also, we note 2 2 2 T T 1 that as increases ( > 0.06), the surface shell spins are driven into the m = state (JS JC JC 2 and D2 are against each other). As the temperature increases, the sublattice magnetizations TC decrease to zero continuously and a second-order phase transition occurs at = 1.1. It is JC TC , the susceptibility increases very rapidly also seen that, when the temperature approaches JC TC and diverges at = 1.1. JC We have plotted, in Fig. 5, the variation of the critical and compensation temperatures D1 D2 as a function of the interface coupling R3 for = = −3.0, R1 = −R2 = 1.0 and for dierJC JC ent values of Rs . It is seen that, as in Fig. 3, depending on the value of RS , three topologies of the phase diagram are obtained. The rst one obtained for RS < 0.56, is presented in Fig. 5(a). It is observed that, the system exhibits two types of transition lines separating the ordered and desordered phases, the rst one is a rst order transition line, it starts from R3 = 0.0 and increases with decreasing R3 to terminate at a tricritical point. The second one is a second

7 order transition line, it extends from the tricritical point and increases with decreasing R3 . The coordinates of the tricritical point shift to lower values of T and higher values of R3 , when increasing RS (T = 0.74, R3 = −0.83 for RS = 0.2 and T = 0.41, R3 = −0.39 for RS = 0.5). The second phase diagram type is obtained for 0.56 ≤ RS ≤ 1.2 and is presented in Fig. 5(b), for RS = 1.0, it shows that the system undergoes two successive phase transitions; one of a rst order between two ordred phases and the second one is of a second order from ordered phase to desordered one. The critical temperature increases with decreasing R3 . The same behavior has been found in Ref [18]. The third type obtained for RS ≥ 1.2 is plotted in g. 5(c), as it is seen, the phase diagram exhibits three successive phase transitions for a lower values of |R3 | ; two of a rst order between three ordered phases and the third one is of a second order from ordered phase to desordered one. We mention also that, the critical temperature increases as R3 decreases. Concerning the compensation phenomenon, it is found that, for RS < 0.45 and in a certain range of R3 the system exhibits a compensation points, this range becomes smaller as increasing R3 . In order to illustrates the calculation of the transition points as well as the characterization of the nature of the transitions, a few explanatory results are plotted in Fig. 6(a)-(d). Fig. 6(a) illustrates the thermal variations of the magnetizations, the total susD1 D2 ceptibility and the internal energy for = = −3.0, R1 = RS = −R2 = 1.0 and R3 = 0.0. JC JC In this gure, as the temperature increases , the magnetizations MC = 0.5 and MI = −1.0 TC = 0.13, while MS = 0.5 decreases to zero continuously decrease to zero discontinuously at JC TC at = 0.57. In this case, the system undergoes two phase transtions; the rst one is of a JC TC1 TC2 rst order at = 0.13 and the second one is of a second order at = 0.57. One can see JC JC TC1 , the non-magnetic phase, MC = MI = 0.0 and that the ferrimagnetic phase exists below JC TC1 TC2 TC2 MS 6= 0.0 exists between and and the paramagnetic phase exists above . Thus, JC JC JC the system passes from the ferrimagnetic phase to non-magnetic phase and then the paramagnetic phase as the temperature increases, the same fact has been found in [16]. Fig. 6(b) shows the temperature dependence of the magnetizations, the total susceptibility and the internal D1 D2 energy for = = −3.0, R1 = RS = −R2 = 1.0 and R3 = −0.6. In this gure, MC = 0.5, JC JC MI = −1.0 and MS = 1.5 at zero temperaure. The magnetization MS decreases discontinuously 1 3 to 0.5 with increasing the temprature; hence a rst order transiton, from ( ,-1, ) state to 2 2 1 1 TC1 ( ,-1, ) state, occurs at = 0.06. Above this temperature, the magnetizations MC = 0.5, 2 2 JC MI = −1.0 and MS = 0.5 decreases continously to zero; hence a second order phase transiTC2 tion, from ferrimagnetic phase to the paramagnetic phase, occurs at = 0.96. Fig. 6(c) JC

8

D1 D2 = = −3.0, R1 = −R2 = 1.0, RS = 1.5 and R3 = −0.05. In this case, the JC JC TC1 system undergoes three successive phase transitions; two of rst order namely at = 0.14 JC TC2 TC3 and = 0.62, the third one is of second order namely at = 0.86, where a peak appears JC JC TC1 in the susceptibility-temperature curve. The internal energy undergoes two jumps at and JC TC2 as the temperature increases. However, it varies relatively slowly in the high temperature JC region. There is inection points on the internal energy curve which corresponds to the rst is plotted for

order phase transition temperature of the system. Fig 6(d) shows the magnetizations, the susceptibility and the internal energy as a function of the temperature for the same paramaters as in Fig. 6(c) except here R3 = −0.15. This gure shows that when we decrease R3 , the TC2 system exhibits two phase transitions; one of a rst oder at = 0.74, the second one is of JC TC3 a second order at = 0.98 as in Fig. 6(c). JC In order to conrm the existence of the compensation phenomenon, we have plotted the total magnetization as a function of the temperature for R1 = 1.0, R2 = −1.0, R3 = −0.8, D1 D2 = = −3.0 and for RS = 0.2 (Fig. 7(a)). It is clear that the system presents the comJC JC pensation phenomenon at TK = 0.45. We have also examined the eects of the surface coupling parameter RS on the comT pensation and the critical behaviors. Fig.8 shows the phase diagrams in the ( ,RS ) plane J D1 D2 for = = −3.0, R1 = 1.0 and R2 = R3 = −1.0. It is seen that, when RS increases, the JC JC TC critical temperature of the system remains constant ( = 0.95) below a critical value of RS JC (RSC = 0.43), above RSC , TC increases linearly with RS . We can also remark that, the system exhibits a compensation phenomenon below the critical value RSC , the compensation temperature increases linearly with RS . A similar behavior has been found for a ferrimagnetic mixed 1 spin and spin 1 Ising nanowire [19], for a hexagonal prismatic nanoparticle consisting of 2 3 a ferromagnetic spin 1 core surrounded by a ferromagnetic spin shell with ferrimagnetic 2 interface exchange coupling [20] and for a hexagonal nanowire consisting of the ferromagnetic 3 spin- core and spin-1 outer shell coupled with ferrimagnetic interlayer coupling [21]. 2 To examine the inuence of the uniaxial anisotropy on the phase diagrams of the system, we have plotted the variations of the critical and compensation temperatures versus the uniaxial anisotropy of the surface (D2 ) for R1 = RS = 1.0, R2 = R3 = −1.0 and for dierent

9

D1 (Fig. 9). It is observed that the phase diagram is very rich; it exhibits two JC types of phase transition and a compensation behavior. The rst one is of a rst order 1 1 1 3 separating the two ordered phases designated by ( , −1, ) and ( , −1, ) which starts 2 2 2 2 T D1 from = 0.0 just for = −3.0 and extends to terminate at an isolated critical point(?) JC JC D2 T ( = −2.78, = 0.6). The second one is of a second order separating the ordered and JC JC the disordered phases, in which the critical temperature remains constant below a threshold D2 D2 value of = −3.67 and then increases with to reach a saturation value, which depends JC JC D1 D1 on the value of (TCSat = 5.48, 5.66 and 5.93 for = −3.0, 0.0 and 3.0, respectively), JC JC D2 for large positive values of . Concerning the compensation behavior, we observe that the JC D2 D2 system exhibits a compensation phenomenon below the threshold value of ( < −3.67) JC JC D1 for ≥ 0.0, a similar behavior has been found in Ref [22]. The compensation temperature JC D2 remains constant below = −5.0. Above this value, TK increases speedily to approach the JC D1 second order transition line for ≥ 0.0. JC Finally, we have investigated the eect of the uniaxial anisotropy of the outer shell of D1 the core ( ) on the critcal behavior of the system. In this context, we have presented JC D1 the variation of the critical and the compensation temperatures versus for R1 = 1.0, JC D2 D2 R2 = R3 = −RS = −0.2 and for dierent values of (Fig. 10). For ≤ 0.0, we notice JC JC that, the system exhibits two phase transition lines and a compensation behavior. The rst 1 1 1 1 one is of a rst order separating the two ordered phases designated by ( , −1, ) and ( , 0, ) 2 2 2 2 D2 T = 0.02 and 0.04 for = −3.0 and 0.0 respectively and extends to apwhich starts from JC JC proch the seond order one. The second one is of a second order separating the ordered and values of

the disordered phases, in which the critical temperature remains constant below a threshold D1 D2 D1 value of which depends on the value of , then increases with to reach a saturaJC JC JC D2 D2 tion value, which depends on value (TCSat = 1.85, 2.0 and 2.27 for = −3.0, 0.0 and JC JC D1 3.0, respectively), for large positive values of . Concerning the compensation behavior, we JC observe that the system exhibits a compensation phenomenon above the threshold value of D1 D2 D1 D2 which depends on the value of ( > −2.5, 1.62 for = −3.0, 0.0, respectively), a JC JC JC JC similar behavior has been found in Ref [22]. The compensation temperature remains constant.

10

4

Conclusion In summary, we have studied within the Monte Carlo Simulation based on the heat

bath algorithm the magnetic properties of the ferrimagnetic nanowire which consisting of 1 3 mixed spin- , spin-1 and spin- Ising model in a crystal eld on the hexagonal lattice. We 2 2 have discussed the inuence of the exchange interactions and the uniaxial anisotropy on the magnetic properties of the system. A number of interesting phenomena have been found, such as the tricritcal point, originating from the competition between the crystal eld and dierent D2 interaction constants. It is shown that, the surface parameters (RS and ) have strong eect JC on the shape of the phase diagram; this result is consistent with the nding of [23], and that D2 depending on the values of RS , R2 , R3 and , the system can exhibit the rst and the seond JC order phase transition. Thus the isolated critical points are also observed. We have also shown D1 that the existence of the compensation phenomenon depends on the values of R2 , R3 , RS , JC D2 and . JC

Acknowledgment This work has been supported by the URAC: 08, the project PPR: (MESRSFC-CNRST).

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References [1] Magnetic Molecular Materials, Vol. 198 of NATO Science Series E: Applied Sciences, edited by D. Gatteschi, O. Kahn, J.S. Miller, and F. Palacio (Kluwer Academic, Dordrecht, 1991). [2] H.P.D. Shieh and M.H. Kryder, Appl. Phys. Lett. 49, 473 (1986). [3] Q.A. Pankhurst, J. Connolly, S.K. Jones, J. Dobson, Journal of Physics D: Applied Physics 36 (2003) R167. [4] A.H. Habib, C.L. Ondeck, P. Chaudhary, M.R. Bockstaller, M.E. McHenry, Journal of Applied Physics 103 (2008) 07A307. [5] N. Sounderya, Y. Zhang, Recent Patents on Biomedical Engineering 1 (2008) 34. [6] Z.D. Li, Q.Y. Li, L. Li, W.M. Liu, Physical Review E 76 (2007) 026605. [7] P.B. He, W.M. Liu, Physical Review B 72 (2005) 064410. [8] O. Kahn, in: E. Coronado, etal. (Eds.), From Molecular Assemblies to the Devices, Kluwer Academic Publishers, Dordrecht, 1996. [9] F. Tanaka, S. Tanaka, N. Imamura, Japan Journal of Applied Physics 26 (1987) 231. [10] I.J. Souza, P.H.Z. de Arruda, M. Godoy, L. Craco, A.S. de Arruda, Physica A 444 (2016) 589. [11] M. Erta³, M. Keskin, Physica B 470-471 (2015) 76. [12] M. šukovi£, A. Bobák, Physica A 436 (2015) 509. [13] M. Boughrara, M. Kerouad, A. Zaim, Physica A 433 (2015) 59. [14] M. Boughrara, M. Kerouad and A. Zaim, Journal of Magnetism and Magnetic Materials 360 (2014) 222. [15] A. Feraoun, A. Zaim, M. Kerouad, Physica B 445 (2014) 74. [16] E. Kantar, M. Keskin, Journal of Magnetism and Magnetic Materials 349 (2014) 165. [17] D.P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge U. Press, Cambridge, 2000.

12 [18] T. Kaneyoshi, Physica E 71 (2015) 84. [19] B. Boughazi, M. Boughrara, M. Kerouad, Journal of Magnetism and Magnetic Materials 363 (2014) 26. [20] Wei Jiang, Jian-Qi Huang, Physica E 78 (2016) 115. [21] Wei Jiang, Fan Zhang, Xiao-Xi Li, Hong-Yu Guan, An-Bang Guo, Zan Wang, Physica E 47 (2013) 95. [22] Wei Jiang, Xiao-Xi Li, An-Bang Guo, Hong-Yu Guan, Zan Wang, Kai Wang, Journal of Magnetism and Magnetic Materials 355 (2014) 309. [23] Wei Jiang, Xiao-Xi Li, Li-Mei Liu, Physica E 53 (2013) 29.

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Figure captions Fig.

1.

Schematic representation of a cross-section of a nanowire. Open circles indicate

magnetic atoms at the surface shell, solid circles are magnetic atoms constituting the core and gray circle constituting centric wire. Fig. 2.

Temperature dependences of the total susceptibility χT for RS = R1 = 0.1,

D1 = 0.0, R2 = R3 = −0.1 and and for dierent nanowire lengths. JC

D2 = −3.03, JC

T D1 D2 , −R2 ) plane for = = −3.0, R1 = −R3 = 1.0 JC JC JC when (a) RS = 0.0, 0.5, (b) RS = 1.0 and (c) RS = 1.5. Fig.

3.

The phase diagram in the (

4. Thermal variations of the magnetizations, susceptibility and internal energy for D1 D2 = = −3.0, R1 = RS = −R3 = 1.0 and R2 = 0.0. JC JC

Fig.

T D1 D2 , −R3 ) plane for = = −3.0 and R1 = −R2 = 1.0 JC JC JC when (a) RS = 0.0, 0.5, (b) RS = 1.0 and (c) RS = 1.5. Fig. 5.

The phase diagram in the (

Fig. 6.

The prole of the magnetizations, the susceptibility and the internal energy as a funcD1 D2 tion of the temperature for R1 = −R2 = 1.0, = = −3.0, (a) R3 = −0.0 and RS = 1.0, JC JC (b) R3 = −0.6 and RS = 1.0, (c) R3 = −0.05 and RS = 1.5 and (d) R3 = −0.15 and RS = 1.5. Fig. 7.

The prole of (a) the total, (b) the bulk and the suface magnetizations and (c) the D1 D2 internal energy for R1 = −R2 = 1.0, R3 = −0.8, RS = 0.2 and = = −3.0. JC JC

T D1 D2 , RS ) plane for R3 = R2 = −R1 = −1.0 and = = −3.0. JC JC JC

Fig. 8.

The phase diagram in the (

Fig. 9.

The phase diagram in the (

T D2 , ) for R1 = RS = −R2 = −R3 = 1.0 and for dierent JC JC

values of

D1 . JC

Fig. 10.

The phase diagram in the (

T D1 , ) for R1 == 1.0, RS = −R2 = −R3 = 0.2 and for JC JC

14 dierent values of

D2 . JC

15

Fig. 1

16

2,5

2,0

1,5

1,0

0,5

0,0 0,1

0,2

0,3

Fig. 2

0,4

17

1,75

1,50

2,0

1,25

1,5

1,00

1,0

0,75

0,5

0,50

0,0 0,0

0,5

1,0

1,5

2,0

0,0

0,5

1,0

3,6

3,5

3,4

3,3

3,2

3,1

3,0 0,0

0,5

1,0

Fig. 3

1,5

2,0

1,5

2,0

18

0,35 1 0,30 0 0,25 -1 0,20 -2

0,15

-3

0,10

0,05

-4

0,00 0,0

0,2

0,4

0,6

0,8

Fig. 4

1,0

1,2

19

3 2,0

1,5

2

1,0

1 0,5

0

0,0 0,0

0,4

0,8

1,2

1,6

0,0

2,0

0,5

1,0

4

3

2

1

0 0,0

0,5

1,0

Fig. 5

1,5

2,0

1,5

2,0

20

0,16

0,8 0,5

0,14

1

0,12 0 0,10

0,0 -1

0,4

0,08 0,06

-2

-0,5

0,04 -3 0,02

-1,0

-4

0,0 0,2

0,4

0,00 0,0

0,6

2

0,5

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0,8 1

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-1

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0,2

-3

-4

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-5

0,00 0,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0

Fig. 6

0,2

0,4

0,6

0,8

1,0

1,2

21

0,04

0,5 0,03

0,02

0,0 0,01

0,00

-0,5

-0,01

-0,02

-0,03

-1,0 0,0

0,2

0,4

0,6

0,8

1,0

0,0

1,2

0,4

0,4

0,0

-0,4

-0,8

-1,2 0,2

0,4

Fig. 7

0,6

0,8

1,0

1,2

0,8

1,2

22

2,4 2,2 2,0 1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,0

0,2

0,4

0,6

Fig. 8

0,8

1,0

1,2

23

6

5

4

3

2

1

0 -12

-10

-8

-6

-4

-2

0

Fig. 9

2

4

6

8

10

12

24

2,5

2,0

1,5

1,0

0,5

0,0 -6

-4

-2

0

Fig. 10

2

4

6

HIGHLIGHTS

•

The mixed cylindrical Ising nanowire is investigated using the MCS.

•

The effects of the crystal field on the phase diagrams have been examined.

•

The surface parameters have strong effect on the shape of the phase diagram.

•

The compensation temperatures have been found for the certain values parameters.