- Email: [email protected]
Magnetizations of a transverse Ising nanowire
Journal of Magnetism and Magnetic Materials 322 (2010) 3410–3415
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Magnetizations of a transverse Ising nanowire T. Kaneyoshi 1 510, Kurosawadai, Midoriku, Nagoya 458-0003, Japan
a r t i c l e in f o
a b s t r a c t
Article history: Received 21 May 2010 Received in revised form 8 June 2010 Available online 15 June 2010
Magnetizations of a cylindrical nanowire described by the transverse Ising model are investigated by the use of the effective field theory with correlations (EFT), since the phase diagrams of the system have been examined in the previous work (J. Magn. Magn. Mater. (2010), in press) by using the two theoretical frameworks of the mean field theory and the EFT. The temperature dependences of longitudinal and transverse magnetizations in the system are strongly affected by the surface situations. Many characteristic phenomena are found in the thermal variations, depending on the ratio of the physical parameters in the surface shell and the core. In particular, the effects of the two transverse fields at the surface shell and in the core to these magnetizations have been firstly clarified. & 2010 Elsevier B.V. All rights reserved.
Keywords: Nanomagnetic material Transverse Ising model Magnetization
1. Introduction Recently, there has been growing interest in the magnetic properties of a material with a nanostructure [1,2]. In particular, magnetic nanowires have been successfully fabricated by various methods. They have been receiving considerable attention experimentally due to their distinctive properties and potential applications, such as ultrahigh density magnetic recording media [3,4]. There are now experimental evidences about deviations from bulk behavior. The magnetic properties are strongly influenced by finite size and surface effects. These facts give a strong motivation for the investigations of magnetic nanowires. Theoretically, the magnetic properties of a cubic Ising nanowire [5], which is consisted of a ferromagnetic spin-1/2 core and a ferromagnetic spin-1shell coupled with an antiferromagnetic interlayer coupling Jinter to the core, have been investigated by the use of the Monte Carlo method. In Ref. [6], the phase diagrams and the temperature dependences of magnetization in a Ising nanowire (or nanotube) with diluted surface are examined by the use of the mean field theory (MFA) and the effective field theory with correlation (EFT). The temperature dependences of magnetization in the two nano-systems where the direction of spins on the surface are directed oppositely to that in the core because of the antiferromagnetic interlayer coupling J1 have been examined, in order to clarify the common behaviors between the two systems. The transverse Ising model is generally believed to describe the phase transition of order–disorder type ferroelectrics. The model has also been applied successfully to the investigation of a
E-mail address: [email protected] Prof. Emeritus at Nagoya University, Japan.
1
0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.06.037
material with a nanostructure. Except the studies of magnetic nanoparticles [7–11] and nano-scaled thin films [12,13], however, only a few related results of the model have been reported for the magnetic properties of a material with a nanostructure. In the previous work [14], the phase diagrams in a transverse Ising nanowire have been examined by the use of the two theoretical frameworks of MFA and EFT [15,16]. As far as we know, the temperature dependences of longitudinal and transverse magnetizations in the nanowire described by the transverse Ising model have not been discussed. In particular, the effects of the transverse fields at the surface shell and in the core to these magnetizations have not been clarified, although the effects of the surface shell to the magnetizations in the two nano-systems with zero transverse fields at surface shell and in the core have been discussed in Ref. [6]. In this work, the longitudinal and transverse magnetizations of the cylindrical nanowire with the three (S¼3) shells, namely one shell of the surface and two shells in the core, described by the transverse Ising model, are examined by the use of the EFT, since the phase diagrams in Ref. [14] obtained by the both theories are very similar to each other and the numerical results of the EFT have improved the corresponding results of the MFA to the reasonable direction. The EFT corresponds to the Zernike approximation [17]. The paper is organized as follows. In Section 2, we define the model and give briefly the formulations of the EFT. In Section 3, the temperature dependences of magnetizations in the nanowire, when the interaction J1 between the surface shell and the core is positive, are examined, in order to clarify the relations between the results of the phase diagrams in Ref. [14] and the present results. In Section 4, the effects of a negative J1 on the thermal variation of longitudinal magnetizations in the system are examined in detail. Section 5 is devoted to a brief discussion.
T. Kaneyoshi / Journal of Magnetism and Magnetic Materials 322 (2010) 3410–3415
2. Formulation We consider a nanowire, as depicted in Fig. 1, in which the wire is consisted of the surface shell and the core. The each site on the figure is occupied by a Ising spin. Each spin is connected to the two nearest neighbor spins on the above and below sections. The surface shell is coupled to the next shell in the core with an exchange interaction J1. In other words, the system is consisted from the three (S¼3) shells, namely one shell of the surface and two shells in the core. The Hamiltonian of the system is given by P Z Z P Z Z P H ¼ JS si sj J sm sn J1 sZi sZim ðijÞ
ðmnÞ
ðimÞ
ð1Þ
OS SsXi OSsXm ðiÞ
mc2 ¼ ½coshðCÞ þ mc1 sinhðCÞ½coshðCÞ þ mc2 sinhðCÞ4 ½coshðBÞ þ mS1 sinhðBÞ½coshðBÞ þmS2 sinhðBÞ2 f ðxÞ9x ¼ 0
ð5Þ
mc1 ¼ ½coshðCÞ þ mc1 sinhðCÞ2 ½coshðCÞ þmc2 sinhðCÞ6 f ðxÞ9x ¼ 0
ð6Þ
where A, B and C are defined by A¼JSr, B¼J1r and C ¼Jr. r ¼q/qx is the differential operator. Here, the functions fS(x) and f(x) are defined by fS ðxÞ ¼ ðx=yS ÞtanhðbyS Þ
ð7Þ
and f ðxÞ ¼ ðx=yÞtanhðbyÞ with yS ¼ ðx2 þ O2s Þ1=2
ðmÞ
3411
ð8Þ
and where sai (a ¼z, x) is the Pauli spin operator with sZi ¼ 71. The JS is the exchange interaction between two nearest-neighbor magnetic atoms at the surface shell and the J is the exchange interaction in the core. OS and O represent the transverse fields at the surface shell and in the core, respectively. The surface exchange interaction JS is often defined as JS ¼ Jð1 þ DS Þ
ð2Þ
in order to clarify the effects of surface on the physical properties in the system. For the nanowire depicted in Fig. 1, there exist two longitudinal magnetizations (mS1 and mS2) on the surface shell and two longitudinal magnetizations (mC1 and mC2) on the core in the z direction, where mC1 represents the center magnetization in the core. Within the framework of the EFT, they can be given by 2
2
mS1 ¼ ½coshðAÞ þmS1 sinhðAÞ ½coshðAÞ þ mS2 sinhðAÞ ½coshðBÞ þmc2 sinhðBÞfS ðxÞ9x ¼ 0
mS2 ¼ ½coshðAÞ þmS2 sinhðAÞ2 ½coshðAÞ þ mS1 sinhðAÞ2 ½coshðBÞ þmc2 sinhðBÞ2 fS ðxÞ9x ¼ 0
ð3Þ
y ¼ ðx2 þ O2 Þ1=2 where b ¼1/kBT and T is a temperature. At this place, one should notice that the phase diagrams of the present system have been discussed in Ref. [14], using the coupled equations from Eqs. (3) to (6). Furthermore, let us define the total longitudinal magnetization per site and the total transverse magnetization per site as follows: mT ¼
1 ½6ðmS1 þ mS2 Þ þ6mC2 þ mC1 19
ð9Þ
1 6ðmXS1 þ mXS2 Þ þ6mXC2 þmXC1 19
ð10Þ
and mXT ¼
where the transverse magnetizations in Eq. (10) are also given by the same equations as those of Eqs. (3)–(6), only replacing the functions fS(x) and f(x) in Eqs. (3)–(6) by the new functions hS(x) and h(x), respectively. The new functions hS(x) and h(x) are defined by hs ðxÞ ¼ ðOs =ys Þtanhðbys Þ
ð4Þ
ð11Þ
and hðxÞ ¼ ðO=yÞtanhðbyÞ For the following discussions, let us define the parameters, h, q and r as h¼
q¼
O J
ð12Þ
Os O
ð13Þ
J1 J
ð14Þ
and r¼
3. Ferromagnetic magnetizations
Fig. 1. Schematic representations of a cylindrical nanowire. The gray circles represent magnetic atoms at the surface shell. The black circles are magnetic atoms constituting the core. The lines connecting the gray and black circles represent the nearest-neighbor exchange interactions (JS, J1 and J).
In this section, let us investigate some typical temperature dependences of longitudinal and transverse magnetizations in the nanowire, using the formulation given in Section 2 and selecting the value of J1 as positive. That is to say, the spins on the surface shell are aliened to the same direction to the spins in the core. In order to clarify the contributions from the surface shell and the core to the transverse magnetization (10), let us here define the transverse magnetizations per site of the surface shell and the core as follows: 1 mXS ¼ ðmXS1 þ mXS2 Þ 2
ð15Þ
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T. Kaneyoshi / Journal of Magnetism and Magnetic Materials 322 (2010) 3410–3415
and 1 mXC ¼ ð6mXC2 þmXC1 Þ 7
ð16Þ
At first, let us discuss some relations between the phase diagrams in Ref. [14] and the present works, selecting the two figures from the results in Ref. [14]. Fig. 2(A) (Fig. 2 in Ref. [14]) shows the variations of transition temperature TC as a function of O in the system with fixed values of DS ¼0.0, r ¼1.0 and q¼ 1.0. Such a variation of TC is also observed in the bulk transverse Ising system [18]. The dashed and solid lines represent the results obtained by the use of the MFA and the EFT, respectively. Here, one should notice that the solid curve is smaller than that of the MFA and hence the EFT improves the numerical result to the more reasonable direction than the MFA. Fig. 2(B) represents the temperature dependences of total longitudinal and transverse magnetizations (mT and mXT ) per site for the system with DS ¼ 0.0, r ¼1.0 and q¼ 1.0, when the two values of h are selected from Fig. 2(A). One should note that each mT reduces to zero at the same temperature as that in Fig. 2(A). Similar variations of mT and mXT can be also observed in the bulk transverse Ising system [18]. As shown in Fig. 2(C), however, the thermal variations of mXS and mXC defined by Eqs. (15) and (16) take the forms different to each other below the transition temperature. The contribution to mXT from the surface shell and the core below its TC is completely inverse, which phenomenon has not been reported. As discussed in Fig. 5 of Ref. [14], the TC curve of the core reduces to zero at the critical value of hC ¼ OC/J¼4.32. As shown in Fig. 7 of Ref. [14], some characteristic features can be seen in the TC curve, when the value of h is given by the critical value hC ¼4.32. In Fig. 3(A) (Fig. 7 in Ref. [14]), the TC curves of the nanowire with q¼ 1.0 and O/J¼4.32 are once again plotted as a function of DS, changing the value of r. From the present studies, we can also confirm that the results are correct. In Fig. 3(B), for instance, the thermal variations of mT and mXT in the nanowire with DS ¼ 0.0, q¼ 1.0 and O/J¼ 4.32 are plotted, changing the value of r. In fact, each mT curve reduces to zero at the corresponding TC point in Fig. 3(A) and the mT for the system with r ¼0.01 is given by mT ¼0.0 in the whole temperature region. In Fig. 3(C), the thermal variations of mXT , mXS and mXC are plotted for the three values of h, namely for h¼0.05, 1.0 and 2.0. The solid, dashed and dotted lines represent the mXT , mXS and mXC curves, respectively. The results labeled r ¼0.05 indicate that the main contribution to mXT comes from the surface shell, since the mXS and mXC curves take the same shape and the nearly same values. With the increase of r, the contribution to the mXT becomes to separate between the surface shell and the core, especially below its transition temperature, as is understood from the curves labeled by r ¼1.0 and 2.0. In this way, we can obtain a lot of information about the thermal variations of mT, mXT , mXS and mXC by selecting the reasonable values for the physical parameters (q, r, h and DS) in the system. As in Fig. 4, one of such results is depicted for the comparison with the results given in the next section. In Fig. 4(A), the temperature dependences of mT (solid line) and mXT (dashed line) are depicted for the system with DS ¼ 0.5, h¼4.0 and r ¼1.0, selecting the three values of q. In Fig. 4(B), the thermal variations of mXT (solid line), mXS (dashed line) and mXC (dotted line) are plotted for the two values of h, namely for h¼0.05 and 2.0.
4. Ferrimagnetic magnetizations As discussed in Ref. [14], one should note that the phase diagrams are independent of the sign of r and they are also valid for the negative values of r. When the value of r is negative, the spins at the surface shell are coupled antiferromagnetically to the
Fig. 2. (A) Phase diagram of the nanowire plotted as a function of O/J, when the ratio q of transverse fields at the surface shell and in the core is fixed at q¼1.0 and the exchange interaction at the surface shell is taken as JS ¼J(DS ¼0.0). The result of the EFT (solid lines) is compared with that of the mean-field approximation (dashed lines). The value of r is fixed at r¼ 1.0. (B) The temperature dependences of mT and mXT in the system are plotted for the two values of h, namely h¼ 2.0 and h¼ 4.0. (C) The thermal variations of mXT , mXS and mXC in the system are plotted for the two values of h.
T. Kaneyoshi / Journal of Magnetism and Magnetic Materials 322 (2010) 3410–3415
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Fig. 4. (A) The temperature dependences of mT (solid line) and mXT (dashed line) in the system with DS ¼ 0.5, O/J ¼ 4.0 and r ¼ 1.0 are plotted, selecting the three values of q. (B) The thermal variations of mXT (solid line), mXS (dashed line) and mXC (dotted line) in the system are plotted for the two values of q.
spins of the core. In this section, let us examine some typical results of total longitudinal and transverse magnetizations in the system with a negative value of r, although we could select a variety of values for the physical parameters (q, r, h and DS) in the system. In order to clarify the contributions from the surface shell and the core to the longitudinal magnetization (9), let us also define the longitudinal magnetizations per site of the surface shell and the core as follows: 1 mS ¼ ðmS1 þ mS2 Þ 2
ð17Þ
and Fig. 3. (A) Phase diagram of the nanowire with O/J ¼4.32 and q¼ 1.0 plotted as a function of DS, when the value of r is changed. (B) The temperature dependences of mT (solid line) and mXT (dashed line) in the system with DS ¼0.0, O/J¼ 4.32 and q ¼1.0 are plotted, changing the value of r. (C) The thermal variations of mXT (solid line), mXS (dashed line) and mXC (dotted line) in the system are plotted for the three values of h.
1 mC ¼ ð6mC2 þ mC1 Þ ð18Þ 7 Fig. 5(A) shows the temperature dependences of mT in the system with DS ¼ 0.5, h¼4.0 and r ¼ 1.0, changing the value of q from 2.0 to 0.05. The curves exhibit some typical thermal variations observed in the bulk ferrimagnetic system [19].
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T. Kaneyoshi / Journal of Magnetism and Magnetic Materials 322 (2010) 3410–3415
valid for Fig. 5 of the system with r ¼ 1.0. The reason is as follows; When the equations of mXS1 , mXS2 and mXC2 are expanded into the polynomial functions of longitudinal magnetizations, they can be expressed as the even functions of mS1 and mS2 and hence they are independent to the change of the sign for the mS1 and mS2. In Fig. 5, the variation of q from q¼2.0 to 0.05 represents the change from the dominant contribution to mT of the core to that of the surface shell. Such a situation can be realized by changing the value of other physical parameters, such as the values of r and h. The two of such cases are given in the following figures. Fig. 6 shows the temperature dependences of mT in the system with DS ¼ 0.5, q¼2.0 and h¼ 1.0, changing the value of r from r¼ 0.05 to 1.0. The curves also express some typical thermal variations observed in the bulk ferrimagnetic system [19]. The curves labeled r ¼ 0.05 and 0.25 may show the P-type behavior. The curves labeled r ¼ 0.5 and 1.0 exhibit the N-type behavior. The compensation temperature may increase with the decrease of r. Fig. 7(A) shows the temperature dependences of mT in the system with DS ¼0.0, q¼ 4.0 and r ¼ 1.0, changing the value of h from h¼2.0 to 1.0. The curves also express some typical thermal variations observed in the bulk ferrimagnetic system [19]; the curve labeled h¼ 2.0 may show the Q-type behavior. The curve labeled h¼1.25 exhibits the P-type behavior. The curve labeled h¼1.0 shows the N-type behavior, showing a compensation point below its transition temperature. In particular, one should notice that the curve labeled h¼1.5 may exhibit a broad minimum and a broad maximum below its transition temperature, which phenomenon has not been predicted in the bulk ferrimagnetic system [19]. Here, one should also note that the similar phenomenon obtained for the curve labeled h¼1.5 in Fig. 7 has been obtained for the transverse Ising nano-scaled thin film [12]. In Fig. 7(B), the temperature dependences of mT in the system with DS ¼0.0, q¼4.0 and r ¼ 1.0 are plotted by changing the value of h from h¼ 1.5 to 1.2. The curve labeled h¼1.4 also exhibits a very shallow minimum and a broad maximum below its transition temperature. But, the curve labeled h¼1.3 does not show such a minimum, which is nothing but the P-type in the bulk ferrimagnetic system [19]. The curve labeled h¼ 1.2 represents the L-type behavior in the bulk ferrimagnetic system [19]. Fig. 5. (A) The temperature dependences of mT in the system with DS ¼ 0.5, O/J¼ 4.0 and r ¼ 1.0 are plotted by changing the value of q. (B) The thermal variations of mT (solid line), mS (dashed line) and mC (dotted line) in the system are plotted for the two values of q, namely q ¼2.0 and 0.75 in Fig. 5(A).
The curve labeled q ¼2.0 may show the Q-type behavior. The curve labeled q¼1.0 exhibits the P-type behavior. The other curves show the N-type behavior, showing a compensation point below its transition temperature. The compensation temperature may increase with the decrease of q. Fig. 5(B) represents the thermal variations of mT (solid line), mS (dashed line) and mC (dotted line) for the two systems in Fig. 5(A), namely for the systems with q¼2.0 and 0.75, in order to clarify how the variation of mT comes from the behaviors of mS and mC. As is understood from the behaviors of the dashed curves in Fig. 5(B), the appearance of a compensation point is closely connected to the characteristic (approximately linear) feature in the intermediate temperature region for the curve labeled q¼0.75. At this place, one should notice that the difference between Figs. 4 and 5 is only the sign of r, namely r ¼1.0 in Fig. 4 and r ¼ 1.0 in Fig. 5. As discussed in Ref. [14], the longitudinal magnetizations in the two systems with the same value of q in Figs. 4 and 5 reduce to zero at the same temperature. The thermal variations of transverse magnetizations depicted in Fig. 4 are also
Fig. 6. The temperature dependences of mT in the system with DS ¼ 0.5, O/J¼ 1.0 and q¼ 2.0 are plotted by changing the value of r.
T. Kaneyoshi / Journal of Magnetism and Magnetic Materials 322 (2010) 3410–3415
3415
transverse Ising model have been examined by the use of the EFT. In the previous work [14], the phase diagrams of the system have been investigated by the use of the two theoretical frameworks, namely the MFA and the EFT, and the EFT has improved the numerical results to the more reasonable direction than the MFA. In particular, the effects of the two transverse fields (OS ¼0.0 and O ¼0.0) to these magnetizations have been firstly clarified in this work, although the effects of the surface shell to the magnetizations in the two nano-systems with zero transverse fields have been discussed in Ref. [6]. As shown in Figs. 2 and 3, the phase diagrams can be obtained from the present work. In Section 3, the thermal variations of longitudinal and transverse magnetizations in the ferromagnetic system with a positive value of r have been examined. In particular, the temperature dependences of mXS and mXC defined by Eqs. (15) and (16) in the system have shown some characteristic features below its transition temperature, as shown in Figs. 2(C) and 3(C). The contribution to mXT from the surface shell and the core is completely different below its TC, which phenomenon has not been reported. In Section 4, the temperature dependences of longitudinal magnetizations in the system with a negative value of r have been examined. As shown in Figs. 5–7, the systems have exhibited some typical ferrimagnetic behaviors observed in the bulk ferrimagnetic system [19]. As shown in Fig. 7, furthermore, the thermal variation of mT in the system with h¼1.5 has exhibited a broad minimum and a broad maximum below its transition temperature, which phenomenon has not been predicted in the bulk ferrimagnetic system [19]. Thus, the study of a cylindrical nanowire described by the transverse Ising model may open a new field in the research of magnetism and also the new type of ferrimagnetism. References
Fig. 7. (A) The temperature dependences of mT in the system with DS ¼ 0.0, q ¼4.0 and r ¼ 1.0 are plotted by changing the value of h. (B) The thermal variations of mT in the system are plotted for the four values of h, namely from h ¼ 1.5 to 1.2 in Fig. 7(A). The arrow lines in the curves labeled h ¼ 1.5 express the broad minimum and the broad maximum.
5. Conclusion In this work, the thermal variations of longitudinal and transverse magnetizations in the nanowire described by the
[1] R. Skomski, J. Phys.: Condens. Matter 15 (2003) R841. [2] X. Zou, G. Xiao, Phys. Rev. B 77 (2008) 054417. [3] J.E. Wegrowe, D. Kelly, Y. Jaccard, P.H. Guittienne, J.P.H. Ansermet, Europhys. Lett. 45 (1999) 626. [4] A. Fert, L. Piraux, J. Magn. Magn. Mater. 200 (1999) 338. [5] A. Zaim, M. Kerouad, Y. El Amraoui, J. Magn. Magn. Mater. 321 (2009) 1077. [6] T. Kaneyoshi, Phys. Status Solidi B (2010), in press, doi:10.1002/pssb. 201046067. [7] E. Konstantinova, J. Magn. Magn. Mater. 320 (2008) 2721. [8] V.S. Leite, W. Figueiredo, Physica A 350 (2005) 379. [9] A.F. Bakuzis, P.C. Morais, J. Magn. Magn. Mater. 285 (2005) 145. [10] T. Kaneyoshi, Phys. Status Solidi B 242 (2005) 2938. [11] T. Kaneyoshi, J. Magn. Magn. Mater. 321 (2009) 3430. [12] T. Kaneyoshi, J. Magn. Magn. Mater. 321 (2009) 3630. [13] T. Kaneyoshi, Phys. Status Solidi B 246 (2009) 2359. [14] T. Kaneyoshi, J. Magn. Magn. Mater. (2010), in press, doi:10.1016/j.jmmm. 2010.05.021. [15] T. Kaneyoshi, M. Jascur, I.P. Eittipaldi, Phys. Rev. B 48 (1993) 250. [16] T. Kaneyoshi, Acta Phys. Pol. A 83 (1993) 703. [17] F. Zernike, Physica 7 (1940) 565. [18] E.F. Sarmento, T. Kaneyoshi, Phys. Rev. B 48 (1993) 3232. [19] L. Neel, Ann. Phys. Paris 3 (1948) 137.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Magnetizations of a transverse Ising nanowire T. Kaneyoshi 1 510, Kurosawadai, Midoriku, Nagoya 458-0003, Japan
a r t i c l e in f o
a b s t r a c t
Article history: Received 21 May 2010 Received in revised form 8 June 2010 Available online 15 June 2010
Magnetizations of a cylindrical nanowire described by the transverse Ising model are investigated by the use of the effective field theory with correlations (EFT), since the phase diagrams of the system have been examined in the previous work (J. Magn. Magn. Mater. (2010), in press) by using the two theoretical frameworks of the mean field theory and the EFT. The temperature dependences of longitudinal and transverse magnetizations in the system are strongly affected by the surface situations. Many characteristic phenomena are found in the thermal variations, depending on the ratio of the physical parameters in the surface shell and the core. In particular, the effects of the two transverse fields at the surface shell and in the core to these magnetizations have been firstly clarified. & 2010 Elsevier B.V. All rights reserved.
Keywords: Nanomagnetic material Transverse Ising model Magnetization
1. Introduction Recently, there has been growing interest in the magnetic properties of a material with a nanostructure [1,2]. In particular, magnetic nanowires have been successfully fabricated by various methods. They have been receiving considerable attention experimentally due to their distinctive properties and potential applications, such as ultrahigh density magnetic recording media [3,4]. There are now experimental evidences about deviations from bulk behavior. The magnetic properties are strongly influenced by finite size and surface effects. These facts give a strong motivation for the investigations of magnetic nanowires. Theoretically, the magnetic properties of a cubic Ising nanowire [5], which is consisted of a ferromagnetic spin-1/2 core and a ferromagnetic spin-1shell coupled with an antiferromagnetic interlayer coupling Jinter to the core, have been investigated by the use of the Monte Carlo method. In Ref. [6], the phase diagrams and the temperature dependences of magnetization in a Ising nanowire (or nanotube) with diluted surface are examined by the use of the mean field theory (MFA) and the effective field theory with correlation (EFT). The temperature dependences of magnetization in the two nano-systems where the direction of spins on the surface are directed oppositely to that in the core because of the antiferromagnetic interlayer coupling J1 have been examined, in order to clarify the common behaviors between the two systems. The transverse Ising model is generally believed to describe the phase transition of order–disorder type ferroelectrics. The model has also been applied successfully to the investigation of a
E-mail address: [email protected] Prof. Emeritus at Nagoya University, Japan.
1
0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.06.037
material with a nanostructure. Except the studies of magnetic nanoparticles [7–11] and nano-scaled thin films [12,13], however, only a few related results of the model have been reported for the magnetic properties of a material with a nanostructure. In the previous work [14], the phase diagrams in a transverse Ising nanowire have been examined by the use of the two theoretical frameworks of MFA and EFT [15,16]. As far as we know, the temperature dependences of longitudinal and transverse magnetizations in the nanowire described by the transverse Ising model have not been discussed. In particular, the effects of the transverse fields at the surface shell and in the core to these magnetizations have not been clarified, although the effects of the surface shell to the magnetizations in the two nano-systems with zero transverse fields at surface shell and in the core have been discussed in Ref. [6]. In this work, the longitudinal and transverse magnetizations of the cylindrical nanowire with the three (S¼3) shells, namely one shell of the surface and two shells in the core, described by the transverse Ising model, are examined by the use of the EFT, since the phase diagrams in Ref. [14] obtained by the both theories are very similar to each other and the numerical results of the EFT have improved the corresponding results of the MFA to the reasonable direction. The EFT corresponds to the Zernike approximation [17]. The paper is organized as follows. In Section 2, we define the model and give briefly the formulations of the EFT. In Section 3, the temperature dependences of magnetizations in the nanowire, when the interaction J1 between the surface shell and the core is positive, are examined, in order to clarify the relations between the results of the phase diagrams in Ref. [14] and the present results. In Section 4, the effects of a negative J1 on the thermal variation of longitudinal magnetizations in the system are examined in detail. Section 5 is devoted to a brief discussion.
T. Kaneyoshi / Journal of Magnetism and Magnetic Materials 322 (2010) 3410–3415
2. Formulation We consider a nanowire, as depicted in Fig. 1, in which the wire is consisted of the surface shell and the core. The each site on the figure is occupied by a Ising spin. Each spin is connected to the two nearest neighbor spins on the above and below sections. The surface shell is coupled to the next shell in the core with an exchange interaction J1. In other words, the system is consisted from the three (S¼3) shells, namely one shell of the surface and two shells in the core. The Hamiltonian of the system is given by P Z Z P Z Z P H ¼ JS si sj J sm sn J1 sZi sZim ðijÞ
ðmnÞ
ðimÞ
ð1Þ
OS SsXi OSsXm ðiÞ
mc2 ¼ ½coshðCÞ þ mc1 sinhðCÞ½coshðCÞ þ mc2 sinhðCÞ4 ½coshðBÞ þ mS1 sinhðBÞ½coshðBÞ þmS2 sinhðBÞ2 f ðxÞ9x ¼ 0
ð5Þ
mc1 ¼ ½coshðCÞ þ mc1 sinhðCÞ2 ½coshðCÞ þmc2 sinhðCÞ6 f ðxÞ9x ¼ 0
ð6Þ
where A, B and C are defined by A¼JSr, B¼J1r and C ¼Jr. r ¼q/qx is the differential operator. Here, the functions fS(x) and f(x) are defined by fS ðxÞ ¼ ðx=yS ÞtanhðbyS Þ
ð7Þ
and f ðxÞ ¼ ðx=yÞtanhðbyÞ with yS ¼ ðx2 þ O2s Þ1=2
ðmÞ
3411
ð8Þ
and where sai (a ¼z, x) is the Pauli spin operator with sZi ¼ 71. The JS is the exchange interaction between two nearest-neighbor magnetic atoms at the surface shell and the J is the exchange interaction in the core. OS and O represent the transverse fields at the surface shell and in the core, respectively. The surface exchange interaction JS is often defined as JS ¼ Jð1 þ DS Þ
ð2Þ
in order to clarify the effects of surface on the physical properties in the system. For the nanowire depicted in Fig. 1, there exist two longitudinal magnetizations (mS1 and mS2) on the surface shell and two longitudinal magnetizations (mC1 and mC2) on the core in the z direction, where mC1 represents the center magnetization in the core. Within the framework of the EFT, they can be given by 2
2
mS1 ¼ ½coshðAÞ þmS1 sinhðAÞ ½coshðAÞ þ mS2 sinhðAÞ ½coshðBÞ þmc2 sinhðBÞfS ðxÞ9x ¼ 0
mS2 ¼ ½coshðAÞ þmS2 sinhðAÞ2 ½coshðAÞ þ mS1 sinhðAÞ2 ½coshðBÞ þmc2 sinhðBÞ2 fS ðxÞ9x ¼ 0
ð3Þ
y ¼ ðx2 þ O2 Þ1=2 where b ¼1/kBT and T is a temperature. At this place, one should notice that the phase diagrams of the present system have been discussed in Ref. [14], using the coupled equations from Eqs. (3) to (6). Furthermore, let us define the total longitudinal magnetization per site and the total transverse magnetization per site as follows: mT ¼
1 ½6ðmS1 þ mS2 Þ þ6mC2 þ mC1 19
ð9Þ
1 6ðmXS1 þ mXS2 Þ þ6mXC2 þmXC1 19
ð10Þ
and mXT ¼
where the transverse magnetizations in Eq. (10) are also given by the same equations as those of Eqs. (3)–(6), only replacing the functions fS(x) and f(x) in Eqs. (3)–(6) by the new functions hS(x) and h(x), respectively. The new functions hS(x) and h(x) are defined by hs ðxÞ ¼ ðOs =ys Þtanhðbys Þ
ð4Þ
ð11Þ
and hðxÞ ¼ ðO=yÞtanhðbyÞ For the following discussions, let us define the parameters, h, q and r as h¼
q¼
O J
ð12Þ
Os O
ð13Þ
J1 J
ð14Þ
and r¼
3. Ferromagnetic magnetizations
Fig. 1. Schematic representations of a cylindrical nanowire. The gray circles represent magnetic atoms at the surface shell. The black circles are magnetic atoms constituting the core. The lines connecting the gray and black circles represent the nearest-neighbor exchange interactions (JS, J1 and J).
In this section, let us investigate some typical temperature dependences of longitudinal and transverse magnetizations in the nanowire, using the formulation given in Section 2 and selecting the value of J1 as positive. That is to say, the spins on the surface shell are aliened to the same direction to the spins in the core. In order to clarify the contributions from the surface shell and the core to the transverse magnetization (10), let us here define the transverse magnetizations per site of the surface shell and the core as follows: 1 mXS ¼ ðmXS1 þ mXS2 Þ 2
ð15Þ
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and 1 mXC ¼ ð6mXC2 þmXC1 Þ 7
ð16Þ
At first, let us discuss some relations between the phase diagrams in Ref. [14] and the present works, selecting the two figures from the results in Ref. [14]. Fig. 2(A) (Fig. 2 in Ref. [14]) shows the variations of transition temperature TC as a function of O in the system with fixed values of DS ¼0.0, r ¼1.0 and q¼ 1.0. Such a variation of TC is also observed in the bulk transverse Ising system [18]. The dashed and solid lines represent the results obtained by the use of the MFA and the EFT, respectively. Here, one should notice that the solid curve is smaller than that of the MFA and hence the EFT improves the numerical result to the more reasonable direction than the MFA. Fig. 2(B) represents the temperature dependences of total longitudinal and transverse magnetizations (mT and mXT ) per site for the system with DS ¼ 0.0, r ¼1.0 and q¼ 1.0, when the two values of h are selected from Fig. 2(A). One should note that each mT reduces to zero at the same temperature as that in Fig. 2(A). Similar variations of mT and mXT can be also observed in the bulk transverse Ising system [18]. As shown in Fig. 2(C), however, the thermal variations of mXS and mXC defined by Eqs. (15) and (16) take the forms different to each other below the transition temperature. The contribution to mXT from the surface shell and the core below its TC is completely inverse, which phenomenon has not been reported. As discussed in Fig. 5 of Ref. [14], the TC curve of the core reduces to zero at the critical value of hC ¼ OC/J¼4.32. As shown in Fig. 7 of Ref. [14], some characteristic features can be seen in the TC curve, when the value of h is given by the critical value hC ¼4.32. In Fig. 3(A) (Fig. 7 in Ref. [14]), the TC curves of the nanowire with q¼ 1.0 and O/J¼4.32 are once again plotted as a function of DS, changing the value of r. From the present studies, we can also confirm that the results are correct. In Fig. 3(B), for instance, the thermal variations of mT and mXT in the nanowire with DS ¼ 0.0, q¼ 1.0 and O/J¼ 4.32 are plotted, changing the value of r. In fact, each mT curve reduces to zero at the corresponding TC point in Fig. 3(A) and the mT for the system with r ¼0.01 is given by mT ¼0.0 in the whole temperature region. In Fig. 3(C), the thermal variations of mXT , mXS and mXC are plotted for the three values of h, namely for h¼0.05, 1.0 and 2.0. The solid, dashed and dotted lines represent the mXT , mXS and mXC curves, respectively. The results labeled r ¼0.05 indicate that the main contribution to mXT comes from the surface shell, since the mXS and mXC curves take the same shape and the nearly same values. With the increase of r, the contribution to the mXT becomes to separate between the surface shell and the core, especially below its transition temperature, as is understood from the curves labeled by r ¼1.0 and 2.0. In this way, we can obtain a lot of information about the thermal variations of mT, mXT , mXS and mXC by selecting the reasonable values for the physical parameters (q, r, h and DS) in the system. As in Fig. 4, one of such results is depicted for the comparison with the results given in the next section. In Fig. 4(A), the temperature dependences of mT (solid line) and mXT (dashed line) are depicted for the system with DS ¼ 0.5, h¼4.0 and r ¼1.0, selecting the three values of q. In Fig. 4(B), the thermal variations of mXT (solid line), mXS (dashed line) and mXC (dotted line) are plotted for the two values of h, namely for h¼0.05 and 2.0.
4. Ferrimagnetic magnetizations As discussed in Ref. [14], one should note that the phase diagrams are independent of the sign of r and they are also valid for the negative values of r. When the value of r is negative, the spins at the surface shell are coupled antiferromagnetically to the
Fig. 2. (A) Phase diagram of the nanowire plotted as a function of O/J, when the ratio q of transverse fields at the surface shell and in the core is fixed at q¼1.0 and the exchange interaction at the surface shell is taken as JS ¼J(DS ¼0.0). The result of the EFT (solid lines) is compared with that of the mean-field approximation (dashed lines). The value of r is fixed at r¼ 1.0. (B) The temperature dependences of mT and mXT in the system are plotted for the two values of h, namely h¼ 2.0 and h¼ 4.0. (C) The thermal variations of mXT , mXS and mXC in the system are plotted for the two values of h.
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Fig. 4. (A) The temperature dependences of mT (solid line) and mXT (dashed line) in the system with DS ¼ 0.5, O/J ¼ 4.0 and r ¼ 1.0 are plotted, selecting the three values of q. (B) The thermal variations of mXT (solid line), mXS (dashed line) and mXC (dotted line) in the system are plotted for the two values of q.
spins of the core. In this section, let us examine some typical results of total longitudinal and transverse magnetizations in the system with a negative value of r, although we could select a variety of values for the physical parameters (q, r, h and DS) in the system. In order to clarify the contributions from the surface shell and the core to the longitudinal magnetization (9), let us also define the longitudinal magnetizations per site of the surface shell and the core as follows: 1 mS ¼ ðmS1 þ mS2 Þ 2
ð17Þ
and Fig. 3. (A) Phase diagram of the nanowire with O/J ¼4.32 and q¼ 1.0 plotted as a function of DS, when the value of r is changed. (B) The temperature dependences of mT (solid line) and mXT (dashed line) in the system with DS ¼0.0, O/J¼ 4.32 and q ¼1.0 are plotted, changing the value of r. (C) The thermal variations of mXT (solid line), mXS (dashed line) and mXC (dotted line) in the system are plotted for the three values of h.
1 mC ¼ ð6mC2 þ mC1 Þ ð18Þ 7 Fig. 5(A) shows the temperature dependences of mT in the system with DS ¼ 0.5, h¼4.0 and r ¼ 1.0, changing the value of q from 2.0 to 0.05. The curves exhibit some typical thermal variations observed in the bulk ferrimagnetic system [19].
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valid for Fig. 5 of the system with r ¼ 1.0. The reason is as follows; When the equations of mXS1 , mXS2 and mXC2 are expanded into the polynomial functions of longitudinal magnetizations, they can be expressed as the even functions of mS1 and mS2 and hence they are independent to the change of the sign for the mS1 and mS2. In Fig. 5, the variation of q from q¼2.0 to 0.05 represents the change from the dominant contribution to mT of the core to that of the surface shell. Such a situation can be realized by changing the value of other physical parameters, such as the values of r and h. The two of such cases are given in the following figures. Fig. 6 shows the temperature dependences of mT in the system with DS ¼ 0.5, q¼2.0 and h¼ 1.0, changing the value of r from r¼ 0.05 to 1.0. The curves also express some typical thermal variations observed in the bulk ferrimagnetic system [19]. The curves labeled r ¼ 0.05 and 0.25 may show the P-type behavior. The curves labeled r ¼ 0.5 and 1.0 exhibit the N-type behavior. The compensation temperature may increase with the decrease of r. Fig. 7(A) shows the temperature dependences of mT in the system with DS ¼0.0, q¼ 4.0 and r ¼ 1.0, changing the value of h from h¼2.0 to 1.0. The curves also express some typical thermal variations observed in the bulk ferrimagnetic system [19]; the curve labeled h¼ 2.0 may show the Q-type behavior. The curve labeled h¼1.25 exhibits the P-type behavior. The curve labeled h¼1.0 shows the N-type behavior, showing a compensation point below its transition temperature. In particular, one should notice that the curve labeled h¼1.5 may exhibit a broad minimum and a broad maximum below its transition temperature, which phenomenon has not been predicted in the bulk ferrimagnetic system [19]. Here, one should also note that the similar phenomenon obtained for the curve labeled h¼1.5 in Fig. 7 has been obtained for the transverse Ising nano-scaled thin film [12]. In Fig. 7(B), the temperature dependences of mT in the system with DS ¼0.0, q¼4.0 and r ¼ 1.0 are plotted by changing the value of h from h¼ 1.5 to 1.2. The curve labeled h¼1.4 also exhibits a very shallow minimum and a broad maximum below its transition temperature. But, the curve labeled h¼1.3 does not show such a minimum, which is nothing but the P-type in the bulk ferrimagnetic system [19]. The curve labeled h¼ 1.2 represents the L-type behavior in the bulk ferrimagnetic system [19]. Fig. 5. (A) The temperature dependences of mT in the system with DS ¼ 0.5, O/J¼ 4.0 and r ¼ 1.0 are plotted by changing the value of q. (B) The thermal variations of mT (solid line), mS (dashed line) and mC (dotted line) in the system are plotted for the two values of q, namely q ¼2.0 and 0.75 in Fig. 5(A).
The curve labeled q ¼2.0 may show the Q-type behavior. The curve labeled q¼1.0 exhibits the P-type behavior. The other curves show the N-type behavior, showing a compensation point below its transition temperature. The compensation temperature may increase with the decrease of q. Fig. 5(B) represents the thermal variations of mT (solid line), mS (dashed line) and mC (dotted line) for the two systems in Fig. 5(A), namely for the systems with q¼2.0 and 0.75, in order to clarify how the variation of mT comes from the behaviors of mS and mC. As is understood from the behaviors of the dashed curves in Fig. 5(B), the appearance of a compensation point is closely connected to the characteristic (approximately linear) feature in the intermediate temperature region for the curve labeled q¼0.75. At this place, one should notice that the difference between Figs. 4 and 5 is only the sign of r, namely r ¼1.0 in Fig. 4 and r ¼ 1.0 in Fig. 5. As discussed in Ref. [14], the longitudinal magnetizations in the two systems with the same value of q in Figs. 4 and 5 reduce to zero at the same temperature. The thermal variations of transverse magnetizations depicted in Fig. 4 are also
Fig. 6. The temperature dependences of mT in the system with DS ¼ 0.5, O/J¼ 1.0 and q¼ 2.0 are plotted by changing the value of r.
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transverse Ising model have been examined by the use of the EFT. In the previous work [14], the phase diagrams of the system have been investigated by the use of the two theoretical frameworks, namely the MFA and the EFT, and the EFT has improved the numerical results to the more reasonable direction than the MFA. In particular, the effects of the two transverse fields (OS ¼0.0 and O ¼0.0) to these magnetizations have been firstly clarified in this work, although the effects of the surface shell to the magnetizations in the two nano-systems with zero transverse fields have been discussed in Ref. [6]. As shown in Figs. 2 and 3, the phase diagrams can be obtained from the present work. In Section 3, the thermal variations of longitudinal and transverse magnetizations in the ferromagnetic system with a positive value of r have been examined. In particular, the temperature dependences of mXS and mXC defined by Eqs. (15) and (16) in the system have shown some characteristic features below its transition temperature, as shown in Figs. 2(C) and 3(C). The contribution to mXT from the surface shell and the core is completely different below its TC, which phenomenon has not been reported. In Section 4, the temperature dependences of longitudinal magnetizations in the system with a negative value of r have been examined. As shown in Figs. 5–7, the systems have exhibited some typical ferrimagnetic behaviors observed in the bulk ferrimagnetic system [19]. As shown in Fig. 7, furthermore, the thermal variation of mT in the system with h¼1.5 has exhibited a broad minimum and a broad maximum below its transition temperature, which phenomenon has not been predicted in the bulk ferrimagnetic system [19]. Thus, the study of a cylindrical nanowire described by the transverse Ising model may open a new field in the research of magnetism and also the new type of ferrimagnetism. References
Fig. 7. (A) The temperature dependences of mT in the system with DS ¼ 0.0, q ¼4.0 and r ¼ 1.0 are plotted by changing the value of h. (B) The thermal variations of mT in the system are plotted for the four values of h, namely from h ¼ 1.5 to 1.2 in Fig. 7(A). The arrow lines in the curves labeled h ¼ 1.5 express the broad minimum and the broad maximum.
5. Conclusion In this work, the thermal variations of longitudinal and transverse magnetizations in the nanowire described by the
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