- Email: [email protected]
Theoretical simulations of magnetic nanotubes using Monte Carlo method
ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 320 (2008) 2721– 2729
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Theoretical simulations of magnetic nanotubes using Monte Carlo method E. Konstantinova Departamento de Fı´sica, ICE, Universidade Federal de Juiz de Fora, 36036-330 Juiz de Fora, MG, Brazil
a r t i c l e in fo
abstract
Article history: Received 27 December 2007 Received in revised form 20 April 2008 Available online 12 June 2008
We present the results of the Monte Carlo simulations of magnetic nanotubes, which are based on the plane structures with the square unit cell at low temperatures. The spin configurations, thermal equilibrium magnetization, magnetic susceptibility and the specific heat are investigated for the nanotubes of different diameters, using armchair or zigzag edges. The dipolar interaction, Heisenberg model interaction and also their combination are considered for both ferromagnetic and antiferromagnetic cases. It turns out that the magnetic properties of the nanotubes strongly depend on the form of the rolling up (armchair or zigzag). The effect of dipolar interaction component strongly manifests itself for the small radius nanotubes, while for the larger radius nanotubes the Heisenberg interaction is always dominating. In the thermodynamic part, we have found that the specific heat is always smaller for the nanotubes with smaller radii. Crown Copyright & 2008 Published by Elsevier B.V. All rights reserved.
PACS: 75.75.+a 75.10.b 61.46.Fg 75.40.Mg 05.10.Ln Keywords: Monte Carlo simulation Magnetic property Nanotube Spin configuration Magnetic susceptibility Specific heat
1. Introduction Since the discovery of carbon nanotubes [1], the electronic, magnetic and mechanical properties of the nanometric scale materials have been attracting considerable attention. During the last few years one could observe a growing interest in the experimental and theoretical investigations of various new structures at nano-scale [2–5]. These structures include different geometric configurations such as fullerenes, nanotubes, nanoparticles, nano-cones, nano-rings and so on. The exploration of different properties of these objects opens wide perspectives for applications. One of the potentially interesting aspects of nanophysics is related to magnetic phenomena. In particular, experimental and theoretical investigations of the magnetism in nanostructures stimulates extensive theoretical studies of different materials at the nanometric scale, such as pure and doped different carbon-based materials [6–12] and semiconductor nanoclusters, ferromagnetic and anti-ferromagnetic nanocrystals [13–16] and thin films [17]. The new experimental works which have been done recently in this area opened the way for the creation of nanotubes based on the composite molecules that
Tel.: +55 32 32174707; fax: +55 32 3229 3312.
E-mail address: konst@fisica.ufjf.br
contain metal atoms [18–20]. In this case, we can consider the nanotube as a tube, composed of rectangular (square, or others) unit cells with spins situated in the vertices of the unit cells. The magnetic properties of nano-scale materials look promising for applications, and this represents a strong motivation for theoretical investigation. In particular, different types of theoretical techniques have been used for the numerical simulations of magnetic properties such as the spin model with dipolar and nearest-neighbor interactions [17,21,22], the nearest-neighbor tight-binding Hamiltonian [11,23–25], the calculations using the ab initio pack with the spin-polarized density functional theory [26] and the Monte Carlo simulations [27]. In the present paper, we start a systematic analysis of the magnetic properties of nanotubes. As a first step, we perform the theoretical simulations of various properties of the model magnetic nanotubes based on the geometrically simplest lattice with the square unit cell. The main target of our study is the dependence of the thermodynamic and magnetic properties on geometry. In particular, we explore the difference between the nanotubes of different diameters and also the dependence on the type of the rolling which affects, in particular, the form of the edge. The results of the analysis look rather natural and one can expect similar dependencies for the more realistic cases of magnetic nanotubes, e.g., similar to those considered in Refs. [18–20].
0304-8853/$ - see front matter Crown Copyright & 2008 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.06.007
ARTICLE IN PRESS 2722
E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
2. Method of calculations
Table 1 The radii of the nanotubes expressed in the units of a (unit cell’s size)
The purpose of present work is to investigate the possible influence of the nanotube’s diameter and the type of the edge on the magnetic properties of nanotubes, based on the plane structure with the square unit cell. In Fig. 1 one can observe the two-dimensional plane with the square unit cell of the lattice. The rolling up corresponding to the armchair nanotubes is indicated by the vector (m,0) while the rolling up corresponding to the zigzag nanotubes is indicated by the vector (m,m). The value of the integer m defined the size of the nanotubes. In this paper, we are using the terms ‘‘armchair’’ and ‘‘zigzag’’ which are traditional for the carbon nanotubes. However, we use these terms in the opposite order, such that they agree with the actual geometry of the nanotubes under consideration. The geometry here means the form of the edge: for the case of the armchair the rolling up vector is passing through the neighboring vertices of the unit cells; for the case of zigzag this vector is passing through the diagonal vertices of the unit cells. After rolling, any nanotube is defined by the pair of integer parameters (m1,m2), which describe its circumference vector on the initial plane, that is ~ L ¼ m1~ a1 þ m2~ a2 , where ~ a1 ; ~ a2 are unit cell’s vectors (later we used a as a lattice constant, which we set to unity, a ¼ 1). It proves useful to introduce the following terminology. The two spins belong to the same ‘‘line’’ if the vector directed from one of them to another is parallel to the axis of the magnetic nanotube. So, the space position of each spin may be characterized by the line and by the layer. In what follows, we will explore the spin configurations obtained using the numerical calculations within the Monte Carlo method for the nanotubes of different diameters, with the armchair or zigzag edges, for dipolar interaction, Heisenberg model interaction and the combinations of these two interactions, in the ferromagnetic and anti-ferromagnetic cases. We will be using the nanotubes of the (4,0), (5,0), (6,0), (7,0), (8,0), (12,0), (4,4), (5,5), (6,6), (7,7), (8,8) and (12,12) types. Obviously, these nanotubes have different diameters and chiralities. Table 1 contains the radii of these nanotubes. Furthermore, the investigated nanotubes had finite length; in our case they had 9 or 13 layers along the axis. In order to illustrate the geometry of the structures under consideration, we present a corresponding picture for the (8,0) and (8,8) cases in Fig. 2. We have investigated the spin configurations, the thermal equilibrium magnetization, the susceptibility and the specific heat for these structures. As a result of this study, one can observe how the spin configurations
1
(m,0)
0.5
a a (m,m)
0
0
0.5
1
Fig. 1. Two-dimensional square unit cell lattice. The indicated vectors are used for rolling up the armchair and zigzag nanotubes.
Structure
Radius (in units of a)
(4,0) (5,0) (6,0) (7,0) (8,0) (12,0) (4,4) (5,5) (6,6) (7,7) (8,8) (12,12)
0.6362 0.7958 0.9549 1.1141 1.2732 1.9099 0.9003 1.1254 1.3505 1.5756 1.8006 2.7010
depend on the diameter of the nanotube, on the form of the edge and also on the type of the interactions between spins. In our simulations we used a Hamiltonian model given by [21] H ¼ J
X hi;ji
~ B Si~ Sj ~
X i
~ Si o
X 3ð~ Si ~ eij Þð~ eij ~ Sj Þ ~ Si~ Sj ioj
r 3ij
In this expression the first sum represents the ferromagnetic (or anti-ferromagnetic) exchange between the nearest-neighbors with a coupling constant J, the second sum stands for the coupling of the spins to an external magnetic field B and the last sum is the dipolar interaction term, where the coupling o describes the strength of the dipole–dipole interaction. The ~ Si are threedimensional magnetic moments of unit length, ~ eij are unit vectors pointed from lattice site i to the lattice site j and rij are the distances between these lattice sites. The correlation between constant values was chosen as o/J ¼ 0.001, in according to Ref. [21]. For the numerical analysis of the magnetic nanotubes described above, we have used the Monte Carlo simulations with the Metropolis algorithm [28,29]. The Metropolis Monte Carlo algorithm enables one to obtain the macro-state equilibrium for a physical system at the given temperature T. The basic idea of this method consists of the following procedure: we start from some randomly chosen initial micro-state and then proceed by performing a very large number of random transformations of the micro-states, until we arrive at the equilibrium macro-state. In our case, we start simulations with an initial configuration in which all spins have parallel directions. Then the direction of one (randomly chosen) of these spins is randomly changed. In this way, we arrive at the new micro- and macro-states and evaluate the change of the overall energy DE compared to the previous configuration. If DEo0, the temporary direction of the spin becomes permanent. If DE40, the temporary direction becomes permanent with the probability exp (DE/kbT). We repeat this procedure n ¼ 10,000 multiplied by the factor equal to the number of sites (spins). The final state corresponds to the stable configuration and is interpreted as equilibrium macro-state. In order to fix the number n, the simulation is firstly performed several times for one particular system. The criterion of the choice of the number n is that the change of the overall energy DE in the last steps (at least 20%) must be negligible. These preliminary calculations show that the equilibrium state is really achieved for 104 Monte Carlo steps per spin and, therefore, this number of steps is adequate for our calculations. After that, all simulations have been performed for this choice of n. In the case of dipolar interaction the spin at the site i was allowed to interact with all other spins of the nanotube. In the Heisenberg interaction case the spin at the site i was allowed to interact only with the nearest-neighboring spins. The same
ARTICLE IN PRESS E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
2723
Fig. 2. Some typical examples of the structures with different edges. The geometries correspond to the armchair (8,0) and zigzag (8,8) nanotubes.
definitions for each interaction type have been applied in the cases when the combination of the two (dipolar and Heisenberg) types of interactions was used. The external magnetic field was directed along the z-axis. The simulations for magnetization and magnetic susceptibility were performed for the values B ¼ 20, 18,y, 0,y, 18, 20. Here, the energy and the applied magnetic field were expressed in units of J. The temperature is expressed in units of J/kb, where J is the magnitude of the coupling constant and kb is Boltzmann’s constant. In all these cases the value for the temperature was chosen to be T ¼ 0.2. In order to study the low-temperature thermodynamics all simulations have been performed for temperatures essentially smaller than the critical temperature. Our choice T ¼ 0.2 provides a rapid convergence of the MC procedure for the system of our interest. We obtain the susceptibility w (in this case, along the Oz-axis) by using the Monte Carlo data, according to the formula
w ¼ N 1
1 ðhm2z i hmz i2 Þ kb T
where N is the number of spins in the system and /mzS is the mean magnetization in the z-direction per spin. The specific heat C is obtained from the energy fluctuation relation C ¼ N 1
1 kb T 2
ðhE2 i hEi2 Þ
where /ES is the mean energy per spin. For calculating the specific heat, we used B ¼ 0 and the values T ¼ 2.0, 1.9, 1.8, 1.7,y, 0.5.
3. Results and discussion Let us start the description of the results from spin orientations for different types of interactions and different values of external
magnetic field. As we have already mentioned above, the simulations have been performed for the longitudinal external magnetic field with the 21 different values of B between 20 and +20. If the external field is sufficiently strong, one can observe that the spins are oriented along the field direction. This picture does change when one weakens the external field. For the sake of simplicity, we start from the B ¼ 0 case. In this case, we can better see how the orientations of the spins depend on the type of interaction between them (dipolar, Heisenberg or both of them together) and also on the geometry of the nanotube (diameter and the edge). In the absence of the external field, considering only the dipolar interaction, we find that the directions of the spins depend on the form of the edge. In the armchair case the spins are directed along the tangent line of the circumference of the nanotube (Fig. 3). Looking at the graphic presentation of the spin distribution one can observe an alternating rule. This means that the directions of spins at layer i are opposite to the directions of spins at the neighbor layers i+1 and i1. For instance, if the even layers have spins oriented clockwise, the odd ones have spins oriented counterclockwise. In the zigzag case, the spins are directed along the tangent lines of the ellipses which pass through the closest neighboring vertices of the unit cells, as it is shown in Fig. 4. The graphic presentation of the spin distribution exhibits an alternating rule which is similar to the armchair case, but the alternation concerns the ellipses which do not coincide with the layers. For Heisenberg interaction, the spin at each site j interacts only with the nearest-neighboring spins. In this case, the geometric distribution of the spin’s directions for the ferromagnetic case is qualitatively the same for all magnetic nanotubes. The spins in all the sites have almost the same direction and this direction is perpendicular to the axis of the nanotubes (Fig. 5). Of course, the choice of this preferred direction in the plane orthogonal to the axis is random. The deviations of the individual spins from the
ARTICLE IN PRESS 2724
E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
Fig. 3. The spin configurations for the dipolar interaction without external magnetic field. The illustrative diagram shows the spins in the three layers for the (6,0) nanotube.
Fig. 5. These diagrams represent the result for the spin distributions for the Heisenberg interaction in the ferromagnetic case. The example given here corresponds to the (6,0) structure.
Fig. 4. The spins distribution for the dipolar interaction without external magnetic field. The illustrative diagram shows the spins in the three layers for the (6,6) nanotube.
preferred direction are quite small and show certain dependence on the nanotube diameter. For the Heisenberg-type interaction in the anti-ferromagnetic cases without the external field the neighbor spins have the opposite directions, as it is shown in Fig. 6a. In the armchair case, the distribution of the spins is such that in each line the spins from the layer i have opposite directions to the ones at the neighbor layers i+1 and i1. For zigzag-type magnetic nanotubes we meet another spin distribution. In order to understand the distributions of spins in this case, let us first notice that the number of lines in a given nanotube is always even in the zigzag case. The spins choose, in apparently random way, one plane which passes through the axis of the tube. Then all the spins orient themselves approximately parallel to this plane and orthogonal to the nanotube axis. Furthermore, the spins in the given line look in the very same direction and the ones in the neighbor lines are directed in the opposite way. We are illustrating this spin distribution in Fig. 6b. Finally, when the dipolar and the Heisenberg interactions are introduced simultaneously, the spin configurations are very similar to the ones for the case of nearest-neighboring interac-
Fig. 6. These illustrative diagrams represent the result of numerical simulation for the spins distribution for the Heisenberg interaction in the anti-ferromagnetic case. (a) Three layers for the (6,0) nanotube and (b) three layers for the (6,6) nanotube.
tions, as we can see in Figs. 5 and 6. In this sense the effect of dipolar interaction on the spins distribution is suppressed compared to the one of the Heisenberg-type interaction. Now we are in a position to discuss the effect of external magnetic field. In general, the effect of external field is to orient
ARTICLE IN PRESS E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
the spins along this field. Of course, this tendency is getting stronger when the external field becomes more intensive, independent on the geometric type of the nanotube. For the pure dipolar interaction, the dependences of magnetization on the applied magnetic field have very similar characters for all types of geometric structures under consideration, as it is shown in Fig. 7. One can observe certain differences between different nanotubes only for the large values of the applied fields, e.g., close to the value B ¼ 10. The effect of applied external magnetic field may be quite different for the Heisenberg interaction case. First of all, there is a big difference between ferromagnetic and anti-ferromagnetic cases. In the anti-ferromagnetic version and for the armchair edge, we meet an essentially stronger dependence on the radii of the tubes. The corresponding plots are presented in Fig. 8. Obviously, the effect of an external magnetic field is stronger for the tubes with larger radii, while the structures with small radii demonstrate weaker reaction on the applied magnetic field. In other words, in order to achieve the same level of spin orientation, one has to use stronger magnetic field for the thinner tubes. It is interesting that the described dependence between magnetization and the radius of the nanotube practically does not take place
2725
for the zigzag edge. The plots of magnetization for this case are presented in Fig. 9. In the ferromagnetic case, with the Heisenberg interaction, the plots of magnetization versus applied magnetic field are very similar for all radii of the nanotubes and moreover they do not depend of the edge types (see Fig. 10). Finally, let us consider the nanotubes where the dipolar and the Heisenberg-type interactions are introduced simultaneously. The plots for the anti-ferromagnetic case and for the armchair nanotube are presented in Fig. 11. It is interesting that this combination of the two interactions results in the much stronger (compared to any one of the pure interaction cases) dependence on the radius of the tube. For the nanotubes with small diameters, we notice that spins orient along the field very weakly even for the large values of the external field, such that the magnetization remains small even for the maximal field intensity. The weakness of magnetization is especially explicit for the (4,0) structure, where the plot has no analogs for the pure interaction cases. The similarity starts only from the (7,0) structure and even in this case the magnetization is much weaker compared to any of the pure interactions. For the zigzag-type nanotubes the plots of magnetizations versus applied magnetic fields are presented in Fig. 12 and look
1 1 (4,0)
0.5
(4,4)
(4,4) (5,5)
0.5
(12,12)
Magnetization
Magnetization
(12,0)
0
-0.5
(6,6) (7,7) (8,8)
0
(12,12)
-0.5 -1 -20
-10
0 Applied Field
10
20 -1 -20
Fig. 7. The plots of magnetization versus applied field for the dipolar interaction. (4.0) and (12.0) cases have armchair edge while (4.4) and (12.12) represent zigzag edge. In this and next figures the value of the applied magnetic field is in the units of J.
(4,4)
0.5 Magnetization
Magnetization
20
(4,0)
(4,0) (5,0) (6,0) (8,0) (12,0)
-0.5
-1 -20
10
1
(7,0)
0
0 Applied Field
Fig. 9. The plots of magnetization versus applied field for the Heisenberg interaction, in the anti-ferromagnetic case for zigzag nanotubes.
1
0.5
-10
(7,0) (7,7)
0
-0.5
-10
0 Applied Field
10
20
Fig. 8. The plots of magnetization versus applied field for the Heisenberg interaction, in the anti-ferromagnetic case for armchair nanotubes.
-1 -20
-10
0 Applied Field
10
20
Fig. 10. The plots of magnetization versus applied field for the Heisenberg interaction (ferromagnetic case) for armchair and zigzag nanotubes.
ARTICLE IN PRESS 2726
E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
1
1 (4,0)
(4,0)
(5,0)
(4,4)
0.5
(6,0) (7,0)
Magnetization
Magnetization
0.5
0
(7,7)
0
-0.5
-0.5
-1 -20
(7,0)
-10
0 Applied Field
10
-1 -20
20
Fig. 11. The plots of magnetization versus applied field for the combination of dipolar and Heisenberg interactions in the anti-ferromagnetic case for armchair nanotubes.
-10
0 Applied Field
10
20
Fig. 13. The plots of magnetization versus applied field for the sum of the dipolar and the Heisenberg interactions in the ferromagnetic case for armchair and zigzag nanotubes.
1
(4,0) (5,0) (6,0) (7,0) (8,0) (12,0)
0.15 Susceptibility
Magnetization
0.5
0.2
(4,4) (5,5) (6,6) (7,7)
0
-0.5
0.1
0.05 -10
0 Applied Field
10
20
Fig. 12. The plots of magnetization versus applied field for the combination of the dipolar and Heisenberg interactions in the anti-ferromagnetic case for zigzag nanotubes.
rather similar to the pure Heisenberg case. However, for the smallest radius (4,4) tube one can observe slightly distinct behavior. One can consider this as an increased influence of the dipolar interaction. In Fig. 13, we show the similar plots for the structures where the dipolar and the Heisenberg interactions coexist. The dependence of magnetization versus applied field in the ferromagnetic case has the same character as in the nearestneighboring interaction. As usual, some influence of the dipolar interactions one can see only for nanotube (4,0). The plots of magnetic susceptibility versus applied field are presented in Figs. 14 and 15 for the dipolar interaction cases. The nanotubes for armchair or zigzag edges produce different forms of these plots. At the same time, one can observe definite similarity between the plots produced for the zigzag edge and different diameters of the tube. This similarity is somehow less explicit in the armchair case. For the Heisenberg interaction and the anti-ferromagnetic case the plots of magnetic susceptibility versus applied field are presented in Figs. 16 and 17. It is easy to see that these plots demonstrate stronger dependence on the radius for the armchair edge (Fig. 16). The plot looks wider for the smaller radius tubes. At
0 -20
-10
0 Applied Field
10
20
Fig. 14. Magnetic susceptibility versus applied field for the dipolar interaction for armchair-type structures.
0.2 (4,4) (5,5) (6,6) (7,7) (8,8) (12,12)
0.15 Susceptibility
-1 -20
0.1
0.05
0 -20
-10
0 Applied Field
10
20
Fig. 15. Magnetic susceptibility versus applied field for the dipolar interaction in the zigzag-type structures.
ARTICLE IN PRESS E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
2
0.2 (4,0) (5,0) (6,0) (7,0) (8,0) (12,0)
(4,0) (4,4)
1.5
(7,0)
Susceptibility
Susceptibility
0.15
0.1
0.05
0 -20
2727
(7,7)
1
0.5
-10
0 Applied Field
10
0 -20
20
Fig. 16. Magnetic susceptibility versus applied field for the Heisenberg interaction in the anti-ferromagnetic case for armchair nanotubes.
-10
0 Applied Field
10
20
Fig. 18. Magnetic susceptibility versus applied field for the Heisenberg interaction in the ferromagnetic case for armchair (4.0), (7,0) and zigzag (4.4), (7,7) nanotubes.
0.2
(4,4)
0.1
0.05
(5,5)
0.15 Susceptibility
0.15 Susceptibility
0.2
(4,4) (5,5) (6,6) (7,7) (8,8) (12,12)
(6,6) (7,7)
0.1
0.05
-10
0 Applied Field
10
20
Fig. 17. Magnetic susceptibility versus applied field for the Heisenberg interaction in the anti-ferromagnetic case for zigzag nanotubes.
the same time, for the zigzag edge (Fig. 17) it is difficult to establish such dependence. In the ferromagnetic case, with the Heisenberg interaction we meet the same form of the plots for all the structures (see Fig. 18). If we compare the plots of magnetic susceptibility versus applied field presented in Figs. 16–18 (for anti-ferromagnetic and ferromagnetic cases), it is obvious that the plots in the ferromagnetic cases are narrower than in the anti-ferromagnetic one. For the coexisting dipolar and Heisenberg-type interactions, in the anti-ferromagnetic case we present, in Fig. 19, only the results for the zigzag form of edge. One can see that the dependence from the radii of the nanotubes is similar to the one in the Heisenbergtype interaction case. The plot is wider for smaller radius of the tube. In the ferromagnetic case for both armchair and zigzag edges we meet wider plots compared to the pure Heisenberg-type interaction (see Fig. 20). Only the (4.0) nanotube gives the plot which is different from the other ones. The thermal equilibrium results obtained by Monte Carlo simulations permit us to obtain the specific heat versus temperature for zigzag form of nanotubes. For all types of interactions the nanotubes with the smaller radius have smaller
0 -20
-10
0 Applied Field
10
20
Fig. 19. Magnetic susceptibility versus applied field for the combination of the dipolar and Heisenberg interactions in the anti-ferromagnetic case for zigzag nanotubes.
2 (4,0) (4,4)
1.5 Susceptibility
0 -20
(7,0) (7,7)
1
0.5
0 -20
-10
0 Applied Field
10
20
Fig. 20. Magnetic susceptibility versus applied field for the sum of the dipolar and Heisenberg interactions in the ferromagnetic case for armchair and zigzag nanotubes.
ARTICLE IN PRESS E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
value of specific heat. The plots of specific heat versus temperatures, for the dipolar interactions and different radii of the tubes are presented in Fig. 21. One can notice that the temperature corresponding to the maximal specific heat depends on whether the number of spins in the layer is even or odd. Typically, for the even numbers (4,4), (6,6), (8,8) and (12,12) the specific heat corresponds to larger temperatures. Let us remark that such difference between even and odd numbers of spins in the layer does not show up for the Heisenberg interactions, which is shown in Fig. 22. As it was already mentioned above, in this case the specific heat is smaller for the tubes with smaller radii. Furthermore, the maximal specific heat is achieved for the slightly higher temperatures in the cases of small radii tubes. Finally, for the nanotubes of the same radius, the maximal values of the specific heat correspond to slightly smaller temperatures in the anti-ferromagnetic case compared to the ferromagnetic case. The plots of specific heat versus temperature for the coexisting dipolar and Heisenberg interactions are presented in Fig. 23. There is no visible difference between anti-ferromagnetic and ferromagnetic tubes with the same radii. Let us note that the MC calculations always present a certain error, such that all results must be given, in general, with the
300 (4,4) (5,5) (6,6) (7,7) (8,8) (12,12)
250
C
200 150 100 50 0 0.5
1
2
1.5 T
Fig. 21. The plots of specific heat versus temperature for the dipolar interaction for zigzag-type structures. In this and next figures the temperature is in units of J/kb and the specific heat is in the units of kb.
150 (4,4) (5,5) (6,6)
100
(7,7)
C
2728
50
0
1
1.5
2
T Fig. 23. The plots of specific heat versus temperature obtained for the combination of the dipolar and Heisenberg interactions in the anti-ferromagnetic (and ferromagnetic) case for zigzag nanotubes.
corresponding error bars. However, as we have already discussed in Section 2, the size of these errors strongly depends on the number n. In our case, for the value n ¼ 10,000, the errors are very small and they cannot influence the qualitative conclusions concerning the dependence on the geometry, in particular on the diameter of the tube and on the rolling-up rule (armchair or zigzag), which we obtain through the MC simulations. In fact, the analysis performed for the selected nanotubes (the procedure explained in the previous section) has shown that the degree of the error compared to the value obtained in the calculations does not exceed the value of 0.01. This aspect of the theory is well known and has been discussed, for instance, in Refs. [14,21,22]. The MC calculations for the systems composed by a large amount of spins require a significant amount of calculation time. The purpose of the present paper is to obtain a qualitative and in part quantitative estimative of how the magnetic proprieties of nanotubes may depend on their geometric form. The detailed analysis of the error bars is, in principle, possible, but it would lead to the unreasonable increase of the computer time and by no means would improve the conclusions, which we can draw from the simulations presented here.
4. Conclusions (4,4) (5,5) (6,6) (7,7) (8,8) (12,12)
C
200
100
0 0.5
1
1.5
2
T Fig. 22. Specific heat versus temperature for the Heisenberg interaction in the anti-ferromagnetic and ferromagnetic cases for zigzag nanotubes. The ferromagnetic plots are marked by *.
In this article, we presented the results of the Monte Carlo simulations of magnetic nanotubes which are based of the plane structures with the square unit cells. We have found that some magnetic and thermodynamic properties of such structures manifest strong dependence on the form of rolling up (armchair or zigzag). Furthermore, we can see the noticeable dependence of these properties on the nanotubes radius. The general conclusion we can draw from our investigation is that the magnetic and thermic properties of the nanostructures are strongly dependant on their geometry. In the present work, we have considered a model based on the square form of the unit cell and, as we have already mentioned above, discovered a strong dependence on the rolling rule and on the diameter of a nanotube. This dependence concerns the spin distribution, magnetic susceptibility, magnetization and specific heat. In particular, the specific heat is always smaller for the nanotubes with smaller radii. One can suppose that similar dependencies on the geometry of magnetic nanotubes will take
ARTICLE IN PRESS E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
place for the more realistic structures including metal compounds [18–20]. We expect to continue investigation of magnetic properties of the nanotubes and in particular explore these realistic structures in the near future.
Acknowledgments The author is indebted to Prof. Julio F. Ferna´ndez for numerous discussions and explanations, especially concerning the application of Monte Carlo simulations to magnetic systems. I am very grateful to A.R. Perreira, S.A. Leonel, P.Z. Coura and Ilya Shapiro for useful discussion. The author is thankful to the Departamento de Fı´sica de Materia Condensada at the Universidad de Zaragoza for the kind hospitality. The work was supported by FAPEMIG (MG, Brazil) and also by the PRONEX project by FAPEMIG and CNPq. References [1] S. Iijima, Nature. London 56 (1991) 354. [2] P.J.F. Harris, Carbon Nanotubes and Related Structures. New Materials for the Twenty-first Century, Cambridge University Press, Cambridge (London, New York), 1991. [3] P. Ball, Designing the Molecular World: Chemistry at the Frontier, Princeton University Press, Princeton, NJ, 1993. [4] M.S. Dresselhaus, G. Dresselhaus, P.C. Eklund, Fullerens and Carbon Nanotubes, Academic Press, San Diego, 1996. [5] T.W. Ebbesen (Ed.), Carbon Nanotubes. Preparation and Properties, CRC Press, Boca Raton, FL, 1997. [6] Y. Shibayama, H. Sato, T. Enoki, Phys. Rev. Lett. 84 (2000) 1744.
2729
[7] T.L. Makarova, B. Sundqvist, R. Ho¨hne, P. Esquinazi, Y. Kopelevich, P. Scharff, V.A. Davydov, L.S. Kashevarova, A.V. Rakhmanina, Nature. London 413 (2001) 716. [8] P. Esquinazi, A. Setzer, R. Ho¨hne, C. Semmelhack, Y. Kopelevich, D. Spemann, T. Butz, B. Kohlstrunk, M. Lo¨sche, Phys. Rev. B 66 (2002) 024429. [9] P. Esquinazi, R. Ho¨hne, J. Magn. Magn. Mater. 290–291 (2005) 20. [10] C.A. Macedo, A.M.C. Souza, Physica B 354 (2004) 290. [11] M. Marganska, M. Szopa, E. Zipper, Phys. Rev. B 72 (2005) 115406. ˜ oz-Sandoval, J.A. Rodrı´guez-Manzo, [12] H. Terrones, F. Lo´pez-Urı´as, E. Mun A. Zamudio, A.L. Elı´as, M. Terrones, Solid State Sci. 8 (2006) 303. [13] J.F. Ferna´ndez, J.J. Alonso, Phys. Rev. B 73 (2006) 024412. [14] J.F. Ferna´ndez, J.J. Alonso, Phys. Rev. B 76 (2007) 014403. [15] A. Franceschetti, S.V. Dudiy, S.V. Barabash, A. Zunger, Phys. Rev. Lett. 97 (2006) 047202. [16] G.M. Dalpian, J.R. Chelikowsky, Phys. Rev. Lett. 96 (2006) 226802. [17] J.J. Alonso, J.F. Ferna´ndez, Phys. Rev. B 74 (2006) 184416. [18] W.A. Alves, R.H.A. Santos, A. Paduan-Filho, C.C. Becerra, A.C. Borin, A.M.D.C. Ferreira, Inorg. Chem. 357 (2004) 2269. [19] W.A. Alves, G. Cerchiaro, A. Paduan-Filho, D.M. Tomazela, M.N. Eberlin, A.M.D.C. Ferreira, Inorg. Chem. 358 (2005) 3581. [20] M.A. Bizeto, W.A. Alves, C.A.S. Barbosa, A.M.D.C. Ferreira, V.R.L. Constantino, Inorg. Chem. 45 (2006) 6214. [21] R. Wieser, U. Nowak, K.D. Usadel, Phys. Rev. B 69 (2004) 064401. [22] G.M. Wysin, A.R. Pereira, I.A. Marques, S.A. Leonel, P.Z. Coura, Phys. Rev. B 72 (2005) 076533. [23] J.A. Rodrı´guez-Manzo, F. Lo´pez-Urı´as, M. Terrones, H. Terrones, Nano. Lett. 4 (2004) 2179. [24] C.C. Tsai, F.L. Shyu, C.W. Chiu, C.P. Chang, R.B. Chen, M.F. Lin, Phys. Rev. B 70 (2004) 075411. [25] C.G. Rocha, A. Latge´, L. Chico, Phys. Rev. B 72 (2005) 085419. [26] P.O. Lehtinen, A.S. Foster, Y. Ma, A.V. Krasheninnikov, R.M. Nieminen, Phys. Rev. Lett. 93 (2004) 187202. [27] O. Iglesias, A. Labarta, Phy. B: Condens. Matter 372 (2006) 247. [28] N.A. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087. [29] K. Binder (Ed.), Monte Carlo Methods in Statistical Physics, Spriger, New York, 1979.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Theoretical simulations of magnetic nanotubes using Monte Carlo method E. Konstantinova Departamento de Fı´sica, ICE, Universidade Federal de Juiz de Fora, 36036-330 Juiz de Fora, MG, Brazil
a r t i c l e in fo
abstract
Article history: Received 27 December 2007 Received in revised form 20 April 2008 Available online 12 June 2008
We present the results of the Monte Carlo simulations of magnetic nanotubes, which are based on the plane structures with the square unit cell at low temperatures. The spin configurations, thermal equilibrium magnetization, magnetic susceptibility and the specific heat are investigated for the nanotubes of different diameters, using armchair or zigzag edges. The dipolar interaction, Heisenberg model interaction and also their combination are considered for both ferromagnetic and antiferromagnetic cases. It turns out that the magnetic properties of the nanotubes strongly depend on the form of the rolling up (armchair or zigzag). The effect of dipolar interaction component strongly manifests itself for the small radius nanotubes, while for the larger radius nanotubes the Heisenberg interaction is always dominating. In the thermodynamic part, we have found that the specific heat is always smaller for the nanotubes with smaller radii. Crown Copyright & 2008 Published by Elsevier B.V. All rights reserved.
PACS: 75.75.+a 75.10.b 61.46.Fg 75.40.Mg 05.10.Ln Keywords: Monte Carlo simulation Magnetic property Nanotube Spin configuration Magnetic susceptibility Specific heat
1. Introduction Since the discovery of carbon nanotubes [1], the electronic, magnetic and mechanical properties of the nanometric scale materials have been attracting considerable attention. During the last few years one could observe a growing interest in the experimental and theoretical investigations of various new structures at nano-scale [2–5]. These structures include different geometric configurations such as fullerenes, nanotubes, nanoparticles, nano-cones, nano-rings and so on. The exploration of different properties of these objects opens wide perspectives for applications. One of the potentially interesting aspects of nanophysics is related to magnetic phenomena. In particular, experimental and theoretical investigations of the magnetism in nanostructures stimulates extensive theoretical studies of different materials at the nanometric scale, such as pure and doped different carbon-based materials [6–12] and semiconductor nanoclusters, ferromagnetic and anti-ferromagnetic nanocrystals [13–16] and thin films [17]. The new experimental works which have been done recently in this area opened the way for the creation of nanotubes based on the composite molecules that
Tel.: +55 32 32174707; fax: +55 32 3229 3312.
E-mail address: konst@fisica.ufjf.br
contain metal atoms [18–20]. In this case, we can consider the nanotube as a tube, composed of rectangular (square, or others) unit cells with spins situated in the vertices of the unit cells. The magnetic properties of nano-scale materials look promising for applications, and this represents a strong motivation for theoretical investigation. In particular, different types of theoretical techniques have been used for the numerical simulations of magnetic properties such as the spin model with dipolar and nearest-neighbor interactions [17,21,22], the nearest-neighbor tight-binding Hamiltonian [11,23–25], the calculations using the ab initio pack with the spin-polarized density functional theory [26] and the Monte Carlo simulations [27]. In the present paper, we start a systematic analysis of the magnetic properties of nanotubes. As a first step, we perform the theoretical simulations of various properties of the model magnetic nanotubes based on the geometrically simplest lattice with the square unit cell. The main target of our study is the dependence of the thermodynamic and magnetic properties on geometry. In particular, we explore the difference between the nanotubes of different diameters and also the dependence on the type of the rolling which affects, in particular, the form of the edge. The results of the analysis look rather natural and one can expect similar dependencies for the more realistic cases of magnetic nanotubes, e.g., similar to those considered in Refs. [18–20].
0304-8853/$ - see front matter Crown Copyright & 2008 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.06.007
ARTICLE IN PRESS 2722
E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
2. Method of calculations
Table 1 The radii of the nanotubes expressed in the units of a (unit cell’s size)
The purpose of present work is to investigate the possible influence of the nanotube’s diameter and the type of the edge on the magnetic properties of nanotubes, based on the plane structure with the square unit cell. In Fig. 1 one can observe the two-dimensional plane with the square unit cell of the lattice. The rolling up corresponding to the armchair nanotubes is indicated by the vector (m,0) while the rolling up corresponding to the zigzag nanotubes is indicated by the vector (m,m). The value of the integer m defined the size of the nanotubes. In this paper, we are using the terms ‘‘armchair’’ and ‘‘zigzag’’ which are traditional for the carbon nanotubes. However, we use these terms in the opposite order, such that they agree with the actual geometry of the nanotubes under consideration. The geometry here means the form of the edge: for the case of the armchair the rolling up vector is passing through the neighboring vertices of the unit cells; for the case of zigzag this vector is passing through the diagonal vertices of the unit cells. After rolling, any nanotube is defined by the pair of integer parameters (m1,m2), which describe its circumference vector on the initial plane, that is ~ L ¼ m1~ a1 þ m2~ a2 , where ~ a1 ; ~ a2 are unit cell’s vectors (later we used a as a lattice constant, which we set to unity, a ¼ 1). It proves useful to introduce the following terminology. The two spins belong to the same ‘‘line’’ if the vector directed from one of them to another is parallel to the axis of the magnetic nanotube. So, the space position of each spin may be characterized by the line and by the layer. In what follows, we will explore the spin configurations obtained using the numerical calculations within the Monte Carlo method for the nanotubes of different diameters, with the armchair or zigzag edges, for dipolar interaction, Heisenberg model interaction and the combinations of these two interactions, in the ferromagnetic and anti-ferromagnetic cases. We will be using the nanotubes of the (4,0), (5,0), (6,0), (7,0), (8,0), (12,0), (4,4), (5,5), (6,6), (7,7), (8,8) and (12,12) types. Obviously, these nanotubes have different diameters and chiralities. Table 1 contains the radii of these nanotubes. Furthermore, the investigated nanotubes had finite length; in our case they had 9 or 13 layers along the axis. In order to illustrate the geometry of the structures under consideration, we present a corresponding picture for the (8,0) and (8,8) cases in Fig. 2. We have investigated the spin configurations, the thermal equilibrium magnetization, the susceptibility and the specific heat for these structures. As a result of this study, one can observe how the spin configurations
1
(m,0)
0.5
a a (m,m)
0
0
0.5
1
Fig. 1. Two-dimensional square unit cell lattice. The indicated vectors are used for rolling up the armchair and zigzag nanotubes.
Structure
Radius (in units of a)
(4,0) (5,0) (6,0) (7,0) (8,0) (12,0) (4,4) (5,5) (6,6) (7,7) (8,8) (12,12)
0.6362 0.7958 0.9549 1.1141 1.2732 1.9099 0.9003 1.1254 1.3505 1.5756 1.8006 2.7010
depend on the diameter of the nanotube, on the form of the edge and also on the type of the interactions between spins. In our simulations we used a Hamiltonian model given by [21] H ¼ J
X hi;ji
~ B Si~ Sj ~
X i
~ Si o
X 3ð~ Si ~ eij Þð~ eij ~ Sj Þ ~ Si~ Sj ioj
r 3ij
In this expression the first sum represents the ferromagnetic (or anti-ferromagnetic) exchange between the nearest-neighbors with a coupling constant J, the second sum stands for the coupling of the spins to an external magnetic field B and the last sum is the dipolar interaction term, where the coupling o describes the strength of the dipole–dipole interaction. The ~ Si are threedimensional magnetic moments of unit length, ~ eij are unit vectors pointed from lattice site i to the lattice site j and rij are the distances between these lattice sites. The correlation between constant values was chosen as o/J ¼ 0.001, in according to Ref. [21]. For the numerical analysis of the magnetic nanotubes described above, we have used the Monte Carlo simulations with the Metropolis algorithm [28,29]. The Metropolis Monte Carlo algorithm enables one to obtain the macro-state equilibrium for a physical system at the given temperature T. The basic idea of this method consists of the following procedure: we start from some randomly chosen initial micro-state and then proceed by performing a very large number of random transformations of the micro-states, until we arrive at the equilibrium macro-state. In our case, we start simulations with an initial configuration in which all spins have parallel directions. Then the direction of one (randomly chosen) of these spins is randomly changed. In this way, we arrive at the new micro- and macro-states and evaluate the change of the overall energy DE compared to the previous configuration. If DEo0, the temporary direction of the spin becomes permanent. If DE40, the temporary direction becomes permanent with the probability exp (DE/kbT). We repeat this procedure n ¼ 10,000 multiplied by the factor equal to the number of sites (spins). The final state corresponds to the stable configuration and is interpreted as equilibrium macro-state. In order to fix the number n, the simulation is firstly performed several times for one particular system. The criterion of the choice of the number n is that the change of the overall energy DE in the last steps (at least 20%) must be negligible. These preliminary calculations show that the equilibrium state is really achieved for 104 Monte Carlo steps per spin and, therefore, this number of steps is adequate for our calculations. After that, all simulations have been performed for this choice of n. In the case of dipolar interaction the spin at the site i was allowed to interact with all other spins of the nanotube. In the Heisenberg interaction case the spin at the site i was allowed to interact only with the nearest-neighboring spins. The same
ARTICLE IN PRESS E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
2723
Fig. 2. Some typical examples of the structures with different edges. The geometries correspond to the armchair (8,0) and zigzag (8,8) nanotubes.
definitions for each interaction type have been applied in the cases when the combination of the two (dipolar and Heisenberg) types of interactions was used. The external magnetic field was directed along the z-axis. The simulations for magnetization and magnetic susceptibility were performed for the values B ¼ 20, 18,y, 0,y, 18, 20. Here, the energy and the applied magnetic field were expressed in units of J. The temperature is expressed in units of J/kb, where J is the magnitude of the coupling constant and kb is Boltzmann’s constant. In all these cases the value for the temperature was chosen to be T ¼ 0.2. In order to study the low-temperature thermodynamics all simulations have been performed for temperatures essentially smaller than the critical temperature. Our choice T ¼ 0.2 provides a rapid convergence of the MC procedure for the system of our interest. We obtain the susceptibility w (in this case, along the Oz-axis) by using the Monte Carlo data, according to the formula
w ¼ N 1
1 ðhm2z i hmz i2 Þ kb T
where N is the number of spins in the system and /mzS is the mean magnetization in the z-direction per spin. The specific heat C is obtained from the energy fluctuation relation C ¼ N 1
1 kb T 2
ðhE2 i hEi2 Þ
where /ES is the mean energy per spin. For calculating the specific heat, we used B ¼ 0 and the values T ¼ 2.0, 1.9, 1.8, 1.7,y, 0.5.
3. Results and discussion Let us start the description of the results from spin orientations for different types of interactions and different values of external
magnetic field. As we have already mentioned above, the simulations have been performed for the longitudinal external magnetic field with the 21 different values of B between 20 and +20. If the external field is sufficiently strong, one can observe that the spins are oriented along the field direction. This picture does change when one weakens the external field. For the sake of simplicity, we start from the B ¼ 0 case. In this case, we can better see how the orientations of the spins depend on the type of interaction between them (dipolar, Heisenberg or both of them together) and also on the geometry of the nanotube (diameter and the edge). In the absence of the external field, considering only the dipolar interaction, we find that the directions of the spins depend on the form of the edge. In the armchair case the spins are directed along the tangent line of the circumference of the nanotube (Fig. 3). Looking at the graphic presentation of the spin distribution one can observe an alternating rule. This means that the directions of spins at layer i are opposite to the directions of spins at the neighbor layers i+1 and i1. For instance, if the even layers have spins oriented clockwise, the odd ones have spins oriented counterclockwise. In the zigzag case, the spins are directed along the tangent lines of the ellipses which pass through the closest neighboring vertices of the unit cells, as it is shown in Fig. 4. The graphic presentation of the spin distribution exhibits an alternating rule which is similar to the armchair case, but the alternation concerns the ellipses which do not coincide with the layers. For Heisenberg interaction, the spin at each site j interacts only with the nearest-neighboring spins. In this case, the geometric distribution of the spin’s directions for the ferromagnetic case is qualitatively the same for all magnetic nanotubes. The spins in all the sites have almost the same direction and this direction is perpendicular to the axis of the nanotubes (Fig. 5). Of course, the choice of this preferred direction in the plane orthogonal to the axis is random. The deviations of the individual spins from the
ARTICLE IN PRESS 2724
E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
Fig. 3. The spin configurations for the dipolar interaction without external magnetic field. The illustrative diagram shows the spins in the three layers for the (6,0) nanotube.
Fig. 5. These diagrams represent the result for the spin distributions for the Heisenberg interaction in the ferromagnetic case. The example given here corresponds to the (6,0) structure.
Fig. 4. The spins distribution for the dipolar interaction without external magnetic field. The illustrative diagram shows the spins in the three layers for the (6,6) nanotube.
preferred direction are quite small and show certain dependence on the nanotube diameter. For the Heisenberg-type interaction in the anti-ferromagnetic cases without the external field the neighbor spins have the opposite directions, as it is shown in Fig. 6a. In the armchair case, the distribution of the spins is such that in each line the spins from the layer i have opposite directions to the ones at the neighbor layers i+1 and i1. For zigzag-type magnetic nanotubes we meet another spin distribution. In order to understand the distributions of spins in this case, let us first notice that the number of lines in a given nanotube is always even in the zigzag case. The spins choose, in apparently random way, one plane which passes through the axis of the tube. Then all the spins orient themselves approximately parallel to this plane and orthogonal to the nanotube axis. Furthermore, the spins in the given line look in the very same direction and the ones in the neighbor lines are directed in the opposite way. We are illustrating this spin distribution in Fig. 6b. Finally, when the dipolar and the Heisenberg interactions are introduced simultaneously, the spin configurations are very similar to the ones for the case of nearest-neighboring interac-
Fig. 6. These illustrative diagrams represent the result of numerical simulation for the spins distribution for the Heisenberg interaction in the anti-ferromagnetic case. (a) Three layers for the (6,0) nanotube and (b) three layers for the (6,6) nanotube.
tions, as we can see in Figs. 5 and 6. In this sense the effect of dipolar interaction on the spins distribution is suppressed compared to the one of the Heisenberg-type interaction. Now we are in a position to discuss the effect of external magnetic field. In general, the effect of external field is to orient
ARTICLE IN PRESS E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
the spins along this field. Of course, this tendency is getting stronger when the external field becomes more intensive, independent on the geometric type of the nanotube. For the pure dipolar interaction, the dependences of magnetization on the applied magnetic field have very similar characters for all types of geometric structures under consideration, as it is shown in Fig. 7. One can observe certain differences between different nanotubes only for the large values of the applied fields, e.g., close to the value B ¼ 10. The effect of applied external magnetic field may be quite different for the Heisenberg interaction case. First of all, there is a big difference between ferromagnetic and anti-ferromagnetic cases. In the anti-ferromagnetic version and for the armchair edge, we meet an essentially stronger dependence on the radii of the tubes. The corresponding plots are presented in Fig. 8. Obviously, the effect of an external magnetic field is stronger for the tubes with larger radii, while the structures with small radii demonstrate weaker reaction on the applied magnetic field. In other words, in order to achieve the same level of spin orientation, one has to use stronger magnetic field for the thinner tubes. It is interesting that the described dependence between magnetization and the radius of the nanotube practically does not take place
2725
for the zigzag edge. The plots of magnetization for this case are presented in Fig. 9. In the ferromagnetic case, with the Heisenberg interaction, the plots of magnetization versus applied magnetic field are very similar for all radii of the nanotubes and moreover they do not depend of the edge types (see Fig. 10). Finally, let us consider the nanotubes where the dipolar and the Heisenberg-type interactions are introduced simultaneously. The plots for the anti-ferromagnetic case and for the armchair nanotube are presented in Fig. 11. It is interesting that this combination of the two interactions results in the much stronger (compared to any one of the pure interaction cases) dependence on the radius of the tube. For the nanotubes with small diameters, we notice that spins orient along the field very weakly even for the large values of the external field, such that the magnetization remains small even for the maximal field intensity. The weakness of magnetization is especially explicit for the (4,0) structure, where the plot has no analogs for the pure interaction cases. The similarity starts only from the (7,0) structure and even in this case the magnetization is much weaker compared to any of the pure interactions. For the zigzag-type nanotubes the plots of magnetizations versus applied magnetic fields are presented in Fig. 12 and look
1 1 (4,0)
0.5
(4,4)
(4,4) (5,5)
0.5
(12,12)
Magnetization
Magnetization
(12,0)
0
-0.5
(6,6) (7,7) (8,8)
0
(12,12)
-0.5 -1 -20
-10
0 Applied Field
10
20 -1 -20
Fig. 7. The plots of magnetization versus applied field for the dipolar interaction. (4.0) and (12.0) cases have armchair edge while (4.4) and (12.12) represent zigzag edge. In this and next figures the value of the applied magnetic field is in the units of J.
(4,4)
0.5 Magnetization
Magnetization
20
(4,0)
(4,0) (5,0) (6,0) (8,0) (12,0)
-0.5
-1 -20
10
1
(7,0)
0
0 Applied Field
Fig. 9. The plots of magnetization versus applied field for the Heisenberg interaction, in the anti-ferromagnetic case for zigzag nanotubes.
1
0.5
-10
(7,0) (7,7)
0
-0.5
-10
0 Applied Field
10
20
Fig. 8. The plots of magnetization versus applied field for the Heisenberg interaction, in the anti-ferromagnetic case for armchair nanotubes.
-1 -20
-10
0 Applied Field
10
20
Fig. 10. The plots of magnetization versus applied field for the Heisenberg interaction (ferromagnetic case) for armchair and zigzag nanotubes.
ARTICLE IN PRESS 2726
E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
1
1 (4,0)
(4,0)
(5,0)
(4,4)
0.5
(6,0) (7,0)
Magnetization
Magnetization
0.5
0
(7,7)
0
-0.5
-0.5
-1 -20
(7,0)
-10
0 Applied Field
10
-1 -20
20
Fig. 11. The plots of magnetization versus applied field for the combination of dipolar and Heisenberg interactions in the anti-ferromagnetic case for armchair nanotubes.
-10
0 Applied Field
10
20
Fig. 13. The plots of magnetization versus applied field for the sum of the dipolar and the Heisenberg interactions in the ferromagnetic case for armchair and zigzag nanotubes.
1
(4,0) (5,0) (6,0) (7,0) (8,0) (12,0)
0.15 Susceptibility
Magnetization
0.5
0.2
(4,4) (5,5) (6,6) (7,7)
0
-0.5
0.1
0.05 -10
0 Applied Field
10
20
Fig. 12. The plots of magnetization versus applied field for the combination of the dipolar and Heisenberg interactions in the anti-ferromagnetic case for zigzag nanotubes.
rather similar to the pure Heisenberg case. However, for the smallest radius (4,4) tube one can observe slightly distinct behavior. One can consider this as an increased influence of the dipolar interaction. In Fig. 13, we show the similar plots for the structures where the dipolar and the Heisenberg interactions coexist. The dependence of magnetization versus applied field in the ferromagnetic case has the same character as in the nearestneighboring interaction. As usual, some influence of the dipolar interactions one can see only for nanotube (4,0). The plots of magnetic susceptibility versus applied field are presented in Figs. 14 and 15 for the dipolar interaction cases. The nanotubes for armchair or zigzag edges produce different forms of these plots. At the same time, one can observe definite similarity between the plots produced for the zigzag edge and different diameters of the tube. This similarity is somehow less explicit in the armchair case. For the Heisenberg interaction and the anti-ferromagnetic case the plots of magnetic susceptibility versus applied field are presented in Figs. 16 and 17. It is easy to see that these plots demonstrate stronger dependence on the radius for the armchair edge (Fig. 16). The plot looks wider for the smaller radius tubes. At
0 -20
-10
0 Applied Field
10
20
Fig. 14. Magnetic susceptibility versus applied field for the dipolar interaction for armchair-type structures.
0.2 (4,4) (5,5) (6,6) (7,7) (8,8) (12,12)
0.15 Susceptibility
-1 -20
0.1
0.05
0 -20
-10
0 Applied Field
10
20
Fig. 15. Magnetic susceptibility versus applied field for the dipolar interaction in the zigzag-type structures.
ARTICLE IN PRESS E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
2
0.2 (4,0) (5,0) (6,0) (7,0) (8,0) (12,0)
(4,0) (4,4)
1.5
(7,0)
Susceptibility
Susceptibility
0.15
0.1
0.05
0 -20
2727
(7,7)
1
0.5
-10
0 Applied Field
10
0 -20
20
Fig. 16. Magnetic susceptibility versus applied field for the Heisenberg interaction in the anti-ferromagnetic case for armchair nanotubes.
-10
0 Applied Field
10
20
Fig. 18. Magnetic susceptibility versus applied field for the Heisenberg interaction in the ferromagnetic case for armchair (4.0), (7,0) and zigzag (4.4), (7,7) nanotubes.
0.2
(4,4)
0.1
0.05
(5,5)
0.15 Susceptibility
0.15 Susceptibility
0.2
(4,4) (5,5) (6,6) (7,7) (8,8) (12,12)
(6,6) (7,7)
0.1
0.05
-10
0 Applied Field
10
20
Fig. 17. Magnetic susceptibility versus applied field for the Heisenberg interaction in the anti-ferromagnetic case for zigzag nanotubes.
the same time, for the zigzag edge (Fig. 17) it is difficult to establish such dependence. In the ferromagnetic case, with the Heisenberg interaction we meet the same form of the plots for all the structures (see Fig. 18). If we compare the plots of magnetic susceptibility versus applied field presented in Figs. 16–18 (for anti-ferromagnetic and ferromagnetic cases), it is obvious that the plots in the ferromagnetic cases are narrower than in the anti-ferromagnetic one. For the coexisting dipolar and Heisenberg-type interactions, in the anti-ferromagnetic case we present, in Fig. 19, only the results for the zigzag form of edge. One can see that the dependence from the radii of the nanotubes is similar to the one in the Heisenbergtype interaction case. The plot is wider for smaller radius of the tube. In the ferromagnetic case for both armchair and zigzag edges we meet wider plots compared to the pure Heisenberg-type interaction (see Fig. 20). Only the (4.0) nanotube gives the plot which is different from the other ones. The thermal equilibrium results obtained by Monte Carlo simulations permit us to obtain the specific heat versus temperature for zigzag form of nanotubes. For all types of interactions the nanotubes with the smaller radius have smaller
0 -20
-10
0 Applied Field
10
20
Fig. 19. Magnetic susceptibility versus applied field for the combination of the dipolar and Heisenberg interactions in the anti-ferromagnetic case for zigzag nanotubes.
2 (4,0) (4,4)
1.5 Susceptibility
0 -20
(7,0) (7,7)
1
0.5
0 -20
-10
0 Applied Field
10
20
Fig. 20. Magnetic susceptibility versus applied field for the sum of the dipolar and Heisenberg interactions in the ferromagnetic case for armchair and zigzag nanotubes.
ARTICLE IN PRESS E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
value of specific heat. The plots of specific heat versus temperatures, for the dipolar interactions and different radii of the tubes are presented in Fig. 21. One can notice that the temperature corresponding to the maximal specific heat depends on whether the number of spins in the layer is even or odd. Typically, for the even numbers (4,4), (6,6), (8,8) and (12,12) the specific heat corresponds to larger temperatures. Let us remark that such difference between even and odd numbers of spins in the layer does not show up for the Heisenberg interactions, which is shown in Fig. 22. As it was already mentioned above, in this case the specific heat is smaller for the tubes with smaller radii. Furthermore, the maximal specific heat is achieved for the slightly higher temperatures in the cases of small radii tubes. Finally, for the nanotubes of the same radius, the maximal values of the specific heat correspond to slightly smaller temperatures in the anti-ferromagnetic case compared to the ferromagnetic case. The plots of specific heat versus temperature for the coexisting dipolar and Heisenberg interactions are presented in Fig. 23. There is no visible difference between anti-ferromagnetic and ferromagnetic tubes with the same radii. Let us note that the MC calculations always present a certain error, such that all results must be given, in general, with the
300 (4,4) (5,5) (6,6) (7,7) (8,8) (12,12)
250
C
200 150 100 50 0 0.5
1
2
1.5 T
Fig. 21. The plots of specific heat versus temperature for the dipolar interaction for zigzag-type structures. In this and next figures the temperature is in units of J/kb and the specific heat is in the units of kb.
150 (4,4) (5,5) (6,6)
100
(7,7)
C
2728
50
0
1
1.5
2
T Fig. 23. The plots of specific heat versus temperature obtained for the combination of the dipolar and Heisenberg interactions in the anti-ferromagnetic (and ferromagnetic) case for zigzag nanotubes.
corresponding error bars. However, as we have already discussed in Section 2, the size of these errors strongly depends on the number n. In our case, for the value n ¼ 10,000, the errors are very small and they cannot influence the qualitative conclusions concerning the dependence on the geometry, in particular on the diameter of the tube and on the rolling-up rule (armchair or zigzag), which we obtain through the MC simulations. In fact, the analysis performed for the selected nanotubes (the procedure explained in the previous section) has shown that the degree of the error compared to the value obtained in the calculations does not exceed the value of 0.01. This aspect of the theory is well known and has been discussed, for instance, in Refs. [14,21,22]. The MC calculations for the systems composed by a large amount of spins require a significant amount of calculation time. The purpose of the present paper is to obtain a qualitative and in part quantitative estimative of how the magnetic proprieties of nanotubes may depend on their geometric form. The detailed analysis of the error bars is, in principle, possible, but it would lead to the unreasonable increase of the computer time and by no means would improve the conclusions, which we can draw from the simulations presented here.
4. Conclusions (4,4) (5,5) (6,6) (7,7) (8,8) (12,12)
C
200
100
0 0.5
1
1.5
2
T Fig. 22. Specific heat versus temperature for the Heisenberg interaction in the anti-ferromagnetic and ferromagnetic cases for zigzag nanotubes. The ferromagnetic plots are marked by *.
In this article, we presented the results of the Monte Carlo simulations of magnetic nanotubes which are based of the plane structures with the square unit cells. We have found that some magnetic and thermodynamic properties of such structures manifest strong dependence on the form of rolling up (armchair or zigzag). Furthermore, we can see the noticeable dependence of these properties on the nanotubes radius. The general conclusion we can draw from our investigation is that the magnetic and thermic properties of the nanostructures are strongly dependant on their geometry. In the present work, we have considered a model based on the square form of the unit cell and, as we have already mentioned above, discovered a strong dependence on the rolling rule and on the diameter of a nanotube. This dependence concerns the spin distribution, magnetic susceptibility, magnetization and specific heat. In particular, the specific heat is always smaller for the nanotubes with smaller radii. One can suppose that similar dependencies on the geometry of magnetic nanotubes will take
ARTICLE IN PRESS E. Konstantinova / Journal of Magnetism and Magnetic Materials 320 (2008) 2721–2729
place for the more realistic structures including metal compounds [18–20]. We expect to continue investigation of magnetic properties of the nanotubes and in particular explore these realistic structures in the near future.
Acknowledgments The author is indebted to Prof. Julio F. Ferna´ndez for numerous discussions and explanations, especially concerning the application of Monte Carlo simulations to magnetic systems. I am very grateful to A.R. Perreira, S.A. Leonel, P.Z. Coura and Ilya Shapiro for useful discussion. The author is thankful to the Departamento de Fı´sica de Materia Condensada at the Universidad de Zaragoza for the kind hospitality. The work was supported by FAPEMIG (MG, Brazil) and also by the PRONEX project by FAPEMIG and CNPq. References [1] S. Iijima, Nature. London 56 (1991) 354. [2] P.J.F. Harris, Carbon Nanotubes and Related Structures. New Materials for the Twenty-first Century, Cambridge University Press, Cambridge (London, New York), 1991. [3] P. Ball, Designing the Molecular World: Chemistry at the Frontier, Princeton University Press, Princeton, NJ, 1993. [4] M.S. Dresselhaus, G. Dresselhaus, P.C. Eklund, Fullerens and Carbon Nanotubes, Academic Press, San Diego, 1996. [5] T.W. Ebbesen (Ed.), Carbon Nanotubes. Preparation and Properties, CRC Press, Boca Raton, FL, 1997. [6] Y. Shibayama, H. Sato, T. Enoki, Phys. Rev. Lett. 84 (2000) 1744.
2729
[7] T.L. Makarova, B. Sundqvist, R. Ho¨hne, P. Esquinazi, Y. Kopelevich, P. Scharff, V.A. Davydov, L.S. Kashevarova, A.V. Rakhmanina, Nature. London 413 (2001) 716. [8] P. Esquinazi, A. Setzer, R. Ho¨hne, C. Semmelhack, Y. Kopelevich, D. Spemann, T. Butz, B. Kohlstrunk, M. Lo¨sche, Phys. Rev. B 66 (2002) 024429. [9] P. Esquinazi, R. Ho¨hne, J. Magn. Magn. Mater. 290–291 (2005) 20. [10] C.A. Macedo, A.M.C. Souza, Physica B 354 (2004) 290. [11] M. Marganska, M. Szopa, E. Zipper, Phys. Rev. B 72 (2005) 115406. ˜ oz-Sandoval, J.A. Rodrı´guez-Manzo, [12] H. Terrones, F. Lo´pez-Urı´as, E. Mun A. Zamudio, A.L. Elı´as, M. Terrones, Solid State Sci. 8 (2006) 303. [13] J.F. Ferna´ndez, J.J. Alonso, Phys. Rev. B 73 (2006) 024412. [14] J.F. Ferna´ndez, J.J. Alonso, Phys. Rev. B 76 (2007) 014403. [15] A. Franceschetti, S.V. Dudiy, S.V. Barabash, A. Zunger, Phys. Rev. Lett. 97 (2006) 047202. [16] G.M. Dalpian, J.R. Chelikowsky, Phys. Rev. Lett. 96 (2006) 226802. [17] J.J. Alonso, J.F. Ferna´ndez, Phys. Rev. B 74 (2006) 184416. [18] W.A. Alves, R.H.A. Santos, A. Paduan-Filho, C.C. Becerra, A.C. Borin, A.M.D.C. Ferreira, Inorg. Chem. 357 (2004) 2269. [19] W.A. Alves, G. Cerchiaro, A. Paduan-Filho, D.M. Tomazela, M.N. Eberlin, A.M.D.C. Ferreira, Inorg. Chem. 358 (2005) 3581. [20] M.A. Bizeto, W.A. Alves, C.A.S. Barbosa, A.M.D.C. Ferreira, V.R.L. Constantino, Inorg. Chem. 45 (2006) 6214. [21] R. Wieser, U. Nowak, K.D. Usadel, Phys. Rev. B 69 (2004) 064401. [22] G.M. Wysin, A.R. Pereira, I.A. Marques, S.A. Leonel, P.Z. Coura, Phys. Rev. B 72 (2005) 076533. [23] J.A. Rodrı´guez-Manzo, F. Lo´pez-Urı´as, M. Terrones, H. Terrones, Nano. Lett. 4 (2004) 2179. [24] C.C. Tsai, F.L. Shyu, C.W. Chiu, C.P. Chang, R.B. Chen, M.F. Lin, Phys. Rev. B 70 (2004) 075411. [25] C.G. Rocha, A. Latge´, L. Chico, Phys. Rev. B 72 (2005) 085419. [26] P.O. Lehtinen, A.S. Foster, Y. Ma, A.V. Krasheninnikov, R.M. Nieminen, Phys. Rev. Lett. 93 (2004) 187202. [27] O. Iglesias, A. Labarta, Phy. B: Condens. Matter 372 (2006) 247. [28] N.A. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087. [29] K. Binder (Ed.), Monte Carlo Methods in Statistical Physics, Spriger, New York, 1979.