Re-orientation Transition in Molecular Thin Films: Potts Model with Dipolar Interaction
We study the low-temperature behavior and the phase transition of a thin film by Monte Carlo simulation. The thin film has a simple cubic lattice structure where each site is occupied by a Potts parameter which indicates the molecular orientation of the site. We take only three molecular orientations in this paper which correspond to the 3-state Potts model. The Hamiltonian of the system includes: (i) the exchange interaction between nearest-neighbor sites and (ii) the long-range dipolar interaction of amplitude truncated at a cutoff distance (iii) a single-ion perpendicular anisotropy of amplitude . We allow between surface spins, and otherwise. We show that the ground state depends on the the ratio and . For a single layer, for a given , there is a critical value below (above) which the ground-state (GS) configuration of molecular axes is perpendicular (parallel) to the film surface. When the temperature is increased, a re-orientation transition occurs near : the low- in-plane ordering undergoes a transition to the perpendicular ordering at a finite , below the transition to the paramagnetic phase. The same phenomenon is observed in the case of a film with a thickness. We show that the surface phase transition can occur below or above the bulk transition depending on the ratio . Surface and bulk order parameters as well as other physical quantities are shown and discussed.
- PACS numbers:64.60.De, 75.10.-b, 75.40.Mg, 75.70.Rf
pacs:Valid PACS appear here
Surface physics has been intensively developed during the last 30 years. Among the main reasons for that rapid and successful development we can mention the interest in understanding the physics of low-dimensional systems and an immense potential of industrial applications of thin films (1); (2); (3). In particular, theoretically it has been shown that systems of continuous spins (XY and Heisenberg) in two dimensions (2D) with short-range interaction cannot have long-range order at finite temperature (4). In the case of thin films, it has been shown that low-lying localized spin waves can be found at the film surface (5) and effects of these localized modes on the surface magnetization at finite temperature () and on the critical temperature have been investigated by the Green’s function technique (6); (7). Experimentally, objects of nanometric size such as ultrathin films and nanoparticles have also been intensively studied because of numerous and important applications in industry. An example is the so-called giant magnetoresistance used in data storage devices, magnetic sensors, etc. (8); (9); (10); (11). Recently, much interest has been attracted towards practical problems such as spin transport, spin valves and spin-torques transfer, due to numerous applications in spintronics.
In this paper, we are interested in the phase transition of the Potts model (12) in thin films taking into account a dipolar interaction and a perpendicular anisotropy. The -state Potts model is very popular in statistical physics and much is known for models with short-range ferromagnetic interactions in 2D and 3D (12). The Potts model with an algebraically decaying long-range interaction has been investigated in 1D (13); (14). Such a monotonous long-range interaction can induce an ordering at finite in one-dimensional systems. The dipolar interaction, however, is very special because it contains two competing terms which yield complicated orderings depending on the sample shape. For example, the dipolar interaction favors an in-plane ordering in films and slabs with infinite lateral dimensions. Many studies have been done with the dipolar interaction in thin films with the Heisenberg spin model (15); (16). The absence of the Potts model for thin films has motivated the present work.
We will consider a thin film made of a molecular crystal where molecular spins can point along the , or axes. The interactions between molecular spins include a dipolar interaction truncated at a distance and an exchange interaction between nearest neighbors (NN). We also take into account a single-ion perpendicular anisotropy which is known to exist in ultrathin films (1). The method we employ is Monte Carlo (MC) simulations. Phase transition in systems of interacting particles is a major domain in statistical physics. Much is now understood with the analysis provided by the fundamental concepts of the renormalization group (17) and with the use of the field theory (18). But these methods encountered some difficulties in dealing with frustrated spin systems (19); (20). MC simulations are therefore very useful to complete theories and to interpret experiments. They serve as testing means for new theoretical developments. Over the years, the standard MC method (21) has been improved by the finite-size scaling theory (22) and by other high-performance techniques such as histogram techniques (23); (24), cluster updating algorithms (25); (26); (27) and Wang-Landau flat-histogram method (28). We have now at hand these efficient techniques to deal with complex systems. We can mention our recent investigations by MC techniques on multilayers (29), on frustrated surfaces (30); (31) or on surface criticality (32); (33).
In section II, we describe our model and the method we employ. Results of MC simulations are shown and discussed in section III for several cases: 2D, homogeneous films and effects of surface interaction. Concluding remarks are given in section IV.
Ii Model and Method
We consider a thin film of simple cubic lattice. The film is infinite in the plane and has a thickness in the direction. The Hamiltonian is given by the following 3-state Potts model:
where is a variable associated to the lattice site . is equal to 1, 2 and 3 if the spin at that site lies along the , and axes, respectively. is the exchange interaction between NN at and . We will assume that (i) if and are on the same film surface (ii) otherwise.
The dipolar Hamiltonian is written as
where is the vector of modulus connecting the site to the site . One has . In Eq. (2), is a positive constant depending on the material, the sum is limited at pairs of spins within a cut-off distance , and is given by the following three-component pseudo vector representing the spin state
where () is the component with values .
The perpendicular anisotropy is introduced by the following term
where is a constant.
Note that the dipolar interaction as applied in our Potts model is not similar to that used in the vector spin model where is a true vector. In our model, each spin can only choose to lie on one of three axes, pointing in positive or negative direction.
We use as the unit of energy. The temperature is expressed in the unit of where is the Boltzmann constant.
In the absence of , the GS configuration is perpendicular to the film surface due to the term . In the absence of , the GS is an in-plane configuration due to . When both and are present, the GS depends on the ratio . An analytical determination of the GS is impossible due to the long-range interaction. We therefore determine the GS by the numerical steepest-descent method which works very well in systems with uniformly distributed interactions. This method is very simple(30); (31) (i) we generate a random initial spin configuration (ii) we calculate the local field created at a given spin by its neighbors using Eqs. (1) and (2) (iii) we change the spin axis to minimize its energy (i. e. we align the spin in its local field) (iv) we go to another spin and repeat until all spins are visited: we say we make one sweep (v) we do a large number of sweeps until a good convergence to the lowest energy is reached.
We shall use MC simulation to calculate properties of the system at finite . Periodic boundary conditions are used in the planes for sample sizes of where is the film thickness. Free symmetric surfaces are supposed for simplicity. Standard MC method (21) is used to get general features of the phase transition. Systematic finite-size scaling to obtain critical exponents is not the purpose of the present work. In general, we discard several millions of MC steps per spin to equilibrate the system before averaging physical quantities over several millions of MC steps. The averaged energy and the specific heat are defined by
where indicates the thermal average taken over several millions of microscopic states at .
We define the order parameter for the -state Potts model by
where is the spatial average defined by
being the value attributed to denote the axis of the spin at the site . The susceptibility is defined by
We did not use the theory of finite-size scaling(22); (23); (24) because the calculation of critical exponents is not the purpose of the present work. However, in order to appreciate finite-size effects, we carried out simulations in the 2D case for sizes from to and in the case of thin films from to . Results for the largest size are not significatively different from those of smaller sizes, excepted for the thickness. We will show therefore in the following results for lateral lattice size for the 2D case, and for thin films with thicknesses and 6. In order to check the first-order nature of a weak first-order transition, the histogram technique is very efficient (23); (24). But in our case as will be seen below, the re-orientation is a very strong first-order transition. The discontinuity of energy and magnetization is clearly seen at the transition. We just use the histogram technique to check the 2D case for a demonstration.
Iii Ground State and Phase Transition
iii.1 Two dimensions
In the case of 2D, for a given , the steepest-descent method gives the ”critical value” of above (below) which the GS is the in-plane (perpendicular) configuration. depends on . Let us take and make vary and in the following. The GS numerically obtained is shown in Fig. 1 for several sets of . For instance, when , we have .
We show in Fig. 2 the energy per site and the specific heat, and in Fig. 3 the order parameter as well as the susceptibility , as functions of in the case of , for and on two sides of . We observe one transition of second order for these values of . Note that the transition for larger is sharper.
It is interesting to examine the region very close to , namely close to the frontier of two different GS. We have seen in the past that many interesting phenomena occur at the boundaries of different phases: we can mention the reentrance phenomenon in frustrated spin systems (19); (20) and the re-orientation transition in the Heisenberg film with a dipolar interaction similar to the present model (16). We have carried out simulation for values close to . We find indeed a transition from the in-plane ordering to the perpendicular one when increases in the region . We show an example at in Fig. 4 where we observe that in the low- phase () the spins align parallel to the axis and in the intermediate- phase () the spins point along the axis perpendicular to the film. The system becomes disordered at . Note that in the disordered phase, each ”state” of the Potts spin (along of one of the three axes) has 1/3 of the total number of spins. This explains why and tend to 1/3 at high in Fig. 4. The transition from the in-plane to the perpendicular configuration is of first order as seen in Fig. 4 by the discontinuity of , , the energy and the magnetization at the transition point. The first-order character has been confirmed by the double-peaked energy histogram at the re-orientation transition temperature as shown in Fig. 5.
We show in Fig. 6 (top) the phase diagram in the space for where the line of re-orientation transition near is a line of first order. Let us discuss about the effect of changing . Increasing will increase the dipolar energy at each site. Therefore, a smaller value of suffices to ”neutralize” the effect of perpendicular anisotropy energy. The critical value of is thus reduced as seen in the phase diagram established with shown in Fig. 6 (bottom) where compared to when (top).
It is interesting to compare the present system using the 3-state Potts model with the same system using the Heisenberg spins (16). In that work, the re-orientation transition line is also of first order but it tilts on the left of , namely the re-orientation transition occurs in a small region below , unlike what we find here for the Potts model. To explain the ”left tilting” of the Heisenberg case, we have used the following entropy argument: the Heisenberg in-plane configuration has a spin-wave entropy larger than that of the perpendicular configuration at finite , so the re-orientation occurs in ”favor” of the in-plane configuration, it goes from perpendicular to in-plane ordering with increasing . Obviously, this argument for the Heisenberg case does not apply to the Potts model because we have here the inverse re-orientation transition. We think that, due to the discrete nature of the Potts spins, spin-waves cannot be excited, so there is no spin-wave entropy as in the Heisenberg case. The perpendicular anisotropy is thus dominant at finite for slightly larger than .
iii.2 Thin films
The case of thin films with a thickness where goes from a few to a dozen atomic layers has a very similar re-orientation transition as that shown above for the 2D case.
Let us show in Fig. 7 the GS obtained by the steepest-descent method with and as before, for two thicknesses and . Changing the film thickness results in changing the dipolar energy at each lattice site. Therefore, the critical value will change accordingly. We note the periodic layered structures at large and for both cases. In the case , for the critical value above which the GS changes from the perpendicular to the in-plane configuration is .
As in the 2D case, we expect interesting behaviors near the critical value . For example, when , and , we find indeed a re-orientation transition which is shown in Fig. 8. The upper curves show clearly a first-order transition from in-plane ordering to perpendicular ordering at . The total magnetization (middle curve) and the energy (bottom curve) show a discontinuity at that temperature. The whole phase diagram is shown in Fig. 9. Note that the line separating the uniform in-plane phase (II) and the periodic single-layered phase (1) is vertical.
To close this subsection, let us show in Fig. 10 the transition at values of far from the critical values of . There is only one transition from the ordered phase to the paramagnetic phase. As seen the transition from the in-plane ordering [phases (II) and (1)] to the paramagnetic phase is sharper than that from the perpendicular one [phase (I)], as in the 2D case.
iii.3 Effect of surface exchange interaction
We have calculated the effect of by taking its values far from the bulk value () for several values of . In general, when is smaller than the surface spins become disordered at a temperature below the temperature where the interior layers become disordered. This case corresponds to the soft surface (or magnetically ”dead” surface layer) (7). On the other hand, when , we have the inverse situation: the interior spins become disordered at a temperature lower that of the surface disordering. We have here the case of a magnetically hard surface. We show in Fig. 11 an example of a hard surface in the case where for with . The same feature is observed for . Note that the surface and bulk transitions are seen by the respective peaks in the specific heat and the susceptibility. In the re-orientation region, the situation is very complicated as expected because the surface transition occurs in the re-orientation zone.
Note that for a given , the effect of the cutoff distance is to move the critical value of as seen in Figs. 1 and 7. At one has 146 neighbors for each interior spin (not near the surface). This huge number makes MC simulations CPU-time consuming. We therefore performed simulations at finite only with two values of in the 2D case. As seen in Fig. 6, the change of does not alter our conclusion on the re-orientation transition. We think that the cutoff is more than a technical necessity, it involves also physical reality. We have in mind the observation that in most experimental systems interaction between faraway neighbors can be neglected. The concept that the interaction range between particles can go to infinity is a theoretical concept. Models in statistical physics limited to interaction between NN are known to interpret with success experiments (1); (18). Rarely we have to go farther than third NN. For example, in our recent paper on the spin resistivity in semiconducting MnTe, we took interactions up to third neighbors to get an excellent agreement with experiments (34). Therefore, we wanted to test in the present paper how physical results depend on in the dipolar interaction. If we know for sure that in a thin film the interaction is dipolar and that a double-layered structure for example is observed, from what is found above we can suggest the interaction range between spins in the system. Finally, we note that if we change , the value of will change. The choice of =0.5 which is a half of is a reasonable choice to make the re-orientation happen. A smaller will induce a smaller but again, the physics found above will not change.
Iv Concluding Remarks
We have shown in this paper MC results on the phase transition in thin magnetic films using the Potts model including a short-range exchange interaction and a long-range dipolar interaction of strength , truncated at a distance . We have also included a perpendicular anisotropy which is known to exist in very thin films.
Among the striking results, let us mention the re-orientation transition which occurs in 2D and in thin films at a finite temperature below the overall disordering. This re-orientation is a very strong first-order transition as seen by the discontinuity of the energy and the magnetization. We emphasize that the re-orientation is possible only because we have two competing interactions: the perpendicular anisotropy and the dipolar interaction.
We would like to acknowledge the financial support from a grant of the binational cooperation program POLONIUM of the French and Polish Governments.
- preprint: APS/123-QED
- A. Zangwill, Physics at Surfaces (Cambridge University Press, Cambridge, England, 1988).
- Ultrathin Magnetic Structures, edited by J. A. C. Bland and B. Heinrich (Springer-Verlag, Berlin, 1994), Vols. I and II.
- H. W. Diehl, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, London, 1986), Vol. 10; H. W. Diehl, Int. J. Mod. Phys. B 11, 3503 (1997).
- N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
- H. Puszkarski, Acta Physica Polonica A 38, 217 (1970); ibid. A 38, 899 (1970).
- Diep-The-Hung, J.C.S. Levy and O. Nagai, Phys. Status Solidi (b) 93, 351 (1979).
- Diep-The-Hung, Phys. Status Solidi (b) 103, 809 (1981).
- M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988).
- P. Grunberg, R. Schreiber, Y. Pang, M. B. Brodsky, and H. Sowers, Phys. Rev. Lett. 57, 2442 (1986); G. Binasch, P. Grunberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 (1989).
- A. Barthélémy et al., J. Magn. Magn. Mater. 242-245, 68 (2002).
- See review by E. Y. Tsymbal and D. G. Pettifor, Solid State Physics (Academic Press, San Diego, 2001), Vol. 56, pp. 113 237.
- R. J. Baxter, Exactly Solved Models in Statistical Physics, Academic Press Inc., London (1982).
- E. Bayong, H. T. Diep and V. Dotsenko, Phys. Rev. Lett. 83, 14 (1999).
- S. Reynal and H. T. Diep, Phys. Rev. E 69, 026169 (2004); S. Reynal and H. T. Diep, Phys. Rev. E 72, 056710 (2005).
- H. Puszkarski, M. Krawczyk and H. T. Diep, Surface Science 602, 2197 (2008).
- C. Santamaria and H. T. Diep, J. Magn. Magn. Mater. 212, 23 (2000).
- K. G. Wilson, Phys. Rev. B 4, 3174 (1971).
- J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed., Oxford Univ. Press (2002); D. J. Amit, Field theory, the renormalization group and critical phenomena, World Scientific, Singapor (1984).
- Frustrated Spin Systems, edited by H. T. Diep (World Scientific, Singapore, 2005).
- M. Debauche, H. T. Diep, P. Azaria, and H. Giacomini, Phys. Rev. B 44, 2369 (1991) and references on other exactly solved models cited therein.
- D. P. Laudau and K. Binder, in Monte Carlo Simulation in Statistical Physics, Ed. K. Binder and D. W. Heermann, Springer-Verlag, New York (1988).
- P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49 435 (1977).
- A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988) ; ibid. 63, 1195(1989) .
- A. M. Ferrenberg and D. P. Landau, Phys. Rev. B44, 5081 (1991).
- J. Hoshen and R. Kopelman, Phys. Rev. B14, 3438 (1974).
- U. Wolff, Phys. Rev. Lett. 62, 361 (1989).
- U. Wolff, Phys. Rev. Lett. 60, 1461 (1988).
- F. Wang and D. P. Landau, Phys. Rev. Lett. 86, 2050 (2001); ibid. Phys. Rev. E 64 056101 (2001).
- See V. T. Ngo, H. V. Nguyen, H. T. Diep, and V. L. Nguyen, Phys. Rev. B 69, 134429 (2004) and references on magnetic multilayers cited therein.
- V. T. Ngo and H. T. Diep, Phys. Rev. B 75, 035412 (2007) and references on surface effects cited therein.
- V. Thanh Ngo and H. T. Diep, J. Phys.: Cond. Mat. 19, 386202 (2007).
- X. T. Pham Phu, V. Thanh Ngo, and H. T. Diep, Surface Science 603, 109 (2009).
- X. T. Pham Phu, V. Thanh Ngo, and H. T. Diep, Phys. Rev. E 79, 061106 (2009).
- Y. Magnin and H. T. Diep, Phys. Rev. B 85, 184413 (2012).