Polarised electromagnetic wave 3 propagation through the 1 0 2 ferromagnet: Phase boundary n a of dynamic phase transition J 4 1 Muktish Acharyya ] Department of Physics, Presidency University h 86/1 College Street, Calcutta-700073, India c e E-mail:[email protected] m - t a Abstract: The dynamical responses of ferromagnet to the propagating elec- t s tromagnetic field wave passing through it are modelled and studied here by . at Monte Carlo simulation in two dimensional Ising ferromagnet. Here, the m electromagnetic wave is linearly polarised in such a way that the direction - of magnetic field is parallel to that of the magnetic momemts (spins). The d n coherent propagating mode of spin-clusters is observed. The time average o magnetisation over the full cycle (time) of the field defines the order param- c [ eter of the dynamic transition. Depending on the value of the temperature 1 andtheamplitude ofthepropagatingmagnetic fieldwave, a dynamical phase v transition is observed. The dynamic transition was detected by studying the 1 temperaturedependencesofthedynamicorderparameter, thevarianceofthe 7 0 dynamic order parameter, the derivative of the dynamic order parameter and 3 the dynamic specific heat. The phase boundaries of the dynamic transitions . 1 were drawn for two different values of the wave lengths of the propagating 0 3 magneticfieldwave. Thephaseboundarywasobservedtoshrink(inward)for 1 lower speed of propagation of the EM wave. The divergence of the releavant : v length scale was observed at the transition point. i X r a PACS Nos:05.50+q, Keywords: Ising ferromagnet, Monte Carlo simulation, Propagat- ing wave, Dynamic phase transition 1 I. Introduction. Thedyamical response ofIsing ferromagnettoatimedependent magnetic field has become an active field of research [1]. The hysteretic responses and the nonequilibrium dynamic phase transitions are two major foci of atten- tion. The scaling behaviour [2] of hysteretic loop area with the amplitude, frequency ofthe sinusoidally oscillating magnetic field isthe main outcome of theresearch. Anotherinteresting aspectisthenonequilibriumdynamicphase transition which yields variety of interesting nonequilibrium behaviours and promted the researchers to take continuous attention in this field. Histor- ically, some important observations like (i) divergences of dynamic specific heat and relaxation time near the transition point [3], (ii) divergence of the scale of length near the transition point [4], (iii) studies regarding the exis- tence of tricritical point [5, 6], (iv) the relation with the stochastic resonance [5] and the hysteretic loss [7], enriched thefield andestablished thetransition asathermodynamic phasetransition. Veryrecently, asurfacedynamic phase transition [8] is observed in kinetic Ising ferromagnet driven by oscillating magnetic field. The dynamic phase transition was studied experimentally[9] also, in ultrathin Co film on Cu(100) system by surface magneto-optic Kerr effect. This dynamic phase transition is also observed in other magnetic mod- els. The dynamic phase transition was observed [10] in anisotropic clas- sical Heisenberg model and in XY model [11]. The multiple (surface and bulk) dynamic transition was observed [12] in Heisenberg model (with bi- linear exchange anisotropy) and it was found[13] also in Heisenberg ferro- magnet driven by polarised magnetic field. The dynamic transition were observed [14] in kinetic spin-3/2 Blume-Capel model and in Blume-Emery- Griffith model[15] studied by meanfield approximation. The dynamic phase transition was studied by Monte Carlo simulation [16] and by meanfield calculation[17] in Ising metamagnet. However, all thestudied mentioned so far, were done bysinusoidally oscil- latingmagneticfieldwhichwasuniformover thespace(lattice)atanyinstant of time. In those studies, the spatio-temporal variation of applied magnetic field was not considered. One such spatio-temporal variation of applied mag- netic field would be the propagating magnetic field wave. In reality, if the electromagnetic wave passes through the ferromagnet, the spatio-temporally varying magnetic field coupled with the spin, would affect the dynamic na- ture of the system. Here also dynamic transition would be observed. Very 2 recently, itisreportedbriefly[18]thatpropagatingmagneticfieldwave would lead to dynamical phase transition in Ising ferromagnet. A pinned phase and a phase of coherent motionof spin-clusters were observed recently [19] inran- dom field Ising model swept by propagating magnetic field wave. Here the nonequilibrium dynamic phase transition is athermal and tunned by random disorder. A rich dynamical phase boundary was also drawn. Here in this paper, the nonequilibrium dynamic phase transition is stud- ied extensively in two dimensional Ising ferromagnet swept by propagating electromagnetic field wave. The technique employed here is Monte Carlo simulation. The phase boundary of the dynamical transition is drawn in this study. The paper is organised as follows: The model and the simulation technique are discussed in section-II, the numerical results are reported in section-III and the papers ends with a summary, in section-IV. II. Model and Simulation. TheHamiltonian(timedependent)representingthetwodimensionalIsing ferromagnet in presence of a propagating electromagnetic field wave (having spatio-temporal variation) can be written as ′ ′ H(t) = −JΣs(x,y,t)s(x,y ,t)−Σh(x,y,t)s(x,y,t) (1) The s(x,y,t) represents the Ising spin variable (±1) at lattice site (x,y) at time t on a square lattice of linear size L. J(> 0) is the ferromagnetic (taken here as uniform) interaction strength. The first sum for Ising spin-spin inter- action is carried over the nearest neighbours only. The h(x,y,t) is the value of the magnetic field (at point (x,y) and at any time t) of the propagating electromagnetic wave. It may be noted here that the electroagnetic wave is linearly polarised in such a way that the direction of magnetic field is parallel to that of the spins. For a propagating magnetic field wave h(x,y,t) takes the form h(x,y,t) = h0cos(2πft−2πy/λ) (2) The h0, f and λ represent the amplitude, frequency and the wavelength respectively of the propagating electromagnetic field wave which propagates along the y-direction. In the present simulation, a L ×L square lattice (in contact with a heat reservoir of temperature T) is considered. The boundary condition, used here, is periodic in both (x and y) direction. The initial (t = 0) configuration, as half of the total number (selected randomly) of spins are 3 up (s(x,y,t = 0) = +1 for all x and y)), is taken here. This configuration of spins, corresponds to the high temperature disordered (paramagnetic) phase. The spins are updated randomly (a site (x,y) is chosen at random) and the spin flip occurs (at temperature T) according to the Metropolis rate[20] W W(s → −s) = Min[exp(−∆E/kT),1], (3) where ∆E is the change in energy due to spin flip and k is Boltzmann con- 2 stant. L such random updates of spins constitutes the unit time step here and is called Monte Carlo Step (MCS). Here, the value of magnetic field is measured in the unit of J and the temperature is measured in the unit of J/k. The dynamical steady state , at any temperature, is reached by cooling the system slowly (in small step of temperature, ∆T = 0.02) from the high temperature configuration. The frequency of the propagating magnetic field was taken f = 0.01 throughout the study. The total length of the simulation is 2×105 MCS and first 105 MCS transient data ware discarded. The data 5 are taken by averaging over 10 MCS. Since the frequency of the propagating 5 field is f = 0.01, the complete cycle of the field requires 100 MCS. So, in 10 3 MCS, 10 numbers of cycles of the propagating EM wave are present. The time averaged data over the full cycle (100 MCS) of the propagating field are further averaged over 1000 cycles. III. Results. In this study, a square lattice of size L = 100 is considered. The steady state dynamical behaviours of the spins are studied here. The amplitude, frequency and the wavelength of the propagating wave are taken h0 = 0.6, f = 0.01andλ = 25respectively. Themagneticfieldispropagatingalongthe y-direction (vertically upward in the graphs shown here). The temperature of the system is taken T = 1.50. The configuration of the spins at any instant of time t = 100100 MCS is shown in Fig-1(a). Here, it is noted that, the clusters of spins (down) are formed in strips and these strips move coherently as time goes on. The propagation of the spin-strips are clear in Fig-1(b), where the snapshot was taken at later instant t = 100125. The similar study was performed at a lower temperature T = 1.26 (with all other parameters of the propagating field remain same). Here, the spin clusters are observed to form but not in the shape of strips (as observed in the case of higher temperature T = 1.50 , mentioned above). This is shown in Fig-1(c), 4 at any instant t = 100100 MCS. These irregularly shaped spin-clusters are observed to propagate (along the direction of propagating magnetic field), which is clear from Fig-1(d) (for t = 100125 MCS). To show the propagations of these spin-clusters, the instantaneous line magnetisation m(y,t) = R s(x,y,t)dx/L was plotted against y at any par- ticular instant t = 100100 MCS. This is shown in Fig-2(a) (compare with Fig-1(a)). The periodic variation of m(y,t) along y-direction is found. This was observed to propagate (see Fig-2(b) and compare with Fig-1(b) when shown at later time t = 100125 MCS. It may be noted here, that the line magnetisation is periodic (with y) at any instant of time t. This is also peri- odicintimetatanypositiony. Inboththecases, theoscillationissymmetric about zero (for higher temperature T = 1.50). Here, the time average mag- netisation over a full cycle of the propagating field is Q = ω R R m(y,t)dydt, 2πL becomes zero (due to symmetric oscillation about zero). This is dynamically symmetric disordered (Q = 0) phase. Now, for lower temperature T = 1.26, the spatio-temporal periodicity is lost. The symmetric-oscillation (about zero) is lost here. As a consequence, the time average magnetisation over a full cycle of the propagating field, be- comes nonzero (Q 6= 0). These are shown in Fig-2(c) and Fig-2(d) (compare with Fig-1(c) and Fig-1(d) respectively). But the spin-clusters propagates, in this case. So, as the temperature decreases, Q becomes nonzero (lower temperature) from a zero value (higher temperature). This is dynamically symmetry-broken disordered (Q 6= 0) phase. The temperature variations of the dynamic order parameter Q, its stan- darddeviation< (δQ)2 >arestudied. ThedynamicenergyisE = ω H H(t)dt 2π and the dynamic specific heat is C = dE. The derivatives are calculated nu- dT merically by using the three points central difference formula[21] All these quantities are calculated statistically (averaged) over 1000 different values. The temperature variations of Q, dQ, < (δQ)2 > and C are studied fow two dT different values of the amplitude of the propagating magnetic field and are shown in Fig-3. As the temperature decreases Q starts to grow from zero and near the transition point Td (depending upon the value of h0 and λ) it 2 becomes nonzero. Near the transition temperatures the < (δQ) > and C show shap peaks and dQ show a sharp dips. From the figure it is also evi- dT dent that the transition occurs at lower temperature (T ) for higher values d of the field amplitude (h0). In this case, for λ = 25, the transitions occur at Td = 1.90 and Td = 1.28 for h0 = 0.3 and h0 = 0.6 respectively. These values 5 of the transition temperatures are obtained from the position of sharp dip of the dQ and corresponding sharp peaks of < (δQ)2 > and C shown in Fig-3. dT Collecting all values of the transition temperatures (depending on the values of h0), the dynamical phase boundary is obtained. This dynamic transition temperature (T ) was observed to depend on the d wave length (λ) of the propagating magnetic field wave. The temperature dependences of Q, dQ, < (δQ)2 > and C are studied and shown in Fig-4, dT for two different values of λ (= 25 and 50). From the figure, it is clear that, transition occurs at higher temperature for higher value of the wavelength (λ). To be precise, for h0 = 0.3, the transitions occur at Td = 1.88 and T = 1.94 for λ = 25 and λ = 50 respectively. Here also, the values of the d transition temperatures are obtained from the position of sharp dip of the dQ and corresponding sharp peaks of < (δQ)2 > and C (shown in Fig-4). So, dT the dynamical phase boundary should shift depending on the value of λ. In Fig-5, the dynamical phase boundaries are drawn for two different valuesofλ(=25and50),intheplaneformedbyTd andh0. Itisobservedthat the boundary shrinks inward as the wavelength of the propagating magnetic field decreases. Since, the speed of propagating EM wave is c = fλ, for fixed f(= 0.01 here), the phase boundary essentially shrinks inward (low teperature) for lower speed of propagation. The dynamic phase transition mentioned above is associated with the 2 2 divergences of releavant length scale. For this reason the L < (δQ ) > is studied as a function of temperature T. Is is found that the peak of 2 2 L < (δQ) > (observed at T ) increases as L increases. This is shown in d Fig-6. This result is quite conclusive to say that there exists the diverging length scale associated with the dynamic phase transition. It may be noted here, that this method was successfully employed [4] to show the diverging length scale associated to the dynamic transition in Ising ferromagnet driven by oscillating (but not propagating) magnetic field. IV. Summary. The dynamical responses of a ferromagnet to a polarised electromagnetic wave is modelled and studied here by Monte Carlo simulation in two di- mensional Ising ferromagnet. In the steady state, the coherent motion (in propagating mode) of spin clusters was observed in dynamically symmetric disordered phase. The time average magnetisation over the full cycle of the 6 propagating EM wave is a measure of a order parameter in the dynamic phase transition observed here. The dynamic phase transition observed here is of continuous type and found to be dependent on the amplitude and the wave length ofthe propagatingpolarised EM wave. Hence, a phase boundary (transition temperature as a function of the amplitude) is drawn for two dif- ferent values of the wavelengths of EM wave. The phase boundary was found to shrink (towards the lower temperature) for lower speed of propagation of EM wave. Another important observation is the divergence of releavant length scale near the dynamic transition. This observation supports the dynamic transi- tionto become a member of thewell known equilibrium critical phenomenon. Acknowledgments: The library facilities provided by the University of Calcutta is gratefully acknowledged. 7 References 1. B. K. Chakrabarti and M. Acharyya, Rev. Mod. Phys., 71, (1999) 847; M. Acharyya, Int. J. Mod. Phys. C, 16 (2005) 1631 2. M.AcharyyaandB.K.Chakrabarti, Phys. Rev. B,52(1995)6550; See also, M. Acharyya and B. K. Chakrabarti, Ising system in oscillat- ing field: Hysteretic response, in Annual reviews of computational physics, Ed. D. Stauffer, (World Scientific, Singapore), Vol. 1 (1994) 107 3. M. Acharyya, Phys. Rev. E, 56 (1997) 2407 4. S. W. Sides et al., Phys. Rev. Lett., 81 (1998) 834 5. M. Acharyya, Phys. Rev. E, 59 (1999) 218 6. G. Korniss, P.A. Rikvold, M.A. Novotny, Phys. Rev. E66, 056127 (2002), 7. M. Acharyya, Phys. Rev. E, 58 (1998) 179 8. H. Park and M. Pleimling, Phys Rev Lett 109 (2012) 175703 9. Q. Jiang, H. N. Yang, G. C. Wang, Phys Rev B 52 (1995) 14911; J. Appl. Phys. 79 (1996) 5122 10. M. Acharyya, Int. J. Mod. Phys. C, 14 (2003) 49 11. H. Jung, M. J. Grimson, C. K. Hall, Phys Rev B 67 (2003) 094411 12. H. Jung, M. J. Grimson, C. K. Hall, Phys Rev E 68 (2003) 046115 13. M. Acharyya, Phys. Rev. E., 69 (2004) 027105 14. M. Keskin, O. Canko, B. Deviren, Phys. Rev. E 74 (2006) 011110 15. U. Temizer, E. Kantar, M. Keskin, O.Canko, J. Magn. Magn. Mater. 320 (2008) 1787 16. M. Acharyya, J. Magn. Magn. Mater., 323 (2011) 2872 8 17. M. Keskin, O. Canko, M. Kirak, Phys. Stat. Solidi B 244 (2007) 3775; B. Deviren, M. Keskin, Phys. Lett. A 374 (2010) 3119 18. M. Acharyya, Physica Scripta, 84 (2011) 035009 19. M. Acharyya, J. Magn. Magn. Mater., cond-mat arXiv:1209.2841 (in press) (2013) 20. K. Binder and D. W. Heermann, 1997, Monte Carlo Simulation in Statistical Physics (Springer Series in Solid State Sciences) (New York: Springer) 21. C. F. Gerald and P. O. Weatley, 2006, Applied Numerical Analysis (Reading, MA: Addison-Wesley); J. B. Scarborough, 1930, Numerical Mathematical Analysis (Oxford: IBH) 9 100 100 80 80 (a)60 (b)60 40 40 20 20 0 0 0 20 40 60 80 100 0 20 40 60 80 100 100 100 80 80 (c)60 (d)60 40 40 20 20 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Fig-1. The motion of spin-clusters of down spins (shown by black dots), swept by propagating magnetic field wave, for different values of (a) Time = 100100 MCS, T=1.5 and h0 = 0.6 (b) Time = 100125 MCS, T = 1.5 and h0 = 0.6 (c) Time = 100100 MCS, T = 1.26 and h0 = 0.6 (d) Time = 100125 MCS, T = 1.26 and h0 = 0.6. 10