Ferroelectrics Characterization and Modeling Part 14 doc

Temperature Dependence of the Dielectric Constant Calculated Using a Mean Field Model Close to the Smectic A - Isotropic Liquid Transition 445 80.0 80.5 81.0 81.5 82.0 82.5 83.0 4.80 4.85 4.90 4.95 5.00 5.05 5.10 5.15 ε T ( o C) Calculated Observed FIGURE 4 Fig. 4. Values calculated from Eqs. (20) and (22) for the smectic A and the isotropic liquid phases, respectively, for the racemic A7. The ψ values used in Eq.(20) were calculated from the mean field theory (Eq. 19). Calculated values were obtained from the fitting Eq.(20) (SmA) and Eq.(22) (I) to the experimental data (Bahr et al., 1987) for the ε ⊥ . The observed data (Bahr et al., 1987) is also plotted here (T c =82 o C). 4. Discussion The temperature dependence of the dielectric constant ε ⊥ was calculated here using our mean field model with the biquadratic coupling P 2 ψ 2 , as given by Eq.(1). For this calculation of ε ⊥ or the inverse dielectric susceptibility 1 χ − (Eq.20), the orientational order parameter ψ was first calculated as a function of temperature from the mean field theory (Eq.19), as plotted in Figs.1 and 2 for the 50% optically active compound and the racemic 4-(3-methyl-2- chlorobutanoyloxy)- 4 ′ -heptyloxybiphenyl, respectively. The temperature dependence of the spontaneous polarization P can also be calculated from Eq.(6) which gives similar critical behaviour as the orientational order parameter ψ. It decreases smoothly with increasing temperature in the SmA phase as the smectic A- isotropic liquid (SmA-I) transition temperature (Table 1) is approached for the 50% mixture and the racemic A7. By using the temperature dependence of the order parameter for the 50% optically active (Fig.1) and the racemic A7 (Fig.2), the dielectric constant ε ⊥ was calculated in these compounds, as plotted in Figs. 3 and 4, respectively. This calculation was carried out for the smectic A phase ( 0 ψ ≠ ) and for the isotropic liquid ( 0 ψ = ) according to Eqs. (20) and (22), respectively, as stated above. Eqs. (20) and (22) were both fitted to the experimental data for the smectic A and the isotropic liquid phases with the coefficients given in Tables 1 and 2. We also calculated the dielectric constant ε ⊥ of the 50% mixture in the isotropic liquid by using the same values of a and the 1 C TT χ − = extracted for the smectic A phase (Table 1). Those calculated ε ⊥ values are also in good agreement with the experimental data in the isotropic phase of the 50% mixture, as plotted in Fig.3. As seen from Fig.3, variation of the dielectric constant ε ⊥ is continuous with the temperature so that the ε ⊥ decreases continuously as the Ferroelectrics - Characterization and Modeling 446 temperature increases from the smectic A to the isotropic liquid phase for the 50% optically active A7. This continuous change in the dielectric constant ε ⊥ indicates a second order transition between the smectic A and the isotropic liquid of the 50% mixture. In regard to the variation of the ε ⊥ with the temperature for the racemic A7 (Fig.4),the smectic A-isotropic liquid transition in this compound is more likely closer to the first order transition. In the smectic A phase, the dielectric constant ε ⊥ varies slightly within the temperature interval of nearly 2K (from about 80.5 to 82K) and then there occurs a kink just above 82K (at around 82.25K) prior to the isotropic liquid (Fig. 4). When Eqs.(20) and (22) were fitted to the experimental data (Bahr et al., 1987) for the smectic A and the isotropic liquid phases, respectively, this kink that occurs in the racemic A7 was not studied in particular, which might be the pretransitional effect. As shown in Fig.4, Eqs. (20) and (22) are adequate to describe the observed behaviour of the smectic A and the isotropic liquid phases of the racemic A7, respectively. Above T c in the isotropic liquid phase of the racemic A7, the dielectric constant ε ⊥ exhibits closely a discontinuous behaviour. It drops more rapidly in a small temperature interval, as shown in Fig. 4. In fact, this first order behaviour is supported by a very large value of ≈-11 or -12 for the ratio 2 /cb α ′′ ′ extracted (Eq.14) by using the coefficients (Table 1 and 2) for the racemic A7 in comparison with the value of 1.9 for the 50% mixture. The first order character of the smectic A- isotropic liquid transition can also be seen from the temperature difference ( 0 c TT TΔ= − ) which is nearly 6K for the racemic A7 compared to the value of 2.5 K for the 50% mixture (Table 2). Another comparison for the first order (racemic A7) and the second order (50% mixture) character of the smectic A- isotropic liquid transition in both compounds can be made in terms of the slope ratio of the SmA/I, as given in Table 3. Again, a very large value of ≈14 for the racemic A7 also indicates a first order SmA-I transition in this compound in comparison to the value of 1 /2  for the 50% mixture which can be considered to exhibit a second order SmA-I transition within the temperature intervals studied. This slope ratio can be used as a criterion to describe a first or second order transition exhibited by the ferroelectric liquid crystals, which was used in particular for the smectic A-smectic C* (AC*) phase transition in A7 (Bahr et al., 1987), and also in general for the ferroelectrics and related materials (Lines & Glass, 1979). The dielectric constant ε ⊥ was calculated using the temperature dependence of the orientational order parameter ψ (Eq.19) from the mean field theory in Eq.(18), as stated above. According to the minimization condition, by taking the derivative of the free energy g (Eq.10) with respect to the ψ ( /0g ψ ∂∂= ), a quadratic equation obtained in ψ 2 can be solved, which also gives a similar functional form of the temperature dependence of ψ with the critical exponent of 1/2 β = from the mean field theory, as given by Eq.(19). This quadratic solution in ψ 2 can also be used to calculate the temperature dependence of the spontaneous polarization P (Eq.6) and of the dielectric constant ε ⊥ (or 1 χ − ) (Eq.18). The calculated ε ⊥ can then be compared with the experimental data (Bahr et al., 1987) for the SmA-I transition of 4-(3-methyl-2-chlorobutanoyloxy)- 4 ′ -heptyloxybiphenyl below T c . As shown in Figs. (3) and (4), the observed behaviour of the dielectric constant ε ⊥ is described satisfactorily by our mean field model with the P 2 θ 2 coupling which considers quadrupolar interactions in the 50% mixture and the racemic A7. Our results for the dielectric constant ε ⊥ (Figs. 3 and 4) indicate that the quadrupolar interaction ( 22 P ψ coupling) is the dominant mechanism for the first order (or a weak first order) transition in the racemic A7 and the second order (or close to a second order) transition in the 50% mixture. Temperature Dependence of the Dielectric Constant Calculated Using a Mean Field Model Close to the Smectic A - Isotropic Liquid Transition 447 5. Conclusions The dielectric constant ε ⊥ of the ferroelectric 50% optically active and the racemic compounds of 4-(3-methyl-2-chlorobutanoyloxy)- 4 ′ -heptyloxybiphenyl was calculated as a function of temperature for the smectic A-isotropic liquid (SmA-I) transition. A mean field model with the biquadratic coupling P 2 ψ 2 between the spontaneous polarization P and the orientational order parameter ψ of the smectic A (SmA) phase was used to calculate ε ⊥ through the temperature dependence of the order parameter ψ. Our mean field model describes adequately the observed behaviour of ε ⊥ for this liquid crystal with high spontaneous polarization close to the smectic A –isotropic liquid transition. It is indicated here that the 50% mixture exhibits a second order (or close to a second order) and that the racemic A7 exhibits a first order (or a weak first order) smectic A- isotropic liquid transition. 6. References Bahr Ch. & Heppke, G. (1986). Ferroelectric liquid-crystals with high spontaneous polarization. Mol. Cryst. Liq. Cryst. Lett., Vol. 4, No.2, pp. 31-37 . Bahr, Ch., Heppke G. & Sharma, N.K. (1987). Dielectric studies of the smectic-C-]-smectic A transition of a ferroelectric liquid-crystal with high spontaneous polarization. Ferroelectrics, Vol. 76, No. 1-2, pp. 151-157. Bahr, Ch., Heppke, G. & Sabaschus, B. (1988). Chiral-racemic phase-diagram of a ferroelectric liquid-crystal with high spontaneous polarization. Ferroelectrics, Vol. 84, pp. 103-118. Bahr, Ch. & Heppke, (1990). Influence of electric-field on a 1st-order smectic-A ferroelectric- smectic-C liquid-crystal phase-transition - a field-induced critical-point. G. Phys. Rev. A , Vol. 41, No. 8, pp. 4335-4342. Bahr, Ch. & Heppke, G. (1991). Critical exponents of the electric-field-induced smectic- alpha ferroelectric-smectic-C liquid-crystal critical-point. Phys. Rev. A, Vol. 44, No. 6, pp 3669-3672. Benguigui, L. (1984). Dielectric-properties and dipole ordering in liquid-crystals. Ferroelectrics, Vol. 58, No. 1-4, pp. 269-281. Blinc, R. (1992). Models for phase transitions in ferroelectric liquid crystals: Theory and experimental results , In: Phase Transitions in Liquid Crystals, Martellucci, S. & Chester , A.N.,. Plenum Press, New York Carlsson, T. & Dahl, I. (1983). Dependence of the tilt angle on external forces for smectic-C and chiral smectic-C liquid-crystals - measurement of the heat-capacity of DOBAMBC. Mol. Cryst. Liq. Cryst., Vol. 95, No. 3-4, pp. 373-400. Carlsson, T., Zeks, B., Levstik, A., Filipic, C., Levstik, I. & Blinc, R. (1987). Generalized landau model of ferroelectric liquid-crystals. Phys. Rev. A, Vol. 36, No. 3, pp. 1484-1487. de Gennes, P. G. (1973) The physics of liquid crystals. Clarendon Press. ISBN: 0 19 851285 6, Oxford Denolf, K., Van Roie, B., Pitsi, G. & Thoen, (2006). Investigation of the smectic-A-smectic-C* transition in liquid crystals by adiabatic scanning calorimetry. J. Mol. Cryst. Liq. Cryst. , Vol. 449, pp. 47-55. Dumrongrattana, S. & Huang, C.C. (1986). Polarization and tilt-angle measurements near the smectic-A-chiral-smectic-C transition of p-(n-decyloxybenzylidene)-p-amino-(2- methyl-butyl)cinnamate (DOBAMBC). Phys. Rev. Lett., Vol. 56, No. 5, pp. 464-467. Ema, K., Yao, H., Fukuda, A., Takanishi, Y. & Takezoe, H. (1996). Non-Landau critical behavior of heat capacity at the smectic-A-smectic-C-alpha(*) transition of the Ferroelectrics - Characterization and Modeling 448 antiferroelectric liquid crystal methylheptyloxycarbonylphenyl octyloxycarbonylbiphenyl carboxylate. Phys. Rev. E, Vol. 54, No. 4, pp. 4450-4453. Garland, C.W. & Nounesis, G. (1994). Critical-behavior at nematic smectic-A phase- transitions. Phys. Rev. E, Vol. 49, No. 4, pp. 2964 -2971. Indenbom, V. L., Pikin, S. A. & Loginov, E. B. (1976). Phase-transitions and ferroelectric structures in liquid-crystals. Kristallografiya, Vol. 21, No. 6, pp. 1093-1100. Johnson, P.M., Olson, D.A., Pankratz, S., Bahr, Ch., Goodby, J.W. & Huang, C.C. (2000). Ellipsometric studies of synclinic and anticlinic arrangements in liquid crystal film. Phys. Rev. E, Vol. 62, No. 6, pp. 8106-8113. Kilit, E. & Yurtseven, H. (2008). Calculation of the dielectric constant as a function of temperature near the smectic AC* phase transition in ferroelectric liquid crystals. Ferroelectrics, Vol. 365, pp.130-138. Lines M. E. & Glass, A. M. (1979). Principles and Applications on Ferroelectrics and Related Materials , Oxford University Press, pp. 71-81, Oxford Matsushita, M. (1976). Anomalous temperature dependence of the frequency and damping constant of phonons near T λ in ammonium halides. J. Chem. Phys., Vol. 65, p. 23. Mercuri, F., Marinelli, M., Zammit, U., Huang, C.C. & Finotello, D. (2003). Critical behavior of thermal parameters at the smectic-A-hexatic-B and smectic-A-smectic-C phase transitions in liquid crystals. Phys. Rev. E, Vol. 68, No. 5, Article Number 051705. Mukherjee, P. K. (2009). Tricritical behavior of the smectic-A to smectic-C-* transition. J. Chem. Phys., Vol. 131, No. 7, Article Number 074902. Musevic, I., Zeks, B., Blinc, R., Rasing, Th. & Wyder, P. (1983). Dielectric study of the modulated smectic-C star uniform smectic-C transition in a magnetic-field. Phys. Stat. Sol. (b), Vol.119, No. 2, pp. 727-733. Safinya, C.R., Kaplan, M., Als-Nielsen, J., Birgeneau, R. J., Davidov, D., Litster, J. D., Johnson, D. L. & Neubert, M. (1980). High-resolution x-ray study of a smectic-A- smectic-C phase-transition. Phys. Rev. B, Vol. 21 No. 9, pp. 4149-4153. Salihoğlu, S., Yurtseven, H. & Bumin, B. (1998). Concentration dependence of polarization for the AC* phase transition in a binary mixture of liquid crystals. Int J. Mod. Phys. B, Vol. 12, No. 20, pp. 2083-2090. Salihoğlu, S., Yurtseven, H., Giz, A., Kayışoğlu, D. & Konu, A. (1998). The mean field model with P-2 theta(2) coupling for the smectic A-smectic C* phase transition in liquid crystals. Phase Trans., Vol. 66, No. 1-4, pp. 259-270. Yurtseven, H. & Kilit, E. (2008). Temperature dependence of the polarization and tilt angle under an electric field close to the smectic AC* phase transition in a ferroelectric liquid crystal. Ferroelectrics, Vol. 365, pp. 122-129. Yurtseven, H. & Kurt, M. (in press) (2011). Tilt angle and the temperature shifts calculated as a function of concentration for the AC* phase transition in a binary mixture of liquid crystals, Int. J. Mod. Phys. B Zeks, B. (1984). Landau free-energy expansion for chiral ferroelectric smectic liquid-crystals. Mol. Cryst. Liq. Cryst., Vol. 114, No. 1-3, pp. 259-270. 0 Mesoscopic Modeling of Ferroelectric and Multiferroic Systems Thomas Bose and Steffen Trimper Martin-Luther-University Halle, Institute of Physics Germany 1. Introduction Motivated by the progress of a multi-scale approach in magnetic materials the dynamics of the Ising model in a transverse field introduced by de Gennes (1963) as a basic model for a ferroelectric order-disorder phase transition is reformulated in terms of a mesoscopic model and inherent microscopic parameters. The statical and dynamical behavior of the Ising model in a transverse field is considered as classical field theory with fields obeying Poisson bracket relations. The related classical Hamiltonian is formulated in such a manner that the quantum equations of motion are reproduced. In contrast to the isotropic magnetic system, see Tserkovnyak et al. (2005), the ferroelectric one reveals no rotational invariance in the spin space and consequently, the driving field becomes anisotropic. A further conclusion is that the resulting excitation spectrum is characterized by a soft-mode behavior, studied by Blinc & Zeks (1974) instead of a Goldstone mode which appears when a continuous symmetry is broken, compare Tserkovnyak et al. (2005). Otherwise the underlying spin operators are characterized by a Lie algebra where the total antisymmetric tensor plays the role of the structural constants. Using symmetry arguments of the underlying spin fields and expanding the driving field in terms of spin operators and including terms which break the time reversal symmetry we are able to derive a generalized evolution equation for the moments. This equation is similar to the Landau Lifshitz equation suggested by Landau & Lifshitz (1935) with Gilbert damping, see Gilbert (2004). Alternatively, such dissipative effects can be included also in the Lagrangian written in terms of the spin moments and bath variables Bose & Trimper (2011). Due to the time reversal symmetry breaking coupling the resulting equation includes under these circumstances a dissipative equation of motion relevant for ferroelectric material. The deterministic equation is extended by stochastic fields analyzed by Trimper et al. (2007). The averaged time dependent polarization offers three modes below the phase transition temperature. The two transverse excitation energies are complex, where the real part corresponds to a propagating soft mode and the imaginary part is interpreted as the wave vector and temperature dependent damping. Further there exists a longitudinal diffusive mode. All modes are influenced by the noise strength. The solution offers scaling properties below and above the phase transition. The results are preferable and applicable for ferroelectric order-disorder systems. A further extension of the approach is achieved by a symmetry allowed coupling of the polarization to the magnetization. The coupling is related to a combined space-time symmetry due to the fact that the magnetization is an axial vector with  m(  x, −t)=−  m(  x, t) whereas 23 2 Will-be-set-by-IN-TECH the polarization is represented by a polar vector  p(−  x, t)=−  p(  x, t). Multiferroic materials are characterized by breaking the combined space-time symmetry. Possible couplings are considered. Introducing a representation of spin fields without fixed axis one can incorporate spiral structures. Different to the previous system the ground state is in that case an inhomogeneous one. The resulting spectrum is characterized by the conventional wave vector q and a special vector Q characterizing the spiral structure. Our studies can be grouped into the long-standing effort in understanding phase transitions in ferroelectric and related materials, for a comprehensive review see Lines & Glass (2004). To model such systems the well accepted discrimination in ferroelectricity of order-disorder and displacive type is useful as discussed by Cano & Levanyuk (2004). Both cases are characterized by a local double-well potential the depth of which is assumed to be V 0 . Furthermore, the coupling between atoms or molecules at neighboring positions is denoted by J 0 . The displacive limit is identified by the condition V 0  J 0 , i.e. the atoms are not forced to occupy one of the minimum. Instead of that the atoms or molecular groups perform vibrations around the minimum. The double-well structure becomes more important when the system is cooled down. The particle spend more time in one of the minimum. Below the critical temperature T c the displacement of all atoms tends preferentially into the same direction giving rise to elementary dipole moments, the average of which is the polarization. The opposite limit V 0  J 0 means the occurrence of high barriers between the double-well structure, i.e. the particles will reside preferentially in one of the minimum. Above the critical temperature the atoms will randomly occupy the minimum whereas in the low temperature phase T < T c one of the minimum is selected. The situation is sketched in Fig. 1. Following de Gennes (1963) the double-well structure can be modeled by a conventional Ising model where the eigenvalues of the pseudo-spin operator S z specify the minima of the double-well potential. The dynamics of the system is described by the kinetic energy of the particles which leads to an operator S x . Due to de Gennes (1963) and Blinc & Zeks (1972; 1974) the situation is described by the model Hamiltonian (TIM) H = − 1 2 ∑ <ij> J ij S z i S z j − ∑ i ΩS x i , (1) where S x and S z are components of spin- 1 2 operators. Notice that these operators have no relation to the spins of the material such as KH 2 P0 4 (KDP). Therefore they are denoted as pseudo-spin operators which are introduced to map the order-disorder limit onto a tractable Hamiltonian. The coupling strength between nearest neighbors J ij is assumed to be positive and is restricted to nearest neighbor interactions denoted by the symbol < ij >. The transverse field is likewise supposed as positive Ω > 0. Alternatively the transverse field can be interpreted as tunneling frequency. In natural units the time for the tunneling between both local minimums is τ t = Ω −1 whereas the transport time between different lattice sites is given by τ i =(¯hJ) −1 . The high temperature limit is determined by τ t < τ i or Ω > ¯hJ. The tunnel frequency is high and the behavior of the system is dominated by tunneling processes. With other words, the kinetic energy is large which prevents the localization of the particles within a certain minimum. The low temperature limit is characterized by a long or a slow tunneling frequency or a long tunneling time τ t > τ i . The behavior is dominated by the coupling strength J. Since already the mean-field theory of the model Eq. (1), see Stinchcombe (1973) and also Blinc & Zeks (1972), yields a qualitative agreement with experimental data, the model was increasingly considered as one of the basic models for ferroelectricity of order-disorder type 450 Ferroelectrics - Characterization and Modeling Mesoscopic Modeling of Ferroelectric and Multiferroic Systems 3 i j J ij Ω Ω V 0 V 0 Fig. 1. Schematical representation of the physical situation in ferroelectric material. J ij is the interaction between the atoms in the double-well potential at lattice site i and j, Ω is the tunneling frequency and V 0 the height of the barrier. as analyzed by Lines & Glass (2004); Strukov & Levnyuk (1998). Whereas the displacive type of ferroelectricity offers a mainly phonon-like dynamics, a relaxation dynamics is attributed to the order-disorder type by Cano & Levanyuk (2004). The Ising model in a transverse field allows several applications in solid state physics. Thus a magnetic system with a singlet crystal field ground state discussed by Wang & Cooper (1968) is described by Eq. (1), where Ω plays the role of the crystal field. The model had been extensively studied with different methods by Elliot & Wood (1971); Gaunt & Domb (1970); Pfeuty & Elliot (1971), especially a Green’s function technique was used by Stinchcombe (1971). It offers a finite excitation energy and a phase transition. A more refined study using special decoupling procedures for the Green’s function investigated by Kühnel et al. (1977) allows also to calculate the damping of the transverse and longitudinal excitations as demonstrated by Wesselinowa (1984). Very recently Michael et al. (2006) have applied successfully the TIM to get the polarization and the hysteresis of ferroelectric nanoparticles and also the excitation and damping of such nanoparticles, compare Michael et al. (2007) and also the review article by Wesselinowa et al. (2010). Despite the great progress in explaining ferroelectric properties based on the microscopic model defined by Eq. (1), the ferroelectric behavior should be also discussed using classical models. Especially, the progress achieved in magnetic systems, see Landau et al. (1980) and for a recent review Tserkovnyak et al. (2005), has encouraged us to analyze the TIM in its classical version capturing all the inherent quantum properties of the spin operators. The classical spin is introduced formally by replacing  S →  S/(¯hS(S + 1)) in the limit ¯h → 0 451 Mesoscopic Modeling of Ferroelectric and Multiferroic Systems 4 Will-be-set-by-IN-TECH and S → ∞. Stimulated by the recent progress in studying ferromagnets reviewed by Tserkovnyak et al. (2005), we are interested in damping effects, too. In the magnetic case the classical magnetic moments obey the Landau-Lifshitz equation, see Landau et al. (1980). It describes the precession of spins around a self-organized internal field, which can be traced back to the interaction of the spins. The reversible evolution equation can be extended by introducing dissipation which is phenomenologically proposed by Landau & Lifshitz (1935) or alternatively the so called Gilbert-damping is introduced by Gilbert (2004). Usadel (2006) has studied the temperature-dependent dynamical behavior of ferromagnetic nanoparticles within a classical spin model, while a nonlinear magnetization in ferromagnetic nanowires with spin current is discussed by He & Liu (2005). Even the magnetization of nanoparticles in a rotating magnetic field is analyzed by Denisov et al. (2006) based on the Landau-Lifshitz equation. The dynamics of a domain-wall driven by a mesoscopic current is inspected by Ohe & Kramer (2006) as well as the thermally assisted current-driven domain-wall was considered recently by Duine et al. (2007). In the present chapter we follow the line offered by magnetic materials to extend the analysis to ferroelectricity accordingly. The main difference as already mentioned above is that in ferroelectric system the internal field is an anisotropic one and therefore, both the reversible precession and the irreversible damping are changed. 2. Model In this section the model and the basic equation will be discussed. Especially the differences between isotropic magnets and anisotropic ferrolectrics are analyzed. 2.1 Hamiltonian In this section we propose a classical Hamiltonian which is dynamical equivalent to the quantum case introduced in Eq. (1). The Hamiltonian is constructed in such a manner that it leads to the same evolution equations for the spins. The most general form is given by H =  d d x  −Ω μ S μ + 1 2 J μνκδ ∂S μ ∂ κ ∂S ν ∂ δ + 1 2 Γ μν S μ S ν  . (2) Here summation over repeated indices is assumed. If the system is symmetric in spin space the coupling tensor J is diagonal in the spin indices J μνκδ = δ μν ˜ J κδ In case that spin and configuration space are independent one concludes the separation J μνκδ = ˆ J μν ˜ J κδ . The anisotropic TIM is obtained by assuming  Ω =(0, 0, Ω), ˆ J μν = Jδ μz δ νz , ˜ J κδ = δ κδ , and Γ μν = Jzδ μz δ νz Here z is the coordination number. The Hamiltonian reads now H f =  d d x  −ΩS x + J 2 (∇S z ) 2 − JzS 2 z  . (3) The last equation represents the TIM on a mesoscopic level, i. e. the spin variables are spatiotemporal fields  S(  x, t). The Hamiltonian Eq.(3) offers no continuous symmetry as the corresponding magnetic one. For that case the magnetic Hamiltonian is written in terms of spin fields  σ(  x, t) as Tserkovnyak et al. (2005) H m = K 2  d d x(∇  σ) 2 . (4) 452 Ferroelectrics - Characterization and Modeling Mesoscopic Modeling of Ferroelectric and Multiferroic Systems 5 Here K designates the exchange coupling. The last Hamiltonian is invariant under spin-rotation. A further difference between the ferroelectric and the magnetic case is the form of the internal field and the underlying dynamics which obeys the mesoscopic equation of motion, compare Hohenberg & Halperin (1977): ∂S α ∂t = {H, S α }. (5) This equation will be discussed in the following subsection. 2.2 Poisson brackets Because the Hamiltonian Eq. (2) is given in terms of classical fields the dynamics of the system has to be formulated using Poisson brackets. They are defined for two arbitrary functionals of an arbitrary field φ by {A(φ), B(φ  )} =  d d xd d x  δA δφ α (x) { φ α (x), φ β (x  )} δB δφ β (x  ) . In case that the arbitrary field φ is realized by the components of spin fields we get due to Mazenko (2003) {A(S), B(S)} =  d d x  αβγ S α (  x) δA δS β (  x) δB δS γ (  x) . (6) Here we have applied the Poisson brackets for angular momentum fields {S α (  x, t), S β (  x  , t)} =  αβγ S γ (  x, t) δ(  x −  x  ) . According to Eq. (6) the spin field satisfies the evolution equation ∂  S ∂t =  B ×  S , (7) where the effective internal field  B is introduced by B α (  x, t)=− δH f δS α (  x, t) . (8) Using the Hamiltonian Eq. (3) the vector of the internal field is given by  B =(Ω,0,[Jz + J∇ 2 ]S z ) . (9) Eqs. (7)-(8) coincide with the quantum mechanical approach based on the Heisenberg equation of motions i¯h ∂S α r ∂t =[H, S α r ] , and the quantum model defined in Eq. (1), compare also the article by Trimper et al. (2007). Because the quantum model is formulated on a lattice we have performed the continuum limit. Eq. (7) describes the precession of the spin around the internal field  B defined in Eq. (8). Notice that the Hamiltonian should be invariant against time reversal. From here we conclude that the tunneling frequency Ω or alternatively the transverse field is changed to −Ω in case of t →−t. As a consequence the self-organized internal field  B is antisymmetric under time 453 Mesoscopic Modeling of Ferroelectric and Multiferroic Systems [...]... Systems 465 17 7 References Blinc, R & Zeks, B (1972) Dynamics of order-disorder-type ferroelectrics and anti -ferroelectrics, Adv Phys 21: 693 Blinc, R & Zeks, B (1974) Soft modes in ferroelectrics and antiferroelectrics, North-Holland, Amsterdam Bose, T & Trimper, S (2010) Correlation effects in the stochastic Landau-Lifshitz-Gilbert equation, Phys Rev B 81(10): 104413 Bose, T & Trimper, S (2011)... 476 Ferroelectrics - Characterization and Modeling p( x , t )n( x , t )  0 (35) Multiplying both sides of Eq (19) by Δp(x,t) or Δn(x,t) and then applying Eq (35), we obtain  1 D( x , t )  p( x , t )  p( x , t )  0 q x   (36)  1 D( x , t )  n( x , t )  n( x , t )  0 q x   (37) and The solutions to Eqs (36) and (37) are p( x , t )  0, p( x , t )  1 D( x , t ) q x (38) and. .. inequalities Ep(0) ≥ 0 and Ep(0) ≤ 0 together imply Ep(0) = 0 Using Kirchoff's loop law (i.e Kirchoff's voltage law) and Eq (6), we obtain L  8Jp V   Ep ( x )dx    9  p 0      1/2 L3/2 (7) Hence,  9  p Jp    8L3   2 2 V ~ V   (8) 470 Ferroelectrics - Characterization and Modeling If n-type instead of p-type free carriers are injected into the dielectric sample, Eqs (2) and (4) should... the total current, could be varying with position, i.e Eq (3) no longer holds (ii) In many ferroelectric and dielectric materials, there exist two opposite types of free charge-carriers, p-type and n-type, with 472 Ferroelectrics - Characterization and Modeling the system behaving like a wide bandgap semiconductor Such double-carrier cases are not considered by the Mott-Gurney law (iii) The Mott-Gurney... free-carrier generation, hence Eq (18) Upon rewriting Eqs (16) and (18) as p( x , t )  1 D( x , t )  n( x , t ) q x (19) and C in ( x )[ p( x , t )  n( x , t )]  p( x , t )n( x , t )  0 (20) respectively, it can be seen that two quadratic equations, for Δp(x,t) and Δn(x,t) respectively, can be obtained: 474 Ferroelectrics - Characterization and Modeling p( x , t )2  Bp ( x , t )p( x , t )  C... Trimper, S & Wesselinowa, J M (2007) Size effects on static and dynamic properties of ferroelectric nanoparticles, Phys Rev B 76: 094107 Mostovoy, M (2006) Ferroelectricity in spiral magnets, Phys Rev Lett 96: 067601 466 18 Ferroelectrics - Characterization and Modeling Will-be-set-by-IN-TECH Ohe, J.-i & Kramer, B (2006) Dynamics of a domain wall and spin-wave excitations driven by a mesoscopic current,... transverse field, physica status solidi (b) 84: 653 Landau, L D & Lifshitz, E M (1935) On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Zeitschr d Sowjet 8: 153 Landau, L D., Lifshitz, E M & Pitaevski, L P (1980) Statistical Physics, Part 2, Pergamon, Oxford Lines, M E & Glass, A (2004) Principles and Applications of Ferroelectrics and Related Materials, Clarendon Press, Oxford... hydrodynamic regimes above (red) and below Tc (blue) The bold and the dashed line indicate the cross-over between the regimes for T ≈ Tc The two other regimes I I+ and I I− are the hydrodynamic regimes relevant for T > Tc and T < Tc , respectively Our model exhibits propagating modes denoted as ε l , Eq (22), and ε h , Eq (24) They play the role of the characteristic energy and obey the scaling form of... enables us to calculate both the excitation energy and its damping The temperature dependence of both quantities is in accordance to the present analysis and also in qualitative agreement with experimental results as shown by Michael et al (2007) 461 13 Mesoscopic Modeling of Ferroelectric and Multiferroic Systems Mesoscopic Modeling of Ferroelectric and Multiferroic Systems 4 Stochastic equation The... (q) l + Jzτ2 τ1 , Jκ ] = − g21 , τ1 Bl q2 iωpz Al − Ω τ2 = Bl q2 g31 Ω2 (34) 462 14 Ferroelectrics - Characterization and Modeling Will-be-set-by-IN-TECH The real correlation function is defined conventionally by ϕα (q, ω ) ϕ† (q, ω ) = Cαβ (q, ω )(2π )d+1 Using the solution for ϕ in terms of Green’s function in Eq (33) and the relation for the ∗ excitation energy ω1,2 = −ω2,1 the correlation function . (τ) −ν with the critical exponent ν and τ =(T − T c )/T c . As well f 1 and f 2 are 458 Ferroelectrics - Characterization and Modeling Mesoscopic Modeling of Ferroelectric and Multiferroic Systems 11 scaling. =  B 1 ×  p +  B 0 ×  ϕ , 456 Ferroelectrics - Characterization and Modeling Mesoscopic Modeling of Ferroelectric and Multiferroic Systems 9 with  B 0 =(Ω ,0,Jzp z ), and  B 1 =(0, 0, J(∇ 2 + z)ϕ z ) see Landau & Lifshitz (1935), the coupling is H fm = λ 1  d d x  S · [  σ ×(∇×  σ)] (38) 462 Ferroelectrics - Characterization and Modeling Mesoscopic Modeling of Ferroelectric and Multiferroic