Ferroelectrics Characterization and Modeling Part 14 doc

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

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

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 (Tc=82oC)

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 (ψ ≠ ) and for the isotropic liquid (0 ψ= ) according to Eqs (20) and (22), 0respectively, 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

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 Tc 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 α′ ′c b/ ′ extracted (Eq.14) by using the coefficients (Table 1 and 2) 2for 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 (Δ =T T0− ) which is nearly 6K for the racemic A7 compared to the value of 2.5 K T c

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 (∂g/∂ = ), a quadratic equation obtained in ψψ 0 2 can be solved, which also gives a similar functional form of the temperature dependence of ψ with the critical exponent of β=1 / 2 from the mean field theory, as given by Eq.(19) This quadratic solution in ψ2 can also be used to calculate the temperature dependence of the spontaneous polarization P (Eq.6) and of the dielectric constant ε⊥ (or χ− 1) (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 Tc

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 P2θ2 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 (P2 2ψ 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

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 P2ψ2 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

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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

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behavior of heat capacity at the smectic-A-smectic-C-alpha(*) transition of the

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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

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structures in liquid-crystals Kristallografiya, Vol 21, No 6, pp 1093-1100

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Ellipsometric studies of synclinic and anticlinic arrangements in liquid crystal film

Phys Rev E, Vol 62, No 6, pp 8106-8113

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temperature near the smectic AC* phase transition in ferroelectric liquid crystals

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Materials, Oxford University Press, pp 71-81, Oxford

Matsushita, M (1976) Anomalous temperature dependence of the frequency and damping

constant of phonons near Tλ in ammonium halides J Chem Phys., Vol 65, p 23

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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

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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.,

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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

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with P-2 theta(2) coupling for the smectic A-smectic C* phase transition in liquid

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under an electric field close to the smectic AC* phase transition in a ferroelectric

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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

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Mol Cryst Liq Cryst., Vol 114, No 1-3, pp 259-270

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 modelfor a ferroelectric order-disorder phase transition is reformulated in terms of a mesoscopicmodel and inherent microscopic parameters The statical and dynamical behavior of the Isingmodel in a transverse field is considered as classical field theory with fields obeying Poissonbracket relations The related classical Hamiltonian is formulated in such a manner that thequantum 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 spinspace and consequently, the driving field becomes anisotropic A further conclusion is thatthe 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 arecharacterized by a Lie algebra where the total antisymmetric tensor plays the role of thestructural constants Using symmetry arguments of the underlying spin fields and expandingthe driving field in terms of spin operators and including terms which break the time reversalsymmetry we are able to derive a generalized evolution equation for the moments Thisequation is similar to the Landau Lifshitz equation suggested by Landau & Lifshitz (1935) withGilbert 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 includesunder 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 phasetransition temperature The two transverse excitation energies are complex, where thereal part corresponds to a propagating soft mode and the imaginary part is interpreted asthe wave vector and temperature dependent damping Further there exists a longitudinaldiffusive mode All modes are influenced by the noise strength The solution offers scalingproperties below and above the phase transition The results are preferable and applicable forferroelectric order-disorder systems

A further extension of the approach is achieved by a symmetry allowed coupling of thepolarization to the magnetization The coupling is related to a combined space-time symmetrydue to the fact that the magnetization is an axial vector withm  ( x, − t ) = − m ( x, t)whereas

23

the polarization is represented by a polar vector p (− x, t ) = − p ( x, t) Multiferroic materialsare characterized by breaking the combined space-time symmetry Possible couplings areconsidered Introducing a representation of spin fields without fixed axis one can incorporatespiral structures Different to the previous system the ground state is in that case aninhomogeneous 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-disorderand 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 V0.Furthermore, the coupling between atoms or molecules at neighboring positions is denoted

by J0 The displacive limit is identified by the condition V0  J0, i.e the atoms are notforced to occupy one of the minimum Instead of that the atoms or molecular groups performvibrations around the minimum The double-well structure becomes more important whenthe 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 samedirection giving rise to elementary dipole moments, the average of which is the polarization

The opposite limit V0  J0 means the occurrence of high barriers between the double-wellstructure, i.e the particles will reside preferentially in one of the minimum Above the criticaltemperature the atoms will randomly occupy the minimum whereas in the low temperature

phase T < T cone 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 zspecify the minima of the double-wellpotential 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)

where S x and S z are components of spin-12 operators Notice that these operators have no

relation to the spins of the material such as KH2P04(KDP) Therefore they are denoted aspseudo-spin operators which are introduced to map the order-disorder limit onto a tractable

Hamiltonian The coupling strength between nearest neighbors J ijis assumed to be positiveand is restricted to nearest neighbor interactions denoted by the symbol< ij > The transversefield is likewise supposed as positive Ω > 0 Alternatively the transverse field can beinterpreted as tunneling frequency In natural units the time for the tunneling between bothlocal minimums isτ t=Ω−1whereas the transport time between different lattice sites is given

byτ i = (¯hJ)−1 The high temperature limit is determined byτ t < τ iorΩ> ¯hJ The tunnel

frequency is high and the behavior of the system is dominated by tunneling processes Withother 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 tunnelingfrequency 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 alsoBlinc & Zeks (1972), yields a qualitative agreement with experimental data, the model wasincreasingly considered as one of the basic models for ferroelectricity of order-disorder type

Fig 1 Schematical representation of the physical situation in ferroelectric material J ijis the

interaction between the atoms in the double-well potential at lattice site i and j,Ω is the

tunneling frequency and V0the 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 fieldallows several applications in solid state physics Thus a magnetic system with a singletcrystal 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 differentmethods 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 excitationenergy and a phase transition A more refined study using special decoupling procedures forthe 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) Veryrecently Michael et al (2006) have applied successfully the TIM to get the polarization andthe hysteresis of ferroelectric nanoparticles and also the excitation and damping of suchnanoparticles, 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 microscopicmodel defined by Eq (1), the ferroelectric behavior should be also discussed using classicalmodels Especially, the progress achieved in magnetic systems, see Landau et al (1980) andfor a recent review Tserkovnyak et al (2005), has encouraged us to analyze the TIM in itsclassical version capturing all the inherent quantum properties of the spin operators Theclassical spin is introduced formally by replacing S →  S/(¯hS(S+1)) in the limit ¯h → 0

and S → ∞ Stimulated by the recent progress in studying ferromagnets reviewed byTserkovnyak et al (2005), we are interested in damping effects, too In the magnetic casethe 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 tracedback to the interaction of the spins The reversible evolution equation can be extended byintroducing 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 nanoparticleswithin a classical spin model, while a nonlinear magnetization in ferromagnetic nanowireswith 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-Lifshitzequation 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 consideredrecently 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 inferroelectric system the internal field is an anisotropic one and therefore, both the reversibleprecession and the irreversible damping are changed

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 = ˆJ μν ˜J κδ Theanisotropic TIM is obtained by assuming Ω = (0, 0,Ω), ˆJ μν = J δ μz δ νz , ˜J κδ = δ κδ, and

Γμν=Jzδ μz δ νz Here z is the coordination number The Hamiltonian reads now

Here K designates the exchange coupling. The last Hamiltonian is invariant underspin-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 ofmotion, compare Hohenberg & Halperin (1977):

∂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 continuumlimit 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 concludethat the tunneling frequencyΩ 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

reversal In the magnetic case, represented by the Hamiltonian in Eq (4), the internal field isisotropic and is defined by



Directly from Eq (7) it follows that the spin length σ2or even S2is conserved which is reflected

in the quantum language by the conservation of the total spin[H,  S2]− =0 Vice versa oneconcludes from the conservation of any vector, that ˙ S ·  S =0, i.e the time derivative of the

vector is perpendicular to the vector itself which is simply fulfilled by assuming ˙ S ∝ A  ×  S

for an arbitrary vector fieldA Insofar Eq (7) is a consequence of the spin conservation The 

same is valid for the spin field σ.

2.3 Dissipation

Eq (7) is a reversible equation, i.e it is invariant against time reversal As demonstrated inthe next section Eq (7) allows pseudo-spin-wave solutions However, the excitation does nottend to restore a continuous symmetry, i.e the Goldstone theorem, for details see Mazenko(2003), is not valid Instead of that a soft mode behavior is observed Normally, the excitationmodes are damped It is the aim of the present section to extend the evolution equation byincluding damping effects From a microscopic point of view the damping can be traced back

to scattering processes of spin-wave excitations with different wave vectors emphasized by

Wesselinowa (1984) in second order of the interaction J In principle this interaction effects are

included in the microscopic Hamiltonian Recently, Wesselinowa et al (2005) have studied theinfluence of layer defects to the damping of the elementary modes in ferroelectric thin films.Likewise the analysis can be generalized for ferroelectric nanoparticles, where the interaction

of those can also lead to finite life-times of the excitation modes as performed by Michael

et al (2007) Otherwise, the damping of pseudo-spin-waves can be originated by a coupling

to lattice vibrations Due to the coupling of the TIM to phonons the spin excitations can bedamped as detected by Wesselinowa & Kovachev (2007) Generally one expects that due toattenuation the spin length is not conserved On a mesoscopic level the inclusion of dampingeffects are achieved by a generalized evolution in the form

∂  S ( x, t)

∂t =  B ( x, t ) ×  S ( x, t ) +  D ( S) (11)The origin of the damping termD is a pure dynamic one, i.e all possible static parts should 

be subtracted From Eq (11) one finds

∂  S2

∂t =  D ·  S <0

A non-trivial damping part is oriented into the direction of the effective field B Following

Hohenberg & Halperin (1977) and using the approach discussed by Trimper et al (2007) wemake the ansatz

In case the coefficient matrixΛαβ ( S)is positive and independent of the spin field Eq (11)corresponds to a pure relaxation dynamics for a non-conserved order parameter field Thisfact reflects another difference to the magnetic case, where the internal field is defined in

Eq (10) The evolution equation for the Heisenberg spins σ reads

∂  σ ( x, t)

∂t =  B (m) ( x, t ) × σ ( x, t) +Λαβ K ∇2 σ

Provided the coefficient matrix Λαβ is independent on σ the damping effects are realized

by spin diffusion and hence the order parameter is conserved according to the classificationintroduced by Hohenberg & Halperin (1977)

To proceed further in the ferroelectric model with non-conserved order parameter let usexpand the coefficient matrix Λαβ ( S) in terms of the spin field  S, where only terms are

included which break the time reversal symmetry Denoting the field independent part as

Λ(0)αβ we get up to second order in S

Λαβ ( S) =Λ(0)αβ +Λ(1)αβγδ S γ S δ+O ( S4) (13)Due to the spin algebra the tensor structure of the coefficientsΛ are given in terms of thestructure constant of the underlying Lie-group, i.e the complete antisymmetric tensor αβγ

and unit tensorδ αβ The zeroth order term is

Λ(0)αβ ∝ αμν βμν.From here we define

in the next section

3 Excitation spectrum

In this section we investigate the spectrum of collective pseudo-spin-wave excitations andtheir damping The starting point is Eq (16) Firstly we study the reversible precession part

3.1 Soft mode

Because no continuous symmetry is broken one expects a soft mode behavior, see Blinc & Zeks(1974); Lines & Glass (2004), which is characterized by the temperature dependent excitationenergyε ( q, T)offering the following behavior

the internal field B defined in Eq (9) into Eq (7) In the frame of linear spin-wave theory the

spin field S ( x, t)is splitted into a static and a dynamic part according to



S ( x, t ) =  p ( x ) +  ϕ ( x, t), (18)where p ( x) = (p x , 0, p z)is a time-independent but temperature-dependent vector in the x − z

plane as suggested in Eq (1) In case that p is independent of the coordinates it describes

the homogeneous polarization whereas for multiferroic material, for a review compare Wang

et al (2009), one finds a spiral structure of the form

 p ( x) = p0[cos( Q ·  x ) e x+sin( Q ·  x ) e y] +p z  e z (19)HereQ characterizes the spiral structure. 

Inserting the ansatz made in Eq (18) into Eq (7) the field ϕ obeys in spin-wave approximation

˙

 ϕ =  B1×  p +  B0×  ϕ ,

with B0 = (Ω, 0, Jzp z), and B1 = (0, 0, J (∇2+z)ϕ z) The direction of the homogeneouspolarization is given by p ×  B0=0 The last relation has two solutions

(i) p z(T ) =0 , p x = Ω

Jz if T ≤ T c

(ii) p z=0 , p x(T ) = Ω

Here the phase transition temperature is obtained by p z(T=T c) =0 Moreover, the relation

p x(T c) = Ω/Jz should be fulfilled, i.e p x remains fixed and is temperature-independent

below T c In a quantum language it means that the quantization axis is oriented within the

x − z-plane in accordance with microscopic investigations, see for instance de Gennes (1963); Kühnel et al (1977) In the frame of a multi scale approach the temperature dependence of p x and p zis calculated based upon the microscopic model Eq (1) In the high temperature regime

p x decreases with increasing temperature which can be estimated to be p x ∝ Ω/T, compare the book by Lines & Glass (2004) The quantity p z offers a behavior like p z ∝(− τ)βwhere

τ= (T − T c)/T cis the relative distance to the phase transition temperature andβ ≤1/2 is thecritical exponent of the polarization The results are shown in Fig 3 The subsequent analysis

Fig 3 Static polarization p z(T)(blue line) and p x(T)(red line) versus the ratio T/T c

Whereas p z vanishes at T caccording to a power law∝(− τ)β , p xremains temperature

Here we have used the abbreviationκ ( q) = z − q2 Notice that the lattice constant a is set

to unity and the approach is valid in the long wave length limit qa 1 Notice that we set

a=1 A non-trivial solution of Eqs (21) is given byϕ α∝ exp[iε ( q)t] The eigenvalue of thecoefficient matrix leads to the excitation energyε l ( q), which is in the low temperature phase

dominated by the coupling J as pointed out already in the introduction It results

whereΦl ( q)is the amplitude of the excitation mode determined by the initial condition

The high temperature phase is characterized by p z = 0 For that case one gets a similar

dispersion relation as for T < T cwhich can be written as

called as stiffness parameter remains finite at T c This result is also different to the magnetic

case, where the stiffness constant tends to zero for T → T c The spin wave field ϕ ( q)exhibits

in the high temperature phase a similar form as for T < T c , but one has to set p z=0 andε l

has to be replaced byε h As expected the field ϕ is continuous at T c

3.2 Dynamic scaling

In the vicinity of a second order phase transition a system is usually characterized by criticalexponents and scaling relations, see Hohenberg & Halperin (1977) Especially there existcharacteristic energies (propagating and relaxation modes) which fulfill

ε c ( q, T) =q z c f1(qξ ) ≡ ξ −z c f2(qξ) (26)

Here z cmeans the dynamic scaling exponent,ξ is the correlation length which behaves near

to T casξ ∝ (τ)−ν with the critical exponentν and τ = (T − T c)/T c As well f1and f2 are

scaling functions which depend only on the combination q ξ The domain of wave vector and

relative distance to the critical temperatureτ is due to Hohenberg & Halperin (1977) shown

in Fig 4 The critical regime denoted as region I is characterized by q ξ >>1 which is relevant

τ

phase transition

I

Fig 4 Domain of wave vector versusτ= (T − T c)/T c , I is the critical regime (green), I I ±are

the hydrodynamic regimes above (red) and below T c(blue) The bold and the dashed lineindicate the cross-over between the regimes

for T ≈ T c The two other regimes I I+ and I I − are the hydrodynamic regimes relevant for

T > T c and T < T c, respectively Our model exhibits propagating modes denoted asε l,

Eq (22), andε h, Eq (24) They play the role of the characteristic energy and obey the scalingform of Eq (26) So we get in the low temperature phase

3.3 Damping effect

For the complete Eq (16) with damping we make the same ansatz as in Eq (18) to find thecomplex dispersion relationω ( q, T) In the low temperature phase T ≤ T cthe system revealsthree complex modes

The spin-wave field ϕ ( q, t)introduced in Eq (18) behaves as

Whereas the modesω1,2describe the propagation of pseudo-spin waves and their damping,the modeω3 = − iω d ( q, T)is a pure imaginary one, influenced only byτ1 Such a situation

is also known for magnets, see Hohenberg & Halperin (1977) with two complex transversemodes and one diffusive longitudinal mode Due to the different physical situation the pureimaginary mode ω3 offers here a dispersion relation different to diffusion The results in

Eq (28) are valid in the long wave length limit q  1 and in first order in τ1,2−1 In thisapproximation the propagating partε l ( q, T) remains unchanged in Eq (22) Higher orderterms give rise to a slightly changed behavior The finite life-time of the excitation is related

to the temperature and wave vector dependent damping terms which read

it depends on T in the low temperature regime via p z which disappears at T c according

to p z(T) ∝ (− τ)β with a critical exponentβ ≤ 1/2 The temperature dependence of theexcitation energy and the relevant life-time(Γ2l)−1of the soft mode at q=0 are depicted inTrimper et al (2007) The damping function can be written in scaling form using the resultsobtained in subsection (3.2) For simplicity we assumeτ1 τ2≡ τ0 Definingγ 1l =Γ1l/2τ o

andγ 2l=Γ1l/2τ o In the long wave length limit it results

γ −1 l ( q, T c ) < γ −1 l ( q, T ≤ T c).When the system is approaching the phase transition temperature, the elementary excitationdecays more rapidly for wave vector q =0

The corresponding damping parametersΓ1handΓ2hcan be also found in the high temperature

limit characterized by p z = 0 and p x → 0, compare Fig 3 The analytical expressions arepresented in Trimper et al (2007) In that case we obtainε h ( q, T → ∞) =Ω, i.e the mode isfrozen in A corresponding analysis for the damping shows that the system is overdampedwithε h Γ1h

The analysis can be performed likewise on a microscopic level using Eq (1), cf Michael et al.(2006) In that case a more refined Green’s function technique enables us to calculate boththe excitation energy and its damping The temperature dependence of both quantities is inaccordance to the present analysis and also in qualitative agreement with experimental results

as shown by Michael et al (2007)

limit characterized by p z = 0 and p x → 0,... enables us to calculate boththe excitation energy and its damping The temperature dependence of both quantities is inaccordance to the present analysis and also in qualitative agreement with experimental... τ0 Definingγ 1l =Γ1l/2τ o

and< i>γ 2l=Γ1l/2τ o In the long wave length