Ultrafast polarization switching in ferroelectrics
aa r X i v : . [ c ond - m a t . o t h e r] M a r Ultrafast polarization switching in ferroelectrics
V.I. Yukalov
1, 2 and E.P. Yukalova Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia Instituto de Fisica de S˜ao Carlos, Universidade de S˜ao Paulo,CP 369, S˜ao Carlos 13560-970, S˜ao Paulo, Brazil Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna 141980, Russia
A method of ultrafast switching of ferroelectric polarization is suggested. The method is based onthe interaction of a ferroelectric sample with the feedback field of a resonator in which the sample isinserted. The polarization reversal time can be of order of femtoseconds. The polarization switchingproduces a coherent electromagnetic pulse.
I. INTRODUCTION
Ferroelectric materials, possessing spontaneous electricpolarization, can be harnessed for various electronic de-vices [1, 2]. For example, they are used in devices reg-ulating tunneling resistance [2] and enabling nonvolatilememory [3], in memristors [4], in neuromorphic networks[5, 6], and in solar cells [7, 8].To regulate the processing of such devices, it is oftennecessary to be able to quickly vary the direction andmagnitude of ferroelectric polarization. There exist twoways of polarization switching that can be called inhomo-geneous (or incoherent) and homogeneous (or coherent).First, the inhomogeneous way of polarization switch-ing has been studied, being realized through the nucle-ation and growth of domains of opposite polarization,with moving domain walls under the influence of staticelectric fields [9–13]. This way, however, provides ratherlong switching times, on the order of nanoseconds, beinglimited by the domain recrystallization time that is typ-ically hundreds of picoseconds [1, 2, 11, 14, 15]. Similarslow switching in the nanoscale volume of a ferroelectriccan be realized by mechanical deformation of a ferroelec-tric sample [16]. Another slow mechanism of polarizationswitching is due to the chemical oxidization at the sur-face of a ferroelectric film [17]. Because of the principalrestriction of the inhomogeneous switching caused by thelimited domain recrystallization time, it has been neces-sary to find other ways that could provide much fasterswitching.The other way that has been developed relatively re-cently is the homogeneous (or coherent) switching pro-cess realized by external alternating fields, applied per-pendicular to the ferroelectric polarization, in the optical[18–21], terahertz [22–28], or infrared [29] regions. Underthis process, the alternating field acts directly on all ionsof a single-domain sample, and polarization switchingoccurs through a continuous homogeneous mechanism,without formation of new domains of opposite polariza-tion. Homogeneous switching is facilitated in films ofnanometer thickness, where inhomogeneous nucleation isstrongly suppressed [30].In order to realize a homogeneous switching, it is neces-sary that, first, all characteristic times of the process bemuch shorter than the domain nucleation-growth time and, second, that the sample be a single-domain ferro-electric. Under the ultrafast switching by means of al-ternating fields, the first condition is easy to accomplish,since the domain nucleation-growth time is sufficientlylong, being of order of nanoseconds. And the preparationof single-domain ferroelectrics is a technical problem hav-ing several solutions [31–34]. For instance, single-domainstates can be made stable by using strain [34] or dop-ing with point defects [35]. Also, there are plenty offerroelectric films of nanometer thickness, where domainnucleation is suppressed [30].The homogeneous switching mechanism under the ac-tion of an alternating field, involving no nucleation andgrowth of oppositely polarized domains, can provide re-versal times of order of picoseconds. However the presentsources do not provide the strength of a pulse sufficientfor completely switching the polarization. Experiments[36] have shown that the reversal can be only 40% of itsequilibrium value. Although the reversal is quite fast,occurring in about 10 − s, but the reversed polariza-tion rapidly, during the same 10 − s, returns to the ini-tial state, similarly to the dynamics induced by terahertzpulses, when the reversal happens over a picosecond timescale, followed by its fast complete retrieval [37].Finally, time-dependent density functional theory sim-ulations show that by strongly exciting electrons via laserpulses it could be possible to change the underlying dy-namical potential energy surface, which could result inthe polarization switching within tens of picoseconds [38].In the present paper, we consider the homogeneous wayof polarization switching involving alternating fields, butwith a rather different setup. The idea of the methodis to put a ferroelectric, subject to an external constantelectric field, into a resonator cavity. Then the polar-ization motion induces a resonator feedback field actingback on the ferroelectric. In such a way, there is no needin additionally imposed external electromagnetic pulses,but the ferroelectric produces the required pulse by itselfthrough the feedback field. The polarization switchingcan be realized in femtoseconds. The suggested methodof switching also uses alternating fields, similar to thetechniques employing oscillating fields with fixed prop-erties. However the principal difference is that here thealternating field is not imposed by external sources, butis self-organized, being created by moving polarizationitself. Such a self-organized feedback field turns out tobe essentially more effective than an externally imposedfield.We use the system of units where the Planck constant ~ is set to one. II. EVOLUTION EQUATIONS
We consider a ferroelectric inserted into a cavity. Gen-erally, if the sample is sufficiently large and especiallywhen it is in contact with other media, say dielectric,then it can be separated into domains [39]. But we con-sider the case of a cavity containing no other materialsinside it, except the ferroelectric itself, which is a single-domain sample.Our main aim is to illustrate the idea of using a self-organized field of a resonator for accelerating polariza-tion switching. We do not claim to treat a specific ma-terial, but we demonstrate the efficiency of the idea by amodel. For this purpose, let us take the Hamiltonian inthe pseudospin representation [40, 41], having the formof an Ising-type model in a transverse field.ˆ H = − Ω X j S xj − X i = j J ij S zi S zj − X j E tot · P j . (1)Here S αj is an α -component of the S = 1 / j , Ω is tunnelingfrequency, J ij = J ji > E tot is the total electric field acting ondipoles, and P j = d S j (2)is a dipolar operator.In what follows, the total electric field E tot will havetwo components that can be called longitudinal andtransverse. It is important that the polarization wouldalso have these two nonzero components.This form of the Hamiltonian provides a good de-scription of the so-called order-disorder ferroelectrics, al-though it can also be a reasonable approximation forother types of ferroelectrics [40, 41]. Among order-disorder ferroelectrics, it is possible to mention such asKH PO , KH AsO , RbH PO , RbH AsO , CsH PO , CsH AsO , NH H PO , NH H AsO , and their deuterated analogs, in which H is replaced byD . Similar Hamiltonians also are used for describingrelaxor ferroelectrics [42].Note that usually in the case of order-disorder fer-roelectrics with spatially symmetric double wells, thevector of polarization possesses only the longitudinal z -component. However, we keep in mind the general case of nonsymmetric potentials (especially with respect to the inversion x → − x at the location of dipoles, because ofwhich the polarization may have a transverse component.Hamiltonian (1) is the standard, widely used, Hamil-tonian for describing macroscopic ferroelectric samples.For finite samples, in general, one should take into ac-count the depolarizing field caused by the charges on thesurfaces of the sample [43]. However, there are ways [44]of compensating surface charges, thus reducing or remov-ing depolarizing fields.Also, considering an external electric field E , appliedin the z -direction, leads to the appearance of the depo-larizing field proportional to − P z , which is of the or-der of ρd S , where ρ is the sample density. The energy,corresponding to the depolarizing field, is d o P z , whichgives ρd S . The latter expression equals the dephasingrate or transverse attenuation denoted by γ = ρd S .The magnitude of the energy, corresponding to the ex-ternal field, is | d E | , which defines the dipole rotationfrequency denoted as ω = | d E | . For what follows, wewill need a sufficiently strong external field, such that ω be much larger than γ . This is necessary for realizingcoherent motion of dipoles, so that the reversal time bemuch shorter than the dephasing time. Under the con-dition ω >> γ , corrections, related to the depolarizingfield, can be omitted in the Hamiltonian. At the sametime, the attenuation rate γ will be taken into accountin the equations of motion.The total electric field consists of two terms, E tot = E e x + E e z , (3)where the first term is a field of the resonator cavity,in which the sample is inserted, and the second term isan external constant electric field. The resonator cavityis chosen such that it supports the TM fundamentalmode, whose electric field is directed along the cavityaxis that is taken to be the x -axis. The resonator cavityelectric field is a feedback field generated by the movingpolarization P = d V X j h S j i , (4)with V being the sample volume.The equation for the feedback field can be derivedin the standard way [45]. From the Maxwell equa-tions inside the cavity with an inserted ferroelectric, it isstraightforward to get the equation for the electric field ∇ E − c ∂ E ∂t − πσc ∂ E ∂t = 4 πc ∂ P ∂t (5)generated by the ferroelectric polarization (4), where σ isconductivity and c is light velocity. It is possible to lookfor the solution to this equation in the form E ( r , t ) = e ( r ) E ( t ) , (6)where e ( r ) is a cavity mode defined by the Helmholtzequation and normalized to the cavity volume V c , so that1 V c Z | e ( r ) | d r = 1 . (7)We are looking for the TM fundamental mode, which,by definition, is the mode directed along the cavity axisthat here is the axis x , which implies the conditions e y ( r ) = 0 , e z ( r ) = 0 . (8)The mode x -component is nonzero inside the cavity,while satisfies the boundary condition e x ( r ) | r = R = 0 (9)on the cavity cylindrical surface of radius R . The expres-sion for the TM fundamental mode is known [45] to bepresented through the Bessel function of the first kind, e x ( r ) = C J (cid:16) ωc r (cid:17) , C = J − (cid:16) ωc R (cid:17) , with C being the normalization constant. The boundarycondition e x ( R ) = 0 corresponds to the first zero of theBessel function J ( ωR/c ) = 0, which defines the cavitynatural frequency ω = 2 . cR . (10)The normalization constant becomes C = 1 J (2 . . . . Thus the TM fundamental mode reads as e ( r ) = C J (cid:16) ωc r (cid:17) e x = e x ( r ) e x , (11)where e x is a unit vector along the cavity axis x . Thenwe substitute expression (6), with mode (11), into Eq.(5), multiply the latter by form (11), take into accountthat P · e ( r ) = P x e x ( r ) , and integrate over the cavity volume. This leads to theequation d Edt + 2 γ dEdt + ω E = − πη f d P x dt , (12)in which γ = 2 πσ is a cavity attenuation, ω is the cavitynatural frequency (10), and η f ≡ V c Z e x ( r ) d r = 0 . funda-mental mode.The evolution equations, following from the Heisenbergequations of motion are dS xi dt = X j J ij S zj + d E S yi , dS yi dt = − X j J ij S zj + d E S xi + (Ω + d E ) S zi ,dS zi dt = − (Ω + d E ) S yi . (13)The observable quantities are given by the statistical av-erages s α ≡ N S X j h S αj i ( α = x, y, z ) , (14)where N is the number of lattice sites and S = 1 / ζ α ≡ N S X j h S αj i loc , (15)where S = 1 / h S αj i loc is defined as the average h S αj i loc = Tr ˆ ρ loc S αj , (16)with the local equilibrium statistical operatorˆ ρ loc = exp( − ˆ H loc /T )Tr exp( − ˆ H loc T )for the ensemble of spins with the Hamiltonianˆ H loc = − Ω X j S xj − s z X i = j J ij S zj − d E X j S zj . Accomplishing explicit calculations for the local average(16), we keep in mind low temperatures, such that T ≪ J , where J ≡ N X i = j J ij . (17)Thus we obtain the low-temperature local equilibriumvalues ζ x = Ω[Ω + ω (1 − As z ) ] / , ζ y = 0 ,ζ x + ζ z = 1 , ζ z = − ω (1 − As z )[Ω + ω (1 − As z ) ] / , (18)where we introduce the notation A ≡ JSω (19)and define the frequency ω ≡ − d E > . (20)The positive value of ω implies that the external electricfield is directed downwards.Thus we come to the mean-field evolution equationsfor the pseudospin variables ds x dt = − ω (1 − As z ) s y − γ ( s x − ζ x ) ,ds y dt = ω (1 − As z ) s x + (Ω + γ h ) s z − γ s y ,ds z dt = − (Ω + γ h ) s y − γ ( s z − ζ z ) . (21)Here γ is the longitudinal relaxation rate due to spin-phonon interactions (see Blinc [41]), while γ = ρd S isthe transverse attenuation caused by dipolar interactions.Usually, γ ≪ γ .By introducing the dimensionless feedback field h ≡ d Eγ ( γ = ρd S ) , (22)where ρ = N/V , and taking account of the expression P x = ρd Ss x , we get the feedback-field equation d hdt + 2 γ dhdt + ω h = − γ f γ ds x dt . (23)Here γ f ≡ πη f ρd S = πη f γ = 2 . γ (24)is the coupling rate characterizing the interaction be-tween the ferroelectric sample and resonator.Equations (21) and (23) define the dynamics of thevariables s α and the feedback field h . The most inter-esting is the behavior of the dimensionless polarization s z = s z ( t ) as a function of time for different parameters,under the given initial polarization s z (0) = s . III. NUMERICAL SOLUTION
The polarization switching is realized in the followingway. Suppose the ferroelectric sample is initially polar-ized along the axis z . The sample is placed inside a res-onator cavity supporting the TM fundamental modeand is subject to an external electric field directed op-posite to the initial polarization. The polarization dy-namics is governed by Eqs. (21) and (23). In order to precisely describe this dynamics, we have to fix realisticparameters typical for ferroelectrics [1, 2, 41].First, let us notice that a ferroelectric characterized byHamiltonian (1), without the last term containing elec-tric fields, acquires spontaneous polarization below thecritical temperature T c = Ω (cid:18) ln J + 2Ω J − (cid:19) − . (25)This temperature is positive for the tunneling frequencyΩ < J T c > . (26)The interaction strength, due to dipolar forces, J ≈ ρd . The electric dipole is d = e l , with l ∼ − cm and the electric charge about a proton charge e =1 . × − C. Keeping in mind that one Coulomb 1 C =2 . × g / cm / / s, we find d ∼ − C cm ∼ D ,where one Debye is 1 D = 3 . × − C cm. For thedensity ρ ∼ cm − , we obtain ρd ∼ − erg, that is ρd ∼ s − . Really, for typical ferroelectrics J ∼ K, that is, J ∼ s − . The dipolar forces induce thetransverse attenuation γ = ρd S . Thus we have JS ≈ γ = ρd S ∼ s − . The longitudinal attenuation, caused by the interactionof pseudospins with phonons, is much smaller than thetransverse attenuation, γ ≪ γ .The cavity is called resonant, since its natural fre-quency ω has to be tuned close to the dipole rotationfrequency ω , satisfying the quasi-resonance condition (cid:12)(cid:12)(cid:12)(cid:12) ∆ ω (cid:12)(cid:12)(cid:12)(cid:12) ≪ ≡ ω − ω ) . (27)While the attenuations are to be smaller than ω , so that γ ω ≪ , γω ≪ . (28)Therefore, for parameter (19), we get A ≡ JSω = γ ω ≪ . (29)Since JS = γ ∼ s − , to satisfy condition (29),we need that ω be at least about 10 s − to 10 s − ,which is in the near infrared or visible light range. Thisgives the wave vector k ≡ ω /c of the order 3 × cm − to 3 × cm − and the wavelength λ ∼ − cmto 10 − cm. Resonator cavities in the range of visiblelight are widespread, and also there exist various cavitiesoperating in the infrared region [48–53].We solve numerically the system of Eqs. (21) and(23), concentrating our attention on the behavior of thepolarization s z = s ( t ) as a function of time, for differ-ent parameters in the admissible range. In the figures,time is measured in units of 1 /γ and the frequency pa-rameters are measured in units of γ . As initial condi-tions, we need to fix the values s x (0) = p − s , s y (0), s z (0) ≡ s (0) ≡ s , h (0), and the time derivative ˙ h (0).The initial polarization is positive, s >
0. If some ofother initial conditions, except s , are not zero, the re-versal begins immediately at t = 0. When all of them(except s ) are zero, there is a time delay. In the figures,we show the results for the initial conditions s y (0) = 0, h (0) = 0, and ˙ h (0) = 0, while s and, respectively, s x (0) = p − s can be varied. The resonance is as-sumed, when ω = ω .Figure 1 demonstrates the polarization reversal at dif-ferent values of the resonator attenuation. For small γ ,the polarization oscillates after the switching. Hence,in order to achieve a steady state after the polarizationreversal, it is necessary to take larger γ . For γ = 10,the after-switching oscillations are suppressed. Thus, toavoid oscillations, it is preferable that the resonator ring-ing time τ ≡ /γ be shorter than the dephasing time T ≡ /γ .Figure 2 shows the dependence of the polarizationswitching on the tunneling frequency. The larger Ω, theshorter the delay time. As is clear from the evolutionequations, this is because the tunneling triggers the po-larization motion.In Fig. 3, the role of the frequency ω is illustrated.The larger ω , the shorter the delay time and better thepolarization inversion. This happens because the largerfrequency makes stronger the coupling between the res-onator cavity and the ferroelectric sample.Figure 4 shows the similar dependence of the polar-ization switching on the frequency ω , as in the previousfigure, but for the initial polarization s = 0 .
5, when s x (0) = p − s is not zero. As is mentioned above, anonzero s x (0) triggers the start of the polarization mo-tion, so that there is no delay time, and the switchingbegins from t = 0.In Fig. 5, the polarization switching for different ini-tial polarizations is compared: s = 0 . s = 1 (dashed-dotted line). For s = 0 .
5, the initialtransverse component s x (0) = p − s is not zero, be-cause of which the process of switching starts from thevery beginning, practically at t = 0, without delay.Figure 6 demonstrates that the transverse polarizationcomponent s x oscillates around zero. The oscillation isfaster for larger ω . The maximal oscillation amplitudecorresponds to the moment of the polarization switching.Since the transverse component s x generates, by meansof relation (23), the dimensionless cavity field h , hencethe dimensional electric field inside the cavity E , the tem-poral behavior of h , as is seen from Fig. 7, is similar tothat of the component s x . IV. ANALYTICAL SOLUTION
Although the numerical solution of the previous sec-tion gives us an accurate description of the process ofpolarization switching, nevertheless it is desirable to haveanalytic solutions that would provide, at least approxi-mately, explicit formulas allowing for the better under-standing of the related physics and for straightforwardestimates of characteristic quantities.It is convenient to pass to the variables u ≡ s x − is y ,w ≡ | u | = s x + s y , s ≡ s z . (30)In terms of these variables, Eqs. (21) transform into theequation for the transverse component dudt = − iω (1 − As ) u − γ u − iγ (cid:18) h + Ω γ (cid:19) s + γ ζ x , (31)for the coherence intensity dwdt = − γ w − iγ (cid:18) h + Ω γ (cid:19) ( u ∗ − u ) s + γ ζ x ( u ∗ + u ) , (32)and for the polarization, dsdt = i γ (cid:18) h + Ω γ (cid:19) ( u ∗ − u ) − γ ( s − ζ z ) . (33)Keeping in mind the case of resonance, defined by Eq.(27), and the existence of the small parameters describedin Eqs. (26), (28), and (29), we notice that the variables u and h can be classified as fast, while the variables w and s , as slow. In that case, for solving the given systemof equations, it is admissible to resort to the averagingtechniques [54, 55]. Below, we follow the variant of themethod described in detail in Refs. [56, 57]. First, wesolve the equations for the fast variables, keeping therethe slow variables as quasi-integrals of motion. Such asolution is straightforward, although cumbersome, sincethe equations for the fast variables become linear with re-spect to the latter, when the slow variables are kept fixed.Then the found solutions for the fast variables are sub-stituted into the equations for the slow variables, withthe averaging of the slow-variable equations over time.This yields the equations for the guiding centers, whichcan be analyzed. All this machinery has been thoroughlydescribed in Refs. [56, 57] and its use has been demon-strated for studying the dynamics of magnetic systems[58–60].To present the solutions for the fast variables, underfixed slow variables, in a compact form, we take accountof the small parameters and introduce several notations.We define the coupling parameter g ≡ γ f ω γγ = 2 . ω γ (34)characterizing the strength of the coupling between theferroelectric sample and resonator, the coupling function α ≡ g (1 − As ) (cid:0) − e − γt (cid:1) (35)describing the dynamics of the ferroelectric-resonator in-teraction, and the effective frequency ω eff ≡ ω (1 − As ) − iγ (1 − αs ) . (36)Thus we obtain the transverse component u = (cid:18) u + Ω s + iγ ζ x ω eff (cid:19) exp( − iω eff t ) − Ω s + iγ ζ x ω eff (37)and the feedback field h = − iα ( u ∗ − u ) . (38)Note that from Eq. (18), we have ζ x ∼ = Ω ω , ζ z ∼ = − . Substituting the fast variables into the equations forthe slow variables, and averaging the resulting equationsover time gives the guiding-center equations for the co-herence intensity, dwdt = − γ (1 − αs ) w + 2 γ (1 − α − αs ) s , (39)and for the polarization dsdt = − γ αw − γ (1 + s − αs ) − γ ( s − ζ z ) , (40)where γ ≡ γ Ω ω . (41)The parameter γ is very small. However, it cannot beneglected, since it plays an important role in triggeringthe polarization motion at the initial stage. At the verybeginning of the process, when t →
0, so that γt ≪ , γ t ≪ , γ t ≪ , γ t ≪ , (42)the coupling function (35) is close to zero. Then, keepingin mind that usually γ ≪ γ , the equations of motionbecome dwdt = − γ w + 2 γ s ,dsdt = − γ (1 + s ) ( t → . (43)Their solutions are w ≃ (cid:18) w − γ γ s (cid:19) e − γ t + γ γ s , s ≃ (1 + s ) e − γ t − , which, in view of inequalities (42), can be simplified to w ≃ w + 2 (cid:0) γ s − γ w (cid:1) t ,s ≃ s − γ (1 + s ) t , (44)where w ≡ w (0) = 1 − s and s ≡ s (0). These arethe solutions at the initial stage, when the motion ofindividual polarizations is not mutually synchronized.The coupling function (35) grows with time, imply-ing the increase of the magnitude of the resonator feed-back field, which collectivizes the individual polariza-tions, forcing them to move coherently. The influenceof the feedback field becomes crucial after the coherencetime t coh , when the coupling function grows so that αs = 1 ( t = t coh ) . (45)This defines the coherence time t coh = τ ln gs (1 − As ) gs (1 − As ) − (cid:18) τ ≡ γ (cid:19) (46)with τ being the resonator ringing time. If theferroelectric-resonator coupling is strong, such that gs ≫
1, then the coherence time is t coh ≃ τgs (1 − As ) . (47)At the coherence time, the solutions (44), that is w coh = w ( t coh ) , s coh = s ( t coh ) , (48)can be written as w coh ≃ w + 2 γ s t coh , s coh ≃ s , (49)where we assume that the coupling parameter g is suf-ficiently large, so that inequalities (42) are yet valid at t coh .After t coh , the coupling function (35) quickly growsreaching the value g (1 − As ) ≃ g . At this stage, theparameters γ and γ , that are much smaller than γ ,and especially than gγ , can be neglected. Then Eqs.(39) and (40) become dwdt = − γ (1 − gs ) w ,dsdt = − γ gw ( t > t coh ) . (50)The latter equations enjoy the exact solutions for thecoherence intensity w = (cid:18) γ s gγ (cid:19) sech (cid:18) t − t τ s (cid:19) (51)and polarization s = − γ s gγ tanh (cid:18) t − t τ s (cid:19) + 1 g . (52)The quantities γ s and t are the integration constantsthat are defined by sewing expressions (51) and (52) withEqs. (49). This gives the switching time τ s ≡ /γ s , inwhich γ s = γ g + ( gγ ) w coh , γ g ≡ γ ( gs − , (53)and the delay time t = t coh + τ s (cid:18) γ s + γ g γ s − γ g (cid:19) . (54)The delay time shows the time, when the switchingstarts, while the switching time is the time during whichthe polarization reversal occurs. The delay time can alsobe written as t = t coh + τ s ln (cid:18) γ s + γ g gγ √ w coh (cid:19) , (55)which demonstrates its explicit dependence on w coh . V. CHARACTERISTIC QUANTITIES
The derived analytic expressions provide a transpar-ent illustration for the role of different system param-eters and their combinations. To study more carefullythese dependencies, let us keep in mind the case of strongferroelectric-resonator coupling, when gs ≫
1. Then thecoherence time (47) takes the form t coh ≃ . ω s , (56)which shows that the larger the frequency ω , the shorterthis time.Quantities (53) read as γ s ≃ gγ (cid:18) . γ Ω ω s (cid:19) , γ g ≃ γ gs . (57)The switching time τ s ≡ /γ s becomes τ s ≃ . γω γ , (58)so that larger ω makes the process of switching faster.The delay time (55) acquires the form t ≃ t coh + τ s ln (cid:18) s √ w coh (cid:19) , (59)which depends on the value w coh . The latter is connectedwith the initial polarization s . When the system at the initial moment of time is wellpolarized, with s = 1 and w = 0, then w coh = 0 . γ Ω ω ( s = 1) . (60)And the delay time turns into t ≃ . ω (cid:20) γ γ ln (cid:18) . ω γ Ω (cid:19)(cid:21) ( s = 1) . (61)While when s = 0 . w = 0 .
75, then the delay timeis t ≃ . ω (cid:18) . γγ (cid:19) ( s = 0 . . (62)To get concrete estimates, let us take the typical val-ues of parameters as have been used when numericallysolving the evolution equations: ω ∼ γ , Ω ∼ . γ , γ ∼ γ , and s ∼
1. Then g ∼ γ s ∼ γ g ∼ γ , and γ ∼ − γ . For the coherence time, we have t coh ∼ − s and for the switching time we get τ s ∼ − s.Diminishing s decreases the delay time. Thus, if s = 1and w = 0, then w coh ∼ − and t ∼ − s. Butif s ≈ .
5, so that w ∼
1, then t ∼ − s. Theseestimates are in good agreement with numerical calcula-tions.An important question is how the switching time islimited in realistic materials. In particular, what is therelation between the switching time τ s and the cavityringing time (delay time τ ≡ /γ ). From the above esti-mates, we find the ratio τ s τ = 0 . γ ω γ . (63)It looks that by varying the system parameters, it is pos-sible to make this ratio rather small. However, there arelimitations for the variation of the parameters. Thus,for a good quality cavity one has γ ≪ ω . But γ can-not be arbitrarily small, since for γ < γ , there appearoscillations in the polarization. In order to realize a sta-ble switching without oscillations, it is necessary to take γ ≫ γ . Therefore ratio (63) lies in the interval γ ω ≪ τ s τ ≪ ω γ ( γ ≪ γ ≪ ω ) . (64)For the infrared region, where ω /γ ∼
10, we have0 . ≪ τ s τ ≪ (cid:18) ω γ ∼ (cid:19) , which actually means that the switching time τ s is oforder of the ringing time τ . In the visible light region,when ω /γ ∼ . ≪ τ s τ ≪ (cid:18) ω γ ∼ (cid:19) . This tells us that again the switching time is correlatedwith the ringing time. For the visible light, it looks ad-missible to reach the shortest switching time of order τ s ∼ − s. VI. COHERENT RADIATION
The motion of electric dipoles has to produce electro-magnetic radiation. If this motion is coherent, the pro-duced radiation should also be coherent. Since the sam-ple is inside a resonator cavity, the radiation can prop-agate only along the cavity axis, that is, along the axis x . The radiation intensity in the direction of n = e x consists of two terms describing incoherent and coherentradiation, I ( n , t ) = I inc ( n , t ) + I coh ( n , t ) . (65)Radiation, produced by moving dipoles can be describedin the following way [45, 57, 61]. The incoherent radiationintensity reads as I inc ( n , t ) = 316 π N ω γ [1 + s ( t )] (66)and the coherent radiation intensity is I coh ( n , t ) = 332 π N ω γ w ( t ) F ( k n ) , (67)with the shape factor F ( k n ) = 4 k L sin (cid:18) k L (cid:19) , (68)where L is the cavity length and γ is the natural width γ = 23 | d | k (cid:16) k = ω c (cid:17) . (69)For the frequency ω ∼ s − , we have the wave-length λ ∼ − cm and the natural width γ ∼ s − .Then the radiation intensities at the maximum, are I inc ( n , t ) ∼ N − W ,I coh ( n , t ) ∼ N F ( k n )10 − W .
The number of dipoles that could radiate coherently canbe estimated as N ∼ ρλ . If we consider a small sample,with the length L smaller than the radiation wavelength λ , then the shape factor (68) is of order one. In that case,for the density ρ ∼ cm − , we get N ∼ . Andfor the radiation intensities, we find I inc ( n , t ) ∼ − W , I coh ( n , t ) ∼ W .
When the frequency is ω ∼ s − , then λ ∼ − cm and γ ∼ − . The radiation intensities are I inc ( n , t ) ∼ N − W ,I coh ( n , t ) ∼ N F ( k n )10 − W .
Considering again coherently radiating dipoles, with thenumber N ∼ ρλ , we have N ∼ . This gives for theradiation intensities I inc ( n , t ) ∼ − W , I coh ( n , t ) ∼ W .
Such a level of radiation can be easily measured. Theprevailing coherent component of radiation shows thatthis radiation is of the type of superradiance.
VII. CONCLUSION
A method of ultrafast polarization switching in ferro-electrics is suggested. The main idea is to place a ferro-electric sample into a resonator cavity. In the presenceof a constant electric field, directed opposite to the fer-roelectric polarization, the sample is in a nonequilibriumstate. As soon as the polarization starts moving, it pro-duces an electric field in the cavity. This field acts backon the sample forcing the polarization to move faster.Thus the ferroelectric itself generates a feedback field ac-celerating the polarization motion, so that there is nonecessity of applying external alternating fields, as oneusually does for realizing polarization switching. It turnsout that the self-organized feedback field is essentiallymore effective for the polarization reversal than an ex-ternally imposed field.The system of equations, describing the ferroelectricpolarization and feedback field is solved numerically andalso analytically by means of averaging techniques. Thismakes it possible to give a detailed description of thewhole procedure, to study the role of the system param-eters, and to estimate the characteristic quantities in-volved in the process. The polarization switching can berealized extremely fast: for the parameters of typical fer-roelectrics, the switching time can reach femtoseconds.This ultrafast polarization reversal generates a coherentelectromagnetic pulse.We stress that the main goal of the paper is to at-tract attention to the possibility of accelerating the po-larization switching in ferroelectrics by using the self-acceleration effect caused by the action of a resonator-cavity feedback field. This is the idea of principle that, toour knowledge, has not been considered for ferroelectricsbefore. It goes without saying that the method is notnecessarily applicable to any particular material. Thusthe method seems to be not applicable for the order-disorder ferroelectrics with spatially symmetric doublewells, where there is only the longitudinal polarization.However the considered model assumes the general caseof asymmetric potentials for which the method is appli-cable. We also hope that, since there are various typesof ferroelectric systems [41, 62–64], there can exist othermaterials for which the suggested idea could work.In order that the suggested method could be realized,the existence of two spin components of polarization,longitudinal and transverse, is required. If one keepsin mind an order-disorder ferroelectric with lattice-sitedouble wells that are ideally symmetric with respect tospatial inversion (especially with respect to the inversion x → − x ), then there is only a longitudinal component,and the sample polarization is expressed through the z -component of the spin operator. But in the general caseof an asymmetric double well, the sample polarizationcontains a term with the x -component of spin. The asym-metry can be induced by stress or by incorporating intothe sample admixtures or vacancies. Thus, the inclusionof vacancies in order-disorder ferroelectrics is attributedto the breaking of spatial inversion symmetry along dif-ferent directions [65]. 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5. Here: ω = 10 (dashed line); ω = 20 (dotted line); ω = 100(dashed-dotted line); and ω = 1000 (solid line). t -1-0.500.51 s z (t) s = 0.5 s = 1 = 100 = 0.1 = 10 FIG. 5. Polarization switching for different initial polarizations: s = 0 . s = 1 (dashed-dotted line). Otherparameters are: ω = 100, Ω = 0 .
1, and γ = 10. For s <
1, the delay time is practically absent. t -0.8-0.400.40.8 s x (t) = 20 = 10 (a) t -1-0.500.51 s x (t) (b) = 100 FIG. 6. Temporal dependence of the transverse polarization component s x , with the parameters Ω = 0 . γ = 10, and s = 1,for different frequencies: (a) ω = 10 (solid line); ω = 20 (dashed-dotted line); (b) ω = 100 (solid line). t -8-4048 h(t) (a) = 10 = 20 0 0.5 1 1.5 t -40-2002040 h(t) (b) = 100 FIG. 7. Dimensionless electric field inside the cavity h ( t ), as a function of time with the parameters Ω = 0 . γ = 10, and s = 1, for different frequencies: (a) ω = 10 (solid line); ω = 20 (dashed-dotted line); (b) ωω