This work was supported by the USA National Science Foundation and the Nebraska Research Initiative.
The defining characteristic of a ferroelectric is the abi- sequently there has been relatively little published discussion lity repeatably to reverse, or switch, the polarization by of intrinsic switching, except to note that it remained out application of a sufficiently large external electric field as of reach. The thin ferroelectric films have indeed shown illustrated by the hysteresis loops in Fig. 1. Yet since the the nucleation-inhibition finite site effect, an increase of the discovery of ferroelectricity eighty years ago by Valasek, coercive field with the decrease of thickness, but the value few published reports have considered the value of the of the measured coercive field was still much lower than intrinsic coercive field connected with collective polarization the intrinsic one and correspondingly the switching is still reversal, a consequence of the instability of the macroscopic controlled by the nucleation and domains. Recently , we polarization state in an opposing electric field. This is succeeded in finding the limiting intrinsic coercive field in because the measured value of the ferroelectric coercive field films as thin as 1 nm made by Langmuir–Blodgett(LB) dehas been invariably much smaller than the intrinsic value position of a two-dimensional ferroelectric copolymer .
predicted by theory, such as the Landau–Ginzburg (LG) mean field model  or by ferroelectric Ising model [2,3].
The low extrinsic coercive field observed in real ferroelectric crystals and films is caused by nucleation of domains with reversed polarization, which then grow and coalesce by domain wall motion [2,4,5]. Nucleation can occur at fixed defects in the crystal or due to a passive (nonferroelectric) surface layer . Nucleation mechanisms and domain wall dynamics must be arbitrarily inserted into these models in order to explain the experimental observations of switching kinetics with much lower coercive fields in real ferroelectric materials . In fact, the study and modeling of extrinsic switching is a great industry in the field and is central to the application of ferroelectric films to nonvolatile randomaccess memories .
It has proven difficult to prevent extrinsic switching by eliminating defects or pinning domain walls. Another way to achieve intrinsic switching is to make a particle small enough or a film thin enough to inhibit nucleation [5,8–10]. Several mechanisms for this finite-size effect have been proposed, including reduction of nucleation volume, introduction of space charge near the electrodes, elimination of passive layers, and domain-wall pinning [4,6,10,11]. Even in the Figure 1. Theoretical ferroelectric hysteresis loops from the firstthinnest ferroelectric films obtained previously, the measured order Landau–Ginzburg model (solid line), second-order Landau– Ginzburg model (dashed line), Ising-Devonshire model (dotted extrinsic coercive field is much smaller than the expected line).
intrinsic value calculated from theoretical models, and conGeneral Features of the Intrinsic Ferroelectric Coercive Field The measured values of the intrinsic coercive field were cive field can be written as in good agreement with the predictions of the LG mean- 3 3 PS field theory [1,14]. Here, we analyze the calculation of the Ec 1 - t (from LG1). (2) 0 25 intrinsic coercive field from a more general perspective and from several specific models of ferroelectricity.
The value of the intrinsic coercive field Ec near the Curie We can estimate the intrinsis ferroelectric coercive field Ec temperature T0 is about 1/10 of the depolarization field.
by calculating the external electric field necessary to over- The Landau–Ginzburg continuous ferroelectric phase transition of the second order (LG2)  is modeled with come the depolarization field PS/0, where PS is the a free energy G = G0 +(1/2)P2 +(1/4)P4 - PE, where spontaneous polarization, is the dielectric constant (More the first coefficient is again =(T - T0)/(0C), and the precisely, the contribution to the dielectric constant from second coefficient > 0. The ferroelectric phase in zero ferroelectricity. The total dielectric constant of the medium electric field exists below the transition temperature Tc = T0, T = B + contains contributions from the background where the spontaneous polarization PS = ± -/ and polarizability and the ferroelectric polarizability) alond the the ferroelectric contribution to the dielectric permittivity direction of spontaneous polarization and 0 is the permitis given by 1/0 = -2. The intrinsic coercive field tivity of free space. The energy density associated with the Ec = 2(-/3)3/2 calculated from the LG2 free energy spontaneous polarization is uP =(1/2)PS /0. Application can be written  of an external field E in a direction opposite to the spontaneous polarization, would contribute to energy density 1 PS Ec = (from LG2). (3) uE = E · PS = -EPS. (This contribution applies to all kinds 3 of ferroelectrics, whether displacive, order-disorder, or other.
Devonshire  applied the Ising model to ferroelectrics For example, in ferroelectric materials consisting of fixed consisting of electric dipoles with dipole moment p0, which dipoles, with the dipole moment µ and density N, that form can achieve a collective ordered state. The two-level macroscopic polarization PS = -N µ, as in ferroelectric (spin-1/2) Ising–Devonshire (ID) model describes a unipolymers, the energy density of the dipoles in the applied axial ferroelectric with an order-disorder phase transition field is uE = -N E · µ, leading to the same result). The with the an order parameter equal to the normalized intrinsic coercive field Ec is then approximately the value of spontaneous polarization S = PS/PS0, where PS0 = Np0.
the applied field that produces an energy comparable to the The order parameter is obtained from the transcendental polarization energy, uE uP, so that the intrinsic coercive equation [2,3,16] S = tanh [(J0S + Ep0)/kT ], where k field is approximately half the depolarization field, is the Boltzmann constant and J0 is the pseudospin Ising interaction constant.
1 PS In the ferroelectric phase near (but below) TC = TEc (from energy). (1) 2 = J0/k, the zero-field spontaneous polarization follows a simple 1/2 power scaling S ± 3(1 - T /TC), The intrinsic coercive field for a ferroelectric-paraelect- and the dielectric constant is given by 1/ric phase transition of the first order (close to second [(kT0)/(Np2)][1 - T/T0]. The ID intrinsic coercive field order) can be calculated from the Landau–Ginzburg phe- near TC is approximately Ec [(2kT0)/(3p0)][1 - T /T0]3/2, nomenology (LG1) [15,16]. The LG1 Gibbs free ener- so the coercive field near TC can be written gy in a uniaxial material is G = G0 + (1/2)P1 PS Ec (ID). (4) +(1/4)P4 +(1/6)P6 - PE, where the first coefficient 3 is the Curie–Weiss form = (T - T0)/(0C), T0 is the Curie temperature, C is the Curie constant, the second Note that the ID and LG2 models are equivalent near T0, coefficient obeys the condition <0, whereas that for the predicting the same temperature dependencies for necessary third coefficient is >0. The parameters T0,, spontaneous polarization, dielectric constant, and coercive field, because near TC the ID order parameter S is small and are assumed independent of temperature and electric and the free energy of the ID model can be expanded in field. From the minima of the free energy density, it is powers of PS, to reproduce the LG2 free energy.
possible to calculate the properties of the ferroelectric phase The value of the coercive field near the Curie temthat exists just below the zero-field transition temperature perature T0 obtained from the three mean-field models TC = T0 +(3/16)0C2/. In the ferroelectric phase, the (Eqs. 2–4) ranges from 10 to 20% of the depolarization spontaneous polarization is PS = ± -(1 + 1 - t)/2, field PS/0, a little less than the value 50% estimated from where t = 4/2, and the ferroelectric contribution to the basic energy considerations (Eq. 1), even as the spontaneous dielectric constant is given by 1/0 = -8+(9/4)(2/).
polarization and dielectric constant vary strongly with temThe LG1 intrinsic coercive field from just below perature. The predictions hold well near T0 for all models, T0(t = 0) up to TC(t = 3/4) is well approximated by as shown in Fig. 2. Therefore, the essential existence and Ec 2(3||/5)3/2(1 - (25/24)t) [12,17], so the coer- approximate value of the intrinsic coercive field does not Физика твердого тела, 2001, том 43, вып. 1270 V.M. Fridkin, Stephen Ducharme exp Typical experimental values of the coercive field EC compared to the depolarization field Ed for BaTiO3, TGS and BL copolymer films (all values obtained near T0) exp exp Material PS, C/m2 EC, MV/m Ed = PS/0, MV/m EC /(1/2)Ed BaTiO3  150 0.26 0.20 196 0.triglycine sulphate (TGS)  43 0.028 0.011 74 0.KD2PO4  43 0.062 0.34 163 0.PZr0.25Ti0.75O3 (PZT)100 nm thin film  200 0.38 10 215 0.polyvinylidene fluoride (PVDF)  11 0.065 55 667 0.# P(VDF : TrFE 65 : 35)  9.3 0.080 45 972 0.# P(VDF : TrFE 75 : 25) 60 nm thin film  10 0.10 125 1129 0.P(VDF-TrFE 70 : 30) 15 nm thin films [13,17] 8 0.10 480 1412 0.Ratio EC / (1/2)Ed (at T0) predicted by Model LG1 0.LG2 0.ID 0.depend on the nature of the ferroelectric transition, whether and Ohigashi  reported that copolymer films thinner it is first order or second order, displacive or order-disorder, than 1 µm showed increasing coercive fields with decreasing from permanent dipoles or induced dipoles. However, thickness (Fig. 3), down to a then-record coercive field of the measured dielectric constant includes the background 125 MV/m in a copolymer film 60 nm thick . But the contribution and near TC the ferroelectric contribution is solvent-spinning techniques used would not yield thinner typically larger than the equilibrium value due to thermal films of sufficient quality and dielectric strength, so the hysteresis (especially in first-order ferroelectrics), particu- measurement of the intrinsic coercive field lay just beyond larly in ferroelectrics with a first-order phase transition.
Therefore, the value of the depolarization field PS/0 The successful fabrication by LB deposition of ultraobtained from direct measurements near TC will likely be thin films of PVDF and copolymers with trifluoroethylene somewhat less than the intrinsic value.
(TrFE) beginning in 1995 [25–27] presented us with highly Measurements of the coercive field in bulk ferroelectric crystalline films with the polymer chains lying in the film crystals has invariably yielded values much smaller than the plane and polarization axis perpendicular to the film. These intrinsic value, as summarized in the Table. While the films, ranging in thickness from 1 nm to over 100 nm, traditional ferroelectrics extremely low coercive fields, the allowed us to study ferroelectricity down to the monolayer ferroelectric polymer polyvinylidene fluoride (PVDF) and level [12,13,25–29]. One of the surprising results from its copolymers showed considerably higher coercive fields, these studies was the existence of two-dimensional ferrotypically 50 MV/m in ”bulk” films, approaching the intrinsic electricity , as the ferroelectric phase persisted in the value predicted by the various models. By 1986, Kimura thinnest films, with no significant decrease in the transition temperature, contrary to the common expectation that finitesize scaling would suppress ferroelectricity. But finite-size effects did appear to suppress nucleation in films thinner than 1 µm [24,28]. As Fig. 3 shows, the rising coercive field with decreasing thickness d observed by Kimura and Ohigashi continues with the thinner LB films to follow a d-0.7 power law scaling down to a thickness of 15 nm , consistent with a finite-thickness suppression of nucleation or domain wall motion.
The films of thickness 15 nm (30 monolayers) or less, on the other hand, had coercive fields of about 500 MV/m at 25C, comparable to the theoretical intrinsic value  calculated with the LG1 model with free-energy coefficients determined in prior studies [30,31]. Because of uncertainties in the LG1 parameters the good coincidence between the measured and expected coercive field did not ensure that it was indeed the intrinsic value. But, cessation of Figure 2. Intrinsic coercive field Ec, normalized to the depolafinite-size scaling below 15 nm implies that polarization rization field PS/0 calculated from LG2 model, LG1 model, and reversal longer initiated by nucleation, that is, nucleation ID model.
Физика твердого тела, 2001, том 43, вып. General Features of the Intrinsic Ferroelectric Coercive Field  K. Dimmler, M. Parris, D. Butler, S. Eaton, B. Pouligny, J.F. Scott, Y. Ishibashi J. Appl. Phys. 61, 5467 (1987).
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Gordon and Breach, Amsterdam (1996). P. 135.
 A.K. Tagantsev. Integrated Ferroelectrics 16, 237 (1997).
 S. Ducharme, V.M. Fridkin, A. Bune, S.P. Palto, L.M. Blinov, N.N. Petukhova, S.G. Yudin. Phys. Rev. Lett. 84, 175 (2000).
 A.V. Bune, V.M. Fridkin, S. Ducharme, L.M. Blinov, S.P. Palto, A. Sorokin, S. G. Yudin, A. Zlatkin. Nature 391, 874 (1998).
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