A thermodynamic analysis of graded ferroelectric films demonstrates that in the equilibrium state the films are subdivided into a single-domain band and a polydomain band which consists of wedge-shape domains. Polarization under an external electrostatic field proceeds through an inter-band boundary movement due to growth or shrinkage of the wedge domains. It is shown how the domain structure and evolution are determined by the principal characteristics of the film: the distribution of the spontaneous polarization and dielectric constant. Graded films exhibit a sharp increase of polarization with the field for weak fields, with a drop of the dielectric constant when the field is increasing. A general approach to finding the dependence of the displacement and the wedge-domain shape on the field as well as analytical solutions for the p{sup 4} Landau-Devonshire and parabolic potentials are presented

Roytburd, A.

Roytburd, V.

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Domain Evolution and Polarization of ContinuouslyGraded Ferroelectric Films(Dedicated to Prof A. Khachaturian on his 75th Birthday)Alexander Roytburda and Victor Roytburdb∗aDepartment of Materials Science and Engineering University of Maryland,College Park, MD 20742; bDepartment of Mathematical Sciences,Rensselaer Polytechnic Institute, Troy, NY 12180AbstractA thermodynamic analysis of graded ferroelectric films demonstrates that in theequilibrium state the films are subdivided into a single-domain band and a polydo-main band which consists of wedge-shape domains. Polarization under an externalelectrostatic field proceeds through an inter-band boundary movement due to growthor shrinkage of the wedge domains.It is shown how the domain structure and evolution are determined by the prin-cipal characteristics of the film: the distribution of the spontaneous polarization anddielectric constant. Graded films exhibit a sharp increase of polarization with the fieldfor weak fields, with a drop of the dielectric constant when the field is increasing.A general approach to finding the dependence of the displacement and the wedge-domain shape on the field as well as analytical solutions for the p4 Landau-Devonshireand parabolic potentials are presented.Keywords: graded ferroelectrics, ferroelectric films, ferroelectric domains∗Corresponding author. Email: roytbv@rpi.edu; Phone: 703-292-2189; Fax: 703-292-903211 IntroductionGraded ferroelectric films and multilayers are highly sensitive to the electric field and tem-perature changes; this gives rise to their use for microelectronic applications [1]. However,after many years of theoretical and experimental studies there is no definitive understandingof mechanisms that determine special functional properties of graded ferroelectrics. Most ofthe theoretical work in this area relates fundamental properties of graded ferroelectrics tothe short-range polarization self-interactions, which are represented by the gradient energyterm in the Landau-Devonshire potential. These interactions are essential if the gradients ofpolarization are comparable to those in the domain walls. The effects of smaller gradientsmay be important in the absence of strong long-range electrostatic interactions, for example,in ferroelectrics with high concentration of free charges [2].A different approach was presented in [3] where the dominant role of the long-rangeinteractions was suggested. It has been shown that minimization of electrostatic energy of agraded multilayer results in the formation of a special wedge-domain structure. Polarizationof multilayers proceeds through movement of domain walls with progressive transformationof polydomain layers into the single-domain ones. The theory of [3] provides the principalexplanation of dielectric properties of graded multilayers, in particular it explains a dramaticincrease of the dielectric constant with the field decrease.This thermodynamic theory does not take into account short-range interactions, thusneglecting the domain walls energy. However the comparison of the theoretical results withphase-field modeling that does include all interactions shows a good qualitative agreement[4]. It confirms that electrostatic interactions do dominate formation and evolution of thedomain structure in graded ferroelectrics.In this paper we extend the concept of wedge domains to continuously graded ferroelectricfilms. We will consider the equilibrium state of a ferroelectric film, with its parameters(spontaneous polarization, electric susceptibility, etc.) changing continuously throughoutthe film thickness. For simplicity we treat the case of a uniaxial ferroelectric.2 TheoryThe equilibrium state of a single-domain film is determined by the minimum of the Gibbsfree energyF (P (X)) =∫ L0[f(P (X), X) + (1/2ε0)(P (X)− P¯)2 − EP (X)] dX, (1)where L is the film thickness; P (X) is the polarization in the layer X; P¯ is the averagepolarization, P¯ = L−1∫ L0P (X) dX; E is the external electrostatic field generated by theelectrodes under the potential U = EL (the short-circuit condition is being considered).The free energy density f(P ) is given by the Landau-Devonshire potential; it describes themicroscopic energy of the ferroelectric for the given value of the order parameter P. Thispotential determines spontaneous polarization in each layer P0(X). To avoid complicationsdue to the elastic interaction between the layers we consider a film clamped by substrate;thus we use the Landau-Devonshire potential that is normalized to include the elastic effect2of the substrate. The other two terms in (1) represent the macroscopic contribution of theelectrostatic field to the Gibbs free energy: the electrostatic energy of the interaction betweenthe layers, and the work of the electric field.For the polydomain film, which is an alternation of two domains with polarizationsof opposite directions, the macroscopic terms preserve their form while the microscopiccomponent becomes a weighted average of the single-domain energies:F (P ) =∫ L0[(1− β) f(P1(X), X) + βf(P2(X), X) + (1/2ε0)(P˜ − P¯)2− EP˜]dX, (2)where β(X) and 1 − β(X) are the fractions of the domains with the opposite directionpolarizations P2 and P1. The macroscopic polarization in the layer X is given by P˜ =(1− β)P1 + βP2 while P¯ here is the average of P˜ .We will show that for the ferroelectric film the equilibrium state has a band structure,with alternating bands of single-domain layers and of polydomain layers. This band struc-ture is completely determined by the microscopic potential f and by the applied field E.Under variation of the external field the film polarization changes due to the motion of theboundary between the bands, which in its turn is caused by the single domain polydomaintransformations of the layers.To analyze the simplest band architecture we consider the spontaneous polarization pro-file P0(X) which increases monotonically from P0(0) = P0 on one surface of the film toP0(L) on the other. We will prove that in this case the film will subdivide into two bands:a single-domain band at 0 < X < L, and a polydomain one at L < X < L, Fig. 1. In thedimensionless form the Gibbs free energy for the two-band film is as follows:F(p, E) = F/ (LP20/ε0) = F1 + F2 = ∫ ξ0[φ(p, x)− Ep(x) + (p(x)− p¯)2 /2] dx+∫ 1ξ[(1− β(x))φ(p1(x)) + β(x)φ(p2(x))− E p˜(x) + (p˜(x)− p¯)2 /2]dx (3)where x = X/L, ξ = L/L, φ = f/ (P20/ε0) , E = E/ (P0/ε0) , p(x) = P (xL)/P0, p1(x) =P1(xL)/P0, p2(x) = P2(xL)/P0, p˜(x) = (1− β(x)) p1(x) + β(x)p2(x), p¯ =∫ ξ0p(x)dx +∫ 1ξp˜(x)dx. Thus the Gibbs free energy depends on ξ and on the following functional pa-rameters: p(x), p1(x), p2(x), and β(x).For the polarization fields p(x), p1(x), and p2(x) to realize the minimum of the Gibbsfree energy they have to satisfy the following variational equations. First of all, in thesingle-domain band (x < ξ) ,δF1/δp = ∂φ/∂p− E + p(x)− p¯ = 0. (4)In the polydomain band (ξ < x < 1)δF2/δβ = −φ(p1, x) + φ(p2, x) + E (p1 − p2)− (p˜(x)− p¯) (p1 − p2) = 0. (5)Variations with respect to p1 and p2 yieldδF2/δp1 = (1− β) [−E + p˜(x)− p¯] = 0 (6)δF2/δp2 = β [∂φ/∂p2 − E + p˜(x)− p¯] = 0 (7)3In view of (5)-(7),∂φ/∂p1 = ∂φ/∂p2 = (φ(p2, x)− φ(p1, x)) / (p2 − p1) = E − p˜(x) + p¯ (8)Geometrically these equations mean that the tangents to φ at p1 and p2 coincide. Since φis the symmetric two-well potential with equal minima at p(x) = ±p0(x), the only values p1and p2 for which equations in (8) hold arep1 = −p2 = p0. (9)From the condition that the variational derivative of the Gibbs free energy with respect toξ is zero, it is easy to show that the polarization should be continuous through the interfacex = ξ and therefore β(ξ) = 0. It immediately yields that for ξ < x < 1,p˜(x) = (1− 2β(x)) p0(x) = (1− 2β(ξ)) p0(ξ) = p0(ξ). (10)Moreover, the common value of all the equations in (8) is 0; in particular, E − p˜(x) + p¯ = 0,i.e., for ξ < x < 1,D(E) = E + p¯ = E +∫ ξ0p(x)dx+ (1− ξ)p0(ξ) = p0(ξ). (11)Given D = p0(ξ), the fraction β is found explicitly from (10):β = 1/2−D/2p0(x) (12)The expression (11) for D is deceivingly simple. What is hidden in this expression isthat the inter-band boundary position ξ is related to E and that p actually depends on E ,p = p(x; ξ(E)) being a solution of Eq. (4), which in view of (11) takes the form∂φ/∂p+ p(x)− p0(ξ) = 0. (13)However, if ξ is taken as a departure point instead of E then D = E + p¯(E) versus E canbe found quite easily as a parametric curve. Suppose ξ is the optimal position of the bandinterface for a given E . Thenp(x) ={p(x; ξ), x < ξp0(ξ), 1 > x > ξ(14)where p(x; ξ) is a solution of Eq. (13), which for an arbitrary φ(p(x), x) can be easily solvednumerically, say, by a Newton’s method. Then the field is determined fromE(ξ) = D − p¯ = p0(ξ)−∫ ξ0p(x; ξ)dx− (1− ξ)p0(ξ) = ξp0(ξ)−∫ ξ0p(x; ξ)dx (15)Eqs. (11)-(15) define D versus E as a parametric curve.So far we considered the case of two bands, i.e., when the band interface ξ < 1. If ξ = 1,then the free energy is given by (1) and in the dimensionless form the variational problemsbecomes as followsF1(p, E) =∫ 10[φ(p, x)− Ep(x) + (p(x)− p¯)2 /2] dx→ min (16)4Then p(x)is found from Eq. (4), which in this case yields∂φ/∂p−D + p(x) = 0 (17)The corresponding electrical field is then equal toE = D −∫ 10p(x;D)dx (18)This latter formula gives the relation between E and D for the one-band case.In the next sections we present analytical solutions for two important cases, a simpleform of the Landau-Devonshire potential, and its parabolic approximation.3 Landau-Devonshire potentialIf φ is a p4 Landau-Devonshire potentialφ(p, x) = A(x)(−p22+p44p20(x)), A(x) =12 (ε(x)− 1) (19)then Eq. (13) is cubic and can be solved explicitly via Cardano’s formulae. Fig. 2 illustratesthe dependence of the displacement on the field for three spontaneous polarization profiles.The curves for the single-domain films are shown for comparison. It is clear that the poly-domain band is responsible for the sharp increase of the dielectric constant at low fields.This results from the absence of the field in the polydomain band and its concentration inthe single-domain one. This effect can be interpreted as the decrease of the effective filmthickness with the field. The apparent singularity at E = ′ is the consequence of the single-domain band thickness tending to zero. In actual graded films the layers with minimumspontaneous polarization which remains in the single-domain state have finite thickness andthe dielectric has a large however finite magnitude.In Fig. 3 we present the wedge-domain shape evolutions that correspond to the polariza-tion of graded film from Fig. 2. This domain shapes are calculated according to Eq. (12)that can be restated as followsβ = 1/2− p0(ξ)/2p0(x) (20)The dependence of β on the coordinate x describes how the thickness of the wedge domainincreases with the distance from the band interface; at the interface, x = ξ, the thicknessgoes to zero. The shape of the domain tip depends on the characteristics of the gradedfilm, in particular on the polarization profile p0(x). Under the weak field, the domain tiphas a rounded shape for p0(x) ∼ x1/2, a cusp for p0(x) ∼ x2, and makes an acute anglefor p0(x) ∼ x. As the filed increases the tip shape differences become less pronounced.Similarly to the crack propagation the domain tip shape is important for the kinetics ofdomain evolution. However, this issue is beyond the scope of the thermodynamic study ofthe present paper; it will be discussed elsewhere.54 Parabolic potentialIn practically important cases of moderate gradients of spontaneous polarization, the poten-tial is well-approximated by a quadratic functionφ = (p(x)− p0(x))2 /2 (ε(x)− 1) . (21)Then the solution is particularly simple:p(x) ={(1/ε(x)) p0(x) + (1− (1/ε(x))) p0(ξ), x < ξ0p0(ξ), 1 > x > ξ(22)For D we get theD = p0(ξ) = E/∫ ξ0dxε(x)+∫ ξ0p0(x)ε(x)dx/∫ ξ0dxε(x)(23)(a version of this formula for graded multilayers was introduced in [3]). Eq. (23) can also berestated as follows:E = p0(ξ)∫ ξ0dxε(x)−∫ ξ0p0(x)ε(x)dx =∫ ξ0(dp0(x)dx∫ x01ε(s)ds)dx (24)It relates ξ and E through the basic characteristics of the graded film, its gradient of spon-taneous polarization and dielectric constant distribution.For example, if the polarization and susceptibility are given by p0(x) = 1 + kxn, n > 0,and 1/ε(x) = 1/εmax(1 + χxm) then ξ(E) is determined from the following equationξn+1 + χξn+m+1 = (εmax/k) E (25)It clearly shows that for relatively weak fields where ξ  1 the first term in (25) domi-nates. Then ξ as well as D are determined by the polarization gradient dp0/dx, and in thisapproximationξ = (εmaxE (n+ 1) /nk)1/(n+1) , D = 1 + k1/(n+1) [εmax (n+ 1) E/n]n/(n+1) (26)In particular, if n = 1, then ξ =√2εmaxE/k and D = 1 +√2εmaxEk.On the other hand, the field Ec corresponding to the completely transformed state (atξ = 1) depends also on the distribution of the dielectric constantEc = (k/εmax) {[n/ (n+ 1)] + χ [n/ (m+ 1) (m+ n+ 1)]} (27)A similar dependence of the dielectric response of the graded film on its characteristics existsfor the p4 potential and is expected for a more general form of Landau-Devonshire energy.In Fig. 4 we compare the D(E) curves for the p4 potential with p0(x) = 1 + 0.5x and twodifferent dielectric constant profiles (the constant is select as the average of the variableprofile). It confirms the above mentioned effects of the polarization and susceptibility on thepolarization curve.65 Concluding remarksThe analysis above has been focused on the two-band structure; however, it can be easilygeneralized to the multi-band structure. By using a non-monotonic spontaneous polariza-tion profile it is possible to obtain multi-band wedge domain structures, with nonsmoothD(E) dependences. Our thermodynamic approach used the homogenization approximation,which is valid if the domain structure period is much smaller than the domain thickness.This corresponds to a relatively small contribution of the domain walls into the graded filmthermodynamics. Since the ratio of the domain wall energy to the characteristic electro-static energy (P20/ε0) is on the order of√l/L, where l is the domain wall thickness, thehomogenization is an adequate approximation for thermodynamics of the films no thinnerthan tens of nm. This approximation does not take into account hysteresis effects due to thegeneration [5] and annihilation of domain walls during the wedge domain evolution. How-ever the wedge domain walls are expected to be highly mobile because they deviate from theclose-packed plane orientation and do not need to overcome energy barriers between discretelayers, in contrast with the graded multilayers [3]. If the complete transformation of the filmto the single-domain state is not attained (in particular, for films with strong polarizationgradients) then the domain evolution can be highly reversible. Thus, the thermodynamicapproach of this paper provides useful insights into understanding the behavior of gradedferroelectrics. Results of this paper open possibilities of designing graded films with desirabledielectric properties.Acknowledgments. A.R. gratefully acknowledges the support of the Israel-USA Bi-national Science Foundation. V.R.’s work was partially supported by the National ScienceFoundation, while working at the Foundation. Part of his work was done while on a sabbat-ical leave at the Lawrence Berkeley National Laboratory.References[1] J. V. Mantese and S. P. Alpay, Graded Ferroelectrics, Transpacitors and Transponents(Springer, New York, 2005)[2] Z.-G. Ban, S. P Alpay, and J. V. Mantese, Phys. Rev. B 67, 184104 (2003)[3] A. L. Roytburd and J. Slutsker, Appl. Phys Lett. 89, 042907 (2006).[4] A. Artemov et al, J. Appl. Phys. 103, 074104 (2008).[5] A. M. Bratkovsky and A. P. Levanyuk, Phys. Rev. B 66, 184109 (2002).7Figure CaptionsFigure 1: Two-band film structure.Figure 2: The polarization curves D(E) for A(x) = 0.005 and for three model polarizationprofiles: p0(x) = 1+ 0.5x (thin line); p0(x) = 1+ 0.5x2 (dot-dashed line); p0(x) = 1+ 0.5√x(dashed line). The level D ≥ 1.5 and the return (higher) curves correspond to the single-domain state.Figure 3: Wedge shapes for the three positions of the band interface ξ = 0, 1/3, 2/3, andfor the three model polarization profiles: (a) p0(x) = 1 + 0.5x; (b) p0(x) = 1 + 0.5x2; (c)p0(x) = 1 + 0.5√x.Figure 4: Comparison of D(E) curves for the p4 potential with p0(x) = 1 + 0.5x; A(x) =0.00625 (dashed line) and A(x) = 0.005 (+0.5x) (solid line); these parameters correspond tothe PZT-film with the composition (1− x)PbTiO3 + xZrTiO3, x = 0.0-0.3.80 1 2 3 4 5 6x 10−311.11.21.31.41.51.6ED0.4 0.5 0.600.20.40.60.81ξ(a)0.4 0.5 0.6(b)0.4 0.5 0.6(c)0 1 2 3 4 5 6 7x 10−311.11.21.31.41.51.6ED

Domain evolution and polarization of continuously graded ferroelectric films

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Domain evolution and polarization of continuously graded ferroelectric films

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