Curie temperature and critical thickness of ferroelectric thin films

The dynamic Ginzburg–Landau theory is applied to establish the critical conditions that control the transition between the paraelectric and ferroelectric states. Analytic expressions of the para-ferroelectric transition temperatures in a thin film under various electromechanical surface conditions are derived via a linear stability analysis of the evolutionary trajectory of the system for both firstand second-order transitions. Explicit expressions are then derived for the critical thickness, below which the thin film is paraelectric for all temperatures. For first-order transitions, the difference between the superheating and supercooling transition temperatures is found to be insensitive to the film thickness and surface boundary conditions. From these expressions, the relative importance on ferroelectricity in thin films due to applied mechanical constraints on the transformation strain and the depolarizing effect of surface charges is discussed and compared with experimental data. ©2005 American Institute of Physics . fDOI: 10.1063/1.1861517 g


I. INTRODUCTION
Ferroelectricity is a collective phenomenon, the characteristics of which depend on the combined effects of many factors, such as the ambient temperature, boundary conditions, sample dimensions, misfit epitaxial stresses, etc.2][3][4][5][6][7] Recently, research in this area has found renewed relevance due to the surge of technological interest in ultraminiaturized electronics, ultrahigh-density memory devices, and nanotechnologies in general.
Characteristics governing the para-ferroelectric transitions in a film have been investigated within the framework of both thermodynamic theory and first-principles calculations.However, various discrepancies between experimental measurements and theoretical calculations and among the theoretical calculations themselves may be the result of incomplete understanding.As an example, the critical thickness of PbTiO 3 thin films at room temperature has been estimated at ϳ5.5 nm, within the phenomenological theory. 7owever, recent experimental measurements show that stable polarization orthogonal to the surface can exist down to a 4-nm-thick lead zinconate titanate ͑PZT͒ film. 3Firstprinciples calculations 2 even found that an orthogonally polarized ferroelectric state can be maintained under shortcircuit electric boundary conditions, even for a film as thin as 1.2 nm ͑three unit cells͒.Despite extensive investigations [1][2][3][4][5][6][7][8] that have been carried out on the size-dependent Curie temperature and critical thickness, its nature and the physical reason of its existence remain uncertain.For example, although it is fairly plausible that the critical thickness origi-nates from the depolarizing effect of the film surface, information on the details of the competing mechanisms, their relative importance, and their relations with various electromechanical surface conditions are still needed.Except for the simplest cases, numerical calculations using either first-principles 2 or thermodynamic approaches 7,8 are the main theoretical tools for explorations in this area.In the latter case, characteristics of the para-ferroelectric transformation have often been obtained from the condition of phase equilibrium at the transition point.This approach has been proven effective for continuous ͑second order͒ phase transitions, where the order parameter is continuous and well defined at the point of transition.However, the case is more complicated for discontinuous ͑first order͒ transitions, where the order parameter is discontinuous and not well defined at the point of transition.
Alternatively, phase transition may also be described in terms of the time evolution of the order parameter as an irreversible dynamical process governed by the evolution equation.The latter can be formulated using the Ginzburg-Landau functional 9 expressed in terms of a set of control parameters ͕͖, including the ambient temperature, boundary conditions, surface characteristics, sample dimensions, misfit epitaxial stresses, etc.Within the dynamical description, 10 the evolution of the system can be pictured as the trajectory of a moving "particle" in the order-parameter space, starting from an initial point ͑i.e., the order parameter at t =0͒ representing the initial state and ending towards a point representing the final state ͑i.e., the order parameter at t = ϱ͒.The trajectory and the final state are functions of ͕͖, obtained by solving the evolution equation.This operation, however, is generally rather difficult because of the nonlinear nature of the problem. 10While the numerical approach has been followed for focused problems involving a relatively small number of control parameters, the comprehensive understanding of a multiparameter, multimechanism system is not easily achievable outside an analytic approach.
It is clear that the final state is a stationary state ͑i.e., with a fixed order parameter͒ that must be asymptotically stable. 10][11] Without loss of generality, we focus on the paraelectric to ferroelectric transformation for easy discussion.In this case, the GLE has two stationary solutions, with order parameters corresponding to the two states.Suppose we start with an initial state that is paraelectric.If the corresponding stationary solution is asymptotically stable, the final state must remain paraelectric, and no transition to the ferroelectric state will occur.This is the case even if both the paraelectric and the ferroelectric states are asymptotically stable at the same time.Thus, a necessary condition for the transition to occur is that the stationary paraelectric solution is unstable.This provides us with the means to determine the range of ͕͖, within which the paraelectric state is stable, and outside which the paraelectric state may transform into the ferroelectric state, i.e., the phase diagram.The foregoing approach allows important characteristics of a ferroelectric thin film, such as the transition temperatures, critical thicknesses, domain morphology, substrate nature, etc., to be related through the instabilities of the initial state, the linear nature of which opens the system to many established analytic techniques.
In the present paper, we follow the analytic approach described in the foregoing to establish conditions of the stability of the paraelectric versus ferroelectric phases in a thin film.The polarizing/depolarizing effects due to the presence of the film surface, namely, the surface lattice relaxation, the restraint of the transformation strain, the electromechanical surface conditions involving the epitaxial stress, and the induced surface charges, are taken into account.Expressions for the para-ferroelectric transition temperatures as a function of film thickness and the corresponding critical thickness are derived, and the complex relation of the critical thickness with the various parameters discussed.Both firstorder and second-order transitions are considered.

II. THE EVOLUTION EQUATION AND STABILITY OF THE STATIONARY STATES
We consider a thin film of ferroelectric material of dimensions of ϱϫϱϫh, h being the film thickness.The origin of the coordinate system is at the center of the cell.We assume that P, the single polarization component normal to the surface of the film, is the order parameter to describe the phase transition.In general, the transition between the cubic and tetragonal phases involves three polarization components related through a system of three coupled nonlinear differential equations, which have to be analyzed together.Nevertheless, this is beyond the scope of the present paper.
Noting that a para-ferroelectric transformation is caused by a transformation between a central symmetric and a noncentral symmetric lattice structure, the transition is always accompanied by a transformation strain.When the ferroelectric state under consideration has a tetragonal lattice structure, the transformation strain in the plane of the film can be expressed in the form of T = xx T = yy T = QP 2 , where Q is the electrostrictive coefficient.Accordingly, the free energy of a ferroelectric thin film is made up of mechanical and electrical components.The mechanical component contains the elastic self energy due to the transformation strain and the interaction energy between the applied stress and the transformation strain.For rigid substrates, the transformation strain is fully constrained, and the elastic self energy is given by F e = GQ 2 ͐͐ V ͐P 4 dV, where G = ͑C 11 + C 12 −2C 12 2 / C 11 ͒, C 11 and C 12 being components of the elastic modulus of the film and V the volume of the film.We note that due to the free upper surface of the films, the component xx T of the transformation strain does not contribute to the elastic energy.Fabrication processes often leave a residual stress distribution in the film due to the lattice mismatch, the epitaxial stress.Its interaction with the transformation strain is given by where xx = yy = r .The electrical component of the free energy can be expressed in terms of the order parameter P and the temperature using the Ginzburg-Landau functional 9 of the bulk material.Including the effects of the depolarization field E ¯d = ͑0,0,E d ͒ and the surface effects, the Ginzburg-Landau free energy in the neighborhood of the transition point can be expressed as where A , B , C, and D are the expansion coefficients of the corresponding bulk material under the zero-stress state.T c0 is the cooling phase-transition temperature of the bulk crystal, S represents the upper and lower surface planes that cover the entire surface of the film, and ␦ is the extrapolation length that measures the effect of lattice relaxation on the surface or the change of the polarization due to the surface effect. 4,12The total Ginzburg-Landau free energy of the cell can be written as a sum of the electrical and mechanical components, 7 Two separate cases may be considered: the more common ␦ Ͼ 0 case, corresponding to a reduction of self-polarization on the surface, and the rarer ␦ Ͻ 0 case, corresponding to an enhancement.We note that in general ␦ is not an absolute constant.Furthermore, effects of the epitaxial stress and the surface lattice relaxation may contribute to or against the transformation.
Another important surface effect is the depolarization field that comes from the polarization-induced surface charges, which is governed by the electric boundary conditions.In this paper, we consider two cases: ͑a͒ the ferroelectric film is sandwiched between two metallic electrodes in short-circuit conditions and ͑b͒ the ferroelectric film is sandwiched between two dielectric substrates.
Although perfect screening is not achieved even under the short-circuit condition, partial compensation by the electrodes still reduces the depolarization field, due to the variation of the spontaneous polarization in the z direction.In this case, the relation between the depolarization field E d and the spontaneous polarization P is given by 12

͑2͒
where is the dielectric constant of the film, and the second term on the right-hand side is due to the compensating charges on the upper and lower metallic electrodes.For dielectric substrates, there are no compensation charges, and the corresponding relation is given by ͑see the Appendix͒ We may write down a general expression that encompasses both contact conditions, Pdz ͪ ,

͑4͒
where takes on the value of 0 or 1 depending on the contact condition as discussed in the foregoing.Using Eqs.͑1͒ and ͑4͒, the time evolution of the system is governed by the time-dependent Ginzburg-Landau equation: 7,8 where M is the kinetic coefficient related to the domain-wall mobility.The surface term in Eq. ͑1͒ yields the boundary conditions, Equation ͑5͒ has a trivial stationary solution, P = 0, representing the paraelectric state.As discussed in the Introduction, to be able to transform from the initial paraelectric ͑P =0͒ state to the ferroelectric ͑P 0͒ state, the paraelectric stationary state must become unstable.The same also applies to the reverse transformation.
The dynamic stability of the stationary states can become probed by applying an infinitesimal perturbation ⌬ to the corresponding stationary solution of Eq. ͑5͒.The dynamics of ⌬ follows from Eq. ͑5͒ by retaining only the terms linear in ⌬, which are given by ‫⌬ץ‬ ‫ץ‬t

͑7͒
The boundary condition is the same as Eq.͑6͒, only with P replaced by ⌬.For a symmetric configuration, the condition d⌬ / dz =0 at z = 0 must also hold.This equation governs the stability of the stationary solution of Eq. ͑5͒.It is a linear equation of ⌬.Its analysis in the context of phase stabilities, without having to resort to a solution of the nonlinear equations ͓Eq.͑1͒ or ͑5͔͒, is our main aim.It is obvious from Eq. ͑7͒ that the equation governing ⌬ in the paraelectric and ferroelectric states are different in general, from which it follows that the stability conditions for the two branches are also different.However, as we shall see, for second-order transitions, P is continuous at the transition point.The ferroelectric and paraelectric branches are both unstable when crossing this point from opposite directions and transform into one another.The Curie temperature and the critical thickness are then independent of the initial state and are thus well defined.
For first-order transitions, on the other hand, P is discontinuous across the transition temperature.The paraelectric ͑P =0͒ and the ferroelectric ͑P 0͒ branches obey different stability conditions due to the difference in the governing equations of ⌬ in Eq. ͑7͒.Suppose the paraelectric branch is dynamically stable for temperatures down to T c and the ferroelectric branch ͑P 0͒ is dynamically stable for temperatures up to T ch , then the two critical temperatures are likely to be unequal.Indeed, depending on the direction of the transformation between the two states, T c is called the supercooling transition temperature and T ch the Curie temperature ͑superheating transition temperature͒.

III. STABILITY OF THE PARAELECTRIC STATE AND CRITICAL CHARACTERISTICS
In view of its simplicity, the case in which the initial state is paraelectric, i.e., P = 0, is considered first.Transition from the ferroelectric state ͑P 0͒ will be considered in Sec.IV, where it will be shown that the two transition temperatures are related by a constant shift.We do not differentiate between the first-and second-order transitions, to which this analysis applies equally.In the following, the cases ␦ Ͼ 0 and ␦ Ͻ 0 are separately considered.

A. Reduced surface polarization: The ␦ > 0 case
In most ferroelectric materials, lattice relaxation weakens the polarization on the surface, and ␦ Ͼ 0. Using the method of separation of variables and taking into account d⌬ / dz =0, at z = 0, Eq. ͑7͒ can be separated into time-dependent and time-independent parts, connected by the eigenvalue.For P = 0, the solution can be written as where c is the eigenvalue and c ͑z͒ the corresponding eigenfunction of Eq. ͑7͒, and R a constant given by The P = 0 solution is unstable when c Ͼ 0, because in this case, ⌬ increases exponentially with time.It can be seen from Eq. ͑9͒ that when the temperature T is sufficiently high, c Ͻ 0, and the paraelectric state is stable.When T is sufficiently low, c turns positive and the paraelectric state is no longer stable, since any small perturbation ⌬ will grow exponentially beyond all bounds.The critical condition c =0, yields, in this case, the supercooling transition temperature T c of the film, where the first term on the right-hand side is related to the bulk ferroelectric property, the second term to the misfit of the substrate, the third term to the induced surface charges, and the fourth term to the lattice relaxation on the surface.While the last two terms act against the transformation by lowering the transition temperature, the second term can act either way, depending on the direction of the misfit, i.e., the sign of r .Here k c depends on the film thickness h, the extrapolating length ␦, and the electric boundary condition through , as the smallest nonzero root of the equation, The general relation between T c and the film thickness h can be obtained by solving Eqs.͑11͒ and ͑12͒ simultaneously.Equation ͑12͒ is a transcendental equation that can easily be solved numerically in general.Furthermore, as we shall show in the following, simple analytic approximations can also be derived in many important cases.The cooling critical thickness ͑CCT͒ of the film h cc , below which ferroelectric transition by cooling is not possible, can be derived by putting T c = 0 in Eq. ͑11͒ , yielding the following expression for the corresponding k c 2 , which we denote by k cc 2 in terms of the material properties of the film: ͑13͒ h cc can then be obtained by solving for h as the smallest positive root of Eq. ͑12͒ corresponding to T c = 0, with k cc given by Eq. ͑13͒.Since h cc varies inversely with k cc according to Eq. ͑12͒, Eq. ͑13͒ then requires that the CCT for these materials increase with decreasing bulk transition temperature and substrate constraint, and with increasing depolarization field.We further note that, for k cc to be real, ͑AT c0 +4Q r ͒ Ͼ 1 must be satisfied, putting a necessary condition on the possibility of the cooling transition from the paraelectric state.

Asymptotic solutions for thin films "i.e., h ™ ␦…
When hk c Ӷ 1, Eq. ͑12͒ can be solved via a series expansion of the left-hand side, yielding from which T c is given by where G c is a function of T c , When h Ӷ ␦, the condition k c h Ӷ 1 is satisfied from Eq.
͑14͒.Thus, T c in Eq. ͑15a͒ can be solved explicitly to give Thus, independent of the contact condition, the critical temperature decreases monotonically as the film thickness h decreases.The corresponding CCT can be obtained by putting T c = 0 in Eqs.͑16a͒ and ͑16b͒, and solving for h, Equations ͑17a͒ and ͑17b͒ show that the CCT decreases with decreasing surface relaxation ͑i.e., increasing extrapolation length ␦͒, for both contact conditions.In addition, the short-circuit boundaries, with a smaller depolarization field, gives a lower CCT than the dielectric boundaries.Furthermore, h cc may become negative if r is negative and large.Both results can be predicted from physical considerations.

Asymptotic solutions for thick films "i.e., h š ␦… with G c ™ 1
In the neighborhood of k c h Ϸ , the left-hand side of Eq. ͑12͒ can be approximated by cot͑x͒Ϸ͑ /2͒ − x.This approximation yields Since the condition k c h Ϸ is satisfied when h ӷ ␦ and G c Ӷ 1, k c in Eq. ͑18͒ is a solution of Eq. ͑12͒ for thick films with a small depolarization field.In this case, an explicit relation of T c can be obtained from Eq. ͑11͒, which, taking into account ␦ Ӷ h and G c Ӷ 1, can be solved for T c to give .

͑19͒
The behaviors of T c in Eqs.͑16a͒, ͑16b͒, and ͑19͒ are similar.
As the film thickness h is reduced, its critical temperature decreases monotonically.
The corresponding CCT is given by It is interesting to note that, even for thick films, the constraint of the substrate and the surface relaxation cannot be neglected as long as there is a surface in the sample.Furthermore, h cc may become imaginary if r is negative and sufficiently large.

B. Enhanced surface polarization: The ␦ < 0 case
In rare cases, polarization may be enhanced, instead of reduced, on the surface of some materials.In this case, ␦ Ͻ 0, and, similar to the ␦ Ͼ 0 case, discussed in the foregoing, the solution of Eq. ͑7͒ can be written as The eigenvalue c is given by The eigenfunction is given by c ͑z͒ = ⌬ 0 ͓cosh͑kz͒ + R͔, with The cooling critical temperature T c is obtained by putting c = 0 in Eq. ͑22͒, where k c is related to the film thickness through the boundary conditions in ͑6͒, as the largest root, corresponding to the highest T c , of the equation where we have defined ␥ ϵ −␦ Ͼ 0. Similar to the ␦ Ͼ 0 case, h and k c can be shown to vary inversely with each other in general.This behavior in Eq. ͑24͒ leads to a value of T c that decreases as the film thickness h increases, i.e., opposite to the ␦ Ͼ 0 case.Physically, the reason for this behavior is clear, as caused by the enhanced polarization on the surface.
The major difference between the ␦ Ͼ 0 and ␦ Ͻ 0 cases lies in the sign for the polarization gradient in the two cases.
In the ␦ Ͻ 0 case, the enhanced surface polarization, as represented by the positive gradient term, together with the misfit interaction of the transformation strain ͑if r Ͼ 0͒, favors the ferroelectric state over the paraelectric state.From Eq. ͑24͒, it is clear that unless the depolarization field caused by the surface-induced charges ͑the −1 term͒ is sufficiently large, T c cannot be zero as long as a solution of k c in Eq. ͑25͒ exists, independent of the film thickness.
The foregoing results show that, if the polarization is enhanced on the surface, i.e., ␦ Ͻ 0, a positive value of the CCT in ferroelectric materials need not exist, and in such a case, the transition to the ferroelectric state by cooling is always possible.In this case, the existence of the CCT depends on whether the depolarization field due to the surfaceinduced charges is sufficiently large.For the more common ␦ Ͼ 0 case, on the other hand, this condition does not have to hold for the CCT to exist.

Asymptotic solutions for thin films, i.e., h ™ ␥
Similar to the case of ␦ Ͼ 0, analytic approximations of the cooling transition temperature and the CCT can also be derived.
When hk c Ӷ 1, expanding the left-hand side of Eq. ͑25͒ to second order in hk c , the solution can be written as so that The validity of solution ͑26͒ is satisfied when h Ӷ ␥.An explicit solution of T c from Eq. ͑27͒ is given by These expressions show explicitly that, independent of the contact condition, the critical temperature increases monotonically as the film thickness h decreases.This is opposite to the ␦ Ͼ 0 case.At the same time, T c for = 1 is positive definite for any film thickness h Ͻ ␥ / 6.In such a case, the CCT does not exist.From Eq. ͑28a͒, when the depolarization field is sufficiently large, i.e., ͑AT c0 +4Q r ͒ Ͻ −1 , a solution of h cc for T c = 0 exists when = 0.This is given by It can be seen from the foregoing that the behavior of h cc depends strongly on the boundary contact conditions in this case of ␦ Ͻ 0.

Asymptotic solutions for thick films "h š ␥… with G c ™ 1
When the solution lies in the neighborhood of k c h Ͼ , the left-hand side of Eq. ͑25͒ can be approximated by coth͑x͒Ϸ1.This approximation yields that satisfies k c h Ͼ when h ӷ ␥ and G c Ӷ 1.An explicit expression of T c can be derived by substituting into Eq.͑24͒, , for = 0 and = 1.͑31͒ In both cases, T c is independent of h, and the critical thickness does not exist.At this point, we note that the analytic results for second-order transitions, 8 obtained using the static Ginzburg-Landau equation, are consistent with the present results.Our results are also applicable to first-order transitions from the paraelectric branch to the ferroelectric branch, yielding, instead of the Curie temperature, the stability limit of the paraelectric phase during cooling, i.e., the supercooling transition temperature.To obtain the Curie temperature and the critical thickness, in the following, we will analyze the stability of the ferroelectric branch and investigate the stability limit of the ferroelectric phase on heating, i.e., transitions from the ferroelectric branch to the paraelectric branch.

IV. STABILITY OF THE FERROELECTRIC STATE AND TRANSITION CHARACTERISTICS
In the ferroelectric state, the stationary polarization field P͑z͒ is nonzero, and the corresponding stability equation ͓Eq.͑7͔͒ is no longer the same as that in the paraelectric state.Treating the P-dependent terms as a perturbation and expanding in terms of the complete orthonormal set of eigenfunctions of the unperturbed equation, Eq. ͑7͒ can be solved within the first-order perturbation approximation.The eigenvalue h in this case is given by where T c is the supercooling temperature obtained in the foregoing section, and ͗P n ͘ is the expectation value of P n given by where c is the eigenfunction of the unperturbed state.
Since both P and peak at z = 0, we may approximate ͗P n ͘ by P 0 n = P n ͑0͒.This approximation is also valid when P is a slowly varying function.Thus, we may write where p 1 2 and p 2 2 are the two roots of Eq. ͑34͒ for h =0.We note that p 1 2 and p 2 2 can be both real or are complex conjugates.In the simplest case that they are real, the stability condition of the ferroelectric state is satisfied if P 0 2 lies between p 1 2 and p 2 2 .In the following, we will consider several specific cases.
Let us first consider the case in which C Ͼ 0. We can write down the solutions as

͑35͒
Here d 2 is the discriminant of the quadratic in Eq. ͑34͒, given by Note that the roots p 1 and p 2 are both functions of temperature through d.Let us define a temperature T ch by For temperatures above T ch , d 2 Ͻ 0 and h has the same sign as A͑T − T c ͒, i.e., positive definite.Thus, the ferroelectric system will be unstable at temperatures above T ch .
In the temperature range T c Ͻ T Ͻ T ch , the product p 1 2 p 2 2 , from Eq. ͑34͒, is equal to A͑T − T c ͒ /5C, which is positive definite ͑C Ͼ 0͒.In this temperature regime, it can be shown that d 2 Ͼ 0, and the roots in Eq. ͑35͒ are real and have the same sign.In addition, if 4GQ 2 + B Ͻ 0, it can be seen from Eq. ͑35͒ that both roots are positive.In this case, both p 1 and p 2 are real, and the ferroelectric state is stable when P 0 lies between p 1 and p 2 .We note that at T = T ch , d 2 = 0 and p 1 = p 2 so that P 0 is uniquely defined.At T = T c , the smaller solution becomes zero.Within this temperature regime, which is above the cooling para-ferroelectric transition temperature T c , the ferroelectric state is still stable, and T ch can be identified as the superheating transition temperature.This difference between the heating-up and cooling-down behaviors is well known for first-order phase transformations near the critical point, and is caused by the restraint on the realization of the transformation strain, in the present case by the rigidity of the substrate through the 4GQ 2 term.
In the case when 4GQ 2 + B Ͼ 0, both p 1 2 and p 2 2 are negative, and P 0 has no real solution in this temperature regime.In this case, the ferroelectric state does not exist for temperatures above T c , and the ferroelectric state cannot be superheated beyond the cooling transition temperature T c .We note that when T = T c , the largest solution is zero.
At temperatures below T c , the product of the two roots is negative, and thus only one of them, the largest one, can be positive.For the ferroelectric state to be stable, P 0 must be smaller than the only real root.The system here then behaves as a typical second-order system.Thus, in the case C Ͼ 0, the order of para-ferroelectric transition in a thin film depends not on the sign of B but on the sign of B +4GQ 2 .Since 4GQ 2 is positive, a film made of a first-order bulk material may exhibit a second-order transition behavior if the constraint due to the substrate is sufficiently large.
Plotted in Figs.1͑a͒ and 1͑b͒ as a function of temperature for various film thicknesses are the regions of stability of P 0 for PbTiO 3 according to the condition p 2 Ͻ P 0 Ͻ p 1 from Eqs. ͑35͒ and ͑36͒.Figure 1͑a͒ shows the case of a freestanding film, whereas Fig. 1͑b͒ shows the film on a SrTiO 3 substrate.It is obvious that the constraint of the substrate weakens the polarization of the ferroelectric state, reduces the Curie temperature, and changes the first-order transition of the ferroelectric material into a second-order one.
It is interesting that the difference between the superheating and supercooling transition temperatures is insensitive to the film thickness h.This behavior is also seen in the numerical solution of the static Ginzburg-Landau equation for the first-order phase transition of a freestanding ferroelectric film. 13These results show that the two temperatures, plotted as a function of film thickness, are practically parallel.
The case of materials with C = 0 can also be easily analyzed using Eq.͑34͒, which becomes If B +4GQ 2 Ͼ 0, h is positive definite for T Ͼ T c , and the system is unstable.For T Ͻ T c , the system is stable as long as P 0 2 Ͻ A͑T c − T͒ /3͑4GQ 2 + B͒.This system behaves like a second-order system.For the rare case in which B +4GQ 2 Ͻ 0, the ferroelectric state is stable in the temperature range T Ͻ T c −3͑B +4GQ 2 ͒ / A, which is above the supercooling transition temperature T c .This system thus behaves like a first-order system, with a superheating temperature T ch = T c −3͑B +4GQ 2 ͒ / A. The case C Ͻ 0 rarely happens and will not be considered.At any rate, the foregoing analysis can be repeated easily for this case also.
Analytic expressions of the Curie temperature and the corresponding critical thickness for first-order transitions can thus be obtained by replacing T c0 with T c0 +9͑B +4GQ 2 ͒ 2 /20AC in the respective equations in Sec.III.

V. DISCUSSIONS
The following discussions refer to the specific examples of PbTiO 3 and BaTiO 3 .Straightly speaking, the foregoing results only apply to uniaxial ferroelectrics and not so much to cubic perovskites such as PbTiO 3 and BaTiO 3 .However, even for these perovskite ferroelectric thin films, the polarization field is often found to be perpendicular to the surface due to the mechanical constraint of the substrate. 14he material parameters for the free energy expressions of PbTiO 3 and BaTiO 3 film are listed in Table I.The critical thickness of the thin film at T c = 0 can be calculated using Eqs.͑11͒ and ͑12͒, following the usual assumption of a reduced polarization field on the surface, i.e., ␦ Ͼ 0.
We consider a BaTiO 3 film, epitaxially grown on a thick SrRuO 3 / SrTiO 3 substrate.In SI units, the elastic compliance components of a BaTiO 3 film are 16 s 11 = s 22 = s 33 = 8.3 ϫ 10 −12 , s 12 = s 13 = s 23 = −2.7ϫ 10 −12 , and Q = −0.043.If stress relaxation during fabrication is neglected, the misfit strain can reach 2%, corresponding to a misfit compressive stress of 3.57 GPa.From the second term of Eq. ͑11͒, this large compressive misfit stress may raise the Curie temperature by an unrealistic value of 900°.Experimentally, Yanase et al. 17 measured a Curie temperature of 350 °C.Compared with an inherent Curie temperature of BaTiO 3 of 130 °C, one may estimate that the Curie temperature is raised only by 220 °C, corresponding to a misfit stress and strain to be 852 MPa and 0.00477, respectively.In the first-principles calculation of Junquera and Ghosez, 1 the critical thickness of BaTiO 3 under short-circuit boundary conditions was found to have a value of about 2.4 nm.The corresponding extrapolating length can be estimated by numerically solving Eqs.͑11͒ and ͑12͒ simultaneously, giving a value of ␦ Ϸ 0.5 nm, which satisfies ␦ Ӷ h.We note that the small extrapolating length physically means that the tetragonal phase on the surface becomes almost central symmetric due to lattice relaxation, resulting in a large reduction in the polarization on the surface.The depolarizing effect so arises does not come from the usual depolarization field due to induced electric charges on the surface.Indeed, Eq. ͑20͒ shows explicitly that this is the case.Thus, even with zero depolarizing charges, i.e., both the third term on the right-hand side of Eq. ͑11͒ and the last term in Eq. ͑12͒ vanish, a solution of h c for these equations may still exist, as can be seen explicitly in Eqs.͑17a͒ and ͑17b͒, which is independent of the depolarization field.This conclusion is independent of the contact conditions.From the numerical solution of Eq. ͑13͒, the relation between the critical thickness and the extrapolation length of BaTiO 3 / SrRuO 3 / SrTiO 3 is shown in Fig. 2. The decrease of the extrapolation length ␦ decreases the surface polarization and causes the critical thickness to increase.The corresponding analytic approximations for the cases of ␦ Ͼ 0, and h c Ӷ ␦ or h c ӷ ␦ are derived in Eqs.͑17͒ and ͑21͒, respectively.
Using values from Table I, these equations give h c ͑nm͒ = 3.5-2␦ and h c ͑nm͒ = 2.2/ ␦, respectively.Despite their simple forms, these expressions give a very good description of the system, as shown by comparing with the numerical solution in Fig. 2.
In the first-principles calculations of Ghosez and Rabe 2 for ͑001͒ PbTiO 3 films, a stable nonferroelectric ground state cannot be observed for films with thicknesses down to Ϸ1.2 nm ͑three unit cells͒, suggesting the possibility that the critical thickness may not exist in this case.Ghosez and Rabe's work 2 also showed that the calculated polarization was enhanced on the surface, implying a value of ␦ Ͻ 0. Our analysis in Sec.III B shows that in such cases, a critical thickness indeed may not exist for the short-circuit boundary conditions.Furthermore, in such a case, the Curie temperature should increase with decreasing thickness.Since surface stress relaxation in this calculation has not been taken into account, it is not unreasonable to speculate that the surface polarization calculated in Ref. 2, may be overestimated.A proper treatment of the relaxation may produce a positive value of ␦.Indeed, Streiffer et al., 14 who measured the Curie temperatures as a function of film thickness up to 50 nm of epitaxial films of PbTiO 3 , grown on a SrTiO 3 ͑001͒ substrate ͑i.e., =0͒, found that the Curie temperature decreased with decreasing film thickness, indicating a reduced polarization on the surface corresponding to a positive value of ␦.
Assuming the film is fully strained epitaxially, with a compressive epitaxial strain of xx = yy = 1.2%, 14 and elastic compliance components of s 11 = 8.3ϫ 10 −12 N/m 2 and s 12 = −2.5ϫ 10 −12 N/m 2 , 18 the epitaxial stresses can be calculated to be r = xx = yy = −2.4293ϫ 10 9 N/m 2 .Substituting the epitaxial stresses into Eq.͑11͒ with the parameters in Table I and using the electrostrictive coefficient Q = −0.026͑in SI unit͒, 18 the Curie temperature is found to increase by about 335 K.The transition temperature is a function of film thickness ͑in nanometers͒, as given by Eqs.͑11͒ and ͑12͒ ͑for =0͒.The results are shown in Fig. 3, together with the experimental points of Streiffer et al. 14 The transition temperature can be calculated analytically using Eqs.͑16b͒ and ͑20͒ for large and small values of h, respectively.For PbTiO 3 these equations can be put into the simple forms T c = 1041-36000/ h 2 for h Ͼ 20 nm, and T c = 1041-1440/ h for h Ͻ 20 nm, h being in units of nanometer, using the parameters in Table I.It can be seen that this simple relationship describes the experimental data very well.Plotted together in the dotted line is the theoretical prediction used for comparison by Streiffer et al. 14 The difference between the two theoretical results in the small film thickness regime is obvious.We note that in the foregoing analysis the elastic strain energy, being proportional to 0 2 ϳ P 4 , is assumed negligible in comparison with the interaction energy between the epitaxial stress and the transformation strain.The calculated Curie temperature versus thickness for PbTiO 3 thin films under the two contact conditions is shown in Fig. 4. For films with the same thickness, those on the dielectric substrates have Curie temperatures slightly lower than on short-circuit electrodes due to the screened depolarization effect in the latter case.

VI. SUMMARY AND CONCLUSIONS
A thermodynamic model has been developed to describe the critical parameters in the ferroelectric transformation in  thin films, such as the Curie temperature and critical thickness.By analyzing the conditions of dynamic instability of the time-dependent Ginzburg-Landau equation, we obtain analytic expressions for both first-order and second-order transitions for the relationship among the epitaxial stresses, the surface-charge-induced depolarization, the electromechanical contact conditions, the film thickness, and the Curie temperature.From this relation, analytic expressions for the critical thickness are also derived.Despite the simple form of these expressions, they give very good description of the system, as shown by comparing with the numerical solutions.
Our analysis yields the following conclusions: ͑1͒ Four factors influence the dependence of the paraferroelectric transition on film thickness in thin films: ͑a͒ surface lattice relaxation, ͑b͒ induced surface charge, ͑c͒ interaction of epitaxial stress with transformation strain, and ͑d͒ restraint of the transformation strain.͑2͒ The restraint of the transformation strain due to the rigidity of the substrate weakens the polarization of the ferroelectric state, reduces the Curie temperature, and may even cause a thin film made of first-order ferroelectric material to undergo a second-order transition.͑3͒ Enhanced or reduced polarization on the surface due to the effects of the lattice relaxation and the epitaxial stress plays an important role in determining the Curie temperature and the critical thickness.In the case where surface polarization is enhanced, the Curie temperature is increased and the critical thickness reduced, as the film thickness decreases.It opposes the effect of the depolarization field due to the induced surface charges, the sole presence of which is not sufficient to guarantee the existence of a positive value of the critical thickness.In the reduced case, the opposite is true.͑4͒ For first-order transitions, the difference between the superheating and supercooling transition temperatures is found to be insensitive to the film thickness and surface boundary conditions.

ACKNOWLEDGMENTS
This project was supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region ͑PolyU Grant Nos.5173/01E, 5309/03E, and 5312/ 03E͒, and the National Science Foundation of China ͑Grant Nos.50232030 and 10172030͒.

APPENDIX: DEPOLARIZATION FIELD IN THE FERROELECTRIC THIN FILM BETWEEN TWO DIELECTRIC SUBSTRATES
The thin film is assumed to be infinite along the x and y directions with a thickness h.Consider a unit positive charge acting at ͑zЈ , rЈ , Ј͒ in the film, the electric potential at ͑Z , R , ͒ in the film has been derived by Wang and Woo as follows: 19 G͑x ជ,x ជЈ͒

͑A4͒
From Eq. ͑A1͒, one can derive the electric potential at ͑Z , R , ͒ in the film if a unit dipole along the z axis acts at ͑zЈ , rЈ , Ј͒ in the film as This result for a single dipole can be readily generalized to a distribution of polarization P͑zЈ͒ as follows: and E 0 is a constant.In fact, the first integral in Eq. ͑A6͒ can be reduced to a surface integral on the upper and lower surfaces of the film.By integrating r from zero to infinity, it becomes a constant.As for the second integral in Eq. ͑A6͒, since ‫ץ‬ 2 G / ‫͑ץ‬zЈ͒ 2 = ␦͑x ជ − x ជ Ј͒ / ͑4␣͒, one can derive the result easily.The depolarization field can be derived as follows: One should bear in mind that the expansion ͑A1͒ is not suitable for the case with upper and lower electrodes for which ␣ 0 , ␣ 2 → ϱ and ␤ =1.

FIG. 1 .
FIG. 1. ͑a͒ The stable area of the ferroelectric to paraelectric transition of a freestanding PbTiO 3 film.͑b͒ The stable area of the ferroelectric to paraelectric transition of a PbTiO 3 film on a SrTiO 3 substrate.

FIG. 4 .
FIG. 4. The Curie temperature vs thickness under different boundary conditions.