Thermodynamic modeling of nanoscale ferroelectric systems

Published by AMSS Press, Wuhan, China.Acta Mechanica Solida Sinica, Vol. 22, No. 6, December, 2009 ISSN 0894-9166

THERMODYNAMIC MODELING OF NANOSCALEFERROELECTRIC SYSTEMS��

Yue Zheng Biao Wang�

(State Key Laboratory of Optoelectronic Materials and Technologies/Institute of Optoelectronic and Functional

Composite Materials, School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, China)

Chung-Ho Woo

(Department of Electronic and Information Engineering, The Hong Kong Polytechnic University,

Hong Kong SAR, China)

Received 1 July 2009; revision received 23 December 2009

ABSTRACT Thermodynamic models formulated based on the Landau free-energy expansion arepopular and well suited to studies involving properties of the ferro/para-electric transition, or nearit. Indeed, the general nature of thermodynamics, from which the strength of the model is derived,allows a valid model to be constructed based on simple functional forms with parameters fittedto experiments, by passing the mechanistic complexity. Despite inaccuracy due to the neglect offluctuations, this approach has been proven effective and powerful for recent research develop-ment of ferroelectrics in nanoscale. Efforts in some important works have recently faced muchchallenge, when free-energy contributions have to be incorporated to account for the presenceof depolarization fields, surfaces and other defects. To minimize the problems with mechanisticobscurity, it is of paramount importance that the electromagnetics, mechanics and thermodynam-ics involved are accounted for explicitly and with full self-consistency. It is important that thefree-energy functional of nanoscale ferroelectric systems, such as ferroelectric thin films (FTF),bilayers (FB), superlattices (FS), nanowires (FNW), nanotubes (FNT) and tunneling junctions(FTJ) etc., must be derived thermodynamically from first principles.

KEY WORDS ferroelectrics, nanoscale, thin film, superlattice, nanowire, nanotube, tunnelingjunction, mechanical load

I. INTRODUCTIONFerroelectricity 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, surface effects and electrostatic interactions, etc.[1–24] This is particularly truefor the state of ferroelectricity itself. Studies of the effects of nanoscale dimensions on the polarization,dielectric constants, Curie temperature and critical sizes of ferroelectrics have been conducted for the lastseveral decades. Recently, research in this area has found new relevance due to the surge of technological

� Corresponding author. E-mail: [email protected]�� Project supported by the National Natural Science Foundation of China (Nos.10732100, 10831160504, 10972239 and10902128), and the Research Grants Council of the Hong Kong Special Administrative Region (G-YX0T, 5322/04E,N53408).

Vol. 22, No. 6 Yue Zheng et al.: Thermodynamic modeling of nanoscale ferroelectric systems · 525 ·

interest in ultra-miniaturized electronics, ultra-high density memory devices, and nanotechnologies ingeneral.

Theoretically, first-principle calculations, atomistic simulations and thermodynamic models havecontributed significantly to understanding properties of nanoscale ferroelectric systems[2,3,19–24]. Tak-ing into account symmetry, Landau theory can provide a reliable and reasonable description of a system’sequilibrium behavior near a phase transition. Thermodynamic models of ferroelectrics formulated basedon the Landau free-energy expansion are popular and well suited to studies involving properties of theferroelectrics[1,2,14]. This phenomenological approach is also a conceptual bridge between microscopicmodels and macroscopic phenomena of materials[1–20]. Model simplicity and easy experimental compat-ibility are among the attractions. Indeed, it is the general validity of thermodynamics from which thestrength of the model is derived. For example, Landau’s symmetry-based treatment of phase transitionswas applied to the case of ferroelectrics by Devonshire and ferroelectrics in finite size by Ginzburg[1].Then many relative theories based on Landau theory (i.e., Landau-Devonshire (LD), Ginzburg-Landau(GL), Landau-Ginzburg-Devonshire (LGD), the time dependent Ginzburg-Landau (TDGL), phase fieldmodel (PFM) etc.) have been widely used for investigating properties of ferroelectrics in different di-mension scales. Despite inaccuracies due to the neglect of stochastic fluctuations of a thermodynamicsystem[1], these approach have been proven powerful and effective, and widely used for investigatingproperties of ferroelectrics[5–20].

Recent modeling efforts of nanoscale ferroelectrics using the Landau approach have faced challengesin the formulation when contributions of the surface effects, inhomogeneous polarization distribution,and depolarization field have to be incorporated in the derivation of the free-energy functional. In thisregard, it is of paramount importance that thermodynamics, mechanics and electromagnetics involvedare considered with full self-consistency[25,26].

When surface effects from polarization gradients, near-surface effect and depolarization field Ed

are added, the accuracy of the Landau approach suffers, particularly when Ed is significant. We knowthat the depolarization field is mainly determined by the surface polarization charge of ferroelectricwith the open-circuit boundary conditions. For the short-circuit boundary conditions, in general, thedepolarization field has two components: one coming from the incomplete charge compensation in theelectrodes, and the other from the inhomogeneity of the polarization field near the interface. Herethe first component can be formulated in terms of the screening length in the electrodes, the secondcomponent is more complex and is determined by several factors, including space and surface charges,the small size effect, interface effect or Schottky barriers etc. Indeed, misunderstandings in this regardhave led to controversies concerning the contribution from the depolarization field, which results inlarge discrepancies in the outcomes of different calculations.

Under open-circuit boundary conditions, very often, the depolarization field calculated based on theusual expression Ed = −PT /ε0 completely overwhelms the total free energy of a typical ferroelectric thinfilm of PbTiO3 or BaTiO3. Here PT and ε0 are the total polarization and permittivity of the vacuum,respectively. Indeed, the depolarization field in such cases reaches strengths of 105 kV/cm, which aretwo orders of magnitude larger than the sum of all other contributions to the total free energy. Theconsequences include a downward shift in the phase transition temperatures of an unphysical amountof 104 K, or an 180◦ domain shrinkage of over 100% at very low temperature. Even under short-circuitboundary conditions (i.e., Ed = −(P T − q)/ε0, here q is the induced surface charge density in theelectrodes), this depolarization contribution to the total free energy is about two orders of magnitudelarger than the sum of all the other contributions. This is true even when the deviation from uniformityPT (z)− 〈P T 〉 is only 1% of the polarization for bulk BaTiO3 at room temperature (0.25 C/m2). Dueto the dominance of this term, a minimized total free energy cannot afford a relative deviation ofPT stronger than 10−4 from a uniform distribution, even in the presence of lattice defects, such asdislocation, cracks and surfaces. In order to minimize the non-uniformity of the polarization distributionunder boundary conditions imposed by the presence of defects with significant surface and interfaceeffects, the action of the Ginzberg term (i.e., gradient term) would force the polarization field to deviatesignificantly from zero. In this way, the effect of the depolarization would be so overpowering thatthe permanent polarization that determines its ferroelectric character would be practically eliminated,buried under its dielectric properties. The dominance of the depolarization field could lead to thequenching of domain patterns and other properties in the presence of strain fields created by interfacial

· 526 · ACTA MECHANICA SOLIDA SINICA 2009

dislocations, point defects, composition gradient, bilayers, multilyaers or superlattices, etc. As a result,no successful application of the Landau approach to the study of multilayered structures has been carriedout thus far. Indeed, without a good understanding of the source of the problem and a satisfactorymethod of resolution, any study in this direction cannot be very meaningful. Importantly, Tagantsevet al.[27] also discussed ‘Landau expansion for ferroelectrics: which variable to use’, and generally giventhe problem of the choice of the order parameter in the Landau expansion of proper ferroelectrics wasdiscussed.

Based on the premise that phase transition from a stable state of a dynamic system occurs only whenthe state becomes unstable, irrespective of the order of the transition, Wang and Woo[10] developeda thermodynamic model to describe critical parameters in ferroelectric transformation in thin filmswith approximately considering effects of the background dielectric material. Instead of adopting anumerical approach, the critical values of the parameters, such as the Curie temperature and criticalthickness, were calculated via a linear stability analysis of the evolution equation of the system. Zheng andWoo[25,26] comprehensively showed that effect of the depolarization field determined by the spontaneouspolarization and background dielectric constant is very important to critical properties of ferroelectricthin film. These works have investigated the source of the foregoing difficulties and suggested thatthe problem is the result of a serious underestimation of the dielectric screening of the electrostaticinteraction between the spontaneous polarization and the depolarization field Ed, and that the simplesolution is to use the spontaneous polarization as the order parameter. This suggestion has also beenadopted by several recent works.

Indeed, although the order parameter PT used in the Landau free-energy expansion is supposed torepresent the total polarization, an approach that we may call the total-polarization-order-parameter(TPOP) approach, in reality the order parameter at this stage is only an unknown of the Euler-Lagrange(E-L) equation derived from the free-energy functional through the variational principle (see e.g., thederivation of the time-dependent Ginzberg-Landau equation in Ref.[10]). The dynamic characteristicof PT , such as its instability that marks the phase transition, is totally determined by the form and theparameters of the E-L equation. Thus, in order for the polarization effects of the depolarization field toenter into the dynamic character of the E-L equation, it must be included explicitly in the free-energyfunctional, and cannot be done implicitly through the order parameter. In the TPOP approach, thepolarization induced by an applied field is only implicitly included in the free-energy functional. Itscontribution is thus not recognized, and independent of what the assigned meaning of PT is supposeto be, its effect will not be present in the dynamical behavior of its solution P T . As a result, there is alack of the dielectric screening in the effect of the depolarization field.

Using the spontaneous polarization alone as the order parameter in the Landau free-energy expansionand including the contribution from the induced polarization explicitly in the free-energy functional,recent works have shown that their effects enter the associated E-L equation properly, and the difficultiesencountered in the TPOP approach can be avoided. For easy description, and to highlight the substantialdifference between the results of the two approaches, particularly in regard to the critical properties,we name this approach the spontaneous-polarization-order-parameter (SPOP) approach.

Straight adherence to thermodynamic principles is emphasized, and the relationship between thepolarization and depolarization field in the free-energy expansion is carefully considered. Using theSPOP in the Landau free-energy functional, the difficulties encountered in the TPOP approach can beavoided. Based on the SPOP approach, the phase transition temperature, polarization and susceptibilityof ferroelectrics are calculated and discussed in following sections. Thermodynamic model of ferroelectricsis derived from first principles is in §II. Moreover, properties of nanoscale ferroelectric thin film (in§III), interface dislocations (in §IV), bilayers and superlattices (in §V), nanowires and nanotubes (in§VI) are discussed. Mechanical loads controlling of nanoscale ferroelectrics are introduced (in §VII).

II. PHYSICAL MODEL2.1. Electrostatics

In the SPOP approach, the spontaneous polarization field P (simply referred to as the polarizationin the following) is used as the order parameter. As mentioned, P is the polarization from the permanentelectric moment formed from spontaneous atomic displacements generated in a dielectric when goingthrough a ferroelectric phase transformation. We also consider another component of the polarization


P E , which is the result of mechanisms such as the electronic polarization, and other non-permanent

displacement of the ionic charge distribution, induced by an applied electric field E, and which accountsfor the dielectric screening of the effect of E.

This description is consistent with Lines and Glass[14] who observed that the total dielectric sus-ceptibility χT of ferroelectrics can be expressed in general as the sum of linear and nonlinear parts

χT = χ0+

(∂χT

∂E

)σ,T

E+

(∂χT

∂σ

)E,T

σ+

(∂χT

∂T

)E,σ

T +

{higher - ordernon - linerities

}≡ χ0+χ (T, E, σ) (1)

according to which the dependence of the susceptibility, Curie-Weiss relation, etc. on the temperatureT , stress σ and electric field E are described by the nonlinear component χ. The total polarizationP T of ferroelectrics in this description is a sum of two components, one due to a permanent electricmoment, which can be identified with P and whose relation to T , σ and E can be expressed via χ. Thisis embedded in a background material with dielectric response to E, which is explicitly describable bya susceptibility χ0 independent of P , and which creates an induced polarization P E proportional toE. The sum of P E and P constitutes the total polarization P T. To a good approximation, one mayconsider χ0 as the dielectric susceptibility of the ferroelectrics in the paraelectric state far away fromthe phase transition (e.g. ∂χT/∂T = ∂χ/∂T = 0 at T � Tc for E = σ = 0, and χT = χ0 for T � Tc).It should thus be accurately measured at T � Tc for E = σ = 0, here Tc is the phase transitiontemperature. To be consistent with the notations, we identify χ0 with the background susceptibilityχb or χd used in our previous works[10,12,25]. Similar discussions have also been reported by otherauthors[28–32]. We note that the response of P to an applied electric field E, and for that matter, thenon-linear component of the total susceptibility, is implicit in this formulation, and is only realizablethrough the thermodynamic relation between P T and E via the total free energy functional, after theassociated E-L equation is solved.

At constant applied stress and temperature, the electric displacement field D can thus be expressedin terms of the spontaneous polarization as,

D = ε0E + P T = ε0E + P E + P = ε0E + χbE + P = εbE + P (2)

where εb is the background dielectric constant. Indeed, in the absence of an external electric field, thetotal electric field has only one contribution that is the depolarization field Ed, i.e., E = Ed, the electricdisplacement is given by,

D = ε0Ed + P T = ε0Ed + P E + P = ε0Ed + χbEd + P = εbEd + P (3)

2.2. Thermodynamics

In Eqs.(2) and (3), the induced component of the total polarization is explicitly taken into account.We consider a crystal as our thermodynamic reference, which is of infinite extent (surfaceless) absentan applied field, i.e., electric, magnetic, or mechanical (internal or external). The general free-energyfunctional of ferroelectrics can be written as,

g(P , E, σ) = g (P , E, σ)|E=0,σ=0 −

∫σ

0

udσ −

∫E

0

DdE = g0 −

∫σ

0

udσ −

∫E

0

DdE (4)

where u and σ are the strain and stress tensor fields, respectively. Here we note that g0 is the free-energyfunctional of the field-free, uniform and infinite crystal (surfaceless and thus with no depolarizationcharges) we use as reference.

In the following discussions, we limit ourselves to consider the Gibbs free energy density (includingthose of the applied fields) in an unstressed sample at constant temperature. If the electric field in theferroelectrics is not zero, the thermodynamic potential can be obtained by starting from the relationas dg/dE = −D, where D is given by Eq.(2). In this case, it follows from Eq.(4) and the fact that P

has no explicit functional dependence on E that,

g(P , E) = g(P , E)|E=0 − P ·E −

∫E

0

εb ·EdE = g0 − P ·E −1

2εbE

2 (5)


here we note that Eq.(5) should be the same as that derived by Landau and Lifshitz[2] with consideringthe linear part of polarization.

As the free-energy functional of the field-free (i.e., E = 0), uniform and infinite (surfaceless) crystal,g0 can be expanded near the phase-transition instability in terms of P in the form of the Landaufree-energy functional. Then for the case in which E and P are parallel, and are both spatially uniform,substituting E = Eext + Ed in Eq.(5) leads to[25,26],

g(T, E) =A

2(T − Tc0)P

2 +B

4P 4 +

C

6P 6 − P (Eext + Ed)−

1

2εb(Eext + Ed)

2 (6)

where A = 1/(ε0C0), C0 being the Curie-Weiss constant and ε0 the permittivity of vacuum. Tc0 is theCurie temperature and B and C the expansion coefficients of the free energy of the reference crystal. Itcan be seen that the corresponding E-L equation will give the correct phase transition characteristicsfor E = 0.

Instead of Eq.(6), a number of authors prefer to express the free energy functional and depolarizationfield in terms of the total polarization and adopt the TPOP approach Refs.[6–13], in which the freeenergy functional is written as,

g (T, E) =A

2(T − Tc0) PT2 +

B

4PT4 +

C

6PT6 − PT Eext −

1

2PT Ed (7)

where the effect of the extrapolation length has been neglected. It can be seen immediately that boththe free energies of the reference state and the electrostatic contributions in Eqs.(6) and (7) are different.Although the free energy expressed by Eq.(7) has the correct form in the limit of E → 0, yet Eq.7(7) no longer satisfies ∂g/∂E = −D, as it should. As mentioned, in order for the effect of the appliedfield E to be fully reflected, it should be explicitly excluded from the order parameter when derivingthe E-L equation from the free-energy functional via the variational principle. Indeed, it can be clearlyseen that the dielectric screening effect due to P E is totally missing from the E-L equation derivedfrom the free-energy expression in Eq.(7). Furthermore, ignoring the interrelation between Ed and PT

could also produce errors in evaluating contributions from the internal energy of the field∫

D(E)dEin Eq.(5), and from missing the interaction term with Eext in the last term of Eq.(6).

More importantly, according to Eq.(1), the application of E produces an indirect effect on P whenthe variational minimum of the free-energy functional is sought. The total susceptibility χT is then asum of the linear part χb and nonlinear part χ, which measures this indirect effect of E on P , and whichcan be obtained by solving the corresponding E-L equation. Accordingly, the dielectric properties offerroelectrics are governed by a Curie-Weiss-type relation expressed as[10,12,25–27]

χT = χ + χb =

⎧⎪⎪⎨⎪⎪⎩

ε0C0

2 |T − Tc|+ χb, for T < Tc

ε0C0

|T − Tc|+ χb, for T > Tc

(8)

It also gives the correct limit of the total susceptibility far away from its phase transition temperatureTc, i.e. χT

∣∣T→∞

= [ε0C0/(T − Tc) + χb]|T→∞ = χb, which should be experimentally determined[11,24].For example, the background susceptibility χb or dielectric constants εb of PbTiO3, BaTiO3, SrTiO3,CaTiO3 are of the order of 50ε0 in the limit of infinite temperature[26,28,33,34] which are also used in ourfollowing calculations. The corresponding relation obtained using the TPOP approach of Eq.(7), on theother hand, the limit of the total susceptibility tends to zero, i.e. χT

∣∣T→∞

= ε0C0/(T − Tc)|T→∞ = 0.

III. FERROELECTRIC THIN FILMCharacteristics governing the para/ferro-electric transitions in a film have been investigated within

the framework of both thermodynamic theory and first-principle calculations. However, a clear pic-ture has yet to be completed, and there’re various discrepancies between experimental measurementsand theoretical calculations. Among the theoretical calculations themselves still remains unresolved.Although it is fairly plausible that the critical thickness has its origin from the depolarizing and in-terface effect on the para/ferroelectric transition, yet the details of the competing mechanisms, their


relative importance, and their relation with various electro-mechanical surface conditions, have yet tobe clarified.

It is self-evident that phase transitions from a stable state of a dynamical system cannot occurwithout the state becoming unstable, independent of the order and the direction of the transition. Theevolution equation of the state of polarization in a thin film can be derived from its free energy, interms of control parameters such as the ambient temperature, boundary conditions, surface character-istics, sample dimensions, and misfit epitaxial stresses, etc. Then important characteristics of the film,such as the transition temperatures, critical thicknesses, domain morphology, substrate nature, etc.,are related through the instabilities of the initial state, which is accessible via a stability analysis, thelinear nature of which opens the system to many established analytic techniques. This is the approachwe have adopted in Ref. [10]. We follow an analytic approach to establish conditions of the stabilityof the paraelectric versus ferroelectric phases in a thin film. The polarizing/depolarizing effects dueto the presence of the film surface, the restraint of the transformation strain, the electromechanicalsurface conditions involving the epitaxial stress and the induced surface charges, are taken into account.Expressions for the pare/ferroelectric transition temperatures as a function of film thickness, and thecorresponding critical thickness, is derived, and the complex relation of the critical thickness with thevarious parameters discussed. Both first-order and second-order transitions are considered.

3.1. Open-circuited and Short-circuited Ferroelectric Film

Firstly, we give details of reason why the TPOP approach have to face challenges in the formulationwhen contributions of the surface effects, inhomogeneous polarization distribution, and depolarizationfield.

Let us consider the simple case of a thick ferroelectric film with parallel surfaces of infinite extendperpendicular to the z-direction, free standing in vacuum in the absence of an electric field (i.e., Eext = 0).Due to translational symmetry in the x and y-directions, all variables here only depends on z, withconstant x- and y- components. Accordingly, only the z-component has to be solved and only in termsof a single variable z. Thus, inside the plate in the dielectric, there are no free charges, so that we alsohave dD/dz = 0 ⇒ constant. Outside the film in the field-free vacuum, D = 0 and continuity of thenormal component of the displacement field gives, for inside the dielectric,

D = P T + ε0Ed = 0, and Ed = −PT /ε0 (9)

Using Eq.(3), we may also write,

D = P + εbEd = 0, and Ed = −P/εb (10)

If Eq.(9) is used, the free-energy density expression with TPOP is given by,

gTPOP =A

2(T − Tc0)P

T2 +B

4PT4 +

C

6PT6 +

PT2

2ε0(11)

Firstly, we only consider a single domain in ferroelectric film, the condition of dynamic instabilitygives the equation of the transition temperature Tc = Tc0 − 1/(ε0A). More importantly, with a valueof 8.85 × 10−12 F/m for ε0, typical systems like KH2PO4, BaTiO3 and PbTiO3 etc. have values of1/(ε0A) from 103 K to 105 K[15,27]. Using Eq.(12), the Curie temperature of these films would be farbelow 0K, and could never have existed in the ferroelectric phase. Moreover, if we consider the 180◦

domain appear due to effect of the large depolarization field in film. This depolarization contributionin Eq.(12) is about three orders of magnitude larger than the rest of the contributions to the total freeenergy. When we minimize the total free energy, this large depolarization field will reduce domain widthabout three orders of magnitude smaller than thickness of film at low temperature (i.e. T � Tc0). Thisresult means that a very thick BaTiO3 film must be in paraelectric state, because domain will shrinkand disappear due to the effect of the depolarization field[14].

If the derived free-energy density with the SPOP approach in Eq.(6) is used instead, then

gSPOP =A

2(T − Tc0)P

2 +B

4P 4 +

C

6P 6 +

P 2

2εb(12)


The corresponding condition of dynamic instability gives, for the transition temperature under theinfluence of the depolarization field, Tc = Tc0 − 1/(εbA), yielding a downward transition temperatureshift of ΔTc = −1/(εbA). Compared with the one calculated using the traditional expression Eq.(8),the temperature shift is reduced by a factor of εb/ε0 for KH2PO4, BaTiO3 and PbTiO3 etc., and fallsinto much more reasonable values. Of course, the depolarization field can induce the 180◦ domains infilm with a reasonable width when temperature is below the Curie temperature.

Another commonly studied system is the ferroelectric capacitor in which the ferroelectric film ofthickness h is sandwiched between perfect metallic electrodes, which mean that the incomplete chargecompensation in the electrodes is neglected. In this case, inside the conducting electrode plates, D = 0, sothatD·n = D = qe on the ferroelectric boundary and inside. Here qe is the induced surface chargedensityin the electrode plates. Using Eq.(3) and SPOP approach, we can write qe/ε0 = D/ε0 = E(z)+P (z)/εb,from which the applied electric field is given by E(z) = −[P (z) − qe]/εb. When a potential gradientΔΦ/h is applied across the upper and lower surfaces, a relation between qe = 0 and ΔΦ/h can be foundby integrating Eq.(14). Thus, for shorted electrodes, ΔΦ = 0, and we obtain Ed(z) = −[P (z)−〈P 〉]/εb.

The TPOP approach, such as model derived by Kretschmer and Binder[15] as cited by many works[26],has the expression of the free energy as,

gTPOP =A

2(T − Tc0)P

T2 +B

4PT4 +

C

6PT6 +

1

2ε0(P T2 −

⟨PT

⟩2) (13)

This can be compared with the corresponding expression derived from Eq.(6),

gSPOP =A

2(T − Tc0)P

2 +B

4P 4 +

C

6P 6 +

1

2εb(P 2 − 〈P 〉2) (14)

We can see that the depolarization contribution in Eq.(13) is about two orders of magnitude largerthan the rest of the contributions to the total free energy, even when the deviation from uniformityPT (z)− 〈P T 〉 is only 1% of the polarization for bulk BaTiO3 at room temperature (0.25 C/m2). As aresult of the dominance of this term, a minimized total free energy cannot afford a relative deviationof P T stronger than 10−4 from a uniform distribution, even in the presence of lattice defects, such asdislocation, cracks and surfaces. Moreover, in order to minimize the non-uniformity of the polarizationdistribution under boundary conditions imposed by the presence of defects with significant surfaceeffects, the action of the Ginzberg term[1] would dictate that the polarization field cannot significantlydeviate from zero. For this reason, the effect of the depolarization would be so much overestimatedthat the ferroelectric nature would be largely eliminated. The dominance of the depolarization fieldwould lead to the quenching domain patterns or other properties induced by interface dislocations, pointdefects, compositionally graded, bilayers, multilyaers or superlattices, etc., which will be discussed infollowing sections.

3.2. Curie Temperature and Critical Thickness of Ferroelectric Thin Film

Based on the SPOP approach, we can use an analytic approach to establish conditions of thestability of the paraelectric versus ferroelectric phases in a thin film grown on a compressive substrate.The polarizing effects due to the presence of the film surface, the restraint of the transformation strain,the electromechanical surface conditions involving the epitaxial stress and the induced surface charges,are taken into account.

A ferroelectric thin film grown on a thick compressive substrate is considered with dimensions as∞×∞×h. The origin of the coordinate system is at the center of the cell. Due to effect of the compressivesubstrate, we suppose P is orthogonal to the surface of the film. Firstly, we only assume that perfectscreening is achieved due to metallic electrodes under the short-circuit condition. Important role onproperties of ferroelectric thin film is the depolarization field, which is from the polarization-inducedsurface charges and governed by the electric boundary conditions. Here we only consider two cases: (1)the ferroelectric film is sandwiched between two metallic electrodes with perfect screening under theshort-circuit condition; (2) the ferroelectric film is sandwiched between two dielectric substrates. Wegive a general expression that encompasses both contact conditions[10],

Ed = −1

εb

(P −

φ

h

∫ h/2

−h/2

Pdz

)(15)


where φ takes on the value of 0 or 1 depending on the contact condition as discussed in the foregoing.The modified thermodynamic potential may be found in an explicit form using the relation G =

G + u1σ1 + u2σ2 + u6σ6 given by Pertsev et al.[19], here G is the Gibbs free energy function. So themodified total free energy Fof ferroelectric thin film grown on substrate including contributions ofelectrical and the mechanical components can be given by,

F =

∫ ∫ ∫V

{[A

2(T − Tc0)−

2Q12um

s11 + s12

]P 2 +

(B

4+

Q212

s11 + s12

)P 4 +

C

6P 6 +

D

2

(dP

dz

)2

+P 2

2εb−

φ

2εb

(1

h

∫ h/2

−h/2

Pdz

)2

+u2

m

s11 + s12

⎫⎬⎭dz + D

∫ ∫S

δ−1

2P 2dxdy (16)

where D is from the Ginzburg term and defined as the gradient coefficient from contribution of theinhomogeneous distribution of the polarization. sij are the elastic compliance tensor. u11 = u22 = um

are the misfit strains between the thin film and substrate. S represents the upper and lower surfaceplanes that cover the entire surface of the film. δ is the so-called extrapolation length that measures theeffect of surface on polarization. Two separate cases may be considered: the more common δ > 0 case,corresponding to a reduction of self-polarization on the surface, and the rarer δ < 0 case, correspondingto an enhancement. Up to now, the extrapolation lengths in thermodynamic model is only approximatelyobtained in experimental works, for future calculations, values obtained from the first-principle-based[35]

or first-principle-phenomenological[36] calculations may provide a more reliable input data base.Using Eq.(16), the time evolution of the system is governed by the time-dependent Ginzburg-Landau

(TDGL) equation

∂P

∂t= −M

δF

δP= M

{−A(T − Tc0)P +

4Q12um

s11 + s12P −

P

εb

−(B +4Q2

12

s11 + s12)P 3 − CP 5 + D

∂2P

∂z2+

φ

εbh

∫ h/2

−h/2

Pdz

}(17)

where M is the kinetic coefficient related to the domain wall mobility. The surface term in Eq.(16)yields the boundary conditions ∂P/∂z = ∓P/δ, for z = ±h/2.

Equation (17) has a trivial solution of P = 0, representing the stationary para-electric state. To beable to transform between the paraelectric (P = 0) and the ferro-electric (P = 0) states, the initial statemust become unstable. The stability of the initial state can be probed by applying a small perturbationΔ to the corresponding stationary solution of Eq.(17). The dynamics of Δ follows from Eq.(17) byretaining only terms linear in Δ, which is given by:

∂Δ

∂t= M

[−A(T − Tc0)Δ +

4Q12um

s11 + s12Δ−

Δ

εb

−3

(B +

4Q212

s11 + s12

)P 2Δ− 5CP 4Δ + D

∂2Δ

∂z2+

φ

εbh

∫ h/2

−h/2

Δdz

](18)

The boundary condition is the same as Eq.(17), only withP replaced by Δ. For a symmetric configuration,the condition dΔ/dz = 0 at z = 0 must also hold.

This equation governs the stability of the stationary solution of Eq.(17). It is a linear equation of Δ. Itsanalysis in the context of phase stabilities, without having to resort to a solution of the nonlinear Eqs.(17)or (18), is our main aim. It is obvious from Eq.(18) that the equation governing Δ in the paraelectricand ferroelectric states are different in general, from which it follows that the stability conditions for thetwo branches are also different. However, as we shall see, for second-order transitions, P is continuousat the transition point. The ferroelectric and paraelectric branches are both unstable when crossingthis point from opposite directions, and transform into one another. The Curie temperature Tc and thecritical thickness hc are then independent of the initial state, and are thus well defined. For first-ordertransitions, on the other hand, P is discontinuous across the transition temperature. The paraelectricand ferroelectric branches obey different stability conditions, due to the difference in the governing


Fig. 1. The stable area of ferroelectric to paraelectric transition.

equations of Δ in Eq.(18). Suppose the paraelectric branch is stable/metastable for temperatures downto Tc, and the ferroelectric branch (P = 0) is stable/metastable for temperatures up to Tch, then thetwo transition temperatures are likely to be unequal. Indeed, two transition temperatures do exist,depending on the direction of the transformation between the two states, Tc and Tch are called thesupercooling and superheating transition temperature, respectively.

Using the method of separation of variables, and taking into account dΔ/dz = 0, at z = 0, Eq.(18)can be separated into time-dependent and time-independent parts, connected by the eigenvalue. Thesolution can be written as Δc(z, t) = eωctφωc

(z) = Δ0eωct[cos(kz) + R][17]. The P = 0 solution is

unstable when ωc > 0, because in this case, Δ increases exponentially with time. It can be seen thatwhen the temperature T is sufficiently high, ωc < 0, and the paraelectric state is stable. When Tis sufficiently low, ωc turns positive and the paraelectric state is no longer stable, since any smallperturbation Δ will grow exponentially beyond all bounds. The critical condition, ω = 0, yields, in thiscase, the supercooling transition temperature Tc of the film,

Tc = Tc0 +4Q12um

A(s11 + s12)−

1

Aεb−

D

Ak∗ (19)

where the first term on the RHS is related to the bulk ferroelectric property, the second term, to themisfit of the substrate, the third term, to the induced surface charges, and the fourth term, to thesurface effect and boundary conditions. While the last two terms act against the transformation bylowering the transition temperature, the second term can act either way, depending on the direction ofthe misfit. k∗ depends on the film thickness h, the extrapolating length δ, and the electric boundaryconditions through φ. If the phase transition of the film is the first order, Tch can be given by Tch =Tc + 9[4Q2/(s11 + s12) + B]2/(20AC)[10].

The following discussions refer to the specific example of a PbTiO3 film with the short-circuitboundary conditions and epitaxially grown on a thick SrTiO3 substrate. For the short circuit bound-ary condition, Fig.1(a) shows the case of a free-standing film, whereas Fig.1(b) shows the film ona SrTiO3 substrate. It is obvious that the con-straint of the substrate increases the Curie tem-perature, and changes the first-order transition ofthe ferroelectric material into a second-order one.Substituting the misfit strains (i.e. −0.012) intoEq.(19), the Curie temperature is found to increaseby about 335 K. Figure 2 show relation betweenCurie temperature andfilm thickness togetherwiththe experimental points of Streiffer et al.[18]. ForPbTiO3 these equations can be put into the simpleforms Tc = 1041 − 36000/h2 for h > 20 nm, andTc = 1041 − 1440/h for h < 20 nm, h being in

Fig. 2 Relation between Curie temperature and film thick-ness with the short-circuit boundary conditions in Ref.[10].


units of nm. It can be seen that this simple relationship describes the experimental data very well.

3.3. Curie-Weiss Law of Ferroelectric Thin Film

Most of bulk ferroelectrics exhibit a first-order phase transition. In such cases, the parameter B isnegative, and the parameter C is positive. For bulk materials in which the ferroelectric transition issecond-order, the parameter B is positive and the higher-order terms can be neglected.

According to theorem for the bifurcation of a simple eigenvalue[12,17], the bifurcating solutions areboth asymptotically stable in the supercritical regime. The phase diagram for this case is shown in Fig.3.Thus, if the bulk ferroelectrics exhibits second-order phase transition, their thin-film counterparts willalso exhibit the same order of phase transition, and the universal critical exponents remain unchanged.On the other hand, if the corresponding transition in the bulk material is first-order, i.e., B < 0, thendepending on the magnitude of the elastic self-energy of the transformation volume 4GQ2, which ispositive definite and has a magnitude of the same order as B, B + 4GQ2 can be either positive ornegative. When 4GQ2 is sufficiently large, B + 4GQ2 is positive definite, as in BaTiO3 and PbTiO3,then a first-order bulk ferroelectrics becomes second-order one in a rigidly constrained thin film.

The explicit solution of the steady-state Ginzburg-Landau equation for a ferroelectric film nearthe transition temperature can be derived using a perturbation approach. The Curie-Weiss relationwith respect to the non-linear part of the total susceptibility near the phase transition temperature

was approximately written in the form as⟨χ−1

s

⟩−1= A−1Θ−1(h)/(T − Tc). The corresponding Curie

parameter A−1Θ−1(h) changes sign discontinuously across the transition temperature. For films offinite thickness, the magnitude of the Curie parameter varies according to the thickness, leading toasymmetric divergences across the transition temperature. Indeed, the Curie parameter is independentof the film thickness (Θ−1 = 1) on the paraelectric side (T > Tc), a well known fact, but dependson the film-thickness on the ferroelectric side. It can be shown that the value of Θ−1 depends on thefilm thickness h, approaching −(1 − 8δ/3h) for thick films (h � δ), and −0.5(1 + h2/3δ2) for thinfilms (h � δ). We consider the example of a well-known ferroelectric PbTiO3. The normalized Curieparameter −[Θ(h)]−1 is plotted against the film thickness h in Fig.4. It can be seen that the magnitudeof the normalized Curie parameter does decrease rapidly from 1 to 0.5 with decreasing thickness. Indeed,for films of PbTiO3 about 40 nm (= 8δ), the ferroelectric transition is accompanied with a decrease ofthe Curie parameter of about 33%.

Fig. 3. Schematic of bifurcation diagram for ferroelectricthin film.

Fig. 4. The normalized Curie-Weiss parameter −[Θ(h)]−1

verses the film thickness h for PbTiO3.

IV. DISLOCATION GENERATIONS AND EFFECTS

IN FERROELECTRIC THIN FILM4.1. Dislocation Generations in Ferroelectric Thin Film

It is well-known that the presence of dislocations in thin films may adversely affect their functionality.The existence of a minimum thickness for non-ferroelectric films, below which misfit dislocations are


absent, has been observed and extensively investigated in many non-ferroelectric thin films. Thus, Zubiaet al.[37] use the equilibrium criterion to establish the critical conditions. Matthews and Blakeslee[38],and People and Bean[39] show that the critical thickness increases with increasing substrate thicknessunder many different conditions. Freund et al.[40] use the so-called energy approach to derive the criticalthickness. The case of misfit dislocations in a ferroelectric thin film on a compliant substrate has beenstudied due to the complication of the polarization in our recent works. Based on Timosheko’s[41] methodfor thermal stresses, the critical thickness for misfit dislocation generation is established. The effectsof the polarization and the film/substrate thicknesses on the critical thickness were analyzed.

For a ferroelectric thin film grown on a compliant substrate of finite-thickness, the total elastic energyof can be derived by following Timosheko’s method for calculating thermal stresses[41,42]. For a coherentinterface between the film and substrate, we define the biaxial in-plane misfit strains in the film. Inaddition to the misfit strain, there is also in-plane electrostrictive strain given by εT

11 = εT22 = εT =

Q12P2. Both the film and the substrate are treated as cubic elastic body with elastic Modulus C11, C12;

C11, C22, respectively. Applying the misfit stresses and the electrostrictive stresses on the ferroelectricthin film σappl = G(εm−QP 2), where G = C11 +C12−2C2

12/C11, keeps the lattice constant of the filmequal to that of the unstressed substrate. The resultant force (per unit length) due to σappl is given by

N1 = N2 = N =∫ h

0σappldz =

∫ h

0G(εm −QP 2)dz.

Equilibrium of the film/substrate system requires an equal and opposite force −N to balance theeffect of σappl. The uniform strain εb produced by the force is given by σb

11 = σb22 = σb = Gεb,

σb11 = σb

22 = σb = Gεb and hσb11 + Hσb

11 = hσb22 + Hσb

22 = −N , where G = C11 + C12 − 2C212/C11, and

the barred quantities refers to the substrate. can εb can be solved as εb = −N/(hG + HG). Then thetotal elastic energy of the film/substrate system can be given by,

Felastic =

∫ h

0

G(εm −QP 2 + εb)(εm −QP 2 + εb)dz +

∫ h+H

h

Gεbεbdz (20)

The stationary polarization state corresponding to the minimum total free energy can be obtainedby solving the variational equation. If P is the stationary value of the polarization, the strains in thefilm and substrate are given respectively by ε11 = ε22 = e = εm −QP 2 + εb, ε11 = ε22 = e = εb. Wecan introduce a dislocation b = {b1, b2, b3} at the coherent interface. The formation energy Ef of thedislocation at the coherent interface is given by[42]

Ef =1

2

∫ ∫Σ

(σdis + σc)(εdis + εc)dxdz −1

2

∫ ∫Σ

σcεcdxdz = Edis + Eint (21)

where Edis = 12

∫∫Σ σdisεdisdxdz and Eint = 1

2

∫∫Σ (σcεdis + σdisεc)dxdz. σdis and εdis are respectively

the stress and strain fields of the dislocation. σc and εc are respectively the stress and strain in thefilm/substrate system, i.e., without the dislocation. Edis and Eint are respectively the self-energy ofthe dislocation and the elastic interaction between the strain fields of the dislocation, the misfit andthe spontaneous polarization. Using contour integration and considering the traction-free conditionalong the surface and the vanishing far-field of the dislocation, Edis can be approximately solved. Thespontaneous formation of the dislocation is energetically viable when the formation energy satisfiesEf < 0. Solutions of Eq.(21) for Ef = 0 then give the minimum film thickness for misfit dislocationformation.

In Fig.5, we consider critical thickness hc of interface dislocation generation as a function of misfitstrain for various values of P in the important case of thick substrates. We consider cases with bothcompressive misfit strains, such as in the BaTiO3/SrTiO3 system, and tensile ones, such as in theBaTiO3/KTaO3 system. In this regard, we note that the polarization depends on variables such as theambient temperature and the external field, while the misfit strain varies with the substrate. Figure 5(a)shows hc in the compressive misfit case. Figure 5(b) shows results for positive misfit strains (tensile). It isfound to depend on the substrate thickness, the polarization state and the misfit strain, particularly itssign. Of particular importance, the generation of misfit dislocations during transitions between differentferroelectric states should be a design concern. This effect is particularly serious for thick substrateswhen the critical thickness. For compliant substrates, the critical thickness increases as the substratethickness decreases and becomes more compliant.


Fig. 5. For (a) compressive and (b) tensile substrates, the critical thickness of interface dislocation generation versus themisfit strain.

4.2. Dislocation Effects on Properties of Ferroelectric Thin Film

We have known that the interface dislocation introduces a large stress field superimposing on theepitaxial stresses to significantly affect the physical properties of the film[43–48]. Related investigationsconstitute an active area of thin-film research. In this section, effects of the interfacial dislocationson the polarization, Curie temperature and the depolarization field are considered as functions oftemperature. The spatial variations of the polarization, the depolarization, and the surface relaxationare specifically taken into account. This is done via numerical solution of the TDGL derived from theLGD functional[45,49,50].

When an interface dislocation is generated in film/substrate system, the free energy is also obtainedfrom a Legendre transformation from a Gibbs free energy[19,49]. Considering the effects of the depo-larization field with short-circuit boundary conditions in foregoing discussions, the total free energy ofa ferroelectric thin film with the interface dislocations can be established. We use periodic boundaryconditions along the x-direction to reflect a configuration in which the interfacial dislocations forman array lying on the z = 0 plane, with CII interface dislocation and line directions parallel to they-direction, through the centers of identical simulation cells repeated ad infinitum along the x-directionas shown in Fig.6.

We can obtain the time evolution of the spontaneous polarization in ferroelectric thin film, whichis governed by TDGL. The electrical and mechanical boundary conditions have a significant effecton properties of ferroelectric thin films. When surface effects are negligible, corresponding to the ex-trapolation lengths is infinite. The calculated polarization field with an interfacial edge dislocation at(0,0,0) for a PbTiO3 film at different temperature between 0 K and 1100 K are shown in Fig.7(a). Thelocal Curie temperature around the dislocation can be obtained by noting the locations at which thepolarization vanishes. In Fig.7(b), the variation of the Curie temperature near a single dislocation isshown. Although most regions experience a drop of the Curie temperature by several hundred degreesdue to the presence of the dislocation, it can also rise by as much as 800 degrees in the compressiveregions and lower by over 1000 degrees in the tensile regions, within 2 nm from the dislocation core,giving a variation over a range of nearly 2000 degrees in the Curie temperatures in this small region.As a result, while the polarization may still persist in the compressive regions at temperatures much

Fig. 6. Schematics of different interface dislocations.


Fig. 7. (a) The stable polarization field in a PbTiO3 film in the presence of an interfacial edge dislocation at (0,0,0) atT = 0 ∼ 1100 K. (b) Curie temperature around a b = a[−1, 0, 0] interfacial edge dislocation at (0,0,0). Arrow indicatesCurie temperature of PbTi3 film grown on LaAlO3 substrate without the interface dislocations.

higher than the Curie temperature of the bulk crystal Tc0, polarization already begins to disappear inthe tensile regions at temperatures much below the bulk Tc0.

The equilibrium thermodynamic theory of the misfit dislocations was developed by Matthews andBlakeslee and well established by Nix[38,51]. The theory can only approximately predict the criticalthickness of the dislocation generation and the dislocation density can be obtained. We must note thatthe interface dislocation density and critical thickness of the interface dislocations generation shouldalso be determined by the polarization in ferroelectric thin film[42]. Figure 8 shows the case with thedepolarization field and the near-surface eigenstrain relaxation neglected. Compared to Fig.7, a similardead zone in the ferroelectric state and residual polarization near the dislocation in the paraelectricstate can be seen.

When effects of the depolarization field and surface effect are taken into account, the differencein the spatial variations of P can be simulated. Effect of the depolarization field is to reduce thepolarization field in the film. At the same time, effect of the depolarization field can also reduce theregion of the dead layer. Under short-circuit boundary conditions, the region of the dead layer is only

Fig. 8. The stable polarization distribution around periodic interface at different temperature of T = 0,300,600,900 Kwithout considering effects of the depolarization field and the surface energy.


about 1/4 compared with it in Fig.8. The relation between the polarization and temperature withtemperature heating-up can be obtained. The presence of the dislocation substantially weaken theoverall polarizability of the film, particularly at higher temperatures (800 K < T < 1265 K)[49], wherethe averaged polarization drops to less than 10% of the dislocation-free case. In the highly strainedcore region of the dislocations, the ferroelectric phase persists at temperatures much higher than theCurie temperature of the bulk material. Thus, when dislocations are present, the spatial distributionof the polarization is inhomogeneous and a sharp cutoff of the polarization at the Curie temperaturedoes not exist, unlike the case without dislocations.

V. FERROELECTRIC BILAYERS AND SUPERLATTICEArtificially structured ferroelectrics are experiencing a rapid surge of research interest, driven by

the demand of new functional materials that provide new opportunities for novel applications. Stud-ies of ferroelectric bilayers (FB), multilayers (FM) and superlattices (FS), such as KNbO3/KTaO3,PbTiO3/SrTiO3, BaTiO3/SrTiO3, PbTiO3/PbZrO3, PbTiO3/BaTiO3 etc.[52–61], have revealed thatproperties of ferroelectric multilayers can be controlled by adjusting structural periodicity, misfit strain,boundary conditions, electrostatic interactions etc.

5.1. Ferroelectric/paraelectric Bilayers

Accompanying the paraelectric to ferroelectric phase transition, the transformation strains will beinduced. By constraining the strain, the substrate will restrict the ferroelectric phase transition of thefilm. Although many ferroelectric materials exhibit the first-order phase transition, some of them maybehavior as the second-order transition materials due to the constraint of the substrate. To reduce thethickness of the substrate, the constraint will be reduced. In such way, one can obtain the first-ordertransition thin film or the second-order transition thin film from the same material simply by changingthe relative thickness of the film and substrate. For PbTiO3 film on SrTiO3 substrate system, the criticalthickness of the substrate, above which the transition becomes second-order due to the constraint ofthe substrate, was determined.

We considered the PbTiO3 film/SrTiO3 substrate system, which bending effects firstly neglecteddue to this system can only shrink or expand under special conditions (i.e., see Ref.[62,63]). The Curietemperature of the film can be adjusted in the range from 0◦K to 1000◦K as show in Fig.9, and theferroelectric phase transition can also be tuned from the first-order as their bulk counterpart to thesecond-order.The interesting case is when B < 0, parameter γ2

[62] may then take either signs for differentfilm and substrate thickness and surface conditions. γ2 is negative, and the ferroelectric transition ofthe film is second-order and continuous, despite being first-order and discontinuous in the bulk. Anexample can be found in a well-known first-order ferroelectric material PbTiO3. Plotted in Fig.10 isγ2/C2

1 vs the ratio of the film thickness over the substrate thickness. One can see that if one can controlthe ratio h/H < 0.6, one can change the first-order transition into the second-order ones.

Fig. 9. The transition temperature of PTO film on STOsubstrate versus the film thickness with different substratethicknesses.

Fig. 10. γ2/C2

1versus the ratio of the film thickness over the

substrate thickness of PbTiO3/SrTiO3 system.


In addition, we also constructed an elastic and thermodynamic model for investigating the piezo-electric and bending response of a bilayer system. Results show that both the static and dynamicpiezoelectric and bending responses due to the inhomogeneous stress in the bilayer can produce verticaldisplacements. At the same time, this displacement, together with the associated polarization, radiusof curvature, and vertical displacement, can be adjusted by varying the thicknesses of the ferroelectricand paraelectric layers.

5.2. Ferroelectric Superlattices

Using the SPOP approach, we calculated crit-ical properties of ferroelectric superlattices as afunction of sample dimensions have been calcu-lated. To be specific, we only calculate PbTiO3/SrTiO3 (A/B) superlattice grown on STO sub-strate as show in Fig.11.

In Fig.12, the resulting ferro/para-electric tran-sition temperature Tc of the FS is plotted as a func-tion of the PTOvolume fraction (here the thicknessof a single-period of the FS is only fixed at 10 nm),

Fig. 11 Schematic structures of (a) an epitaxial superlatticeon a substrate, and (b) the single-period superlattice A andB.

compared with the corresponding experimental measurements using X-ray diffraction[60]. The agreementbetween the theory and model is excellent. It can be seen that the minimum volume fraction for ferro-electricity (MVFF), below which ferroelectricity disappears permanently, for PTO in the PTO/STOsuperlattice is 16%, similar to experimental measurements[58–60]. With the same thickness and boundaryconditions, we also solve the evolution equation corresponding to the TPOP approach. The results areplotted in the red triangles in Fig.12. It can be seen that the MVFF for PTO becomes 70%, over 400%over-estimated. The substantially lower transition temperatures predicted by the TPOP approach isconsistent with our speculation, as stated earlier, that the depolarization effects are very large and hadbeen double-counted.

Figure 13 shows the calculated averaged total polarization P T (calculated as the sum of Pi and PEi )

at 0 K as a function of the thickness of the PTO layer relative to the STO layer. Here the thicknessesare measured in terms of number of atomic planes np (for PTO) and ns (for STO). To facilitate thecomparison with the results of ab initio calculations[61], the thickness of the STO layer is kept at ns = 3,and those of the PTO layer is varied between np = 1, · · · , 48. The comparison clearly shows very goodagreement between the two sets of results. Repeating the calculation with the TPOP approach, theresults are plotted as the red triangles in Fig.13. Comparison with the ab initio calculations showsobviously worse agreement. The modification using the electrostatic model helps push the results tobetter agreement in the regime of small np/ns. However, the polarization does not seem to tend to thecorrect limit, when the FS is dominated by the PTO layer. In this regard, the present results are alsocorrect in that limit. Figure 13 also shows a critical ratio of np/ns, below which there is no ferroelectricity.Consistent with the results of recent work[61], for a STO thickness of ns = 3, the critical thickness of

Fig. 12. The relation between Tc and PTO volume fraction. Fig. 13. The relation between the average total polarizationand the ratio np/ns.


the PTO layer is 1.2 nm. The corresponding critical thickness calculated using the traditional modelis much larger, at 7 nm.

VI. FERROELECTRIC NANOWIRES AND NANOTUBESTheory and experiments on nanoparticles (NP), nanowires (NW), nanorods (NR), nanodisks (ND)

and nanotubes (NT) all indicate strong surface effects on their polarization[64–75]. This is expectedbecause the large surface to volume ratio in such cases naturally leads to strong surface effects. Indeed,the tendency to minimize the surface energy introduces a large intrinsic surface stress, i.e., surfacetension, in samples of nano-dimension, and is one of the most important surface effects considered.

In this section, the transition temperature and polarization of ferroelectric NWs and NTs are derivedand analyzed, paying particular attention to the effects of the surface tension and surface relaxation.The transition temperature and polarization as a function of wire and tube dimensions is derivedthrough the actions of the surface tension, polarization gradient coefficient, extrapolation length, andelectrostriction coefficient. The implications of the results will be considered and discussed.

6.1. Surface Tension of Ferroelectric Nanowires and Nanotubes

Nanocrystals generally have a large surface to volume ratio. The tendency to minimize the surfaceenergy density is often sufficiently large to significantly affect the properties of the nanocrystal. Exper-imentally, Ma et al.[69] and Uchino et al.[72] noted that the effect of surface tension on a nanocrystalis analogous to a hydrostatic pressure on a bulk single crystal.

For ferroelectric NW, NR or NPs, surface effects expressed in terms of the extrapolation lengthcan change the polarization near surfaces and associated properties. At the same time, the surfacetension produces a radial pressure, which is expected to affect properties. Based on the LD equation,Morozovska et al.[73] found that the surface tension in finite-size NR/NW/ND plays a role similar toan epitaxial stress in thin films. The uniform radial stress due to the surface tension μR in a NR, withsurface energy density μ and radius R, compresses the NR in the transverse direction and stretchesit along the polar axis z. Recently, Zhou et al.[70] also illustrated the effect of the surface tension andinternal compressive stress on the properties of ferroelectric NW. By a similar reasoning, surface tensionin a NT should also experience similar effects.

The linear elastic problem of a thick-wall tube under radial pressures pa and pb at the inner andouter surface (see Fig.14) has been solved by Lame[41]. The resulting stress field can be expressed incylindrical coordinates as follows:

σrr(r) =a2

b2 − a2

(1−

b2

r2

)pa −

b2

b2 − a2

(1−

a2

r2

)pb

σϕϕ(r) =a2

b2 − a2

(1 +

b2

r2

)pa −

b2

b2 − a2

(1 +

a2

r2

)pb

σrϕ = 0, σrz = 0, σzz = 0, σzϕ = 0

(22)

For the NW, we note that the inner radius a is zero, Eqs.(22) reduces to the stress field of a NW(i.e. a = 0 and b = R, and p = −μ/R), which has been discussed in Refs.[69–75]. For the NT, the radialstresses pa and pb in Eqs.(22) can be approximately expressed in terms of the surface energy densitiesμa and μb, i.e., pa = −μa/a and pb = μb/a.

Fig. 14. Schematics of the stress field in the nanotube problem under applied inner and outer radial pressures.


For the NW, assuming a simple elastic-plastic model with a yield stress σY , an effective stress dueto the surface tension should be defined as[73],

σeff =

⎧⎨⎩−

μ

Rfor R > RY

−σY for R ≤ RY

(23)

where RY is the radius of the NW below which plastic yield occurs.For the NT, an effective stress due to the surface tension should be defined as[74]

σeff =

⎧⎨⎩−

μ

w

(1 +

wΔμ

2Rμ

)for w > wY

−σY for w ≤ wY

(24)

where μ is the average surface energy density and Δμ = (μb − μa)/2. For NT, we define the averagetube radius R is (a + b)/2. wY is the thickness below which plastic yield occurs. The use of σeff helpsus avoid the unphysical divergence at R→ 0 or a ≈ b.

6.2. Ferroelectric Nanowires

The FNW is assumed to be a perfect insulator with short-circuited electrodes, with length andradius denoted by h andR, respectively. To simplify the formulation and computation, axial symmetryis assumed, so that all vector fields in the NW are directed in the z-direction, and all the foregoingvectors can be represented only in terms of their magnitudes. In this regard, we note that the directionsof the polarization are not necessarily all along the z-direction, and radial components should beconsidered in general. Nevertheless, under short-circuit electric condition, the radial compression dueto the surface tension, and the uniaxial loading along the z-direction, the polarization is most easilyaligned along the z-direction. Under short-circuit boundary conditions, the depolarization field canbe estimated by EFE ≈ −η(R, h)[P (z, r) − qe]/εb, where η(R, h) can be estimated using the relationη(R, L) ≈ 1/{1 + [h/(2R)]2}[73].

The study is conducted for a PbTiO3 NW which has been successfully fabricated and implementedrecently. Since typical values of the effective surface tension coefficient μ vary between 5 to 50 N/m.One can determine the relation between the phase transition temperature and the radius of the FNWby using linear stability analysis. The phase transition temperature Tc as a function of the radius R isas shown in Fig.15(a). The effect of the surface tension can be immediately discerned from the peakof the transition temperature T max

c ≈ 1.4Tc0 at R ≈ 3 nm. From Fig.15(a), the critical radius of theNW at which Tc vanishes, i.e., ferroelectricity disappears permanently, can also be obtained. Similarobservations for the existence of the critical ferroelectric size in NWs and thin films have been discussedin Ref.[10,12,73]. Interestingly, besides increasing the transition temperature, the surface tension alsoaffects the polarization in the NW significantly. We obtain the average polarization of a PbTiO3 NWat room temperature as a function of radius R, which we plot in Fig.15(b). The similarity between

Fig. 15. (a) The phase transition temperature Tc , and (b) the average polarization versus the radius of NW.


Figs.15(a) and (b) is clear. Figure 15(b) also demonstrates the existence of a critical radius Rc of about1.2 nm, below which the polarization disappears.

6.3. Ferroelectric Nanotubes

We only consider the case of a long tube with length h much larger than its inner/outer radius, so thatthe depolarization field can be neglected[74], thus allowing us to concentrate on the effects of the surfacetension. The analytical and numerical calculations are for BaTiO3 NT. Phase-transition temperaturesTc are calculated for various wall thicknesses w of an NT with inner radius a = 10 nm. The resultsare plotted in Fig.16(a), as a function of w. The important feature that can immediately be discernedis the maximum shown by Tc at T max

c ≈ 580 K for a wall-thickness of w ≈ 8.5 nm. Furthermore, forwall thickness of larger than 4 nm, Tc is actually higher than that of the bulk material at 400 K. Forcomparison, the calculation was repeated without the surface tension contributions, i.e., by settingμ = 0 N/m, and plotted as the blue line in Fig.16(a). It can be seen from the comparison that thesurface tension is responsible for the raise of the transition temperature, with a magnitude that tendsto increase with the increasing surface tension as w decreases. As the compression due to the surfacetension stops to increase beyond the elastic limit, the effects of the near-surface eigen-strain relaxationdescribed by the extrapolation length starts to dominate and lower the transition temperature as wdecreases. As a result, the rise in Tc shows a maximum of T max

c ≈ 580 K at a wall-thickness of w ≈ 8.5nm.

Fig. 16. Relation between (a) Tc and the wall thickness w, and (b) Tc and the average radius R.

In pure materials, the surface energy densitiesbetween the inner and outer wall surfaces of the NTare the same, so that it is reasonable to assume thatΔμ ≈ 0. However, practical considerations some-times require this condition to be relaxed, such as incases where the environments encountered by theinner and outer surfaces are different. We considerthat properties of ferroelectric NT can be changedby adjusting difference of the surface tension co-efficients between the inner and outer surface. InFig.16(b), we plot the calculated transition tem-perature Tc as a function of the average radius Rfor different values of Δμ. It can be seen that pos-

Fig. 17 The Curie-Weiss relation of BTO nanotube.

itive values of Δμ causes an increase of Tc, which increases as the average tube radius R decreases.The increase is most rapid for the small-radii tubes with an average tube radius of less than 10 nm.In addition to raising the transition temperature, the enhancement of ferroelectricity in NTs via theeffects of the surface tension can also be seen directly from the strength of the average polarization.The average polarization for a BTO NT is as a function of wall thickness w. The enhancement effect


of the polarization due to the surface tension is obvious, and follows the same trend as the transitiontemperature, reaching a maximum near w = 8 nm. A Curie-Weiss-type relation of the dielectric permit-tivity can be obtained for a BTO NT with different wall thickness near the transition temperature asshown in Fig.17. It can be seen that when the ferroelectric NT is in the paraelectric state (T > Tc) theCurie-Weiss relation is independent of the geometry of the tube. When ferroelectric NT is ferroelectricstate, the Curie-Weiss relation is a function of the wall thickness, radius and extrapolation length.

VII. MECHANICAL LOADS CONTROLLINGOF NANOSCALE FERROELECTRICS

7.1. Electricity Generation from Ferroelectric Nanowires

Under the short-circuit boundary conditions, we can investigate the electricity response of a singleFNW to an alternating tensile and compressive load, as shown in Fig.18. Two time-dependent loadingpatterns are considered[76],

(a) σz(t) = σΔ(t) =

⎧⎪⎪⎨⎪⎪⎩

σ02(t− nT Δ)

T Δ, nT Δ < t ≤

(n +

1

2

)T Δ

σ02[(n + 1)T Δ − t]

T Δ,

(n +

1

2

)T Δ < t ≤ (n + 1)T Δ

(25)

and

(b) σz(t) = σ∩(t) = σ0

∣∣∣∣sin(

2πt

T∩

)∣∣∣∣ (26)

Here σ0 and T∩ = 2T Δ are the amplitude and period of the external applied stress, respectively.Naturally, the corresponding applied force is given by fz(t) = πR2σz(t). The evolution of the polarizationP can then be calculated by solving the corresponding TDLG. Under an external axial load, the averagecharge density in the electrodes qe is governed by 〈P 〉 as a function of the surface tension and externalmechanical load. The current i can be described according to i = dQe/dt = πR2dqe/dt, where Qe isthe total charge in the electrodes. Here dqe/dt is the current density.

Figure 19 shows the responses of 〈P 〉 (Figs.19(a) and (c)) and current density (Figs.19(b) and (d))of the NW system to the cyclic mechanical loads. Here the external mechanical force fz are only from 0μN to 0.314, 0.628, 0.942 μN, respectively. The response can be seen to be highly sensitive to externalmechanical force. An ac with current density can be generated using a periodic tensile load on the orderof micro-Newtons. The mechanically-induced electric current is not only dependent on the applied force,but also on the dimension of the nanowire. These results show that ferroelectric NWs (or NRs andNBs) have sufficient strength to be potentially useful in applications such as a nanoscale ac generatorfor energy harvesting and as mechanical sensors of ultra-high sensitivity.

Fig. 18. Schematics of a ferroelectric NW under the effective radial pressures and external mechanical stress load withdifferent boundary condition.


Fig. 19. Under a tensile stress load as type of σz = σΔ and σz = σ∩, the current density of NW system.

7.2. Giant Piezoelectric-Resistance (GPR) in Ferroelectric Tunnel Junctions

Ferroelectric materials in the nano-scale are interesting not only from an academic point of view,they are also potentially useful in many applications, such as nano-transducers, nano-sensors, tunnelingjunctions and nonvolatile random access memories, etc.[77–85] Most tunnel junctions are constructedwith dielectric tunnel barriers such as Al2O3, MgO and NdGaO3 etc.[77]. However, recent interest in theconcept of ferroelectric tunneling junctions (FTJs) using ferroelectric tunnel barriers (FTBs) begins toexpand for reasons of both scientific curiosity and prospects of practical applications.

Considering the structure of most FTBs sandwiched between two compressive substrates as shownin Fig.20[86], we assume in the present paper that all vector fields in the FTB, such as polarizationand electric fields, are directed in the z-direction. In general, the depolarization field Ed has twocomponents, one coming from the incomplete charge compensation in the electrodes, and the otherfrom the inhomogeneity of the polarization field near the interface. So a general expression of thedepolarization field Ed can be given by[86],

Ed(z) = −P (z) + qe

εb= −

1

εb

[P (z)−

H/εb

2ls/εe + H/εb〈P 〉

](27)

Fig. 20. Schematic structures of (a) NFTJ with applied tensile and compressive stress loads, (b) charge distribution in apoled FTJ, and (c) distribution of the electrostatic potential.


The effect of mechanical stress on the tunneling conductions (TCs) of a FTJ sandwiched betweensubstrates is investigated by considering its influence on the polarization P and depolarization fieldsEd in the FTB. We are particularly interested in the GER-like effect due to the change of the TC whenthe applied stress is reversed from compression to tension. Denoting the TCs of the two polarizationstates by C±[P (σz)], the effect of a mechanical stress on the TC can be expressed in terms of the stressmodulation factors F±(σz) = C±[P (σz)]/C±[P (0)].

We consider as an example a BaTiO3(BTO)FTB sandwiched between SrTiO3 substrates. Thedissimilar electrodes can be made from SrRuO3

and noble metals (i.e. Pt) on substrates. Using athermodynamic model and taking into account po-larization charge screening in the electrodes andthe near-surface inhomogeneous polarization dis-tribution, the tunneling conductance (TC) of FTJsis modeled and calculated as a function of the ap-plied mechanical stress (as shown in the Fig.21).Calculations show that the conductance can bemodulated not only via the polarization orienta- Fig. 21 TC of the FTJ as functions of the applied stress.

tions, but also by the applied stress giving rise to a GER-like effect, which we call giant piezoelectric-resistance or GPR effect. These results show that FTJs have sufficient strength to be potentially usefulin applications of electronic and mechanical sensors, memories or other nano-devices. However, theseresults from our simple model need confirmation either experimentally or by more rigorous treatment ofthe temperature dependence, the interface effect and dielectric properties etc. of the FTB and electrodes.7.3. Piezo-photovoltaic Effects in Ferroelectric Nano-cylinders

Ferroelectrics like BaTiO3, LiNbO3, PbTiO3, PbZrTiO3(PZT), LiTaO3, (PbLa)(ZrTi)O3(PLZT),BiFeO3(BFO) or (PbTiO3)x(BiFeO3)1-x(PTOxBFO1-x) etc.[87–99], are photovoltaic, and can produce asteady voltage or direct current when illuminated with ultraviolet (UV) or near-ultraviolet (NUV) light.These materials have potential applications in many areas, such as optical micro-sensing, wireless actua-tion, micro-actuation in micro-electro-mechanical systems (MEMS), high-sensitivity photo-mechanicalsensors (HSPMS), switchable nano-devices, nondestructive read out of memory, optoelectronic devicesand optical information storage, etc.

Photovoltaic effects occur generally in systems with asymmetric interfaces, such as semiconducting p-n junctions or metal-semiconductor interfaces with Schottky barriers. Both experimental and theoreticalworks have shown that the ferroelectric photovoltaic (FPV) effect is different from that in conventionalp-n junctions or Schottky barriers of the semiconductors. Although a complete understanding of thephysical mechanisms involved in the FPV effect in metal-ferroelectric-metal (MFM) structures withthe short-circuit boundary conditions has not yet been accomplished, a widely accepted explanationof the photoelectric current is the separation of the photon-generated electron-hole pairs (excitons) bythe internal electric field, which also provides the potential difference that drives the photocurrent inthe external circuit connected to the ferroelectrics.

It is clear that FPV effects may become very small when the average internal electric field in FNCis zero. In this case, the photocurrent is only due to charge transport via diffusion. For the FNCwith the short-circuit boundary conditions, the internal electric field is mainly due to the spontaneouspolarization and interface effects, which is known to be very much affected by an applied mechanicalload, particularly near the para/ferroelectric transition region. It then follows that the FPV effects in aFNC should also be sensitive to the external applied stress, and give rise to a piezo-photovoltaic (PPV)effect that couples the photocurrent with mechanical loading. A FNC sample is sandwiched betweenshort-circuited (SC) metal electrodes, and with length H and radius Ras shown in Fig.22. To facilitatediscussion, we classify the FNC according to H < 0.5R (ND), 0.5R ≤ H < 4R (NR) and H > 4R(NW). Expressing the depolarization field as functions of length, radius and external applied stress,the photocurrent for the FNC can be approximately written as,

Isc ≈πR2βαϕTele[

τnμn

(qEd − καkT

)]−1+ κα/q

exp(−αH) (28)


Fig. 22. Schematic illustration of the ferroelectric nano-cylinder, (a) sandwiched between the top electrode (TP) andbottom electrode (DP) with the short-circuit boundary conditions, (b) the mechanism of the photocurrent generation forthe two possible orientations of the polarization ‘down’, and ‘up’ relative to the top electrode, (c) the photocurrent canbe adjusted under the external compressive and tensile stress loads.

where Ed is the mean depolarization field. A proven effective approach is via a thermodynamic for-mulation based on a Ginzberg-Landau-type free energy functional. Interface effects should take intoaccount effects on the distribution of the polarization due to space and surface charges, interface effector Schottky barriers etc. of the top and bottom interfaces.

The study in this paper is conducted for (Pb0.97

La0.03)(Zr0.5Ti0.5O3) (PLZT) which is one of themost promising candidates for ferroelectric-basedphotovoltaic applications. We consider a PLZTFNC sandwiched between Au top-bottom elec-trodes with the short-circuit boundary conditions.The normalized photocurrents Isc/I0

sc are plot-ted in Fig.23 as functions of the applied uniaxialstress, here I0

sc is the diffusion component of thephotocurrent, i.e. I0

sc = Isc|Ed=0. It can be seen

that in all cases the photocurrent increases as theapplied stress becomes more tensile. The piezo-photovoltaic (PPV) effect is found to be highlysensitive to the applied stress in a region of about

Fig. 23 The normalized photocurrent of the ferroelectric ND,NR and NW as function of the applied stress loads, respec-tively.

100 MPa near σc. An order-of-magnitude change in the photocurrent can be produced in this regionwith a change of stress of just a few percent. Although the present study is based on PLZT which is oneof the most promising candidates for ferroelectric-based photovoltaic applications, the BFO system mayprovide a much larger photocurrent than PLZT thin films[93]. In this regard, BFO or PTOxBFO1-x etc.may be used to make FNCs in photovoltaci or piezo-photovoltaic nano-devices of even higher sensitivity.

VIII. CONCLUSIONSIn summary, modeling ferroelectrics in nano-scale using the Landau free-energy functional approach

requires incorporating contributions from effects of the surface, interfacial and depolarization fieldetc. We have shown that derivation of the free-energy functional based on the Landau expansion hasto be performed in straight adherence to thermodynamic principles, i.e. the SPOP approach. Failingthat, expressions of the depolarization field and free energy by using the TPOP approach may haveeffectively double-counted the depolarization field, and thus result in a serious overestimation of itseffects on properties of ferroelectric superlattices. Based on a more rigorously derived free energy offerroelectrics by using the SPOP approach, we comprehensively calculated and discussed the propertiesof nano-scale ferroelectrics, such as the phase transition temperature, critical thickness, polarization field


and Curie-Weiss-type relation etc., and investigated mechanical controlling of nanoscale ferroelectricsystems.

Nevertheless, despite the more reasonable thermodynamic model of ferroelectrics in nano-scale ad-vanced in this paper, we still consider that there also still are many other outstanding challenges to thethermodynamic models based on the Landau-type theory. A comprehensive treatment of the surface,interface, background dielectric properties etc. of ferroelectrics still requires a better understandingof the materials behavior, which may require the combined efforts of the phenomenological theory,molecular dynamics and first-principle-based calculations etc.

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Thermodynamic modeling of nanoscale ferroelectric systems