Ferroelectric Thin Films

Ferroelectric thin films presently are the object of intensive fundamental and applied studies, which to a great extent have been stimulated by the horizons of their applications as functional materials in non-volatile memories in micro-electronics, mechanical sensors, actuators in micromechanics, pyroelectric detectors, and in tunable microwave and electro-optical devices. When speaking of ferroelectric thin films, one usually means films of thicknesses ranging from a few tens to hundreds of nanometers deposited on a thick (typically 0.5 mm or thicker) substrate. This determines mixed mechanical boundary conditions with the fixed in-plane components of the deformation while the film is free in the out-of-plane direction. The anisotropy of these conditions plays a decisive role in many properties of ferroelectric thin films.

1. 1.

   In the case where the substrate is pseudocubic.

2. 2.

   Actually, this may not always be the case; see papers by Shuvalov et al. (1987) and Dudnik and Shuvalov (1989).

3. 3.

   From this type of image one cannot distinguish the variants that have identical orientation of the intersection between their domain walls and the free surface of the film.

4. 4.

   In the case for low-temperature deposition in amorphous phase with subsequent crystallization at a high temperature one should speak about the crystallization temperature. In the book, as shorthand, we will use the term ‘deposition temperature’ instead of the crystallization temperature.

5. 5.

   Presently, the Matthews–Blakeslee theory is not the only one available on the dislocation-assisted stress release in thin films (see, e.g., papers by People and Bean (1985) and Holec et al (2008)). To the best of our knowledge the results of these theories have never been applied to the description of ferroelectric thin films, though they may be relevant.

6. 6.

   For h larger than the distance between the dislocations \(1/\rho\), strictly speaking, Eq. (9.3.10) should be modified: h under “ln” should be replaced with \(1/\rho\). The impact of this replacement on the final results is small. Hereafter, for simplicity, in this and similar “ln” factors we will ignore the difference between h and \(1/\rho\), keeping h for any film thicknesses.

7. 7.

   According to the accepted accuracy of our calculations, in the obtained expression, the ratio aL(h)/h has been substituted for a _s L(h)/h.

8. 8.

   One should mention that the choice of the appropriate potential matters for derivation of the phase diagram of a system only in the case where the phase diagram contains lines of first-order phase transitions. If it is not the case, all treatments based on different thermodynamic potentials will result in the same phase diagram. This relates to the fact that all the potentials result in equivalent equations of state.

9. 9.

   In the cited papers, the term ‘misfit strain’ was used for this quantity. We had to change the original terminology to avoid confusion.

10. 10.

    The condition that the electric field in the film is zero, strictly speaking, is not satisfied in a short-circuited capacitor because of incomplete screening of the bond charge with the free charges in the electrodes. This effect, which may be of importance for ultrathin films, will be discussed in Sect. 9.4.

11. 11.

    This result can also be presented in terms of invariant tensors of the cubic symmetry \(g_{ijkl}^{(4)}\) and \(g_{ijklmn}^{(6)}\) whose components in the cubic reference frame equal ‘1’ when all their indices are equal, otherwise the components equals ‘0’. Namely \(I_1 = g_{ijkl}^{(4)} P_i P_j P_k P_l\) and \(I_2 = g_{ijklmn}^{(6)} P_i P_j P_k P_l P_m P_n\) calculated in the film reference frame.

12. 12.

    An (ɛ _P, T) diagram for (001) BaTiO₃, which is different from that shown in Fig. 9.3.5a was given in the paper of Pertsev et al. (1998). This diagram should be neglected as obtained based on a set of thermodynamic parameters of bulk BaTiO₃, which did not properly correspond to the system.

13. 13.

    Let us recall, that, in the bulk BaTiO₃, a sequence of the first-order phase transitions separating four (cubic–tetragonal–orthorhombic–rhombohedral) phases occurs whereas, in PbTiO₃, only one cubic–tetragonal first-order phase transition does.

14. 14.

    As was specified in Sect. 9.1, when indicating the (001) orientation of a tetragonal ferroelectric film we imply the orientation of the film in the paraelectric cubic phase. Thus, the films of such orientation being in the ferroelectric phase can contain both a- and c domains.

15. 15.

    In this and ongoing sections we are not taking into account the contribution of the macroscopic electric field that can arise in ferroelectric multidomain structures (depolarizing field). We will address the impact of this contribution later on in this chapter.

16. 16.

    The elastic energy of the substrate can be neglected. See Sect. 9.3.1 for the justification.

17. 17.

    For the moment Eq. (9.52) is written in analogy with Eq. (9.40), a general proof for it will be given in Sect. 9.3.4.

18. 18.

    The approximation addresses the case where the elastic properties of the substrate are identical to that of the film. For this reason, elastic parameters of the substrate do not enter Eq. (9.3.54).

19. 19.

    The third hypothetical possibility that, for h < h _o, the domain period always stays of the order of the film thickness we exclude as containing an excessive density of domain walls.

20. 20.

    One should note that for smaller thicknesses, where the situation is on the limit or out of the limit of applicability of the mean-strain approximation, its predictions are still qualitatively correct. First, like in the exact theory, it is predicted that the a/c pattern becomes unstable for the film thicknesses below a certain critical value. Second, at small \(h/h_{0_{c/a} }\), taking into account the thickness dependence of α (given by Eq. (9.3.63)) in Eq. (9.3.57), one arrives at a \(W(h)\) curve deviating up from the square-root law similar to the numerically calculated curves in Fig. 9.3.12 do.

21. 21.

    It is proper to indicate that Eqs. (9.3.80) and (9.3.81) give actually the results of the third modification of Landau theory for this kind of materials. Speaking about three modifications we mean the versions of Landau theory for (i) mechanically free material (Sect. 2.3.4), (ii) single-domain film clamped by a thick substrate (Sect. 9.3.2), and (iii) film clamped by a thick substrate but containing a dense ferroelectric domain pattern. Similar to cases (i) and (ii), in case (iii), the stresses are mainly homogeneous throughout the sample. In terms of the stress level, case (iii) is intermediate between (i) and (ii).

22. 22.

    The materials parameters used in these calculations are listed in Table 2.3.1.

23. 23.

    One readily checks that at \(\phi = 0\), i.e., at \(\varepsilon _{\rm{P}} = Q_{12} P_{\rm{S}}^{\rm{2}}\), \(P_{\rm{S}}^{\rm{*}} = P_{\rm{S}}^{}\).

24. 24.

    Equations (9.3.85) and (9.3.86) are written to within the approximation of small \(\varepsilon _{\rm{T}}\) and \(\delta\).

25. 25.

    In principle it may not be the case when the thermal expansion of the substrate is not isotropic in the plane of the film or the epitaxy relation in the cubic phase lowers the in-plane symmetry of the film. We will not cover these situations which have not been addressed in the literature.

26. 26.

    In the approximation of elastic isotropy of the material of the film and the substrate, which is currently used in the problems of this kind.

27. 27.

    In the corresponding equation from the paper of Kopal et al. (1999), Eq. (2), there is a misprint: V/2 instead of V.

28. 28.

    The range of validity of the shown curve is actually limited by conditions \(h>W\) for large h and \(d>W\) for small h. For \(h_{\rm{M}} /d = 3\) and \(\kappa _{\rm{a}} = \kappa _{\rm{c}}\), that gives the range of validity \(0.2<h/h_{\rm{M}}<45\).

29. 29.

    Equations (9.4.9) and (9.4.10) with \(\kappa _{\rm{d}} = 1\) become identical to those obtained by Kopal et al. (1997) for the problem of the domain pattern in a ferroelectric slab in vacuum (see Chap. 5).

30. 30.

    In this relation, following the original paper, we do not make any difference between the dielectric constant at the temperature of the electrochemical equilibrium (at which the depletion layer forms) and that at the measuring temperature (at which the trapped creates the built-in field). The incorporation of this difference in the theory will not affect the qualitative conclusions of the analysis. A more advance discussion of the depletion effect can be found in Subsect. 9.5.3.

31. 31.

    The gap seen on the calculated loops is an artifact of calculations which use an experimental set of loops with gaps (like shown in Fig. 9.5.6b).

32. 32.

    Here, it is worth to remind that, in general, the slope of the loop at coercitivity is different from the differential susceptibility measured at this point since the latter is not influenced, in contrast to the slope, by the irreversible contribution to the polarization response.

33. 33.

    We remind that the situation of a thin passive layer is treated \((d <\ \ <h)\). This is taken into account in the following equations.

34. 34.

    Note that, in this case, the total (measured) polarization of the system is not given by the charge on the electrode, which is equal to \(P_{\rm{f}} - \sigma\), since the variation of the latter is not fully controlled by the current in the external circuit. We remind that, in the used approximation where the difference between the polarization and the electrical displacement is neglected, the current in the external circuit is liked to the polarization by the relation \(I_{{\rm{ext}}} = A \cdot {\rm{d}}P/{\rm{d}}t\), where A is the electrode area.

35. 35.

    As everywhere in this book we neglect the difference between the polarization and electric displacement of ferroelectric systems.

36. 36.

    This analysis is related to the case of positive \(E_{\rm{d}}\); in the general case, Eq. (9.5.39) should be taken with \(|E_{\rm{d}} |\) instead of \(E_{\rm{d}}\). This obviously does not affect the results of our analysis.

37. 37.

    This kind of solution is well known in the theory of transient currents in dielectrics (see, e.g., Baginskii and Kostsov, 1985; Lundstrom and Svensson, 1972).

38. 38.

    We neglect the sign in the expression for \(V_{{\rm{off}}}\) since, in practice, the sign of the voltage applied to a capacitor is fixed by convention. For a given convention, the sign of \(V_{{\rm{off}}}\) can be determined from simple electrostatic arguments.

39. 39.

    An important difference between the piezoelectric and flexoelectric effects is that the sign of the piezoelectric effect in ferroelectrics (with the rare exception for ferroelectrics with a piezoelectric paraelectric phase) is controlled by that of the ferroelectric polarization. For this reason, the strain can control (via the piezoelectric effect) only the spatial axis along which the ferroelectric polarization is directed, but not its sign.

40. 40.

    As everywhere in the book we consider all strains as small, thus here we neglect the difference between \(\varepsilon _{\rm{M}}^{ * 2} /a_{\rm{S}}\) and \(\varepsilon _{\rm{M}}^{ * 2} /a\).

41. 41.

    The upper limit of this range correspond to the situation where, after full charge compensation in the ‘up’ state the sample is brought to the ‘down’ state without any redistribution of the free charge.

42. 42.

    These relations follow from simple electrostatics on the standard assumption that the charge density in the depletion layers is homogeneous.

43. 43.

    In the paper by Dimos et al. (1994), a relation Eq. (1), which predicts a linear thickness dependence of \(V_{{\rm{off}}}\), was used in the discussion. This relation is valid only in the absence of charges on the electrodes. The formulae, taking into account these charges, Eqs. (9.5.27) and (9.5.35) of this book, do not predict such dependence.

44. 44.

    Making allowance for the background permittivity becomes often important when the depolarizing effect is involved, cf. Sects. 2.2.3 and 2.3.6.

45. 45.

    Here, as mainly in this book, we neglect the difference between D and P in ferroelectrics.

46. 46.

    The following relations correspond to those from paper by Tagantsev et al. (2002a) to within the difference in the definition of \(Q_{44}\).

47. 47.

    This relation was also later obtained by Bratkovsky and Levanyuk (2000a). However, in this paper it was argued that it holds not only for \(d>> W\). This statement contradicts the results of detailed calculations given in the original paper by Kopal et al. (1999).

48. 48.

    This effect is similar to the formation of a-domains in ferroelectric films under tensile misfit strain (c.f. Sect. 9.3.3)

49. 49.

    In (111) Pb(Zr_0.45Ti_0.55)O₃ films discussed, the spontaneous polarization may not be directed normal to the electrodes so that the real situation may be more complicated than the considered model. However, this complication does not affect the qualitative conclusions of the consideration given below in the text.

Authors and Affiliations

1. EPFL, LC IMX STI Station 12, 1015, Lausanne, Switzerland

   Alexander K. Tagantsev

2. Department of Electrical Engineering, Pennsylvania State University, University Park, PA, 16802, USA

   L. Eric Cross

3. Department of Physics, Technical University of Liberec, Halkova 6, 461 17, Liberec 1, Czech Republic

   Jan Fousek

Corresponding author

Correspondence to Alexander K. Tagantsev .

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