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"Physics of ferroelectrics PBLittlewood January 27, 2002 Contents 1 Introduction and scope 2 Macroscopic properties 2.1 What is a ferroelectric? . . . . . . . . 2.1.1 Examples of ferroelectrics . . 2.1.2 Ferroelectric phase transitions 2.2 Landau theory . . . . . . . . . . . . . 2.2.1 Coupling to strain . . . . . . 2.3 Domains . . . . . . . . . . . . . . . . 1 2 2 4 5 6 9 12 14 14 14 16 18 19 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Microscopic properties 3.1 Phonons . . . . . . . . . . . . . . . . . . . 3.1.1 One-dimensional monatomic chain 3.1.2 One-dimensional diatomic chain . . 3.1.3 Phonons in three-dimensional solids 3.2 Soft modes . . . . . . . . . . . . . . . . . . 3.3 A microscopic mean ﬁeld theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction and scope These notes are designed to accompany those lectures of the course covering the physics of ferroelectrics. While there is a fair amount of algebra here and there, none of it is terribly taxing — moreover, the algebra is here largely to bolster arguments that can be made mostly in pictures, so if you are happy with the pictures, you have probably got the point. The notes divide into two parts. In the ﬁrst, we are concerned with the macroscopic description of ferroelectrics, namely the study of the electrical 1 polarisation on length scales much longer than the separation between the atoms. On this scale the polarisation of a solid can be regarded as a continuous degree of freedom, just as one would look at the density of a ﬂuid if one averaged over the short interatomic length scales. This view will enable us to understand the onset of ferroelectricity as a phase transition like any other, to discuss the types of phase transition, and to begin to address the issues of domains, switching, and hysteresis. The second chapter introduces the study of ferroelectricity from the perspective of atomic scale physics. The reason that a particular material happens to be ferroelectric is of course that the chemistry and physics on an atomic scale favour a particular atomic rearrangement. As well as understanding the development of the macroscopic polarisation, we need also to understand how the microscopic degrees of freedom arrange themselves. In this chapter, we will particularly discuss how lattice vibrations (phonons) give a signature of the transition, and are aﬀected by it. Because the lattice vibrations are directly observable by inelastic neutron scattering, and in certain cases also by infra-red absorption or Raman scattering, there are experimental probes that allow one access to details of the transition. Most of this chapter actually consists of an introduction to lattice dynamics (in a linear chain of atoms, which is all we shall need) for those who have not come across it before. For further references on the physics (as opposed to the technology or the materials science) of ferroelectrics, one of the best books is an old one (F.Jona and G.Shirane, Ferroelectric Crystals, Dover 1993 (republication of Pergamon edition of 1962)). The ﬁrst chapters of J.F.Scott , Ferroelectric Memories, AP, 2000 also cover most of the material on macroscopic properties of ferroelectrics that you will need for this course. Phonons and lattice dynamics are covered well in several solid state texts, for example C. Kittel, Introduction to Solid State Physics, 7th Edition, Wiley, 1996. If you can ﬁnd it (not in print, but in some college libraries), I’d also recommend W.Cochran, The Dynamics of Atoms in Crystals, Edward Arnold, 1973. 2 2.1 Macroscopic properties What is a ferroelectric? A ferroelectric material has a permanent electric dipole, and is named in analogy to a ferromagnetic material (e.g. Fe) that has a permanent magnetic dipole. One way to understand how ferroelectricity can arise is to start by looking 2 at small molecules. A molecule that is symmetric, such as methane (CH4 ) has no dipole, but many simple molecules are not symmetric (e.g. H2 O) and have a dipole moment. The formal deﬁnition of a dipole moment is p= dV ρ(r)r (1) where ρ(r) is the charge density in the molecule - which consists of both the positive nuclear charge and the negative electronic charge density. Provided the molecule is overall neutral, this deﬁnition is conveniently independent of the choice of origin1 If the atoms can be treated as point charges Qi at positions Ri , then this just becomes (2) Qi Ri p= i It is clear from thinking about examples that ferroelectricity is prohibited if there is a centre of symmetry. If a centre of symmetry is not present, the remaining crystal classes2 have one or more polar axes. Those that have a unique polar axis are ferroelectric and have a spontaneous electrical polarisation. The others show the piezoelectric eﬀect, wherein an electrical polarisation is induced by application of an elastic stress; extension or compression will induce electrical polarisation of opposite signs. A ferroelectric solid can be made up by adding together large numbers of molecules with their dipoles aligned, so that the total dipole moment is then p= molecules pmolecule (3) but now it is more convenient to deﬁne the polarisation P as the dipole moment per unit volume P = p T otal volume pmolecule = V olume per molecule Dipole moment per unit cell = V olume of unit cell (4) There are some technical problems about extending this deﬁnition to an inﬁnite solid, that I won’t go into. 2 Actually with one exception 1 3 Figure 1: Crystal structure of N aN 02 . Atoms of the bent nitrite group are joined by lines; the coordinates in the ﬁgure are the heights of the atoms along the axis perpendicular to the page 2.1.1 Examples of ferroelectrics One of the simplest examples of a ferroelectric is N aN O2 (Fig. 1), which (in one of its several structures) can be viewed just as the prescription above as an array of aligned dipoles. Unlike in a molecule, where the dipole moment can be oriented in any direction by rotating the molecule in free space, here the dipole moment points along a special axis or axes, aligned with the crystal. This is called the polar axis. Usually there is more than one polar axis, and this is what makes ferroelectrics useful for devices, because on application of an electric ﬁeld, the polarisation can be switched from one direction into another. We will come back to this in a moment. The properties of ferroelectrics can be understood by reference to a (ﬁctitious) one-dimensional crystal made up of two atoms of opposite charge shown in Fig. 2. In this crystal, it is clear that we can orient the dipoles to point all to the right, or all to the left. The two structures are completely equivalent, except that they have an opposite sign to the dipole moment. They must therefore have exactly the same energy. We could transform one into the other by dragging one type of atom toward the other. As we do this, the bulk polarisation will reduce in magnitude, and change sign at the point where the atoms are equally spaced and ﬁnally switch to the opposite direction. Given that we know the crystal is stable in either of the two polarised states, there must be an energy barrier between the two states, and we can sketch a curve (Fig. 3) for the energy as a function of the polarisation. 4 Figure 2: Model (ﬁctitous) crystal How can we switch between the two states? In an electric ﬁeld E the two stable states no longer have the same energy because of the electric polarisation energy −P · E. The wells are tilted by the electric ﬁeld. It is also clear from this ﬁgure that a small ﬁeld will not necessarily immediately switch the polarisation from one direction to the other because there is a barrier to be overcome. In an ideal (and ﬁctitious) case where all the dipoles have to be overturned together — as in the ﬁgure — there will now be hysteresis, schematically demonstrated in Fig. 4. While this ﬁgure demonstrates the origins of the hysteresis phenomenon seen in real ferroelectrics, it is much too simple a description, because in a real material not all the microscopic dipoles will uniformly switch together. 2.1.2 Ferroelectric phase transitions The description above is limited to low temperatures. It is common to observe that as the temperature is raised, the bulk polarisation decreases and vanishes abruptly at a temperature Tc . This is a phase transition, just as in Figure 3: Schematic potential well 5 Figure 4: Schematic picture of hysteresis in an idealised ferroelectric a ferromagnet raised above its Curie temperature, or a solid raised above its melting point. It arises microscopically because as temperature is raised the thermal vibrations of the atoms in the solid cause ﬂuctuations which overcome the potential barrier between the two (or more) wells. It is most easily understood in a molecular crystal such as N aN O2 , where one can imagine that each molecule can ﬂuctuate between two conﬁgurations. each of which has a double potential well as in Fig. 3, and some interactions between the dipoles that tend to align them. The detailed microscopic theory of how this happens will be diﬀerent from material to material, but the macroscopic properties of the phase transition will be similar across many diﬀerent classes of materials. We will not discuss details of the chemistry and interactions, but instead present a macroscopic theory that provides a very useful description of many diﬀerent ferroelectrics. This is the Landau theory of phase transitions. 2.2 Landau theory Any crystal is in a thermodynamic equilibrium state that can be completely speciﬁed by the values of a number of variables, for example temperature T , entropy S, electric ﬁeld E, polarisation P , stress σ and strain s. Usually we are in a situation where we are applying externally electric ﬁelds E and elastic stresses σ, so we can regard the polarisation and strain 6 Figure 5: Free energy as a function of polarisation for (a) a para-electric material, and for (b) a ferroelectric material as "internal" or dependent variables. A fundamental postulate of thermodynamics is that the free energy F can be expressed as a function of the ten variables (three components of polarisation, six components of the stress tensor, and temperature), and our goal here is to write down an ansatz for the free energy. The second important thermodynamic principle is that the values of the dependent variables in thermal equilibrium are obtained at the minimum of the free energy. The approximation we make is just to expand the free energy in powers of the dependent variables, with unknown coeﬃcients (which can be ﬁt to experiment). If we are lucky, we may be able to truncate thus series with only a few terms. To be speciﬁc, let us take a simple example where we expand the free energy in terms of a single component of the polarisation, and ignore the strain ﬁeld. This might be appropriate for a uniaxial ferroelectric. We shall choose the origin of energy for the free unpolarised, unstrained crystal to be zero, and hence write 1 1 1 (5) FP = aP 2 + bP 4 + cP 6 + ... − EP . 2 4 6 Here E is the electric ﬁeld, and the unknown coeﬃcients a, b, c, etc. are in general temperature-dependent, and may have any sign. The equilibrium conﬁguration is determined by ﬁnding the minima of F, where we shall have ∂F =0 . (6) ∂P If a, b, c are all positive, the free energy (for E = 0) has a minimum at the origin (Fig. 5) In this case we can ignore the higher order terms than quadratic to estimate the polarisation induced by an electric ﬁeld from ∂F = aP − E = 0 , ∂P 7 (7) Figure 6: Second order phase transition. (a) Free energy as a function of the polarisation at T > To , T = To , and T < To ; (b) Spontaneous polarisation Po (T ) as a function of temperature (c) Inverse of the susceptibility χ, where χ = ∂P/∂E|Po is evaluated at the equilibrium polarisation Po (T ) and so we have a relationship between the polarisability and the ﬁeld (in linear response, for small electric ﬁeld) which deﬁnes the dielectric susceptibility 1 P = (8) χ= E a The dielectric susceptibility is proportional to the capacitance you would measure by putting the (insulating) ferroelectric in an electrical circuit. On the other hand, if the parameters are such that a < 0, while b, c > 0, then the free energy will look like the second ﬁgure in Fig. 5, which has a minimum at a ﬁnite polarisation P . Here, the ground state has a spontaneous polarisation and is thus a ferroelectric. The demarcation between these two curves comes if a changes continuously with temperature, and changes sign at a temperature To . This suggests a simple description of the ferroelectric transition might be obtained by assuming that a(T ) varies linearly with temperature, say of the form a × (T − To ). A little bit of thought (see also the question sheet) will then show that this phenomenological description will predict the behaviour of the free energy, polarisation, and susceptibility shown in Fig. 6 This is an example of a second-order, or continuous, phase transition where the order parameter (here the spontaneous polarisation) vanishes continuously at the transition temperature Tc = To . 8 Figure 7: First order phase transition. a) Free energy as a function of the polarisation at T > Tc , T = Tc , and T = To < Tc ; (b) Spontaneous polarisation Po (T ) as a function of temperature (c) Susceptibility χ. Logically (and practically as it turns out), we should consider the case of b < 0 (while c remains positive). This is sketched in Fig. 7. With the quartic coeﬃcient negative it should be clear that even if T > To (so the quadratic coeﬃcent is positive) the free energy may have a subsidiary minimum at non-zero P . As a is reduced (the temperature lowered), this minimum will drop in energy to below that of the unpolarised state, and so will be the thermodynamically favoured conﬁguration. The temperature at which this happens is the Curie temperature Tc (by deﬁnition), which however now exceeds To . At any temperature between Tc and To the unpolarised phase exists as a local minimum of the free energy. The most important feature of this phase transition is that the order parameter jumps discontinuously to zero at Tc . This type of phase transition is usually called a ﬁrst-order or discontinuous transition. Other common examples of this type of transition are solid- liquid transitions. 2.2.1 Coupling to strain An important feature of ferroelectric materials is often their great sensitivity to elastic stress. To understand why this happens we can again take recourse to Landau theory, by adding in strain dependent terms to Eq. (5). For a uniaxial ferroelectric, the leading order terms will be of the following form 1 Fs = Ks2 + dsP 2 + ... − sσ , 2 9 (9) Figure 8: Sketch of a volume strain, and two types of shear strain – that are volume preserving Here s is (a component of) the strain ﬁeld, and the ﬁrst term represents Hooke’s law – that the elastic energy stored in a solid is quadratically dependent on the distortion, so K is (one of) the elastic constant(s). The second term is a coupling between the elastic strain and the polarisation; the fact that this is linear in the strain and quadratic in the polarisation depends on the special symmetry of the transition (see more below) 3 . I have chosen not to complicate things by introducing umpteen components of the stress and strain tensors, but for completeness (though not for examination) I will expand a bit on strain ﬁelds. The strain in a solid is measured by how the displacement u of a point in the solid varies with position r, and since this is the dependence of a vector upon a vector, the answer is a tensor: the strain is usually deﬁned as sij = 1 2 ∂ui ∂uj + ∂rj ∂ri . (10) here i, j mean the x, y, z components of the vectors. s is therefore a 3x3 symmetric matrix, with six independent components. In materials that are cubic, or nearly so, there will be three independent components to the strain — the volume strain (uniform in all three directions, and two kinds of shear). Rather than do the mathematics, these are best understood in pictures (Fig. 8) In general, the polarisation will couple to one or more types of strain, and which types can generally be seen by inspection. Consider a cubic crystal (e.g.BaT iO3 ) that undergoes a ferroelectric phase transition to a state where the polarisation can point along one of the six orthogonal cubic directions. Now it is clear that there is a special axis (one of the six directions after While this is the leading term in pseudocubic materials, there are other materials (e.g. KH2 P O4 ) where the symmetry is lower, and the coupling can be of the form sP – linear in both strain and polarisation. Materials with a linear relation between stress and polarisation are called piezoelectric 3 10 Figure 9: Strain in a polarised crystal: the spontaneous polarisation chooses an axis, and distorts the crystal from cubic to tetragonal. By symmetry, if the polarisation is reversed, the strain stays the same, so the allowed coupling term must be quadratic in the polarisation the symmetry has been broken) and so it would no longer be expected that the crystal as a whole will remain cubic — one expects a distortion into a tetragonal crystal, which can be described by a tetragonal strain st . The fact that the lowest order coupling allowed in this case is of the form sP 2 (and not, for example, sP or s2 P ) can be seen by a thought experiment based on Fig. 9. Returning to the free energy, which now consists of the terms in Eq. (5) and Eq. (9), F = FP + Fs . we should now determine the properties in equilibrium by minimising with respect to both P and s, viz ∂F(P, s) ∂F(P, s) = =0 ∂P ∂s Let us take the second of these equations ﬁrst: (11) ∂F(P, s) = Ks + dP 2 − σ (12) ∂s There are a few diﬀerent limits to look at. Firstly, note that if the polarisation is zero, we get Hooke’s law s = σ/K. The second – apparently trivial case – is when a stress is applied to force the strain to be exactly zero at all times. This is not as absurd as it seems, because often crystals can be considered to be clamped by their surroundings so that no strain is allowed at all. One common situation is of a thin epitaxial ﬁlm which is forced to have the lattice constants matched to the substrate, and is free to relax only in the third direction. In the case of perfect “clamping s = 0, and the free energy is just as before. The third case to consider is when no external stresses are applied (σ = 0), and we then have s = −dP 2 /K (13) 11 Figure 10: Surface charge density generated by a bulk polarisation at an interface so that a spontaneous (tetragonal strain) occurs proportional to the square of the polarisation. Notice now that we can substitute for the strain as a function of polarisation, and we have a free energy 1 1 1 F(P, s(P )) = aP 2 + (b − 2d2 /K)P 4 + cP 6 + ... − EP . 2 4 6 (14) In comparison with the clamped system, the only change is to reduce the quartic coeﬃcient (notice that the result is independent of the sign of d). This means that in the case of an already ﬁrst-order transition (b < 0) the transition is driven even more strongly ﬁrst order, and Tc is raised. It is also possible to have 2d2 /K > b > 0, in which case a second-order transition in a clamped system becomes ﬁrst-order when the strain is allowed to relax. 2.3 Domains So far we have pretended that the polarisation in a ferroelectric can be treated as entirely uniform, and this is far from ..."

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