International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 3.1, pp. 381-392

Section 3.1.5. Microscopic aspects of structural phase transitions and soft modes

J. F. Scottc*

3.1.5. Microscopic aspects of structural phase transitions and soft modes

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3.1.5.1. Introduction

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Phase transitions in crystals are most sensitively detected via dynamic techniques. Two good examples are ultrasonic attenuation and internal friction. Unfortunately, while often exquisitely sensitive to subtle second-order phase transitions [e.g. the work of Spencer et al. (1970[link]) on BaMnF4], they provide no real structural information on the lattice distortions that occur at such phase transitions, or even convincing evidence that a real phase transition has occurred (e.g. transition from one long-range thermodynamically stable ordered state to another). It is not unusual for ultrasonic attenuation to reveal a dozen reproducible anomalies over a small temperature range, none of which might be a phase transition in the usual sense of the phrase. At the other extreme are detailed structural analyses via X-ray or neutron scattering, which give unambiguous lattice details but often totally miss small, nearly continuous rigid rotations of light ions, such as hydrogen bonds or oxygen or fluorine octahedra or tetrahedra. Intermediate between these techniques are phonon spectroscopies, notably infrared (absorption or reflection) and Raman techniques. The latter has developed remarkably over the past thirty years since the introduction of lasers and is now a standard analytical tool for helping to elucidate crystal structures and phase transitions investigated by chemists, solid-state physicists and materials scientists.

3.1.5.2. Displacive phase transitions

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3.1.5.2.1. Landau–Devonshire theory

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Landau (1937[link]) developed a simple mean-field theory of phase transitions which implicitly assumes that each atom or ion in a system exerts a force on the other particles that is independent of the distance between them (see Section 3.1.2.2[link]). Although this is a somewhat unphysical crude approximation to the actual forces, which are strongly dependent upon interparticle spacings, it allows the forces of all the other particles in the system to be replaced mathematically by an effective `field', and for the resulting equations to be solved exactly. This mathematical simplicity preserves the qualitative features of the real physical system and its phase transition without adding unnecessary cumbersome mathematics and had earlier been used to great advantage for fluids by Van der Waals (1873[link]) and for magnetism by Weiss (1907[link]). Landau's theory is a kind of generalization of those earlier theories. In it he defines an `order parameter' x, in terms of which most physical quantities of interest may be expressed via free energies. In a ferromagnet, the order parameter corresponds to the net magnetization; it is zero above the Curie temperature [T_c] and increases monotonically with decreasing temperature below that temperature. In a liquid–gas phase transition the order parameter is the difference in density in the gas and liquid phases for the fluid.

Devonshire independently developed an equivalent theory for ferroelectric crystals around 1953 (Devonshire, 1954[link]). For ferroelectrics, the order parameter is the spontaneous dielectric polarization P. In both his formalism and that of Landau, the ideas are most conveniently expressed through the free energy of the thermodynamic system: [F(P, T) = A(T - T_c)P^2 + BP^4 + CP^6, \eqno (3.1.5.1a)]where A and C are positive quantities and B may have either sign. Scott (1999[link]) shows that C changes sign at ferroelectric-to-superionic conducting transition temperatures. As shown in Fig. 3.1.5.1[link], minimization of the free energy causes the expectation value of P to go from zero above the Curie temperature to a nonzero value below. If B is positive the transition is continuous (`second-order'), whereas if B is negative, the transition is discontinuous (`first-order'), as shown in Fig. 3.1.5.2[link]. The coefficient B may also be a function of pressure p or applied electric field E and may pass through zero at a critical threshold value of p or E. Such a point is referred to as a `tricritical point' and is marked by a change in the order of the transition from first-order to second-order. The term `tri-critical' originates from the fact that in a three-dimensional graph with coordinates temperature T, pressure p and applied field E, there are three lines marking the ferroelectric–paraelectric phase boundary that meet at a single point. Crossing any of these three lines produces a continuous phase transition (Fig. 3.1.5.3[link]).

[Figure 3.1.5.1]

Figure 3.1.5.1 | top | pdf |

Free energy [F(P,T)] and order parameter [P(T)] from the Landau–Devonshire theory [equation (3.1.5.1a)[link]] for a continuous second-order ferroelectric phase transition [coefficient B positive in equation (3.1.5.1a)]. The insert shows the temperature dependence of the order parameter, i.e. the expectation value of the displacement [x(T)].

[Figure 3.1.5.2]

Figure 3.1.5.2 | top | pdf |

Free energy [F(P,T)] and order parameter [P(T)] from the Landau–Devonshire theory [equation (3.1.5.1a)[link]] for a discontinuous first-order ferroelectric phase transition [coefficient B negative in equation (3.1.5.1a)[link]]. [T_1] is the temperature (see Fig. 3.1.2.6[link]) below which a secondary minimum appears in the free energy.

[Figure 3.1.5.3]

Figure 3.1.5.3 | top | pdf |

Three-dimensional graph of phase boundaries as functions of temperature T, pressure p and applied electric field E, showing a tricritical point where three continuous phase boundaries intersect.

3.1.5.2.2. Soft modes

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Minimization of the free energy above leads to the dependence of spontaneous polarization P upon temperature given by [P(T) = P(0) [(T_c - T)/T_c]] for continuous transitions. In the more general case discussed by Landau, the polarization P is replaced by a generic `order parameter' [\varphi(T)] with the same dependence. Cochran's contribution (1960[link], 1961[link]) was to show that for continuous `displacive' (as opposed to `order–disorder') transitions, this order parameter is (or is proportional to) a normal mode of the lattice. One normal mode of the crystal must, in Cochran's theory, literally soften: the generalized force constant for this mode weakens as a function of temperature, and its frequency consequently decreases. This soft-mode theory provided an important step from the macroscopic description of Landau and Devonshire to a microscopic theory, and in particular, to vibrational (phonon) spectroscopy.

Cochran illustrated this theory using a `shell' model in which the electrons surrounding an ion were approximated by a rigid sphere; shell–shell force constants were treated as well as shell–core and core–core terms, in the general case. The initial application was to PbTe and other rock-salt cubic structures that undergo ferroelectric structural distortions.

For this simple case, the key equations relate the optical phonon frequencies of long wavelength to two terms: a short-range force constant [R_0'] and a long-range Coulombic term. It is important that in general neither of these terms has a pathological temperature dependence; in particular, neither vanishes at the Curie temperature. Rather it is the subtle cancellation of the two terms at [T_c] that produces a `soft' transverse optical phonon.

The longitudinal optical phonon frequency [\omega_{\rm LO}(T)] is positive definite and remains finite at all temperatures: [\mu \omega_{\rm LO}^2 = R_0' + {8\pi Z^2e^2\over 9\varepsilon V(T)}, \eqno (3.1.5.1b)]where [\mu] is a reduced mass for the normal mode; Ze is an effective charge for the mode, related to the valence state of the ions involved; [\varepsilon] is the high-frequency dielectric constant and [V(T)] is the unit-cell volume, which is a function of temperature due to thermal expansion.

By comparison, the transverse optical phonon frequency [\mu \omega_{\rm TO}^2 = R_0' - {4\pi Z^2e^2\over 3\varepsilon V(T)} \eqno (3.1.5.1c)]can vanish accidentally when [V(T)] reaches a value that permits cancellation of the two terms. Note that this does not require any unusual temperature dependence of the short-range interaction term [R_0']. This description appears to satisfy all well studied ferroelectrics except for the `ultra-weak' ones epitomized by TSCC (tris-sarcosine calcium chloride), in which the Coulombic term in (3.1.5.1b[link]) and (3.1.5.1c[link]) is very small and the pathological dependence occurs in [R_0']. This leads to a situation in which the longitudinal optical phonon is nearly as soft as is the transverse branch.

Subsequent to Cochran's shell-model developments, Cowley (1962[link], 1964[link], 1970[link]) replaced this phenomenological modelling with a comprehensive many-body theory of phonon anharmonicity, in which the soft-mode temperature is dominated by Feynman diagrams emphasizing renormalization of phonon self-energies due to four-phonon interactions (two in and two out). This contrasts with the three-phonon interactions that dominate phonon linewidths under most conditions.

It is worth noting that the soft optical phonon branch is necessarily always observable in the low-symmetry phase via Raman spectroscopy in all 32 point-group symmetries. This was first proved by Worlock (1971[link]), later developed in more detail by Pick (1969[link]) and follows group-theoretically from the fact that the vibration may be regarded as a dynamic distortion of symmetry [\Gamma_i] which condenses at [T_c] to produce a static distortion of the same symmetry. Hence the vibration in the distorted phase has symmetry given by the product [\Gamma_i \times \Gamma_i], which always contains the totally symmetric representation [\Gamma_1] for any choice of [\Gamma_i]. If [\Gamma_i] is non-degenerate, its outer product with itself will contain only [\Gamma_1] and there will be a single, totally symmetric soft mode; if [\Gamma_i] is degenerate, there will be two or three soft modes of different symmetries, at least one of which is totally symmetric.

Since the totally symmetric representation is Raman-active for all 32 point-group symmetries, this implies that the soft mode is always accessible to Raman spectroscopy at least in the distorted, low-symmetry phase of the crystal.

3.1.5.2.3. Strontium titanate, SrTiO3

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Among the perovskite oxides that are ferroelectric insulators, barium titanate has received by far the most attention from the scientific community since its independent characterization in several countries during World War II. The discovery of a ferroelectric that was robust, relatively inert (not water-soluble) and without hydrogen bonding was a scientific breakthrough, and its large values of dielectric constant and especially spontaneous polarization are highly attractive for devices. Although not ferroelectric in pure bulk form, strontium titanate has received the second greatest amount of attention of this family over the past thirty years. It also provides a textbook example of how optical spectroscopy can complement traditional X-ray crystallographic techniques for structural determination.

Fig. 3.1.5.4[link] shows the structure of strontium titanate above and below the temperature (T0 = 105 K) of a non-ferroelectric phase transition. Note that there is an out-of-phase distortion of oxygen ions in adjacent primitive unit cells (referred to the single formula group ABO3 in the high-temperature phase). This out-of-phase displacement approximates a rigid rotation of oxygen octahedra about a [100], [010] or [001] cube axis, except that the oxygens actually remain in the plane of the cube faces. We note three qualitative aspects of this distortion: Firstly, it doubles the primitive unit cell from one formula group to two; this will approximately double the number of optical phonons of very long wavelength ([q = 0]) permitted in infrared and/or Raman spectroscopy. Secondly, it makes the gross crystal class tetragonal, rather than cubic (although in specimens cooled through the transition temperature in the absence of external stress, we might expect a random collection of domains with tetragonal axes along the original [100], [010], [001] cube axes, which will give macroscopic cubic properties to the multidomain aggregate). Thirdly, the transition is perfectly continuous, as shown in Fig. 3.1.5.5[link], where the rotation angle of the oxygen octahedra about the cube axis is plotted versus temperature.

[Figure 3.1.5.4]

Figure 3.1.5.4 | top | pdf |

Structure of strontium titanate above (undisplaced ions) and below (arrows) its anti-ferrodistortive phase transition at ca. 105 K. Below this temperature, the cubic primitive cell undergoes a tetragonal distortion and also doubles along the [001] cubic axis (domains will form along [100], [010] and [001] of the original cubic lattice). The ionic displacements approximate a rigid rotation of oxygen octahedra, out-of-phase in adjacent unit cells, except that the oxygens actually remain on the cube faces, so that a very small Ti—O bond elongation occurs.

[Figure 3.1.5.5]

Figure 3.1.5.5 | top | pdf |

Rotation angle versus temperature for the oxygen octahedron distortion below 105 K in strontium titanate described in Fig. 3.1.5.4[link]. The solid curve is a mean-field least-squares fit to an [S = 1] Brillouin function.

Fig. 3.1.5.4[link] does not correspond at all to the structure inferred earlier from X-ray crystallographic techniques (Lytle, 1964[link]). The very small, nearly rigid rotation of light ions (oxygens) in multidomain specimens caused the X-ray study to overlook the primary characteristic of the phase transition and to register instead only the unmistakable change in the [c/a] ratio from unity. Thus, the X-ray study correctly inferred the cubic–tetragonal characteristic of the phase transition but it got both the space group and the size of the primitive cell wrong. The latter error has many serious implications for solid-state physicists: For example, certain electronic transitions from valence to conduction bands are actually `direct' (involving no change in wavevector) but would have erroneously been described as `indirect' with the structure proposed by Lytle. More serious errors of interpretation arose with the microscopic mechanisms of ultrasonic loss proposed by Cowley based upon Lytle's erroneous structure.

The determination of the correct structure of strontium titanate (Fig. 3.1.5.4[link]) was actually made via EPR studies (Unoki & Sakudo, 1967[link]) and confirmed via Raman spectroscopy (Fleury et al., 1968[link]). The presence of `extra' [q = 0] optical phonon peaks in the Raman spectra below [T_0] (Fig. 3.1.5.6[link]) is simple and unmistakable evidence of unit-cell multiplication. The fact that two optical phonon branches have frequencies that decrease continuously to zero (Fig. 3.1.5.7[link]) as the transition temperature is approached from below shows further that the transition is `displacive', that is, that the structures are perfectly ordered both above and below the transition temperature. This is a classic example of Cochran's soft-mode theory discussed above.

[Figure 3.1.5.6]

Figure 3.1.5.6 | top | pdf |

Raman spectra of strontium titanate below its cubic–tetragonal phase transition temperature. These features disappear totally above the phase transition temperature, thereby providing a vivid indication of a rather subtle phase transition.

[Figure 3.1.5.7]

Figure 3.1.5.7 | top | pdf |

Temperature dependence of phonon branches observed in the Raman spectra of tetragonal strontium titanate.

3.1.5.2.4. Lanthanum aluminate, LaAlO3

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A structural distortion related to that in strontium titanate is exhibited in lanthanum aluminate at approximately 840 K. As in strontium titanate, the distortion consists primarily of a nearly rigid rotation of oxygen octahedra. However, in the lanthanide aluminates (including NdAlO3 and PrAlO3) the rotation is about the [111] body diagonal(s) of the prototype cubic structure. The rotation, shown in Fig. 3.1.5.8[link], is out-of-phase in adjacent cubic unit cells, analogous to that in strontium titanate.

[Figure 3.1.5.8]

Figure 3.1.5.8 | top | pdf |

(a) Structure of lanthanum aluminate above (undistorted) and below (arrows) its cubic–rhombohedral phase transition near 840 K. As in strontium titanate (Figs. 3.1.5.4[link]–3.1.5.7[link][link][link]), there is a nearly rigid rotation of oxygen octahedra (the oxygen ions actually remain on the cube faces); however, in the lanthanide aluminates (Ln = La, Pr, Nd) the rotation is about a cube [111] body diagonal, so that the resulting structure is rhombohedral, rather than tetragonal. The primitive unit cell doubles along the cubic [111] axis; domains will form with the unique axis along all originally equivalent body diagonals of the cubic lattice. (b) Optical phonon frequences versus temperature in lanthanum aluminate.

Historically, this phase transition and indeed the structure of lanthanum aluminate were incorrectly characterized by X-ray crystallography (Geller & Bala, 1956[link]) and correctly assigned by Scott (1969[link]) and Scott & Remeika (1970[link]) via Raman spectroscopy. The causes were as in the case of strontium titanate, namely that it is difficult to assess small, nearly rigid rotations of light ions in twinned specimens. In the case of lanthanum aluminate, Geller and Bala incorrectly determined the space group to be [R{\bar 3}m] ([D^5_{3d}]), rather than the correct [R{\bar 3}2/c] ([D^6_{3d}]) shown in Fig. 3.1.5.8[link], and they had the size of the primitive unit cell as one formula group rather than two.

3.1.5.2.5. Potassium nitrate, KNO3

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Potassium nitrate has a rather simple phase diagram, reproduced in Fig. 3.1.5.9[link]. Two different structures and space groups were proposed for the ambient temperature phase I: Shinnaka (1962[link]) proposed [D^6_{3d}] ([R{\bar 3}2/c]) with two formula groups per primitive cell ([Z =2]), whereas Tahvonen (1947[link]) proposed [D^5_{3d}] ([R{\bar 3}m]) with one formula group per primitive cell. In fact, both are wrong. The correct space group is that of Nimmo & Lucas (1973[link]): [D^6_{3d}] ([R{\bar 3}2/c]) with one formula group per primitive cell. Again, Raman spectroscopy of phonons shows that the Tahvonen structure predicts approximately twice as many spectral lines as can be observed. Balkanski et al. (1969[link]) tried creatively but unsuccessfully to account for their spectra in terms of Tahvonen's space-group symmetry assignment for this crystal; later Scott & Pouligny (1988[link]) showed that all spectra were compatible with the symmetry assigned by Nimmo and Lucas. In this case, in contrast to the perovskites strontium titanate and lanthanum aluminate, the confusion regarding space-group symmetry arose from the large degree of structural disorder found in phase I of KNO3. The structures of phases II and III are unambiguous and are, respectively, aragonite [D^{16}_{2h}] ([Pnma]) with [Z = 4] and [C^{5}_{3v}] ([R3m]) with Z = 1.

[Figure 3.1.5.9]

Figure 3.1.5.9 | top | pdf |

Phase diagram of potassium nitrate, KNO3.

3.1.5.2.6. Lanthanum pentaphosphate

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The lanthanide pentaphosphates (La, Pr, Nd and TbP5O14) consist of linked ribbons of PO4 tetrahedra. In each material a structural phase transition occurs from a high-temperature [D^7_{2h}] ([Pncm]) point-group symmetry orthorhombic phase to a [C_{2h}] ([P2_1/c]) monoclinic phase. The macroscopic order parameter for this transition is simply the monoclinic angle [\varphi], or more precisely ([\varphi - 90^\circ]). In this family of materials, the X-ray crystallography was unambiguous in its determination of space-group symmetries and required no complementary optical information. However, the Raman studies (Fox et al., 1976[link]) provided two useful pieces of structural information. First, as shown in Fig. 3.1.5.10[link], they showed that the phase transition is entirely displacive, with no disorder in the high-symmetry phase; second, they showed that there is a microscopic order parameter that in mean field is proportional to the frequency of a `soft' optical phonon of long wavelength ([q = 0]). This microscopic order parameter is in fact the eigenvector of that soft mode (normal coordinate), which approximates a rigid rotation of phosphate tetrahedra.

[Figure 3.1.5.10]

Figure 3.1.5.10 | top | pdf |

(a) `Soft' optical phonon frequency versus temperature in LaP5O14, showing displacive character of the phase transition. Large acousto-optic interaction prevents the optical phonon frequency from reaching zero at the transition temperature, despite the second-order character of the transition. (b) Lanthanum pentaphosphate structure, showing linked `ribbons' of phosphate tetrahedra.

3.1.5.2.7. Barium manganese tetrafluoride

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BaMnF4 is an unusual material whose room-temperature structure is illustrated in Fig. 3.1.5.11[link](a). It consists of MnF6 octahedra, linked by two shared corners along the polar a axis, with ribbons of such octahedra rather widely separated by the large ionic radius barium ions in the b direction. The resulting structure is, both magnetically and mechanically, rather two-dimensional, with easy cleavage perpendicular to the b axis and highly anisotropic electrical (ionic) conduction.

[Figure 3.1.5.11]

Figure 3.1.5.11 | top | pdf |

(a) Structure of barium metal fluoride BaMF4 (M = Co, Mn, Mg, Zn, Ni) at ambient temperature (300 K). (b) Raman spectroscopy of barium manganese fluoride above and below its structural phase transition temperature, ca. 251 K. (c) Temperature dependence of lower energy phonons in (b).

Most members of the BaMF4 family (M = Mg, Zn, Mn, Co, Ni, Fe) have the same structure, which is that of orthorhombic [C_{2v}] ([2mm]) point-group symmetry. These materials are all ferroelectric (or at least pyroelectric; high conductivity of some makes switching difficult to demonstrate) at all temperatures, with an `incipient' ferroelectric Curie temperature extrapolated from various physical parameters (dielectric constant, spontaneous polarization etc.) to lie 100 K or more above the melting point (ca. 1050 K). The Mn compound is unique in having a low-temperature phase transition. The reason is that Mn+2 represents (Shannon & Prewitt, 1969[link]) an end point in ionic size (largest) for the divalent transition metal ions Mn, Zn, Mg, Fe, Ni, Co; hence, the Mn ion and the space for it in the lattice are not a good match. This size mismatch can be accommodated by the r.m.s. thermal motion above room temperature, but at lower temperatures a structural distortion must occur.

This phase transition was first detected (Spencer et al., 1970[link]) via ultrasonic attenuation as an anomaly near 255 K. This experimental technique is without question one of the most sensitive in discovering phase transitions, but unfortunately it gives no direct information about structure and often it signals something that is not in fact a true phase transition (in BaMnF4 Spencer et al. emphasized that they could find no other evidence that a phase transition occurred).

Raman spectroscopy was clearer (Fig. 3.1.5.11[link]b), showing unambiguously additional vibrational spectra that arise from a doubling of the primitive unit cell. This was afterwards confirmed directly by X-ray crystallography at the Clarendon Laboratory, Oxford, by Wondre (1977[link]), who observed superlattice lines indicative of cell doubling in the bc plane.

The real structural distortion near 250 K in this material is even more complicated, however. Inelastic neutron scattering at Brookhaven by Shapiro et al. (1976[link]) demonstrated convincingly that the `soft' optical phonon lies not at ([0,1/2,1/2]) in the Brillouin zone, as would have been expected for the bc-plane cell doubling suggested on the basis of Raman studies, but at ([0.39,1/2,1/2]). This implies that the actual structural distortion from the high-temperature [C^{12}_{2v}] ([Cmc2_1]) symmetry does indeed double the primitive cell along the bc diagonal but in addition modulates the lattice along the a axis with a resulting repeat length that is incommensurate with the original (high-temperature) lattice constant a. The structural distortion microscopically approximates a rigid fluorine octahedra rotation, as might be expected. Hence, the chronological history of developments for this material is that X-ray crystallography gave the correct lattice structure at room temperature; ultrasonic attenuation revealed a possible phase transition near 250 K; Raman spectroscopy confirmed the transition and implied that it involved primitive cell doubling; X-ray crystallography confirmed directly the cell doubling; and finally neutron scattering revealed an unexpected incommensurate modulation as well. This interplay of experimental techniques provides a rather good model as exemplary for the field. For most materials, EPR would also play an important role in the likely scenarios; however, the short relaxation times for Mn ions made magnetic resonance of relatively little utility in this example.

3.1.5.2.8. Barium sodium niobate

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The tungsten bronzes represented by Ba2NaNb5O15 have complicated sequences of structural phase transitions. The structure is shown in Fig. 3.1.5.12[link] and, viewed along the polar axis, consists of triangular, square and pentagonal spaces that may or may not be filled with ions. In barium sodium niobate, the pentagonal channels are filled with Ba ions, the square channels are filled with sodium ions, and the triangular areas are empty.

[Figure 3.1.5.12]

Figure 3.1.5.12 | top | pdf |

Structure of the tungsten bronze barium sodium niobate Ba2NaNb5O15 in its highest-temperature [P4/mbm] phase above 853 K.

The sequence of phases is shown in Fig. 3.1.5.13[link]. At high temperatures (above [T_c = 853] K) the crystal is tetragonal and paraelectric ([P4/mbm = D^5_{4h}]). When cooled below 853 K it becomes ferroelectric and of space group [P4bm = C^2_{4v}] (still tetragonal). Between ca. 543 and 582 K it undergoes an incommensurate distortion. From 543 to ca. 560 K it is orthorhombic and has a `[1q]' modulation along a single orthorhombic axis. From 560 to 582 K it has a `tweed' structure reminiscent of metallic lattices; it is still microscopically orthorhombic but has a short-range modulated order along a second orthorhombic direction and simultaneous short-range modulated order along an orthogonal axis, giving it an incompletely developed `[2q]' structure.

[Figure 3.1.5.13]

Figure 3.1.5.13 | top | pdf |

Sequence of phases encountered with raising or lowering the temperature in barium sodium niobate.

As the temperature is lowered still further, the lattice becomes orthorhombic but not incommensurate from 105–546 K; below 105 K it is incommensurate again, but with a microstructure quite different from that at 543–582 K. Finally, below ca. 40 K it becomes macroscopically tetragonal again, with probable space-group symmetry [P4nc] ([C^6_{4v}]) and a primitive unit cell that is four times that of the high-temperature tetragonal phases above 582 K.

This sequence of phase transitions involves rather subtle distortions that are in most cases continuous or nearly continuous. Their elucidation has required a combination of experimental techniques, emphasizing optical birefringence (Schneck, 1982[link]), Brillouin spectroscopy (Oliver, 1990[link]; Schneck et al., 1977[link]; Tolédano et al., 1986[link]; Errandonea et al., 1984[link]), X-ray scattering, electron microscopy and Raman spectroscopy (Shawabkeh & Scott, 1991[link]), among others. As with the other examples described in this chapter, it would have been difficult and perhaps impossible to establish the sequence of structures via X-ray techniques alone. In most cases, the distortions are very small and involve essentially only the oxygen ions.

3.1.5.2.9. Tris-sarcosine calcium chloride (TSCC)

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Tris-sarcosine calcium chloride has the structure shown in Fig. 3.1.5.14[link]. It consists of sarcosine molecules of formula CH3NHCH2COOH in which the hydrogen ion comes off the COOH group and is used to hydrogen bond the nitrogen ion to a nearby chlorine, forming a zwitter ion. As is illustrated in this figure, this results in a relatively complex network of N—H...Cl bonds. The COO ion that results at the end group of each sarcosine is ionically bonded to adjacent calcium ions. The resulting structure is highly ionic in character and not at all that of a `molecular crystal'. The structure at ambient temperatures is [Pnma] ([D^{16}_{2h}]) with [Z = 4]; below 127 K it distorts to [Pna2_1] ([C^9_{2v}]) with Z still 4.

[Figure 3.1.5.14]

Figure 3.1.5.14 | top | pdf |

Structure of tris-sarcosine calcium chloride, (CH3NHCH2COOH)3CaCl2. The hydrogen ion (proton) on the COOH group is relocated in the crystal onto the N atom to form a zwitter ion, forming an H—N—H group that hydrogen bonds to adjacent chlorine ions. Each nitrogen forms two such hydrogen bonds, whereas each chlorine has three, forming a very complex network of hydrogen bonding. The phase transition is actually displacive, involving a rather rigid rolling of whole sarcosine molecules, which stretches the N—H bonds; it is not order–disorder of hydrogen ions in a Cl...H—N double well. (The Cl...H—N wells are apparently too asymmetric for that.)

It had been supposed for some years on the basis of NMR studies of the Cl ions, as well as the conventional wisdom that `hydrogen-bonded crystals exhibit order–disorder phase transitions', that the kinetics of ferroelectricity at the Curie temperature of 127 K in TSCC involved disorder in the proton positions along the N—H...Cl hydrogen bonds. In fact that is not correct; even the NMR data of Windsch & Volkel (1980[link]), originally interpreted as order–disorder, actually show (Blinc et al., 1970[link]) a continuous, displacive evolution of the H-atom position along the H...Cl bond with temperature, rather than a statistical averaging of two positions, which would characterize order–disorder dynamics. In addition, as shown in Fig. 3.1.5.15[link], there is (Kozlov et al., 1983[link]) a lightly damped `soft' phonon branch in both the paraelectric and ferroelectric phases. TSCC is in fact a textbook example of a displacive ferroelectric phase transition. The hydrogen bonds do not exhibit disorder in the paraelectric phase. Rather, the transition approximates a rigid rotation of the sarcosine molecules, which stretches the N—H...Cl bond somewhat (Prokhorova et al., 1980[link]).

[Figure 3.1.5.15]

Figure 3.1.5.15 | top | pdf |

`Soft' optical phonon frequencies versus temperature in both ferroelectric and paraelectric phases of tris-sarcosine calcium chloride.

3.1.5.2.10. Potassium dihydrogen phosphate, KH2PO4

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Potassium dihydrogen phosphate, colloquially termed `KDP', has probably been the second most studied ferroelectric after barium titanate. It has been of some practical importance, and the relationship between its hydrogen bonds, shown in Fig. 3.1.5.16[link], the perpendicular displacement of heavier ions (K and P) and the Curie temperature has fascinated theoretical physicists, who generally employ a `pseudo-spin model' in which the right and left displacements of the hydrogen ions along symmetric hydrogen bonds (O...H...O) can be described by a fictitious spin with up ([+1/2]) and down ([-1/2]) states.

[Figure 3.1.5.16]

Figure 3.1.5.16 | top | pdf |

The structure of potassium dihydrogen phosphate, KH2PO4, showing the O...H...O hydrogen bonds.

Unlike TSCC, discussed above, KDP has perfectly symmetric hydrogen bonds. Therefore, one might expect that above a sufficiently high temperature the protons can quantum-mechanically tunnel between equivalent potential wells separated by a shallow (and temperature-dependent) barrier. Below [T_C] the protons order (all to the right or all to the left) in spatial regions that represent ferroelectric domains. This model, initially proposed by Blinc (1960[link]), is correct and accounts for the large isotope shift in the Curie temperature noted for deuterated specimens. The complication is that the spontaneous polarization arises along a direction perpendicular to these proton displacements, so the dipoles do not arise from proton displacements directly. Instead, the proton coupling (largely Coulombic) to the potassium and phosphorus ions causes their displacements along the polar axis. This intricate coupling between protons along hydrogen bonds, which undergo an order–disorder transition, and K and P ions, which undergo purely displacive movements in their equilibrium positions, forms the basis of the theoretical interest in the lattice dynamics of KDP. Following Strukov & Levanyuk (1998[link]), we would say that arguments over whether this transition is displacive or order–disorder are largely semantic; the correct description of KDP is that the thermal change in occupancy of the O...H...O double wells modifies the free energy in such a way that the K and P ions undergo a displacive rearrangement.

The difficulty comes in recognizing that the normal-mode coordinate x corresponding to the soft mode in this case involves protons (H ions) and K and P ions. Therefore, the free-energy description (as in Fig. 3.1.5.17[link]) will have partly displacive character and partly order–disorder. If the transition were purely displacive (as in TSCC, discussed above), all the important temperature changes would be in the shape of the free energy [F(x)] with temperature T. Whereas if the transition were purely order–disorder (as in NaNO2, discussed below), the shape of the free-energy curves [F(x)] would be quite independent of T; only the relative populations of the two sides of the double well would be T-dependent. KDP is intermediate between these descriptions. Strictly, it is `displacive' in the sense that its normal mode is a propagating mode, shown in Fig. 3.1.5.18[link] by Peercy's pressure-dependence Raman studies (Peercy, 1975a[link],b[link]). If it were truly order–disorder, the mode would be a Debye relaxation with a spectral peak at zero frequency, independent of pressure or temperature. Only the width and intensity would depend upon these parameters.

[Figure 3.1.5.17]

Figure 3.1.5.17 | top | pdf |

Double-well models [circled letters show the time-averaged expectation values of the position [x(T)] of the order parameter at each temperature]. (a) For purely order–disorder systems, the depth and separation of the wells is temperature-independent; only the thermal populations change, due to either true quantum-mechanical tunnelling (which only occurs for H or D ions) or thermally activated hopping (for heavier ions). (b) For purely displacive systems, all the temperature dependence is in the relative depths of the potential wells. [For mixed systems, such as KH2PO4, both well depth(s) and thermal populations change with temperature.]

[Figure 3.1.5.18]

Figure 3.1.5.18 | top | pdf |

Pressure dependence of the `soft' optical phonon branch Raman spectra in potassium dihydrogen phosphate (after Peercy, 1975b[link]), showing the displacive character of the phase transition [purely order–disorder phase transitions cannot exhibit propagating (underdamped) soft modes].

As a final note on KDP, this material exhibits at ambient pressure and zero applied electric field a phase transition that is very slightly discontinuous. Application of modest pressure or field produces a truly continuous transition. That is, the tricritical point is easily accessible [at a critical field of 6 kV cm−1, according to Western et al. (1978[link])].

3.1.5.2.11. Sodium nitrite, NaNO2

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Sodium nitrite exhibits a purely order–disorder transition and has been chosen for discussion to contrast with the systems in the sections above, which are largely displacive. The mechanism of its transition dynamics is remarkably simple and is illustrated in Fig. 3.1.5.19[link]. There is a linear array of Na and N ions. At low temperatures, the arrow-shaped NO2 ions (within each domain) point in the same direction; whereas above the Curie temperature they point in random directions with no long-range order. The flopping over of an NO2 ion is a highly nonlinear response. Therefore the response function (spectrum) associated with this NO2 flip-flop mode will consist of two parts: a high-frequency peak that looks like a conventional phonon response (lightly damped Lorentzian), plus a low-frequency Debye relaxation (`central mode' peaking at zero frequency). Most of the temperature dependence for this mode will be associated with the Debye spectrum. The spectrum of sodium nitrite is shown in Fig. 3.1.5.20[link].

[Figure 3.1.5.19]

Figure 3.1.5.19 | top | pdf |

Structure of sodium nitrite, NaNO2. The molecularly bonded NO2 ions are shaped like little boomerangs. At high temperatures they are randomly oriented, pointing up or down along the polar b axis. At low temperatures they are (almost) all pointed in the same direction ([+b] or [-b] domains). Over a small range of intermediate temperatures their directions have a wave-like `incommensurate' modulation with a repeat length L that is not an integral multiple of the lattice constant b.

[Figure 3.1.5.20]

Figure 3.1.5.20 | top | pdf |

Raman spectra of sodium nitrite, showing diffusive Debye-like response due to large-amplitude flopping over of nitrite ions [note that the high-frequency phonon-like response is due to the small-amplitude motion of this same normal mode; thus in this system N ions give rise not to 3N (non-degenerate) peaks in the spectral response function, but to [3N+1]].

Particularly interesting is its phase diagram, relating structure(s) to temperature and `conjugate' field applied along the polar axis. As Fig. 3.1.5.21[link] illustrates somewhat schematically, there are first-order phase boundaries, second-order phase boundaries, a tricritical point and a critical end point (as in a gas–liquid diagram). If the electric field is applied in a direction orthogonal to the polar axis, a Lifshitz point (Fig. 3.1.5.22[link]) may be expected, in which the phase boundaries intersect tangentially. The ionic conductivity of sodium nitrite has made it difficult to make the figures in Figs. 3.1.5.21[link] and 3.1.5.22[link] precise.

[Figure 3.1.5.21]

Figure 3.1.5.21 | top | pdf |

Phase diagram for sodium nitrite for `conjugate' electric fields applied along the polar b axis, showing triple point, tricritical point and critical end point. (a) Schematic; (b) real system.

[Figure 3.1.5.22]

Figure 3.1.5.22 | top | pdf |

Phase diagram for sodium nitrite for electric fields applied perpendicular to the polar b axis. In this situation, a Lifshitz point is possible where phase boundaries `kiss' (touch tangentially).

3.1.5.2.12. Fast ion conductors

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As exemplary of this class of materials, we discuss in this section the silver iodide compound Ag13I9W2O8. This material has the structure illustrated in Fig. 3.1.5.23[link]. Conduction is via transport of silver ions through the channels produced by the W4O16 ions (the coordination is not that of a simple tetrahedrally coordinated WO4 tungstate lattice).

[Figure 3.1.5.23]

Figure 3.1.5.23 | top | pdf |

(a) Crystal structure of silver iodide tungstate (Ag13I9W2O8); (b) showing conduction paths for Ag ions (after Chan & Geller, 1977[link]).

This crystal undergoes three structural phase transitions (Habbal et al., 1978[link]; Greer et al., 1980[link]; Habbal et al., 1980[link]), as illustrated in Fig. 3.1.5.24[link]. The two at lower temperatures are first-order; that at the highest temperature appears to be perfectly continuous. Geller et al. (1980[link]) tried to fit electrical data for this material ignoring the uppermost transition.

[Figure 3.1.5.24]

Figure 3.1.5.24 | top | pdf |

Evidence for three phase transitions in silver iodide tungstate: (a) dielectric and conductivity data; (b) specific heat data; (c) Raman data. The lower transitions, at 199 and 250 K, are first order; the upper one, at 285 K, is second order.

As in most of the materials discussed in this review, the phase transitions were most readily observed via optical techniques, Raman spectroscopy in particular. The subtle distortions involve oxygen positions primarily and are not particularly well suited to more conventional X-ray techniques. Silver-ion disorder sets in only above the uppermost phase transition, as indicated by the full spectral response (as in the discussion of sodium nitrite in the preceding section).

Infrared (Volkov et al., 1985[link]) and Raman (Shawabkeh & Scott, 1989[link]) spectroscopy have similarly confirmed low-temperature phase transitions in RbAg4I5 at 44 and 30 K, in addition to the well studied [D_3^7\hbox{--}D_3^2] (R32–P321) transition at 122 K. The two lower-temperature phases increase the size of the primitive cell, but their space groups cannot be determined from available optical data. The 44 K transition is signalled by the abrupt appearance of an intense phonon feature at 12 cm−1 in both infrared and Raman spectra.

3.1.5.2.13. High-temperature superconductors

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It is useful to play Devil's Advocate and point out difficulties with the technique discussed, to indicate where caution might be exercised in its application. YBa2Cu3O7−x (YBaCuO) provides such a case. As in the case of BaMnF4 discussed in Section 3.1.5.2.7[link], there was strong evidence for a structural phase transition near 235 K, first from ultrasonic attenuation (Wang, 1987[link]; Laegreid et al., 1987[link]) and then from Raman studies (Zhang et al., 1988[link]; Huang et al., 1987[link]; Rebane et al., 1988[link]). However, as years passed this was never verified via neutron or X-ray scattering. Researchers questioned (MacFarlane et al., 1987[link]) whether indeed a phase transition exists at such a temperature in this important material. At present it is a controversial and occasionally contentious issue. A difficulty is that light scattering in metals probes only the surface. No information is obtained on the bulk. Ultrasonic attenuation and internal friction probe the bulk, but give scanty information on mechanisms or structure.

In the specific case of YBaCuO, the `extra' phonon line (Fig. 3.1.5.25[link]) that emerges below 235 K is now known not to be from the superconducting YBa2Cu3O7−x material; its frequency of 644 cm−1 is higher than that of any bulk phonons in that material. However, this frequency closely matches that of the highest LO (longitudinal optical) phonon in the semiconducting YBa2Cu3O6+x material, suggesting that the supposed phase transition at 235 K may be not a structural transition but instead a chemical transition in which oxygen is lost or gained at the surface with temperature cycling.

[Figure 3.1.5.25]

Figure 3.1.5.25 | top | pdf |

Raman spectra of YBa2Cu3O7−x below an apparent phase transition at ca. 235 K (Zhang et al., 1988[link]).

3.1.5.3. Low-temperature ferroelectric transitions

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It has historically been difficult to establish the nature of ferroelectric phase transitions at cryogenic temperatures. This is simply because the coercive fields for most crystals rise as the temperature is lowered, often becoming greater than the breakdown fields below ca. 100 K. As a result, it is difficult to demonstrate via traditional macroscopic engineering techniques (switching) that a material is really ferroelectric. Some authors have proposed (e.g. Tokunaga, 1987[link]) on theoretical grounds the remarkable (and erroneous) conjecture that no crystals have Curie temperatures much below 100 K. A rebuttal of this speculation is given in Table 3.1.5.1[link] in the form of a list of counterexamples. References may be found in the 1990 Landolt–Börnstein Encyclopedia of Physics (Vol. 28a). The original work on pure cadmium titanate and on lead pyrochlore (Hulm, 1950[link], 1953[link]) did not demonstrate switching, but on the basis of more recent studies on mixed crystals Ca2−2xPb2xNb2O7 and CaxCd1−xTiO3, it is clear that the pure crystals are ferroelectric at and below the stated temperatures.

Table 3.1.5.1| top | pdf |
Low-temperature ferroelectrics

FormulaCurie temperature [T_c] (K)Curie constant C (K)Entropy change [\Delta S] (cal mol−1 K−1)
NH4Al(SO4)2·12H2O 71 ? ?
NH4Fe(SO4)2·12H2O 88 400 0.15
(NH4)2Cd(SO4)3 95 ? ?
CdTiO3 55 [4.5\times 10^4] ?
Pb2Nb2O7 15.3 ? ?
LiTlC4H4O6·H2O 10.5 ? ?
K3Li2Nb5O15 7 ? ?

Hence, in Table 3.1.5.1[link] we see examples where X-ray structural studies may establish the symmetries requisite for ferroelectricity without the macroscopic switching being demonstrated. This is the converse case to that primarily emphasized in this section (i.e. the use of techniques complementary to X-ray scattering to determine exact crystal symmetries); it is useful to see these reverse cases to demonstrate the full complementarity of X-ray crystallography and dynamic spectroscopic techniques.

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