Roger Johnson Structure and Dynamics: Displacive phase transition Lecture 9

Transcription

1 9.1. Summary In this Lecture we will consider structural phase transitions characterised by atomic displacements, which result in a low temperature structure that is distorted compared to a higher temperature, higher symmetry structure. Such phase transitions are intimately linked to the breaking of symmetry, and hence they can be described in terms of space groups, and measured using diffraction techniques. Any component of the crystal symmetry may change: the crystal system, point group, centring translations, and lattice translational symmetries may be broken upon cooling through the phase transition. A displacive phase transition is driven by the thermodynamics of the system, which we will discuss in detail in Lecture 10. As a result, it is very unusual for symmetry to be increased or reinstated upon cooling through a displacive phase transition, so in general the low temperature phase has the lower symmetry Ferroelectric phase transitions At high temperature PbTiO 3 crystallises in the well known perovskite crystal structure. This structure has a cubic unit cell with the lead atoms at the corners, the titanium atoms at the body centre, and the oxygen atoms at the face centres (see Figure 1). At 766 K PbTiO 3 undergoes a displacive structural phase transition. The titanium ion moves by a small amount along the z direction, and the oxygen atoms move by a small amount in the opposite direction, also along z. These atomic displacements break the cubic symmetry of the high temperature phase, and the crystal system becomes tetragonal. Accordingly, the lattice will distort due to strain, constrained by the tetragonal symmetry of the new crystal system i.e. a c = b c = c c a t = b t c t, where subscripts c and t refer to the cubic and tetragonal crystal systems, respectively. Note that at a ferroelectric phase transition the periodicity of the direct lattice remains the same, i.e. the atomic displacements are the same in every unit cell of the high temperature crystal structure. One can visually identify that the atomic displacements in PbTiO 3 generate a net microscopic dipole moment in the unit cell, as the centre of mass of the negative charges (O 2 ) is no longer coincident with the centre of mass of the positive charges (Pb 2+, Ti 4+ ). The bulk crystal will therefore exhibit a macroscopic electric polarisation, as every unit cell is the same by the translational symmetry of the lattice. In this case an electrical polarisation appears spontaneously, and the material becomes ferroelectric. Figure 1: The crystal structure of PbTiO 3 below (left) and above (right) the ferroelectric displacive phase transition (n.b. distortions are not drawn to scale). Page 1 of 7

2 9.3. Neumann s principle We can deduce the presence of a spontaneous electric polarisation from the space group of the distorted structure. The high temperature space group is cubic (P m3m, non-examinable), and has 48 symmetry operators including the inversion. At the displacive phase transition 40 of the 48 symmetries are broken, and the crystal space group becomes tetragonal P 4mm. Firstly, P 4mm does not contain inversion symmetry. Therefore, by symmetry alone we know that it could support a macroscopic polar vector. Secondly, P 4mm has a unique axis, the four-fold axis, and all other symmetry operators (1, m x, m y, m xy, m xy ) leave a polar vector parallel to this axis invariant. The absence of inversion symmetry is a necessary but not sufficient condition for a space group to allow a spontaneous polarisation. For example, P 422 does not have inversion symmetry, but it also does not have a unique rotation axis. In this case a polarisation along the four-fold axis, say P z, transforms into its inverse, P z, by the two orthogonal two-fold axes, therefore P z = 0 by symmetry. Neumann s Principle states that the physical properties of a crystal must be symmetric under the symmetry operations of the space group. Typically, the measurement of physical properties occurs on length scales that effectively average over the translational symmetry of the crystal (diffraction is an exception). Therefore, we can generally state that the physical properties of the crystal must be symmetric under the symmetry operations of the point group of the crystal. It follows from Neumann s principle that we can define a subset of the 32 point groups that can support a macroscopic electric polarisation, they are 1, 2, m, mm2, 4, 4mm, 3, 3m, 6, and 6mm. In P 4mm it is a requirement of symmetry that the bulk electric polarisation is aligned parallel to the 4-fold symmetry element. In this example we have used the same real space basis to describe both high and low temperature structures. This is not essential, and is often inconvenient. However, the symmetry elements of the low temperature space group are necessarily present in the high temperature space group, and in the same relative orientation. One can therefore immediately determine the direction of the polarisation with respect to the undistorted structure s basis by identifying the orientation of the symmetry element that defines the unique axis, regardless of any change of basis Domains In the cubic structure of PbTiO 3 there are three 4-fold axes parallel to x, y, and z. At the displacive structural phase transition the atomic displacement can in principle occur along any of the three symmetry equivalent directions. This leads to the formation of six domains (positive and negative displacements along x, y, or z). The domains of the distorted structure are related by symmetry operations broken at the phase transition. The number of domains can therefore be calculated by dividing the number of symmetry operators in the high temperature point group by the number of operators in the low temperature point group. For PbTiO 3 we get 48 / 8 = 6, consistent with the discussion above. As the atomic displacements are small they can be influenced by applying an electric field. In a ferroelectric, the effect of an electric field is to grow domains of one polarisation, at the expense of shrinking domains of the other, giving rise to ferroelectric hysteresis (analogous to ferromagnetism), as shown in figure 2. Page 2 of 7

3 Figure 2: Ferroelectric hysteresis loop 9.5. Antiferroic phase transitions In a ferroelectric phase transition the atomic displacements occur within a single unit cell, hence the translational symmetry of the lattice is fully preserved below the transition. This case is in fact rare. It is typical that atomic displacements in one unit cell of the high temperature structure are balanced by atomic displacements in a neighbouring unit cell. Then the unit cell of the low temperature structure is larger, increased by an integer multiple depending on the number of neighbouring cells that balance the distortion. This gives rise to a new real space lattice with new translational symmetry. This type of phase transition is known as antiferroic. If opposite dipole moments are created in neighbouring cells, then the phase transition is considered to be antiferroelectric, in analogy to a ferroelectric phase transition but with an exact cancellation of the electric polarisation. If the atomic displacements do not result in the creation of electric dipoles, then the transition is known just as antiferroic. Like PbTiO 3, SrTiO 3 is a cubic perovskite crystal at high temperature. Unlike PbTiO 3, SrTiO 3 undergoes an antiferroic displacive phase transition at 105 K, as illustrated in figure 3. The atomic displacements correspond to a rotation of the TiO 6 octahedra about the z axis, and the sense of rotation alternates in neighbouring unit cells within the xy plane (and also between adjacent unit cells along z). The low temperature crystallographic unit cell is tetragonal with a volume times larger than the high temperature cubic cell. The basis vectors of the tetragonal cell are related to the cubic axes by [[1, 1, 0], [ 1, 1, 0], [0, 0, 2]], where the axis of octahedral rotation is parallel to z (see figure 3). Accordingly, the reciprocal lattice unit cell is times smaller. At the displacive phase transition the space group symmetry is lowered from cubic (P m3m) to tetragonal I4/mcm, which has 16 symmetry operations in the primitive unit cell. We therefore expect 48 / 16 = 3 domains, which correspond to a TiO 6 octahedra rotation axis along x, y, or z, respectively. Page 3 of 7

4 Figure 3: The crystal structure of SrTiO 3 below (left) and above (right) the antiferroic displacive phase transition. Arrows indicate the direction of oxygen displacements compared to the high temperature phase. Figure 4: X-ray powder diffraction pattern for SrTiO 3 in the cubic phase Diffraction Displacive phase transitions can be observed by diffraction. Figure 4 shows the powder diffraction pattern for SrTiO 3 in the high temperature phase. Figure 5 shows the same powder data, but with the data for the low temperature phase overlaid. The two patterns are similar, but there are two key differences: a) Figure 6 shows the two data sets plotted on a logarithmic y-axis. One can see that in the low temperature phase additional, well isolated, weak diffraction peaks have appeared, marked with an asterisk. These diffraction peaks arise due to the increase in the direct space lattice periodicity, and the respective decrease in the reciprocal lattice periodicity. For example, there are twice as many reciprocal lattice points along the a direction for a crystal with a direct lattice of periodicity 2a, compared to a crystal with periodicity a. b) Figure 7 shows the splitting of the (2,0,0) reflection below the phase transition. This arises due to the distortion of the lattice from cubic to tetragonal. This peak has multiplicity 6 in the cubic crystal, and splits into two peaks, one of multiplicity 4, and one of multiplicity 2, in Page 4 of 7

5 Figure 5: X-ray powder diffraction pattern for SrTiO 3 in the cubic phase (blue) with the tetragonal phase pattern overlaid (red). Figure 6: X-ray powder diffraction pattern for SrTiO 3 in the cubic phase (blue) with the tetragonal phase pattern overlaid (red), plotted on a log scale. the tetragonal case. One can see that the ratio of multiplicities is consistent with the ratio of intensities. The degree and direction of the splitting is dependent upon the a/c ratio of the tetragonal lattice The order parameter The magnitude of the individual atomic displacements quantifies the phase transition at a microscopic level. The problem can be simplified as the pattern of atomic displacements remains the same at all temperatures below the phase transition. In analogy to the vibrational modes of molecules, or the phonon modes of the crystal, the pattern of atomic displacements that appear below the structural phase transition is a displacive mode of the high temperature crystal. The mode has an independent amplitude, and the displacement pattern is determined solely by the microscopic force constants. The mode amplitude (global atomic displacement magnitude) can be used as a single order parameter to quantify the phase transition. The order parameter can sometimes be represented as a macroscopic observable. For example, in PbTiO 3 the net ferroelectric polarisation is directly proportional to the ferroelectric order parameter. In other cases the order parameter can be represented microscopically, for example, the angle of rotation of the TiO 6 octahedra in SrTiO 3. Page 5 of 7

6 Figure 7: The (200) x-ray powder diffraction reflection of SrTiO 3 in the cubic phase (blue) with the tetragonal phase overlaid (red). If upon warming out of the low temperature distorted phase the order parameter varies smoothly down to zero at the phase transition, then the transition is known as continuous, or second order. If, upon warming, the order parameter abruptly drops to zero at the phase transition, the transition is discontinuous, or first order. The names for the phase transitions come from thermodynamics, which we will study in detail in Lecture 10. If there is a discontinuous change in the first derivative of the free energy as a function of temperature (the entropy) at the phase transition, then the transition is labelled first order. If the first derivative is continuous, but the second derivative (heat capacity) exhibits a discontinuity, then the transition is labelled second order Soft phonon modes If a displacive phase transition occurs in a crystal, then the energetic potential of an atom that becomes displaced, say along x, must have a double well shape either side of the high temperature equilibrium position ( x = 0), as shown below. As stated above, the static atomic displacements that appear at low temperature correspond to a displacive mode of the high temperature crystal structure. Above the phase transition, there will exist a phonon mode with the same pattern of atomic displacements. In the presence of double well potentials, the frequency of this phonon mode will reduce, or soften (in analogy with a softening of the atomic force constants), as the crystal is cooled. At a certain temperature the phonon frequency will become zero, and the crystal will then undergo a displacive phase transition. On further cooling, the frequency of the mode will increase, or harden, Page 6 of 7

7 now that the crystal structure has distorted into its energetic ground state. A ferroelectric phase transition can be described by the softening of an optic mode at the centre of the Brillouin zone. The fact that the phonon mode is at the zone centre means that the translational symmetry of the lattice is not broken at the phase transition a phonon with zero momentum is a standing wave with the same periodicity as the lattice, and hence the respective atomic displacements are the same in every unit cell. The fact that the phonon is an optic mode tells us that it is associated with some atoms moving in opposite directions, which gives rise to a net dipole moment if the atoms/ions have different charge. Modes at the Brillouin zone boundary are also standing waves, but with finite momentum. Here, the softening of a mode will result in a change in the translational symmetry of the lattice, and an increase in the unit cell by an integer multiple of the high temperature cell. For example, the softening of a mode with wavevector ( 1 2, 0, 0) will result in a doubling of the direct lattice periodicity along a (note that the wave vector of the soft mode corresponds exactly to the new set of diffraction peaks observed below the phase transition, discussed above). Figure 8 shows the calculated phonon dispersion for SrTiO 3 at the displacive phase transition. One can see that the acoustic phonon modes at the A-point on the Brillouin zone boundary have softened. Figure 8: The calculated phonon dispersion for SrTiO 3 at the displacive phase transition. Page 7 of 7