Condensed Matter A Week 2: Crystal structure (II)

Transcription

1 QUEEN MARY, UNIVERSITY OF LONDON SCHOOL OF PHYSICS AND ASTRONOMY Condensed Matter A Week : Crystal structure (II) References for crystal structure: Dove chapters 3; Sidebottom chapter. Last week we learnt about some common lattice types, including simple cubic, body-centred cubic, and face-centred cubic. This week we will list all the possible types of lattice in three dimensions. In order to do this, though, we first need to consider in more detail the possible symmetries of a crystal structure. Recall that a symmetry of some object is a geometric transformation that leaves the object for instance, a crystal structure unchanged. We have already seen that crystals have translational symmetry: specifically, that translation by any lattice vector leaves the crystal structure unchanged. But you know from everyday usage that there are many other types of symmetry. In particular, we will now discuss point symmetry: symmetry operations that, unlike translation, leave at least one point in space fixed. One point symmetry operation is rotation about some axis. Note the words we use, which are conventional: rotation is a symmetry operation; the axis about which we rotate is a symmetry element. If repeating the rotation n times will give a full revolution or in other words, if the rotation is through 36 /n we call this an n-fold rotation axis. Another familiar operation is reflection in some mirror plane. Here reflection is the operation; the mirror plane is the corresponding element. An operation with which you may be less familiar is inversion (the operation) through a point called a centre of symmetry (the element). If the centre of symmetry is at the origin, inversion is the transformation that takes a point at (x,y,z) to ( x, y, z). This is equivalent to reflection through three perpendicular mirror planes. Finally, we have the operation of improper rotation, consisting of rotation about some axis followed by inversion. In fact, reflection is equivalent to twofold improper rotation rotation through 8 followed by inversion about an axis perpendicular to the mirror plane. Similarly, inversion is trivially equivalent to onefold improper rotation, rotation through 36 which of course does nothing, followed by inversion. Thus we can describe all point symmetry operations as proper or improper rotation. As an example of some of these points, consider a set of atoms forming a tetrahedron, as seen, for example, in the diamond structure: Some of the point symmetry elements of this collection of atoms are illustrated in the figure. There are four threefold rotation axes, of which one is More generally, inversion through a centre of symmetry at (p,q,r) takes (x,y,z) to (p x,q y,r z). We can call rotation proper rotation if we want to explicitly distinguish it from improper rotation.

2 shown in blue, and three mirror planes of which one is shown in red. There is no proper fourfold rotation axis, but there are three improper fourfold rotation axes, of which one is shown in green. There is no centre of inversion. We will continue occasionally to note symmetry in the structures we study this week. Symmetry is important in many different parts of condensed matter physics, some of which are well beyond the scope of this module. For our purposes there are two main consequences of the symmetry of a structure. The first is that point symmetry operations are closely related to the lattice underlying a given crystal structure. Consider the graphene structure once more. By inspection we concluded that the lattice parameters were related by a = b and γ =. Looking for symmetry elements, though, we see that there are clear sixfold rotation axes, shown by hexagons in the diagram below: A little consideration of possible lattice vectors a and b will show that only these lattice parameters are compatible with this symmetry. 3 More generally, and also in three dimensions, the particular symmetries of a crystal structure limit the possibilities for the lattice parameters. Taking these possibilities into account, there are only seven different lattice systems containing fourteen Bravais lattices, which are listed in the table below: simple base-centred body-centred face-centred Triclinic Monoclinic α = γ = 9 one twofold axis Orthorhombic α = β = γ = 9 three twofold axes Rhombohedral a = b = c, α = β = γ one threefold axis Tetragonal a = b, α = β = γ = 9 one fourfold axis Hexagonal a = b, α = β = 9, γ = one threefold or sixfold axis Cubic a = b = c, α = β = γ = 9 four threefold axes In particular the three different cubic lattices and the hexagonal lattice are familiar from the crystal structures we have already studied. This diagram shows a further reason why it is useful to consider centred cells: as well as making calculations easier, the conventional cells also make the symmetry 3 There are also twofold and threefold rotation axes in the graphene structure; putting these in the correct spots is left as an exercise!

3 requirements of the lattice clearer. It is much easier, for instance, to see the symmetry of the face-centred cubic conventional unit cell than the corresponding primitive unit cell with α = β = γ = 6. The second reason why symmetry is so important is that it has a direct impact on the physical properties of a material. For instance, in many materials electric polarisation is important. In ferroelectrics, the polarisation can be switched back and forth by applying an external electric field, much like the magnetisation in ferromagnets; this property finds applications, for instance, in computer memory ( FeRAM ). However, certain symmetry elements destroy this property. For instance, a centre of symmetry will mean that any dipole moment is exactly cancelled out by an identical one pointing in the opposite direction; thus centrosymmetric materials those with a centre of symmetry can never have a permanent electric dipole moment. Deeper analysis of a material s symmetry can similarly rule out or constrain many of its physical properties. Having established the different lattice systems, we will continue our study of common crystal structures. Last week we considered elemental structures; this week we will look at materials containing two or three elements. All of the common structures we will learn about are based on cubic lattices. We have already seen materials in which the dominant interactions were covalent or metallic. Diatomic structures will also include materials in which the major bonding type is ionic. Thus we are building up a repertoire of different types of material to discuss the energies associated with crystal packing next week. Similarly, we have examples of metals, insulators, and semiconductors for when we discuss electronic properties later in the module. The simplest diatomic crystal structure is the sodium chloride structure, also called halite or rock salt (both of which are names for sodium chloride as a naturally occurring mineral). This is very common in simple ionic materials, and simply consists of a checkerboard pattern of the two different ions: It is important to note that, although the arrangement of atoms looks similar to that in a primitive cubic elemental structure, in fact closer inspection will show that this is based on a face-centred cubic lattice with a motif consisting of one type of atom at (,,) and the other at (,,). The coordination number of both cations (positively charged ions) and anions (negatively charged ions) is 6. Note also that, although the primitive cubic elemental structure is unstable with respect to shearing, the sodium chloride structure is not, because in this case shearing would bring pairs of ions with the same charge together, which is unfavourable by Coulomb s law: 3

4 Simple cubic Rock salt Another common diatomic structure is the caesium chloride structure, common in ionic materials with bigger cations: Again, although this looks similar to the body-centred cubic structure, it is based on a simple cubic lattice with a motif of one type of atom at (,,) and the other at (,, ). This time the coordination number for both cations and anions is 8. Just as we saw for the elements, the coordination number of diatomic materials tends to be lower where the bonding is more covalent than when the bonding is more ionic. An example of such a structure is the cubic zinc sulfide structure, also known as the zinc blende or sphalerite structure. (Again, both of these are names for zinc sulfide as a mineral. The cubic is important because there is also a form of zinc sulfide that has a hexagonal lattice.) 3 3 This is also based on the face-centred cubic lattice, but with a motif of one type of atom at (,,) and the other at (,, ). In fact, this is identical to the diamond structure, except that the two motif atoms are of different types. Like the atoms in the diamond structure, both cations and anions have a coordination number of four. Closely related to this is the calcium fluoride or fluorite structure, which simply has twice the number of anions, so that while the anions remain tetrahedrally coordinated, the cations now have a coordination number of eight:

5 , 3, 3, 3, 3 Again, this has a face-centred cubic lattice, with a motif of a cation at (,,) and two anions at ±,±,±. One important difference between the fluorite and zinc sulfide structures is that the fluorite structure is centrosymmetric whereas the zinc sulfide structure is not. Finally, the perovskite structure is the only triatomic structure that we will consider as part of this module. It has general formula ABX 3, where A and B are usually metal ions and X oxide. However, part of its interest comes from the fact that there is substantial compositional flexibility, and materials of this general structure can also be made with fluoride or other linker anions, and with many different mono- and polyatomic cations on the A and B sites. The B site, shown in black below, is octahedrally coordinated to six X atoms (the coordination octahedron is itself shown for clarity in the 3D picture below); the A site is surrounded by X atoms: An alternative unit cell in which the B atoms are at the corners, with the A atom at the centre, is also a common representation of this structure. If, as previously, we model the ions as rigid spheres, in a perfectly fitting structure the B and X ions would just touch along an edge, in the [] direction, so that r B + r X = a. On the other hand, the A and X ions would just touch along a face diagonal, in the [] direction, so that r A +r X = a. In general, given arbitrary A, B, and X, it is not possible to find an a that solves both of this equations. Instead, the perovskite structure is able to distort to accommodate mismatch in the size of the component ions. If the A ion is too small, the BX 6 octahedra will rotate so that the cavity in which the A ion sits is smaller: 5

6 This, for instance, happens in strontium titanate, SrTiO 3. On the other hand, if the B ion is too small, it will tend to be displaced from the centre of the BX 6 octahedra: (Note that in both of these figures the scale of the distortion has been exaggerated for clarity.) The perovskite structure is centrosymmetric. However, displacement of the B ion as shown above breaks the symmetry so that there are no longer any centres of symmetry in the material, allowing a net polarisation. Lead titanate, PbTiO 3, is a material with this structure used, as mentioned above, in ferroelectric RAM. Other well-known perovskites include lithium niobate, used in the telecommunications and photonics industries; magnesium silicate, found in the Earth s lower mantle, and methylammonium lead iodide, a promising solar cell material. Next week: What forces between atoms give rise to these different crystal structure types? How can we calculate the energy associated with crystal packing? Can you find some centres of symmetry? There are several. 6

7 Revision questions for week A9. What is translational symmetry? State how a crystal lattice shows the translational symmetry of the structure. A. Give the relationship between the lattice parameters of the following lattice types: (a) Cubic (b) Hexagonal (c) Tetragonal (d) Orthorhombic (e) Monoclinic (f) Triclinic A. Calculate the volume of the unit cells of the following crystal structures: (a) An orthorhombic structure with lattice parameters a =.Å, b = 5.Å, and c = 6.Å. [ Å 3 ] (b) A hexagonal structure with lattice parameters a =.Å and c = 6.Å. [83 Å 3 ] (c) A tetragonal structure with lattice parameters a = 8.Å and c = 5.5Å. [35 Å 3 ] (d) A monoclinic structure with lattice parameters a = 5.Å, b = 9.Å, c = 6.7Å, and β =.. [9. Å 3 ] A. Define the following: (a) Mirror plane (b) Rotation axis (c) Centre of symmetry A3. State the coordination number of the cations in the following simple ionic structure types: (a) NaCl [6] (b) CsCl [8] (c) Cubic ZnS [] (d) CaF [8] A. Draw D projections viewed down the z axis of the following structure types, showing the outline of the unit cell and the atomic positions marked with their z fractional coordinate: (a) NaCl (b) CsCl (c) Cubic ZnS (d) CaF A5. (a) Draw a D projection of the cubic perovskite structure of SrTiO 3 viewed down the z axis, showing the outline of the unit cell, and the atomic positions marked with their z fractional coordinate. In this material Ti is the cation with octahedral coordination. (b) Give expressions for the Sr O, Ti O and O O distances in terms of the lattice parameter a. (c) How many atoms are in the unit cell? [5] (d) What are the coordination numbers of the Sr and Ti cations? [, 6] B6. (a) Explain two consequences of the point symmetry of a crystal. (b) What is the difference between a primitive and a centred unit cell? Explain two reasons why using a centred unit cell may be more convenient. (c) Explain why the simple cubic structure is unstable for metals, but the sodium chloride structure is stable for ionic compounds. B7. Identify, with the aid of a suitable sketch or diagram, as many symmetry operations as you can, including centres of symmetry, rotation axes, roto-inversion axes, and mirror planes, for atoms arranged in the following geometries. Remember that angles between different rotation and roto-inversion axes need not be 9. (a) A group of seven atoms arranged in the form of a perfect octahedron (six atoms at the vertices, one at the centre). (b) A group of five atoms arranged in the form of a perfect tetrahedron (four atoms at the vertices, one at the centre). B8. Sketch a D projection of the unit cell of the caesium chloride structure, looking down the c axis and as usual labelling each atom with its c fractional coordinate. On your diagram, show one example of each of the following symmetry elements: a translation vector; a fourfold rotation axis; a twofold rotation axis that is not a fourfold axis; a mirror plane; and a centre of symmetry. Be careful to show the positions of each unambiguously (e.g., give the c coordinate where necessary). 7