Crystal Structure and Dynamics Paolo G. Radaelli, Michaelmas Term 2014
Part 1: Symmetry in the solid state Lectures 1-5
Web Site: http://www2.physics.ox.ac.uk/students/course-materials/c3-condensed-matter-major-option
Bibliography
◦ C. Hammond The Basics of Crystallography and Diffraction, Oxford University Press (from Blackwells). A well-established textbook for crystallography and diffraction.
◦ D.S. Sivia, Elementary Scattering Theory for X-ray and Neutron Users, Oxford University Press, 2011. An excellent introduction to elastic and inelastic scattering theory and the Fourier transform.
◦ Martin T. Dove, Structure and Dynamics: An Atomic View of Materials, Oxford Master Series in Physics, 2003. Covers much the same materials as Lectures 1-11.
◦ T. Hahn, ed., International tables for crystallography, vol. A (Kluver Academic Publisher, Dodrecht: Holland/Boston: USA/ London: UK, 2002), 5th ed. The International Tables for Crystallography are an indispensable text for any condensed-matter physicist. It currently consists of 8 volumes. A selection of pages is provided on the web site. Additional sample pages can be found on http://www.iucr.org/books/international-tables.
◦ C. Giacovazzo, H.L. Monaco, D. Viterbo, F. Scordari, G. Gilli, G. Zanotti and M. Catti, Fun- damentals of crystallography (International Union of Crystallography, Oxford University Press Inc., New York)
◦ Paolo G. Radaelli, Symmetry in Crystallography: Understanding the International Tables , Oxford University Press (2011). Contains a discussion of crystallographic symmetry using the more abstract group-theoretical approach, as described in lectures 1-3, but in an extended form.
◦ ”Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher”, W.H.Freeman and Company, 1990. A collection of symmetry drawings by M.C. Escher, also to be found here: http://www.mcescher.com/Gallery/gallery-symmetry.htm
◦ Neil W. Ashcroft and N. David Mermin, Solid State Physics, HRW International Editions, CBS Publishing Asia Ltd (1976) is now a rather old book, but, sadly, it is probably still the best solid-state physics book around. It is a graduate-level book, but it is accessible to the interested undergraduate.
1 Contents
1 Lecture 1 — Translational symmetry in crystals: unit cells and lattices 3 1.1 Introduction: crystals and their symmetry ...... 3 1.2 Lattices and coordinates ...... 3 1.2.1 Bravais lattices ...... 5 1.2.2 Choice of basis vectors ...... 6 1.2.3 Centred lattices ...... 6 1.3 Distances and angles ...... 6 1.4 Dual basis and coordinates ...... 8 1.5 Dual basis in 3D ...... 10 1.6 Dot products in reciprocal space ...... 10 1.7 A very useful example: the hexagonal system in 2D ...... 11 1.8 Reciprocal lattices ...... 11 1.8.1 Centring extinctions ...... 12
2 Lecture 2 — Rotational symmetry 16 2.1 Introduction ...... 16 2.2 Generalised rotations expressed in terms of matrices and vectors ...... 16 2.2.1 Rotations about the origin ...... 16 2.2.2 Rotations about points other than the origin ...... 17 2.2.3 Roto-translations ...... 18 2.2.4 Generalisation of symmetry to continuous functions and patterns ...... 18 2.3 Some elements of group theory ...... 18 2.3.1 Symmetry operators as elements of crystallographic groups ...... 19 2.3.2 Optional: the concept of group class ...... 20 2.4 Symmetry of periodic patterns in 2 dimensions: 2D point groups and wallpaper groups 21 2.4.1 Point groups, wallpaper groups, space groups and crystal classes ...... 22 2.4.2 Point groups in 2D ...... 23 2.4.3 Bravais lattices in 2D ...... 23
3 Lecture 3 — Symmetry in 3 dimensions 26 3.1 New symmetry operators in 3D ...... 26 3.1.1 Inversion and roto-inversion ...... 26 3.1.2 Glide planes and screw axes in 3D ...... 26 3.2 Understanding the International Tables of Crystallography ...... 29 3.2.1 ITC information about crystal structures ...... 29 3.2.2 ITC information about diffraction patterns ...... 29
4 Lecture 4 — Fourier transform of periodic functions 33 4.1 Fourier transform of lattice functions ...... 33 4.2 Atomic-like functions ...... 33 4.3 Centring extinctions ...... 34 4.4 Extinctions due to roto-translations ...... 34 4.5 The symmetry of |F (q)|2 and the Laue classes ...... 35
5 Lecture 5 — Brillouin zones and the symmetry of the band structure 37 5.1 Symmetry of the electronic band structure ...... 38 5.2 Symmetry properties of the group velocity ...... 40 5.2.1 The oblique lattice in 2D: a low symmetry case ...... 41 5.2.2 The square lattice: a high2 symmetry case ...... 42 5.3 Symmetry in the nearly-free electron model: degenerate wavefunctions ...... 44 1 Lecture 1 — Translational symmetry in crystals: unit cells and lattices
1.1 Introduction: crystals and their symmetry
In 1928, Swiss physicist Felix Bloch obtained his PhD at University of Leipzig, under the supervision of Werner Heisenberg. In his doctoral dissertation, he presented what we now call the quantum theory of crystals, and what a mo- mentous occasion that was! Much of the technology around us — from digital photography to information processing and storage, lighting, communication, medical imaging and much more is underpinned by our understanding of the behavior of electrons in metals (like copper) and semiconductors (like silicon). Today’s contemporary physics also focuses on crystals, in particular on prop- erties that cannot be described in terms of simple one-electron physics, such as high-temperature superconductivity.
One may ask what is special about a crystal as compared, for example, to a piece of glass. From previous introductory courses, you already know part of the answer: crystals have a lattice, i.e., atoms are found regularly on the corners of a 3-dimensional grid. You also know that some crystals have a base, comprising several atoms within the unit cell, which is replicated by placing copies at all nodes of the lattice.
At a more fundamental level, crystals differ from glasses and liquids because they have fewer symmetries. In a liquid or glass, properties look identical on average if we move from any point A to any point B (translation) and if we look in any direction (rotation). By contrast in a crystal, properties vary in a periodic manner by translation, and are only invariant by certain discrete rotations. The theory of symmetries in crystals underpin all of the crystal physics we have mentioned and much more. Here, we will start by describing translational symmetry in a more general way that you might have seen so far, suitable for application to crystals with non-orthogonal lattices. We will then look at ways of classifying rotational symmetries systematically, and we will learn how to use the International Tables of Crystallography (ITC hereafter), an essential resource for condensed matter physics research.
1.2 Lattices and coordinates
Let us start by looking at the cubic, tetragonal and orthorhombic lattices you should already be familiar with. The nodes (i.e., unit cell origins) of an or-
3 thorhombic lattice are expressed by means of vectors 1 of the form:
ˆ ˆ ˆ rn = a i nx + b j ny + c k nz (1) where a, b, c are the lattice constants (a = b for the tetragonal lattice, a = b = c for the cubic lattice) and ˆi,ˆj, kˆ are unit vectors along the x, y and z axes, while nx, ny and nz are arbitrary integers. We can interpret these vectors as invariant translation operators — if points A and B are separated by one such vector, the properties of the crystal are identical in A and B.
The position of points in the crystal that are not on lattice nodes can be ex- pressed as
r = aˆi x + bˆj y + c ˆk z (2)
One can see the relation between x, y and z (which are dimensionless) and the usual Cartesian coordinates, which have dimensions. We can define the basis vectors (not to be confused with the unit cell base) a = aˆi, b = bˆj and c = c ˆk. In the cubic, tetragonal and orthorhombic lattices, the basis vectors are orthogonal, but this is not so in other most important cases — for example, that of hexagonal crystal like graphite, where the angle between a = b 6= c, α = β = 90◦ and γ = 120◦. We can, however, generalise the notion of basis vectors so that it applies to cases such as this and even the most general (triclinic) lattice. In synthesis:
• In crystallography, we do not usually employ Cartesian coordinates. In- stead, we employ coordinate systems associated with appropriate ba- sis vectors.
• Basis vectors have the dimension of a length, and coordinates (position vector components) are dimensionless.
• We can denote the basis vectors as ai , where the correspondence with the usual crystallographic notation is
a1 = a; a2 = b; a3 = c (3)
1Here, we will call vector an object of the form shown in eq. 2, containing both components and basis vectors. In this form, a vector is invariant by any coordinate transformation, which affect both components and basis vectors. Both basis vectors and components (coordinates) are written as arrays — see below. For a concise introduction to this kind of notation, see ITC volume B, chapter 1.
4 • We will sometimes employ explicit array and matrix multiplication for clarity. In this case, the array of basis vectors is written as a row 2, as in [a] =
[a1 a2 a3].
• With this notation, a generic position within the crystal will be defined by means of a position vector, written as
X i r = aix = ax + by + cz (4) i
Points within the unit cell at the origin of the lattice have 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 — the so-called fractional coordinates.
• We can also define other vectors on the lattice, for example, the instanta- neous velocity of a vibrating atom. The components of a generic vector v will be denoted as vi, where
1 2 3 v = vx; v = vy; v = vz; (5)
v1 • Components will be expressed using column arrays, as in [v] = v2 v3
• A generic vector is then written as
X i 1 2 3 v = aiv = a1v + a2v + a3v (6) i
1.2.1 Bravais lattices
From eq. 4, we can deduce the position of the lattice points in the most general case:
X i rn = ain = anx + bny + cnz (7) i
As you already know, in 3 dimensions there are 14 Bravais lattices (from 19- century French physicist Auguste Bravais) — see fig. 1. The lattices have distinct rotational symmetries, belonging to one of the 7 crystal systems (see
2By writing basis vectors as rows and coordinates or components as columns, vectors become the usual product of a row and a column array. Moreover, upon coordinate trans- formations, the transformation matrices for basis vectors and components are inverse of one another. The subscript and superscript notation also emphasises this — note that vectors have fully “contracted” indices.
5 below), and distinct topologies, since some crystal systems admit both primi- tive and centred lattices.
1.2.2 Choice of basis vectors
There can be many choices of basis vectors. In order to describe the same lattice, the two transformation matrices converting one basis set to the other and vice versa must be integer matrices, which implies that their determinant is exactly 1 or -1 (depending on the handedness of the basis sets). All these choices are equivalent from the point of view of the translational symmetry. Whenever possible, the basis vectors are chosen so that the corresponding unit cell has the full rotational symmetry of the lattice.
1.2.3 Centred lattices
Making such a choice is not possible for the so-called centred lattices, so one has to make a choice of either using a primitive cell or using a non-primitive cell (the conventional cell) with the full rotational symmetry of the lattice. You have already encountered some important cases of centred lattices, such as the BCC and FCC cubic lattices of many metals — the others are shown in fig. 1. Body centred cells (e.g., cubic BCC) and cells with one pair of centred faces (A, B or C centring) have a 2-atom base. Cells with all faces centred (such as cubic FCC) have a 4-atom base. The conventional hexagonal cell of rhombohedral (trigonal) lattices have a 3-atom base. All lattice points de- fine invariant translations as before, regardless of whether points are at the corners of the unit cell or at ”centring positions”. For these lattices, we can always choose to use either primitive or conventional basis vectors:
• When primitive basis vectors are used, coordinates of points related by translation will differ by integral values of x,y and z.
• When conventional basis vectors are used, coordinates of points related by translation will differ by either integral or simple fractional (either n/2 or n/3) values of x,y and z.
1.3 Distances and angles
• The dot product between two position vectors is given explicitly by
6 Figure 1: The 14 Bravais lattices in 3D. The R-centred hexagonal cell can be used as a conventional cell for the trigonal lattice.
7 r1 · r2 = a · a x1x2 + b · b x1x2 + c · c z1z2 +
+a · b [x1y2 + y1x2] + a · c [x1z2 + z1x2] + b · c [y1z2 + z1y2] = 2 2 2 = a x1x2 + b x1x2 + c z1z2 + ab cos γ [x1y2 + y1x2]
+ac cos β [x1z2 + z1x2] + bc cos α [y1z2 + z1y2] (8)
• More generically, the dot product between two vectors is written as
X i X j X i j v · u = aiv · ajv = [ai · aj] u v (9) i j i,j
• The quantities in square bracket represent the elements of a symmetric matrix, known as the metric tensor. The metric tensor elements have the dimensions of length square.
Gij = ai · aj (10)
Calculating lengths and angles using the metric tensor
• You are generally given the lattice parameters a, b, c, α, β and γ. In terms of these, the metric tensor can be written as
a2 ab cos γ ac cos β G = ab cos γ b2 bc cos α (11) ac cos β bc cos α c2
• To measure the length v of a vector v:
v1 2 2 v = |v| = [ v1 v2 v3 ]G v2 (12) v3
• To measure the angle θ between two vectors v and u:
v1 1 cos θ = [ u1 u2 u3 ]G v2 (13) uv v3
1.4 Dual basis and coordinates
• Let us assume a basis vector set ai for our vector space as before, and let us consider the following set of new vectors.
8 i X −1 ki b = 2π ak(G ) (14) k
From Eq. 10 follows:
j X −1 ki X −1 kj j ai · b = ai · 2π ak(G ) = 2π Gik(G ) = 2πδi (15) k k
i −1 • Note that the vectors b have dimensions length . Since the bi are linearly independent if the ai are, one can use them as new basis vectors, form- ing the so-called dual basis. This being a perfectly legitimate choice, can express any vector on this new basis, as
X i q = qib (16) i
• We can write any vector on this new basis, but vectors expressed us- ing dimensionless coordinates on the dual basis have dimensions length−1, and cannot therefore be summed to the position vectors.
• We can consider these vectors as representing the position vectors of a separate space, the so-called reciprocal space.
• The dot product between position vectors in real and reciprocal space is a dimensionless quantity, and has an extremely simple form (eq. 17):
X i q · v = 2π qix (17) i
• In particular, the dot product of integral multiples of the original basis vec- tors (i.e., direct or real lattice vectors), with integral multiples of the dual basis vectors (i.e., reciprocal lattice vectors) are integral multi- ples of 2π. This property will be used extensively to calculate Fourier transforms of lattice functions.
9 Recap of the key formulas for the dual basis
• From direct to dual bases (eq. 14)
i X −1 ki b = 2π ak(G ) k
• Dot product relation between the two bases (eq. 15)
j j ai · b = 2πδi • Dot product between vectors expressed on the two different bases (eq. 17)
X i q · v = 2π qix i
1.5 Dual basis in 3D
• In 3 dimensions, there is a very useful formula to calculate the dual basis vectors, which makes use of the properties of the vector product:
a2 × a3 b1 = 2π (18) a1 · (a2 × a3)
a3 × a1 b2 = 2π a1 · (a2 × a3)
a1 × a2 b3 = 2π a1 · (a2 × a3)
Note that
2 2 2 1/2 v = a1·(a2 × a3) = abc 1 − cos α − cos β − cos γ + 2 cos α cos β cos γ (19) is the unit cell volume.
• In crystallographic textbooks, the dual basis vectors are often written as a∗, b∗ and c∗.
1.6 Dot products in reciprocal space
• As we shall see later, it is very useful to calculate the dot product between two vectors in reciprocal space. This can be tricky in non-Cartesian coordinate systems.
10 • A quick way to do this is to determine the reciprocal-space metric tensor, which is related to the real-space one.
• The reciprocal-space metric tensor is G˜ = (2π)2G−1, so that, for two reciprocal- space vectors q and r:
X ij q · r = G˜ qirj (20) i,j
1.7 A very useful example: the hexagonal system in 2D
• The hexagonal system in 2D has a number of important applications in con- temporary solid-state physics problems, particularly for carbon-based materials such as graphene and carbon nanotubes.
• By crystallographic convention, the real-space basis vectors form an angle a ∧ b of 120◦ (this is also true in 3D).
• The real-space metric tensor is therefore:
1 −1/2 G = a2 (21) −1/2 1
• From eq. 20, we can find the reciprocal-space metric tensor:
(2π)2 4 1 1/2 G˜ = (22) a2 3 1/2 1
• It follows, for example, that the length of a vector in reciprocal space is given by:
2π r4 q = (h2 + hk + k2) (23) a 3
1.8 Reciprocal lattices
Having defined the dual basis, we can use it to construct a lattice in reciprocal space, known as the reciprocal lattice.
• Reciprocal-space vectors are described as linear combinations of the re- ciprocal or dual basis vectors, (dimensions: length−1) with dimen- sionless coefficients.
11 • Reciprocal-lattice vectors (RLV ) are reciprocal space vectors with inte- gral components. These are known as the Miller indices and are usu- ally notated as hkl.
It can be easily shown that all equivalent primitive cells in real space de- fine equivalent primitive dual bases and reciprocal space lattices 3. However, a problem arises when dealing with conventional bases, since the recipro- cal lattice they generate is patently different from that of any of the primitive bases. This is illustrated with an example in fig. 2. One can see that the ∗ conventional reciprocal-space unit cell (associated with the dual basis ac and ∗ ∗ ∗ bc) is smaller than the primitive reciprocal-space unit cell (ap and bp), and therefore generates more reciprocal lattice points. Bragg scattering is never observed at these points, for the reasons explained here below — we will say that they are extinct by centering.
1.8.1 Centring extinctions
• The dot product of real and reciprocal space vectors expressed in the usual coordinates is
X i q · v = 2π qiv (24) i
• The dot product of real and reciprocal lattice vectors is:
- If a primitive basis is used to construct the dual basis, 2π times an integer for all q and v in the real and reciprocal lattice, respectively. In fact, as we just said, all the components are integral in this case. - If a conventional basis is used to construct the dual basis, 2π times an integer or a simple fraction of 2π. In fact the components of the centering vectors are fractional.
• Therefore, if a conventional real-space basis is used to construct the dual basis, only certain reciprocal-lattice vectors will yield a 2πn dot product with all real-lattice vectors — a necessary (bot not suffi- cient, see below) condition to observe Bragg scattering at these points. These reciprocal-lattice vectors are exactly those generated by the corresponding primitive basis.
3This is due to the fact that the transformation matrix of the dual basis is the inverse matrix of the transformation matrix of the real-space basis. In this course, I will shy away from a full treatment of crystallographic coordinate transformations — a huge subject in itself, but one that at this level does not add much to the comprehension of the subject.
12 Real Space
Conven onal u.c. Primi ve u.c.
bp*
bc bp ap bc*
ac*
ap* ac
Reciprocal Space
bp*
bc*
ac*
ap*
Figure 2: C-centred cells: construction of reciprocal basis vectors on the real-space lattice (top) and corresponding reciprocal lattice (bottom). One can see that the primitive reciprocal basis vectors generate fewer points (closed circles) than the conventional reciprocal basis vectors (open an closed points). Only closed points correspond to observed Bragg peaks, the others are extinct by centring.
13 • Each non-primitive lattice type has centring extinctions, which can be ex- pressed in terms of the Miller indices hkl (see table 1).
Table 1: Centering extinction and scattering conditions for the centered lat- tices. The “Extinction” columns lists the Miller indices of reflections that are extinct by centering, i.e., are “extra” RLV generated as a result of using a conventional basis instead of a primitive one. The complementary “Scatter- ing” column corresponds to the listing in the International Tables vol. A, and lists the Miller indices of “allowed” reflections. “n” is any integer (positive or negative).
Lattice type Extinction Scattering
P none all
A k + l = 2n + 1 k + l = 2n B h + l = 2n + 1 h + l = 2n C h + k = 2n + 1 h + k = 2n
F k + l = 2n + 1 or k + l = 2n and h + l = 2n + 1 or h + l = 2n and h + k = 2n + 1 h + k = 2n
I h + k + l = 2n + 1 h + k + l = 2n
R −h + k + l = 3n + 1 or −h + k + l = 3n −h + k + l = 3n + 2
• The non-exctinct reciprocal lattice points also form a lattice, which is natu- rally one of the 14 Bravais lattices. For each real-space Bravais lattice, tab. 2 lists the corresponding RL type.
14 Table 2: Reciprocal-lattice Bravais lattice for any given real-space Bravais lattice (BL).
Crystal system Real-space BL Reciprocal-space BL Triclinic P P
Monoclinic C C
Orthorhombic P P A or B or C A or B or C I F F I
Tetragonal P P I I
Trigonal P P R R
Hexagonal P P
Cubic P P I F F I
15 2 Lecture 2 — Rotational symmetry
2.1 Introduction
While the first experimental confirmation of translational symmetry in crystals was provided by the X-ray diffraction experiments performed in 1914 by W.H and W.L. Bragg (after and son), the existence of rotational symmetry had been known for centuries, simply by the observation of crystal shapes. In the presence of rotational symmetry, some atoms will have “companions”, related to them by rotation around a certain axis or mirroring by certain planes. Other atoms, positioned on top of certain symmetry elements, will have fewer or no companions. The number of symmetry-related companions is known as the multiplicity of that atomic site. It is therefore evident that knowing the full crystal symmetry (translation + rotation) allows a simplified description of the crystal structure and calculations of its properties in terms of fewer unique atoms.
2.2 Generalised rotations expressed in terms of matrices and vectors
2.2.1 Rotations about the origin
In Cartesian coordinates, a rotation about the origin is expressed as:
x(2) x(1) y(2) = R y(1) (25) z(2) z(1) where x(1), y(1), z(1) are the coordinates of the original point, x(2), y(2), z(2) are the coordinates of the companion point and R in an orthogonal matrix with det(R) = +1. Matrices with det(R) = −1 represent the so-called improper rotations, which include mirror reflections (we will see other examples later).
In crystallographic (non-Cartesian) coordinates, eq. 25 is modified as:
x(2) x(1) y(2) = D y(1) (26) z(2) z(1) where in general D is a non-orthogonal matrix, which, however, still has det(D) = ±1
16 2.2.2 Rotations about points other than the origin
In general, the internal rotational symmetry of crystals will include rotations around points that are not the origin, although, unlike the case of a liquid or glass, only a discrete set of such rotations exists in crystals. In Cartesian coordinates, the expression for a proper of improper rotation through the point x0, y0, z0, is
x(2) x(1) − x0 x0 y(2) = R y(1) − y0 + y0 z(2) z(1) − z0 z0 x(2) x(1) tx = y(2) = R y(1) + ty (27) z(2) z(1) tz where
tx x0 x0 ty = y0 − R y0 (28) tz z0 z0 for example, a mirror plane perpendicular to the x axis and located at x = 1/4 will produce the following transformation:
x(2) −x(1) + 1/2 y(2) = y(1) (29) z(2) z(1)
Symmetry operators written in the form of eq. 27 are said to be in normal form. They are written as the composition (application of two operators in succession) of a rotation (proper or improper) — the rotational part, followed by a translation — the translational part.
Note that when using non-Cartesian coordinates the normal form of symme- try operators remains the same:
x(2) x(1) tx y(2) = D y(1) + ty (30) z(2) z(1) tz
In general, D is not orthogonal, but its determinant is still ±1.
17 2.2.3 Roto-translations
A special class of symmetries in crystallography is represented by proper or improper rotations followed by a non-lattice translation. A typical example of this is the screw axis, which produces a companion point by rotating (e.g., by 120◦) and then translating along the rotation axis (e.g., by 1/3 of a lattice translation — the notation of this operation is 31, see below). Clearly we can still use the normal form in eq. 30 to express roto-translations, it is sufficient to add the additional translation to the vector t.
2.2.4 Generalisation of symmetry to continuous functions and patterns
Up to now, we have discussed the symmetry of crystals in terms of atoms and their companions. However, the symmetry of crystals is much deeper than this: the electron density in a crystal (a continuous function) has the same symmetry as the atoms. Continuous functions in reciprocal space (e.g., phonon or electron dispersion relations) also have crystallographic symme- tries that are related to the symmetry of the crystal (see below). In the spirit of describing real experiments, this means that smaller regions of the recip- rocal space need to be measured (a quadrant, an octant etc.) to reconstruct the whole dispersion. Symmetry also restricts, for example, certain compo- nents of the group velocity. We will use the word pattern to mean, generically, sets of points (e.g., atoms) or continuous functions, and with reference to the “wallpaper” patterns we will discuss below.
In describing the symmetry of isolated objects or periodic systems, one de- fines operations (or operators) that describe transformations of the pattern. We create the new pattern from the old pattern by associating a point p2 to each point p1 (as shown in eq. 30 in terms of coordinates) and transferring the “attributes” (for example, the electron density) of p1 to p2. This trans- formation preserves distances and angles, and is in essence a combination of translations, reflections and rotations. If the transformation is a symmetry operator, the old and new patterns are indistinguishable.
2.3 Some elements of group theory
If two symmetry operators are known to exist in a crystal, say O1 and O2, then we can construct a third symmetry operator by applying O1 and O2 one after the other. This means that the set of all symmetry operators is closed by “application in succession” (a form of composition — see below). Likewise,
18 we can always create the inverse of an operator, e.g., by reversing the sense of rotation and the direction of translations etc. These considerations indicate that the set of all symmetry operators of a crystal has the mathematical struc- ture of a group. A group is defined as a set of elements with a defined binary operation known as composition, which obeys the following rules.
• A binary operation (usually called composition or multiplication) must be defined. We indicated this with the symbol “◦”.
• Composition must be associative: for every three elements f, g and h of the set
f ◦ (g ◦ h) = (f ◦ g) ◦ h (31)
• The “neutral element” (i.e., the identity, usually indicated with E) must exist, so that for every element g:
g ◦ E = E ◦ g = g (32)
• Each element g has an inverse element g−1 so that
g ◦ g−1 = g−1 ◦ g = E (33)
• A subgroup is a subset of a group that is also a group.
• A set of generators is a subset of the group (not usually a subgroup) that can generate the whole group by composition. Infinite groups (e.g., the set of all lattice translations) can have a finite set of generators (the primitive translations).
2.3.1 Symmetry operators as elements of crystallographic groups
Note that, in the case of crystallographic operators:
• Composition of two symmetry operators is the application of these one
after another. When denoting the composition O1 ◦ O2, we mean that the operator to the right is applied first.
• Composition is not commutative. As shown in the example in fig. 3, the order of application of rotations and translations does matter. Therefore, crystallographic groups are discrete non-Abelian groups.
19 + + 4 ◦m10 m10◦4
m11
4+ 4+ 45º m10 m10
45º
m11
+ Figure 3: Left: A graphical illustration of the composition of the operators 4 and m10 to + give 4 ◦m10 = m11. The fragment to be transformed (here a dot) is indicated with ”start”, and the two operators are applied in order one after the other, until one reaches the ”end” position. + + Right: 4 and m10 do not commute: m10 ◦ 4 = m11¯ 6= m11.
• Graphs or symmetry elements are sets of invariant points of a pattern un- der one or more symmetry operators.
Another symmetry element or graph, related to the other mirror plane by 4-fold rota on. Belongs to the same class. Symmetry element or graph: mirror plane. A set of invariant points under the symmetry operator
Symmetry operator: transforms one part of the pa ern into another (in this case by reflec on)
Figure 4: Symmetry operators and symmetry elements (graphs). A symmetry operator is the point by point transformation of one part of the pattern into another (arrow). In the language of coordinates, the attributes of point x(1), y(1), z(1), e.g., colour, texture etc., will be transferred to point x(2), y(2), z(2). Symmetry elements (shown by lines) are points left invariant by a given set of transformations.
2.3.2 Optional: the concept of group class
Group classes are an optional and non-examinable topic. However, they are hugely important for a more advanced understanding of symmetry in crystal-
20 lography, so I include here a few basic notions.
• In group theory, classes are subset of a group (not in general subgroups) related by are conjugation, which is the equivalent of similarity trans- formations in matrices. h0 is conjugated with h if there is a g ∈ G so that:
h0 = g ◦ h ◦ g−1 (34)
Figure 5: Left. A showflake by by Vermont scientist-artist Wilson Bentley, c. 1902. Right The symmetry group of the snowflake, 6mm in the ITC notation. The group has 6 classes, 5 marked on the drawing plus the identity operator E. Note that there are two classes of mirror planes, marked “1” and “2” on the drawing. One can see on the snowflake picture that their graphs contain different patterns.
• In crystallography, there is an extremely important relation between conju- gation of symmetry operators and symmetry elements: if two symmetry −1 operators O1 and O2 are conjugated as O2 = H ◦ O1 ◦ H , then their symmetry elements (graphs) are transformed into one another by the symmetry of H.
2.4 Symmetry of periodic patterns in 2 dimensions: 2D point groups and wallpaper groups
We will start our description of the crystallographic symmetry groups by ex- amining the problem in 2 dimensions. This is much simpler than the full 3D case, and it is amenable to be explored both by the more conventional method
21 Equivalent points for Primary symmetry direc on general posi on (here 12 c) Graph representa on Secondary symmetry direc on Ter ary symmetry direc on
Site symmetry Wychoff le er Site mul plicity
Figure 6: An explanation of the most important symbol in the Point Groups subsection of the International Tables for Crystallography. All the 10 2D point groups (11 if one counts 31m and 3m1 as tow separate groups) are reproduced in the ITC (see lecture web site). The graphical symbols are described in fig. 10. Note that primary, secondary and tertiary symmetries never belong to the same class. of matrices and vectors and with the ore abstract and visual method of the symmetry elements.
2.4.1 Point groups, wallpaper groups, space groups and crystal classes
• Point groups (in any dimensions) are finite groups of pure rotations/reflections around a fixed point (origin).
• Space groups in 3D describe the full translational and rotational symme- try of crystals. They are infinite discrete groups, and they contain the infinite group of translations as a subset.
• Plane (or wallpaper) groups are the equivalent to space groups in 2D. They are so called because they describe the symmetry of all possible wallpapers.
• Crystal classes (in any dimension, not to be confused with conjugation
22 classes — see above) are point groups obtained from a space or wall- paper group by removing the translational part of the operators4. It is easy to see how this is done by writing the operators in normal form, eq. 30
Table 3: The 17 wallpaper groups. The symbols are obtained by combining the 5 Bravais lattices with the 10 2D point groups, and replacing g with m systematically. Strikeout symbols are duplicate of other symbols. crystal system crystal class wallpaper groups 1 p1 oblique 2 p2 m pm, cm,pg, cg rectangular 2mm p2mm, p2mg (=p2gm), p2gg, c2mm, c2mg, c2gg 4 p4 square 4mm p4mm, p4gm, p4mg 3 p3 hexagonal 3m1-31m p3m1, p3mg, p31m, p31g 6 p6 6mm p6mm, p6mg, p6gm, p6gg
2.4.2 Point groups in 2D
There is an infinite number finite point groups in 2D and above. Examples are groups or rotations about an axis by angles that are rational fractions of 2π.
However, only a small number of these groups are relevant for crystallogra- phy, since all others are incompatible with a lattice. There are 10 crystallo- graphic point groups in 2D (corresponding to the 10 crystal classes in tab. 3) and 32 in 3D (see web site for a complete list of crystallographic point groups and their properties from the ITC)5.
2.4.3 Bravais lattices in 2D
In 2D, there are 5 types of lattices: oblique, p-rectangular, c-rectangular, square and hexagonal (see Fig. 7 — more on this topic in the Supplementary
4 In fact, it can be easily shown that the rotational part of g ◦ f is RgRf . Therefore, the rotational parts of the operators of a wallpaper group or a space group form themselves a group, which is clearly one of the 10 point groups in 2D. This is called the crystal class of the wallpaper group. 5A very interesting set of “sub-periodic” groups is represented by the so-called frieze groups, which describe the symmetry of a repeated pattern in 1 dimension. There are only 7 frieze groups, which are very simple to understand given the small number of operators involved. For a description of the frieze groups, with some nice pictorial example, see the Supplementary Material.
23 Material). In the c-rectangular lattice, no primitive cell has the full symmetry of the lattice, as in the 3D centred lattices. It is therefore convenient to adopt a non-primitive centred cell having double its volume. As in the 3D case, the origin of the unit cell is to a large extent arbitrary. It is convenient to choose it to coincide with a symmetry element.
Oblique
p c
Rectangular
Square Hexagonal
Figure 7: The 5 Bravails lattices in 2 dimensions.
• Although there are an infinite number of symmetry operators (and symme- try elements), it is sufficient to consider elements in a single unit cell.
• Glides. This is a composite symmetry (roto-translation — see above), which combines a translation with a parallel reflection, neither of which on its own is a symmetry operator. In 2D groups, the glide is indicated with the symbol g. Twice a glide translation is always a symmetry trans- lation: in fact, if one applies the glide operator twice as in g ◦ g, one ob- tains a pure translation (since the two mirrors cancel out), which there- fore must be a lattice (symmetry) translation.
24 • Walpaper groups. The 17 Plane (or Wallpaper) groups describe the sym- metry of all periodic 2D patterns (see tab. 3). The decision three in fig. 8 can be used to identify wallpaper groups. 6 cm Has Has glides? pm pg 1 Has Has mirrors? Has glides? Has c2mm p1 off mirros? off Has rotations Has mirrors? p2mm Has orthogonal Has p2mg p2gg Has Has mirrors? Has Has glides? p2 p3m1 All axes All p31m on mirrors? on 3 2 Axis of highestAxis order p3 Has Has mirrors? p4mm 4 mirrors? Axes on on Axes p4gm Has Has mirrors? p4 p6mm 6 p6 Has Has mirrors?
Figure 8: Decision-making tree to identify wallpaper patterns. The first step (bottom) is to identify the axis of highest order. Continuous and dotted lines are ”Yes” and ”No” branches, respectively. Diamonds are branching points.
6For a more complete description of wallpaper groups and of the symmetry of the underly- ing 2D lattices, see the Supplementary Material. Examples of Escher drawings with different wallpaper group symmetries will be shown in the lectures (see lecture slides on the web site).
25 3 Lecture 3 — Symmetry in 3 dimensions
Symmetry in 3 dimensions is not dissimilar from the 2D case, but it is richer, having more Bravais lattices (14 instead of 5) and many more space groups (219 instead of 17 — the ITC lists 230 because it distinguishes left- and right- handed versions of otherwise identical groups). The rotation axes in 3D are the same as in 2D, but they are combined in different ways, most notably, in the cubic groups.
3.1 New symmetry operators in 3D
3.1.1 Inversion and roto-inversion
In 3D, we can also distinguish between proper and improper rotations, be- cause the inversion operator I, which has matrix representation −1 (minus the identity matrix) in all coordinate systems, has a negative determinant. All improper rotations can therefore be obtained by composing a proper rota- tion with the inversion — itself an improper rotation — as I ◦ r. The identity commutes with all other operators. Here below are some more properties of improper operators in 3D:
• I ◦ 2 = m⊥, so the mirror plane is the improper operator corresponding to a 2-fold rotation axis perpendicular to it.
• There are three other significant improper operators in 3D, known as roto- inversions. They are obtained by composition of an axis r of order higher than two with the inversion, as I ◦r. These operators are 3¯ ( ), 4¯ ( ) and 6¯ ( ), and their action is summarized in Fig. 9. The sym- bols are chosen to emphasize the existence of another operator inside the ”belly” of each new operator. Note that 3¯◦3¯◦3¯ = 3¯3 = I, and 3¯4 = 3, i.e., symmetries containing 3¯ also contain the inversion and the 3-fold rotation. Conversely, 4¯ and 6¯ do not automatically contain the inversion. In addition, symmetries containing both 4¯ (or 6¯) and I also contain 4 (or 6).
3.1.2 Glide planes and screw axes in 3D
Glide planes and 2-fold screw axes in 3D are the improper and proper version (respectively) of the same 2D operator, which we have indicated with the
26 - + - +/-
- + + + +/-
+ - +/- -
Figure 9: Action of the 3¯ , 4¯ and 6¯ operators and their powers. The set of equivalent points forms a trigonal antiprism, a tetragonally-distorted tetrahedron and a trigonal prism, respec- tively. Points marked with ”+” and ”-” are above or below the projection plane, respectively. Positions marked with ”+/-” correspond to pairs of equivalent points above and below the plane. symbol g. Glide planes in 3D consist in the sequential application of a mirror plane and of a non-lattice translation parallel to the plane (any perpendicular component being equivalent to a translation of the plane itself). Glide planes have different symbols, depending on the orientation of the glide vector with respect to the basis vectors and/or to the plane of the projection (fig. 10).
There is also a new type of operator, resulting from the compositions of proper rotations (including two-fold axes) with translation parallel to them. These are known as screw axes. For an axis of order n, the nth power of a screw axis is a pure translation, and must therefore be a lat- m tice translation t . Therefore, the translation component must be n t, where m is an integer. We can limit ourselves to m < n, all the other operators be- ing composition with lattice translations. Roto-translation axes are therefore indicated as nm, as in 21, 63 etc. (fig. 10).
27 2 3 4 6 m g
1 3 4 6
m a,b or c n e d
3 8 1 8
3 2 21 1 62 64 41 43 6 1 65
42 32 63
Figure 10: The most important graph symbols employed in the International Tables to de- scribe 3D space groups. Fraction next to the symmetry element indicate the height (z coordi- nate) with respect to the origin.
28 3.2 Understanding the International Tables of Crystallography
The ITC have been designed to enable scientists with minimal understanding of group theory to work with crystal structures and diffraction patterns. Most of the ITC entries, except for the graphical symbols that we have already introduced, are self-explanatory (see fig. 11 and 12).
3.2.1 ITC information about crystal structures
By using the ITC, one can (amongst other things):
• Given the coordinates of one atom, determine the number and coordinates of all companion atoms. This greatly simplifies the description of crystal structures.
• Determine the local point-group symmetry around each atom, which is the same as the symmetry of the electron density and is connected to the symmetry of the atomic wavefunction. This is particularly important for certain types of spectroscopy.
Atomic positions are listed in order of decreasing multiplicity, starting from the most general position (see 12). Positions of lower multiplicity are located on symmetry elements (graphs), and therefore have a special site symmetry. Each unique site is aligned a letter, known as the Wyckoff letter, starting with a for the site of lowest multiplicity. It is quite possible for sites to have the same site symmetry and yet not be companions of each other — in this case, these sites are assigned different Wyckoff letters (see fig. 12, Wyckoff letters a, b, c, d corresponds to set of sites having the same point symmetry 1¯). In the lecture, we will discuss the case of Ag2O2, a simple oxide that crystallise in space group P 21/c (no. 14) — see fig. 13 .
3.2.2 ITC information about diffraction patterns
The ITC lists centring extinction conditions as well as extinctions associated with glide planes and screw axes — all these extinctions conditions are known as “General” because they affect the contribution to scattering from all atoms. Also, atoms on certain sites do not contribute to certain Bragg peaks — these extinctions are known as “Special” and are specific to each Wyckoff site.
29 IT numbering SG symbols (Hermann-Mauguin nota on) Crystal class (point group)
SG symbol (Schoenflies nota on) Crystal system
SG diagrams projected along the 3 direc ons b is orthogonal to a and c. Cell choce 1 here means that the origin is on an inversion centre.
A generic posi on and its equivalent. Dots with and without commas are related by an improper operator (molecules located here would have opposite chirality)
Asymmetric unit cell: the smallest part of the pa ern that generates the whole pa ern by applying all symmetry A symmetry operator notated as follows: a 2-fold axis with an operators. associated transla on of ½ along the b=axis, i.e., a screw axis. This is located at (0,y,¼) (see SG diagrams, especially top le ). Only a subset of the symmetry operators are indicated. All others can be obtained by composi on with transla on operators.
Figure 11: Explanation of the most important symbols and notations in the Space Group entries in the International Tables. The example illustrated here is SG 14: P 21/c .
30 Refers to numbering of operators on previous page
Group generators
Primi ve transla ons
Equivalent posi ons, obtained by applying the operators on the previous page (with numbering indicated). Reflects normal form of operators.
4 pairs of equivalent inversion centres, not equivalent to centres in other pairs (not in the same class). They have their own dis nct Wickoff le er.
Reflec on condi ons for each class of reflec ons. General: valid for all atoms (i.e., no h0l reflec on will be observed unless l=2n; this is due to the c glide). Special: valid only for atoms located at corresponding posi ons (le side), i.e., atoms on inversion centres do not contribute unless k+l=2n
Figure 12: IT entry for SG 14: P 21/c (page 2).
31 Figure 13: Crystal structure data for silver (I,III) oxide (Ag2O2), space group P 21/c (no. 14). Data are from the Inorganic Crystal Structures Database (ICSD), accessible free of charge from UK academic institutions at http://icsd.cds.rsc.org/icsd/.
32 4 Lecture 4 — Fourier transform of periodic functions
4.1 Fourier transform of lattice functions
• It can be shown (see Supplementary Material on the web site) that the Fourier transform of a function f(r) (real or complex) with the periodicity of the lattice can be written as:
1 P i Z X −2πi i qin −iq·x F (q) = 3 e d(x)f(x)e 2 u.c. (2π) ni Z v0 P i P i X −2πi i qin i i −2πi i qix = 3 e dx f(x )e (35) 2 u.c. (2π) ni
where v0 is the volume of the unit cell
• The triple infinite summation in ni = nx, ny, nz is over all positive and nega-
tive integers. The qi = qx, qy, qz are reciprocal space coordinates on the
dual basis. The xi = x, y, z are real-space crystallographic coordinates i (this is essential to obtain the qin term in the exponent). The integral is over one unit cell.
• F (q) is non-zero only for q belonging to the primitive RL. In fact, if q be- longs to the primitive reciprocal lattice, then by definition its dot product to the symmetry lattice translation is a multiple of 2π, the exponential factor is 1 and the finite summation yields N (i.e., the number of unit cells). If q does not belong to the primitive reciprocal lattice, the expo- nential factor will vary over the unit circle in complex number space and will always average to zero.
• It is the periodic nature of f(r) that is responsible for the discrete nature of F (q).
4.2 Atomic-like functions
Although the result in eq. 35 is completely general, in the case of scattering experiments the function f(x) represents either the density of nuclear scat- tering centres (neutrons) or the electron density (x-rays, electrons). These functions are sharply pointed around the atomic positions, and it is therefore convenient to write:
33 X f(x) = fj(x) (36) j where the summation if over all the atoms in the unit cell, and fj(x) is (some- what arbitrarily) the density of scatterers assigned to a particular atom (see next lectures for a fully explanation). With this, eq. 35 becomes:
Z v0 P i P i P i i X −2πi i qin X −2πi i qixj i i −2πi i qi(x −xj ) F (q) = 3 e e dx fj(x )e = 2 u.c. (2π) ni j v0 P i P i X −2πi i qin X −2πi i qixj = 3 e e fj(q) (37) 2 (2π) ni j
When evaluated on reciprocal lattice points, the rightmost term is now the structure factor Fhkl.
4.3 Centring extinctions
In particular, we can show that those “‘conventional”’ RLV that we called ex- tinct by centring are indeed extinct, that is, F (q) = 0. For example, in the case of an I-centrel lattice, we have:
X −2πi(hx+ky+lz) −2πi( h + k + l Fhkl = e 1 + e 2 2 2 fj(q) (38) j and this is = 0 for h + k + l 6= 2m, which precisely identifies the non-primitive RL nodes.
4.4 Extinctions due to roto-translations
Similarly, the presence of roto-translation operators (glide planes and screw axes) produces extinction conditions, which, however, do not affect whole RL sub-lattices but only certain sections of it. In the case of glides, there are planes of hkl’s in which some of the reflections are extinct. In the case of screws, there are lines of hkl’s in which some of the reflections are extinct. Here, I will show only one example, that of a glide plane ⊥ to the x axis and with glide vector along the y direction, which generates the two companions 1 x, y, z and −x, y + 2 , z. We have:
34 X −2πi(ky+lz) −2πihx −2πi(−hx+ k ) Fhkl = e e + e 2 fj(q) (39) j
We can see that, within the plane of reflections having h = 0, all reflections with k 6= 2m are extinct.
4.5 The symmetry of |F (q)|2 and the Laue classes
• It is very useful to consider the symmetry of the RL when |F (x)|2 is asso- ciated with the RL nodes. In fact, this corresponds to the symmetry of the diffraction experiment, and tells us how many unique reflections we need to measure.
• Translational invariance is lost once |F (x)|2 is associated with the RL nodes.
• Let R be the rotational part and t the translational part of a generic symme- try operators. One can prove that
Z N −i(R−1q)·x −iq·t −1 −iq·t F (q) = 3 d(x)f(x)e e = F (R q)e (40) (2π) 2 u.c.
Eq. 40 shows that the reciprocal lattice weighed with |F (q)|2 has the full point-group symmetry of the crystal class.
• This is because the phase factor e−iq·t clearly disappears when taking the modulus squared. In fact, there is more to this symmetry when f(x) is real, i.e., f(x) = f ∗(x): in this case
Z ∗ N ∗ iq·x F (q) = 3 dxf (x)e (41) (2π) 2 u.c. Z N iq·x = 3 dxf(x)e = F (−q) (2π) 2 u.c.
• Consequently, |F (q)|2 = F (q) F (−q) = |F (−q)|2 is centrosymmetric. As we shall shortly see, the lattice function used to calculate non-resonant scattering cross-sections is real. Consequently, the |F (q)|2-weighed RL (proportional to the Bragg peak intensity) has the symmetry of the crys- tal class augumented by the center of symmetry. This is necessarily one of the 11 centrosymmetryc point groups, and is known as the Laue class of the crystal.
35 Fridel’s law
For normal (non-anomalous) scattering, the reciprocal lattice weighed with |F (q)|2 has the full point-group symmetry of the crystal class supplemented by the inversion. This symmetry is known as the Laue class of the space group. In particular, for normal (non-anomalous) scattering, Fridel’s law holds:
|F (hkl)|2 = |F (h¯k¯¯l)|2 (42) Fridel’s law is violated for non-centrosymmetric crystals in anomalous conditions. Anomalous scattering enables one, for example, to determine the orientation of a polar crystal or the chirality of a chiral crystal in an absolute way.
36 5 Lecture 5 — Brillouin zones and the symmetry of the band structure
• The electronic, vibrational and magnetic phenomena occurring in a crystal have, overall, the same symmetry of the crystal.
• However, individual excitations (phonons, magnons, electrons, holes) break most of the symmetry, which is only restored because symmetry-equivalent excitations also exist and have the same energy (and therefore popula- tion) at a given temperature.
• The wavevectors of these excitations are generic (non-RL) reciprocal-space vectors.
• When these excitations are taken into account, one finds that inelastic scat- tering of light, X-rays or neutrons can occur. In general, the inelastic scattering will be outside the RL nodes.
• Various Wigner Seitz constructions 7 are employed to subdivide the recip- rocal space, for the purpose of:
Classifying the wavevectors of the excitations. The Bloch theo- rem states that “crystal” wavevectors within the first Brillouin zone (first Wigner-Seitz cell) are sufficient for this purpose. Perform scattering experiments. In general, an excitation with wavevec- tor k (within the first Brillouin zone) will give rise to scattering at all points τ + k, where τ is a RLV . However, the observed scattering intensity at these points will be different. The repeated Wigner- Seitz construction is particularly useful to map the “geography” of scattering experiments. Construct Bloch wave functions (with crystal momentum within the first Brillouin zone) starting from free-electron wave-functions (with real wavevector k anywhere in reciprocal space). In partic- ular, apply degenerate perturbation theory to free-electron wave- functions with the same crystal wavevector. Here, the extended Wigner-Seitz construction is particularly useful, since free-electron wavefunctions within reciprocal space “fragments” belonging to the same Brillouin zone form a continuous band of excitations. 7A very good description of the Wigner-Seitz and Brillouin constructions can be found in the book by Ashcroft and Mermin (see bibliography). See also the Supplementary Material for a summary of the procedure.
37 • The different types of BZ constructions were already introduced last year. They are reviewed in the lecture and in the supplementary material on the web site.
5.1 Symmetry of the electronic band structure
• We will here consider the case of electronic wavefunctions, but it is important to state that almost identical considerations can be applied to other wave-like excitations in crystals, such as phonons and spin waves (magnons).
• Bloch theorem: in the presence of a periodic potential, electronic wave- functions in a crystal have the Bloch form:
ik·r ψk(r) = e uk(r) (43)
where uk(r) has the periodicity of the crystal. We also recall that the crystal wavevector k can be limited to the first Brillouin zone (BZ). In ik0·r 0 fact, a function ψk0 (r) = e uk0 (r) with k outside the first BZ can be rewritten as
ik·r h iτ ·r i ψk0 (r) = e e uk0 (r) (44)
where k0 = τ +k, k is within the 1st BZ and τ is a reciprocal lattice vector (RLV ). Note that the function in square brackets has the periodicity of the crystal, so that eq. 44 is in the Bloch form.
• The application of the Bloch theorem to 1-dimensional (1D) electronic wave- functions, using either the nearly-free electron approximation or the tight-binding approximation leads to the typical set of electronic disper- sion curves (E vs. k relations) shown in fig. 14. We draw attention to three important features of these curves:
• Properties of the electronic dispersions in 1D
They are symmetrical (i.e., even) around the origin. The left and right zone boundary points differ by the RLV 2π/a, and are also related by symmetry. The slope of the dispersions is zero both at the zone centre and at the zone boundary. We recall that the slope (or more generally the gradient of the dispersion is related to the group velocity of the wavefunctions in band n by:
38 k
Figure 14: A set of typical 1-dimensional electronic dispersion curves in the reduced/repeated zone scheme.
1 ∂En(k) vn(k) = (45) ~ ∂k • Properties of the electronic dispersions in 2D and 3D
They have the full Laue (point-group) symmetry of the crystal. This applies to both energies (scalar quantities) and velocities (vector quantities) Zone edge centre (2D) or face centre (3D) points on opposite sides of the origin differ by a RLV and are also related by inversion sym- metry. Otherwise, zone boundary points that are related by symmetry do not not necessarily differ by a RLV . Zone boundary points that differ by a RLV are not necessarily related by symmetry. Group velocities are zero at zone centre, edge centre (2D) or face centre (3D) points. Some components of the group velocities are (usually) zero at zone boundary points (very low-symmetry cases are an exception — see below).
39 Group velocities directions are constrained by symmetry on symme- try elements such as mirror planes and rotation axes.
• The dispersion of the band structure, En(k), has the Laue symmetry of the crystal. This is because:
En(k) is a macroscopic observable (it can be mapped, for example, by Angle Resolved Photoemission Spectroscopy — ARPES) and any macroscopic observable property of the crystal must have at least the point-group symmetry of the crystal (Neumann’s principle — see later). ik·r If the Bloch wavefunction ψk = e uk(r) is an eigenstate of the Schroedinger equation
2 − ~ ∇2 + U(r) ψ = E ψ (46) 2m k k k
† −ik·r † then ψk = e uk(r) is a solution of the same Schroedinger equa- tion with the same eigenvalue (this is always the case if the poten- † tial is a real function). ψk has crystal momentum −k. Therefore, the energy dispersion surfaces (and the group velocities) must be inversion-symmetric even if the crystal is not.
5.2 Symmetry properties of the group velocity
• The group velocity at two points related by inversion in the BZ must be opposite.
• For points related by a mirror plane or a 2-fold axis, the components of the group velocity parallel and perpendicular to the plane or axis must be equal or opposite, respectively.
• Similar constraints apply to points related by higher-order axes — in partic- ular, the components of the group velocity parallel to the axis must be equal.
ik·r i(k+τ)·r −τ·r • Since e uk(r) = e {e uk(r)} and the latter has crystal momen-
tum k + τ, Ek must be the same at points on opposite faces across the Brillouin zone (BZ), separated by τ.
• If the band dispersion Ek is smooth through the zone boundary, then the
gradient of Ek (and therefore vn(k)) must be the same at points on opposite faces across the Brillouin zone (BZ), separated by τ. 8
8 This leaves the possibility open for cases in which vn(k) jumps discontinuosly across the
40 Figure 15: the first and two additional Wigner-Seitz cells on the oblique lat- tice (construction lines are shown). Significant points are labelled, and some possible zone-boundary group velocity vectors are plotted with arrows.
5.2.1 The oblique lattice in 2D: a low symmetry case
• The point-group symmetry of this lattice is 2, and so is the Laue class of any 2D crystal with this symmetry (remember that the inversion and the 2-fold rotation coincide in 2D).
• Fig. 15 shows the usual Wigner-Seitz construction on an oblique lattice.
• Points on opposite zone-boundary edge centres (A-A and B-B) are related by 2-fold rotation–inversion (so their group velocities must be opposite) and are also related by a RLV (so their group velocities must be the same). Follows that at points A and B, as well as at the Γ point (which is on the centre of inversion) the group velocity is zero.
• For the other points, symmetry does not lead to a cancelation of the group
velocities: For example, points a1 and b1 are related by a RLV but not by inversion, so their group velocities are the same but non-zero.
Similarly, points c1 and c2 are related by inversion but not by a RLV , so their group velocities are opposite but not zero.
• Points a1, a2, b1 etc. do not have any special significance. It is therefore customary for this type of lattice (and for the related monoclinic and
BZ boundary and is therefore not well defined exactly at the boundary. This can only happen if bands with different symmetries cross exactly at the BZ boundary.
41 Figure 16: Relation between the Wigner-Seitz cell and the conventional reciprocal-lattice unit cell on the oblique lattice. The latter is usually chosen as the first Brillouin zone on this lattice, because of its simpler shape.
triclinic lattices in 3D) to use as the first Brillouin zone not the Wigner- Seitz cell, but the conventional reciprocal-lattice unit cell, which has a simpler parallelogram shape. The relation between these two cells is shown in fig. 16.
5.2.2 The square lattice: a high symmetry case
• A more symmetrical situation is show in fig. 17 for the square lattice (Laue symmetry 4mm). A tight-binding potential has been used to calculate constant-energy surfaces, and the group velocity field has been plotted using arrows.
• By applying similar symmetry and RLV relations, one finds that the group velocity is zero at the Γ point, and the BZ edge centres and at the BZ corners.
• On the BZ edge the group velocity is parallel to the edge.
• Inside the BZ, the group velocity of points lying on the mirror planes is parallel to those planes.
42 Figure 17: Constant-energy surfaces and group velocity field on a square lattice, shown in the repeated zone scheme. The energy surfaces have been calculated using a tight-binding potential.
43 5.3 Symmetry in the nearly-free electron model: degenerate wave- functions
• one important class of problems involves the application of degenerate per- turbation theory to the free-electron Hamiltonian, perturbed by a weak periodic potential U(r):
2 H = − ~ ∇2 + U(r) (47) 2m
• Since the potential is periodic, only degenerate points related by a RLV are allowed to “interact” in degenerate perturbation theory and give rise to non-zero matrix elements.
• The 1D case is very simple:
Points inside the BZ have a degeneracy of one and correspond to travelling waves. Points at the zone boundary have a degeneracy of two, since k = π/a and therefore k − (−k) = 2π/a is a RLV . The perturbed solutions are standing wave, and have a null group velocity, as we have seen (fig. 14).
• The situation is 2D and 3D is rather different, and this is where symmetry can help. In a typical problem, one would be asked to calculate the energy gaps and the level structure at a particular point, usually but not necessarily at the first Brillouin zone boundary.
• The first step in the solution involves determining which and how many de- generate free-electron wavefunctions with momenta differing by a RLV have a crystal wavevector at that particular point of the BZ.
• Symmetry can be very helpful in setting up this initial step, particularly if the symmetry is sufficiently high. For the detailed calculation of the gaps, we will defer to the “band structure” part of the C3 course.
44 Figure 18: Construction of nearly-free electron degenerate wavefunctions for the square lattice (point group 4mm). Some special symmetry point are la- belled. Relevant RLV s are also indicated.
Nearly-free electron degenerate wavefunctions
• Draw a circle centred at the Γ point and passing through the BZ point you are asked to consider (either in the first or in higher BZ — see fig. 18). Points on this circle correspond to free-electron wavefunctions having the same energy. • Mark all the points on the circle that are symmetry-equivalent to your BZ point. • Among these, group together the points that are related by a RLV . These points represent the degenerate multiplet you need to apply degenerate perturbation theory. • Write the free-electron wavefunctions of your degenerate multiplet in Bloch form. You will find that all the wavefunctions in each multiplet have the same crystal momentum. Functions in different multiplets have symmetry-related crystal momenta.
• This construction is shown in fig. 18 in the case of the square lattice (point group 4mm) for boundary points between different Brillouin zones. One can see that:
• The degeneracy of points in the interior of the first BZ is always one, since no two points can differ by a RLV . This is not so for higher zones though (see lecture). For points on the boundary of zones
or at the interior of higher zones, the formula DN /Mn = D1/M1
45 holds, where DN is the degeneracy of the multiplet in the higher
zone (the quantity that normally we are asked to find), MN is the
multiplicity of that point (to be obtained by symmetry) and D1 and
M1 are the values for the corresponding points within or at the boundary of the 1st BZ. X-points: there are 4 such points, and are related in pairs by a RLV . Therefore, there are two symmetry-equivalent doublets of free-electron wavefunctions (which will be split by the periodic po- tential in two singlets). Y-points: there are 8 such points, and are related in pairs by a RLV . Therefore, there are four symmetry-equivalent doublets of free- electron wavefunctions (which will be split by the periodic potential in two singlets). M-points: there are 4 such points, all related by RLV s. Therefore, there is a single symmetry-equivalent quadruplet of free-electron wavefunctions (which will be split by the periodic potential in two singlets and a doublet).
X2-points: these are X-point in a higher Brillouin zone. There are 8 such points, related by RLV s in groups of four. Therefore, there are two symmetry-equivalent quadruplets of free-electron wave- functions (each will be split by the periodic potential in two singlets and a doublet). These two quadruplets will be brought back above (in energy) the previous two doublets in the reduced-zone scheme.
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