Crystal Structure and Dynamics - University of Oxford Department of

Crystal Structure and Dynamics
Paolo G. Radaelli, Michaelmas Term 2013
Part 1: Symmetry in the solid state
Lectures 1-4
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.
◦ Paolo G. Radaelli, Symmetry in Crystallography: Understanding the International Tables ,
Oxford University Press (2011). Contains much of the same materials covering lectures
1-3, but in an extended form.
◦ 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, Fundamentals of crystallography (International Union of Crystallography, Oxford University
Press Inc., New York)
◦ ”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 — Symmetry of simple patterns
1.1 Introduction to group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Symmetry of periodic patterns in 2 dimensions: 2D point groups and wallpaper
groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Combinations of rotations and translations — normal form of symmetry operators
2 Lecture 2 — Coordinates and calculations
2.1 Crystallographic coordinates . . . . . . . . . . . . .
2.2 Distances and angles . . . . . . . . . . . . . . . . .
2.3 Dual basis and coordinates . . . . . . . . . . . . . .
2.4 Dual basis in 3D . . . . . . . . . . . . . . . . . . . .
2.5 Dot products in reciprocal space . . . . . . . . . . .
2.6 A very useful example: the hexagonal system in 2D
2.7 Symmetry in 3 dimensions . . . . . . . . . . . . . . .
3
3
5
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12
12
13
14
15
15
16
16
3 Lecture 3 — Fourier transform of periodic functions
3.1 Centring extinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Fourier transform of lattice functions . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The symmetry of |F (q)|2 and the Laue classes . . . . . . . . . . . . . . . . . . .
22
22
23
24
4 Lecture 4 — Brillouin zones and the symmetry of the band structure
4.1 Symmetry of the electronic band structure . . . . . . . . . . . . . . . . .
4.2 Symmetry properties of the group velocity . . . . . . . . . . . . . . . . .
4.2.1 The oblique lattice in 2D: a low symmetry case . . . . . . . . . .
4.2.2 The square lattice: a high symmetry case . . . . . . . . . . . . .
4.3 Symmetry in the nearly-free electron model: degenerate wavefunctions
26
27
29
30
31
31
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1
1.1
Lecture 1 — Symmetry of simple patterns
Introduction to group theory
• Previous courses have already illustrated the important of translational symmetry (lattice periodicity), especially in the context of the Bloch theorem.
Here, we will extend this by illustrating the important of rotational symmetry:
• 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).
Diffraction experiments can help to identify the crystal symmetry (albeit not uniquely).
Symmetry applies to continuous functions (e.g., electron density in a
crystal) as well as to collections of discrete objects (atoms).
The crystal symmetry generates a corresponding symmetry of the
dispersion relations (phonons, electrons) in reciprocal space. In
the spirit of describing real experiments, this means that smaller
regions of the reciprocal space need to be measured (a quadrant,
an octant etc.) to reconstruct the whole dispersion. Symmetry also
restricts, for example, certain components of the group velocity.
• In describing the symmetry of isolated objects or periodic systems, one
defines 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 and transferring the “attributes” of p1
to p2. This transformation 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.
• Symmetry operators can be applied one after the other, generating new operators. Taken all together they form a finite (for pure rotations/reflections)
or infinite (if one includes translations) consistent set.
• The set of operators describing the symmetry of an object or pattern conforms to the mathematical structure of a group. We will only deal with
sets of operators, not with the more abstract mathematical concept of
group. A group is a set of elements with a defined binary operation
known as composition, which obeys certain rules.
3
A binary operation (usually called composition or multiplication)
must be defined. We indicated this with the symbol “◦”. When
group elements are operators, the operator to the right is applied
first.
Composition must be associative: for every three elements f , g and
h of the set
f ◦ (g ◦ h) = (f ◦ g) ◦ h
(1)
The “neutral element” (i.e., the identity, usually indicated with E) must
exist, so that for every element g:
g◦E =E◦g =g
(2)
Each element g has an inverse element g −1 so that
g ◦ g −1 = g −1 ◦ g = E
(3)
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).
• Composition of two symmetry operators is the application of these one after
another. we can see that the rules above hold.
• Composition is not commutative (fig. 1).
• Graphs or symmetry elements are sets of invariant points of a pattern under one or more symmetry operators.
• Sets of symmetry-related graphs form a class. In group theory, they correspond to classes of symmetry operators, defined as follows: two symmetry operators belong to the same class when they are conjugated. h0
is conjugated with h if there is a g ∈ G so that:
h0 = g ◦ h ◦ g −1
(4)
• Conjugation of graphs defines set of equivalent points with special properties. Wychoff letters are used to label these points.
4
m10◦4+
4+◦m10
m11
4+
4+
45º
m10
m10
45º
m11
Figure 1: 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+ = m1̄1 6= m11 .
Another symmetry element or graph, related to the other mirror plane by 4-­‐fold rota5on. 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 reflec5on) Figure 2: 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.
1.2
Symmetry of periodic patterns in 2 dimensions: 2D point
groups and wallpaper groups
• Point groups are finite groups of rotations/reflections around a fixed point.
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 crystallog5
Figure 3: 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.
Table 1: 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
raphy, since all others are incompatible with a lattice. There are 10
crystallographic point groups in 2D and 32 in 3D (see web site for a
complete list of crystallographic point groups and their properties from
the ITC)1 .
• Periodic patterns are invariant by a set of translations, which also form a
group. They may also be invariant by rotations/reflections and combi1
A 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.
6
Equivalent points for general posi+on (here 12 c) Primary symmetry direc+on Graph representa+on Secondary symmetry direc+on Ter+ary symmetry direc+on Site symmetry Wychoff le8er Site mul+plicity Figure 4: An explanation of the most important symbol in the Point Groups subsection of the
International Tables for Crystallography. All the 11 2D point group entries are reproduced in
the long version of the notes (se web site, Lecture 1). The graphical symbols are described in
fig. 9. Note that primary, secondary and tertiary symmetries never belong to the same class.
nations of translations/rotations.
• Lattice is an alternative (and visual) concept to the translation set.
• Unit cells are parts of the pattern that reproduce the whole pattern by translation.
• There is a variety of ways to choose primitive (=smallest possible) unit cells.
The most convenient choice is the most symmetric one.
• In 2D, there are 5 types of lattices: oblique, p-rectangular, c-rectangular,
square and hexagonal (see Fig. 5).
• In the c-rectangular lattice, no primitive cell has the full symmetry of the
lattice. It is convenient to adopt a non-primitive centred cell.
• The origin of the unit cell is to a large extent arbitrary. It is convenient to
choose it to coincide with a symmetry element.
7
Oblique c
p
Rectangular Square Hexagonal Figure 5: The 5 Bravails lattices in 2 dimensions.
• Although there are an infinite number of symmetry operators (and symmetry elements), it is sufficient to consider elements in a single unit cell.
• Glides. This is a composite symmetry, 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 translation: in fact, if one applies the
glide operator twice as in g ◦ g, one obtains a pure translation (since the
two mirrors cancel out), which therefore must be a symmetry translation.
• Walpaper groups. The 17 Plane (or Wallpaper) groups describe the symmetry of all periodic 2D patterns (see tab. 1). The decision three in fig.
6 can be used to identify wallpaper groups.
2
2
For a more complete description of wallpaper groups and of the symmetry of the underlying 2D lattices, see the Supplementary Material.
8
1
4
6
3
2
Axis of highest order
Has mirrors?
Has mirrors?
Has mirrors?
Has mirrors?
p6
p6mm
p4
Has mirrors?
Has glides?
All axes
on mirrors?
Axes on
mirrors?
p3
p3m1
p31m
p4gm
p4mm
Has glides?
p2mg
p1
Has rotations
off mirros?
p2gg
p2
Has orthogonal
mirrors?
pg
Has glides?
cm
pm
c2mm
p2mm
Figure 6: 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.
1.3
Combinations of rotations and translations — normal form
of symmetry operators
• Our aim here is to show that all symmetry operators can be written as the
composition of a rotation (first) followed by a translation second. We
will do this by employing Cartesian coordinates (we will generalise to
9
the 3D case).
• We have so far seen two combinations or rotations and translations:
Glides are an example of roto-translations. In Cartesian coordinates,
roto-translations can be written as:



 

x(2)
x(1)
tx
 y(2)  = R  y(1)  +  ty 
z(2)
z(1)
tz
(5)
where R is a proper (det=1) or improper (det=-1) rotation (orthogonal) matrix and t is the glide vector.
for example, a glide perpendicular to the x axis and with glide
vector along the y axis will produce the following transformation:

 
−x(1)
x(2)
 y(2)  =  y(1) + 1/2 
z(1)
z(2)

(6)
Operators with graphs that do not cross the origin (however it is
chosen). We show now that these operators can be written in the
same form. If the operator graph goes through the point x0 , y0 , z0 ,
the general form of such operator in Cartesian coordinates 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 
z(2)
z(1)
tz
(7)
where

 



tx
x0
x0
 ty  =  y0  − R  y0 
tz
z0
z0
(8)
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)
z(2)
z(1)
10
(9)
• Symmetry operators written in the form of eq. 5 are said to be in normal
form. They are written as the composition (in this order) of a rotation
(proper or improper) — the rotational part, followed by a translation —
the translational part.
• Note that when using non-Cartesian coordinates (see next lesson), the
normal form of symmetry operators remains the same:



 
x(2)
x(1)
tx
 y(2)  = D  y(1)  +  ty 
z(2)
z(1)
tz

(10)
In general, D is not orthogonal, but its determinant is still ±1.
• It can be easily shown that the rotational part of g ◦ f is R g R f . Therefore,
the rotational parts of the operators of a wallpaper group (and a space
group late on) form themselves a group, which is clearly one of the 11
point groups in 2D. This is called the crystal class of the wallpaper
group.
11
2
2.1
Lecture 2 — Coordinates and calculations
Crystallographic coordinates
• In crystallography, we do not usually employ Cartesian coordinates. Instead, we employ coordinate systems with basis vectors coinciding
with either primitive or conventional translation operators.
• When primitive translations are used as basis vectors points of the pattern related by translation will differ by integral values of x,y and z.
• When conventional translations are used as basis vectors, {points of
the pattern related by translation will differ by either integral or simple
fractional (either n/2 or n/3) values of x,y and z.
• Basis vectors have the dimension of a length, and coordinates (position
vector components) are dimensionless.
• We will denote the basis vectors as ai , where the correspondence with the
usual crystallographic notation is
a1 = a; a2 = b; a3 = c
(11)
• We will sometimes employ explicit array and matrix multiplication for clarity.
In this case, the array of basis vectors is written as a row, as in [a] =
[a1 a2 a3 ].
• Components of a generic vector v will be denoted as v i , where
v 1 = vx ; v 2 = vy ; v 3 = vz ;
(12)

v1
• Components will be expressed using column arrays, as in [v] =  v 2 
v3

• A vector is then written as
v=
X
ai v i = a1 v 1 + a2 v 2 + a3 v 3
(13)
i
• A position vector is written as
r=
X
ai xi = ax + by + cz
i
12
(14)
2.2
Distances and angles
• The dot product between two position vectors is given explicitly by
r1 · r2 = a · a x1 x2 + b · b x1 x2 + c · c z1 z2 +
+a · b [x1 y2 + y1 x2 ] + a · c [x1 z2 + z1 x2 ] + b · c [y1 z2 + z1 y2 ] =
= a2 x1 x2 + b2 x1 x2 + c2 z1 z2 + ab cos γ [x1 y2 + y1 x2 ]
+ac cos β [x1 z2 + z1 x2 ] + bc cos α [y1 z2 + z1 y2 ]
(15)
• More generically, the dot product between two vectors is written as
v·u=
X
i
ai v i ·
X
aj v j =
j
X
[ai · aj ] ui v j
(16)
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
(17)
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
b2
bc cos α 
G =  ab cos γ
ac cos β bc cos α
c2

(18)
• To measure the length v of a vector v:

v1
v 3 ]G  v 2 
v3

v 2 = |v|2 = [ v 1 v 2
(19)
• To measure the angle θ between two vectors v and u:
 1 
v
1
cos θ =
[ u1 u2 u3 ]G  v 2 
uv
v3
13
(20)
2.3
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.
bi = 2π
X
ak (G−1 )ki
(21)
k
From Eq. 17 follows:
ai · bj = ai · 2π
X
ak (G−1 )ki = 2π
X
k
Gik (G−1 )kj = 2πδij
(22)
k
• Note that the vectors bi have dimensions length−1 . Since the bi are linearly
independent if the ai are, one can use them as new basis vectors, forming the so-called dual basis. This being a perfectly legitimate choice,
can express any vector on this new basis, as
q=
X
qi bi
(23)
i
• We can write any vector on this new basis, but vectors expressed using 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. 24):
q · v = 2π
X
qi xi
(24)
i
• In particular, the dot product of integral multiples of the original basis vectors (i.e., direct or real lattice vectors), with integral multiples of the
dual basis vectors (i.e., reciprocal lattice vectors) are integral multiples of 2π. This property will be used extensively to calculate Fourier
transforms of lattice functions.
14
Recap of the key formulas for the dual basis
• From direct to dual bases (eq. 21)
bi = 2π
X
ak (G−1 )ki
k
• Dot product relation between the two bases (eq. 22)
ai · bj = 2πδij
• Dot product between vectors expressed on the two different bases (eq. 24)
q · v = 2π
X
qi xi
i
2.4
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
a1 · (a2 × a3 )
a3 × a1
= 2π
a1 · (a2 × a3 )
a1 × a2
= 2π
a1 · (a2 × a3 )
b1 = 2π
b2
b3
(25)
Note that
v = a1 ·(a2 × a3 ) = abc 1 − cos2 α − cos2 β − cos2 γ + 2 cos α cos β cos γ
1/2
(26)
is the unit cell volume.
• In crystallographic textbooks, the dual basis vectors are often written as a∗ ,
b∗ and c∗ .
2.5
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.
15
• 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 G̃ = (2π)2 G −1 , so that, for two reciprocalspace vectors q and r:
q·r=
X
G̃ij qi rj
(27)
i,j
2.6
A very useful example: the hexagonal system in 2D
• The hexagonal system in 2D has a number of important applications in contemporary 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:
G=a
2
1
−1/2
−1/2
1
(28)
• From eq. 27, we can find the reciprocal-space metric tensor :
(2π)2 4
G̃ =
a2 3
1 1/2
1/2 1
(29)
• It follows, for example, that the length of a vector in reciprocal space is
given by:
2π
q=
a
2.7
r
4 2
(h + hk + k 2 )
3
(30)
Symmetry in 3 dimensions
• In 3D, there is a distinction between proper and improper rotations.
• All improper rotations can be obtained by composing a proper rotation with
the inversion — itself an improper rotation — as I ◦ r.
• In all coordinate systems, the matrix representation of the inversion (located at the origin) is minus the identity matrix. From this, it is easy to
see that the inversion commutes with all operators.
16
• 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 rotoinversions. They are obtained by composition of an axis r of order
higher than two with the inversion, as I ◦ r. These operators are 3̄ (
4̄ (
) and 6̄ (
),
), and their action is summarized in Fig. 7. The sym-
bols are chosen to emphasize the existence of another operator inside
the ”belly” of each new operator. Note that 3̄ ◦ 3̄ ◦ 3̄ = 3̄3 = I, and 3̄4 = 3,
i.e., symmetries containing 3̄ also contain the inversion and the 3-fold
rotation. Conversely, 4̄ and 6̄ do not automatically contain the inversion.
In addition, symmetries containing both 4̄ (or 6̄) and I also contain 4 (or
6).
+ -­‐ -­‐ -­‐ + + -­‐ +/-­‐ + + -­‐ +/-­‐ +/-­‐ Figure 7: Action of the 3̄ , 4̄ and 6̄ operators and their powers. The set of equivalent points
forms a trigonal antiprism, a tetragonally-distorted tetrahedron and a trigonal prism, respectively. 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.
• Glide planes in 3D have different symbols, depending on the orientation of
the glide vector with respect to the plane of the projection (fig. 9).
• There is also a new type of operator, resulting from the compositions of
proper rotations 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 primitive translation t . Therefore, the translation component must be
m
n t,
where m
is an integer. We can limit ourselves to m < n, all the other operators
being composition with lattice translations. Roto-translation axes are
therefore indicated as nm , as in 21 , 63 etc. (fig. 9).
17
Figure 8: Bravais lattices in 3D. For a full explanation of their symmetry, and on convention
for point/space groups in 3D, see Supplementary Material.
18
2
3
4
3
1
6
4
a,b or c
m
n
e
d
3
8
2
21
31
41
43
42
32
g
m
6
1
8
62
64
61
65
63
Figure 9: The most important graph symbols employed in the International Tables to describe 3D space groups. Fraction next to the symmetry element indicate the height (z coordinate) with respect to the origin.
19
IT numbering SG symbols (Hermann-­‐Mauguin nota6on) Crystal class (point group) SG symbol (Schoenflies nota6on) Crystal system SG diagrams projected along the 3 direc6ons b is orthogonal to a and c. Cell choce 1 here means that the origin is on an inversion centre. A generic posi6on and its equivalent. Dots with and without commas are related by an improper operator (molecules located here would have opposite chirality) A symmetry operator notated as follows: a 2-­‐fold axis with an associated transla6on of ½ along the b=axis, i.e., a screw axis. This is located at (0,y,¼) (see SG diagrams, especially top leO). Only a subset of the symmetry operators are indicated. All others can be obtained by composi6on with transla6on operators. Asymmetric unit cell: the smallest part of the paGern that generates the whole paGern by applying all symmetry operators. Figure 10: 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 .
20
Refers to numbering of operators on previous page Group generators Primi4ve transla4ons Equivalent posi4ons, 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 dis4nct Wickoff leEer. Reflec4on condi4ons for each class of reflec4ons. General: valid for all atoms (i.e., no h0l reflec4on will be observed unless l=2n; this is due to the c glide). Special: valid only for atoms located at corresponding posi4ons (leI side), i.e., atoms on inversion centres do not contribute unless k+l=2n Figure 11: IT entry for SG 14: P 21 /c (page 2).
21
3
3.1
Lecture 3 — Fourier transform of periodic functions
Centring extinctions
• Reciprocal-space vectors are described as linear combinations of the reciprocal or dual basis vectors, (dimensions: length−1 ) with dimensionless coefficients.
• Reciprocal-lattice vectors (RLV ) are reciprocal space vectors with integral components. These are known as the Miller indices and are usually notated as hkl.
Dot products
• The dot product of real and reciprocal space vectors expressed in the usual coordinates is
q · v = 2π
X
qi v i
(31)
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. These reciprocal-lattice vectors are exactly those generated by the corresponding primitive basis.
A conventional basis generates more RL vectors that a corresponding primitive basis.
As we shall see, the “extra” points are not associated with any scattering intensity —
we will say that they are extinct by centering.
• Each non-primitive lattice type has centring extinctions, which can be expressed in terms of the Miller indices hkl (see table 2).
• The non-exctinct reciprocal lattice points also form a lattice, which is naturally one of the 14 Bravais lattices. For each real-space Bravais lattice,
tab. 3 lists the corresponding RL type.
22
Table 2: Centering extinction and scattering conditions for the centered lattices. 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 “Scattering” 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
3.2
Extinction
Scattering
P
none
all
A
B
C
k + l = 2n + 1
h + l = 2n + 1
h + k = 2n + 1
k + l = 2n
h + l = 2n
h + k = 2n
F
k + l = 2n + 1 or
h + l = 2n + 1 or
h + k = 2n + 1
k + l = 2n and
h + l = 2n and
h + k = 2n
I
h + k + l = 2n + 1
h + k + l = 2n
R
−h + k + l = 3n + 1 or
−h + k + l = 3n + 2
−h + k + l = 3n
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:
F (q) =
1
(2π)
=
X
3
2
v0
(2π)
i
i qi n
P
Z
d(x)f (x)e−iq·x
u.c.
ni
X
3
2
e
−2πi
e−2πi
i
i qi n
P
Z
dxi f (xi )e−2πi
P
i qi x
i
(32)
u.c.
ni
where v0 is the volume of the unit cell
• The triple infinite summation in ni = nx , ny , nz is over all positive and negative integers. The qi = qx , qy , qz are reciprocal space coordinates on the
dual basis. The xi = x, y, z are real-space crystallographic coordinates
(this is essential to obtain the qi ni term in the exponent). The integral is
over one unit cell.
23
Table 3: Reciprocal-lattice Bravais lattice for any given real-space Bravais
lattice (BL).
Crystal system
Triclinic
Real-space BL
P
Reciprocal-space BL
P
Monoclinic
C
C
Orthorhombic
P
A or B or C
I
F
P
A or B or C
F
I
Tetragonal
P
I
P
I
Trigonal
P
R
P
R
Hexagonal
P
P
Cubic
P
I
F
P
F
I
• F (q) is non-zero only for q belonging to the primitive RL. In fact, if q belongs 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). Conversely, if q does not belong to the primitive reciprocal lattice,
the exponential factor will vary over the unit circle in complex number
space and will always average to zero. This is true, in particular, for
those “‘conventional”’ RLV that we called extinct by centring.
• It is the periodic nature of f (r) that is responsible for the discrete nature of
F (q).
3.3
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 associated 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.
24
• Let R be the rotational part and t the translational part of a generic symmetry operators. One can prove that
F (q) =
Z
N
(2π)
3
2
−1
d(x)f (x)e−i(R
q)·x −iq·t
e
= F (R−1 q)e−iq·t
(33)
u.c.
Eq. 33 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
∗
F (q) =
Z
N
3
(2π) 2
=
Z
N
(2π)
3
2
dxf ∗ (x)eiq·x
(34)
u.c.
dxf (x)eiq·x = F (−q)
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 crystal 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.
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 (h̄k̄¯l)|2
(35)
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.
25
4
Lecture 4 — 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 population) 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 scattering of light, X-rays or neutrons can occur. In general, the inelastic
scattering will be outside the RL nodes.
• Various Wigner Seitz constructions
3
are employed to subdivide the recip-
rocal space, for the purpose of:
Classifying the wavevectors of the excitations. The Bloch theorem 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 wavevector 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 WignerSeitz 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 particular, apply degenerate perturbation theory to free-electron wavefunctions 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.
• 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.
3
A very good description of the Wigner-Seitz and Brillouin constructions can be found in [?].
See also the Supplementary Material for a summary of the procedure.
26
k Figure 12: A set of typical 1-dimensional electronic dispersion curves in the
reduced/repeated zone scheme.
4.1
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 wavefunctions in a crystal have the Bloch form:
ψk (r) = eik·r uk (r)
(36)
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
0
fact, a function ψk0 (r) = eik ·r uk0 (r) with k0 outside the first BZ can be
rewritten as
h
i
ψk0 (r) = eik·r eiτ ·r uk0 (r)
(37)
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. 37 is in the Bloch form.
27
• The application of the Bloch theorem to 1-dimensional (1D) electronic wavefunctions, using either the nearly-free electron approximation or the
tight-binding approximation leads to the typical set of electronic dispersion curves (E vs. k relations) shown in fig. 12. 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:
vn (k) =
1 ∂En (k)
~ ∂k
(38)
• 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 symmetry.
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).
Group velocities directions are constrained by symmetry on symmetry 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:
28
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).
If the Bloch wavefunction ψk = eik·r uk (r) is an eigenstate of the
Schroedinger equation
~2 2
−
∇ + U (r) ψk = Ek ψk
2m
(39)
then ψk† = e−ik·r u†k (r) is a solution of the same Schroedinger equation with the same eigenvalue (this is always the case if the potential 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.
4.2
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 particular, the components of the group velocity parallel to the axis must be
equal.
• Since eik·r uk (r) = ei(k+τ )·r {e−τ ·r uk (r)} and the latter has crystal momentum 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 .
4
4
This leaves the possibility open for cases in which vn (k) jumps discontinuosly across the
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.
29
Figure 13: the first and two additional Wigner-Seitz cells on the oblique lattice (construction lines are shown). Significant points are labelled, and some
possible zone-boundary group velocity vectors are plotted with arrows.
4.2.1
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. 13 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
triclinic lattices in 3D) to use as the first Brillouin zone not the WignerSeitz cell, but the conventional reciprocal-lattice unit cell, which has a
30
Figure 14: 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.
simpler parallelogram shape. The relation between these two cells is
shown in fig. 14.
4.2.2
The square lattice: a high symmetry case
• A more symmetrical situation is show in fig. 15 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.
4.3
Symmetry in the nearly-free electron model: degenerate wavefunctions
• one important class of problems involves the application of degenerate perturbation theory to the free-electron Hamiltonian, perturbed by a weak
periodic potential U (r):
31
Figure 15: 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.
32
H=−
~2 2
∇ + U (r)
2m
(40)
• 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. 12).
• 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 degenerate 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.
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. 16). 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. 16 in the case of the square lattice (point
group 4mm) for boundary points between different Brillouin zones. One
can see that:
33
Figure 16: Construction of nearly-free electron degenerate wavefunctions for
the square lattice (point group 4mm). Some special symmetry point are labelled. Relevant RLV s are also indicated.
• 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
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 potential 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 freeelectron 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
34
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 wavefunctions (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.
35