Group Theory in Solid State
Physics II
Preface
This lecture expands on the fundamental group theoretical concepts and
methods introduced in Part I. Again the main aim is to show how to use
symmetry arguments for solving problems in atomic, molecular and solid
state physics. Particular attention will be paid to double groups, as they
are pivotal in understanding problems involving the spin of electrons and
related topics in other field of Physics such as particle Physics. In particular, we will search for applications of the Clebsch-Gordan coefficients and
of the main theorem governing group theoretical methods in Physics – the
Wigner-Eckart-Koster theorem. Furthermore, we will provide a deep discussion of space (double) groups – the true symmetry groups of crystals – and
their relation to the more common point groups. Finally, we will discuss the
application of group theoretical methods within the Landau theory of phase
transitions. The relevant literature for the topics presented in this lectures
is:
L.D. Landau, E.M. Lifshitz, Lehrbuch der Theor. Pyhsik, Band III, ”Quantenmechanik”, Akademie-Verlag Berlin, 1979, Kap. XII and Band V, ”Statistische Physik”, Teil 1, Akademie-Verlag 1987, Kap. XIII.
Zürich, October 2003
D. Pescia
ii
Contents
Preface
ii
1 The Kronecker Product
1.1 Kronecker product of representations . . . . . . . . .
1.1.1 An example: Product representations of SO(3)
1.2 The direct product of groups . . . . . . . . . . . . . .
1.3 The Clebsch-Gordan coefficients . . . . . . . . . . . .
1.3.1 Clebsch-Gordan coefficients for point groups .
1.4 The Wigner-Eckart theorem . . . . . . . . . . . . . .
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1
1
3
5
7
9
18
2 Energy bands in solids
2.1 The translation group . . . . . . . . . . . . . . . . . . . . . .
2.2 Space groups . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The irreducible representations of space groups . . . .
2.3 Symmetry adapted plane waves . . . . . . . . . . . . . . . . .
2.3.1 The one-dimensional lattice . . . . . . . . . . . . . . .
2.3.2 The square lattice . . . . . . . . . . . . . . . . . . . . .
2.3.3 Free electron energy bands in three-dimensional lattices
25
25
30
34
39
39
41
47
3 Landau theory of phase transitions
48
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Chapter 1
The Kronecker Product
So far we have considered elementary groups. In this chapter we treat two
complementary subjects. In the first one we consider the Kronecker product
of representations, find out which irreducible components it contains and develop methods to find the symmetry adapted basis functions. In the second
one we deal with enlarging the symmetry group to account for extra symmetries. Example: consider a system whith SO(3) as symmetry group. Having
noticed that the inversion I is also a symmetry element, we ask how the symmetry group can be expandend and which are the irreducible representations
of the expanded group.
1.1
Kronecker product of representations
The Kronecker (or direct or tensor) product of vector
spaces
Consider the two spaces Ln and Lm . The direct-product space is a space of
dimension p = n · m defined by the p basis vectors
e1 ⊗ i1 , e1 ⊗ i2 , ..., e1 ⊗ im , e2 ⊗ i1 , ...en ⊗ im
with Lp = Ln ⊗ Lm . The scalar product in the spaces Ln and Lm produces a
scalar product in Lp according to
(u1 ⊗ v1 , u2 ⊗ v2 ) = (u1 , u2) · (v1 , v2 )
As any vector of Lp can be expressed as u = i,j uij ei ⊗ ij , the scalar product
between any vector in Lp can be calculated according to
(u, v) = (
i,j
uij ei ⊗ ij ,
l,m
1
vlm el ⊗ im ) =
∗
uij vij
i,j
CHAPTER 1. THE KRONECKER PRODUCT
2
(because ei ⊗ ij , el ⊗ ik ) = δil δjk ). The action of the operator A ⊗ B on u ⊗ v
is defined as (A ⊗ B)u ⊗ v = Au ⊗ Bv. With this definition one can find the
action of A ⊗ B on each vector of Lp as (A ⊗ B)u = i,j uij Aei ⊗ Bij . The
matrix representation of A ⊗ B is the direct product of the two matrices. Let
[Alm ] be a matrix of the order LxM and [Bpq ] a matrix of order P xQ. The
direct (tensor or Kronecker) product of these two matrices is a matrix C of
the order L · P × M · Q, which can be written as








A11 B A12 B ... A1M B
A12 B A22 B ... A2M B
.
.
.
.
.
.
.
.
AL1 B AL2 B ... ALM B








where the ’element’ Alm · B stands for a matrix of order PxQ given by








Alm B11 Alm B12 ... Alm B1Q
Alm B21 Alm B22 ... Alm B2Q
.
.
.
.
.
.
.
.
Alm BP 1 Alm BP 2 ... Alm BP Q








If A1 , A2 and B1 , B2 are any matrices whose dimensions are such that the
ordinary matrix product A1 A2 and B1 B2 are defined, then the direct product
has an important property:
(A1 ⊗ B1 )(A2 ⊗ B2 ) = (A1 A2 ) ⊗ (B1 B2 )
Further, if F is the direct product of a number of matrices A, B,C, ..., then
trF = (trA) · (trB) · .....
Exercise: prove this.
The operation of the direct product of matrices is associative, so that
A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C = A ⊗ B ⊗ C
The operation is distributive with respect to the matrix addition,
A ⊗ (B + C) = A ⊗ B + A ⊗ C
Moreover, (AB)⊗k = A⊗k · B ⊗k , where A⊗k = A ⊗ A ⊗ A ⊗ A... (k-times).
CHAPTER 1. THE KRONECKER PRODUCT
3
Consider two matrix representations T1 and T2 (reducible or irreducible)
of a group G. Let us take the direct product of the corresponding matrices
for the two representations and denote it by
T (a) = T1 (a) ⊗ T2 (a)
Theorem: T(a) is also a representation of the group.
Proof:
T (a)T (b) =
=
=
=
(T1 (a) ⊗ T2 (a)(T1 (b) ⊗ T2 (b)
(T1 (a)T1 (b)) ⊗ (T2 (a)T2 (b))
T1 (ab) ⊗ T2 (ab)
T (ab)
Thus, we see that the set of matrices constructed by taking the direct product
of two representations also form a representation. An important property of
the direct product representation is that
χ(a) = χ1 (a) · χ2 (a)
i.e. the characters of the direct product representations are the product of the
characters of the individual representations. In general, the direct product
representation is reducible, certainly if T1 or T2 are reducible. For example,
for the reduction of the direct product of irreducible representations Ti and
Tj we expect an expansion of the type
Ti ⊗ Tj = ⊕k ni,j
k Tk
where ni,j
k are non-negative integers. A sum of this type is also known as the
Clebsch-Gordan series.
Exercise: work out all direct products of all irreducible representations of
C4v .
1.1.1
An example: Product representations of SO(3)
According to the rules worked out in Part I, the reduction of a representation
D in irreducible components relies on the knowledge of the characters alone.
Let now G be SO(3) and D with corresponding characters χ(φ) the characters
of the representation D. Then the coefficients cj in D = ⊕cj Dj are given by
the general formula
cj =
1
2π 2
do · dφ sin[(2j + 1)φ]χ(φ) sin φ
CHAPTER 1. THE KRONECKER PRODUCT
4
which is equivalent of expanding the function χ(φ) sin φ in sin(2j + 1)φ, j ∈
N . We now apply this equation to the product representation Dm ⊗ Dl of
two irreducible representations Dm , Dl (without limitation of generality, we
consider m ≥ l. Dm ⊗ Dl has dimension (2m + 1) · (2l + 1) and its character
is given by χ = χl · χm . Rewriting χl (φ) · χm (φ) sin φ by using the identity
sin a cos b = 1/2[sin(a + b) + sin(a − b)], as
(l |integer
half int. ) =
=
+
+
+
1/2
2(cos 2lφ + cos(2(l − 1)φ + .....+ |cos φ ) sin(2m + 1)φ
sin[2(m + l) + 1]φ + sin[2(m − l) + 1]φ +
sin[2(m + l − 1) + 1]φ + sin[2(m − l + 1) + 1]φ +
.......
sin(2m+1)φ
|sin[2(m+1/2)+1]φ+sin[2(m−1/2)+1]φ
we recognize that the coefficients of the representation Dm+l , ....Dm−l are just
equal one, all others are vanishing. Thus
Dm ⊗ Dl = Dm+l + Dm+l−1 + ... + Dm−l
This results is the so called Clebsch-Gordan series for the reduction of direct
product representations of SO(3).
A practical method for reducing any representation D of SO(3) without
the explicit use of the characters and for finding the invariant subspaces is
given by Cartan.
First Cartan Algorithmus
This is a practical way to work out the irreducible representations Dj and
the numbers nj in D = ⊕j nj Dj
1. Let D be a representation of SO(3) and I3D the corresponding matrix
representing the generator I3 .
2. Among the diagonal matrix elements (which are called Cartan’s weights
or multiplicators) lets pick up the largest one jk . Then the reduction
of the representation D contain Djk at least once.
3. The representation Djk uses the corresponding 2jk + 1 diagonal matrix
elements.
4. the remaining eigenvalues are sorted out starting from the largest remaining one and repeating the above procedure until all eigenvalues
are used
CHAPTER 1. THE KRONECKER PRODUCT
5
An important application of this algorithm is the reduction of Dm ⊗ Dl .
Without limitation of generaliy, we take m ≥ l. To perform the reduction, we
consider only rotation about the z axis, for which the matrix representations
are










e−imϕ
0
0
e−i(m−1)ϕ
0
0
0
0
0
0
0
0










.
.
.
.
.
.
e−ilϕ
0
−i(l−1)ϕ
0
e
0
0
0
0
0
0
0
0
.
0
0
.
0
0
.
0
0
.
0
0
. e−i(−m+1)ϕ
0
.
0
e−i(−m)ϕ
.
.
.
.
.
.
.
0
0
.
0
0
.
0
0
.
0
0
−i(−l+1)ϕ
. e
0
−i(−l)ϕ
.
0
e




















The matrix representing I3 for Dm ⊗ Dl is given by
−1 d
(Dm ⊗ Dl ) |ϕ=0 = I3,m ⊗ E + E ⊗ I3,l
i dϕ
This matrix has the following form:













m+l
0
0
m+l−1
0
0
0
0
0
0
0
0
0
0
.
.
.
.
.
.
. m−l
.
.
.
.
.
.
0 0
0
0 0
0
.
.0
. .
0
.
0
. .
.
0 0 −m − l













The top diagonal elements m+l, ..., m−l represent all the largest eigenvalues
and can be used to classify the irreducible representations into which Dm ⊗Dl
reduces. The result is the Clebsch-Gordan series that we have worked our
above using the theory of characters.
1.2
The direct product of groups
Let H = e, h1 , h2 , ... and K = e, k1 , k2 , ... be two groups such that all the
elements hj commute with kj . If we multiply each element of H with each of
CHAPTER 1. THE KRONECKER PRODUCT
6
K we obtain a new set of elements.
Theorem: This set build a group: the group is called the direct product of
H and K and is denoted by G ≡ H ⊗ K.
Proof. With G = e, g11 , g12 , ...g1k , g21 , g22 , ..., gij , ...., where gij = hi kj and
hi hm = hp and kj kn = kq , then
gij gmn = (hi kj )(hm kn ) = hi hm kj kn = hp kq = gpq
Exercise: Prove that H and K are subgroups of G.
Let Th be a representation of H and Tk a representation of K, i.e.
Th (hi )Th (hm ) = Th (hp )
Tk (kj )Tk (kn ) = Tk (kq )
Theorem: the direct product Th ⊗Tk of the representations of two commuting
groups is a representation of the direct product group.
Proof:
(Th ⊗ Tk )(gij )(Th ⊗ Tk )(gmn )
= (Th (hi ) ⊗ Tk (kj ))(Th (hm ) ⊗ Tk (kn ))
= Th (hp ) ⊗ Tk (kq )
= Th ⊗ Tk (gpq )
Theorem: if Th and Tk are irreducible representations of H and K, then
Th ⊗ Tk ≡ Tg is an irreducible representation of G.
Proof: Th and Tk are irreducible representations, i.e.
Mhi∈H χ∗h (hi )χh (hi ) = 1
Mki∈K χ∗k (ki)χk (ki ) = 1
Taking the product of both sides of these equations leads to
1 = Mhi ,kj χ∗h (hi )χh (hi )χ∗k (kj )χk (kj ) = Mgij χ∗g (gij )χg (gij )
which proves that Tg is indeed an irreducible representation of the product
group.
Theorem: All irreducible representations of G are the direct product of an
irreducible representation of H and one of K.
Proof (for finite groups): Let the number of irreducible representations
of H be nh , their dimensions be lih , the number of irreducible representations
CHAPTER 1. THE KRONECKER PRODUCT
7
of K be nk and their dimensions ljk :
nh
(lih )2 = h
i
ng
(ljk )2 = k
j
Taking the product of both sides gives
h·k =g =
(lih )2 (ljk )2
i,j
The irreducible representations of G obtained by direct product will have
g
the dimensions lih · ljk = lij
. The sum over all such irreducible representations
gives
g
(lij )2 = (lih )2 (ljk )2 = g
i,j
i,j
so that the direct product of all irreducible representations exhausts the
all irreducible representations of G. The number of such representations is
ng = nh · nk . This is a very important result as it helps in constructing all
irreducible representations of a bigger group from those of smaller groups.
Example of group product. The most famous example of group product is
SO(3) ⊗ SU(2). This is the full symmetry group of an electron with spin
embedded in a spherically symmetric potential. If spin-orbit coupling is neglected, then the symmetry group consists of separate rotations in real and
spin space, and the energy levels can be classified according to the irreducible
representations D l ⊗D 1/2 . The dimension of these representations is (2l+1)·2.
If spin-orbit coupling terms are allowed, then the symmetry group consists
of simoultaneous rotations in real and spin space. This group is thus essentially SU(2), and the Kronecker product representations split into irreducible
representations of SU(2), according to the Cartan algorithm.
1.3
The Clebsch-Gordan coefficients
Once the irreducible components of the product representations are worked
out (if any exists), one can ask about the invariant subspaces belonging to
the irreducible components. In other words, we would like to construct out of
the product functions the symmetrized linear combinations. The coefficients
of the linear combinations with proper symmetry are called Clebsch-Gordan
coefficients.
CHAPTER 1. THE KRONECKER PRODUCT
8
In the case of SO(3) one proceeds, typically, by means of the second
Cartan Algorithm.
1. find all basis vectors transforming according to the same Cartan weight
jk . This space Vjk has, in general, more than one dimension.
2. construct the operator I+ for the representation D and solve the equation I+ x = 0 within the space Vjk . The vector x transforms, by construction, according to the top weight of the representation Djk , i.e.
when I3 (Djk ) is applied to x, x get multiplied by jk .
3. The remaining vectors transforming according to Djk are found by applying the lowering operator I− to x.
Example. The Clebsch-Gordan coefficients. Let us calculate, as an example
of this second algorithmus, the invariant subspaces of Dl ⊗ D1/2 . This representations is important for describing the phenomenon of spin-orbit coupling
in a single electron atom with spherically symmetric potential. We recall
briefly the important matrix elements for the operators I± . The irreducible
representation Dj has dimension 2j + 1 and the non-vanishing matrix elements of the generators are given by (m = j, j − 1, ... − j) (Condon-Shortley
convention)
(ψj,m , Iz ψj.m ) = m
1
(ψj,m±1 , Ix ψj,m ) =
(j ∓ m)(j ± m + 1)
2
i
(j ∓ m)(j ± m + 1)
(ψj,m±1 , Iy ψj,m ) = ∓
2
For I± = 12 [Ix ± i · Iy ] we have
1 I± Xj,m = √ · j(j + 1) − m(m ± 1)Xj,m±1
2
Because of the Clebsch-Gordan series,
Dl ⊗ D1/2 = Dl+1/2 ⊕ Dl−1/2
Let the basis functions, carrying the irreducible representation Dl , be Xm ,
with m = l, ... − l and let call the spin functions Yz , z = ±1/2. The only basis
function transforming according to the Cartan weight l + 1/2 is Xl ⊗ Y1/2 .
This is also the only solution of the equation L+ x = 0 within the space transforming according to this Cartan weight. Applying the operator L− to this
CHAPTER 1. THE KRONECKER PRODUCT
9
function we can find all functions transforming according to Dl+1/2 .
Exercise: find the function within the invariant space of Dl+1/2 which transforms according to the l − 1/2 Cartan weight.
We now consider the space which carries the Cartan weight l −1/2. There are
two functions in this space, namely Xl ⊗ Y−1/2 and Xl−1 ⊗ Y1/2 . One possible
solution of the equation
is α =
I+ (αXl ⊗ Y−1/2 + βXl−1 ⊗ Y1/2 ) =
α[(I+ Xl ) ⊗ Y−1/2 + Xl ⊗ (I+ Y−1/2 )]
+β[(I+ Xl−1 ) ⊗ Y1/2 + Xl−1 ⊗ (I+ Y1/2 )]
√
1
= [ √ α + β l]Xl ⊗ Y1/2 = 0
2
2l
2l+1
and β = −
1
.
2l+1
This means that the function
2l
Xl ⊗ Y−1/2 −
2l + 1
1
Xl−1 ⊗ Y1/2
2l + 1
belongs to the invariant subspace of Dl−1/2 .
Exercise: check that the two functions transforming according to the same
Cartan weight l − 1/2 but belonging to different irreducible representations
are orthogonal.
Exercise: find all symmetry adapted functions in the reduction of D1 ⊗ D1/2
(these are the functions belonging to the spin orbit split levels 2 P3/2 and
2
P1/2 .)
The coefficients determining the symmetry adapted product functions, i.e.
those basis functions that transform according to a given weigth of a given
irreducible representation in the reduction of Dl ⊗Dk are called the ClebschGordan coefficients:
ψ(j, mj ) =
c(l, k, j, ml , mk , mj )ψ(l, ml ) ⊗ ψ(k, mk )
ml ,mk
The Clebsch-Gordan
coefficent
c(l, k, j, ml , mk , mj ) is often written as the 3j
l
k
j
.
symbol
ml mk mj
Exercise: show that ml +mk = mj , otherwise the CC-coefficient is vanishing.
1.3.1
Clebsch-Gordan coefficients for point groups
For some physical problems, the full rotational group is not a symmetry
group of the system. Let us consider a single electron atom embedded – say–
CHAPTER 1. THE KRONECKER PRODUCT
10
in a cubic cell. The potential experienced by the electron has a symmetry
group with much less elements than SO(3). In fact, the symmetry group of
the Hamiltonian is the cubic group Oh , which transforms a cube into itself.
Oh is a subgroup of SO(3) with discrete number of elements. Its elements are
summarized in the follwing table as the result of operating on a coordinate
system placed in the center of the cube. With the aid of the figure, we can
illustrate all 48 symmetry elements of Oh .
Figure 1.1: The 48 operations of Oh : the identity e, 3 rotations by π about
the axes x, y, z (3C42 ), 6 rotations by ±π/2 about the axes x, y, z (6C4 ), 6
rotations by π about the bisectrices in the planes xy, yz, xz (6C2 ), 8 rotations
by ±2π/3 about the diagonals of the cube (8C3), the combination of the
inversion I with the listed 24 proper rotations
The question is now: what happens to the level scheme of the electron
when such a symmetry breaking potential is switched on? From the point of
view of group theory, the answer to this question is quite straightforward. An
irreducible representation of SO(3) might no longer be irreducible
when it is limited to the elements of a subgroup of SO(3). This leads,
in general, to a level splitting, i.e. a reduction of the degeneracy, which is often encountered in the band structures of crystals.
Consider for instance the point Γ in the Brillouin zone of the GaAs crystal.
The point group symmetry of a GaAs crystal at the Γ-point is Td , which is a
subgroup of Oh . The top of the valence band is described (we initially neglect
spin) by the single group irreducible representation Γ15 of the group Td . The
Γ15 irreducible representation is a three dimensional one, and it coincides
with D1 . Thus, the top of the valence band behaves exactly as if the atom
would have the full rotational symmetry. This is why we can use the CG
coefficients derived from the full rotational group when calculating the spin
polarization of electrons excited in GaAs. This is, however, a coincidence:
CHAPTER 1. THE KRONECKER PRODUCT
Type
C1
C42
C4
C2
C3
Operation Coordinate
C1
xyz
C2z
x̄ȳz
C2x
xȳ z̄
C2y
x̄yz̄
−1
C4z
ȳxz
C4z
y x̄z
−1
C4x
xz̄y
C4x
xz ȳ
−1
C4y
zy x̄
C4y
z̄yx
C2xy
yxz̄
C2xz
z ȳx
C2yz
x̄zy
C2xȳ
ȳx̄z̄
C2x̄z
z̄ ȳx̄
C2yz̄
x̄z̄ ȳ
−1
C3xyz
zxy
C3xyz
yzx
−1
C3xȳz
zx̄ȳ
C3xȳz
ȳz̄x
−1
C3xȳz̄
z̄ x̄y
C3xȳz̄
ȳzx̄
−1
C3xyz̄
z̄xȳ
C3xyz̄
yz̄x̄
Type
I
IC42
IC4
IC2
IC3
11
Operation Coordinate
I
x̄ȳz̄
IC2z
xyz̄
IC2x
x̄yz
IC2y
xȳz
−1
IC4z
y x̄z̄
IC4z
ȳxz̄
−1
IC4x
x̄z ȳ
IC4x
x̄z̄y
−1
IC4y
z̄ ȳx
IC4y
z ȳ x̄
IC2xy
ȳx̄z
IC2xz
z̄y x̄
IC2yz
xz̄ ȳ
IC2xȳ
yxz
IC2x̄z
zyx
IC2yz̄
xzy
IC3xyz
z̄x̄ȳ
IC3xyz
z̄x̄ȳ
−1
IC3xȳz
z̄xy
IC3xȳz
yzx̄
−1
IC3xȳz̄
zxȳ
IC3xȳz̄
yz̄x
IC3xyz̄
zx̄y
IC3xyz̄
ȳzx
Table 1.1: Coordinate transformations under the action of the elements of
the group Oh
CHAPTER 1. THE KRONECKER PRODUCT
12
for instance, the D2 irreducible representation of SO(3), which carries the
d-electrons, does indeed split at Γ into two irreducible representations of Td .
In addition, as soon as we move away from the Γ-point things change dramatically. We consider now, as an example, the Λ direction in the Brillouin
zone. The symmetry group of this direction is C3v .
The single group C3v
The single group C3v contains six elements:
−1
, IC2xȳ , IC2x̄z , IC2yz̄ , all transforming the k-vector along the
E, C3xyz , C3xyz
Λ-direction into itself (kΛ = π/a(λ, λ, λ) with 0 < λ < 1). The symbols C3xyz
−1
and C3xyz
, for instance, means rotations by an angle 2π/3 around an axis
with director cosines on the ratio 1 : 1 : 1, see the table and figure for Oh . To
construct the character table of the group one must find first the classes of
conjugate elements, because we then know that, for finite groups, the number
of irreducible representations exactly equals the number of classes. There are
three classes: the identity element, the two there-fold rotations and the three
reflections. We summarize now some rules for constructing the character table of a finite
group.
1. It is convenient to display in table form the characters of the irreducible representations. Such a table gives less information than a complete set of matrices, but it
is sufficient for classifying the electronic states
2. The number of irreducible representations is equal to the number of classes in the
group
3. The sum of the squares of the dimensions lα of the irreducible representations is
equal to the number of elements g of the group
lα2 = g
α
4. The characters of the irreducible representation must be mutually orthogonal and
normalized to the order of the group:
χα (a)∗ χβ (a) = g · δαβ
a
5. Every group admits the one-dimensional identical representation in which each
element of the group is represented by the number 1. The orthogonality relation
between characters then shows that for any irreducible representation other than
the identity representation
χ(a) = 0
a
With the help of these rules we can easily construt the character table of
the single point group C3v . We are now able to investigate the fate of the top
CHAPTER 1. THE KRONECKER PRODUCT
Irr.Rep.
Λ1
Λ2
Λ3
C3v
Γ15
E
1
1
2
E
3
2C32
1
1
-1
2C32
0
13
3σv
1
-1
0
3σv
1
valence band eigenvalue at Γ. Γ15 , restricted to the elements of C3v , is also a
representation of C3v . It is, however, not irreducible, and its characters are
given in the Table. Using the theorem for finding the direct sum expansion
of reducible representations, we obtain Γ15 = Λ1 ⊕ Λ3 . This means that we
expect the theree fold degenerate eigenvalue Γ15 to split into a once degenerate Λ1 -band and a doubly degenerate Λ3 band along the Λ direction. Using
the projection operator method we can construct symmetrized linear combinations transfroming according to each particular irreducible representation.
D
The double group C3v
This group has twice the number of elements and twice the number of classes
in the D 1/2 repreas C3v . The matrices representing the 2C3 group elements
sentation can be constructed by using l = m = n = 13 and a rotation angle
±ϕ = 2π
:
3
1−i
1+i
−
2
2
C3xyz = 1−i
1+i
+
2
2
Similarly, one can obtain the matrices IC2xȳ , IC2x̄z , IC2yz̄ by considering that
the inversion is followed by the rotation of 2π/2 around
√ an axis with
√ director
cosines in the ration 1 : 1̄ : 0, i.e. ϕ = π and l = / 2, m = −1/ 2 n = 0.
Thus, for instance,
1−i
√
0
2
IC2xȳ = I ·
√
− 1+i
0
2
The remaining matrices can be constructed in a similar way.
E=
C3xyz = 1/2
1 0
0 1
1 − i −1 − i
1−i 1+i
; Ē =
1̄ 0
0 1̄
; C̄3xyz = 1/2
−1 + i 1 + i
−1 + i −1 − i
CHAPTER 1. THE KRONECKER PRODUCT
Irr.Rep.
Λ1
Λ2
Λ3
Λ4
Λ5
Λ6
−1
C3xyz
IC2xȳ =
0
1−i
−1 − i
0
1/2I
IC2x̄z =
1/2I
IC2yz̄ =
Ē 2C3
1
1
1
1
2
-1
-1 -1
-1 -1
-2
1
1+i 1+i
−1 + i 1 − i
= 1/2
E
1
1
2
1
1
2
1/2I
−i i
i i
i −1
1 −i
2c̄3
1
1
-1
1
1
-1
14
3σv
1
-1
0
i
-i
0
3σ̄v
1
-1
0
-i
i
0
;
−1
C̄3xyz
= 1/2
−1 − i −1 − i
1 − i −1 + i
; I C̄2xȳ =
1/2I
; I C̄2x̄z =
1/2I
; I C̄2yz̄ =
1/2I
0
−1 + i
1+i
0
i −i
−i −i
−i 1
−1 i
The 12 elements are thus divided into 6 classes, and the equation α (lα )2 =
12 has the solution 12 + 12 + 22 + 12 + 12 + 22 = 12. Thus, there are a
total of four one dimensional representations and two two-dimensional representations. The three irreducible representations of C3v can be extended
D
as irreducible representations of C3v
by representing Ē with E and the C̄
elements as the non-C̄ elements. There are three extra additional irreducible
representations to be found. We notice that the two dimensional D 1/2 representation is irreducible: in fact with the characters
χ(E) = 2, χ(Ē) = −2, χ(C3 ) = 1, χĒC3 = −1, χ(σv ) = 0
we have A χ(A)2 = 12, which is a necessary and sufficient condition for
an irreducible representation. The remaining two-extra one dimensional representations are found by applying the unitarity conditions. The character
table is given above. (Λ6 is the name of D 1/2 when restricted to this point
group).
We now would like to know what happens to the crystal split bands Λ3
and Λ1 when spin-orbit coupling is switched on. In the absence of spin-orbit
coupling, the Hamiltonian is invariant with respect to separate rotations in
orbit and spin space, i.e. the symmetry group is the direct product of two
groups, namely G × SU(2). We consider a subspace spanned by the eigenfunctions f1 , ..., flα which act as basis for representation D α a the single
CHAPTER 1. THE KRONECKER PRODUCT
Irr.Rep.
Λ1
Λ2
Λ3
Λ4
Λ5
Λ6
D 1/2
Λ1 ⊗ D 1/2
Λ2 ⊗ D 1/2
Λ3 ⊗ D 1/2
E
+1
+1
2
+1
+1
2
2
2
2
4
Ē 2C32
+1 +1
+1 +1
2
-1
-1
-1
-1
-1
-2 +1
-2 +1
-2
1
-2
1
-4
-1
2C̄32
+1
+1
-1
+1
+1
-1
-1
-1
-1
1
15
3σv
+1
-1
0
i
-i
0
0
0
0
0
3σ̄v
+1
-1
0
-i
i
0
0
0
0
0
group of the Hamiltonian, i.e, when the spin is neglected. The product of the
functions fi with u+ , u− which are the basis functions for the representation
D 1/2 , constitutes a basis for the representation D α ⊗ D 1/2 of the direct product group G × SU(2), with characters χα (R) · χ1/2 (R). If the the symmetry
group of the Hamiltonian consists of independent operations in coordinate
and spin space, then D α ⊗D 1/2 is irreducible. Should the Hamiltonian contain
spin-orbit coupling, then the symmetry group of the Hamiltonian contains
only simultaneous operations in coordinate and spin space, i.e. it is GD : as
a consequence, the direct product representation, when limited to simoultaneous operations, is, in general, reducible. Using known theorems, we can
reduce the product representation to a sum of irreducible representations of
the double group, and we can also find the linear combinations of product
functions building the basis for the irreducible representations of the double
group (Clebsch-Gordan coefficients). The number of irreducible representations contained in the sum and the dimension of the representation gives
the splitting of the eigenvalue and the degeneracy of the sublevels when spin
terms – like spin-orbit interaction – are included in the Hamiltonian. ConD
sider now the Λ direction in k space again. The character table of C3v
is
1/2
extended to include the characters of Λi ⊗ D . From the characters of the
additional representations one sees that
Λ1 ⊗ D 1/2 = Λ6
Λ2 ⊗ D 1/2 = Λ6
Λ3 ⊗ D 1/2 = Λ4 + Λ5 + Λ6
Thus, the Λ3 band split into three subbands, each carrying one irreducible
representation of the double group. The band Λ1 remains degenerate and
CHAPTER 1. THE KRONECKER PRODUCT
16
carries the irreducible representation Λ6 of the double group.
The symmetry adapted basis functions
We might ask now what are the symmetry adapted wave functions belonging
to these bands. The answer to this question is a nice piece of application of
the CGC for point groups. As the groups involved are no longer the continuos
group SO(3), the method of Cartan can no longer be applied. We have to
resort to a different method, which is called the projection operator method.
This method is based on the following theorem:
Theorem (without proof): The operator
P (p) =
lp ∗(p)
χ (T )OT
g T
projects out of any normalizable wave function the sum of all basis functions
transforming according to the columns of the p-th irreducible representation
of any group G.
Having determined φ(p) , we can then apply to φ(p) the operations of the symmetry group to obtain exactly lp -linear independent functions which can be
made orthonormal and thus can be used as a basis set for the irreducible
representation D p (the application of g operation of the group to a function
belonging to an invariant subspace carrying the irreducible representation
D p cannot bring us outside this subspace). Alternatively, one can apply the
projector operator to a different function with the hope that the resulting
function is a further linearly independent basis function transforming as D p .
Notice that, if the irreducible representation is one dimensional, this operator
produce the only required symmetry adapted basis function.
The important aspect of this theorem is that it is only necessary to know
the characters of a given representation in order to obtain the whole set
of symmetry adapted basis functions transforming according to it. Notice
that, after having generated the basis functions transforming according to
the irreducible representation D p , it is possible to construct the matrix representation of D p itself (in practice, this is the procedure most adopted to
construct matrix representations of the irreducible representations), using
OT φ(p)
n =
p
Dnm
φpm
m
Thus, in is possible to generate a complete matrix representation knowing
only the characters.
CHAPTER 1. THE KRONECKER PRODUCT
17
Example 1.
We would like to find, within the subspace Y00 , Y11 , Y10 , Y1−1 , the symmetry
adapted wave functions transforming according to Λ1 and Λ3 . We start for
instance from Y00 and we apply the projection operator:
PΛ1 Y00 =
1
1 · Y00 = Y00
6
This means that Y00 transforms according to Λ1 . The lowest conduction band
along the Λ direction of GaAs or Ge is indeed a s-state with Λ1 symmetry.
Applying the projector operator to the same eigenfunctions in oder to project
out the wave functions transforming according to Λ3 will inevitably leads to
zero, as Y00 does not contain parts transforming like Λ3 . Starting from Y10
we obtain (for convenience, we set the z axis along the Λ direction)
PΛ1 Y10 =
1
1 · Y10 = Y10
6
i.e. Y10 transforms according to Λ1 as well. The deep valence Λ1 band along
the Λ- direction in GaAs and Ge has mostly z-character. In order to compute
the transformation properties of x, y we use
Oϕ f (x, y) = f (x cos ϕ + y sin ϕ, −x sin ϕ + y cos ϕ)
The projector applied to x respectively y gives x respectively y: thus they
are basis functions carrying the representation Λ3 . The highest valence band
along Λ has this symmetry. Thus, the p-states, which are degenerate at the
Γ-point, split into two components, the one containing Y10 (Λ1 ) and the one
containing Y1±1 (Λ3 ).
Example 2.
So far we have neglected spin-orbit coupling. This interaction leads to further
splitting of bands, as we shall illustrate now. We would like to find the linear
combinations of
X1 Y1/2 , X1 Y−1/2 , X31 Y1/2 , X31 Y−1/2 , X3−1 Y1/2 , X3−1 Y−1/2
transforming according to the irreducible representations Λ6 , Λ4 and Λ5 appearing in Λ1 ⊗ D1/2 and Λ3 ⊗ D1/2 . We will use as basis functions for Λ3
Y±1 : the matrices representing rotations by an angle ϕ aroung the z axis in
this basis are diagonal, with diagonal elements exp(∓iϕ).
Exercise: prove this last statement.
CHAPTER 1. THE KRONECKER PRODUCT
18
The matrices representing such rotations in the D1/2 representation are also
diagonal, their matrix elements being exp(∓i ϕ2 ).
Exercise: prove this.
The symmetry adapted basis functions build up from orbital wave function
of the type X1 and spin fuctions Y±1/2 can be found from
PΛ6 X1 Y±1/2 ∝ X1 Y±1/2
Thus, the Λ6 band originating from the coupling of a Λ1 band with spin is
doubly degenerate and its basis functions are simpy X1 Y±1/2 . The symmetry
adapted basis functions transforming according to the Λ6 irreducible representation within the subspace containing the four functions X3±1 Y±1/2 are
found again by the projector method.
3
PΛ6 X31 Y1/2 ∝ 4 · X31 Y1/2 + 4 cos ϕX31 Y1/2 = 0 = PΛ6 X3−1 Y−1/2
2
so that the wave functions X3±1 Y±1/2 do not transform as Λ6 . In contrast,
PΛ6 X3±1 Y∓1/2 ∝ 4 · X3±1 Y∓1/2 + 4 cos
ϕ ±1
X Y∓1/2 ∝ X3±1 Y∓1/2
2 3
i.e. the two wave functions X3±1 Y∓1/2 belong to the Λ6 representation. Thus,
the Λ6 band derived from the coupling of Λ3 orbital wave functions with the
spin is doubly degenerate and contains the functions X31 Y−1/2 and X3−1 Y1/2 .
The remaining two functions must be in the subspace transforming according
to Λ4 and Λ5 . We notice that the time reversal operation is also a symmetry
operation of the Hamiltonian, and states which are transformed into each
other by the time reversal operator must belong to the same energy eigenvalue. One can show that the two states X3±1 Y±1/2 are exactly such states,
i.e. states related by time reversal symmetry, so that the bands belonging to
Λ4 and Λ5 symmetry are degenerate when it comes to their energy (although
they behave differently under the operation of the point group). In the band
structure of GaAs and Ge the Λ3 band is found to generate only two spinorbit split bands, namely Λ6 and Λ4+5 . The level scheme, used for instance
in band structure theory or to study optical transitions along the Λ direction
in GaAs or Ge is given in the following figure.
1.4
The Wigner-Eckart theorem
The construction of the symmetry adapted wave functions is one of the main
applications of of group theory to quantum mechanical problems. The second
CHAPTER 1. THE KRONECKER PRODUCT
19
Figure 1.2: Level scheme Λ-direction. The arrows show allowed optical transition (see later in this chapter)
one relies on the Wigner-Eckart-Koster theorem. The WE (Koster) theorem
is the consequence of the following Lemma:
Lemma: Let G be a unitary matrix group with measure and φ(k, mk , r) be a
basis function transforming according to the mk -th column of the irreducible
representation D k of the group G.
The index r must be introduced because functions having the same symmetry under the
operations of G might differ on aspects other then their symmetry. In the case of SO(3), for
example, functions which have a certain symmetry under rotations might have a different
radial dependence (i.e. they might have a different quantum number n).
Let also φ(l, ml , s) be a basis function transforming according to the ml -th
column of the irreducible representation D l . Then
(k)
φ(k, mk , r), φ(l, ml , s) = δkl δmk ml Crs
i.e. the scalar product of two basis functions is vanishing if the basis functions
belong to different irreducible representations or to different columns of the
same irreducible representation. Else is the scalar product a constant which
does not depend on mk but only on k, r, s.
Proof: Since the scalar product of two functions is invariant under unitary
transformations, for any operation T ∈ G
φ(k, mk , r), φ(l, ml , s)
φ(k, mk , r), φ(l, ml , s)
=
= (
lk
mp
=
OT φ(k, mk , r), OT φ(l, ml , s) =⇒
k
Dm
(T )φ(k, mp , r),
p mk
mp ,mq
φ(k, mk , r), φ(l, ml , s)
=
mp ,mq
mq
l
Dm
(T )φ(l, mq , s))
q ml
∗k
l
Dm
(T )Dm
(T )(φ(k, mp , r), φ(l, mq , s))
p mk
q ml
Taking the measure over both sides we obtain
ll
∗k
l
(φ(k, mp , r), φ(l, mq , s)) · MT ∈G Dm
(T )Dm
(T )
p mk
q ml
CHAPTER 1. THE KRONECKER PRODUCT
20
1
δkl δmk ml
δmp mq (φ(k, mp , r), φ(l, mq , s))
lk
mp mq
1
= δkl δmk ml
(φ(k, mp , r), φ(l, mp , s))
lk mp
=
This proves the orthogonality of the basis functions. As the right hand side is
independent of mk and ml , provided mk and ml are equal, the scalar product
does not depend on the column index but only on k, r and s. The remaining
constant i.e. l1k · mp (φ(k, mp , r), φ(l, mp , s)) must, in general, be explicitly
computed, as symmetry arguments no longer help.
This orthogonality Lemma for basis functions can be applied for calculating the matrix elements of operators (this application goes often under the
name of Wigner-Eckart theorem).
Definition: The set of ll operators Ql1 (r), Ql2 (r), ...., Qlll (r) transform according to the columns of the ll -dimensional irreducible representation D l of the
group G if for every T of G
OT Qli (r)OT−1 =
l
Dpi
Qlp (r)
p
This equation has to be understood as an operator equation. That is, both
sides must produce the same result when acting on any scalar function. We
also assume that in the left-hand side every operator acts on everything to
its right. We consider now matrix elements of one of these operators between scalar functions transforming according to the columns of irreducible
representations:
OT φ(k, mk ) =
mq
OT φ(j, mj ) =
mr
and
k
Dm
φ(k, mq )
q mk
j
Dm
φ(j, mr )
r mj
φ(j, mj ), Qli φ(k, mk )
As both Qli and φ(k, mk ) transform according to representations of the symmetry group, the wave function Qli φ(k, mk ) will be in the subspace transforming according to the direct product representation D l ⊗ D k . The representation constructed over the set of basis function Qli φ(k, mk ), i = 1, ..., ll ,
mk = 1, ..., lk will be in general reducible, according to D l ⊗ D k = ⊕s ns D s ,
and the function Qli φ(k, mk ) will occur in the subspace of one or more of
these irreducible representations. Thus, according to our previous theorem,
CHAPTER 1. THE KRONECKER PRODUCT
21
this function will have non-vanishing matrix element only with wave functions transforming according to the same irreducible representations. This
means that a matrix element involving a final state wave function transforming according to a representation D j which is not contained in the direct
sum decomposition of the representations D l ⊗ D k , is certainly vanishing
(selection rule theorem). Notice that, to apply this selection rule theorem to a concrete example, only the Clebsch-Gordan series is needed, so
that the vanishing of a matrix element can be determined with certainty
and using symmetry arguments only. Thus, group theory forbid, with absolute certainty, certain matrix elements. Those which are not forbidden by
symmetry can still be vanishing by some reason other than symmetry. This
selection rule theorem is, however, the weakest form of a more general matrix element theorem, known as the Wigner-Eckart-Koster theorem (G.F.
Koster, Phys. Rev. 109, 227 (1958)). In fact, if we consider more closely the
procedure adopted to analyze the matrix element, we will notice that the
wave function Qli φ(k, mk ) enters the subspace transforming according to D j
exactly nj times, nj being specified by the general formulas for the decomposition of product representations. The coefficients of the linear combination determining the various basis functions, for which the representation
D l ⊗ D k is in block form, are the Clebsch-Gordan coefficients and can be
determined by group theoretical arguments (for SO(3) see Cartan’s algorithmus). Clearly, the basis function Qli φ(k, mk ) will enter these basis functions
with given Clebsch-Gordan coefficients. Thus, the matrix element will be a
linear combination of nj Clebsch-Gordan coefficients C(l, k, j, ml , mk , mj , r).
Notice that for SO(3) nj = 1 and only one coefficient is needed to describe
matrix elements. In summary: if the irreducible representation D j occurs nj
times in the direct sum decomposition of D l ⊗ D k , then
φ(j, mj ), Qli φ(k, mk )
= a1 · C(l, k, j, ml , mk , mj , 1)
+ a2 · C(l, k, j, ml , mk , mj , 2)
+ ......
+ anj · C(l, k, j, ml , mk , mj , nj )
The coefficients ai must be calculated explicitely.
Example 1: Spin polarized excitation in a spherically symmetric potential (or at the Γ-point of a cubic lattice) by circularly polarized light.
Consider the optical transition between p and s states. The operator produc · r). If we choose lineraly
ing a transition in an atom is proportional to (A
polarized light along z and circularly polarized light in the (x − y)-plane, we
obtain the operators A0 ≈ z ≈ Y10 and A± ≈ x ± iy ≈ Y1±1 . These operators
CHAPTER 1. THE KRONECKER PRODUCT
22
transform according to the representation D 1 of SO(3). As a consequence,
p → s, p → p and p → d transitions are optically allowed. The p → p transition is forbidden if one consider that D 1 -wave functions are uneven with
respect to the parity operation, so that an optical transition is only allowed
between states of different parity (if the inversion is a symmetry element).
The sought for matrix elements are
< 0, 0 | Y1,i | Y1,j >
i = 1(−1) for right(left)-hand circularly polarized light and j = 1, 0, −1.
The states on the right-hand side are all possible initial states and the state
Y0,0 is the final state of the p → s transition. To find the value of these
matrix elements we search for the linear combinations of product functions
| 1, m >| 1, m > which transfom according to the D 0 . There is only one
linear combination, which can be read out from the table for Clebsch-Gordan
coefficients given e.g. in the book by E.U. Condon and G.H. Shortley, ”The
Theory of Atomic Spectra”, Cambridge Univeristy Press, 1935. From Table
2 on p.76 we obtain
| 0, 0 >=
1
1
1
| 1, −1 >| 1, 1 > −
| 1, 0 >| 1, 0 > +
| 1, 1 >| 1, −1 >
3
3
3
From this last equation we deduce that the only non-vanishing matrix elements are
< 0, 0 | Y1,−1 | Y1,1 >=< 0, 0 | Y1,1 | Y1,−1 >=
1
3
In the spirit of the Wigner-Eckart theorem, these non-vanishing matrix elements are determined up to a common constant, which includes the details
of the radial part of the wave function. If we consider now the spin-orbit
split states p3/2 and p1/2 constructed by means of the linear combination of
p = 1 wave functions multiplied by spin functions, we obtain the scheme
shown in the figure. Accordingly, for e.g. left hand circularly polarized light
(Y1,−1 ) there are two possible transitions from the p3/2 states and one from
the p1/2 state. The relative strength of the transition from the p3/2 leading to
the wave function with ”up” spin and of the one leading to ”down” spins is
3. Accordingly, the spin polarization of the electrons excited into the s- state
is
(3 − 1)
< σz >=
= 50%
(3 + 1)
If these electrons are in some way extracted, we will have a spin polarized
electron beam. The spin polarization is an ideal quantity for applying the
CHAPTER 1. THE KRONECKER PRODUCT
23
Figure 1.3: Level scheme for an optical transition at the Gamma point of
GaAs.
WE-theorem, because all details other than symmetry cancel out completely.
Example 2: spin polarization of the optical transitions along the Λ-direction.
We refer to the figure above describing the spin-split bands along the Λdirection. The opical transitions indicated for circularly polarized light produces polarized electrons, as one can easily derive using the WEK-theorem.
There is, however, something very unphysical in this level scheme: the spin
polarization achieved is 100%, albeit with opposite sign when starting from
the Λ6 and Λ4+5 band. Upon approaching the Γ-point, both bands come
close, so that one must assume that both are excited. As the matrix element
for the excitation is the same, we obtain the unphysical result that the net
spin polariztion goes to zero when the Γ point is approached, although our
calculation at the Γ-point gives a spin polarization of 50%. Clearly, there is
something wrong in the level scheme. The solution to this contradiction is
that bands with the same symmetry – in this case Λ6 symmetry, are allowed
to hybridize, in particular when they are close in energy. This means that
they are allowed to contain orbital components coming from different single
group symmetries, provided these orbital components end up to have the
same double group symmetry when the spin is introduced. As we approach
the Γ-point, the two Λ6 bands originating from the single group irreducible
representation Λ1 and Λ3 come also close together, so that we might hybridize
them. The recipe for determining the degree of hybridization is given by the
requirement that the net spin polarization achievd when both Λ6 and Λ4+5
are simoultaneouly excited is 50%. This is obtained by modifying the level
scheme as in the following figure.
CHAPTER 1. THE KRONECKER PRODUCT
24
Figure 1.4:
Figure 1.5: Level scheme for optical transitions including hybridized bands.
Chapter 2
Energy bands in solids
So far we have assumed that the discrete symmetry observed in crystals can
be dealt with using point groups, i.e. symmetry groups which leave one point
invariant. We will soon see that this approach is a useful one in many cases.
However, strictly speaking, a crystal consists of an infinite regular array of
identical unit cells, and translations – which move all points in space – might
be symmetry operations as well. Thus, we must consider so called space
groups.
2.1
The translation group
A crystal has the fundamental property of remaining unchanged under the
translation tn = n1a1 + nea2 + n3a3 , where the subscript n on t indicates
a collection of three integers n1 , n2 , n3 and ai are the primitive translation
vectors. The translation operation can be indicated by a symbol introduced
by Seitz, (E | tn ), where R = E indicate the identity operation in the general
mapping r → rt = Rr + tn . In order for the functions defined over the crystal
to be normalizable (square integrable), we consider the crystal to have finite
dimensions and to terminate at the planes r ·a1 = N1 , r ·a2 = N2 , r ·a3 = N3 ,
Ni being large but finite integer numbers. In order to preserve translational
symmetry, we impose the cyclic (Born-von Karman) boundary conditions
(E | t(N1 ,0,0) ) = (E | t(0,N2 ,0) ) = (E | t(0,0,N3 ) ) = (E | 0)
The collection of elements with this boundary condition constitutes a finite
group T with as many elements as the number of unit cells of the crystal,
i.e. g = N1 · N2 · N3 : in fact (E | tn ) + (E | tm ) = (E | tn + tm ), which is
again an element of the set, even if the sum of the two vectors lie outside the
crystal, in virtue of the boundary condition. The group is Abelian, because
25
CHAPTER 2. ENERGY BANDS IN SOLIDS
26
all transformations consisting of pure translations commute, so that each element is a class on its own and the group has only one-dimensional irreducible
representations. To find the irreducible representations, notice that the full
translation group T is the direct product of the three translation groups T1 ,
T2 and T3 . This groups have order Ni and elements n1ai , n2ai , ......nsai , ...Niai ,
with nsai = (ai )ns , i.e. they form a cyclic groups. The matrix element of the
irreducible representation D pi of the element nsai is,
e
−i·2π
pi ·ns
Ni
with pi = 0, ..., Ni−1 labeling the irreducible representations. As a consequence, the translation (E | tn ) is represented by
2π
2π
2π
−i( N
p1 ·n1 + N
p2 ·n2 + N
p3 ·n3 )
e
1
2
3
the set of integers p1 , p2 , p3 labeling an irreducible representation. We can
simplify this formula by introducing the following notation. Define the basic
vectors of a fictional reciprocal lattice b1 , b2 , b3 by aibj = 2πδij , i, j = 1, 2, 3,
so that explicitly
a2 × a3
a1 · (a2 × a3 )
a3 × a1
= 2π
a1 · (a2 × a3 )
a1 × a2
= 2π
a1 · (a2 × a3 )
b1 = 2π
b2
b3
then define the so called allowed k-vectors
k = k1b1 + k2b2 + k3b3
with kj = pj /Nj . Because of the special definition of the basis vectors of the
reciprocal lattice, we have the important equation
k · tn = 2π p1 · n1 + 2π p2 · n2 2π p3 · n3
N1
N2
N3
so that the element (E | tn ) is represented in the k-irreducible representation
by
D (k) (E | tn ) = e−i·k·tn
where now the N irreducible representations are labeled by the N allowed k
vectors. These representations are clearly unitary. The N irreducible representations of T are described by the allowed k vectors. These k vectors can
CHAPTER 2. ENERGY BANDS IN SOLIDS
27
Figure 2.1:
be imaged as laying on a very fine lattice within and upon three faces of the
parallelepiped having edges b1 , b2 , b3 that is shown in the figure. With the
basis vectors of the reciprocal space one can define a set of lattice vector in
m = m1b1 + m2b2 + m3b3 where mi are integers.
the reciprocal lattice by K
Notice that
e±i·Km ·tn = 1
so that two k-vectors differing by a reciprocal lattice vector label the same
irreducible representation. Using this property, it is possible to redefine the
allowed k-region into a more symmetric polyhedron in reciprocal space, which
is called the first Brillouin zone, and consists of all points of k space that
lie closer to k = 0 than any other reciprocal lattice points. Its boundaries
are therefore the planes that are perpendicular bisectors of the lines joining
the point k = 0 to the nearer reciprocal lattice points. Thus, only the first
Brillouin zone is required to label completely all irreducible representation
of T (and ultimately all electronic states of a crystal). In the following, we
give the first Brillouin zones of important lattices and indicate the label used
to indicate special point in the Brillouin zone, which we will discover later
to have special extra symmetries (the present labeling was introduced by
Bouckaert et al., Phys. Rev. 50, 58 (1936).
Body centered cubic lattice.
For this lattice,
b1 = 2π (0, 1, 1)
a
2π
b2 =
(1, 0, 1)
a
b3 = 2π (1, 1, 0)
a
The Brillouin zone is shown in the figure. The position vectors of the high
CHAPTER 2. ENERGY BANDS IN SOLIDS
28
Figure 2.2:
symmetry points (see later) marked are as follows:
k = (0, 0, 0) → Γ
k = π (2, 0, 0) → H
a
k = π (1, 1, 0) → N
a
k = π (1, 1, 1) → P
a
Face centered cubic lattice.
For this lattice,
b1 = 2π (−1, 1, 1)
a
2π
b2 =
(1, −1, 1)
a
b3 = 2π (1, 1, −1)
a
The Brillouin zone is shown in the figure. The position vectors of the high
symmetry points (see later) marked are as follows:
k = (0, 0, 0) → Γ
k = π (2, 0, 0) → X
a
π
k = (3/2, 3/2, 0) → K
a
k = π (1, 1, 1) → L
a
CHAPTER 2. ENERGY BANDS IN SOLIDS
29
Figure 2.3:
Basis functions
Let φ(r) be a scalar function in the Hilbert space of the crystal and (E |
t) be a transformation mapping any vector r into rt = r + t. Under this
transformation, the scalar function φ will be transformed into the scalar
.
function φt = O(E|t) φ, defined by
O(E|t) φ(r) = φ(r − t)
If φ is a basis function of a irreducible representations of T , φ will obey the
following equation:
O(E|tn ) φ(r) = φ(r − tn ) = e−ik·tn φ(r)
By multiplying on the both sides with e−ik(r−tn ) we obtain
e−ik(r−tn ) φ(r − tn ) = e−ik·r φ(r)
.
which shows that e−ik·r φ(r) = u(r) is a function invariant under translations
(a periodic function). Thus, the basis functions of the irreducible representation of the translation group can be written in the familiar form first obtained
by Bloch
φ(r) = eik·r u(r)
where u(r) = u(r − tn ). This equation is known as Bloch theorem, which
in this case has been derived from symmetry properties alone. The wave
functions are called Bloch functions. As the Hamiltonian H of a crystal commute with the operator O(E|tn ) , its eigenfunctions must be Bloch functions. In
CHAPTER 2. ENERGY BANDS IN SOLIDS
30
general, there is an infinite set of energy eigenfunctions and eigenvalues corresponding to each irreducible representation of the group of the Schroedinger
equation and hence to each allowed k vector. Eigenfunctions and eigenvalues
corresponding to the vector k can be indicated by φn (k, r) and En (k), where
the index n labels the members of the set of energies corresponding to the
wave vector k. The set of energy eigenvalues En (k) corresponding to a particular n are said to form the n−th-energy band, and the set of energy bands
is said to constitute the band structure. To visualize the band structure, it
is convenient to consider one at a time the axes of the Brillouin zone that
join the high symmetry points and for every allowed k vector on each axis
to plot the energy levels En (k). A typical example of such a plot is shown
in the figure, which gives the energy bands along ∆ and Λ in fcc Cu. The
occurrence of degenerate values and the apparent opening of gaps within a
band of a given index is a consequence of the rotational symmetry which has
been so fare neglected.
In the one electron approximation on which the whole theory is based, the
Pauli exclusion principle implies that no two electrons can ’occupy’ the same
electron state, here specified by an allowed k vector, a band index n and
a spin quantum number that can only take two possible values. It follows
that each energy level En (k) can ’hold’ two electrons and hence each energy
band can hold 2N electrons. If there are V valence (or conduction) electrons
per atoms, and A atoms per unit cell, there will be NV A electrons in the
large basis block of the crystal which will therefore require 12 V A bands to
hold them. In the ground state of the system, all energy levels En (k) will be
doubly occupied up to a certain energy EF , the Fermi energy and all levels
above this energy will be unoccupied. The surface in k space defined by
EF = En (k)
is called the Fermi surface. If one and only one band contains the Fermi
energy, and all other are entirely above or below it, then the Fermi surface
merely consist of one sheet. If no band contains the Fermi surface, as happened for insulators and semiconductors, there is no Fermi surface. In all
other cases, the Fermi surface consists of several sheets, and it is the distribution of energy levels near the Fermi energy that largely determines the
electronic properties of a solid.
2.2
Space groups
A crystal structure is fully specified by the primitive vectors a1 ,a2 ,a3 of the
translation group and by the basis of vectors d1 , d2, ..., dj which determine the
CHAPTER 2. ENERGY BANDS IN SOLIDS
31
Figure 2.4:
position of the atoms in the unit cell. There are symmetry operations which
leaves the crystal unchanged. Besides the translation symmetry operations,
proper or improper rotations, followed by an appropriate translation, which
is not necessarily a primitive translation, can also transform a crystal into
itself. The symmetry elements can be denoted by (R | a), where R is a real
orthogonal matrix indicating a proper or improper rotational part, and a is
an appropriate translation. All possible translation vectors associated with R
have the form a = tn + fR , where tn is the translation defined in the previous
section and fR is a fractional translation required for some rotations. The
multiplication of two elements is defined as
(R | a)(S | b) = (RS | Rb + a)
CHAPTER 2. ENERGY BANDS IN SOLIDS
32
(R | a)−1 = (R−1 | −R−1a)
Provided the result of the multiplication also leaves the crystal unchanged,
the set of symmetry transformation form a group, which is called the space
group. There is a restriction on the possible rotations R that can occur in
the operations of the space group, namely that if (R | tn ) is a member
of the space group and tn is a primitive translation, then Rtn must also
be a primitive translation. This follows because if (R | tn ) is a member of
the group, so is (R | tn )(E | tn )(R | tn )−1 = (E | Rtn ) which is a pure
translation and therefore must be a primitive one. Let us find out which
rotational symmetries are compatible with the translational symmetry. Let
A (see the Figure) be a point of a crystal lattice, and let us assume that it
Figure 2.5:
has a n-fold rotational axis passing though it. In B there is another lattice
point, and A and B can be joined by a primitive translation vector. Let, for
simplicity, the rotational axis be perpendicular to the plane formed by A and
B. We perform now a rotation by ϕ = 2π/n about A. This bring B into B .
Because B is equivalent to A there is also a n fold axis passing though B,
and a rotation by −ϕ brings A into A . As the rotation is a symmetry of
the lattice, A and B are two lattice point, and their distance must be pa, p
being some integer and a the lattice constant. From the figure we determine
the following trigonometric equation:
a + 2a sin(ϕ −
π
) = a − 2a cos ϕ = pa
2
i.e.
1−p
2
As | cos ϕ |≤ 1, we can only allow p to be 3, 2, 1, 0. These values lead to the
possible n values being 2, 3, 4, 6. A crystal can only have two-three-fouror six fold rotational axes. Notice that the rotational part of a space group
constitute a group by itself: it is therefore one of the point groups. Because of
the restriction on the rotational symmetry, only 32 point groups are allowed
cos ϕ =
CHAPTER 2. ENERGY BANDS IN SOLIDS
33
as rotational part of a space group. Space groups G having the same point
group G0 belong to the same crystal class, and there are 32 different crystal
classes. In the classification of Schönflies, a space group is denoted by the
Schönflies symbol for its point group together with a superscript. Thus, for
example the space group of the face centered cubic structure with point group
Oh is denoted Oh5 . The diamond structure belong to the same class and is
denoted Oh7 (the superscript is rather arbitrary). Notice that the restrictions
on rotations impose also corresponding restrictions on possible lattice vectors,
and, in fact only 14 different lattices are allowed. They are known as the
Bravais lattices. A more detailed description of the crystallography of space
group is given in the book by Landau Lifshitz on ’Statistical Physics’, Part
I.
Example: The operations of the space group Oh7 .
The diamond lattice, see figure, is invariant for translations (E | tn ) with
Figure 2.6:
tn = i niai and
a
a1 = (0, 1, 1)
2
a
a2 = (1, 0, 1)
2
a
a3 = (1, 1, 0)
2
where a is the length of the cube edge. The first Brillouin zone is the one of
the face centered cubic lattice. There are two equal atoms in the unit cell in
the positions
d1 = (0, 0, 0)
a
d2 = (1, 1, 1)
4
CHAPTER 2. ENERGY BANDS IN SOLIDS
34
The diamond lattice can be thought as consisting of two interpenetrating
face-centered cubic sublattices displaced with respect to each other by the
vector d2 ≡ f. The point group of the diamond lattice is the cubic group Oh .
Notice that some of the symmetry operations of the cubic group appear in
the space group associated with the fractional translation
1
a
f = (1, 1, 1) = (a1 + a2 + a3 )
4
4
By choosing a lattice point as the origin of a cubic coordinate system we obtain for the symmetry operations the explicit expressions given in the table.
2.2.1
The irreducible representations of space groups
We now set up to construct all irreducible representations of a space group.
The problem is complicated by two facts: first, the presence of fractional
translations associated with some rotation. If all fractional translations are
zero, the space group is said to by symmorphic, otherwise the space group
is non-symmorphic. We will see that constructing the irreducible representations of symmorphic space groups is somewhat easier. The second is that even
in the case of symmorphic space groups, the operations of the point group
G0 do not necessarily commute with the primitive translations, so that in
general G = T ⊗ G0 .
We now look at an invariant subspace carrying an irreducible representation of the space group. If G coincides with T , the basis functions are Bloch
functions, and can be written as φ(k, r) = u(k, n)eikr . u(k, n) are invariant
under primitive translations. The index n signifies the possibility of the existence of linearly independent periodic functions having the same symmetry
under translations k (n is the band index). In the case of G = T all irreducible representations are one-dimensional. If we introduce extra symmetry
elements, the one dimensional spaces might no longer be invariant under the
operation of these new elements, i.e. the various Bloch functions belonging to
different k and n are mixed by these extra symmetry operations, and essential
degeneracies occur. These enlarged spaces are the new invariant subspaces
sought for. For instance, the operation (R | a) transform the Bloch function
of vector k into a Bloch function of vector Rk. In fact,
−1
−1
O(R|k) φ(k, r) = eik·(R r−R a) u(R−1r − R−1a)
−1
a
= eiRk·r e−ikR
= eiRk·r u1 (r)
u(R−1 (r − a))
CHAPTER 2. ENERGY BANDS IN SOLIDS
Type
C1
C42
C4
C2
C3
Operation
(E | 0)
C2z
C2x
C2y
−1
(C4z | f)
(C4z | f)
−1
(C4x
| f)
(C4x | f)
−1
(C4y
| f)
(C4y | f)
(C2xy | f)
(C2xz | f)
(C2yz | f)
(C2xȳ | f)
(C2x̄z | f)
(C2yz̄ | f)
−1
C3xyz
C3xyz
−1
C3xȳz
C3xȳz
−1
C3xȳz̄
C3xȳz̄
−1
C3xyz̄
C3xyz̄
Coordinate
xyz
x̄ȳz
xȳ z̄
x̄yz̄
a
ȳ + 4 x + a4 z +
y + a4 x̄ + a4 z +
x + a4 z̄ + a4 y +
x + a4 z + a4 ȳ +
z + a4 y + a4 x̄ +
z̄ + a4 y + a4 x +
y + a4 x + a4 z̄ +
z + a4 ȳ + a4 x +
x̄ + a4 z + a4 y +
ȳ + a4 x̄ + a4 z̄ +
z̄ + a4 ȳ + a4 x̄ +
x̄ + a4 z̄ + a4 ȳ +
zxy
yzx
zx̄ȳ
ȳz̄x
z̄ x̄y
ȳzx̄
z̄xȳ
yz̄x̄
Type
I
IC42
a
4
a
4
a
4
a
4
a
4
a
4
a
4
a
4
a
4
a
4
a
4
a
4
IC4
IC2
IC3
35
Operation
(I | f)
(IC2z | f)
(IC2x | f
(IC2y | f
−1
IC4z
IC4z
−1
IC4x
IC4x
−1
IC4y
IC4y
IC2xy
IC2xz
IC2yz
IC2xȳ
IC2x̄z
IC2yz̄
−1
(IC3xyz
| f)
(IC3xyz | f)
−1
(IC3xȳz
| f)
(IC3xȳz | f)
−1
(IC3xȳz̄
| f)
(IC3xȳz̄ | f)
(IC3xyz̄ | f)
(IC3xyz̄ | f)
Coordinate
x̄ + a4 ȳ + a4 z̄ +
x + a4 y + a4 z̄ +
x̄ + a4 y + a4 z +
x + a4 ȳ + a4 z +
y x̄z̄
ȳxz̄
x̄z ȳ
x̄z̄y
z̄ ȳx
z ȳ x̄
ȳx̄z
z̄y x̄
xz̄ ȳ
yxz
zyx
xzy
a
z̄ + 4 x̄ + a4 ȳ +
ȳ + a4 z̄ + a4 x̄ +
z̄ + a4 x + a4 y +
y + a4 z + a4 x̄ +
z + a4 x + a4 ȳ +
y + a4 z̄ + a4 x +
z + a4 x̄ + a4 y +
ȳ + a4 z + a4 x +
a
4
a
4
a
4
a
4
a
4
a
4
a
4
a
4
a
4
a
4
a
4
a
4
CHAPTER 2. ENERGY BANDS IN SOLIDS
36
where u1 (r) is some periodic function, in general different from u(r).
The general strategy to find the invariant subspaces starts with picking out of
the point group those elements which rotate k into itself or into an equivalent
one
Rkk = k + h
h being a reciprocal lattice vector (including 0). This group is a subgroup of
G0 and is called the small point group of the wave vector k and is indicated by
G0 (k). All the symmetry operations in the space group of the form (Rk | a)
constitute a group Gk called the little group of k or simply the group of k.
Notice that at a general point of the Brillouin zone only the identity satisfies
the above condition. However, there are special points, lines and planes of
the Brillouin zone where G0 (k) is larger than the identity operation. These
locations are so called high-symmetry locations (or simply symmetry locations). For example, the Brillouin zone of the body centered cubic lattice has
symmetry points Γ, H, N and P , symmetry axes ∆, Σ, Λ, and G and the
symmetry planes are those containing two symmetry axes. Among the basis
functions for an irreducible representation of the space group, we chose the
subset of functions with a given k vector. As a consequence of the transformation property under the action of (R | a), this subset form a basis of a
irreducible representation of the group of k. The irreducible representations
of the group of the k vector (whose basis functions are Bloch functions of
vector k ) allow the classification of the eigenvalues at any given k vector.
case 1: k inside the first Brillouin zone.
In this case the small point group of k is selected from those operations of
G0 for which Rk = k.
Theorem: From any irreducible representation D (p) (Rk ) of the small point
group G0 (k) an irreducible representation D k,p of the little group of k is obtained by associating to every element (Rk | a) of the group Gk the matrix
e−ik·a D p (Rk ).
Proof: we have to show that these matrices follow the same multiplication
rule as the elements of the group Gk . The matrix product corresponding to
the left-hand side of
(Rk | a)(Sk | b) = (Rk Sk | Rkb + a)
is
e−ik·a D p (Rk )e−ik·b D p (Sk ) = e−ik·(a+b) D p (Rk Sk )
The matrix corresponding the right-hand side is
−i(R−1kb+k·a)
e−ik·(Rk b+a) D p (Rk Sk ) = e
k
D p (Rk Sk )
CHAPTER 2. ENERGY BANDS IN SOLIDS
37
The phase factors of the above expressions are identical because Rk−1k = k.
Case 2: k at the surface of the Brillouin zone. The same procedure can be
applied if all f vectors are zero, i.e. for symmorphic space groups. However,
for non-symmorphic space groups, it might happens that some points at
the surface are transformed to k + h. e−ih·b is one if b is some primitive
translation vector, but, in general e−ih·f = 1. Then the phase factors in the
expressions above are different and the above matrix is not a representation.
The procedure of finding the irreducible representations in this special cases
is outside the scope of this lecture.
In both cases, considering the group of the wave vector, we are able to
construct a number of irreducible representation of this group which obey
the equation
(lp )2 = gk
p
with gk being the order of the small group of k, i.e. G0 (k). So far, we have
considered only those symmetry operations which transform k into itself.
In general, the effect of a point group symmetry is to send k into a star
of vectors k1 , k2 , ..ks . The eigenvalues at all points of the star are the same
and the eigenfunctions at all points of the star are obtained by applying the
symmetry operations of the space group to the eigenfunctions of the vector
k. It is convenient to define the restricted volume inside the Brillouin zone
as a volume containing all points that do not belong to the same star. The
volume of this restricted region is the volume of the Brillouin zone divided by
g0 , the order of G0 . The basis functions φ(k, (1)), φ(k, (2), ..., φ(k, (lp ) of the
irreducible representation D k,p of Gk and the corresponding eigenfunctions at
the vectors of the star of k constitute a basis for the irreducible representation
k,p
Dspace
of the space group G. Its dimension is sk · lp , where sk is the number
of points in the star of k. We now show that the so constructed irreducible
representations represent all possible irreducible representations of the space
group. From
(sk · lp )2 =
(sk )2 · gk
k∈res.vol.,p
k∈res.vol.
and sk gk = g0 we obtain
k∈res.vol.
sk · g 0 = g 0 ·
= N · g0
k∈Brill.zone
which is exactly the order of the space group.
Summarizing: This construction the invariant subspace shows that the irreducible representations of G are specified by an allowed k-vector and a label
CHAPTER 2. ENERGY BANDS IN SOLIDS
38
p, specifying an irreducible representation of Gk . To classify the energy levels
along one specific k, it is enough to know the irreducible representation of
Gk . This group consists of element of the type (Rk | fRk ) and the multiplication of these elements with the pure primitive translations. The lp × lp
representation matrices of the elements (Rk | fRk ) within the Brillouin zone
can be easily calculated following our discussion. Those for k on the surface
of the Brillouin zone require more work. Once these matrices are known,
their characters χ(p,k) (Rk | fRk ) can be obtained taking the trace of the diagonal matrix element. The matrices for the elements (Rk | fRk + tn ) obey
the following equation:
D p,k (Rk | fRk + tn ) = e−ik·tn D p,k (Rk | fRk )
Correspondingly, their characters can be obtained by multiplying the char
acters χ(p,k) (Rk | fRk ) with e−ik·tn . Thus, in order to specify the character
table of the group Gk is enough to give the character table of the h elements
(Rk | fRk ). For symmorphic space groups, this is the character table of the
point group G0 (k).
Exercise. Show that the representations D p,k of Gk (Case 1) are irreducible.
Example: some irreducible representations of Oh7 .
Point Γ. This is the point at the origin of the first Brillouin zone. The small
point group of k is the entire point group Oh . Since we are inside the Bril louin zone and e−ik·f = 1, the characters of the irreducible representations
at Γ are simply given by the characters of the point group Oh , without any
modification.
Line Λ. k = πa (λ, λ, λ), λ < 1. This is a point on the line joining the center
of the Brillouin zone to the midpoint of an hexagonal face. The small point
group of k is made up by the operations whose rotational parts interchange
(x, y, z) among themselves, because these are the only operations which leave
k invariant. The small point group of k is thus C3v . Since no fractional translation is associated with the elements of C3v , the characters of the irreducible
representations at Λ are the characters of the group C3v without any modification.
Line ∆ (k = 2π
(δ, 0, 0), with δ < 1). This is a point on the line joining the
a
center of the Brillouin zone to the midpoint of the square face. the small
point group of k is given by the operations whose rotational part leave x
unchanged. The small point group of k is thus C4v . The irreducible representations at ∆ are obtained by multiplying the characters of the irreducible
π
representations of the group C4v by the phase factor e(−ik·f ) = e−i( 2 ·δ) .
CHAPTER 2. ENERGY BANDS IN SOLIDS
Irr.Rep.
(E | 0)
∆1
∆1
∆2
∆2
∆5
1
1
1
1
2
2.3
−1
(C2x | 0) (C4x
| f)
(C4x | f)
1
1
1
1
-2
39
(IC2x | f)
(IC2y | f)
π
e−i( 2 ·δ)
π
e−i( 2 ·δ)
π
−e−i( 2 ·δ)
π
−e−i( 2 ·δ)
0
(IC2xy | 0)
(IC2yz̄ | 0)
π
e−i( 2 ·δ)
π
−e−i( 2 ·δ)
π
e−i( 2 ·δ)
π
−e−i( 2 ·δ)
0
1
-1
-1
1
0
Symmetry adapted plane waves
We would like now to address the problem of calculating, at least in a qualitative way, the band structure of real crystals. A possible approach is to
start with symmetry adapted plane waves and introduce a crystal potential.
This will not be very precise in general, but will give at least a feeling for the
difficulties and tasks lying behind a real band structure calculation.
2.3.1
The one-dimensional lattice
We consider a chain of N-atoms with lattice constant a. The fundamental
translation group has N elements n · a, with n = 0, ..., N − 1. The group has
N irreducible representations. In the representation with label p the n-th
−i·2π·p·n
element has the character e N , with p = 0, 1, ..., N − 1. This establishes a
reciprocal lattice vector of length 2π
hosting the allowed k-vectors k = N2πa · p.
a
, π ].
The first Brillouin zone of the 1-dimensional lattice is the intervall [ −π
a a
The point group G0 of the system is Ci , while the small point group of any
vector inside the intervall is just E. This means that – at general k inside
the intervall – k is the only quantum number labeling the energy levels and
the symmetry adapted wave functions are of the form eik·x u(x), where u(x)
is a periodic function. As possible basis functions we might decide to choose
1 ik·x iG·x
e ·e
Na
with G = n · 2π
, n ∈ Z being a reciprocal lattice vector. For point inside
a
the intervall, the inversion operation tranfsorms k into −k, which together
with k forms the star of k in the one dimensional lattice. As the energy levels
belonging to the star of k are degenerate, we obtain that energy levels in the
one dimensional lattice are at least double degenerate.
CHAPTER 2. ENERGY BANDS IN SOLIDS
40
The representation theory of the k point at the end of the intervall (including k = 0) must be modified because the small group of k contains both
operations E and I. For this reason, the energy levels at the edge of the one
dimensional Brillouin zone are classified according to the two one-dimensional
irreducible representations of Ci , namely the one carrying symmetric and the
one carrying anti-simmetric wave functions.
We are now able to calculate the empty lattice band structure (also
free electron band structure), which sets the crystal potential to zero and
only consider the kinetic energy of the electrons. Depending of the choice of
G we can generate a set of empty lattice energy bands with energy
E=
h̄2
(k + G)2
2m
with G = 0, ± 2π
, ... The first band is the one characterized by G = 0, see
a
Figure 2.7:
the figure. The G = 0 band starts at Γ and ends at the point
π
a
with a dou-
CHAPTER 2. ENERGY BANDS IN SOLIDS
41
ble degenerate energy level with wave functions N2a cos πa x and N2a sin πa x.
as the next G-vector we produce a further band taking off from
Taking − 2π
a
the Brillouin zone edge, see the figure. This
band ends at k
= 0 with a doux and N2a sin 2π
x. The
ble degenerate band with wave functions N2a cos 2π
a
a
2π
general expression for the
next band is generated by G = a , and so on. The
wave functions at k = 0 is N2a cos n·2π
· x and N2a sin n·2π
· x, n = 0, 1, 2, ....
a
a
From this expression it is clear that the deepest lying level (k = G = 0) is
not degenerate, while the degeneracy of the higher lying
levels it two. The
π
2
]x and
general expression for the wave function at k = a is N a cos[ πa + n 2π
a
2
Na
sin[ πa + n 2π
]x, n = 0, 1, .....
a
The band structure calculated for the empty lattice using symmetry
adapted wave function is exact. However, the same functions can be used,
within in a perturbational approach, to find the first order correction caused
by a small crystal potential V (x). The most remarkable feature of switching
a small crystal potential is the lifting of the accidental double degeneracy at
the center and at the edge of the Brillouin zone. In fact, from general symmetry considerations, we expect that the double degenerate level splits into
two single degenerate energy levels, thus opening a gap between the various
bands at k = 0 and at k = πa . The energy of the levels at these two singular
points of the BZ can be calculated, in first order perturbation theory, to be
h̄2 2π 2 2
1 Na
( ) ·n +
dx · V (x)
2m a
4Na 0
1 Na
2π
±
dx · V (x) · cos(2n · )x
4Na 0
a
E± (k = 0) =
(the - sign-level is suppressed for n = 0) and
Na
h̄2 2π 2 2n + 1 2
π
1
( ) ·(
) +
dx · V (x)
E± (k = ) =
a
2m a
2
4Na 0
Na
1
2π
±
dx · V (x) · cos[(2n + 1) · ]x
4Na 0
a
(n = 0, 1, 2, ....). Thus, the gap is twice the Fourier-transform of the crystal
potential perturbing the empty lattice band structure.
2.3.2
The square lattice
The Figure shows the Brillouin zone of a plane square lattice of lattice constant a. The special points and lines of symmetry and the corresponding
small point group are indicated in the table.
CHAPTER 2. ENERGY BANDS IN SOLIDS
42
Figure 2.8:
k
(kx , ky )
kx , kx )
1
, kx )
2
kx , 0
1
,0
2
1 1
,
2 2
(0, 0)
Symbol G0 (k)
general
C1
Σ
C1,h
Z
C1,h
∆
C1,h
X
C2,v
M
C4,v
Γ
C4,v
Table 2.1: The special point and the lines of symmetry in the Brillouin zone
.
of the planar square lattice, in units of 2π
a
The next figure gives the stars of the various k vectors for a plane square
lattice. Every dashed vector is related to one solid vector by a reciprocal
lattice vector and is therfore not counted separately. The number of distinct
vectors in the star is given by s. Notice that the full point group G0 is C4v
and that the space group is symmorphic.
We now proceed toward mapping the empty lattice energy bands. The energy
versus k relation for a two-dimensional lattice may be represented by surfaces in the three-dimensional space kx , ky , E. Typically, one shows the cross
sections of these surfaces along variuos lines of symmetry of the Brillouin
zone. The first task is to construct symmmetry adapted wave functions from
the plane waves
1
ei(k+G)·r
φk,G = √
2
Na
CHAPTER 2. ENERGY BANDS IN SOLIDS
43
Figure 2.9:
is some reciprocal lattice vector with coordinates 2π (m, n), where m, n ∈
G
a
Z. We consider for instance, the ∆-direction. Ths small group of k consist
of [E, σx ] and thus has two irreducible representations, which we call ∆1 and
∆2 . Upon reflection with respect to the x axis in the BZ, states belonging to
∆1 remains invariant, while states belonging to ∆2 change sign. The enegy
bands generated by the numbers (m, n) have energy
h̄2 2π 2
( ) [(κ + m)2 + n2 ]
E(m, n) =
2m a
and the corresponding plane waves are
e
i·2π
[(κ+m)·x+n·y]
a
The lowest lying band is the one with m = n = 0 and has ∆1 symmetry. The
next band is charactereized by m = −1, n = 0 and has, again ∆1 symmetry.
The degeneracy of these bands is 1. These two bands are followed by a double
degenerate band corresponding to the indices (m = 0, n = 1) and (m = 0, n =
−1). This band has an accidental degeneracy involving a wave function with
∆1 symmetry and a wave function with ∆2 symmetry. The m = 1, n = 0 band
has ∆1 symmetry and has higher energy than the (0, 1)-band, except at the
Γ-point where the bands meet. The next reciprocal lattice vectors are the four
(1, 1) vectors. The (−1, 1) and (−1 − 1) bands are degenerate along ∆, their
energy being equal to the energy of the 0, 1, 0, −1 band at the X-point. The
resulting free electron band structure, compared with a calculation including
a realistic crystal potential, is given in the Figure. We now proceed analyzing
CHAPTER 2. ENERGY BANDS IN SOLIDS
44
Figure 2.10:
the Γ and X-points. Both have a larger symmetry group and a larger number
of irreducible representations. The plane wave corresponding to the lowest
lying energy level E = 0 is (0, 0)Γ . It is a constant and transforms according
to the irreducible representation Γ1 . The next fours waves (1, 0)Γ , (1̄, 0)Γ ,
(0, 1)Γ and (0, 1̄)Γ are degenearte with energy 1. They carry some irreducible
represenations of C4v , which can be found by decomposing the representation
constructed over these four plane waves. We find Γ1 ⊕ Γ3 ⊕ Γ5 . Using the
projector operator technique one can easily find the symmetryized linear
combination corrsponding to these components. For instance, the function
CHAPTER 2. ENERGY BANDS IN SOLIDS
Irr.Rep.
∆1 , Γ1
∆1 , Γ2
∆2 , Γ3
∆2 , Γ4
∆5 , Γ5
Γ1
∆1
X1
e C4 , C43
1
1
1
1
1
-1
1
-1
2
0
Γ2
∆2
X2
Γ3
∆1
X3
C2
1
1
1
1
-2
Γ4
∆2
X4
45
2m 2σ
1
1
-1 -1
1
-1
-1
1
0
0
Γ5
∆1 ∆2
Table 2.2: Compatibility table for the square lattice along the ∆-direction
with ∆1 symmetry is clearly
√
2π
2π
2π
2π
1
[ei a (x) + ei a (y) + ei a (−x) + ei a (−y) ]
2
4Na
The next four plane waves (1, 1)Γ, (1̄, 1)Γ, (1, 1̄)Γ and (1̄, 1̄)Γ belong to the four
fold degenerate energy level E = 2 and transform according to Γ1 ⊕ Γ4 ⊕ Γ5 .
The symmetry group at X is C2v = [E, mx , my , C42 ] and has four irreducible representations which we denote X1 , X2 , X3 , X4 for the square lattice.
We obtain the following set of degenerate plane waves at X:
1. the set (0, 0)X and (0, 1̄)X generating the represenations X1 ⊕ X3 with
E = 0.25
2. the set (0, 1)X , (0, 1̄)X , (1̄, 1)X , (1̄, 1̄)X transforming according to X1 ⊕
X2 ⊕ X3 ⊕ X4 with E = 1.25
3. and so on
The connection between the band structure along ∆ with the end-points
of the BZ can be directly read out in general terms from the compatibility
relations given in the Table.
We turn now to explore the aspects of the band strucure when a crystal
potential is switched on. The key to solve the full problem including crystal
potential is to use symmetrized linear combinations. Even few symmetrized
plane waves are quite successful in predicting the real band structure, although in some materials the real wave functions require many of them, see
the figure. In fact, it turns out that the energy bands of many simple metals,
CHAPTER 2. ENERGY BANDS IN SOLIDS
46
particularly the one-electron metals such as the alkalis, are not significantly
different from the empty lattice band structure, except for some general features wich can be easily worked out from symmetry arguments alone.
In fact, as we learned from the one-dimensional band strucure, accidental
degeneracies are removed when a potentiat is switched on. This mechanism
introduces typical gaps in the band structure. In addition, group theory also
allows to treat special crossing points (such as the one around E = 1.5 in
the empty lattice band structure). When the crystal potential is taken into
account, there will be, in general, non-vanishing matrix elements between
symmetrized wave function belonging to the same representation. This
means that the proper eigenfunctions must be constructed by taking linear
combinations of as many functions having the same symmetry as practically
possible. This phenomenon is known as hybridization and has the obvious
effect of lifting the degeneracy of band having the same symmetry. A hybridization gap arises. The ”birth” of an hybridization gap is illustrated in
the figure. Close to the crossing point the bands strongly hybridize, i.e. the
wave function close to the crossing point will contain a sizeable amount of
the waves belonging to the crossing bands. Away from the crossing point, according to perturbation theory, the hybridization decreases and the original
wave funtion will be restored. The consequence of the lifting of the crossing
is that, moving along the ”new” bands will produce a change of wave function from one type to the other. As the rule that bands having the same
symnmetry do not cross is a quite general one, we expect such hybridization
gaps to be introduced upon switching on the spin-orbit interaction as well,
when two bands belonging to two different single group representations end
up into the same extra-representation of the double group.
Figure 2.11:
CHAPTER 2. ENERGY BANDS IN SOLIDS
2.3.3
47
Free electron energy bands in three-dimensional
lattices
Using the same methods we can produce and label the empty lattice bands in
three-dimensional lattices, as illustrated in the following figure. More results
can be found in the monumental work by J.C. Slater, Quantum Theory of
Molecules and Solids, Volume 2, Mc Graw-Hill book company, New York,
1965.
Figure 2.12: Empty-lattice band structure for a bcc-lattice (right) and for
Na (left).
Chapter 3
Landau theory of phase
transitions
The Van der Waals theory of real gases contains a most important feature:
liquid and vapour can transform into each other along a well defined phase
transition line in the P − T -phase diagram. It is also well known experimentally that the solidification establishes a phase transition line as well, and
that there are important phenomena which are also realized by a phase transition, such as supercoductivity and magnetism. Landau established in 1937
(L. Landau, Phys.Z.Sowjet.11, 26 (1937) and ibidem, 545) that in some cases
the phase transitions proceeds by a continuous change of the thermodynamic
potential when the phase transition line is crossed. In this case the phase transition line is called the Curie line and the transition is a continuous one (also
called 2nd order phase transition, in contrast to first order phase transitions
where there is a discontinuous change of the thermodynamic potential and
of other relevant quantities). The main idea of Landau is to characterize the
system at the Curie point by mean of a symmetry group G0 containing those
symmetry operations which leave the system invariant. Landau showed that
G0 determines quite univocally the structure of the thermodynamic potential
during the phase transition. Thus, in this respect, Landau theory of phase
transition is a non-trivial application of group theory. In this third chapter
we will discuss the essential elements of this theory.
General arguments
To be concrete, we describe the system by a spatially dependent density
ρ0 (x, y, z) (in the case of magnetic system the quantity describing the system
is the effective current distribution j0 (x, y, z)). Let G0 be the symmetry group
48
CHAPTER 3. LANDAU THEORY OF PHASE TRANSITIONS
49
at the transition point (G0 can be one of the 230 space groups, but also some
continuous group like the full group of translations in a liquid or the full group
of rotations O3 in an Heisenberg exchange ferromagnet). We will consider only
those transitions where the state of the system changes continuously, i.e. ρ
changes continuously. A clear example of transition of this kind is represented
in Fig.1.
A small decrease of the maxima, which can occur in a continuous way
upon varying some parameter like the temperature, produces a drastic change
of symmetry, in this case a doubling of the translation period of the lattice.
Thus, while ρ changes continuolsly, the symmetry of the system can change
abruptly. A general feature of any function ρ(x, y, z) is that it can be decomposed into a linear combination of invariant subspaces transforming according
to an irreducible representation of G0 :
ρ(x, y, z) =
ηi,n ϕi,n
n,i
The index n runs over all irreducible representations Γn , the index i labels the
variuous partners transforming according to the n-dimensional representation
Γn . Without loosing generality, we require that the functions ϕi,n are real
functions and thus all matrix elements of Γn are real. Among the functions
ϕi,n there is one – which we call ρ0 – transforming according to the 1 representation of G0 . In other words: ρ0 is invariant with respect to any
operation of G0 . Thus, we can write
ρ(x, y, z) = ρ0 + (
) ηi,n ϕi,n
n,i
. where δρ = n,i ηi,n ϕi,n contains all irreducible representations with the exception of the 1 one. Notice that the function δρ has a lower symmetry than
G0 , because not all transformations of G0 leaves δρ invariant. We call the
symmetry group of δρ G. G is some subgroup of G0 . The thermodynamic
potential Φ of the body is some functional of ρ, Φ = Φ[ρ] and depends on
T and P . When P and T are given, Landau suggests that the form of the
CHAPTER 3. LANDAU THEORY OF PHASE TRANSITIONS
50
function ρ is determined from the condition that Φ[ρ] should have a minimum. This also determines the symmery G of the crystal at any moment,
because once the coefficients of the linear combination are established the
functions appearing in the equilibrium ρ are established and thus G. At the
transition point we require that the symmetry group is G0 : this means that
exactly at the transition point ρ = ρ0 and δρ = 0. It the transition has to
proceed in a continuous way, δρ – which is exactly zero at the transition point
– must assume very small values close to the transition point. Therefore the
thermodynamic potential can be expanded in powers of δρ close to the transition point. Individual terms of this expansion are integral operators of δρ.
To determine which terms appear in the expansion, we notice that Φ, as a
qunatity characterizing any physical property of the body, should remains
invariant under all possible coordinate transformations, in particular those
belonging to G0 :
Φ[ρ0 + δρ] = Φ[ρ0 + δρ ]
ρ0 is left invariant by G0 but δρ is changed. During this transformation,
the functions belonging to the same irreducible representation trasform into
themselves, i.e. the space remains invariant. Thus, we can consider a transfomation of G0 as acting on the coefficients of the linear combination in
the same way as it acts on the functions themselves. As a consequence, the
invariance property of Φ with respect to G0 means that the expansion of
Φ in powers of ηn,i should contain invariants of the relevant degrees constructed over the coefficients ηn,i . As one cannot construct linear invariants
from quantities transforming according to a given irreducible representations
(other than the 1 representation) (Exercise 1: prove this statement),
the expansion of Φ will start with a scalar and with invariant polynomials
of the second degree. We will prove later that the only polynomial of second degree build up from the coefficients transforming according to a given
irreducible representation is the sum of the squares of the coefficients. Thus,
Φ = Φ0 +
n
An (P, T )
2
ηn,i
i
the coefficients An depending on T and P . At the transition point the coefficienst ηn,i are all vanishing. For the minimum of Φ to be at ηn,i = 0 all An
must be non negatives. If at the transition point all coefficients are positive
they would remain positive in the vicinity of the transition point and no
symmetry change occurs. Thus, at least one of the coefficients must change
sign at the transition point and vanish at the transition point. Notice that
it is very improbable that more than one coefficients vanishes along a line
in the P − T plane. Such a simoultaneoous vanishing of two coefficients can
CHAPTER 3. LANDAU THEORY OF PHASE TRANSITIONS
51
only occur in an isolated point of the plane which represents the intersection of two line of continuous phase transitions. The change of sign of the
coefficients means that at one side of the transition point all coefficients are
positive and the value minimizing Φ is ηn,i = 0. At this side, the symmetry
group G0 is realized. On the other side, instead, a set of coefficients assume
finite – albeit small in the vicinity of the transition point – values, and the
subgroup G is realized. In the following, we will set all positive coefficients to
zero and retain only the irreducible representation which is realized on the
low symmetry side of the transition point. By introducing the quantities
η · γi = ηi
with i γi2 = 1 we define a unit vector with ”amplitude” η within a ndimensional space. This vector, carrying the Γn -irreducible representation of
the group G0 , appears below the transition point and is called the order
parameter of the system and the corresponding space is the order parameter
space. Depending on the dimension of Γn , the OP can have more than one
component. The appearance of an OP below the transition point establishes a
new symmetry group G. Which group G is realized below the transition point
depends on the dierction of the OP. The expansion of the thermodynamic
potential can be continued with the third, fourth,...,order terms and can be
written in general as
Φ = Φ0 + A(T, P )η 2 + η 3
α
Cα (P, T )fα(3) (γi ) + η 4
Bα (P, T )fα(4) (γi ) + .....
α
where fα(3) , fα(4) , .... are homogeneous polynomials of degree 3,4,..., which are
invariant under the operations of G0 . The sum over α takes into account
that there might be more than one invariant polynomial of a given degree.
However, terms of order 3 can be vanishing in some cases. At least one term
of order 4 do always exist, and is given be ( i γi2 )2 .
Before we address more technical problems related to group theoretical
aspects we would like to investigate the role of the third and fourth order
terms. Consider a thermodynamic potential that includes third order terms.
At Tc , the minimum corresponds to ηi = 0. Thus, for any value of ηi close
to ηi = 0 Φ − Φ0 must be positive. However, a third order term cannot be
always positive for any set of infinitesimal ηi : if f3 is positive for some set
corresponding to a given direction in the OP space, it will be negative for
the opposite direction. Thus, in the presence of a third order term, the value
ηi = 0 does not correspond to a minumum but to a saddle point of the potential. Hence, in order to ensure that the equilibrium state of the system at
Tc is indeed a minimum with values ηi = 0, there must be no third-degree
CHAPTER 3. LANDAU THEORY OF PHASE TRANSITIONS
52
term in the expansion of the potential. The presence of a third order invariant leads to the mimimum at Tc being at a finite value for ηi and thus to a
discontinuous (first order) phase transition. Thus, the Landau condition for
a phase transition being continuous is the absence of any third order invariant in the expansion of the thermodynamic potential. The absence of third
order invariants can be due to the impossibility to find 3rd degree invariant
polynomials for the group G0 . This absence, due to symmetry reasons, will
hold at any temperature and pressure. It might also be possible at some isolated point of the P − T -plane, that the coefficient of the third order term
vanishes ”accidentally”. In this case the Curie point is the intersection of the
lines A(T, P ) = 0 and C(T, P ) = 0 and is therfore an isolated point in the
P − T -plane.
In contrast to the third order terms, a fourth order term can be positive in
any direction and therefore can sustain a second order phase transition. The
fourth degree terms are essential in determining the actual direction of the
OP in the OP space: as the 2nd order term is independent of γi , these quantities must be determined by the requirement that the fourth oder terms be
minimized with respect to the γi . The amplitude η of the OP is determined
by minimizing globally Φ.
Invariant polynomials
Invariant polynomials play a central role in the Landau theory of phase transitions. The procedure for costructing invariant polynomials starts by considering the f -dimensional irreducible representation Γ of G0 which carry the
invariant subspace Λ defined by φ1 , φ2 , ..., φf . We discuss in depth the construction of the 2nd degree polynomials, the costruction of the higher degree
polynomials proceed in a similar way. The first step is the realization of the
Kronecker product representation Γ ⊗ Γ by the n2 basis functions φi φk . Its
character is
χ(Γ ⊗ Γ)(g) = (χ(g))2
The Kronecker product representation can be reduced to the direct sum of
two representations constructed over the f (f + 1)/2 functions φi φj + φj φi
and the f (f − 1)/2 functions φiφk − φk φi (i = k). Both sets of functions
are invariant with respect to the operations of the group and thus carry a
representation of the group. The first representation is called the symmetrized
2nd power representation of Γ and is denoted by [Γ2 ]. The corresponding
space is denoted by [Λ2 ] and the character of the representation is [χ2 ](g).
CHAPTER 3. LANDAU THEORY OF PHASE TRANSITIONS
53
[χ2 ](g) can be determined from the following equations:
ĝ(φi φk + φk φi ) =
gli gmk (φl φm + φm φl )
l,m
=
1
(gligmk + gmi glk )(φl φm + φm φl )
2 l,m
The character is
[χ2 (g)] =
< (φi φk + φk φi ), ĝ(φi φk + φk φi >=
i,k
1
(gii gkk + gkigik )
2 i,k
Because
gii = χ(g)
i
gik gki = χ(g 2 )
i,k
we arrive at the important formula
1
[χ2 (g)] = (χ(g)2 + χ(g 2 ))
2
which allows to express the characters of the symmetrized 2nd power representation of Γ by means of the characters of Γ. A similar expression can
be found for [χn ](g), i.e. for the character of the symmetrized n-th power
representation: for instance
1
1
1
[χ3 (g)] = χ(g 3 ) + χ(g 2 ) · χ(g) + χ3 (g)
3
2
6
1
1
1
1
1
[χ4 (g)] = χ(g 4 ) + χ(g 3) · χ(g) + χ2 (g 2 ) + χ(g 2 )χ2 (g) + χ4 (g)
4
3
8
4
24
These group theoretical results can be used to calculate the invariant polynomials – and in particular to determine the occurrence of third power terms
in the expansion of the theormodynamic potential – because the monomials of a given degree costructed over the components ηi transform as the
corresponding symmetric product functions. Thus, to find the presence of
invariant polynomials of the n-th degree one has to check for the occurrence
of the 1 -representation in the reduction of [Γn ]. If the 1 -representation is
absent, the corresponding invariant polynomial does not exist. Should the
1 representation occur nα times, then there are nα different invariant polynomials of the given degree. To find them, one can use the usual projection
CHAPTER 3. LANDAU THEORY OF PHASE TRANSITIONS
54
operator technique.
Exercise 2: Show that there is only one second degree invariant polynomial
and that it takes the form i ηi2 .
Exercise 3: Show that if the OP corresponds to a one dimensional nontotally symmetric IR, then the Landau condition is necessarily verified.
Exercise 4: For the group C3,v and C4,v find the invariant polynomials of
the third degree constructed with the basis functions belonging to the 2dimensional IR Λ3 .
Magnetic phase transitions
The most famous case for the application of the Landau theory of phase
transitions is the classical Ising magnet on a lattice (specified by an index i)
described by the Hamilton function
H = −J
σi · σj
i,j
where the variable σ takes the value 1 or −1. The symmetry group of this
system consists of two operations: the identity operation E and the inversion
I which transforms simoultaneously all σi into −σi . This group – called often
Z2 – has two irreducible representations, the non-trivial one establishing a
one-dimensional OP space. Third order invariants are absent. The only fourth
order invariant is η 4 . The Landau functional Φ[η] is
Φ[T, P, η] = A(T, P )η 2 + B(T, P )η 4
At one side of the Curie point (which we take to be the high temperature
side) A > 0: provided B > 0 the functional has a minimum at η = 0, i.e.the
system has the full Z2 symmetry. At the Curie point A = 0 and below it
A < 0. From the minimization of Φ, i.e. from the equation ∂Φ/∂η = 0 we
find
Aη + 2Bη 3 = 0
A
with two solutions: η = 0 and η = ± − 2B
. The solution η = 0 represents,
below the Curie point, a maximum of Φ, see the figure. The other two solutions are minima of the Landau functional and both realize a symmetry
broken state below the phase transition. If the system assume one of the
minima, its symmetry group will consist of only the identity operation. The
remaining operation of the full symmetry group transforms one minimum
into the other. Thus, the existence of a larger symmetry group above the
phase transitiuon manifests itself below the phase transition by the existence
CHAPTER 3. LANDAU THEORY OF PHASE TRANSITIONS
55
of different states which are related by operations of the full symmetry group
Z2 .
The application of Landau Theory to the Heisenberg exchange ferromagnet follows the same arguments. The Heisenberg classical ferromagnet is defined by the Hamiltonian
H = −J
ni · nj
i,j
where the variables n are unit (pseudo) vectors in a three dimensional Euclidean space. The symmetry group of this Hamiltonian is the full rotation
group O3 , which includes improper rotations. We search for a phase transition where the IR D1+ is realized below the Curie point. Third order degree
polynomials – which contain odd power of the basis functions of D1+ – cannot
be excluded from the O3 symmetry alone. However, the time reversal symmetry represents a further symmetry element of the Hamiltonian, and odd
order polynomials of the order parameter η·(nx , ny , nz ) are not invariant with
respet to time reversal symmetry and cannot appear in Φ. We have at least
one invariant of the fourth degree – η 4 , which is independent on the direction
of the OP. In fact, as the group O3 contains rotations by arbitrary angles, we
come to the conclusion that this is the only fourth degree invariant polynomial. Thus, below the phase transition we will have a continuous degeneracy
of the system corresponding to the surface of a sphere of radius 1, as any
direction minimize the fourth order invariant. This continuous degeneracy
reflects again the continuous symmetry of the system above the transition
point. The expansion of Φ contains only the amplitude of the OP and its
minimization proceeds along the same lines encountered for the Ising model.
Once the system has assumed one of the many degenerate states corresponding to some direction of the OP, the symmetry of the system is lowered from
O3 to O2 , because the state of the system below the transition point will only
be invariant with respect to rotations around the direction of the OP.
Landau’ s model of the liquid-solid phase transition
The symmetry group of the liquid is primarily the full group T of continuous
translations in space. Since the liquid is isotropic, ρ0 = const. At the point
of transition ρ becomes ρ0 + δρ, where δρ will, in general, assume the form
of a Fourier integral
δρ(r) = η(k)e−ik·r dk
CHAPTER 3. LANDAU THEORY OF PHASE TRANSITIONS
56
(to ensure reality of δρ we impose η(−k) = η ∗ (k)). In the case of a continuous
group of translations, the integral extends over all k-vectors in the threedimensional space, while for a discrete translations the integral is restricted to
a fine mesh within the first Brillouin zone. The fully symmetric representation
of T is the one dimensional one with k = 0. We proceed now to expanding
Φ. Near the tansition point Φ will be expanded in different powers of the
coefficients ηk . Different terms of this expansion will contain the integrand
ηk1 · ηk2 · ηk3 ......
The condition that the polynomials are invariant with respect to translations
is
k1 + k2 + ..... = 0
Terms linear in ηk do not exist. The quadratic contribution to Φ writes
A(T, P, k)dk | ηk |2
where we have used the symmetry between ηk and η−k . Because of the symmetry of the liquid with respect to the full rotation group we can take
A = A(T, P, | k |) and perform the integral over spherical surfaces of varying
radius. As usual in the Landau theory, we select that isolated vector k0 for
which the coefficient A vanishes at the transition point. In general, this is
an isolated point as it is very difficult that the coefficients of different k vanishes simultaneously. Thus, we restrict the expansion to an integral over the
surface of a sphere of radius | k0 | and call, the coefficient A(k0 ) = A. Below
the transition point the second order coefficient is negative and the stability
of the solution is determined by the high-degree terms in the Landau expansion. As the third degree terms are not required by symmetry to be absent,
we will consider them with the requirements | ki |= k0 and
k1 + k2 + k3 = 0
i.e. the three vectors build an equilateral triangle. In all third order terms
these triangles have equal size and differ only by their orientation in space.
Again, because of the rotational symmetry, the coefficients of the third order
terms depend only on the size, not on the orientation of the triangles. Therfore, all coefficints in the third order terms are equal: their common value
will be denoted by B(T, P, k0 ). Thus, the third order contribution writes
f3 = B(T, P, k0 ) · η 3 · [
|kli |=k0
γ(kl1 ) · γ(kl2 ) · γ(kl3 )]
CHAPTER 3. LANDAU THEORY OF PHASE TRANSITIONS
57
with the usual condition i | γ(kli ) |2 = 1. the integral has been changed to
a sum due to the discrete nature of the mesh in k-space in a crystal. The
stable phase obtained upon solidification corresponds to the minimum of the
third order term with respect to the γ(kli ). Notice that if f3 is positive for
a certain direction in OP space, it is negative for the opposite one. As both
directions have the same invariance group, the symmetry below Tc does not
depend on the sign of B. Thus, the stable phase corresponds to a set of γ(kli )
associated with the maximum value of the expression
ϕ3 = [
γ(kl1 ) · γ(kl2 ) · γ(kl3 )]
|kli |=k0
The question about which restriction is imposed on the solid by the geometrical requirement of equilateral triangels was answered by Alexander and
McTague , Phys.Rev.Lett. 41, 702 (1978). They pointed out that such vectors cab be essentially seen as terminating at the edges of three-dimensional
polyhedra with faces consisting of identical triangles, see the figure. The octahedron on the left generates a fcc Brillouin zone, which corresponds to a bcc
lattice. The one on the right is a regular icoshedron and generates icosahedral
quasi-crystals. One can show that ϕ3 (bcc) > ϕ3 (icosahedron).