Fundamentals of the Physics of Solids / 06-Consequences of Symmetries

6.1 Quantum Mechanical Eigenvalues and Symmetries

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according to the representations Γ7 and Γ8, therefore the energy level is split into a twofold and a fourfold degenerate level. Similarly, the j = 7/2 level is split into two twofold (Γ6 and Γ7) and a fourfold degenerate (Γ8) level. The full spectrum thus contains three twofold and two fourfold degenerate levels. The resulting level structure is shown in Fig. 6.2.

		D7/2	(8)		8 (4)		25 (6)
					7 (2)
l=3 (14)					6 (2)				l=3 (14)
					8 (4)		15 (6)
		D5/2 (6)					2' (2)
					7 (2)

Fig. 6.2. Splitting of the 14-fold degenerate energy level of a single electron on the f shell in a cubic crystal field. On the left-hand side, spin–orbit interaction is taken into account first, and crystal-field splitting second; this order is reversed on the right-hand side. Numbers in parentheses show the degree of degeneracy

The right part of the diagram shows another approach: the 14-fold degenerate state of quantum numbers S = 1/2 and L = 3, which initially transforms according to the representation D3, is first split by the crystal field, and then the spin–orbit interaction is also turned on. In a cubic crystal field the representation D3 is decomposed into the sum of the representations Γ2, Γ15, and Γ25 – which correspond to a twofold and two sixfold degenerate levels (when spins are also counted). To account for further splitting caused by the spin–orbit interaction, spin is associated with a wavefunction that transforms according to D1/2. To reduce the product of the spatial and spin wavefunctions, one has to exploit the relations

Γ2	D1/2 = Γ7 ,
Γ15	D1/2 = Γ6 Γ8 ,	(6.1.19)
Γ25 D1/2 = Γ7 Γ8 .

This leads to the same results as above, where the e ects of the two interactions were taken into account in reversed order. The reason for this is that group-theoretical considerations say nothing about the order of split levels. The choice of one approach or the other is dictated by the relative strengths of the interactions: the stronger is taken into account first, and then the weaker is treated as a perturbation.

As we have seen, when an ion with a single f -electron is placed into a lattice with cubic symmetry, crystal-field splitting reduces the sixfold degeneracy of the ground state to twofold. Because of this, at low temperatures the

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magnetic properties of the atom are similar to those of a spin-1/2 particle. At higher temperatures, where thermal energies are comparable with the energy splitting, each of the six J = 5/2 states are excited thermally. The temperature dependence of susceptibility therefore deviates from the simple Curie law.

6.1.4 Kramers’ Theorem

The previous calculation of the possible states of a single electron showed that whatever kind of splitting is studied and however low the symmetry of the local environment, if spin is also taken into account then even spin–orbit interaction fails to split atomic energy levels completely – and the degree of degeneracy will be an even number for each state. This finding is generalized by Kramers’ theorem:3 In the absence of a magnetic field – which could break time-reversal symmetry – each energy level is at least doubly degenerate in any system that contains an odd number of electrons.

To verify this statement it should be recalled that the quantum mechanical operator of time reversal T is antiunitary, that is, it can be written as the product of a unitary operator U and the operator K0 of complex conjugation.4 For spinless particles T e ectively transforms the wavefunction into its complex conjugate:

T ψ(r1, r2, . . . , t) = ψ (r1, r2, . . . , −t) .

(6.1.20)

For spin-1/2 particles the components of the wavefunction represented by a two-component spinor become mixed as time reversal changes the sign of spin:

T sT −1 = −s .

(6.1.21)

Following the customary quantization conventions, the eigenvalues of the spin z component are chosen as spin states:

K0sxK0 = sx, K0sy K0 = −sy , K0sz K0 = sz .

(6.1.22)

Then the unitary part U of the time-reversal operator has to rotate the spin through π around the y-axis. One possible choice is

T = ie−iπsy K0 = σy K0 ,

(6.1.23)

where σy is the corresponding Pauli matrix. When time reversal is performed on the eigenstates of the spin z components, the spin component changes its sign and the wavefunction receives a further spin-dependent factor. A convenient choice is

3H. A. Kramers, 1930.

4For antiunitary operators T φ, T ψ = ψ, φ , where (, ) denotes the scalar product, and T (aφ + bψ) = a T φ + b T ψ.

6.1 Quantum Mechanical Eigenvalues and Symmetries		183
T ψ(r, sz , t) = (−i)2sz ψ (r, −sz, −t) .	(6.1.24)

These expressions can be generalized to several particles. Either the form

T = σ1y σ2y . . . K0

(6.1.25)

is used, or, when the particles are in some eigenstates of the spin z component,

T ψ(r1, s1z , r2, s2z , . . . , t) = (−i)2(s1z +s2z +... )ψ (r1, −s1z , r2, −s2z , . . . , −t) .

(6.1.26)

This implies that for the eigenstates of a system with n electrons,

T 2ψ = (−1)nψ .

(6.1.27)

As T contains a rotation of the spin by π, T 2 corresponds to a 2π rotation. However, a spinor is not transformed into itself but into its negative by such a rotation. This explains the factor (−1)n.

Time reversal transforms a Hamiltonian that contains position, momentum, and spin variables ri, pi, and si into

T H(ri, pi, si)T −1 = H(ri, −pi, −si) .

(6.1.28)

In the absence of a magnetic field, even when the spin–orbit interaction is taken into account, the Hamiltonian is invariant under time reversal:

H(ri, pi, si) = H(ri, −pi, −si) .

(6.1.29)

So, besides ψ, T ψ is also an eigenfunction, with the same energy. However, as we shall prove it below, when the number of electrons is odd, (6.1.27) implies the orthogonality of the two eigenfunctions – and so the state is at least doubly degenerate, as asserted by Kramers’ theorem.

To prove their orthogonality, consider the scalar product of two functions f and g. The properties of the product imply

(K0f, K0g) = (f, g) = (g, f )

(6.1.30)

for the complex conjugate functions. By taking time reversal instead of complex conjugation, the involved spin factors do not spoil the relation, therefore

(T f, T g) = (g, f ) ,

(6.1.31)

in agreement with the antiunitarity of time reversal. Applying this relation to the states ψ and T ψ, and using (6.1.27)), we have

(ψ, T ψ) = (T 2ψ, T ψ) = (−1)n(ψ, T ψ) .

(6.1.32)

The last equation shows that for n odd, the two states are orthogonal indeed, and so besides ψ, T ψ is another independent eigenfunction, with the same energy.

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6.1.5 Selection Rules

In solid-state physics the following questions are often addressed: How does the initial state of a system change under some external perturbation? What new states may arise? Quantum mechanics tells us that when the system is initially in an eigenstate ψi(r) of the unperturbed Hamiltonian and perturbation is given by the interaction term Hint then the probability for a transition into a state with wavefunction ψf(r) is proportional to the absolute square of the matrix element

Mi→f = ψf (r)Hintψi(r) dr . (6.1.33)

For the system of electrons and for the atoms (ions), as well as the crystalline state built up of them, specification of the set of transitions that are allowed or forbidden for symmetry reasons is of particular interest. Rules that determine through symmetry considerations the set of states into which transition from a particular state is possible under a given perturbation are called selection rules. (A more rigorous mathematical formulation is given in Appendix D.) Here we just note that selection rules arise because of the possibility to classify the eigenstates of the unperturbed system that are involved in the transition according to the irreducible representations of the symmetry group of the unperturbed Hamiltonian. Since transition probabilities have to be invariant under symmetry operations, transitions are allowed only between states with specific symmetry properties. As an application, below we shall examine the restrictions imposed by translational symmetry.

6.2 Consequences of Translational Symmetry

When studying crystalline systems one is frequently concerned with the determination of the energy eigenvalues and eigenstates of a lattice-periodic Hamiltonian H: just like the crystal itself, the Hamiltonian must also be invariant under translations through Bravais lattice vectors in an ideal crystal. In other words

H(r + tn) = H(r)

(6.2.1)

is required, where tn is a translation vector of the form (5.1.1). The explicit form of the Hamiltonian is irrelevant now – therefore the following results will equally apply to electron states, phonon states of lattice vibrations, or magnetic excitations.

Rigorously speaking, translation symmetry is violated in finite lattices, even though invariance under discrete translations highly facilitates the mathematical description of the properties of crystalline materials. As surface phenomena can be usually ignored in the calculation of macroscopic properties of bulk materials, infinitely large samples seem to be the natural choice in theoretical studies. However, the mathematical discussion is far more complicated when the number of symmetry elements is infinite rather than finite.

6.2 Consequences of Translational Symmetry

185

A suitable choice of the boundary condition allows for a reduction to a fi- nite number of symmetry elements without losing the advantages of strict translational invariance.

6.2.1 The Born–von Kármán Boundary Condition

Max Born5 and T. von Kármán (1912) proposed a particular boundary condition for the description of finite samples that opens the way to exploiting the consequences of translational symmetry. The periodic or Born–von Kármán boundary condition can be formulated in two ways:

1.Consider a crystal that is built up of identical groups of atoms repeated N1 times along direction a1, N2 times along direction a2, and N3 times along direction a3. Imposing periodic (Born–von Kármán) boundary condition amounts to formally identifying points on opposite faces of the crystal with each other. Thus translation through Nj aj (j = 1, 2, 3) takes any point r of the lattice into itself. Intuitively this can be imagined as folding one end of the sample back to the other. One-dimensional chains with two free ends are thus transformed into rings, and two-dimensional finite sheets into tori. No intuitive picture can be given for three-dimensional crystals – therefore preference is given to the second formulation in this case.

2.The periodic boundary condition can be regarded as if the originally finite, parallelepiped-shaped sample were translated through all integral linear combinations of the vectors Nj aj . The obtained pattern fills the entire space, and the point at r is rigorously equivalent to all points

3
r = r + pj Nj aj ,	(6.2.2)

j=1

where the pj are integers.

The two formulations of the Born–von Kármán boundary condition are illustrated in Fig. 6.3 for a one-dimensional chain of atoms.

When only nearest neighbors interact, the original chain of N atoms contains N − 1 bonds – one less than the ring. The energy spectra of the two systems are thus not identical. Deviations are expected to be on the order of 1/N , and are thus negligible for macroscopic samples. When the chain with free ends is replaced by a ring, other choices for the additional bond are also possible. For example, when studying a system of spins localized on atoms, in some cases it might be advantageous to choose the interaction between the two spins at the two ends to di er in sign or a phase factor from the

5Max Born (1882–1970) was awarded the Nobel Prize in 1954 “for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction”.

186 6 Consequences of Symmetries

1 2 3 . . .		N-1 N
		N-1		3
		N	1	2
N-1 N 1 2 3		N-1 N 1 2 3	N-1 N 1 2 3

Fig. 6.3. Two di erent formulations of the Born–von Kármán boundary condition on a finite one-dimensional chain of atoms. The chain is either wrapped into a ring or repeated indefinitely. Consequences of translational symmetry can be exploited in both cases

interaction within the chain. This boundary condition is called antiperiodic or twisted. For electrons moving along a one-dimensional chain, the choice of periodic (antiperiodic) boundary condition is more practical when the number of fermions is odd (even).6

The restriction imposed by the Born–von Kármán boundary conditions can be formulated in terms of the quantum mechanical wavefunction as

ψ(r) = ψ(r + Nj aj ) , j = 1, 2, 3 .

(6.2.3)

Its counterpart for antiperiodic boundary conditions is

ψ(r) = −ψ(r + Nj aj ) .

(6.2.4)

6.2.2 Bloch’s Theorem

The lattice-periodic character of the Hamiltonian may not be exhibited by the wavefunction, which is not observable in itself. However, Bloch’s theorem7 applies to the solutions of the Schrödinger equation with a lattice-periodic potential. A general formulation of the theorem reads: Solutions to the Schrödinger equation for a lattice-periodic Hamiltonian can be characterized by a vector quantum number k defined in the primitive cell of the reciprocal lattice. This “wave vector” governs the behavior under translations in the sense that values of ψk (r) – i.e., the wavefunction associated with the quantum number k – taken in di erent primitive cells di er by a simple phase factor:

ψk (r + tn) = eik·tn ψk (r) .

(6.2.5)

6This must be so, since if the number of electrons is even, moving an electron from one end of the chain to the other equivalent end requires exchanging it with an odd number of electrons.

7F. Bloch, 1928. The one-dimensional version of this theorem is also known as Floquet’s theorem in other branches of physics and mathematics.

6.2 Consequences of Translational Symmetry

187

Alternatively, as it will be discussed in Chapter 17, Bloch’s theorem states that the eigenfunctions of a lattice-periodic Hamiltonian may be written in the form

ψk (r) = eik·r uk(r) ,

(6.2.6)

where uk(r) is lattice periodic. As we shall see, k can take only well-defined discrete values determined by the specific boundary condition.

To prove Bloch’s theorem, we shall introduce the linear operator O(tn) of the translation through tn. By definition, for any function f (r)

O(tn)f (r) = f (r − tn) .

(6.2.7)

First, we shall demonstrate that O(tn) commutes with the lattice periodic Hamiltonian H(r). By operating O(tn) on H(r)f (r) and making use of (6.2.1) and (6.2.7),

O(tn)H(r)f (r) = H(r − tn)f (r − tn)

(6.2.8)

= H(r)f (r − tn) = H(r)O(tn)f (r)

is obtained, that is, the Hamiltonian commutes with any operator associated with a translation through a lattice vector:

O(tn)H(r) = H(r)O(tn) .

(6.2.9)

As H(r) and O(tn) commute, they can be diagonalized simultaneously. The obtained wavefunctions are then denoted by ψα(r), and the eigenvalues by εα and tα(tn),

H(r)ψα(r) = εαψα(r) ,

(6.2.10)

O(tn)ψα(r) = tα(tn)ψα(r) .

Here the label α serves only to distinguish the states. Its possible values will be determined later.

The succession of two translations through tn and tn is the same as a

single translation through tn + tn :
O(tn) O(tn ) = O(tn + tn ) .	(6.2.11)

This equation defines group multiplication in the group of translations. As the result is independent of the order of the two translations, group multiplication is commutative,

O(tn) O(tn ) = O(tn ) O(tn) ,

(6.2.12)

and so translations form a commutative or Abelian group. Irreducible representations of Abelian groups are known to be one-dimensional, thus the quantum number α also serves to index them.

When the translation operator acts on the basis function of the onedimensional representation of index α, (6.2.11) implies

188 6 Consequences of Symmetries

tα(tn)tα(tn ) = tα(tn + tn )

(6.2.13)

for the eigenvalues. Translation through tn can be built up of translations through primitive vectors, in the form given by (5.1.1). Thus the translation operator O(tn) can also be expressed in terms of the operators that correspond to the elementary translation vectors,

O(tn) = [O(a1)]n1 [O(a2)]n2 [O(a3)]n3 .

(6.2.14)

Consequently, the eigenvalues of the translation operator satisfy the equation

tα(tn) = [tα(a1)]n1 [tα(a2)]n2 [tα(a3)]n3 .

(6.2.15)

The eigenvalues tα(tn) can be determined from the requirement that the boundary conditions imposed on the wavefunctions should also be satisfied. For a general parallelepiped-shaped sample the Born–von Kármán boundary conditions (6.2.3) may be written as

ψ(r − Nj aj ) = ψ(r) ,	j = 1, 2, 3 .	(6.2.16)
This implies
O(N1a1) = O(N2a2) = O(N3a3) = 1 ,		(6.2.17)
leading to
[tα(a1)]N1 = [tα(a2)]N2 = [tα(a3)]N3 = 1 .		(6.2.18)
The solutions of these equations are the roots of unity,
tα(aj ) = e−2πipj /Nj ,		(6.2.19)

where pj is an integer. The Nj di erent roots are most simply obtained by restricting pj to the range

0 ≤ pj ≤ Nj − 1 .	(6.2.20)
Substituting the above form of tα(ai) into (6.2.15) gives
tα(tn) = exp[−2πi(n1p1/N1 + n2p2/N2 + n3p3/N3)] .	(6.2.21)

This eigenvalue can thus be expressed in terms of the three quantum numbers p1, p2, p3. The precise specification of the state may require further quantum numbers that appear in the wavefunction and the energy eigenvalues – but not in the eigenvalues of the translation operator. Below, we shall focus on the triplet p1, p2, p3 and ignore these further quantum numbers; their importance will transpire only later.

The expression for tα(tn) may be written in a more concise form using the

notation				p2			p3
k =	p1	b1	+		b2	+		b3 ,	(6.2.22)
	N1			N2			N3

6.2 Consequences of Translational Symmetry

189

where b1, b2, and b3 are the primitive vectors of the reciprocal lattice. Because of the restrictions on pj , the k are within the primitive cell spanned by the vectors bi. When the pj run over their entire range, the resulting set of vectors fill this primitive cell evenly.

Using the multiplication rule (5.2.12) for the product of a directand a reciprocal-lattice vector, the expression in (6.2.21) can be written as

tk (tn) = e−ik·tn .

(6.2.23)

Returning to (6.2.10), the solutions ψk(r) to the Schrödinger equation are found to have the following property:

ψk(r + tn) = O(−tn)ψk (r) = tk (−tn)ψk (r) = eik·tn ψk (r) . (6.2.24)

This completes the proof of Bloch’s theorem. As a by-product, we have also derived the allowed values of the quantum number specifying the behavior under translations, the wave vector (also called wave-number vector) k.

The phase factor contains the scalar product of the discrete translation vector and the wave vector. For continuous translations a Taylor expansion and subsequent summation of the wavefunction at the position shifted through a, ψ(r + a) would lead to

ψ(r + a) = ψ(r) +			∂	ψ(r)aα + 1
				2
		α ∂rα
= exp	∂	· a ψ(r) .
	∂r

∂2

αβ ∂rα∂rβ ψ(r)aαaβ + . . .

(6.2.25)

This can be expressed in terms of the canonical momentum p as

ψ(r + a) = eip·a/ ψ(r) .

(6.2.26)

This formula suggests that k plays the same role in discrete translations as momentum in continuous translations. This analogy justifies calling k the crystal momentum. Nevertheless it should be stressed that the state ψk (r) is not necessarily an eigenstate of the momentum operator, and its momentum is not necessarily k.

6.2.3 Equivalent Wave Vectors

The number of di erent k vectors allowed by the boundary condition is N = N1 × N2 × N3, i.e., the number of primitive cells in the finite crystal. The choice of the wave vectors is, however, not unique. In the previous subsection the quantum numbers pi (that appeared in the definition of k) were chosen according to the condition (6.2.20). This particular choice implied that the wave numbers fill a primitive cell of the reciprocal lattice.8 However, this

8Throughout the present subsection “primitive cell” should be understood as “primitive cell spanned by the primitive vectors of the reciprocal lattice”.

190 6 Consequences of Symmetries

choice is not unique. As the ks were determined from the requirement that (6.2.18) should be satisfied, the same eigenvalues are found when pj is replaced by pj + hj Nj in (6.2.19) – where hj is an arbitrary integer. The behavior of the system under translations is then characterized by the vector

k =	p1 + h1N1	b1	+	p2 + h2N2	b2	+	p3 + h3N3	b3
									(6.2.27)
	N1			N2			N3

= k + h1b1 + h2b2 + h3b3

instead of k. The two di er by a reciprocal-lattice vector,

k = k + G .

(6.2.28)

It is readily seen that it makes no di erence in Bloch’s theorem whether the phase factor is determined using k or k . For any translation through a lattice vector,

tk (tn) = tk (tn) .

(6.2.29)

Thus k and k – which di er by a vector of the reciprocal lattice – can be equivalently used as a quantum number of the states. In this respect, the two vectors are equivalent.

As it was mentioned in Chapter 5, the object obtained with Dirichlet’s construction in the reciprocal lattice is called the Brillouin zone. By appropriately cutting the primitive cell and translating the parts through suitably chosen reciprocal-lattice vectors, the Brillouin zone is totally covered without overlap in an unambiguous way. The parallelepiped-shaped primitive cell, the Brillouin zone, and the parts that have to be translated to make the two overlap are shown in Fig. 6.4 for a two-dimensional reciprocal lattice.

Fig. 6.4. Equivalence of the primitive cell spanned by the primitive vectors and the Brillouin zone of a two-dimensional reciprocal lattice. Regions with identical shading are displaced through some lattice vector

Since each vector k within the primitive cell can be unambiguously associated with a wave vector inside the Brillouin zone, the states of the latticeperiodic system can be specified using vectors k in the Brillouin zone. More often than not, this path is taken in solid-state physics. The reason for this