Marder M.P. Condensed Matter Physics

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Translational

Symmetry—Bloch 's

Theorem

183

Since for a given value of k, there is still the possibility of many energy eigen-

values, one must introduce the band index n. The classification of eigenfunctions

made possible by periodicity is completed by writing

n^nÙ

Λ>

(7.43a)

fÎ\ï[) -) =

'^\lp

y). Combine Eqs. (7.39) and (7.42). In the posi- (7.43b)

R "* "*■ tion representation, this reproduces Eq. (7.36).

From Eq. (7.43b) the previous forms of Bloch's theorem can be obtained. Acting

with

(7\

gives

Ψώ?+κ)

Ρ'

Ψώ?)·

(

)

One can also proceed to define

U y (7) ΞΞ \JN e~~ ih y (7). When ψ is normalized over the volume (7.45)

™ ™ V, u is normalized over the unit cell Ω.

It follows from Eq. (7.44) that u is periodic;

u(7 + R) = u(7). For all R in the Bravais lattice. (7.46)

7.2.4 Additional Implications of Bloch's Theorem

Effective Hamiltonian. By virtue of Bloch's theorem, the original Hamilto-

nian, (7.2), can be transformed into an effective Hamiltonian whose solution no

longer needs to be sought over the whole crystal. This Hamiltonian is easily found

by searching for the equation to be obeyed by u. Writing ψ in terms of u from

Eq. (7.45) and writing out (Ä - Ε)ψ = 0 yields

- -

5kw(r) = —

[-V

-2i*·

V +

}u(7)

+ U(7)u(r) = Eu(r). (7.47)

Because u is periodic, this equation can be restricted to a single unit cell, subject to

the boundary conditions that whenever 7 lies on one boundary of the unit cell, and

7 +Ris another boundary point of the cell, then

u(7) = u(7 + R) R is a Bravais lattice vector. (7.48a)

and

h(7)-Vu(7) = -h(7 + R)-'S7u(7+R). n(f) is a unit normal to the cell (7.48b)

boundary at 7.

Using the relation (7.45) between u and ψ, one can equally well write down Schrö-

dinger's equation, and solve it in a single unit cell, subject to the boundary condi-

tions

(7) =

(7+R)

(7.49a)

and

n{7)

■

V^(r) = -n{7+R)-V^

R). (7.49b)

184

Chapter 7. Non-Interacting Electrons in a Periodic Potential

In a crystal with

atoms, Eq. (7.47) requires 10

times less numerical effort to

solve than Eq. (7.2).

Counting k. The demonstrations of Bloch's theorem relied on periodic boundary

conditions. As a consequence, as discussed in Section 6.3, there are restrictions

on the permitted values of the Bloch wave vector k. In a cubic crystal of volume

V = L?,k takes the form given in (6.7), just as for free electrons. The density of

allowed k states is therefore still given by Eq. (6.13).

These results can all be generalized to crystals that are not cubic. Consider any

lattice described by three primitive vectors a\, 02, and 03 as in Eq.

(A.

17).

The

vectors k consistent with periodic boundary conditions are of the form

](_

= \ Jy, 0 < mi < Ml

l g'

ves th

e total number of lattice points along (7.50)

*--' Mi '

—

' direction /; the total number of lattice points is

'=1

}

where b\ . . . ^3 are primitive vectors of the reciprocal lattice and are chosen to

satisfy

bi -aii = 2πδ[ΐι. (7.51)

The primitive vectors b\ . . . £3 describe the boundaries of a primitive cell in

reciprocal space, and the k of (7.50) form a fine network of vectors filling this prim-

itive cell. It is possible to adopt a convention where k takes values outside of this

cell, allowing the integers ni\ to continue on to values greater than M\. However,

the resulting k will be redundant, in the sense that they must differ from some k

within the primitive cell by a reciprocal lattice vector. Any two k that differ pre-

cisely by a reciprocal lattice vector K share the eigenvalue e\p[ik-R] in Eq. (7.43b),

since exp[/^

1 ·

Because k is nothing but a label for this eigenvalue, two such

k can be regarded as physically identical.

According to Eq. (7.50), the total number of k states in a primitive cell in k

space is

M1M2M3,

which is also the number of lattice points N in the original real

space lattice. So one has the convenient and general result that the

Number of physically distinct

Bloch wave vectors k = Number of lattice sites, N.

Brillouin Zone. The primitive cell in k space spanned by arbitrary primitive

vectors

t>\

. . . b^ is not convenient to use, because it is not uniquely specified,

nor does it have the full symmetry of the crystal. A primitive cell that overcomes

these defects is the Wigner-Seitz cell of the origin in reciprocal space. As in the

one-dimensional case, it is called the Brillouin zone, or first Brillouin zone. Sums

such as in Eq. (7.57) over k or q are always carried out over the net of vectors in

Eq. (7.50) filling the first Brillouin zone. However, any sum over K is carried out

over a very different set, the reciprocal lattice vectors given by Eq. (3.24d).

Density of States. As in the case of free electrons, it is useful to be able to convert

sums over k into integrals. To carry out a calculation analogous to Eq. (6.10), one

Translational

Symmetry—Bloch 's

Theorem 185

needs to know the volume of reciprocal space associated with each point k. From

Eq. (7.50), this volume is

■

£3)

The volume of any paralleliped whose (7.53)

M1M2M3

edges are given by three vectors A,

C is

A-(BxC).

2-7Γ

■

(p2

x (a\ x aj))

ατ,

■

(ai

Ü2) M1M2M3

(2π)

Use

Eq.

(3.24c). (7.54)

Using the identity (7.55)

M1M2M3Ö1 -(22X03) Âx(BxC)=B(Â-C)-C(Â-B)

(2π)

V is given by the number of lattice points.

A/2M3,

times the volume of each

Because the total volume of the crystal (7.56)

is given by the

' "-«■

------

M1A/2AÎ3,

timi

primitive cell.

Therefore any sum that needs

be carried out over the states

k in

the first

Brillouin zone can use the relation

Σ^

= ν j[dk]F

(7.57)

using the density

states in Eq. (6.15) just as for free electrons. The density of

energy states in the nth band is similarly given by

D„(£)= f[dk]8{E-E

i). Compare with Eq. (6.19). (7.58)

Energy Bands and Group

Velocity.

One of the most important pieces of physical

information

be obtained from the energy functions £

the velocity

υ^ of

electrons in the

rath

band with wave number k:

T» _V7^P

^ Vj

means that one takes derivatives with respect

/η cq\

7j k nk

to components of X.

Without embarking on the careful but lengthy justification

this relation to be

provided in Chapter 16, one can note that Eq. (7.59) is nothing but the statement

that solutions of wave equations generally have a group velocity

= θω/dk.

the context

wave mechanics,

particle

actually

wave packet, which

is a

sum over wave functions that is contrived so as to have a large peak at position

and simultaneously a dominant wave vector k. The flavor of calculations for wave

packets is easiest to share if one imagines solutions of Schrödinger's equation with

energy £^ and for which u^(r) varies only very weakly in space. Then define

Here

is a Bloch wave, and

[di'] w(k'

e-^'^th,

(?),

is a

fimcti

sharply peaked

(7.60)

about k

—

0; thus W

mainly built out of waves with

wave vector

186

Chapter 7. Non-Interacting Electrons

in a

Periodic Potential

njk-r-iw*

{

sr]

Çk")

/M-v

i£i

,/^

w;;(?)j

»^x

(761)

J exponent

first order

in k",

because

w is

peaked about

jfc"

^ fk-7-iE-

t/Hcr(-;_f]

e ,/t\

Assuming the spatial variation

(r)

can be

ne-

,_ „.

e J

> glected. The value

the integral

does

not

mat- \'·

ΌΔ

ter; just the fact that

it is in

the form

a traveling

wave moving

Vj£j//i

= Vju^.

7.2.5 Van Hove Singularities

In the course of computing the energy densities of states defined by Eq. (7.58), it

is common to encounter divergences called van Hove singularities.

In one dimension, suppressing the index n on fir. gives

D(£) = ί dk (2/2π)ί (£ - £*) (7.63)

— / § (£, — £t) An

additional factor

two comes generally

(7 64)

\d£

/dk\

from the fact that

£* = £_*.

When explicit solutions

Bloch's equation

are

found,

as in

Figure 7.7,

will

emerge that £„&

is a

periodic function

of k and

that dZ

kjdk vanishes

at the

edge

the Brillouin zone.

As a

result,

the

density

states must diverge there. Assuming

that dE,

k/dk vanishes linearly

as k

approaches

π/α, the

divergence

is of

the form

D(E)

-. .

Taking (7.66)

k-π/α V£max-£ 2*~e

x-C(*-7r/

)

,with

C some constant near the zone

edge.

The only

way for

dE/dk

vanish

in one

dimension

is at a

maximum

or a

mini-

mum. Therefore,

van

Hove singularities

in one

dimension tend

occur only

at the

beginning

or the end of a

range

allowed energies.

In two and

three dimensions,

van Hove singularities typically occur

in the

midst

energy intervals

well—as

for example

Figures 13.9

18.4.

The

generalization

Eq. (7.65)

- 2

D(E)

= I dk δ(Ε — £r).

Thenumberof dimensions

might

be 2 or

(7.67)

Integrals in the form of Eq. (7.67) can also be expressed as integrals over the en-

ergy surface on which δ = £^. To obtain the correct expression, begin with the

observation that

0(£-£j)-0(£-£

-£/£)

~ d~Ë

δ(Ε

— ε vi = — —

FordE very small.

V nk

(7.68)

As shown

Figure 7.3,

Eq.

(7.67)

therefore given

integrating over

the

energy surface

£ = £ r

with surface measure

άΣ,,

Translational

Symmetry—Bloch

Theorem 187

•

multiplying by the normal distance to the surface £ = £^ + dE, and

•

dividing by dE.

What is the normal distance between the two surfaces? First, note (Problem

)

that

Vt£r

ΚΓΤΜ

69)

is a unit normal between the two surfaces. Next, let dk be any k vector such that

^l+dk

£%

dE.

The normal distance between the surfaces is therefore dk

■

But

£-l

l = £-l + dk- V^^ Taylor expansion. (7.70)

=>dk-'V

= d£.

(7.71)

^dk-ri = —— (7.72)

λΙ

Figure 7.3. Two energy surfaces are separated by distance d£, with a vector dk running

from one to the other, the normal direction between the surfaces given by V^£.

Therefore, denoting by άΣ an integral over the energy surface, Eq. (7.67) can

be rewritten as

2 f dE

D(ß)

= / The integral is a (d

— 1 )

dimensional integral (7 73)

fc-space

carried out over the energy surface

Comparing with Eq. (7.59), this means that the density of states has a singularity

at points in reciprocal space where the electron velocity υ^ vanishes. The origin

of this singularity, in a one-dimensional context, is illustrated in Figure 7.4(A).

In addition to the maxima and minima of the energy function, there can also be

singularities at saddle points, [Figure 7.4(B)] which are at a maximum along some

direction but a minimum along another.

In two and three dimensions, van Hove singularities behave as

D(E)

~ In

|£/£n—1|

or $(i£) In two

dimensions;

the first expression at sad-(7.74)

die,

second at maximum or minimum.

VE for £ > 0, 0 else,

D(£)~<

In three dimensions, at maximum or mini-(7.75)

—

mum. At saddle, have square roots meeting

-c for t < 0, 0 else,

cusp.

188 Chapter 7. Non-Interacting Electrons in a Periodic Potential

Figure 7.4. (A) If the states that contribute to some process are those that lie between £

and E+dE, then

the

number of

states

becomes singular when

coincides with an extremal

point on the energy surface. Compare with Figure

16.11.

(B) A saddle point.

These claims are verified in Problem 3.

The van Hove singularities manifest themselves in a variety of physical con-

texts.

They appear in calculations of electronic densities of states, shown through-

out Section 10.4, in calculations of density of vibrational states as shown in Section

13.3.1,

and are measured to a limited extent in experiments involving optical ab-

sorption, as shown in Section 23.4.

Uniqueness of Label

It may appear that one now has many more wave functions

than one had in the case of free electrons. All states of free electrons can be labeled

by the index k. For electrons in a periodic potential, one has an additional index n

as well. In fact, the number of eigenstates has not multiplied; the apparent difficulty

is caused by a certain arbitrariness in how to label states.

Given any k, one can find a set of wave functions

with differing energy eigenvalues, but with the same eigenvalues exp[ik-R] when

acted upon by 7"J. Choosing K in the reciprocal lattice, consider now the eigenval-

ues of ψ

£. Because

i{UKyR

ik-R^

(7J7)

it follows that

φ j i n = ^ ,7. For some integer n' φη, since ψ^ are defined (7.78)

n,k+K n ,k

e a

tne waV

g functions with translational

eigenvalue exp[ik

■

R\.

There now are two possible ways to ensure that the set of all φ ^ will be a com-

plete and linearly independent set of wave functions. The first, called the reduced

zone scheme is to limit the collection of k vectors. In this convention, k is permitted

only if it belongs to the first Brillouin zone. Choosing k's in this way guarantees

that no two of them can satisfy k = k'+ K.

The second way to avoid overcounting states is to drop the index n but allow k

to range through all of reciprocal space; this is the extended zone scheme. A state

that was labeled φ^ in the reduced zone scheme will be labeled Φι

^ (for some

reciprocal lattice vector

K„)

in the extended zone scheme.

Translational

Symmetry—Bloch 's

Theorem 189

Example: Free Electron. One can choose to regard a free electron as periodic

over any lattice desired. This gives great freedom in indexing the states. Moving

to the one-dimensional case for the purpose of illustration, suppose one uses the

fact that a free electron Hamiltonian is periodic subject to translations through R.

Bloch's theorem says in this case to index the wave function in the form

L·

= e e'

One can regard

either as the band index, (7.79)

or as describing the multiple of the recipro-

cal lattice vector K to add to k.

where K is a primitive vector of the reciprocal lattice, and —K/2 < k < K/2 is the

first Brillouin zone. In the reduced zone scheme, the name of the wave function in

Eq. (7.79) is ψ^, with n a band index. In the extended zone scheme, the wave func-

tion is instead named φ^, with k' = k + nK. These two ways of viewing Eq. (7.79)

are drawn in in Figure 7.5, together with a third convention, called the repeated

zone scheme, which shows what would happen if both

and n were allowed to

range freely over all possible values.

Summary. To recapitulate, first index solutions of Schrödinger's equation by

vector k that lies in the first Brillouin zone. For every value of k there is a count-

ably infinite number of energy bands, which can be labeled by band index n (re-

duced zone scheme) or by adding reciprocal lattice vectors K

to k (extended zone

scheme). Allowing k to range over all reciprocal space and also retaining the band

index n duplicates each eigenstate infinitely often (repeated zone scheme).

7.2.6 Kronig-Penney Model

Kronig and Penney

(

1931 ) found an exactly soluble model that illustrates the nature

of energy bands. Suppose that in each unit cell of a one-dimensional lattice with

lattice points R = na and reciprocal lattice vectors K, there is a potential of the

form

UoaÔ(x),

a is

the lattice

spacing. (7.80)

where i/o has dimensions of energy. Then

as defined in Eq. (7.26) is simply

= U

, (7.81)

and Eq. (7.33) becomes

0 =

(ε°-ε)φ(ς)

ΣνοΨ(ς-Κ).

(7.82)

Define

Ö, = £>(<?-*)· (7-83)

Then Eq. (7.82) becomes

190

Chapter 7. Non-Interacting Electrons in a Periodic Potential

3π/α

Figure 7.5. Three indexing schemes for labeling k states are illustrated by plotting £5

for the free electron. The first Brillouin zone is taken to extend from — π/α to π/α. The

extended zone scheme allows k to range throughout all reciprocal space. In the repeated

zone scheme, one plots £j, £j

^ , £^_^ · · ·· Finally, in the reduced zone scheme, all

wave vectors are mapped into the first Brillouin zone, so that energy levels originating

from k + nK are now regarded as belonging to an nth band. The extended zone scheme

is the most natural way to plot free electron energy levels; however, in the presence of a

periodic potential,

E-

is discontinuous in the extended zone scheme and is continuous in

the reduced zone scheme.

#?) + girrgßi =

Note from its definition (7.83) that

for all reciprocal lattice vectors K. So

i/o

Qq = Qq-K

1p(k-K)

+ -g Q

-K = 0

Just

evaluate Eq. (7.84) alq = k-K.

Σ k(k-K)+

i/o

^k-K £

z= 0 Sum on K and use Eq. (7.85).

β*+Σ

i/o

PO _ c

K ^k-K °

= 0. Definition Eq. (7.83).

(7.84)

(7.85)

(7.86)

(7.87)

(7.88)

Translational Symmetry—Bloch 's Theorem 191

Figure 7.6. In order to solve Eq. (7.89), choose a value of k and then find when the sum

S*(£) = Σκ l/(£*-if

£) equals

—

1/f/o by varying £. The sums are carried out for a

chain of spacing a. In the upper portion of the figure, £ lies along the vertical axis, and

5jt(£) is plotted on the horizontal axis for k = 0.2ΛΌ, where

= 2π/α. In the lower portion

of the figure, this procedure is carried out for 10 consecutive values of k. For each plot of

Sk(£),

one checks to see where

equals —I/UQ, shown as a vertical line. The intersection

point is a solution of Eq. (7.89), and the collection of intersection points gives £^. By

varying

UQ,

one varies the shapes of the bands £/;. When

is small, they are free-electron

like,

while when i/o is large they are very nearly fiat.

Assuming that

does not vanish, one has finally that

-—

Y-ΊΑ

(£). (7.89)

Equation (7.89) is easy to solve numerically. The method is illustrated graphically

in Figure 7.6.

Energy levels for the periodic array of delta-function potentials appear in Fig-

ure 7.7. This figure should be compared with Figure 7.5. The effects of the po-

tential are especially strong at the edges of the first Brillouin zone. It is crucial to

note that

£&

is now discontinuous in the extended zone scheme but continuous in

the repeated and reduced zone schemes. The origin of the gaps between the energy

levels will be the subject of Section 8.2.

The motivation for creating the repeated and reduced zone schemes should

now be evident. These index energy states so that energy levels remain continuous

functions of k in the presence of a periodic potential. The band energy £„£ is a

continuous, periodic function of k.

192

Chapter 7. Non-Interacting Electrons in a Periodic Potential

Extended Zone

Repeated Zone

-3π/α -π/α k π/α

Reduced Zone

3π/α

Figure 7.7. Bands resulting from Eq. (7.89) with U

= -l0h

K$/(2m) and for a one-

dimensional lattice of lattice constant a, shown in the reduced, repeated, and extended

zone schemes. The calculation sketched in Figure 7.6 produces the bands in the reduced

zone scheme

directly;

by translating these

bands

through various multiples of

the

reciprocal

lattice vector

= 2π/α, the upper two plots can be produced.

7.3 Rotational Symmetry—Group Representations

Consider a lattice with a collection of point group operations {G} leaving it invari-

ant. All the possible operations are represented by 3 x 3 unitary matrices and have

been listed in Table 2.10; rotations through 60°, 90°, or 120°as well as reflections.

The stereograms of

Table

2.9 allow one to read off the sets of symmetry operations

that correspond to the 32 point groups.

The theory of group representations describes the consequences of these sym-

metries for solutions of Schrödinger's equation. These consequences are not as

dramatic or important as those that follow from translational freedom, as reflected

in Bloch's theorem. Bloch's theorem takes a problem involving 10

atoms and

reduces it to a problem involving only one or two. By comparison, the simplifi-

cations provided by rotational symmetries may only reduce computational effort

by a factor of something like 3 or 24. The significance of group representations

lies elsewhere. By carefully describing the possible symmetries of electron states,

one can obtain selection rules that predict when light or other external agents are