y À1=2

eÀiq Á l ay ; À1=2

eiq Á l aq :

al ¼ al ¼ N

N q

q q

The sum over q is restricted as for the case of phonons to N distinct allowed

values, such as are contained within the ¬rst Brillouin zone de¬ned in Section

3.6. We then ¬nd that

02 X 0 0 0 0

0

½aq ay 0 ei°q Á lÀq Á l Þ þ ay aq 0 eÀi°q Á lÀq Á l Þ

s°lÞ Á s°l Þ ™ q q

2N q;q 0

0 0 0

À ay aq 0 °eÀi°qÀq Þ Á l þ eÀi°qÀq Þ Á l Þ þ 1 02 :

q 4

If we de¬ne

X 0

eiq Á °l ÀlÞ Jll 0

Jq ¼

l0

we have for the Heisenberg Hamiltonian (3.11.1)

X y X hÀ Á y3i

Áy À

H ¼ 0!0 aq aq À 2 þ 2 0

12

J0 À Jq a q aq þ J0 À JÀq aq a q À 2 J0 :

1

q q

120 Boson systems

For a Bravais lattice it will be true that Jq ¼ JÀq , giving us

X

0!q ay aq ;

H ¼ E0 þ q

q

where E 0 is a constant and

!q ¼ !0 þ 0°J0 À Jq Þ:

Within the approximations we have made we can thus consider the magnet

as a system of independent bosons. Because

X ‚1 Ã

q Á °l À l 0 Þ

J0 À Jq ¼ 2 Jll 0 sin2 2

l0

the magnon frequency ! will always increase as q2 for small values of q, in

agreement with the classical approach of Section 1.4.

This simple theory is adequate to account for a number of the low-

temperature properties of ferromagnets, when only a few magnons are

excited. The total magnetization, Mz , for instance, is given by

! !

eX X X

1 1

e e0

Mz ¼ sz °lÞ ¼ N0 À nl 0 ¼ NÀ nq ; °3:11:3Þ

2 2

mc l mc mc

l q

showing that each magnon carries a magnetic moment of °e0=mcÞ. The Bohr

magneton B is de¬ned as e0=2mc, and so we can write the deviation from

saturation of the magnetization as

X

M0 À Mz ¼ 2B nq :

q

Because the magnons behave as bosons the average number present in any

mode will be given by (3.3.2), from which

X X 1

"

nq ¼ : °3:11:4Þ

exp °0!q =kTÞ À 1

q q

At low temperatures only magnons of low energy will be present, and so in

a cubic crystal in the absence of an applied ¬eld !q may be approximated

by q2 , where is a constant. Writing the summation over q as an integral in

121

3.11 Magnons

q-space we ¬nd

°

X q2 dq

"

nq / :

exp °0q2 =kTÞ À 1

q

Changing the variable of integration from q to x ¼ 0q2 =kT gives the low-

temperature relation

M0 À Mz / T 3=2 ; °3:11:5Þ

which is a result well veri¬ed experimentally. Mean ¬eld theories predict

either a linear or exponential variation, according to whether a classical or

quantized picture of the magnetic moment is used.

As the temperature is increased and the magnetization begins to deviate

from its saturation value, the approximation of replacing (1 À nl Þ1=2 by 1 will

become less valid. If we expand this expression binomially, writing

°1 À nl Þ1=2 ¼ 1 À 1 nl À 1 n2 À Á Á Á

8l

2

1Xy 0

a q aq 0 eÀi°qÀq Þ Á l À Á Á Á

¼1À

2N q;q 0

we can interpret the exact Hamiltonian as describing a magnon system with

interactions. We could then use perturbation theory to calculate a better

estimate of how the magnetization should vary with temperature at low T.

However, we know that there exists a Curie temperature, TC , at which the

magnetization vanishes. It will thus be a fruitless task to pursue the perturba-

tion approach too far in this direction, as convergence will become very slow

as soon as T becomes comparable to TC . There are also complications that

arise from the upper limit !max to the frequencies !q over which one sums in

Eq. (3.11.4). This introduces terms of the form eÀ0!max =kT , which are not

expressible as any sort of power series in T, and makes comparison with

experiment very di¬cult.

As a ¬nal note on magnons it should be mentioned that more complicated

magnetic structures than the ferromagnet also have elementary excitations in

the form of spin waves. Simple, helical, and canted antiferromagnetism and

ferrimagnetism are examples of phenomena that arise when various inter-

actions occur between localized spins in various crystal structures. All these

exhibit magnon excitations of one form or another, and show a variety of

forms of !°qÞ.

122 Boson systems

Problems

3.1 Verify that for the Bogoliubov operators de¬ned in Section 3.4

y

½k ; k 0 ¼ kk 0 :

3.2 Calculate the excitation spectrum of a gas of charged bosons interacting

through the Coulomb potential.

3.3 An alternative approach to the Bogoliubov theory of interacting bosons

¬rst expresses the Hamiltonian (3.4.1) in terms of and y . One then

argues that the ground-state energy is found by evaluating hjHji,

where ji is the vacuum state such that k ji ¼ 0 for all k. Show that

minimization of this ground-state energy with respect to the k leads to

the same results as the approach given in the text.

3.4 Optical and Acoustic Modes The problem of the chain of masses and

springs is modi¬ed by the introduction of extra springs connecting every

second particle. Then, with l ¼ na,

!

X 121

H¼ pl þ K1 °yl À ylþa Þ2

2M 2

all n

X1

þ K2 °yl À ylþ2a Þ2 :

2

even n

Find the phonon frequencies for this system. [Hint: First make the

transformations:

X Àiql X iql

°1Þ °2Þ

yq ¼ e yl ; yq ¼ e yl :

n even n odd

3.5 A particle is bound in a one-dimensional potential, V°xÞ, which can be

approximated for small x by

V ¼ 1 m!2 x2 À x3 :

2

Ð

Show how the mean position of the particle, *x dx, changes with

the energy of the eigenstates when is small. [Hint: Use perturbation

theory on the harmonic oscillator states by writing x3 and x in terms of

ay and a.] This illustrates the fact that the thermal expansion of a crystal

is due to anharmonic terms in the potential energy.

123

Problems

3.6 Magnon“phonon Interactions If we allow the spins in the Heisenberg

Hamiltonian (3.11.1) to be displaced by the presence of phonons in

the lattice, then we must allow the constants Jll 0 to be functions of

the displacements yl and yl 0 . At low temperatures these displacements

will be small and one can put

Jll 0 °yl ; yl 0 Þ ¼ Jll 0 °0; 0Þ þ yl Á Kll 0 À yl 0 Á Kll 0 :

Rewrite the total Hamiltonian in terms of magnon and phonon anni-

hilation and creation operators, a°magnonÞ , ay °magnonÞ , a°phononÞ , and

q q q

y °phononÞ

.

aq

3.7 Show that the Hamiltonian for magnon“phonon interactions derived in

Problem 3.6 exhibits conservation of the total number of magnons, in

that

!

X

ay aq ¼ 0

H; q

q

when ay and aq are magnon operators.

q

3.8 The result of Problem 3.7 is no more than an expression of the

conservation of total angular momentum in the z-direction. Noncon-

servation of total magnon number can occur when there is interaction

between the electron spin and the spin of the nucleus at a particular site.

Express in terms of magnon operators the Hamiltonian of a Heisenberg

ferromagnet interacting with a nuclear spin of 1 0 at one particular

2

site.

y

3.9 Evaluate the expectation value of nk ¼ a k ak in the ground state ji of

the Bogoliubov picture of helium. [Hint: Express nk in terms of the

-operators and then make use of the fact that k ji ¼ 0 for all k.]

3.10 Express the ground-state ji of the Bogoliubov picture of helium in

y

terms of the operators a k and the vacuum state j0i.

ij

3.11 Prove the statement preceding Eq. (3.6.3) which says that the matrix Vq

has three mutually perpendicular eigenvectors.

124 Boson systems

3.12 Consider a large number N of spinless interacting bosons of mass m in a

large one-dimensional box of length L. There are periodic boundary

conditions. The particles interact via a delta-function potential, and so

the Hamiltonian is

X X

y y y

H¼ Ak2 a k ak þ °V=2LÞ a kÀq a k 0 þq ak 0 ak

k;k 0 ;q

k

with A and V constants. The sums proceed over all permitted values of

k, k 0 , and q. That is, the terms with q ¼ 0 are not excluded from the

sum.

(a) Calculate the energy of the ground state of the noninteracting

system.

(b) Calculate the energy of the ground state of the interacting system in

the Hartree approximation.

(c) Estimate the speed of low-frequency sound waves in this system.

Chapter 4

One-electron theory

4.1 Bloch electrons

The only model of a metal that we have considered so far has been the gas of

interacting electrons. A real metal, of course, contains ions as well as elec-

trons, and we should really include the ionic potentials in the Hamiltonian,

rather than the uniform background of positive charge that we used to

approximate them. The di¬culty of the many-electron problem is such, how-

ever, that the loss of translational invariance caused by adding an ionic

potential V°r) to the Hamiltonian proves disastrous. Even in the Hartree“

Fock approximation, for example, it becomes impossible to write the energy

in a closed form. When there was no ionic potential we could write the

wavefunction of the noninteracting system as

Y y

jÈi ¼ ck j0i; °4:1:1Þ

jkj<kF

y

where the operators ck created electrons in plane-wave states. But if there is

an additional potential applied to the system we might ¬nd the energy to be

lower in the state È if we replaced the cy by operators cy that create electrons

k

in states that are not plane waves. The de¬nition of the Hartree“Fock

approximation in this general case is then taken to be that È must be a

Slater determinant and must make the expectation value of the

Hamiltonian a minimum. It becomes very laborious to work out what states

the cy should create to ful¬ll this condition.

We are saved from what seems to be an impossible task by three fortunate

features of the problem. The ¬rst of these is the fact that we can initially

ignore the thermal motion of the lattice, and study the motion of the elec-

trons in the potential of a stationary array of ions. This is known as the

adiabatic or Born“Oppenheimer approximation. Its justi¬cation appears when

125

126 One-electron theory

we apply the motion of the ions as a perturbation, and calculate the e¬ects of

the gas of phonons interacting with the electrons. This is discussed in detail

in Chapter 6. There we ¬nd the corrections to the electron energies to be

comparatively small, and of importance only in special circumstances like

superconductivity.

The second useful fact is that the electrons in the core of an atom are so

tightly bound that they are not signi¬cantly perturbed by the motion of

electrons at the Fermi energy. The e¬ects of the core electrons on the proper-

ties of a solid are limited to their repulsion of other electrons through elec-

trostatic forces, and to the e¬ective repulsion that arises from the demands of

the Exclusion Principle that no two electrons of the same spin occupy the

same location. The consequences of the requirement that the wavefunctions

of the higher-energy electrons be orthogonal to those of the core states will be

discussed in Section 4.4.

The third happy feature of the problem is the most important. It is the fact

that the properties of a system of interacting electrons in a static lattice

potential can be found by solving a set of related one-electron Schrodinger

¨

equations. The formalism by means of which this equivalence can be proved

is known as density functional theory, and is the topic of Chapter 5. For now

we assume this result, and turn our attention to the solution of the

Schrodinger equation for a single electron moving in the potential V°r) due

¨

to a periodic crystal lattice. That is, we need to solve the equation

02 2

À r °rÞ þ V°rÞ °rÞ ¼ E °rÞ; °4:1:2Þ

2m

where the lattice potential has the property that, for all lattice vectors l,

V°rÞ ¼ V°r þ lÞ.

If the potential were zero, the wavefunctions would be of the form dis-

cussed in Section 2.1,

k ¼ À1=2 eik Á r ;

with energies given by

02 k2

Ek ¼ :

2m

If we now slowly switch on the lattice potential the wavefunctions are per-

turbed to a new form

¼ À1=2 uk °rÞeik Á r : °4:1:3Þ

k

127

4.1 Bloch electrons

Because the lattice potential is periodic, this modi¬cation of the wavefunction

is also periodic, and so

uk °rÞ ¼ uk °r þ lÞ:

The fact that the wavefunctions can be written in the form (4.1.3) is known

as Bloch™s theorem, and can be proved as follows. Let us consider any solu-

tion of the Schrodinger equation

¨

H°rÞ °rÞ ¼ E °rÞ:

On relocating the origin of r we ¬nd

H°r þ lÞ °r þ lÞ ¼ E °r þ lÞ:

But since V°r þ lÞ ¼ V°rÞ it follows that

H°rÞ °r þ lÞ ¼ E °r þ lÞ:

Thus any linear combination of the °r þ lÞ for di¬erent l gives a valid

eigenstate of energy E. Let us in particular choose the combination

X

°r þ lÞeÀik Á l

k °rÞ ¼

l

X

ik Á r

°r þ lÞeÀik Á °rþlÞ :

¼e

l

Because the sum is over all l it must be a periodic function of r with the

period of the lattice, and can be identi¬ed with the function uk °rÞ of

Eq. (4.1.3). If we now impose the cyclic boundary conditions that we used

in Section 2.1 for free electrons, and demand that

k °rÞ ¼ k °r þ LÞ

for three di¬erent large lattice vectors L, we shall clearly have the condition

that all the components of k must be real.

An electron having a wavefunction of the form (4.1.3), where uk °rÞ is

periodic in the lattice, is known as a Bloch electron. In terms of uk °rÞ the

Schrodinger equation (4.1.2) becomes

¨

02

À °r þ ikÞ2 þ V°rÞ uk °rÞ ¼ E k uk °rÞ; °4:1:4Þ

2m

128 One-electron theory

which we write as

Hk uk °rÞ ¼ E k uk °rÞ:

The reciprocal lattice vectors, g, de¬ned in Section 3.6, have the property

that eig Á l ¼ 1 for all g and l. It follows that any function that may be written

P

in the form g ag eig Á r is periodic with the periodicity of the lattice. The

converse may also be shown to be true for any well behaved periodic func-

tion, which allows us to expand V°rÞ and uk °rÞ in Fourier series of the plane

waves eig Á r . We can thus write

X X

ig Á r

ug °kÞeig Á r ;

V°rÞ ¼ Vg e ; uk °rÞ ¼

g g

where

° °

À1 ÀigÁr À1

eÀig Á r uk °rÞ dr:

Vg ¼ ug °kÞ ¼

e V°rÞ dr;

If we substitute these expressions into (4.1.4) and equate the various Fourier

components we ¬nd the in¬nite set of equations

X

02

°g þ kÞ ug °kÞ þ

2

Vg 0 ugÀg 0 °kÞ ¼ E k ug °kÞ;

2m 0

g

which can in principle be solved for E k and ug °k). The graph of E k against k is

known as the band structure, for reasons that will soon be apparent. It is

interesting to note that it is only certain Fourier components of the atomic

potentials that contribute to these equations which determine the E k . If the

lattice potential is supposed to be due to a superposition of atomic potentials

Va °rÞ, so that

X

V°rÞ ¼ Va °r À lÞ;

l

one can write the Fourier transform of V°rÞ as

°

À1

eÀiq Á r V°rÞ dr

V°qÞ ¼

X °

0

À1 Àiq Á l

Va °r 0 ÞeÀiq Á r dr 0 :

¼ e

l

129

4.1 Bloch electrons

Thus the Fourier transform of the lattice potential is expressed in terms of the

Fourier transform, Va °qÞ, of the atomic potential Va °rÞ. However, because

the sum over l vanishes unless q is equal to a reciprocal lattice vector, g, the

energies of the Bloch electrons depend only on these particular terms in V°qÞ.

This is illustrated in Fig. 4.1.1.

If one were to take a free electron and slowly switch on the lattice potential,

the wavefunction would be gradually transformed from a plane wave to a

Bloch wave of the form (4.1.3). In general the value of k is then well de¬ned,

since it does not change from its original value, and the plane wave merely

becomes modulated by the function uk °rÞ. It is, however, possible to write the

Figure 4.1.1. V°qÞ is the Fourier transform of the potential V°rÞ due to a single atom.

When one sums the contributions of all the atoms in the lattice the only parts of V°qÞ

that do not vanish occur when q is equal to a reciprocal lattice vector g.

130 One-electron theory

Bloch wavefunction in the form

¼ À1=2 ½uk °rÞeig Á r ei°kÀgÞ Á r :

k °rÞ

Because both uk °rÞ and eig Á r are periodic, so is their product. This means that

the wavefunction can be considered as obeying the Bloch condition (4.1.3)

not just for one value of k, but for any value such as k À g. This is not the

same thing as the equivalence that was found in Section 3.6 of the coordinate

yq to yqþg , for here we are dealing with a set of wavefunctions that are all

di¬erent although they happen to obey the Bloch equation with the same k.

The function uk °rÞeig Á r is not the same as ukþg °rÞ. The di¬erence arises from

the fact that the phonon problem has only a limited number of degrees of

freedom, while the Schrodinger equation has an in¬nity of solutions. The

¨

Bloch theorem allows these solutions to be classi¬ed either according to the

original value of k before the potential was switched on, or else according to

the value of k after some reciprocal lattice vector has been added to it. The

¬rst Brillouin zone was de¬ned in Section 3.6 as the volume in k-space con-

taining all those points for which jkj jk þ gj for all g. It is sometimes

convenient to classify Bloch states by specifying the value of k reduced to

lie in the ¬rst Brillouin zone. To de¬ne the state completely it is then also

necessary to de¬ne a band index which is related to the value of g necessary to

achieve this reduction. In general the band index is de¬ned as the number of

di¬erent values of g (including zero) for which jkj ! jk þ gj. This is illustrated

in Fig. 4.1.2 for a hexagonal lattice in two dimensions. The point a cannot be

Figure 4.1.2. In this two-dimensional hexagonal lattice the point a lies in the ¬rst

Brillouin zone. The point b lies in the second Brillouin zone while the point c lies in

the third zone.

131

4.1 Bloch electrons

brought closer to the origin À by addition of any nonzero reciprocal lattice

vector. It thus has band index 1, and is said to be in the ¬rst Brillouin zone.

The point b can be brought closer to À only by addition of the vector g that

takes it to b 0 . It thus has band index 2 and is in the second Brillouin zone.

Point c is in the third Brillouin zone, since it may be reduced either to c 0 or c 00 ,

and so on. One thus has two alternative schemes for depicting the band

structure E k . The ¬rst scheme, in which k is allowed to take on any value

consistent with the boundary conditions, is known as the extended zone

scheme. In the second, the band structure is written in the form E °nÞ , where

k

n is the band index, and now k is the reduced wavenumber, restricted to lie

within the ¬rst Brillouin zone. This is known as the reduced zone scheme. The

usefulness of the reduced zone scheme is a consequence of the most impor-

tant property of Bloch electrons, namely that surfaces of discontinuity in

general exist in E k in the extended zone scheme at all the boundaries between

Brillouin zones. In the reduced zone scheme, however, E °nÞ is always a con-

k

tinuous function of k.

This may be made plausible by the following argument. When k does not

lie on a Brillouin zone boundary, so that there is no nonzero g for which

jkj ¼ jk þ gj, then k is certainly di¬erent from its complex conjugate * . k

Now a complex wavefunction always carries a current, since then * r k À k

k r * cannot vanish, and so the application of an electric ¬eld to the system

k

will change the energy of the electron even if the ¬eld is vanishingly small.

This energy change comes about by the mixing of k with states of neighbor-

ing wavenumber whose energies are arbitrarily close to E k . If k lies on a zone

boundary, however, k may be real and still satisfy the Bloch condition.

There is then no current carried by these states, and so there are not neces-

sarily states whose energies are arbitrarily close to E k both from above and

below. One may then ¬nd ranges of energy over which there are no states.

One may de¬ne a density of states in energy D°E) by the relation

X

D°EÞ ¼ °E À E k Þ;

k

where the sum proceeds over all values of k in the extended zone scheme. For

an in¬nite crystal this spectrum of delta-functions becomes a continuum, and

can be plotted to give a curve that might, for example, be of the form shown

in Fig. 4.1.3. The states for which k is in the ¬rst Brillouin zone contribute to

the shaded part of D°EÞ, while states in the second and higher Brillouin zones

give the rest. The fact that there are bands of energy for which D°EÞ vanishes

justi¬es the naming of E°k) as the band structure of the crystal.

132 One-electron theory

Figure 4.1.3. In this model all the states in the valence band (shown shaded) are

occupied, while all those in the conduction band are empty. This system would be an

insulator, since a ¬nite amount of energy would be required to create a current-

carrying elementary excitation.

4.2 Metals, insulators, and semiconductors

If a crystal is composed of N atoms then each Brillouin zone contains N

allowed values of k. As an electron has two spin eigenstates each zone may

then contain 2N electrons, which is equivalent to saying that there are two

states per zone per atom in a Bravais lattice. In some elements it happens that

the number of electrons that each atom possesses is just su¬cient to populate

all the states below one of the gaps in D°EÞ. This has a very important

e¬ect on a number of the properties of these elements, and especially on

the electrical properties. At zero temperature all the states below one of the

band gaps are ¬lled, while all the states above it are empty. One thus is

required to provide an energy equal to the width, 2Á, of the band gap if

one is to excite an electron from the lower band (known as the valence band)

to the upper one (known as the conduction band). Thus the crystal will not

absorb electromagnetic radiation of frequency ! if 0! < 2Á. In particular the

crystal will not absorb energy from a weak electric ¬eld of zero frequency “

that is, it is an insulator. In a metal, on the other hand (in which term we

include such good conductors as ReO3 or RuO2 ), the number of electrons is

such that at zero temperature there are occupied and unoccupied states

di¬ering in energy by an arbitrarily small amount. The energy of the most

energetic electron (the Fermi energy, E F , which is the chemical potential

at zero temperature) does not coincide with a gap in the density of states.

The crystal can then absorb radiation of low frequency, which makes it a

conductor.

133

4.2 Metals, insulators, and semiconductors

The Fermi surface of a metal of Bloch electrons is de¬ned as the locus of

values of k for which E°kÞ ¼ E F . Whereas for free electrons this surface was a

sphere of radius 0À1 °2mE F Þ1=2 , for Bloch electrons it is distorted to a greater

or lesser extent from spherical shape, particularly by the e¬ects of the dis-

continuities in E°kÞ. Because the discontinuities of E°kÞ occur only at the zone

boundaries, the Fermi surface remains continuous in the reduced zone

scheme. If, however, there are states with di¬erent band indices contributing

to the Fermi surface then the surface will consist of two or more sheets. In

experiments on metals in magnetic ¬elds one can sometimes separately dis-

tinguish the e¬ects of the various parts of the Fermi surface.

We started our discussion of Bloch electrons by considering the e¬ects of

the lattice potential as a perturbation of the free-electron wavefunction. We

might, however, have approached the problem from the opposite extreme,

and looked at Bloch states as perturbations of atomic orbitals. A useful

picture of this viewpoint is given in Fig. 4.2.1(a), which shows the degenerate

energy levels of a crystal of widely spaced atoms broadening into bands as the

lattice spacing a is reduced from some initial large value. In the case of a

simple monovalent metal like sodium or potassium, with only one electron in

its outer shell, the lower band shown in Fig. 4.2.1(b) would always be only

half-¬lled, since there are two possible spin states for each value of k. The

Figure 4.2.1. Atomic energy levels found when the interatomic spacing a is

large broaden into bands when a is reduced (a). Monovalent atoms have half-¬lled

bands (b). Divalent atoms have ¬lled valence bands at large a but become

conductors when a is small and the bands overlap (c).

134 One-electron theory

electrical conductivity would then be high, provided a were su¬ciently small

to allow passage from one atom to another.

For a crystal of divalent atoms like magnesium, the lower band will be

¬lled, as in Fig. 4.2.1(c). For large a the crystal would be an insulator, as by

symmetry the occupied states would carry zero net total current. That mag-

nesium is a metal must be a consequence of a being small enough that two

bands overlap, allowing each to be partially ¬lled. Magnesium does indeed

crystallize in the hexagonal close-packed structure, and is a good metal.

Carbon, on the other hand, can crystallize into the much more open diamond

structure. Being tetravalent it then has a ¬lled valence band, and is an excel-

lent electrical insulator, with a band gap of over 5 eV.

Silicon and germanium also crystallize in the diamond structure, but have

much smaller band gaps, of the order of 1.1 eV for Si and 0.67 eV for Ge.

This makes them semiconductors, as it is possible at room temperature to

excite a few electrons from the valence band into the conduction band.

Thermal energies at room temperature are only about 0.03 eV, giving a

probability of thermal excitation of only about eÀ22 , which is 10À10 , in

germanium. Intrinsic semiconductors, like pure Ge and Si, thus have quite

low conductivities.

A much larger conductivity, and hence more technological usefulness, can

be obtained by adding impurities to produce an extrinsic semiconductor.

Adding small amounts of a pentavalent element such as arsenic to silicon

increases the number of electrons available for conduction. This doping with

donor atoms moves the chemical potential up from its previous position in

the middle of the energy gap, and into the conduction band. The material is

now known as n-type silicon. The converse process of doping with trivalent

acceptor atoms produces p-type silicon, whose chemical potential lies in the

valence band.

The union of p-type and n-type material to form a p“n junction enables a

wealth of useful phenomena to occur. In a photovoltaic cell, electrons fed

into the valence band of p-type material may be elevated into the conduction

band by a photon of sunlight. They then emerge from the n-type material at a

higher potential and can do useful work before being returned to the lower

potential of the p-type material. In a light-emitting diode, or LED, the

reverse process occurs. Electrons at a high enough potential to be fed into

the conduction band of the n-type material can move to the p-type side and

then fall into the valence band, emitting a photon of light as they do so. From

more complex arrangements of p-type and n-type semiconductors and

metals, that ubiquitous foundation of our technological society “ the tran-

sistor “ can be constructed.

135

4.3 Nearly free electrons

4.3 Nearly free electrons

There are now powerful computer codes that can produce credible forms for

E°kÞ for any given periodic potential. In materials in which there are several

atoms in each unit cell these band structures may be exceedingly complex,

and so it is useful to start by looking at some much simpler situations. In this

way we can appreciate some of the concepts that play important roles in

determining material properties. The simplest approximation one can make

is to neglect the lattice potential altogether, except in as much as to allow the

existence of in¬nitesimal discontinuities in E°kÞ at the zone boundaries. The

Fermi surface one obtains then consists of portions of the free-electron

sphere reduced to lie in the ¬rst Brillouin zone. This is known as the

remapped free-electron model. An example of this construction in two dimen-

sions is shown in Fig. 4.3.1 for a hexagonal reciprocal lattice. The circle of the

extended zone scheme is reduced to a central portion, A, derived from the

second Brillouin zone and a group of small regions, B, derived from the third

Brillouin zone. The occupied electron states are always on the concave side of

the boundaries, and so the surface A is seen to contain unoccupied states. It is

consequently known as a ˜˜hole surface.™™ The portions B, on the other hand,

contain electrons. We note that the use here of the term ˜˜hole™™ is quite

Figure 4.3.1. The circle represents the free-electron Fermi surface in the extended

zone scheme. In the remapped representation of the reduced zone scheme it forms a

central ˜˜hole™™ surface and a group of small electron surfaces.

136 One-electron theory

distinct from that of Chapter 2, where a hole was simply the absence of an

electron from a state below the Fermi energy. In the context of band struc-

tures, a hole is also an unoccupied state, but one that has an additional

property: the energy of the state decreases as one moves away from the

interior of the constant-energy surface, as is the case for the surface A in

Fig. 4.3.1. A hole state in band-structure parlance may have an energy either

below or above the Fermi energy. We shall see some examples of hole states

in later sections of this chapter.

The way in which the remapped free-electron Fermi surface is derived from

the Fermi surface in the extended zone scheme is seen most easily when the

¬rst Brillouin zone of Fig. 4.3.1 is repeated periodically to form the scheme

shown in Fig. 4.3.2. In this repeated zone scheme one sees that the various

parts of the Fermi surface are formed in the ¬rst Brillouin zone when a free-

electron sphere is drawn around each point in the reciprocal lattice.

The simplest approximation that can be made that includes the e¬ect of the

lattice potential is known as the model of nearly free electrons. Here one

assumes that only a certain small number of di¬erent plane waves combine

to form the Bloch wave k . The relative coe¬cients of these plane waves

Ð

* H k dr, and this gives an approximation for

are then varied to minimize k

E°kÞ and the wavefunction in the ¬rst Brillouin zone. The form of E°kÞ in the

second zone is found by minimizing the integral by varying wavefunctions

restricted to be orthogonal to those in the ¬rst zone, and so on. In practice

Figure 4.3.2. This picture shows the model of Fig. 4.3.1 in the repeated zone scheme,

and is formed by periodically repeating the ¬rst Brillouin zone of the reduced zone

scheme.

137

4.3 Nearly free electrons

the labor involved can be greatly reduced if one chooses to include in k only

those plane waves that one thinks will enter with large coe¬cients. Thus if a

point k in the extended zone scheme is much nearer to one zone boundary

than any other, then one might approximate k by a mixture of eik Á r and

ei°kþg1 Þ Á r where g1 is that reciprocal lattice vector that makes jkj close to

jk þ gj. Physically this is equivalent to saying that the plane wave eik Á r will

only have mixed with it other plane waves whose energies are close to its own.

Thus, if we write

¼ À1=2 ½u0 °kÞeik Á r þ u1 °kÞei°kÀg1 Þ Á r °4:3:1Þ

k

and

X

Vg eig Á r

V°rÞ ¼

g

with the zero of potential energy de¬ned to make V0 vanish, then

°

02

*H dr ¼ ½u* u0 k2 þ u* u1 °k À g1 Þ2 þ u* u1 Vg1 þ u* u0 VÀg1 :

2m 0 1 0 1

The normalization condition is

°

* dr ¼ u*u0 þ u* u1 ¼ 1:

0 1

We now minimize the energy by varying the wavefunction. According to

Lagrange™s method of undetermined multipliers we can take account of the

normalization condition by writing

°

°

*H dr À * dr ¼ 0:

But now we can immediately identify the multiplier with the energy E, for

this is the only way we can ensure that just multiplying by a constant will

leave the term in brackets unchanged. On di¬erentiating partially with

respect to u* and u* and putting ¼ E we ¬nd

0 1

02 k2

u þ Vg1 u1 ¼ Eu0

2m 0

02 °k À g1 Þ2

u1 þ VÀg1 u0 ¼ Eu1 :

2m

138 One-electron theory

For these equations to be consistent the determinant of the coe¬cients must

vanish, and so

22

0 k

2m À E Vg 1

¼0

0 °k À g1 Þ

2 2

VÀg1 ÀE

2m

and thus

(

02 k2 02 °k À g1 Þ2

1

E¼ þ

2 2m 2m

s¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬)

2 2

0 °k À g1 Þ2 02 k2

Æ À þ 4Vg1 VÀg1 : °4:3:2Þ

2m 2m

The two possible signs of the square root correspond to the nonuniqueness of

the wave vector k that characterizes the Bloch state. That is to say, this

expression tells us the energy of the electron in the ¬rst and second energy

bands. If k is chosen to lie in the ¬rst Brillouin zone then the negative square

root will give the energy of the state that is formed from the wavefunction

eik Á r when the lattice potential is turned on slowly. The positive square root

will refer to the state formed from ei°kÀg1 Þ Á r , and which was originally in the

second Brillouin zone.

It is interesting to note that this formula (4.3.2) for the energy of a Bloch

state is identical to the one we should obtain from the use to second order of

the Brillouin“Wigner perturbation theory described in Section 2.5. This result

is peculiar to the two-plane-wave assumption of expression (4.3.1), and

should not be looked upon as indicating that an approach using perturbation

theory is necessarily equivalent to a variational approach.

The general expression of the model of nearly free electrons is found when

any ¬nite sum of plane waves is chosen as the trial wavefunction. Then if

X

¼ À1=2 ug °kÞei°kþgÞ Á r ;

k

g

minimization of the energy leads to the series of equations

X

02

EÀ °k þ gÞ2 gg 0 À Vg 0 Àg ug °kÞ ¼ 0:

2m

g

139

4.3 Nearly free electrons

For these to be consistent the determinant of the coe¬cients of the ug °k) must

vanish, and so

02

EÀ °k þ gÞ gg 0 À Vg 0 Àg ¼ 0:

2

°4:3:3Þ

2m

This polynomial in E has as many solutions as there are plane waves in the

expansion of k , and reduces to expression (4.3.2) when that number is only

two.

We can observe some of the e¬ects of the lattice potential in the simplest

three-dimensional model, which is known as ˜˜sandwichium.™™ Here the lattice

potential is just 2V cos gx, and so the loci of the points jkj ¼ jk þ gj are just

the planes de¬ned by kx ¼ Æ°n=2Þg. The two-plane-wave version of the

nearly-free-electron model then gives expression (4.3.2) for the energy,

which in the neighborhood of kx ¼ 1 g becomes

2

(

0 2

1 02 k2 02 °kx À gÞ2

x

E¼ °ky þ kz Þ þ

2 2

þ

2m 2 2m 2m

s¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬)

2 2

0 °kx À gÞ2 02 k2 x

Æ À þ 4V 2 ; °4:3:4Þ

2m 2m

which we write as

02 2

E¼ °ky þ k2 Þ þ E x °kx Þ:

z

2m

The form of E x °kx ) is shown in Fig. 4.3.3 for two di¬erent values of V. The

sign of the square root has been chosen so that E x ! 02 k2 =2m as V ! 0,

x

which means that we are using the extended zone scheme. One sees that it

is only when kx is in the vicinity of 1 g that E x deviates appreciably from its

2

free-electron value, and that a discontinuity in E x does indeed occur when

kx ¼ 1 g. The magnitude of this discontinuity is 2V.

2

The shapes of the surfaces of constant energy are shown in Fig. 4.3.4,

where their intersections with the plane kz ¼ 0 are plotted. For low energies

the surfaces are close to spherical; then as kx approaches the zone boundary

E x °kx ) starts to fall below the free-electron value, and the magnitude of

kx becomes correspondingly greater for a given energy. One says that the

constant-energy surfaces are ˜˜pulled out™™ towards the zone boundary. The

140 One-electron theory

Figure 4.3.3. The variation of energy with kx in the sandwichium model is shown by

the solid line for the case where V is small compared with 02 g2 =8m. As V is

increased, the discontinuity at the zone boundary becomes larger, as illustrated by

the dashed curve.

Figure 4.3.4. The variation of energy with the xÀy component of wavenumber in the

sandwichium model is shown in this picture, in which the lines of constant energy are

drawn in the plane in which kz ¼ 0.

141

4.3 Nearly free electrons

lowest-energy surface to meet the zone boundary is, in fact, pulled out to a

conical point at the place where it does so. We can verify this by writing

k ¼ ° 1 g À x ; ky ; kz Þ

2

and expanding the square root in (4.3.4). For the negative root we ¬nd

2

02 h2 2 02 2 02 g2

1

E™ ÀV þ °k þ kz Þ À

2

À1

g

2m y 2m x 4mV

2m 2

02 2 02 2 2E g

x

¼ Eg À V þ °ky þ k2 Þ À À1 °4:3:5Þ

z

2m 2m V

where we have used the abbreviation

2

02 1

¼ Eg:

g

2m 2

Thus when

E ¼ Eg À V

the energy surfaces are given by

2E g

k2 þ k2 ™ À 1 2 :

y z x

V

This equation de¬nes a cone whose axis is in the x-direction.

When the energy is greater than E g þ V one also ¬nds energy states in the

second band. Then the positive square root is chosen in Eq. (4.3.4) and

02 2 h2 2 2E g

x

E ™ Eg þ V þ °ky þ kz Þ þ

2

þ1 :

2m 2m V

The constant-energy surfaces are thus approximately spheroidal in this

region of k-space. If it were not for the factor of (2E g =VÞ þ 1 that multiplies

the term in 2 the surfaces would be spherical, and the free-electron band

x

structure that one ¬nds near k ¼ 0 would merely be repeated at the bottom of

the second band. It is possible to exploit this similarity by considering the

electron energy to be given by the free-electron relationship, with the excep-

tion that the inverse of the electron mass must now be considered a tensor.

142 One-electron theory

Thus we can write

E ™ E g þ V þ 1 02 k Á MÀ1 Á k;

2

with the understanding that the origin of k is taken to be the point ( 1 g; 0; 0Þ

2

and that

0 1

1 2E g

Bm V þ 1 0 0C

B C

B C

B 0 C:

1

MÀ1 B C

0

B C

m

@ 1A

0 0

m

The inverse-e¬ective-mass tensor in this problem is thus anisotropic in that

the energy increases more rapidly as a function of kx than of ky or kz . If the

lattice potential is weak enough, then (M À1 Þxx may be many times larger than

(M À1 Þyy . In this case one says that it is a light electron for motion in the

x-direction.

It is also possible to interpret the band structure in the ¬rst band in these

terms by writing Eq. (4.3.5) in the form

E ™ E g À V þ 1 02 k Á MÀ1 Á k:

2

In this case (M À1 Þxx is negative while (M À1 Þyy is still positive. The electron is

said to exhibit hole-like behavior for motion in the x-direction. If there were

also a periodic potential 2V cos gy then there would also be the possibility of

(M À1 Þyy being negative for some points in k-space [Problem 4.5], and for a

three-dimensional crystal the states at the corners of the ¬rst Brillouin zone

will be completely hole-like.

The particular constant-energy surface that represents the boundary

between ¬lled and empty states in a metal is again known as the Fermi

surface. The shape of the Fermi surface depends on the crystal structure,

the lattice potential, and the electron density, and is di¬erent for every

metal. For some, such as sodium or potassium, the lattice potential is

weak and the Fermi surface deviates little from a sphere. For others, and

in particular the polyvalent metals, the Fermi surface is far from

spherical, and may be formed from regions in several di¬erent Brillouin

zones.

143

4.4 Core states and the pseudopotential

4.4 Core states and the pseudopotential

In using the nearly-free-electron approximation we have con¬ned our interest

for the most part to the ¬rst few Brillouin zones. We have tacitly assumed

that the conduction-electron states can be found by solving the Schrodinger

¨

equation for electrons moving in a weak potential that is composed of the

Coulomb attraction of the nuclei screened by the presence of the electrons in

the ¬lled atomic shells.

This picture is not justi¬able on two counts. Firstly, we must remember

that the Exclusion Principle demands that the wavefunctions of the conduc-

tion electrons be orthogonal to those of the electrons in the ¬lled atomic

shells, or core states as we shall call them. Secondly, we can calculate that

for real solids the lattice potential is too strong for the nearly-free-electron

approach to be valid when only a few plane waves are used. That is to say,

the lowest Fourier components Vg of the lattice potential are not small

compared with 02 g2 =8m for the smallest reciprocal lattice vectors.

However, while either one of these considerations alone would prevent us

from using the nearly-free-electron approximation, it happens that taken

together they present a tractable situation. Because we are now going to

take account of the Bloch states of the core electrons, the determinant

(4.3.3) must now be much larger than the 2 ‚ 2 form that we have just

been using. If we wish to apply this method to potassium, for instance, we

must calculate that with an atomic number of 19 this metal has enough

electrons to ¬ll 9 1 Brillouin zones. This means that a large number of

2

Fourier components of the lattice potential must be included if we are to

¬nd energy discontinuities at all the relevant zone boundaries. But if the

conduction states and the Fermi surface are to be located in higher

Brillouin zones than the ¬rst few, then their k-vectors in the extended zone

scheme must be very large. This means that if they are to be scattered by the

lattice to a state of approximately equal energy, then it will be mostly large

reciprocal lattice vectors that will describe the di¬erence in wavenumbers of

the two states. Consequently it will be the Fourier components Vg of the

lattice potential corresponding to large g that will describe the energy dis-

continuities at the zone boundaries. Because these components are much

smaller than those corresponding to small g, the validity of the nearly-

free-electron approximation is restored as a means of calculating the band

structure of the conduction bands.

The use of plane-wave expansions of all the electron states in solids has,

however, one big disadvantage. We have not so far made use of the fact that

the core states are very highly localized around the nuclei of the atoms. If the

144 One-electron theory

core level in the free atom is tightly bound, then its kinetic energy starts to

become negative at a short distance from the nucleus, and the wavefunction

decays rapidly outside this distance. When many of these atoms are brought

together to form a solid, the wavefunctions of the core states of di¬erent

atoms do not overlap appreciably, and the tendency of the degenerate core

states to broaden into a band is very small. This suggests that it would be

more appropriate to expand the k , not in terms of plane waves, but in terms

of the atomic wavefunctions, i °rÞ. Because we know that Bloch™s theorem

must still be obeyed, we ¬rst form linear combinations of normalized atomic

wavefunctions centered on di¬erent atoms by writing

X ik Á l

À1=2

i °rÞ ¼ N e i °r À lÞ: °4:4:1Þ

k

l

We then expand in terms of these, and write

k

X

°rÞ ¼ ui °kÞk °rÞ:

i

k

i

This formalism is known as the Linear Combination of Atomic Orbitals, or

LCAO, method.

For an exact solution we should include not only the bound atomic states,

but also states of positive energy, so that we have a complete set in which to

expand k . In practice, however, this method is still useful when only a few

atomic states are assumed to contribute. We take matrix elements of the

Hamiltonian between the states k °rÞ and write a secular equation analogous

i

to Eq. (4.3.3) of the form

jDj ¼ 0

where

°

Ã

Dij ¼ k °rÞ°H À EÞk °rÞ dr:

i j

We note that the nonorthogonality of the k must be taken into account.

i

The LCAO method is widely used for practical computations, as are varia-

tions of it in which k is expanded in eigenstates of other spherically sym-

metric potentials. Our goal in this section, however, is not to provide detailed

instructions for performing these calculations. It is rather to point out the

physical signi¬cance of the presence of the core states in reducing the e¬ective

lattice potential. To this end we make the most drastic simpli¬cation possible,

which is known as the method of tight binding, and assume that only the

diagonal elements contribute to this determinant. The energy is then given

145

4.4 Core states and the pseudopotential

simply by the expectation value of H in the state k °rÞ. Now

i

X ik Á l

Hk °rÞ ¼ N À1=2 e Hi °r À lÞ;

i

l

and it is convenient to consider the lattice potential V°r) that acts on i °r À lÞ

as the sum of two separate terms “ the potential due to an atom located at l

and that due to all the other atoms. We thus write

02 2

H¼À r þ Va °r À lÞ þ W°r À lÞ

2m

where Va °rÞ is the atomic potential and W°rÞ is the di¬erence between the

lattice potential and the atomic potential (Fig. 4.4.1). We expect W°rÞ to be

Figure 4.4.1. In the method of tight binding the periodic lattice potential V°rÞ is

considered to be the sum of an atomic potential Va °rÞ and a correction W°rÞ that is

small in the neighborhood of the origin. The potential W°rÞ is then treated as a

perturbation acting on the known atomic wavefunctions.

146 One-electron theory

small when jrj is less than half the interatomic distance. If we de¬ne the

energy of the atomic state i °rÞ as E i , then

X ik Á l

À1=2

Hi °rÞ ¼ N e ½E i þ W°r À lÞi °r À lÞ

k

l

X

¼ E i k °rÞ þ N À1=2 eik Á l W°r À lÞi °r À lÞ;

i

l

so that

° °

Ã Ã

k Hk dr ¼ Ei k k dr

i i i i

°

X 0

þ N À1 eik Á °lÀl Þ *°r À l 0 ÞW°r À lÞi °r À lÞ dr:

i

l;l 0

If we assume that the i °r À lÞ overlap appreciably only when they are

centered on adjacent atoms, the double summation reduces to a sum over

pairs of neighboring atoms. With the further approximation that this overlap

is small we ¬nd the k to be normalized, and so

i

°

E ¼ E i þ *°rÞW°rÞi °rÞ dr

i

°

X

eÀik Á L

þ *°r À LÞW°rÞi °rÞ dr;

i

L

where L are the di¬erent lattice vectors connecting nearest neighbors. As the

integrals are just constants, one ¬nds for a Bravais lattice a result of the form

X

E ¼ E0 þ W cos k Á L: °4:4:2Þ

L

The tight-binding method is suitable only when the overlap between atomic

wavefunctions is small, and this is appropriate only for states whose energies

are well below the Fermi energy. Metals whose energy bands are composed of

low-lying core states, which may be approximated by tight-binding wave-

functions, well separated from conduction states, which may be described

in the nearly-free-electron approach, are known as simple metals. For these it

is possible to reformulate the nearly-free-electron description of the conduc-

tion states by using as basis functions plane waves that have been modi¬ed so

as to be automatically orthogonal to the tight-binding states of the core

electrons. This is known as the method of orthogonalized plane waves (the

147

4.4 Core states and the pseudopotential

OPW method). One ¬rst de¬nes a set of OPW functions which are formed

from plane waves by subtracting their projections on the Bloch waves of the

occupied core states. In the tight-binding approximation for the core states

Xk °k

À1=2 ik Á r ik Á r

k °rÞ ¼ À c °rÞ c *°rÞe dr ;

e

OPW

c

or, in a briefer notation,

X

jk i ¼ jki À jk ihk jki:

OPW c c

c

We note that the k were de¬ned in the repeated zone scheme, since by (4.4.1)

c

k ¼ kþg ;

c c

while the wavenumber k in the term eik Á r , is allowed to take on all values, and

is thus considered to be in the extended zone scheme. One then again takes

matrix elements of H À E between the various OPW functions and sets

jDj ¼ 0

where

0

Dgg 0 ¼ hkþg jH À Ejkþg i:

OPW OPW

The most noticeable di¬erence between this equation and Eq. (4.3.3), in

which matrix elements were taken between pure plane waves, is in the o¬-

diagonal elements. Because the OPWs are not mutually orthogonal, we now

¬nd terms involving the energy as well as terms involving the tight-binding

Bloch energies, E k . When g 6¼ g 0

c

X

°E À E k Þhk þ gjk ihk jk þ g 0 i:

Dgg 0 ¼ Vgg 0 þ c c c

c

Because E > E k , the summation over core states has a tendency to be positive,

c

while Vgg 0 , which is just the Fourier transform of the lattice potential, tends

to be negative. The o¬-diagonal elements of D, and hence the energy dis-

continuities at the zone boundaries, are thus smaller than we should expect

from using the model based on plane waves.

We can see this another way by explicitly separating the core functions

from the sum of OPWs that form the complete wavefunction. Let us

¬rst abbreviate the operator that projects out the core Bloch states by the

148 One-electron theory

symbol Pk . Thus

X

Pk jk ihk j

c c

c

and

jkþg i ¼ °1 À Pk Þjk þ gi:

OPW

The exact wavefunction is a sum of OPWs, and so

X OPW

j ki ¼ ug °kÞ°1 À Pk Þjk þ gi;

g

which we write as

j ki ¼ °1 À Pk Þjk i

where

X

jk i ¼ uOPW °kÞjk þ gi:

g

g

Then

°H À EÞj ki ¼ °H À EÞ°1 À Pk Þjk i ¼ 0 °4:4:3Þ

and we may look upon the problem not as one of ¬nding the states j k i that

are eigenfunctions of H À E, but as one of ¬nding the states jk i that are

eigenfunctions of (H À EÞ°1 À Pk Þ. Now H is composed of kinetic energy T

and potential V, so that k must be an eigenfunction of

°T þ V À EÞ°1 À Pk Þ ¼ T À E þ V°1 À Pk Þ À °T À EÞPk :

Thus (4.4.3) can be written

°T þ Uk Þjk i ¼ Ejk i;

where the operator

Uk ¼ V°1 À Pk Þ À °T À EÞPk °4:4:4Þ

is known as a pseudopotential operator. We can argue that we expect it to

have only a small e¬ect on the pseudo-wavefunction k by noting two points.

149

4.4 Core states and the pseudopotential

Firstly we expect V°1 À Pk ) to have small matrix elements, since it is just

what is left of V after all the core states have been projected out of it. The

strongest part of V will be found in the regions near the atomic nuclei, and so

the core states, which are concentrated in the same regions, will be suitable

functions in which to expand V. The combination T À E, on the other hand,

is not so drastically a¬ected by the operation 1 À Pk . On the contrary, it

becomes reasonable to assume that (T À EÞPk has only a small e¬ect, for

there will be little overlap between k and the core states if k is indeed just

acombination of a few plane waves of small wavenumber. It thus is self-

consistent to assume that the pseudopotential is weak and that k is a

smoothly varying function. One should remember, however, that although

Uk may be small it remains an operator rather than a simple potential, and

has a dependence on energy that must sometimes be treated carefully.

Yet another way of looking at the pseudopotential is obtained by de¬ning

a new Hamiltonian H 0 formed by adding (E À HÞPk to the original

Hamiltonian. Then

H 0 ¼ H þ °E À HÞPk

X

¼Hþ °E À E k Þjk ihk j:

c c c

c

The extra terms added to H have arti¬cially raised the energies of the core

states to be equal to E, as can be seen by letting H 0 act on the k . Now since

c

0

the lowest energy levels of H are degenerate, we can state that any linear

combination of k and the k are eigenstates of H 0 , and we are at liberty to

c

choose that combination k that is most smoothly varying, and hence which

can be best approximated by the fewest plane waves. This expresses the fact

that the pseudo-wavefunction k is not uniquely de¬ned by (4.4.3), which

only says that the part of k that is orthogonal to the core states must be

equal to k .

When pseudopotentials are used in numerical calculations, their character

as operators makes itself felt. One must then deal with a nonlocal form of the

pseudopotential in which the interaction between an electron and a nucleus

depends on their coordinates separately, and not only on their relative co-

ordinates. Fortunately, the pseudopotential can usually be split into factors,

each of which depends on only one separate coordinate. This greatly reduces

the memory requirements for computer calculations. Pseudopotentials

have been developed in which the normalization of the pseudo-wavefunction

jk i has been relaxed in favor of making the pseudopotential as soft as

possible. While this leads to a slight complication in calculating the electron

150 One-electron theory

charge density, the advantage of these so-called ultrasoft pseudopotentials is

that many fewer plane waves are required in expansions of the electron

valence states.

In summary, then, pseudopotential theory serves to show that the band

structure of simple metals may be much closer to that of the remapped free-

electron model than one would be led to believe by considering the strength

of the lattice potential alone.

4.5 Exact calculations, relativistic effects, and the structure factor

Although pseudopotential theory provides a useful short cut for the calcula-

tion of band structures and Fermi surfaces of simple metals, there remain

many cases for which it is di¬cult to implement. In transition metals, for

example, the electron states of interest are formed from atomic s-states and

d-states, and thus mix core-like and free-electron-like behavior. To account

correctly for the magnetic properties of transition metals, care has to be

taken to include adequately the interactions between bands formed from

3d and 4s states and deeper-lying bands formed from atomic 3s and 3p states.

For these cases a variety of ways of solving the Schrodinger equation have

¨

been derived, and these are discussed in great detail in the many books now

available that are devoted solely to band structure calculations. Here we shall

outline just one such method which follows fairly naturally from Eq. (2.5.6),

the starting point of Brillouin“Wigner perturbation theory.

Equation (2.5.6) may be written in the form

j i ¼ aji þ °E À H0 ÞÀ1 Vj i;

where a is a constant whose value is determined by the condition that the

presence of the term aji ensure that j i reduces to the unperturbed state ji

as V tends to zero. For the present problem we take H0 to be the kinetic

energy of a single electron and V the lattice potential, so that for the Bloch

state j k i

¼ ajki þ °E À H0 ÞÀ1 Vj k i

j ki

X À1

02 k 02

jk 0 ihk 0 jVj

¼ ajki þ EÀ k i:

2m

k0

But since k is a Bloch state and V°rÞ is a periodic function, the matrix

element hk 0 jVj k i must vanish unless, for some reciprocal lattice vector g,

151

4.5 Exact calculations, relativistic effects, and the structure factor

we ¬nd that k 0 ¼ k þ g. This follows from the direct substitution

°

0

hk 0 jVj eÀik Ár

V°rÞeik Á r uk °rÞ dr

ki /

°

0

ei°kÀk Þ Á °rÀlÞ V°r À lÞuk °r À lÞ dr

¼

°

1 X Ài°kÀk 0 Þ Á l 0

ei°kÀk Þ Á r V°rÞuk °rÞ dr

¼ e

Nl

¼ 0 unless k À k 0 ¼ g:

Thus

X

02 °k þ gÞ2 À1

j k i ¼ ajki þ EÀ jk þ gihk þ gjVj k i:

2m

g

Let us de¬ne an operator Gk °EÞ by writing

X À1

02 °k þ gÞ2

Gk °EÞ EÀ jk þ gihk þ gj:

2m

g

That is, Gk °EÞ is just the operator (E À H0 ÞÀ1 restricted to act only on states

that are of the Bloch form with wavenumber k. Then

j ki ¼ ajki þ Gk °EÞVj k i:

We can verify by making use of the normalization condition hkj k i ¼ 1

that in this case the constant a can be put equal to zero (Problem 4.15), so

that

j ki ¼ Gk °EÞVj k i: °4:5:1Þ

It then follows that if one de¬nes a quantity Ã by

Ã¼h k jVj k i Àh k jVGk °EÞVj k i; °4:5:2Þ

then from (4.5.1) we ¬nd that Ã vanishes when is a solution of the

k

Schrodinger equation. That is, since

¨

ki ¼ °E À H0 Þj ki

Vj

152 One-electron theory

we are in e¬ect writing

1

Ã ¼ h k jVj k i À h k j°E À H0 Þ °E À H0 Þj ki

E À H0

¼h k j°E À HÞj k i:

But now we not only know that Ã vanishes, but also that we may determine

k by a variational approach that minimizes Ã.

In terms of integrals in r-space, expression (4.5.2) may be written

°

Ã¼ * °rÞVrÞ k °rÞ dr

k

°°

* °rÞV°rÞGk °r À r 0 ÞV°r 0 Þ 0