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X X
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 ފ 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
½ k ; k 0 Š 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 :Š
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 Šei°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 Š °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 Š 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 lފi °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
° °
à 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 à by

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

then from (4.5.1) we ¬nd that à 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
à 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 à 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
°
ü * °rÞVrÞ k °rÞ dr
k
°°
* °rÞV°rÞGk °r À r 0 ÞV°r 0 Þ 0

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