<<

. 6
( 14)



>>

Þ dr dr 0
À k °r °4:5:3Þ
k



where

X 
02 °k þ gÞ2 À1 i°kþgÞ Á °rÀr 0 Þ
0 À1
Gk °r À r Þ ¼  EÀ :
e
2m
g


An advantage claimed for this method is that Gk °r À r 0 Þ depends only E, k,
r À r 0 and the positions of the reciprocal lattice sites g, and may thus be
computed once and for all for any particular crystal structure. One may
then use this function in conjunction with whatever lattice potential V°rÞ is
appropriate to the material under consideration. This approach is known as
the Korringa“Kohn“Rostoker (KKR) or Green™s-function method.
One e¬ective approach of this type is known as the Linear Mu¬n-Tin
Orbital method, or LMTO method. If we look back to Eq. (4.5.1) we may
be reminded of equations used to describe the scattering of a particle by a
single spherically-symmetric scatterer. Particle-scattering theory describes the
perturbed wavefunction in terms of phase shifts for the various angular-
momentum components of the scattering. This suggests that the operator Gk
°EÞ could be recast in a basis of spherical-harmonic components that are
solutions of the Schrodinger equation for a single spherically-symmetric
¨
potential. It is convenient to choose a potential that approximates the actual
e¬ective potential of the atom, but which vanishes outside a certain radius
(hence the term ˜˜mu¬n-tin™™!). This reduces the computational e¬ort needed,
and makes possible the calculation of band structures in crystals having bases
of hundreds of atoms per unit cell.
153
4.5 Exact calculations, relativistic effects, and the structure factor
Before leaving the topic of the calculation of band structures, we should
glance brie¬‚y at the question of when it is valid to ignore relativistic e¬ects.
The Fermi energies of metals, by which we generally mean the energy di¬er-
ence between the lowest and highest ¬lled conduction state, are of the order
of a few electron volts. As this ¬gure is smaller than the rest-mass energy by a
factor of about 10À5 , it might at ¬rst be thought that we could always neglect
such e¬ects. However, one must remember that the potential wells near the
nuclei of heavy atoms are very deep, and that only a small change in energy
may sometimes cause qualitative di¬erences in band structure in semiconduc-
tors in which the band gaps are small.
Accordingly, we turn to the Dirac equation, which describes the motion of
a relativistic electron in terms of a four-component wavefunction. Because
the Dirac Hamiltonian, like the Schrodinger Hamiltonian, has the periodicity
¨
of the lattice, each component of the wavefunction obeys Bloch™s theorem,
and one may associate a wavenumber k with each state. It is thus formally
possible to recast the OPW method in terms of the orthogonalization of four-
component plane waves to the four-component tight-binding core states, a
procedure that was ¬rst carried through for thallium. However, because the
relativistic e¬ects usually contribute only a small amount to the total energy
of a Bloch electron, it is often possible to treat them as a perturbation of the
nonrelativistic band structure. In Section 3.10 we noted that the Dirac equa-
tion could be reduced by means of the Foldy“Wouthuysen transformation to
an equation of the form

H ¼E

where now is a two-component wavefunction describing an electron with
spin 1, and H, in the absence of a magnetic ¬eld, is given by
2


p2 p4
H¼ þ V°rÞ À 3 2
2m 8m c
02
1
þ 2 2 s Á ½rV°rÞ ‚ pŠ þ 2 2 r2 V þ Á Á Á :
2m c 8m c

The third term in this expression simply re¬‚ects the relativistic increase in
mass of the electron, and is known as the mass“velocity term. The next term
contains the spin angular momentum s of the electron, and is the spin“orbit
coupling term. It may be qualitatively understood as the energy of alignment
of the intrinsic magnetic moment of the electron in the magnetic ¬eld caused
154 One-electron theory
by its own orbital motion. The ¬fth term is known as the Darwin term, and
can be thought of as a correction due to the ¬nite radius of the electron.
If these three correction terms are to be treated as perturbations to the
Schrodinger Hamiltonian, then it is necessary to calculate not only the ener-
¨
gies of the Bloch states in the various bands, but also their wavefunctions,
which involves considerably more labor. Some calculations have con-
sequently been made using the tight-binding approximation to describe the
wavefunctions. Because the relativistic terms are important only in the vici-
nity of the atomic nuclei, the tight-binding model provides a wavefunction
whose shape is a very good approximation to that of the true Bloch state in
the region that is important. However, the amplitude of the wavefunction
and consequently the size and k-dependence of these e¬ects may be less
accurately predicted.
Since the Darwin, mass“velocity, and spin“orbit terms give energy shifts of
comparable magnitude, they must all be considered in semiconductors such
as PbTe in which such small perturbations may qualitatively change the band
structure. Because their e¬ect is strongest close to the atomic nuclei, the
Darwin and mass“velocity terms tend to lower the energies of s-states relative
to p- and d-states. Some of the di¬erences in properties between copper,
silver, and gold arise in this way. In the hexagonal metals it is the spin“
orbit term which, because of its lack of symmetry, most often causes obser-
vable e¬ects. The detailed study of the e¬ect of the spin“orbit term on
band structure is a di¬cult topic which requires some knowledge of group
theory, but the nature of the e¬ects can be seen from the following simple
examples.
Let us consider ¬rst the zone boundary at kx ¼ g in the sandwichium
model used in Section 4.3 and in Problems 4.1 and 4.2. If we were to use
the nearly-free-electron approximation with only the two plane waves eikx
and ei°kÀ2gÞx we should ¬nd no discontinuity in the energy at kx ¼ g. This
would be a consequence of the vanishing of the matrix element
°
eÀikx x 2V cos gx ei°kx À2gÞx dr:
V2g /


If, however, we were to use the three plane waves eikx x , ei°kx ÀgÞx , and ei°kx À2gÞx ,
then we should ¬nd a discontinuity in the energy at kx ¼ g (Problem 4.9). In
physical terms we could say that the electron is scattered by the lattice ¬rst
from kx to kx À g, and then from kx À g to kx À 2g. Accordingly the discon-
tinuity in energy is proportional to V 2 , rather than to V as was the case at the
zone boundary at kx ¼ 1 g.
2
155
4.5 Exact calculations, relativistic effects, and the structure factor
In contrast to this we now consider a square lattice of side a in which there
are two identical atoms per unit cell, one at (1 a; 1 a) and one at °À 1 a; À 1 aÞ,
4 4 4 4
as shown in Fig. 4.5.1(a). The ¬rst Brillouin zone is then a square of side
2=a, while the second Brillouin zone is contained by a square of side
p¬¬¬
2 2=a, as shown in Fig. 4.5.1(b). Once again we ¬nd that certain Fourier
components of the potential vanish, so that, for example, Vg ¼ 0 when
g ¼ °0; 2=a) or (2=a; 0Þ. More generally one may suppose the lattice potential
to be composed of atomic potentials Va centered on the various sites, so that
X
V°rÞ ¼ Va °r À l À bÞ; °4:5:4Þ
l;b




Figure 4.5.1. In this two-dimensional model the unit cell in r-space (a) is a square of
side a containing two atoms at the points Æ° 1 a; 1 aÞ. The ¬rst Brillouin zone (b) is
4 4
then a square of side 2=a.
156 One-electron theory
where the l describe the positions of the centers of the unit cells and the b
describe the positions of the atoms within the cell, so that in this case

b1 ¼ °1 a; 1 aÞ; b2 ¼ °À1 a; À 1 aÞ:
4 4 4 4

Then
°
1
eÀig Á r V°rÞ dr
Vg ¼

°
N X Àig Á b
eÀig Á r Va °rÞ dr:
¼ e
b

The summation
X
eÀig Á b
Sg ¼
b

is known as the structure factor, and in this case vanishes when
2
gx þ gy ¼ °2n þ 1Þ
a
for all integral n.
The vanishing of the structure factor, and hence of Vg , for g ¼ °0; 2=a)
and (2=a; 0) means that there is no discontinuity in energy to ¬rst order in
the lattice potential at the boundaries of the ¬rst Brillouin zone. But this is
not all. In this model we should ¬nd that to all orders in the lattice potential,
there is no discontinuity at these zone boundaries. This fact becomes obvious
if we merely tilt our heads on one side and notice p¬¬¬ in fact, we are really
that,
just considering a square Bravais lattice of side a= 2 (Fig.¬4.5.2) whose ¬rst
p¬¬
Brillouin zone is bounded by the same square of side 2 2=a that was the
boundary of the second Brillouin zone in our ¬rst way of looking at the
model. We note the distinction between the vanishing of the structure factor,
which is a property only of the crystal structure, and the vanishing of V2g in
sandwichium, which was an accident of our choice of potential.
This possibility that the energy discontinuity may vanish identically at
some Brillouin zone boundaries is not con¬ned to such arti¬cial models as
the present one. In such common structures as hexagonal close-packed,
in which more than a dozen elements crystallize, and in the diamond and
graphite structures, this very phenomenon occurs. This makes it reasonable
to de¬ne a new set of zones that are separated by planes on which energy
discontinuities do occur. These are known as Jones zones. The construction
by which one de¬nes which Jones zone a particular state is in is the following.
157
4.5 Exact calculations, relativistic effects, and the structure factor




Figure 4.5.2. The model of which a unit cell was shown in Fig. 4.5.1(a) is here seen to
be merely a square lattice of side 2À1=2 a. This explains why no energy discontinuities
were found at the boundaries of the ¬rst Brillouin zone shown in Fig. 4.5.1(b).



A straight line is drawn from the origin of k-space to the point k in the
extended zone scheme. If this line passes through n discontinuities in energy,
then k is in the (n þ 1)th Jones zone.
The relevance of spin“orbit coupling to these considerations lies in the fact
that the lack of symmetry in this term in the Hamiltonian can cause the
reappearance of energy discontinuities within the Jones zones of some crystal
structures. The hexagonal close-packed structure is a particularly important
example of a structure in which such e¬ects have been observed. Although
this particular lattice is rather complicated to investigate here, we can under-
stand the way in which the energy gaps are restored by the spin“orbit inter-
action by considering a modi¬cation of the square lattice shown in Fig. 4.5.1.
We retain the square cell of side a, but this time we place the two identical
atoms at ( 1 a; 1 a) and (À 1 a; À 1 a), as shown in Fig. 4.5.3. The structure
6 4 6 4
factor will now be

 
agx agy
Sg ¼ 2 cos þ ;
6 4
158 One-electron theory




Figure 4.5.3. In this modi¬cation of the model shown in Fig. 4.5.1(a) the atoms are
now placed at the points Æ° 1 a; 1 aÞ:
6 4




Figure 4.5.4. Although the Brillouin zone for the lattice shown in Fig. 4.5.3 is a
square, no discontinuities in energy occur at the dashed lines when a nonrelativistic
Hamiltonian is used.



which will still vanish for g ¼ °0; 2=aÞ but no longer for g ¼ °2=a; 0). In
¬rst order the energy discontinuities will then occur at the solid lines of
Fig. 4.5.4. Let us now suppose that we use the nearly-free-electron approxi-
mation to ¬nd the wavefunctions that result from considering the Fourier
component Vg of the lattice potential for g ¼ °2=a; 0). We then could write
159
4.5 Exact calculations, relativistic effects, and the structure factor
the wavefunctions in the form

¼ À1=2 ½u0 eik Á r þ u1 ei°k Á rÀ2x=aÞ Š
k


with u0 and u1 a pair of real coe¬cients which we could determine explicitly
in terms of k, if we so wished. We might now look for a second-order dis-
continuity along the lines ky ¼ =a (the dashed lines in Fig. 4.5.4) by seeing
whether the lattice potential can mix the states of wavenumbers k and k þ g
with g ¼ °0; À2=a). We thus form
°
W¼ * V°rÞ k dr
kþg
°
¼ À1 °u0 þ u1 e2ix=a Þe2iy=a V°rÞ°u0 þ u1 eÀ2ix=a Þ dr:


A substitution of the form (4.5.4) then serves to show that W vanishes
because of the form of the structure factor; there are terms in u2 and in u2
0 1
which vanish because Sg ¼ 0 for g ¼ °0; 2=a) and two terms in u0 u1 which
cancel because the value of Sg when g ¼ °2=a; À2=a) is the negative of its
value when g ¼ °À2=a; À2=aÞ. If, however, we add to V°rÞ the spin“orbit
term we shall ¬nd a di¬erent result. Then

°
1
W¼ sÁ * ½rV°rÞ ‚ pŠ dr:
kþg k
2m2 c2

The terms in u2 and u2 still vanish, but the cross term leaves a contribution
0 1
from the di¬erent values p takes when acting on the two plane-wave compo-
nents of k . One ¬nds

°
0u u
W ¼ À 20 21 s Á e2i°yÀxÞ=a rV ‚ °2=a; 0Þ dr:
2m c 

The integral is proportional to Sg for g ¼ °2=a; À2=aÞ, which does not
vanish. The degenerate states k and kþg are thus mixed by the spin“
orbit interaction, and energy discontinuities reappear at the Brillouin zone
boundaries. Although these splittings are usually small they are still su¬cient
to alter the topology of the Fermi surface, and thus cause e¬ects which are
readily observable.
160 One-electron theory

4.6 Dynamics of Bloch electrons
In what we have considered so far, the wavenumber k has been little more
than a label for the Bloch states. Experiments, however, are concerned with
such measurable properties as the electric current carried by a system of
electrons in the presence of applied ¬elds. We accordingly now turn to a
consideration of the velocity of Bloch electrons and the modi¬cation of this
quantity by applied electric and magnetic ¬elds.
The velocity of an electron in the absence of a magnetic ¬eld is propor-
tional to the expectation value of its momentum
°
1
v¼ *p dr
m
°
i0
¼À °4:6:1Þ
*r dr:
m

We can relate this to the band structure by returning to the Schrodinger
¨
equation written in the form (4.1.4)

Hk uk °rÞ ¼ E k uk °rÞ °4:6:2Þ

where

02
Hk  À °r þ ikÞ2 þ V°rÞ:
2m

We di¬erentiate (4.6.2) with respect to k (that is, we take the gradient in
k-space) to ¬nd
 
@ @
°Hk À E k Þ u °rÞ ¼ À °Hk À E k Þ uk °rÞ
@k k @k
 
@E k
i0
¼ °r þ ikÞ þ u °rÞ:
@k k
m

But since

¼ eik Á r uk °rÞ;
k

then from (4.6.1)
°
@E @
0vk ¼ k À u* °rÞ°Hk À E k Þ u °rÞ dr:
@k @k k
k
161
4.6 Dynamics of Bloch electrons
The integral vanishes because of the Hermitian nature of Hk , as can be seen
by integrating by parts, leaving the result

1 @E k
vk ¼ : °4:6:3Þ
0 @k

This result appears more familiar if we de¬ne a frequency !k by writing
E k ¼ 0!k . Then vk ¼ @!k =@k, which is the usual result for the group velocity
of a wave of angular frequency !k in a dispersive medium.
We now know the total electric current carried by the conduction electrons
if we know which k-states are occupied. The current density due to a single
electron in the state k will be evk =, so that the total current density is
X
À1
j¼ nk evk : °4:6:4Þ
k


In this independent-particle model the occupation number nk takes on only
the values 0 or 1.
In equilibrium j, which is of course a macroscopic quantity, vanishes, and if
we are to set up a current ¬‚ow we must ¬rst apply an electric ¬eld by, for
example, adding to the Hamiltonian a potential ÀeE Á r. There are now two
paths open to us in investigating the e¬ect of the electric ¬eld “ the time-
dependent approach and the time-independent approach. At ¬rst it seems
that one should treat the applied ¬eld as a perturbation and look for the
eigenstates of the perturbed system. Because the Hamiltonian is constant in
time there appears no reason to use time-dependent methods. Unfortunately,
however, this approach is a very di¬cult one, the chief di¬culty arising from
the fact that no matter how small E is, the potential ÀeE Á r cannot be treated
as a perturbation in an in¬nite system because r then becomes inde¬nitely
large. A similar di¬culty arises when one applies a magnetic ¬eld, the vector
potential then becoming large at large distances. We shall consequently leave
the question of the eigenstates of Bloch electrons in applied ¬elds and turn to
the time-dependent approach.
A wave packet traveling with velocity vk in a uniform force ¬eld eE might
be expected to increase its energy at the rate eE Á vk . On the other hand, if this
change in energy re¬‚ects a change in the wavenumber of the Bloch states
forming the wave packet, we could write

dE k @E k dk dk
¼ Á ¼ 0vk Á
@k dt
dt dt
162 One-electron theory
from (4.6.3). For these two pictures to be equivalent we must have

dk
0 ¼ eE: °4:6:5Þ
dt

This result is not quite correct, as it is really only the kinetic energy that we
should expect to increase at the rate eE Á vk , and the potential energy of the
Bloch state will also be changing if k is changing. To see this more clearly we
can consider the time-dependent Schrodinger equation for an electron initi-
¨
ally in a Bloch state of wavenumber k. Then
ÀiEt=0
Ék °r; tÞ ¼ k °rÞe

¼ eik Á r uk °rÞeÀiEt=0

satis¬es the Schrodinger equation in the absence of the applied ¬eld. If we
¨
now add the potential ÀeE Á r to the Bloch Hamiltonian H0 , then at t ¼ 0

@Ék
¼ °H0 À eE Á rÞÉk
i0
@t
¼ °E k À eE Á rÞÉk : °4:6:6Þ

But if the only change in Ék is to be a change in k at the rate given by (4.6.5)
we should ¬nd at t ¼ 0
   
@Ék @Ék @Ék dk
¼ i0 þ Á
i0
@t @t k @k t dt
 
@
¼ E k Ék þ ieE Á eik Á r ir þ u °rÞ
@k k
 
@ ln uk °rÞ
¼ E k À eE Á r þ ieE Á Ék : °4:6:7Þ
@k

When the third term in (4.6.7) is neglected, this expression becomes identical
to (4.6.6) and one may say that the wavenumber of a Bloch electron is
changed by the ¬eld at just the same rate as that of a free electron. We
note, however, that the rate of change of the velocity of the electron bears
no similarity to that of the free particle, in that as k approaches a zone
boundary the velocity may fall to zero. This would be the case in sandwi-
chium for k ¼ °kx ; 0; 0Þ, as shown in Fig. 4.6.1. The discontinuity in slope of
vk at the zone boundary draws attention to the fact that we do not expect a
weak steady ¬eld to be able to provide the energy to enable the electron to
163
4.6 Dynamics of Bloch electrons




Figure 4.6.1. When the energy (a) varies in the kx direction in sandwichium in the
usual way, the velocity (b) in this direction falls to zero at the zone boundaries.


move from the ¬rst to the second Brillouin zone. That is, we cannot interpret
(4.6.5) in the extended zone scheme, but must look more closely at the Bloch
states for which k lies directly on the zone boundary.
In the two-plane-wave approximation for sandwichium, for example,

uk °rÞ ¼ u0 °kÞ þ u1 °kÞeig Á r ;

and as k approaches ( 1 g; 0; 0) we choose for g the reciprocal lattice vector
2
164 One-electron theory
(Àg, 0, 0). It may be veri¬ed by solving the equations of Section 4.3 for u0 and
u1 that when k lies on the zone boundary, then u0 ¼ Àu1 for the solution of
lowest energy when V > 0, so that

ug=2 °rÞ ¼ 2À1=2 °1 À eÀigx Þ:

When k approaches (À 1 g; 0; 0Þ, on the other hand, we choose g ¼ °þg; 0; 0Þ.
2
Again we ¬nd that also on this zone boundary u0 ¼ Àu1 , and

uÀg=2 °rÞ ¼ 2À1=2 °1 À eigx Þ
¼ Àeigx ug=2 °rÞ:

We now note that these two wavefunctions are identical, in that when we
multiply by eik Á r (with the appropriate k) we ¬nd just the same wavefunction
k , apart from an unimportant constant factor. Thus the action of the
electric ¬eld is to cause the wavenumber of the electron to change at a
constant rate until the zone boundary is reached, at which point the
wavenumber is ambiguous. The electron may then be considered to have
wavenumber k À g, and so the whole process may be repeated, with k
increasing until the same zone boundary is again reached. Alternatively
we may use the repeated zone scheme, and say that k is changing steadily
with time, although the electron always remains in the ¬rst band. This is
illustrated in Fig. 4.6.2(a) which shows the variation of the various com-
ponents of uk °r) with k. In Fig. 4.6.2(b) the electron velocity vk in the ¬rst
band is plotted in the repeated zone scheme. The fact that it is a periodic
function of k shows that the electron would exhibit oscillatory motion in a
crystal so perfect that no scattering occurred. In the region near the zone
boundary vk becomes more negative with increasing k as a consequence of
the hole-like behavior characterized by the negative curvature of the function
E x °kx Þ.
The term in (4.6.7) that we neglected was of the form

@
ieE Á ln uk °rÞ:
@k

It is only when this term is small that the approximation (4.6.5) is valid, and
this will only be the case when uk °r) is a slowly varying function of k. Now if
the lattice potential is very weak then the Bloch wave is very similar to a
plane wave over most of the Brillouin zone. At the zone boundary, however,
u0 and u1 will always be of equal magnitude irrespective of the strength of the
165
4.6 Dynamics of Bloch electrons




Figure 4.6.2. As the electron is accelerated by a weak electric ¬eld, its wave number
kx changes uniformly with time and the amplitudes (a) of the various plane wave
components of uk °rÞ, the periodic part of the Bloch wave function, also change.
Because the electron remains in the ¬rst Brillouin zone, the velocity (b) then changes
periodically and not in the way shown in Fig. 4.6.1.



lattice potential. Thus the derivative @ °ln uÞ=@k is greatest when the lattice
potential is weak, and it is then that the picture of the electron moving in a
single band breaks down. This extra term that appears in (4.6.7) must be
subtracted from the Hamiltonian if the electron is to remain in one band.
This term is a function of uk °rÞ, and is thus periodic with the period of the
lattice. We may estimate its magnitude very simply by a glance at Fig. 4.6.3
which shows the band structure near the zone boundary. Since u1 °kÞ is of
order unity at the zone boundary, and has become very small by the time it
166 One-electron theory




Figure 4.6.3. When the lattice potential is weak one may estimate with the aid of this
diagram the range Ák of kx -values over which the energy departs signi¬cantly from
its free-electron value.



has reached a distance Ák away, where

@E
V™ Á Ák
@k

we may write

@u 02 k
1
$ $ ;
@k Ák mV

and the extra term is of order eE02 k=mV. When this term is of the order of
the lattice potential it may cancel the lattice potential and allow the electron
to make a transition to another band. The condition for this not to occur is
then

02 k
( V:
eE
mV

Since the Fermi energy E F is roughly 02 k2 =2m and k is of the order of 1=a,
where a is the lattice spacing, we may write

V2
eEa ( :
EF

The condition for (4.6.5) to be valid is thus that the energy gained by the
electron in being accelerated through one lattice spacing should be small
167
4.6 Dynamics of Bloch electrons
compared with V 2 =E F . When this condition is not obeyed Zener breakdown is
said to occur. While it is di¬cult to reach such high ¬elds in homogeneous
materials, the junction between n- and p-type semiconductors naturally con-
tains a steep potential gradient which permits observation of these e¬ects and
as a result of which a variety of device applications are possible.
In the case of an applied magnetic ¬eld we might suppose that the Lorentz
force would tend to change k in the same way as the force of the electric ¬eld,
so that

dk e
0 ¼ v ‚ H: °4:6:8Þ
dt c

This in fact turns out to be true within limitations similar to those imposed in
the case of the electric ¬eld, although the demonstration of this result is a
little more involved. Let us ¬rst choose the gauge so that the vector potential
is

A ¼ 1 °H ‚ rÞ; °4:6:9Þ
2


and write the Hamiltonian (as in Eq. (3.10.8)) as

H ¼ 1 mv2 þ V°rÞ
2
 2
1 e
¼ p À A þ V°rÞ;
2m c

where V°rÞ is now the lattice potential. Then

dv
¼ ½v; HŠ
i0
dt
¼ 1 m½v; v2 Š þ ½v; V°rފ
2
e
¼ i0 v ‚ H þ ½v; V°rފ °4:6:10Þ
mc

as may be veri¬ed by substituting (p À eA=cÞ for v and using the explicit form
(4.6.9) for A.
If we had performed a similar manipulation in the case where an electric
¬eld was applied we should have found

dv eE
¼ i0 þ ½v; V°rފ;
i0
dt m
168 One-electron theory
and since we were able to identify approximately eE with 0 dk=dt we could
then have written
 
dv 0 dk
½v; V°rފ ¼ i0 À : °4:6:11Þ
dt m dt

This relation describes the rate of change of velocity of a Bloch electron
whose wavenumber is changing, but which is remaining in a single band. It
does not discuss the agency that causes k to change, but merely states the
consequent change in velocity. It is thus more general than the case of an
applied electric ¬eld, and can be used in combination with (4.6.10) when a
magnetic ¬eld is applied, the commutator ½A°rÞ; V°rފ vanishing. Substitution
of (4.6.11) in (4.6.10) then gives the expected result, (4.6.8).
While the electric ¬eld caused the wavenumber k to move in a straight line
with uniform velocity in the repeated zone scheme, the e¬ect of the magnetic
¬eld is more complicated. Equation (4.6.8) states that dk=dt is always per-
pendicular to both the electron velocity and the magnetic ¬eld. But since the
velocity is proportional to dE=dk the energy of the electron must remain
constant, and k moves along an orbit in k-space which is de¬ned by the
two conditions that both the energy and the component of k in the direction
of the magnetic ¬eld remain constant.
As an illustration we consider the possible orbits of an electron in sandwi-
chium when the magnetic ¬eld is applied in the z-direction. For states of low
energy the constant-energy surfaces are approximately spherical, and their
intersections with the planes of constant kz are nearly circular. The electrons
thus follow closed orbits in k-space with an angular frequency, !, close to the
cyclotron frequency, !0 , of a classical free electron, which is given by eH=mc.
Such an orbit is labeled in Fig. 4.6.4. In real space the path of such a
classical electron would be a helix with its axis in the z-direction. For an
electron of slightly higher energy the orbit passes closer to the zone bound-
ary, and the electron velocity is reduced below the free-electron value. A
circuit of the orbit labeled in Fig. 4.6.4 thus takes a longer time than a
circuit of , and one says that the cyclotron frequency of the orbit is less
than !0 , and would be the same as for a free particle of charge e and mass
greater than m in the same magnetic ¬eld. One sometimes de¬nes a cyclotron
mass m* in this way for a particular orbit by means of the relation
m!0
m* ¼ :
!
The cyclotron mass, which is a function of the electron velocity at all points
on an orbit, must be distinguished from the inverse e¬ective mass de¬ned in
169
4.6 Dynamics of Bloch electrons




Figure 4.6.4. In a magnetic ¬eld in the z-direction an electron of low energy in
sandwichium will travel the almost circular orbit in k-space, while one of slightly
larger energy will follow the distorted orbit . At still higher energies the electron
may either follow the second-zone orbit  or the periodic open orbit
that lies in the
¬rst Brillouin zone.




Figure 4.6.5. This diagram shows Fig. 4.6.4 replotted in the repeated zone scheme.
The periodic open orbits
carry a current that does not average to zero over a
period of the motion.



Section 4.3 which characterized the band structure in the neighborhood of a
single point in k-space.
If the electron energy is greater than E g À V and kz is su¬ciently small
there will be some orbits, such as
in Fig. 4.6.4, that meet the zone boundary.
The path of the electron in k-space is then a periodic open orbit in the repeated
zone scheme, as shown in Fig. 4.6.5. Such orbits are particularly important in
determining the conductivities of metals in magnetic ¬elds in that the electron
velocity does not average to zero over a period of the orbit. For energies
170 One-electron theory
greater than E g þ V there will also be orbits in the second band, such as those
labeled  in Fig. 4.6.4. In the repeated zone scheme these appear as the small
closed orbits in Fig. 4.6.5. Because the velocity may be close to its free-elec-
tron value (in the extended zone scheme) over much of these orbits while their
perimeter is much smaller, the time taken to complete an orbit may be very
small. The cyclotron mass is then stated to be correspondingly small.
The range of validity of Eq. (4.6.8) may be deduced in a similar way to our
estimate in the case of an electric ¬eld, and we ¬nd that

@
e
v ‚ HÁ ln uk °rÞ
@k
c

must be small compared with the lattice potential. When we write

@u 02 k 0k eH
$
@k mV ; v$ ; ¼ !0
m mc

we ¬nd the condition to be

V2
0!0 ( :
EF

When this is violated magnetic breakdown is said to occur. The electron then
has a ¬nite probability of making a transition from a
-orbit to a -orbit in
Fig. 4.6.4, and the conductivity may be qualitatively a¬ected.


4.7 Scattering by impurities
We have now seen how the application of an electric ¬eld causes the wave-
number of a Bloch electron to change, and hence how the electric current
grows with time in a perfect periodic lattice. We know, however, that for
moderate electric ¬elds the current rapidly becomes constant and obeys
Ohm™s law in all normal metals. The current does not grow and then oscillate
in the way that our simple dynamics predict, because the electron is scattered
by some departure of the lattice from perfect periodicity. The two most
important mechanisms that limit the magnitude the current attains in a
particular ¬eld are scattering by lattice vibrations and scattering by impuri-
ties. The topic of the interaction of Bloch electrons with phonons is a major
part of the theory of solids, and Chapter 6 is devoted to a discussion of such
processes. The theory of alloys, in which the problem is to calculate the
properties of partially disordered systems, is also a topic of some importance.
171
4.7 Scattering by impurities
For the present, however, we shall just consider the problem of a single
impurity center in an otherwise periodic lattice. This avoids the statistical
problems of the theory of alloys, but still allows us to formulate an expres-
sion for the probability per unit time that an electron is scattered from one
Bloch state to another. We shall then have all the ingredients we need for the
formulation of a simple theory of the conductivity of metals.
The customary approach to the scattering theory of a free particle involves
the expansion of the wavefunction in spherical harmonics and the discussion
of such quantities as phase shifts and cross sections. This approach is not so
useful for Bloch electrons because of the reduced symmetry of the problem
when the lattice potential is present. Instead we consider an electron initially
in some Bloch state, k , and then apply the perturbing potential, U. The
wavefunction will then be transformed into some new function, k . We
interpret the scattering probability between the two Bloch states, k and k 0 ,
as being proportional to the amount of k 0 contained in k . That is, we form the
integral hk 0 j k i to measure the amplitude that tells us how much of the state
that was originally k has been transformed to k 0 . The square of the modulus
of this quantity will then be proportional to the probability Q°k; k 0 Þ that in
unit time an electron is scattered between these states, i.e.,

Q°k; k 0 Þ / jhk 0 j k ij :
2


We may use the starting point of perturbation theory to rewrite this expres-
sion in a more useful form. We ¬rst write

H0 k ¼ E k k

and

°H0 þ UÞ ¼ Ek k;
k

and note that the perturbed and unperturbed energies will be very close to
each other provided no bound states are formed, since the impurity causing
U only perturbs a negligible portion of our large volume . We next note
that these Schrodinger equations are satis¬ed by
¨

¼ jk i þ °E k À H0 þ iÞÀ1 Uj
j ki ki °4:7:1Þ

when  ! 0. Then because hk 0 jk i vanishes we ¬nd

Q°k; k 0 Þ / jhk 0 j°E k À H0 þ iÞÀ1 Uj k ij
2


¼ ½°E k À E k 0 Þ2 þ 2 ŠÀ1 jhk 0 jUj k ij :
2
172 One-electron theory
Because the scatterer has no internal degrees of freedom in this model the
energy of the electron must be conserved, and only elastic scattering can take
place. This is expressed by the term in brackets. Since
 
Ek À Ek 0
1d
2 À1
½°E k À E k 0 Þ þ  Š ¼
2
arctan
 dE k 

and arctan ½°E k À E k 0 Þ=Š becomes a step function as  ! 0, we can interpret
the derivative of the step function as a -function. The constant of propor-
tionality can be found from time-dependent perturbation theory, which in
lowest order gives the result

2
Q°k; k 0 Þ ™ jhk 0 jUjk ij2 °E k À E k 0 Þ: °4:7:2Þ
0

In order for our result to reduce to this when the potential is weak so that k
may be replaced by k we must choose the same constant of proportionality,
and write

2
Q°k; k 0 Þ ¼ jhk 0 jUj k ij
2
°E k À E k 0 Þ: °4:7:3Þ
0

The approximation (4.7.2) is known as the Born approximatiom, and may
be thought of as neglecting multiple scattering by the impurity. The exact
formula (4.7.3) might be rewritten by repeatedly substituting for from
(4.7.1). We should then have a series of terms in which U appeared once,
twice, three times, and so on. These could be interpreted as single, double,
triple, and higher-order scattering by the impurity (Fig. 4.7.1).
It is sometimes useful to ask what the potential T would be that, if the
Born approximation were exact, would give the scattering predicted by
(4.7.3) for the potential U. That is, we ask for the operator T such that

hk 0 jUj ki ¼ hk 0 jTjk i:

This operator is known as the transition matrix (or sometimes just as the
T-matrix), and does not in general have the form of a simple potential. It can
be seen from (4.7.1) that

T ¼ U þ U½E À H0 þ iŠÀ1 T:

Also

½E À H0 þ iŠÀ1 T ¼ ½E À H þ iŠÀ1 U;
173
4.7 Scattering by impurities




Figure 4.7.1. The Born approximation is a result of ¬rst-order perturbation theory,
and can be diagrammaticaliy represented as a single scattering event (a). The T-
matrix includes multiple scattering (b).



as may be seen by operating on both sides with [E À H þ i]. Thus

T ¼ U þ U½E À H þ iŠÀ1 U:

Since the potential U is real, the only di¬erence between T and its Hermitian
conjugate Ty will be that the term i will be replaced by Ài. We could thus
have equally well used Ty in calculating Q°k; k 0 Þ. But since by the de¬nition
of the Hermitian conjugate

hk 0 jTjk i* ¼ hk jTyjk 0 i

we see that the scattering probability must be the same in either direction,
and

Q°k; k 0 Þ ¼ Q°k 0 ; kÞ: °4:7:4Þ

We could also argue this from the starting point of the principle of micro-
reversibility, which states that the transition probability will be una¬ected by
time reversal. The time reversal of the state k will be Àk , and so

Q°k; k 0 Þ ¼ Q°Àk 0 ; ÀkÞ: °4:7:5Þ

However, Àk ¼ * , as can be seen from the Schrodinger equation in the
¨
k
form (4.1.4), and we do not expect a real transition probability to depend on
our convention as to complex numbers. We thus deduce (4.7.4) to be a
consequence of (4.7.5).
We also note that the perturbation of the electron wavefunctions changes
the density of electrons, and hence of electric charge, in the vicinity of an
174 One-electron theory
impurity in a metal. If the impurity represents an added electric charge the
change in electron density will screen the ¬eld of the impurity. One can thus
equate the excess charge of the impurity with the excess charge of the elec-
trons that are in the process of being scattered. The formulation of this
concept is rather complicated for Bloch electrons, but reduces to a simple
form for free electrons, where it is known as the Friedel sum rule. It is a useful
condition that all models of impurity potentials must approximately satisfy.


4.8 Quasicrystals and glasses
Our study of band structure so far has been built on the concept of the Bloch
waves that we have proved to exist in perfectly periodic structures. In the real
world, however, nothing is perfectly periodic, and so we should ask ourselves
what the consequences are of deviations from perfect periodicity. In Chapter 6
we shall look at the e¬ect of the weak deviations from perfect order that are
introduced by phonons. There we shall see that this type of motion in a three-
dimensional crystal does not destroy the long-range order. That is to say,
when X-rays or neutrons are scattered by a thermally vibrating three-dimen-
sional lattice there will still be sharp Bragg peaks, although in one or two
dimensions this would not be the case. We now look at some other systems
that lack perfect order, and examine whether the concept of band gaps will
survive. The ¬rst of these is a remarkable family of structures known as
quasicrystals. These are a form of not-quite-crystalline solid that was discov-
ered experimentally as recently as 1984, although similar structures had been
studied as mathematical constructs much earlier.
An example of a quasicrystal in two dimensions is given in Fig. 4.8.1. It
clearly depicts an ordered array, but closer inspection shows it not to be a
Bravais lattice. The telltale sign is the fact that it has a ¬ve-fold rotational
symmetry. This is forbidden for Bravais lattices in two dimensions, as one
cannot completely cover a plane using pentagonal tiles. One can, however,
tile a plane using two types of diamond-shaped Penrose tile, one of which has
an acute angle of =5, the other tile having an angle of 2=5 (Fig. 4.8.2). In
three dimensions the task becomes much harder to accomplish, and nearly
impossible to illustrate. Nevertheless, experiment shows that if a molten
mixture of aluminum and manganese in an atomic ratio of 4 : 1 is cooled
ultrarapidly ($ 1 megakelvin/second!) then small pieces of solid are produced
that give di¬raction patterns having the ¬ve-fold symmetry characteristic of
an icosahedron. These materials are thus clearly not crystalline (this is
deduced from the ¬ve-fold symmetry) but do have long-range order (deduced
from the existence of sharp Bragg peaks).
175
4.8 Quasicrystals and glasses




Figure 4.8.1. A quasicrystal in two dimensions.




Figure 4.8.2. Two types of tile can cover a plane with quasicrystalline symmetry.



Figure 4.8.3. A Fibonacci chain is built from atoms separated by either long or short
spacers placed in a special order.


We can gain some insight into the nature of quasicrystals by looking at the
one-dimensional chain of atoms shown in Fig. 4.8.3. The spacing between
atoms is either long (L) or short (S), with L=S an irrational number. If the
arrangement of L and S spacings were random, then the chain would have no
long-range order, and would give rise to no sharp Bragg di¬raction peaks.
176 One-electron theory
But it is not random. It is a Fibonacci chain, built according to the following
prescription. We start with a single spacing S, and then repeatedly apply the
operation that each S is turned into L and each L is turned into the pair LS.
In this way S ! L ! LS ! LSL ! LSLLS ! LSLLSLSL and so on. (An p¬¬¬
important special case occurs when L=S ¼ 2 cos°=5Þ ¼ 1 °1 þ 5Þ, a number
2
known as the golden mean.) Although this sequence does not at ¬rst sight
appear to have any long-range order, one can, with the aid of some ingenious
arguments, calculate the Fourier transform of the atomic density exactly. One
¬nds that there are large, sharp, Bragg peaks at various wavenumbers. The
chain is clearly not periodic in the sense of a Bravais lattice, but it does have
some sort of long-range order. Evidently there are some hidden repeat lengths
that are disguised by local deviations from periodicity. An invisible hand is
placing the L and S segments in just such a way as to retain the Bragg peaks.
We ¬nd a clue to what is happening by looking at a strip cut from a true
Bravais lattice in a higher dimension. In Fig. 4.8.4 we see a square lattice
across which two parallel lines have been drawn with a slope equal to the
reciprocal of the golden mean and passing through the opposite corners of
one unit cell. We then project all the lattice points included in this strip onto
the lower line to form a one-dimensional array. This array turns out to be
precisely the special-case Fibonacci chain. We have thus made a connection




Figure 4.8.4. The Fibonacci chain also appears as a projection of a regular square
lattice.
177
4.8 Quasicrystals and glasses
between a quasiperiodic array in one dimension and a Bravais lattice in a
higher dimension. This idea may be extended to show that ¬ve-fold rota-
tional symmetry may be found in spaces of six or more dimensions. In
particular, icosahedral symmetry may be found in a cubic lattice in six dimen-
sions. An icosahedron has 20 identical faces, each of which is an equilateral
triangle. Five of these faces meet at each of the 12 vertices, and so there are
six ¬ve-fold symmetry axes. This symmetry is clearly seen experimentally in
single grains of some quasicrystals which form beautiful structures resem-
bling ¬ve-petaled ¬‚owers. It is truly remarkable that this obscure crystallo-
graphic niche is actually occupied by real materials.
In one dimension, the existence of sharp Bragg peaks will always lead to
gaps in the electronic density of states, but in three dimensions this is not
assured. Thus the density of states for electrons in the potential due to a
Fibonacci chain of atoms will always have band gaps. These chains would
then be good insulators if there were two electrons per atom. In three dimen-
sions the long-range order characteristic of quasicrystals will not necessarily
cause gaps in the density of states, so that even if the number of electrons
were two per atom, the material might still be a metallic conductor.
As we moved from considering crystalline lattices to the less-ordered qua-
sicrystals, we have found that the continued existence of long-range order
was the factor that made plausible the sustained presence of band gaps. If we
move further in this direction we ¬nd amorphous or glassy solids, in which
no long-range order remains. The structure factor revealed by X-ray scatter-
ing shows no sharp peaks, but only broad maxima. Surely these materials
should not have band gaps in their electronic density of states? Surprisingly,
band gaps persist in amorphous materials. In silicon, the e¬ective band gap is
even greater in amorphous material than it is in a crystal.
It was only in 1966 that a demonstration was given of how band gaps could be
proved to persist in one simple model of an amorphous solid. In this model the
potential has the mu¬n-tin form, in which identical spherically symmetric
attractive potential wells are separated by regions of constant potential V0 .
No two wells overlap or have their centers closer together than a distance we
de¬ne as 2. We consider the real wavefunction describing an eigenstate
of energy E < V0 . In units in which 0 ¼ 2m ¼ 1, the Schrodinger equation is
¨

r2 ¼ °V À EÞ : °4:8:1Þ

If we multiply by and integrate over the volume of the container we ¬nd
°
f°V À EÞ 2 þ °r Þ2 g d ¼ 0; °4:8:2Þ
178 One-electron theory
provided is equal to zero over the surface of the box. Let us de¬ne a cell as
the region closer to one particular well than to any other (this is sometimes
known as a Voronoy polyhedron). Then we can certainly ¬nd a cell such that
°
f°V À EÞ 2 þ °r Þ2 g d 0 °4:8:3Þ
cell


when the integrations are con¬ned to the volume of the cell. Because both
parts of the integrand are positive at distances greater than  from the center
of the well the integral will furthermore be negative when the integration is
restricted to a sphere of radius . If S is the surface of this sphere it then
follows that
°
r Á dS < 0: °4:8:4Þ
S


Taking spherical polar coordinates with the center of this well as the origin,
we expand in spherical harmonics, writing
X
¼ cl;m Yl;m °; ÞRl °rÞ °4:8:5Þ
l;m


and substitute in the inequality to obtain
X d
c2 fRl °rÞg2 jr¼ < 0: °4:8:6Þ
l;m
dr
l;m


If this inequality holds for the sum of terms, it must also be true for at least
one term of the sum, and so an l must exist for which

d
fRl °rÞg2 jr¼ < 0: °4:8:7Þ
dr

If there are bands of energy for which no l can be found such that this
inequality is satis¬ed, then the existence of gaps in the density of states is
proved.
The presence of band gaps in the electronic structure is central to many of
the most important properties of solids. It is thus satisfying that we can
calculate band structures and band gaps in a variety of structures provided
that the one-electron model is a satisfactory approximation. Our next step
must be a more careful look at this assumption, and an exploration of the
elegant analysis with which it can be justi¬ed.
179
Problems

Problems
4.1 In sandwichium metal the lattice potential is 2V cos gx. Investigate, in
the nearly-free-electron model, the electron velocity in the neighbor-
hood of the point (g=2; 0; 0) in reciprocal space.

4.2 Investigate qualitatively the density of states of the sandwichium
de¬ned in Problem 4.1 in the regions near E ¼ 02 g2 =8m Æ V, and sketch
the overall density of states.

4.3 Another type of sandwichium has a lattice potential

X
1
V°rÞ ¼ Va °x À naÞ:
n¼À1


Investigate its band structure in the nearly-free-electron model, using
two plane waves.

4.4 Apply the nearly-free-electron approach using four plane waves to the
band structure of a two-dimensional crystal whose lattice potential is

V°rÞ ¼ 2V½cos gx þ cos gyŠ:

Under what conditions will this crystal be an insulator if there are two
electrons per ˜˜atom™™? (An ˜˜atom™™ is assumed to occupy one unit cell of
dimensions 2=g ‚ 2=g.)

4.5 What are the possible forms of the inverse-e¬ective-mass tensor in the
model of Problem 4.4 at the point (1 g; 1 gÞ in k-space?
2 2


4.6 In the Kronig“Penney model a one-dimensional electron moves in a
potential

X
1
V°xÞ ¼ À Va °x À naÞ:
n¼À1


Contrast the exact solution for the width of the lowest band with that
given by the method of tight binding when V is very large. Assume
overlap only of nearest neighbors in the tight-binding approach.
180 One-electron theory
4.7 Examine the inverse e¬ective mass of the states at the bottom of the
third band of the model in Problem 4.6, again assuming V to be large.
Solve this problem in the following ways.
(1) Exactly.
(2) In the two-plane-wave NFE approximation.
(3) In the OPW method, treating the ¬rst band as core states in the
tight-binding approximation. [Use two OPW™s, and neglect the k-
dependence of E k “ i.e., take E k as the energy of the ˜˜atomic™™ bound
c c
state.]

4.8 Evaluate the Korringa“Kohn“Rostoker Gk °r À r 0 ) for the sandwichium
Ð
of Problem 4.1. [Hint: C cosec zf °zÞ dz may be a helpful integral to
consider.]

4.9 Calculate an approximate value for the energy discontinuity and e¬ec-
tive inverse masses in the neighborhood of k ¼ °g; 0; 0Þ in sandwichium
by using the nearly-free-electron approximation with three plane waves.

4.10 Draw the Jones zone for a square lattice of side a with four identical
atoms in each cell at the points Æ°a=8; Àa=8) and Æ°3a=8; 3a=8Þ.

4.11 In the limit of vanishingly small size of an orbit the cylotron mass m*
and the inverse-e¬ective-mass tensor (M À1 Þij are related. What is this
relationship between m*, (M À1 ), and the direction x of the applied
^
magnetic ¬eld? [It is helpful to consider the area A of an orbit, and
its variation with energy, dA=dE.]

4.12 A magnetic ¬eld is applied in the z-direction to sandwichium. How,
qualitatively, does m* vary for orbits with kz ¼ 0 as E ! E g À V?

4.13 A Bloch electron in sandwichium is scattered from (kx ; ky ; kz Þ to
(Àkx ; ky ; kz ) by the potential U exp ½À°gr=4Þ2 ]. Investigate qualitatively
how the transition probability for this process varies with kx . [Use the
Born approximation for Q°k; k 0 Þ and the two-plane-wave approxima-
tion for k and k 0 . Sketch the variation of Q as kx varies from 0 to 1 g.]
2


4.14 When the Coulomb interaction is included in the Hamiltonian of
an insulator it becomes possible for an electron in the conduction
band and a hole in the valence band to form a bound state together;
this elementary excitation of the crystal is known as an exciton. In the
181
Problems
simple model of an insulator in which the lattice potential is
2V°cos gx þ cos gy þ cos gz) such a state can be formed if we allow an
interaction e2 =jre À rh j to exist between the electron and hole states at
the corner of the ¬rst Brillouin zone. Investigate the possible energies of
such an excitation by solving a Schrodinger equation analogous to that
¨
describing a hydrogen atom, but in which the proton and electron are
replaced by an electron and a hole having the appropriate e¬ective
masses.

4.15 Verify that the constant a of Section 4.5 vanishes, as claimed in the
sentence preceding Eq. (4.5.1).

4.16 In the model illustrated in Fig. 4.5.3 it was shown that spin“orbit
coupling introduces energy discontinuities at the zone boundaries
shown as dashed lines in Fig. 4.5.4. Does (a) the mass“velocity term
or (b) the Darwin term cause a similar e¬ect?
Chapter 5
Density functional theory




5.1 The Hohenberg“Kohn theorem
In Chapter 2 we explored some of the consequences of electron“electron
interactions, albeit in some simple perturbative approaches and within the
random phase approximation. There we found that the problem of treating
these interactions is exceedingly di¬cult, even in the case where there is no
external one-particle potential applied to the system. We have also explored
some of the properties of noninteracting electrons in an external potential, in
this case the periodic lattice potential. This led to the concepts of electron
bands and band structure, subjects of fundamental importance in under-
standing the physics of metals, insulators, and semiconductors. Of course,
in the real world, electrons in matter are subjected both to electron“electron
interactions and to external potentials. How to include systematically and
correctly the electron“electron interactions in calculations of real systems is
truly a formidable problem.
Why that is so is easily demonstrated. Suppose that we want to solve the
problem of N electrons interacting in some external potential. The N-electron
wavefunction can be expanded in Slater determinants of some suitable single-
particle basis such as plane waves. We can describe the Slater determinants
by occupation numbers in our second-quantized notation. Suppose further-
more that we have a basis of a total of Nk plane wave states at our disposal.
Here Nk must be large enough that all reasonable ˜˜wiggles™™ of the many-
body wavefunction can be included. The size of our Hilbert space and
hence the size of the Hamiltonian matrix to be diagonalized can then be
found by using combinatorics: the size of the Hilbert space is given by the
number of ways that we can put N ˜˜balls™™ in Nk ˜˜boxes,™™ with only one ball
per box. This number is a binomial factor, Nk !=N!°Nk À NÞ!, which has the
unfortunate property that it grows factorially. Careful use of symmetry may

182
183
5.1 The Hohenberg“Kohn theorem
help us reduce the size of the Hamiltonian by a factor of ten or so, and the
increasing power of computers allows us to consider ever-larger systems, but
it remains stubbornly the case that current state-of-the-art exact numerical
diagonalizations have di¬culty handling more than a few tens of electrons.
Also, even though the computer power at our disposal grows exponentially
with time, the size of the Hilbert space of our N-electron problem
grows much faster than exponentially with N. We may therefore, some-
what pessimistically, conclude that we may never have enough computer
resources available to solve a problem with a macroscopic number of
electrons.
This draws attention to the urgent need for some alternative way to include
electron“electron interactions in our calculations. Virtually the only way to
do so in realistic calculations is provided by density functional theory (DFT).
Since its formulation in the mid 1960s and early 1970s, DFT has been used
extensively in condensed matter physics in almost all band-structure and
electronic structure calculations. It has also been widely adopted in the quan-
tum chemistry community, and has led to a computational revolution in that
area. Density functional theory was conceived by Walter Kohn, who also led
many of the successive developments in this ¬eld.
What makes density functional theory so powerful to use is a deceptively
simple-looking theorem, the Hohenberg“Kohn theorem, which has profound
implications. This theorem allows for the systematic formulation of a many-
body problem “ interacting electrons in an external potential “ in terms of the
electron density as the basic variable. It is worth spending a moment to
re¬‚ect on this. Consider the Schrodinger equation for N interacting
¨
electrons. This is a di¬erential equation for a complex quantity, the
Schrodinger wavefunction, which in three dimensions is a function of 3N
¨
variables. This large number makes it impractical to solve even for just the
ground-state wavefunction, which will generally be insu¬cient, as we
also need information about the excited states. Finally, the physical quanti-
ties in which we are interested have to be extracted from the wavefunctions
that we have laboriously obtained. This in itself may be technically very
di¬cult. It is clear that if we can instead work with just the electron density
as the basic variable, this will lead to an enormous simpli¬cation, since
the density of a three-dimensional system is a scalar ¬eld of only three vari-
ables. What is truly remarkable is, as we shall see, that all physical properties
of the system can in principle be determined with knowledge only of the
ground-state density! That is precisely the statement of the Hohenberg“
Kohn theorem, as we now prove for systems with nondegenerate ground
states.
184 Density functional theory
Let

H ¼ T þ Vext þ V

be the nonrelativistic, time-independent Hamiltonian of a system of N
electrons. Here, T is the kinetic energy, Vext is an external potential which
couples to the density (an example being that from the nuclei in a solid), and
V is the two-body electron“electron interaction (usually the Coulomb inter-
action). In second-quantized notation we write

X 02 k2 y X 1X
y
Vq cy cy 0 þq;s 0 ck 0 ;s 0 ck;s :
H¼ c k;s ck;s þ Vext °qÞck;s ckþq;s þ kÀq;s k
2m 2 k;k 0 ;q;s;s 0
k;s k;q;s



The Hohenberg“Kohn theorem then states that the expectation value O of
any operator O is a unique functional O½n0 °rފ of the ground-state density
n0 °rÞ, by which we mean that the value of O depends on the value of n0 °rÞ at
all points r.
What does this imply? Well, we already know that if we could solve the
Schrodinger equation for the Hamiltonian H and ¬nd all the many-body
¨
eigenstates É , we could then calculate the expectation value of any operator.
The Hamiltonian therefore determines the expectation value of any operator,
and, in particular, the Hamiltonian determines the ground-state density, since
this is just the ground-state expectation value of the density operator. We can
be even more speci¬c: since the kinetic energy operator T and the interaction
V are universal, meaning that they are the same for all nonrelativistic inter-
acting N-electron systems, it is really only the external potential Vext that
characterizes the Hamiltonian, and thus the eigenstates and the ground-state
density. This is straightforward. What the Hohenberg“Kohn theorem states
is that this mapping from external potential to ground-state density is inver-
tible. Given any density n°rÞ, which is speci¬ed to be the ground-state density
for some N-electron system, the Hamiltonian of that system is then uniquely
determined, and so then are all the eigenstates and the expectation value
of any operator. So with knowledge of only the ground-state density of an
N-electron system, we can (in principle, at least) determine everything about
that system, including excited states, excitation energies, transport properties,
etc.
The proof of this theorem is simple. We ¬rst show that two potentials, Vext
0
and Vext , that di¬er by more than a trivial constant (a constant is unimpor-
tant since we can always shift the reference point of the potential energy),
185
5.1 The Hohenberg“Kohn theorem
0
necessarily lead to di¬erent ground states É0 and É0 . The Schrodinger equa-
¨
0
tions for É0 and for É0 are

°T þ V þ Vext ÞÉ0 ¼ E 0 É0 °5:1:1Þ
0 0 00
°T þ V þ Vext ÞÉ0 ¼ E 0 É0 ; °5:1:2Þ

0
where E 0 and E 0 are the respective ground-state energies. We prove the ¬rst
0
part of the theorem by contradiction. Suppose now that É0 and É0 are the
same. We then subtract Eq. (5.1.1) from Eq. (5.1.2) to obtain
0 0
°Vext À Vext ÞÉ0 ¼ °E 0 À E 0 ÞÉ0 :
0
But E 0 and E 0 are just real numbers, so this means that the two potentials Vext
0
and Vext can di¬er at most by a constant, in contradiction to our hypothesis.
0 0
We have thus shown that if Vext 6¼ Vext then É0 6¼ É0 .
At this point we pause to note the relation between n0 °rÞ, Vext °rÞ, and
hÉ0 jVext jÉ0 i. We recall that
° X
N
n0 °rÞ ¼ É*°r1 ; r2 ; . . .Þ °r À ri Þ É0 °r1 ; r2 ; . . .Þ dr1 ; dr2 ; . . . ;
0
i


which allows us to write

hÉ0 jVext jÉ0 i
° X
N
¼ É*°r1 ; r2 ; . . .Þ Vext °ri Þ É0 °r1 ; r2 ; . . .Þ dr1 ; dr2 ; . . . ; drN
0
i
° X
N
¼ É*°r1 ; r2 ; . . .Þ °rp À ri ÞVext °rp Þ É0 °r1 ; r2 ; . . .Þ dr1 ; dr2 ; . . . ; drN ; drp
0
i
°
¼ n0 °rÞVext °rÞ dr:

0 0
Now we can prove that if Vext 6¼ Vext (so that consequently É0 6¼ É0 ), then we
0
must also have n0 °rÞ ¼ n0 °rÞ. Again, we prove this assertion by contradiction.
6
Assume that n0 °rÞ ¼ n0 °rÞ, and that H and H 0 are the two Hamiltonians
0
0
corresponding to Vext and Vext , respectively. According to the Rayleigh“
Ritz variational principle, we have
0 0
E 0 ¼ hÉ0 jHjÉ0 i < hÉ0 jHjÉ0 i;
186 Density functional theory
and
°
0 0
hÉ0 jH 0
0 0 0 0 0 0
hÉ0 jHjÉ0 i ¼ þ Vext À Vext jÉ0 i ¼ E0 þ n0 °rÞ½Vext °rÞ À Vext °rފ dr;


so that
°
0 0 0
E0 < E0 þ n0 °rÞ½Vext °rÞ À Vext °rފ dr: °5:1:3Þ


An analogous argument, obtained by interchanging primed and unprimed
quantities, yields
°
0 0
E0 < E 0 þ n0 °rÞ½Vext °rÞ À Vext °rފ dr: °5:1:4Þ

0
Adding Eqs. (5.1.3) and (5.1.4), and using our assumption that n0 °rÞ ¼ n0 °rÞ
then leads to the expression
0 0
E0 þ E0 < E0 þ E0;

which appears unlikely. We have thus established that two di¬erent, nonde-
generate ground states necessarily lead to di¬erent ground-state densities. It
follows that two identical ground-state densities must stem from identical
external potentials, and with that our proof of the Hohenberg“Kohn theo-
rem is complete.
There is also an important variational principle associated with the
Hohenberg“Kohn theorem. Since the expectation value of any operator O
of a system is a unique functional of the ground-state density n0 °rÞ, this
certainly applies to the ground-state energy. We write this functional as

E½nŠ  hÉ0 ½nŠjT þ Vext þ VjÉ0 ½nŠi; °5:1:5Þ

where Vext is the speci¬c external potential of a system with ground-state
density n0 °rÞ and ground-state energy E 0 . For the case where the density n°rÞ
equals the ground-state density n0 °rÞ corresponding to the external potential
Vext , the functional E½nŠ then takes on the value E 0 . Since the ground-state
energy is uniquely determined by n0 °rÞ, the Rayleigh“Ritz principle estab-
lishes that

E 0 < E½nŠ for n 6¼ n0 :
187
5.2 The Kohn“Sham formulation
We shall ¬nd that this is a very useful property. The ground-state energy can
be found by varying the density to minimize the energy, provided we know
the form of the functional E½nŠ, or at least have a good approximation for it.
In fact, we can write the ground-state energy functional as
°
E½nŠ ¼ FHK ½nŠ þ Vext °rÞn°rÞ dr; °5:1:6Þ


where FHK ½nŠ ¼ hɽnŠjT þ VjɽnŠi is a unique functional. By that we mean
that FHK ½nŠ is the same functional of the density n°rÞ for all interacting N-
electron systems. We thus need to determine it only once, and can then apply
it to all systems.
We have here discussed the Hohenberg“Kohn theorem only for nonde-
generate ground states. The theorem can also be extended to include the
case of degenerate ground states, which is formally very important. There
are also many other extensions that are important for practical calculations,
such as extensions to polarized systems, and to systems at ¬nite temperatures.
For example, we might consider a spin-polarized system with a ¬xed quanti-
zation axis, which we take to be the z-axis. The system may then have a net
magnetization along this axis. In this case, we can de¬ne up- and down-spin
densities n" and n# , or, equivalently, total density n and polarization , with

n ¼ n" þ n#
n" À n#
¼ :
n" þ n#

A Hohenberg“Kohn theorem can then be formulated in terms of n" and n#
(or in terms of n and ). It turns out that calculations formulated in this way
are usually much more accurate than calculations cast in terms of density
alone, even if the system itself has no net polarization.


5.2 The Kohn“Sham formulation
While the Hohenberg“Kohn theorem rigorously establishes that we may use
the density, and the density alone, as a variable to ¬nd the ground-state
energy of an N-electron problem, it does not provide us with any useful
computational scheme. This is provided by the Kohn“Sham formalism.
The idea here is to use a noninteracting ˜˜reference,™™ or auxiliary, system,
and to look for an external potential Vs such that the noninteracting system
has the same ground-state density as the real, interacting system. Once we
188 Density functional theory
have obtained this density, we can use it in the energy functional Eq. (5.1.5),
or in some approximation of it. The ground-state of a noninteracting system

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