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 ‚ p 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

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 ‚ p 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; H

i0

dt

¼ 1 m½v; v2 þ ½v; V°rÞ

2

e

¼ i0 v ‚ H þ ½v; V°rÞ °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°rÞ;

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°rÞ 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°rÞ 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 À1 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 ¼

2

arctan

dE k

and arctan ½°E k À E k 0 Þ= 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 iÀ1 T:

Also

½E À H0 þ iÀ1 T ¼ ½E À H þ iÀ1 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 iÀ1 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 gy:

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 °rÞ 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 °rÞ dr;

so that

°

0 0 0

E0 < E0 þ n0 °rÞ½Vext °rÞ À Vext °rÞ 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 °rÞ 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

E½n hÉ0 ½njT þ Vext þ VjÉ0 ½ni; °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 E½n 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 < E½n 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 E½n, or at least have a good approximation for it.

In fact, we can write the ground-state energy functional as

°

E½n ¼ FHK ½n þ Vext °rÞn°rÞ dr; °5:1:6Þ

where FHK ½n ¼ hÉ½njT þ VjÉ½ni is a unique functional. By that we mean

that FHK ½n 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