R?ssler U. Solid State Theory: An Introduction

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126 5 Electrons in a Periodic Potential

The considerations of this section apply to any interacting fermion system

and external potential. The DFT-LDA has b een and is currently applied to

atoms, molecules, and condensed matter. With the available computer power

it has become the dominant tool for solving quantum many-body problems.

The input with respect to the exchange–correlation energy comes from the

homogeneous electron gas (which accounts also for the replacement made in

(

5.24)), which we have studied in detail in Chap. 4.

The DFT concepts have been reﬁned to include spatial inhomogeneity

of the density in the xc energy (in the generalized gradient approximation),

spin polarization (in the spin or SDFT), and time dependent perturbations

(in the DFT perturbation theory, mentioned in Chap.

3)[100, 123]. Also, the

restriction to the system ground state has been overcome by taking care of

the discontinuity of the correlation energy in its dependence on the quasi-

particle energy across the Fermi energy (quasi-particle corrections in the GW

approximation) for which we refer also to Chap.

7 [123, 129, 130]. But even

without the latter, the eigenvalues E

of the Kohn–Sham equations with the

single particle potential (

5.30) are usually taken as single particle energies not

only for states occupied in the ground state, but also for unoccupied states.

This identiﬁcation is widely supported by the agreement between experimen-

tal mapping of the energy bands from photoelectron spectroscopy (PES) and

results from DFT calculations [

131]. It should be noted, however, that this

agreement is not found for optical properties in semiconductors and insulators

due to the b a nd gap problem: as it turns out, the separation between conduc-

tion and valence band states, as obtained from DFT-LDA calculations is too

small [

130]. This discrepancy can be resolved systematically by considering

diﬀerent xc energies for conduction and valence band states, as is done in the

already mentioned GW approximation.

As we have seen in this section, the Hartree, the Hartree–Fock, and the

eﬀective LDA potential (

5.30) depend o n the solutions of the correspond-

ing Schroedinger equations. For this typ e of problem an iterative procedure

applies: for the given external potential one has to choose a single-particle

density n

(r) to start with, calculate the Hartree and LDA contributions and

solve the Schroedinger equation. The occupied states of this solution deﬁne a

density n

(r), which in general will be diﬀerent from n

(r)andistakentocre-

ate a modiﬁed Hartree and LDA potential. With these modiﬁed potentials (or

a mixture with the potential of the previous step) the Schroedinger equation

is solved again and the procedure is repeated until the calculated densities (or

energy eigenvalues) for two successive iteration steps are reproduced within

desired limits of accuracy. At this level, the obtained single-particle energies,

wave functions and the potential to which they contribute via the occupied

states are self-consistent.

5.2 Bloch Electrons and Band Structure 127

5.2 Bloch Electrons and Band Structure

In the previous section we have reduced the many-body problem of the

electrons in a solid to the single-particle problem

Hψ

= E

,H= −

¯h

∆ + V

eﬀ

(r) (5.31)

with the periodic eﬀective potential

eﬀ

(r + R

)=V

eﬀ

(r). (5.32)

The periodicity of the potential means invariance of the Hamiltonian under

lattice translations

[H, T

]=0, with T

−

¯h

p·R

(5.33)

and the momentum operator p. The properties

(r)=ψ

(r − R

)and|T

(r)|

= |ψ

(r)|

(5.34)

of the translation op erator allow one to write

(r)=e

−ik·R

(r) (5.35)

where the phase factor, the eigenvalue of the translation operator, is charac-

terized by the wave vector k (note that ¯hk is the eigenvalue of the momentum

operator p). Due to the property (

5.35) the eigenfunctions of (5.31)canbe

split into an exponential and a lattice periodic function

(r)=e

ik·r

(r) ,u

(r + R

)=u

(r). (5.36)

This form, a modulated plane wave, is characteristic of electrons in a peri-

odic lattice: the Bloch

function. It is schematically depicted in Fig. 5.1.The

ψ ( )r

Fig. 5.1. Schematic view of a Bloch function (solid line) and its plane wave part

(dashed line), the dots mark the lattice points

Felix Bloch 1905–1983, Nobel prize in Physics 1952.

128 5 Electrons in a Periodic Potential

characterization of the Bloch function by the wave vector is unique within

the 1st Brillouin zone, because the phase factor in (

5.35)isthesameforallk

diﬀering by a recipro cal lattice vector.

For a given k, the eigenvalue equation (

5.31) has in general an inﬁnite

set of indep en dent solutions, which are distinguished by an energy quantum

number, the band index n.Thusthesolutionsof(

5.31) are classiﬁed by the

complete set of quantum numbers α =(n, k) (a spin quantum number can be

added where required)

−

¯h

∆ + V

eﬀ

(r)

(r)=E

(k)ψ

(r) (5.37)

and the energy eigenvalues E

(k) for all n and k from the 1st Brillouin zone

deﬁne the energy band structure of electrons in a periodic potential. By apply-

ing periodic bounda ry conditions, one can easily verify, that there are as many

diﬀerent k in the Brillouin zone as there are unit cells in the crystal volume.

Thus, each energy band can accommodate one electron of either spin per

Wigner–Seitz cell (Problem 5 .3).

Supplement: Symmetry of E

(k):

Besides the translation symmetry E

(k)=E

(k+G) considered already by restrict-

ing k to the ﬁrst Brillouin zone, there are other symmetries of the problem according

to which the band structure repeats within the Brillouin zone.

1. Due to time reversal invariance the Hamiltonian is hermitian, H = H

†

, i.e., it is

real and w e may write the complex conjugate equation to (

5.37)

Hψ

∗

= E

(k)ψ

∗

. (5.38)

The complex conjugate Bloch function ψ

∗

,writtenintheformof(

5.36), is

characterized by a phase factor with −k and belongs to the solutions of (

5.37)

in the form

Hψ

n−k

= E

(−k)ψ

n−k

. (5.39)

Both these equations are eigenvalue equations for the same operator and yield

the same spectrum. If w e consider the electron spin (whose direction changes

under time inversion) as additional quantum number, we may write

n↑

(k) , all n} = {E

n↓

(−k) , all n}, (5.40)

where the curly brackets denote the whole set of eigenvalues. This degeneracy,

follow ing from time reversal symmetry, is known as Kramers degeneracy.

2. We mentioned already in Chap.

1 the symmetry of the periodic lattice under the

operations S of the point group, which now means in variance of the Hamiltonian

of (

5.37)

[H, S ]=0, or SH S

−1

= H. (5.41)

Application of the point group operation S to the wave function ψ(r) changes r

into S

−1

r and we may write

5.3 Almost Free Electrons and Pseudo-Potentials 129

Sψ

(r)=ψ

−1

r)=e

ik·(S

−1

r) (5.42)

which because of k ·(S

−1

r)=Sk ·r means that the Blo ch function (

5.42)isone

with wa ve vector Sk and we conclude, as for the time reversal symmetry, that

(Sk) , all n} = {E

(k) , all n}. (5.43)

Thus, the energy band structure reﬂects completely t he point group symmetry

of the crystal structure. This property of the band structure can be exploited

in performing sums over k, which can be restricted to the so-called irreducible

w edge of the Brillouin zone.

3. Consider a combination of 1 and 2: if time reversal invariance combines with the

p oint group symmetry, we have for operations S with Sk = −k that

n↑

(k) , all n} = {E

n↓

(k) , all n}. (5.44)

In contrast to (

5.40), this situation with the up and down spin states of a given

k having the same energy, is called spin degeneracy.

Let us have a look back to the beginning of this chapter with the N

electron Hamiltonian (

5.2) as starting point. Within the DFT concepts we

have reduced this many-particle problem to a single-particle one (of course

with the restrictions already mentioned). Nevertheless, we can formulate H

in an approximate way (compare with (

4.94))

≃



(k)c

†

(5.45)

by making use of the band structure and by introducing fermion creation and

annihilation op erators c

†

and c

with

†

′

} = c

†

′

+ c

′

†

= δ

′

. (5.46)

In this approximate formulation, the electron–electron interaction is incor-

porated in the single-particle properties and we may address these Bloch

electrons as the quasi-particles of the density functional theory.

5.3 Almost Free Electrons and Pseudo-Potentials

Starting with this section we present concepts of band structure calculations

which are outlined in many textbooks of Solid State Theory [

11, 121, 132–134].

The task is to solve the Schroedinger equation (

5.37)

−

¯h

∆ + V

eﬀ

(r)

(r)=E

(k)ψ

(r) (5.47)

for Bloch electrons with

(r)=e

ik·r

(r). (5.48)

The simplest possible concept is to make use of the periodicity of u

(r)

and expand it in the complete set of normalized plane waves with reciprocal

130 5 Electrons in a Periodic Potential

lattice vectors G

(r)=



(G)

√

i(k+G)·r

. (5.49)

The problem of solving (

5.47) co nsists in ﬁnding the expansion coeﬃcients

(G). For this purpose we u se the expansion (

5.49)in(5.47), multiply

from left with a normalized plane wave with k + G

′

, and integrate over the

crystal volume V

. This leads to a set of coupled homogeneous linear equations







¯h

(k + G)

− E



′

+ V (G − G

′

)



(G)=0. (5.50)

The ﬁrst term results from the operator of kinetic energy and the orthogonal-

ity of the plane waves, the second is the Fourier component o f the eﬀective

potential

V (G − G

′



eﬀ

(r)e

i(G−G

′

)·r

r. (5.51)

The energy eigenvalues are obta ined from the secular problem



¯h

(k + G)

− E



′

+ V (G − G

′

)

=0. (5.52)

The eﬃciency of this concept depends essentially on the convergence of the

plane wave expansion (

5.49) or on the strength of the periodic potential in

terms of its Fourier coeﬃcients. When looking at Fig.

5.1 it is conceivable

that a strong mo dulation of the Bloch function (caused by a strong potential)

requires more terms in the plane wave expansion than a Bloch function with

a weak modulation. In general, one can say that the plane wave expansion is

expected to work well if in the secular problem (

5.52) the kinetic energy terms

(the diagonal terms of the matrix) dominate over the (non-diagonal) potential

terms. This is the case for almost free electrons, for which the periodic p oten-

tial acts as a weak perturbation. In fact (

5.52) represents an expression of ﬁrst

order perturbation theory applied to the free electron states and leads to a

Brillouin–Wigner perturbation expansion. The anticipation of a weak periodic

potential and the occupation of the lowest bands deriving from a free electron

parabola with the valence electrons, is consistent with the idea put forward

at the beginning of this book that the constituents of the solid are ions and

valence electrons. But we have not yet shown how this concept is considered

in the eﬀective potential.

Intuitively, the potential of an ion seen by a valence electron is that of a

screened uncomp ensated charge outside of the closed shell radius but approx-

imately zero (due to charge compensation) inside of this shell. This leads to

the model of an empty core potential

(r)=



0 r<R

−

4πε

−k

r>R

(5.53)

5.3 Almost Free Electrons and Pseudo-Potentials 131

V (r)

v (q)

r q

Fig. 5.2. Empty core pseudo-potential and its Fourier transform (the arrows

indicate the length of reciprocal lattice vectors)

where Z is the ion charge, R

the core radius, and k

the inverse

Thomas–Fermi screening length (see Problem 4.11). This potential is depicted

in Fig.

5.2 together with its Fourier transform

(q)=−

+ k

)

cos qR

ion

(q)

ε(q)

, (5.54)

which can be expressed as the ion p otential v

ion

(q)=−Ze

/4πε

divided

by the dielectric constant ε(q) in the Thomas–Fermi approximation. Charac-

teristic values of the parameters Z, R

, and k

, yield values for the Fourier

coeﬃcients of the periodic potential of a few tenths of the Rydberg energy,

which can in fact be considered as a perturbation on the scale of the free

electron energies (which are of the order of 1 Ry).

The empty core potential is a prototype pseudo-p otential adapted to the

anticipated construction of the solid out of io ns and valence electrons [

134,

139]. If we had started from nuclei and a l l electrons, then the eﬀective single-

particle po tential would have the characteristic form depicted in Fig.

2.1 with

strong attractive parts close to the nuclei. Such a potential would have strong

Fourier coeﬃcients up to large r eciprocal lattice vectors, which are required

to obtain a converging representation, especially of the most strongly bound

electrons in the inner shell. But as already discussed, these electrons are not

relevant for solid state properties and we are interested here only in the valence

electrons. In view o f the all-electron problem, the valence el ectrons are in

states which are orthogonal to those in closed shells and we can understand the

problem of calculating the band structure as that of looking for eigensolutions

of the all-electron Hamiltonian in that part of the Hilb ert space, which i s

orthogonal to the core states. This view opens the principle access to pseudo-

potentials.

Let us expand for this purpose the Bloch function ψ

(r)=r|nk in

terms of plane waves r|k + G which by construction are made orthogonal

to the core states

|nk =(1−P)|nk

, |nk



(G)|k + G, (5.55)

132 5 Electrons in a Periodic Potential

where

P =



ν∈core

|νkνk| (5.56)

is the projection operated onto the core Bloch states (see (

5.67)). Mak-

ing use of this expansion in the eigenvalue problem with the single-particle

Hamiltonian but with all electrons considered in the periodic potential V

all

one ﬁnds

all

|nk =



+ V

all

−



ν∈core

|νkνk|



|nk

= E



1 −



ν∈core

|νkνk|



|nk

. (5.57)

An eigenvalue equation for this expansion is obtained by rearranging (

5.57)

in the form



+ V

all



ν∈core

(E − E

)|νkνk|



|nk

= E|nk

(5.58)

with the nonlo cal pseudo-potential op erator

psp

= V

all



ν∈core

(E −E

)|νkνk|. (5.59)

By construction, the solutions of (

5.58) are orthogonal to the core electron

states and yield the electron states of valence and conduction bands. When

taking the exp ectation value of the nonlocal operator with |nk

one ﬁnds

(note, that we calculate valence electron states with E>E

)



ν∈core

(E −E

)|νk|nk

> 0, (5.60)

i.e., the additional potential cancels (partially) the attractive potential of the

nuclei and converts the all electron potential into a weak pseudo-potential. In

this sense, the eﬀective potential in (

5.47) is to be und erstood as a pseudo-

potential and we can safely assume, that the plane wave expansion converges

with reasonable eﬀort.

Initially, the Fourier coeﬃcients of the pseudo-potentials have been used as

empirical parameters [

134, 139] which were determined by ﬁtting a calculated

band structure to experimental data, such as Fermi surface parameters o r

energy gaps. In the course of time, the pseudo-potential concepts have been

developed and it is now possible to perform ab initio band calculations which

make use of so-called norm conserving a nd soft pseudo-potentials, which are

free of adjustable parameters [

140, 141].

Instead of treating the full problem of solving (

5.52),wemaylookfor

situations where the free electron picture, with the parabolic dispersion of the

5.3 Almost Free Electrons and Pseudo-Potentials 133

Ist BZ

4π

2π

4π

Fig. 5.3. Free electron energy bands in one dimension: extended and repeated zone

scheme

energy, is only slightly perturbed by the periodic potential. Figure 5.3 shows

the free-electron band structure for a one-dimensional system without peri odic

potential but with the periodicity taken into account by introducing Brillouin

zone boundaries. Due to the translation symmetry, the energy dispersion is

periodic in k with periods 2π/a, i.e., the free-electron parabola may start at

each value 2nπ/a with integer n . In order to avoid the redundancy in this

repeated zone scheme, it is suﬃcient to consider the dispersion only in the

ﬁrst Brillouin zone (reduced zone scheme). This picture of the reduced zone

scheme can be obtained also from the parabola starting at 0, by shifting those

parts, which are outside the ﬁrst Brillouin zone, by multiples of 2π/a to bring

them back to this zone. This concept is not restricted to the free electron

dispersion in 1D but applies as well to any realistic band structure not only

of electrons (see e.g., the situation in Problem 3.1 for the linear chain, when

the two masses become equal).

Coming back to the secular problem (

5.52) we recognize in Fig. 5.3 degen-

eracies of the free-electron energy dispersion at the Brillouin zone bou n daries

and in its center, e.g., the parabolas starting at 0 and at 2π/a cross at k = π/a.

Writing the secular problem for the corresponding states one obtains

E(k) − EV(G)

V (G) E(k − G) − E

=0,G=

2π

, (5.61)

134 5 Electrons in a Periodic Potential

where E(k) is the kinetic energy of free electrons, with the zeros

(k)=



E(k)+E (k − G)





(E(k) − E (k − G))

+ |V (G)|



1/2

. (5.62)

For k = π/a one has E(k)=E(k − G)=¯h

/2m



π/a



and





¯h





2π

. (5.63)

The eﬀect of the periodic p otential is to remove the degeneracy o f the free

electron states (here at the boundary of the Brillouin zone) and to create

an energy gap or band gap in the otherwise continuous spectrum. The gap is

determined by a Fourier coeﬃcient of the potential for the reciprocal lattice

vector connecting the degenerate plane waves. The dispersion o f the lowest

energy bands E

(k) around k = π/a is shown in Fig.

5.4 together with the

corresponding free electron dispersion (thin d ashed lines).

In general, the reciprocal lattice vectors of degenerate plane waves fulﬁll

the Bragg condition (Problem 5.4), thus the plane waves being reﬂected at the

crystal planes is characterized by t he reciprocal lattice vector form standing

waves. This becomes evident by looking at the eigenfunctions. In our simple

example with the energies of (

5.63), they take the form

(x) ∼ isin

πx

and ψ

−

(x) ∼ cos

πx

(5.64)

and their modulus shows minima (maxima) at the lattice points, which leads

to the raising (lowering) of the energy due to the periodic potential in com-

parison with the free electron case. (Note that the potential is attractive for

electrons in the vicinity of the ion positions.) The energy gaps and the hori-

zontal slope of energy bands (preferentially) at the boundary of the Brillouin

π/2a π/ak

Fig. 5.4. Opening of a gap at the Brillouin zone b oundary due to a periodic potential

5.3 Almost Free Electrons and Pseudo-Potentials 135

Fig. 5.5. Brillouin zones of the tw o -dimensional square lattice and Fermi contours

zone, which corresponds to a va nishing group velocity, are the most prominent

features of the energy band structure.

Let us now proceed to the situat ion of a 2D square lattice and look at the

Fermi contours as depicted in Fig.

5.5. In the left part, we recognize the recip-

rocal lattice points and the construction lines for the Brillouin zones: the solid

lines mark the central square of the ﬁrst Brillouin zone (ﬁrst BZ); the dashed

lines, augmenting the ﬁrst BZ to a square of double size, mark the second

Brillouin zone. The four tr iangles outside of the ﬁrst BZ can be r earranged

by translations with reciprocal lattice vectors to form a square reproducing

the 1st BZ as indicated in the right upper part of Fig.

5.5.ThethirdBZis

obtained from the dash-dotted smaller triangles which again can b e rearranged

to a square of the size of the ﬁrst BZ around the M point (see lower right part

of Fig.

5.5). The free-electron dispersion in the extended zone scheme (Fig. 5.5,

left) is a paraboloid with its minimum at the Γ point or k =(0, 0). In the

presence of a 2 D periodic potential, (causing the square lattice with lattice

constant a) this continuous energy dispersion will be deformed in the vicin-

ity of the Brillouin zone bound aries in connection with the opening of gaps

(Problem 5.5).

Instead of looking at the band dispersion, we want to discuss here the con-

sequences of the Fermi circle shown in the left part of Fig.

5.5 for an a ssumed

electron density, i.e., the a rea inside this c ircle deﬁnes the occupied states of

the free-electron disp ersion at T = 0 K. When constructing the Brillouin zones

the Fermi circle, breaks into pieces which by rearrangement of the extended

to the reduced zone scheme, give the grey areas indicating occupied states in

diﬀerent b ands. The ﬁrst BZ or the lowest energy band is completely occupied,

the second and third band (plotted for a BZ around the point M) are only par-

tially ﬁlled and the grey areas are only faintly reminiscent of deriving from a

circle. In the presence of a weak periodic potential, these contours, which sep-

arate the empty from the occupied states, are distorted when the gaps open

at the BZ boundaries. These Fermi contours (which in 3D become Fermi