Marder M.P. Condensed Matter Physics

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References

173

(d) Show by expanding 1/(1 + e

) in a Taylor series that

Σ^4-

83)

ι=ι

Densities of states in all dimensions:

(a) Find the volume of a sphere V

(R) of radius Rind dimensions. In order to

accomplish this task, first find the surface area A

of a sphere of unit radius in

d dimensions by considering the integral

t, = /*,*,... a« .-«-M-**. (6.84)

On the one hand,

/</

is easy to evaluate as a product of d separate integrals. On

the other hand, it can also be evaluated in polar coordinates, where it equals

[ drr

e-'

. (6.85)

By comparing Eqs. (6.84) and (6.85), find A

. As a consequence, find

(R)=

/ drr

. (6.86)

(b) Find a general expression for the density of k and energy states for an ideal

noninteracting Fermi gas in all dimensions.

8. Fermi function:

(a) Consider the electron density appropriate to aluminum. Assuming the con-

duction electrons to be free noninteracting fermions, find the chemical poten-

tial numerically to two place accuracy at 0 K, 300 K, and

000 K.

(b) Compare these results with the corresponding results of the Sommerfeld ex-

pansion at lowest nontrivial order.

the chemical potential.

References

L. D. Landau and E. M. Lifshitz

(

1980),

Statistical

Physics,

Part 1, 3rd ed., Pergamon Press, Oxford.

R. A. Millikan (1913), On the elementary electrical charge and the Avogadro constant, Physical

Review, 2, 109-143.

A. Sommerfeld (1928), On the electron theory of metals based on Fermi statistics, Zeitschrift für

Physik,

47, 1-32; 43-60. In German.

G. R. Stewart (1983), Measurement of low-temperature specific heat, Review of Scientific Instru-

ments, 54, 1-11.

G. R. Stewart (1984), Heavy-fermion systems, Reviews of Modern Physics, 56, 755-787.

J. J. Thomson (1907), The Corpuscular Theory of Matter, Charles Scribner's Sons, New York.

7. Non-Interacting Electrons in a

Periodic Potential

7.1 Introduction

While Sommerfeld's theory of the free-electron gas provides an excellent starting

point for the study of metals, it leads to a new fundamental difficulty. "The main

problem was to explain how the electrons could sneak by all the ions in a metal

so as to avoid a mean free path of the order of atomic distances. Such a distance

was much too short to explain the observed resistances, which even demanded that

the mean free path become longer and longer with decreasing temperature" [Bloch

(1976),

p. 26]. Bloch (1928) had provided the solution of

the

problem in his thesis.

Analyzing a single electron moving in a perfectly periodic potential, he says, "... I

found to my delight that the wave differed from the plane wave of free electrons

only by a periodic modulation. This was so simple that I didn't think it could

be much of a discovery, but when I showed it to Heisenberg he said right away:

'That's it!' " [Bloch (1976),p. 26.].

Immediately, the problem was altered from a hard one: "Why is the mean free

path so long?" to an easier

one:

"Why is the mean free path so short?" Scattering of

electrons is caused not by the lattice itself but by defects in the lattice, due either to

thermal fluctuations or to impurities. At low temperatures, electron mean free paths

in metals should increase up to a limit that is set by impurity density (Chapter 18).

This behavior is observed experimentally and gives great plausibility to Bloch's

great idea to study crystals through single electrons in periodic potentials.

7.2 Translational Symmetry—Bloch's Theorem

The free Fermi gas of Chapter 6 treats solids as if they are empty boxes filled

with electrons that interact neither with nuclei nor each other, apart from effects

due to the Pauli exclusion principle. Bloch (1928) added back some some of the

ingredients missing from this simplified view of solids. He first added back the

interaction of electrons with the nuclei, treated as a static external potential.

For general potentials U(r) the problem is still nearly intractable, and Bloch

settled on an additional simplification. He posed the problem of electrons moving

about in a periodic potential U(r), one which obeys

U(7 + R) = U(7). For all R in a Bravais lattice. (7.1)

175

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

176 Chapter 7. Non-Interacting Electrons in a Periodic Potential

Figure

7.1.

The setting for Bloch's theorem in one dimension is a potential U(x) of period

a on a periodic domain of length L.

The Hamiltonian is

. P

K=—

+ U(R). (7.2)

Now the model contains just the right balance of simplicity and realism. The sym-

metry (7.1) permits remarkable analytical progress and makes numerical solutions

Eq.

(6.3) tractable. The model is realistic enough to predict many physical prop-

erties of crystals such as the fact that some are metals and others are electrical

insulators.

7.2.1 One Dimension

The basic features of

wave

functions in periodic potentials are easiest to understand

in one dimension. Writing out Schrödinger's equation in the position representa-

tion gives

-^■^ψ{χ) + υ(χ)ψ(χ) = εψ(χ) (7.3)

Take U(x) to be periodic with period a as shown in Figure 7.1.

The one-dimensional space where ψ is defined is of length L. Take ψ to be

periodic so that

Periodic boundary conditions simplify the calculations considerably

ih(x 4- / ) =

i/i(r)

without substantially affecting any feature of the solutions. The pe- ιη Λ\

Ψ\ > ) Ψ\ )■ riodicity of ψ over distance L should not be confused with the peri- ^ ' '

odicity of U over distance a.

Suppose that the potential U(x) was just U(x) = 0. Then it would be easy to

write down all the solutions. They would be

rl>k(x)

= 4= (7.5)

When the potential U(x) is not zero, the solutions retain the same basic structure,

but change to

ilex ( \

, i \ & U\X) While φ is normalized over the .,

Wk\X)

7FE -, whole system, u is normalized ('·")

V^» over a single unit cell.

where u(x) is a function that like U(x) is periodic with period

and where N = L/a

is the number of cells in the full periodic system. That is, as shown in Figure 7.2,

the solutions are plane waves exp[/fcc] modulated by a periodic function u{x).

Translational

Symmetry—Bloch 's

Theorem

177

Periodic function u(x)

φ(χ) =

exç[ikr]u{x)

Figure 7.2. Bloch wave functions are periodic functions u(r) modulated by a plane wave

of longer period. The lower portion of

the

figure

displays the real part of ψ(χ).

The following derivation shows how this form arises, and shows that each and

every solution of Schrödinger's equation has the form of

Eq.

(7.6).

Fourier's theorem [for example, Tolstov (1978)] says that every periodic func-

tion can be written as a sum of all the complex exponential functions exp[/fcc] that

share the same period. Because ψ is periodic with period L, φ(χ) can be written

as a sum of Fourier components expD'g'x] where q' is of the form q' = 2irl'/L and

(—oo . . .

—

1, 0, 1 . . . oo) is any integer:

ψ(χ)

= —j= ^ 1p(q')e'

The sum over q' is thi

L ~f /' with q' =

ΙπΙ'/L.

normalization.

the

same as summing over (7.7)

The

factor of \/L fixes

U is periodic with period a = L/N, and it can be written as a sum of Fourier com-

ponents exp[/ÄJt] where the reciprocal lattice vector K is of the form K

—

2πΙ/α,

and / is an integer:

U(x) = Σ

iKx

(7.8)

Substitute Eqs. (7.7) and (7.8) into Eq. (7.3), finding

This equation must hold separately for each Fourier component exp[/gx],a condi-

tion imposed formally by choosing q = 2TTI/L, multiplying

Eq.

(7.9) by exp[—iqx]/L

and integrating from 0 to L. It is easy to verify that

-t

L Jo

dx e

i(q'-q)x

I Γ dxe^'

= ô

q,q'

;

LJo

'Jq',q-K-

Therefore

2^2

■i,{q) +

Y^5

^{q')U

= ^{q).

q'K

(7.10)

(7.11)

178 Chapter 7. Non-Interacting Electrons in a Periodic Potential

=>(E°

-E)tp(q)

+ J2

Φ(

)

K =

ε» is defined in Eq. (6.8). (7.12)

Equation (7.12) is a restatement of Schrödinger's equation in Fourier space and

it has

remarkable property. Each solution as

function of q is zero except at

discrete and evenly spaced set of q values, like a comb. Here is why.

Suppose one has

solution. There must be at least one

= lirm/L for which

ip(k) is not equal to zero. The equation for ip(k) involves ip(k

—

K) for all K of the

form

2-rrl/a.

Pick any of these wave function components, say

xp(k —

K'), and ask

what Eq. (7.12) implies. It says

(E°

,-E)t(k-K')

J2i;(k~K

-K)U

= 0 (7.13)

(££_*-,

~E)ïp(k-K')+y 1p{k-K)

<=0.

Send

-AT'as

(7.14)

~^ the sum index.

Equation (7.14) is a different equation from Eq. (7.12) with different coefficients,

but it involves exactly the same discrete set of wave function components: compo-

nents of the form ip(k-K) where K = 2πΙ/α is in the reciprocal lattice. Therefore,

Eq. (7.12) can be viewed as a collection of algebraic equations for this set of wave

function components, and all other components can be zero. This structure of the

matrix Hamiltonian is illustrated in Table 7.1.

So solutions of Eq. (7.12) have the form

γ-Λ The

are constants. The independent variable is q, and

1p{q) —

/ Oq^^K U/(.

is some value of q for which ip(q) is not zero. Do not (7.15)

„ worry about signs of the index K, since both

and —K

are in the reciprocal lattice.

Block's Theorem. Transform Eq. (7.15) back into real space by inserting

into

Eq. (7.7) to find

4τΣ <W+*

eiq

=-7f

«*

{h+K)x

)

ikx

u(x)

=> ψ(

) =

-±-L

where u(x)

— V

N = L/a is the

(7.17)

\Ja ^f

number of unit cells.

Equation (7.17)

Bloch's theorem.

says that when

single particle moves

a periodic potential the solutions are products

two pieces (Figure 7.2).

The

first piece is just the plane wave

lkx

the particle would adopt

empty box

(Eq. (6.7)). The second piece

is a

periodic function u{x) with the same period

as the potential U{x). Since u(x)

built entirely from plane waves

the form

exp[/ÄJt],

it must be periodic with period a.

To capture all solutions

Eq. (7.12) the notation must be generalized.

index

all

solutions

φ,

one begins

specifying some Fourier component

for

which the solution is not zero. It is customary to affix the index both to ψ(χ) and

u(x),

denoting by ι/^(*) a solution such that

(χ)

^^^

(

a) = Mx)e

,ka

. (7.18)

Translational

Symmetry—Bloch 's

Theorem 179

Table 7.1. The structure of Bloch's Hamiltonian in reciprocal space

UK_

-υ«,

v«

-M

(

L_,+^o/

(£-1+UK

UK,

ε°.

'*ι+«Ί

Φ{Κι)

ψ(Κ

)

■Φ{κ

-Λ

Ψ(*ι+*ι)

This matrix contains blocks that link together wave function components

ip(k

+ Ki)

for

a given m, where k

= 2wm/L and

= 2πΙ/α. There are no matrix elements connecting

■ip(q) 's

when the q's do not differ

reciprocal lattice vectors. The dimension of each block

is M, the number of reciprocal lattice vectors retained in the calculation, while the total

number of

blocks

is equal to the total number of unit

cells,

N = L/a.

The Fourier component k is called the wave number and

is called the crystal

momentum.

Selecting k does not specify a single unique solution V^(*)· As shown in Table

7.1,

choosing k means selecting one of the blocks that links together wave func-

tions φ with Fourier components of

the

form q = k+K. The block matrix as shown

is of dimension M, and has M eigenvectors. Each of these eigenvectors is a wave

function ψ with crystal momentum k. Cataloging these solutions can be done with

an additional index n called the band index. Actually, M is only finite if the po-

tential U(x) can be constructed from a finite number M of Fourier terms; since in

general the number of Fourier terms needed for U is infinite, M

—>

oo.

Choosing k specifies a set of Fourier components q = k + K from which the

wave function ψ^ will be constructed. Choosing k + K' picks out exactly the same

set. From this point of view, two wave numbers k are physically distinct only if

they do not differ by any reciprocal lattice vector K. This means that indices k

should be chosen from

2nm π π,

— whence

[--,-]

Taking k in the interval [0, 2π/α] would do (7.19)

just as well.

This collection of k

—

2irm/L, m e

[—N/2,

N/2

—

1] is called the first Brìllouin

zone, and will be defined in greater generality in Section 7.2.4.

Thus a complete set of solutions to Eq. (7.12) is

^nk(x)

lkx

{x)

with band energy £„&.

(7.20)

180 Chapter 7. Non-Interacting Electrons in a Periodic Potential

where k lies in the first Brillouin zone, and the band index runs from 0 to oo.

The significance of band energies £„* in explaining the behavior of solids can-

not easily be overstated. They contain information on whether a solid is a metal,

semiconductor, or insulator. Their slopes give electron velocities, and therefore

they predict electrical transport properties. Details of their shapes can be used to

calculate minimum energy crystal structures, and even magnetic properties. Sev-

eral of

the

following chapters will be devoted to learning how they can be calculated

and to examining their consequences.

Since there is still an infinite number of unknown solutions, of what use is

Bloch's theorem? Finding that solutions are of the form shown in Eq. (7.20) re-

duces the computational cost of solving Eq. (7.12) by a factor of N, where N is

the number of unit cells in the crystal. In short, the computational time has been

reduced by about a factor of 10

. By solving a problem in a single unit cell one ob-

tains a solution that applies to a crystal of arbitrary size. Bloch's theorem provides

a scaling theory relating microscopic computations to macroscopic phenomena.

7.2.2 Bloch's Theorem in Three Dimensions

Extending Bloch's theorem to three dimensions involves no substantially new ideas,

but the notation becomes a little more involved than in one dimension. Gener-

alize to a box of volume V = L

in which the wave function φ{?) is periodic:

ip(x + L, y, z) = ψ(χ, y, z), and the same holds for the y and z components. Thus,

choosing q as in Eq. (6.7),

^ = -i Σ ^)e

rqrr

(7.21)

The three-dimensional periodic potential U(r) described in Eq. (7.1) is com-

posed entirely of Fourier components exp[iK

■

R],

where K are reciprocal lattice

vectors (Section

3.2.5)

of the Bravais lattice R. This claim is easily proved, since if

a Fourier component is unchanged after translation by a Bravais lattice vector, then

=e . A function can only be periodic over S if

each

(J .11)

its Fourier coefficients is periodic in the

same

fashion.

But Eq. (7.22) is nothing other than the condition in Eq. (3.17) for a wave vector

k = K to belong to the reciprocal lattice. So

i/(r) = Ç^

. (7.23)

A more formal demonstration of

the

same fact is obtained by taking the Fourier

transform of U(r), explicitly:

ί df

e-^

U{r)

= Σ L

e-^

U{r

+ R)e-^

Use

U(7+R) =

u{7).

(7.24)

^ cell

= Ω5>-'*%

, (7.25)

Translational

Symmetry—Bloch

Theorem 181

where Ω is the volume of the unit cell, and

df —-

subscript on the integral Ω means that

£/-

= /

—e~

lqr

U(r).

the integral

over

unit cell. Substituting (7.26)

K for q gives a specific prescription for U^ if

U{r) is known.

The sum over Bravais lattice vectors appearing in Eq. (7.25) obeys the conve-

nient identity, derived as Eq. (A.29) in Appendix A,

it K

Therefore, applying an inverse Fourier transformation to Eq. (7.27) gives

\ Σ ^

**-%■

(

)

k k

Given the forms (7.28) and (7.21) for wave function and potential, one can

write Schrödinger's equation in Fourier space. Acting upon ψ with 'K

—

£, using

the Hamiltonian of

Eq.

(7.2) gives

0 = - Σ [

ψ -

(rj\ Ψ$)^''

Recall that

E\,

/2m.

(7.29)

[

(£

+ Σ ^

+?,)

·^] V>tf) (7.30)

^°=E/f[(4-

)^'^

)

''+E^

^]v'(?

) (7.31)

[(4

)<%' +

h-k«>

1>&)

(7-32)

=Φ

(ε

-ε)ψ{ξ)

ΣυζΦ(3-Κ)

= 0. (7.33)

Equation (7.33) generalizes Eq. (7.12) to three dimensions. It has similar implica-

tions.

Suppose that ψ(£) is nonzero. Then all the components of

ip(q)

involved in

Eq. (7.33) are of the form

<φ(ί

+ K), where K is in the reciprocal lattice, and

(a) = N S-- - Up.

The coefficients «^ are again constants (7.34)

Thus similar to Eq. (7.17)

M-

(r)

= V/Q equals the number of points in the

ib-('r) =

lattice and is convenient to introduce to nor-

'X'>

^Ί<\

' /Tj

malize uj(r) within each unit cell.

where u^(r + R)

u^(r)

is a

periodic function on the Bravais lattice. Mirroring

the one-dimensional case, there is an infinite number of solutions

i/j^(r)

for each

182

Chapter 7. Non-Interacting Electrons in a Periodic Potential

k. Now

is called the Bloch wave vector, and hk is again called the crystal mo-

mentum. The many solutions for each k are again distinguished by the band index

which also decorates

as in

u ^(r)e

7+R

VV^+tf) =

VV^V*·*

JÈ

^ ·

(7-36)

Saying that -φ^ solves Schrödinger's equation means that it solves the eigen-

value equation (7.33), and the corresponding band energy is denoted £^.

It is also valuable to restate Eq. (7.33) in terms of Dirac notation; it is

Λ = Σ \q')£%{q'\ + £ tf)Ufrtf -K'\. Compute <$|Α-ε|ψ>

(7.37)

_,-,

to regain Eq. (7.33).

, 6

7.2.3 Formal Demonstration of Bloch's Theorem

Since Bloch's theorem follows from symmetry of the potential U, there must be

some way to derive

that focuses on symmetry and does not require grinding

through long computations. Indeed there is, but the derivation is fairly abstract.

Recall that the operator generating a translation through a vector R is

—

g-'^/'i

See, forexample, Landau and Lifshitz (1977),

38)

45.

where P is the momentum operator and R a Bravais lattice vector. Since the com-

ponents of momentum commute, all of these operators, for allowed values of R,

commute with one another and with the Hamiltonian (7.2) (shown in Problem 2).

Therefore [Landau and Lifshitz (1977), pp. 14 and 34; Schiff (1968), p. 154]

all of these operators and the Hamiltonian can be diagonalised simultaneously.

Any eigenvector of the Hamiltonian can be taken as an eigenfunction of all the

translation operators as well:

Τΐ\ψ) =

\tp)

= Cs

ψ)

■

Here C

is a

constant that is yet to be determined. (7.39)

Operating with the bra vector (r\ (eigenfunction of position) on Eq. (7.39) gives

1p(?

+ R) =

Cfi1p(r). Because φ{7)

(r\ip) and {r\f!

(r + R\.

(7.40)

On the other hand, operating with the bra vector (k\ (eigenfunction of momentum)

on Eq. (7.39) gives

€

*(ϊ\ψ)=ε

(Ϊ\ψ) (7.41)

=*►

either C

ilâ

or (k\ip) = 0. (7.42)

Therefore, \φ) can have nonzero overlap with only a single (k\, and that value of

is used as an index to label the eigenfunction ψ^.