Marder M.P. Condensed Matter Physics

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Part II

ELECTRONIC STRUCTURE

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

6. The Free Fermi Gas and Single

Electron Model

6.1 Introduction

Much of condensed matter physics lies within a Hamiltonian that one easily can

write down in a single line. It is

K=Y-^+-T

iqi

. (6.1)

^2M

2fy\R

-R

The sum ranges over all electrons and nuclei in a solid; M; is the mass of

electron

or nucleus, and qi is its charge. The simplicity is deceptive. Equation (6.1) can be

attacked directly by computer for little more than 10 to 20 particles. Dealing with

the 10

particles in actual solids requires a series of approximations, few of which

are particularly well controlled. One has to replace the original equations with

simpler ones, not always possible to justify, but capturing essential features of the

system. A number of these approximations is indicated in Figure

6.1.

In the study

of electronic structure about to begin, the goal will be to start at the lower left hand

side of the sketch and work back up toward the top.

Since Eq. (6.1) so completely intractable in its original form as to be almost

useless, progress comes about by posing and solving a series of model problems.

In these model problems, one makes drastic simplifications that make it possible

to solve the resulting equations analytically or numerically. The arbiter of success

is partly comparison with experiment, and partly the ability to gain qualitative in-

sights. There is now more than 70 years' experience with approximation schemes,

and their strengths and weaknesses are fairly well understood.

The simplest model of

metal is the free

Fermi

gas.

In this model, the Coulomb

interactions between electrons and electrons and electrons and nuclei are all turned

off. One considers a collection of electrons moving freely around in a box, and the

only nontrivial aspect of reality to be maintained is the Pauli exclusion principle.

In applying this model to a metal, one considers just a subset of the material's

electrons, the conduction electrons. The free Fermi gas model captures the features

of some metals remarkably well, particularly the alkali metals such as sodium.

It simply cannot be employed for any solid that happens to be an insulator or a

magnet.

Some important features of the solution of the free Fermi gas do not depend

upon details of the solution. They depend upon the fact that energy of the system

of electrons can be written as the sum of energies of individual noninteracting

155

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

156 Chapter 6. The Free Fermi Gas and Single Electron Model

*=Y.ê,+

Pf , i v- mi

Ù\

''\

Figure

6.1.

Some of

the

approximations used to make

Eq.

(6.1) tractable.

electrons. Thus some of the results can quickly be extended to the more general

single electron model.

The Pauli principle alone is able to resolve the main paradox in the electron

theory of solids prior to quantum mechanics. The problem was pointed out by

Thomson (1907), and it occurs in trying to account for the specific heat of silver.

Silver has one conduction electron per atom, leading to 6 · 10

electrons cm

-3

Thomson overestimated this number by a factor of 100, partly because his mea-

surements of the charge on the electron were several times greater than the correct

values to be obtained by Millikan (1913) several years later, but the conclusions

he drew still correctly captured the difficulty. The equipartition of energy should

Starting Hamiltonian 157

contribute kgT jl to the mean energy for each of the three kinetic and potential

degrees of freedom for the free electrons plus another 3kgT for the nuclei of the

atoms, leading to a specific heat of

c\>

= 6nkß = 1.19 cal cm~

K '. The exper-

imental specific heat is, however, half this value, as if the electrons are somehow

immune to the demands of statistical mechanics. Thomson concluded that " We

thus get to a contradiction. The value of the specific heats of the metals shows that

the corpuscles [electrons] cannot exceed a certain number, but this number is far

too small to produce the observed conductivities...." [Thomson (1907), p. 85] The

resolution of the contradiction was not found for 20 years, and lies (as explained

below) in the fact that only a tiny fraction of the electrons is permitted by the Pauli

principle to participate in the process of absorbing energy.

The free Fermi gas and single electron models are such a crude approximations

that it may seem silly to study them at all. Two ideas eventually explained why even

the free Fermi gas can have great quantitative success for some metals. Neglect of

the periodic potential generated by the lattice of ions is justified by the pseudopo-

tential (Section 10.2.1), which shows how a redefinition of the electrons' wave

functions turns them into particles interacting with weak potentials. Neglect of the

interactions of electrons with one another

justified by the idea of the Fermi liquid

(Section 17.5), which shows that properly chosen linear combinations of electron

states behave like noninteracting particles. But these ideas ultimately provide only

partial justification for approximation schemes, so there is no choice but to suspend

disbelief and begin to calculate.

6.2 Starting Hamiltonian

The single-electron model is defined by the Hamiltonian

N fc2

It describes N conduction electrons, each of which interacts with an external po-

tential U but does not interact with the other conduction electrons. Equation (6.2)

is called the single-electron model because if one finds the eigenfunctions φι(7ι)

for single electrons, obeying

H V

+U(r)\ ψ,(η = ε

{?), (6.3)

y 2m

then the eigenfunctions describing many particles are simply obtained from prod-

ucts of the one-particle functions. The energy of the many-electron system is just

a sum of the energies of the one-electron functions that make it up (Problem 1).

That is, although the equation can be used to study large numbers of electrons,

their properties can be obtained one electron at a time.

Writing down Eq. (6.2) requires severe approximations outlined in Figure 6.1.

However for a general potential U it is still impossible to solve in general. To start

158 Chapter 6. The Free Fermi Gas and Single Electron Model

making progress, throw away the potential U too. The free Fermi

gas,

is described

-h

— T

(

■

· ·

") =

εφ

(

■

· · **)· (

)

It describes N conduction electrons, interacting neither with nuclei nor each other.

The eigenvalues and eigenfunctions of this Hamiltonian can be found exactly.

No differential equation is completely specified without naming its boundary

conditions. A natural choice would be to take Φ to vanish whenever any of its

arguments reaches the boundaries of the system, but this choice is not convenient

for calculations. Instead, one conventionally chooses a square box of side length

= V and imposes periodic boundary conditions;

Φ(χι + L, y

z\ . . . ,

ZN)

= y{xi,y\,zi . - .

ZN)

Φ(χι,)>ι +

zi ... , ΖΝ) = Ψ(*Ι,)Ί,ΖΙ -

■ ■

ZN)-

This

boundary condition is what one obtains

demanding that the system repeat

indef-

initely

with period L along the x, y, and z

directions.

(6.5)

This mathematical condition cannot possibly be realized experimentally. However,

almost any physical quantity one might calculate using Eq. (6.5) alters at most by

an amount proportional to

/L if

one

adopts more realistic boundary conditions. So

long as L is macroscopic, such corrections are negligible. It is sufficient to impose

some condition on the wave function at the system boundaries.

One Free Fermion. For the free Fermi gas, obeying Eq. (6.4), one can find the

one-electron solutions explicitly. Eigenstates of (6.3) are all of the form

Φ^ = ~j=

Because of the factor of l/VV, this function (6.6)

V is normalized.

with k of the form

- 2π

k=—{l

,ly,l

(6.7)

, /y, and l

are integers ranging from —oo to oo. Only if k is chosen in this way will a

function

of the form Eq. (6.6) satisfy the boundary condition Eq.

(6.5).

This enumeration

allowed values of k is more general than may now appear and it will survive unchanged

when

electrons interact with periodic potentials.

The eigenvalue corresponding to the eigenfunction (6.6) is

£9

= Simply insert Eq. (6.6) into Eq.

(6.4).

Setting (6.8)

hk = p = mv, the energy

just ^mv

Ground States of Many Free Fermions. The ground state of electrons obeying

Eq. (6.4) is constructed from products of the one-electron wave functions (6.6).

The Pauli exclusion principle forbids any given state from being occupied more

than once, and therefore any given state indexed by k is able to host no more than

two electrons, one for each value of spin.

Densities of States 159

The ground state of N electrons is built by first putting two electrons into the

single-particle state of lowest energy, which is \k\ = 0. Next one puts electrons

two at a time into all the states with \k\ = 2-n/L, and so on. The early stages of

this process are the subject of Problem 6.2. Because the energy Eq. (6.8) increases

with k, as one keeps adding electrons, one always wants to fill states with the lowest

available values of

£®.

Since £^ is proportional to k

, this means that successive

electrons get added to the ground state in successive spherical shells.

Define the occupation number f^ of a state indexed by k to be 1 if this one-

electron state is part of the ground state, and 0 otherwise. For a large number N of

electrons, the ground state corresponds to setting the occupation number _/j of all

states with k less than a certain wave vector kp, the Fermi wave vector, to 1 and

setting all others to 0. That is, the ground state is built out of one-electron wave

functions occupying a sphere in &-space, as shown in Figure 6.2. Naturally, one

wants to relate the number of electrons within the Fermi sphere to the value of kf,

and for this purpose it is necessary to recall Eq. (6.7) and analyze it more closely.

Figure 6.2. The ground state of the free electron gas is constructed by occupying vertices

in a cubic grid of states that approximate a sphere of radius k

in k space. Any other shape

would miss opportunities to put electrons as close as possible to the origin in k space, and

thus would not minimize the total energy.

6.3 Densities of States

The k states described by Eq. (6.7) occupy a cubic lattice in k space, also called

reciprocal space, with neighboring points separated by distances of Ιπ/L, as shown

in Figure 6.2. One can draw a Wigner-Seitz cell around each of these points and

can see as a result that the volume of the region of reciprocal space associated

with k in this way is (2TT/L)

. Not only is this statement true for the free Fermi

160 Chapter 6. The Free Fermi Gas and Single Electron Model

gas,

it is also true, as shown in Chapter 7, for electrons moving in general periodic

potentials, and as shown in Chapter 13 for phonons. Therefore, all the terminology

developed in this section to keep track of k states will recur constantly throughout

the remainder of this volume, and the densities of states D defined here will have

broad physical significance.

In order to calculate the total number of electrons N, or their energy £, or any

other thermodynamic quantity, one needs to carry out sums of the type

(6-9)

where F is some function of wave vector k, with the k vectors summed over values

allowed by (6.7). Mathematical convenience requests that one convert these sums

into integrals over a continuous function

F^.

Integration is defined by partitioning

space into a large number of volume elements and summing a function over these

elements, multiplied by the volume of each one. Because each k point is associated

with a volume (2π/£)

, one obtains

27Γ

Ft = y ( ) Fr

This expression constitutes a long wavelength ap- (6.10)

*-^* L "·

proximation that is only accurate if

varies slowly

from one k to its neighbor, a condition that usually

obtains in practice.

V f -*

> Fr = -^ / dk Fr.

Recall V

= Ü.

Although this formula has only (6.1

TT"'

(^ττ) J

been derived for large volumes in the shape of a

k cube,

can be proved to hold for large volumes

of arbitrary shape.

On some occasions, the function

Fr,

may involve

delta function, such as

δ^.

Viewed as

function of

continuous variable, this function clearly transforms

into some multiple of ô(k

—

q). In order to remain consistent with Eq. (6.11), the

correspondence must be

δ-^—yì

'—ßß

—

Only in this way do both sides of Eq. (6.11) (6.12)

Ύ*

equal

when these delta function are inserted

6.3.1 Definition of Density of States D

The sum in Eq. (6.11) is over all wave vectors allowed for free electrons. Because

for each wave vector Pauli's exclusion principle allows two electrons, one with

spin up and the other with spin down, it is conventional to define the density of

electronic states, also sometimes called the density of levels

The factor of 2 accounts for spin. (6_

(2π)

which is defined to be the quantity allowing one to write

£F; =

dkD

lFl

(6.14)

Densities of States

161

To avoid perpetually writing Dp which does not even depend upon k, the notation

|Σ=7'

dîD

jè)s

i5)

will be adopted in the equations that follow.

Energy Density

States. Several separate functions

D are all

referred

to as

densities

states. They are distinguished by their arguments. The most important

D(£), the

energy density

states which

useful

dealing with sums over

functions that depend upon

only through an energy function

£p

such

Electrons in periodic potentials will differ from

dED(E)F(E)

their free-electron counte^arts largely because

16)

/ \ /

the energy

fcj

will differ from the tree-electron

ka value given byEq. (6.8). This definition looks

ahead to that possibility.

To find D(£), note that

J2F(E

)=V

([dk}F(E$ First use of

Eq.

(6.15).

)

Ισ

/· Integrating

the

function with respect

to E

[dk]

δ(Ε-

Er)F(E)

ust

roduces

··

However,

one chooses

(

J integrate with respect to k first, something new

comes

out.

=^D(E)

= I

[dl] 0{E

E-

)

Obtained by comparing Eq. (6.16) with (6.18). (6.19)

The units

densities

states are able

change without much warning. Often

they

are

expressed

units

l/[eV atom], which means they are related

to the

function defined by Eq. (6.19) by a factor

density

6.3.2 Results for Free Electrons

For the free Fermi gas, one has that

D(E)

f[dk}S(E-E^) „„«,«,.

■

(

°)

' J

UseEq. (6.15), then switch

polar

roo

coordinates

in k

space to do the

4π-

Tj / dkk

S(E-E^)

integral, because

£2

depends upon the (6.21)

' "

magnitude but not the direction

of k.

f°° dE°

2mE°

_£(£_£°) Use Eq. (6.8)

express

k in

terms

(6.22)

\dE°/dk\

2 y

εΡ.

VïmÊ (6.23)

From Eq. (6.8) one

has

6.812 · 10

Λ/ε/eVeV"

cm'

àS?/dk

k/m; then use Eq. (6.8) (6.24)

V again

write

k in

terms

of £.

The number of electrons that can

fit

into

sphere of radius kp

]\j

\ f-

Because/j

only

the state

occupied. (6.25)

1σ

162 Chapter 6. The Free Fermi Gas and Single Electron Model

= V [ [dk] fl, Application of Eq.

(6.11 )

(6.26)

= V / \dk\ 9(kp—it) Because the occupation number vanishes for (6.27)

J k > kf.

is a Heaviside step function that

vanishes when its argument is less than 0 and

otherwise.

V 47Γ 3 _V#

= Ιζί = — Using the formula for the volume of

sphere, (6 28)

47Γ

3 3π

' and Eq. (6.15).

which implies that the Fermi wave vector is related to the density of electrons

n = N/V by

= (3τΛ)

1/3

= 3.09 [n-A

A~\ (6.29)

The density of electrons is sometimes described by finding the volume per electron

V/N and defining the free-electron sphere to be one with the same volume, so that

4π , V

3_V

1/3

(6.30)

The energy of electrons in the most energetic occupied state is called the Fermi

energy or Fermi level and denoted by

F =

36.46[n-À

]

eV. (6.31)

L J

The Fermi surface is the collection of k such that electrons with wave number k

have energy Ef. Finally, the Fermi velocity is defined to be

Vf = Hk

/m = 3.58

[n-À

]

1/3

· IO

cm s"

. (6.32)

The density of states D(E) has broad physical significance. Usually, it is not the

whole function that is important, but just D(Ep), the density of states at the Fermi

surface. Almost every electronic transport property of a solid will be proportional

to D(Ef-), whether it be the ability of electrons to absorb heat, or their response to

applied electric fields. Electrons lying far in energy below the Fermi surface cannot

contribute to transport properties, because all states around them are occupied, and

if they cannot change their state, they cannot respond to perturbations. States far

above the Fermi surface are not occupied at low temperatures and cannot play any

role in responding to external fields. The electron number density right at the Fermi

surface is therefore the crucial quantity to determine. For the free Fermi gas, one

gets the result

D(£

) = ^=4.11-10-

[n-Â

]eV-

À"

. (6.33)

2 c/r

Note that n = J

d£D(e) = f

dt D(E

)

y/t/t