--
Chapter4
Electrical properties
of metals
4.1 Introduction
How does an electric current flow through a material? A current is a flow of charged
particles, which implies that either ions or electrons are involved. In liquids, both of
these types of particles take part in electrical conduction, but in solids the ions occupy
fixed positions in the crystal lattice and so are not tree to move. (Afew solids do exhibit some ionic conduction. but the magnitude of this type of conductivity is usually
very smalL) Consequently, we can conclude that the electrical conductivity of solids is
due almost entirely to the movement of the electrons. This implies that metals should
be good electrical conductors because they have a plentiful supply of essentially tree
electrons, whereas all other materials should be insulators because there are no tree
electrons. This picture is actually rather oversimplified, and in the next chapter we
will look at how some non-metals can be reasonably good electrical conductors.
However. in this chapter we will concentrate on the process of electrical conduction
in metals.
What happens when we apply a voltage to a metal object? The negatively charged
electrons in the sample are attracted towards the positive terminal and are repelled
by the negative terminal. We can represent this by tilting the energy bands to indicate
that the electrons at the positive end have a lower energy than those at the negative
end, as shown in Fig.4.1. Since the valence electrons in a metal are essentially tree, we
would expect them all to rush down the slope towards the lower energy states near
the positive terminal. However, we will see that this is far trom the truth.
4.2 Drude's classical theory of electrical conduction
The first attempt to explain the behaviour of electrons in a metal was made by P.
Drude in 1900, only three years after the discovery of the electron. The theory is
incorrect in many respects, despite the fact that it gives several rather convincing
results, but it is worth examining in detail because it introduces some useful
..
88
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O~UDiE'S CLASSICAL THEORY Of lEliE(1IUI::Al (0111l:»U(110111 89
average thermal velocity, Vt,of a molecule with mass m at temperature T (measured in
kelvin) is given by the equation
1
z3
"2mVt ="2
kT
B
/'
(4.1)
where kBis Boltzmann's constant. Ifwe apply the same formula to the electrons in a
metal and replace m by the electron mass methen we obtain a value for the speed of
the electrons <:itroom temperature of about 105m S-l(see Question 4.1).
Using the ahalogy with kinetic theory we can also define the mean free path '11.which is the average distance that an electron travels between collisions-and the relaxation time T-which is the average time duration between collisions. These
quantities are related by the equation
A,
Vt=-::r
concepts which we will use later in a more sophisticated treatment of electrical
conduction.
Drude assumed that a metal is composed of ions, which are stationary, and valence
electrons, which are free to move. He treated the electrons as small solid objects
which behave rather like the ball in a pinball machine, colliding with one ion after another so that a typical trajectory looks something like Fig.4.2. If there is no voltage applied to the metal then at each collision the electron is deflected in a different
direction so that the overall motion is quite random. Of course, this is just what we
expect when there is no applied voltage, because any net movement of the electrons
in a particular direction would constitute a flow of current.
Since the random motion of the valence electrons appears to be similar to that of
the molecules in an ideal gas, Drude suggested borrowing some of the concepts used
in the kinetic theory of ideal gases to describe the properties of the electron 'gas'. Let
us see what we can determine by this approach. (If you are unfamiliar with kinetic
theory, it may be worth consulting an introductory general physics text-see Further
reading.)
We can begin by estimating the velocity of the electrons. In kinetic theory the
(4.2)
If we assume that the mean free path is of the order of a few atomic spacings, i.e.
about 1 nm, then the relaxation time is typically about 10-14S.
Now let us consider what happens when we apply an electric field (i.e. a voltage) to
the sample. The electrons are attracted towards the positive end of the sample, and so
we expect a net flow of electrons in this direction. Using some simple arguments we
can make an estimate of the magnitude ofthis effect. If the electric field is e, then the
force on each electron is ee. Dividing by the electron mass, me,we find that the acceleration on each electron in the direction of the field is
a=ee
me
(4.3)
This acceleration produces a change in velocity Llvalong the direction of the electric
field. Since the acceleration is constant, the value of Llvat time t after a collision is
given by
Llv=at= ee t
me
(4.4)
If the average time between collisions is denoted by T, then the average velocity in the
direction
of the field is
Llv=ee T
me
(4.5)
This quantity is referred to as the drift velocity and represents the average velocity of
the valence electrons in the direction of the electric field. Ifwe substitute typical val-
ues into this equation we find that in a moderate electricfield of 10Vm-1 the drift velocity is only about 0.02 m S-l(see Question 4.2).This is ofthe order ofl0 million times
smaller than the thermal velocity of the electrons.
Why is the drift velocity so small? It is because even when an electric field is applied
to the metal, the electrons still collide with the ions and so they continue to follow a
more or less random trajectory (as in Fig.4.2) with just a very slight tendency to move
.
90
Of
~: flf(r~I(Al
towards
the positive
end
ing result that when
of the sample.
a current
flows
Consequently,
in a metal,
we
the valence
arrive at the rather
electrons
COIIIDl.IlI::flOI\l
91
surpris-
are not all mov-
......
......
ing towards
many
the
electrons
What
positive
terminal.
travelling
in the opposite
evidence
do we
use it to provide
Let us assume
that when
electron
has gained
vibration
of the
other
words,
is a familiar
pass
an
effect because
electric current
the wire
sure
of the
amount
turn
can
also
show
theory
is correct?
collides with
to an increase
of the electrons
it is used
through
to the collisions
Drude's
model
are nearly
as
~v
......
wire
that
first of all we
an ion, the excess
energy
to the ion. This
in the temperature
resistance
and
between
the electrons
of electrical
heat
and
In
This
light bulbs:
if we
can be thought
into
the
energy.
it gets hot. If the resistance
is converted
......
......
that the
increases
of the sample.
into thermal
of the wire
of as a mea-
energy,
which
in
the ions.
conduction
is consistent
where
we
given
material,
with
0"
Ohm's
law. First of all we
is a measure
will define
of the ease with
which
a quantity
called the electron
the electron
moves
through
mobility
J1-which
the lattice of ions. If
we write
(4.6)
then by comparing
eqns (4.5) and(4.6) we can see that
have
substituted
we
can
for
,dv
from
the
product
express
eqn
(4.6). Since
of these
n, J1-and
three
terms
e are
as
constants
a single
for
(4.9)
cr is referred
which
is the
to
as the
inverse
conductivity.
of the
conductivity,
Alternatively,
we
can
define
the
resistivity,
i.e.
1
1
P = 0" = nj.ie
Consequently,
we
(4.10)
can
write
eqn
(4.8) as
c
j.i
=a/me V
(4.7)
Ifwe assume that the relaxation time has a fixed value (i.e. independent oftemperature and electric field strength) for'a particular material, then the mobility is also a
constant. (In practice this is not quite true. It is found that T, and therefore J1-,vary
slightly with temperature and electric field strength, but these effects can usually be
neglected. )
Let us then consider the behaviour of the valence electrons in a metal wire in which
the electric field is directed along the axis of the wire. The analysis is simplest if we
neglect the random thermal motion of the electrons-we can do this because we
know that the average thermal velocity in any particular direction is zero-and
assume that each electron travels at a velocity ,dv along the wire, as shown in Fig.4.3.
This means that in a time interval t each electron moves a distance ,dvt along the wire.
As a result, the number of electrons passing through the cross-Section A inthis time
is equal to n ,dvt A,where n is the number of valence electronj per unit volume. Since
the current is defined as the amount of charge passing through the wire per unit time,
and as each electron carries a charge e, the current I flowi~g in the wire is n .avAe.So,
finally, we arrive at an expression for the current density J:
J = l = n,dve
=
nj1£e
(4.8)
J = O"C=
a
constant
=nj.ie
where
p,
,dv=j1£
......
......
can
of electrical resistance.
is converted
gets hotter. Consequently,
that
there
Well,
in electric kettles, toasters
a metal
of electrical energy
be attributed
can
so leads
energy
in time
of the phenomenon
the electric field is transferred
ion, and
the kinetic
instant
direction!
that Drude's
explanation
an electron
from
is increased,
We
have
a qualitative
In fact, at any
(4.11)
P
-
This isOhm's law,or at leastthe microscopic
form of Ohm's law. We can put it in a
more familiar form (interms ofcurrent, voltageand resistance)by substitutingJ =IJA,
c= VJL and R = pLJA (see Question
4.3).
Table 4.1 Conductivity and resistivity data at 273 Kfor a selection of metals. n is the
number of valence electrons per unit volume, which is equal to the number of atoms per
unit volume multiplied by the valence ofthe atom.
Valence
()
(107 n-l
n
(1028m-3)
..................................................................................
........................
Sodium
Copper
Silver
Gold
Magnesium
Zinc
Aluminium
Tin
Lead
m-l)
p
(10-8nm)
1
1
1
1
2
2
3
4
4
2.38
6.45
6.80
4.88
2.54
1.82
4.00
0.87
0.52
4.20
1.55
1.47
2.05
3.94
5.50
2.50
11.5
19.2
2.65
8.50
5.86
5.90
8.60
13.2
18.1
14.5
13.2
..
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Finally, if we use the expression for the conductivity from eqn (4.9)and assume that
the mean free path is of the order of a few atomic spacings then we find that the predicted value of conductivity is in good agreement with the experimental data in Table
4.1 (see Example 4.1).
So we have some quite convincing evidence that Drude's model of electrical conduction is correct. We have managed to show that Drude's model is:
. consistent with Ohm's law;
. qualitatively explains the phenomenon of electrical resistance; and
. gives good values for the conductivity.
However, as we shall see in the next section, Drude's model has some severe
limitations.
I
fAlumu
Of nu:. nASSI(:AI.. MODEl.
93
highest electron concentration tend to have the lowest conductivities (with the exception of aluminium) and vice versa!
Another problem occurs if we consider the electrical properties of alloys (which are
mixtures of two or more metallic elements). The classical model seems to suggest that
the resistivity of an alloy should be intermediate between the values for the corresponding pure materials. However, many alloys have resistivities which are considerably larger than those of either of the pure constituents. For example, nichrome,
which is an alloy of 80%nickel and 20%chromium, has a resistivity which is 10 times
higher than that of either nickel or chromium (see Fig. 4.5).
The dependence of resistivity on temperature is also at odds with experimental evidence. Ifwe assume that the mean free path is approximately constant with temperature, then we can use Drude's model to predict that the resistivity should be
proportional to r112 (see Question 4.8). But experimental measurements show that the
resistivity is actually proportional to r over a wide range of temperatures.
Finally, let us consider the molar specific heat capacity, which is the amount of energy required to raise the temperature of 1 mole of solid by 1 K. Ifwe again use the
analogy with an ideal gas, then we can see from Example 4.2 that Drude's theory predicts a value for a monovalent metal of9R/2. A similar argument suggests values of6R
for a divalent metal, 15R/2for a trivalent metal, and so on (see Question 4.9). However,
experimental results show that the molar specific heat capacity at room temperature
is approximately 3R, regardless of the valency of the metal.
There are many other characteristics that cannot be explained using Drude's
model, but I am sure that by now you are more than convinced that we need to improve this model if we want to obtain a better understanding of electrical conduction.
4.3 Failuresof the classical model
Despite the successes ofDrude's theory, there are many features of the electrical conductivity of solids that cannot be explained by this model.
For instance, let us first of all compare the conductivities of different metals.
Equation (4.9) shows that the conductivity is proportional to the number density of
valence electrons n and to the mobility f1 and Table 4.1 gives values of n for various
different metals, but how does the mobility vary from one metal to another? From
eqns (4.7) and (4.2)we can see that the mobility varies inversely with the mean free
path, and we would expect the mean free path to be dependent on the size of the ions
and on the percentage of the total volume that is occupied by the ions. Since most
metallic ions tend to be of a more or less uniform size (with an io'nic radius of about
0.15 nm) and form close-packed crystal structures, we can assume that the mean free
path-and therefore the mobility-does not vary substantially between different
metals. This suggests that the conductivity should be directly proportional to the valence electron concentration, n. However, this is not in good agreement with the experimental data plotted in Fig.4.4. In fact, it appears that those metals which have the
..
94
4: EU!C'I"RI(AI..
Of
THEORY Of UIi:("I"III(AI. (Ol\llnltl'lON
4.4 Bloch's quantum theory of electrical conduction
We should not be too surprised that Drude's theory does not give us a complete picture of electrical conduction in solids. After all, we have no justification for assuming
that eqn (4.1)can be applied to the electrons in a metal. Also we have treated the ions
as inert objects that simply get in the way of the electrons, ignoring the electrostatic
interactions that occur between the valence electrons and the ions (electron-ion
interactions) and between the valence electrons themselves (electron-electron interactions). The latter turns out to be fairly insignificant in most cases, so we can safely
ignore this effect, but the interactions between the electrons and the ions are of great
importance and substantially alter our view of electrical conduction.
Let us first consider the behaviour of an electron approaching an isolated positive
ion. The electron is attracted to the ion when the separation between them is relatively large, but as the electron gets close to the ion it is repelled by the outer filled
shell of electrons (Fig.4.6).We are therefore not dealing with 'collisions' between the
electrons and ions, as we assumed in the previous sections, rather the electrons are
deflected or 'scattered' by the ions. Todescribe the motion of an electron in a solid we
need to extend this idea to a large number of ions in close proximity.
Another.failing ofDrude's model is that it treats the electron as a classical particle.
However, we have already seen in Chapter 1 that we cannot explain the behaviour of
an electron in an atom using classical arguments. It follows that in order to describe
the interactions between the valence electrons and the ions we must use a quantum
theory treatment.
How does this affect our understanding of elect\ical conduction? The first person to
tackle this problem was F. Bloch in 1925. By solving the quantum mechanical equations for an electron in a perfect crystal lattice, Bloch showed that an electron moves
through the crystal without being deflected at all. We can picture this behaviour by
imagining the electron weaving through the lattice of ions, as shown in Fig. 4.7.
.
96
~: nlE(1~I(Al
!3ANO 1HEORY
PRO~E~1IU Of
However, this result poses a riddle. Since it is the scattering of the electrons by the ions
that gives rise to an electrical resistance, if the electrons are not scattered then a metal
should offer no resistance to an electric current. Since this is clearly at odds with our
everyday observations, we might be inclined to scrap this model and look for an alternative approach. It turns out that the model is quite correct. The flaw in our reasoning
is that we have assumed that the crystal is perfect. However, any imperfections in the
crystal will cause the electrons to be deflected. Consequently, the mean free path of an
imperfect crystal is not infinite, and so the material has a non-zero resistance.
There are essentially three different types of imperfection which produce electron
scattering, and therefore can be thought of as the origin of electrical resistance in a
metal. Firstly, the ions are not stationary, but are in a state of continual thermal vibration about their equilibrium positions. Consequently, at any instant in time the
ions do not occupy the positions of a perfect lattice. This leads to scattering of the electrons, as shown schematically in Fig.4.8, which in turn gives rise to a resistance.
A second cause of scattering is the presence of impurities. This is particularly important if the impurity ion is significantly larger or smaller than the host ions, or if it
has a different valence. For example, in brass, which is a copper-zinc alloy, the zinc
ions are about 9%larger in diameter than the copper ions and have a charge of +2e
compared with +1e on the copper ions. The effect on scattering is shown schematically in Fig.4.9.
Thirdly, imperfections in the crystal structure also disrupt the lattice and so cause
scattering. We illustrate this effect in Fig.4.10 by considering one of the simplest imperfections-a missing atom, or vacancy.
To summarize, we have shown that the processes which result in the scattering of
the electrons are very different to the simple picture of electrons colliding with fixed
ions that we assumed in Drude's model. However, the trajectories of the electrons are
quite similar in the two cases: the electrons are frequently deflected and so follow an
almost random path. Consequently, the concepts that we introduced in the classical
§OUI)§
97
theory (such as the mean free path, the drift velocity and the mobility) are all relevant
in a quantum theory of conduction. However, the values of these parameters may be
quite different, as we shall see in Section 4.10.
4.5 Band theory of solids
We now have a qualitative description of electrical conduction from a quantum viewpoint. lfwe wish to obtain quantitative results then we also need to consider the energies of the electrons in a solid.
.
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(b)
(a)
35
;.,
C1
"-
OJ
c:
LLJ
.
..--
15
. .0
- - - - - - - -- - 35 band capacity 2N,
..,,'
2p. . . . . . ..""
25
99
'"
7///77//j
partiallyfilled
...
'/ / / / / / / //
~p band capacity 6N,
filled
"
..
25 band capacity 2N,
" / / / / / / / / /
filled
.::' , , , , , , "
15 band capacity 'lN,
filled
To begin with let us remind ourselves of the restrictions which govern the energy
levels in an isolated atom (see Chapter 1).These can be summarized by the following
statements:
1. The properties ofthe electrons in an atom are determined by four parameters,
or quantum numbers, which are denoted n, I,mland ms.n is the principal
quantum number, Itakes integer values from 0 to (n-l), mlis allowed integer
values from -I to +1and mscan be - !or +!.
2. The electrons are only allowed to occupy certain discrete energy levels, which
we label using the quantum numbers n and I (since-the..e.n.ergyof an electron
does not usually depend on mlor ms).The letters s, p, d and f are used to denote
the states I = 0, 1, 2 a~ely.
3. The occupancy ofthese levels is determined by the exclusion principle which
states that each electron must possess a unique set of quantum numbers.
Ifwe apply these rules to a sodium atom, which has 11 electrons, we find that two
electrons occupy each of the 1s and 2s levels, six occupy the 2p level, leaving one electron in the 3s level, as shown in Fig. 4.11(a).However, in a solid the atoms are not isolated. In fact they are in close proximity to a large number of other atoms. What
happens in this case?
Let us first of all consider the situation with just two sodium atoms. When the
atoms are far apart-they behave like isolated atoms, but as we move them closer together, the outer (Le. 3s) electrons begin to interact, which affects the energies of
these electrons. Ifwe plot the positions of the energy levels as a function of the separation between the atoms we obtain a picture as in Fig. 4.12. At the equilibrium
separation, ao,we therefore have two states, one of which is at a lower energy than the
3s state in an isolated atom, and one which is at a higher energy. We have to be very
careful how we interpret this result. The energy levels can no longer be considered to
belong to a specific atom, rather they are a property of the pair of atoms considered
as a whole. Consequently, this picture does not imply that the 3s electron attached to
one atom has a higher energy than that on the other atom. In fact, since there are two
allowed values of ms for each state, both of the electrons can occupy the lowest energy
level (Fig.4.13).
Ifwe extend this same argument to a system of N sodium atoms, we should get N
discrete 3s energy levels, as shown in Fig. 4.14(a). However, in a macroscopic crystal
the number of atoms, N, is typically of the order of 1024and the energy levels span a
range of only a few electron volts. Consequently, the spacing between adjacent levels
is so small (see Question 4.11) that we effectively have a continuous band of energies,
as shown in Fig. 4.14(b).Similar interactions occur for the 1s, 2s and 2p electrons, but
the interactions are much weaker because these electrons are closer to their respective nuclei, and so the corresponding bands cover a smaller range of energies (Fig.
4.15).
.
100
#,
4: ELECTRICAL PROPERTIES OF METALS
'
.
DISTRIBUTION
OF THE ELECTRONS
BETWEEN
THE ENERCY
STATES
101
.111111
Illil
""
"'/
/
.
3s
,,,
,
,,
~
,,
,,
Isolated
Naa~~m
. .
,,
, ,-
Two
Na,atoms
,,
,.
,,
.
3s
illill
3s
Isolated
Na atom
>-
~
Qj
c:
LIJ
Figure 4.13 Ina system of two sodium atoms. the two 3s electrons
occupy a lower energy levelthan in the isolated atoms. (Note the
similarity with the diagram of a hydrogen molecule. Fig. 1.13.)
2s
1s
>en
...
Qj
c:
LIJ
°0
~
Separation
>-
~
Figure 4.15 The formation of 1s, 2s, 2p and 3s energy bands in sodium.
The bands become progressively narrower for electrons which are more
tightly bound to the parent nucleus.
Qj
c:
LIJ
1III
II
°0
Separation
°0
Separation
4.6 Distribution of the electrons between the energy
states-the Fermi-Diracdistrib~tion
Figure 4.14 (a) The energy levels for the 3s electrons in a group of Nsodium atoms as a
function of the separation between the atoms. (b) For large values of Nthe states are so
closetogether that we effectively have a continuous band of allowed energies.
Although the quantum states in a solid are so closely spaced that we can treat them
as a continuous band of energies, we still need to be aware of the existence of the discrete energy levels when we consider how many electrons can occupy each band. If
the crystal contains N atoms then in the is band we have N discrete states, which are
capable of accommodating a total of 2N electrons. Since there are two is electrons in
each sodium atom, we have precisely the right number of is electrons needed to occupyall of the states in the is band. Similarly, we find that all2N and 6Navailable positions in the 2s and 2p bands are occupied. Now we come to the 3s band. This also has
a capacity to accommodate 2Nelectrons, but since there is only a single 3s electron in
each sodium atom, there are only N electrons available to occupy these spaces.
Consequently, the 3s band is just partially occupied. We therefore obtain an overall
picture for the energy bands of sodium as shown in Fig.4.11{b).
Before we consider how the presence of the energy bands <¥fectsour model of electrical conduction in a metal we need to examine this partially filled band in more detail. In particular, we want to know which quantum states are filled and which are
vacant. We would also like to have some information about the energies of these
quantum states. We will address these questions in the following sections.
If an energy band is only partially filled, such as the 3s band in sodium, which states
do the electrons occupy? We lmow that electrons tend to occupy the lowest available
energy levels, so if there areN electrons in the band then as a first approximation we
can assume that the lowest Nf2 states are occupied and the higher energy states are vacant. This suggests that there is an abrupt cut-offbetween those states which are occupied and those which are vacant. The corresponding energy is called the Fermi
energy, Ep.We can state this another way by saying that the pr06al5i1ity-thara-state is
occupied is equal to 1.0 if the energy of the state is less than the Fermi energy, and
equal to 0.0 if the energy is greater than the Fermi energy, as shown in Fig. 4.16.
Whilst the above analysis is correct at a temperature of absolute zero, at any finite
temperature some of the electrons gain thermal energy and are excited into higher
energy states. Consequently, at any instant in time there are a number of electrons
which have energies greater than the Fermi energy, and a corresponding number of
vacant spaces below the Fermi energy. This is shown schematically in Fig.4.17.
The statistical distribution of the electrons in this case was first determined by
Enrico Fermi and Paul Dirac in 1926.Accordingto their theory, the probabilityf{E)
that a state with energy E is occupied at temperature T is given by
I
I
1
1
f(E)= e (E-E,J/k.T +1
(4.12)
.
102
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103
Fermi energy now represents a sort of average maximum energy. In fact, we can see
trom Example 4.3 that the Fermi energy can be defined as the energy for which the
probability of occupation is equal to one-half.
4.7 The density of states
This is the Fermi-Dirac distribution. We will not attempt to justifY this relationship
(if you want more details then refer to Beiser-see Further reading) but we are interested in what the distribution looks like. At T = 0 Kthe equation predicts the same distribution as shown in Fig. 4.16 (see Question 4.12), but at higher temperatures the
abrupt step in the distribution becomes more gradual, as shown in Fig. 4.18. In this
case it is no longer meaningful to refer to a maximum energy of the electrons, but the
In the previous section we introduced the Fermi-Dirac equation. This describes how
the electrons are distributed as a function of energy, but it does not tell us how many,
if indeed any, electrons exist at a particular energy. To find out this information we
need to determine how many quantum states exist at the relevant energy.
Since we are dealing with a continuous band of energies, we should really determine how many quantum states occur in a small energy range, e.g. between E1and
E1+ oE.By repeating this procedure over many such intervals we can construct a histogram showing how the number of quantum states varies as a function of energy (see
Fig. 4.19). Ifwe make the energy range oEsufficiently small, the histogram becomes a
smooth curve. This curve is referred to as the density of states function and can be determined either by experimental methods or by calculation. The density of states for
a typical energy band is shown in Fig. 4.20. A characteristic feature is that the majority of states occur in the middle of the band, with comparatively few states available
near the extremes of the band.
A calculation of the density of states is in general rat~er complicated, but for a simple metal the density of states can be obtained, at least for the lower part of the energy band, without too much difficulty. We will consider this problem in the next
section. Readers with no previous experience of quantum theory may wish to omit
this section.
.
104
4: ELECTRICAL PROPERTIES OF METALS
, .'"
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E1
-----THEFREEELECTRONMODEL 105
comparison, the average thermal energy of the electrons is equal to kBT,which at
room temperature is approximately 0.025 eV. Consequently, the chances of an electron escaping from the metal under normal conditions are very small, and so for
mathematical convenience we will assume that the box containing the electrons is of
infinite depth.
To simplify matters further, let us begin by restricting our attention to a onedimensional case. We will assume that the box is oflength L,where L is the size of the
sample. According to quantum theory, the energy, E, of an electron in such a system
is given by the differential equation
VI
G.I
...
fa
...
VI
....
0
...
G.I
.'
EI+0E
E,+20E
fi2
Energy
d21f!
(4.13)
- 2m. dx2 =Elf!
Figu re 4.19 Ahistogram of the number of quantum states with energies between £1and £,
+ 8E,E, +8Eand £, + 28E, and so on.
where If!is known as the eigenfunction. (Ifyou are familiar with quantum theory, you
may recognize the above equation as the Schr6dinger equation for a particle in a onedimensional potential well of infinite depth.) From Example 4.4, we can see that
Equation (4.13)is satisfied by eigenfunctions of the form
(4.14)
If! = A sin(kx x)
VI
2fa
...
VI
where A is a constant. These eigenfunctions can be represented graphically, as shown
in Fig. 4.21, and the corresponding energies are given by
'0
...
G.I
..c
E
::J
Z
fi2k2
fi21l2n2
2m.
2m. L2
E=-2£=
x
(4.15)
where nx is an integer.
EL
Eu Energy
Fi,gure4.20 Thedensityof states for a typical energy band.ELandEu indicate the lower and
upper edges of the energy band.
--
>-
~
In order to calculate the density of states, we first of all need to determine the positions ofthe energy levels in the crystal. To do this we will use the so-called free electron model. In this model it is assumed that the average potential inside the metal due
to the valence electrons and ions is constant throughout the sample, but that at the
edge of the sample there is a large potential which stops the electrons escaping from
the metal. The depth of the box is equal to the amount of energy required for an
electron to escape from the metal. This quantity is known as the work function and
can be determined from measurements of the photoelectric effect (see Serway in
Further reading). The work function is usually of the order of a few electron volts. In
nx= 2
G.I
C
LU
4.8 The free electron model *
nx= 1
L
I
Figu re 4.21 In theJree electron model it is assumedthat theelectrogsare trapped in a
deep potential well. The figure,showsthe form of the first feweigenfuottions fora one..
dimensionalcase. Thecorrespondingenergiesare givenbyeqn (4.15).
.
L