Chapter 8
Insulators
Semiconductors
Conductors
Superconductors
What is electrical conductivity?
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Electrical Conductivity
When the band is not completely full, the electrons close to the Fermi surface can
easily be promoted to nearby empty levels. As a result, they are mobile, and can move
relatively freely through the solid. The substance is an electronic conductor. One way of
seeing that this is so is to think of the individual orbitals in a band as being
standing waves. As we saw when discussing the real and complex representations of
atomic p orbitals (Section 1.4), standing waves can be regarded as superpositions of
traveling waves corresponding to motion in opposite directions. In the absence of a
potential difference, the two directions of travel are degenerate and are equally
populated up to the Fermi level. (Fig. 3.28a) However, when a potential difference is
applied, electrons tralveling in the opposite direction, and the two sets of orbitals are no
longer equally populated (Fig. 3.28b). Consequently, there are now more electrons
traveling in one direction that in the other, and an electric current flows through the
solid.
Fig. 3.28 Another way of representing the bands is to draw the orbitals for motion to the right and
motion to the left separately. (a) In the absence of a potential difference applied across the metal, the
corresponding orbitals are degenerate. However, when a field is applied, (b) one 'half band' has a
lower energy than the other. it is more heavily populated, and the net result is a f|ow of electrons.
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FIGURE 12.6 Density-of-states diagrams for electrons moving to the left (upper curves)
and to the right (lower curves). (a) The diagram for a metal wire in the absence of an
electric held. (b) The diagram for a metal wire subjected to an electric filed; + at the left
end, and - at the right end.
When the temperature of a metal is increased, lattice irregularities become more
pronounced because of increased atomic vibrations. These irregularities scatter the
electrons, thus reducing the electrical conductivity. This effect of temperature on
conductivity may also be explained in terms of MO theory. In a highly regular metal,
the electron orbitals extend for great distances through the lattice, and the electrons are
highly mobile because of the delocalization. The introduction of irregularities increases
the amount of localized bonding and hence decreases the conductivity.
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The electrical conductivity of a filled band is zero because, in such a band, the
number of electrons moving in one direction must always be equal to the number
moving in the opposite direction. Even though application of an electric field can shift
the relative positions of the left-moving and right-moving bands on the energy scale,
because each band is full it is impossible for electrons to spill out of the higher-energy
band into the lower-energy band. For this reason one might expect an alkaline-earth
metal such as calcium, which has a filled valence s shell in the free atomic state, to be a
nonconductor. However, at the interatomic distances found in this metal, the valence s
and p bands overlap, as shown in Fig 12.7. Consequently the density-of-states diagram
is as shown in Fig. 12.8, and the metal exhibits typical metallic electrical conductivity.
Li
Li
Ca
Ca
2s
2s
Ef
a half-filled band
a fully-filled band
FIGURE 12.8 Density of states as a
function of energy for an alkalineearth metal such as calcium.
FIGURE 12.7 The valence band of an alkaline-earth
metal such as calcium-The dashed line indicates the
equilibrium internuclear distance in me actual
metal.
4
Fig. 13.1. Dependence on internuclear separation of sodium
atomic levels. Notice how as r decreases the collection of s
and p orbitals broaden into bands. At the equilibrium
internuclear distance (ro) the s and p bands overlap.
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In general, the energetic relationship of the energy bands of a solid material and how
many electrons are contained in each has an extremely important bearing on the
properties of the system. If the highest occupied band (the valence band) is full then the
solid is and insulator or semiconductor, depending on whether the energy gap, Eg, (the
band gap) between the valence band and the lowest empty band (conduction band) is
respectively large or small (13.4). If the valence band is only partially full, or full and
empty bands overlap, then a typical metal results. In the notation used in 13.4 weimply
that all the electrons in the occupied levels are paired. The case of 13.5 where the band
is full of unpaired electrons gives rise to a magnetic insulator. The energetic
considerations that control the stability of the alternatives 13.5 and 13.6 are very similar
indeed to those used in Section 8.8 to view high and low spin arrangements in
molecules.
Fig. 3.29 A typical density of states in a
metal
Fig. 3.29 A density of states typical of a
semimetal
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Insulators
If enough electrons are present to fill a band
completely, and there is a considerable energy gap
before an empty orbital becomes available (Fig. 3.34),
the substance is an insulators. In an NaCl crystal, for
instance, the N Cl- ions are nearly in contact and their
3s and 3p valence orbitals overlap to form a narrow
band of 4N levels. The electronegativity of chlorine is
so much greater than that of sodium that the chlorine
band lies well below the sodium band, and the band
gap is about 7 eV. A total of 8N electrons are to be
accommodated (7 from each chlorine atom, one from
each sodium atom). These enter the lower chlorine
band, fill it, and leave the sodium band empty. Since
kT ≈ 0.03 eV at room temperature, electrons cannot
easily be promoted into empty orbitals.
We normally think of an ionic or molecular solid as consisting of discrete ions or
molecules, yet according to the picture we have just described it appears that they
should be regarded as having a band structure. The two pictures can be reconciled in
much the same way as we showed that a delocalized bonding description of discrete
molecules is virtually equivalent to a description in terms of localized bonds. In this
case, it is possible to show that a full band is equivalent to a sum of localized
electron densities. In NaCl, a full band built from Cl orbitals is equivalent to a
collection of discrete Cl- ions, As with molecules, the delocalized band picture is needed
for description of spectra where processes involve one electron at a time such as
photoelectron spectra and X-ray spectra.
The density of state diagram for an insulator looks like that in Fig. 12.10, in which the
high-energy empty band corresponds to nonbonding or antibonding levels. Sometimes it
is possible to convert such a material into a metallic conductor by the application of
high pressure (for example, 100 kbars). From Fig. 12.11 it can be seen that, if the
interatomic distance is sufficiently reduced by increased pressure, the bands will overlap,
yielding a density-of-states diagram similar to that of calcium (Fig. 12.8). In Table 12.4
the electrical resistivities of the elements of the first long row of the periodic table are
listed. The metals with partially filled d bands generally have higher resistivities than
other metals such as K, Ca, Cu
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Figure 12.10 Density of states as a function of energy for an insulator
.
Figure 12.11 The bands of an insulator which can be converted into a metal by high pressure.
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3.6 Semiconduction
The characteristic physical property of a semiconductor is that its electrical
conductivity increase strongly with temperature. at room temperature, conductivities of
semiconductors are typically intermediate between those of metals and insulator (in the
region of 103 Scm-1). The dividing line between insulator and semiconductors is a matter
of the size of the band gap, and even diamond is now being considered as a promising
semiconductor material. The actual value of the conductivity is and unreliable criterion
because, as its temperature is increased, a given substance may have a low, intermediate,
or high conductivity. the band gap and conductivity that are taken as indicating
semiconduction rather than insulation depend on the application being considered.
Fig. 3.20 The variation of the electrical conductivity of a substance with temperature.
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Intrinsic semiconductors
In an intrinsic semiconductor, the band gap is so small that
the Fermi distribution results in some electrons populating the
empty upper band (Fig. 3.35). This introduces negative carriers
into the upper level and positive holes into the lower and the solid
is conducting. The upper band is broader than the lower, more
localized band. A semiconductor at room temperature is generally
much less strongly conducting than a conventional metal because
only electrons and holes in the exponentially decaying tail of the
Fermi distribution are active as charge carriers. The strong
temperature dependence follows from the exponential Bolzmannlike temperature dependence of the electron population in the
Fig 3.35 In an intrinsic
semiconductor, the band gap is so
small that the fermi distribution
results in some electrons
populating the empty upper band
upper band.
We can anticipate that the conductivity of a semiconductor will
show an arrhenius-like temperature dependence of the form
σ = Ae − E
a
/ kT
on account of the exponential dependence of the population of
charge carriers. But how is the activation energy related to the
band gap Eg ? We can answer this by deciding how E − E F ,
which appears in the high temperature form of the Fermi-Dirac
distribution (eqn 3) depends on Eg. This hinges on knowing the
1
location of the Fermi energy (the energy at which P = ). In a
2
Fig. 7.24 Thermal excitation of
electrons in an intrinsic
semiconductor. The x's represent
electrons and the o' holes.
simple picture of the band structure, the Fermi energy is
approximately half-way between the upper and lower bands (Fig.
3.36), and the energy of the lowest level of the upper and lower bands (Fig. 3.36), and
the energy of the lowest level of the upper band E- is related to EF and Eg by
1
E− − E F = E g
2
It follows that the number of charge carriers in the upper band, and hence the
conductivity, is approximately propotional to
n∝e
− E g / 2 kT
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Hence, the temperature dependence of the conductivity of a semiconductor can be
expected to be Arrhenius-like with an activation energy equal to half the band gap,
1
Ea ≈ E g . This is found to be the case in practice.
2
Example 3.7: Determining the band gap
The conductance G of a sample of germanium varied with temperature as indicated
below. Estimate the value of E g .
T/K
G/S
312
354
0.0847 0.429
420
2.86
Answer. Form eqn. 4 we see that the analysis is similar to that used to obtain the
activation energy of a chemical reaction. Since the conductance is proportional to the
conductivity σ, we can write
σ = bG = Ae − Ea / kT
Taking logarithms gives
ln b + ln G = − Ea / kT
therefore, a plot of ln G against 1/T should yield a straight line of slope -Ea/k. The data
give a slope of -4.26 × 103, and since k = 8.614 eVK-1, we find Ea = 0.367 eV.
Since Eg = 2Ea , this gives Eg = 0.73eV.
Exercise. What is the conductance of the sample at 370K?
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The resistance R of a sample is measured in ohm, Ω. The inverse of the resistance
is called the conductance G and is measured in siemens S, where 1 S = 1 Ω-1 . The
resistance of a sample increase with its length l and decreases with its cross-sectional
area A, and we write
R=
ρl
A
where ρ is the resistivity of the substance. The units of resistivity are Ωm. The
conductivity σ is the reciprocal of the resistivity, and its units are S m-1 or S cm-1 .
Table 12.4 Electrical resistivities at 22°C of the elements of the first
long row of the periodic table.
Typical band-gap and conductivity values for insulator (diamond,C), semiconductors
(Si, Ge), and an " almost metal ", gray tin.
It is interesting to consider the trend in electrical conductivity of the tetrahedrally
bonded elements of main group IV: C, Si, Ge, and Sn. Diamond is an insulator, silicon
and germanium are semiconductors (corresponding to a narrow gap between the filled
and empty bands), and tin is a good energy between the atoms on going down the
decrease in the covalent bond energy between the atoms on going down the family. The
stronger the bonds, the greater the separation between the filled bonding bands and the
empty antibonding bands. In diamond, the band gap is high; in tin, the bands actually
overlap and cause metallic behavior.
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