MCEN 5024. Fall 2003.
Basic types of solid materials.
Overview
The theory of bands provides a basis for understanding the classification and
physical properties of solid materials such as electrical conductivity, optical
behavior and cohesive energy.
One of the most useful classification schemes for electrical properties
involves grouping materials based upon the following two criteria.
• If the solid is characterized by a partially filled valence band at 0 K, it is
a conductor.
• If the valence band is completely filled at 0 K, the solid is either a
semiconductor or an insulator.
The distinction between a semiconductor and an insulator is based on the
magnitude of the energy gap.
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Conductors.
Metals are good electrical conductors.
The most important fact about metals is that the valence band is not
completely full of electrons, i.e. the valence band of metals contains empty
states above and immediately adjacent to the Fermi level into which
electrons can be excited.
These states require only an infinitesimal energy excitation from the filled
states at the Fermi level.
Most of the thermal, electrical and optical properties of metals depend on this
low excitation energy.
The energy band diagram for metals fall into two general classes:
For monovalent alkali metals such as sodium, the 3s energy band contains
twice as many levels as there are Na atoms; however, only half of these
levels are filled.
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For divalent alkaline earth metals such as magnesium the 3s and 3p energy
bands overlap; this provides many empty states into which electrons can be
easily excited.
It is tempting to conclude that all metals are conductors and that all
conductors are metals.
While the first generalization is essentially correct, the second is not.
Consider for example the transition metal oxides such as CrO2.
This solid has a partially filled and extended energy band involving overlap of
d and f levels resulting in the ability for conduction.
Thus there are both metallic and ceramic conductors.
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Semiconductors.
The most common semiconducting materials are the covalently bonded
solids such as Si, Ge, and GaAs.
There are many other band gap solids with Eg equal to or greater than 2.5 eV
including ceramics such as SiC and organic materials such as anthracene
(coal).
Consider the 1s22s22p63s23p2 electronic structure of Si.
Si is covalently bonded with each atom having four sp3 hybridized bonds.
Therefore, each atom contributes four levels to the valence and conduction
bands,
At 0K, the valence band is completely filled, the conduction band is
completely empty, and the Fermi energy level falls in the gap.
Given the small gap, an increase in temperature enables electron to move to
the conduction band.
However, the gap allows only a limited number of electrons to contribute to
the conduction process.
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A pure semi conductor such as Si is about 1010 times a poorer conductor at
room temperature than typical metals.
The controlled addition of impurities (doping) has a significant effect on the
electronic properties of the semiconductors.
If atoms with fewer than 4 sp3 electrons are added as impurities (e.g. Ga in
Ge), then additional states appear in the forbidden gap near the upper edge
of the valence band.
If atoms with more than 4 sp3 electrons are added as impurities (e.g. As in
Ge), then additional states appear in the forbidden gap near the bottom edge
of the conduction band.
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Insulators
Examples of materials that are generally considered insulators include most
pure metal oxides such as Al203, the silicate ceramics, and the common
organic polymers.
As in the case of semiconductors, the Fermi energy falls inside the energy
gap; however, the energy gap for insulators is so large that in most cases it
cannot easily be overcome by thermal energy or the application of an electric
field.
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In summary, classification of materials may be represented as shown below:
Simplified energy band diagrams for conductors, insulators, and
semiconductors.
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Use of the band theory as a means for the classification of solids.
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Electrical conduction.
Basic concepts.
If a bar of material of length L and cross-sectional area A is subjected to an
applied voltage V, then the bar's resistance to current flow R (ohms) is given
by V/I.
Here R is an extrinsic property, i.e. it depends upon the geometry of the
sample.
The corresponding intrinsic property, i.e. geometry independent property is
the resistivity, ρ (ohm-m) where:
ρ = (A/L)R or σ = 1 / ρ
where σ is the conductivity in ohm/m.
Like ρ, σ is a materials property which varies over approximately 23 orders of
magnitude from 6 x 105 (Ω-cm)-1 for silver to 1 x 10-18 (Ω-cm)-1 for PTFE (Poly
Tetra Fluoro Ethylene).
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Since conduction occurs by the motion of electrical charge through a solid,
the magnitude of σ will depend upon three factors:
•
The number of mobile charge carriers per unit volume N with
units (#/m3 or m-3);
•
The charge per unit carrier q with units of coulombs C;
•
The mobility of the charge carriers µ with units (m2/ V s)
Electron mobility can be interpreted as the ease with which charge carriers
move through the atomic scale structure of the material in response to an
applied electric field.
The relationship among these factors is given by σ = N q µ
Although the electrical conductivity of all materials can be explained using
this equation, the nature of the terms for each material can vary significantly.
In metals N and q are essentially constant so the influence of both internal
and external variables on the conductivity of a metal can be understood in
terms of their influence on µ.
In contrast, the conductivity of semiconductors is dominated by the factors
that alter N.
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Factors influencing electrical conductivity.
Charge per carrier
In many materials such as metals and some covalent solids, electrons carry
the electrical charge.
For these materials q is the charge on an electron qe (1.6 x 10-19C).
In ionic solids the ions themselves can contribute to electrical conduction.
Since ionic solids contain more than one type of ion and each may contribute
to the overall conductivity:
Where the summation includes all types of charge carriers.
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Charge mobility
The two major types of charge carriers move in different ways.
Ions move through the solid by diffusion whereas the much smaller size of
electrons permits them to move through the solid relatively unimpeded.
Although a complete description of electron motion requires quantum
mechanics, many important characteristics of this motion may be described
using classical mechanics and treating the electrons as rigid particles.
Under the influence of an electric field, electrons undergo acceleration and
ion-core collision cycles such that:
Where
is the average of drift velocity, a is the acceleration due to the
applied field, and is the mean time between collisions.
As temperature is increased, atoms gain thermal and kinetic energy and
begin to vibrate about their equilibrium positions.
The magnitude of the atomic vibrations increases with temperature.
This, in turn, results in an enhanced perturbation of the crystal lattice, a
corresponding decrease in F, and ultimately a decrease in the electron
mobility.
Therefore,
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Number of charge carriers.
When the energy band is sparsely populated, the number of charge carriers
N is equal to the number of electrons in the partially filled band.
When the band is completely filled, N = 0
When the band is nearly filled, N equals the number of empty levels in the
partially filled band.
An empty level corresponds to an electron hole, a charge carrier,, whose
motion is equivalent to the motion of electrons in the opposite direction.
Note that the hole has a charge equal in magnitude but opposite in sign to
that of an electron.
In general, the mobility of a hole is less than the mobility of an electron.
The number of electrons in the conduction band Ne is given by:
Ne = N0 exp (-Eg/2kT) where N0 is a materials constant.
Hence, the number of electrons in the conduction band of a band gap
material increases exponentially with temperature.
In contrast, Ne is independent of temperature in materials with partially filled
valence bands.
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Conductor characteristics.
Metals are good electrical conductors for two reasons:
• They have partially filled valence bands and therefore a large number
of charge carriers;
• Since the charge carriers are electrons, which are not localized to
specific nuclei, the mobility is reasonably high.
The corresponding form of the conductivity equation is:
σ = Ne qe µe
Since the product Ne qe is effectively independent of temperature, the
influence of variables such as temperature on the conductivity of a conductor
can be understood in terms of its effect on electron mobility.
Simply stated, since the mobility decreases with increasing temperature,
electrical conductivity decreases with increasing temperature.
In ionic solids, charge transport may occur by the motion of ions.
The relative contribution of the carriers including electrons, holes and ions
depends upon several factors including the band structure of the solid,
temperature and defect density.
When an ionic solid has a partially filled valence band, the contribution from
electrons is significant.
If the solid has a small band gap, Eg < 2.5 eV, then electrons and holes may
both contribute to the overall conductivity.
If the bandgap is large, then the conductivity will be dominated by the motion
of ions.
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