The development of electron theory of solids was gradually started early in 20th century with the enunciation of Drude-Lorenze theory, the Sommerfeld modification and zone theory etc. The electron theory, in the initial stages was only a model for metals but later on with further development it became applicable to metals and non-metals.
Metallic Bonding:
In metallic crystals the atoms are ionized and the metal is an assemblage of positive ions immersed in a cloud of electrons. The electrons of this cloud are relatively free and majority are not bound to any particular ion but move rapidly through the metal in such a way that there is an approximately uniform density of them throughout the interior between the ions. Metallic crystals are held together by the electrostatic attraction between this gas of negative electrons and positively charged ions.
The binding forces in metals are thus of different character from those in non-metallic substances in which the predominating forces are between neighbouring atoms or between +ve and -ve ions and are strongly dependent upon the interatomic distances. The modern picture of metallic bond assumes that the metallic bond is more closely related to the covalent or electron-pair bond and resembles, somewhat, the ionic bond.
The features of the metallic bond, however, can be summarised as follows:
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(i) Essentially, the metallic bond is unsaturated covalent bond allowing a large number of atoms to be held together by mutual sharing of valence electrons,
(ii) The density of electrons between the atoms is much lower than is allowed by Pauli exclusion principle. This allows electrons to move fairly freely from point to point without a significant increase in their energy.
Cohesive and Repulsive Forces:
In metals the atoms and molecules are held together by certain force of attraction or cohesive forces. These forces of metallic crystal together are due to attraction of the positively charged ions and the cloud of negative charge lying between them.
Repulsive forces also exist between ions of like charge and between the electrons, caused by interaction of their electron shells. Since the valency electrons lie some distance away from the nucleus and their shells are not filled, therefore, these orbits can overlap the orbits of other atoms without producing repulsive forces resulting in attractive forces only.
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When nuclei of two atoms are brought into close contact their positive charges repel each other until a balance is reached between the attractive forces that pull the atoms together and repulsive forces that hold the nuclei apart. The attractive as well as repulsive forces operate at much closer range.
The two forces balance each other at a distance and this becomes the equilibrium distance between the two atoms, since in order to increase or decrease their atomic spacing energy must be supplied. The atoms may be separated by thermal, electrical and mechanical energy. Conversely compression is resisted by solid because greater and greater repulsive forces must be overcome by the compressive forces.
Drude Theory:
Drude was the man who pronounced initial electron theory of metals. It was based upon the idea that the conduction of electric and thermal current in the metals is by electrons. According to his postulation the valency electrons of an atom in a solid can move freely within the solid.
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It appears that they are held due to some type of force. Actually their motion is not restricted within the metal. Drude computed equations of electrical and thermal conductivity by treating these electrons as an electron gas which however, did not tally when applied to metals practically.
Sommerfeld Theory:
The introduction of wave mechanics, which takes account of the fact that a moving electron behaves as if it is a system of waves, led to the quantum free electron theory proposed by Sommerfeld.
This theory entails the following assumptions:
1. In a metal crystal the valence electrons are free.
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2. The valence electrons are confined to move within the boundaries of a crystal, that is they move back and forth inside the crystal but do not leave it. This indicates that the electrons have a lower potential energy inside than outside of the crystals.
3. Inside a crystal the potential energy of the electron is uniform and the free electrons have the same potential energy anywhere within the crystal.
Because the potential energy of electron within the crystal is assumed to be constant, it may be taken as zero arbitrarily, the total energy is then equal to the kinetic energy.
The kinetic energy of the electrons may be expressed as:
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E = h2 k2 / 82m, where h, k, m are Planck’s constant, wave number and mass of an electron 871 m respectively.
The relation between energy and wave number is parabolic in nature.
The improvements which this theory provided over Drude theory were the explanation of the -specific heat anomaly and the conclusions that all the free electrons are conduction electrons. The equation developed for electrical and thermal conductivities were, however, no better than those developed by Drude.
Zone Theory of Solids:
The free electron theory of Sommerfeld explains the phenomenon like electrical conductivity, electron emission as well as certain magnetic properties. This theory assumes that electrons move in a region of constant potential. The introduction of the concept of a periodic potential field with the electrons moving in it gave rise to a theory known as Zone theory of solids.
This theory is briefly explained as follows:
Within any real metal there is a periodic arrangement of positively charged ions through which the electrons move. Consequently, the potential along a line such as aa, which passes through the centre of the ions, must vary as shown in Fig. 11.2.
It can be shown that the movement of the electrons in a crystal cannot occur under conditions which satisfy the Bragg’s diffraction law:
Brillouin Zone:
Refer to Fig. 11.3. It is defined as the area in a wave number space enclosed by planes which gives the value of normal component of the wave number k as /d.
Consider a simple cubic lattice of two dimensions. The Bragg’s condition is first satisfied by the (100) plane of this lattice. As the energy of electrons increases a point is reached when the Bragg’s condition with respect to (100) planes influences the motion of the electrons.
In a two dimensional lattice there are two (100) planes, (0, 0) horizontal and (100) vertical. The Bragg’s condition is satisfied for these two types of planes by all k vectors corresponding to the line ABCD. The area in wave number space enclosed by the line is known as Brillouin zone. The zone obtained with the help of sets of planes are generally termed as first Brillouin zone, second Brillouin zone and so on. The different Brillouin zones for a given crystal have the same area.
Electron Energies in Metal:
According to Sommerfeld free electron theory the electrons in a metal move with extremely high velocities and at 0°K all electrons occupy the lowest possible energy states. By the Paul’s exclusion principle only two electrons can occupy any state, one with spin ‘up’ and the other with spin ‘down’. Consequently at 0°K the electrons fill all the states upto a certain maximum energy level known as the Fermi level and none have the energies above this.
The level at which the probability of occupation is 50% is Fermi level EF. The average energy lies at 3/5 EF. If an electron is required to be removed from the Fermi level and take it out of the metal some energy is required to do so. This is called work function (denoted by ɸ) and is equal to the energy which is normally increased when an electron is removed from the surface of the metal.
Fig. 11.4 illustrates the above statements. Vs is the difference in potential between a point inside and outside the metal. The volume of the metal has been considered to be in a potential well. For silver the value of ɸ determined experimentally is 4.46 electron volts while EF of silver is 5.51 electron volts, this means 9.97 electron volts (i.e., 5.51 + 4.46) is the difference in potential (Vs) between a point inside and outside the metal.
Conductors, Insulators and Semiconductors:
Zone theory for determining which are conductors and which are insulators can be applied as follows:
To determine the electrical conductivity of a solid it is necessary to know how completely the Brillouin zones are filled. No flow of electrons takes place in the absence of an external electric field. On application of external field electrons flow in the direction of the field, provided the field can actually accelerate some of the electrons, that is, if some unoccupied states are available with energies just above the occupied levels. In a metal there are always unfilled states at energy levels very slightly higher than the highest energy of filled states, and there is no difficulty in raising the electron energies by applying an external field.
In real metals, two zones can overlap. It may be assumed the first zone is filled and second one is unfilled. Due to overlap in energy levels, higher energy electrons can be moved to the other levels in the second zone and there will be conduction.
In a monovalent metal the valence electrons fill one half the energy levels of first Brillouin zones, while in a divalent metal such as magnesium the first zone is nearly full and the lower energy levels of the second zone are occupied.
In case of an insulator there is an energy gap, just above the filled levels, large enough to prevent electrons from being raised to the next allowed energy level by an externally applied electric field. An important example is that of a diamond. It has four valence electrons per atom in each zone.
This is first the number of valence electrons needed to completely fill the first zone in the diamond structure. Since considerable energy gap exists between first and second zones, diamond behaves like an insulator. In general, insulators are those materials which have more than 3 eV (electron volts) of energy gaps.
Semiconductors are a group of materials in which the energy gap between the filled and unfilled zones is sufficiently small so that the electrons may be excited by thermal energy to move from the filled zones to empty zones, thus providing some current to flow. The energy gap in a semiconductor is usually about 2 to 3 eV or less. A semiconductor would be a nonconductor or insulator at 0°K since there would be no energy available to excite an electron across the gap.
Factors Affecting Electrical Resistance of Materials:
The factors which affect electrical resistance of materials are explained as follows:
The resistance of metallic conductors increases with temperature. It is so because thermal vibrations of the atoms cause more oscillation of the crystal lattice, this makes the atomic spacing less regular and consequently the mobility of electrons through the metals is decreased.
A pure metal has a more regular structure as compared to an alloy. Alloying elements or impurities greatly increase the lattice imperfections which decrease the conductivity (i.e., increase resistivity) of an alloy in comparison with that of a pure metal.
Any mechanical process that increases the number of dislocations will result in an increase in the electrical resistance of the metal. For example, strain hardening results in higher resistivity than annealed samples of the same metal.
The electrical resistivity of an alloy increases due to age hardening. During this process crystal lattice undergoes distortion due to which electron mobility is decreased and eventually resistivity of the alloy is increased.