Conductivity of Metals | Metallurgy

The significant characteristics properties of metals are:

(i) High electrical conductivities, and

(ii) High thermal conductivities.

Example:

The electrical conductivity at room temperature of silver is 0.6 x 10⁸ ohm^-1 m as compared to 10^-10 (approximately) for a good insulator and 0.02 for a semiconductor such as germanium.

Adequate evidence is available to support the notion that the high conductivity in metals is associated with the presence of ‘free’ or ‘conduction’ electrons. The electrons are free to move in the lattice and do not belong to particular atoms. The only electrons which have this degree of freedom are those corresponding to the valence electrons in the atoms.

Thus we may visualize a metal consisting of a lattice of positive ions held in the lattice structure with the electron gas consisting of freely moving valence electrons. Although the properties of the electron gas should be discussed on the basis of wave mechanics but here, for sake of clarity and simplicity of approach, we take up only the classical discussion. This may no doubt involve some inaccuracy but we arrive at a reasonable qualitative picture concerning the mechanism of conductivity.

Ohm’s Law:

Ohm’s law is most fundamental to the electrical engineer and is used every day in electrical engineering and consequently it seems proper to investigate how this law can be interpreted in terms of an atomic picture. In order to arrive at an atomic interpretation of ohm’s law, we shall formulate the law in the form.

J = σE (or J ∝ E)

Where, J = Current density,

E = Field strength, and

σ = Electrical conductivity of the metal.

This ohm’s law states that when a conductor is subjected to an electric field E, the resulting current density J is proportional to the electric field E.

The reciprocal of conductivity σ is the resistivity r, i.e.-

r = 1 / σ

One important feature of ohm’s law is the fact that current density remains constant in time as long as E remains constant.

Electron Theory of Gases — Ohm’s Law:

According to kinetic theory of gases (also referred to as electron gas theory of gases) the valence electrons are supposed to be completely detached from the atoms. The basis for this theory is the picture of a conductor as a lattice of positive ions, through which an electron cloud or gas can move. The number of electrons in such a gas is equal to the number of valence electrons. When not affected by external electric fields the valence electrons oscillate equally in all directions among the atoms just like the molecules in a gas.

When electric field is not present, the random velocities of the electrons will be determined by the temperature of conductor. The temperature of electrons need not be the same as that of the conductor. This temperature is a measure of the kinetic energy of the electrons. The temperature of the lattice system and the electron system will try to equalize.

When an electric field is applied to the conductor, the electrons acquire a systematic velocity which can be calculated.

Thus there are two components of motion, as follows:

(i) Random motion, due to thermal effect (order 10⁶ m/sec),

(ii) Directed motion, the direction being determined by the polarity of the electric field.

Mobility:

We have noted [equation (iv)] that average drift velocity varies in direction proportional to the field, through a proportionality factor eτ/m, called mobility of the electrons which is denoted by μ_e. The mobility may thus be defined as the magnitude of the average drift velocity per unit field. The mobility and the conductivity are related by the equation- σ = ne μ_e.Thus the mobility of the electrons can be determined by knowing the conductivity of the material and estimating the number of free electrons.

The relaxation time is closely related to the mean time of flight between collision and also to the mean free path of the conduction electrons, which is the average distance of undistributed motion between collision. All the collision processes occurring in the electron gas can be explained through relaxation time.

The collisions are caused by thermal or structural imperfections in the lattice. The relaxation time is introduced as the characteristics time governing the establishment of equilibrium through collision, from an initial disturbed situation in which v_x ≠ 0.

The mean free path of an electron λ, is defined as- λ = v τ_e where v is an appropriate average electron velocity and τ_e, is some constant with the dimensions of time.

The velocity of an electron with fermi energy, v_F, is given by:

Heat Developed in a Current-Carrying Conductor (Joule’s Law):

It was determined experimentally by the Joule that the heat developed in a conducting wire is given by PR.

Thermal Conductivity of Metals:

The heat is conducted by all solids, however, the metals are the best heat conductors. Among the metals the best electrical conductors are also best heat conductors. In such conductors, like electrical conductivity, the heat conduction is mostly through valance (free) electrons.

Thermal conductivity, which denotes the transport of thermal energy through a material, generally consists of two terms:

(i) Lattice term,

(ii) Electron term.

In the case of insulating materials in which electrons are held tightly by individual atoms or molecules, absorption or transport of thermal energy takes place only through lattice vibrations, and the electrons do not make a contribution. On the other hand, electrons in metals are relatively free from specific attraction to individual ion cores, and therefore, electrons in addition to thermal vibrations absorb and transmit thermal energy.

In insulating solids, the heat is carried by the lattice vibrations. This, in part is also the case in the metals, but the thermal conductivity due to the conduction electrons predominates in both insulators and conductors. The electrons at the hot end have a higher kinetic energy.

They move to the cold end where the excess energy is released to the atoms whereby the thermal agitation of the atoms and the temperature increase. The electrons of the cold end have less kinetic energy; so in passing to the hot end they decrease the thermal agitation and the temperature. Since the same electrons also conduct electric current, the transfer of heat and the conduction of current must be closely related processes.

Table 7.12 gives thermal conductivities of different materials.

Contact Potential:

The phenomenon of contact potential arises from the fact that the work function varies from metal to metal. It is known fact that when two metals are placed in contact with each other there is a potential difference established between the two. The contact potential established between two metals is equal to the difference of their respective work functions.

The alignment of Fermi levels entails a transfer of electrons from the metal that has a higher Fermi energy (before contact) to the other metal. Such a transfer of electrons leaves the former metal positively charged while the latter becomes negatively charged.

Thus an electric dipole layer is set up between the metals. As the separation between the metals diminishes, so does the contact potential. In the limit when the contact is perfect, the contact potential disappears because of an infinitesimally thin dipole layer between the metals.

Temperature Effect on Electrical Conductivity of Metals:

The resistivity of most metals increases with an increases in the temperature. This is due to the fact that as the temperature is increased there is a greater thermal motion of the atoms which decreases the regularity in the atom spacings with a consequent decrease in the mobility of the electrons.

On the basis of free electron model the electrical conductivity (a) of metal is given by:

The value of v corresponds to fermi energy because only those electrons which are at the top of the Fermi distribution curve can be accelerated and can gain energy. Since the number and energy of the electrons at the top of the Fermi distribution curve vary insignificantly with temperature, the change in temperature must be associated with a change in the mean free path.

Ideally the mean free path of an electron in perfectly regular lattice of atoms should be infinite. In a perfectly regular lattice, each electron will exist in a particular energy state and thus will have a fixed velocity indefinitely. Practical metals do not have a perfect lattice because of impurities and because of the deviation of atoms about their mean position due to lattice oscillations. Thus the mean free path for an imperfect lattice is finite. This accounts for the lower conductivity of alloys which have a disordered lattice.

As the temperature approaches absolute zero, the conductivity increases rapidly since the lattice oscillations decrease at low temperature the scattering of electron waves falls. There is a limiting value beyond which the conductivity will not increase. In general, the more pure a substance, the higher is limiting conductivity.

The conductivity of many metals decreases linearly as the temperature is increased above the room temperature but below this temperature the conductivity increases markedly with a higher power of the absolute temperature (7⁵).