§ 8. Classical electronic theory of electrical conductivity of metals

8.1. Fundamentals of classical theory and its experimental confirmation

To identify the nature of the current in metals several experiments were done. In 1901, Rike took two copper and one aluminum cylinder, polished butt ends, weighed them and put in the following order: copper - aluminum - copper. Thus, he formed a conductor. Through it an electric current continuously passed in one direction during the year. For the whole period the electrical charge passed through the conductor was 3,5 ∙ C.

Then the cylinders were weghed again, and it turned out that their weight did not change. Under the microscope metal compounds were studied. Penetration of one metal to another did not happen.

Thus it was found that the charge was not transfer by atoms of the cylinder, but by other particles (electrons). It was necessary to determine the sign and value of the specific charge of current carrier to make sure that these were the electrons.

The experiments, which were conducted later, were based on the following

Presume the conductor moves with a speed . We begin to brake it with acceleration ω. Continuing inertial motion carriers will have a relatively conductor acceleration ω in the opposite direction (-ω). The same acceleration can provide carriers in a fixed conductor, if you create in it electric field of strengthE.

That is to apply the difference of potentials to the end of the conductors.

w

e -w

v₀

fig. 8.1

where m - charge carrier mass, l –length of the conductor, е – charge of the carrier.

In this case, current flows through the conductor, where R – conductor resistance. During the timedt the following charge will pass through the conductor

The product of acceleration ω and time dt is the speed:

The charge that passed through the conductor during the time of braking may be calculated:

(8.1)

The charge will be positive, when it is passed in the direction of the conductor movement.

Thus, after having measured the length of the conductor l, velocity V and the conductor resistance R and calculating the charge that passed through during the time of braking, specific charge of the carrier may be found. The direction of the current impulse will be shown by the sign of the charged carrier.

The first experiment of this type was carried out by German physicist Mandelshtamp and Russian physicist Topalevsky in 1913. The numerical result was received by Tolman and Stewart in 1916. They took the coil with the length of 500 m and put it in motion, at which the linear speed of coils were 300 m/s. Then they rapidly braked the coil and by means of ballistic galvanometer measured the value of the charge that passed through the circuit during the whole time of breaking. The calculated by formula (8.1) value was close to the value of charge-to-mass ratio of the electron. In this way, it was determined that the carriers of the current in metals are electrons.

The current in metals can be created with the help of small potentials difference. That is why we may assume that carriers of the current move freely in metals.

The existence of electrons in metals may be also explained by the following: during the formation of the lattice the less connected electrons (valent) split off, and then become ‘collective property’ of the whole piece of metal. If one more electron is spitted from each atom, the concentration of free electrons will have the value

(8.2)

where ρ - density of the metal. Then n = 1028 ... 1029 m-3 will be obtained.

Let us consider the classical theory of elementary metals (Drude-Lorentz theory).

Drude believed that the conduction electrons of metals behave like ideal gas molecules. That is, in the intervals between collisions, electrons move freely and pass in average the distance λ. The electrons collide mostly not with each other, but with the ions that form the lattice of the metal. These clashes lead to the establishment of thermal equilibrium between the electron gas and the crystal lattice.

Taking into account, that the results of molecular kinetic theory of gases may be applied to electron gas, evaluation of thermal velocity of the electrons can be carried out as follows:

м/с. (8.3)

To this thermal motion of electrons in metals ordered movement of electrons with velocity is imposed. The value of this velocity can be found from the formula

(8.4)

Thus, even for very large values of current density, the average velocity of ordered motion of electrons is times smaller than the thermal velocity of motion:

Therefore, in calculation the resulting velocity can be replaced by speed of thermal motion module.

Let us find the change in the average value of the kinetic energy Ек. According to the theory of probability, two events which can be describe as follows: the rate of thermal motion of the electrons take the value and ordered movement speed – valueare statistically independent. Therefore, by product of probabilities theorem,

But = 0, so. Hence, the ordered movement increases the kinetic energy of the electrons on average

. (8.5)

8.2. Ohm's law in terms of the classical theory of electrical conductivity

Drude believed that under collision of electrons and ions in the crystal lattice, additional energy acquired by electrons (equation (8.5)) is transmitted to ion and consequently the velocity in the result of such collision equals to zero.

Imagine that the field that accelerates the electrons is homogeneous. Then under the influence of the field electron gets constant acceleration, which is numerically equal to and till the end of the run, ordered movement speed reaches the maximum value.

, (8.6)

where τ - the average time between two consecutive collision of electrons and ions of the crystal lattice.

Drude did not take into account the distribution of electron velocities and attributed to all electrons the same speed value in this approximation. But as it was shown, alove the speed rate of heat and ordered movement is approximately equal to the thermal motion. Then substitute the value τ in equation (8.6):

(8.7)

The speed during the run varies linearly, so its average value for each run close to half the maximum:

.

If we substitute the last expression in equation (8.4), we will get:

.

According to Ohm's law (in differential form) the current density is proportional to the field strength by a factor of proportionality

, (8.8)

where - conductivity.

If the electrons are not in contact with the ions of the lattice, the value of λ and σ would be infinitely large. Thus, according to classical conceptions resistance of metals is due to collisions of free electrons and ions in the crystal lattice of metals.