Basic principles of electronic theory. Elementary classical theory of electrical conductivity of metals

LABORATORY WORK - No. 217

STUDYING THE DEPENDENCE OF THE RESISTANCE OF METALS AND SEMICONDUCTORS ON TEMPERATURE

PURPOSE OF THE WORK: Study of the temperature dependence of the resistance of metals and semiconductors, determination of the temperature coefficient of metal resistance and the band gap of the semiconductor.

ACCESSORIES: Samples - copper wire and semiconductor, electric heater, thermometer, combined digital device Shch 4300 or electronic digital voltmeter VK7 - 10A.

Basic principles of the classical theory of electrical conductivity of metals

From the standpoint of classical electronic theory, the high electrical conductivity of metals is due to the presence of a huge number of free electrons, the movement of which obeys the laws of classical Newtonian mechanics. In this theory, the interaction of electrons with each other is neglected, and their interaction with positive ions is reduced only to collisions. In other words, conduction electrons are considered as an electron gas, similar to a monatomic, ideal gas. Such an electron gas must obey all the laws of an ideal gas. Consequently, the average kinetic energy of the thermal motion of an electron will be equal to , where is the mass of the electron, is its root-mean-square velocity, k is Boltzmann’s constant, T is the thermodynamic temperature. Hence, at T = 300 K, the root-mean-square speed of thermal motion of electrons is » 105 m/s.

The chaotic thermal movement of electrons cannot lead to the emergence of an electric current, but under the influence of an external electric field, an ordered movement of electrons occurs in a conductor at a speed of . The value can be estimated from ratios, for j - current density, where - electron concentration, e - electron charge. As the calculation shows, "8×10-4 m/s. The extremely small value of the value compared to the value is explained by the very frequent collisions of electrons with lattice ions. It would seem that the result obtained for contradicts the fact that the transmission of an electrical signal over very long distances occurs almost instantly. But the fact is that the closure of an electrical circuit entails the propagation of an electric field at a speed of 3 × 108 m/s (the speed of light). Therefore, the ordered movement of electrons at speed under the influence of the field will occur almost immediately along the entire length of the circuit, which ensures instantaneous signal transmission. On the basis of classical electronic theory, the law of electric current was derived - Ohm's law in differential form, where g is the specific conductivity, depending on the nature of the metal. Conduction electrons, moving in a metal, carry with them not only an electric charge, but also the kinetic energy of random thermal motion. Therefore, those metals that conduct electricity well are good conductors of heat. The classical electronic theory qualitatively explained the nature of the electrical resistance of metals. In an external field, the ordered movement of electrons is disrupted by their collisions with positive ions of the lattice. Between two collisions, the electron moves at an accelerated rate and acquires energy, which it gives back to the ion during a subsequent collision. We can assume that the movement of an electron in a metal occurs with friction similar to internal friction in gases. This friction creates resistance in the metal.

Based on the concept of free electrons, Drude developed the classical theory of electrical conductivity of metals, which was then improved by Lorentz. Drude suggested that conduction electrons in a metal behave like molecules of an ideal gas. In the intervals between collisions they move completely freely, covering a certain distance on average. True, unlike gas molecules, the range of which is determined by the collisions of molecules with each other, electrons collide primarily not with each other, but with ions that form the crystal lattice of the metal. These collisions lead to the establishment of thermal equilibrium between the electron gas and the crystal lattice. Assuming that the results of the kinetic theory of gases can be extended to electron gas, the average speed of thermal motion of electrons can be estimated using the formula. For room temperature (300K) calculation using this formula leads to the following value: . When the field is turned on, the chaotic thermal movement occurring at a speed is superimposed on the ordered movement of electrons with a certain average speed. The magnitude of this speed is easy to estimate based on the formula relating the current density j with the number n of carriers per unit volume, their charge e and average speed:

(18.1)

The maximum current density allowed by technical standards for copper wires is about 10 A/mm 2 = 10 7 A/m 2. Taking for n=10 29 m -3 , we get

On the basis of classical electronic theory, the basic laws of electric current discussed above were derived - Ohm's and Joule-Lenz's laws in differential form and. In addition, the classical theory provided a qualitative explanation of the Wiedemann-Franz law. In 1853, I. Wiedemann and F. Franz established that at a certain temperature the ratio of the thermal conductivity coefficient l to the specific conductivity g is the same for all metals. Wiedemann-Franz law has the form , where b is a constant independent of the nature of the metal. The classical electron theory explains this pattern as well. Conduction electrons, moving in a metal, carry with them not only an electric charge, but also the kinetic energy of random thermal motion. Therefore, those metals that conduct electricity well are good conductors of heat. The classical electronic theory qualitatively explained the nature of the electrical resistance of metals. In an external field, the ordered movement of electrons is disrupted by their collisions with positive ions of the lattice. Between two collisions, the electron moves at an accelerated rate and acquires energy, which it gives back to the ion during a subsequent collision. We can assume that the movement of an electron in a metal occurs with friction similar to internal friction in gases. This friction creates resistance in the metal.

However, the classical theory encountered significant difficulties. Let's list some of them:

1. A discrepancy between theory and experiment arose when calculating the heat capacity of metals. According to kinetic theory, the molar heat capacity of metals should be the sum of the heat capacity of atoms and the heat capacity of free electrons. Since atoms in a solid body perform only vibrational movements, their molar heat capacity is equal to C=3R (R=8.31 J/(mol×K) - molar gas constant); free electrons move only translationally and their molar heat capacity is equal to C=3/2R. The total heat capacity should be C»4.5R, but according to experimental data C=3R.

2. According to calculations of electronic theory, the resistance R should be proportional to , where T is the thermodynamic temperature. According to experimental data, R~T.

3. The experimentally obtained values of electrical conductivity g give for the average free path of electrons in metals a value of the order of hundreds of interstitial distances. This is much more than according to the classical theory.

The discrepancy between theory and experiment is explained by the fact that the movement of electrons in a metal obeys not the laws of classical mechanics, but the laws of quantum mechanics. The advantages of classical electronic theory are the simplicity, clarity and correctness of many of its qualitative results.

The classical electronic theory of metals was developed by Drude, Thomson and Lorentz. According to this theory, the electron gas in a metal is treated as an ideal gas, and the laws of classical mechanics and statistics are applied to it. In the absence of an external electric field, free electrons in the metal undergo chaotic thermal motion, which does not create a directed transfer of electric charge. When applying an electric field E a force acts on each electron

directed against the field and leading to the generation of electric current. The movement of an electron in a crystal is a complex movement due to its constant collision with ions at the nodes of the crystal lattice. Between two collisions the electron accelerates. At the end of the free path λ, under the influence of force F, the electron acquires a speed of directed motion

where m is the electron mass; A - its acceleration; τ is the time of electron motion between two collisions. τ is called free run time . As a result of a collision with an ion, the speed of the electron becomes zero. Therefore, the average speed of ordered motion is:

Because ,

That ,

where is the average speed of thermal motion of electrons.

Magnitude called mobility . Mobility is equal to the speed acquired by an electron in an electric field whose strength is E = 1 V/m.

In an electric current, the motion of an electron is a complex motion, representing a superposition of chaotic thermal motion with ordered motion with speed in an electric field. The electrical resistance of a metal is caused by the collision of electrons with the nodes of the crystal lattice and their release from the general flow. The more often an electron collides with nodes, the higher the electrical resistance of the metal.

At an average speed of ordered motion, all the electrons contained in a parallelepiped with an edge will pass through an area of 1 m 2 located perpendicular to the flow in 1 second. The volume of this parallelepiped is , the number of electrons in it is , n is the concentration of electrons in the metal. These electrons will carry a charge equal to . Then the current density in the conductor will be equal to

For specific conductivity we have

Substituting into formula (1) the value u for the conductivity of the metal we obtain the expression:

Thus, according to the classical theory, the conductivity of a metal is determined by the average free path of an electron in the crystal and the average speed of thermal motion. The mean free path is approximately equal to the interatomic distance in the lattice. To determine the validity of this assumption, let us estimate the value for silver using experimental data on conductivity. We determine the average speed of thermal motion of electrons from the relationship:

Then for temperature T~300 K we obtain . This value is two orders of magnitude larger than the interatomic distance for silver. Therefore, experimental values for the conductivity of metals can be explained by assuming that the electron mean free path is much greater than the average distance between atoms. During its movement, the electron does not collide with ions at the sites of the crystal lattice as often as classical theory assumes. Before experiencing a collision, an electron flies a fairly large distance, equal to approximately 100 interatomic distances in a crystal. The classical theory is unable to explain this fact.

The next difficulty of the classical theory comes down to the temperature dependence of electrical resistance. According to the classical theory, the mean free path does not depend on temperature and is equal to the average interatomic distance in the crystal. Therefore, according to formula (2), the temperature dependence of resistance is determined by the temperature dependence of the speed of thermal movement. Then the resistivity, according to classical theory, is determined by the expression . However, experimental data show that for metals, resistance in a wide range increases linearly with increasing temperature.

They also knew that the carriers of electric current in metals are negatively charged electrons. All that remained was to create a description of electrical resistance at the atomic level. The first attempt of this kind was made in 1900 by the German physicist Paul Drude (1863-1906).

The meaning of the electronic theory of conductivity comes down to the fact that each metal atom gives up a valence electron from the outer shell, and these free electrons spread throughout the metal, forming a kind of negatively charged gas. In this case, the metal atoms are combined into a three-dimensional crystal lattice, which practically does not interfere with the movement of free electrons inside it ( cm. Chemical bonds). As soon as an electrical potential difference is applied to a conductor (for example, by shorting two terminals of a battery at its two ends), free electrons begin to move in an orderly manner. At first they move uniformly accelerated, but this does not last long, since very soon the electrons stop accelerating, colliding with lattice atoms, which, in turn, begin to oscillate with increasing amplitude relative to the conditional rest point, and we observe the thermoelectric effect of heating the conductor.

These collisions have a retarding effect on electrons, similar to how, say, it is difficult for a person to move at a sufficiently high speed in a dense crowd of people. As a result, the speed of electrons is set at a certain average level, which is called migration rate, and this speed, in fact, is by no means high. For example, in ordinary household electrical wiring, the average speed of electron migration is only a few millimeters per second, that is, electrons do not fly along the wires, but rather crawl along them at a pace worthy of a snail. The light in a light bulb comes on almost instantly only because all these slow electrons start moving. simultaneously, as soon as you press the switch button, the electrons in the coil of the light bulb also begin to move immediately. That is, by pressing the switch button, you produce an effect in the wires similar to what would happen if you turned on a pump connected to a watering hose filled to capacity with water - a stream at the end opposite to the pump will rush out of the hose immediately.

Drude took the description of free electrons very seriously. He assumed that inside a metal they behave like an ideal gas, and applied to them the ideal gas equation of state, quite fairly drawing an analogy between the collisions of electrons and the thermal collisions of molecules of an ideal gas. This allowed him to formulate the formula for electrical resistance as a function of the average time between collisions of free electrons with atoms of the crystal lattice. Like many simple theories, the electronic theory of conductivity is good at describing some basic phenomena in the field of electrical conductivity, but is powerless to describe many of the nuances of this phenomenon. In particular, it not only does not explain the phenomenon of superconductivity at ultra-low temperatures ( cm. The theory of superconductivity, on the contrary, predicts an unlimited increase in the electrical resistance of any substance as its temperature tends to absolute zero. Therefore, today the electrically conductive properties of matter are usually interpreted within the framework of quantum mechanics ( cm.