It is well known that Fermi distribution function for electrons reads as

where E is the electron energy, m is the chemical potential, k is the Boltzmann’s constant, and T is the absolute temperature. This function means the probability that energy E level is occupied by electron.

The total density of valence (free) electrons n in conduction band is

where g(E) is the density of states in the conduction band.

The conductivity and the diffusion coefficient D of charge carriers are related by the following expression [16]

By simple calculation of the derivative n on parameter (chemical potential) μ we have

On the other hand, it can be shown that the expression in the angle brackets can be replaced by

Thus, the Equation (4), accounting Equation (5), can be presented in the following form

where

This expression in comparison to Equation (2) can be named as effective density of randomly moving charge carriers. The Equation (6) unambiguously shows that conductivity in all cases is determined by the effective density of randomly moving charge carriers (Equation (7)), but not by the total free electron density in the conduction band (Equation (2)). The total free electron density for determination of conductivity can be used only for materials with non-degenerated electron gas.

Thus, the conductivity (Equation (6)) can also be presented as

where m_drift is the drift mobility of randomly moving charge carriers in homogeneous materials with one type of charge carriers (electrons or holes). From this general relationship (Equation (8)) we get the Einstein’s relation

This expression is true for both degenerated and nondegenerated homogeneous materials with one type of free randomly moving charge carriers (for electrons or for holes). The same conclusion also follows from the thermal noise investigations. The spectral density of the current fluctuations of the thermal noise at low frequencies, here τ is the relaxation time of charge carriers) can be presented as [14,17,18]

where R is the resistance of the sample of the test material; A and L are respectively the cross-section and the length of the sample. This relation is right for all homogeneous materials at equilibrium state. The equalities of Equation (10) also can be obtained from the general Kubo formula for conductivity [2,15]

i.e. in general the conductivity is defined by the correlation function on time of the current density fluctuation due to random motion of charge carriers. Considering the Wiener-Khintchine theorem and the displacement of charge carriers due to Brownian motion during time t, it is possible to obtain both expressions of Equation (10) (See Appendix).

From Equation (7) directly follows that effective density of randomly moving electrons n_eff and which causes the conductivity depends not only on the density of states and Fermi function that the energy level E is occupied by electron, but it also depends on the probability that any of such electron can leave the occupied energy level at a given temperature.

Usually for calculation different kinetic processes in materials instead of probability function ones use the derivative of the Fermi distribution function (here) [2-6], but

The probability function is more understandable in the case of the interpretation of the effective density of randomly moving charge carriers (Equation (7)). Function means that only such electrons can be scattered and change its energy for which.

Illustrations of Equations (1), (2) and (7) are presented in Figure 1. From this figure it is clearly seen that the effective density of randomly moving electrons n_eff (7) is many times smaller than the total density of free electrons n.

Now let ones to evaluate the Equation (7). For non-degenerated materials the probability, because, and, therefore, all electrons in the conduction band n participate in the random motion and in the conduction process

This is the case where the classical statistics is applicable.

In the case of high degeneracy and considering that Equation (12) has a sharp maximum at the energy E equal to the chemical potential m, the Equation (7) can be written as

where at.

The chemical potential m and the Fermi energy E_F are related by the following relation [5]

and the difference between these quantities is only about

0.01% even at room temperature. So, in many cases for calculation of different quantities ones use the Fermi distribution function in the following form

The density of states at Fermi energy can be determined from of the experimental results of the electron heat capacity [5,6]

where

and

Thus, for metals and other materials with high degenerated electron gas

i.e. n_eff is proportional to temperature. Experimental results of conductivity and electron heat capacity and calculated effective density n_eff of randomly moving electrons from the Equation (20) at T = 295 K for different metals are presented in Table 1. The ratio obviously shows that n_eff amount in the total free electron density n is about a few percents and smaller. From Equation (20) it is seen that the effective density of randomly moving charge carriers n_eff is completely defined quantity for materials with high degeneracy of electron gas and can be simply obtained from electron heat capacity measurements.

From Equations (8) and (20) for highly degenerated materials we obtain that diffusion coefficient of randomly moving electrons

Thus, on the ground of experimental results of the conductivity and the electron heat capacity we can evaluate the diffusion coefficient D (from Equation (21)) and drift mobility (from Equation (8) or (9)) for randomly moving charge carriers in materials with highly degenerated electron gas. The calculated results for different metals at T = 295 K are presented in Table 2.

The diffusion coefficient usually also is determined by the average of square velocity of charge carriers and by their average relaxation time (for metals free pass time)

For highly degenerated electron gas

i.e. the diffusion coefficient is determined by the square velocity and free pass time of randomly moving charge carriers at the Fermi level. From the Equations (21) and (23) follows that conductivity

Table 1. Conductivity and heat capacity parameters of different metals.

Conductivity and electron heat capacity data are taken from books [5,6,7,19]. Conductivity and n_eff are presented for T = 295˚ K.

Table 2. Kinetic parameters of different metals.

The data for calculation at T = 295 K were used from Table 1. The data of Tables 1 and 2 give the very clear view, why some of metal conductivity is larger than other one. For example, the effective density of randomly moving charge carriers n_eff of Pb is about two times larger than one of Au, but the conductivity of Au is about ten times larger than that of Pb. It is because the Fermi energy of Au is about twenty times larger than that of Pb, and, as a consequence, the drift mobility of randomly moving charge carriers in Au is about twenty times higher than one in Pb.

This expression is well known for metals and is obtained by solving the kinetic equation [2,3].

The correctness of presented diffusion coefficient and drift mobility values in Table 2 also follows from the electron heat conductivity for randomly moving electrons in metals [1-6]

From Equations (24) and (25) one obtains the well known Wiedemann-Franz law

The diffusion coefficient calculated from Equation (25) gives the same results as from Equation (21). In behalf that Einstein relation also is valid for metals shows this fact that Einstein relation for metals can be obtained from the Wiedemann-Franz law

and now

.

From Equations (9) and (22) follows that drift mobility

Including into this relation the effective mass m^* of charge carriers the drift mobility can be presented in the following form

or

; (30)

where is the average kinetic energy of randomly moving charge carriers. Thus, the Equation (29) is the general relation for drift mobility of randomly moving charge carriers in homogeneous materials with one type of charge carriers (electrons or holes). This relation is true for degenerated and for non-degenerated materials, including metals. The Equation (28) shows that drift mobility of randomly moving electrons in materials is proportional to the square of the electron random velocity, i.e. the electrons with high random velocity contribution into conductivity is larger than that of slow electrons. The factor

shows, how many times the mean kinetic energy of randomly moving charge carriers is larger than. For non-degenerated material and

but for highly degenerated semiconductors, metals and superconductors in the normal state

i.e. the drift mobility (Equation (30)) of random moving charge carriers in metals at room temperature in many cases is about a hundred times larger (see the ratio values in Table 2) than it follows from the classical Equation (32).

From the Equation (29) follows that the drift velocity of electrons in the electric field E can be presented as

A very important behavior of electrons in materials follows from this expression: the drift velocity of electrons in the electric field E is proportional to the square velocity of randomly moving electrons. This result in principle can not be explained by application of Newton’s law for calculation of acceleration of electrons in the electric field E in materials.

From the Equations (8) and (30) follows that the general expression for conductivity

For non-degenerated material (and)

but for highly degenerated semiconductors and metals

Now compare the obtained drift mobility results with that from the Hall effect measurements. On the ground of the Hall effect for highly degenerated materials the Hall mobility [20]

where is the Hall electric field strength in y direction perpendicular to the current flow direction; is the applied external electric field strength in x direction; is the Hall factor, which depends on charge carrier scattering mechanism, but for high degenerated charge carriers. From Equation (37) it is seen that Hall mobility is determined by average relaxation time of electrons at Fermi level.

According to investigations in the work [21], for all homogeneous metals and highly degenerated semiconductors the charge carrier relaxation time at Fermi level at room temperature is general and equal

; (39)

where is the Plank constant. At T = 295 K. So, at T = 295 K for m^* = m₀, while the drift mobility depends also on the kinetic energy of the random moving charge carriers. Now from Equation (23) or from Equation (24) it is possible to find the electron velocity v_F at the Fermi level and the Fermi energy E_F considering that the effective mass m^* of electron is equal to the free electron mass m₀, i.e.. The calculation results for different metals are presented in Table 2. In this table for different metals there also are presented the free pass length of electron, the latter obtained values are close to experimental data [22].

Usually for evaluation of the Fermi energy it is considered that Fermi surface is spherical, i.e. [5,15]

and

But for many metals and superconductors in normal state the density of states has composite view (Figure 1(b)), and there is no definite relation between the total density of charge carriers n and the Fermi energy E_F. So, as it was mentioned above in order to evaluate the Fermi energy the sure way is from Equation (24) to find the electron velocity at the Fermi level v_F and than the Fermi energy E_F.

The obtained mobility results clearly show that for high degenerated materials the drift mobility can tens or hundred times be larger than the Hall mobility.

From Equations (37) and (38) the conductivity and Hall mobility for metals are related with the following expression

Thus, the charge carrier density in highly degenerated materials, determined by using the Hall mobility

The charge carrier density n_H obtained from Hall effect measurements only in particular cases is equal to the total free charge density for metals with spherical Fermi surfaces, but even in these cases the factor characterizes not the charge density, but the drift mobility. For multivalent metals and superconductors in the normal state with composite density of states dependency on energy there is no definite relation between n and n_H.

Equation (14) shows that the randomly moving charge carrier density in high degenerated materials is not constant but is proportional to the temperature and the relaxation time close to room temperature, so in this temperature range the Hall mobility, while the drift mobility. Hence, one can be very careful in the interpretation of the Hall effect and conductivity measurement results. For example, there are many strange interpretations of charge carrier density and drift mobility of superconductors in the normal state from Hall effect and conductivity temperature measurements [9-13].

Considering that Pauli principle and Fermi distribution is known over 80 years, and knowing that only the electrons, which energy is close to the Fermi energy, can be scattered and can change their energy and take part in random moving and in kinetic processes, it is strange that till now in many books of solid state physics the conductivity of metals and superconductors in the normal state is presented in the form of classical expression (Equation (36)? As it is presented in Tables 1 and 2 sometimes the charge carrier density of metals obtained from Hall measurement n_H is close to the total density of electrons n, and the conductivity is near to experimental results, but for multivalent metal there is no such coincidence. This conductivity expression gives the increased values of randomly moving charge carriers and decreased values of their drift mobility. Sometimes these two errors compensate one other. In spite of these coincidences the expression (Equation (36)) for metals from the view of the Pauli principle and the Fermi statistics is completely wrong, because the electrical field has no effect to electrons, which are well below the Fermi level.