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It is shown that the linear resistivity dependence on temperature for metals above the Debye’s temperature mainly is caused by electron-electron scattering of randomly moving electrons. The electron mean free path in metals at this temperature range is in inverse proportion to the effective density of randomly moving electrons,
*i.e.* it is in inverse proportion both to the temperature, and to the density-of-states at the Fermi surface. The general relationships for estimation of the average diffusion coefficient, the average velocity, mean free length and average relaxation time of randomly moving electrons at the Fermi surface at temperatures above the Debye’s temperature are presented. The effective electron scattering cross-sections for different metals also are estimated. The calculation results of resistivity dependence on temperature in the range of temperature from 1 K to 900 K for Au, Cu, Mo, and Al also are presented and compared with the experimental data. Additionally in temperature range from 1 K to 900 K for copper, the temperature dependences of the mean free path, average diffusion coefficient, average drift mobility, average Hall mobility, average relaxation time of randomly moving electrons, and their resultant phonon mediated scattering cross-section are presented.

The foundations of the electron theory of metals were laid on the preposition of free electrons which are assumed to be of the same order of valence electrons per unit volume. In order to obtain the correct order of the magnitude for conductivity and correct temperature variation of the mean free path of electrons at room temperature range, it was assumed that electron mean path is caused by thermal vibrations of lattice [_{F}, and the Fermi energy, because the Fermi surfaces for many of metals are not spherical [_{eff}_{ }of the randomly moving electrons in metals is presented in [

This study mainly is addressed to using the stochastic description of the effective density of randomly moving electrons to investigation of the electron scattering mechanism at temperatures above the Debye’s temperature. Additionally we estimated the resistivity dependence on temperature for Au, Cu, Mo, and Al in the temperature range from 1 K to 900 K. The mentioned problems are very important in solid state physics for relevant understanding and description of kinetic phenomena in metals, and semiconductors with highly degenerate electron gas.

It is well known that the Fermi distribution function f(E) for electrons is expressed as

f ( E ) = 1 / [ 1 + exp ( E − η ) / k T ] . (1)

It specifies a probability that a level with the energy E on average is occupied by an electron. Here, E is the electron energy; η is the chemical potential; k is the Boltzmann’s constant, and T is the absolute temperature. Considering that the difference between the chemical potential and Fermi energy E_{F} is only about 0.01% for room temperature [

f ( E ) = 1 / [ 1 + exp ( E − E F ) / k T ] . (2)

Thus, the total density of the free valence electrons n in conduction band is described by an integral on the density-of-states g(E) as

n = ∫ 0 ∞ g ( E ) f ( E ) d E . (3)

The electrical conductivity depends not only on the density-of-states g(E) in conduction band and Fermi distribution function f(E), but it also depends on the probability f 1 ( E ) = 1 − f ( E ) that any electron with the definite energy E at a given temperature T can be thermally scattered or can change his energy under influence of the external fields. Thus, the effective density of electrons n_{eff} which take part in random motion and in conductivity is described by the probability

h ( E ) = f ( E ) [ 1 − f ( E ) ] . (4)

Therefore, the effective density of randomly moving electrons n_{eff} is described as [

n eff = ∫ 0 ∞ g ( E ) f ( E ) [ 1 − f ( E ) ] d E = k T ∫ 0 ∞ g ( E ) ( − ∂ f ( E ) / ∂ E ) d E . (5)

From this expression follows that the term ( − ∂ f ( E ) / ∂ E ) is the probability density function p(E) that electron with energy E is available to change his energy due to thermal and electrical influence:

p ( E ) = ( − ∂ f ( E ) / ∂ E ) = h ( E ) / k T , (6)

and the integral probability distribution function is

F ( E ) = ∫ 0 E p ( E 1 ) d E 1 = 1 − f ( E ) . (7)

The functions p(E) and F(E) meet the requirements of the probability theory. The probability density dependence on energy of randomly moving electrons is presented in

n eff = n = ∫ 0 ∞ g ( E ) f ( E ) d E . (8)

This is the case when the classical statistics is applicable. In the case of highly degenerate electron gas, and considering that probability density function ( − ∂ f ( E ) / ∂ E ) has a sharp maximum at E = E F , the Equation (5) can be presented in the following form:

n eff = g ( E F ) k T ≪ n , (9)

where g ( E F ) = g ( E ) at E = E F . The density-of-states (DOS) at Fermi energy g(E_{F}) can be obtained from the experimental results of the electron heat capacity measurements [

c V = ( π 2 / 3 ) g ( E F ) k 2 T = γ T . (10)

Now we estimate the energy fluctuation of randomly moving electrons on the Fermi surface:

〈 ( E − 〈 E 〉 ) 2 〉 = 〈 ( E − E F ) 2 〉 = ∫ 0 ∞ ( E − E F ) 2 g ( E ) f ( E ) [ 1 − f ( E ) ] d E , (11)

or

〈 ( E − 〈 E 〉 ) 2 〉 = g ( E F ) ( k T ) 3 ∫ 0 ∞ ( ε − ε F ) 2 f ( ε ) [ 1 − f ( ε ) ] d ε ≅ 3.2899 g ( E F ) ( k T ) 3 ≅ π 2 3 g ( E F ) ( k T ) 3 . (12)

From Equations (10) and (12) follows that

〈 ( E − 〈 E 〉 ) 2 〉 = k T 2 c V . (13)

It is in accordance with the theory of energy fluctuation of free randomly moving particles [

The volume for one of randomly moving electron with the Fermi velocity for different metals can be estimated as:

V 1el = 1 / n eff = 1 / [ g ( E F ) k T ] , (14)

This volume increases with temperature decrease as 1/T.

For metals and other materials with highly degenerate electron gas the effective density of scattered electrons is described by Equation (9), and it is proportional to temperature. The DOS g(E_{F}) for various metals are presented in _{F}) for metals is described as [

g ( E F ) = m ∗ ( 3 n ) 1 / 3 π 4 / 3 ℏ 2 , (15)

where m* is the effective mass of the electron, and ħ = h/2p is the Plank’s constant. This relation is presented in _{F}) for some of alkali and noble metals, but for other metals there are very large differences. Considering that energy structures of DOS for many metals are composite [_{F} and the total density of free electrons n, and also due to the same reason there is no definite relation between the Fermi energy E_{F} and the density-of-states at the Fermi surface.

The electric conductivity σ of the homogeneous materials and diffusion coefficient D of randomly moving carriers can be described by such general expression [

σ = q 2 D ( ∂ n ∂ η ) T . (16)

After simple calculation ones obtain [

σ = q 2 D ∫ 0 ∞ g ( E ) f ( E ) [ 1 − f ( E ) ] d E = q 2 D k T n eff = q μ drift n eff , (17)

where μ_{drift} is drift mobility of randomly moving charge carriers. These relations are valid for all homogeneous materials with a single type of charge carriers at any their degeneracy degree and it confirms that the electric conductivity for metals is determined by the effective density of randomly moving charge carriers, but not by the total density of free charge carriers. The distribution of experimental values of electric conductivity [

For metals the Equation (17) can be rewritten as

σ = q 2 g ( E F ) D = ( 1 / 3 ) q 2 g ( E F ) v F 2 τ F , (18)

This expression is well known for metals, and it also can be obtained by solving Boltzmann’s kinetic equation [^{1/2}(E_{F}) (^{3/2}(E_{F}) (

From Equation (18) follows that diffusion coefficient of randomly moving electrons is related minimum with the five transport parameters of randomly moving electrons:

D = σ q 2 g ( E F ) = 1 3 v F 2 τ F = 1 3 l F v F , (19)

where l F = v F τ F is the mean free path of the electrons at the Fermi surface. What parameters mostly cause the spread of diffusion coefficient of electrons on density-of-states for metals at E = E_{F}?

The very important parameter characterizing the scattering mechanism of randomly moving charge carriers is their mean free path. In the case of electron-phonon scattering mechanism, the cross-section of scattering σ_{el-ph} increases with temperature due to lattice ions vibration amplitude increasing [_{eff}. According to [^{−15} cm^{2}. All randomly moving electrons n_{eff} with the average Fermi velocity v_{F} are scattered. Then the mean free path of scattered electrons in metals can be described as [

l F = v F τ F = 1 / ( σ eff n eff ) = 1 / [ σ eff g ( E F ) k T ] , (20)

and the average relaxation time as

τ F = 1 / ( σ eff n eff v F ) = 1 / ( σ eff g ( E F ) v F k T ) . (21)

There it must be pointed that effective electron scattering cross-section σ_{eff} doesn’t depend on temperature at temperatures above the Debye’s temperature. So, the statement that scattering cross-section at this temperature range is proportional to temperature contradicts to experimental resistivity data of metals. The Equation (20) directly shows that electron mean free path is in inverse proportion not only to temperature, but also to density-of-states at the Fermi surface. Besides the product l F σ eff = 1 / n eff = V 1 el (here V_{1el} is the volume for one randomly moving electron (Equation (15)). Considering that D = ( 1 / 3 ) l F v F ~ 1 / g 3 / 2 ( E F ) , and l F ~ 1 / g ( E F ) , it means that electron Fermi velocity v F ~ 1 / g 1 / 2 ( E F ) , i.e. electron velocity at Fermi surface on average decreases with density-of-states increasing.

In paper [

l F ( T ) = 1.39 ( D ( T 0 ) ) 2 / 3 ( T 0 / T ) , (22)

here l_{F} is in nm, and D(T_{0}) is in cm^{2}/s at T_{0} = 295 K. A small spread of the data in _{F} for Cu is 55 nm, for Al is 22 nm, and for W is 19.1 nm, while for the same metals in the paper [

From Equation (20) the effective electron-electron scattering cross-section can be estimated as

σ eff = 1 / ( l F n eff ) = 1 / [ l F g ( E F ) k T ] . (23)

The effective electron scattering cross-sections are distributed in the range (0.40 - 3.6) 10^{−15} cm^{2}, and they do not depend on temperature for that temperature range where the resistivity is proportional to temperature T. The effective electron-electron cross-section for every metal has a particular value, which depends on the Fermi surface structure. Considering that mean free path data for different metals are different due to specific effective electron scattering cross-section values, the spread of those values causes also the spread of both the conductivity (_{D} is caused by the effective randomly moving charge carrier density dependence on temperature.

From Equations (19)-(22) it is possible to design the following important pattern (

v F = 2.165 ( D ( T 0 ) ) 1 / 3 , (24)

where v_{F} is in units 10^{5} m/s and does not depend on temperature, D(T_{0}) is in cm^{2}/s at T_{0} = 295 K, and similarly the average relaxation time of electrons at the Fermi surface in the linear resistivity dependence on temperature range as

τ F ( T ) = 0.64 ( D ( T 0 ) ) 1 / 3 ( T 0 / T ) , (25)

where τ_{F} is in units 10^{−14} s, and D(T_{0}) in cm^{2}/s at T_{0} = 295 K.

There must be pointed that statement that average relaxation time for all metals in the range of the linear resistivity dependence on temperature can be expressed as τ F ≅ ℏ / k T [^{−14} s (^{−14} s (horizontal dash line in

In the case of electron-phonon scattering the cross-section must increase with temperature increasing [^{2} in the range of temperatures above the Debye’s temperature, but it does not take place. Heretofore the linear resistivity dependence on temperature was explained by electron-phonon scattering keeping that all valence electrons take part in current transport, but it is inconsistent with the Pauli’s principle and Fermi distribution function.

Accounting the real effective density of randomly moving electrons let us to conclude that electron-electron scattering is the main mechanism that causes the linear resistivity dependence on temperature of metals above the Debye’s temperature.

It was shown that the effective electron scattering cross-section σ_{eff} doesn’t depend on temperature at temperatures above the Debye temperature, but also it was shown that average electron relaxation time at Fermi surface τ_{F}~1/T, and mean free path l_{F} = v_{F}τ_{F}~1/T. Considering that at temperatures below the Debye’s temperature the resistivity decreases with temperature more steeply then 1/T, there arise the question: What is the role of phonons? In

The independent resistivity part on temperature below 10 K is the residual electric resistivity due to scattering of electrons from chemical and structural imperfections in the investigated samples.

It seems that effective electron scattering cross-section depends on the exchange of thermal energies of lattice and randomly moving electrons, i.e. depends on the ratio of the thermal energy of the phonon to thermal energy of the electron. Though the kinetic energy of randomly moving electron is equal to E k1 = m v F 2 / 2 , but its thermal energy is only

Metal | ρ_{0}, 10^{−8} Ωm at T = 1 K | ρ, 10^{−8} Ωm at T_{h} = 700 K | v_{F}, 10^{6} m/s | Θ, K used in this work | Θ, K cited in references | σ_{eff}, 10^{−15} cm^{2} |
---|---|---|---|---|---|---|

Au | 0.0220 | 5.816 | 1.16 | 200 | 165 [ | 0.53 |

Cu | 0.0020 | 4.514 | 1.14 | 343 | 310 [ | 0.40 |

Mo | 0.00070 | 15.81 | 0.603 | 410 | 375 [ | 0.66 |

Al | 0.00010 | 7.322 | 0.874 | 410 | 380 [ | 0.50 |

Note: the resistivities ρ_{0} at T = 1 K, and ρ at T_{h} = 700 K are taken from [_{F} is calculated from Equation (24); σ_{eff} is calculated from Equation (23).

E e1 = ( π 2 / 6 ) ( k T ) 2 g ( E F ) / n eff = ( π 2 / 6 ) ( k T ) ≈ 1.64 k T . (26)

The average phonon thermal energy at temperatures over Debye’s temperature Θ is about 3 kT [

E ph1 = 3 k T ( T Θ ) 4 ∫ 0 Θ / T 4 x 5 ( e x − 1 ) ( 1 − e − x ) d x . (27)

Thus, the ratio of the average thermal energies between phonon to electron can be described as

E ph1 E e 1 = 3 k T 1.64 k T ( T Θ ) 4 ∫ 0 Θ / T 4 x 5 ( e x − 1 ) ( 1 − e − x ) d x ≈ 1.83 η ( T / Θ ) , (28)

where

η ( T / Θ ) = ( T Θ ) 4 ∫ 0 Θ / T 4 x 5 ( e x − 1 ) ( 1 − e − x ) d x (29)

is the phonon mediation factor. So, the phonon mediated resultant electron scattering cross-section σ_{res} can be described as

here the quantity σ_{eff} accounts the constant multiplier 1.83, because at _{eff} independent on T. The resultant average relaxation time τ_{res} is described as

where_{imp} is the average cross-section of the impurity defect, N_{imp} is the impurity density.

The metal resistivity in the overall temperature range can be described as

where _{h} = 700 K). The resistivity dependence on temperature for Au, Cu, Mo and Al are presented in

Considering that calculation data by using Equation (32) very well agree to the measurement results for Cu (

dependency on temperature for Cu. The diffusion coefficient has been evaluated from Equation (18). The comparison of the mean free path l_{mean} and the resultant electron relaxation time τ_{res} of the randomly moving electrons dependencies on temperature for Cu in temperature range from 1 K to 900 K is presented in _{mean} was evaluated from relation D = (1/3)l_{mean}v_{F}, and the resultant electron relaxation time τ_{res} was evaluated as τ_{res} = l_{mean}/v_{F}.

The resultant phonon mediated electron-electron scattering cross-section σ_{res} is shown in ^{−15} cm^{2} [_{imp} we take σ_{imp} » 0.7・10^{−15} cm^{2}). The effective density n_{eff}, drift mobility μ_{drift}, and Hall mobility μ_{H} of randomly moving electrons dependencies on temperature from 1 K to 900 K are presented in ^{2}, and at temperature below 20 K the Hall mobility is constant, but drift mobility changes as 1/T. Thus, considering that density-of-states and velocity of electrons at Fermi surface don’t depend on temperature, it is easy from resistivity (conductivity) data to find the basic electron transport characteristics for metals.

The investigation results of the fundamental transport characteristics of free electrons on the base of the effective density of randomly moving electrons in metals have been presented. It let to estimate the real average characteristics of randomly moving electrons in metals, such as electron diffusion coefficient, drift mobility, free path, velocity, relaxation time, and to show that these quantities in homogeneous metals above the Debye temperature mainly are due to electron-electron scattering. It is shown that the effective electron scattering cross-section does not depend on temperature in the linear resistivity dependence on temperature range.

The calculation results of resistivity dependence on temperature in the range of temperature from 1 K to 900 K for Au, Cu, Mo, and Al are presented and compared with the experimental data. Additionally for Cu at first time the mean free path, average diffusion coefficient, average drift mobility, average relaxation time of randomly moving electrons dependences on temperature from 1 K to 900 K are presented.

The authors declare no conflicts of interest regarding the publication of this paper.

Palenskis, V. and Žitkevičius, E. (2018) Phonon Mediated Electron-Electron Scattering in Metals. World Journal of Condensed Matter Physics, 8, 115-129. https://doi.org/10.4236/wjcmp.2018.83008