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Technical Physics

, Volume 64, Issue 5, pp 593–595 | Cite as

Self-Induction of a Fine Cylindrical Metallic Wire Versus a Mechanism of Electron Surface Scattering

  • E. V. ZavitaevEmail author
  • K. E. Kharitonov
  • A. A. Yushkanov
THEORETICAL AND MATHEMATICAL PHYSICS
  • 2 Downloads

Abstract

The self-induction of a thin cylindrical metallic wire has been calculated. A general case when the ratio of the electron free path to the wire radius may take an arbitrary value has been considered. The dependence of the regular reflectance on surface defect density and angle of incidence of electrons on the wire’s inner surface has been taken for boundary conditions of the problem.

Much interest has recently been heightened in the electromagnetic properties of systems including fine metallic wires [1–5]. The electromagnetic behavior of these systems depends on such properties of fine wires as conductance and inductance. For wires with a radius much greater than the electron free path in a sample, these parameters may be found in terms of macroscopic electrodynamics [6].

However, the electrical properties of wires the radius of which is comparable to the electron free path differ greatly from those of “massive” wires [7–9].

Methods to calculate the conductivity of fine cylindrical metallic wires are discussed in [7, 9], and magnetic induction in such a wire is determined in [9]. In the articles cited, an approach based on solving the Boltzmann kinetic equation for electrons in a metal is applied.

In this paper, we consider a cylindrical wire of a nonmagnetic metal (magnetic permeability μ ≈ 1) with radius R and length D (let DR). To the ends of the wire, a variable voltage with frequency ω is applied. It is assumed that the electric field direction is aligned with the cylinder axis. The skin effect is ignored (it is supposed that R < γ, where γ is the skin depth).

A uniform time-periodic electric field with strength vector E = E0exp(–iωt) acts on conduction electrons inside the wire and generates a high-frequency current with density j.

Let us calculate the self-induction of the wire caused by the magnetic field in it.

It is known that self-induction L is a proportionality coefficient between magnetic flux Φ and current I generating a magnetic field [6]:

$$\Phi = LI \Rightarrow L = \frac{\Phi }{I}.$$
(1)

Having cut a fine metallic wire along the OO' axis (Fig. 1), we will see elementary surface area dS, over which integration should be to calculate a magnetic flux in the inner domain. Crosses in Fig. 1 indicate the direction of magnetic lines of induction B provided that the current density vector is directed along the Z axis, which is aligned with the wire’s axis.

Fig. 1.

Integration domain used to calculate magnetic flux Φ inside the wire.

Magnetic flux Φ through surface element dS of a conductor is given by
$$\Phi = \int {BdS,} $$
where dS = Ddr = DRdδ and δ = r/R is the “dimensionless radius of induction” (r is the radial coordinate of an electron).

Then,

$$\Phi = \int\limits_0^1 {B(\delta )dS = DR\int\limits_0^1 {B(\delta )d\delta .} } $$
(2)
In [10], an expression for magnetic induction B inside a conductor was derived in which the dependence of regular reflectance q on surface defect density H and angle β of incidence of electrons on the wire’s inner surface is taken for boundary conditions (Soffer model) [10]. This boundary condition can be written in the form
$$q(H,\cos \theta ) = \exp ( - {{(4\pi H)}^{2}}{{\cos }^{2}}\theta ),$$
where
$$\cos \theta = \rho \cos \alpha ,\quad H = \frac{{{{h}_{s}}}}{{{{\lambda }_{{\text{F}}}}}},$$
hs is the rms surface roughness, and λF is the electron wavelength on the Fermi surface.
Then,
$$\begin{gathered} B = \frac{{3{{\mu }_{0}}n{{e}^{2}}{{R}^{2}}{{E}_{z}}}}{{\pi {{{v}}_{{\text{F}}}}m\delta }}\int\limits_0^\delta {\int\limits_0^1 {\int\limits_0^\pi {\frac{{\rho \sqrt {1 - {{\rho }^{2}}} }}{\nu }} } } \\ \times \left( {\frac{{({\text{exp}}( - {{{(4\pi H)}}^{2}}{\text{co}}{{{\text{s}}}^{2}}\theta ) - 1){\text{exp}}( - \nu \eta {\text{/}}\rho )}}{{1 - {\text{exp}}( - {{{(4\pi H)}}^{2}}{\text{co}}{{{\text{s}}}^{2}}\theta ){\text{exp}}( - \nu {{\eta }_{0}}{\text{/}}\rho )}}\, + \,1} \right)\xi d\xi d\rho d\alpha , \\ \end{gathered} $$
(3)
where
$$\xi = \frac{r}{R},\quad \rho = \frac{{{{{v}}_{ \bot }}}}{{{{{v}}_{{\text{F}}}}}},\quad \nu = \left( {\frac{1}{\tau } - i\omega } \right)\frac{R}{{{{{v}}_{{\text{F}}}}}},$$
$$\begin{gathered} \eta = \xi \cos \alpha + \sqrt {1 - {{\xi }^{2}}{{{\sin }}^{2}}\alpha } , \\ {{\eta }_{0}} = 2\sqrt {1 - {{\xi }^{2}}{{{\sin }}^{2}}\alpha } , \\ \end{gathered} $$
μ0 is the permeability of vacuum; n, e, and m are, respectively, the concentration, charge, and mass of an electron; \({{{v}}_{ \bot }}\) is the radial component of the electron velocity; \({{{v}}_{{\text{F}}}}\) is the Fermi velocity; and τ is the electron relaxation time.
Having substituted expression (3) into (2), we obtain a formula for magnetic flux Φ:
$$\Phi = \frac{{{{\mu }_{0}}n{{e}^{2}}{{R}^{3}}{{E}_{z}}D}}{{{{{v}}_{{\text{F}}}}m}}{{J}_{\Phi }},$$
(4)
where
$$\begin{gathered} {{J}_{\Phi }} = \frac{3}{\pi }\int\limits_0^1 {\int\limits_0^\delta {\int\limits_0^1 {\int\limits_0^\pi {\frac{{\xi \rho \sqrt {1 - {{\rho }^{2}}} }}{{\delta \nu }}} } } } \\ \times \left( {\frac{{({\text{exp}}( - {{{(4\pi H)}}^{2}}{\text{co}}{{{\text{s}}}^{2}}\theta )\, - \,1){\text{exp}}( - \nu \eta {\text{/}}\rho )}}{{1 - {\text{exp}}( - {{{(4\pi H)}}^{2}}{\text{co}}{{{\text{s}}}^{2}}\theta ){\text{exp}}( - \nu {{\eta }_{0}}{\text{/}}\rho )}}\, + \,1} \right)d\delta d\xi d\rho d\alpha . \\ \end{gathered} $$
A formula that expresses current I through the cross-sectional area of a fine cylindrical wire with Soffer boundary conditions was derived in [8]:
$$I = \frac{{n{{e}^{2}}{{R}^{3}}{{E}_{z}}}}{{{{{v}}_{F}}m}}{{J}_{l}},$$
(5)
where
$$\begin{gathered} {{J}_{I}} = \int\limits_0^1 {\int\limits_0^1 {\int\limits_0^\pi {\frac{{\xi \rho \sqrt {1 - {{\rho }^{2}}} }}{\nu }} } } \\ \times \left( {\frac{{(\exp ( - {{{(4\pi H)}}^{2}}{\text{co}}{{{\text{s}}}^{2}}\theta ) - 1)\exp ( - \nu \eta {\text{/}}\rho )}}{{1 - \exp ( - {{{(4\pi H)}}^{2}}{{{\cos }}^{2}}\theta )\exp ( - \nu {{\eta }_{0}}{\text{/}}\rho )}} + 1} \right)d\xi d\rho d\alpha . \\ \end{gathered} $$

Dividing (4) by (5) we come to desired self-induction L due to a magnetic field inside a conductor:

$$L = {{\mu }_{0}}D\frac{{{{J}_{\Phi }}}}{{{{J}_{l}}}}.$$
(6)

It should be emphasized that self-induction is a complex quantity for which not only the magnitude (absolute value) but also the argument (phase) have a physical meaning (the latter determines an angle between the radius vector of a given point and the positive real plane on the “complex plane of self-induction.”

The significance of self-induction for different objects is explained by the fact that such an important parameter as the energy of the current-induced magnetic field [6] can be expressed through it. This parameter is necessary, for example, to describe surface plasmons in cylindrical semiconductors.

It should be noted that self-induction due to a magnetic field outside a wire is independent of the current distribution over its cross section and has a logarithmic dimension.

If wire radius R is much larger than electron free path Λ (R ≫ Λ), expression (6) yields the macroscopic asymptotics of self-induction Lm due to a magnetic field inside a wire. After simple computation, we have

$${{L}_{m}} = \frac{{{{\mu }_{0}}D}}{{4\pi }}.$$

Figure 2 plots magnitude M of dimensionless self-induction L0D of a fine cylindrical wire versus “dimensionless roughness parameter” H for fixed dimensionless frequency Ω = ωR/\({{{v}}_{{\text{F}}}}\) of an external field and different ratios of the wire radius to reciprocal dimensionless electron free path Ψ = R/Λ. From the run of the curves it follows that with an increase in the roughness parameter, the self-induction magnitude smoothly grows, reaching its asymptotic value, which depends on reciprocal dimensionless electron free path Ψ.

Fig. 2.

Magnitude M of dimensionless self-induction vs. dimensionless surface roughness parameter H for fixed dimensionless frequency Ω = 1 of the external field and dimensionless reciprocal electron free path length Ψ = (1) 0.1, (2) 1, and (3) 3.

Figure 3 plots argument A of self-induction L of a fine cylindrical wire against dimensional surface roughness parameter H. Dimensional frequency Ω of the external field and dimensional reciprocal electron free path Ψ remain the same as in the previous figure. If the radius of the wire does not exceed electron free path (R ≤ Λ), the argument of self-induction has a minimum, which smoothes out with increasing radius. When the radius of the wire exceeds the reciprocal electron free path length (R > Λ), the argument changes sign.

Fig. 3.

Argument A of self-induction L vs. dimensionless surface roughness parameter H for fixed dimensionless frequency Ω = 1 of the external field and dimensionless reciprocal electron free path length Ψ = (1) 0.1, (2) 1, and (3) 3.

Thus, considering that the regular reflectance depends on surface defect density and angle of incidence of electrons on the wire’s inner surface reveals a number of features in the behavior of self-induction that are not typical of macroscopic wires.

Notes

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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  • E. V. Zavitaev
    • 1
    Email author
  • K. E. Kharitonov
    • 1
  • A. A. Yushkanov
    • 2
  1. 1.State University of Humanities and TechnologyOrekhovo-ZuevoRussia
  2. 2.Moscow Region State UniversityMoscowRussia

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