Influence of Specularity Coefficients on the Interaction of Electromagnetic H-Wave with the Thin Metal Film is Disposed on the Dielectric Substrate

Alexey Utkin Igorevich^*, Alexander Yushkanov Alexeevich

Theoretical Physics Department, Moscow State Regional University, Moscow, Russia

Abstract

Interaction of electromagnetic H-wave with thin metal film is located between two dielectric environments ε₁, ε₂ in the case of different incident angles of H-wave θ and in the case of different reflection coefficients q₁ и q₂ is calculated in this article. Behavior analysis of reflection coefficient R, transmission coefficient T and absorption coefficient A in the case of its frequency dependence y and variation dielectric permeability of its environments is done.

Keywords

The Thin Metal Film, Electromagnetic H-Wave, Dielectric Environments, Reflection Coefficient, Transmission Coefficient, Absorption Coefficient

Received: March 1, 2015
Accepted: March 11, 2015
Published online: March 20, 2015

@ 2015 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY-NC license. http://creativecommons.org/licenses/by-nc/4.0/

1. Introduction

Currently microelectronics, optoelectronics and thin-film technology are actively developing. In particular the greatest interest represents researching of interaction electromagnetic radiation with thin conductive films in the different frequency range [1-6]. This interest is related not only with extensive practical importance of thin conductive films, but with some unresolved theoretical tasks.

In our case thickness of the thin metal film a is not more, than thickness of skin-layer δ and this thickness comparable with the average free path of electrons Λ. For this reason skin-effect is not considered. Skin-effect was researched in [8] in the case of the thin metal cylindrical wire. Quantum effects are not taken into account. This effects were researched in [9] in the case of quantum film in the dielectric environment.

2. Problem Definition and Methods

Consider the thin metal layer thickness of a is located between two dielectric (non-magnetic) environments, with dielectric permeability ε₁ (the first environment) and ε₂ (the second environment) with reflection coefficients q₁ and q₂ in the case of falling electromagnetic H-wave (from the first environment) at the θ angle. Reflection coefficients q₁ and q₂ are associated with reflection of electrons from top and lower surfaces layer. Clarify, if electric field vector is parallel the surface of the thin layer, then this wave is called H-wave. Electric field of electromagnetic wave is parallel the thin metal layer and directed along Y-axis, while X-axis is directed into the layer.

Then behavior of electromagnetic field inside the thin metal layer is described by the equation system [10]:

   (1)

k = ω/c – wave number, с – speed of light, j – electric current density.

We have reflections coefficient R, transmission coefficient T and absorption coefficient A of thin metal film, when H-wave falling on this film [11]:

(2)

Expression (2) contains P⁽¹⁾ and P⁽²⁾ [12] :

(3)

Z⁽¹⁾ and Z⁽²⁾correspond to the impedance of lower surface the layer. In particular Z⁽¹⁾corresponds antisymmetric, in electric field, configuration of the external field: E_y(0) = -E_y(a), H_z(0) = H_z(a), and Z⁽²⁾ – corresponds symmetric configuration: E_y(0) = E_y(a), H_z(0) = -H_z(a) [11].

Expression for surface impedance in the case of interaction H-wave with the thin metal film, were obtained in [11] in the case when wavelength much more thickness of the thin layer:

(4)

For σ_a expression (this is electrical conductivity of the thin metal layer, with average thickness of this layer) we used results [13]. In this article we compared our results with experiment data [14]. Our σ_a expression have look:

(5)

x = a/(v_Fτ) – the dimensionless frequency of bulk electron collision, y = aω/v_F – the dimensionless frequency of the electric field, λ = x/(x-iy), σ₀ =ω²_pτ/4π  – the static electrical conductivity, v_F  – Fermi speed, τ – electron relaxation time, ω_p – plasma frequency, q₁ и q₂ – reflection coefficients.

Finally, reflection coefficient R, transmission coefficient T and absorption coefficient A (expression (2)) will have look [12]:

(6)

ε_1,2 =ε₂/ε₁, .

Now we will begin to analyze behavior of this coefficients (expression (6)).

3. Results and Discussion

Let us consider behavior of coefficients R, T and A in the case of their frequency dependence with variation dielectric permeability value of the second environment ε₂ and in the case of different reflection coefficients q₁ and q₂. Clarify some parameters of potassium for further calculations: ω_p = 6.5 10¹⁵1/s, v_F  = 8.52 ∙ 10⁵ m/s, τ = 1.54 ∙ 10^-13 s, a = 10 nm.

Figure 1. The dependence of reflection coefficients R on the dimensionless frequency of the electric field y. Curve 1: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 40, q₁ = 0.5, q₂ = 0.6; curve 2: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 30, q₁ = 0.5, q₂ = 0.6; curve 3: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 15, q₁ = 0.5, q₂ = 0.6; curve 4: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 5, q₁ = 0.5, q₂ = 0.6; curve 5: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 1, q₁ = 0.5, q₂ = 0.6.

Figure 2. The dependence of transmission coefficients T on the dimensionless frequency of the electric field y. Curve 1: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 1, q₁ = 0.5, q₂ = 0.6; curve 2: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 5, q₁ = 0.5, q₂ = 0.6; curve 3: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 15, q₁ = 0.5, q₂ = 0.6; curve 4: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 30, q₁ = 0.5, q₂ = 0.6; curve 5: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 40, q₁ = 0.5, q₂ = 0.6.

Figure 3. The dependence of absorption coefficients A on the dimensionless frequency of the electric field y. Curve 1: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 1, q₁ = 0.5, q₂ = 0.6; curve 2: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 5, q₁ = 0.5, q₂ = 0.6; curve 3: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 15, q₁ = 0.5, q₂ = 0.6; curve 4: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 30, q₁ = 0.5, q₂ = 0.6; curve 5: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 40, q₁ = 0.5, q₂ = 0.6.

Figure 4. The dependence of reflection coefficients R on the dimensionless frequency of the electric field y. Curve 1: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 1, q₂ = 1; curve 2: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 0.8, q₂ = 0.9; curve 3: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 0.5, q₂ = 0.6; curve 4: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 0.2, q₂ = 0.3; curve 5: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 0, q₂ = 0.

Figure 5. The dependence of transmission coefficients T on the dimensionless frequency of the electric field y. Curve 1: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 0, q₂ = 0; curve 2: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 0.2, q₂ = 0.3; curve 3: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 0.5, q₂ = 0.6; curve 4: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 0.8, q₂ = 0.9; curve 5: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 1, q₂ = 1.

Figure 6. The dependence of absorption coefficients A on the dimensionless frequency of the electric field y. Curve1: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 0, q₂ = 0; curve 2: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 0.2, q₂ = 0.3; curve 3: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 0.5, q₂ = 0.6; curve 4: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 0.8, q₂ = 0.9; curve 5: x = 0.002, θ = 20⁰, ε₁ = 1, ε₂ = 4, q₁ = 1, q₂ = 1.

4. Conclusions

In figure 1 we can see that the descending velocity of the curve increases with increasing values of the dielectric permittivity of the second environment ε₂.

In figure 2 we can see that the increase velocity of the curve increases with increasing values of the dielectric permittivity of the second environment ε₂.

In figure 3, in the case of not large value of the dielectric permittivity (ε₂ < 30) we can see, that coefficient A increases, reaches maximum and descents. Cleary visible absorption maxima. In the case of large value of the dielectric permittivity (ε₂ > 30) coefficient A begin immediately descents.

In figure 4, 5, 6 we can see, that variation of the thin metal layer reflection coefficients q₁ and q₂ (from diffuse q₁ = q₂ = 0 to reflection q₁ = q₂ = 1 cases) affects to R, T, A coefficients. It is obvious that reflection coefficients will change, when the thin metal film borders with different environments. In particular, in figure 6, in the reflection case, coefficient A immediately descent. In all other cases coefficient A increases reaches its maximum and descents.

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