Ohmic and Diamagnetic Currents Contribution on the Electromagnetic Penetration Depth of a Conducting Surface

Due to its conducting electron, metal is a good reflector for electromagnetic wave. An electromagnetic wave penetrating a metallic surface has a finite penetrating depth. There are two limit that are well studied in the physics textbooks. They are high frequency electromagnetic wave penetrating a metal with small conductivity and a static (low frequency) field penetrating a superconductor (metal with infinitely large conductivity). In this article we study the intermediate regime between these two limits. By setting the electric current density as the total sum of both Ohmic and Diamagnetic currents, we derive the penetration depth in the intermediate regime., we show the transition between these two limits.


INTRODUCTION
Due to its conducting nature, electromagnetic waves are reflected by metallic surface. The reflection mechanism can be mathematically described by substituting the following Ohmic current into the Maxwell equation [1].
= , (1) where is electric current density, is conductivity and is the electric field. In the limit of small conductivity, the skin depth of the penetrating electromagnetic wave is inversely proportional to the conductivity of the metal [2].
Here is the permittivity and is the permeability. Because of that, one can expect that the skin depth of superconductor is zero. However, Meisner effect shows that a penetrating magnetic field has a finite skin depth, also known as London penetration depth , is not zero [3], [4].
Here is the electron mass, − is the electron charge and is the carrier density. London penetration can be derived by substituting the following London diamagnetic current into the Maxwell equation and taking a static limit [5]- [7]. The reflection of electromagnetic field in superconductor is mainly caused by the London diamagnetic current Here is the vector potential.
and are two limiting cases that are well studied in the physics textbooks [1], [8]. In this article, we focus on the intermediate regime where the effect of both ohmic and diamagnetic current are important. The effect of Ohmic current has been widely studied in surface designs [9], [10]. On the other hand, the study of diamagnetic current is gaining increasing attention in physics [11], [12] and related areas [13], [14].

METHOD: MATHEMATICAL FORMALISM
The dynamics of electric field and magnetic field in a material is best described by Maxwell equations. The Maxwell equations can be written in term of scalar potential and vector potential .
= × , (5b) By using Lorentz gauge condition, one can obtain the partial differential equations for and vector potential [15] Al Here = 1/ √ is the speed of light in the material. We will solve Eqs. (6a) and (b) by choosing = 0. In this case one can show that for electromagnetic wave propagating in direction, the solution of , and that have the following form Here is the wavenumber and is the frequency. Using this choice of and vector potential , we can derive the penetration depth by examining the imaginary part of because it creates an exponentially decaying term in the electromagnetic fields. The inverse of this imaginary term is the penetration depth.
In the following subsections we illustrate the derivation of these two limits: high frequency electromagnetic wave penetrating a metal with small conductivity and a low frequency field penetrating a metal with large conductivity, by consider Ohmic and diamagnetic current, respectively. In the next Section we will combine both currents to see the penetration depth for intermediate regimes.

Reflection by Ohmic current
To study the limit of high frequency electromagnetic wave penetrating a metal with small conductivity, we solve Eq. (6b) by substiting Eq. (1).
Substituting Eq. (7) to Eq. (8) we arrive at the following equation The dispersion ( ) can then be derived as follows.
For small conductivity and large frequency ≪ , one can show that the imaginary part of is proportional to The inverse of this imaginary term is the penetration depth in Eq. (2).

Reflection by Diamagnetic current
To study the limit of low frequency electromagnetic wave penetrating a metal with large conductivity, we solve Eq. (6b) by substituting Eq. (4).
The dispersion ( ) can then be derived as follows.

RESULT AND DISCUSSION: REFLECTION BY OHMIC AND DIAMAGNETIC CURRENT
To study the intermediate regime, we set the electric current density as the total sum of both Ohmic and Diamagnetic currents.
In terms of and : We can solve the expression for by substituting Eq. (8)  Substituting Eq. (7a) to Eq. (12) we arrive at the following equation The dispersion ( ) can then be derived as follows.
To discuss the penetration depth i.e. the inverse of the imaginary part of k, we need to look at two frequency regimes separated by the critical frequency 0 = / High frequency > / In this case, the dispersion ( ) can be written as follows Its imaginary part is as follows. Low frequency < / In this case, the dispersion ( ) can be written as follows Its imaginary part is as follows. Figure 1. Penetration depth as a function of / and / 0 . (a)Penetration depth normalized to . By normalizing to we can clearly see that in the limit of low frequency, approach . (b)Penetration depth normalized to . By normalizing to we can clearly see that in the limit of high frequency, approach . (c) Illustration of penetration of electromagnetic wave.
Using Eq. (26), we can illustrate penetration depth as a function of / and / 0 . In Fig. 1a and 1b, is normalized to and , respectively. We can clearly see that our result converges to the following well-known limits. lim Furthermore, we can see that when = , ≈ = .

CONCLUSION
To summarize, we study the intermediate regime between two well-studied limits: high frequency electromagnetic field penetrating a metal with small conductivity and a static field penetrating a superconductor. The former limit can be derived by consider Maxwell equations with Ohmic current. On the other hand, the later limit is obtained by focusing on the diamagnetic current.
By setting the electric current density as the total sum of both Ohmic and Diamagnetic currents, we derive the penetration depth in the intermediate regime. The penetration depth that include both contribution of Ohmic and diamagnetic currents depends on the ratio of / and / 0 (see Eq. (26) ). Figure 1 show that our result converges to the following well-known limits. When = , the penetration depth does not depend strongly to frequency.