Consider an array of infinitely long, parallel and very thin metallic wires of radius r placed periodically at a distance a in a square lattice with a r as shown in Figure 3-5. The electric field is considered to be applied parallel to the wires (along the Z axis). The electrons are localized to move only within the wires, as which has the first effect of reducing the effective electron density since the radiation cannot sense the individual wire structure but only the average charge density. The effective electron density is immediately represented by
neff=
2 2
πr n
a (3.8) where n is the actual density of conduction electrons in the metal.
There is a second equally important effect to consider. The thin wires have a large inductance, and it is not easy to change the currents flowing in these wires. Thus, it appears as
Figure 3-5. An array of infinitely long thin metal wires of radius r, and a lattice period of behaves as a low frequency plasma for the electric field oriented along the wires and E.M.
wave oriented normal to the wires. the figure is imaged from [8].
if the charge carriers, namely the electrons, have acquired a tremendously large mass. By symmetry there is a point of zero field between the wires, and hence the magnetic field along the line between two wires can be estimated by
H(ρ)= ˆφI 1 1
( - )
2π ρ a-ρ (3.9) The vector potential associated with the field of a single infinitely long current-carrying conductor is non-unique, unless the boundary conditions are specified at definite points. In our case, there is a periodic medium that sets a critical length of a/2. The vector potential
associated with a single wire can thus be approximated by This choice avoids the vector potential of one wire overlapping with another one, and thus the mutual induction between any two adjacent wires is addressed to some extent. Noting that r a by about three orders of magnitude in the present model, and the current I = πr2nev, where v is the mean electron velocity, the vector potential can be written as
A(ρ)=
2
μ r nev0 ln[ ]za ˆ
2 ρ (3.12) This is a very good approximation in the mean field limit, when considering only two wires, but in fact the lattice has actually a four-fold symmetry. The actual deviations from this expression are much smaller than in this case. It is noted that the canonical momentum of an electron in an electromagnetic field is p + eA. Thus assuming that the electrons flow on the surface of the wire (assuming a perfect conductor), one can express the momentum per unit length of the wire as
Thus, assuming a longitudinal plasmonic mode for the system, one derives to present the plasmon frequency. It is noted that reducing effective electron density as well as tremendously increasing effective electronic mass would immediately reduce the plasmon frequency for this system. As an example, for aluminum wires (n = 1029 m−3) with r = 1 μm
and a = 10 mm, the calculated effective mass value was
meff=2.67x10-26kg , (3.16) was almost 15 times heavier than that of a proton, The derived plasma frequency is about 2 GHz, i.e. the structure is a negative dielectric material at microwave frequencies. Note that the final expression for the plasma frequency in equation (3.15) is independent of the microscopic quantities such as the electron density and the mean drift velocity. It only depends on the radius of the wires and the spacing, implying that the entire problem can be recast in terms of the value combination of capacitances and inductances. This approach was used in references [42-43], and it is presented below for the sake of completeness. Consider the current induced by the electric field along the wires, which is related to the total inductance (self and mutual) per unit length (L):
Ez= iωLI=iωLπr nev2 , (3.17) and the polarization per unit volume in the homogenized medium is
P=-neffer=n eveff
iω =- 2 2Ez
ω a L (3.18) where neff = πr2n/a2 as before, one can estimate the inductance, L, by calculating the magnetic flux per unit length, ϕ, passing through a plane between the wire and the point of symmetry between itself and the next wire where the field is zero: ϕ,is presented by
The Equation 3.19 is identical to the relation obtained in the plasmon picture. However, one loses the physical interpretation of a low frequency plasmonic excitation here. It is simple to add the effects of finite conductivity in the wires, which had been neglected in the above
discussion. The electric field and the current would now be related by
Ez=iωLI+σπr I2 (3.21) where σ is the conductivity, which modifies the expression for the dielectric permittivity to
Thus the finite conductivity of the wires contributes to the dissipation, showing up in the imaginary part of ε. For aluminum, the conductivity is σ = 3.65 × 107Ω−1m−1, which yields a values of γ = 0.1ωp that is comparable to the values in real metals [44]. Thus the low frequency plasmon is sufficiently stable against absorption to be observable.
In the above discussion, only wires pointing in the z-direction were considered. This made the medium anisotropic with negative ε only for waves with the electric field along the z direction. The medium could be constructed to have a reasonably isotropic response by considering a lattice of wires oriented along the three orthogonal directions as shown in figure3-6. In the limit of large wavelengths, the effective medium appears to be isotropic as the radiation fails to resolve the underlying cubic symmetry yielding a truly three-dimensional low frequency plasma.
For very thin wires, the polarization in the direction orthogonal to the wires is small and can be neglected. Thus the waves only sense the wires parallel to the electric field, and correspondingly have a longitudinal mode. The effects of the connectivity of the wires along different directions at the edges of the unit cell have also been examined [9]. A wire mesh with non-intersecting wires was shown to have a negative ε at low frequencies below the plasma frequency, but had strong spatial dispersion for the transverse modes above the plasma frequency. This issue of spatial dispersion has been studied more recently by Belov et al [43].
Figure 3-6. A three-dimensional lattice of thin conducting wires behaves like an isotropic low frequency plasma. The figure is imaged from [8].
The continuous wire structure behaves like a high-pass filter, which means that the effective permittivity will take negative values below the plasma frequency. However, for discontinuous wire structures, the negative permittivity region does not extend to zero frequency, and there appears a stop-band around the resonance frequency. In contrary to the continuous wire structures that exhibit a stop-band with no lower frequency edge, the present configuration exhibits a stop-band with a well-defined lower edge due to the discontinuous nature of the wires.