The plasmon is a well established collective excitation of metals
in the visible and near UV, but at much lower frequencies
dissipation destroys all trace of the plasmon and typical Drude
behavior sets in. We propose a mechanism for depression of the
plasma frequency into the far infrared or even GHz band:
Periodic structures built of very thin wires dilute the average
concentration of electrons and considerably enhance the
effective electron mass through selfinductance. Computations
replicate the key features and confirm our analytic theory. The
new structure has novel properties not observed before in the
GHz band, including some possible impact on superconducting
properties. [S00319007(96)004917]
PACS numbers: 61.85.+p, 41.20.Jb, 77.22.–d, 84.90.+a
Much of the fascination of condensed matter turns on our
ability to reduce its apparent complexity and to summarize
phenomena in terms of a new excitation that is, in fact, a
composite put together from the elementary building blocks of
the material but behaves according to its own simplified
dynamics. One of the earliest and most celebrated of these
composites occurs in metals and is known as a plasmon [1,2]: a
collective oscillation of electron density. In equilibrium the
charge on the electron gas is compensated by the background
nuclear charge. Displace the gas and a surplus of uncompensated
charge is generated at the ends of the specimen, with opposite
signs at opposite ends supplying a restoring force resulting in
simple harmonic motion, v 2 p ne2 « 0meff
. (1)
The plasma frequency, vp, is typically in the ultraviolet
region of the spectrum: around 15 eV in aluminum.
The plasmons have a profound impact on properties of metals,
not least on their interaction with electromagnetic radiation
where the plasmon produces a dielectric function of the form
2
v
p
«svd 1 2 (2)
vsv1igd ,
which is approximately independent of wave vector, and the
parameter g is a damping term representing dissipation of the
plasmon’s energy into the system. In simple metals, gis small
relative to vp. For aluminum,
vp 15 eV, g 0.1 eV . (3)
The significant point about Eq. (2) is that it is essentially
negative below the plasma frequency, at least down to
frequencies comparable to g.
Why is negative epsilon interesting? Cut a metal in half and
the two surfaces created will be decorated with surface plasmons
[3,4]: collective oscillations bound to the surface whose
frequency is given by the condition
« 1svsd 1«2svsd 0, (4)
where « 1 and « 2 are dielectric functions for material on
either side of the interface. Choosing vacuum on one side
and metal on the other gives
p
vs vpy2 (5)
if we neglect dissipation. It is of course an essential
precondition that « for the metal be negative. Shape the
metal into a sphere and another set of surface modes appears.
Two spheres close together generate yet another mode structure.
Therefore negative « gives rise to a rich variety of
electromagnetic structure decorating the surfaces of metals with
a complexity controlled by the geometry of the surface.
In fact, the electromagnetic response of metals in the
visible region and near ultraviolet is dominated by the negative
epsilon concept. Ritchie and Howie [5], Echenique et al. [6–8],
Howie and Walsh [9], and many other researchers have shown how
important the concept of the plasmon is in the response of
metals to incident charged particles. However, at lower
frequencies, from the near infrared downwards, dissipation
asserts itself, and the dielectric function is essentially
imaginary. Life becomes rather dull again.
In this Letter we show how to manufacture an artificial
material in which the effective plasma frequency is
depressed by up to 6 orders of magnitude. The building
blocks of our new material are very thin metallic wires of
the order of 1 mm in radius. These wires are to be assembled
into a periodic lattice and, although the exact structure
probably does not matter, we choose a simple cubic lattice shown
below in Fig. 1.
Sievenpiper, Sickmiller, and Yablonovitch [10] have
independently investigated metallic wire structures. Our
work differs from theirs in one important respect: We
suggest that very thin wires are critical to applying the
concept of plasmons to these structures.
We now derive the plasma frequency for collective oscillations
of electrons in the wires. Consider a displacement of
electrons along one of the cubic axes: The active wires will be
those directed along that axis. If the density of electrons in
these wires is n, the density of these active electrons in the
structure as a whole is given by the fraction of space occupied
by the wire,
2
pr
neff n . (6)
a2
Before we rush to substitute this number into formula
(1) for the plasma frequency, we must pause to consider
another effect which is at least as important: Any restoring
force acting on the electrons will not only have to work against
the rest mass of the electrons, but also against selfinductance
of the wire structure. This effect is not present in the
original calculation of the plasma frequency, but in our
structure it is the dominant effect. It can be represented as a
contribution to the electron mass. The important point is that
the inductance of a thin wire diverges logarithmically with
radius. Suppose a current I flows in the wire creating a
magnetic field circling the wire, 2
prnye
HsRd , (7)
2pR
where R is the distance from the wire center. We have also
reexpressed the current in terms of electron velocity, y, and
charge density, ne. We write the magnetic field in
FIG. 1. The periodic structure is composed of infinite wires
arranged in a simple cubic lattice, joined at the corners of the
lattice. The large selfinductance of a thin wire delays the
onset of current mimicking the effect of electron mass.
terms of a vector potential,
21
HsRd m =3AsRd . (8)
0
For an isolated wire the vector potential is illdefined
until boundary conditions are specified. However, for a
threedimensional array of wires, the mutual inductance
actually simplifies the problem and introduces the lattice
spacing as a natural cutoff,
2
m0prnye
AsRd lnsayRd , (9)
2p
where a is the lattice constant. We shall derive (9) in a
subsequent paper. Here we ask the reader to take the result on
trust, and offer the agreement with computer calculations
presented in Fig. 2 as justification.
We note that, from classical mechanics, electrons in a
magnetic field have an additional contribution to their momentum
of eA, and therefore the momentum per unit
FIG. 2. Numerical simulations of the band structure: real
(top) and imaginary (bottom) parts of the wave vector for a
simple cubic lattice, a 5 mm, with wires along each axis
consisting of ideal metal wires, assumed 1 mm in radius. The
wave vector is assumed to be directed along one of the cubic
axes. The full lines, largely obscured by the data points,
represent the ideal dispersion of the longitudinal and
transverse modes defined above, assuming a plasma frequency of
8.2 GHz. The light cone is drawn for guidance. Note that the two
degenerate transverse modes in free space are modified to give
two degenerate modes that are real only above the plasma
frequency of 8.2 GHz. The new feature in the calculation is
the longitudinal mode at the new plasma frequency.
length
of the wire is
2422
m0preny
2
pr2enAsrd lnsayrd meffprny,
2p
(10) where meff is the new effective mass of the electrons
given by
22
m0pren
meff lnsayrd . (11)
2p
This new contribution is dominant for the parameters we have
in mind. For instance, for aluminum wires
63
r 1.0 3102m, a 5 3102m,
23
n 1.806 31029 msaluminumd (12)
gives an effective mass of
meff 2.4808 310226 kg
2.7233 3104me 14.83mp . (13)
In other words, by confining electrons to thin wires we have
enhanced their mass by 4 orders of magnitude so that they are
now as heavy as nitrogen atoms.
Having both the effective density, neff, and the effective
mass, meff, on hand we can substitute into (1),
2 neffe22pc02
v øs8.2 GHzd2. (14)
p « 0meff a2 lnsayrd
Here is the reduction in the plasma frequency promised.
Note in passing that, although the new reduced plasma
frequency can be expressed in terms of electron effective
mass and charge, these microscopy quantities cancel, leaving
a formula containing only macroscopic parameters of the system:
wire radius and lattice spacing. It is possible to formulate
this problem entirely in terms of inductance and capacitance of
circuit elements. However, in doing so, we miss the analogy with
the microscopic plasmon. Our new reduced frequency plasma
oscillation is every bit the quantum phenomenon as is its high
frequency brother.
One remaining worry: How stable is the plasmon? The plasmon
may decay through electronhole pair creation, or through
generation of phonons in the wires, depending on the
temperature. Either way, this mechanism acts entirely through
the electrical resistance of the wires. A more careful
calculation including resistance gives the following expression
for an effective dielectric function of the structure:
2
v
p
« eff 1 2 ³´ , (15)
2 r2s
vv1i« 0a2vyp
p
where sis the conductivity of the metal. Typically, for
aluminum,
21 21
s 3.65 3107 Vmsaluminumd (16)
and
2
v
p
¡
« eff ø 1 2 ¢ saluminumd . (17)
vv1i0.1vp
Thus our new plasmon is about as well defined relative to its
resonant frequency as the original plasmon.
To what extent is our theory confirmed by detailed calculations?
We have developed a method for computing dispersion
relationships in structured dielectrics [11,12], and we use this
to check our analytic predictions. Figure 2 shows our numerical
computations of dispersion in our lattice. We choose the most
critical case of infinitely conducting lossless wires.
Also shown in Fig. 2 is the result for dispersion of
transverse light obtained by applying our effective dielectric
function taken from Eq. (15) with g 0,
q
pv2 2vp 2 « eff
K v , (18)
c0 c0
which gives real bands only above the plasma frequency of 8.2
GHz, imaginary bands below. In addition, we show the analytic
prediction of a dispersionless longitudinal plasmon.
So accurately does our formula reproduce the computed result
that the points obscure the analytic line. The computed
longitudinal mode agrees very well at K 0, but shows a
small degree of dispersion towards the Brillouin zone boundary.
Computations for other directions in the Brillouin zone show a
similar picture, and Eq. (18) appears to give a good description
of the results, at least when K is less than the free space wave
vector. It is worth emphasizing that at the plasma frequency
of 8.2 GHz the free space wavelength of light is about 35
mm, much greater than the lattice spacing of 5 mm. In other
words, as far as external electromagnetic radiation is
concerned, this structure appears as an effectively homogeneous
dielectric medium whose internal structure is only apparent
insofar as it dictates « eff. In this respect it is important
that the structure be made of thin wires. Equation (14) shows
that the function of the small radius is to suppress the plasma
frequency. In a thick wire structure in Eq. (14),
lnsayrdø 1 (19)
so that the plasma frequency corresponds to a free space
wavelength of approximately twice the lattice spacing.
Therefore Bragg diffraction effects would interfere with our
simple plasmon picture. Choosing a small radius ensures that
diffraction occurs only at much higher frequencies.
We are currently investigating the possibilities for the
manufacture of this structure. In one alternative we plan to
exploit technology developed for constructing spark chamber
particle detectors which happen to employ wires of approximately
the dimensions we require. In another alternative we have
considered winding helices of wires on rods of square cross
section such that, when the rods are stacked in an ordered way,
the wires intersect as required. Calculations show that it is
not necessary to have precisely a simple cubic structure to
observe the effect. In this manner it is possible to produce
structures of the order of
0.2 m 3 0.2 m 3 0.2 m, that is to say, much larger than the
wavelength of any relevant radiation at vp. In its ideal
dissipationless form the structure has the novel feature that
below the plasma frequency all electromagnetic modes are
excluded from the structure. At sufficiently low frequencies,
dissipation must take charge in a normal metal, but, if
superconducting material were employed for the wires and kept
well below the transition temperature, dissipation could be
small down to zero frequency. In the context of
superconductivity it should be noted that plasma frequencies in
these structures can be well below the gap energy of a
conventional superconductor. Anderson [13] has stressed the role
of the plasmon in the electromagnetic properties of
superconductors where it appears as a “Higgs boson” but with
a very large mass relative to the superconducting gap. In our
new material the Higgs boson is now well within the gap, giving
rise to speculation about a more active role for the Higgs boson
in the superconducting mechanism itself. This theme will be
pursued elsewhere.
Another conclusion to be drawn is in regard to the doping of
semiconductors. It is plain from Fig. 2 that in the GHz
frequency range the electromagnetic spectrum is very severely
modified. This has been achieved with an extremely small amount
of metal; the average density of metal in the structure is less
than 1 part 106, comparable to doping levels in a semiconductor.
The interest in this new material derives from the analogy to
be made with the role of the plasmon at optical frequencies.
Objects constructed from the new material will support GHz
plasmons bound to the surface which can be controlled by the
local geometry. Here are possibilities for novel waveguides.
Such material is also a very effective band stop or band pass
filter. Below the plasma frequency very little can be
transmitted; above, and especially in the visible, the structure
is transparent.
Another aspect is coupling to charged particles [14]. It is
well known from electron microscope studies that metals, metal
spheres, and colloids are all efficient at extracting
electromagnetic energy from an electron. The mechanism is
essentially .
Cerenkov radiation into the almost dispersionless plasma
modes. In our materials the energy scale is much smaller, and it
is possible to imagine ballistic electrons with a few eV energy
injected into our new material where they would have a rather
fierce interaction with the low frequency plasmon which could
conceivably be exploited in microwave devices.
We have demonstrated that a very simple metallic
microstructure comprising a regular array of thin wires
exhibits novel electromagnetic properties in the GHz region,
analogous to those exhibited by a solid metal in the UV. We
trust that the analogy will prove a powerful one and lead to
further novel effects and applications.
This work has been carried out with the support of the
Defense Research Agency, Holton Heath.
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