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 self-inductance. 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. [S0031-9007(96)00491-7]

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 self-inductance 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 self-inductance 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 ill-defined
until boundary conditions are specified. However, for a
three-dimensional 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 electron-hole 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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