ISSN 1063-7397, Russian Microelectronics, 2008, Vol. 37, No. 3, pp. 157–165. © Pleiades Publishing, Ltd., 2008.
Original Russian Text © A.V. Latyshev, A.A. Yushkanov, 2008, published in Mikroelektronika, 2008, Vol. 37, No. 3, pp. 181–189.
The problem concerning the behavior of a plasma in
an external alternating electric ﬁeld was ﬁrst solved by
Landau in the case of a collisionless plasma occupying
a half-space, an electric ﬁeld perpendicular to the
boundary, and mirror boundary conditions . Later,
Keller et al.  and Gokhfel’d et al. [3, 4] considered
more general boundary conditions but again assumed
the plasma volume to be inﬁnite.
Latyshev et al.  addressed the response from the
electron gas of a thin metal layer under general condi-
tions, but did not investigate the solution on account of
its complicated character. Latyshev and Yushkanov 
examined the zeros of the dispersion function, or the
existence of plasma modes as dependent on electron
collision rate, ﬁeld frequency, and plasma frequency.
In the present paper, the solution by Latyshev et al.
 is used to make some quantitative predictions in the
case of a small ﬁeld frequency
, with the ﬁeld taken as
. Specifically, we investigate the behavior
of an electric ﬁeld inside a thin metal slab, analyzing
the respective contributions by Drude modes and
Debye modes (associated with a discrete spectrum) and
by Van Kampen waves (associated with a continuous
spectrum). In addition, we ﬁnd and investigate the dis-
sipation in the metal, using analytical calculations.
The problem of an electron plasma in a thin metal
slab arises, e.g., in the context of electromagnetic radi-
ation interacting with ﬂat-particle aerosols. Other
examples are found in submicrometer microelectron-
ics, such as the propagation of long-wavelength surface
plasmons [7, 8]. The reason is that the plasma response
to an alternating electric ﬁeld cannot be treated macro-
scopically on the submicrometer scale . Instead, one
must construct a detailed kinetic description.
FORMULATION OF THE PROBLEM
Consider a metal slab that is thin in the sense of its
being much less than its length and width;
this means that the length and width can be considered
inﬁnite. Let an alternating electric ﬁeld
applied to the slab along its normal. We are to ﬁnd the
response of the electrons in the metal to the ﬁeld, the
behavior of the electric ﬁeld in the metal, and the dissi-
We ﬁrst introduce Cartesian coordinates with the ori-
gin located at the center of the slab and the
normally. Assume that the ﬁeld is sufﬁciently weak for a
linear approximation to be valid . We will seek a dis-
in the form
is the Fermi distribution function and
is the Dirac delta function,
is the kinetic energy of an electron,
is the Fermi kinetic energy, and
is the Fermi velocity.
x v t,,() iωt–()δε
ε–()ψx µ,(), µexp
Response of the Electron Plasma in a Thin Metal Slab
to a Low-Frequency External Electric Field
A. V. Latyshev and A. A. Yushkanov
Moscow State Region University, Moscow, Russia
e-mail: email@example.com, firstname.lastname@example.org
Received June 13, 2007
—The response of the electrons in a thin metal slab to an external alternating electric ﬁeld is studied
theoretically in the case of the ﬁeld frequency being much less than the plasma frequency. Dissipation in the
metal is calculated.
PACS numbers: 81.40.Wx