Thermal properties in mesoscopics: physics and ... - ResearchGate

for refrigeration purposes. For instance, this may happen
in thermionic transport over a potential barrier as well as
in energy-dependent tunneling through a barrier. Since
an exhaustive analysis of all possible predicted refrigeration
methods is far beyond the limits set to this Review,
in the following we give a brief description of a few examples
that we believe are relevant in the present context.
In such devices, several different effects may contribute
to the refrigeration process (e.g., thermionic transport,
quantum tunneling as well as thermoelectric effects), so
that making a proper ”classification” of the refrigeration
principle is, strictly speaking, rather difficult. As a consequence,
we shall try mainly to follow the definitions as
they were introduced in the original literature.
1. Thermionic refrigerators
A vacuum thermionic cooling device consists of two
electrodes separated by a vacuum gap. Cooling occurs
when highly energetic electrons overcome the vacuum
barrier through thermionic emission, thus reducing
the electron temperature of one of the two electrodes.
In such a situation, the refrigerator operation
is mainly affected and limited by radiative heat transfer
between the electrodes. The thermionic emission current
density (jR) is given by the Richardson equation,
jR = A0T 2 exp(− Φ
kBT
), where Φ and T are the work func-
tion and the temperature of the emitting electrode, respectively,
A0 = 4πemkB/h 3 is the Richardson constant,
and m is the electron mass. From the expression above,
a strong reduction of jR is expected upon lowering the
temperature. Mahan (1994) developed a simple model
for thermionic refrigeration, and demonstrated that its
efficiency can be as high as 80% of the Carnot value. Vacuum
thermionic refrigerators are generally characterized
by a higher efficiency as compared to thermoelectric coolers,
and are considered to be an attractive solution for
future refrigeration devices (Nolas and Goldsmid, 1999).
However, the high Φ values typical of currently available
materials make thermionic cooling efficient, at present,
only above 500 K.
Several ideas on how to increase the cathode emission
current and to improve the operation of these refrigerators
have been proposed (Hishinuma et al., 2001, 2002;
Korotkov and Likharev, 1999; Purcell et al., 1994). Korotkov
and Likharev suggested to cover the emitter with a
thin layer of a wide-gap semiconductor, and to exploit the
resonant emission current to cool the emitter (Korotkov
and Likharev, 1999). The analysis of such a thermionic
cooler predicted an efficient refrigeration down to 10 K.
The issue of high Φ values can be overcome by a reduction
of the distance between the electrodes (in the
submicron range) and by the application of a strong
electric field. Following this scheme, Hishinuma et al.
(2001) theoretically analyzed a thermionic cooler where
the two electrodes where separated by a distance in the
nanometer range. In such a situation, the potential
44
FIG. 34 (a) Schematic diagram of the potential barrier profile
V (x) for Φ = 1 eV and with electrode separation of 60
˚A. (b) Heat flow distribution at T = 300 K. Adapted from
(Hishinuma et al., 2001).
barrier is essentially lowered (see Fig. 34(a)), allowing
both thermionic emission and energy-dependent tunneling.
As a consequence, rather small (if compared to vacuum
thermionic devices) external voltages (∼ 1...3 V) are
required in order to produce significant electric currents.
For suitable values of the applied voltage and distances
between the electrodes, electrons above the Fermi level
dominate the electric transport (both thermionic emission
over the barrier and tunneling through the barrier),
thus leading to cooling of the emitter. As it can be inferred
from Fig. 34(b), the contribution of the energydependent
tunneling to total cooling is essential, and this
refrigerator could be classified as a vacuum tunneling
device. The cooling power surface density in this combined
thermionic-tunneling refrigerator was predicted to
obtain values as high as 100 W/cm 2 at room temperature.
However, in the only experimental demonstration
of this device, a moderate emission current (below 10
nA) was reported at room temperature, with an observed
temperature reduction of about 1 mK (Hishinuma et al.,
2003). So far, no experimental demonstration of vacuum
thermionic refrigeration at cryogenic temperatures has
been reported.
2. Application of low-dimensional systems to electronic
refrigeration
The exploitation of low-dimensional systems gives additional
degrees of freedom in order to engineer materials
that may lead to enhanced operation of thermoelectric
and thermionic devices (Hicks et al., 1993; Hishinuma
et al., 2002; Sales, 2002; Sofo and Mahan, 1994).
Some of the limitations intrinsic to vacuum thermionic
refrigerators can be overcome with solid state thermionic
coolers (Mahan and Woods, 1998; Shakouri and Bowers,
1997). As a matter of fact, modern growth techniques
easily allow one to control both the barrier height
and its width within a wide range of values. One disadvantage
of solid state thermionic coolers stems from
the thermal conductivity of the barrier which is essentially
absent in vacuum devices. Nevertheless, a large