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

A. General principles
The physical principle at the basis of thermoelectric
cooling is that energy transfer is associated with quasiparticle
electric current, as shown in Sec. II. Under suitable
conditions thermal currents can be exploited for heat
pumping, and if heat is transferred from a cold region to
a hot region of the system, the device acts as a refrigerator.
The term refrigeration is associated throughout
this Review to a process of lowering the temperature of
a system with respect to the ambient temperature, while
cooling, in general, means just heat removal from the
system. It is noteworthy to mention that the maximum
cooling power of a refrigerator is achieved at a vanishing
temperature gradient, while the maximum temperature
difference is achieved at zero cooling power. The efficiency
of a refrigerator is normally characterized by its
coefficient of performance (η), i.e., the ratio between the
refrigerator cooling power ( ˙ Qcooler) and the total input
power (Ptotal):
η = ˙ Qcooler
. (77)
Ptotal
Irreversible processes (e.g., thermal conductivity and
Joule heating) degrade the efficiency of refrigerators, and
are essential elements that need to be carefully evaluated
for the optimization of any device.
The basic principles of the design are in general similar
for different types of refrigerators. The increase of
temperature gradient can be achieved by realizing a multistage
refrigerator. In this case, the stage operating at
a higher temperature should be designed for larger cooling
power in order to efficiently extract the heat released
from the lower-temperature stage (”pyramid design”).
The enhancement of the refrigerator cooling power can
be achieved by connecting several refrigerators in parallel.
The parallel design is more effective both for an
efficient heat evacuation from the hotter regions of the
device and for the application of higher electric currents
to the refrigerator. It also allows more freedom to properly
bias all cooling elements.
The temperature dependence of the electric and thermal
properties of the active parts in the refrigerator may
limit their exploitation at low temperatures. The reduction
of thermal conductivity by lowering the temperature
has both positive (better thermal insulation between cold
and hot regions) and negative (difficulty in removing heat
from the system) effects.
At cryogenic temperatures different types of superconductors
can be efficiently exploited. They can be used
both as passive and active elements: in the former case,
owing to their low thermal conductivity and zero electric
resistance (e.g., as one of the two arms in Peltier
refrigerators), in the latter as materials with an energy
gap in the density of states for energy-dependent electron
tunneling (e.g., in NIS coolers).
Currently, the development of refrigerating techniques
follows two main directions: search of new materials with
p - type
e +
T cold
e -
Heat sink Heat sink
I
n - type
FIG. 23 (Color in online edition) Basic Peltier thermoelement.
32
improved characteristics suitable for existing refrigeration
schemes, and development of new refrigeration methods
and principles.
B. Peltier refrigerators
Thermoelectric (Peltier) refrigeration is widely used for
cooling different electronic devices (Nolas et al., 2001;
Phelan et al., 2002; Rowe and Bhandari, 1983). Nowadays
Peltier refrigerators providing temperature reduction
down to 100...200 K and cooling power up to 100
W are available. Peltier cooling (or heating) occurs
when an electric current is driven through the junction of
two different materials. The heat released or absorbed,
˙QP eltier, depending on the direction of the electric current
at the junction, is proportional to the electric current
(I) driven through the circuit, ˙ QP eltier = ΠABI,
where ΠAB = αABT , and ΠAB and αAB are the Peltier
and Seebeck coefficients of the contact, respectively (see
also Eq. (39)). In order to obtain enhanced values of
the Peltier coefficient, conventional Peltier refrigerators
generally consist of p- and n-type semiconductors with
opposite sign of Π coefficients (see Fig. 23). The efficiency
of a Peltier refrigerator is not determined only
by the coefficient ΠAB = ΠA − ΠB, but also by thermal
conductivities (κ) of both materials across the contact.
Furthermore their electric resistances (ρ) are responsible
of Joule heating affecting the coefficient of performance.
The maximum temperature difference (∆Tmax) achievable
with a Peltier refrigerator is given by (Nolas et al.,
2001) ∆Tmax = ZT 2 α2AB
cold /2, where Z = ρκ is a figure of
merit of the refrigerator, and Tcold is the temperature of
the cold junction (see Fig. 23). More often, however,
the refrigerator efficiency is characterized by the dimensionless
figure of merit ZT . We recall that ZT ∝ ( kBT
EF )2 .
Most of the materials used in thermoelectric applications
have ZT ∼ 1 at room temperature (DiSalvo, 1999; Min
and Rowe, 2000). In general, at low temperatures only
one single thermoelectric material is needed, because a
superconductor can be used as one of the two arms of the
refrigerator (Goldsmid et al., 1988; Nolas et al., 2001).
In spite of the drastic reduction of cooling efficiency
at low temperature, there is, however, some development
of new materials and devices suitable for operation at