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

that for a ”cross” system without the central wire (i.e.,
R5 = 0), this term dominates V 0
1/2 . Further corrections
to the result (47) are discussed by Kogan et al. (2002)
and Virtanen and Heikkilä (2004a).
The above theoretical results explain the observed
magnitude and temperature dependence of the thermopower
and also predict an induced voltage oscillating
with the phase φ. However, the thermopower calculated
by Seviour and Volkov (2000b), Kogan et al. (2002) and
Virtanen and Heikkilä (2004a,b) is always an antisymmetric
function of φ, and it vanishes for a vanishing supercurrent
in the junction (including all the correction
terms). Therefore, the symmetric oscillations of α cannot
be explained with this theory.
The presence of the supercurrent breaks the timereversal
symmetry and hence the Onsager relation Π =
T α (see Eqs. (39) and below) need not to be valid for
φ = 0. Heikkilä, et al. predicted a nonequilibrium
Peltier-type effect (Heikkilä et al., 2003) where the supercurrent
controls the local effective temperature in an
out-of-equilibrium normal-metal wire. However, it seems
that the induced changes are always smaller than those
due to Joule heating and thus no real cooling can be realized
with this setup.
Recently, the thermoelectric effects in coherent SNS
Josephson point contacts have been analyzed in (Zhao
et al., 2003, 2004). In such point contacts, the heat transport
is strongly influenced by the Andreev bound states
forming between the two superconductors.
G. Heat transport by phonons
When the electrons are thermalized by the phonons,
they may also heat or cool the phonon system in the film.
Therefore, it is important to know how these phonons
further thermalize with the substrate, and ultimately
with the heat bath on the sample holder that is typically
cooled via external means (typically by either a dilution
or a magnetic refrigerator). Albeit slow electron-phonon
relaxation often poses the dominating thermal resistance
in mesoscopic structures at low temperatures, the poor
phonon thermal conduction itself can also prevent full
thermal equilibration throughout the whole lattice. This
is particularly the case when insulating geometric constrictions
and thin films separate the electronic structure
from the bulky phonon reservoir (see Fig. 1). In the
present section we concentrate on the thermal transport
in the part of the chain of Fig. 1 beyond the sub-systems
determined by electronic properties of the structure.
The bulky three-dimensional bodies cease to conduct
heat at low temperatures according to the well appreciated
κ ∝ T 3 law in crystalline solids arising from Debye
heat capacity via
κ = CvSℓel,ph/3, (49)
where κ is the thermal conductivity and C is the heat capacity
per unit volume, vS is the speed of sound and ℓel,ph
15
is the mean free path of phonons in the solid (Ashcroft
and Mermin, 1976). Thermal conductivity of glasses follows
the universal ∝ T 2 law, as was discovered by Zeller
and Pohl (1971), which dependence is approximately followed
by non-crystalline materials in general (Pobell,
1996). These laws are to be contrasted to ∝ T thermal
conductivities of pure normal metals (Wiedemann-Franz
law, c.f., below Eq. (39)). Despite the rapid weakening
of thermal conductivity toward low temperatures, the dielectric
materials in crystalline bulk are relatively good
thermal conductors. One important observation here is
that the absolute value of the bulk thermal conductivity
in clean crystalline insulators at low temperatures does
not provide the full basis of thermal analysis without a
proper knowledge of the geometry of the structure, because
the mean free paths often exceed the dimensions
of the structures. For example, in pure silicon crystals,
measured at sub-kelvin temperatures (Klitsner and Pohl,
1987), thermal conductivity κ 10 Wm −1 K −4 T 3 , heat
capacity C 0.6 JK −4 m −3 T 3 and velocity vS 5700
m/s imply by Eq. (49) a mean free path of 10 mm,
which is more than an order of magnitude longer than the
thickness of a typical silicon wafer. Therefore phonons
tend to propagate ballistically in silicon substrates.
What makes things even more interesting, but at the
same time more complex, e.g., in terms of practical
thermal design, is that at sub-kelvin temperatures the
dominant thermal wavelength of the phonons, λph
hvS/kBT , is of the order of 0.1 µm, and it can exceed
1 µm at the low temperature end of a typical experiment
(see discussion in Subs. II.C.2). A direct consequence of
this fact is that the phonon systems in mesoscopic samples
cannot typically be treated as three-dimensional, but
the sub-wavelength dimensions determine the actual dimensionality
of the phonon gas. Metallic thin films and
narrow thin film wires, but also thin dielectric films and
wires are to be treated with constraints due to the confinement
of phonons in reduced dimensions.
The issue of how thermal conductivity and heat capacity
of thin membranes and wires get modified due to
geometrical constraints on the scale of the thermal wavelength
of phonons has been addressed by several authors
experimentally (Holmes et al., 1998; Leivo and Pekola,
1998; Woodcraft et al., 2000) and theoretically (see, e.g.,
(Anghel et al., 1998; Kuhn et al., 2004)). The main conclusion
is that structures with one or two dimensions
d < λph restrict the propagation of (ballistic) phonons
into the remaining ”large” dimensions, and thereby reduce
the magnitude of the corresponding quantities κ and
C at typical sub-kelvin temperatures, but at the same
time the temperature dependences get weaker.
In the limit of narrow short wires at low temperatures
the phonon thermal conductance gets quantized, as was
experimentally demonstrated by Schwab et al. (2000).
This limit had been theoretically addressed by Angelescu
et al. (1998), Rego and Kirczenow (1998) and Blencowe
(1999), somewhat in analogy to the well-known Landauer
result on electrical conduction through quasi-one-