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

tacts may be more preferable than thermometers applying
tunnel contacts. In an SNS system, one may again
employ either the supercurrent or quasiparticle current as
the thermometer. For a given phase φ between the two
superconductors, the former can be expressed through
(Belzig et al., 1999)
IS(φ) = 1
eRN
∞
0
dEjS(E; φ)(1 − 2f(E)), (56)
where RN is the normal-state resistance of the weak link,
jS(E; φ) is the spectral supercurrent (Heikkilä et al.,
2002), and the energies are measured from the chemical
potential of the superconductors. Hence, the supercurrent
has the form of Eq. (54). In practice, one does
not necessarily measure IS(φ), but the critical current
IC = maxφ IS(φ). In diffusive junctions, this is obtained
typically for φ near π/2, although the maximum point
depends slightly on temperature.
The problem in SNS thermometry is in the fact that
the supercurrent spectrum jS(ε) depends on the quality
of the interface, on the specific geometry of the system,
and most importantly, on the distance L between
the two superconductors compared to the superconducting
coherence length ξ0 = D/(2∆) (Heikkilä et al.,
2002). Therefore, IC(T ) dependence is not universal.
However, the size of the junction can be tuned to meet
the specific temperature range of interest. In the limit of
short junctions, L ξ0 (Kulik and Omel’yanchuk, 1978),
the temperature scale for the critical current is given by
the superconducting energy gap ∆ and for kBT ≪ ∆,
the supercurrent depends very weakly on the temperature.
In a typical case ξ0 ∼ 100 . . . 200 nm, and thus
already a weak link with L of the order of 1 µm, easily
realisable by standard lithography techniques, lies in
the ”long” limit. There, the critical current is IC =
c(kBT ) 3/2 exp(− √
2πkBT/ET )/(eRN ET ) (Zaikin and
Zharkov, 1981). This equation is valid for kBT 5ET .
Here ET = D/L2 and the prefactor c depends on the
geometry (Heikkilä et al., 2002), for example for a twoprobe
configuration c = 64 √ 2π3 /(3 + 2 √ 2). The exponential
temperature dependence and the crossover between
the long- and short-junction limits were experimentally
investigated by Dubos et al. (2001) and the
above theoretical predictions were confirmed.
Using a four-probe configuration with SNS junctions of
different lengths, Jiang et al. (2003) exploited the strong
temperature dependence of the supercurrent to measure
the local temperature. The device worked in the regime
where part of the junctions were in the supercurrentcarrying
state and part of them in the dissipative state.
In the limit where supercurrent is completely suppressed,
there is still a weaker temperature dependence of the
conductance, due to the proximity-effect correction (c.f.,
Eq. (46)) (Charlat et al., 1996). This can also be used for
thermometry (Aumentado et al., 1999), but due to the
much smaller effect of temperature on the conductance,
it is less sensitive.
Besides being just a thermometer of the electron gas,
20
FIG. 12 A typical CBT sensor for the temperature range 20
mK - 1 K. The structure has been fabricated by electron beam
lithography, combined with aluminium and copper vacuum
evaporation. Both top view and a view at an oblique angle
are shown; the scale indicated refers to the top view. Figure
adapted from (Meschke et al., 2004).
critical current of a SNS Josephson junction has been
shown to probe the electron energy distribution (Baselmans
et al., 1999, 2001b; Giazotto et al., 2004b; Huang
et al., 2002) as indicated by Eq. (56).
B. Coulomb blockade thermometer, CBT
Single electron tunnelling (SET) effects were foreseen
in micro-lithographic structures in the middle of 1980’s
(Averin and Likharev, 1986). Such effects in granular
structures had been known to exist already much earlier
(Giaever and Zeller, 1968; Lambe and Jaklevic, 1969;
Neugebauer and Webb, 1962). The first lithographic SET
device was demonstrated by Fulton and Dolan (1987).
Since that time SET effects have formed a strong subfield
in mesoscopic physics. Typically SET devices operate in
the full Coulomb blockade regime, where temperature is
so low that its influence on electrical transport characteristics
can be neglected; at low bias voltages current
is blocked due to the charging energy of single electrons.
Coulomb blockade thermometer (CBT) operates in a different
regime, where single electron effects still play a role
but temperature predominantly influences the electrical
transport characteristics (Bergsten et al., 2001; Farhangfar
et al., 1997; Meschke et al., 2004; Pekola et al., 1994).
CBT is a primary thermometer, whose operation is based
on competition between thermal energy kBT at temperature
T , electrostatic energy eV at bias voltage V , and
charging energy due to extra or missing individual electrons
with unit of charging energy EC = e 2 /(2C ∗ ), where
C ∗ is the effective capacitance of the system. Figure 12
shows an SEM micrograph of a part of a typical CBT
sensor suitable for the temperature range 0.02. . . 1 K.
This sensor consists of four parallel one-dimensional arrays
with 40 junctions in each.
In the partial Coulomb blockade regime the I −V characteristics
of a CBT array with N junctions in series do
not display a sharp Coulomb blockade gap, but, instead,
they are smeared over a bias range eV ∼ NkBT . The
asymptotes of the I − V at large positive and negative
voltages have, however, the same offsets as at low T , de-