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

optimal bias voltage (Vopt), Vopt ≈ (∆ − 0.66kBTe,N )/e,
as well as for the maximum cooling power at Vopt,
˙Qopt ≈ ∆2
e2 kBTe,N
[0.59( RT ∆
)3/2
2πkBTe,S
− ∆ exp(− ∆
kBTe,S )].
(78)
Equation (78) is useful for getting quantitative estimates
on the performance of realistic coolers. In the
same temperature limit and for V = Vopt the cur-
rent through the NIS junction can be approximated as
I ≈ 0.48 ∆
eRT
kBTe,N
∆ . The NIS junction coefficient of
performance η is given by η(V ) = ˙ Q(V )/[I(V )V ]. For
V ≈ ∆/e and in the low temperature limit, η thus obtains
the approximate value
ηopt ≈ 0.7 Te,N
, (79)
where we assumed ∆ = 1.764 kBTc, and Tc is the critical
temperature of the superconductor. Equation (79) shows
that the efficiency of a NIS junction is around or below
20% at the typical operation temperatures. The full behavior
of η versus temperature calculated at each optimal
bias voltage is displayed in Fig. 26(c). The simple results
presented above point out how the optimized operation of
a superconducting tunnel junction as a building block of
microrefrigerators stems from a delicate balance among
several factors such as the contact resistance, the operation
temperature, the superconducting gap ∆ as well as
the bias voltage across the junction.
The first observation of heat extraction from a normal
metal dates back to 1994 (Nahum et al., 1994), where
cooling of conduction electrons in Cu below the lattice
temperature was demonstrated using an Al/Al2O3/Cu
tunnel junction. A significant improvement was made
two years later, still in the Al/Al2O3/Cu system, by
Leivo et al. (1996), which recognized that using two NIS
junctions in series and arranged in a symmetric configuration
(i.e., in a SINIS fashion) leads to a much stronger
cooling effect. This fact can be understood keeping in
mind that ˙ Q is a symmetric function of V so that, at
fixed voltage across the structure, quasi-electrons above
∆ are extracted from the N region through one junction,
while at the same time quasi-holes are filled in N below
−∆ from the other junction (see Fig. 26(d)). In this configuration,
a reduction of the electron temperature from
300 mK to about 100 mK was obtained. Later on, several
other experimental evidences of electron cooling in SI-
NIS metallic structures were reported (Arutyunov et al.,
2000; Clark et al., 2004; Fisher et al., 1995, 1999; Leivo
et al., 1997; Leoni et al., 1999, 2003; Luukanen et al.,
2000; Pekola et al., 2000a, 2004a, 2000b; Tarasov et al.,
2003; Vystavkin et al., 1999). In these experiments NIS
junctions are used to alter the electron temperature in the
N region as well as to measure it. In order to measure
the temperature, the N region is normally connected to
additional NIS contacts (i.e., ”probe” junctions) acting
as thermometers (previously calibrated by varying the
Tc
34
bath temperature of the cryostat), and operating along
the lines described in Sec. III.A.1. Moreover, the differential
conductance of the probe junctions gives also
detailed information about the actual quasiparticle distribution
function in the N region (Pekola et al., 2004a;
Pothier et al., 1997b).
Figure 27(a) shows the SEM micrograph of a typical
Al/Al2O3/Cu SINIS refrigerator including the superconducting
probe junctions. The schematic of a commonly
used experimental setup for electron refrigeration
and temperature measurement is shown in Fig. 27(b).
The voltage bias Vrefr across the SINIS structure allows
to change the electron temperature in the N region;
at the same time, a measure of the voltage drop
across the two probe junctions (Vth) at a constant bias
current (I0) yields the electron temperature Te,N in the
normal electrode (Rowell and Tsui, 1976). Figure 27(c)
illustrates the experimental data of Leivo et al. (1996)
of the measured electron temperature T ≡ Te,N versus
Vrefr, taken at different bath temperatures (i.e., those
at Vrefr = 0). As can be readily seen, the electron
temperature rapidly decreases by increasing the voltage
bias across the SINIS structure, reaching the lowest value
e 2 R T /Δ 2 (0)
Q&Q&
-0.05
η (%)
0.10
0.05
0.00
15
10
5
(a)
k B T/Δ(0)
0
0.0 0.1 0.2 0.3 0.4 0.5
kBT/Δ(0) 0.05
0.1
0.2
0.25
0.4
0.5
e 2 R T /Δ 2 (0)
Q&Q &
T/Tc 0.0
0.08
0.5 1.0
0.06
0.04
Δ(T) = Δ BCS (T)
Δ(T) = Δ(0)
NIN
0.02
-0.10
0.0 0.5 1.0 1.5
eV/Δ(0)
(b)
0.00
T/T
0.0 0.3 0.6
c
0.0 0.2 0.4 0.6 0.8
kBT/Δ(0) 30
Electric
(c)
25
current
20
Q •
(d)
FIG. 26 (Color in online edition) (a) Calculated cooling power
˙Q of a NIS junction vs bias voltage V for several temperatures
T = Te,N = Te,S. Also shown is the behavior of a
NIN junction. (b) ˙ Q calculated at the optimal bias voltage
as a function of temperature, assuming both a temperatureindependent
energy gap (dash-dotted blue line) and the real
BCS dependence (black line). Topt indicates the temperature
value that maximizes ˙ Q. (c) Coefficient of performance η calculated
at the optimal bias voltage versus temperature. (d)
Scheme of the energy band diagram of a voltage biased SINIS
junction. The electric current flows into the normal region
through one tunnel junction and out through the other, while
the heat current ˙ Q flows out of the N electrode through both
tunnel junctions.
T opt