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

E F
eV
(a)
e
E
Δ
Δ
N I S
E F
forbidden
levels
Q&
A A(nW/μm2
)
(b)
20
15
10
5
0.15
0.2
k BT/Δ (0) = 0.3
0
0 3 6 9
T (10-2 )
FIG. 4 (Color in online edition) (a) Sketch of the energy band
diagram of a voltage biased NIS junction. Upon biasing the
structure, the most energetic electrons (e) can most easily
tunnel into the superconductor. As a result the electron gas
in the N electrode is cooled. (b) Maximum cooling power
surface density ˙ QA vs interface transmissivity T at different
temperatures calculated for a NS contact.
can hence be typically ignored. Therefore, the Peltier refrigerators
discussed in Subs. V.B rely on materials with
a low EF .
2. Superconducting tunnel structures
Consider a NIS tunnel structure, coupling a large superconducting
reservoir with temperature Te,S to a large
normal-metal reservoir with temperature Te,N via a tunnel
junction with resistance RT . Let us then assume
a voltage V applied over the system. In this case, the
heat current (cooling power) from the normal metal is
given by Eq. (30) with NL(E) = 1, NR(E) = NS(E),
fL(E) ≡ feq(E − eV, Te,N ) and fR(E) ≡ feq(E, Te,S).
For small pair breaking inside the superconductor, i.e.,
Γ ≪ ∆, ˙ QNIS is positive for eV < ∆, i.e., it cools
the normal metal. It is straightforward to show that
˙QNIS(V ) = ˙ QNIS(−V ). This is in contrast with Peltier
cooling, where the sign of the current determines the direction
of the heat current. For eV > ∆, the current
through the junction increases strongly, resulting in Joule
heating and making ˙ QNIS negative. The cooling power is
maximal near eV ≈ ∆.
In order to understand the basic mechanism for cooling
in such systems, let us consider the simplified energy
band diagram of a NIS tunnel junction biased at voltage
V , as depicted in Fig. 4(a). The physical mechanism
underlying quasiparticle cooling is rather simple: owing
to the presence of the superconductor, in the tunneling
process quasiparticles with energy E < ∆ cannot tunnel
inside the forbidden energy gap, but the more energetic
electrons (with E > ∆) are removed from the N electrode.
As a consequence of this ”selective” tunneling of
hot particles, the electron distribution function in the N
region becomes sharper: the NIS junction thus behaves
as an electron cooler.
The role of barrier transmissivity in governing heat
flux across the NIS structure was analyzed by Bardas
hQ /(2 ∆ )
2
.
(a)
0.1
0.08
0.06
0.04
0.02
0
- 0.02
- 0.04
- 0.06
- 0.08
- 0.1
0 0.5 1 1.5
eV/∆
(b)
2.5
hI /(2e ∆ )
2
1.5
1
0.5
11
0
0 0.5
eV/∆
1 1.5
FIG. 5 (Color in online edition): (a) NS point contact heat
current from the N side as a function of voltage for different
transparencies T (from top to bottom, tunneling limit T → 0,
T = 0.005, T = 0.01, T = 0.05, T = 0.5 and T = 1) at
kBT = 0.3∆. For T 0.05, the heat current is positive, corresponding
to cooling. (b) NS point contact current-voltage
characteristics for the same values of T and T as in (a) (now
T increases from bottom to top). The first four curves lie essentially
on top of each other. The corresponding NIN curves
are shown with the dashed lines.
and Averin (1995). They pointed out the interplay between
single-particle tunneling and Andreev reflection
(Andreev, 1964a) on the heat current. In the following it
is useful to summarize their main results.
The cooling regime requires a tunnel contact. The effect
of transmissivity is illustrated in Fig. 4(b), which
shows the maximum of the heat current density (i.e., the
specific cooling power) ˙ QA versus interface transmissivity
T at different temperatures. This can be calculated
for a generic NS junction using Eqs. (9), (10a) and (31b).
The quantity ˙ QA is a non-monotonic function of interface
transmissivity, vanishing both at low and high values of
T . In the low transparency regime, ˙ QA turns out to
be linear in T , showing that electron transport is dominated
by single particle tunneling. Upon increasing barrier
transmissivity, Andreev reflection begins to dominate
quasiparticle transport, thus suppressing heat current extraction
from the N portion of the structure. The heat
current ˙ QA is maximized between these two regimes at
an optimal barrier transmissivity, which is temperature
dependent. Furthermore, by decreasing the latter leads
to a reduction of both the optimal T and of the transmissivity
window where the cooling takes place. In real
NIS contacts used for cooling applications, the average T
is typically in the range 10 −6 ...10 −4 (Leivo et al., 1996;
Nahum et al., 1994) corresponding to junction specific
resistances Rc (i.e., the product of the junction normal
state resistance and the contact area) from tens to several
thousands Ω µm 2 . This limits the achievable ˙ QA
to some pW/µm 2 . From the above discussion it appears
that exploiting low-Rc tunnel contacts is an important requirement
in order to achieve large cooling power through
NIS junctions. However, in real low-Rc barriers, pinholes
with a large T appear. They contribute with a large Joule
heating (see Fig. 5, which shows the heat and charge currents
through the NIS interface as functions of voltage for
different T ), and therefore tend to degrade the cooling