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

f N (E) (a)
1
0.8
0.6
0.4
0.2
0
−0.6 −0.4 −0.2 0
E/∆
0.2 0.4 0.6
f N (E)
1
0.8
0.6
0.4
0.2
K e−e =
0
1
10
100
0
−0.6 −0.4 −0.2 0
E/∆
0.2 0.4 0.6
FIG. 6 (Color in online edition): (a) Nonequilibrium distribution
inside the SINIS island, calculated from Eq. (42)
with Icoll = 0, T = 0.1Tc, and using a depairing strength
Γ = 10 −4 ∆ inside the superconductors. The voltage runs
from zero to V = 3∆/e in steps of ∆/(2e) and spans three
regimes: (i) anomalous heating regime (eV = 0 . . . ∆, solid
lines, widening with an increasing V ) discussed in (Pekola
et al., 2004a), (ii) cooling regime (eV = 1.5∆ and eV = 2∆,
dashed lines, narrowing with an increasing V ) and the strong
nonequilibrium heating regime (eV = 2.5∆ and eV = 3∆,
dotted lines, widening with an increasing V ). (b) Distribution
at eV = 2.5∆ for different strengths of the electron–
electron interaction, parametrized by the parameter Ke−e
from Eq. (16) with RD replaced by RT and E ∗ by ∆, as
in this case these describe the collision integral.
performance at the lowest temperatures, and to overheat
the superconductor at the junction region due to strong
power injection. Barrier optimization in terms of both
materials and technology seems to be still nowadays a
challenging task (see also Sec. VI.F.1).
In a SINIS system in the quasiequilibrium limit, the
temperatures Te,N and Te,S of the N and S parts may
differ. If the superconductors are good reservoirs, Te,S
equals the lattice temperature. In the experimentally
relevant case when the resistance RD of the normal-metal
island is much lower than the resistances RL and RR of
the tunnel junctions, the normal-metal temperature Te,N
is determined from the heat balance equation
(b)
˙QSINIS(V ; Te,S, Te,N ) = Pcoll, (40)
where ˙ QSINIS = 2 ˙ QNIS and Pcoll describes inelastic scattering
due to phonons (Eq. (24)) and/or due to the electromagnetic
environment (Eq. (25)).
For further details about NIS/SINIS cooling in the
quasiequilibrium limit, we refer to Sec. V.C.1 and
(Anghel and Pekola, 2001).
Let us now consider the limit of full nonequilibrium,
neglecting the proximity effect from the superconductors
on the normal-metal island. 2 Then the distribution function
inside the normal metal may be obtained by solving
the kinetic equation Eq. (3) in the static case along with
the boundary conditions given by Eq. (8). For simplicity,
2 This is justified in the limit RL, RR ≫ RD.
12
let us assume RL = RR ≡ RT ≫ RD. In this limit, the
distribution function fN (E) in the normal-metal island
is almost independent of the spatial coordinate. Then,
in Eq. (3) we can use ∂ 2 xf ≈ (∂xf(L) − ∂xf(0))/L, where
L is the length of the N wire. From Eq. (11) we then get
RD
RT
(NR(fN − fR) − NL(fL − fN)) = τDIcoll, (41)
where τD = L2 N /D is the diffusion time through the island.
As a result (Giazotto et al., 2004b; Heslinga and
Klapwijk, 1993), the distribution function in the central
wire can be expressed as
fN = fRNR + fLNL + RIVe 2νF Icoll
. (42)
NR + NL
Here V = LA is the volume of the island. In the presence
of inelastic scattering, this is an implicit equation, as Icoll
is a functional of fN . Examples of the effect of inelastic
scattering have been considered in (Heslinga and Klapwijk,
1993) in the relaxation time approach, and in (Giazotto
et al., 2004b) including the full electron–electron
scattering collision integral. The distribution function
fN in Eq. (42) is plotted in Fig. 6 for a few values of the
voltage for Icoll = 0 and for a few strengths of electron–
electron scattering at eV = 2.5∆.
3. Superconductor-normal-metal structures with transparent
contacts
If a superconductor is placed in a good electric contact
with a normal metal, a finite pairing amplitude is induced
in the normal metal within a thermal coherence
length ξT = D/(2πkBT ) near the interface. This
superconducting proximity effect modifies the properties
of the normal metal (Belzig et al., 1999; Lambert and
Raimondi, 1998) and makes it possible to transport supercurrent
through it. It also changes the local distribution
functions by modifying the kinetic coefficients in
Eq. (5). Superconducting proximity effect is generated
by Andreev reflection (Andreev, 1964b) at the normalsuperconducting
interface, where an electron scatters
from the interface as a hole and vice versa. Andreev
reflection forbids the sub-gap energy transport into the
superconductor, and thus modifies the boundary conditions
for the distribution functions. In certain cases, the
proximity effect is not relevant for the physical observables
whereas the Andreev reflection can still contribute.
This incoherent regime is realized if one is interested in
length scales much longer than the extent of the proximity
effect. Below, we first study such ”pure” Andreevreflection
effects, and then go on to explain how they are
modified in the presence of the proximity effect.
In the incoherent regime, for E < ∆, Andreev reflection
can be described through the boundary conditions
(Pierre et al., 2001)
f(µS + E) = 1 − f(µS − E) (43a)
ˆn · ∇[f(µS − E) − f(µS + E)] = 0, (43b)