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

two superconducting contacts and δG is a positive temperature
and voltage-dependent correction to the conductance.
Its magnitude for typical geometries is at maximum
some 0.1GN. The proximity-induced conductance
correction is widely studied in the literature and we refer
to (Belzig et al., 1999; Lambert and Raimondi, 1998) for
a more detailed list of references on this topic.
The thermal conductance of Andreev interferometers
has been studied very recently. The formulation of the
problem is very similar as for the conductance correction.
For E ≪ ∆, there is no energy current into the superconductors.
Therefore, it is enough to solve for the energy
current jL = DL∂xf L in the wires 1, 2 and 5. This yields
˙Q = GN /e 2
dEEDL(E)(f L eq(E; T2) − f L eq(E; T1)),
where f L eq(E; T ) = tanh[E/(2kBT )] and the thermal
conductance correction is obtained from DL(E) =
( L dx
0 DL(x;E) /L)−1 . Here the spatial integral runs along
the wire between the two normal-metal reservoirs. In
the general case, DL has to be calculated numerically.
The thermal conductance correction has been analyzed
by Bezuglyi and Vinokur (2003) and Jiang and Chandrasekhar
(2005b). They found that it can be strongly
modulated with the phase φ: in the short-junction limit
where ET ≫ ∆, for φ = 0, DL almost vanishes, whereas
for φ = π, DL approaches unity and thus the thermal
conductance approaches its normal-state value. For a
long junction with ET ∆, the effect becomes smaller,
but still clearly observable. The first measurements
(Jiang and Chandrasekhar, 2005a) of the proximityinduced
correction to the thermal conductance show the
predicted tendency of the phase-dependent decrease of κ
compared to the normal-state (Wiedemann-Franz) value.
Prior to the experiments on heat conductance in proximity
structures, the thermoelectric power α was experimentally
studied in Andreev interferometers (Dikin
et al., 2002a,b; Eom et al., 1998; Jiang and Chandrasekhar,
2004; Parsons et al., 2003a,b). The observed
thermopower was surprisingly large, of the order of 100
neV/K — at least one to two orders of magnitude larger
than the thermopower in normal-metal samples. Also
contrary to the Mott relation (c.f., below Eqs. (39)),
this value depends nonmonotonically on the temperature
(Eom et al., 1998) at the temperatures of the order of a
few hundred mK, and even a sign change could be found
(Parsons et al., 2003b). Moreover, the thermopower was
found to oscillate as a function of the phase φ. The symmetry
of the oscillations has in most cases been found to
be antisymmetric with respect to φ = 0, i.e., α vanishes
for φ = 0. However, in some measurements, the Chandrasekhar’s
group (Eom et al., 1998; Jiang and Chandrasekhar,
2004) have found symmetric oscillations, i.e.,
α had the same phase as the conductance.
The observed behavior of the thermopower is not completely
understood, but the major features can be explained.
The first theoretical predictions were given by
α SN,2 ( µV/K)
4
3
2
1
T0 , V= 0 T0 , V= 0
φ S S φ
3
5
4
2
2
1 2
N N
T1, V1
T2, V2 0
0 5 10
k T / E
B 2 T
15
14
FIG. 7 Supercurrent-induced N-S thermopower αNS for the
system depicted in the inset. Solid line: αNS at T1 ≈ T2. Dotted
line: approximation (47). The correction (48) accounts
for most of the difference. Dashed line: αNS at kBT1 = 3.6ET
with varying T2. Dash-dotted line: the corresponding approximation.
Inset: Andreev interferometer where thermoelectric
effects are studied. Two superconducting reservoirs with
phase difference φ are connected to two normal-metal reservoirs
through diffusive normal-metal wires. Adapted from
(Virtanen and Heikkilä, 2004b).
Claughton and Lambert (1996), who constructed a scattering
theory to describe the effect of Andreev reflection
on the thermoelectric properties of proximity systems.
Based on their work, it was shown (Heikkilä et al., 2000)
that the presence of Andreev reflection can lead to a violation
of the Mott relation. This means that a finite
thermopower can arise even in the presence of electronhole
symmetry. Seviour and Volkov (2000b), Kogan et al.
(2002), and Virtanen and Heikkilä (2004a,b) showed that
in Andreev interferometers carrying a supercurrent, a
large voltage can be induced by the temperature gradient
both between the normal-metal reservoirs and between
the normal metals and the superconducting contacts.
Virtanen and Heikkilä (2004b) showed that in long
junctions, the induced voltages between the two normalmetal
reservoirs and the superconductors can be related
to the temperature dependent equilibrium supercurrent
IS(T ) flowing between the two superconductors via
V 0
1/2
= 1
2
R5(2R4/3 + R5)R3/4 [IS(T1) − IS(T2)]
. (47)
RNNNRSNS
Here RNNN = R1 + R2 + R5, RSNS = R3 + R4 + R5
and Rk are the resistances of the five wires defined in
the inset of Fig. 7. This is in most situations the dominant
term and it can be also phenomenologically argued
based on the temperature dependence of the supercurrent
and the conservation of total current (supercurrent
plus quasiparticle current) in the circuit (Virtanen and
Heikkilä, 2004b). A similar result can also be obtained
in the quasiequilibrium limit for the linear-response thermopower
(Virtanen and Heikkilä, 2004a). In addition to
this term, the main correction in the long-junction limit
comes from the anomalous coefficient T an ,
eV 1
1/2 = ∓R1/2 〈T
RNNN
an
1/2 〉 ∓ R3/4R5 〈T
RNNNRSNS
an
5 〉. (48)
Lk
0 dx ∞
0
Here 〈T an
1
an
k 〉 ≡ dET Lk
k (E)[f L eq(E, T1) −
f L eq(E, T2)] and Lk is the length of wire k. One finds