Interference effect heat conductance in a Josephson junction and its detection in an rf SQUID
Glen D. Guttman, Eshel Ben-Jacob, and David J. Bergman
School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University,
Ramat-Aviv 69978, Tel-Aviv, Israel
Received 4 August 1997
The energy current through a superconductor-insulator-superconductor Josephson junction consists of a
quasiparticle current, an interference current, and a pair current. The quasiparticle part represents the normal
dissipative heat current. This part is shown to have a unique temperature dependence. The other two parts
depend on the phase drop across the junction . When the junction is biased by a fixed temperature drop, the
interference current can flow in either direction, depending on the sign of cos. This gives rise to an effect in
which the total heat current oscillates with the phase drop across the junction. We suggest an experimental
setup involving an rf superconducting quantum interference device, which is designed to measure these effects.
S0163-18299803506-1
The total electrical current through a Josephson junction
is usually described by two independent currents:
1,2
one is a
normal dissipative current which gives rise to thermoelectric
transport—it corresponds to the BCS quasiparticles; the
other is an equilibrium supercurrent known as the Josephson
current. However, as pointed out by Josephson, there exists a
third electrical current that flows through a Josephson junc-
tion. This is referred to as the interference current, which is
understood as a superposition of tunneling of normal elec-
trons and of tunneling of superconducting pairs.
3
In previous publications
4,5
we studied thermoelectric and
thermal transport in superconductor-insulator-supercon-
ductor SIS Josephson junctions. Usually, thermoelectric ef-
fects are attributed to quasiparticle transport. However, we
found that the interplay or interference between quasiparti-
cles and pairs in the superconducting electrodes comprising
the junction gives rise to new thermal and thermoelectric
transport phenomena in these systems. In particular, in Ref. 5
we found that the heat current through the junction can be
regulated by controlling the superconducting phase differ-
ence across the junction. This effect ensues from an anoma-
lous energy transfer term that is analogous to the electrical
interference current. In this paper we propose an experiment
designed to detect this phase-dependent heat current. We
also demonstrate the anomalous temperature dependence of
the heat current.
The analytical calculation of the energy current in a Jo-
sephson junction, presented in Ref. 5, was based on micro-
scopic theory. In order to calculate the total energy transfer
across the SIS junction we employed perturbation theory,
similar to the derivation of the electrical current presented by
Ambegaokar.
6
The total energy current, flowing from left to
right, will be denoted by Q
tot
l
. The result can be written as a
sum of three parts:
Q
tot
l
=Q
qp
l
+Q
qp-pair
l
cos +Q
pair
l
sin , 1
where is the superconducting phase difference across the
junction. The full expressions are given in Ref. 5.
The form of Eq. 1 is analogous to the expression for the
total electric current in a Josephson junction.
3
The first term
is just the normal heat current which is carried by the quasi-
particles. This is the current that is derived by employing the
golden rule.
7
The other two terms in Eq. 1 are related to the
occurrence of pair tunneling in the junction, and thus depend
on the phase drop across the system. The last term on the
right-hand side RHS of Eq. 1 is analogous to the Joseph-
son current, whereas the middle term resembles the interfer-
ence current and will be referred to as the interference energy
current. The pair related energy currents produce no dissipa-
tion in the system. This issue is explained in detail in Ref. 4.
In this paper we focus on the case where the junction is
biased by a temperature drop across the junction T , but the
voltage across it vanishes. We are interested in the behavior
of the heat conductance of a Josephson junction. Note that
we do not include the contribution of lattice vibrations to the
heat conductance. It turns out that there is only a contribu-
tion from Q
qp
and Q
qp-pair
henceforth we omit the super-
script l in the notation of the heat current. According to Ref.
5 the total heat current is then
Q
tot
=Q
qp
+Q
qp-pair
cos
=
8 N
l
N
r
| T
lr
|
2
T
T
T
dw
-
df
dw
w
2
w
2
+
2
cos
w
2
-
2
T + T w
2
-
2
T
, 2
where f ( w ) is the Fermi distribution function of the quasi-
particles. The density of states N
l
, N
r
and the tunneling
matrix element | T
lr
|
2
were taken at the Fermi energy.
The first term in the square brackets on the RHS of Eq.
2 leads to the quasiparticle contribution to the heat current
in the superconducting state Q
qp
. The temperature depen-
dence of Q
qp
is illustrated in Fig. 1. In order to calculate this
contribution we approximated 4 k
B
T
c
1 -T / T
c
, where
k
B
is the Boltzmann coefficient, and where T
c
is the super-
conducting phase transition temperature. This is a good ap-
proximation even at low temperatures due to the exponential
dependence on temperature of the integrands in Eq. 2. For
comparison, we also plotted in Fig. 1 the heat current for the
PHYSICAL REVIEW B 1 FEBRUARY 1998-I VOLUME 57, NUMBER 5
57 0163-1829/98/575/27173/$15.00 2717 © 1998 The American Physical Society