Photon-assisted parity change and Andreev tunneling
Ulrik Hanke and Magnus Gisselfa ¨lt
Department of Physics, Norwegian Institute of Technology, The University of Trondheim, N 7034 Trondheim, Norway
and Department of Applied Physics, Chalmers University of Technology and Go ¨teborg University, S-412 96 Go ¨teborg, Sweden
K. A. Chao
Department of Physics, Norwegian Institute of Technology, The University of Trondheim, N 7034 Trondheim, Norway
Received 24 January 1996
A microscopic theory has been constructed to investigate the tunneling current in a normal-superconductor-
normal single-electron tunneling transistor with an oscillating potential coupled to the grain. The oscillating
potential produces a photon-assisted Andreev tunneling and causes a photon-assisted parity change of the
grain. S0163-18299611827-0
When an electron tunnels through a potential barrier in
the presence of an oscillating potential V
˜
cos(t), it may emit
or absorb n photons with energy n . When k
B
T ,
such processes can be detected experimentally as steps in the
current-voltage ( I -V ) characteristics or as peaks in the
conductance-voltage curve. The pioneering theory of Tien
and Gordon
1
explains qualitatively the photon-assisted PA
electron tunneling observed in superconducting diodes.
2
Fol-
lowing the recent advancement of technology to fabricate
samples of nanometer size, PA tunneling in semiconducting
or metallic nanostructures of different geometries has been
studied both theoretically and experimentally.
3–12
In a quan-
tum dot where single-electron tunneling events are correlated
due to Coulomb blockade, the observed PA tunneling
process
13,14
has been explained with a combination of the
Tien-Gordon theory
1
and the orthodox theory
15
of single-
electron tunneling SET.
The normal-superconductor-normal NSN SET transis-
tor, the equivalent circuit of which is shown in Fig. 1, has a
superconducting grain connected to two normal-metal leads
through two tunnel junctions, which are characterized by the
capacitance and tunneling conductance ( C
s
, G
s
) and
( C
d
, G
d
), respectively. The Coulomb blockade on the grain
can be controlled by the gate voltage V
g
and the gate capaci-
tance C
g
. The transport properties of the NSN SET are very
sensitive to whether the number of excess electrons on the
grain is even or odd. In the absence of an ac potential, when
the gate charge Q C
g
V
g
/e is an odd integer, at low dc bias
voltage the tunneling current is due to the Andreev process
where, effectively, two electrons tunnel coherently through
the barrier between the superconducting grain and a normal-
metal lead. Hence, the number of excess electrons on the
grain is even. As the dc bias increases to the threshold value
V
th
for quasiparticle tunneling, the parity changes from even
to odd, and the Andreev current is drastically suppressed.
The I -V characteristics of an NSN SET transistor under a dc
bias have been studied both experimentally
16,17
and
theoretically.
18
When an additional oscillating potential
V
˜
cos(t) is applied, besides the possible enhancement of
Andreev current, the most interesting phenomenon is that
under proper conditions, one expects the PA quasiparticle
tunneling to occur before the dc bias reaches the threshold
value V
th
. It then results in a PA parity change of the I -V
characteristics of an NSN SET transistor.
The purpose of this paper is to investigate these PA pro-
cesses, which requires the calculation of not only the second-
order PA quasiparticle tunneling, but also the fourth-order
PA Andreev tunneling. To our knowledge, the present paper
is the first attempt to study such higher-order effect. Other
higher-order elastic and inelastic cotunneling
19
will be ne-
glected because their contributions to the current are unim-
portant. In general, the total current consists of both tunnel-
ing current and displacement current. Since we will consider
only the time-averaged current, the contribution of displace-
ment current is zero.
10
We will show that under the high-
frequency condition , where is the supercon-
ducting gap and the tunneling rate, the PA parity change
can be observed.
The system we consider is illustrated in Fig. 1, where
each voltage consists of a dc term and an ac term: V
l
( t ) =
V
l
0
+V
˜
l
cos t with l =s , d , and g . The Hamiltonians H
s
and
H
d
for the normal leads can be expressed as
H
l
=
p
l , p
+eV
l
t a
l , p
†
a
l , p
; l =s , d , 1
where the single-electron energy
l , p
is measured relative to
the chemical potential. The Hamiltonian for the supercon-
ducting grain,
H
g
=
q
q
q
†
q
+E
n
t , 2
FIG. 1. Equivalent circuit of a SET transistor.
PHYSICAL REVIEW B 15 JULY 1996-I VOLUME 54, NUMBER 3
54 0163-1829/96/543/15294/$10.00 1529 © 1996 The American Physical Society