Flux flow and vortex tunneling in two-dimensional arrays of small Josephson junctions
C. D. Chen, P. Delsing, D. B. Haviland, Y. Harada, and T. Claeson
Department of Physics, Chalmers University of Technology and University of Go ¨teborg, 412 96 Go ¨teborg, Sweden
Received 12 October 1995
We have measured the temperature dependence and magnetic field dependence of the zero-bias resistance
( R
0
) as well as the current-voltage ( I -V ) characteristics for several two-dimensional arrays of small aluminum
Josephson junctions. R
0
( T ) decreases with decreasing temperature, which can be described in terms of two
types of vortex motion: flux, flow, and vortex tunneling. At temperatures higher than the Kosterlitz-Thouless
transition temperature ( T T
c
) or at a bias current greater than the current corresponding to the onset of the
nonlinear I -V characteristics ( I I
d
), the effective damping resistance which characterizes flux-flow motion is
found to be approximately equal to the junction normal-state resistance R
N
. At low temperatures and at small
bias current, R
0
is temperature independent and remains finite down to our minimum attainable temperature.
This finite resistance is found to be dependent on the array size as well as the junction parameters.
S0163-18299602630-6
I. INTRODUCTION
Vortex dynamics and superconducting and insulating
transitions in two-dimensional systems have attracted much
interest both experimentally
1–10
and theoretically.
6,11–17
In
Josephson-junction arrays, a superconductor-insulator transi-
tion can be achieved
2–4
by changing the junction normal-
state resistance R
N
and the junction capacitance C . The tran-
sition is found to occur near a critical point at R
N
R
Q
and
E
J
/ E
C
1, where R
Q
h /4e
2
=6.45 k is the quantum re-
sistance, E
C
e
2
/2C is the charging energy associated with
the junction capacitance C , and E
J
( /2)( R
Q
/ R
N
) is the
Josephson coupling energy with the superconducting en-
ergy gap. The charge soliton dynamics, which is comple-
mentary to the vortex dynamics, is important to the interpre-
tation of the insulating state and has been investigated
previously.
18
There we reported the existence of a finite re-
sistance induced by quantum fluctuation of charge at low
temperatures. In the present work, we study two-dimensional
2D arrays exhibiting superconducting behavior zero bias
resistance R
0
decreases upon cooling, emphasizing the im-
portant role played by quantum tunneling of the vortex.
In studying vortex motion, one has to compare three en-
ergies: the thermal energy k
B
T , the Josephson coupling en-
ergy E
J
( T ), and the charging energy E
C
. For E
J
E
C
and
temperatures k
B
T E
J
, thermal fluctuations of the phase
lead to finite sample resistance. As the temperature is low-
ered, E
J
( T ) grows and a superconducting state is possible,
depending on the strength of the dissipation. If E
C
is not
negligible compared to E
J
, the vortices behave as quantum
particles
19
with a mass inversely proportional to E
C
, and one
can expect strong quantum-mechanical effects which may
destroy the global superconducting state, leading to a finite
sample resistance. A systematic experimental study of super-
conducting behavior for samples with different parameters
can test theoretical predictions
7,20–22
and help to understand
vortex motion in superconductors. Two-dimensional arrays
of Josephson junctions provide a model system for this pur-
pose where the sample parameters can be controlled. These
parameters are the array size N , the unit cell area a
0
2
, and the
junction parameters: the normal-state resistance R
N
, E
J
, and
E
C
.
In a superconducting junction array, the Josephson cou-
pling between the islands presents a pinning force for the
vortex motion, whereas the Magnus force
23
induced by a bias
current represents a driving force. Depending on the magni-
tude of these two forces, the vortex transport mechanisms
can be classified into two types. In the first type, the pinning
force is small compared to the Magnus force and vortices
flow continuously. Vortices are accelerated by the Magnus
force and the terminal drift velocity is determined by a
damping coefficient which can be expressed by an effective
damping resistance R
D
. The measured flux-flow resistance
R
ff
is related to the density of free vortices n
f
as opposed to
the bound vortex-antivortex pairs, the size of a unit cell a
0
2
and R
D
, as
7,24
R
ff
=2 n
f
a
0
2
R
D
. 1
There are three sources of free vortices: Above the
Kosterlitz-Thouless KT transition temperature T
C
, free
vortices are created from thermal energy by unbinding of
vortex-antivortex pairs.
11–13
Free vortices can also be in-
duced by an applied magnetic field. The number of field-
induced free vortices per unit cell is denoted as f ,
f =Ba
0
2
/
0
, with B the applied magnetic field. Furthermore,
as emphasized by Halperin and Nelson,
13
free vortices can
also appear in samples of finite size even at T T
c
.
In the second type of vortex motion, the pinning force
dominates over the Magnus force, and the dynamics can be
described by tunneling of vortices through a barrier. In the
minimum potential-energy configuration, the vortex is situ-
ated in the center of a loop. In the presence of a small bias
current, a free vortex may change its position by either quan-
tum tunneling through the barrier or thermal hopping over
the barrier. The latter dominates at high temperature,
whereas the former is essentially temperature independent.
The most simple type of hopping which can occur is a hop to
PHYSICAL REVIEW B 1 OCTOBER 1996-I VOLUME 54, NUMBER 13
54 0163-1829/96/5413/94499/$10.00 9449 © 1996 The American Physical Society