Physica C 366 (2002) 135±146
www.elsevier.com/locate/physc
Reduction of low-frequency noise in high-Tc SQUIDs
by arti®cial defects
Roger W
ordenweber *, Peter Selders
Institut f
ur Schichten und Grenz¯achen (ISG), Abt. SL, Forschungszentrum Julich, D-52425 Julich, Germany
Received 23 March 2001; received in revised form 18 May 2001; accepted 21 May 2001
Abstract
We demonstrated that (i) quite simple arrangements of a few `strategically positioned' antidots can lead to signi®cant
reduction of the low-frequency noise in high-Tc SQUIDs and (ii) that a careful analysis of the unlocked SQUID signal
can be used to identify the number and position of vortices that penetrate the superconducting device. By using only
two antidots in the vicinity of the Josephson junction and a ring of antidots at the position of the ®rst penetrating
vortices, the onset ®eld at which the low-frequency noise starts to increase is shifted from Bon 8 lT without antidots
to Bon 40 lT in ®eld-cooled experiments and to Bon 25 lT in zero-®eld cooled measurements. The exact position
for the antidots is obtained by careful analysis of the impact of the position of moving vortices on the low-frequency
noise, the driving forces for vortex motion and the position of the ®rst vortices that penetrate the SQUID washer. The
attained improvement of the sensitivity of the superconducting device demonstrates the potential of antidots in applications which not necessarily are restricted to high-Tc SQUIDS. Ó 2002 Elsevier Science B.V. All rights reserved.
Keywords: Antidots; Low-frequency noise; SQUIDs; Vortex penetration
1. Introduction
The sensitivity of cryogenic active elements is
limited by the frequency dependent noise level of
the device. Additional to the contribution of the
electronics, in superconducting active devices two
di€erent sources are considered to be responsible
for the noise, i.e. the contribution of the active part
of the device, which usually consists of Josephson
junctions, and the noise of the passive component,
*
Corresponding author. Tel.: +49-24-6161-2365; fax: +4924-6161-2470.
E-mail
address:
r.woerdenweber@fz-juelich.de
(R.
W
ordenweber).
the superconducting thin ®lm. Studies and understanding of the noise mechanisms in Josephson
junctions are well established [1±4] and the noise
reduction by electronic means has successfully
been demonstrated [5±10]. The contributions of
the superconducting thin ®lm to the noise of active
devices are basically understood. They are attributed to the low-frequency noise and are ascribed to
motion of quantized ¯ux (vortices) in the superconducting thin ®lms. One indication for this
mechanism is given by the scaling of the spectral
noise density SU at low frequencies f with the applied magnetic induction B according to
p Bn
SU f ; B† / m
f
0921-4534/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 0 8 3 7 - 1
1†
136
R. Wordenweber, P. Selders / Physica C 366 (2002) 135±146
with n ˆ m ˆ 0:5. This contribution to the noise
still represents a serious limitation for the application active devices especially if they are used in
unshielded environment.
Various remedies to reduce the low-frequency
noise by vortex motion have been suggested and
tested [11±15], which in principle can be classi®ed
into two categories: (i) either vortex penetration of
the superconductor has to be avoided [11,12] or (ii)
vortices have to be pinned by suciently strong
pinning sites in the superconductor [13±15]. In case
of ¯ux avoidance extremely good screening or alternatively relatively small superconducting geometrical structures are required. Both solutions are
technically dicult to realize and/or very costly.
For instance the patterning of the complete washer
into a grid of striplines of linewidths w < 6 lm [16]
would be necessary for application of these devices
in earth ®eld of up to 50 lT. This turns out to be
technically quite complicated considering the requirements of the patterning. As we will demonstrate below, vortex pinning by a few small defects,
that are strategically positioned in the device, is
technically easier to handle and might be a better
solution for the problem in a number of SQUID
applications.
One of the most e€ective ways to create arti®cial pinning sites in thin ®lms is provided by the
preparation of submicrometer holes, so called antidots [17±20]. These defects can be placed arbitrarily in superconducting thin ®lm devices and, in
contrast to other pinning defects, which have to
be of the size of the superconducting coherence length n, holes with sizes much larger than n
will trap ¯ux very e€ectively [17,18]. In previous
work [14,19,20], we demonstrated that antidots
of sizes down to 250 nm in diameter can be patterned into YBa2 Cu3 O7 d (YBCO) thin ®lms and
rf-SQUIDs without deterioration of the superconducting and SQUID properties. Commensurability e€ects (demonstrating the attractive interaction
between vortices and antidots) and reduction of
the low frequency 1=f -noise of rf-SQUIDs in ambient magnetic ®elds have been demonstrated
[14,19,20]. However, it has been seen ± and is obvious from theoretical considerations ± that regular arrays of antidots lead to noise reduction only
at discrete values of the magnetic ®eld (matching
®elds) whereas in case of non-commensurability
between the vortex and antidot lattice even an
increase of the noise is observed [14,20]. Therefore
it is important to use only a few, `strategically well
positioned' antidots in the superconducting device,
which trap the vortices, that attribute strongly to
the low-frequency noise, and leave the vortex lattice free to arrange itself within the device.
In this paper we report on two very simple arrangements of a few, strategically positioned antidots in high-Tc (HTS) rf-SQUIDs that lead to
considerable reduction of the 1=f -noise in ambient
®eld down to the level of zero-®eld noise. The
geometric arrangements of the antidots are motivated by analysis of the current distribution in the
SQUID washer and the position of penetrating
vortices. The latter can be obtained by careful
analysis of the SQUID signal. Di€erent geometries
of antidots are necessary for noise reduction in
®eld-cooled (fc) experiments and in zero-®eld
cooled (zfc) experiments. By combining the two
geometries the onset ®eld Bon is shifted from
Bon 8 lT without antidots to Bon 40 lT with
antidots for FC measurements and to Bon 25 lT
for zfc experiments which in both cases is sucient
for a number of applications, e.g. for SQUID applications in slightly screen magnetic ®eld B <
B earth† 50 lT. It should be stated, that we
used a standard SQUID lay-out for magnetometer
applications. Due to technical reasons (among
others: rapid ®eld sweeps within the magnetic
shielding), the SQUID system shows average noise
properties. Equipped with an optimized rf resonator for read-out and a superconducting ¯ux focus this type of SQUID has been able to show
noise data of 24 fT/Hz1=2 (white noise) and 83 fT/
Hz1=2 (1 Hz) [21,22]. However, such a low noise
level is not necessary for the demonstration of the
advantage of antidots in SQUIDs which is characterized by the onset of the increase of 1=f noise.
2. Experimental setup
Planar washer type rf-SQUIDs with outer diameter of c ˆ 3:5 mm, SQUID holes of 100 100
lm2 and 3 lm wide step-edge junction are patterned via optical lithography and Ar ion milling
R. Wordenweber, P. Selders / Physica C 366 (2002) 135±146
into magnetron sputtered YBCO ®lms on 2 in.
LaAlO3 substrates. Step height and ®lm thickness
are h 270 nm and t 320 nm, respectively. Due
to the wafer scaling 48 SQUIDs are fabricated simultaneously, 90% of the devices show SQUID
signals with suciently large amplitudes of the
transfer function. The noise measurements are
executed in liquid-nitrogen cryostat shielded with
four l-metal layers which are characterized by
a residual static magnetic induction Bres < 5 nT.
Inside the shielding ambient magnetic inductions
perpendicular to the plane of the SQUIDs can be
applied using Helmholtz coils with lead acid batteries as power supply. For all frequencies and
®elds the spectral noise density of the coils for the
magnetic ®eld was more than one magnitude
smaller than the spectral noise density of the rfSQUIDs (Fig. 1). During noise measurements the
rf-SQUIDs are operated in a ¯ux locked loop
using a 600 MHz rf-SQUID electronics. Due to
the ac-mode of the electronics the 1=f -noise due to
¯uctuations in the resistance or critical current
of the Josephson junction is automatically eliminated.
The ®eld-to-¯ux coecient of the SQUIDs is
typically dB=d/ 9 nT=/0 , the amplitude of the
transfer function and the voltage-to-¯ux coecient
is ampli®ed by the SQUID electronic to typically
1.2 V and 1.65 V//0 , respectively. The noise level
in zero ®eld is typically about 35 l/0 =Hz1=2 (corresponding to 0.3 pT/Hz1=2 ) at 1 kHz (white noise)
and 200 l/0 =Hz1=2 (1.8 pT/Hz1=2 ) at 1 Hz (1=f -
Fig. 1. Comparison to the spectral ¯ux noise density of the
Helmholtz coils and a typical 3.5 mm rf-SQUID at 1 Hz.
137
Fig. 2. Comparison of spectral noise density of one SQUIDs
before and after the two additional patterning processes (see
Fig. 3).
noise). The corner frequency in zero ®eld is about
25 Hz. No degradation of the SQUID performance due to the subsequent patterning of the
antidots could be detected (see Fig. 2).
For the characterization of the impact of
strategically positioned antidots upon the lowfrequency noise, one typical rf SQUID is characterized before and after the patterning processes
of the two di€erent antidot con®gurations with
respect to transfer function, spectral noise density,
and penetration of vortices into the washer of the
SQUID by analyzing the unlocked SQUID-signal.
The latter is described in more detail in Ref. [23].
These di€erent characterization techniques provide a comprehensive set of data which allows a
detailed analysis of the vortex penetration and
motion in a washer type SQUID in the fc and zfc
experiments and their impact on the noise performance in an applied magnetic ®eld.
The di€erent SQUID con®gurations are sketched in Fig. 3:
(1) Con®guration #0 represents the bare, planar
washer type rf-SQUIDs (outer diameter of 3.5
mm, 100 100 lm2 SQUID hole and 3 lm wide
step-edge junction) without antidots.
(2) In con®guration #1 only two antidots
(diameter 1:5 lm) are patterned into the bare
rf-SQUID (see also SEM image in Fig. 4). The
antidots are positioned at both sides of the stepedge junction with a distance between junction and
138
R. Wordenweber, P. Selders / Physica C 366 (2002) 135±146
Fig. 3. Sketch of the rf-SQUID with di€erent antidot con®gurations: without antidots (conf. #0), with two antidots in the
vicinity of the junction (conf. #1), and with an additional ring
of antidots at r ˆ c=2 (conf. #2).
antidot center of 8 lm and a distance between
antidot center and the edge of the washer of 2.5
lm. Transfer function and noise level remained
unchanged before and after patterning of the antidots (Fig. 2).
(3) In con®guration #2 an additional ring of
antidots, positioned at the radial position r ˆ c=2
of the ring-shaped SQUID washer, is patterned
into the SQUID (see Figs. 3 and 4). The optimal
ring position and it's diameter are derived from the
analysis of the vortex penetration in zfc measurements and will be described below. An antidot±
antidot distance of 50 lm and antidot diameter of
2.5 lm have been chosen. The choice of the antidot distances is based on simple estimations of
the antidot±vortex interaction and technical reasons. Transfer function (amplitude 1.05 V), voltage to ¯ux coecient (1.73 V//0 ) and noise level at
zero ®eld are not a€ected by the patterning process
(see Fig. 2).
3. Experimental results and discussion
In this paragraph the experimental results of zfc
and fc measurements of the low-frequency noise
of and the vortex penetration into our HTS rf
SQUID with various antidot con®gurations will
be presented and discussed. For comparability
reasons, we will show the results obtained for one
typical SQUID which has subsequently been provided with antidots, i.e. modi®ed from conf. #0 to
conf. #1 and, ®nally, to conf. #2.
3.1. Field cooled noise measurements
First, fc measurements of the low-frequency
noise of the di€erent con®gurations were executed.
The magnetic induction B is applied perpendicular to the plane of the SQUID, noise spectra are
recorded at liquid nitrogen temperature. For all
inductions and con®gurations, fc measurement
reveal 1=f -type noise spectra at low frequencies.
Typical fc noise spectra of the bare rf SQUID
(conf. #0) for di€erent magnetic inductions are
given in Fig. 5. The 1=f -type low-frequency noise
increases at large magnetic induction, whereas the
white noise is not a€ected by the ®eld. The corresponding transfer function (inset of Fig. 5) is only
slightly reduced by the magnetic ®eld within a ®eld
range of B < 130 lT, which is larger than the ®elds
considered in this publication.
A 1=f noise spectrum in HTS ®lms is generally
ascribed to an incoherent superposition of many
thermally activated microscopic ¯uctuators, which
are given by moving or hopping vortices in case of
superconducting material in a magnetic induction.
According to the Dutta±Dimon±Horn (DDH)
model [24,25], this frequency dependence is evidence for a distribution of activation energies for
vortex hopping processes. Since our measurements
of 1=f noise are sensitive only to small activation
Fig. 4. SEM images of the two antidots in the vicinity of the junction (left) and the ring of antidots (right).
R. Wordenweber, P. Selders / Physica C 366 (2002) 135±146
Fig. 5. Field-cooled measurement of the noise of the bare rfSQUID (conf. #0) for two di€erent magnetic inductions. The
1=f -type low-frequency noise increases at large magnetic induction, whereas the white noise is not a€ected by the ®eld. The
transfer function of fc experiments (see SQUID signal in the
inset) is not reduced by the magnetic ®eld within a ®eld range
B < 130 lT.
energies, we probe the low-energy part of this
distribution [26].
A comparison of the spectral noise density in
the 1=f regime of all con®gurations is given in Fig.
6. Each data point represents an average value
obtained from three independent fc measurements.
Fig. 6. Spectral noise density obtained for fc measurements at 1
Hz as function of the applied magnetic induction B for a typical
rf-SQUID with di€erent con®gurations of antidots. Each data
point represents an average value obtained from independent fc
measurements. The onset of the increase of the low-frequency is
characterized by the onset ®eld Bon . The proportionality S / B
of the low-frequency noise for B > Bon (dashed lines) agrees
with the theoretical expectation in Eq. (1). The onset ®eld for
conf. #1 and #2 are comparable to the magnetic induction of
the earth.
139
The double logarithmic plot of the spectral noise
density at 1 Hz as function of B shows the interesting aspects.
In principle, the ®eld dependence of the lowfrequency noise is identical for all SQUID con®gurations: (a) at low ®elds B < Bon the spectral
1=2
noise density is ®eld independent (i.e. SU (1 Hz,
1=2
B < Bon ) 180±200 l/0 /Hz
for the examined
SQUID) and (b) at high ®elds B > Bon the spectral
noise densities increase linear with increasing ®eld
according to the theoretical expectation given in
Eq. (1). However, the transition from ®eld-independent to ®eld-dependent spectral noise density is
strongly shifted from Bon 8 lT without antidots
(conf. #0) to Bon 40 lT with antidots (conf.
#1 and #2). The signi®cant increase of the onset
®eld Bon by the arrangement of strategically positioned antidots is sucient for most applications
of SQUIDs in unshielded environment. The e€ect,
which has been discussed in parts in Ref. [16], has
to be ascribed only to the two antidots positioned
at the vicinity of the junction. The choice of the
position for these ®rst two antidots in the SQUID
is a result of the following considerations:
(a) First, due to the radial dependence of the
¯ux coupled by a vortex into a SQUID, the lowfrequency noise is generally expected to be dominated by motion (with a radial component) of
vortices in the superconducting ®lm, which are
close to the SQUID hole. 1 A vortex at radial
position r couples magnetic ¯ux /v r† into the
SQUID that can be calculated in analogy to the
method of image charges in two-dimensional
electrostatics [27]:
"
r a
U0
/v r† ˆ
ln
c r
ln a=c†
1
X
a ai
i
‡
1† ln
c ai
iˆ1
#
1
X
a ci
i
‡
1† ln
:
2†
c ci
iˆ1
1
Exceptions could be present in case of morphologically
strongly inhomogeneous ®lms, which could lead to large
variations in the hopping ranges of vortices at di€erent
positions.
140
R. Wordenweber, P. Selders / Physica C 366 (2002) 135±146
Fig. 7. Normalized ¯ux inducted by a vortex at normalized
radial position r=c according to Eq. (2) for a SQUID with inner
radius a (indicated by the arrow) of the washer. The inset
sketches the di€erent situations for two vortices at di€erent
radial positions. The data points represent the position of the
®rst three vortices which enter the washer in zfc measurements
obtained reproducible for di€erent experiments (see last section
of this chapter).
Here, U0 represents the ¯ux quantum, a and c the
inner and outer radius of the SQUID washer, and
ai ˆ b a=c†i and ci ˆ b c=a†i with b ˆ r (i even)
and b ˆ ca=r (i odd). The resulting coupling
strength of the ¯ux is plotted in Fig. 7. According
to Eq. (2), /v r† decreases strongly with increasing
distance between vortex and the SQUID hole.
Thus, vortex motion will contribute stronger to the
low-frequency noise if the vortex is positioned
close to SQUID hole. As a consequence the antidots should be positioned close to the SQUID
hole, in order to reduce vortex motion in this area.
(b) Second, the motion of vortices is based on a
driving force, which can be supplied by e.g. ther-
mal energy, a magnetic ®eld gradient or a current.
In fc experiments the ®eld gradient is expected to
be negligible. However, the screening current and,
especially, the rf current that is induced by the rf
tank coil can result in a driving force (Lorentz
force) for vortex motion. Fig. 8 represents a sketch
of the shielding current and of the distribution of
the inducted rf current. The latter is simulated via
the program Sonnet. Both ®gures demonstrate
that a crowding of shielding and induced currents
is expected at the vicinity of the Josephson junction, which would add to the activation of vortex
motion in this area.
(c) Finally, according to the arguments in (a)
and (b) vortex motion within the junction stripline
on which the Josephson junction is positioned
(see Fig. 4) would be most likely (due to the
large driving currents) and relevant for the lowfrequency noise (position very close to the SQUID
hole). However, it can be shown easily, that this
stripline is too small (width of 3 lm) to contain
vortices for the magnetic induction used in the
experiments. Considering the Gibbs free energy of
a vortex in a superconducting strip of width w
during cooling down below the critical temperature Tc , vortices can only be trapped in the superconductor for perpendicular magnetic ®elds
exceeding an exclusion ®eld Bex [16]:
Bex ˆ
pU0 1
:
4 w2
3†
Accordingly, extremely small exclusion ®elds of
Bex 0:5 nT are expected for the washer (linewidth is equivalent to c a 1:7 mm), whereas
Fig. 8. Sketch of shielding current and rf current induced by the resonant circuit. The latter is simulated via Sonnet, the dotted white
line represents the position of the tank coil in the simulation.
R. Wordenweber, P. Selders / Physica C 366 (2002) 135±146
large values of Bex 175 lT are expected for the
small junction stripline (w ˆ 3 lm). As a consequence, in our fc measurements vortices should
easily and exclusively penetrate the washer but not
the junction stripline. That is, due to the ®nite
residual magnetic induction Bres 5 nT of our
magnetic shielding, vortices should populate the
washer even if no external magnetic ®eld is applied. With increasing magnetic induction the
density of vortices increases linearly with the applied ®eld in fc experiments. However, no vortices
are expected to be trapped in the stripline for the
®elds considered in the experiment. Therefore,
antidots in the stripline would not e€ect the lowfrequency noise.
In conclusion, the ®rst antidots are patterned at
positions, at which vortices are expected, the e€ect
of vortex motion on the low-frequency noise is
largest, and the driving force due to induced and
screening current are largest. The resulting `strategic' positions for the ®rst antidots (conf. #1 and
#2) is situated on both ends of the junction strip
line but in the washer (see Fig. 4). The increase of
Bon in fc experiments is a consequence of these two
antidots (see Fig. 3), the additional antidots in
conf. #2 do not lead to a further increase of the
onset ®eld. Furthermore, due to the repulsive
vortex±vortex interaction vortex penetration into
the junction area is energetically not favorable up
to a ®eld for which the vortex±vortex spacing
a0 ˆ 1:075 /0 =B†1=2 is roughly equivalent to half
of the distance d between the antidots [28], i.e.
Bon 4:6/0 =d 2 . For the examined geometry with
d ˆ 15 lm we obtain an expected onset ®eld
Bon 41 lT which is very close to the experimentally observed value.
141
experiments. In the following, zfc measurements
of the low-frequency noise and the penetration of
vortices into the washer are presented. The latter is
discussed in details in Ref. [23]. In analogy to the
fc experiments we will demonstrate, that well positioned antidots also lead to a reduction of the
low-frequency noise in zfc experiments. The exact
analysis of vortex penetration motivates the position of the additional antidots of conf. #2, which
cause an increase of Bon in zfc experiments, too.
The zfc measurements are performed by cooling
the SQUID in zero-®eld, ramping the magnetic
induction to a given value B and back to zero-®eld,
where the measurement takes place. The ®eld value
characterizing the di€erent data sets is the induction B, to which the ®eld has been ramped.
Typical zfc noise spectra are displayed in Fig. 9.
Although these data are taken at di€erent ®elds for
conf. #1, similar spectra are observed for the other
con®gurations. At most ®elds and low frequencies
a 1=f power spectrum is observed which (identically to the spectra for fc experiments) can be ascribed to an incoherent superposition of many
thermally activated vortex hopping processes.
Only at intermediate ®elds (close to Bon of zfc experiment) contributions of a Lorentzian type noise
spectrum become visible. These contributions are
characteristic for random telegraph noise (RTN)
signals [29]. RTN in HTS thin ®lms is believed to
4. Zero-®eld cooled measurements
In the previous chapter the signi®cant improvement of the low-frequency noise in fc experiments due to a pair of strategically positioned
antidots was demonstrated. However, in many
applications the magnetic ®eld at the SQUID
changes during the measurement. This situation is
comparable to zfc experiments, which are therefore as relevant for application of SQUIDs as fc
Fig. 9. Flux noise spectra of zfc experiments for di€erent
magnetic inductions recorded for conf. #1 after a relaxation
time. In contrast to fc experiments, contribution of a Lorentzian spectrum are recorded for magnetic inductions close to Bon
(see inset).
142
R. Wordenweber, P. Selders / Physica C 366 (2002) 135±146
be originated by hopping of a single vortex or a
bundle of several vortices between two energetically favored positions, e.g. pinning sites [26]. At
higher ®elds the Lorentzian spectrum vanishes
again and a 1=f -type noise characteristic is measured after a relaxation time of up to 2 h after the
®eld ramp was completed [23].
An overview of the ®eld dependence of the zfc
low-frequency noise for the con®gurations with
antidots is given in Fig. 10a. It represents the
1=2
spectral noise density SU B† at 1 Hz as function of
the applied magnetic ®eld. In comparison to fc
measurements the increase of low-frequency noise
starts at lower ®elds Bon (18 lT for conf. #1 and
24 lT for conf. #2) in these experiments. Furthermore, in contrast to the fc experiments, the
®eld dependence in zfc measurements can be divided into three di€erent regimes:
Fig. 10. (a) Spectral noise density at 1 Hz and (b) number of
penetrating vortices as a function of the applied magnetic induction for zfc measurements of conf. #1 (circles) and conf. #2
(triangles). The solid and open symbols in (a) represent the 1=
f -type and Lorentzian-type noise spectra, respectively. The
arrows mark the onset ®eld for both con®gurations. The experimental details of the determination of the number of penetrating vortices is given in the last section of this chapter.
(i) Obviously, at low magnetic ®elds the applied
magnetic ®eld does not in¯uence the low-frequency noise, i.e. if ¯ux penetrates the washer (see
Fig. 10b) it does not contribute to the noise. The
low-frequency noise spectra recorded in this ®eld
regime resemble 1=f -type spectra according to Eq.
(1) with m ˆ 0:5. (ii) In the intermediate ®eld regime starting at B ˆ Bon , the low-frequency noise
shows a steep increase with increasing magnetic
®eld. For both con®gurations #1 and #2 this increase can be ®tted according to Eq. (1) however
with an unusually large exponent n 3:3, i.e.
SU1=2 / B3:3 . (iii) At high magnetic ®elds the increase
of low-frequency noise becomes weaker. However,
the exponent n, which ®ts the experimental data,
1=2
is still larger than theoretically expected, i.e. SU /
Bn with n 0:85±1. In this regime 1=f noise spectra are observed for all con®gurations.
For all con®gurations the onset of the increase
of low-frequency noise generally takes place at
smaller magnetic inductions for zfc measurements
compared to fc measurements (see Figs. 6 and
10a). Nevertheless, the introduction of antidots
leads to a signi®cant improved onset ®eld in zfc
measurements, too. Onset ®elds of Bon 18 lT
(conf. #1) and, especially, Bon 24 lT (conf. #2)
compared to Bon < 8 lT for the bare SQUID are
sucient for many applications. Furthermore, the
zfc measurements reveal the advantages of conf.
#2 with respect to conf. #1. The onset ®eld is increased strongly by the additional ring of antidots
in the middle of the washer. This indicates, that
vortex motion in the vicinity of these antidots i.e.
in the center of the washer seem to be responsible
for the ®rst increase low-frequency noise for conf.
#1. This assumption is also supported by the fact
that di€erent type of spectra are observed in the
intermediate ®eld above Bon for the di€erent con®gurations. Whereas conf. #1 and conf. #0 reveal
Lorentzian spectra in this ®eld regime (see Fig.
10a), 1=f -type low-frequency noise is observed for
conf. #2. The Lorentzian spectra arise from single
¯uctuators, i.e. single vortices hopping between
two energetically favored positions. The antidots
in the middle of the washer seem to abandoning
the motion of theses vortices resulting in the
modi®cation of the spectra to a pure 1=f -type
low-frequency noise spectrum. This explanation of
R. Wordenweber, P. Selders / Physica C 366 (2002) 135±146
the di€erence in the onset ®eld and the type of
low-noise spectra is con®rmed by measurements
of vortex penetration. The penetration measurements, which also provide the motivation for the
development of conf. #2, is presented in the following sections.
4.1. Detection of vortex motion and penetration
A change in the position of a vortex in the
washer is accompanied by a change of magnetic
¯ux in the SQUID. The dependence of the magnetic ¯ux on the radial position of the vortex in a
ring-shaped SQUID washer is given by Eq. (2) and
shown in Fig. 7. Thus, changes in the unlocked
SQUID signal have to be attributed either to
changes in the applied ®eld or to changes of the
radial position of vortices within the washer. In
our experiment we can distinguish between both
contributions as long as the hopping distances for
motion within the SQUID is small. For these
measurements the applied ®eld is cycled around a
mean value which itself is changed continuously.
By subtraction of the unlocked SQUID signal for
reducing and increasing ®eld during one cycle,
changes in the SQUID signal due to the vortex
motion within the superconducting device are detected. Two di€erent mechanisms of vortex motion
can be distinguished:
(i) A di€usion-like motion of vortices within the
washer can be resolved in some of the zfc mea-
143
surements. It results in a continuous shift of the
phase of the SQUID signal. Fig. 11b represents a
typical result of such a measurements. The ®eld
has been ramped to a 21 lT and reduced again.
The SQUID signals for increasing and decreasing
®eld show a continuous phase shift, which can be
converted to a continuous change of the magnetic
induction DB in the SQUID due to vortex motion
in the washer. The sign of the phase shift is correlated to the direction of vortex motion. In the
measurement shown in Fig. 11b, a vortex moves
from the inner edge to the outer edge of the
washer. The total change in magnetic induction is
8±9 nT, which is equivalent to the drift of one ¯ux
quantum from one to the other edge of the washer.
The gradual shift is followed by a small discontinuous change of the phase (not shown in the
®gure), which can be interpreted in terms of the
vortex passing the surface barrier [23] at the inner
edge of the washer. In contrast to SQUIDs without antidots, for which thermal activated motion
of vortices has been assumed to be responsible for
hysteretic SQUID signals [30], these continuous
shifts are rarely observed in our SQUIDs with
antidots. Especially for con®guration #2 with additional antidots in the washer, no continuous
shifts could be resolved.
(ii) The most interesting irregularity in the
SQUID signal is given by a discrete change of the
phase. Fig. 11a displays a typical example for such
a discontinuous change of the phase observed in
Fig. 11. Experimental determination of the vortex motion in and penetration into the superconducting washer: (a) discrete phase shift
of the unlocked SQUID signal due to a penetrating vortex and (b) di€usion-like motion of a vortex in the washer.
144
R. Wordenweber, P. Selders / Physica C 366 (2002) 135±146
zfc measurements. The abrupt change of the phase
has to be ascribed to vortex penetration into or
expulsion out of the washer if we assume, that the
hopping distances for motion within the SQUID is
small. Thus, the number of abrupt phase changes
provides a upper limit of the number of vortices
penetrating the superconducting ®lm (i.e. the
washer). The sign of the phase shift is correlated
to the direction of vortex motion. In contrast to
the di€usion like motion of vortices within the
washer, which was discussed above, in case of
penetration or expulsion the vortex has to overcome the surface barrier which is formed by the
edge of the superconducting washer. There are two
interesting aspects connected to these measurements:
(1) The experimental data of vortex penetration
or expulsion provide information about the surface barrier at extremely low ®elds and the mechanism of vortex motion across this barrier. This
aspect is discussed extensively in Ref. [23].
(2) In case of vortex penetration, the radial
position to which the vortex penetrates can be
obtained from the experiments. This yields (a)
more insight into the potentials (e.g. ¯ux pinning)
that are encountered by the penetrating vortex and
(b) provides indications for `strategical' positions
of antidots, that could reduce the low-frequency
noise of the device in zfc measurements. The latter
was actually the method to chose the positions of
additional antidots in conf. #2 and will be discussed below.
The radial position of a penetrating vortex can
be derived from the experimental data. Inserting
the ®eld-to-¯ux coecient dB=d/ ˆ 9 nT=U0 , the
discrete phase shifts are translated into discrete
¯ux changes D/. Assuming that vortices (antivortices are not considered) are responsible for
these ¯ux changes, which are either penetrating
into or expelled out of the washer, the radial position can be derived according to Eq. (2). Fig. 12
represents the resulting radial positions of vortices
for conf. #1 and #2 obtained from independent zfc
experiments. For these measurements the magnetic ®eld is increased continuously and extremely
slowly with 0.1 nT/s up to 30 lT. All discontinuous phase shifts of the SQUID signal seem to
be caused by vortices penetrating the washer, i.e.
Fig. 12. Radial position of vortices after penetration of the
washer derived from two independent experiments (circles and
triangles) for (a) conf. #1 and (b) conf. #2. The corresponding
onset ®eld for the increase of low-frequency noise are taken
from Fig. 10a. The reproducibility of the measurement indicates, that vortex hopping within the SQUID can be ruled out
to be the reason for the abrupt phase changes at least for small
®elds (B < 12 lT).
D/ > 0. For both con®gurations two di€erent ®eld
regimes can be distinguished:
At low ®elds, only a few vortices penetrate the
washer, i.e. up to three vortices for ®elds up to 13
or 18 lT for conf. #1 or conf. #2, respectively (see
also Fig. 10b). The penetrations occur very reproducible with respect to the radial position r=c
to which the vortex penetrates and the ®eld at
which it penetrate. The ®nal radial position of
these vortices after penetration is always close to
the center of the washer at r c=2. This observation was the main reason for the development of
conf. #2 with additional antidots at exactly this
radial position.
At higher ®elds, the number of discrete phase
changes increases strongly. However, the ¯ux
change connected to these events are not identical
R. Wordenweber, P. Selders / Physica C 366 (2002) 135±146
and in most cases smaller than observed for the
low-®eld regime. It is not clear, whether vortex
penetration into the washer or vortex hopping
within the washer is responsible for each individual
recorded change of ¯ux in the SQUID. Therefore,
the given radial positions should not be taken too
seriously and indications for further need of antidots in the washer cannot be derived from the
data of this ®eld regime directly. In contrast to the
low-®eld regime the events do not occur at identical ®elds and with identical ¯ux change. Nevertheless, considering the total change of ¯ux due to
the events, a quite reproducible behavior is also
observed for this ®eld regime (see also the comparable high-®eld behavior of conf. #1 and #2 in
Fig. 10b). Furthermore, these more statistical
events of ¯ux penetration and/or reorganization of
the vortex arrangement in this ®eld regime seem to
be connected to the increase of low-frequency
noise, which is indicated by the onset ®eld Bon in
the ®gure. The onset of increased vortex penetration and/or reorganization in the washer seem to
trigger the increase of low-frequency noise at Bon .
The additional antidots of conf. #2 lead to a
considerable shift of this onset from 13 lT (for
conf. #1 with onset ®eld Bon 18 lT) to 19 lT
(for conf. #2 with onset ®eld Bon 24 lT).
Finally, the total number of penetrated vortices
can be estimated from these experiments. The ®eld
dependence of the number of vortices that penetrate the washer in zfc measurements (see Fig.
10b) resembles the ®eld dependence of the lowfrequency noise (see Fig. 10a). At small ®eld only
few vortices (exactly three vortices) enter, whereas
at intermediate ®elds the number of vortices increases strongest. Finally, at large ®eld the increase
scales according to a power law, nv / B2:5 . At the
onset ®eld Bon of the increase of low-frequency
noise, the total number of penetrated vortices is
8 for conf. #1 and 13 for conf. #2.
5. Conclusions
We demonstrated that (i) quite simple arrangements of a few `strategically positioned'
antidots can lead to signi®cant reduction of the
low-frequency noise in HTS SQUIDs in ambient
145
®elds and (ii) that a careful analysis of the unlocked SQUID signal can be used to identify the
number and position of the ®rst penetrating vortices. The geometric arrangements of the antidots
are obtained by analysis of the current distribution in the SQUID washer, of the impact of the
position of moving vortices on the noise and of
the position of penetrating vortices. The latter can
be obtained by analysis of phase changes in the
unlocked SQUID signal. Di€erent geometries of
antidots are necessary for noise reduction in fc
experiments and zfc experiments. By combining
the two geometries, the onset ®eld is shifted from
Bon 8 lT without antidots to Bon 40 lT for
fc measurements and to Bon 25 lT for zfc experiments. This seems to be sucient for most
applications. The attained improvement of the
sensitivity of the superconducting device demonstrates the potential of antidots in applications which not necessarily are restricted to HTS
SQUIDs.
Acknowledgements
The authors like to acknowledge the cooperation and assistance of H.P. Bochem, A.I. Braginski, S. Bunte, A. Castellanos, P. Dymachevski, R.
Gross, N. Klein, R. Kleiner, D. Koelle, R. Kutzner, P. Lahl, R. Ott, R. Straub and M. Vaupel.
This work was supported in part by the DFG
Grant no. WO549/3-1 and ESF scienti®c program
VORTEX.
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