Optimizing the spin sensitivity of grain boundary junction nanoSQUIDs -- towards detection of small spin systems with single-spin resolution

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by  R. Wölbing

Optimizing the spin sensitivity of grain boundary junction nanoSQUIDs – towards
detection of small spin systems with single-spin resolution
R. W¨olbing,1 T. Schwarz,1 J. Nagel,1 M. Kemmler,1 D. Koelle,1 and R. Kleiner1
1 Physikalisches Institut – Experimentalphysik II and Center for Collective Quantum Phenomena in LISA+,
Universita¨t Tu¨bingen, Auf der Morgenstelle 14, D-72076 Tu¨bingen, Germany
We present an optimization study for improving the spin sensitivity of nanoSQUIDs based on
resistively shunted grain boundary Josephson junctions. In addition the dc SQUIDs contain a
narrow constriction onto which a small magnetic particle can be placed with efficient coupling of
3
1 its stray magnetic field to the SQUID loop. The separation of the location of optimum coupling
0 fromthejunctionsallowsforanindependentoptimization ofcouplingandjunctionproperties. Our
2 analysisofthedependenceofthefluxnoiseoftheSQUIDandofthecouplingofamagneticparticle
onthegeometricparametersoftheSQUIDlayoutshowsthataspinsensitivityS1/2ofafewµ /√Hz
n µ B
should be feasible for optimized parameters, respecting technological constraints.
a
J
PACSnumbers: 85.25.CP,85.25.Dq,74.78.Na,74.72.-h74.25.F-74.40.De
7

]
n I. INTRODUCTION placed close to the constriction type junctions in order
o to achieve optimum coupling. However, this means that
c the junction properties and the coupling factor φ can-
- Miniaturized direct current (dc) superconducting µ
notbeoptimizedindependently,whichhampersacareful
r quantuminterferencedevices(SQUIDs)withdimensions
p optimization of the spin sensitivity.
in the sub-micrometer range (nanoSQUIDs) are promis-
u
s ing devices for the sensitive detection and investigation We should note here, that very sensitive Nb thin film
. of small spin systems.1 The basic idea behind this is to nanoSQUIDs basedonconstrictiontype junctions, resis-
t
a attach a small (nanometer-sized) magnetic particle di- tively shunted with a thin W layer, have been realized.7
m
rectly to the SQUID and trace out magnetic hysteresis However, in this case, the devices show optimum per-
- loopsofthe particle. Thisshallbe donebydetecting the formance only in a narrow range of temperature T not
d
change of the stray magnetic field of the particle with toofarbelowthetransitiontemperatureT ofNb,which
n c
o magnetic moment ~µ via the change of the magnetic flux makes them less interesting for applications. With re-
c Φ coupled to the SQUID loop.2–4. In order to meet the specttotheapplicationofnanoSQUIDsforthedetection
[ ultimate goalof detecting the flipping ofonly a few elec- of the magnetization reversal of nanomagnets, the most

v1 torpotnimspiziends,5catrheefusplliynvsieansrietdivuictyinSgµ1t/h2e=spSeΦ1c/t2r/aφlµdhenassittoyboef ainntderaesttivnegryrehgiigmhemofagonpeetriactifioenldiss iant tThe≈te1sKlaarnadngbee.1loIwt
9 flux noise SΦ of the SQUID and increasing the coupling has been demonstrated that Nb thin film nanoSQUIDs
18 fbayctsohrriφnµkin≡g tΦh/eµsi(zweitohf tµhe≡SQ|~µU|)I.DSloΦopc,ananbdehreendcuecietds bimasperdesosinvecobnasctkrigcrtoiounndtyfipeeldjusnucptiotnos7caTn.8bHeoowpeevraetre,dthine
1 inductance L, and φ can be increased by placing the upper critical field of typical Nb thin films ( 1T) re-
. µ ∼
1 particle on a narrow constriction inserted in the SQUID quires to use very thin Nb films with thicknesses of only
0 loop,whichmotivatestheneedtoimplementsub-micron a few nm, i.e. well below the London penetration depth
3 SQUID structures. λL of the Nb films, if such SQUIDs shall be operated in
1
tesla fields. This leads to a large kinetic inductance con-
: Until now,the mostcommonapproachfor the realiza-
v tribution to the SQUID inductance, and hence a large
tionofnanoSQUIDsistouseconstrictiontypeJosephson
i flux noise of such SQUIDs, which does not allow to use
X junctionsintersectingsmallSQUIDloops(seee.g.Ref.[6]
the huge potential for the realization of ultralow-noise
r published in a special issue on nanoSQUIDs and related
a articles therein). However, this approach comes with nanoSQUIDs.
several drawbacks: Constriction type Josephson junc- In order to circumvent the above mentioned draw-
tions often show hysteretic current-voltage characteris- backs, we recently started to develop dc nanoSQUIDs
tics (IVCs). This does not allow to operate the SQUID based on YBa2Cu3O7 (YBCO) thin films with submi-
magnetometers continuously, which is required for the cron wide bicrystal grain boundary Josephson junctions
investigation of the magnetization dynamics of the sam- (GBJs).9 Due to the huge upper critical field of YBCO,
pleunderinvestigation. Furthermore,thisdoesnotallow such SQUIDs can be realized with film thicknesses on
touseadvancedSQUIDread-outelectronics,whichhave the order of λ and above and operated in tesla fields.
L
been developed for SQUIDs with non-hysteretic Joseph- Furthermore,duetothelargecriticalcurrentdensitiesof
son junctions and which are required for the read-out of the YBCO GBJs (several 105A/cm2 at T = 4.2K and
◦
ultra-sensitive dc SQUIDs. Also, the noise properties of below for a grain boundary misorientation angle of 24 )
suchjunctionsarenotwellunderstoodandhencehardto submicronjunctions still yield reasonablylarge values of
optimize. And, finally, the magnetic particles have to be the criticalcurrentI0. Inordertoachievenon-hysteretic
2
IVCs, the GBJs are shunted by a thin Au film. Due resistanceRandcapacitanceC. Iftheconstrictioncould
to the fact that the barrier of the GBJs is oriented per- be made not only arbitrarily thin and narrow, but also
pendicular to the YBCO thin film plane, it is possible arbitrarilyshort,onecouldenvisionascenario,whereφ
µ
to apply tesla magnetic fields in the plane of the film, reaches a value around 0.5Φ0/µB,4 while, at the same
without a significant reduction of I0.10 And finally, im- time, the inductance of the constriction remains small
plementing anadditionalnarrowconstriction(whichcan (Φ0 is the magnetic flux quantum). Then, SΦ could be
be much narrower than the GBJs) in the SQUID loop, optimized independently by proper choice of the SQUID
the optimization of the coupling factor is possible with- size and the junction properties. For the type of device
out affecting the junction properties. wediscusshere,thisiscertainlynotthecaseandwethus
Here, we present a detailed optimization study of look for an optimization, which is compatible with tech-
the spin sensitivity of such grain boundary junction nological limitations. A large coupling φ demands as
µ
nanoSQUIDs by analyzing the dependence of the flux narrow and thin as possible constrictions. On the other
noise SΦ and the coupling factor φµ on the geometry of hand, for too narrowconstrictions,givena fixed value of
ourdevices. WefindthatforanoptimizedSQUIDgeom- d,itsinductance L andthus alsothetotalinductance L
c
etry a continuous detection of magnetic moments down of the SQUID may become too large,possibly degrading
toaspinsensitivityS1/2 ofafewµ /√Hzisfeasible(µ thefluxnoise. Thismaybecounterbalancedbychoosing
µ B B
is the Bohr magneton). adifferentfilmthicknessandchanging,e.g.,the junction
width w .
J
Inthefollowingsections,wederiveexplicitexpressions
II. NANOSQUID DESIGN for the dependence of φµ (Sec. III) and SΦ (Sec. IV) on
variousgeometricandelectricSQUIDparameters,which
then allows us to optimize S (Sec. V).
The layout of the nanoSQUID (top view) is shown in µ
Fig. 1. The SQUID structure is patterned in a YBCO
thin film of thickness d, covered by a thin Au film with
III. COUPLING FACTOR
thickness dAu. The two bridges straddling the grain
boundaryhaveawidthw andlengthl . Theupperpart
J J
of the SQUID loop contains a constriction of width w We numerically calculate the coupling factor φ =
c µ
and length l . An applied bias current I is flowing from Φ/µ, i.e. the flux Φ coupled into the SQUID loop by
c
top to bottom across the two GBJs. A small magnetic a point-like particle with magnetic moment µ, using
particle can be placed on top of the constriction, and the softwarepackage3D-MLSI, whichsolves the London
an in-plane magnetic field (perpendicular to the grain equations.11 Details on the calculation procedure can be
boundary)canbeappliedwithoutsignificantsuppression found in Ref. [9]. In brief, for a given SQUID geometry
of the critical current I0 of the two GBJs. (cf. Fig. 1), one calculates the magnetic field distribu-
Optimizing the SQUID for spin sensitivity means to tion B~(~r) generated by a current J circulating around
minimize the ratio SΦ/φ2µ. The coupling factor φµ is es- the SQUID hole. The coupling factor is obtained from
sentiallydeterminedbythegeometryoftheconstriction, φ = e B~(~r)/J. Here, e is the direction of the mag-
µ µ µ
i.e., its width wc and thickness d. SΦ depends on the netic m−oment ~µ at position~r.
SQUIDinductanceLandonthejunctionparametersI0, For a particle with its magnetic moment oriented in
b b
the plane of the thin film SQUID perpendicular to the
l grain boundary, we find a maximum φµ if the particle
c constriction is placed directly on top of and as close as possible to
w 10
c the constriction. Assuming that a particle is placed
L L L at a distance of 10nm above the constriction (without
e c e
w an Au layer, which can be removed without affecting
L L J l the junction properties), we calculate φµ in the range
J J J 40nm<w <200nm and 40nm< d<500nm and find
c
grain boundary
the dependence
L w0 0.7 d0
b φµ γµ (12.3+ ), (1)
≈ w d
(cid:18) c(cid:19)
junctions
with d0 =320nm, w0 =1nm and γµ =23nΦ0/µB.
As expected, we find a decreasing φ with increas-
µ
FIG. 1: (Color online) Schematic view of the nanoSQUID ing width w of the constriction. Our simulations yield
c
layout, divided (by white dotted lines) into the constriction φ w−0.7,i.e.,achangeinφ byafactorof3withinthe
(inductance Lc, length lc, width wc), two corners (each with siµm∝ulatcion range used for wc.µFurthermore, φµ also de-
inductanceLe), thetwo junctions (each with inductanceLJ, creaseswithincreasingfilmthicknessd. However,within
length l , width w ) and thebottom part (inductanceL ).
J J b the(larger)simulationrangeford,wefindamuchweaker
3
dependence on d; in particular, for d>∼d0, the coupling We next determine the dependence of the SQUID in-
starts to saturate. We note that some uncertainty in de- ductance L on the various geometrical parameters. We
termining φ arises from the fact, that 3D-MLSI uses separatetheSQUIDintotheconstriction(inductanceL ,
µ c
supercurrents that flow only within a 2D layer. Hence, length l , width w ), the two (symmetric) bridges con-
c c
in the constriction region – in particular in the case of a taining the junctions (inductance L , length l , width
J J
large aspect ratio d/w – there may be a nontrivial 3D w ),the twocornersconnecting the constrictionandthe
c J
currentflow. As a consequence,the prediction of a satu- junction arms (inductance L ), and the bottom part of
e
rationofφ withincreasingdmaynotbecorrect. Onthe theSQUID(inductanceL ),asindicatedinFig.1. Then,
µ b
other hand, we will show below, that SΦ shows clearly a L is given by
stronger dependence on d than φ does. Hence, for the
µ L=L +2L +2L +L , (3)
optimization of S the uncertainty in φ (d) is not very c J e b
µ µ
critical. We should find L (w ,l ,d), L (w ,l ,d), L (w ,w ,d)
c c c J J J e c J
and L (l ,w ,d). Trivially, l and l should be as small
b c J c J
aspossible. Fromsimulations,using3D-MLSIwithλ =
L
IV. FLUX NOISE 335nm,19 we find the parametrization Lc(wc,lc,d)
′ ′ ≈
L l /(w d), where L = 143pHnm. This expression
c c
· ·
fits the simulated L well, within the parameter range
In order to determine the flux noise of the SQUID, we c
40nm < w < 300nm and 40nm < d < 500nm, cov-
use the theoretical expression obtained from Langevin c
ered by the simulations. We use the same parametriza-
simulations,
tion for L (w ,l ,d). For the corners we find, within a
J J J
15% variation with respect to w and w , the expres-
SΦ =f(βL)Φ0kBTL/I0R, (2) sion L L′/d, with L′ = 10J0pHnm.c Finally, we
e ≈ ′e ′′e ′·
find L L l /w d+L /d, with L = 45pHnm and
which is valid for a Stewart-McCumber parameter ′′ b ≈ b c J b b ·
βC ≡ 2πI0R2C/Φ0<∼1 and ΓβL < 0.1.12 Here, Γ ≡ yLibel=ds150pH·nm. InsertingtheseexpressionsintoEq.(3)
2πkBT/I0Φ0 is the noise parameter, and βL 2LI0/Φ0
is the screening parameter. For β > 0.4≡, f(β ) L′ l (2l +bl )
L L c j c
≈ L + +r , (4)
4(1+βL). For lower values of βL, SΦ increases. ≈ d (cid:26)wc wJ (cid:27)
The first factor to be discussed is I0R. The junction ′ ′ ′ ′′ ′
with b L /L =0.315 and r (2L +L )/L =2.45.
resistanceR canbe variedto someextentby varyingthe ≡ b ≡ e b
FortheminimizationofS ,wewilluseβ asavariable
thickness dAu of the Au layer covering the YBCO film; µ L
parameter. Since both, L and w are not independent
the maximum achievable value is the unshunted junc- J
tion normal state resistance RN (for dAu = 0). For 24◦ of each other and are related to βL, we express both as
functions of β . This will allow us to eliminate L and
YBCOgrainboundaryjunctions,I0RN values 2 3mV L
are achievable at 4.2K.13 However, such junc∼tion−s typi- wJ in the final expression for Sµ which has to be opti-
cally have hysteretic IVCs. We thus demand βC<∼1 to mized. With βL = 2I0L/Φ0 and I0 = j0wJd, we obtain
avoid hysteresis. Ideally, one would like to derive an wJ(βL,L) = Φ0βL/2j0dL. Inserting this into Eq. (4)
yields
expression for I0R as a function of wJ, d and dAu us-
ing the constraint βC<∼1 and assuming certain values for L′ l κ −1
c
the critical current density j0, unshunted normal junc- L(βL) +r 1 (5)
≈ d w − β
tion resistance times area ρ R w d and capacitance (cid:18) c (cid:19)(cid:26) L(cid:27)
N J
per junction area C′. Howe≡ver, the scaling of R with with κ(lJ,lc,j0) 2(2lJ +blc)j0L′/Φ0.
≡
wJ, d and dAu is currently not known. Furthermore, an InsertingEq.(5) intoEq.(2)andusing f(βL)=4(1+
estimate of C′ as a function of w and d, based on vari- β ) finally yields
J L
ous scaling laws available in the literature,14–16, is quite
difficult, in particular since it is difficult to determine C SΦ ≈SΦ,0dd0 wlc +r 11+βκL (6)
for underdamped YBCO GBJs and since the stray ca- (cid:18) c (cid:19) − βL
mpaacyitapnlacye dauneimtoptohretacnotmrmoloe.n1l7yOusnedthSerTotiOhe3rshuabnstdr,atwees with SΦ1/,02 ≡ 2 Φ0Ik0RBdT0L′ = 7.1nΦ0/√Hz at T = 4.2K
◦
have fabricated 24 YBCO GBJs with different junction and with I0R =q1mV. The most important result here
width and film thickness (wJ = 450nm, d = 50nm and is the scaling SΦ 1/d. This is due to the fact that the
∝
w = 330nm, d = 300nm), using the focused ion beam SQUID inductance L 1/d within the simulation range
J
(FIB) milling technique as described in10. Those junc- for d because of the i∝ncrease of the kinetic inductance
tions had only weakly hysteretic IVCs but I0R 1mV contribution with decreasing d below λL. For d>∼2λL we
≈
at T = 4.2K. Below we will find an optimum junction expectasaturationofL(d)andhenceofSΦ(d). However,
width w 260nm (for d=300nm), which is not much we will neglect this for the optimization of S , since val-
J µ
∼
below the width of those junctions. Thus, rather than ues for d>∼500nm are outside the simulation range for
introducing an ill-defined scaling of I0R with wJ and d, L(d) and since we cannot expected to produce high-
below we fix I0R and use 1mV as a realistic value. quality GBJs for such large values of d.
4

we obtain SΦ1/2 33nΦ0/√Hz, φµ = 20nΦ0/µB and
wc = 50 nm 5 0 d n m Sµ1/2 ≈ 1.6µB/√≈Hz. Corresponding SQUID parameters
are listed in Table I (“opt. device 1”).
3
This is a drastic improvement, compared to the ex-
1/2 z) 100 nm pbeersitmeexnptearlimvaelnutealSdµ1e/v2ic≈e w6i0thµBp/a√raHmzetfeorrs odur=, s6o0nfamr,,
H
/B 200 nm wc =90nm, wJ =130nm, βL =0.65, L=36pH, I0R=
2 ( 2 0.13mV, SΦ1/2 = 1.3µΦ0/√Hz and φµ = 21nΦ0/µB.10
1/ S 300 nm The main improvementof Sµ1/2 comes from SΦ1/2. Before
discussing the various factors improving this quantity,
500 nm
we should say, however, that standard YBCO SQUIDs
1/2
often have a S , which is a factor of 2–10 higher than
Φ
1 thetheoreticalpredictions18 (fortheexperimentaldevice
0 1 2 3 mentioned above S1/2 was a factor 6 higher than pre-
Φ
L ∼
dicted by Eq. (2)). In this respect we expect the theo-
FIG. 2: (Color online) Spin sensitivity Sµ1/2(βL), calculated retical S1/2 to be by a factor 2–10 too low, even if all
f(rinom50Enqm. (s7t)epast),Tw=ith4.c2oKnstfroirct1io0nvawluidetshowfcfil=m5th0incmkne(sasndd factors eΦntering SΦ1/2 are feasib∼le.
lc =200nm, lJ =400nm, j0 =3mA/µm2). One major improvement of SΦ1/2 arises from the to-
tal SQUID inductance L, which is almost an order of
magnitude lower than the inductance of our experimen-
V. OPTIMIZATION OF SPIN SENSITIVITY
tal device. This is mainly because of the much larger
VIA IMPROVED SQUID GEOMETRY
valueofd. ThelowvalueofLseemsrealistic;intermsof
geometricalfactors,w =50nmandlengthsl andl are
c c J
WithEq.(1)and(6)wecanexpressthespinsensitivity
chosensportive,butnotimpossible. Thisissupportedby
as
preliminary tests, which show that 300nm thick YBCO
films can be patterned properly by FIB milling.
d 0.7
Sµ1/2 ≈Sµ1/,0212.q3dd0d0+1(cid:18)ww0c(cid:19) rwlcc +rs11−+ββκLL opAtimnoitzhedervbailgueimofprIo0vRe,mwenhticohfisSΦ1b/y2 acomfacetsorfroomf ∼th8e
(7) higher than that of our experimental device. For nar-
with Sµ1/,02 ≡ SΦ1/,02/γµ = 0.31µB/√Hz at T = 4.2K and trhowis w(wouJld∼be1i0m0pnomss)ibalnedtothacinhie(vde.9∼H5o0wnemve)r,jtuhnectoipotnis-
with I0R=1mV.
mization yielded much wider (w 260nm) and thicker
The evaluation of Eq. (7) shows that the minimiza- J
∼
1/2 (d 300nm)junctions, forwhich we havealreadyfound
tion of Sµ (βL,d,wc,lc,lJ) requires to make lc and lJ tha∼t I0R 1mV is achievable.
assmallaspossible. Duetotechnologicallimitations,we ∼
Finally, we note that, if we take more easily achiev-
usel =200nmandl =400nmbelow. Theconstriction
widthc enters Eq. (7)Jvia the factors w0.7(l /w +r)1/2, able values d = 200nm and wc = 100nm (other input
c c c parameters are the same as for the initial optimization),
demanding w to be made as small as possible. Fur-
thermore, S1/c2 decreases with increasing d within the we still get Sµ1/2 = 2.6µB/√Hz, with parameter values
µ
simulation range; for large enough values of d, we find wJ = 375nm, L = 4.8pH, I0 = 225µA and R = 4.44Ω
Sµ1/2 ∝ 1/√d. Hence, the film thickness should be as (ssuelteinTgabSle1/.2Iifsorlepssartahmanetearsfaocfto“ropotf.2dedvifficeere2n”t).frTomhetrhee-
largeaspossible,respectingtechnologicalconstraints. As µ
value obtained for the initial input parameters, so that
mentioned at the end of Sec. IV, we expect L and hence
we are optimistic that the design discussed in this paper
SΦ tosaturateford>∼2λL. Inthiscase,whichliesoutside
our simulation range, we expect S1/2 to increase again is indeed capable of obtaining a spin sensitivity in the
µ range of a few µ /√Hz.
with increasing d, due to the degradation of φ . Hence, B
µ
we should expect a minimum in S1/2 for an optimum d,
µ
which however lies above our simulation range and also VI. CONCLUSIONS
above technologically feasible values for d.
The dependence S1/2(β ) at T = 4.2K is shown in
µ L In summary, we have performed a detailed analysis
Fig. 2 for variousvalues ofd, with fixed w =50nm. All
c of the coupling factor φ and the spectral density of
µ
cbuercvaelscuexlahtiebditfraomshaElqlo.w(7m)ainsimβum at=aκv(a1l+ue√w1h+ichκ−c1a)n. flux noise SΦ, and hence of the spin sensitivity Sµ1/2 =
With the (fixed) values for lcL,amnind lJ and with j0 = SΦ1/2/φµ of a grain boundary junction dc nanoSQUID.
3mA/µm2, we obtain κ = 0.36 and βL,min = 1.05. For Basedonthecalculationofφµ andSΦ,wederivedanex-
a reasonable value d = 300nm (cf. thick line in Fig. 2) plicitexpressionforthespinsensitivityS1/2asafunction
µ
5
d lc lj wc wJ βL L I0 R I0R j0 Lc LJ Le Lb SΦ1/2 φµ Sµ1/2
units nm nm nm nm nm pH µA Ω mV mA/µm2 pH pH pH pH nΦ0/√Hz nΦ0/µB µB/√Hz
opt.
300 200 400 50 260 1.05 4.7 233 4.30 1 3 1.9 0.74 0.33 0.62 32.7 19.9 1.65
device1
opt.
200 200 400 100 375 1.05 4.8 225 4.44 1 3 1.4 0.76 0.50 0.87 33.3 12.7 2.62
device2
TABLE I: Summary of geometric and electric nanoSQUID parameters (as defined in the text). The values for “opt. device
1” are calculated for an optimized device with film thickness d = 300nm and constriction width w = 50nm. The values for
c
“opt. device 2” are calculated for more relaxed values d=200nm and w =100nm and otherwise identical input parameters
c
for lc, lJ, I0R and (optimum) βL.
ofthegeometricandelectricalparametersofourdevices. for using these grain boundary junction nanoSQUIDs.
1/2
This allowsfor anoptimizationofS ,which predicts a
µ
spin sensitivity of a few µ /√Hz. Such a low value for
B
1/2 Acknowledgments
S can be achieved by realization of very low induc-
µ
tancenanoSQUIDswithultra-lowfluxnoiseontheorder
of only a few tens of nΦ0/√Hz. This poses severe chal- J. Nagel and T. Schwarz acknowledge support by
lenges on proper readout electronics for such SQUIDs. the Carl-Zeiss-Stiftung. This work was funded by
It remains to be shown whether or not the readout of the Nachwuchswissenschaftlerprogramm of the Univer-
such ultralow-noise SQUIDs is feasible and whether or sit¨atTu¨bingen,bytheDeutscheForschungsgemeinschaft
not the envisaged values for the spin sensitivity can also (DFG) via the SFB/TRR 21 and by the European Re-
be achievedin high fields, which is a major driving force search Council via SOCATHES.
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(2011). 19 ForYBCOSQUIDs,3D-MLSIisknowntopredicttoolow
10 T. Schwarz, J. Nagel, R. W¨olbing, M. Kemmler, inductances if a proper value for λ ( 180nm) is used.
L
≈
R. Kleiner, and D. Koelle, ACS Nano (2012), in press; Thereasonisunclearbutmayhavetodowiththelayered
DOI:10.1021/nn305431c. structure and anisotropy of YBCO. In our simulations we
11 M. Khapaev,M. Kupriyanov,E.Goldobin, and M.Siegel, haveincreasedλ tobeconsistentwiththeexperimentally
L
Supercond.Sci. Technol. 16, 24 (2003). determined values of L
12 B. Chesca, R. Kleiner, and D. Koelle, in