Self-field effects in window-type Josephson tunnel junctions

1.9 MB·English

by  Roberto Monaco

Self-field effects in window-type Josephson tunnel junctions
Roberto Monaco∗
Istituto di Cibernetica del CNR, Comprensorio Olivetti, 80078 Pozzuoli, Italy
and Facolta` di Scienze, Universita` di Salerno, 84084 Fisciano, Italy
Valery P. Koshelets
Kotel’nikov Institute of Radio Engineering and Electronics,
Russian Academy of Science, Mokhovaya 11, Bldg 7, 125009 Moscow, Russia.
Anna Mukhortova
Kotel’nikov Institute of Radio Engineering and Electronics,
3
Russian Academy of Science, Mokhovaya 11, Bldg 7, 125009 Moscow, Russia and
1
Moscow Institute of Physics and Technology, Institutskii per.,
0
9, 141700 Dolgoprudny, Moscow Region, Russia.
2

n
Jesper Mygind
a
J DTU Physics, B309, Technical University of Denmark, DK-2800 Lyngby, Denmark
(Dated: December 11, 2013)
7
ThepropertiesofJosephsondevicesarestronglyaffectedbygeometricaleffectssuchasthoseasso-

] ciatedwiththemagneticfieldinducedbythebiascurrent. ThegenerallyadoptedanalysisofOwen
n
andScalapino[Phys. Rev.164,538(1967)]forthecriticalcurrent,I ,ofanin-lineJosephsontunnel
o c
junctioninpresenceofanin-planeexternalmagneticfield,H ,isrevisitedandextendedtojunctions
c e
- whose electrodes can be thin and of different materials. We demonstrate that the asymmetry of
r themagneticdiffractionpattern,I (H ),isascribedtothedifferentelectrodeinductancesforwhich
p c e
we provide empirical expressions. We also generalize the modeling to the window-type junctions
u
used nowadays and discuss how to take advantage of the asymmetric behavior in the realization of
s
. some superconducting devices. Further we report a systematic investigation of the diffraction pat-
t
a terns of in-line window-type junctions having a number of diverse geometrical configurations and
m madeofdissimilarmaterials. Theexperimentalresultsarefoundinagreementwiththepredictions
and clearly demonstrate that the pattern asymmetry increases with the difference in the electrode
-
d inductances.
n
o PACSnumbers: 74.50.+r,85.25.Cp,74.25.N-
c
[
CONTENTS Acknowledgments 12
1
v
References 13
4 I. Introduction 1
7
2 II. Modeling 2
1 A. Background review 2 I. INTRODUCTION
.
1 B. Boundary conditions 3
0 C. Symmetric biasing 4 In the first years after the discovery of the Josephson
3 D. Asymmetric biasing 5 effect it was realized that, as the size or the critical cur-
1
E. Normalized units 5 rent of a planar Josephson tunnel junction increases, the
:
v 1. Magnetic diffraction pattern 5 influence of the magnetic field induced by the Joseph-
Xi 2. Current diffraction pattern 6 son current itself becomes more and more important1–4.
Significantadvancesinthemodelingoftheso-calledself-
r
a III. Window-type junctions 6 field effects were made by Owen and Scalapino5 (OS)
A. Normalized units 8 who considered the geometrical configuration most suit-
abletoanalyzetheself-field,namely,theone-dimensional
IV. Experiments 9 in-line geometry shown in Fig. 1(a) where the externally
A. Samples 9 applied bias current, I, flows in the direction of the long
B. Magnetic diffraction patterns 9 dimension, L. The figure also shows the coordinate sys-
C. Current diffraction patterns 9 tem used in this work. Long Josephson tunnel junctions
D. Comments 11 (LJTJs),i.e.,junctionswhosedimensionLperpendicular
E. Flux diffraction patterns 11 toanexternallyappliedmagneticfield, H , islargecom-
e
pared to the Josephson penetration depth, λ , behave
j
V. Conclusions 12 like an extreme type-II superconductor; they exhibit a
2
Meissnereffectinweakmagneticfields,andvortexpene-
tration starts at a critical field H . This behavior can be
c
probed into by studying the magnetic field dependence
of the maximum tunneling supercurrent, I , because the
c
Meissner region and the vortex structure are reflected
in the I versus H curve in a very characteristic way.
c e
The strength of the current-induced field is measured
by the ratio of L/λ. The OS analysis was restricted
j
to the ideal case in which the junction electrodes had
the same width and were made of the same bulky ma-
terial. Despite these severe restrictions the OS modeling
has been used without further consideration the design
ofanydevicebasedonLJTJs. Oneofthepurposeofthe
present work is to generalized the OS theory taking into (a)
account geometrical or electrical differences in the elec-
trodes. Wewillextendourinterestalsotojunctionsfab-
ricated with the so-called whole wafer processes, which
make use of a tri-layer structure from which individual
junctions are later defined. Due to the large diffusion of
these reliable processes, window junctions have become
widespread while, in contrast, naked (or bare) junctions
are a rarity. While the interaction of the Josephson tun-
neljunctionwithitsembeddingcircuitryhasreceivedan
(b)
exhaustive and adequate attention6–9, the changes due
to the self-field effects still remain an unexplored topic,
FIG. 1. (a) Schematic of an in-line junction with symmetric
albeittheexactknowledgeofthecurrentinducedfieldis
current bias, I, in the presence of an external magnetic field,
highlydesirablefortherealizationofsuperconductorde- H , applied along the Y-axis.; (b) Simplified vertical cross
e
vices based on LJTJs such as, for example, oscillators10, section (drawn not to scale) of a in-line Josephson tunnel
magnetic sensors11 and rectifiers12. Besides modeling, junction. Thejunctiontopelectrodeisingray,whilethebase
a thorough experimental investigation of the self-field electrodeisinblack;thetunnelinginsulatinglayerinbetween
effects has been carried out for window-type Nb-based is hatched. Also shown is the coordinate system used in this
LJTJs having in-line configuration and electrodes of dif- work.
ferent widths, thicknesses and materials. The results are
well aligned with the expectations and demonstrate that
window-type LJTJs can be designed with a greater flex-
flowing,respectively,inthebottomandtopjunctionelec-
ibility in the control of the self-field effects. The issue of
trodes within a distance of the order of the penetration
the asymmetry in the threshold curves of Josephson de-
depthfromthefilmsurfacesandparalleltotheinsulating
viceswasintroducedlongagoanditsimportancewasrec-
layer. X ∈ [−L/2,L/2] is the laboratory spatial coordi-
ognized in determining the performances of amplifiers13
nate. Throughout the paper the subscripts b and t refer
andmagnetometers14. Currently,thesearchfornewand
to the base and top electrode, respectively. Further the
betterwaystoachieveandcontroltheasymmetryisstill
currents are positive when they flow from the left to the
on-going15,16.
right. Inthevastmajorityofpracticalcasesthejunction
electrodes are comparable or thinner than their penetra-
tion depths and the currents can be well assumed to be
II. MODELING
uniformly distributed over the film cross section. I and
b
I alsotakeintoaccountthescreeningcurrents, I , that
t sc
A. Background review circulate to expel the magnetic field from the interior of
the junction.
In its simplest form, the area of a planar Josephson
For in-line LJTJs, it is important to distinguish between
tunneljunctionisdefinedbytheoverlapoftwosupercon-
the symmetric and asymmetric biasing: in the former,
ducting films with rectangular cross section and weakly
the bias current, I, enters at one extremity and exits at
coupled through a thin tunneling barrier. Fig. 1(b) de-
theother5,17,18: I (L/2)=I (−L/2)=I andI (−L/2)=
picts the vertical cross section of an in-line Josephson t b t
I (L/2) = 0, while in the latter, the bias current enters
tunneljunction. Anin-planeexternalmagneticfield,H , b
e and exits from the same extremity1,17,19,20: I (−L/2) =
is applied in the Y-direction, i.e., perpendicular to the t
−I (−L/2)=I and I (L/2)=I (L/2)=0.
junction length. The very thin insulating layer between b b t
thesuperconductingfilmshaslengthLandwidthW (not The gauge-invariant phase difference φ of the order pa-
shown). I (X) and I (X) denote the local supercurrents rameters of the superconductors on each side of the tun-
b t
3
nel barrier obeys the Josephson equations21 L ≥ πλ /2). As reported by several authors17,19,24, the
J
largest supercurrent carried by a very long in-line LJTJ
is4I ,whereI ≡J Wλ isacharacteristicjunctioncur-
0 0 c J
JZ(X)=Jcsinφ(X), (1a) rent (generally from a fraction of a milliampere to a few
κ∇φ(X) =H×nˆ, (1b) milliamperes) and depends on the junction normalized
length, (cid:96)≡L/λ , as I ((cid:96))=I tanh(cid:96)/2.
J 0 0
in which JZ is the Josephson current density and Jc is As already stated, for small fields a LJTJ behaves as a
its maximum (or critical) value, which is assumed to be perfect diamagnet by establishing circulating screening
uniform over the barrier area. Eq.(1b) states that the currents which maintain the interior field at zero. This
phase gradient is everywhere proportional to the local Meissner regime is reflected by a linear decrease of I
c
magnetic field H and parallel to the barrier plane. Here with H . The threshold curves, I (H ), have been the
e c e
nˆ is the versor normal to the insulating barrier separat- subject of many analytical, numerical and experimen-
ing the superconducting electrodes and κ ≡ Φ0/2πµ0dm tal works25–27. It is well known that the critical mag-
has the dimension of a current (Φ0 is the magnetic flux netic field is7,28 Hc =Φ0/πµ0λJdm =2JcλJdj/dm. The
quantumandµ0isthevacuumpermeability). Neglecting gradual crossover of Hc from short ((cid:96) << 2π) to long
the insulating barrier thickness22, ((cid:96) > 2π) junctions has been numerically computed29
and it was found to be well described by the empirical
relationship: H ((cid:96)) = H coth(cid:96)/π in very good agree-
d d c c
d =λ tanh b +λ tanh t , (2) ment with the experimental findings reported in Ref.28.
m b 2λ t 2λ
b t Note that in the small junction limit, (cid:96) → 0, we re-
isthejunctionmagneticthicknessinandλb,tanddb,tare, cover Hc =Φ0/µ0Ldm and the Fraunhofer-like magnetic
respectively, the bulk magnetic penetration depths and diffraction pattern (MDP).
thicknesses of the films. Eq.(2) reduces to d =λ +λ
m b t
in the case of thick superconducting films (d > 4λ ).
b,t b,t
The net current, I, crossing the tunnel barrier is given B. Boundary conditions
by
In this section we will derive the boundary conditions
for an in-line junction. Let us define the sheet induc-
(cid:90) L/2
tances:
I ≡W J (X)dX. (3)
Z
−L/2
Throughout this paper we will limit our interest to the T ≡ µ0λb,t tanh db,t ; (6a)
zero-voltagetime-independentstate;thiscanbeachieved b,t 2 2λ
b,t
as far as I is smaller than the junction critical cur- µ λ d
rent, Ic. We also assume that the junction width, W, Cb,t ≡ 02b,t coth2λb,t , (6b)
is smaller than the Josephson penetration depth1,21,23, b,t
(cid:112)
λ = Φ /2πµ d J ,settingthelengthunitofthephys-
J 0 0 j c and observe that for any value of the ratio d /λ , it is
a a
ical processes occurring in the Josephson junction; here
C ≥ µ λ /2 ≥ T with a = b,t; in the thin film limit
dj is the junction current thickness22 (see later) Ta = µ0da/4 andaC = µ λ2/d is the film kinetic30
a 0 a a 0 a a
sheetinductanceduetotheinertialmassofmobilecharge
dj =λbcothλdb +λtcothλdt ≥dm. (4) cµa0rλraiecrosth(Cdoao/pλear,wphaiilres)C.a−FTuart=heµr0mλoarces3c1h,dCa/aλ+a. TTahen=,
b t
according to Eqs.(2) and (4), µ d ≡ 2(T + T ) and
0 m b t
For thick film junctions, d = d = λ + λ . By its µ d ≡T +T +C +C .
j m b t 0 j b t b t
definition, κ≡(d /d )J λ2. In the case of a Josephson structure made by two films
j m c J
It is well known1,5 that combining Eqs.(1a) and (b) of the same width, W, Weihnact22 provided a general
withthestaticMaxwell’sequations,astaticsine-Gordon expression for the phase derivative as the sum of three,
equation is obtained that describe the behavior of a one- in general, independent terms
dimensional in-line LJTJ
Φ dφ I I
0 =µ d H + t (C +T )− b(C +T ). (7)
λ2 d2φ =sinφ(X). (5) 2π dX 0 m e W t b W b t
JdX2
Moreprecisely,ifweintroducethemagneticfluxchanges,
Equation(5) was first introduced by Ferrel and Prange1 ∆Φ , in the Josephson barrier proportional to the
j
in 1963 in the analysis of asymmetrically biased in-line changes of the Josephson phase32, ∆Φ = Φ ∆φ/2π,
j 0
LJTJs; few years later, OS5 reported an extensive study each term in Eq.(7) represents a magnetic flux per unit
of its analytical solutions in terms of elliptic functions length along the X-direction. Let us assume first that
for symmetrically biased in-line junctions (provided that the junction is unbiased, so that the screening current,
4
I , only circulate in the electrodes to expel any exter- term is the magnetic energy stored in the opposite elec-
sc
nal field, H , from the interior of the junction; charge trode. For a thin-film (symmetric) transmission line the
e
conservation requires I (X)=−I (X)=I (X). Then: ratiooftheinternalmagnetictothekineticenergygoes26
t b sc
as(2d/λ)2/12,i.e.,forthinfilmsthekineticenergydom-
inates. At this stage we are still allowed to neglect the
Φ dφ I
0 =µ d H +µ d s. (8) magnetic energy stored in the very thin insulating tun-
2π dX 0 m e 0 jW
nel barrier. It is worth to stress that Eqs.(9) are valid
as far as the energy of the magnetic fields outside the
By resorting to the definition of inductance as flux per
unit current33, we can identify L ≡ µ d /W as the transmission line is negligible.
JTL 0 j
inductance per unit length for the screening current23, ThecaseofasuperconductingstripofwidthW carrying
a current I over a a superconducting shield was consid-
where the subsript JTL stands for Josephson Transmis-
sion Line. The same expression33–36 was obtained for a eredalongago39. Byresortingtothetheoremofimages,
itwasshownthat,ifthestripisclosetoathicksupercon-
strip-type superconducting transmission line (STL) con-
ducting shield, the field between them is approximately
sisting of a dielectric layer sandwiched between two su-
uniform and equals to H = I/W and, outside the edge
perconductors of width W; here, the current fed into the
of the film, it falls to zero within a distance of the or-
topelectrodereturnsbackinthebaseelectrodeactingas
der of the strip-to-shield distance. The shield screen-
agroundplane,namely,I (X)=−I (X). Insuchaway,
b t
ing current, whose integral is equal to I, must therefore
all fields are identically zero outside the waveguide (the
be uniformly distributed over that portion of the shield
only difference being that, for JTLs, the dielectric thick-
surface covered by the film. On the other side of the
nesscanbeneglected). ThescreeningcurrentinEq.(8)is
current-carrying strip the field is zero. In the absence of
negative for positive external field, and vice versa. Since
a shield, it is easy to show that the field on both sides
µ H istheexternallyappliedmagneticfluxdensity, B ,
0 e e
of a freestanding strip will be equal and opposite and of
and µ I /W is the magnetic flux density, B , induced
0 sc sc
mean magnitude H = I/2W. Hence, by bringing a su-
in the barrier by the current I , Eq.(8) help us to un-
sc
perconducting plane close to a current-carrying strip, we
derstand why we named d and d as the junction mag-
m j
double the field between the plane and the strip and re-
netic and current thickness, respectively. Likewise µ d
0 m
duceiteverywhereelse. Thiseffectcanbeusedtoreduce
and µ d can be seen as, respectively, the magnetic and
0 j
the effective inductance of superconducting elements by
current sheet inductances of the junction. To find the
depositing them on top of another insulated supercon-
boundaryconditionsforEq.(5)whenthein-linejunction
ductor. These considerations allow the use of Eqs.(9)
is biased is not an easy task, because it requires the sep-
when a current locally flows either in the bottom or in
arate knowledge of the properties of each electrode. All
previous analytical approaches33–35 dealt with transmis- the top junction electrode, as it occurs in a symmetri-
cally biased in-line junction. However, if in a given place
sion line structures in which the current flowing into the
I (X) and I (X) are comparable and flow in the same
top electrode returns back in the lower electrode acting b t
direction, then a considerable magnetic field is induced
as a ground plane; in such cases the inductance per unit
in the outer space (unless the films are thin).
length of the transmission line is the only required pa-
rameter. However, for our purposes we have to find the
inductances, L and L per unit length of of the bottom
b t
C. Symmetric biasing
and top electrodes, respectively. Resorting to Eq.(7), we
easily identify them as:
Toobtainthe local magneticfield, H (X), wehaveto
Y
divide Eq.(7) by µ d ; in particular, at the boundaries
0 m
C +T
L ≡ b,t t,b, (9) of a symmetrically biased LJTJ we have
b,t W
so that L +L =L : this is a well known identity in (cid:12) (cid:18) (cid:19)
the theoryb of ttwo-coJnTdLuctor transmission lines23,26,37. If κddXφ(cid:12)(cid:12)(cid:12) ≡HY −L2 =He−µLbdI =He−ddj LLb WI ,
both films are thick, L = L = µ (λ +λ )/2W; in the X=−L 0 m m JTL
b t 0 b t 2
opposite case, L = µ λ2 /Wd , that is when both
b,t 0 b,t b,t
films are thin their inductance is essentially of kinetic
(cid:12) (cid:18) (cid:19)
ohraivgeinb30e.enCmlaesrsgiecdalliyn, tthheeirboptatroamllelancodmtboipnaitnidonu2c3t,a3n8cseos κddXφ(cid:12)(cid:12)(cid:12)X=L≡HY L2 =He+ µL0dtIm=He+ ddmj LLJTtLWI .
2
that the role played by each supercurrent separately (10)
was lost; however, to correctly describe a biased in-line For L = L = L /2 (and d = d ), we recover the
b t JTL j m
Josephson junction it is mandatory to keep the distinc- symmetric OS boundary conditions5 that were generally
tion, since, each electrode transports its own supercur- adopted thereafter for untrue symmetry reasons. In the
rent, and, in general, I (X) (cid:54)= −I (X). The first term early eighties2,18, the reported asymmetric behavior of
b t
in Eqs.(9) takes into account the magnetic and kinetic samples that were believed to be symmetric led many
energy in the film carrying the current, while the second experimentaliststoabandonthein-linegeometryinfavor
5
of the overlap one. Indeed, the last term in Eqs.(10) is the above reasons, in the rest of this paper we will only
the magnetic field induced in the tunnel barrier by the consider the more interesting case of the symmetrically
current I flowing in the lower or in the upper electrodes. biased in-line junction with no ground plane.
Ampere’s law applied along the barrier perimeter in the
X-Y plane requires that the magnetic fields at the two
endsofthejunctionsdifferbytheamountoftheenclosed
E. Normalized units
current: I = W[H (L/2) − H (−L/2)]. It is easy to
Y Y
showthatEqs.(1a),(3),(5)and(10)fulfillAmpere’slaw.
From the boundary conditions (10), it follows that In normalized units of x ≡ X/λJ, the differential equa-
tion (5) becomes:
(cid:18) (cid:19) (cid:18) (cid:19)
L L d I L −L
H − +H =2H + j t b.
Y 2 Y 2 e d W L
m JTL φ =sinφ(x), (12)
xx
The last term, vanishing when L = L , has been omit-
b t
with x ∈ [−(cid:96)/2,(cid:96)/2]. Normalizing the currents to I ≡
ted in all previous analysis of LJTJs. The difference, 0
J Wλ and the magnetic fields to H /2 ≡ J λ d /d ,
L −L , in the self-inductances of the upper and lower c J c c J j m
t b
the boundary conditions (10) for a symmetrically biased
junction electrodes is responsible of the distortion of
the experimental curve2; it equals µ [λ csch(d /λ ) − LJTJ read
0 t t t
λ csch(d /λ )] and reduces to µ (λ − λ ) in case of
b b Lb 0 t b
thick electrodes. We identify the dimensionless parame-
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:96) L i (cid:96) L i
terα≡(Lt−Lb)/LJTL asadirectmeasureoftheasym- φ − ≡h =h − b ; φ ≡h =h + t .
metry of the system, with −1 ≤ α ≤ 1. The asymmetry x 2 l e LJTL x 2 r e LJTL
canbeascribedtodifferencesintheelectrodethicknesses (13)
and/ormaterials, thatis, itcanhavegeometricaland/or With these notations, the normalized critical magnetic
electric origins. field h of a short Josephson junction is 2π/(cid:96).
c
We like to point out that Eqs.(10) are very general and
canbeusedtocorrectlydescribetheso-calledself-fieldef-
fectsoccurringinanysubstantiallyin-lineLJTJinwhich
1. Magnetic diffraction pattern
the bias current or a part of the bias current enters or
leaves the junction from one or even both its ends. Un-
fortunately, their implementation requires the separate Setting hl and hr at their extreme values ±2 in
knowledge of the bottom and top electrode inductances Eqs.(13), we obtain the normalized MDP, ic(he), in the
per unit length (rather than just their sum). Meissner regime
(cid:40)
D. Asymmetric biasing i (h )= Lb2/+LhJeTL for −2≤he ≤hmax (14)
In the case of asymmetric bias, the boundary condi- c e Lt2/−LhJeTL for hmax ≤he ≤2,
tions are
with h = 2α being the field value which yields the
max
H (cid:18)−L(cid:19)=H − dj Lb+Lt I , maximum critical current ic(hmax)=4. The second-last
Y 2 e d L W equalityturnsouttobeveryusefulintheexperimentsto
m JTL
determinetheasymmetryparameterfromtheanalysisof
the junction MDP:
(cid:18) (cid:19)
L
H =H , (11)
Y 2 e
h H
α= max = max. (15)
and, recalling that Lb+Lt =LJTL, we end up with the 2 Hc
boundaryconditionsfirstfoundbyFerrelandPrange1for
thick electrode LJTJs. We observe that the difference in Fig. 2 shows the theoretical I (H ) for a very long in-
c e
theinductancesdoesnotplayanyroleinthiscase. Many line Josephson junction in the Meissner regime and for
pioneeristic experiments19,40 were also carried out with different values of α; the junction critical current, I , is
c
in-line junctions built over a large and thick supercon- normalized to I and the critical magnetic field, H , is
0 c
ducting ground plane that makes the current I to flow thetheoreticalfieldvaluethatfullysuppressesthecritical
b
in the lower skin layer of the bottom electrode provided current. Weremarkthatthepatternsareantisymmetric,
itisathickfilmandisshieldedbytheupperpart. When I (−H ) = −I (H ) and piecewise linear. They have
c e c e
thisisthecase,L mustbesettozeroinEqs.(10)or(11), different absolute slopes |dI /dH | on the left and right
b c e
sothatthebiasingconfigurationbecomesirrelevant. For branches, respectively, equal to µ d /L and µ d /L .
0 m b 0 m t
6
CDP slope with a current gain41,42 g ≡dI /dIˆ=di /dˆι.
c c
From Eq.(17), it follows
(cid:40)
g = LLbt for −ˆιc ≤ˆι≤ˆιmax (18)
−1 for ˆι ≤ˆι≤ˆι .
max c
With L > L , then ˆι is negative and so, in zero
t b max
external field, we have a unitary current gain. However,
large current gains can be achieved with samples having
a large L /L ratio by control current biasing the upper
t b
electrode to have ˆι<ˆι .
max
III. WINDOW-TYPE JUNCTIONS
FIG. 2. Theoretical magnetic diffraction patterns I (H ) Weihnact’s Eq.(7) was derived considering that the
c e
of a very long in-line symmetrically current-biased Joseph- thickness of the tunnel barrier is much smaller that typ-
son junction in the Meissner regime for different values of ical penetration depths so that the magnetic field in be-
Λ ≡ L /L . The critical current, I , is normalized to tween the electrodes is in the Y-direction, is uniform
t t JTL c
I0 = JcWλJ and the externally applied magnetic field, He, along Z, and only depends on the X coordinate. Ac-
to the critical magnetic field Hc. cordingly, there are no fringing effects and, at the same
time, the magnetic energy stored in the dielectric layer
can be neglected. In addition, it was assumed that the
2. Current diffraction pattern base and top electrode match the width of the barrier,
W = W = W. However, these conditions are not
b t
Let us consider now the case when the junction criti- fulfilled in nowadays window-type planar tunnel junc-
cal current is modulated by the transverse field induced tions whose electrodes have quite different widths; typi-
by a stationary current entirely flowing in either the top cally, W > W > W, that is, two strips, of total width
b t
or bottom electrode. This so-called control current tech- W = W −W, exist along the junction sides where the
i t
nique has been used to produce local magnetic fields for top electrode overhangs the base electrode, but no tun-
digital applications of Josephson circuits since 196940. If neling is possible due to the thick insulating layer. In
the control current, Iˆ, is injected into, say, the top elec- case W > W , then W = W −W; however, through-
t b i b
trode, then the magnetic field at the inner surfaces is out this paper we will assume that the base electrode
Hˆ =L Iˆ/µ d , as in Eq.(10). The value of the control is wider or at most matches the top electrode. The
t t 0 m
currentforwhichthejunctioncriticalcurrentI vanishes so-called idle region is formed after the patterning of
c
is Iˆ =Φ /πλ L and will be named the critical control the wiring film which provides the electrical connection
c 0 J t
current. Likewise, Iˆ will be that value of Iˆ which to the junction top electrode. In this region surround-
max
maximizes the critical current, I . In normalized units, ing the tunnel area the insulation between the bottom
c
hˆ ≡2Hˆ /H =L ˆι/L ,whereˆι≡Iˆ/I . Replacingh and top electrode is provided by an oxide layer typi-
t t c t JTL 0 e
with hˆ , Eqs.(14) become cally made of a native anodic oxide of the base electrode
t and/or a deposited SiO layer. The total thickness of
x
this passive layer, d , is comparable or even larger than
ox
ic(ˆι)=(cid:40)LLbt(ˆιc+ˆι) for −ˆιc ≤ˆι≤ˆιmax (16) tbheeceolmecptaroradbelepewniethtratthieonjudnecpttiohns,wλibd,tt,h,anWd.mTighhetnatlhsoe
ˆι −ˆι for ˆι ≤ˆι≤ˆι ,
c max c screening currents in the upper and lower electrodes dis-
tribute over an effective width larger than the junction
where ˆι ≡ Iˆ/I = 2L /L and ˆι ≡ Iˆ /I = width. This leads to a new smaller junction inductance
c c 0 JTL t max max 0
ˆι (L −L )/L . It is worth mentioning that per unit length L(cid:48) corresponding to the so-called6–8
c b t JTL JTL
inflation of the Josephson penetration depth occurring
in window-type Josephson tunnel junctions. In Refs.6,7,
Iˆmax = Hmax (17) each electrode was modeled as a parallel combination of
Iˆc Hc two stripes having quite different oxide thicknesses re-
sulting in a rather involved expression of the effective
indicating that the degree of asymmetry is the same for current thickness. From a phenomenological point of
the magnetic and the current diffraction pattern (CDP). view, a unidimensional junction with a lateral idle re-
In other words, as far as the modulation of the critical gion behaves as a bare junction having the same width,
current concerns, a control current is equivalent to an W, and length, L, but with an effective current thick-
externallyappliedfield. Itisthenpossibletoidentifythe ness, d(cid:48), given by a proper combination of the current
j
7
thicknesses of the naked junction, d and of the idle re- where K are fringing field factors that remain to be
j b,t
gion, d , namely: d(cid:48) =d /[1+(d /d )(W /W )]<d , so determined: with W ≥ W we expect K ≥ K ≈ 1.
i j j j i i j j b t b t
(cid:113) Eqs.(20a) and (b) rely on the fact that a current I (I )
that L(cid:48) ≡ µ d(cid:48)/W and λ(cid:48) ≡ Φ /2πµ d(cid:48)J > λ . t b
JTL 0 j J 0 0 j c J flowinginthetop(bottom)electrodeinducesamagnetic
The prime symbol (’) labels the parameters relative to field I /W (I /W ) between the electrodes. The last
t t b b
the window junction. Taking into account the thickness, termsineachofthepreviousequationstakeintoaccount
dw, of the wiring layer, it is: the magnetic energy stored in the thick oxide layer. In
Eq.(19)wealsohaveconsideredthatthemagneticthick-
d (d +d ) nessoutsidethetunnelingareaisd(cid:48) =d +d (butthis
d =λ coth b +λ coth t w , m m ox
j b λ t λ is a secondary field focusing effect26,43). Consequently,
b t
for any window-type Josephson tunnel junction the crit-
d d icalfieldislowerthanthatofanakedone; forawindow-
di =λbcothλb +λtcoth λw +dox. type LJTJ it is
b t
Outside the junction area the idle region takes the
form of a microstrip-line made by two electrodes of fi- H(cid:48) =Φ /πµ d(cid:48) λ(cid:48) <H . (21)
c 0 0 m J c
nite width and thickness. Chang35 considered the case
ofasuperconductingstriptransmissionline,i.e.,astruc- Ultimately, the boundary conditions Eqs.(10) still hold,
ture consisting of a finite-width superconducting film of if we replace d , d , L and L for the naked junc-
m j b,t JTL
thickness h over an infinite (and thick) superconducting tion with their respective counterparts, d(cid:48) , d(cid:48), L(cid:48) and
m j b,t
ground plane. As far as the strip linewidth, w, exceeds L(cid:48) , for the window junction. It should be clear that,
JTL
abouttheinsulationthickness,t,theinductanceperunit this time, the inductance per unit length, L(cid:48) , of the
STL
length, L , was analytically derived as superconducting (non-Josephson) transmission line out-
STL
sidetheJosephsonareaisnotgivenbythesumL(cid:48) +L(cid:48) ,
b t
(cid:18) (cid:19) otherwise the magnetic energy stored in the dielectric
µ d w t
LSTL = 0wjK−1 h,h , layer would be counted twice. Naturally, in general,
L(cid:48) +L(cid:48) not even matches L(cid:48) . For the samples fab-
b t JTL
with the fringing-field functional K, first introduced in ricated by means of a tri-layer process, the top electrode
Ref.39, being always larger than unity; the fringing fields to be considered in Eqs.(20a) and (b) is the wiring layer
have the effect to increase the system energy, i.e., to re- which does not necessarily cover the whole barrier area,
duce the inductances per unit length. K decreases with so that it could as well be W <W.
t
the the ratio w/h and increases with t/h. Definitely, Indeed, the in-line approximation fails, if W or W
b t
Chang’sresultscanbeusedwhenWb >>Wt,but,unfor- are much larger than W, because part of the bias cur-
tunately, no analytical expression is available for a STL rent enters the tunnel barrier also along the long junc-
whenbothelectrodeshavefiniteandcomparablewidths. tiondimension,L,andamixedin-line-overlapmodel44,45
ThesituationcomplicatesifIb(X)(cid:54)=−It(X). Therefore, shouldbeadoptedinwhichtheJosephsonphaseφobeys
we resort again to Eq.(7) to determine the flux per unit the time-independent perturbed sine-Gordon equation
length just outside the tunneling area
d2φ(X) I (X)
Φ2π0ddXφ =µ0d(cid:48)mHe+L(cid:48)tIt−L(cid:48)bIb. (19) sinφ(X)=λ2J dX2 + JovcW .
In other words, for a window-type LJTJ the left (right) I (X) is the distribution of the externally applied bias
ov
boundary condition is determined by the property of current, I = (cid:82)L/2 I (X)dX, giving the fraction of
the longitudinal idle region to the left (right) of the left ov −L/2 ov
the bias current I that enters the junction through the
(right) junction end. The longitudinal idle region also
long dimension (overlap component). Nevertheless, the
acts on the junction as a capacitive load, which does not
junctionconfigurationisstillsubstantiallyin-lineand,as
play any role, as far as the static properties concern. We
such,theboundaryconditionsdiscussedsofarstillapply
propose that, provided that W slightly exceeds W , the
b t with the only caveat that the in-line component of the
inductances per unit length of the bottom and top elec-
bias current is I −I . The consequences resulting from
trodes at the extremities of a symmetrically biased in- ov
the asymmetric boundary conditions imposed by a non-
line junction (where the bias current, I, at the junction
uniformexternalmagneticfieldattheextremitiesofboth
extremities either flows is the bottom or in the top elec-
short and long Josephson junctions have been recently
trode) are
investigated46,47;thefieldasymmetryisresponsiblefora
degeneracy of the critical field, H , that was numerically
c
L(cid:48) ≈ Ct+Tb+µ0dox, (20a) demonstratedandexperimentallyverified. Tobetterun-
t K W derstand this point, we sketch in Fig. 3(a) the geometry
t t
C +T +µ d of a 100µm long and 1.2µm wide, symmetrically biased
L(cid:48) ≈ b t 0 ox. (20b)
b KbWb window-type Nb-AlN-NbN junction with Wt = 4µm
8
(cid:18) (cid:96)(cid:48)(cid:19) (cid:18)(cid:96)(cid:48)(cid:19)
φ − ≡h(cid:48) =h(cid:48) −Λ(cid:48)i; φ ≡h(cid:48) =h(cid:48) +Λ(cid:48)i,
x(cid:48) 2 l e b x(cid:48) 2 r e t
(22)
wherewehaveintroducedthereducedinductancesΛ(cid:48) ≡
b,t
(a) L(cid:48) /L(cid:48) and, in general, Λ(cid:48) +Λ(cid:48) (cid:54)=1. Setting h(cid:48) and
b,t JTL b t l
h(cid:48) at their extreme values ±2 in Eqs.(22), we obtain the
r
MDP i(cid:48)(h(cid:48)) in the Meissner regime lobe
c e
(cid:40)2+h(cid:48)e for −2≤h(cid:48) ≤h(cid:48)
i(cid:48)c(h(cid:48)e)= 2−Λh(cid:48)b(cid:48)e for h(cid:48) ≤eh(cid:48) ≤m2a,x (23)
Λ(cid:48) max e
t
with h(cid:48) ≡ 2(L(cid:48) −L(cid:48) )/(L(cid:48) +L(cid:48) ) being the normal-
max b t b t
ized field value yielding the maximum critical current
i(cid:48)(h(cid:48) ) = 4/(Λ(cid:48) +Λ(cid:48)) = 4L(cid:48) /(L(cid:48) +L(cid:48) ). Still we
c max b t JTL b t
can define the asymmetry parameter
(b) α(cid:48) ≡ L(cid:48)t−L(cid:48)b = Hm(cid:48) ax (24)
L(cid:48) H(cid:48)
JTL c
FIG. 3. (a) Electrode configuration of a 100µm long and
1.2µm wide, window-type Josephson tunnel junction hav- and express the ratio of the inductances per unit length
ing quite different electrode widths: W = 4µm and W = as
t b
30µm; (b) Its experimental magnetic diffraction patterns,
I (H ),whensymmetricallybiased: I =1.10mA,H =
c e c,max c1 L(cid:48) 1−α(cid:48)
±580A/m and Hc2 =±830A/m. t = ; (25)
L(cid:48) 1+α(cid:48)
b
when α(cid:48) is positive then L(cid:48) <L(cid:48) and vice versa. If the
t b
modulating magnetic field is the transverse field induced
and W =30µm. Fig. 3(b) shows theMDP in which the
b byacontrolcurrent,Iˆ,flowinginthetopelectrode,then
two critical fields, H and H , are quite evident. Since
c1 c2 Hˆ(cid:48) = L(cid:48) Iˆ/µ d(cid:48) = d(cid:48)Λ(cid:48)Iˆ/d(cid:48) W. In normalized units,
in real measurement Ic never vanishes before the next t t 0 m j t m
lobe ”grows up”, the critical field values are obtained by hˆ(cid:48)t ≡ 2Hˆ(cid:48)t/Hc(cid:48) = Λ(cid:48)tˆι(cid:48), where ˆι(cid:48) ≡ Iˆ/I0(cid:48) is the reduced
extrapolatingto zero the linearbranches of the principal control current. Replacing h(cid:48) with Λ(cid:48)ˆι(cid:48) in Eq.(23), we
e t
lobes; this is indicated by the gray dashed lines in this get the CDP of a window-type junction
andforthcomingplots. Asimilarpatternasymmetrywas
first reported in Ref.4 and was erroneously ascribed to a
laonwexutnriefomremcitayseoifnthwehbicahrrtiheerapsryompemrteitersy. iHsomwaeivnelry,dthuiestios i(cid:48)c(ˆι(cid:48))=(cid:40)22−+ΛΛΛ(cid:48)b(cid:48)t(cid:48)tˆιˆι(cid:48)(cid:48) foforr−ˆι(cid:48)ˆι(cid:48)c ≤≤ˆι(cid:48)ˆι≤(cid:48) ≤ˆι(cid:48)mˆιa(cid:48)x, (26)
Λ(cid:48) max c
the very different widths of the electrodes. The experi- t
mentalfindingsthatwillbepresentedinthenextSection where ˆι(cid:48) ≡ 2/Λ(cid:48) and ˆι(cid:48) ≡ h(cid:48) /Λ(cid:48) = 2(Λ(cid:48)/Λ(cid:48) −
c t max max t b t
refer to window junctions having Wb ≥ 1.5Wt for which 1)/(Λ(cid:48) +Λ(cid:48)). Thecurrentgain, g(cid:48) ≡di(cid:48)/dˆι(cid:48), forain-line
b t c
thedegeneracyofthecriticalfields,althoughobservable, window-type junction is
is small enough to be ignored.
(cid:40)Λ(cid:48)t = L(cid:48)t for −ˆι(cid:48) ≤ˆι(cid:48) ≤ˆι(cid:48)
g(cid:48) = Λ(cid:48)b L(cid:48)b c max (27)
−1 for ˆι(cid:48) ≤ˆι(cid:48) ≤ˆι(cid:48).
max c
Since most real samples have W > W , in the case of
A. Normalized units b t
electrodeshavingthesamepenetrationdepth,thenL(cid:48) >
t
L(cid:48) , suggesting that, for applications it is preferable to
b
The boundary conditions for a window junction are ob- inject the control current into the top electrode which,
tained dividing Eq.(19) by µ d(cid:48) . In normalized units of typically, with respect to the bottom electrode, has a
0 m
x(cid:48) ≡ X/λ(cid:48) , with x(cid:48) ∈ [−(cid:96)(cid:48)/2,(cid:96)(cid:48)/2] where (cid:96)(cid:48) ≡ L/λ(cid:48) is smallerwidthandsoalargerinductanceperunitlength.
J J
the normalized length of the window junction; normaliz- Furthermore,L(cid:48) canbeincreasedbysimplyreducingthe
t
ing the currents to I(cid:48) ≡ J Wλ(cid:48) and the magnetic fields wiring width at the junction extremities. This can be an
0 c J
to H(cid:48)/2=J λ(cid:48) d(cid:48)/d(cid:48) , Eqs.(13) become advantage for some applications.
c c J i m
9
IV. EXPERIMENTS Sample J d d d λ H d(cid:48) d(cid:48) λ(cid:48) H(cid:48)
c i m j J c m j J c
kA/cm2 nm nm nm µm A/m nm nm µm A/m
The findings of the previous Section can be experi-
Nb-Nb 11 400 160 180 3.7 890 380 89 5.3 260
mentally verified by measuring the magnetic and cur-
Nb-NbN 3.6 780 270 530 3.6 520 470 210 5.9 190
rent diffraction patterns of symmetrically biased in-line
window-type LJTJs. In particular, Eqs.(24) and (25) al-
TABLE I. Relevant electric parameters (at 4.2K) for the
lowthedeterminationoftheratiooftheinductancesper
Nb/Al /Nb and Nb/AlN/NbN Josephson tunnel junctions.
ox
unit length which can be compared with the expected Thevaluesareapproximatedtotwosignificantdigitsandthe
value from Eqs.(20a) and (b): prime symbol (’) denotes the parameters relative to the win-
dowjunctions. Allsampleshavethesamelength,L=100µm,
width, W =1.2µm, and idle region width, W =2.8µm.
L(cid:48) W C +T +µ d i
t =σ b t b 0 ox (28)
L(cid:48) W C +T +µ d
b t b t 0 ox
carriedoutat4.2K.Theexperimentalset-uphasalready
whereσ ≡K /K isafactorwhichtakesintoaccountthe
film asymmetbry36t and is equal to 1 when the electrodes been described elsewhere29.
havethesamegeometricalandelectricalparameters. We
will test the validity of the Eq.(28), for samples having
B. Magnetic diffraction patterns
W =W and W =1.5W .
b t b t
The upper panel of Fig. 4 shows the geometry of a in-
A. Samples line junction realized by equal width electrodes, W =
b
W = 7µm. The electrodes are vertically shifted, but
t
In order to have a solid benchmark of data, we have they preserve the symmetry with respect to the junction
compared the static properties of thin-film Nb-Nb and axis. In Fig. 4(b) we report the MDP of such junction
Nb-NbN samples having the same geometrical config- when fabricated with the all-Nb technology. The small
uration. The NbN films used in this study were de- positive asymmetry, α(cid:48) = Hm(cid:48) ax/Hc(cid:48) ≈ 7%, can be fully
posited by dc magnetron sputtering of a pure Nb tar- ascribed to the difference in the electrode thicknesses,
get in an argon and nitrogen sputtering gas. Although dw (cid:39)2.5db, and implies that L(cid:48)t is sligthly smaller than
the samples were not heated intentionally, the substrate L(cid:48)b. From Eqs.(28) with σ =1, we have L(cid:48)t/L(cid:48)b =0.90,
temperature slightly rose during deposition; neverthe- in quite good agreement with the value of about 0.88
less surface temperature never exceeds 120oC allowing obtained from Eq.(25). Fig. 4(c) is the counterpart of
for a lift-off process. The superconducting transition Fig. 4(b) for a Nb-NbN junction having the same ge-
temperature (determined as the resistivity midpoint by ometry. Now the asymmetry parameter, α(cid:48) ≈ −0.53,
a four-point method) for samples deposited under op- is negative indicating that L(cid:48)t > L(cid:48)b. The factor σ in
timal conditions is 15.5K. Polycrystalline Niobium Ni- Eqs.(28) must be set to 1.9 to reproduce the measured
tride has a dirty-limit penetration depth48,49, λNbN(T = ratio L(cid:48)t/L(cid:48)b =3.3.
4.2K) = 370nm, several times larger than that of epi- Figs. 5(b) and (c) compare the MDPs of two symmet-
taxially grown Niobium, λ (T =4.2K)=90nm. rically biased junctions having the geometrical configu-
Nb
Allsampleswerelongwindow-typein-linejunctionshav- ration, depicted in Fig. 5(a) (Wb = 1.5Wt = 6µm), but
ing physical length L = 100µm and width W = 1.2µm madebydifferentmaterials,respectively,Nb-NbandNb-
with a 1.4µm wide idle region on each side, so that NbN. We now have asymmetries of α(cid:48) ≈ −30% and
W = 2.8µm. They were fabricated with the same pa- −89%, respectively. For the former sample the asym-
i
rameters for the deposition and anodization of the base metry is mainly ascribed to the difference in the elec-
electrodes, as well as for the deposition of the passive trode widhts, while for the latter it is further enhanced
layer: d =190±10nmandd =220±10nm. Further, by the large penetration depth of the upper (NbN) elec-
b ox
the top and wiring layers had quite similar thicknesses: trode. To reproduce these values, we had to set σ in
d = d = 65±5nm, d = 470±10nm and Eqs.(28) to 1.4 and 3.3, respectively. From Eqs.(25) we
t,Nb t,NbN w,Nb
dw,NbN = 390 ± 10nm. Therefore, the material used have L(cid:48)t,NbN ≈4.6L(cid:48)t,Nb ≈8.5L(cid:48)b.
for the top (and wiring) electrode was the only notable
difference. More importantly the wiring layers were Nb-
Nb samples, d ≈ 6λ , and thin for the Nb-NbN C. Current diffraction patterns
w,Nb Nb
ones, d ≈ λ , in which case the inductance is
w,NbN NbN
predominantly kinetic. We also note that the thickness, Fig. 6(a) shows the CDP of the same Nb-NbN sam-
d , of the passive layer is comparable to the electrode ple of Fig. 5(c) when the control current flows in the top
ox
thicknesses. Table I reports the relevant electric param- (wiring)electrodeandinabsenceofanexternallyapplied
eters (at 4.2K) for the Nb/Al /Nb and Nb/AlN/NbN field. Comparing it to the MDP in Fig. 5(c), we observe
ox
Josephson tunnel junctions used for this work. For all asimilarqualitativebehaviorand,asexpectedaccording
samples it was L >> λ(cid:48) < W. All measurements were to Eq.(17), the measured asymmetry parameters are the
j
10
(a)
(a)
(b)
(b)
(c)
(c)
FIG. 5. Experimental magnetic diffraction patterns, I (H ),
c e
for two symmetrically biased in-line long Josephson tunnel
FIG. 4. Experimental magnetic diffraction patterns, I (H ),
c e
junctions having the same geometrical configuration (W =
for two symmetrically biased in-line long Josephson tunnel t
4µm and W = 6µm), but made of different materials:
junctions having equal width electrodes (W = W = 7µm), b
b t
(a) electrode configuration; (b) Nb-Nb; I = 2.23mA,
but made of different materials: (a) electrode configuration; c,max
H = −175A/m and H = ±585A/m and (c) Nb-NbN;
(b) Nb-Nb sample: I = 2.28mA, H = 50A/m and max c
c,max max
I =1.03mA, H =−545A/m and H =±680A/m.
H =±710A/mand(c)Nb-NbN sample: I =1.08mA, c,max max c
c c,max
H =−465A/m and H =±870A/m.
max c
agree with Eq.(29). Other results in agreement with the
same within the experimental uncertainty of a few per-
cent. Furthermorethecurrentgains,g(cid:48),measuredbythe theory (and not reported here) where obtained for the
Nb-Nb sample of Fig. 5(a). The wide range of linear-
slopes of the left and right branches of the Meissner lobe
ity of the CDPs is very attractive for the realization of
are in agreement with the expectation of Eq.(27).
cryogenic current amplifiers with a large dynamic range
Ifthecontrolcurrentisfedtothebottom,ratherthanthe
especially because large slopes can be achieved in the
top electrode, the normalized inductance per unit length
of the junction bottom electrode, Λ(cid:48), must replace Λ(cid:48) in [−ˆιc,ˆιmax] interval. The asymmetry in the electrode in-
b t ductance can help to significantly improve the gain of
the numerator of the fractions in Eq.(26), so that
a current amplifier over a device with symmetric induc-
tances. The most practical way to achieve this is to in-
(cid:40)
1 for −ˆι(cid:48) ≤ˆι(cid:48) ≤ˆι(cid:48) jectthesignalcurrentintheelectrodehavingthelargest
g(cid:48) = c max (29)
−L(cid:48)b for ˆι(cid:48) ≤ˆι(cid:48) ≤ˆι(cid:48). inductance per unit length. In practice, the top one is
L(cid:48)t max c chosen which can be made thin, narrow and of a ma-
wherenowˆι(cid:48) ≡2/Λ(cid:48) andˆι(cid:48) =2(Λ(cid:48)/Λ(cid:48)−1)/(Λ(cid:48)+Λ(cid:48)). terial with a large penetration depth, such as polycrys-
c b max t b b t
Fig.6(b)istheCDPofthesamesamplewhenthecontrol tallineNbN;possibly,onthecontrary,thebaseelectrode
current is injected into the base electrode. The asymme- shouldbethick,moderatelywiderandmadeofamaterial
try parameter is still unchanged and the pattern slopes having a low penetration depth, such as epitaxial Nb.