Beyond the effective length How to analyze magnetic interference patterns of thin film planar Josephson junctions with finite lateral dimensions R. Fermin1B. de Wit1and J. Aarts1

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Beyond the effective length: How to analyze magnetic interference patterns of thin
film planar Josephson junctions with finite lateral dimensions
R. Fermin,1B. de Wit,1and J. Aarts1
1Huygens-Kamerlingh Onnes Laboratory, Leiden University,
P.O. Box 9504, 2300 RA Leiden, The Netherlands.
(Dated: February 13, 2023)
The magnetic field dependent critical current Ic(B) of a Josephson junction is determined by the
screening currents in its electrodes. In macroscopic junctions, a local vector potential drives the
currents, however, in thin film planar junctions, with electrodes of finite size and various shapes,
they are governed by non-local electrodynamics. This complicates the extraction of parameters such
as the geometry of the effective junction area, the effective junction length and, the critical current
density distribution from the Ic(B) interference patterns. Here we provide a method to tackle this
problem by simulating the phase differences that drive the shielding currents and use those to find
Ic(B). To this end, we extend the technique proposed by John Clem [Phys. Rev. B, 81, 144515
(2010)] to find Ic(B) for Josephson junctions separating a superconducting strip of length Land
width Wwith rectangular, ellipsoid and rhomboid geometries. We find the periodicity of the inter-
ference pattern (∆B) to have geometry independent limits for LWand LW. By fabricating
elliptically shaped S−N−S junctions with various aspect ratios, we experimentally verify the L/W
dependence of ∆B. Finally, we incorporate these results to correctly extract the distribution of
critical currents in the junction by the Fourier analysis of Ic(B), which makes these results essential
for the correct analysis of topological channels in thin film planar Josephson junctions.
I. INTRODUCTION
Planar Josephson junctions are ubiquitous in mod-
ern solid state physics research, with examples ranging
from topological junctions[1–3], high Tc(grain boundary)
junctions[4,5], gated-junctions that control supercurrent
flow[6,7], graphene-based junctions[8,9], magnetic field
sensors[10–12] and, junctions with a ferromagnetic weak
link[13–15]. A major tool in analysing these junctions
experimentally is the magnetic interference pattern ob-
served in the critical current (Ic(B)), the shape and pe-
riodicity of which can reveal, using Fourier transform,
information about the underlying distribution of critical
current in the weak link[16]. Often this Fourier analysis is
carried out in terms of an effective junction length, given,
for macroscopic junctions, by 2λ+d, where λis the Lon-
don penetration depth and dthe thickness of the weak
link. This effective length originates from the Meissner
effect. However, when the junction is formed between two
superconducting thin films, with a thickness below λ, the
shielding currents running along the junction, responsible
for the shape and periodicity of the magnetic interference
of the critical current Ic(B), are no longer determined by
the Meissner effect in its macroscopic form (i.e., by the
local vector potential). Rather they are determined by
non-local electrodynamic effects[17–20].
In numerous theoretical and experimental studies, it
was found that in thin film planar junctions, Ic(B) be-
comes completely independent of λand is solely deter-
mined by the geometry of the sample[20–24]. Moreover,
John Clem provided a method to calculate Ic(B) for
planar junctions that are also restricted in their lateral
size (i.e., a Josephson junction separating a rectangu-
lar superconducting strip of width Wand length Lin
two halves)[23]. As experimental studies often deal with
finite-size geometries, his theory is highly topical at the
moment.
This paper bridges the gap between predicting the
Ic(B) of thin film planar junctions featuring finite lateral
geometry, and the correct analysis of the experimental
interference patterns used to extract the current density
distribution. First we review the technique proposed by
Clem and extend on his work by covering two more ge-
ometries: the ellipse and the rhomboid. We calculate
Ic(B) for these geometries, extract the periodicity of the
interference pattern (∆B) for different ratios of L/W ,
and find ∆Bto have two geometry independent limits for
LWand LW. By fabricating elliptically shaped
S−N−S junctions with different ratios of L/W , we ex-
perimentally verify the geometry dependence of ∆B. Fi-
nally, we adapt the well-known Fourier relation between
Ic(B) and the critical current density distribution for use
on laterally finite thin film planar junctions. We find that
altering the Fourier transform is crucial for predicting the
location of possible current channels in thin film planar
junctions.
II. REVIEW OF THE CLEM MODEL
We consider a normal metal Josephson junction (di-
mensions WJJ and d) that divides a symmetric super-
conducting thin film, having dimensions Land W, into
two halves. Figure 1shows a schematic of three of such
films, having different geometries. The junction, colored
red in Figure 1, is running along the y-direction from
−W/2 to W/2 (i.e., WJJ =W). Since we examine the
thin film limit, the screening current density is assumed
uniform along the thickness of the film, which effectively
reduces the problem to a 2D one. We specifically con-
arXiv:2210.05388v3 [cond-mat.supr-con] 10 Feb 2023
2
FIG. 1. Schematics of the three geometries used for calcu-
lating Ic(B), being (a) the rectangle, (b) the ellipse and (c)
the rhombus. The schematics resemble superconducting thin
films of width Wand length L, which are separated by a nor-
mal metal junction of width d(colored red). By numbers we
indicate different sections of the right electrode edge. The
boundary conditions of the calculations for these are summa-
rized in Table I. In (d) we show a zoom of the junction area
under the magnetic induction B
B
B=Bˆz. The dark blue path
is used as loop integral to determine Ic(B).
sider the junction to be in the short junction limit, as
the model by Clem treats an infinitesimally thin insulat-
ing tunnel junction. Furthermore, it is assumed that the
electrode the electrode dimensions are smaller than the
Pearl length, given by:
Λ = 2λ2
tfilm
(1)
Where tfilm the thickness of the superconducting films.
This implies that the self fields originating from the
screening currents are far smaller than the applied ex-
ternal field. Additionally we assume that the junction is
in the narrow limit, meaning that the junction is less wide
than the Josephson penetration length, which for planar
junctions in the thin film limit is the given by[20,21,23]:
l=Φ0tjuncW
4πµ0λ2Ic(0) (2)
Here tjunc is the thickness of the junction (not necessar-
ily equal to the thickness of the film), Ic(0) its critical
current at zero magnetic field, µ0is the vacuum perme-
ability, and Φ0is the magnetic flux quantum.
In order to calculate Ic(B), we assume a sinusoidal
current-phase relation Jx=Jcsin ϕ(y), where ϕ(y) is
the gauge-invariant phase difference over the junction,
Boundary (∇
∇
∇γ)·ˆn
ˆn
ˆnΩ
12πB
Φ0y
2−2πB
Φ0y
3 0
42πB
Φ0
W xy
Lq(W x
L)2+( Ly
W)2
5−2πB
Φ0
W y
√W2+L2
TABLE I. The Neumann boundary conditions for each elec-
trode boundary, listed by the numbering used in Figure 1.
which depends on the location along the junction. It can
be evaluated within the framework of Ginzburg-Landau
theory by considering the second Ginzburg-Landau equa-
tion, which is given as:
J
J
J=−Φ0
2πµ0λ22π
Φ0
A
A
A+∇γ=Φ0
2πµ0λ2θ(3)
Here A
A
Ais the vector potential corresponding to the ap-
plied magnetic field (B
B
B=∇
∇
∇ × A
A
A), and γis the gauge
covariant phase of the wavefunction describing the super-
conducting order parameter (given by Ψ = Ψ0eiγ [25]).
Finally, θis the gauge-invariant phase gradient (required
by the fact that J
J
J is a gauge-invariant property). ϕ(y) is
then given by integrating θacross the junction:
ϕ(y) = γ(−d
2, y)−γ(d
2, y)−2π
Φ0Zd/2
−d/2
Ax(x, y) dx(4)
In Figure 1d, we sketch a zoom of a junction, where we
specify an integration contour under a magnetic induc-
tion of B
B
B=Bˆz. By integrating ∇
∇
∇γalong this contour
and realizing that RC∇
∇
∇γdl
l
l= 2πn, where nis an integer
and sin (ϕ+ 2πn) = sin (ϕ), we find:
ϕ(y) = ϕ(0) + 2π
Φ0ydB + 2µ0λ2Zy
0
Jy(d
2, y0) dy0
(5)
Here we have used Stokes theorem to evaluate the flux
entering the contour and used the fact that the elec-
trodes are mirror symmetric (Jy(d
2, y)=-Jy(−d
2, y)).
For macroscopic junctions Jy,R(d
2, y0) = Bµ0
λLresulting
from the the Meissner effect, leading to ϕ(y) = ϕ(0) +
2π(2λ+d)B
Φ0y, where we recognize the effective junction
length. Since the junctions considered here are in the
thin film limit, we take a different approach in evaluating
3
FIG. 2. (a) Gauge-covariant phase simulated in the right electrode for a disk-shaped planar Josephson junction, normalized to
the applied magnetic field and width of the junction γΦ0/BW 2. The junction is shown as a green line. This result allows for
extracting the gauge-covariant phase along the junction. It follows the scaling of Eq. 14, and it is determined by a dimensionless
function, which is plotted in (b). (c) Shows the interference pattern calculated using the result in (a) by numerically evaluating
Equation 12 for different values of B. The typical interference pattern looks like a Fraunhofer pattern at first sight. However,
the peak height decreases less strongly than 1/B, and the width of the side lobes is larger than half of the middle lobe, which
is 10.76 mT wide. Furthermore, the width of the nth side lobe increases and reaches an asymptotic value for large values of n,
which is evident from the inset of (c), where we plot the width of the nth side lobe. The width of the fifth side lobe is used for
comparisons between simulations and experiments.
Jy(d
2, y0). First note that the supercurrent is conserved
and therefore ∇
∇
∇ · J
J
J= 0. By choosing the convenient
gauge A
A
A=−yBˆx, we find ∇ ×A
A
A=Bˆzand ∇ ·A
A
A= 0.
Therefore, the divergence of the second Ginzburg-Landau
equation (Eq. 3) reduces to:
∇2γ= 0 (6)
Therefore, we mapped the second Ginzburg-Landau
equation onto the Laplace equation. With sufficient
boundary conditions, it can be solved for a unique solu-
tion, which allows us to calculate Jy(d
2, y). The boundary
conditions arise from the prerequisite that no supercur-
rent can exit the sample at its outer boundaries. Fur-
thermore, we assume a weak Josephson coupling, mean-
ing that the shielding currents in the electrodes are far
larger than the Josephson currents between the elec-
trodes, which we approximate as Jx(d
2, y) = 0. There-
fore, we can write:
J
J
J·ˆn
ˆn
ˆnR= 0 (7)
Where ˆn
ˆn
ˆnRis the unit vector, normal to the outer edges of
the right electrode. Combined with the second Ginzburg-
Landau equation, this leads to a set of Neumann bound-
ary conditions:
4
(∇
∇
∇γ)·ˆn
ˆn
ˆnR=−2π
Φ0
A
A
A·ˆn
ˆn
ˆnR(8)
Which is sufficient to solve for γ(x, y). Next, Eq. 5allows
us to find the gauge-invariant phase difference over the
junction ϕ(y). Note that we have conveniently chosen
Ay= 0. We then find:
2µ0λ2Zy
0
Jyd
2, y0dy0= 2γd
2, y(9)
Therefore, ϕ(y) is given by the simple expression:
ϕ(y) = ϕ(0) + 2πdB
Φ0
y+ 2γd
2, y(10)
Next, the current across the junction is given by RJ
J
JdS
S
S,
yielding:
I(B) = ZW/2
−W/2
tjuncJcsin ϕ(0) + 2πdB
Φ0
y+ 2γd
2, ydy
(11)
We assume that the critical current density at zero field
is distributed uniformly over the junction, yielding Jc=
Ic(0)
tjuncW. Also, note that ϕ(0) is independent of yand
therefore merely is a phase factor. The critical current is
reached if we current-bias the junction by setting ϕ(0) =
π/2, from which follows:
Ic(B)
Ic(0) =1
W
ZW/2
−W/2
cos 2πdB
Φ0
y+ 2γd
2, ydy
(12)
We see that finding Ic(B) becomes equal to a boundary
condition problem of solving the Laplace equation in the
geometry of the electrodes. Indeed, the solution is com-
pletely determined by the geometry of the sample and is
independent of λ.
III. COMPARING DIFFERENT GEOMETRIES
As it is not trivial to find a general analytical solution
to the boundary problem of Eq. 6for the ellipsoid and
rhomboid geometries, we solve the Laplace equation nu-
merically using COMSOL Multiphysics 5.4. We define
the right electrode geometry in 2D, divided into a trian-
gular grid. Crucial for correctly solving Eq. 6, is a grid
size that is small enough to capture small changes in γ
and, on the edges, ˆn
ˆn
ˆnR. We found a maximum element size
(i.e., the grid edge size) of 0.01 ln (1 + L/W ) nanometer
to be a good compromise between computation time and
precision. Using trigonometry we evaluate A
A
A·ˆn
ˆn
ˆnRfor each
geometry and list the corresponding boundary conditions
in Table I(here the numbering corresponds to the num-
bers in Figure 1). In the Appendix, we provide a full
derivation of each of the boundary conditions.
A. Simulation results
Clem showed that the analytical solution for the rect-
angular geometry is an infinite series of sines and hyper-
bolic tangents[23]. For the rectangle, this leads to the
maximum in γd
2, yto occur at W/2, which can be ap-
proximated as:
FIG. 3. Dimensionless measure of the period ∆B(the width
of the fifth side lobe) of the calculated interference pattern
Ic(B) for the three geometries. In (a) we plot this value on
log-log scale versus the aspect ratio L/W , in (b) it is plot-
ted versus the total electrode area A(i.e, combined area of
left and right electrode), scaled by the W2. Figure (b) re-
veals two limits for ∆Bfor LWand LW. The
first corresponds to the limit of an infinite superconduct-
ing strip ∆B= 1.842Φ0/W 2, whereas in the latter we find
∆B= 2Φ0/A. Contrary to ∆B,Ic(B) itself is not geometry
independent in this limit.
摘要：

Beyondthee ectivelength:Howtoanalyzemagneticinterferencepatternsofthin lmplanarJosephsonjunctionswith nitelateraldimensionsR.Fermin,1B.deWit,1andJ.Aarts11Huygens-KamerlinghOnnesLaboratory,LeidenUniversity,P.O.Box9504,2300RALeiden,TheNetherlands.(Dated:February13,2023)Themagnetic elddependentcritical...

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