Properties of field induced Josephson junction(s)

Properties of ﬁeld induced Josephson junctions
Krzysztof Pomorski
University of Warsaw, Nagoya University
kdvpomorski@gmail.com
December 14, 2016
Krzysztof Pomorski (UW,UN) Properties of FIJJ December 14, 2016 1 / 54

Overview
1 Macroscopic quantum states
2 Essence of Josephson eﬀect
3 Concept of simpliﬁed FIJJs
4 Generalization of FIJJs
5 Mathematical description of Josephson junctions
6 Numerical method and results

Macroscopic quantum states:superconductivity and
superﬂuidity
Figure: Transport without dissipation (R → 0) [Onnes 1911], Meissner eﬀect
[Wikipedia], movement of liquid without viscosity [Wikipedia].

Josephson eﬀect: tunneling junction
Figure: Tunneling Josephson junction [Nature 47, J.You and F.Nori, 2011] and its
electrical circuit and weak-link Josephson junction. Diﬀerent of phase of SCOP
Θ = ΘR − ΘL determines transport properties. Most simple model assumes
ψL =
√
ρLeiΘL
, ψR =
√
ρR eiΘR
.
H = HL + HR + HT , |ψ = ψL |L + ψR |R (1)
H = EL |L L| + ER |R R| + ET (|L R| + |R L|) (2)
I(t) = I0 sin(Θ) +
1
R 2e
dΘ
dt
+
C
2e
d2Θ
dt2
(3)

Weak link Josephson junction systems

Tunneling vs weak link Josephson junctions
Figure: I-V characteristic of tunneling JJ [left] vs weak-link JJ [C-center] and I-V
characteristics in microwave ﬁeld for weak-link [R-right], [C,R] from L.Gomez.
Tunneling vs weak-link Josephson junctions:
Two quantum coherent quantum systems interacting in perturbative
vs non-pertubative way.
sinusoidal vs non-sinusoidal relation between phase diﬀerence and
electric current.
no-current presence vs continous electric current for certain voltageKrzysztof Pomorski (UW,UN) Properties of FIJJ December 14, 2016 6 / 54

Central motivation
Figure: Deﬁnition of Field Induced Josephson Junctions (FIJJ).

[PSS B, K.Pomorski and P.Prokopow, 2012]

Concept of simplified field induced Josephson junctions
Figure: Physical system 1 and its simplification.
Figure: Physical system 2 and its simplification
[’Towards robust coupled field induced JJs’,Arxiv:1607.05013]

Generalization of ﬁeld induced Josephson junction
Figure: Stage I: deformation of sc cable. Stage II: Deformed sc cable + arbitrary
shaped polarizing cable [ArXiv:1607.05013].
Figure: Stage III:Coupled sc cables in any net of polarizing cables. Stage IV:
hybrid quantum system [ArXiv:1607.05013].

Magnetic ﬁeld entangling superconducting lattice
Figure: [Upper ﬁgures]: Electrical ways of controlling topologies of magnetic
entangler placed in superconducting lattice of cables (BdGe cables). [Picture
below]: 2 dim BdGe cables and polarizing cable lattice [ArXiv:1607.05013].

Asymptotic states and scattering region in FIJJ/uJJ

Analytic formulas for FIJJ with insulator
From Ginzburg-Landau we can write the equation for electric current
density in the form (c1 > 0) as
jx,(y,z)(x, y, z) = −c1Ax,(y,z)(x, y, z)|ψ(x, y, z)|2
. (4)
Using Maxwell equation we obtain for time independent vector potential
B = × A equation of the following structure
× ( × Ax,(y,z)(x, y, z)) = µ0jx,(y,z)(x, y, z) (5)
Using the relation a × (b × c) = b(ac) − c(ab) we obtain
×( ×Ax,(y,z)(x, y, z)) = ( Ax,(y,z)(x, y, z))− 2
Ax,(y,z)(x, y, z) (6)
that can be written after using Maxwell equation as
( Ax,(y,z)(x, y, z))− 2
Ax,(y,z)(x, y, z) = −c1Ax,(y,z)(x, y, z)|ψ(x, y, z)|2
.
(7)

( A1(2)x,(y,z)(x, y, z)) − 2
A1(2)x,(y,z)(x, y, z) =
−c1A1(2)x,(y,z)(x, y, z)|ψ0|2
.
( [A1 + A2]x,(y,z)(x, y, z)) − 2
[A1 + A2]x,(y,z)(x, y, z) =
−c1[A1 + A2]x,(y,z)(x, y, z)|ψ0|2
.
System with translational symmetry has current ﬂow as
2
A1(2)x (y, z) = +c1A1(2)x (y, z)|ψ0|2
. (8)
since ( x A1(2)(y, z)x ) = 0. Ax(y) that has translational symmetry is as
Ax (y) = a1cosh(k1y) + b1sinh(k1y), where k1 =
√
c1|ψ0|,y is from -d to d.
I0 =
+d
−d
jx (y)dy = −c1|ψ0|2
+d
−d
(a1cosh(k1y) + b1sinh(k1y))dy

Class of structures to be considered
Figure: Important message: FIJJs has built-in shielding current what implies that
they are α Josephson junctions so CPR is shifted by arbitrary phase. In general
they are weak-links and non-sinusoidal Josephson junctions.

Simple FIJJ system in Ginzburg-Landau formalism
Figure: [Left]: Simplest case of FIJJ. [Right]: Case on FIJJs network.
London relation gives jx = I0 = −c1Ax (x)|ψ(x)|2 = constants and
Az(x) =
k1Ip
(x2+a2
0)1/2 since A(r ) ≈ j(r)dr/|r − r’|. Ginzburg-Landau
equation has the structure [|ψ0| = −α
β for bulk sc]:
α(x)ψ(x, t) + β|ψ(x, t)|2ψ(x, t) + 1
2m ( i
d
dx − 2e
c Ax )2ψ(x, t) = γ d
dt ψ(x, t) .

View of FIJJs in Ginzburg-Landau formalism
We introduce two functions f1(x) and f2(x) that we can engineer.
f1(x) = (Ay (x)2
+ Az(x)2
), f2(x) = (∆y
d
dx
Ay (x) + ∆z
d
dx
Az(x)), (9)
so effective α(x) = α + 1
2m (2e
c )2f1(x) −
2
2m a2
0f2(x)2 and effective GL
equations becomes modified and real-valued for
f (x) = |ψ(x)|-superconducting order parameter.
where (a1, a2, a3, a4) are positive constants so GL is extended
−(
d2f
dx2
) + βf 3
+ α1(x) + a1f1(x) − a2(f2(x))2
f + a3f2(x)
I0
f
+ a4
I2
0
f 3
= 0(10)

Mathematical description of FIJJs-time dependent case
TDGL-Time Dependent Ginzburg-Landaua equation is considered. We
have a time dependent vector potential ﬁelds
Ax (x, y0, z0, t), Ay (x, y0, z0, t), Az(x, y0, z0, t) and I0(t). We need to
calculate
γ
d
dt
|ψ| + V (x, t)|ψ| + ia0(
x
x0
d
dt
Ax (x , y, z, t)dx + (∆y
d
dt
Ay (x, y, z, t) + ∆z
d
dt
Az (x, y, z, t)))|ψ| =
= (α + β|ψ|
2
)|ψ| + e
−i(Θx +Θy +Θz ) 1
2m
(
i
d
dx
−
2e
c
Ax )
2
(|ψ|e
i(Θx +Θy +Θz )
+
1
2m
(
2e
c
)
2
(A
2
y + A
2
z )|ψ| (11)
and
I0(t) =
dAx (x, y0, z0, t)
dt
σ − c1Ax (x, y0, z0, t)|ψ(x, y0, z0, t)|2
(12)
and
Ex (x, y0, z0, t) = −
dAx (x, y0, z0, t)
dt
σ − φ(x, y0, z0, t) (13)
where V (x, t) =
x
x1
Ex (x , y0, z0, t)dx .

Scheme of relaxation method
Gradient method is basing on the following iterations
xn+1
i = xn
i − i
dF(x)
dxn
i
|x=(xn
1 ,...,xn
k ), (14)
where i are constants and vector (xn
1 , ..., xn
k ) gives value of physical ﬁelds in n-th
steps. Relaxation method is basing on the following iteration scheme
δ
δXi
F[Xi (x)] = ηi
dXi (x)
dt
. (15)
In discretized form we have
Xi (tn+1) =
∆tn
ηi
δ
δXi
F[Xi (tn)] + Xi (tn), (16)
where ηi are constants.

Figure: SCOP obtained by the relaxation method [left] and assumed distribution
of α coeﬃcient [right].
Figure: Free energy functional F [left] and average error with iterations [right].

Ginzburga-Landaua model-thermodynamical derivation
Figure: Superconducting order parameter and minimization of functional F.
F[ψ, A] =
1
2m
|(
i
d
dx
−
2e
c
Ax (x))ψ(x)|2
+
α
2
|ψ(x)|2
+
β
4
|ψ(x)|4
, (17)
Setting functional derivatives δ
δXi
F = 0 to zero with respect to
Xi = (ψ, A) we have the following equations of motion
0 =
1
2m
(
i
d
dx
−
2e
c
Ax (x))2
ψ(x) + αψ(x) + β|ψ(x)|2
ψ(x),
j(r) =
e
m
(ψ†
(r)(
i
d
dx
−
2e
c
Ax (x)) + c.c).

Boundary conditions in GL theory
(
i
d
dx
−
2e
c
Ax )ψ(x) = 0 (18)
(
i
d
dx
−
2e
c
Ax )ψ(x) =
1
b(y)
ψ(x) (19)
(bases for non-abrupt uJJ). ’Boundary conditions on the GL eqns for
anisotropic superconductors’, E.A.Shapoval, Sov. Phys. JETP 61, 1985
’General boundary conditions for quasiclassical theory of Superconductivity in the
diﬀusive limit:application to strongly spin-polarized systems’ Eschrig et al.

Concept of unconventional Josephson junction.
Figure: Distribution of SCOP ψ(x, y, 1
2 (zmax + zmin)). Geometrical dimension
Lx : Ly : Lz = 0.4(0.6) : 20 : 20, in terms of units of superconducting coherence
lenght.
[PSS B, K.Pomorski and P.Prokopow, 2012]

Cylindrical/spherical uJJ(FIJJ)
[Perspective on basic architectures and properties of unconventional and
ﬁeld induced Josephson junction devices, K.Pomorski, P.Prokopow, 2013,
International Journal of Microelectronics and Computer Science]

Extended Ginzburg-Landaua formalism
F = Fs + FM + Fs−M,
FM = dr(a(T)|M(r)|2
+
b(T)
2
|M(r)|4
+ C| M(r)|2
), (20)
Fs = dr(
α
2
|ψ|2
+
β
4
|ψ|4
+
1
2m
|(
i
−
2e
c
A)ψ|2
+
(curlA)2
4π
) (21)
Fs,M = dr(γ|ψ(r)|2
|M(r)|2
+ (| M(r)|2
|ψ(r)|2
)
+
µ
2m
|(
i
−
2e
c
A)ψ(r)|2
|M(r)|2
+ curl(A)M) (22)
It is quite essential to tract K.Kubokiego and K.Yano derivation of GL
from extended Hubbard model [Journal of the Physical Society of Japan,
’Microscopic Derivation of Ginzburg-Landau Equations for Coexistent
States of Superconductivity and Magnetism’, 2013].

Figure: Transition between tunneling Josephson junction and weak-link JJ
obtained by extended GL model [PSSB, K.Pomorski, P.Prokopow, 2012].

Bogoliubov-de Gennes equations (BdGe)
Hamiltonian H of free particle [no superconducting order parameter]
H =
1
2m i
d
dx
−
2e
c
Ax (x, y, z)
2
+
i
d
dy
−
2e
c
Ay (x, y, z)
2
+
1
2m i
d
dz
−
2e
c
Az (x, y, z)
2
+ V (x, y, z).
From BCS theory we have
+Hun(x, y, z) + ∆(x, y, z)vn(x, y, z) = nun(x, y, z)
−H†
vn(x, y, z) + ∆(x, y, z)†
un(x, y, z) = nvn(x, y, z)
∆(x, y, z) = −V1
n
un(x, y, z)v†
n (x, y, z)(1 − 2f ( n)),

Local density of states (LDOS) for uJJ
Figure: Local density of state (LDOS) for diﬀerent temperatures T1 and T2
(T1 < T2), [K.Pomorski et al., PSSB 2012]
N(r, E) = −
n
(f ( n − E)|un(r)|2
+ f ( n + E)|vn(r)|2
) (23)

Topological defects in Sc-Fe system
Both Abrikosov and Josephson vortices can be induced in FIJJ as given in
[K.P EJTP 2010, K.P. PhD thesis 2015, G. Carapella et al, Nature 2016 ].

Triangle and tunability of FIJJ properties: critical current,
CPR, α continuous shift in CPR, Density of States, heat
capacity, transmission coeﬃcient, conversion between
singlet and triplet current [EJTP, KP 2010]

Andreev reﬂection at interface between normal and
superconducting state

Andreev bound states in JJ

Andreev reﬂection in 2 dimensions

Full Counting Statistics (FCS) for FIJJs/uJJs
Figure: Effective reflection coefficient can be determined so Full Counting
Statistics can be determined.
In such case we can obtain the scattering matrix tunned by properties of
uJJ/FIJJ and tune its properties in continous way. Having scattering
matrix we can get cummulant generating function F(χ) and use
Lesovik-Levitov formula for getting cummulants of noise.

RCSJ in description of uJJ (FIJJ)
Figure: We are biasing guJJ (granular unconventional Josephson junction) via
V(t) between A i B or electric current I(t). (I20,I21): s=(I0,I0), s1 = (I21 = 0.6I0),
s2 = (I21 = 0.1I0), s3=(I20 = 0.1I0 = I21).

Figure: (I20,I21): s=(I0,I0), s1 = (I21 = 0.6I0), s2 = (I21 = 0.1I0),
s3=(I20 = 0.1I0 = I21), C = 0, basing by electric current.

Figure: Vortices in short Josephson junction:no-self ﬁeld eﬀects

Basic concept of Rapid Single Quantum Flux electronics
Figure: Way of pushing of magnetic ﬂux out of superconducting loop [left] and
concept of Josephson transmission line [right]. The logical gate NOT [left] and its
implementation [right] is given below.

RAM cell for Rapid Single Flux quantum electronics

Scattering vector potential Ay (x, y), Az(x, y) in RAM cell
Figure: Vector potential in Scenario (I, II)[Ay], I[Az] in states(up—down=1—0).

References
[1]. B.D.Josephson, Possible new effects in superconductive tunnelling,
PL, Vol.1, No. 251, 1962
[2]. K.Likharev, Josephson junctions Superconducting weak links, RMP,
Vol. 51, No. 101, 1979
[3]. K.Pomorski and P.Prokopow, Possible existence of field induced
Josephson junctions, PSS B, Vol.249, No.9, 2012 [4]. K.Pomorski, PhD
thesis: Physical description of unconventional Josephson junction,
Jagiellonian University, 2015
[5]. K.Pomorski, H.Akaike, A.Fujimaki, Towards robust coupled field
induced Josephson junctions, arxiv:1607.05013, 2016
[6]. K.Pomorski, H.Akaike, A.Fujimaki, Relaxation method in description
of RAM memory cell in RSFQ computer, Procedings of Applied
Conference 2016 (in progress)
[7]. J.Gelhausen and M.Eschrig, Theory of a weak-link
superconductor-ferromagnet Josephson structure, PRB, Vol.94, 2016
[8]. K.K. Likharev, Rapid Single Flux Quantum Logic
(http://pavel.physics.sunysb.edu/RSFQ/Research/WhatIs/rsfqre2m.html)

[9]. Proceedings of Applied Superconductivity Confence 2016, plenary talk
by N.Yoshikawa, Low-energy high-performance computing based on
superconducting technology
[10]. A.Y.Herr and Q.P.Herr, Josephson magnetic random access memory
system and method, International patent nr:8 270 209 B2, 2012
[11]. J.A.Blackburn, M.Cirillo, N.Gronbech-Jensen, A survey of classical
and quantum interpretations of experiments on Josephson junctions at
very low temperatures, arXiv:1602.05316v1, 2016
[12]. Current driven transition from Abrikosov-Josephson to Josephson-like
vortex in mesoscopic lateral S/S/S superconducting weak links, G.
Carapella, P. Sabatino, C. Barone, S. Pagano and M. Gombos, Nature,
2016
[13].Fluxon Propagation on a Josephson Transmission Line, A. Matsuda
and T. Kawakami Phys. Rev. Lett. 51, 694,1983

Publikacje wlasne
1 K.Pomorski, P.Prokopow, International Journal of Microelectronics and Computer Science (2013), Vol.4, No.3, strony:
110-115, Perspective on basic architecture and properties of unconventional and ﬁeld induced Josephson junction
devices
2 K.Pomorski, P.Prokopow, Physica status solidi B 249, No 9 (2012), strony: 1805-183, Possible existence of ﬁeld
induced Josephson junctions + backover
3 K.Pomorski, P.Prokopow, Electronic Journal of Theoretical Physics (2010), Vol.7, No. 23, strony 85-121, Towards the
determination of properties of the unconventional Josephson junction made by putting non-superconducting strip on
the top of superconducting strip
4 K.Pomorski, P.Prokopow, Bulletin de la societe et des sciences et des lettres de Lodz, Recherches sur les deformations
(2012), Numerical solutions of nearly time independent Ginzburg-Landau equations for various superconducting
structures, part I
(2013), Numerical solutions of nearly time independent Ginzburg-Landau equations for various superconducting
structures, part II
(2011), vol. LXI, no. 2, Numerical solutions of time-dependent Ginzburg-Landau equations for various
superconducting structures
7 K.Pomorski, M.Zubert, P.Prokopow, Transport properties of dirty unconventional Josephson junction devices in RCSJ
model, pierwsze miejsce na konferencji ICSM2014
8 K.Pomorski, M.Zubert, P.Prokopow, Numerical solutions of nearly time-independent Ginzburg-Landau equation for
various superconducting structures: III. Analytical solutions and improvement of relaxation method, Bulletin de la
societe et des sciences et des lettres de Lodz, Recherches sur les deformations

Cytowana literatura
1 K. Kuboki, Microscopic derivation of the Ginzburg-Landau equations for coexistent states of superconductivity and
magnetism, arXiv:1102.3329 (2011)
2 A. Maeda, L. Gomez, Experimental Studies to Realize Josephson Junctions and Qubits in Cuprate and Fe-based
Superconductors, Journal of Superconductivity and Novel Magnetism 23, (2010).
3 K.K. Likharev, Superconducting weak links, Review of Modern Physics 51, (1979)
4 T. Clinton, Advances in the development of the magnetoquenched superconducting valve: Integrated control lines and a
Nb-based device, Journal of Applied Physics 91 (2002)
5 B.D. Josephson,Possible new eﬀects in superconductive tunnelling, Physics Letters 1 (1962)
6 J.S. Reymond, P. SanGiorgio, Tunneling density of states as a function of thickness in superconductor/strong
ferromagnet bilayers, Physical Review B 73 (2006)
7 X.B. Xu, H. Fanohr, Vortex dynamics for low-k type superconductors, Physical Review B 84 (2011)
8 M. Thinkham, Introduction to superconductivity, Dover Publications, (2004)
9 J.Q. You, F. Nori, Superconducting Circuits and Quantum Information, Physics today (2005)
10 N. Cassol-Seewald, G. Krein, Numerical simulation of GinzburgLandau-Langevin equations, Brazilian Journal of Physics
37 (2007)
11 J.J.V. Alvarez, C.A. Balseiro, Vortex structure in d-wave superconductors, Physical Review B 58 (1998)

Cytowana literatura-metody numeryczne
1. Metoda: Variable link.
’Numerical solution of the time-dependent Ginzburg-Landau equation for a
superconducting mesoscopic disk:Link variable method’, J.Barbara-Ortega
et al. , IOP, 2008
2. Algorytm relaksacyjny zastosowany w jednym wymiarze dla GL.
3. Metoda wygrzewania (annealing).
4. Split-step method stosowana w r´ownaniu Grossa-Pitaeveskiego (GP).

Congratulates
Mr. Krzysztof Pomorski,
As the presenting author of
Transport Properties of Dirty Unconventional Josephson
Junction Devices in RCSJ Model
For the first place in the
“Best Poster Award”
To be continued ...!!!

Research plan
1 Design of superconducting RAM cell for RSFQ computer of small
dimensions
2 Extension and validation of M.Eschrig [PhysRevB.94.104502] results
(He cites my PSSB 2012 work)
3 Creation platform supporting last Nature publication on crossover
from Abrikosov vortex to Josephson vortices [G. Carapella et al,
Nature 2016]
4 Determination of properties of robust field induced Josephson junction
5 Application of canonical quantization procedure to one dimensional
field induced Josephson junction [continuation of work with dr hab.
Adam Bednorz (FUW),arXiv:1502.00511 and its generalization
published in IOP paper by K.P and A.B, 2016]
6 Determination of Current Phase Relation for FIJJs with Fe strip on
the top of superconductor with insulator in-between [already some
analytical results are known.]
7 Determination of properties of topological Meissner effect
8 Validation of canonical procedure for systems showing topological JJs

Research plan

Essence of topological Meissner eﬀect

Stages of topological Meissner eﬀect

Properties of field induced Josephson junction(s)

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