Download S–I–S its S–I transition C.D. , Kwangmoo Kim

Physica C 498 (2014) 21–29
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Physica C
journal homepage: www.elsevier.com/locate/physc
S–I–S Josephson junction with a correlated insulator below
its S–I transition
C.D. Porter a, Kwangmoo Kim a,b, D. Stroud a,⇑
a
b
Department of Physics, The Ohio State University, Columbus, OH 43210, USA
School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Republic of Korea
a r t i c l e
i n f o
Article history:
Received 9 September 2013
Received in revised form 20 December 2013
Accepted 31 December 2013
Available online 9 January 2014
Keywords:
Josephson junction array
Superconductor–insulator transition
Proximity effect
Mean-field theory
Quantum Monte Carlo
a b s t r a c t
We consider a Josephson junction composed of two superconducting (S) regions separated by an insulating (I) region, but with the special property that the S and the I regions are superconducting films respectively above and below the superconducting–insulating (S–I) transition. To calculate the properties of this
junction, we describe the system using an inhomogeneous quantum rotor Hamiltonian with a coupling
energy J and spatially varying charging energy U. The ratio J=U is chosen so that it is above the critical
value for an S–I transition in the two superconducting regions, but below it in the insulating regime.
Using both mean-field theory and perturbation theory, we show that the phase order parameter is finite
in the S region and decays exponentially into the I region. Thus, the order parameter, which would be
zero in the I region in isolation, is instead rendered nonzero by the adjacent S region, because of a proximity effect. As a result, there is a nonzero coupling energy between the two S regions. We show, using
both mean-field theory and a quantum Monte Carlo calculation, that the phase stiffness constant, or
helicity modulus, of this junction is nonzero, and falls off exponentially with separation of the two superconductors. We also analytically estimate the dependence of the coupling energy on the properties of the
S and I regions, and suggest an analogy with conventional S–N–S junctions. Our results support the conclusion that this S–I–S sandwich structure, with a correlated insulating region, can be viewed as a single
effective Josephson junction.
Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction
Josephson junctions [1] are commonly divided into two classes:
S–N–S (superconductor–normal–superconductor) and S–I–S
(superconductor–insulator–superconductor). In an S–N–S junction,
the non-superconducting region is a normal (N) metal, the capacitive energy plays no role, and the remaining important energy
term is the Josephson coupling energy between superconducting
islands. In the absence of a vector potential, the Josephson energy
term coupling the two superconductors in the junction is usually of
the form J cosð/1 /2 Þ, where /1 and /2 are the phases of the
superconducting order parameter on superconductors 1 and 2,
and J is the coupling energy. Such an S–N–S junction will have a
current-phase relation (in the absence of a vector potential),
I ¼ Ic sinð/1 /2 Þ, where Ic ¼ 2eJ=
h. In an S–I–S junction, one must
consider, besides the Josephson energy, the charging energy; it has
the form Q 2 =ð2C 0 Þ, where C 0 is the junction capacitance and Q is
the charge on one of the superconductors. The charging and
⇑ Corresponding author. Tel.: +1 614 292 8140; fax: +1 614 292 7557.
E-mail
(D. Stroud).
addresses:
[email protected],
[email protected]
0921-4534/$ - see front matter Ó 2014 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.physc.2013.12.009
Josephson energies tend to be in competition, since charge and
phase are quantum–mechanically conjugate variables.
In this paper, we consider a special type of S–I–S junction,
namely, one in which the insulating region is itself made out of
superconducting material, but below its superconductor-to-insulator (S–I) transition, while the superconducting regions are made of
similar material but above the S–I transition. (By ‘‘below’’ and
‘‘above’’ the S–I transition, we mean in the insulating state and
superconducting state, respectively.) Such S–I transitions can occur
in a variety of such quasi-two-dimensional (2D) materials, as functions of film thickness, magnetic field, composition, or disorder.
They may also occur in three-dimensional (3D) materials. They
can take place in either granular materials or materials which are
disordered on an atomic scale, and even in single CuO2 layers of
high-T c superconductors [2–10].
One example of such an S–I–S junction could be a system in
which the central insulating (I) region is a thin superconducting
film, but below the critical thickness at which the film undergoes
a transition from insulating to superconducting, while the S regions are the same superconducting film but above that thickness.
Another possible realization is a ‘‘sandwich’’ junction, in which the
central insulating region of the sandwich is again a superconducting material below its S–I transition, while the S regions are
C.D. Porter et al. / Physica C 498 (2014) 21–29
composed of that same material above the transition. In this case,
the insulating regime is likely to be more than one intergrain thickness and hence three-dimensional. The film-type junctions could,
in principle, be deliberately designed by having a film deposited
by sputtering or chemical vapor deposition (CVD), so that the film
thickness varies in a controlled way in space. The sandwich-type
junction might be prepared by controlling the composition of the
deposited material so that it is first in the S composition regime,
then in the I regime, and finally again in the S regime as more
and more material is deposited. Film-type junctions might also
come into existence unintentionally in real superconducting films
whose thickness varies in space, and whose average thickness is
close to the critical value for an S–I transition. In such cases, the
thicker regions may be in the S part of the phase diagram, while
the thinner regions lie in the I part. A part of the film consisting
of two S regions with an I region sandwiched between them (either
in 2D or in 3D) may thus act as the type of junction we are considering. Another example would be a junction in which I region consists of many small superconducting islands separated by
insulating regions. Thus, the I region would itself be effectively a
Josephson junction array, which is ordered or, perhaps more plausibly, disordered. Such films have been discussed recently, for
example, by Nguyen et al. [11]. The I region in this case has sometimes been described as a Cooper pair insulator [12,13] or a Cooper
pair glass [14], where the thickness or stoichiometry variations, or
other sources of disorder, are very important.
To model this kind of a junction, we will treat both the I region
and the two S regions using the so-called quantum rotor Hamiltonian. Such a model, also known as the quantum XY model, is a standard Hamiltonian for treating an array of small Josephson
junctions [15]. It has also been used by a number of workers to describe a superconducting film which is disordered on an atomic
scale (e.g., by composition or thickness fluctuations), on either side
of the S–I transition [16]. In order to produce an S–I–S Josephson
junction from this model, its parameters must be such that the
material on the two sides of the junction lies in the S part of the
quantum rotor phase diagram, while that in the middle lies in
the I region. We believe that this type of model should be suitable
for either a ‘‘granular’’ insulating region (such as the Cooper pair
glass described in the previous paragraph) or a film which is disordered on a more atomic scale (e.g., by composition fluctuations)
below the S–I transition. Our actual choice of parameters is described below.
The properties of the quantum rotor model with positionindependent parameters have already been extensively studied.
It has been found that phase transitions in such arrays can be driven by temperature [17–19], voltage bias [20,21], magnetic field
[19], dissipation or shunting resistance [22], and current bias
[23], and the critical junction parameters have been accurately calculated for models of these systems [16,17,24–26]. Recent experimental work has shown that phase transitions analogous to those
seen in Josephson arrays can also be observed in arrays of boson
atoms or in a Bose–Einstein condensate confined to an optical lattice [27,28]; thus these transitions, too, may be described by a
quantum rotor model. From this analogy, the particular S–I–S junction we consider here could be viewed as a composite Josephson
junction array made up of three distinct regions, two of which
are above the S–I transition (S–I-T) (that is, in the superconducting
state), and one below that transition (i.e., in the insulating state).
The region below the S–I-T could be interpreted as a granular
superconductor or Cooper pair insulator, as already mentioned,
in which the individual grains are Josephson-coupled. The region
above the S–I-T could also be granular, or alternatively, it could
be an atomically disordered superconducting film, the quantum
fluctuations within which can be treated using the quantum rotor
model [16].
We expect that the special junction composed of S, I, and S
regions described above will have some unusual properties. In particular, there will be a proximity effect analogous to that seen
when a normal metal is placed next to a superconductor. For a single S–I interface of the type just described, the superconducting order parameter of the S region will decay exponentially into the I
region. For the S–I–S structure (consisting of a stack of three regions: an S, an I, and another S region, each described by the quantum rotor model), the two S regions will be coupled together by
this proximity effect which takes place through the I region. Because of this coupling, the two S regions can mutually phase order,
just like a conventional Josephson junction. Thus, this S–I–S sandwich itself behaves like a single Josephson junction, and might be
called a ‘‘superjunction,’’ since it is a junction formed by the coupling together of two S regions separated by an I region which consists of a superconductor below its S–I transition, rather than two
conventional bulk superconductors separated by a conventional
normal metal or insulator. We expect, as discussed further below,
that the current-phase relation for such a junction, in zero magnetic field, will be of the conventional form, i.e., I ¼ Ic sinðD/Þ,
where D/ is the phase difference across the junction. But despite
this similarity to conventional S–N–S or S–I–S junctions, there
are substantial differences. For example, even though the order
parameter decays exponentially into the I region, as in an S–N–S
junction, our junction will behave more like an S–I–S junction since
the non-superconducting region in isolation would be insulating at
zero temperature. Also, the composite junction we consider will
differ from a conventional S–I–S junction in that the insulating region exhibits quantum superconducting fluctuations, even at zero
temperature, because it contains many small regions containing
Cooper pairs which, in the absence of the S regions, are not coherent with one another. Given these substantial differences between
these S–I–S sandwiches and conventional junctions, it is reasonable to consider that they may also have rather different behavior
in a magnetic field than conventional S–I–S or S–N–S junctions. In
the quantum rotor model, the two parameters whose ratio governs
the low-temperature properties are the coupling energy J and the
charging energy U. In Fig. 1, we show an example of this ratio,
a J=U, as a function of z in a system that includes two S regions
and one I region, as just described. The schematic shows the
1
0.8
α=J/U
22
0.6
S
I
S
20
30
40
Layer in z-direction
50
0.4
0.2
0
0
10
60
Fig. 1. Schematic of the ‘‘superjunction’’ considered in this work. Each region
consists of many layers of sites connected by bonds. Some of these layers are in the
superconducting (S) parameter regime, having a ratio J=U > ðJ=UÞc , where ðJ=UÞc is
the critical value of the parameter ratio. The leftmost and rightmost 20 layers in the
example shown are in the S regime. The 20 middle layers are in the nonsuperconducting (I) regime, and have a parameter ratio J=U < ðJ=UÞc . In calculations,
we use periodic boundary conditions in the z direction as well as the transverse
direction(s), thus connecting the two S regions.
23
C.D. Porter et al. / Physica C 498 (2014) 21–29
variation of a with z, which is taken as the direction perpendicular
to the insulating slab. The S region itself may be 2D, in which case
the ‘‘layers’’ refer to one-dimensional (1D) arrangements of superconducting elements, or 3D, in which case the ‘‘layers’’ represent
2D slab arrangements, which are the usual sandwiched junctions.
An important but difficult question is how to translate the ‘‘lattice constant’’ of the quantum rotor model in the I region into
lengths or thicknesses of the insulating material in real space. This
question needs to be approached differently depending on whether
the insulating material is granular or disordered on a more atomic
scale. If the insulating material is granular, then this lattice constant would presumably be a typical distance between neighboring
superconducting grains. If, on the other hand, the insulating material is strongly disordered on a more or less atomic scale, as in the
case of some superconducting films below the critical thickness
[2], then this length would need to be obtained differently. One
possible mechanism whereby a disordered material of this type
can self-generate granularity has been discussed by, e.g., Ghosal
et al. [29]. This approach suggests a method to estimate the size
and spacing of the effective grains, and hence the lattice spacing
in an effective quantum rotor model, as functions of the parameters of the model Hamiltonian, which includes microscopic disorder. One might also be able to fit experimental measurements
(e.g., penetration depth into the insulating region) to a suitable
effective grain size. The model we use here, of course, actually
takes the Josephson network in the insulator to be ordered, but
the model could be generalized to allow a disordered Josephson
network.
We will describe three simple approximate ways to treat this S–
I–S ‘‘superjunction.’’ The first two are slightly different versions of a
conventional mean-field treatment of the quantum rotor model,
generalized to allow the relevant order parameter to vary with position; the third is a quantum Monte Carlo treatment. All three
methods allow one to calculate the coherence length n within
the I region, the effective coupling energy between the two S regions, and the helicity modulus of the junction.
We turn now to the body of the paper. The relevant formalism is
described in Section 2. Following this, we present the results of
applying the three methods in Section 3 and compare them to
one another and to other work. Finally, in Section 4, we discuss
the results further, and in particular give arguments why we expect this ‘‘superjunction’’ to have a conventional current-phase
relation of the form I ¼ Ic sinðD/Þ in the absence of a vector potential. We also present some concluding remarks, including speculations about how to extend this model to a finite magnetic field.
2. Formalism
2.1. Mean-field and perturbation theories
The quantum rotor [16,17,25,30,31] model Hamiltonian, in the
absence of a vector potential, takes the form
H¼
X
X
U i n2i J
cosð/i /j Þ:
i
ð1Þ
hiji
Here the angled brackets h i indicate summation over distinct
nearest-neighbor pairs, and /i is the phase on the ith site. In order
that this Hamiltonian describe an S–I–S sandwich, the coupling energy J is assumed to be homogeneous throughout the array,
whereas the charging energy U i is assumed to vary with position.
ni is the number operator describing the number of Cooper pairs
on the ith site, and is given by
ni ¼ i
@
:
@/i
ð2Þ
One convenient approximate treatment of the Hamiltonian (1) is to
use mean-field theory (MFT) [31–33]. A simple way of formulating
MFT for this Hamiltonian is to write
1
Ui Uj þ c:c:
2
i
1h
¼ ðhUi i þ dUi ÞðhUj i þ dUj Þ þ c:c: ;
2
cosð/i /j Þ ¼
ð3Þ
where ‘‘c:c.’’ denotes the complex conjugate, h i denotes a
quantum–mechanical average (with respect to the ground state,
at T ¼ 0), Ui ¼ expði/i Þ, and dUi ¼ Ui hUi i. With this separation,
the cosine term in H can be written as the sum of terms of the zeroth, first, and second order in the dUi ’s. The mean-field approximation consists of replacing the second order terms by their quantum–
mechanical average [32]. This makes H just a sum of one-body
operators, and may be written
H
X
JX U i n2i ðUi hUj i þ hUi iUj þ c:c:Þ þ const:
2
i
hiji
ð4Þ
Its eigenfunctions are thus products of one-body eigenfunctions
wi ð/i Þ’s, and the quantum–mechanical averages of any operator
are then readily computed.
The expectation values can be written hwi i gi expðihi Þ, where
gi is real. For the Hamiltonian (1), we expect that the hi ’s are all
equal to the same value h, i.e., that the phases /i ’s are all parallel,
on average. Without loss of generality, we can take h ¼ 0, or
hexpði/i Þi ¼ hcos /i i. The effective Schrödinger equation for wi can
then be written as
U i n2i J cos /i
X
!
gj wi ð/i Þ ¼ Ei wi ð/i Þ;
ð5Þ
j–i
where gj is a measure of the superconducting order parameter on
the jth site given by
gj ¼
Z 2p
0
wj ð/j Þ cos /j wj ð/j Þd/j
ð6Þ
and the sum in Eq. (5) runs over the nearest-neighbors to the site i.
We first consider a system with only one S–I interface. Suppose
we have a simple cubic lattice of sites, which are a typical sandwiched junction (we will also examine a simple square lattice in
the 2D case) with the axes of the lattice parallel to the x; y, and z
axes. Suppose further that the values of the charging energy are given by U ¼ U S for z 6 0 and U ¼ U I for z > 0, such that U is a constant within a given xy plane of the lattice and varies along the z
axis. We also assume that J=U S > ac and J=U I < ac , where
ac ¼ ðJ=UÞc is the critical value of this ratio at which a homogeneous system with this Hamiltonian undergoes an S–I transition
at T ¼ 0. In this geometry, gi depends only on the layer index in
the z-direction, which we denote k. Then the sum in Eq. (5) beP
comes j–i gj ¼ gk1 þ N gk þ gkþ1 , where N is the number of nearest-neighbors in directions transverse to z. In the case of the cubic
lattice, N ¼ 4. This notation will allow for simple application of the
results to both two and three dimensions.
We now calculate gk inside the insulating region under the
assumption that it is small (this assumption is expected to be accurate deep inside the insulator). In this case, wk can be approximated
using perturbation theory, because the potential energy term in
the Schrödinger equation will be small. The relevant solution of
the Mathieu equation (discussed below) will satisfy
wk ð/Þ ¼ wk ð/ þ 2pÞ. Using the first-order perturbation theory, we
may write
wk ð/Þ wk0 ð/Þ þ
h1jVj0i
h1jVj0i
w ð/Þ þ
w ð/Þ;
Ek0 Ek1 k1
Ek0 Ek;1 k;1
ð7Þ
24
C.D. Porter et al. / Physica C 498 (2014) 21–29
where wkn ð/Þ ¼ p1ffiffiffiffi
expðin/Þ; Ekn ¼ n2 U k , and thus
2p
Z 2p
1
hnjVj0i ¼ Jðgk1 þ N gk þ gkþ1 Þ expðin/Þ
2p 0
cos /d/:
ð8Þ
This integral can readily be carried out with the result
J
hnjVj0i ¼ ðdn;1 þ dn;1 Þðgk1 þ Ngk þ gkþ1 Þ:
2
ð9Þ
Substituting Eq. (9) into Eq. (7) gives
1
cos / J
wk ð/Þ ¼ pffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffi
ðgk1 þ Ngk þ gkþ1 Þ:
2p
2p U k
ð10Þ
Finally, to make the above result consistent with Eq. (6), we require
gk ¼
Z 2p
0
wk ð/Þ cos /wk ð/Þd/;
ð11Þ
or, on using Eq. (10),
gk ¼
J
ðg þ Ngk þ gkþ1 Þ:
U k k1
ð12Þ
Inside the insulator, U ¼ U I . We assume that gk decays exponentially, so that
gk K expðk=nÞ;
ð13Þ
where K is a constant and n is the coherence length in units of the
lattice constant. Substituting this form into Eq. (12), we get
1¼
J
½expð1=nÞ þ N þ expð1=nÞ;
UI
ð14Þ
or
n¼
1
cosh
1
1 UI
N
:
2 J
ð15Þ
In the limit where n 1 (i.e., when U I in the insulating region is
very close to its critical value U c ), we may expand the exponential
in Eq. (14) in powers of 1=n to obtain JðN þ 2 þ 1=n2 Þ=U I 1, or
1
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
ðU I =JÞ ðU=JÞc
ð16Þ
where ðU=JÞc ¼ N þ 2 is the critical value for an S–I transition in this
approximation.
In summary, the mean-field theory we have described, when
combined with perturbation theory, is consistent with the exponential decay of the order parameter deep within an insulating region. Furthermore, the coherence length n describing this
exponential decay diverges when the parameter ratio U I =J approaches the critical value ðU=JÞc according to the power law
n / 1=½ðU I =JÞ ðU=JÞc 1=2 .
2.2. Numerical solutions of the mean-field equations
Rather than using perturbation theory, we can solve Eq. (5)
numerically to determine the order parameters in different layers,
given a choice of J=U. One fast way to solve these equations is to
express them as coupled Mathieu equations of the form
fi00 ðxi Þ þ ½ai 2qi cosð2xi Þfi ðxi Þ ¼ 0;
ð17Þ
P
by choosing xi ¼ /i =2; f i ðxi Þ ¼ wi ð/i Þ; qi ¼ ð2J=U i Þ j–i gj , and
ai ¼ 4Ei =U i . We carried out this calculation using standard numerical methods as follows. First, the order parameters in each layer
were initialized at independent random values between 0 and 1.
These random order parameters and the energy ratios ðJ=U i Þ’s in
the ði 1Þth, ith, and ði þ 1Þth layers determine the initial values
of the qi ’s. The scaled energy eigenvalue ai defined above was then
obtained numerically, as were the corresponding eigenfunctions
fi ðxi Þ. Once the eigenfunctions are known, the order parameters
can be re-evaluated using Eq. (11). These new order parameters
are saved and passed to the next round of iterations. This recursive
process typically converges within 100 iterations. All results presented in this work were carried through 500 or more iterations.
For a system with a single S–I interface, these solutions could be
used to confirm the exponential decay of the order parameter deep
within the insulating regime, as found in the previous subsection.
Rather than doing this, however, we looked at the properties of an
S–I–S sandwich within the mean-field approximation, with the
aim of showing that this system behaves very much like a single
Josephson junction. To carry out this calculation, we study the
geometry shown in Fig. 1. In this two-interface geometry, the perturbation approximation may not be accurate, so we use the solutions of the full mean-field equations.
We have calculated two quantities associated with this S–I–S
junction. The first is the coupling energy, as described below. The
second is the so-called helicity modulus, c, of the junction, which
is basically the stiffness of the junction against a phase twist. c is
defined, as is standard, by applying a fictitious vector potential
A ¼ A^z perpendicular to the junction face. Since both the parameter ratio ðJ=UÞi and the order parameters gi ’s vary along the z-direction, the face of this junction lies in the xy plane. Then c is given by
[34]
"
c¼
#
h2 c2
@2F
ð2eÞ2 ad Nb
@A2
!
;
ð18Þ
T;A¼0
where a is the lattice constant, d is the dimensionality of the system
(d ¼ 2 or 3), 2e is the charge of a Cooper pair, c is the speed of light,
Nb is the total number of bonds perpendicular to the interface, T is
the temperature, which we take equal to zero, and F is the Helmholtz free energy (which equals E, the internal energy, at T ¼ 0).
For scaling, it is convenient to rewrite c as
c¼C
@2F
@A2
!
;
ð19Þ
T;A¼0
2
c2 =½ð2eÞ2 ad N b is a constant (for fixed N b ) with units of
where C ¼ h
momentum squared.
Thus, we need to find the energy of the system at T ¼ 0 when it
is subjected to the fictitious vector potential. In the presence of A,
the Hamiltonian (1) is modified by making the replacement
ð/i /j Þ ! ð/i /j Aij Þ;
ð20Þ
for bonds in the z direction, where Aij ¼ 2eaA=ð
hcÞ. For bonds perpendicular to ^z, the terms in the Hamiltonian are unchanged.
We can readily obtain an expression for c in mean-field theory.
Analogously
to
the
case
A ¼ 0,
we
write
cosð/i /j Aij Þ ¼ ½Ui Uj expðiAij Þ þ c:c:=2. We again express
Ui ¼ hUi i þ dUi , and replace terms of the second order in the
dUi ’s by their quantum–mechanical averages, so that they are just
c-numbers. The remaining terms are the first order in the Ui ’s, and
are the sum of single-particle Hamiltonians. The corresponding
eigenfunctions are products of single-particle eigenfunctions. Since
we desire the second derivative of the free energy with respect to A
at A ¼ 0, we need to retain terms in the Hamiltonian only through
the second order in the Aij ’s. To this order in Aij , the Schrödinger
equation for Ui is found to be
(
U i n2i
)
X 1 2
J cos /i
gj 1 Aij
Ui ð/i Þ ¼ Ei Ui ð/i Þ;
2
j–i
ð21Þ
where the sum runs over the nearest-neighbors to the site i and
Aij ¼ 2eaA=ð
hcÞ for bonds parallel to A and 0 otherwise. This expres-
25
C.D. Porter et al. / Physica C 498 (2014) 21–29
sion can be manipulated into a Mathieu equation of form (17), with
the same substitutions as before, except that now
2J
1
ðgiþ1 þ gi1 Þ 1 A2ij þ Ngi :
U
2
0.8
ð22Þ
αS=(J/U)S
The previously described recursive technique can be used to determine the mean-field energy of the system in the presence of several
small values of the fictitious vector potential, and hence the second
derivative needed to evaluate Eq. (18) can be obtained numerically
using a simple Euler method calculation.
0.6
0.2
0.3
0.4
0.5
0.6
η
qi ¼ 1
0.4
0.2
3. Results and comparisons
0
0
10
20
30
40
Layer in z-direction
3.1. Critical energy ratios
60
Fig. 3. Similar to Fig. 2, but for a 3D simple cubic system. Here ðJ=UÞI is kept fixed at
0.15.
1
0.9
0.8
0.7
0.6
η
The order parameters were calculated numerically as a function
of layer for various choices of J=U in both the I and S regions. Figs. 2
and 3 show the calculated gi ¼ hcos /i i in each layer for the 2D
square lattice and 3D cubic lattice, which is a typical sandwiched
junction, for various values of ðJ=UÞS . ðJ=UÞI was held constant in
each case, at 0.24 and 0.15 for 2D and 3D, respectively. Increasing
ðJ=UÞS increases the order parameter in the S region, as expected,
but has little effect within the I region, where the order parameter
falls off rapidly regardless of its value in the S region, consistent
with the exponential decay assumption in Section 2.1.
To understand how changing ðJ=UÞS or ðJ=UÞI influences the
order parameter when these ratios are near the critical value of
J=U, this critical value, which we denote ðJ=UÞc , must be precisely
determined. The results of Section 2.1 suggest that
ðJ=UÞc ¼ 1=ðN þ 2Þ ¼ 1=4 or 1=6 in 2D or 3D, respectively. To calculate this critical value numerically within the mean-field approximation, we studied homogeneous arrays of sites in which all
layers had the same value of J=U. The order parameter was calculated numerically across the entire array and then averaged. We
did this calculation for many values of J=U. Figs. 4 and 5 show
the resulting order parameter as a function of J=U for the 2D and
3D lattices. In both cases, the order parameter increases sharply
with increasing J=U above a critical value. The insets show clearly
that the mean-field perturbative results of Section 2.1 and these
numerical results are in nearly perfect agreement; the mean-field
50
0.16
0.5
0.12
0.4
0.08
0.3
0.04
0.2
0
0.1
0.245
0.25
0.255
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
α =(J/U)
Fig. 4. g versus J=U for a homogeneous system of quantum rotors, as calculated
using the mean-field theory of Section 2.2. g is shown for a 2D square ‘‘superjunction’’ consisting of 30 layers, with periodic boundary conditions imposed. g ¼ 0 for
J=U < 0:25 ¼ ðJ=UÞc within the mean-field theory. The inset is a blowup of the
region near a ¼ 1=4.
1
1
0.9
0.8
0.8
0.7
αS=(J/U)S
0.6
0.6
η
0.4
η
0.3
0.4
0.5
0.6
0.7
0.16
0.5
0.12
0.4
0.08
0.3
0.04
0.2
0.2
0
0.16
0.1
0.165
0.17
0
0
0
0
10
20
30
40
Layer in z-direction
50
60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
α=(J/U)
Fig. 5. Same as Fig. 4, but for a 3D system. Here g ¼ 0 for J=U < 1=6 ¼ ðJ=UÞc .
Fig. 2. Mean-field order parameter g plotted as a function of layer index for a 2D
square junction considered here, with S and I regions as shown in Fig. 1. g ¼ hcos /i
is large and almost constant in the S region, but drops off exponentially with
distance away from an S face inside the I region. This behavior is shown for several
choices of ðJ=UÞS . The parameter ratio inside the insulating region is kept fixed at
ðJ=UÞI ¼ 0:24. The lines are merely guides to the eye, and the calculation is discrete
by layer.
values fall within 1% of the values of 1=4 in 2D and 1=6 in 3D,
respectively, predicted by the mean-field perturbation theory.
ðJ=UÞc has been computed by various other workers in both 2D
and 3D. In Ref. [30], for example, a value ðJ=UÞc ¼ 0:5 was obtained
C.D. Porter et al. / Physica C 498 (2014) 21–29
in a slightly different mean-field theory than that used here. The
discrepancy is due to an error of a factor of two in Ref. [30]; specifically the error is in the substitutions made between Eqs. (7) and
(8) in that reference. When this is corrected, their results agree
with ours. Ref. [25] used two methods which gave values of
ðJ=UÞc of 0.25 and 0.4, respectively, the first of which agrees with
the value obtained here in 2D. ðJ=UÞc has also been calculated for
a 2D square array using a quantum Monte Carlo technique in
Ref. [16], with the result of ðJ=UÞc ¼ 0:408. This result is considerably higher than the mean-field value. But this is expected, since
one expects that mean-field theory would overestimate the ability
of a system to order.
Given ðJ=UÞc , in mean-field theory, we can investigate what
happens when ðJ=UÞI approaches this value from below. Figs. 6
and 7 show gi versus i for a fixed ðJ=UÞS and several values of
ðJ=UÞI . gi falls off rapidly with distance from either of the two S regions, reaching a minimum at the midpoint of the I region. When
ðJ=UÞI is close to ðJ=UÞc , the order parameter penetrates deeply into
the insulating region, while a value of ðJ=UÞI much less than ðJ=UÞc
results in a drop of the order parameter almost to zero by the second insulating layer. In all these calculations in both 2D and 3D,
ðJ=UÞS was kept fixed at 0.3.
3.2. Coherence length
Once values for the order parameter have been determined
numerically, we can calculate the coherence length n. If there were
only one interface, we could obtain n by assuming exponential decay according to
n ¼ 1= lnðgi =giþ1 Þ;
ð23Þ
where gi is calculated within the I region, and increasing i corresponds to increasing distance from the interface. Because an S–I–S
arrangement actually has two S–I interfaces, one would only expect
the decay to be roughly exponential so long as the order parameter
feels the influence of only one interface. If we are considering only
the left-hand interface, then this relation will be true, at most, within the first half of the insulating region. Thus, one would only expect
Eqs. (15) and (16) to be valid well inside the insulating region
(where perturbation theory should be accurate), but before the
exponential increase due to the second superconducting region becomes apparent. To calculate n, we chose to use an S–I–S arrangement of 40 layers, the center 20 of which were insulating. For
0.9
0.8
αI=(J/U)I
0.7
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.6
0.5
η
26
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
Layer in z-direction
30
35
40
Fig. 7. Same as Fig. 6, but for a 3D system. Results are shown for several values of
ðJ=UÞI , and with ðJ=UÞS still fixed at 0.3. The lines are merely guides to the eye.
given values of ðJ=UÞI and ðJ=UÞS , the assumption of exponential decay in the calculation of n was taken to be valid in the 11th through
17th layers (meaning the first through seventh insulating layers).
Values of n were calculated according to Eq. (23) in each layer in
that range, and the results were averaged. This calculation was carried out for many values of ðJ=UÞI in both the 2D and 3D cases.
Figs. 8 and 9 show the coherence length as a function of ðJ=UÞI .
The empty squares represent discrete numerical calculations
according to Eq. (23) as just described. Note the increase in coherence length as one approaches ðJ=UÞc , as is expected. The dashed
line represents the coherence length as determined by the second-order expansion of Eq. (16). Since this is an expansion around
small 1=n, one would expect it to be most valid when the coherence length is large, near ðJ=UÞc . Indeed, the curve deviates from
the numerical results somewhat at low ðJ=UÞI , and becomes a better fit as ðJ=UÞI increases toward ðJ=UÞc . The solid line is the exact
expression for the coherence length within the mean-field approximation, and the perturbative treatment, as given by Eq. (15). This
curve seems to fit the numerical results perfectly. Both the solid
3
2.5
0.5
ξ/a
0.4
αI=(J/U)I
0.1
0.12
0.14
0.16
0.18
0.20
0.22
0.24
η
0.3
0.2
0
5
10
15
20
25
Layer in z-direction
1.5
1
0.5
0
0.1
0
num. av.
-1
cosh
expansion
2
30
35
40
Fig. 6. Dependence of g on J=U for a 2D square ‘‘superjunction’’ with S and I regions
as shown in Fig. 1. g is large and nearly constant in the S region, but drops off
exponentially with distance inside the I region. This behavior is shown for several
values of ðJ=UÞI and a fixed ðJ=UÞS = 0.3. The lines are merely guides to the eye. The
calculation is discrete by layer, and is carried out using mean-field theory.
0
0.05
0.1
0.15
α I=(J/U)I
0.2
0.25
Fig. 8. Coherence length n (in units of the layer spacing a) for a 2D square system,
plotted as a function of ðJ=UÞI . n has been calculated in three ways: using the
average of the numerical results of g in each layer and applying Eq. (23); using the
1
asymptotic expression of Eq. (16); and using the more exact cosh function of Eq.
(15). These methods are discussed in greater detail in Section 2.1. ðJ=UÞS is kept
fixed at 0.3. The calculation was carried out on a 40-layer array, the middle 20
layers of which were insulating. Coherence lengths were averaged over the 11th to
17th layers (the first through seventh insulating layers). Beyond the 17th layer, the
order parameter has fallen to such small values that numerical errors become nonnegligible in the calculation of n.
27
C.D. Porter et al. / Physica C 498 (2014) 21–29
1.2
0.3
num. av.
cosh-1
expansion
0.8
ξ/a
Jeff = [E(NI=21)-E(NI)]/J
0.25
1
0.6
0.4
0.2
0
αI=(J/U)I
0.14
0.19
0.24
0.2
0.15
0.1
0.05
0
0.02
0.04
0.06
0.08
0.1
α I=(J/U)I
0.12
0.14
-0.05
0.16
Fig. 9. Same as Fig. 8, but for a 3D simple cubic system. ðJ=UÞS is kept fixed at 0.6.
curve and the symbols are generated using the same mean-field
approximations, but their agreement demonstrates the validity of
the perturbative treatment used to obtain Eq. (15). Note that the
coherence length is slightly larger for a given ðJ=UÞI in the 3D lattice than in the 2D lattice, presumably because a given value of
ðJ=UÞI is farther away from ðJ=UÞc in 3D than in 2D.
5
10
15
Insulating Layers NI
0.3
ð24Þ
with NImax ¼ 21, and Jeff is the effective coupling energy per unit area
in an xy plane. If the area of a small junction bond with the coupling
2
energy J is b , and the entire area of the xy plane is A, then the entire
2
effective coupling energy for the junction would be Jeff A=b .
Figs. 10 and 11 show this J eff =J (where J is the coupling energy
for a bond in the superjunction) for the 2D and 3D cases, respectively. For these figures, ðJ=UÞS was kept fixed at 0.5 for the both
cases, although more detailed investigation showed that there
was little to no dependence of J eff on ðJ=UÞS . J eff was found to depend quite strongly on the number of insulating layers in both
2D and 3D. Clearly, for ðJ=UÞI very close to the critical value, the order parameter can remain nonzero throughout even a large insulating region. Hence, for large ðJ=UÞI one sees nonzero J eff in a
system with as many as 17 or 18 insulating layers in 2D. For smaller ðJ=UÞI , the order parameters vanish very quickly and J eff seems
to be nonzero only if the insulating thickness is no more than three
or four layers.
3.4. Helicity modulus
3.4.1. Mean-field approximation
The helicity modulus was calculated numerically according to
Eq. (18) for both the square and cubic arrays, using the mean-field
Jeff = [E(NI=21)-E(NI)]/J
0.25
To calculate the coupling energy, we compute the total energy
of a system with a given separation between the faces of the two
superconductors, and then subtract from this the energy of the system when the insulating region is very thick, so that the two S regions are effectively decoupled. The difference between these two
energies is the desired coupling energy. In practice, we have assumed that the thickness of 21 insulating layers is sufficient to
decouple the two superconducting regions, although this is possibly not a sufficient separation when ðJ=UÞI is too close to ðJ=UÞc .
Also, nothing has been specified about the cross-sectional area of
the layers (slices in the xy plane). Clearly the total coupling energy
will scale with this cross-section. Thus, we write for the effective
coupling per island along an xy plane
20
Fig. 10. Effective coupling energy Jeff =J per island in the transverse direction as a
function of the number of insulating layers N I . To obtain these curves, the total
energy E of the 2D square system is calculated and is subtracted from that of a
system with very well-separated superconducting regions (N I ¼ 21). All energies
have been divided by J, the coupling energy of a single bond. This calculation is
shown for several values of ðJ=UÞI , in all cases with ðJ=UÞS ¼ 0:5. The number of S
layers was kept constant at 20 layers. Solid and dashed lines simply connect the
calculated points.
3.3. Effective coupling energy for the junction
J eff ¼ EðN Imax Þ EðNI Þ;
0
αI=(J/U)I
0.04
0.10
0.16
0.2
0.15
0.1
0.05
0
-0.05
0
5
10
15
Insulating Layers NI
20
Fig. 11. Same as Fig. 10, but for a 3D cubic Josephson array. Again ðJ=UÞS is kept
fixed at 0.5 and ðJ=UÞI takes several values below ðJ=UÞc as indicated.
approach as described below Eq. (18). Fig. 12 shows the resulting
helicity moduli for the square and cubic arrays at fixed values of
ðJ=UÞS and ðJ=UÞI , as functions of the number of insulating layers.
In both cases, ðJ=UÞS was set to 0.4, and ðJ=UÞI was chosen to be just
below the corresponding critical value (i.e., to 0.24 in 2D, and 0.16
in 3D).
In both cases, c clearly decreases with increasing number of
insulating layers, N I . This is expected, since the two S regions are
decoupled at sufficiently large N I , implying that the phase difference between the two S regions will have no energy cost.
3.4.2. Quantum Monte Carlo evaluation
For comparison, we have also computed the helicity modulus c
of a 2D array using a quantum Monte Carlo (QMC) technique at
T ¼ 0. The QMC calculation is carried out using the method of
Ref. [16], implemented as described in Secs. IIB, IIC, and IIIA of
Ref. [24]. The results of the QMC calculation are shown in Fig. 13
for a 2D system and two different values of ðJ=UÞI , as indicated in
the figure caption. The other parameters are described in the caption. In both cases, ðJ=UÞS is kept fixed at 0.5. For both values of
ðJ=UÞI , as in the mean-field calculations, c falls off with increasing
28
C.D. Porter et al. / Physica C 498 (2014) 21–29
0.5
2D
3D
γ/CJ
0.4
0.3
0.2
0.1
0
0
5
10
15
20
Insulating Layers
25
30
Fig. 12. Helicity modulus c calculated using mean-field theory as a function of N I in
a 2D square and a 3D cubic Josephson array. All energy values used in the
calculation of the second derivative are divided by J, the coupling energy of a single
bond. Here ðJ=UÞS ¼ 0:4 for both the 2D and 3D calculations. ðJ=UÞI ¼ 0:24 in the 2D
case, and ðJ=UÞI ¼ 0:16 in the 3D case. The number of S layers was kept constant at
20 layers.
0.35
0.3
0.25
γ/C
0.2
αI=(J/U)I
0.40
0.20
0.15
0.1
0.05
0
0
0.1
0.2
0.3
MI/M
0.4
0.5
Fig. 13. Helicity modulus, c, as calculated using quantum Monte Carlo, for a 2D S–I–
S system at two different values of aI , is plotted as a function of M I =M, the fraction
2
of the layers which are insulating. Here C ¼ h c2 =½ð2eÞ2 a2 N b as described in the
text. In both cases, the total number of layers M is taken as 20, and periodic
boundary conditions are used in both spatial directions. ðJ=UÞS was kept fixed at 0.5
for both curves. In one curve, ðJ=UÞI ¼ 0:4 is very near the quantum Monte Carlo
critical value of 0.408, and c falls off slowly with depth into the insulator. In the
other curve, ðJ=UÞI ¼ 0:2, well below that critical value, and c falls off more rapidly.
separation of the two superconductors. Also as in the mean-field
approximation, c falls off more rapidly as ðJ=UÞI moves farther
away from its critical value (not explicitly shown in Fig. 12).
4. Discussion
The present results indicate that superconducting order can be
transmitted from one S region to another through a region of correlated insulator I, where the I region is made up of a superconducting material below its insulator-to-superconductor transition. We
have demonstrated this, by showing that the superconducting order parameter exponentially decays into the I region with a characteristic correlation length nI which depends on the ratio ðJ=UÞI .
Thus, as we have shown, if the two S regions are not too far apart,
there is a finite coupling energy between them which varies
roughly exponentially with separation. Furthermore, under these
conditions, the system of two S regions separated by an I region
has a finite helicity modulus c, as we have shown using both a
mean-field calculation and a quantum Monte Carlo approach.
All these properties indicate that this S–I–S structure behaves
very much like a single Josephson junction. In order to confirm that
this system truly is a Josephson junction, we would need to prove
that the coupling energy may be written Ecoup ¼ E0 cosðD/Þ,
where D/ is the phase difference across the junction and E0 is
the characteristic coupling energy. Since c / ½@ 2 Ecoup =@ðD/Þ2 D/¼0 ,
the non-vanishing of c shows that the energy does increase quadratically with D/ for small D/. We also know that Ecoup must be
periodic in D/ with period 2p. These two facts together strongly
suggest that Ecoup varies as cosðD/Þ even for larger D/. If so, the corresponding critical current of this junction would be Ic ¼ 2eEcoup =
h.
We propose that the structure described here will behave like a
single Josephson junction, i.e., it will have a Josephson supercurrent IJ ¼ Ic sinðD/Þ like a conventional Josephson junction.
Our numerical results presented here show that superconductivity can be transmitted to another superconductor through an
insulator which is actually a superconducting material below its
S–I transition – that is, the insulator contains Cooper pairs which
are localized rather than phase-correlated throughout the insulator. We have also shown that this S–I–S structure can form an unusual type of Josephson junction. The junctions we consider here
appear to be of 0 junctions rather than p junctions, that is, they
have a current-phase relation of the form I ¼ Ic sinðD/Þ. They also
should generally have critical currents which fall off monotonically
with increasing insulator thickness.
As mentioned earlier, we believe that our system could be realized in experiments. Using advanced experimental techniques such
as sputtering and/or vapor deposition, we can deposit an S–I transition material in a line or in a slab with an appropriate thickness
(above or below the S–I transition) for the specific region whether
S or I. By repeating this procedure we can build up a structure
which we want. The thicknesses of metal films were varied over
the S–I transition in Ref. [35], so our system could be realized with
the help of techniques used in Ref. [35].
There are numerous other questions about the properties of
these junctions which are deserving of study. For example, it
would be of interest to investigate the response to a magnetic field,
to see if it differs in any way from that of a conventional S–I–S
junction because of many internal degrees of freedom in the insulating layer. It would also be interesting to study the dependence of
junction properties on temperature, current bias, and voltage bias.
Of these, perhaps the magnetic field dependence would be the
most revealing. It seems possible that this type of junction would
have a large positive magnetoresistance. The reason is that, when
a magnetic field is applied, the critical value of J=U will probably
increase. Thus, the fluctuation conductivity at a given finite temperature would be expected to decrease as the magnetic field increases, i.e., the resistance would increase, leading to a positive
magnetoresistance. This speculation would need to be confirmed
by calculating the fluctuation resistivity as a function of magnetic
field and temperature. The effect might be more conspicuous if
the insulating material were actually a Cooper pair insulator,
which is likely to have a great deal of disorder.
To summarize, we have described the properties of an S–I–S
junction, in which both the insulating and the superconducting regions are made up of a correlated superconducting material, the I
region lying below and the S regions lying above the S–I transition.
We describe both S and I regions using a quantum rotor model.
Based primarily on the mean-field solutions to this model, we find
that this type of system behaves like a single Josephson junction,
and suggest that it has a conventional energy-phase relation of
the form E ¼ E0 cosðD/Þ. The phase rigidity of this junction is confirmed by a quantum Monte Carlo calculation of its helicity modulus. This type of junction could be prepared in a thin
C.D. Porter et al. / Physica C 498 (2014) 21–29
superconducting film with a controlled spatial variation in thickness. Alternatively, it could consist of a typical sandwich junction,
in which the two S regions are separated by a Cooper pair insulator.
In the latter case, the I region would really consist of a disordered
Josephson array below its S–I transition.
Acknowledgments
This work was supported by the Center for Emerging Materials
at the Ohio State University, an NSF MRSEC (Grant No. DMR0820414).
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