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Consequences of unconventional order parameter symmetry
-High critical temperature structures-
A. Baronex, J. R. Kirtleyy, F. Tafurizy, and C. C. Tsueiy
xUniversita di Napoli Federico II, Napoli, Italy
yIBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA
zINFM-Dip. Ingegneria dell'Informazione, Seconda Universita di Napoli, Aversa (CE) Italy
(February 12, 2002)
We review some aspects of the physics of high temperature superconductivity
(HTS) related to coherent phenomena and unconventional pairing symmetry.
We discuss the role of the Josephson eect as a very powerful probe of the
underlying physics of HTS, concentrating on phase-sensitive experiments. We
then proceed to some consequences of d-wave symmetry, including possible
broken time-reversal symmetry, Andreev bound states, and the presence of
an imaginary component of the order parameter (OP ) in the presence of surfaces and interfaces. Finally we discuss some aspects of HTS grain boundary
Josephson junctions which allow fundamental studies and potential applications, in particular the possibility of employing grain boundary Josephson
junctions for -circuitry and d-wave qubit arrays.
[email protected]
Fifteen years after the discovery of high critical temperature superconductors, we are
faced with a fascinating mosaic of theoretical and experimental results. Indeed, the push
to understand the underlying physics of this class of superconductors has stimulated basic
research in many dierent elds, raising a variety of challenging questions. The temperaturedoping phase diagram (Fig. 1) is an example which encompasses several open issues. While
no one claims the existence of a universally accepted theory explaining all of the nature of
superconductivity in these materials, there are some rmly established aspects that shed
light on this fundamental issue. Of paramount importance is the general consensus that the
pairing symmetry in cuprates is predominantly d-wave.
The main intent of the present work is to review some of the aspects of the physics of
HTS, mainly related to coherent phenomena and the unconventional symmetry of the order
parameter, and considering, in particular, some consequences and implications for future
research developments. We shall discuss the role of the Josephson eect as a very powerful
probe of fundamental aspects of the underlying physics governing the superconducting state,
referring mainly to phase-sensitive experiments. We will rst focus on the half-ux quantum
eect and the relation between unconventional order parameter symmetry and spontaneous
currents within the framework of the Josephson eect. These experiments, along with those
on quantum interference, established the universal nature of d-wave symmetry of HTS [1].
The second part of this contribution is mainly devoted to some consequences of a d-wave
order parameter symmetry. Experimental and theoretical issues on time reversal symmetry
breaking, Andreev bound states and the possible presence of an imaginary component of
the order parameter near surfaces and interfaces will be discussed. Time reversal symmetry
breaking phenomena may be manifested as spontaneous magnetization, in particular as
vortices with ux values which are not a multiple of the half-ux quantum 0 =2 (fractional
vortices) (0 = hc=2e is the superconducting ux quantum, where h is Planck's constant and
e is the charge on the electron.) An intriguing topic of general interest in the eld of solid
state physics concerns a possible correlation between spontaneous magnetization, vortices,
and topological arguments. Some aspects of this possible correlation can be found in spincharge fractionalization concepts in cuprates as an expression of the topological order [2]
and in the process of the formation of topological defects in phase transitions [3,4]. This last
issue has even stimulated some analogies between cosmology and condensed matter physics
We shall nally discuss those aspects of HTS grain boundary Josephson junctions which
allow us both to perform fundamental experiments and to envisage new applications. In
particular, the possibility of employing grain boundary Josephson junctions for -circuitry
will be considered, along with various proposals for possible d-wave qubit arrays.
A Superconducting QUantum Interference Device (SQUID) is a superconducting ring
interrupted by one or more Josephson weak links. The remarkable properties of conventional
SQUIDs (0-SQUIDs) arise from: 1) the intimate relationship between the phase ' of the
complex superconducting order parameter =j j e and the vector potential A~ threading
through the SQUID loop; 2) the Josephson current-phase relationship I = I0 sin(') (where
' is the change in phase across the Josephson weak link; and 3) the requirement that the
order parameter be single valued. In particular, for SQUIDs this requirement means that the
changes in the phase ' in a closed loop around the SQUID must sum to an integral multiple
of 2. Bulaevskii and co-workers [6] raised the possibility of SQUIDs with a phase change
of in the order parameter built into the ring (a -SQUID). This work predicted that a
-shift could occur through the interaction between tunneling Cooper pairs and magnetic
impurities. Such eects have been reported [7], but for the purposes of this paper the -phase
shifts built into the -SQUIDs will be assumed to be due to the momentum dependence of
the order parameter in unconventional superconductors, as rst proposed by Geshkenbein,
Larkin, and Barone [8,9]. Consider a single junction SQUID with loop inductance L, critical
current I0, and phase-shift ( = 0 for a 0-SQUID, and = for a -SQUID). The free
energy of the SQUID is given by:
2 ( 2
L j I0 j
cos[ + ] ;
U (; ) =
where is the ux threading through the ring, is the externally applied ux, 0 = hc=2e
is the superconducting ux quantum, and =0, for a 0 and a -ring respectively.
The minimum energy values from Eq. 1 for the ux are plotted as a function of the
applied eld in Fig. 2 for a 0-SQUID (a) and a -SQUID (b) for various values of the
parameter = 2LI =0 . For the 0-SQUID there are a ladder of minimum energy states
with periodicity 0 , and with values approaching = N 0 , N an integer, as becomes
large. This is the familiar integer ux quantization of superconducting rings. -SQUIDs
also have a ladder of minimum energy states, still with periodicity 0 , but oset from that
for 0-SQUIDs by 0 =2. In the limit of large this ladder has minimum energy values at
= (N +1=2)0, N an integer. In particular, a -SQUID with large , when cooled in zero
applied ux, will spontaneously generate a circulating supercurrent suÆcient to generate
0 =2 of magnetic ux threading through the ring. This is the half-integer ux quantum
A second remarkable property of -SQUIDs is that the dependence of the SQUID critical current on applied ux is periodic, with period 0 , but with the dependence on ux
shifted by 0 =2 from that of a conventional 0-SQUID. Consider a symmetric d.c. SQUID,
a superconducting loop of total inductance L interrupted by two Josephson weak links with
critical currents I0 , with two current leads on opposite sides of the junctions. The total
supercurrent from one lead to the other is given by
Is = I0 (sin'a + sin'b );
where ' and ' are the superconducting phase drops across the two junctions (labelled
a and b). The requirement that the superconducting wave function be single valued means
that the sum of the phase drops around the ring must be an integer multiple of 2:
2n = '
' + + 2 =0 ;
where is the intrinsic phase shift in the loop, either 0 for a 0-SQUID, or for a SQUID. The critical current of the SQUID can be calculated by combining Eq. 2 with Eq.
3 [10]. Typical results for a symmetric dc SQUID are shown in Fig. 3. Plotted there is
the critical current I
of the SQUID, normalized by the single junction critical current
I0 , as a function of the normalized externally applied magnetic ux =0 , for values of
= 2LI0 =0 =0.5,1,2,5,10 and 100.
The rst phase-sensitive measurements of the symmetry of the pairing wave function in
the cuprates was made by Wollman et al. [11{13] using the SQUID interference techniques
outlined above [14], as well as related single-junction interference measurements. These
experiments used junctions between single crystals of the high critical temperature superconductor YBa2Cu3 O7 (YBCO) and the conventional superconductor Pb. The junctions
were on adjacent faces of the crystal for one set of SQUIDs. These should be -SQUIDs
because the normal component of the order parameter has a phase shift of between adjacent crystal faces in a tetragonal crystal of a d-wave superconductor. The control set of
SQUIDs had two junctions on the same face of the crystal. Wollman et al. reported shifts
of approximately 0 =2 in the magnetic interference patterns between the 0- and -SQUIDs,
as expected for a d-wave superconductor. Similar results were reported by Brawner and Ott
Interpretation of these experiments was complicated by the inuence of ux trapping and
self-eld eects caused by SQUID asymmetries. However, recently these eects have been
eliminated by Schulz et al. [16] in a cleverly designed thin-lm, grain boundary junction
geometry [17]. Results from these experiments are shown in Fig. 4. These show nearly ideal
interference patterns, with a shift of 0 =2 in the extrema of the 0-SQUID relative to that of
the -SQUID, as expected for a d-wave superconductor in this geometry. Resonance current
steps have been observed in the same kind of conguration, proving the eects of the d-wave
symmetry in the voltage state of a Josephson device [17].
The half-integer ux quantum eect was rst observed by Tsuei et al. [18] in ring samples
of YBa2Cu3 O7 (YBCO) in a tricrystal geometry designed to have a -phase shift for a
d-wave superconductor (Fig. 5a). In these experiments thin-lms were epitaxially grown
on specially designed tricrystal substrates of SrTiO3 and patterned into rings. The central
ring, designed to be a -ring for a d-wave superconductor, showed the half-ux quantum
eect as determined by imaging with a scanning SQUID microscope (Fig. 5b). Three
control rings, designed to be 0-rings, showed the integer ux quantum eect. Subsequent
experiments [19,20] showed that this eect did not occur in two other geometries, ruling out
symmetry independent causes of the -phase shift (such as proposed by Bulaevskii et al.
[6]), and "g-wave" pairing symmetry, with a momentum dependence of the gap proportional
to (cos(k )+cos(k )). Further, experiments in unpatterned rings also showed the half-ux
quantum eect, with a Josephson vortex with total ux 0 =2 spontaneously generated at
the tricrystal point even when the sample was cooled in zero magnetic eld [21].
Experiments in the \tetracrystal" geometry, an adaptation of one proposed by Walker
and Luettmer-Strathmann [22] (Fig. 6a) makes a clear distinction between d-wave and swave symmetries. The observation of a spontaneously generated half-ux quantum Josephson vortex at the tricrystal point in this geometry (Fig. 6b) in the tetragonal single layer
cuprate superconductor Tl2Ba2CuO6+ [23] provides evidence for pure d-wave superconductivity in this material. Tricrystal and tetracrystal experiments on a number of hole-doped
[1,24] and electron-doped [25] superconductors always show the half-ux quantum eect in
geometries designed to produce -rings for d-wave superconductors. Further, at least in
YBCO, the half-ux quantum eect in the appropriate geometry for a d-wave superconductor persists at all temperatures, with total ux of 0 =2 at the tricrystal point to within
experimental uncertainties, from 0.5K to T [26]. These experiments provide evidence that
the predominance of d-wave pairing symmetry is universal in the cuprates. This is a remarkable result, given the wide variation in other properties of the cuprate superconductors.
There has been an enormous amount of work on tests of the pairing symmetry in the
cuprate superconductors using various phase insensitive techniques [27,28,1]. Although this
work has resulted in a general consensus that the pairing wavefunction has lines of nodes,
these techniques, with the exception of angle resolved photoemission, are relatively insensitive to the momentum dependence of the nodal structure. This makes it diÆcult for them
to distinguish, for example, between d
symmetry (with four lines of nodes), and extended s-wave symmetry (with eight lines of nodes). There has recently been support for
the latter symmetry from analysis of various phase insensitive tests of pairing symmetry
[29,30]. These authors attempt to reconcile this conclusion with the phase-sensitive experiments, which denitively favor d
symmetry, by arguing that the cuprates can have
dierent bulk and surface symmetries [31]. In this view the phase-insensitive techniques
probe the bulk extended s-wave symmetry, while the phase-sensitive techniques, which depend on surface sensitive Josephson tunneling, probe d-wave pairing symmetry. This view
seems untenable for several reasons. First, the consistency of the tricrystal experiments [1],
both in that samples with a -ring geometry for a d-wave superconductor always show the
half-ux quantum eect, and that the half-ux quantum eect can be turned on and o by
changing the tricrystal geometry in a way that is consistent with d pairing symmetry,
make it appear extremely unlikely that these experiments are controlled by surface eects.
Second, the tricrystal experiments have shown that the half-ux quantum eect persists,
with no change in the total ux within experimental accuracy, to within a degree of T
[26]. At these temperatures the ab plane coherence length can be ten times the unit cell
dimension, meaning that these experiments are not surface sensitive but rather probe the
order parameter deeply into the bulk: nevertheless the results strongly favor d-wave pairing
symmetry. Further, although the I R product for the grain boundary junctions used in
the tricrystal experiments are relatively small, experiments with -SQUIDs and facetted
-junctions in YBCO-Nb ramp junctions produce convincing evidence for d-wave symmetry
with junction I R products within a factor of four of the Ambegaokar-Barato limit [32].
This means that the superconducting gap is not signicantly reduced by surface eects in
x2 y 2
xy y 2
x2 y 2
these samples. The tetracrystal experiments [23] have provided strong evidence for pure
d-wave pairing symmetry in tetragonal Tl-2201. It also rules out anisotropic s-wave pairing
symmetry. The tetragonal crystal structure of the Tl-2201 epitaxial lms used in this experiment were checked and conrmed with three techniques for microstructural analysis [23].
It is well established that Tl-2201 can be fabricated with either orthorhombic or tetragonal
crystal structure with identical superconducting properties (including T ' 80K) [33{35].
Furthermore, it has been shown that there is no temperature-dependent structural phase
transition in the temperature range 50K (well below T ) -300K in either form of Tl-2201 [33].
This observation, as well as the results of neutron scattering and x-ray diraction, leads to
the conclusion that the orthorhombic to tetragonal structure change is due to static compositional eects, and is not driven by any thermal structural instability. The combination
of the observations above lead us to conclude with condence that the pairing symmetry in
the cuprate superconductors has d symmetry.
x2 y 2
Josephson junctions (JJ) are not only the basic tools used in many of the phase-sensitive
experiments demonstrating the d-wave order parameter symmetry, but also the structures
where d-wave OP related eects can be potentially exploited. In this section we will briey
review some signicant works in the eld, which include possible implementations of -JJ
circuitry. Furthermore we will introduce some concepts on Andreev bound states in HTS
systems, which provide additonal insights into the physics of the junctions and on phenomena
related to spontaneous currents in various systems.
A. Andreev bound states
As rst pointed out by Hu [36,37], a Josephson junction with d-wave electrodes can
have zero energy or \midgap" states (MGS ). These MGS's are Andreev bound states [38]
which are peculiar to superconductors with unconventional (such as d-wave) symmetries. An
Andreev bound state is created when an electron (hole) Andreev scatters from a superconducting interface, transmitting a Cooper pair, and reecting a hole (electron). For certain
trajectories in a d-wave superconductor the Andreev reection connects superconducting
gaps with a phase dierence . The results are analogous to the case of an SNS junction
with current carrying interface bound states with phase dependent energies. Several approaches to the Josephson and tunneling problems based on MGS s have been reviewed in
Ref. [40]. The formation of MGS provides a unied framework for several remarkable effects, such as 0 ! crossover with temperature, zero bias conductance peaks, paramagnetic
currents, time reversal symmetry breaking, spontaneous interface currents, and resonance
features in subgap currents. The existence of Andreev bound states has been conrmed by
dierent experiments, especially those based on tunneling spectroscopy. These experiments
revealed the expected peak of the density of states at zero energy (a wide collection of experimental data is reviewed in [40]). The half-periodic Josephson eect can be easily understood
in terms of the combined eects of conventional and -junctions. The Josephson current
through the -junctions will be a 2-periodic function of phase, shifted by with respect
to the current carried by zero levels [39]. This property has been exploited in developing
concepts for quantum computation, as will be discussed below.
B. Grain boundary Josephson junctions vs
While in conventional superconductors the archetypical Josephson junction sandwich
structure, made by two superconducting layers separated by a thin dieletric tunneling barrier,
can be made with high reproducibility and reliability, this is a very diÆcult task in the highT superconductors. This is due to the very short coherence length in these materials.
To obtain a good interface rigorous lm structural growth as well as superb control of
the local oxygen stoichiometry to avoid degradation of the superconducting properties is
required. To avoid such degradation a vast zoology of junction congurations have been
proposed. Junctions based on grain boundary (GB ) barriers (bicrystal [41], biepitaxial
[42{44] and step-edge structures [45]) have been proved to be very useful for basic studies
and applications.
Grain boundary Josephson junctions oer simple ways to change the relative orientation
of the main crystallographic axes in the two electrodes. By suitably selecting the misorientation angle, traditional Josephson junctions with Fraunhofer-like patterns and -loops can
be obtained. In both these extreme cases, the nature of the barrier is not completely known.
A general feature characterizing the junctions' behavior seems to be high dissipation. Furthermore, despite the fact that each barrier and GB microstructure depends strongly on
the type of junction and its conguration, faceting, occurring with dierent scaling lengths,
and a depression of superconducting properties near the barrier are general features of HT S
junctions. They typically depress the quality of the junctions, and a current crucial issue
concerns the reduction and/or the control of these eects.
However, in this paper we will focus our interest on techniques for fabricating -loops.
The bicrystal technique can be used to make any kind of GB and misorientation angle, but
it is constrained by the bicrystal line [41]. Tri- and tetra-crystal substrates have been used
for the fundamental experiments described above, but they are diÆcult to use for device
concepts that involve more than one loop. The biepitaxial technique, even if limited in
terms of possible misorientation angles, allows the placement of either ordinary junctions or
-loops by photolithographic means [42{44]. In principle, several -loops can be placed on
the same substrate, and their properties can even be tuned by properly adjusting the interface direction with respect to the electrodes. This possibility allows further exploitation of
the anisotropic properties of HTS and of tunneling in HTS junctions [44]. It turns out that
biepitaxial junctions allow a range of misorientation angles useful for spontaneous uxes and
-loops. Another way to obtain -loops is with junctions with a natural or articial (generally a normal metal such as Au) barrier and a low critical temperature counter-electrode
(such as Nb or Pb). These types of junctions, in particular those using high quality single
crystals and natural barriers, have been used for fundamental experiments (see section III).
We will refer to some of the basic intermediate steps which have been made towards 10
circuitry. Even if the experiments we refer to are still addressing fundamental issues and are
somewhat limited by the quality and the control of the junctions, they represent a real rst
step towards possible applications of these junctions within the framework of -circuitry.
The existence of some simple device concepts and capabilities have been demonstrated.
First, we briey recall some theoretical proposals of circuitry, and in particular of novel
device concepts for quantum computation.
A superconducting qubit operates either with charge- or ux- phase-states. The superconducting phase-qubit (\quiet qubit") has been considered as a very promising device
concept. The information is mainly expressed through the phase degree of freedom. The
phase-state itself carries no charge and no current. An interesting proposal, made by the
ETH Zurich group [46], for a superconducting phase-qubit involved mesoscopic junctions
combining s- and d-wave superconductors, so called S D S 0 -junctions (Fig 7). A simpler
design was proposed later, involving superconducting loops with ve Josephson junctions,
four conventional ones and a -junction (Fig. 8). In this design, the quantum degrees of
freedom involve only the conventional junctions, while the -junction, acting primarily as a
phase-shifter, is a macroscopic device and thus more easily fabricated.
Degeneracy of the ground state and spontaneous currents and uxes in equilibrium are
unique properties that could favour novel designs for solid state qubits [47,48]. Alternative
concepts are based on the occurrence of Andreev bound-states in dierent types of Josephson
junctions [49].
Some evidence of doubly degenerate states required by the qubit has been provided
by measurements on the the critical current (I ) vs phase relation in 45Æ asymmetric GB
junctions [50]. More recently the same behavior has been observed in the symmetric conguration for sub-micron junctions, where phenomena due to faceting are reduced. In this
conguration eects due to d-wave symmetry are more neat and less mediated by extrinsic features. In principle the symmetric conguration can violate time-reversal symmetry
without producing local magnetic elds.
The work done at the University of Napoli on GB biepitaxial Josephson junctions is
promising in terms of -circuitry [44]. The junctions are based on particular types of GB s
where the usual 45Æ in-plane rotation of the a-b planes around the c-axis is accompanied by
a 45Æ rotation of the c-axis. The GB is a consequence of the two dierent growths along the
(103) direction on the bare substrate (SrT iO3) and along the (001) direction on the CeO2
seed layer respectively. If MgO is used as a seed layer, no in-plane rotation occurs and high
quality conventional Josephson junctions are obtained. Since the seed layer is patterned by
using only photolithographic means, on the same sample the GB interface can be arbitrarily
oriented with respect to the two electrodes. Despite the complicated morphology of the
resulting GB , this technique also oers the possibility of basal plane GB s in the tilt limit, and
as a consequence very controlled interfaces with reduced faceting. Transport measurements
have been made recently as a function of misorientation angle on these junctions employing
CeO2 as seed layers. Oscillations of the critical current as a function of the misorientation have been observed for the rst time, in remarkable agreement with the expected theoretical
values [51]. This result conrms the relevance of eects of d-wave OP symmetry on the
properties of the junctions, and proves that intrinsic d-wave eects can predominate over
extrinsic eects, such as faceting. Furthermore it demonstrates that these junctions can be
considered as a rst step towards the realization of simple circuits with both traditional
and loops to test concepts for circuitry. The biepitaxial junctions oer the possibility
of obtaining the doubly degenerate state required for a qubit as in the proposals in [46],
and to realize all d-wave qubit arrays, including those based on the half-integer quantum
eect [48]. An example of how to apply the biepitaxial technique to the realization of the
designs proposed in ref. [46,48] is shown in Fig. 9. This technique would combine the
possibility of placing the ordinary 0 junctions (where no additional -phase shift arise) in
correspondence to the MgO seed layer, and to exploit the possible doubly degenerate state
of asymmetric 45Æ asymmetric GB junctions in correspondence of the CeO2 seed layer to
replace the S D S 0 system of Fig. 7 or the junctions, respectively. These developments
represent an important step towards the realization of the complementary Josephson logic
(memory) circuits [52].
Another signicant contribution towards -circuitry is represented by YBCO-Nb zig-zag
shaped ramp-type Josephson junctions developed at the University of Twente [32]. Their
study retraces the path of similar studied on 45Æ asymmetric bicrystal [53] and biepitaxial junctions in a very reproducible and controlled system. Multiple 0- and -junctions
are placed controllably at arbitrary positions. The magnetic pattern features strongly support the hypothesis that the anomalous characteristics of the GB junctions are due to the
occurrence of regions with additional -phase shifts along the interface, i.e. the faceted
microstructure of the grain boundaries.
We have been mainly considering S-N-S (N normal metal) and grain boundary JJs.
Although these techniques have been proved to be successful for dierent aims, several
experimental approaches are currently still limited by the absence of low dissipation S-I-Slike HTS junctions. If this is an intrinsic or extrinsic feature is still an open and challenging
issue. This limit prevents the straightforward application of the advantages of intrinsic
frustration of d-wave HTS in suitable systems to applications based on this eect, including
quantum circuit designs. This consideration has to be extended at the moment to other more
sophisticated types of HTS JJ. Nevertheless, apart from nding the appropriate material
science recipe to fabricate the ideal traditional tri-layer structure [54], new "frontiers" can be
opened by nanolithography techniques properly applied to HTS systems. On one hand this
may lead to a better control of intrinsic junctions, i.e. junctions between dierent planes
in the same lattice. On the other hand the behavior of the various HTS junctions, and
especially of those where eects due to d-wave symmetry are more relevant (45Æ asymmetric
GB congurations), could take advantage of reduced width in terms of reproducibility and
uniformity. HTS junction properties are still mostly unexplored in the limit of sub-micron
widths and very low temperatures, which seem to be the more appropriate conditions for
quantum circuitry.
Once the predominant d-wave nature of the order parameter (OP ) in the bulk of HT S is
established, a crucial issue concerns the possible existence of an imaginary component of the
order parameter at interfaces and surfaces [55]. This problem has been addressed in several
dierent theoretical approaches and experiments and is very relevant for the physics of HT S
junctions. The existence of such a component of the OP would imply the existence of a
gapped region near the surface or interface with several important consequences. Theoretical
predictions seem to invoke the presence of such a region, but only some experimental results
seem to support this hypothesis. The common idea is that if this component exists it is
quite small and localized in a very narrow region close to interfaces and surfaces. The most
remarkable phenomena associated with such an OP conguration would be the presence of
spontaneous currents along interfaces and surfaces [55,58]. These currents can apparently be
detected through tunneling measurements [57] or more directly through Scanning SQUID
Microscopy [1]. The drawback is that they are extremely localized. These currents can
also be the consequence of a more general process of time reversal symmetry breaking not
necessarily related to the presence of an out-of-phase component of the OP . For instance it
has been shown that the splitting of MGS by spontaneous establishment of a phase dierence
= =2 across the junction may be responsible for symmetry breaking [40]. This has been
considered as an evidence that only measurements at surfaces and not at interfaces can be
interpreted as clear benchmark of the existence of a sub-dominant component of the OP
[40,56]. Furthermore, dierently from the other case of time reversal symmetry breaking of
a doubly connected geometry, such as a ring with -junctions, geometry at interfaces and
surfaces is simply connected. In this case no quantization condition for the spontanous ux
holds and the ux can have arbitrary values [53,55,58{60].
Phase-sensitive symmetry tests have provided convincing evidence in favor of predominantly d-wave pairing symmetry in a number of optimally doped cuprates, providing one of
the few established basic features of HTS. The eects of the d-wave order parameter symmetry on the phenomenology of HTS Josephson junctions are remarkably systematic and
\reproducible", especially if one considers the complexity of HTS and of HTS junctions and
the presence of related extrinsic eects. The possibility of exploiting the advantages of the
presence of d-wave induced spontaneous currents and of time reversal symmetry breaking
recently stimulated research in the eld of -circuitry and even of d-wave qubit arrays. A
current crucial challenge in this eld is further improvement in the quality of the junctions,
especially the achievement of low dissipation junctions. We also speculate that, as in the
case of the symmetry tests of HTS, eects related to the Josephson eect, spontaneous magnetization, vortices and topological eects may play a mayor role in studying crucial issues
such as topological order and phase transitions.
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FIG. 1. Schematic phase diagram for the cuprate high temperature superconductors.
Anti-ferromagnetic (AFM) regions with Neel temperatures TN , superconducting regions (SC) with
critical temperatures Tc , and pseudogap regions with pseudogap temperatures T exist for both
hole-doped (right side of phase diagram) and electron-doped (left side). The details of the phase
diagrams are much dierent for electron- vs hole-doped cuprates, but both have predominantly
pairing symmetry.
FIG. 2. Minimum free-energy solutions for the ux , normalized by the superconducting ux
quantum 0 = hc=2e, as a function of the normalized externally applied ux a =0 , for values of
= 2LIc =0 = 0.5,1,2,5,10,100, for a superconducting ring with inductance L, interrupted by a
Josephson weak link with critical current I0 , for a 0-SQUID (no intrinsic phase shift) (a), and a
with an intrinsic phase shift of (b).
FIG. 3. Critical current ISQU ID , normalized by the single junction critical current I0 , for a
symmetric two-junction 0-SQUID (a) and -SQUID (b), as a function of the normalized externally
applied ux a =0 , for = 2LI0 =0 =0,0.5,1,2,5,10, and 100.
FIG. 4. Magnetic interference patterns for a thin lm, spatially distributed junctions, low inductance -SQUID (a,b), and a 0-SQUID (c,d). The characteristics show a nearly ideal interference
pattern over a small eld range (b,d), with a shift of 0 /2, as expected. An interference envelope
due to the nite size of the junctions is apparent over a larger eld range (a,b). From Schultz et
al., Ref. 16.
FIG. 5. Experimental geometry for the tricrystal pairing symmetry experiments (a), and three
dimensional rendering of a scanning SQUID microscope image of a ring sample cooled in zero eld.
The central ring, designed to be a -ring for a superconductor with dx2
pairing symmetry, has
spontaneous magnetization with 0 =2 total ux threading through the it. The three outer control
rings, designed to be 0-rings, have no ux in them.
FIG. 6. Tetracrystal geometry (a). (b) SSM image of a lm of Tl2201 epitaxially grown on a
SrTiO3 substrate with the geometry of (a), cooled in nominally zero eld, and imaged at 4.2K with
an 8.2m square pickup loop. (c) Cross-sections through a bulk Abrikosov vortex, and through
the half-vortex of (b), along the directions indicated in (a). The dots are experimental data, and
the solid lines are modeling, assuming the Abrikosov vortex has hc/2e ux trapped in it, and the
vortex at the tetracrystal point has hc/4e ux.
FIG. 7. The design of the qubit structure proposed by the Zurich group is based on quenching
the lowest order coupling by arranging a junction with its normal aligned with the node of the
d-wave order parameter, thus producing a doubly periodic current-phase relation. The ground
states of the S-D-S' junctions are degenerate and carry no current, while still being distinguishable
from one another. From Ioe et al., Ref. 46a.
FIG. 8. A ve junction loop including a junction uses a =2 junction with two degenerate
minima with phase drops +=2 or
The four conventional junctions are grouped into two
pairs (a) and the energy-phase relation (shown in (b)) is calculated by minimizing the energy of
each pair. The thin lines are the individual energies of the two pairs with a relative phase shift by
the thick line is their sum for a symmetric conguration producing minima at +=2 or
The dotted lines represent the case of symmetry breaking with degenerate minima dierent from
+=2 or
From Blatter et al., Ref. 46b.
FIG. 9. Schematic diagrams of the qubit structures proposed in Ref.s 46a and 46b designed
using biepitaxial grain boundaries. These junctions congurations may mimic the designs of Fig.s
7 and 8 in some conditions, generating the desired doubly degenerate state. From Tafuri et al.,