We theoretically investigate the crystalline anisotropy of topological phase transitions in phase-controlled planar Josephson junctions (JJs) subject to spin-orbit coupling and in-plane magnetic fields. It is shown how topological superconductivity (TS) is affected by the interplay between the magnetic field and the orientation of the junction with respect to its crystallographic axes. This interplay can be used to electrically tune between different symmetry classes in a controlled fashion and thereby optimize the stability and localization of Majorana bound states in planar Josephson junctions. Our findings can be used as a guide for achieving the most favorable conditions when engineering TS in planar JJs and can be particularly relevant for setups containing non-collinear junctions which have been proposed for performing braiding operations on multiple Majorana pairs.
We study vortex structure in three different models of long Josephson junctions: exponentially shaped Josephson junction and Josephson junctions with resistor and shunt inhomogeneities in barrier layer. Numerical calculations of the possible magnetic flux distributions and corresponding bifurcation curves have done. For these three models the critical curves ``critical current-magnetic field'' are constructed. We develop an idea of the equivalence of exponentially shaped Josephson junction and rectangular junction with distributed inhomogeneity and demonstrate that at some parameters of shunt …
Operation of a Superconducting Quantum Interference Device (SQUID) made of stacked Josephson junctions is analyzed numerically for a variety of junction parameters. Due to a magnetic coupling of junctions in the stack, such a SQUID has certain advantages as compared to an uncoupled multi-junction SQUID. Namely, metastability of current-flux modulation can be reduced and a voltage-flux modulation can be improved if junctions in the stack are phase-locked. Optimum operation of the SQUID is expected for moderately long, strongly coupled stacked Josephson junctions. A possibility of making a stack…
The properties of Josephson junction including a ferromagnetic insulating layer are studied theoretically. The calculated current shows the anomalous dependence on the magnitude of the exchange interaction at the interface. The Josephson junction changes from 0-junction to $π$-junction with the increase of the magnitude of the exchange interaction.
Magnetotransport measurements were done on $\Nb/\Al_2Ø_3/\Cu/\Ni/\Nb$ superconductor-insulator-ferromagnet-superconductor Josephson tunnel junctions. Depending on ferromagnetic $\Ni$ interlayer thickness and geometry the standard (1d) magnetic field dependence of critical current deviates from the text-book model for Josephson junctions. The results are qualitatively explained by a short Josephson junction model based on anisotropy and 2d remanent magnetization.
We clarify the applicability of the quasistatic approximation used in Ref. [\onlinecite{VE}], where coupled spin and Josephson plasma waves have been predicted to exist in SIFS Josephson junctions. We show, contrary to the claim of the authors of Ref. [\onlinecite{Maekawa}], that this approximation is very accurate in realistic systems studied experimentally.
Josephson radiation is a useful signature for detecting Majorana zero modes in topological superconductors. We study the Josephson radiation of the underdamped topological Josephson junction within the quantum resistively and capacitively shunted junction model. We show that the quantum dynamics of the Majorana two-level system induce oscillatory patterns in the Josephson emission spectra. With the Floquet method, we obtain analytical results for these oscillatory patterns and find that they are well described by Bessel functions. We perform numerical simulations to verify the analytical resul…
We highlight the reproducibility and level of control over the electrical properties of YBa$_2$Cu$_3$O$_7$ Josephson junctions fabricated with irradiation from a focused helium ion beam. Specifically, we show the results of electrical transport properties for several junctions fabricated using a large range of irradiation doses. At the lower end of this range, junctions exhibit superconductor-normal metal-superconductor (SNS) Josephson junction properties. However, as dose increases there is a transition to electrical characteristics consistent with superconductor-insulator-superconductor (SIS…
We calculate the current phase relation of a planar Josephson junction with a ferromagnetic weak link located on top of a thin normal metal film. Following experimental observations we assume transparent superconductor-ferromagnet interfaces. This provides the best interlayer coupling and a low suppression of the superconducting correlations penetrating from the superconducting electrodes into the ferromagnetic layer. We show that this Josephson junction is a promising candidate for an experimental ϕ junction realization.
The spontaneous flux generation and magnetic field modulation of the critical current in a 0-pi Josephson junction are calculated for different ratios of the junction length to the Josephson penetration depth, and different ratios of the 0-junction length to the pi-junction length. These calculations apply to a Pb-YBCO c-axis oriented junction with one YBCO twin boundary, as well as other experimental systems. Measurements of such a junction can provide information on the nature of the c-axis Josephson coupling and the symmetry of the order parameter in YBCO. We find spontaneous flux even for …
The effect of microwave irradiation on the phase dynamics of intrinsic Josephson junctions in high temperature superconductors is investigated. We compare the current-voltage characteristics for a stack of coupled Josephson junctions under external irradiation calculated in the framework of CCJJ and CCJJ+DC models.
We observed splitting of the low-bias vortex-flow branch in a dense-Josephson-vortex state into multiple sub-branches in current-voltage characteristics of intrinsic Josephson junctions (IJJs) of Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+x}$ single crystals in the long-junction limit. Each sub-branch corresponds to a plasma mode in serially coupled Josephson junctions. Splitting into low-bias linear sub-branches with a spread in the slopes and the inter-sub-branch mode-switching character are in good quantitative agreement with the prediction of the weak but finite inter-junction capacitive-coupling mod…
We study magnetization reversal in a $φ_0$ Josephson junction with direct coupling between magnetic moment and Josephson current. Our simulations of magnetic moment dynamics show that by applying an electric current pulse, we can realize the full magnetization reversal. We propose different protocols of full magnetization reversal based on the variation of the Josephson junction and pulse parameters, particularly, electric current pulse amplitude, damping of magnetization and spin-orbit interaction. We discuss experiments which can probe the magnetization reversal in $φ_0$-junctions.
We discuss how to control dc Josephson current by influencing the structure and nonequilibrium population of Andreev levels via external electrostatic gates, external current injection and electromagnetic radiation. In particular we will consider the "giant" Josephson current in "long" SIS tunnel junctions and the regular and anomalous nonequilibrium Josephson currents in three terminal SNS junctions. We will briefly discuss applications to the Josephson field effect transistor (JOFET) and to the newly invented Josephson interference transistor (JOINT).
Quantum fluctuations in a current biased Josephson junction, described in terms of the RCSJ-model, are considered. The fluctuations of the voltage and phase across the junction are assumed to be initiated by equilibrium current fluctuations in the shunting resistor. This corresponds to low enough temperatures, when fluctuations of the normal current in the junction itself can be neglected. We used the quantum Langevin equation in terms of random variables related to the limit cycle of the nonlinear Josephson oscillator. This allows to go beyond the perturbation theory and calculate the widths …
The most important practical characteristic of a Josephson junction is its critical current. The shape of the junction determines the specific form of the magnetic-field dependence of the its Josephson current. Here we address the magnetic diffraction patterns of specially shaped planar Josephson tunnel junctions. We focus on a wide ensemble of generalized ellipses, called superellipses, which retain the second order symmetry. We analyze the implications of this type of isometry and derive the explicit expressions for the threshold curves of superelliptic Josephson junctions. A detailed study …
We consider theoretically and numerically magnetic field dependencies of the maximum supercurrent across Josephson tunnel junctions with spatially alternating critical current density. We find that two flux-penetration fields and one-splinter-vortex equilibrium state exist in long junctions.
The development of superconducting memory and logic based on magnetic Josephson junctions relies on an understanding of junction properties and, in particular, the dependence of critical current on external magnetic flux (i.e. Fraunhofer patterns). With the rapid development of Josephson junctions with various forms of inhomogeneous barrier magnetism, Fraunhofer patterns are increasingly complex. In this paper we model Fraunhofer patterns for magnetic Josephson junctions in which the barrier magnetic susceptibility is position- and external magnetic field dependent. The model predicts anomalou…
The physics of Josephson tunnel junctions drastically depends on their geometrical configurations. The shape of the junction determines the specific form of the magnetic-field dependence of the its Josephson current. Here we address the magnetic diffraction patterns of specially shaped planar Josephson tunnel junctions in the presence of an in-plane magnetic field of arbitrary orientations. We focus on a wide ensemble of junctions whose shape is invariant under point reflection. We analyze the implications of this type of isometry and derive the threshold curves of junctions whose shape is the…
We investigate stationary charge imbalance effect in the system of coupled overdamped Josephson junctions. We show that coupling between junction and nonzero stationary charge imbalance in the resistive state bring to a decrease of the Josephson frequency in the Josephson junctions of the stack. The formed difference in Josephson frequency leads to the nonuniform switch to the Shapiro step regime in the presence of external electromagnetic radiation and appearance of kinks of voltage on the IV-characteristics of the stack. We also show that stationary charge imbalance brings to the slope of th…
We report systematic measurements of the critical current versus magnetic field patterns of annular Josephson junctions in a wide magnetic field range. A modulation of the envelope of the pattern, which depends on the junction width, is observed. The data are compared with theory and good agreement is found.
We confirmed the existence of the collective transverse plasma modes excited by the motion of the Josephson vortex lattice in stacked intrinsic Josephson junctions of Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ by observing the multiple subbranches in the Josephson-vortex-flow current-voltage characteristics. We also observed the symptom of the microwave emission from the resonance between the Josephson vortex lattice and the collective transverse plasma modes, which provides the possibility of developing Josephson-vortex-flow electromagnetic oscillators.
A novel way to realize a pi Josephson junction is proposed, based on a weak link in an unconventional d-wave superconductor with appropriately chosen boundary geometry. The critical current of such a junction is calculated from a fully selfconsistent solution of microscopic Eilenberger theory of superconductivity. The results clearly show, that a transition to a pi Josephson junction occurs for both low temperatures and small sizes of the geometry.
We have observed a dissipative phase diffusion branch in arrays of hysteretic high-Tc intrinsic Josephson junctions. By comparing the data with a thermal activation model we extract the impedance seen by the junction in which phase diffusion is occurring. At the plasma frequency this junction is isolated from its environment and it sees its own large (~ kilo Ohm) impedance. Our results suggest that stacks of Josephson junctions may be used for isolation purposes in the development of a solid state quantum computer.
Tunnel junction between two superconductors is considered in the vicinity of the critical temperature. Superconductive fluctuations above Tc give rise to the noise of the ac Josephson current although the current itself is zero in average. As a result of fluctuations, current noise spectrum is peaked at the Josephson frequency, which may be considered as precursor of superconductivity in the normal state. Temperature dependence and shape of the Josephson current noise resonance line is calculated for various junction configurations.
Josephson vortices in naturally stacked Bi-2212 tunneling junctions display rich dynamic behavior that derives from the coexistence of three basic states: static Josephson vortex lattice, coherently moving lattice, and incoherent quasiparticle tunneling state. Rich structure of hysteretic branches observed in the current-voltage characteristics can be understood as combinatorial combinations of these three states which are realized in different junctions and evolve separately with magnetic field and bias current. In particular, the multiple Josephson-vortex-flow branches at low bias currents a…
We examine the magnetic interference patterns of Josephson junctions with a region of 0- and of pi-phase shift. Such junctions have recently been realized as c-axis YBCO-Pb junctions with a single twin boundary in YBCO. We show that in general the junction generates self-fields which introduces an asymmetry in the critical current under reversal of the magnetic field. Numerical calculations of these asymmetries indicate they account well for the unexplained features observed in single twin boundary junctions.
Transport properties of high quality Nb/semiconductor/Nb long Josephson junctions based on metamorphic In_0.75Ga_0.25As epitaxial layers are reported. Different junction geometries and fabrication procedures are presented that allow a systematic comparison with quasiclassical theory predictions. The impact of junction transparency is highlighted and a procedure capable of yielding a high junction quality factor is identified.
We have studied both theoretically and experimentally how an LC series circuit connected in parallel to a Josephson junction influences the Josephson dynamics. The presence of the shell circuit introduces two energy scales, which in specific cases can strongly differ from the plasma frequency of the isolated junction. Josephson junctions were manufactured using Nb/Al-AlOx/Nb fabrication technology with various on-chip LC shunt circuits. Spectroscopic measurements in the quantum limit show an excellent agreement with theory taking into account the shunt inductance and capacitance in the Resisti…
The $φ$ Josephson junction has a doubly degenerate ground state with the Josephson phases $\pmφ$. We demonstrate the use of such a $φ$ Josephson junction as a memory cell (classical bit), where writing is done by applying a magnetic field and reading by applying a bias current. In the "store" state, the junction does not require any bias or magnetic field, but just needs to stay cooled for permanent storage of the logical bit. Straightforward integration with Rapid Single Flux Quantum logic is possible.
We report on the noise properties of ion-irradiated YBa$_2$Cu$_3$O$_7$ Josephson junctions. This work aims at investigating the linewidth of the Josephson oscillation with a detector response experiment at $\simeq$132 GHz. Experimental results are compared with a simple analytical model based on the Likharev-Semenov equation and the de Gennes dirty limit approximation. We show that the main source of low-frequency fluctuations in these junctions is the broadband Johnson noise and that the excess ($\frac{1}{f}$) noise contribution does not prevail in the temperature range of interest, as report…
A magnetic field induced electric polarization and the corresponding change of an effective junction capacitance are considered within a 3D model of disordered Josephson junction arrays. At some threshold field (near the Josephson network critical field), the effective junction charge and the related capacitance are shown to reach a maximum and to change a sign, respectively. A possibility to observe the predicted effects in artificially prepared arrays of superconducting grains is discussed.
The Josephson effect in p-wave superconductor / diffusive normal metal / p-wave superconductor junctions is studied theoretically. Amplitudes of Josephson currents are several orders of magnitude larger than those in s-wave junctions. Current-phase (J-ϕ) relations in low temperatures are close to those in ballistic junctions such as J\propto\sin(ϕ/2) and J\proptoϕeven in the presence of random impurity potentials. A cooperative effect between the midgap Andreev resonant states and the proximity effect causes such anomalous properties and is a character of the spin-triplet superconductor juncti…
We derive the qualitative behavior of superconductor-antiferromagnet- superconductor (SAS) Josephson junctions described by Zhang's SO(5) theory. The main differences between these junctions and conventional SIS junctions arise from the non-sinusoidal current-phase relation derived by Demler et al. for thin SAS junctions. Using a simple approximation to this non-sinusoidal function, the current voltage relation, Shapiro steps, thermal fluctuation effects and the diffraction pattern in a magnetic field are obtained.
I consider a Josephson junction crossing the middle of a thin rectangular superconducting strip of length L and width W subjected to a perpendicular magnetic induction B. I calculate the spatial dependence of the gauge-invariant phase difference across the junction and the resulting B dependence of the critical current Ic(B).
We investigate the dynamics of a resistively shunted Josephson junction. We compute the Josephson frequency and the generalized impedances for a variety of the parameters, particularly with relevance to predicting the measurable effects of zero-temperature current noise in the resistor.
Many experiments on high-temperature superconductors have shown paramagnetic behavior when the sample is field cooled. The paramagnetism was first attributed to a d-wave order parameter creating pi-junctions in the samples. However, the same effect was later discovered in traditional low-temperature superconductors and conventional Josephson-junction arrays which are s-wave. By simulating both conventional and mixed pi/conventional Josephson-junction arrays we determine that differences exist which may be sufficient to clearly identify the presence (or absence) of pi-junctions. In particular t…
We present a unified description of a junction between $s$-wave (BCS) superconductors and a junction between $p$-wave superconductors in a topologically nontrivial phase, which relies on a scattering state expansion. We compute Josephson current and thermal noise in the two kinds of junction and exhibit some characteristic features for a junction of two topological superconductors hosting Majorana zero-energy modes.
We report on anomalous conductance in a resistively and capacitively shunted Josephson junction which is simultaneously driven by ac and dc currents. The dependence of the voltage across the junction on the frequency of the ac current shows windows of absolute negative conductance regimes, i.e. for a positive (negative) dc current, the voltage is negative (positive).
We report the electrical transport in vertical Josephson tunnel junctions (area 400 $μm$$^2$) using GdBa$_2$Cu$_3$O$_7$$_{-δ}$ electrodes and SrTiO$_3$ as an insulating barrier (with thicknesses between 1 nm and 4 nm). The results show Josephson coupling for junctions with SrTiO$_3$ barriers of 1 nm and 2 nm. The latter displays a Josephson of 8.9 mV at 12 K. This value is larger than the usually observed in planar arrays of junctions. Our results are promising for the development of superconducting electronic devices in the terahertz regime.
The phase dynamics of the stack of long Josephson junctions has been studied. Both inductive and capacitive couplings between Josephson junctions have been taken into account in the calculations. The IV--curve, bias current dependence of radiation power and dynamics of each JJs of the stack have been investigated. The coexistence of the charge traveling wave and fluxon states has been observed. This state can be considered as a new collective excitation in the system of coupled Josephson junctions. We demonstrate that the observed collective excitation leads to the decreasing of radiation powe…
Traditionally, since the discovery of the Josephson effect in 1962, the magnetic diffraction pattern of planar Josephson tunnel junctions has been recorded with the field applied in the plane of the junction. Here we discuss the static junction properties in a transverse magnetic field where demagnetization effects imposed by the junction geometry and configuration of the electrodes are important. Measurements of the critical current versus magnetic field in planar Nb-based high-quality junctions with different geometry, size and critical current density show that it is advantageous to use a t…
We demonstrate experimentally the existence of Josephson junctions having a doubly degenerate ground state with an average Josephson phase ψ=\pmϕ. The value of ϕ can be chosen by design in the interval 0<ϕ<π. The junctions used in our experiments are fabricated as 0-π Josephson junctions of moderate normalized length with asymmetric 0 and π regions. We show that (a) these ϕ Josephson junctions have two critical currents, corresponding to the escape of the phase ψ from -ϕ and +ϕ states; (b) the phase ψ can be set to a particular state by tuning an external magnetic field or (c) by using a prope…
The sin-Gordon equation for Josephson junctions with arbitrary misaligned anisotropic banks is derived. As an application, the problem of Josephson vortices at twin planes of a YBCO-like material is considered. It is shown that for an arbitrary orientation of these vortices relative to the crystal axes of the banks, the junctions should experience a mechanical torque which is evaluated. This torque and its angular dependence may, in principle, be measured in small fields, since the flux penetration into twinned crystals begins with nucleation of Josephson vortices at twin planes.
In this letter, we present the study of the high-frequency mixing properties of ion irradiated YBa2Cu3O7 Josephson nano-junctions. The frequency range, spanning above and below the characteristic frequencies fc of the junctions, permits clear observation of the transition between two mixing regimes. The experimental conversion gain was found to be in good agreement with the prediction of the three ports model. Finally, we discuss the potential of the junctions to build a Josephson mixer operating in the terahertz frequency range.
Within the framework of the microscopic model of tunneling, we modelled the behavior of the Josephson junction shunted by the Superconductor-Insulator-Normal metal (SIN) tunnel junction. We found that the electromagnetic impedance of the SIN junction yields both the frequency-dependent damping and dynamic reactance which leads to an increase in the effective capacitance of the circuit. We calculated the dc I-V curves and transient characteristics of these circuits and explained their quantitative differences to the curves obtained within the resistively shunted junction model. The correct oper…
We consider Majorana zero modes in a Josephson junction on top of a topological insulator in transverse magnetic field. Majorana zero modes reside at periodically located nodes of Josephson vortices. We find that hybridization of these modes is prohibited by symmetries of the problem at vanishing chemical potential, which ensures better protection of zero modes and yields methods to control the tunnel coupling between Majorana modes for quantum information processing applications.
We predict a novel type of Josephson effect to occur in triplet superconductor - ferromagnet - triplet superconductor Josephson junctions. We show that the Josephson current, I_J, exhibits a rich dependence on the relative orientation between the ferromagnetic moment and the d-vectors of the superconductors. This dependence can be used to build several types of Josephson current switches. Moreover, we predict an unconventional temperature dependence of I_J in which I_J changes sign with increasing temperature
We demonstrate that shunting of Superconductor-Insulator-Superconductor Josephson junctions by Superconductor-Insulator-Normal metal (S-I-N) structures having pronounced non-linear I-V characteristics can remarkably modify the Josephson dynamics. In the regime of Josephson generation the phase behaves as an overdamped coordinate, while in the superconducting state the damping and current noise are strikingly small, that is vitally important for application of such junctions for readout and control of Josephson qubits. Superconducting Nb/AlO${_x}$/Nb junction shunted by Nb/AlO${_x}$/AuPd juncti…
Although Josephson junction qubits show great promise for quantum computing, the origin of dominant decoherence mechanisms remains unknown. We report Rabi oscillations for an improved phase qubit, and show that their coherence amplitude is significantly degraded by spurious microwave resonators. These resonators arise from changes in the junction critical current produced by two-level states in the tunnel barrier. The discovery of these high frequency resonators impacts the future of all Josephson qubits as well as existing Josephson technologies. We predict that removing or reducing these res…
We propose a method to realize a $ϕ$ Josephson junction by combining alternating 0 and $π$ parts (sub junctions) with an intrinsically non-sinusoidal current-phase relation (CPR). Conditions for the realization of the $ϕ$ ground state are analyzed. It is shown that taking into account the non-sinusoidal CPR for a "clean junction with a ferromagnetic (F) barrier, one can significantly enlarge the domain (regime of suitable F-layer thicknesses) of the $ϕ$ ground state and make the practical realization of $ϕ$ Josephson junctions feasible. Such junctions may also have two different stable solutio…
Triplet supercurrents in multilayer ferromagnetic Josephson junctions with misaligned magnetization can penetrate thicker ferromagnetic barriers compared to the singlet component. Although the static properties of these junctions have been extensively studied, the dynamic characteristics remain largely unexplored. Here we report a comprehensive electrodynamic characterization of multilayer ferromagnetic Josephson junctions composed of Co and Ho. By measuring the temperature-dependent current-voltage characteristics and the switching current distributions down to 0.3 K, we show that phase dynam…
Intrinsic Josephson-junction stacks realized in mesas fabricated out of high-temperature superconductors may be used as sources of coherent electromagnetic radiation in the terahertz range. The major challenge is to synchronize Josephson oscillations in all junctions in the stack to get significant radiation out of the crystal edge parallel to the c axis. We suggest a simple way to solve this problem via artificially prepared lateral modulation of the Josephson critical current identical in all junctions. In such a stack phase oscillations excite the in-phase Fiske mode when the Josephson freq…
We present ferromagnetic Nb/Al2O3/Ni60Cu40/Nb Josephson junctions (SIFS) with an ultrathin Al2O3 tunnel barrier. The junction fabrication was optimized regarding junction insulation and homogeneity of current transport. Using ion-beam-etching and anodic oxidation we defined and insulated the junction mesas. The additional 2 nm thin Cu layer below the ferromagnetic NiCu (SINFS) lowered interface roughness and ensured very homogeneous current transport. A high yield of junctional devices with jc spreads less than 2% was obtained.
Employing mutual-inductance measurements we study the magnetic field dependence of complex AC susceptibility of artificially prepared highly ordered (periodic) two-dimensional Josephson junction arrays of unshunted Nb-AlO_x-Nb junctions. The observed behavior can be explained assuming single-plaquette approximation of the overdamped model with an inhomogeneous critical current distribution within a single junction.
We treat theoretically Shapiro steps in tunnel Josephson junctions with spatially alternating critical current density. Explicit analytical formulas for the width of the first integer (normal) and half-integer (anomalous) Shapiro steps are derived for short junctions. We develop coarse-graining approach, which describes Shapiro steps in the voltage-current curves of the asymmetric grain boundaries in YBCO thin films and different superconductor-ferromagnet-superconductor Josephson-type heterostructures.
Preliminary results are presented concerning static properties of a small Josephson junction under the influence of strong microwave radiation. We discuss the correspondence between a Brownian particle moving in a periodic potential and superconducting phase difference in a small Josephson junction. Next, we describe an experimental method of determining the amplitude of microwave current flowing across the junction. Typical examples of static characteristics of the junction are presented, including its dynamical resistance as a function of microwave power. We discuss also the influence of an …
Extended Josephson junction driven by dc bias current is studied numerically. Two types of solutions, symmetrical and asymmetrical, are found. The current-voltage characteristic (IVC) is calculated. The symmetrical solutions form main histeretic IVC and asymmetrical one create an additional branch. Depending on the bias current value periodic, quasiperiodic and chaotic modes of the junction motion was observed. Dynamics of the junction affected by thermal fluctuations was analyzed. Stability of different states of the junction is discussed.
We report experimental and numerical analysis of expontentially shaped long Josephson junctions with lateral current injection. Quasi-linear flux flow branches are observed in the current-voltage characteristic of the junctions in the absence of magnetic field. A strongly asymmetric response to an applied magnetic field is also exhibited by the junctions. Experimental data are found in agreement with numerical predictions and demonstrate the existence of a geometry-induced potential experienced by the flux quanta in nonuniform width junctions.
