Here we discuss the results of our model, which include the effects of energy on the voltage-current (V-I) curve, due to the magnetic field B* inductively coupling vortices and antivortices. This magnetostatic energy leads to a sharp threshold current for vortex pair creation, and causes vortex tunneling to become correlated in time above threshold. Simulation results are compared with experimental measurements of critical currents in HTS-coated conductors and with voltage-current characteristics of cuprate HTS grain boundary junctions and iron-pnictide bicrystal junctions.

#### 2.1. Weber Blockade Threshold Current for Vortex Pair Creation

An HTS grain boundary behaves as a JJ with a periodic Josephson coupling energy vs.

φ. Since a 2π change in

φ(

x) generates a circulating current and vector potential encompassing one flux quantum,

Ф_{0}, a 2π soliton (antisoliton) in

φ is equivalent to a Josephson vortex (antivortex), e.g., see [

6,

8]. A sufficiently high applied current enables nucleation of a bubble bounded by solitons [

22], as shown in

Figure 1. Once nucleated, the lower-energy bubble expands as the Lorentz-like force from the current drives the vortex and antivortex apart. Though spatially extended, Josephson vortices are extremely light. Using the expression provided by Grosfeld and Stern [

27], the mass of a Josephson vortex spanning a 1-µm-thick film is estimated to be ~10

^{−2}m_{e}. If, in a layered superconductor, we view a full vortex as a composite of pancake vortices [

20] (see

Figure 1b), then the mass of each pancake vortex within a single layer is orders of magnitude smaller still.

The two-terminal Weber blockade mechanism proposed here leads to a threshold current for vortex pair creation and is dual to the Coulomb blockade effect that blocks tunneling of electrons [

28] or charge solitons [

11,

12,

13,

14] below a threshold field (see [

29] for a three-terminal Weber blockade effect.)

Figure 2 illustrates vortex pair creation in a zero externally applied field. The vortices generate a magnetic field with an average value,

${B}^{*}~\beta {\Phi}_{0}/{\lambda}^{2}$, linking the vortex and anti-vortex. Here

$\lambda =\sqrt{{\lambda}_{J}{\lambda}_{L}}$ or

$\lambda ={\lambda}_{L}$ for a Josephson or Abrikosov vortex pair, respectively, and

β is a dimensionless factor to account for effects of non-uniform field [

30], partial field penetration, and/or any vortex overlap. When the film thickness

d becomes small,

λ will be replaced by the thickness-dependent Pearl penetration length Λ, to be discussed later.

An applied current density

**J** in a film of thickness

d creates an additional self-field,

${B}_{J}\cong {\mu}_{0}Jd/2$, which runs just outside the film perpendicular to

**J** in opposite directions above and below the film, as shown in

Figure 2, and can partially cancel

B*. Once nucleated, the vortices are driven apart by the Lorentz-like force between flux and current.

Energy conservation prevents vortex pair creation for arbitrarily small applied currents because of the vortices’ inductive magnetic energy. Even though

B_{J} and

B* partially cancel, pair creation is blocked when the integrated magnetic energy between vortices with the pair,

$\propto {\left({B}_{J}-{B}^{*}\right)}^{2}/2{\mu}_{0}$, exceeds that without the pair,

$\propto {B}_{J}^{2}/2{\mu}_{0}$. Equivalently, pair creation cannot occur when the energy difference,

${\left({B}_{J}-{B}^{*}\right)}^{2}/2{\mu}_{0}-{B}_{J}^{2}/2{\mu}_{0}=\left({B}^{*}/{\mu}_{0}\right)\left[{B}^{*}/2-{B}_{J}\right]$, is positive, i.e. if

${B}_{J}<{B}^{*}/2$. When the applied current

J and self-field

B_{J} are sufficiently large that

${B}_{J}>{B}^{*}/2$ however, or when

$\theta \equiv 2\mathsf{\pi}{B}_{J}/{B}^{*}>\mathsf{\pi}$, the former global minimum (vs.

φ) becomes a metastable state. A bubble bounded by 2π solitons (

Figure 1) can thus nucleate when

θ, proportional to

J below threshold, exceeds

π. In the limit,

d >>

λ, this yields a threshold pair creation current per unit width given by:

When

d becomes comparable to or smaller than

λ, Equation (1) will be replaced by an expression that accounts for the thickness dependence of the penetration length, to be discussed shortly. If the nucleated vortex and antivortex are initially separated by a distance

$\mathsf{\Delta}x$, then the pair creation critical current between vortices can be written as:

where

$L~{\mu}_{0}{\lambda}^{2}/2\beta \mathsf{\Delta}x$ is the inductance of the flux toroid coupling the vortices. This is essentially the dual of the Coulomb blockade voltage

V_{c} =

e/2

C for a small capacitance tunnel junction. Nucleation of a vortex near one edge and antivortex near the other can be treated similarly using the image vortex concept [

31].

The critical current vs. thickness,

d, of HTS-coated conductors usually shows sub-linear behavior and often reaches a plateau, causing

J_{c} to decrease with thickness [

32,

33,

34]. This is readily explained by the Weber blockade mechanism. For a strip of width

w, the total pair creation current becomes:

${I}_{pc}^{0}={j}_{pc}^{0}w=\beta {\Phi}_{0}w/\left({\mu}_{0}{\lambda}^{2}\right)$, which is independent of

d when

d >>

λ. Eventually, however, it is likely that vortex pair creation will be superseded by anisotropic ring nucleation when the film or bulk material becomes extremely thick. As

d becomes small the vortices acquire a quasi-2D character and their diameters increase, as predicted by Pearl [

35] and directly imaged by Tafuri et al. [

36]. When

$d\ll \lambda $, the size of each vortex is governed by an effective length, Λ, known as the Pearl penetration length [

35,

36],

$\mathrm{\Lambda}=2{\lambda}^{2}/d$. Since the relevant length scale becomes

$\mathrm{\Lambda}=\lambda $ when

$d\gg \lambda $, we use the approximation:

$\mathrm{\Lambda}\cong \lambda +2{\lambda}^{2}/d$. Using Λ to replace

λ in Equation (1) then yields the following expression, as a function of

d, for the pair creation critical current,

I_{p}_{c} =

j_{pc}w:

Figure 3 shows a favorable comparison of Equation (3) with measured critical currents vs. thickness of HTS-coated conductors, consistent with the observed plateau effect.

The introduction of insulating CeO

_{2} spacer layers into HTS-coated conductors has been reported to improve the critical current [

32,

37] vs. thickness behavior by reducing the plateau effect. Our model provides a straightforward interpretation consistent with [

37], sufficiently thick spacer layers decouple vortex pair nucleation events on adjacent HTS layers, enabling total critical current to scale with the number of layers. This multi-layer approach is potentially advantageous for HTS magnets constructed with tapes in a pancake geometry, since field lines along the

ab-plane would be constrained by spacer layers. The next section discusses the dynamics of vortex tunneling above threshold, following a previous model of time-correlated soliton tunneling [

11,

12,

13,

14] in charge and spin density waves.

#### 2.2. Time-Correlated, Coherent Tunneling of Josephson Vortices

In time-correlated single-electron tunneling [

28], an applied voltage

V across a capacitive tunnel junction increases the displacement charge,

Q =

CV, until it reaches the critical value,

Q =

e/2, needed to overcome the Coulomb barrier,

${\left(Q\pm e\right)}^{2}/2C-{Q}^{2}/2C$. This enables an electron to tunnel through the junction, causing a drop in voltage followed by another increase, and the process repeats many times to cause jerky flow of electric current,

$I=dQ/dt$. After

n tunneling events, the total displacement charge, driven across and through the junction, becomes:

$Q=CV+ne$. Time-correlated soliton tunneling in CDWs [

12,

13,

14] is similar, except that the soliton charge

Q_{0} replaces the electron charge

e. The model combines a sine-Gordon potential, representing CDW pinning, with the solitons’ electrostatic energy [

26,

38]. Following a classic paper on the massive Schwinger model [

38], the soliton tunneling model also relates the ‘vacuum angle’

θ(

t) to the total displacement charge

Q(

t) and soliton charge

Q_{0} using:

$\theta =2\pi \left(Q/{Q}_{0}\right)$.

In the model proposed here,

θ scales with the applied current

J and the resulting self-field

B_{J}, when

J is below threshold, and is proportional to displaced

flux when

J is above threshold. We define the total displacement flux

Ф, after

n vortex tunneling/nucleation events, using

$\Phi ={\lambda}^{2}{B}_{J}/\beta +n{\Phi}_{0}$. The relation between

θ and

Ф then becomes,

$\theta =2\pi \left(\Phi /{\Phi}_{0}\right)$, which highlights the duality between flux and charge. We take

φ_{k} to represent phase differences across a junction for individual layers (e.g.,

Figure 1b), as well as incorporating spatial degrees of freedom. The potential energy for each microscopic degree of freedom

k can then be written as:

where the first term is the Josephson coupling energy and the second, quadratic term is the magnetostatic contribution.

Figure 4a,b shows how the global minimum for the phases switches from the potential well near

φ ~ 0 to the one at

φ ~ π when

θ crosses above the critical value of π.

In

Figure 4c, Equation (4) is minimized by setting

$\varphi ~2\pi n$ (dropping the subscript

k) so that

φ is sitting in a potential well (assuming

${u}_{M}\ll {u}_{J}$). This leads to a series of piecewise parabolic magnetostatic energy plots,

$u\left(\theta \right)\propto {\left(\theta -2\pi n\right)}^{2}\propto {\left(\Phi -n{\Phi}_{0}\right)}^{2}/2L$, as shown in

Figure 4c. These are dual to the piecewise charging Coulomb energy curves in time-correlated soliton tunneling in CDWs [

11,

12,

13,

14], and should not be confused with the Josephson coupling energy. The ‘vacuum angle’

θ continually increases with time as the system evolves. Each time a parabola in

Figure 4c crosses the next, at an instability point

θ = 2π(

n + ½), the phases

φ_{k} tunnel coherently into the next well, causing the overall phase to advance by 2π.

This process repeats itself many times, leading to Josephson-like oscillations and Shapiro steps due to mode-locking with a microwave source. In this picture, although the overall behavior is somewhat jerky, the microscopic degrees of freedom nonetheless flow through the barrier in a continuous fashion, as discussed in [

12,

13,

14]. The advance with time, of

$\theta \left(t\right)$ and the phase expectation value

$\u27e8\varphi \left(t\right)\u27e9$, are closely related. Although they only approximately track within a cycle, their correlation becomes precise when time-averaged. Regardless of the detailed shape of the periodic Josephson coupling energy, which may or may not be sinusoidal and may include disorder, the emergent behavior as

$\theta \left(t\right)$ that evolves is often non-sinusoidal.

We now follow [

12,

13,

14] to model dynamics. We hypothesize that the amplitudes

ψ_{n} and

ψ_{n} _{+ 1} for the system to be on branches

n and

n + 1 in

Figure 4c, respectively, are coupled via coherent, Josephson-like tunneling of microscopic quantum solitons [

39,

40]. We use a matrix element

${T}_{\phi}$ to represent this coupling, motivated by Feynman’s intuitive derivation [

41] of the dc and ac Josephson effects. Advancing

φ_{k}(

x) by 2π within a finite ‘bubble’ is equivalent to creating a pair of microscopic solitons. These can either be pancake Josephson vortices [

20] with very small masses [

27] or even deformations of the Cooper pair wavefunctions within the condensate (see Discussion). We treat the phases

φ_{k} as comprising a quantum fluid, within which they are capable of coherent, Josephson-like tunneling. This picture can presumably be extended to coherent tunneling of Abrikosov pancake vortices [

42], which also have small masses [

19], or to deformations of their resulting pair wavefunctions.

We use a slightly modified resistively shunted junction (RSJ) model [

6], shown in

Figure 5, to simulate voltage-current characteristics of an HTS grain boundary junction. This is discussed in Materials and Methods, which also defines the parameters used.

Figure 6 shows resulting theoretical voltage vs. current plots for various parameter values. Some of the

V-I plots exhibit piecewise linear behavior, which fits neither the classical RSJ model nor a straightforward thermally activated flux flow model. Piecewise quasi-linear behavior is often seen in HTS grain boundary junctions and has largely eluded explanation. Here this behavior is readily explained via the proposed model and two-terminal Weber blockade mechanism. In

Figure 6b the main effect of increasing

q_{0}, which corresponds to increasing barrier height and decreasing

λ_{J} as temperature goes down, is to increase the degree of rounding in the

V-I curve.

Figure 7 shows comparisons between theory and measured

V-I characteristics of an YBa

_{2}Cu

_{3}O

_{7} (YBCO) [

43] grain boundary junction at several temperatures. The simulated 86 K plot in

Figure 7 (top) is obtained using the classical resistively shunted junction (RSJ) model in the overdamped limit:

$V/\left({I}_{c}{R}_{n}\right)=\sqrt{{\left(I/{I}_{c}\right)}^{2}-1}$, without invoking thermal activation. The fact that the 86 K data fit the classical RSJ model almost perfectly suggests that the effective Josephson penetration length is comparable to or longer than the junction width (short junction limit) due to the small Josephson coupling energy at this temperature. Even at this high temperature there is little or no evidence for concave rounding caused by thermal activation.

The theoretical plots for the remaining temperatures in

Figure 7, as well as for

Figure 8a,b, are obtained via the time-correlated vortex tunneling model (see Materials and Methods), using parameters in

Table 1. The theoretical plots (solid lines) show nearly precise quantitative agreement with the experimental data (open symbols). As the temperature is reduced, from 82.5 K down to 70.0 K, both the experimental and theoretical

V-I curves show

more concave rounding, providing convincing evidence that this is

not due to a thermally activated process. The Josephson coupling energy increases with decreasing temperature, causing the characteristic force

F_{0} (proportional to

q_{0}) to increase. This increase in

F_{0} affects the matrix element

${T}_{\phi}$ (Equation (11) of Materials and Methods), and leads to rounded, Zener-like theoretical plots at lower temperatures, in agreement with experiment. Similar quantum fluidic effects may play a role in magnetic relaxation of Abrikosov vortices in bulk YBCO [

21], for which

V-I plots also tend to be quite rounded at low temperatures.

Figure 8a shows excellent agreement between our quantum model and the

V-I curve of an iron pnictide superconducting bicrystal [

44], consisting of coupled SrFe

_{1.74}Co

_{0.26}As

_{2} and Ba

_{0.23}K

_{0.77}Fe

_{2}As

_{2} crystals, each ~300 µm wide, and thus in the long junction limit. The piecewise linear

V-I behavior in

Figure 8a also occurs frequently in cuprate grain boundary junctions (e.g.,

Figure 7, 77.2 K data, and references [

1,

45]). This behavior is analogous to the piecewise linear

I-V curve of an ideal Coulomb blockade tunnel junction.

Figure 8b plots our simulation as compared to the measured

V-I characteristic of a thallium-based cuprate grain boundary [

46], showing good agreement with the data.

Table 1 in Materials and Methods shows the parameters used for the simulations and plots.

The time-dependent Schrödinger equation (Materials and Methods) and generalized tunneling matrix element

${T}_{\phi}$, coupled with the Weber blockade mechanism, provides a simple, yet powerful approach to modeling the dynamics of vortex tunneling. For a uniform junction, the piecewise parabolic curves in

Figure 4 and simulations discussed above often predict non-sinusoidal voltage oscillations. The top plot in

Figure 9 shows such non-sinusoidal voltage oscillations for a uniform junction. In the remaining plots of

Figure 9, we model non-uniformities by representing the junction as 100 domains with a spread in parameters related to coupling energy (see Materials and Methods). The bottom three plots in

Figure 9 represent various levels of disorder. The middle two plots show reduced voltage amplitudes and an apparent amplitude modulation effect, consistent with reported subharmonic Shapiro steps [

47]. The bottom plot, depicting the most disorder for this series, shows further reduced amplitudes. Voltage oscillations and Shapiro steps are thus expected to become immeasurably small in a sufficiently non-uniform junction.