Josephson junctions in a local inhomogeneous magnetic field

A Josephson junction can be subjected to a local, strongly inhomogeneous magnetic field in various experimental situations. Here this problem is analyzed analytically and numerically. A modified sine-Gordon type equation in the presence of time-dependent local field is derived and solved numerically in static and dynamic cases. Two specific examples of local fields are considered: induced either by an Abrikosov vortex, or by a tip of a magnetic force microscope (MFM). It is demonstrated that time-dependent local field can induce a dynamic flux-flow state in the junction with shuttling, or unidirectional ratchet-like Josephson vortex motion. This provides a mechanism of detection and manipulation of Josephson vortices by an oscillating MFM tip. In a static case local field leads to a distortion of the critical current versus magnetic field, Ic(H), modulation pattern. The distortion is sensitive to both the shape and the amplitude of the local field. Therefore, the Ic(H) pattern carries information about the local field distribution within the junction. This opens a possibility for employing a single Josephson junction as a scanning probe sensor with spatial resolution not limited by its geometrical size, thus obviating a known problem of a trade-off between field sensitivity and spatial resolution of a sensor.

A Josephson junction can be subjected to a local, strongly inhomogeneous magnetic field in various experimental situations. Here this problem is analyzed analytically and numerically. A modified sine-Gordon type equation in the presence of time-dependent local field is derived and solved numerically in static and dynamic cases. Two specific examples of local fields are considered: induced either by an Abrikosov vortex, or by a tip of a magnetic force microscope (MFM). It is demonstrated that time-dependent local field can induce a dynamic flux-flow state in the junction with shuttling, or unidirectional ratchet-like Josephson vortex motion. This provides a mechanism of detection and manipulation of Josephson vortices by an oscillating MFM tip. In a static case local field leads to a distortion of the critical current versus magnetic field, I c (H), modulation pattern. The distortion is sensitive to both the shape and the amplitude of the local field. Therefore, the I c (H) pattern carries information about the local field distribution within the junction. This opens a possibility for employing a single Josephson junction as a scanning probe sensor with spatial resolution not limited by its geometrical size, thus obviating a known problem of a trade-off between field sensitivity and spatial resolution of a sensor.
Another motivations of this work is related to a recent proposal to use a single planar JJ as a scanning probe sensor [28]. The leading superconducting scanning probe technique today is the scanning SQUID (superconducting quantum interference device) microscopy [29,30]. Despite many advantages, SQUID's suffer from a trade-off problem between field sensitivity and spatial resolution. SQUID's, as well as most other superconducting magnetic sensors, are measuring flux with a resolution δΦ determined by the flux quantum Φ 0 . Therefore, field sensitivity is inversely proportional to the sensor (pickup loop) area S , δH = δΦ/S . On the other hand, spatial resolution is determined by the sensor size δx ∼ S 1/2 . Consequently, the better is spatial resolution, the worse is field sensitivity, δH ∼ 1/δx 2 . In Ref. [28] it was argued that a sensor based on a single planar JJ would be able to obviate the trade-off problem at least in one spatial direction. Similar to a SQUID, the field sensitivity of a planar JJ is also inversely proportional to the area. Therefore, obviation of the trade-off problem would require independence of spatial resolution on the junction size.
That is, the junction should be able to resolve spatial variation of magnetic field at a scale significantly smaller than the junction length. This brings us again to the problem of a JJ in a local spatially inhomogeneous magnetic field.
In this work I consider analytically and numerically a response of a single JJ to a local inhomogeneous and timedependent magnetic field. First, a modified sine-Gordon equation for this case is derived. The equation is then solved numerically both for short and long junctions, and both in static and dynamic cases. Two specific examples (without loosing generality) are considered with a local field induced either by an Abrikosov vortex, or by a tip of MFM. It is demonstrated that a time-dependent local field can induce a dynamic flux-flow phenomenon with either shuttling or unidirectional ratchet-like motion of Josephson vortices in the junction, which provides a mechanism for detection of Josephson vortices by MFM [27]. Analysis of the static case shows how the critical current versus magnetic field, I c (H), modulation patterns are distorted with introduction of the local inhomogeneous field. Importantly, the shape of distorted I c (H) patterns depends both on the shape, amplitude and position of the local field. Therefore, the I c (H) pattern carries detailed information about the local field and it should be possible to extract field distribution within the junction using proper mathematical treatment. This would open a possibility for making a scanning probe sensor based on a single planar Josephson junction with spatial resolution not limited by its geometrical size, thus obviating the trade-off problem between sensitivity and resolution. field H in y-direction and a local nonuniform field B * . The bias current is applied from one electrode to another in zdirection. I start with derivation of transport equations. Although main parts of the derivation are well known, I will show it in some details both for the sake of pedagogical completeness and because sometimes different equations have been introduced in literature without proper substantiation.

II. THEORETICAL ANALYSIS
According to the two-fluid model, current through a Josephson junction has superconducting and quasiparticle (normal) components. The supercurrent density is given by the DC-Josephson relation, where J c0 is the critical current density at zero magnetic field and ϕ is the Josephson phase difference. The quasiparticle current density is equal to J n = V/R n A, where V is voltage across the junction, R n is the normal resistance of the junction and A is the junction area. Using the AC-Josephson relation V = (Φ 0 /2πc)(∂ϕ/∂t), where c is the speed of light in vacuum, it can be written as Note that for planar JJ's the junction area A (in the (x, y)-plane, see Fig. 1) is much smaller than the sensor area S (in the (x, z)plane) [28,31]. Using Maxwell equation rotB = 4π c J + 1 c ∂D ∂t we can write the total current through the junction as: Here the last term represents the displacement current density where C are the junction capacitance. Both x and y components of magnetic induction in Eq. (3) are essential. The x-component, parallel to the junction, has contributions from the bias current, B xb , and the local nonuniform field B * x , B x = B xb + B * x . The bias contribution is related to the bias current density in the z-direction, J b , as This is how the bias term, which plays the role of the driving force for junction dynamics, enters the sine-Gordon equation. From Eqs. (1-5) we obtain The y-component of magnetic field, going through the junction, induces a phase gradient in the junction, Here d e f f is the so-called magnetic thickness of the junction. B y is the total (screened) induction in the junction subjected both to the applied uniform field H and the local nonuniform field B * . It is generally not known and should be determined.
To do so we separate the phase shift ϕ * caused solely by B * y .
ϕ * (x) is a known function, determined (up to an integration constant) by integration of Eq. (9) along the junction length. Using Eqs. (7-10) we can write Note that the second term in the right-hand-side represents z-component of rotB * . Since B * does not induce vacuum currents, rotB * = 0, this term vanishes. Substituting Eqs. (1,2,4) in Eq. (11) we obtain the desired modified sine-Gordon-type equation: Here space,x = x/λ J , is normalized by the Josephson pen- Eq. (12) should be solved with respect to φ for the known ϕ * (x, t) with boundary conditions at the junction edges x = 0, L x : Note that thanks to separation of variables, Eq. (8), the local nonuniform field drops out from the boundary conditions. This occurs because at the junction edges B y (0, In what follows we will normalize magnetic field by , t i is the junction interlayer width, d 1,2 are the widths and λ 1,2 are the London penetration depths of the two electrodes (for details see Ref. [32]).

III. STATIC CASE
In the static case the only current component is the supercurrent J s , Eq. (1), and Eq. (12) is reduced to According to Eqs. (8)(9)(10), the local supercurrent density J s (x) directly depends on the local field B * (x). Experimentally measurable quantity, however, is the critical current I c , which represents the maximum value of the integral of J s along the junction length. The value of I c alone does not disclose the B * (x) distribution. However, as we will show below, the I c (H) modulation pattern does carry information about distribution of magnetic induction in the junction.

III A. Short junctions
First we consider the simplest case of a short junction L x λ J . In this case we may neglect magnetic field screening in the junction, i.e. set the second derivative term in the lefthand side of Eq. (14) to zero. From Eq. (13), we explicitly obtain φ(x) (2πd e f f H/Φ 0 )x + φ 0 , where φ 0 is the integration constant. The total supercurrent is calculated directly by integration of sin[φ(x) + ϕ * (x)]dx. The critical current is obtained by maximization with respect to the integration constant φ 0 .
To demonstrate how local inhomogeneous field distorts the I c (H) pattern we consider the case when the local field is created by stray fields from an Abrikosov vortex. This case has been described in details in a recent work [25], in which it was shown that vortex-induced Josephson phase shift is well described by the equation: where V is the vorticity (+1 for a vortex, -1 for an antivortex), x v is the coordinate of the vortex along the junction and z v is the distance to the junction. Figure 2 shows (a) vortex stray fields, −B * (x), in the junction and (b) corresponding Josephson phase shifts ϕ * (x) for an Abrikosov vortex, V = 1, at four different distances z v to the JJ along the junction middle line The closer is the vortex to the junction, the sharper and larger is the local stray field B * (x). Note that the sign of the stray field is opposite to that in the vortex, leading to the minus sign in Eq. (15) [25].  [28] that spatial resolution of a scanning probe sensor based on a single planar junction is potentially not limited by its size. Such a device could obviate a trade-off problem between field sensitivity and spatial resolution inherent for scanning SQUID sensors [29,30], as mentioned in the Introduction.  This demonstrates that the effect of local inhomogeneous magnetic field is uniquely encoded in the shape of the I c (H) pattern. Therefore, it should be possible to reconstruct spatial distribution B * (x) from the analysis of I c (H) modulation.

III B. Long junctions
For long JJ's, L x λ J , screening of magnetic field by the junction becomes significant. Simultaneously, Josephson vortices (JV's) appear and start to affect junction properties. To obtain I c with static B * either an ordinary differential equation (14), or a dynamic partial differential equation (12) with timeindependent ϕ * should be solved with boundary conditions, Eq. (13). Eq. (14) is solved by a finite difference method with successive iterations and I c is determined as a maximum bias current at which a solution converges. Eq. (12) is integrated explicitly using a central difference approximation and I c is determined using a threshold criterium for voltage. In case of a significant nonlinearity of ϕ * (x), the iterative solution of Eq. (14) may be quite sensitive to the initial approximation. On the other hand, the damping term in Eq. (12) allows less strict requirements to the initial approximation and usually provides faster convergence because for the considered here static case one can use a large α 1 to speed up calculations. Therefore, all simulations shown below are obtained by solving partial differential Eq. (12).  Fig. 3  (a), it has a broad triangular central lobe, corresponding to the Meissner state [1]. Beyond it JV's penetrate into the junction. Edge pinning of JV's, due to interaction with their own images [25], leads to metastability and multiply-valued I c . Some of the metastable states are seen in Figure 4 (a). To obtain those states simulations are done by sweeping magnetic field backand-forth in different field intervals.
Next we consider the case with a local field. Here we keep in mind another relevant case, when B * is induced by the MFM tip [26,27]. A standard MFM tip is covered by a thin ferromagnetic layer. Therefore B * from the MFM sensor has a sharp dipole-type peak, originating from the end of the tip, and a broad background from the ferromagnetic layer at the cantilever. To mimic it we approximate the tip-induced B * by two Gaussian peaks: a narrow one with the width ∆x 1 = 0.1λ J containing a total flux of 0.5Φ 0 and a broad ∆x 2 = 5λ J with the total flux 5Φ 0 , as shown in Fig. 4 (b). Fig. 4 (c-e) show simulated I c (H) patterns for the same JJ with the MFM tip at different positions x t along the junction, as indicated in insets. It is seen that the I c (H) in a long JJ is also distorted by the local field. However, there are certain differences with respect to the short junction case, Fig. 3. This occurs because I c in long JJ's has a different nature: it can be considered as a depinning current for JV's. Bias current exerts a Lorentz force on JV's and leads to appearance of a flux-flow state with finite voltage. For the considered geometry, Fig. 1, positive J b pushes JV's to the left and negative -to the right, as indicated by red arrows in Figure 4 (f). In the absence of local field, B * = 0, JV's are pinned only at the edges of the junction due to attraction to image antivortices [25]. In this case I + c and I − c correspond to depinning from the right and left edges, respectively, which are equal in the absence of physical nonuniformity of the JJ, as in Fig. 4 (a).
The local inhomogeneous field with a finite gradient, ∂B * /∂x 0, exerts an additional magnetic force on the JV, as indicated by the blue arrow in Fig. 4 (f), which will remove the symmetry between left and right edges and lead to I + c (H) I − c (H), except for the case when the symmetric local field B * is placed symmetrically in the middle of the junction, as in Fig. 4 (e). The local field creates also an additional pinning cite for JV's inside the JJ. This leads to a more profound metastability of I c (H) patterns as can be seen from comparison of Figs. 4 (a) and (c-e). So far we considered perfectly uniform JJ's. However, real JJ's often contain some nonuniformities, e.g., they may have spatial variation of intrinsic junction parameters, such as the critical current density, electrode thickness, bias current density, self-field effect, e.t.c. In this case the I c (H) pattern in the absence of local field must only be centrosymmetric, [5]. The latter is the consequence of spacetime symmetry: simultaneous reversal of field (space) and current (time) is equivalent to flipping the junction upsidedown, which should not affect the output of the experiment.

IV. DYNAMIC CASE
The local field can be time-dependent, as for example in case of MFM in the taping mode [27]. The time-dependent local field B * (t) provides an additional driving force for junction dynamics, given by the last two terms in the right-hand side of Eq. (12). This can cause a flux-flow phenomenon induced by the oscillating MFM tip, as recently reported [27]. Figure 5 shows a time sequence of solutions of Eq. (12) with an oscillating MFM tip. Simulation parameters correspond to Fig. 4 (c) : L x = 10 λ J , x t /L x = 0.01 at H = −0.55. The damping parameter is α = 0.5. The top panels show spatial distributions of the Josephson current density, the middle panels -of voltage, and the bottom panels -magnetization B − H. We assume that the tip is oscillating harmonically, but the tip field is anharmonic due to the non-linear distance dependence of the dipole-like tip   5, or a more complex ratchet-like unidirectional motion with entrance of the JV from one side and exit from the other side of the junction. An example of such ratchet-like motion can be found in the Supplementary material to Ref. [27]. In that case every cycle four JV's enter a junction from the left edge but only three leave from that side while one exits through the right edge, thus creating a ratchet effect, i.e. a net rectified unidirectional flux-flow motion induced by a periodic (or aperiodic) perturbation [5,6]. The back action of the tip-induced flux-flow motion leads to an additional damping of MFM tip oscillations, which can be detected in experiment. As discussed in Ref. [27], this provides a mechanism for detection of Josephson vortices by the MFM technique.

CONCLUSIONS
To conclude, we derived and analyzed numerically equations describing behavior of a Josephson junction in local inhomogeneous magnetic field. As discussed in the Introduction, such situation may have many different reasons and experimental realizations [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]27]. It was demonstrated that time-dependent local field provides an additional driving force, which may induce flux-flow type dynamics in long junctions. This provides a mechanism for detection and manipulation of Josephson vortices by taping-mode magnetic force microscope [27]. Local inhomogeneous field removes the space-time symmetry of the junction and leads to a distortion of I c (H) modulation patterns. Importantly, the distortion uniquely depends on the spatial distribution of local field B * (x) within the junction. Therefore, the information about local field profile is encoded into the shape of the I c (H) pattern and may in principle be reconstructed using an appropriate mathematical analysis. This strengthens an earlier argument that a single planar junction can be advantageously used as a scanning probe sensor [28]. The field sensitivity of such sensor would depend on the area, similar to SQUID, but the spatial resolution would not be limited by the junction size. Therefore, a planar junction sensor can obviate the trade-off problem between field sensitivity and spatial resolution inherent in scanning SQUID microscopy.