Vortex in superconducting thin-film strips of arbitrary width

Thin film superconducting strips are ubiquitous elements in superconducting electronics. In particular, they are used in Single Photon Superconducting Detectors, which are basically made of these strips fed with a transport current close to critical when the condensation energy is small and even a single photon can cause transition to the normal state, recorded as a voltage pulse. In this set up, however, vortices can nucleate at the strip edges and cross the strip also producing dissipation, see e.g. [1, 2] and references therein. To study these so-called dark counts, one has to know properties of vortices in thin film strips, called Pearl vortices after J.Pearl who studied them in infinite films [3].

Unlike the field of Abrikosov vortex in the bulk, localized within the area of a size $\lambda$ of the London penetration depth, the large part of the Pearl vortex energy is stored in the stray field outside the film that decays as a power law. Pearl vortices with their stray fields are in fact three-dimensional (3D) entities. The stray fields distributions in vacuum are not bound either by the London $\lambda$ or by the Pearl length $\Lambda=2\lambda^{2}/d$, but rather determined by persistent current distributions and the film sample shape. Hence, the structure of vortices in thin-film strips may differ substantially from the case of infinite films. To our knowledge this question - at first sight rather academic - has not been addressed in literature. In this publication we attempt to fill this “white spot”.

We argue that for narrow strips of a width $W\ll\Lambda$, the self-field of the vortex can be disregarded, so that the current distribution can be calculated first after which the field evaluation is straightforward. For the general case of arbitrary widths both currents and fields should be found simultaneously by solving an integral equation, a more involved procedure. Also, we calculate the magnetic flux through the strip and the vortex energy as functions of the vortex position.

Our approach is based on the London equation averaged over the film thickness $d$,

$$h_{z}+2\pi\Lambda\,{\rm curl}_{z}{\bm{g}}/c=\phi_{0}\delta({\bm{r}}-{\bm{a}})\,,$$

(1)

where ${\bm{g}}({\bm{r}})$ is the sheet current density, ${\bm{r}}=(x,y)$, ${\bm{a}}=(a,0)$ is the vortex position, and $\Lambda=2\lambda^{2}/d$ is the Pearl length. Equation (1) is valid everywhere at the film except the vortex core and a narrow belt of a width $\xi$ adjacent to the edges, where the London equations break down [4].

Currents ${\bm{g}}({\bm{r}})$ and the field $h_{z}({\bm{r}})$ at the film plane $z=0$ can be found by solving Eq.(̇1) combined with the continuity equation and the Biot-Savart integral which relates the field $h_{z}$ to the surface current:

$${\rm div}{\bm{g}}=0\,,\,\,\,\,\,h_{z}({\bm{r}})c=\int d^{2}{\bm{r}}^{\prime}[{\bm{g}}({\bm{r}}^{\prime})\times{\bm{R}}/R^{3}]_{z}\,;$$

(2)

${\bm{R}}={\bm{r}}-{\bm{r}}^{\prime}$. The specific feature of the thin film limit should be noted: since the derivatives $\partial/\partial z$ in the film are large relative to $\partial/\partial{\bm{r}}$, the Maxwell equation ${\rm curl}{\bm{h}}=4\pi{\bm{j}}/c$ is reduced to conditions relating the sheet current to discontinuities of the tangential field [5]:

$$2\pi g_{x}/c=-h_{y}(+0),\,\,\,2\pi g_{y}/c=h_{x}(+0)\,.$$

(3)

Here, $h_{x,y}(+0)=-h_{x,y}(-0)$, and $\pm 0$ stand for the upper and lower faces of the film. The field $h_{z}$ is related to currents by an integral (2), rather than by a differential equation.

Equations (1) and (2) along with Maxwell equations for the stray field form a complete set for determination of currents and fields. Instead of $\bm{g}$, it is convenient to deal with a scalar stream function $S({\bm{r}})$ such that ${\bm{g}}={\rm curl}S{\hat{\bm{z}}}$:

$$g_{x}=\partial_{y}S\,,\,\,\,g_{y}=-\partial_{x}S\,.$$

(4)

Then the first of Eqs. (2) is satisfied.

The kernel ${\bf R}/R^{3}$ of the Biot-Savart integral is strongly singular. To reduce the degree of singularity one can write ${{\bf R}/R^{3}}=\nabla^{\prime}(1/R)$ (the prime specifies ${\bf r}^{\prime}$ as the variable of differentiation) and integrate by parts. Equation (2) then becomes

	$\displaystyle h_{z}c=\int_{\rm strip}\frac{d^{2}{\bm{r}}^{\prime}}{R}{\rm curl}_{z}{\bf g}({\bm{r}}^{\prime})$
	$\displaystyle+\int_{-\infty}^{\infty}dy^{\prime}\frac{g_{y}({\bm{r}}^{\prime})}{R}\Big{|}_{x^{\prime}=0}+\int_{\infty}^{-\infty}dy^{\prime}\frac{g_{y}({\bm{r}}^{\prime})}{R}\Big{|}_{x^{\prime}=W}\,,\qquad$		(5)

where $W$ is the strip width. Eq. (1) can now be written as an equation for $S(\bm{r})$:

	$\displaystyle\int_{\rm strip}d^{2}{\bm{r}^{\prime}}\frac{\nabla^{2}S({\bm{r}}^{\prime})}{R}+\int_{-\infty}^{\infty}dy^{\prime}\frac{\partial_{x^{\prime}}S({\bm{r}}^{\prime})}{R}\Big{|}_{x^{\prime}=0}$
	$\displaystyle+\int_{\infty}^{-\infty}dy^{\prime}\frac{\partial_{x^{\prime}}S({\bm{r}}^{\prime})}{R}\Big{|}_{x^{\prime}=W}+2\pi\Lambda\nabla^{2}S({\bm{r}})=-c\phi_{0}\delta({\bf r}-{\bf a}).$
			(6)

This is to be solved for $S(x,y)$ at the strip $0<x<W$ subject to boundary conditions ${\bf g(\infty)}=0$ and the vanishing normal component of the current at the film edges $g_{x}(0,y)=g_{x}(W,y)=0$. These conditions imply a constant $S$ at the film edges; one can set $S=0$ at the film boundaries since only the derivatives of $S$ have physical meaning.

The integro-differential Eq. (6) holds for any $W$ and in general can be dealt with numerically. The situation simplifies for $W\ll\Lambda$: as is seen from the Biot-Savart integral (2) $h_{z}\sim g/c$, whereas $2\pi\Lambda\,{\rm curl}_{z}{\bm{g}}/c\sim(\Lambda/W)g/c\gg h_{z}$. Hence, $h_{z}$ in the basic Eq. (1) can be disregarded for narrow strips, so that one can first solve the truncated Eq. (6),

$\displaystyle 2\pi\Lambda\nabla^{2}S({\bm{r}})=-c\phi_{0}\delta({\bf r}-{\bf a}),$

(7)

for $S({\bm{r}})$ and obtain currents. Then one can use the converted Biot-Savart Eq. (5) to evaluate the field.

The problem of solving for $S$ is equivalent to one in 2D electrostatics with a known solution [7, 8, 9, 6]:

$$S(\bm{r})=\frac{c\phi_{0}}{8\pi^{2}\Lambda}\ln\frac{\cosh\pi y-\cos\pi(x+a)}{\cosh\pi y-\cos\pi(x-a)}\,,$$

(8)

where we use $W$ under the log-sign as unit length (in these units $0<x<1$ at the strip). According to Eq. (4) the currents in units of $c\phi_{0}/4\pi\Lambda W$ are

	$\displaystyle g_{x}=\frac{\sin\pi a\sin\pi x\sinh\pi y}{[\cos\pi(x-a)-\cosh\pi y][\cosh\pi y-\cos\pi(x+a)]},\qquad$
	$\displaystyle g_{y}=\frac{\sin\pi a(\cos\pi a-\cos\pi x\cosh\pi y}{[\cosh\pi y-\cos\pi(x-a)][\cosh\pi y-\cos\pi(x+a)]}.\qquad$		(9)

The current flow lines are contours $S=$ const; an example is shown in Fig. 1.

Given the currents, Eq. (9), the Bio-Savart law of Eq. (2) provides $h_{z}(\bm{r})$ everywhere at the plane $z=0$. The calculation implies the double integral over the strip, a heavy numerical task done with the help of the Mathematica package. Figure 2 shows $h_{z}(x,0)$ for $a=0.2$ in units of $W$.

Without going to formal details, we note that $h_{z}(x,0)$ goes as $1/|x-a|$ near the vortex core.

The regions on the left and right of the strip of negative $h_{z}$ are due to the flux going round the edges from the space above the film to one under it. The most interesting, however, is that the sign change of $h_{z}$ happens not at the edges per se, but in their vicinity at the strip. In the example of Fig. 2, the sign change positions are at $x\approx 0.05$ and $x\approx 0.8$ at $y=0$. The region of positive $h_{z}$ is strongly elongated along the strip; for $a=0.2$, this region extends up to $|y|\approx 1.1$.

Now we recall that in infinite isotropic films, $h_{z}>0$ everywhere [3]. However, it was shown in [10] that $h_{z}$ can change sign at some parts of infinite anisotropic thin films. We see now that this may happen also in isotropic thin fim strips. The domain of negative $h_{z}$ is clearly seen in the 2D contour plot of Fig. 3. The white patches of this plot correspond to $|h_{z}(x,y)|>1$. In fact, $h_{z}\to\infty$ at the vortex position $(0.2,0)$. The divergence of the type $1/r$ is an artifact of the London theory and in reality is truncated at distances of the order of coherence length $\xi$.

One may say that the major “flux sinks” of dark colors in Fig. 3 are situated along the edges within the strip and outside it. It is reasonable to expect that such patches exist when vortices reside in finite film samples of any shape.

For the sake of completeness, we mention here other known results for narrow strips:

For the general width $W$, we retain the $h_{z}$ term in Eq. (1), and then solve the coupled equations Eqs. (1) and (2) numerically. For the strip geometry, we expand $S(\bm{r})$ as a series of eigenfunctions of the 2D Laplacian

$$\chi_{n}^{k_{y}}(\textbf{r})=e^{ik_{y}y}\sqrt{\frac{2}{W}}\sin\frac{\pi nx}{W},\ 0\leq x\leq W,\;n=1,2,\cdots,$$

(14)

that vanish at the edges. In particular, the narrow strip solution (8) is expandable as

	$\displaystyle S_{\rm narrow}(\bm{r})$	$\displaystyle=\frac{c\phi_{0}}{2\pi\Lambda}\sum_{n}\int_{-\infty}^{\infty}\frac{dk_{y}}{2\pi}\frac{\chi_{n}^{k_{y}}(\bm{r})\chi_{n}^{k_{y}}(\bm{a})}{(n\pi/W)^{2}+k_{y}^{2}}$
		$\displaystyle\equiv\frac{c\phi_{0}}{2\pi\Lambda}\overline{G}(\bm{r},\bm{a}),$		(15)

where $\overline{G}(\bm{r},\bm{r}^{\prime})$ acts as the Green’s function of the strip problem, see Eq. (7).

The term $h_{z}$ in (2) can be considered as extra source, yielding the relationship

$$S(\bm{r})=\frac{c}{2\pi\Lambda}\Big{[}\phi_{0}\overline{G}({\bm{r}},{\bm{a}})-\int_{\rm strip}d\bm{r}^{\prime}\overline{G}({\bm{r}},{\bm{r}}^{\prime})h_{z}({\bm{r}}^{\prime})\Big{]},\qquad\qquad$$

(16)

which can be coupled with Eq. (2) to solve for $S$ after eliminating $h_{z}$. Unlike $\overline{G}(\bm{r},\bm{r}^{\prime})$, the integration kernels of Eq. (2), i.e. $\bm{R}/R^{3}$, have off-diagonal elements with respect to the $\chi$ basis. Hence, our task is to solve the coupled equations in the matrix form by means of truncation. Numerical results can be obtained for arbitrary sets of parameters $W/\Lambda$ and $a/W$. The actual numerical process requires to work around the divergence of the solutions at the vortex position, as outlined in the Appendix in some detail.

Typical stream functions $S(x,y)$ obtained numerically are shown in Fig. 4. It is worth noting that for $W/\Lambda<1$ the narrow strip solution (the left panel of the figure) does not change much. Since this is the case for strips commonly used in practice, the narrow strip description may often suffice.

This trend is reflected also in the calculated $h_{z}(\bm{r})$. Figures 5 show examples, exhibiting resemblance in the field patterns for $W/\Lambda<1$ for the same vortex position.

Fig. 6 exhibits a qualitative difference between the strip and the infinite film. The comparison between the two cases shows the origin of the difference, i.e. the presence and absence of the flux return paths, or in our plots at $z=0$ the clear evidence of the change of $h_{z}$ sign. Although the wide strip and the infinite film have nearly identical near-the-core field behavior, the field patterns differ globally, the strip field is stretched in the strip direction and exhibits the strong edge effects.

The flux return affects the integrated flux calculation. As stated, the total flux through the entire $xy$ plane ($z$=const) is 0 for a strip, while it is $\phi_{0}$ for the infinite film. The flux through the strip is shown in Fig. 7 versus vortex position. The total flux through the strip surface is well below $\phi_{0}$ in particular due to the negative $h_{z}$ regions at the strip surface. The positive flux, i.e. the restricted integral over the sub-region of the $xy$ plane ($z$=const) where $h_{z}\geq 0$, is greater than the total flux, though still short of $\phi_{0}$. Unlike the infinite film, the positive flux decreases when $z>0$ increases, and in fact the lift-off effect is substantial.

The energy (magnetic plus kinetic) is obtained from the general result for thin films $\epsilon=\phi_{0}S[\bm{r}\to(\bm{a}+\bm{\xi})]/2c$ where $\bm{\xi}$ is a vector of the size of the vortex core (the coherence length) that should be added because $S(\bm{a})$ diverges [6], excluding also small regions of the strip near the edges. The results for $\epsilon(\bm{a})$ are shown in Fig. 8 for a set of strip widths choosing $\Lambda/\xi=1000$. The wide-strip energy in the strip middle approaches the value in the infinite film, albeit slowly.

As stated, the common trend has been observed in the field patterns which are nearly independent of the width for $W\lesssim\Lambda$. The similarity may be viewed in their strength as well. To this end, consider dimensionaless quantities $\tilde{h}_{z}$ and $\tilde{S}$

$$h_{z}=(\phi_{0}/W^{2})\tilde{h}_{z},\ S=(c\phi_{0}/2\pi\Lambda)\tilde{S}.$$

(17)

In general, they are functions of $W/\Lambda$, in addition to $\{\bm{r}/W,a/W\}$. However, the numerical results show that their magnitudes appear to exhibit simple trends, namely $\tilde{h}_{z}$ is roughly proportional to $W/\Lambda$, while $\tilde{S}$ is mostly width-independent, for given $a/W$ and in the said width range. Such approximate trends could be useful in analyzing experimental data with samples of various widths.

In this work we study the currents and field distributions of Pearl vortices at thin-film superconducting strips. The new feature which - to our knowledge - was not discussed in literature is that the vortex magnetic field lines crossing the film from the lower half-space ($z<0$ under the film) to the upper one ($z>0$) return back not only via a free space out of the strip edges, but also through some patches of the film per se, see examples in Figs. 3 and 6. As far as we know, these features of the field distribution have never been experimentally observed. However, with improved techniques of measuring local fields such as nitrogen-vacancy centers in diamonds sensitive to the magnetic field, these observations become possible [15].

Also, we evaluate the magnetic flux through the strip carried by a vortex and its energy as functions of its position and their dependencies on the strip width $W$ and the Pearl length $\Lambda$ of the film. The vortex energy is relevant for the potential barrier the vortex should overcome while crossing the strip, the thermally activated or the quantum tunneling processes [13].

The field distribution of vortices in films has been studied before, see, e.g., the work of H. Brandt [16] and references therein. This was done, however, for thick films, without going to the thin film limit, which has many peculiar features (as this work shows) and which is relevant for applications in superconducting electronics.