Edge superconductivity in Nb thin film microbridges revealed by electric transport measurements and visualized by scanning laser microscopy

R Werner; A Yu Aladyshkin; I M Nefedov; A V Putilov; M Kemmler; D Bothner; A Loerincz; K Ilin; M Siegel; R Kleiner; D Koelle

doi:10.1088/0953-2048/26/9/095011

1. Introduction

The concept of localized superconductivity in bulk superconductors was introduced in 1963 by Saint-James and de Gennes [1]. They demonstrated that superconductivity in a semi-infinite sample with an ideal flat surface in the presence of an external magnetic field H (with amplitude H) parallel to its surface can survive in a thin surface layer, even above the upper critical field H_c2, when bulk superconductivity is completely suppressed. Based on the phenomenological Ginzburg–Landau theory, the critical field H_c3 for surface superconductivity, localized near superconductor/vacuum or superconductor/insulator interfaces, can be calculated as [2, 3]

$\begin{eqnarray} &&1.6 9 5 \hspace{0.167em} {H}_{\mathrm{c}2}\simeq {H}_{\mathrm{c}3}={H}_{\mathrm{c}3}^{(0)}\left (1-T/{T}_{\mathrm{ c}0}\right ), \end{eqnarray} \tag{ 1 }$

where ${H}_{\mathrm{c}3}^{(0)}$ is the upper critical field for surface superconductivity at temperature T = 0, and T_c0 is the superconducting critical temperature for H = 0. This theory predicts that in the regime of the surface superconductivity the order parameter wavefunction Ψ decays exponentially with increasing distance from the surface on the length scale of the coherence length ξ.

Experimental evidence for surface superconductivity has been found by dc transport [4–7] or inductive measurements [8] shortly after the theoretical prediction [1]. Later on, other methods such as ac-susceptibility and permeability measurements [8–11], magnetization measurements [12, 13], surface impedance measurements [14] and tunneling spectroscopy [15] confirmed the existence of surface superconductivity when H was applied parallel to the surface. The evolution of the resistance R versus H, depending on the orientation of H relative to the surface was also investigated [4]. While two different regions for bulk and surface superconductivity were clearly observed for fields parallel to the surface, no signature for surface superconductivity was reported when H was applied perpendicularly. The in-plane-field dependence of the critical current I_c(H) in the regime of surface superconductivity for H parallel to the bias current was described by Abrikosov [16] and studied experimentally [17–19]. Park described theoretically the evolution of I_c(H) in the state of surface superconductivity when the in-plane-field H is applied perpendicular to the bias current flow [20]. He predicted an asymmetry in the critical surface current, resulting from the superposition of surface screening currents and external currents. Such an asymmetry in nominally symmetrical superconducting systems above the upper critical field has not been observed experimentally yet, although asymmetry of the critical current and rectification of the oscillating excitations were observed experimentally for different superconducting systems with broken symmetry, e.g. [21–26].

Similar to surface superconductivity, localized superconductivity can also nucleate near the sample edge in a thin semi-infinite superconducting film, in a thin superconducting disk of very large diameter or around holes in a perpendicular magnetic field [27–32]. It should be mentioned that surface superconductivity and localized states at the sample edges in perpendicular field (called edge superconductivity (ES)) are qualitatively and quantitatively the same. While surface superconductivity has been investigated in several compounds like Pb-based alloys [8, 12, 33], Nb and Nb-based alloys [7, 11, 12, 14], polycrystalline MgB₂ [34], Pb [8, 15, 35], UPt₃ whiskers [36], NbSe₂ [37], experimental studies on ES in thin film structures are rare [38, 39]. Recently, the first real space observation of ES was obtained by scanning tunneling microscopy on Pb thin film islands [40].

Localized states do not only occur at sample boundaries but can also be induced by an inhomogeneous magnetic field as it appears e.g. above domain walls in superconductor/ferromagnet hybrids. This localized state is therefore called domain wall superconductivity (DWS) [41–43]. Recently, a Pb/BaFe₁₂O₁₉ superconductor/ferromagnet hybrid has been investigated by low-temperature scanning laser microscopy (LTSLM) and the inhomogeneous current distribution of the sample in the DWS state has been visualized [44]. LTSLM is therefore a valuable tool to visualize the redistribution of the current at the crossover from bulk to edge superconductivity.

In this paper we present our investigations on the evolution of edge superconductivity in plain and perforated Nb microbridges in perpendicular magnetic field. Measurements of R(H) were performed to compose an experimental phase diagram and to identify the regions of bulk and edge superconductivity. Then we use LTSLM to visualize the current distribution at the transition from the superconducting to the normal state in both bridges. In addition, we used a time-dependent Ginzburg–Landau (TDGL) model to compare our experimental findings with theoretical predictions.

2. Sample fabrication and experimental details

A Nb thin film with thickness d = 60 nm was deposited on a single crystal Al₂O₃ substrate (r-cut sapphire) at T ≈ 800 °C using magnetron sputtering. Two Nb bridges with width W = 40 μm and length L = 660 μm were patterned by e-beam lithography and reactive ion etching into a bridge geometry⁴ as shown in figure 1. One of these bridges was patterned with circular microholes (antidots (ADs) with 580 nm diameter) in a triangular lattice with a period of 1.5 μm.

The samples were electrically characterized in a helium cryostat at 4.2 K ≤ T ≤ 10 K and |H| ≤ 20 kOe using a conventional four-terminal scheme (cf figure 1). For both investigated Nb microbridges we found T_c0 = 8.5 K. H was always applied along the z-direction, i.e. $\mathbf{H}=H{\hat {\mathbf{e}}}_{z}$ was perpendicular to the thin film surface and the applied bias current I. We performed isothermal measurements of voltage V(I) characteristics for different T and H values out of which we determined the dependence of the dc resistance R = V/I on H. The data presented below were obtained with I = 1 mA, unless stated otherwise. This corresponds to a bias current density j ≡ I/(dW) ≈ 40 kA cm⁻².

To visualize the current distribution for different bias points in the T–H phase diagram, we used LTSLM [44, 46–48]. For imaging by LTSLM, the sample was mounted on a cold finger of a helium flow cryostat, which is equipped with an optical window to enable irradiation of the sample in the (x,y) plane by a laser beam (with wavelength 680 nm). We expect that the laser beam is focused within an area of diameter ∼2 μm [46, 47]. The incident laser power on the sample surface is about 25 μW yielding a maximum power density of the order of 10 μW μm⁻². The amplitude modulated laser beam (at frequency f ≈ 10 kHz) induces a local increase of temperature centered at the hot spot position (x₀,y₀) at the surface of the superconducting bridge. The diameter of this hot spot is larger than the 'bare' beam diameter due to quasiparticle excitations and thermal diffusion and, correspondingly, it depends on T,H and I. We argue below that in our experiments under certain conditions the size of the hot spot can be larger than 10 μm, and this value is less than the bridge width (40 μm). During imaging, the Nb bridge is biased at a constant I, and the beam-induced change of voltage ΔV(x₀,y₀) is recorded by a lock-in technique as a function of the beam coordinates (x₀,y₀). The LTSLM voltage signal can be interpreted as follows: if the irradiated part of the sample was in the normal state with resistivity ρ_n, the laser beam induces a very small voltage signal ΔV ∝ ∂ρ_n/∂T. However, if the irradiated part of the bridge took part in the transfer of a substantial part of the superconducting currents, the beam-induced suppression of superconductivity might switch the whole sample from a low-resistive state to a high-resistive state. Details of the LTSLM signal interpretation can be found in [44, 46–48].

3. Results and discussion

3.1. Magnetoresistance data and Ginzburg–Landau simulations

Figure 2 shows R(H) measurements of the resistive transition at T = 4.2 K for different values of I for the plain (figure 2(a)) and the AD bridge (figure 2(b)). All curves are normalized to the normal state resistance R_n at H = 9.0 kOe. Except for the AD bridge at the highest current value of 10 mA, all R(H) curves reach R_n at the same field value |H| ≈ 8 kOe. However, we observe a pronounced dependence of the shape of the R(H) curves on I, which we describe and discuss below.

For the plain bridge (cf figure 2(a)), at the highest current value of 10 mA, we observe with increasing |H| an onset of dissipation (appearance of a finite R) at H ∼ 3 kOe. Upon further increasing |H|, the slope dR/d|H| steadily increases, yielding a rather steep R(|H|) transition curve up to R ∼ 0.9R_n. At R ∼ 0.9R_n (|H| ∼ 4.4 kOe) a kink in R(|H|) appears, i.e. with further increasing |H|, the slope dR/d|H| is significantly reduced. Upon reducing I, the field value where the kink appears stays almost constant; however, the resistance at the kink steadily decreases, and becomes zero for I < 0.5 mA, i.e. the kink disappears. Similar shapes of the R(H) curves (including the above described kink) and their current dependence as shown in figure 2(a) for the plain bridge have been found in [4, 7] when H was applied parallel to the sample surface.

The R(H) measurements of the AD bridge (cf figure 2(b)) show similar behavior upon variation of I as compared to the plain bridge in the following sense: within the same range of (high) bias currents, the onset of dissipation appears almost at the same H value (for the same value of I) as for the plain bridge. Upon further increasing |H|, a similar steep transition (with slightly smaller slope as for the plain bridge) appears, up to the kink in R(|H|), which is also present for the AD bridge within the same range of (high) bias currents.

However, we also observe distinct differences by comparing the AD and plain bridge. The resistance at the kink is lower for the AD bridge. This deviation increases with increasing bias current. Furthermore, for the two highest I values the AD bridge shows a second kink in the R(H) curve where the slope dR/d|H| suddenly increases again with increasing H; this feature is absent at smaller I and is not seen for the plain bridge for all values of I. Finally, for the two lowest I values, the onset of dissipation (upon increasing |H|) is shifted to larger |H| values for the AD bridge, as compared to the plain bridge.

In the following, we present an interpretation of the R(H) curves described above, starting with the discussion of the results obtained for the plain bridge. At the highest I = 10 mA, upon increasing the external magnetic field from H = 0, vortices will enter the sample when H is larger than the field of first vortex entry, which is rather small for thin film structures in perpendicular magnetic field. The onset of energy dissipation can then be attributed to the onset of motion of vortices, when the bias current density j exceeds the depinning current density j_depin at a given T and H. In this case, upon further increasing H, the flux-flow resistance will strongly increase, i.e., the rather large slope dR/dH should correspond to the bias-current-stimulated motion of the vortex lattice in the presence of a strong pinning potential. The kink in the R(H) curve where the slope dR/dH substantially decreases (upon increasing H), can be assigned to the transition from the resistive flux-flow regime to a resistive regime with fully suppressed bulk superconductivity and surviving ES at H_c2 (and above). This interpretation is the same as given in [7] (for surface superconductivity with H parallel to the sample surface). However, in contrast to our observation, a more gradual transition to R_n already at H_c2 without any kinks and no signature of ES was observed in [4, 7] when H was applied perpendicular to the sample surface.

We would like to emphasize that the position of the kink should be close to the upper critical field H_c2 but not identical to it, since the destruction of bulk superconductivity is a thermodynamical property of a material, but the kink can be observed only under strong non-equilibrium conditions upon the bias current injection. Still, below we use the field value where the kink appears as the experimentally determined H_c2 value.

Obviously, in our case the edge states form continuous channels with enhanced conductivity, which reduce the overall resistance to a value below R_n. The observed reduction of the resistance at the kink feature in R(H) with decreasing I can be explained by the strengthening of ES upon decreasing I, until at small enough currents the injected bias current flows entirely as a dissipationless supercurrent along the edge channels at H = H_c2, leading to a disappearance of the kink feature.

As described above, the full normal resistance R_n is reached (for all I values) at the same field, which we now associate with the upper critical field H_c3 ≈ 1.7H_c2 for ES. An analysis of the T dependence of H_c2 and H_c3 will be presented in section 3.2.

In the AD bridge, the holes lead to additional 'edges' in the sample interior, which results in a higher volume fraction of ES and more effective pinning. This explains the lower R value (as compared to the plain bridge) at the kink when bulk superconductivity becomes suppressed at H_c2. The origin of the second kink at H_c2 < |H| < H_c3, developing at rather large bias current (figure 2(b)), might be associated with a slightly reduced H_c3 value at the AD edges, as compared to the edges of the bridge, due to the different edge geometry. However, further investigations are required to provide a more conclusive explanation of this feature. Similarly, we cannot yet provide an explanation for the observed shift of the onset of dissipation to larger H, for the AD bridge (as compared to the plain bridge) for the lowest values of I.

In the following, we compare the experimental results with theoretical calculations based on the TDGL model described in the appendix. Such an approach was successfully applied for the modeling of superconducting properties of one-, two- and three-dimensional microbridges and the interpretation of experimental results [49]. We calculated for a rectangular plain superconducting thin film (W = 30ξ₀,L = 60ξ₀; ξ₀ is the Ginzburg–Landau (GL) coherence length at T = 0) the spatial distribution and time dependence of the normalized order parameter (OP) wavefunction ψ(x,y,t) and the voltage drop V(t) along the rectangle for different values of H and normalized bias current density j at a reduced temperature T/T_c = 0.47 (corresponds to T = 4.2 K for Nb with T_c = 9 K). We note that the real dimensions of the investigated sample considerably exceed the dimensions used in our modeling. Nevertheless, the model correctly captures the essential physics behind the discussed effects for H > H_c2. Figure 3(a) shows the spatial distribution |ψ(x,y)| for a rather small value of j = 5 × 10⁻⁴ for five different values of H/H_c2 from 0.19 to 1.50. We note that the chosen value for j is several orders of magnitude below the GL depairing current density j_GL = 0.386 (cf the appendix) at T = 0. In all cases, the OP distributions are time-independent ('stationary case') corresponding to zero resistance. At |H| < H_c2, a regular vortex structure appears and the density of vortices increases with increasing H. However, even when bulk superconductivity is depleted at |H| > H_c2, superconducting channels with finite and time-independent |ψ| running along the edges of the rectangle are still present and can provide a non-dissipative current transfer. If H is further increased, the superconductor turns to a non-stationary regime with finite resistance and reaches its normal value at the upper critical field for ES at H = H_c3. The calculated R/R_n versus H/H_c2 curves for different j are shown in figure 3(b). The numerical simulations reproduce the shift of the curves to smaller H and the decrease of the slope dR/dH in the interval H_c2 < |H| < H_c3 as j increases. This is qualitatively the same as observed experimentally in figure 2.

**Figure 3.** Numerical GL-simulation results for a superconducting rectangular thin film (W = 30ξ₀,L = 60ξ₀) biased at normalized current density j at variable magnetic field H and T/T_c = 0.47. (a) Spatial distribution of the modulus of the normalized order parameter wavefunction |ψ(x,y)| with j = 0.5 × 10⁻³ (lowest value in (b)). The five panels show simulation results for different values of H/H_c2; j is flowing from top to bottom. (b) R/R_n versus H/H_c2 for different j. Two vertical dashed lines depict the upper critical field H_c2 and the critical field of ES H_c3 = 1.695H_c2.
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However, for large enough j (corresponding to finite R at H_c2) our TDGL model is unable to describe the kink in R(H) close to H_c2 and to explain the disappearance of R for H < H_c2 (cf curve for j = 8 × 10⁻³ in figure 3(b)) by trivial development of bulk superconductivity. Indeed, at H < H_c2 the Abrikosov vortex lattice appears (figure 3(a)), the vortices enter (or exit) the bridge and move easily across it under the action of the Lorentz force ${\mathbf{F}}_{\mathrm{L}}=({\Phi }_{0}/c)[\mathbf{j}\times {\hat {\mathbf{e}}}_{z}]$ , where j = j(x,y) is the local current density at a vortex position, $\mathbf{H}=H{\hat {\mathbf{e}}}_{z}$ .

3.2. Superconducting phase diagram for the plain bridge

Figure 4(a) shows the results of the R(H) measurements for T = 4.2–8.7 K. With increasing T, the deviation from R = 0 and the kink, both shift to smaller H values, and the resistance at the kink shifts to a higher R/R_n ratio, while the change in the slope dR/dH at the kink becomes less pronounced. In order to experimentally determine H_c2 and H_c3 we use the field value at the kink and a criterion of 0.98R_n, respectively. The determined transition lines for H_c2(T) and H_c3(T) for the above-mentioned criteria are shown in figure 4(b). The experimental transition line for H_c3(T) can be fitted with equation (1) and T_c0 = 8.5 K which extrapolates to ${H}_{\mathrm{c}3}^{(0)}=1 5.3~\mathrm{kOe}$ . Plotting the transition line for H_c2(T) with the relation H_c3(T) = 1.695 H_c2(T), we find that the experimentally determined H_c2 values are close to the calculated transition line for bulk superconductivity. This result gives convincing evidence that depending on T,I and H (perpendicular to the sample surface), our sample can be either in the state with developed bulk superconductivity and pinned vortex lattice (H < H_c2), or in the resistive state, controlled by ES (H_c2 < H < H_c3).

3.3. Visualization of the current distribution by LTSLM

We used LTSLM to visualize the current distribution in the Nb bridges during the transition from bulk superconductivity to the normal state. As the maximum H was limited to ∼2 kOe in this setup, the LTSLM measurements were performed at rather high T values, T = 7.0–7.5 K.

Figure 5(a) shows an H-series of beam-induced voltage images, ΔV(x,y), at T = 7.5 K for various superconducting states of the plain Nb bridge, oriented vertically in all these images (cf optical image (left panel) in figure 5(a)). For a more quantitative analysis, we show an H-series of linescans, ΔV(y), across the bridge in figures 5(b) and (c). The insets in figures 5(b) and (c) show the corresponding R(H) curve, from which we estimate H_c2 ≈ 1.1 kOe and H_c3 ≈ 1.8 kOe. At H = 0.67 kOe in figure 5(a), the LTSLM signal is zero, which means that the beam-induced perturbation is not strong enough to suppress superconductivity and to induce a voltage signal. Upon increasing H, the first signal appears at H ≈ 0.8 kOe which corresponds to the onset of the resistive transition (see inset in figure 5(b)). With further increasing H, the signal at the edges is enhanced, but also a signal from the inner part of the bridge appears. The latter can be attributed to the depletion of bulk superconductivity with increasing H (below H_c2), which leads to an increasing voltage response to the perturbation by the laser beam with a maximum beam-induced voltage signal at H = 1.06 kOe, which is very close to the estimated H_c2 value. The pronounced edge signal below H_c2 can be explained by the suppression of the edge barrier for vortex entry/exit by the laser spot. Hence one can expect that irradiation at the edges of the bridge should strongly affect the vortex pattern and the resulting current distribution. In contrast, laser irradiation of the interior of the bridge does not change the energy barrier and the modification is probably less pronounced and the signal in the interior is much smaller.

Figure 5(c) shows linescans for H ≥ H_c2 across the bridge (i.e. perpendicular to the long side of the bridge). This is done to show the evolution of the LTSLM signal at the edges as compared to the signals appearing inside the bridge. All scans presented in figure 5 are taken from left to right. However, upon inverting the scan direction (i.e. scanning from right to left) no changes in the signals were observed, i.e., the asymmetry does not depend on scan direction. We show below that a slight difference in the amplitudes of the beam-induced voltage between the left and right edges can be inverted by changing either current or field direction without changing scanning direction. For fields larger than H_c2, the beam-induced voltage in the center of the bridge drops almost to zero while large peaks are still observed at the edges of the bridge. This apparently reflects the fact that above H_c2, the bulk is no longer superconducting and therefore does not lead to a voltage signal, while the edges still contribute to a strong LTSLM signal due to ES. The rather large width of these edge peaks in the state of ES can be explained by the fact that the edge states are not only perturbed when the laser beam spot is centered right at the edges, but also when the tail of the beam-induced heat distribution leads to a suppression of the edge states when the beam is centered slightly off the edges. A further increase in H leads to a gradual decrease of the edge peaks which finally disappear at H = 1.76 kOe which is close to H_c3. Above H_c3, the sample is completely in the normal state and the effect of the laser beam on the resistive state is negligible. Examining the linescans presented in the panels (b) and (c) in figure 5 and considering the left and right wings of the dV = dV(y) dependences outside the bridge interior, we could conclude that the radius of the hot spot is not less than 10 μm (for I = 1 mA,T = 7.5 K and H ≃ 0.8–1.1 kOe).

In summary, the linescan series in figure 5(c) indicate, that above H_c2, the dominant part of the current is flowing at the edges of the sample. Thus, LTSLM seems to be capable of visualizing the ES states and identifying the different regimes in the R(H) curves for the plain bridge.

For comparison, we show a linescan series (variable H) for the plain (figure 6(a)) and AD bridge (figure 6(b)) at T = 7.0 K. The corresponding R(H) curves with the bias points of the linescans are shown in figure 6(c). The H_c2 value for this temperature is ∼1.6 kOe. We note, that the beam-induced signal of the AD bridge is higher than for the plain bridge, which we ascribe to the higher current density in the AD bridge due to its reduced cross section because of the holes. For the lowest field value, H = 1.20 kOe, the beam-induced heating of the laser has no effect, while at H = 1.34 kOe the whole cross section of both bridges leads to a LTSLM signal. As H increases further, the LTSLM signal from the edges becomes larger than the signal from the interior and the overall signal increases up to H = H_c2 = 1.60 kOe. For the plain bridge the overall signal gets strongly reduced above H_c2, and the signal from the central part of the bridge almost vanishes. The key difference between the plain and AD bridge is that for the latter sample the voltage signal is much less reduced and the whole cross section of the AD bridge gives a measurable signal. This means that the current is distributed across the entire width of the bridge even for H > H_c2. This observation is consistent with the R(H) measurements shown in figure 6(c), where the additional edges inside the AD bridge lead to a different shape in R(H) and a lower R for any value of H within the interval H_c2 ≤ H ≤ H_c3.

3.4. Bias-current-induced asymmetry: LTSLM response and Ginzburg–Landau simulations

According to figures 5(b) and (c), the LTSLM signal ΔV(y) is asymmetric with respect to the bridge center (axis y = 0) for several H values around H_c2, i.e. the right maximum is slightly higher than the left one. This asymmetry in the beam-induced voltage response can be explained by an asymmetry in the supercurrent density distribution j_s,x(y) close to the left and right edge. Based on the time-dependent GL model⁵, we calculate the time-averaged quantities for the OP distribution 〈|ψ|²〉(y) and the x-components of the superfluid current density 〈j_s,x〉(y) and the normal current density 〈j_n,x〉(y).

Figure 7 shows results of such calculations for H = 1.3 H_c2 and T/T_c0 = 0.47, which were obtained for zero bias current density j = j_s,x + j_n,x (figure 7(a)), for j close to the critical current density at H = H_c2 (figure 7(b)) and for j which is larger than the critical current density for ES within the entire field range H_c2 < H < H_c3 (figure 7(c)).

According to our calculations, even in the resistive ES state, there is a finite superfluid flow localized within the ES channels. These supercurrents are circulating in opposite direction within each of the two edge channels, which is due to the applied magnetic field H.

For further analysis, we determined the net currents i_L,i_R and i_n. Here, i_L and i_R are the integrals of j_s,x across the left and right edge channel, respectively (shaded areas in figure 7); i_n is the integral of j_n,x across the entire width of the rectangular film. Hence, for the normalized bias current ${i}_{\mathrm{b}}\equiv j\frac{W}{{\xi }_{0}}$ we have i_b = i_L + i_R + i_n.

For j = 0 (i_b = 0) (cf figure 7(a)), the net currents i_L and i_R in the right and left edge channel have the same finite amplitude, but differ in sign, and i_n = 0. For j > 0 (i_b > 0) (cf figures 7(b) and (c)) the steady-state distribution of the superconducting parameters differs from the case j = 0. Now, i_L and i_R do have the same (positive) sign, but different amplitudes. Thus, analyzing only the large-scale details in the supercurrent distribution (spatially averaged over length scales much larger than the coherence length ξ₀), one can think in terms of a combination of two parallel currents flowing along the sample edges with different amplitudes depending on both the I and H direction. It should be noted that a very similar situation—the asymmetry of the critical current density—was described by Park [20] within a stationary Ginzburg–Landau model. Since the mentioned asymmetry results from the superposition of the bias current I and the currents induced by the applied H field, the asymmetry can therefore be changed either by changing the current direction or the sign of H.

In order to prove whether the asymmetry of the LTSLM signal can be related to the bias-current-induced asymmetry, we calculated the normalized beam-induced voltage Δv(y), i.e., linescans across a rectangular superconducting thin film with the geometry as in figures 3 and 7, biased at j = 4 × 10⁻³. Details of the calculation can be found in the appendix. Assuming a Gaussian shape of the laser-beam-induced increase in T with a maximum amplitude ΔT and a full width at half maximum of σ = 7ξ₀, we obtain the linescan series for different values of H/H_c2 shown in figure 8. These simulations clearly show that the voltage signal has maxima near the left and right edges, and that their amplitudes are different, with this asymmetry being most pronounced at H = H_c2. This is in good agreement with experimental LTSLM results.

**Figure 8.** Calculated normalized LTSLM beam-induced voltage $\Delta v={\bar {v}}_{\mathrm{on}}-{\bar {v}}_{\mathrm{off}}$ versus y/ξ₀ across a rectangular thin film (W = 30ξ₀, L = 60ξ₀; cf figures 3 and 7) for different values of H/H_c2 at T/T_c = 0.47 and j = 4 ×10⁻³. The vertical dashed lines indicate the position of the edges.
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**Figure 8.** Calculated normalized LTSLM beam-induced voltage $\Delta v={\bar {v}}_{\mathrm{on}}-{\bar {v}}_{\mathrm{off}}$ versus y/ξ₀ across a rectangular thin film (W = 30ξ₀, L = 60ξ₀; cf figures 3 and 7) for different values of H/H_c2 at T/T_c = 0.47 and j = 4 ×10⁻³. The vertical dashed lines indicate the position of the edges.
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To proof experimentally that the asymmetry depends on signs of H and I, we performed a series of LTSLM linescans on the plain Nb bridge. The reversal of the asymmetry of the measured LTSLM signal upon the inversion of the I and H signs is illustrated in figures 9(a) and (b). We find that the right peak is larger for I > 0 while the left peak is larger for I < 0 and vice versa. The slightly larger amplitudes of the peaks in figure 9(b) are probably due to the residual field (in the 10 Oe range) in the cryostat at the sample position. To the best of our knowledge, this is the first direct experimental verification of asymmetry in the current density in the ES state, as predicted by Park for surface superconductivity.

**Figure 9.** LTSLM linescans ΔV(y) across the plain Nb bridge (T = 7.2 K,H_c2 = 1.37 kOe,|H| = 1.40 kOe,|I| = 1 mA) for different sign of I and (a) negative H and (b) positive H. Vertical dashed lines indicate the position of the edges of the bridge.
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4. Conclusion

In this paper we studied experimentally and numerically the peculiarities of the resistive transition in thin film Nb microbridges with and without antidots (ADs) in perpendicular magnetic field H. From integral R(H) measurements we find that the transition from bulk to edge superconductivity (ES), and finally to the full normal state, can be identified by a pronounced change in slope dR/dH, which, however, strongly depends on the applied bias current density. The additional edges induced by the holes in the AD bridge lead to a different shape of the R(H) curves as compared to the plain bridge. The ES state as well as the evolution of superconductivity upon sweeping H was imaged by low-temperature scanning laser microscopy (LTSLM). For the ES state, LTSLM revealed an asymmetry in the currents flowing along the left and right edges, depending on the relative direction of applied current and external field, as proposed a long time ago [20]. Our calculations based on the time-dependent Ginzburg–Landau theory confirm essential features of the experimental results.

Acknowledgments

This work was supported by the Russian Fund for Basic Research (grants nos 12-02-00509, 13-02-01011 and 13-02-97084), Russian Academy of Sciences under the Program 'Quantum physics of condensed matter', Russian Agency of Education under the Federal Target Program 'Scientific and educational personnel of innovative Russia in 2009–2013', Ministry of Education and Science of Russian Federation (project 8686), Deutsche Forschungsgemeinschaft (DFG) via grant no. KO 1303/8-1. R Werner acknowledges support by the Cusanuswerk, Bischöfliche Studienförderung, D Bothner acknowledges support by the Evangelisches Studienwerk Villigst eV and M Kemmler acknowledges support by the Carl-Zeiss Stiftung. The authors thank A I Buzdin and D Yu Vodolazov for valuable discussions.

Appendix:

In order to describe the general properties of the resistive state in a mesoscopic superconducting thin film sample and to compare them with the experiment, we use a simple time-dependent Ginzburg–Landau (TDGL) model [50]. Strictly speaking, TDGL equations are well grounded for gapless superconductors [50], however they can be used for the modeling of conventional superconductors with finite gap (like Nb and Nb-based superconducting systems [51–54]).

For simplicity we assume that the effect of the superfluid currents on the magnetic field distribution is negligible and consider the internal magnetic field B equal to the external magnetic field H (perpendicular to the thin film plane). This assumption seems to be valid for the following cases; (i) for mesoscopic thin film superconductors with lateral dimensions smaller than the effective magnetic penetration depth Λ = λ²/d (λ_L is the London penetration depth, d is the thickness); (ii) for superconductors with a large Ginzburg–Landau (GL) parameter κ = λ_L/ξ placed in a moderate magnetic field [2]; (iii) for superconductors for large H and/or T (i.e. close to the phase transition line), when the superfluid density tends to zero. Taking into account the values for clean Nb (ξ^cl ≃ 38 nm and ${\lambda }_{\mathrm{L}}^{\mathrm{cl}}\simeq 3 9~\mathrm{nm}$ at T = 0) and the coherence length ξ ≃ 18.8 nm extrapolated to T = 0, estimated for our sample from the H–T diagram (figure 4(b)), we obtain the estimate for the magnetic field penetration depth in our sample (λ_L ≃ 76 nm) and the Ginzburg–Landau parameter (κ ≃ 4) at low temperatures. Thus, in our experiment screening should not be important for moderate H and negligible at H > H_c2.

As a result of negligible screening in thin film superconductors, the TDGL equations take the form

$\begin{eqnarray} &&u\left (\frac{\partial }{\partial t}+\mathrm{i}\varphi \right )\psi =\tau (\psi -\vert \psi \vert ^{2}\psi )+(\nabla +\mathrm{i}\mathbf{A})^{2}\psi , \end{eqnarray} \tag{ 2 }$

$\begin{eqnarray} &&\tau =1-T(\mathbf{r})/{T}_{\mathrm{c}0}, \end{eqnarray} \tag{ 3 }$

$\begin{eqnarray} &&{\nabla }^{2}\varphi =\mathrm{div}\hspace{0.167em} {\mathbf{j}}_{\mathrm{ s}},\tqs {\mathbf{j}}_{\mathrm{s}}=-\frac{\mathrm{i}}{2}\tau \hspace{0.167em} \{ {\psi }^{\ast }(\nabla +\mathrm{i}\mathbf{A})\psi -\text{c.c.}\} , \end{eqnarray} \tag{ 4 }$

where ψ is the normalized order parameter (OP) wavefunction, φ is the dimensionless electrical potential, A is the vector potential $[\mathrm{rot}\hspace{0.167em} \mathbf{A}=H \hspace{0.167em} {\hat {\mathbf{e}}}_{z}],T(\mathbf{r})$ is local temperature (potentially position-dependent), j_s is the density of the supercurrent, u is the rate of the OP relaxation, c.c. stands for complex conjugate, $({\hat {\mathbf{e}}}_{x},{\hat {\mathbf{e}}}_{y},{\hat {\mathbf{e}}}_{z})$ is the Cartesian reference system. We use the following gauge for the vector potential: A_x = (C − 1)yH, A_y = CxH and A_z = 0, that gives us $\mathrm{rot}\hspace{0.167em} \mathbf{A}=H \hspace{0.167em} {\hat {\mathbf{e}}}_{z}$ for any choice for constant C. Usually we use the symmetrical gauge with C = 1/2. We use the following units: ${m}^{\ast }{\sigma }_{\mathrm{n}}\beta /(2{e}^{2}\tilde {\alpha })$ for time (so called GL relaxation time), the coherence length ξ₀ at temperature T = 0 for distances, Φ₀/(2πξ₀) for the vector potential, $\hbar e\vert \tilde {\alpha }\vert /({m}^{\ast }{\sigma }_{\mathrm{n}}\beta )$ for the electrical potential, and $4 e{\tilde {\alpha }}^{2}{\xi }_{0}/(\hbar \beta )$ for the current density, where $\alpha =-\tilde {\alpha }\tau$ and β are the conventional parameters of the GL expansion, e and m* are charge and the effective mass of carriers, σ_n is the normal state conductivity. In these units the Ginzburg–Landau depairing current density j_dep at T = 0 is equal to 0.386 and its temperature dependence is given by j_dep = 0.386(1−T/T_c)^3/2.

We apply the boundary conditions in the following form

$\begin{eqnarray} &&\left (\frac{\partial }{\partial \mathbf{n}}+\mathrm{i}{A}_{\mathrm{n}}\right )_{\Gamma }\psi =0,\tqs \left (\frac{\partial \varphi }{\partial \mathbf{n}}\right )_{\Gamma }={j}_{\mathrm{ext}}, \end{eqnarray} \tag{ 5 }$

where n is the normal vector to the sample's boundary Γ,j_ext is the normal component of the inward (outward) flow of the bias current density j (with |j| ≡ j). Formally equation (5) describes an injection of normal current into a narrow superconducting thin film bridge located between two macroscopic superconducting leads (which are not considered in the model). For the main part of the presented numerical results we took four points per ξ₀, although some calculations (figure 3(a)) were made for a smaller grid (down to ten points per ξ₀). In the present paper we do not consider structural inhomogeneities (bulk pinning), since the number of additionally required parameters (describing the spatial distribution of pinning sites and their pinning strength) would be too large, although it is possible.

The temporal step for the treatment of the steady-state solutions of the TDGL equation was equal to 0.001–0.1 of the GL relaxation time. The initial condition for the order parameter wavefunction at t = 0 is ψ(x,y) = 1; the initial condition for the electrical potential φ is not necessary, since φ is determined from the ψ distribution at the initial moment. Upon changing the main parameters of the problem (T,I,H), we use the distributions of ψ and φ realized after the evolution of superconducting characteristics with old parameters as initial conditions for new parameters, which slightly differ from the old ones. This corresponds to a real situation when the typical rate of the variation of the parameters is much less than the Ginzburg–Landau relaxation time. For our modeling we use u = 1, which is smaller than the typical value accepted for gapless superconductors (u = 12), in order to describe fast recovery of the OP wavefunction in the edge superconducting channels after entry/exit of vortices. We found that larger u values cannot describe the low-resistive state observed experimentally for the superconducting bridges at H_c2 < H < H_c3 and small bias currents, since the effect of edge superconductivity in the resistive state due to slow recovery of the OP is effectively suppressed. The influence of the u parameter on the current (I)–voltage (V) characteristics was systematically analyzed in [55].

We calculate (see footnote 5) the instant value of the normalized voltage drop v(t) = 〈φ₁(t)〉 − 〈φ₂(t)〉 and analyze the dependence of v(t) on H and j_ext. Here

$\begin{eqnarray} &&\langle {\varphi }_{i}(t)\rangle =\frac{1}{{S}_{i}}{\iint }_{{S}_{i}}{\varphi }_{i}(x,y,t)\hspace{0.167em} \mathrm{d}x \hspace{0.167em} \mathrm{d}y \end{eqnarray} \tag{ 6 }$

is the time-dependent electrical potential averaged over the region S_i ('virtual electrodes', i = {1,2}). These regions have the same width as the sample width and they are shifted from the physical edges towards the sample interior (see figure A.1) for eliminating the effect of the sample edges. In addition we formally consider an inhomogeneous sample, containing two areas at the left and right edges (cf figure A.1) with a critical temperature T_c1 (at H = 0) and upper critical field ${H}_{\mathrm{c}2,1}^{0}$ (at T = 0) considerably exceeding T_c0 and ${H}_{\mathrm{c}2}^{0}$ in the rest of the sample. The reason for that is a purely technical one. This approach guarantees that the injected normal current i_b is fully converted into a supercurrent within these enhanced superconducting areas at any temperature and any value of H.

**Figure A.1.** Schematic drawing (top view) of a rectangular superconducting thin film bridge considered for GL simulations. Arrows indicate injection and extraction of the bias current. Shaded areas S₁ and S₂ are virtual electrodes.
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For the stationary regime all the calculated parameters, after transient processes induced by changes in the external parameters, tend to their time-independent values, pointing to the absence of energy dissipation for the established state and R → 0. For larger T,H or i_b the relaxation to the stationary case becomes impossible and all parameters oscillate in time. Calculating the mean normalized voltage drop $\bar {v}$ , averaged over a very large time interval (including up to 10² of the voltage oscillations), one can determine the normalized beam-induced LTSLM voltage signal $\Delta v={\bar {v}}_{\mathrm{on}}-{\bar {v}}_{\mathrm{off}}$ , where ${\bar {v}}_{\mathrm{on}}$ and ${\bar {v}}_{\mathrm{off}}$ are the time-averaged normalized voltage signals if the laser beam is on or off, respectively.

The effect of the focused laser beam can be treated as a quasistatic perturbation of the superconducting properties of the bridge, since the timescales of this perturbation are much longer than the GL time constant. In the most simple form this perturbation can be modeled as a Gaussian-like increase in local temperature in equation (3):

$\begin{eqnarray} &&T(\mathbf{r})={T}_{0}+\Delta T\cdot {\mathrm{e}}^{[-(x-{x}_{0})^{2}-(y-{y}_{0})^{2}]/{\sigma }^{2}}. \end{eqnarray} \tag{ 7 }$

Here, T₀ is the sample temperature if the laser beam is off or far from the beam spot centered at (x₀,y₀),ΔT is the amplitude of the local heating, depending on the beam intensity and on the rate of heat dissipation due to the thermal conductivity of the superconducting film and the substrate and on the thermal boundary resistance between the film and the substrate; σ is the full width at half maximum of the beam-induced temperature profile [56].

Edge superconductivity in Nb thin film microbridges revealed by electric transport measurements and visualized by scanning laser microscopy

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Abstract

1. Introduction

2. Sample fabrication and experimental details

3. Results and discussion

3.1. Magnetoresistance data and Ginzburg–Landau simulations

3.2. Superconducting phase diagram for the plain bridge

3.3. Visualization of the current distribution by LTSLM

3.4. Bias-current-induced asymmetry: LTSLM response and Ginzburg–Landau simulations

4. Conclusion

Acknowledgments

Appendix:

Footnotes

Edge superconductivity in Nb thin film microbridges revealed by electric transport measurements and visualized by scanning laser microscopy

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Author e-mails

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Dates

Abstract

1. Introduction

2. Sample fabrication and experimental details

3. Results and discussion

3.1. Magnetoresistance data and Ginzburg–Landau simulations

3.2. Superconducting phase diagram for the plain bridge

3.3. Visualization of the current distribution by LTSLM

3.4. Bias-current-induced asymmetry: LTSLM response and Ginzburg–Landau simulations

4. Conclusion

Acknowledgments

Appendix:

Footnotes