Asymmetry in the effect of magnetic field on photon detection and dark counts in bended nanostrips

Current crowding in the bends of superconducting nanostructures not only restricts measurable critical current in such structures, but also redistributes local probabilities for the appearance of dark and light counts. Using structures in the form of a square spiral, where all bends have the same symmetry with respect to the directions of the bias current and external magnetic field, we have shown that areas around the bends largely contribute to the rate of dark counts and to the rate of light counts at small photon energies. The minimum in the rate of dark counts reproduces the asymmetry of the maximum in the critical current as a function of the magnetic field. Contrarily, the minimum in the rate of light counts demonstrates opposite asymmetry. The rate of light counts becomes symmetric at large currents and fields. Comparison of the computed local absorption probabilities for photons and the simulated local threshold detection current reveal the areas near bends that deliver the asymmetric rate of light counts. Asymmetry in count rates is absent in circular spirals without bends.

Figure 1

SEM images of (a) a circular spiral and (b) a square spiral. Dark color represents strips and surrounding fields from NbN film. The outer diameter of the Archimedean spiral is 7.3 μm. The size of the square spiral is $6.5\times 6.5\mu {\mathrm{m}}^2$. (c) Corner rounding in bends of the square spiral. The distance between two vertical cursor lines is 112 nm. Irregularities at the edge of the NbN field that surrounds the spiral do not affect the critical current of the strip.

Figure 2

Schematic of the multilayer structure with the top electrode.

Figure 3

Bend schematics and positive directions of the external magnetic field $\left(B\right)$ and bias current $\left(I_b\right)$. These directions obey left field-current symmetry for which an increase in the magnetic field causes the increase in the superconducting current density at the inner corner of the bend. Pictograms in the left box denote two possible configurations that produce this effect. Pictograms in the right box denote configurations having right symmetry and, consequently, opposite effect on the current density at the inner corner. The pictogram in the bend corresponds to the positive directions of the field and current shown in the figure. The inner corner has coordinates (0; 0; 0) in the system shown here. It will be used through the paper.

Figure 4

Density of the superconducting current in the bend and in the adjacent straight parts of a strip (b) without magnetic field and for the field $0.005B_{C2}$ with (c) the positive and (a) negative directions. The current density is normalized to its density far from the bend at $B=0$. The current distribution was computed for the strip width of 20ξ where $\xi =5\mathrm{nm}$ is the coherence length for our NbN films [14, 18]. A gray circle labeled with the letter A in the common bisector of the bend corners [panel (b)] marks the position where the local current density does not change with the magnetic field.

Figure 5

(a) Relative critical current of the square spiral in magnetic field for positive (open symbols) and negative (closed symbols) current directions. Pictograms depict combination of the field and current directions for each section of the plot. Solid straight line extrapolates to zero field the linear decrease of the critical current with the magnetic field in the right symmetry. Vertical dashed lines show zero field and positions of the maxima on the field axis. (b) Relative critical current of the circular spiral for different current directions. The same convention is used to mark symmetries and current directions.

Figure 6

(a) Rate of dark counts in magnetic field for two directions of the bias current with the magnitude 35 μA. The DCR for the positive current direction is shown with open symbols and for the negative direction with closed symbols. Pictograms depict combination of the field and current directions for each section of the plot. Dashed vertical lines are to guide the eyes; they show field positions of the minima in the DCR and zero field. (b) Magnetic field dependencies of the DCR for different positive bias currents. Values of the bias current are specified in the legend. Vertical dashed line shows the location of the minimum on the field axis.

Figure 7

Rate of light counts in magnetic field for different wavelengths: (a) 1400 nm, (b) 800 nm, and (c) 500 nm. Magnitudes and directions of the bias current are specified in the legends. Conventionally, the PCR for the positive current direction is shown with open symbols, and closed symbols correspond to the negative direction. Vertical dashed lines in the panel (a) guide the eyes to the locations of the minima in the PCR on the field axis. Pictograms depict combinations of the field and current directions for each section of plots.

Figure 8

Relative critical current at different magnetic fields for the sharp bend in the strip with the width $w=20\xi$ (closed symbols) and for the straight part of such a strip with defects (open symbols). The inset shows the geometry used for modeling: A, fragment of a straight strip with defects (not in scale); B, sharp bend. Solid line underlines currents that appear as critical currents of a spiral consisting from bends and straight strips. Vertical lines guide eyes to zero field and to the expected maximum in the critical current of the whole spiral. Horizontal line shows the bias current that was applied to measure the DCR [Fig 6].

Figure 9

Detection current as a function of the positions $\left(x\right)$ of the hotspot at three different distances [(a) $y=w/4$; (b) $y=0$; (c) $y=-w]$ from the inner corner of the bend without magnetic field and for the magnetic field $B=0.005B_{C2}$ with opposite directions. Positive magnetic field corresponds to the left symmetry. Coordinate system is shown in the inset in the panel (a). The inset in panel (a) sketches the bend and the area (gray spot) where $I_{\det }$ is increased/decreased by small magnetic field of the left/right symmetry. The cut through this area at $y=w/4$ is marked with a blue dashed circle in the panel (a). Horizontal straight lines in panel (a) show boundaries for bias currents within which the effect exists. Horizontal lines $I_b=0.5I_{\mathrm{dep}}$ in panels (b) and (c) show the nominal value of the bias current used to measure the PCR for the wavelength 1400 nm [Fig. 7].

Figure 10

Relative probability of photon absorption in the bend and adjacent portions of the straight strips for the wavelength 1400 nm and different polarizations. Red (dark) color corresponds to the largest local probability. Blue curved lines circle the area that delivers the PCR with the inverted asymmetry. Black arrows show directions of currents that are excited in the bends by continuous electromagnetic waves. Polarization directions of the incident waves are shown with the empty arrows.