Microscopic examination of hot spots giving rise to nonlinearity in superconducting resonators

We investigate the microscopic origins of nonlinear rf response in superconducting electromagnetic resonators. Strong nonlinearity appearing in the transmission spectra at high input powers manifests itself through the emergence of jumplike features near the resonant frequency that evolve toward lower quality factor with higher insertion loss as the rf input power is increased. We directly relate these characteristics to the dynamics of localized normal regions (hot spots) caused by microscopic features in the superconducting material making up the resonator. A clear observation of hot-spot formation inside a Nb thin film self-resonant structure is presented by employing the microwave laser scanning microscope, and a direct link between microscopic and macroscopic manifestations of nonlinearity is established.

Figure 1

Global microwave transmission response from a $D_o=6$ mm spiral with 40 turns and $\mathit{w}=\mathit{s}=10$ $\mu \text{m}$ for a set of rf input powers at $T=4.4$ K. Right inset: a closer look at the linear and nonlinear regimes evolving with increasing input power. Black, Lorentzian-type curves corresponding to power values from $-21$ to $-$2 dBm overlap, progressively evolving into the red curves for power values between $-$1.5 and $+$0.6 dBm then blue curves for powers from $+$0.7 to $+$1.1 dBm, orange curves from $+$1.2 to $+$2 dBm, and brown curves from $+$2.2 to $+$3.6 dBm. Top left inset: an optical image of a spiral resonator with $D_o=6$ mm and 40 turns. Bottom left inset: zoom in the area marked in the top left inset showing the Nb spiral windings in detail. Dark strips are Nb turns in the picture.

Figure 2

LSM $PR$ images of the nucleation and rf power evolution of a hot spot. (a) Detection of the first microscopic defect causing the hot-spot formation on the Nb spiral resonator whose details are discussed in Fig. 1 at $+$10 dBm, 4.8 K. (b) The hot spot shown in (a) heats the neighboring area and creates a resistive response coming from the entire spiral at $+$20 dBm. The length scale bar in (a) is the same for (b). (c)–(f) Zoom in of the $30\times 30$ $\mu {\text{m}}^2$ area marked in (a) at progressively higher input powers. The color-scale bar in (d) applies to all LSM images shown in Fig. 2, the maximum LSM $\mathit{PR}$ corresponds to 0.3346 mV for (a), 0.1857 mV for (b), 3.6 $\mu \text{V}$ for (c), 11.4 $\mu \text{V}$ for (d), 23.2 $\mu \text{V}$ for (e), and 12.4 $\mu \text{V}$ for (f). The length scale bar in (c) applies to the images in (c)–(f).

Figure 3

Frequency and space dependence of LSM $\mathit{PR}$ at (a) 0, (b) $+$4, (c) $+$5, and (d) $+$6 dBm on a Nb spiral with $D_o=5$ mm, 45 turns, $\mathit{w}=\mathit{s}=10$ $\mu \text{m}$ over a 20 $\mu \text{m}$ $x$ scan. The frequency is swept between 86.5 and 89.5 MHz through the fundamental resonant frequency $f_0=88.2$ MHz. The images in (a)–(c) show only forward frequency sweeps whose direction is indicated with blue arrows. In (d), different shade color maps point out both direction sweeps, shown with blue (forward) and red (backward) arrows. Corresponding $PR(f,x)$ profiles at a fixed position, $x=x_0$ of the laser probe are shown in (e)–(h). The blue dots are the data collected through forward frequency sweep whereas the red dots are those through backward sweep. The $|S_{21}{\left(f\right)|}^2$ at low power, 0 dBm is shown in (e). The color-scale bar in Fig. 2d applies to all LSM images shown in Fig. 3 as well, the maximum LSM $\mathit{PR}$ corresponds to (forward/back sweep)-(1.61/1.62) mV for (a), (1.49/1.5) mV for (b), (1.54/1.55) mV for (c), and (1.12/1.4) mV for (d).

Figure 4

Evolution of global transmission with small increments in rf input power (a), corresponding LSM $\mathit{PR}$ averaged over 100 $\mu \text{m}$ line scans across the center of a hot spot. Note that $|S_{21}\left(f\right)|$ is measured simultaneously with LSM $\mathit{PR}\left(\mathit{f}\right)$ when frequency is changed very slowly. A nearly exact correlation between $|S_{21}\left(f\right)|$ and $\mathit{PR}$ data is evident.