Model for hot spots and $Q$-slope behavior in granular niobium thin film superconducting rf cavities

We propose a model to explain power dissipation leading to the formation of hot spots in the inner walls of niobium thin film superconducting rf cavities. The physical mechanism that we explore is due to the constriction of surface electrical current flow at grain interface boundaries. This constriction creates an additional electrical contact resistance which induces localized punctual heat dissipation. The temperature at these spots is derived; and the electrical contact resistance is shown to depend on the magnetic field, on the grain contact size over which dissipation occurs, and on other key parameters, including the effective London penetration depth and the frequency. The surface resistance and the quality factors are determined using our model and are shown to be in excellent agreement with experimental data.

Figure 1

(a) Holm model geometry showing constriction of current lines between two grains, creating a hot spot at the contact boundary. The voltage drop $U(T)$ between $T_s$ and $T_0$ in each grain is equal to $ϵ/2$. (b) Schematics of our model representation of a square constriction of length $a$ between two grains of size $d$ in a Nb thin film. (c) Lattice of constriction sites forming a model square cluster of size $\varphi$.

Figure 2

Evolution of hot spot temperatures as the surface magnetic field is increased from 0 to 250 mT for different cluster sizes of 25, 40, 50 and $75\mu \mathrm{m}$. Here, the electrical contact size is set to $a=100\mathrm{nm}$ for each cluster. The number of grain boundary constrictions $N_c$ increases with the cluster size. The vertical dashed lines highlight the shift of $B_{\max }$ to lower fields as $\varphi$ increases. The purple dashed curve corresponds to the theoretical transition from the Meissner state to the mixed state for pure niobium. It shows that early quench is to be expected even for the smaller clusters considered in this study.

Figure 3

The temperature of a cluster of electrical contacts at superconducting grain boundaries forming a hot spot of $\varphi =50\mu \mathrm{m}$. The orange, red, blue and green curves correspond respectively to electrical contact lengths of 30, 50, 100 and 200 nm. The purple dashed curve corresponds to the theoretical transition from the Meissner state to the mixed state for pure niobium.

Figure 4

The BCS surface resistance [Eq. (12)] as a function of $B_z$ for a cluster size of $50\mu \mathrm{m}$ and for three different grain electrical contact lengths, namely, 30, 50 and 100 nm.

Figure 5

Power dissipation in three clusters, each $50\mu \mathrm{m}$ in size and having uniform grain electrical contact lengths of 30, 50 and 100 nm respectively. The inset is a zoom near the origin.

Figure 6

The blue dots are surface resistance measurements of 1.3 GHz $\mathrm{Nb}/\mathrm{Cu}$ cavity measured at 2.23 K, given in Ref. [19] (dataset (e) in Table 1). The blue curve through data points is a fit obtained with our model at the same initial temperature.

Figure 7

The red dots and blue squares correspond to quality factor data given in Ref. [20] for temperatures 1.7 and 4.2 K respectively. The green and orange curves are fits to these data using the parameters values given Table 1. The shaded areas are obtained with the error bars given in Table 1.

Figure 8

Two sets of data at 1.7 K and at 1.5 GHz. The blue squares are data given in Ref. [21] and the red dots are from Ref. [20]. Both curves fitting the datasets are determined using our model.

Figure 9

Red dots correspond to data taken from Ref. [21]. The blue curve is a fit using our model. The cluster size, electrical contact lengths and residual resistance determined from the fit are given in Table 1. The error bars given in Table 1 define the shaded area.

Figure 10

Blue dots are experimental values at 2.23 K of the quality factor as a function of the magnetic field for a Nb/Cu cavity functioning at 1.3 GHz (see Ref. [19]). The red curve is a fit determined using our model. Refer to Table 1 for the fitting parameters.

Figure 11

Evolution of the temperature increase of a hot spot from its initial value $T_0=1.7\mathrm{K}$ as the quality factor decreases with the increase in the magnetic field. The red dashed curve corresponds to the magnetic field and the blue curve to the temperature increase. The plot is done for dataset (d) in Table 1.