# Enhancing Precision in SQUID Sensors: Analyzing Washer Geometry Dependence at the Microscale
## 4 Results
### Self-Inductance  
In the preceding theoretical sections, we explored the influence of washer inductance on different device parameters, emphasizing the delicate balance between quantum system dynamics and noise considerations. This equilibrium sets a maximum limit for the inductance in superconducting loops, capped at 5900 pH for devices operating at low temperatures. The optimal performance demands lower inductance values to ensure sensitivity. Figure 10 shows that, under the same inner diameter conditions, the inductance values of octagonal and circular washers are smaller than the traditional square washer design.  
The self-inductance of a washer is given by  
\[L=\frac{\Phi_{h}^{f}}{I_{h}}, \tag{26}\]  
where \(I_{h}\) is the current that circulates around the hole, and \(\Phi_{h}^{f}\) is the fluxoid for a path enclosing the hole [37], assuming a total current of 1 mA circulates in a clockwise direction within the ring, the simulation results are shown in Figure 10.  
Linear regression analysis of the inductance variation with the washer diameter (D) yields different slopes for each washer shape: for circular washers, the slope is 2.0829; for octagonal washers, it is 2.1969; and notably higher for square washers, with a slope of 2.8391. The analysis results indicate that the inductance of square washers significantly increases with the diameter enlargement, compared to circular and octagonal washers. This heightened sensitivity to size variations could potentially impact the sensor's sensitivity and noise levels adversely. Conversely, circular and octagonal washers exhibit a more controlled and less pronounced increase in inductance with size, indicating their superior ability to maintain stable inductance across different sizes. This characteristic enables them to strike an optimal balance between enhancing the device sensitivity and minimizing device noise.  
Figure 11 depicts a comparative study between inductance values calculated using conventional empirical formulas and those derived from incorporating an externally applied circulating current for washers of square, circular, and octagonal shapes. The slopes of inductance versus washer diameter (D), obtained from empirical formulas for square, circular, and octagonal washers, are, respectively, 1.57, 1.32, and 1.26. In contrast, the  
Figure 10: This figure elucidates the relationship between washer diameter and inductance, with the inner diameter of the washers varying from 150 μm to 250 μm. The blue line represents the inductance of square washers, the green line corresponds to the inductance of circular washers, and the red line denotes the inductance of octagonal washers.  
inductance slopes derived in this study for washers of all shapes markedly exceed the predictions of traditional empirical formulas.  
To assess the deviation of inductance values between various washer shapes and those predicted by empirical formulas, we calculated the average relative error between the simulated inductance and the empirical inductance for each shape. The results revealed that square washers exhibited an average relative error of 80.8%, circular washers showed a deviation of 65.31%, and octagonal washers presented an error of 66.43%. Notably, the discrepancy is most pronounced for square washers, with circular and octagonal washers also showing variations, albeit to a lesser extent, yet still surpassing conventional estimates. This discrepancy is attributed to significant edge effects induced by the acute angles of polygonal washers, which substantially impact inductance. Additionally, the washer shape influences the distribution of superconducting currents within the loop. The observed higher inductance values across the board are speculated to arise from the inability of traditional formulas to fully account for external interferences, suggesting that these conventional approaches may not accurately capture the intricate magnetic response processes of the washers.