# Enhancing Precision in SQUID Sensors: Analyzing Washer Geometry Dependence at the Microscale
## 2 Numerical Calculations
### The Energy Resolution  
Another crucial parameter of DC SQUIDs is the noise energy per unit bandwidth, commonly known as energy resolution. This metric directly influences the SQUID's capability to detect weak signals. A higher energy resolution enables the SQUID to operate effectively even in environments with elevated noise levels. Essentially, enhancing the energy resolution enables the SQUID to more accurately differentiate genuine signals from background noise, thus enhancing measurement precision and reliability. Moreover, a higher energy resolution directly enhances the SQUID's sensitivity to changes in the magnetic fields, making it more adept at detecting subtle variations. Therefore, energy resolution is one of the core indicators of SQUID system performance, which has a direct impact on its signal detection capability, noise limit, measurement sensitivity, and application range [32].  
Presenting as noise energy per unit bandwidth offers a simplified characterization of flux noise, using Equations (1) and (2), which can be reformulated as the following equation:  
\[\varepsilon=\frac{S_{\Phi}(f)}{2L} \tag{5}\]  
The energy resolution may then be written as  
\[\varepsilon=\frac{k_{B}T}{L}\left\{\frac{L^{2}}{R}+\frac{2R_{\text{dyn}}}{( \partial V/\partial\Phi)^{2}}\right\} \tag{6}\]Disregarding the self-screening effects, the total flux passing through the SQUID can be approximated as essentially equivalent to the externally applied flux. To optimize the SQUID's energy resolution, it is crucial to establish correlations between the noise equation and the SQUID's design parameters, simplifying the design process. Here, we introduce the parameter \(\beta=\frac{2L}{\Phi_{0}}\), where L represents the superconducting loop inductance, and \(I_{c}\) is the critical current of each junction. From the previous researcher's simulation [33], the above equation has the best value at \(\beta\approx 1\). \(R_{\mathrm{dyn}}\) and \(\frac{\partial\mathcal{V}}{\partial\Phi}\) are taken as approximations, and the energy resolution can be written at a low temperature of 4.2K:  
\[\varepsilon=k_{B}T\frac{L}{R}\Bigg{\{}1+\frac{\sqrt{2}\big{(}1+\beta\big{)}^{ 2}}{\beta^{2}}\Bigg{\}} \tag{7}\]  
Defining the Stewart-McCumber parameters as \(\beta_{c}=(2\pi/\Phi_{0})I_{c}R^{2}C\), where \(C\) is the junction capacitance, the above equation can be rewritten as  
\[\varepsilon=k_{B}T\sqrt{LC}\sqrt{\frac{\pi\beta}{\beta_{c}}}\Bigg{\{}1+\frac{ \sqrt{2}\big{(}1+\beta\big{)}^{2}}{\beta^{2}}\Bigg{\}}\equiv\gamma k_{B}T \sqrt{LC} \tag{8}\]  
Therefore, in this case, the superconducting tunneling effect can be expressed by the equivalent inductance \(L_{\mathrm{J}}=\frac{\Phi_{0}}{2\pi I_{c}}\).  
The quality factor \(Q_{\mathrm{J}}\) in the context of magnetic sensors, such as SQUIDs, is a measure of the sensor's performance and efficiency in capturing and processing the magnetic flux, which is influenced by several factors. Under the RCSJ model, the quality factor can be written as \(Q_{\mathrm{J}}=R\big{(}C/L_{\mathrm{J}}\big{)}^{1/2}\). In order to minimize the mixing effect in nonlinear systems, the quality factor of the resonant circuit should be kept below 1. Therefore, in practice, the best energy resolution results are obtained when \(\beta_{c}\approx 0.7\) and \(\beta=1\); at this point, the energy resolution can be rewritten as  
\[\varepsilon\approx 16k_{\mathrm{B}}T(\mathrm{LC})^{1/2} \tag{9}\]