# Enhancing Precision in SQUID Sensors: Analyzing Washer Geometry Dependence at the Microscale
## 1 Introduction  
Low-frequency magnetic fields are omnipresent and offer crucial insights across a broad spectrum of applications. The multitude of sensing scenarios and the prevalence of numerous physical systems demonstrating significant magnetic responses have catalyzed the development of diverse magnetometer technologies [1]. Ultrasensitive magnetic sensors [2; 3; 4] play a vital role in various technical fields [3]. The non-contact measurement characteristics of magnetic sensors enable them to work without disturbing the observation system [2; 3]. This characteristic inspires an increasing number of scientists to dedicate themselves to magnetic sensing research [4] and thus develop various precision measurement instruments, including Superconducting Quantum Interference Devices (SQUIDs), Giant Magnetoresistance (GMR), Tunneling Magnetoresistance (TMR), Hall Effect Sensors [4], Diamond Magnetometers based on Nitrogen Void (NV) Centers [5], and Atomic Magnetometers [6]. Among them, due to their ultra-high sensitivity, wide dynamic range, frequency response, minimal measurement disturbance, and broad application areas, SQUIDs have emerged as one of the most popular choices for ultrasensitive magnetic sensors [3; 7; 8; 9; 10]. The current research focuses on the further miniaturization of SQUIDs and efforts to enhance their sensitivity and field resolution. Researchers are actively pursuing strategies to increase the accessibility of SQUIDs for various applications by reducing their design and manufacturing costs. DC Superconducting Quantum Interference Devices (DC SQUIDs) [11] are the world's best-performing magnetic flux detectors [9; 10]. These magnetic sensors utilize flux quantization [12] and Josephson tunneling effects [13; 14]. Specifically, flux quantization dictates that, in a superconducting loop, the magnetic flux enclosed by the loop must be quantized in terms of the flux quantum \(\phi_{0}=\frac{k}{2e}\). Conversely, the Josephson effect [15] refers to the quantum tunneling phenomenon of a Cooper pair across a thin potential barrier between two superconductors linked by a non-superconducting electric layer. SQUID devices, constructed utilizing the aforementioned physical phenomena, are inductively [16] coupled to suitable input circuits to measure a range of physical quantities [17], encompassing magnetic fields, magnetic field gradients, voltages, and magnetization. They serve as pivotal tools in diverse fields such as medicine, geology [18], materials testing [19], magnetic induction communication [7], cosmology and astrophysics [20], condensed matter physics [21], and quantum computing [1; 22; 23; 24]. SQUIDs are classified into two categories, depending on the bias mode: Direct Current (DC) SQUIDs and Radio Frequency (RF) SQUIDs. The first type is a DC SQUID, comprising two Josephson junctions connected in parallel on a superconducting loop. When a constant bias current slightly exceeding the critical current flows through the device, the Josephson junctions become resistive. Consequently, there exists a periodic relationship between the voltage across the device and the applied magnetic flux, with a period of one flux quantum \(\Phi_{0}\). This configuration enables the detection of magnetic flux changes as small as \(10^{-6}\Phi_{0}\)[25]. Figure 1 shows the schematic diagram of the DC SQUID. The second type is the RF SQUID, featuring a sensitive element comprising a single Josephson junction incorporated within a superconducting loop [25; 26], This junction is inductively coupled to a LC resonant circuit. The device is driven by a high-frequency signal [27], typically ranging from tens to hundreds of megahertz, which is applied to an energy storage circuit magnetically coupled to the SQUID [28]. The discovery of the DC SQUID by [PERSON] et al. occurred only a few years after [PERSON] predicted superconducting tunneling [15]. Subsequently, the RF SQUID was introduced shortly thereafter [25].