# Enhancing Sensitivity of Double-Walled Carbon Nanotubes with Longitudinal Magnetic Field
## 1 Introduction  
The advent of carbon nanotubes has created a new revolution in engineering sciences and medical technology. Carbon nanotubes (CNTs) are essential in improving the functioning of nanodevices, such as nanoelectromechanical system (NEMS) nanoresonators and sensors [1]. CNT-based nanosensors have been used extensively in different applications, such as ultrasensitive mass and force sensing, cooling and quantum measurement, and chemical and biological sensing, to detect the masses of bacteria, cells, and viruses [2; 3; 4; 5]. In the case of mass detection, two methods (static and dynamic) are typically used to detect added mass in nano-mass sensors [6; 7; 8]. In the static method, the added mass is determined by measuring the deflection of a nano-beam resulting from the attached mass on the surface. The dynamic method analyzes the shift in the system's frequency due to the attached mass [9]. A higher frequency shift shows the superior sensitivity of a mass sensor [10], which is affected by various parameters, such as geometry [11], vibration modes [12], the characteristics of the structure [13], and surface functionalization [14]. Additionally, external parameters, such as temperature and environmental conditions, can affect the performance of nano-mass sensors [15]. Research has been conducted to enhance and assess the sensitivity of mass nanosensors and has primarily been driven by these factors. [PERSON] et al. [6] conducted a study on carbon nanotube-based biosensors in which they proposed a straightforward analytical relationship using a linear approximation of the non-linear sensor equation. Their approach aimed to detect the masses of biological samples through the analysis of frequency shifts in bothcantilevered and bridged carbon nanotubes. [PERSON] et al. [16] studied the transverse vibrations of supported and bridged nano-beams, which had attached masses and rotational and transverse springs at their ends. The effects of the geometric parameters of mechanical elements were also considered by [PERSON] et al. [11]; in particular, they investigated the impacts of the various geometric parameters of microbeams on mass sensors and realized that the sensitivity of a microsensor increases with microbeam rigidity. In another work, [PERSON] et al. [17] compared micro-CNTs and beam mass sensors and found that the CNT-based sensor was more sensitive than the microbeam sensor. [PERSON] et al. [18] studied the position and mass identification in a single-walled carbon nanotube (SWCNT) mass sensor. [PERSON] et al. [19] investigated the sensitivity of an SWCNT mass sensor resting on an elastic foundation using non-local elasticity theory and proposed the parametric study of elastic foundation parameters. [PERSON] et al. [20] analyzed the vibrations of DWCNTs to sense biological objects, considering six different types of DWCNTs to determine the best type for mass sensors. They concluded that cantilevered configurations were four times more sensitive than bridged configurations. As enhancing the sensitivity of mass sensors is necessary to measure smaller masses, which is necessary in the fields of biotechnology and life sciences, higher mass sensitivity facilitates the detection of molecular interactions and biomolecular events with greater precision [21]. Therefore, researchers have tried to enhance the sensitivity of current mass sensors through the application of modifications or the introduction of new methods. For example, [PERSON] et al. [22] used a system of gold layers of varying thicknesses and showed a linear increase in mass sensitivity using higher vibration modes. They concluded that the sensitivity experiences an enhancement of two orders of magnitude when transitioning from mode 1 to mode 7. [PERSON] et al. [23] investigated a new way to increase the mass sensitivity of a magnetoelastic resonator.