As one of the most widely utilized instruments for magnetic characterization, the VSM is renowned for its versatility, robustness, and adaptability to diverse sample forms, including bulk solids, powders (Frandsen et al., 2021), thin films (Ke et al., 2021), and liquids, whether in the form of single crystals, polycrystals, hard magnetic materials, or soft magnetic ones. This technique was first realized by Simon Foner in 1955 at the Massachusetts Institute of Technology (MIT) Lincoln Laboratory, where he constructed the first prototype using readily available materials, including a paper cup, paper straw, and a loudspeaker (Foner, 1996). The central idea was to mechanically vibrate a magnetized sample within a uniform magnetic field, thereby inducing an alternating magnetic flux through a set of stationary pickup coils. According to Faraday’s law of electromagnetic induction, this time-varying magnetic flux generates an induced voltage ( V e m f ) in the coils that is directly proportional to the sample’s magnetic moment. The corresponding experimental results were published in 1956 (Foner, 1956), and further mechanical refinements followed in 1959. Subsequently, he granted the patent for the VSM design to Princeton Applied Research Corp (PARC), which marked the beginning of the commercial development of the VSM.

Over the decades, continued engineering and electronic advancements have significantly enhanced the sensitivity and stability of VSM systems. Modern commercial VSMs typically consist of three primary subsystems—the magnetic field generation and control unit, the vibration and sample mounting unit, and the signal detection and processing unit (as schematically illustrated in Figure 1)—and are capable of achieving sensitivities in the range of 10⁻⁷ to 10⁻⁸ emu with typical acquisition times of 10 s per measurement point. The magnetic field generation and control unit comprises an electromagnet and a gaussmeter. The electromagnet provides a uniform magnetic field to magnetize the sample, while the gaussmeter monitors the field intensity in real time. This feedback enables precise closed-loop control, ensuring the field remains stable and accurate throughout the measurement process. The vibration and sample mounting unit typically includes a vibration head, a sample rod, and the sample holder. The vibration head, driven by a mechanical oscillator, induces sinusoidal motion in the sample at a fixed frequency (commonly around tens of hertz). The sample, mounted at the lower end of the rod, is positioned within the uniform magnetic field. Vibration can occur either parallel or perpendicular to the field direction, depending on the system configuration (Mészáros, 2007). The signal detection and processing unit is mainly composed of a detection coil and a lock-in amplifier (LIA). The pickup coil system typically adopts a four-coil configuration arranged symmetrically along the vibration axis (Mallinson, 1966). This differential arrangement enhances the detection sensitivity and effectively cancels out background noise and environmental interference, thereby significantly improving the signal-to-noise ratio. As the magnetized sample oscillates within the magnetic field, it causes a time-varying magnetic flux through the pickup coils, thereby inducing a voltage via electromagnetic induction. The LIA, phase-locked to the vibration frequency, selectively amplifies and extracts the signal corresponding to the sample’s magnetic response, effectively suppressing noise at other frequencies. By integrating these subsystems, the VSM enables accurate and efficient magnetic moment measurements across a broad range of material systems and sample geometries.

Further, digging into the working principle, the induced voltage V e m f obtained by the pick-up coils of the VSM can be given below (Dodrill and Lindemuth, 2021): V e m f = m A f S where, m denotes the magnetic moment of the sample, and A and f represent the amplitude and frequency of vibration, respectively, and S is the sensitivity function of the detection coils. It is clear from formula that increasing the amplitude A , the frequency f , or the sensitivity S will enhance measurement accuracy. However, in practical implementations, each of these parameters must be carefully optimized to avoid adverse effects. Excessive vibration frequency can induce substantial eddy currents in conductive samples, which in turn distort the magnetic response and generate undesirable heat. To mitigate these effects, the vibration frequency f is typically maintained below 100 Hertz (Hz). Similarly, to minimize geometric distortions and prevent signal loss due to non-uniform field coupling, the vibration amplitude A should be kept moderate, generally smaller than the diameter of the pickup coil. Furthermore, the sensitivity factor S can be enhanced through optimization of the pickup coil design, for example, by increasing the number of coil turns or modifying the coil geometry to better match the sample’s magnetic field distribution. While increasing the number of turns improves inductive sensitivity, it also raises coil resistance, which elevates thermal noise and may reduce the effective signal-to-noise ratio. As an alternative, improving the magnetic coupling between the sample and the pickup coils, such as by reducing the coil-to-sample distance, offers a promising route to increasing S . It is important to note, though, that the spacing cannot be arbitrarily minimized, as sufficient room must be preserved for the sample to oscillate freely. Therefore, striking the perfect balance is crucial.

The distinctive advantage of the VSM over other magnetometers resides in its multifaceted applicability, enabling it to characterize samples across diverse morphological forms. For instance, VSM has been employed to characterize powder-form nanomaterials, such as Ni_xCo_xMg_xCu_xZn_1−4xO, where doping-dependent transitions from diamagnetic to mixed diamagnetic-ferromagnetic behavior were observed (Thien et al., 2024). It also enables the evaluation of perpendicular magnetic anisotropy (PMA) in thin film structures by analyzing their hysteresis loops (Smith et al., 2023). Furthermore, through modular integration of cryogenic and high-temperature accessories, the VSM facilitates magnetic measurements under controlled thermal environments, a capability particularly critical for studying temperature-dependent magnetic transitions. For example, Feng et al. (2023) employed a VSM to measure the temperature-dependent magnetization and hysteresis loops of tetragonal and hexagonal Mn₃O₄ nanosheets. These measurement results were used to determine the magnetic phase transition, Curie temperature, and identify the ferromagnetic ordering and the origin of magnetism. These applications collectively highlight VSM’s versatility in probing static magnetic properties across a broad range of sample types and environmental conditions.

Despite its widespread use, the measurement precision of conventional VSM systems presents limitations when characterizing ultra-weak magnetic signals, particularly in the context of micro- and nano-electronic materials. In response, recent efforts have been devoted to improving system performance through the development of enhanced vibration drivers and more sensitive detection coil designs (Niazi et al., 2000; Nizhankovskii and Lugansky, 2007; El-Alaily et al., 2015; Jordán et al., 2018; Lopez-Dominguez et al., 2018; Dodrill and Lindemuth, 2021). While these customized, home-built VSM systems offer advantages in cost and user flexibility, they often fall short in achieving the high precision required for advanced magnetic analysis. Therefore, improving measurement precision and enabling versatile testing functions has emerged as a central challenge for the advancement of VSM technology.

As another key technique for magnetic moment measurement, the AGM offers significantly higher precision and sensitivity compared to the VSM, making it especially well-suited for characterizing materials with ultra-weak magnetic signals. Unlike VSM, which relies on detecting induced voltages via sample vibration, the AGM functions as a magnetic force balance, detecting vibration amplitude of a sample subjected to an alternating magnetic field gradient. The original concept of AGM emerged in the 1970s, where the sample was mounted on an elastic rod and subjected to an alternating magnetic field gradient. At resonance, the magnetic force induced a measurable deflection of the rod, visible under a microscope. The deflection amplitude was found to be proportional to the magnetic moment of the sample, establishing the foundation of the AGM technique (Zijlstra, 1970). Subsequent developments in AGM technology primarily focused on enhancing the vibration detection methods and refining the sample holder design. In 1980, the integration of piezoelectric bimorphs and lock-in amplifiers markedly improved detection precision, reaching a sensitivity of 10⁻¹⁰ emu (Roos et al., 1980). Building upon these advancements, Richter et al. (1988) refined the system through optimizing signal acquisition, introducing background noise compensation, and implementing strategies to suppress external interference, thereby achieving a remarkable sensitivity of 10⁻¹¹ emu. Most notably, Todorovic and Schultz (1998) employed a quartz tuning fork as the piezoelectric sensor in combination with a magnetic field gradient of 5 kOe/cm (50 T/m), attaining a noise floor as low as 10⁻¹² emu, one of the most sensitive performances reported to date in quantitative magnetic characterization.

Currently, commercial AGMs, based on Flanders’ research (Flanders, 1988; 1990), utilize piezoelectric bimorphs as sensors, offering advantages such as high sensitivity, improved signal stability, and reduced noise interference, with a typical sensitivity of 10^–8 emu at a sampling rate of one point per second. The Composition is schematically depicted in Figure 2A, and additional detail of probe is shown in Figure 2B. The core sensing element of a typical AGM probe is a piezoelectric bimorph, which serves as a high-sensitivity vibration detector. The bimorph is mechanically coupled to the sample via two fused silica extensions, and the sample itself is mounted on a glass sample holder. The magnetic field system comprises a field-controlled electromagnet that provides a constant direct current (DC) magnetic field, along with gradient coils that generate an AC magnetic field. Under the excitation of the AC gradient field, the sample experiences a force at the same frequency as the alternating field, which causes the bimorph to generate a voltage signal proportional to the applied force. This signal is detected and extracted by a LIA. When a DC magnetic field is applied, the vibration amplitude of the sample is proportional to its magnetic moment. By measuring the vibration amplitude as a function of the applied DC field, a hysteresis loop can be obtained. For an AGM, the force on the sample in the non-uniform magnetic field is given by the following formula (Dodrill and Reichard, 2021): F x = m x ∂ H x ∂ x + m y ∂ H y ∂ x + m z ∂ H z ∂ x In general, a force in the X direction results from the gradient of the X component of the gradient field H X interacting with the X component of the magnetic moment in the sample. The first term in equation is this force. The magnetic moment m X can be estimated by measuring the magnitude of the field gradient and the alternating force. The second and third terms in equation should be decreased because they are undesirable. In normal situations, this is accomplished by lowering the field gradient.

The AGM has gained widespread recognition for its high sensitivity and precision in characterizing weak magnetic materials, making it a reliable tool for detecting subtle magnetic responses. For example, Şuan et al. (2020) utilized AGM to examine the impact of Al₂O₃ addition on Fe₃O₄ nanoparticles, demonstrating its capability to sensitively detect variations in saturation magnetization despite compositional changes. Similarly, Amir et al. (2023) used AGM to characterize superparamagnetic iron oxide nanoparticles (SPIONs) used in magnetic abrasives, extracting key parameters such as coercivity and saturation magnetization, and confirming their superparamagnetic behavior, which is critical for magnetically assisted polishing applications.

Through continuous optimization of mechanical sensing mechanisms, structural design, and gradient coil configurations, researchers have developed various AGM system variants tailored to different measurement demands. These advancements include integration with the MOKE (Hill et al., 1996), the development of vector magnetometers (Thomas et al., 2003), and specialized implementations for measuring reversible parallel and transverse susceptibilities (Barbic, 2004). Additional adaptations have enabled operation under cryogenic conditions (O’Grady et al., 1993) and the development of reed-type AGM systems for compact, high-sensitivity applications (Frey et al., 1988). Despite its high measurement precision and wide adaptability, AGM still faces several technical limitations in practice. Its sensitivity is highly susceptible to environmental disturbances, which may degrade performance by an order of magnitude or more. Common external interferences include acoustic noise, air flow fluctuations, ambient temperature variation, electronic noise, and mechanical vibrations from nearby equipment or human activity. In addition, the mechanical fragility of AGM probes, particularly the fused silica extensions, necessitates careful sample mounting to avoid damage. To address these challenges and enhance detection accuracy, Truth Instruments Co., Ltd., has recently introduced a novel variant known as the Laser Alternating Gradient Magnetometer (LAGM). This system employs laser Doppler vibrometry to capture the oscillatory motion of the sample induced by the alternating magnetic gradient. Leveraging the ultra-high displacement resolution of laser interferometric sensing, on the order of picometer, the LAGM achieves higher measurement precision than traditional AGMs, with magnetic moment noise levels as low as 10⁻⁹ emu. In addition, the LAGM features a decoupled design in which the vibration generation and detection subsystems are physically separated. This modularity facilitates integration with external systems, such as low-temperature cryostats or electrical transport measurement platforms, enabling the construction of a multifunctional physical field characterization system. Such versatility positions the LAGM as a promising tool for future high-precision magnetometer applications.

The Magneto-Optical Kerr Microscope is a specialized instrument renowned for its ability to provide real-time, high-resolution imaging of magnetic domain structures in a non-invasive and surface-sensitive manner. Its working principle is based on the magneto-optical Kerr effect, wherein the polarization state of reflected light is modulated by the magnetization of the sample. This effect was first observed by John Kerr in the 1870s (Kerr et al., 1877), and has since laid the foundation for a range of magneto-optical measurement techniques. Based on the magneto-optical Kerr effect, two main types of instruments have been developed: Kerr magnetometers for hysteresis loops measuring, and Kerr microscopes for magnetic domain structures imaging. Especially, the Kerr microscope stands out for its capability to visualize spatially resolved magnetic phenomena, making it particularly valuable in both fundamental magnetism studies and spintronic research.

The schematic sketch and the mechanism of the MOKE are shown in Figure 3A. By measuring slight changes in the polarization of a polarized laser beam upon reflection from the sample surface, the magnetic information of the sample can be obtained (McCord, 2015). We can use this phenomenological model to describe the detailed origin of this effect. First, the linearly polarized light can be seen as a superposition of left and right circularly polarized components. Upon interacting with a magnetized sample, two distinct effects occur: (i) the two polarized components will travel with different velocities and they emerge at the end of the media with different phase shifts, which will lead to the Kerr rotation θ k ; (ii) the absorption coefficients of the two components for the magnetic media is different and they emerge at the end of the media with different intensities, which will lead to the Kerr ellipticity ε k of the outgoing light (Ciprian et al., 2018). A further underlying reason for this phenomenological model is the Zeeman exchange splitting together with spin-orbit interaction, which is explained in detail in papers (Argyres, 1955; Bruno et al., 1996). Depending on the relative directions of magnetization m and plane of incidence of the light, MOKE can be categorized into three geometries: polar MOKE, longitudinal MOKE, and transverse MOKE. Among them, polar and longitudinal MOKE are the most commonly used configurations. Polar MOKE is sensitive to the out-of-plane magnetization, where the magnetization vector is oriented normal to the sample surface. Longitudinal MOKE, on the other hand, detects the in-plane magnetization component that lies within the plane of incidence. In longitudinal MOKE measurements with oblique incidence, unwanted polar MOKE contributions may also arise, potentially affecting the accuracy of in-plane magnetization detection. To minimize the influence of the polar component and isolate the longitudinal response, several suppression strategies have been developed. These include a dual-configuration approach that swaps the laser source and detector to extract angular-dependent components via addition and subtraction (Ding et al., 2000), a quadrant-detector-based method that separates signals through spatially resolved analysis (Celik et al., 2019), and a recently developed mirror and quarter-wave plate configuration that leverages rotational symmetry to cancel the polar response (Greening et al., 2025). In addition, by combining any two or all of them and using some specified techniques (Vavassori, 2000; Ding et al., 2001), i.e., generalized magneto-optical ellipsometry, MOKE can provide the vectorial magnetic information of the sample. In summary, it should also be noted that, unlike techniques directly measuring magnetic flux or moment, MOKE signals depend also on the material-specific magneto-optical coupling strength.

The basic setup for detecting magnetism using MOKE typically consists of a light source, a polarizer, an analyzer, and a detector (Qiu and Bader, 2000). Linearly polarized light is incident on the surface of a magnetic sample; the reflected light is then collected by the detector after passing through the analyzer. To enable spatially resolved magnetic domain imaging, the Kerr microscope integrates a microscope objective and a camera into the optical path, as shown in Figure 3B. Additionally, to observe different types of MOKEs (e.g., longitudinal, transverse, or polar Kerr effects), aperture or fiber optic entry method can be used to adjust the incident light direction. While the use of a camera enables real-time visualization of magnetic domains, the quantitative accuracy of Kerr rotation angle measurements is generally lower than that of conventional point-detection MOKE magnetometers. This is primarily due to illumination instability, camera noise, and limited detector sensitivity, which reduce the precision of angle-resolved Kerr signal extraction.

Owing to its high spatial resolution, surface sensitivity, and non-destructive nature, magneto-optical Kerr microscope has become a powerful characterization platform for the investigation of magnetic domain structures and their evolution (Domenichini et al., 2019). One of its fundamental applications lies in defect detection and quality assessment of magnetic thin films, where high-resolution Kerr imaging enables precise magnetization mapping to identify local inhomogeneities and pinning sites (Adam et al., 2009; Adam et al., 2010). Similarly, MOKE has been widely employed to visualize magnetic skyrmions, nanoscale magnetic quasiparticles with topological protection, critical for next-generation low-power spintronic devices such as racetrack memories (Kato et al., 2023). Beyond static imaging, magneto-optical Kerr microscope plays a crucial role in the real-time study of magnetic domain wall dynamics, offering insights into the underlying mechanisms governing domain nucleation and propagation. In advanced configurations, MOKE systems can be integrated with electrical probe stations to facilitate the observation of domain wall motion driven by spin-transfer torque (STT) and spin-orbit torque (SOT) effects (Ryu et al., 2013).

To extend conventional MOKE techniques toward ultrafast magnetization dynamics, time-resolved magneto-optical Kerr effect (TR-MOKE) systems employ a pump-probe configuration to achieve picosecond and even femtosecond temporal resolution (Neudert et al., 2005; Mozooni et al., 2014). In this technique, two temporally separated laser pulses are used: a high-energy pump pulse excites the sample or initiates a specific magnetic event, while a delayed probe pulse monitors the resulting changes. By using a high-precision displacement stage to systematically change the transmission distance of the probe pulse, the time delay between the pump and probe pulses can be gradually and precisely adjusted. This allows for the recording of time-resolved Kerr signals to reconstruct the dynamic evolution of the sample’s magnetization (Schäfer and McCord, 2021). This enables the time resolution step of TR-MOKE to be as short as sub-picoseconds. Gradually, TR-MOKE microscope has become the main ultrafast magnetic dynamic measurement method in the field of spintronics, especially it plays an extremely important role in the field of optical-magnetic coupling such as all-optical switching (AOS) (Lalieu et al., 2019; Peng et al., 2024; Peng et al., 2023a; b; c). Researchers have employed TR-MOKE measurements to verify the ultrafast AOS dynamics in materials such as GdFeCo alloys (Kirilyuk et al., 2010), TbFe (Hassdenteufel et al., 2013), and Co/Gd multilayers (Lalieu et al., 2017), as well as the unique phenomena observed in antiferromagnetic materials like Mn₂Au (Bhattacharjee et al., 2018) and IrMn (Guo et al., 2024) under femtosecond laser excitation. Subsequently, TR-MOKE has also been utilized to explore the integration of AOS with magnetic tunnel junctions (MTJs) (Wang et al., 2022), as well as laser-induced magnetization switching in ferromagnetic spin valves (Igarashi et al., 2023; Igarashi et al., 2024). These studies offer novel approaches for the optical-electrical-magnetic integration and pave the way for future information writing and storage technologies. TR-MOKE is one of the most representative applications of ultrafast time-resolved pump-probe techniques. Beyond that, the advantages brought by pump-probe methods—such as ultrafast temporal resolution, multidimensional information, and non-contact measurement—have also been applied in various fields, including transient absorption, lattice and structural dynamics such as time-resolved X-ray diffraction (TR-XRD) and time-resolved Raman spectroscopy (TR-Raman), time-resolved ferromagnetic resonance (TR-FMR), and time-domain thermoreflectance for studying thermal properties.

MOKE technology has seen continuous advancements in recent years, driven by its exceptional sensitivity and scalability. Although quantitative characterization remains technically challenging, some researchers have proposed leveraging magnetic domain displacement as a potential solution (Hrabec et al., 2014; Magni et al., 2022). Moreover, in magnetic chip fabrication, wafer-level MOKE inspection has become the most prevalent method for quality control. Given these developments, MOKE-based techniques are expected to play an increasingly important role in both future research and industrial production.

As another pivotal magnetic imaging technique, MFM has been widely adopted in both materials science research and industrial applications, owing to its ability to achieve nanoscale spatial resolution in the characterization of magnetic structures (Hartmann, 1999). It operates by detecting the magnetic force gradient between a magnetized probe tip and the sample surface, allowing for indirect imaging of magnetic field distributions with high spatial precision. Since its introduction, MFM has undergone continuous technological advancements and methodological refinements, establishing itself as a powerful tool for the high-resolution investigation of localized magnetic phenomena, including domain structures, magnetic coupling, and stray field distributions.

As illustrated in Figure 4A, MFM mainly consists of four key components: a cantilever, a ferromagnetic tip, a detection system, and a scanning control unit. The cantilever is usually made of silicon or silicon nitride, with a ferromagnetic tip attached to its end. The magnetic tips are typically fabricated from polycrystalline metal wires, such as nickel, iron, or cobalt. The detection system is responsible for measuring the subtle deflections of the cantilever induced by magnetic interactions between the tip and the sample. Common detection approaches include optical beam deflection methods (Erlandsson et al., 1988; Rugar et al., 1988), capacitive sensors (Göddenhenrich et al., 1988), and in some high-sensitivity implementations, differential optical interferometry (Schönenberger and Alvarado, 1989). The scanning control unit precisely regulates the relative position between the sample and the tip, enabling high-resolution, point-by-point surface scanning and mapping of the magnetic force distribution at the nanoscale.

MFM operates based on the dipolar interaction between the ferromagnetic tip and the magnetic stray field of the sample surface. This interaction enables MFM to acquire both magnetic and topographical information of the sample simultaneously. As the ferromagnetic tip approaches the sample surface, the stray magnetic field of the sample exerts a localized force on the tip, leading to a measurable deflection of the cantilever. Highly sensitive detection systems are employed to accurately measure the slight displacements of the cantilever. These deflections provide quantitative information about the magnetic force distribution, thereby reflecting the underlying magnetization of the sample surface. The detected signal, typically proportional to the magnetic force or its spatial derivative, is integrated into a servo feedback loop that accurately regulates the tip-sample distance. During the scanning process, this distance is continuously monitored and recorded. As a result, an image representing contours of constant interaction force or force gradient is constructed, revealing critical magnetic features such as the size, shape, and orientation of magnetic domains. In addition to magnetic imaging, MFM can also operate in specific modes to acquire topographical information. For example, in the electrostatic control method, a controlled Coulomb force is applied between the tip and the sample in addition to the magnetic interaction force. By precisely adjusting the magnitude of this electrostatic force, the tip-sample distance is regulated. While keeping the total interaction force constant, the vertical displacement of the tip during scanning is recorded. This displacement data corresponds to the topographical variations of the sample surface, thus enabling topographical imaging. Figures 4B,C give typical domain images measured by MFM.

The diameter and geometry of the MFM probe tip have a critical impact on the instrument’s spatial resolution. A larger tip radius increases the magnetic interaction volume, leading to signal averaging and a diminished ability to resolve fine-scale magnetic features. To address this limitation, several tip miniaturization techniques have been developed. Among them, electron beam deposition (EBD) (Jumpertz et al., 1997), focused ion beam (FIB) milling (Vasile et al., 1991), and the attachment of multi-walled carbon nanotubes (MWNTs) to microfabricated silicon cantilevers (Dai et al., 1996) have shown the most promising results. In addition, researchers have explored strategies to reduce magnetic coating coverage, such as selectively coating only the apex of the tip or thinning the magnetic layer, with the aim of enhancing magnetic sensitivity and spatial resolution. Beyond geometric and coating modifications, further advancements have led to the development of specialized probe types, including double-exchange tips (Han et al., 2007), antiferromagnetic tips (Liu et al., 2002), and synthetic antiferromagnetic tips (Wu et al., 2003). Measurement results indicate that these advanced probes can significantly improve the accuracy and fidelity of magnetic force measurements (Schwarz and Wiesendanger, 2008).

MFM leverages its sensitivity to local magnetic interactions to enable direct imaging of ferromagnetic domain structures, offering valuable insights into both hard and soft magnetic materials. It can resolve individual domain boundaries and reveal the fine internal features of magnetic domains, which are essential for understanding intrinsic magnetic properties (Göddenhenrich et al., 1988). In industrial applications, particularly within the magnetic recording industry, MFM serves as a key diagnostic tool (Martin et al., 1987). It is widely used to analyze the stray magnetic fields generated by recording heads, providing critical information for assessing their performance and structural integrity. Irregularities in stray-field distribution, for instance, may indicate mechanical defects or signal functional degradation (Van Schendel et al., 2001). Furthermore, MFM is applied to investigate written domains in both magneto-optical storage films and longitudinal magnetic recording media, thereby facilitating process optimization and contributing to higher data storage density and improved system reliability (Schoenenberger et al., 1991; Porthun et al., 1995). In recent years, MFM has served as a powerful tool for investigating a wide variety of magnetic structures. MFM has been widely applied to the study of patterned magnetic media (Rodríguez et al., 2016), artificial spin ice (Wang et al., 2016), domain walls (McCord, 2015), nanowires (Wohlhüter et al., 2015), multiferroic structures (Henrichs et al., 2016), and topological magnetic textures such as skyrmions (Pham et al., 2024). In particular, researchers have used MFM to demonstrate various spin current-driven operations (Pham et al., 2024; Mallick et al., 2024) of skyrmions, including their nucleation (Sun et al., 2023), motion (Pham et al., 2024), and annihilation (Mishra et al., 2025), which has significantly facilitated the development of skyrmion-based magnetic tunnel junctions (Chen et al., 2024) and racetrack memory devices (Wang et al., 2020). Meanwhile, quantitative MFM has also continuously evolved over the past 2 decades, employing a variety of approximation methods to quantitatively describe the two-dimensional stray field distribution of magnetic probes. These methods include parameter-free tip transfer function (TTF) approaches (Vock et al., 2011), the use of nitrogen-vacancy (NV) centers in diamond to measure the stray field and derive the cantilever calibration function (Sakar et al., 2021), and the approximation of the tip using a tip-equivalent magnetic charge model to decouple the stray fields of the sample and the probe (Feng et al., 2022a), among others. The two-dimensional stray field distribution of the sample obtained via quantitative MFM can reveal high-resolution, quantitative magnetic information such as the domain structure and the distribution of exchange bias fields (Zingsem et al., 2017). In addition, Monte Carlo (MC) methods have been introduced to analyze uncertainty propagation in quantitative MFM, providing a statistical framework for evaluating the reliability of measured magnetic parameters (Marschall et al., 2022). Moreover, an implementation that integrates specially engineered low-stiffness cantilevers, high-aspect-ratio magnetic nanowire tips, and multi-mode vibration excitation and control has demonstrated significant advantages in studying complex magnetic samples (Freitag et al., 2023). In the future, it is expected to become an important tool for high-resolution magnetic measurements.

MFM offers a series of significant advantages that make it a powerful tool in the study of magnetic materials. It exhibits high environmental adaptability, capable of operating under diverse conditions with minimal sample-preparation needs, which simplifies experimental workflows across various fields. Moreover, its ability to concurrently acquire topographical and magnetic stray-field information provides a comprehensive view of the sample in a single scan, enhancing research efficiency. However, MFM also faces several drawbacks. Its resolution and performance are limited by the cantilever’s sensitivity, which may be insufficient to detect subtle forces. In addition, environmental noise, such as temperature fluctuations and mechanical vibrations, and tip-induced stray fields can further interfere with measurements, especially for soft magnetic samples. These limitations collectively hinder the high-precision measurement capabilities of MFM and must be carefully addressed. With ongoing technological advancements, MFM continues to evolve toward higher precision and broader applicability, including capabilities such as quantitative magnetic moment analysis, image reconstruction and denoising, instrumental integration with complementary techniques, and exploration of emerging application fields (Kazakova et al., 2019; Feng et al., 2022b).

Among various magnetic measurement techniques, SQUID magnetometer stands out as the most sensitive and precise instrument currently available. Utilizing the principles of superconductivity and the Josephson effect, SQUID can measure the magnetic moment with an accuracy of 10⁻⁸ emu in a strong magnetic field (7 T and above) and ultra-low temperature (1.8 K) environment (Schmelz et al., 2016). Due to its reliance on a superconducting state, SQUID magnetometer operation necessitates cryogenic environments, typically maintained by liquid helium or closed-cycle refrigeration systems, which leads to relatively high operational and maintenance costs. Nevertheless, SQUID remains unparalleled in its ability to perform ultra-high-precision magnetic moment measurements across a wide variety of materials and under complex experimental conditions (Weinstock, 2002), making it a critical tool in the study of low-dimensional quantum systems, superconductors, and spintronic materials.

The measurement structure and operating principle of the SQUID magnetometer are illustrated in Figure 5A. Each component in the system is carefully designed to serve the goal of ultra-high-precision magnetic moment detection. For example, the magnetic signal from the sample is first sensed by a pickup coil, designed according to Faraday’s law of electromagnetic induction: as the sample moves relative to the coil, a time-varying magnetic flux induces a current that reflects the sample’s magnetic moment. To further enhance sensitivity and suppress environmental noise, the system employs a balanced second-order gradiometer configuration in the coil design. In this design, both the upper and lower coils are wound clockwise, while the central coil is wound counterclockwise with twice the number of turns. This symmetric winding geometry effectively cancels out uniform magnetic field fluctuations and external magnetic interference, thereby significantly improving the signal-to-noise ratio of the system. Similar to the inductive coil design in VSM, the physical dimensions of the pickup coil in SQUID systems must also be optimized to balance sensitivity and geometric accuracy. On one hand, the gradiometer should ideally be small relative to the sample, so that magnetic flux lines do not return the pickup loop, maximizing magnetic coupling efficiency. On the other hand, to minimize artifacts caused by sample geometry and size variation, the detection coils are often made larger than the sample itself. This trade-off requires precise engineering, as improper coil-sample matching may introduce systematic errors in the measured moment, particularly if the sample deviates from the geometry used during system calibration (Kirtley et al., 1995; Stamenov and Coey, 2006).

In a closed superconducting circuit, the total magnetic flux enclosed by the circuit remains constant, a fundamental property governed by flux quantization. This means that any magnetic flux variation caused by the motion of a magnetic sample within the pickup coil must be compensated by an equivalent response elsewhere in the circuit. Through the design of a flux transformer, this flux variation can be coupled into the detection circuit, enabling indirect but highly precise measurement of the sample’s magnetic moment. The SQUID is currently the most sensitive technology for magnetic flux detection. It can convert extremely small changes in magnetic flux into measurable voltage signals, and its core components are one or more Josephson junctions embedded in a superconducting loop. A Josephson junction is a classic “sandwich” structure consisting of two superconducting electrodes separated by an ultrathin insulating barrier, The behavior of the Josephson junction is governed by the Josephson effect, which manifests when the barrier thickness is less than the coherence length of the Cooper pairs in the superconductor. As illustrated in Figure 5B, when the current flowing through the junction is below the critical current I C , the junction remains in the superconducting state, and no voltage drop is observed across it. Above the threshold current, the junction transitions into a resistive state and exhibits Ohmic behavior. The transition region near I = I C is particularly significant: small variations in current within this region result in large voltage responses. Based on the steepness of the slope in the transition region, the SQUID enables ultra-sensitive magnetic moment measurements (Fagaly, 2006).

Another key reason for the extremely high measurement accuracy of SQUID is the quantization characteristic when it measures magnetic flux. When the SQUID senses an external magnetic flux change Φ a , it induces a compensating magnetic flux Φ i n d such that the total enclosed flux in the loop satisfies the condition n Φ 0 = Φ a + Φ i n d , where Φ 0 is the magnetic flux quantum and n is an integer. As shown in Figure 5C, the induced current changes as a periodic function with the external magnetic flux (Fagaly, 2006). To obtain an absolute and linear measurement of the flux value, an additional structure, known as a flux-locked loop (FLL), is introduced, as shown in Figure 5A. This configuration includes an external feedback voltage source that couples into the pickup coil and generates a compensating flux to actively cancel the flux change produced by the sample (Dumas and Hogan, 2021). The final output of the measurement is obtained by summing the feedback signal applied by the external voltage source and the residual high-precision signal extracted from the Josephson junction response, thus achieving both linearity and ultra-high sensitivity.

Depending on how the sample interacts with the pickup coil, SQUID magnetometers support two primary modes for magnetic moment measurement: the traditional DC scan and the SQUID-VSM mode. The key distinction between DC-SQUID and SQUID-VSM lies in their measurement modes: DC-SQUID operates in a static (DC) scanning mode, while SQUID-VSM employs a dynamic (AC) vibrating mode. This fundamental difference leads to diverging characteristics in terms of measurement targets, spatial resolution, sensitivity, and application focus. The traditional DC scan drives the sample completely through the second-order gradiometer. By fitting the recorded magnetic signal with position, the magnetic moment of the sample can be obtained. For example, Wu et al. (2021) utilized the DC measurement mode of SQUID to characterize the magnetic response of a superconducting sample, confirming its superconducting properties and magnetic shielding effect. In contrast, the SQUID-VSM mode keeps the sample oscillating sinusoidally at a fixed frequency near the center of the gradiometer, rather than translating it through the entire detection region. The time-varying magnetic response induces an AC voltage, which is measured using LIA, and the magnetic moment is extracted from the amplitude of the AC signal at the drive frequency (Hurt et al., 2013). For example, Adanlété Adjanoh et al. (2024) employed the SQUID-VSM mode to measure quasistatic magnetic parameters of Ta/Pt/Co/Pt ultrathin films, revealing that annealing significantly reduces coercivity and enhances magnetic domain structure. Due to the differences in underlying principles and measurement procedures, the two SQUID modes are suited to different types of samples and experimental needs. The DC scan mode determines the magnetic moment by fitting the spatial variation of the output voltage to a theoretical response function (Bouchiat, 2009), making it well suited for large or geometrically irregular samples, where positional correction through fitting improves accuracy. In contrast, the SQUID-VSM mode measures the amplitude of the induced AC voltage and is therefore more appropriate for high-precision measurements of small, geometrically well-defined samples, where stable harmonic motion ensures consistent signal generation.

Although SQUID magnetometer is renowned for its exceptional sensitivity and precision, it presents several practical challenges that must be carefully addressed during operation. The need for cryogenic temperatures increases both system complexity and operational costs, while also limiting measurement throughput and accessibility. Additionally, measurement accuracy can be compromised by flux creep, magnet hysteresis, and instrumental drift, necessitating systematic calibration and stabilization (Sawicki et al., 2011; Buchner et al., 2018). Furthermore, SQUID magnetometer is generally not suitable for large-volume or strongly magnetic samples, as these may exceed the dynamic range of the pickup coils or introduce nonlinearities and flux instabilities in the detection process. Nevertheless, due to its unparalleled capability for detecting ultra-weak magnetic signals and enabling absolute magnetic moment quantification, SQUID magnetometry remains one of the most widely adopted techniques in magnetic characterization. Ongoing advances such as cryogen-free cooling technologies, miniaturized SQUID-on-chip systems, and enhanced real-time feedback electronics continue to improve its usability and reliability, thereby expanding its applications in both fundamental studies and applied magnetism research.

AC susceptometry is a fundamental technique in magnetism research and plays a central role in characterizing the dynamic magnetic responses of materials (Hartshorn, 1925). By quantifying the frequency-dependent magnetic susceptibility under alternating electromagnetic fields, this method provides crucial insights into phenomena such as superconducting phase transitions (Bałanda, 2013), magnetic relaxation processes (Mydosh, 1993), and energy dissipation mechanisms (Gatteschi et al., 2006). As a result, it offers a powerful tool for bridging the gap between a material’s microscopic magnetic dynamics and its macroscopic functional properties.

The theoretical foundation of AC susceptometry lies in the response of materials to alternating fields, described by the complex magnetic susceptibility χ ω χ ω = χ ′ ω + i χ ″ ω Here, real part χ ′ ω represents the in-phase magnetization component, while imaginary part χ ″ ω quantifies out-of-phase energy dissipation arising from hysteresis, eddy currents, or spin-relaxation processes. This formulation assumes the weak-field approximation, under which the system exhibits harmonic responses confined to the fundamental frequency ω . A central framework for interpreting AC susceptibility is the Debye relaxation model (Topping and Blundell, 2018), which incorporates a damped dynamic equation to relate magnetization response to the characteristic relaxation time. This model reveals the frequency-dependent behavior of susceptibility (as shown in Figure 6A): at low frequencies, the magnetic response is fully developed; at high frequencies, the response is constrained by damping; and an energy dissipation peak emerges at intermediate frequencies, allowing precise interpretation of magnetic relaxation mechanisms in materials.

The mutual inductance method is a prevalent technique for AC susceptibility measurement, ideally suited for low-frequency regimes where magnetic relaxation and dynamic responses dominate, offering a simple yet robust solution for fundamental material characterization (as shown in Figure 6B). The system operates through a drive coil connected to an AC current source, generating a stable alternating magnetic field whose frequency and amplitude stability directly impact measurement precision. A sense coil array, typically configured with dual reverse-wound coils around the sample, detects magnetization-induced flux changes, converting magnetic responses into voltage signals and rejecting environmental noise. A lock-in amplifier then processes these signals using the drive coil current as a reference, applying phase-sensitive detection to extract in-phase and out-of-phase components from the fundamental frequency ω (Edgar and Quilty, 1993; Youssif et al., 2000). In practical measurements, the demagnetization effect must be considered. The internal magnetization of a material generates a demagnetizing field H d opposing the external field H e x , resulting in the internal field: H = H e x + H d . This leads to a discrepancy between the measured extrinsic susceptibility ( χ e x t ) and the intrinsic susceptibility ( χ ) of the material:   χ e x t = χ 1 + N χ . Here, N denotes the demagnetization factor, determined by the sample geometry (Blundell, 2001).

AC susceptometry is widely employed to investigate the temperature- and field-dependent magnetic relaxation behavior of materials. In systems such as paramagnetic salts, it enables analysis of the spin relaxation process by applying the Debye relaxation model, which relates susceptibility responses to frequency under both isothermal and adiabatic conditions. By modifying this model to fit experimental data, researchers can elucidate the energy exchange mechanisms between spin systems and the lattice, thereby uncovering the underlying magnetic relaxation dynamics (Casimir and Du Pré, 1938). Furthermore, AC susceptibility has proven to be a powerful tool for probing superconducting phase transitions. For instance, by analyzing the temperature dependence of both complex and wide-band AC susceptibility under various flux pinning regimes, researchers have successfully extracted key physical parameters of high-T_C superconductors, including the critical temperature, magnetic penetration depth, and pinning potential, thereby providing deeper insights into the nature and dynamics of the superconducting transition (Gömöry, 1997). In recent years, AC susceptometry has been increasingly applied to study dynamic magnetic properties in advanced systems such as magnetic nanoparticles, molecular magnets, and low-dimensional heterostructures (Ghigo et al., 2022; Borgohain and Borah, 2021; Fernández-García et al., 2022). With sensitivity to both reversible and dissipative magnetic responses, it serves as a powerful tool for investigating spin dynamics and magnetic relaxation behavior. Notably, the development of high-frequency AC susceptometry has enabled time-resolved analysis of fast magnetic processes, including spin relaxation (Lah et al., 2020), quantum tunneling (Riordan et al., 2019), and high-speed switching relevant to spintronic devices (Cafolla-Ward, 2024). These advancements open new pathways for characterizing materials with short relaxation times and frequency-dependent losses.

However, this technology faces several challenges: during measurements, Joule heating and eddy current heating easily induce sample temperature drift, and existing temperature control measures show limited effectiveness under extreme conditions (e.g., high frequencies). Additionally, inductive and capacitive coupling noise in the environment interferes with detection signals, making interference suppression extremely difficult in complex electromagnetic environments. Moreover, the absence of unified calibration standards reduces the comparability of results from different devices, while the diversity of standard samples also requires enhancement. In the future, AC susceptibility technology may further benefit from integration with multimodal magnetic characterization methods (e.g., Magneto-Optic Kerr Effect, X-ray Magnetic Circular Dichroism), enabling more comprehensive and element-specific analysis.

FMR is a widely used technique for probing the high-frequency magnetization dynamics of magnetic materials. In contrast to AC susceptometer, which characterizes low-frequency magnetic relaxation, FMR-based system investigates the resonant precession of magnetic moments induced by a combination of a static magnetic field and a microwave-frequency oscillating field. When the excitation frequency matches the natural precession frequency of the magnetization vector, resonance occurs, enabling quantitative investigation of properties such as magnetic anisotropy, damping constants, and gyromagnetic ratios. This makes FMR a vital tool in the study of spintronic materials, multilayer heterostructures, and magnetization relaxation phenomena under GHz-range excitations. The early exploration of FMR began in 1911, when V. K. Arkad’yev first observed microwave absorption phenomena in nickel and iron wires exposed to centimeter-wave radiation (Vonsovskii, 2013). In 1923, J. Dorfmann contributed to the theoretical interpretation of these observations and proposed experimental tests to investigate the influence of strong external magnetic fields on ferromagnetic resonance behavior (Dorfmann, 1923). The experimental realization of FMR advanced significantly in 1946, with pioneering work by James Griffiths (Griffiths, 1946), who is widely credited with its discovery. Soon after, Charles Kittel (Kittel, 1948) provided a theoretical explanation for Griffiths’ results, laying the groundwork for the modern interpretation of FMR and its application across a wide range of magnetic systems.

To explain the fundamental principle of FMR, we need to start with a discussion on magnetic dynamics. Within the framework of the macrospin approximation, the time evolution of the magnetization vector ( m ) of a ferromagnet in the presence of an effective magnetic field ( H e f f ) can be effectively described by the Landau-Lifshitz-Gilbert (LLG) equation (Landau and Lifshitz, 1935), ∂ m ∂ t = ‐ γ μ 0 m × H e f f + α m × ∂ m ∂ t where γ , μ 0 , and α are the gyromagnetic ration, vacuum permeability and the Gilbert damping parameterm respectively. The LLG equation includes two torque terms on the right-hand side. The first term, ‐ γ μ 0 m × H e f f , describes precession and involves the rotation of both the magnetization vector ( m ) and the effective magnetic field ( H e f f ). The second term, α m × ∂ m ∂ t , represents damping terms which aligns the magnetization along H e f f . The value of α determines the damping rate, with larger values causing faster alignment and smaller values leading to slower alignment of m with H e f f . The amplitude of the precessions gradually decreases due to damping in the ferromagnet. FMR occurs when an external alternating magnetic field ( H R F ) is applied to the ferromagnet, compensating for the energy dissipation caused by damping. This results in a forced precession of m . By tuning the frequency of H R F to match the natural precession frequency of m , the amplitude of the forced precession reaches its maximum due to resonance. This phenomenon is known as FMR, and the frequency of precession during resonance is referred to as the resonance frequency f ( f = ω r e s / 2 π ). Kittel’s relation establishes a connection between the resonant field H 0 and the resonance frequency f, f = γ 2 π H 0 H 0 + 4 π M e f f where 4 π M e f f is the effective demagnetization field of the ferromagnet. According to the equation, the resonance condition can be matched by fixing H 0 and tuning the frequency of H R F or vice versa.

Early FMR measurements were primarily performed using resonant microwave cavities (Patton and Kohane, 1972), shown in Figure 7A, where the resonance condition was achieved by sweeping a static magnetic field while monitoring microwave absorption within a fixed-frequency cavity. These cavity-based FMR setups offer high sensitivity due to their large quality factors, but their applicability is limited to discrete frequencies, requiring cavity replacement for frequency variation. To overcome this limitation, broadband FMR techniques were developed based on planar transmission lines (as shown in Figure 7B), such as coplanar waveguides (CPW) (Lee et al., 2009; Schäfer et al., 2012) and microstrip lines (Kalarickal et al., 2006; Kennewell et al., 2010), enabling continuous frequency coverage across a wide range (up to 65 GHz) (Wu et al., 2019; Khodadadi et al., 2020; Srivastava et al., 2020). These methods offer enhanced versatility, though they introduce challenges such as impedance matching, field nonuniformity, and increased microwave loss.

The Vector Network Analyzer (VNA)-FMR measurement is the most commonly used method for conducting broadband FMR experiments (Neudecker et al., 2006; Godsell et al., 2010). This method offers a simple experimental setup because the signal is generated and measured by the VNA in the same device. Figure 7C shows the configuration of the VNA-FMR experiment. The VNA typically operates in frequency-swept mode, while field-swept VNA-FMR measurements have also been reported in (Sharma and Kuanr et al., 2018; Tamaru et al., 2018). A key advantage of VNA-FMR lies in its ability to extract both the real and imaginary components of the complex susceptibility, enabling comprehensive magnetic characterization. However, careful full two-port calibration of the VNA system is essential to ensure measurement accuracy. As an alternative, broadband FMR can also be performed without a VNA using a modular scheme known as Phase-FMR (Montoya et al., 2014), illustrated in Figure 7D. This technique relies on direct power reflection measurements that are sensitive to frequency-dependent impedance changes. Therefore, these experiments typically require field sweeping at a fixed frequency to resolve resonance features accurately. Additionally, electrical detection of FMR has emerged as a sensitive method for probing resonance in magnetic structures (Tsoi et al., 2000; Kiselev et al., 2003). It relies on the generation of DC voltage at resonance, driven by effects such as spin-transfer torque and spin pumping (Sankey et al., 2006). Signals are acquired using voltmeters, with lock-in detection improving the signal-to-noise ratio. Owing to its high sensitivity, this method is well suited for studying nanoscale systems, including individual nanomagnets, and shows promise for integration into spintronic applications (Tulapurkar et al., 2005).

Ferromagnetic resonance (FMR) plays a central role in magnetic materials research by enabling the precise determination of key magnetic parameters. It allows accurate extraction of saturation magnetization via resonance conditions (Beik Mohammadi et al., 2019), and in-plane angular-dependent FMR measurements are widely used to analyze magnetic anisotropy, particularly in exchange-biased systems (Beik Mohammadi et al., 2017). In addition, broadband FMR techniques are essential for quantifying Gilbert damping constants, providing a powerful means to validate theoretical models such as Kamberský’s torque correlation model, and to trace the microscopic origins of damping, including mechanisms governed by spin-orbit coupling. In recent years, FMR has also been extended to the in situ characterization of spin-orbit torque (SOT) devices, offering valuable insights into current-induced magnetization dynamics (Coester et al., 2023; Zhan et al., 2024). In such measurements, a radio-frequency current is applied to the device to excite magnetization precession via spin-orbit torques, and the resulting voltage signal—often referred to as the rectification voltage—is analyzed to extract key parameters (Kim et al., 2021; Chen et al., 2021). Specifically, by analyzing variations in resonance lineshape and linewidth induced by the applied current and magnetic field, researchers can determine the strength and angular dependence of spin-orbit torques, as well as quantify spin Hall efficiencies in multilayer structures (Hibino et al., 2024). This approach, known as spin-torque FMR (ST-FMR), has become a critical technique for optimizing spintronic device performance and for evaluating materials with strong spin–orbit interactions in magnetic random access memory (MRAM) and related applications (Chen et al., 2021). It has also been increasingly employed to investigate magnetization dynamics in two-dimensional magnetic materials, enabling the study of spin relaxation, damping, and anisotropy at the atomic thickness limit (Tang et al., 2023).

Furthermore, researchers have extended FMR by integrating it with complementary techniques to enhance spatial, spectral, and element-specific resolution. One such novel technique is X-ray detected FMR (XFMR) (Van der Laan and Hesjedal, 2023), which makes use of the XMCD effect to produce element-specific magnetic contrast in order to examine the dynamics of magnetization in individual layers. Variants such as diffraction-based FMR (DFMR) and reflectivity-based FMR (RFMR) extend XFMR by enabling the detection of magnetization dynamics through modulations in X-ray scattering and reflectivity signals, thereby enhancing structural and interfacial sensitivity (Burn et al., 2021). In addition, alternative non-microwave-based detection approaches have emerged, including the use of a MFM tip (Nazaretski et al., 2007a; Nazaretski et al., 2007b) and micro-focus Brillouin light scattering (BLS) techniques (Demidov et al., 2010; Demidov et al., 2011), both offering high spatial resolution and local FMR sensitivity. These hybrid and indirect detection methods broaden the experimental versatility of FMR, enabling investigation of complex magnetic phenomena in nanoscale systems.