Micrometer-sized magnetic particles have been widely used in magnetic force microscopy, magnetic resonance force microscopy, and bio-sensing. To quantitatively interpret the data obtained with magnetic particles, it is important to know the magnetic properties of the particles. However, the magnetic moment of individual particle is usually too small to be measured by common instruments for samples with large volume. Here, we present a method to characterize magnetic microspheres using patterned FePt thin films as standard samples. The FePt thin film in the L 1 0 phase has perpendicular magnetic anisotropy, and the patterned features can be magnetized to near single-domain magnets, which make them suitable standards for magnetic sphere calibration with magnetic force microscopy. Multiple linear regression is used to analyze the frequency shift images and obtain the effective dipole moment of the spheres. The position of the dipole moment is obtained by minimizing the residuals in multiple linear regression with a gradient descent algorithm. Three NdFeB spheres of different diameters were measured. It was found that the magnetization increases with the increase in the diameter of the sphere, possibly due to the weakening of ferromagnetism on the surface.

Magnetic particles have been used as magnetic tweezers1 as well as probes in magnetic force microscopy (MFM)2 and magnetic resonance force microscopy.3–6 They have also been widely used in biomedical applications, such as biological sensing,7,8 drug delivery,9 and agents for magnetic imaging.10 Among those applications, it is critical to know the magnetic properties of the particles, especially in the application of quantitative measurements. However, the magnetic moment of individual particle is very tiny; for example, the magnetic moment of a ferromagnetic particle with a diameter of 1  μm is on the order of 10 13 A m 2. This is far below the sensitivity of commonly used magnetic measuring instruments, such as superconducting quantum interference device (SQUID) magnetometry and vibrating sample magnetometry.

Many methods have been developed for magnetic measurement of small samples.11 The magnetic moment of small samples can be measured with cantilever magnetometry,12–18 where the torque exerting on the sample in a uniform magnetic field is measured with a sensitive cantilever. A sensitivity better than 10 4 μ B was achieved with a field of 60 kOe.15 Instead of applying a uniform magnetic field, the alternating-gradient magnetometer measures the dynamic force on a vibrating magnetic sample in an alternating gradient field, and a sensitivity of 10 15 A m 2/ Hz is achievable.19,20 A novel magnetometer using a current flowing force sensor was also developed to measure the z-component of the micro-particles’ magnetic moment.21 

The calibration of MFM probes has been conducted using well-defined current carrying wire loops, which were fabricated with lithography.22–29 The current-driven wires create inhomogeneous stray fields, which can be calculated precisely based on the geometry and the applied current. The force on the probes is proportional to the magnetic field gradient; thus, the magnetic field can be imaged by MFM. With this method, the magnetic properties of the probe can be characterized as a function of an external magnetic field.23 The MFM probes can also be calibrated with standard samples with well-known magnetization, such as dispersed magnetic nanoparticles on silicon substrate,30 patterned exchanged biased magnetic layer stripes,31 array of rectangular ferromagnetic thin film elements,32 and single-domain uniformly magnetized CoPt dots.33 Among them, CoPt dots have 100% remnant magnetization and large coercivity so that their magnetization is robust and can be well characterized using SQUID magnetometry.

In this paper, we present a method to characterize the effective magnetic moment of ferromagnetic microspheres using specially designed calibration samples. We intend to develop a method that can be used in an atomic force microscope (AFM) without the need for additional modules. In our application, the magnetic microsphere is attached on the end of a cantilever and used as a probe to search for new interactions beyond the standard model of particle physics.34 The calibration sample is made of an array of micrometer-sized FePt squares, which is fabricated with photolithography. The FePt thin film is in its L 1 0 phase and exhibits perpendicular magnetic anisotropy so that it can be magnetized in a direction normal to the sample surface. We analyze the MFM image by multiple linear regression to obtain the magnetic moment of the sphere and obtain the position of the effective magnetic dipole by the gradient descent algorithm.

Here, we measure the magnetic interaction through the frequency shift ( Δ f) of the cantilever’s resonance in the frequency modulation AFM mode (see Fig. 1). The frequency shift Δ f is related to the force gradient F / z by
Δ f = f 0 2 k F z ,
(1)
where k is the cantilever’s spring constant and f 0 is its resonant frequency. Assuming that the magnetic sphere is represented effectively by a magnetic dipole moment m at position r m = ( x m , y m , z m ), the magnetic potential energy between the probe and the sample can be written as U = m B , where B is given by the integral
B ( r m ) = μ 0 4 π 3 r ^ ( r ^ M ) M r 3 d V ,
(2)
where μ 0 is the vacuum magnetic permeability, M is the magnetization of the standard sample, r is the displacement vector between the dipole moment m and the volume element d V of the sample, and r ^ = r / r is the unit displacement vector. The magnetic force gradient is then given by
F z = 2 B x z m 2 m x + 2 B y z m 2 m y + 2 B z z m 2 m z ,
(3)
which is a linear combination of the components of the dipole moment. Therefore, the frequency shift can be written as
Δ f = G x m x + G y m y + G z m z + Δ f 0 ,
(4)
where G l = ( f 0 / 2 k ) ( 2 B l / z m 2 ), with l = x , y , z. Δ f 0 being the frequency shift of non-magnetic origins, such as the Casimir force, electrostatic force, and measurement offset. According to Eq. (4), the dipole moment can be obtained by analyzing the Δ f image with multiple linear regression.
FIG. 1.

Schematic diagram of the experiment.

FIG. 1.

Schematic diagram of the experiment.

Close modal

In order to minimize the contribution of the electrostatic force and Casimir force, the probe-sample distance is set to greater than 200 nm, where the magnitude of the electrostatic force and Casimir force is negligible compared to the magnetic force. To set the distance, we first approach the probe to the surface area without magnetic structures with the Z feedback on. The setpoint of Δ f used for the tip-approach defines the initial spacing d 0. The probe is then lifted up for certain height Δ d for imaging.

The initial spacing d 0 is determined by the electrostatic force calibration method. When an voltage V is applied between the sphere and the sample, the magnetic sphere is subjected to an electrostatic force given by35 
F e ( d ) = 2 π ε 0 ( V V 0 ) 2 n = 1 coth ( α ) n coth ( n α ) sinh ( n α ) ,
(5)
where V 0 is the contact potential difference between the sphere and the sample. cosh ( α ) = 1 + d / d R R, where R is the radius of the sphere and d = d 0 + Δ d is the separation between the surfaces. The gradient of the electrostatic force causes a change in the cantilever resonance frequency, which can be expressed as
Δ f ( V , d ) = B ( d ) ( V V 0 ) 2 + Δ f n e ( d ) ,
(6)
where Δ f n e is from other contributions unrelated to electrostatic force and independent of V.

The frequency shift is measured by sweeping the voltage V at different lift height Δ d from d 0. The B ( Δ d ) at every Δ d is obtained by fitting the Δ f V curve to Eq. (6). The initial spacing d 0 is then obtained by fitting the B ( Δ d ) Δ d curve to the theoretical formula derived from Eq. (5).

To quantitatively measure its magnetic moment, the microsphere is first glued on the end of a tip-less cantilever as a probe. The surface tilt of the standard sample is then obtained by imaging the area without magnetic structures using the probe. In this area, we also obtain the initial spacing d 0 using the electrostatic calibration procedure described above. Afterward, the probe is lifted up from the initial spacing for a certain distance and then scans over the magnetic structures with Δ f recorded simultaneously. To keep the probe-sample distance constant, the surface tilt is compensated during scanning. A typical image is shown in Fig. 4(b), where nearly half of the image is free of magnetic structures. These areas are used for measuring d 0 and the surface tilt.

A suitable standard sample is critical for a reliable calibration of the magnetic moment of magnetic spheres. The standard sample should have well-known magnetization so that the stray field can be accurately calculated. Its magnetization should be less susceptible to stray field generated by the magnetic spheres. To fulfill these requirements, we chose FePt films as the base material for making standard samples. The FePt films with a thickness of 10 nm were deposited on the MgO(001) substrate by magnetron sputtering. The FePt films are in the L 1 0 phase, which has perpendicular magnetic anisotropy so that we can magnetize the sample normal to the surface.36,37 Figure 2(a) shows its out-of-plane magnetic hysteresis loop, which shows high squareness and near 100% remanent magnetization. The coercive field is 1265 Oe that is larger than the stray field generated by a sphere at a measurement separation of 446 nm.

FIG. 2.

(a) The out-of-plane magnetic hysteresis loop of the FePt film. (b) The AFM topographic image of the standard sample. (c) The MFM image of the standard sample.

FIG. 2.

(a) The out-of-plane magnetic hysteresis loop of the FePt film. (b) The AFM topographic image of the standard sample. (c) The MFM image of the standard sample.

Close modal

The FePt film was patterned to arrays of squares by photolithography and ion beam etching. At the final step, the structures were coated with a 53-nm-thick gold layer to prevent oxidation of FePt films and make the surface isoelectronic. The thickness of the gold film was measured with AFM on a control sample mounted with the standard sample during gold deposition. A gold film structure was fabricated on the control sample using photo-lithography and a lift-off method, which was then used for thickness measurement. The dimension of squares as well as the distance between the squares can be optimized according to the size of the magnetic sphere to improve its spatial resolution. For our application, the length of the square was designed to be 4  μm, and the distance between them is 8  μm. Figure 2(b) shows a topographic image of the standard sample prior to magnetization, taken with a commercial atomic force microscope (Dimension Edge, Bruker). Afterward, the standard sample was magnetized in a magnetic field of 5 kOe perpendicular to the surface, and then magnetic force microscopy measurements were taken in the two-pass technique using the same commercial AFM, where the phase shift of the resonance was measured at the second pass [see Fig. 2(c)]. The MFM image is consistent with the single domain nature of the FePt squares with magnetization along the normal direction.

In this work, we measured the magnetic moment of three NdFeB spheres with different diameters. The magnetic spheres are used as probes to search for exotic interactions beyond the standard of particles.34 The magnetic properties of individual spheres are important information for the data analysis in those experiments.

The commercially available NdFeB powder (MQP-S-11-9, Magnequench, Inc.) are made of micrometer spheres of different sizes as shown in the scanning electron microscopy (SEM) image [see Fig. 3(a)]. The magnetization curve of NdFeB powder was measured with a SQUID magnetometer [see Fig. 3(c)]. The remanent magnetization is measured to be 548 kA/m. NdFeB sphere of suitable sizes is selected and glued onto single crystalline silicon cantilevers (NSG11/tipless, NT-MDT Spectrum Instruments) under an optical microscope. Figure 3(b) shows a SEM image of a magnetic probe. Its resonance frequency is measured to be 134.64 kHz with a spring constant of 8.13 N/m. The magnetic probe was coated with a gold layer of about 100 nm thick. The sphere was magnetized by applying a magnetic field of 3 T perpendicular to the cantilever.

FIG. 3.

(a) The SEM image of the NdFeB powder shows magnetic spheres of different sizes. (b) The SEM image of a magnetic probe. (c) The magnetic hysteresis loop of NdFeB powder.

FIG. 3.

(a) The SEM image of the NdFeB powder shows magnetic spheres of different sizes. (b) The SEM image of a magnetic probe. (c) The magnetic hysteresis loop of NdFeB powder.

Close modal

Figure 4(a) shows a Δ f image of a theoretical simulation to present the expected characteristics of the measurements. The frequency shift has a dip in the center of the square and show a hump around the edge of the square. Asymmetry can be observed around the center, which is due to the tilt of the sphere’s magnetization from the vertical direction [see Fig. 4(e)]. The asymmetry is found to be reduced for spheres with larger diameters due to insufficient spatial resolution [see Fig. 4(f)]. This implies that the method is capable of quantitatively measuring all three components of the dipole moment with the optimal dimensions of FePt patterns.

FIG. 4.

(a) The expected Δ f image as calculated with m = 6.97 × 10 10 A m 2, a separation of 446 nm, and a sphere diameter of 6.24  μm. (b)–(d) MFM Δ f images obtained with magnetic spheres of 6.24, 9.49, and 11.96  μm in diameter at separations of 446, 421, and 553 nm, respectively. (e) The line plot of Δ f along the dashed line indicated in (a). (f) A calculated line plot of Δ f for a sphere with a diameter of 9.49  μm. (g)–(i) The line plots of the experimental data and the fitting curve along the dashed line indicated in (b)–(d).

FIG. 4.

(a) The expected Δ f image as calculated with m = 6.97 × 10 10 A m 2, a separation of 446 nm, and a sphere diameter of 6.24  μm. (b)–(d) MFM Δ f images obtained with magnetic spheres of 6.24, 9.49, and 11.96  μm in diameter at separations of 446, 421, and 553 nm, respectively. (e) The line plot of Δ f along the dashed line indicated in (a). (f) A calculated line plot of Δ f for a sphere with a diameter of 9.49  μm. (g)–(i) The line plots of the experimental data and the fitting curve along the dashed line indicated in (b)–(d).

Close modal

The experiment was performed on a home-built scanning probe microscope.38  Figure 4(b) shows an image taken at a separation of 446 nm with a magnetic sphere of 6.24  μm in diameter. The image size is around 40 × 40 μm 2 ( 128 × 128 pixels). The data acquisition time is 30 ms for every pixel. The image presents typical features as expected from the theoretical simulation, such as the periodicity, the dips in the centers, humps around the edges, and the asymmetry around centers.

We obtain the effective dipole moment m of the sphere by performing multiple linear regression based on two-dimensional Δ f images. The position z of the magnetic dipole can also be optimized by minimizing the residuals in the multiple linear regression analysis using the gradient descent algorithm. In the optimization, the starting position of the dipole is set to the center of the magnetic sphere. Figure 4(g) shows a cut-line plot of the two-dimensional fitting of the Δ f image to the theoretical model, and the fitting results are m = ( 1.89 , 0.031 , 2.71 ) × 10 11 A m 2 and z = 407 nm. The z position of the effective magnetic dipole is 407 nm higer than the center of the sphere. The magnetization is estimated to be 260 kA/m, which is much smaller than the average remanent magnetization measured by the SQUID magnetometer. The measurements of the other two magnetic spheres are presented in Fig. 4, and the fitting results are listed in Table I. We found that the magnetization increases with the diameter of the sphere. As the NdFeB powder is composed of spheres with different diameters around tens of micrometers [Fig. 3(a)], it is reasonable that its average remanent magnetization is 548 kA/m according to our measurements. The diameter dependence of magnetization may be due to oxidation or chemical modification on the surface layer where ferromagnetism is destroyed. Since the ratio of the surface area to volume is large for small microspheres, the average magnetization of the microspheres is expected to decrease as the diameter decreases.

TABLE I.

The magnetic dipole moments and magnetization as measured for different magnetic spheres.

ProbeDiameter ( μm)mxmymzM (kA/m)
(×10−10 A m2)
6.24 0.189 0.003 0.271 260 
9.49 1.008 0.134 1.476 400 
11.96 3.887 −0.741 3.888 619 
ProbeDiameter ( μm)mxmymzM (kA/m)
(×10−10 A m2)
6.24 0.189 0.003 0.271 260 
9.49 1.008 0.134 1.476 400 
11.96 3.887 −0.741 3.888 619 

In conclusion, we have developed a method to measure the magnetic dipole moment of magnetic particles in micrometer sizes using patterned FePt thin film structures as standard samples. In this method, the FePt features are magnetized to near single-domain magnets in a direction normal to the surface. The standard sample is scanned at constant distance with a cantilever glued with the magnetic particle to be measured, and the frequency shift image is recorded. Multiple linear regression analysis is used to determine the magnetic dipole moment of the particle based on the image. The position of the magnetic dipole moment is further optimized by reducing the residuals in the multiple linear regression analysis. We have performed such measurements on three NdFeB microspheres of different diameters. These spheres will be used as probes to search for new interactions beyond the standard model of particle physics. We found that the magnetization of NdFeB sphere increases with increasing diameter. The method developed in this paper will provide an option to quantitatively characterize magnetic particles using AFM without additional upgrades.

This work was supported by the National Key R&D Program of China (Grant No. 2022YFC2204100) and the National Natural Science Foundation of China (NSFC) (Grant Nos. 11875137 and 91736312).

The authors have no conflicts to disclose.

Rui Luo: Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Qian Wang: Formal analysis (equal); Methodology (equal). Yu Lu: Methodology (supporting); Resources (supporting). Feng Xu: Resources (supporting). Zhe Guo: Resources (supporting). Fei Xue: Resources (supporting). Long You: Resources (supporting). Jinquan Liu: Validation (supporting); Writing – review & editing (supporting). Pengshun Luo: Conceptualization (lead); Supervision (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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