Direct measurement and microscale mapping of nanoNewton to milliNewton magnetic forces

Ongoing technological advancements in microscale devices motivate the need for magnetic force metrology at small spatial scales. Atomic force microscopy (AFM) provides force sensing capability with sub-micron spatial resolution¹ and two variants of scanning probe microscopy are capable of measuring forces produced by magnetic fields: magnetic force microscopy (MFM) ^2–11 and magnetic resonance force microscopy (MRFM).¹² While these techniques are well suited for measuring forces at the micrometer and nanometer length scales, they are generally limited to forces on the order of ∼1 nN or smaller.¹³ 

For measurements of larger forces in the nN to μN range, microfabricated microelectromechanical systems (MEMS) force-sensing approaches are well suited.^14,15 Contact-based microscale force measurements have been made using vision-based (optical)^16,17 and capacitance sensing mechanisms.^18,19 However, non-contact magnetic force sensing at the microscale has not been widely explored, but rather forces are inferred from magnetic field measurement methods using Hall probe²⁰ or magneto-optical imaging (MOI)²¹ techniques. To infer forces from magnetic field measurements it is necessary to calculate the magnetic field gradient (F=m$⋅⁣∇$B), which requires multiple scans at different heights (at least two). Furthermore, noise and/or resolution limitations in the measurement of B can lead to significant errors in calculating the gradient, and hence force, especially if the field gradients are small. Therefore, in certain applications a direct force measurement is desired to decrease the measurement time and potentially increase the accuracy.

This work reports the combination of a microforce capacitive sensing probe (FemtoTools FT-S1000)¹⁹ with a microfabricated permanent magnet as a sensing probe yielding uniaxial magnetic force mapping capabilities at any point in space, with microscale spatial resolution and nanometer positional accuracy. A similar construction for sensing forces using MEMS devices and magnetic materials was previously explored,^22,23 but not intended as a force mapping tool. To validate the construction of the proposed magnetic force sensing technique, an analytical model of the force between two magnets was implemented and compared with experimental measurements.

As shown in Fig. 1, an L1₀ CoPt micromagnet²⁴ (∼150 μm x150 μm x 8 μm) with out-of-plane magnetization serves as the “probe magnet” and is attached to a microforce sensing probe (FemtoTools FT-S1000).¹⁹ The force probe is factory calibrated to sense forces in the z-direction in a ±1000 μN range with 50 nN resolution and cross-axis rejection factor of >30. The micromagnet is detached from the silicon substrate on which it was formed and glued on the end tip of sensing probe using UV-curable glue. The micromagnet is oriented such that attractive or repulsive forces acting on the magnet are directed along the sensing direction of the force probe (along the probe shank). The probe is then rastered at different vertical heights (from 25 μm to 200 μm) above a magnetic sample via a micromechanical testing station (FemtoTools FT-MTA02) that facilitates a three-dimensional scanning range of 26 mm and minimum step size of 1 nm in all three directions. Microforce sensing probes can be exchanged in order to extend the force range from ±100 μN (5 nN resolution) up to 100 mN (5 μN resolution). This means that the proposed technique can potentially measure magnetic forces ranging from 5 nN to 100 mN.

The size, shape, magnetization direction, and material properties of the probe micromagnet determine the magnitudes of the forces on the force sensor, as well as the spatial resolution of the measurement. Fig. 2a shows the micromagnet topology as measured using an optical profilometer (Bruker Contour GT-I). Measurements of the stray magnetic field at the surface of the micromagnet are made using a MOI method (Fig. 2b).²¹ The magnetization curves (Fig. 2c) are measured using a vibrating sample magnetometer (VSM) (ADE Technologies EV9) evidencing: B_r=0.65 T, H_c=405.8 kA/m, H_ci=726 kA/m, BH_max=52 kJ/m³ and remanent magnetic moment m=74 μemu.

To create an “interesting” test sample for field mapping purposes, a 3 mm × 3 mm x 0.4 mm N52 grade NdFeB magnet (BJA - Bob Johnson Associates, Inc.) was used, exhibiting preferential out-of-plane magnetic properties, measured by VSM to be B_r=1.16 T, H_ci=899 kA/m, BH_max=170 kJ/m³. The magnet sample was mechanically polished to a surface roughness (R_a) of ∼20 nm and then selectively magnetized using the procedures described by Velez, et al²⁵ to imprint stripe-like alternating negative and positive poles in the out-of-plane direction (perpendicular magnetization). In brief, the magnet was initially magnetized upwards at 7 T and then selectively reversed (downwards) using a 1-mm-thick low-carbon steel sheet (ASTM A108) magnetizing mask subjected to a 1.5 T reversal field. The result of this process is a patterned magnetic substrate where the imprinted poles have an estimated magnetization of $μ0$M_r=±0.27 T. The stray magnetic field (B_z) generated by the selectively magnetized test sample was characterized using magneto-optical image (MOI) measurements at three different heights: 0 μm, 50 μm and 300 μm (Fig. 3).

An analytical model is used for comparison and validation of the measurements made with the proposed force measurement technique. For calculating the 3D forces acting on the force probe, the Akoun-Yonnet model²⁶ is employed in a publicly available computational tool.²⁷ The force calculation assumes uniform magnetization of two cuboidal magnets (⁠$μ0$M_r=0.65 T for micromagnet and $μ0$M_r=0.27 T for sample magnets) and determines the gradient of the energy equation to find the force expression. At any given point in space, the force acting on the probe micromagnet is calculated to be the superposition of the forces from the three stripe poles from the magnet test sample with zero gap between the poles.

Multiple scans were made using 100 $μm$ steps (in the xy direction) at different z heights (25 μm, 100 μm, and 200 μm) above the test sample. Fig. 4 illustrates the measured forces in the z direction (F_z) in both 2D and 3D representations. At 25 μm from the surface, forces ranging between -80 to 50 μN meanwhile at 200 μm, the forces range between -10 and 10 μN. Distinct spatial variations are observed when nearer the test sample at 25 $μm$⁠, whereas at 200 $μm$ the fields look much smoother due to spatial averaging. The spatial asperities with spatial size scales of ∼100 $μm$⁠, are also evident in the MOI field measurements (Fig. 3). These local field variations are attributed to a combination of two factors: 1) magnetic microstructures created by the magnetocrystalline material grains of the NdFeB, and 2) imperfect magnetization due to the selective magnetization process (i.e. not saturating the poles in the test sample).

Fig. 5 shows a comparison between the micromagnetic force measurements (performed with 50 μm step size for maximum resolution) and an analytical calculation.²⁷ The measurements show good general agreement with the force model and prove the concept of direct magnetic force measurement.

The limits of the proposed technique are calculated by hypothetically varying the micromagnet size by changing the edge length from 1 μm to 1 mm, while keeping the magnet thickness constant at 8 μm and assuming a constant out of plane magnetization of the magnetic material (M_r=426 kA/m). The force equation, F=m$⋅⁣∇$(B), reduced in one dimension to dB_z/dz=F_z/(V·M_r), where V is the volume of the micromagnet. Fig. 6 shows the range of measureable field gradients (dB_z/dz) versus magnet size. The red region indicates the maximum and minimum possible force resolvable by the microforce sensor platform using different probes (5 nN to 100 mN). The purple region indicates the max and min force by the specific probe used in the results presented here (50 nN to 1 mN). The field gradient range obtained from the fabricated magnet (150 μm edge length) is also highlighted from 0.7 T/m to 13,500 T/m). This plot highlights the tradeoff between spatial resolution and field gradient resolution.

A magnetic force sensing technique for force range of ±1000 μN with 50 nN resolution and with a minimum step size of 50 μm has been demonstrated. Additional capability for extending forces sensing range between 5 nN to 100 mN and step size of 1 nm (relaying on the size of the micromagnet) are implied. The actuation principle is by attaching a micromagnet (∼150 μm x 150 μm x 8 μm) to a commercial micro-force measuring probe (FemtoTools FT-S1000) and controlled by a micromechanical testing station (FemtoTools FT-MTA02). Resulting magnetic forces measured using this method were compared with the analytical solution of the forces between two magnets (with one fixed and the other raster the surface). Measurements and analytical model present a good agreement. The results demonstrate this technology could be a valuable contribution to close the technological gap for microscale magnetic force sensing. The general platform could also be modified to measure magnetic forces between permanent magnets and other materials (i.e. paramagnetic, ferromagnetic). It is also possible to use different ferromagnetic or diamagnetic materials in the probe tip, rather than a micromagnet, to have direct measuring of variety of magnetic forces, thereby expanding the possible measurement modalities.

The authors especially thank Jonas Henricksson from Femto-Tools AG for his valuable contributions and discussion as well as the staff of University of Florida’s Research Service Centers (RSC) at the Herbert Wertheim College of Engineering for their assistance in the microfabrication and materials characterization.