Probing of multiple magnetic responses in magnetic inductors using atomic force microscopy

Even though nanoscale analysis of magnetic properties is of significant interest, probing methods are relatively less developed compared to the significance of the technique, which has multiple potential applications. Here, we demonstrate an approach for probing various magnetic properties associated with eddy current, coil current and magnetic domains in magnetic inductors using multidimensional magnetic force microscopy (MMFM). The MMFM images provide combined magnetic responses from the three different origins, however, each contribution to the MMFM response can be differentiated through analysis based on the bias dependence of the response. In particular, the bias dependent MMFM images show locally different eddy current behavior with values dependent on the type of materials that comprise the MI. This approach for probing magnetic responses can be further extended to the analysis of local physical features.

(e) and S1(f). As described in the methods section, the MI was composed of soft magnetic particles and resin, with a Cu coil in the middle. As shown in Fig. S1(g), the coil is wound up with an ellipse shape and the inner major and minor axes of the coil are 1100 and 810 μm, respectively. (g). The circular particles and the filled space between the particles depict the soft magnetic particles and resin, respectively. The soft magnetic particles are composed of amorphous Fe-Cr-Si-B-C compounds with two kinds of powder particles, which are coarse and fine particles as shown in Fig. S1(f). The coarse and fine particle sizes were 10 ~ 20 and 1.5 ~ 3.5 μm, respectively. The current flow through the coil induces a magnetic field inside the MI. Thus, it is necessary to clarify the relationship between the applied voltage and the current flow in the coil. As the dc voltage applied to the coil increases, the current flow through the coil also increases according to Ohm's law. The linear coefficient obtained from the fit is 2.20, which reflects the 0.46 Ω of resistance across the coil. Thus, 1.00 V dc corresponds to about 2.20 A.    Fig. 5(a). and (b) a spatial map of the offset coefficient in Fig. 6(c). Scale bar is 0.5 μm.
The MMFM amplitude image with 0 V ac in Fig. 5(a) is modified to have the same color scale as the spatial map as shown in Fig. S6(a). It is apparent that both images are similar as they depict the same information, i.e. static magnetic domains.

IV. EFM analysis under the MMFM setup
In order to analyze the effect of electrostatic contribution to the MMFM response in our experimental setup, the electrostatic response was obtained at the same location as the data shown in Figs. 4 and 5 using electrostatic force microscopy (EFM). Since the experimental setup for obtaining EFM responses is almost the same as that of MMFM, with the exception of using a Pt-coated conductive tip (Multi75E-G, BudgetSensors) instead of a magnetic tip, the obtained EFM response can provide electrostatic contribution to the MMFM response. As shown in Fig. S7, there is no significant difference in the EFM response with respect to V ac . In other words, the electrostatic force exerted between the tip and sample is negligibly small. This might be due to the different pathways for the two ac voltages, which are applied separately to the coil and to the piezo dither to mechanically excite the cantilever.
Alternatively, it may be because the physical distance between the tip and coil is fairly large as shown in Fig. S1. Figure S7. Histograms of EFM amplitudes with respect to V ac .

V. Theoretical calculation of the magnetic forces acting on the tip with different origins
Since there are two magnetic dipoles in our experimental setup, namely the magnetized tip and the magnetic field associated with the sample, coil and eddy currents, the magnetic force can be theoretically calculated with the following assumptions: 1) the magnetized tip is fixed at a constant height with respect to the surface of the sample, and its magnetic moment is also constant, 2) the magnetic field on the sample surface, which results from either the soft magnetic particles in the sample or the coil and eddy currents, is at the center of the circular coil, and the magnetized tip is located along the z axis with respect to the center and 3) the obtained signal contains only the magnetic force exerted along the z direction. Then, the force between the two magnetic dipoles, F magnetic , acting on the tip can be written as follows where M tip , B z , i, j, and k are the tip magnetization, the magnetic field along the z axis, and the unit vectors along the x, y, and z axes, respectively. The equation can be simplified with assumption 3) and the fact that the angle between the two dipoles is 180 degrees.
For the magnetic force associated with the sample, eq. S2 becomes a constant with respect to V ac because the magnetic field of the sample is independent of V ac in the present case. Thus, the magnetic force of the sample, F sample , is written as follows Note that the term, F', stands for the other terms except V ac . The magnetic field generated by a coil current can be written as follows 1 Since the current flow through the coil is directly proportional to the applied ac voltage, the term, I, can be transformed into an expression containing the resistance and V ac according to Ohm's law. Then, similar to the magnetic force of the sample, eq. S2 for the magnetic force related to the coil can be rearranged as follows The magnetic force for the eddy current can be derived from the equation obtained with the other model system, that is, a moving monopole along the x direction at a constant height from the sample with a specific velocity 2 . Then, the lift and drag forces exerted on the monopole, F L and F D , respectively, are theoretically calculated as follows where μ 0 , q, z 0 , v, and ω are the vacuum permeability, the pole strength, the constant height, the uniform velocity of the moving monopole, and the parameter related to the thin conducting plate with velocity as the unit, respectively. Then, eq. S6 can be applied in our experimental setup based on the following assumptions: 1) the conducting plate sample and the moving monopole are considered the magnetized tip and the coil following application of ac voltage.
2) The velocity of the moving monopole is regarded as the frequency of the ac voltage. Thus, eq. S6 can be modified with the parameters associated with our model system as follows where μ r , m coil , and d 0 are the relative permeability, the magnetic moment of the coil and the distance between the coil and the tip along the z axis, respectively. Since the magnetic moment of a solenoid is directly proportional to the current flowing through it, eq. S7 can be further arranged as follows