On the influence of reference sample properties on magnetic force microscopy calibrations

TL;DR

The paper addresses the challenge of quantitative MFM (qMFM) calibration by analyzing how reference-sample feature distributions and measurement parameters shape the tip transfer function (TTF) used to reconstruct stray fields. It combines forward simulations with NV-center benchmarks to show that spectral overlap in $k$-space between reference patterns and the sample under test ($SUT$) largely governs reconstruction fidelity. The key finding is that broad Fourier content in the reference improves agreement with NV-derived tip fields, while narrow spectral content causes information loss, making multi-scale reference patterns essential for extending qMFM across length scales. This work guides the development of reference samples spanning multiple length scales to enhance qMFM accuracy and connect with other imaging modalities like NV-based magnetometry.

Abstract

Magnetic force microscopy (MFM) allows the characterization of magnetic stray field distributions with high sensitivity and spatial resolution. Based on a suitable calibration procedure, MFM can also yield quantitative magnetic field values. This process typically involves measuring a reference sample to determine the distribution of the tip's stray field or stray field gradient at the sample surface. This distribution is called the tip transfer function (TTF) and is derived through regularized deconvolution in Fourier space. The properties of the reference sample and the noise characteristics of the detection system significantly influence the derived TTF, thereby limiting its validity range. In a recent study, the tip stray field distribution, and hence the TTF, of an MFM tip was independently measured in real space using a nitrogen vacancy center as a quantum sensor, revealing considerable discrepancies with the reference-sample-based TTF. Here, we analyze the influence of the feature distribution of the reference sample and the MFM measurement parameters on the resulting TTF. We explain the observed differences between quantum-calibrated stray field distributions and the classical approach by attributing them to a loss of information due to missing or suppressed spectral components. Furthermore, we emphasize the importance of the spectral coverage of the TTF. Our findings indicate that for high-quality reconstruction of the stray field of a sample under test (SUT), it is more critical to ensure a strong overlap of frequency components between the reference sample and the SUT than to achieve an accurate real-space reconstruction of the tip stray field distribution.

Figures (8)

• Figure 1: (a) MFM measurements represented as the convolution of the stray field from the tip with the sample magnetization. (b) MFM phase shift data for a 5.12 µm x 5.12 µm scan of a CoPt multilayer stack, displaying a maze domain pattern. A cross-section along white dashed is shown in black color. The corresponding wavelength density spectrum in Fourier space is shown in red color.
• Figure 2: Comparison between $\mu_0 H_z^{\text{tip}}$ measured by NV magnetometry (red) and qMFM calibration (black). The line profiles display a cross-section through $\mu_0 H_z^{\text{tip}}$ as indicated by the dashed line in the insets.
• Figure 3: Description of forward MFM simulation resulting in (d) by convolving the field of a simulated sample (a) with the tip's field obtained from an NV measurement (b), and incorporating artificial noise (c). A line plot (along the dashed line) of the resulting simulated MFM signal is shown in (e).
• Figure 4: Comparison of spectral densities (left column) for different-sized reference patterns (middle column). The right column compares the $\mu_0 H_z^{\text{tip}}$ obtained from qMFM calibration using relevant reference patterns with the $\mu_0 H_z^{\text{tip}}$ measured by NV magnetometry. In (a), the spectral density of the CoPt sample's reference pattern, as measured with MFM, is compared to the spectral density of the magnetic tip measured by NV magnetometry. The $\mu_0 H_z^{\text{tip}}$ plotted in purple represents the field distribution obtained from a forward simulation using the original reference pattern as a control. From top to bottom, domain sizes of the reference patterns are increased by factors of 2 (b), 3 (c), and 4 (d). The blue, orange, and green plots correspond to different pixel resolutions and image sizes, see Table \ref{['table1']}.
• Figure 5: Comparison of TTFs derived from calibrations based on MFM measurements of two different reference samples. The CoPt reference sample (a) and the TiPtCo reference sample (b) exhibit distinct domain pattern widths, which are also reflected in their circularly averaged Fourier spectra (c). The TTF (here visualized by a cross-section) derived from the TiPtCo sample is notably wider than that obtained from the CoPt sample (d).