Quantum calibrated magnetic force microscopy

We report the quantum calibration of a magnetic force microscope (MFM) by measuring the two-dimensional magnetic stray-field distribution of the MFM tip using a single nitrogen vacancy (NV) center in diamond. From the measured stray-field distribution and the mechanical properties of the cantilever a calibration function is derived allowing to convert MFM images to quantum calibrated stray-field maps. This approach overcomes limitations of prior MFM calibration schemes and allows quantum calibrated nanoscale stray-field measurements in a field range inaccessible to scanning NV magnetometry. Quantum calibrated measurements of a stray-field reference sample allow its use as a transfer standard, opening the road towards fast and easily accessible quantum traceable calibrations of virtually any MFM.

Figure 1

Experimental setup. (a) NV-based stray-field measurements of an MFM tip. When scanning the MFM tip over the sample an ODMR spectrum is measured at every point. (b) Coordinate system of the measurements. The orientation of the NV axis is given by the polar angle θ and the azimuthal angle φ. (c) Typical ODMR spectrum for two different tip stray fields (fluorescence signal as function of microwave frequency). The curves are offset for clarity. The magnitude of the tip stray field at the position of the NV center and field angle β with respect to the NV center axis are derived from the splitting and offset of the resonance lines. (d) Principle of MFM measurements comprising a topography scan in intermittent contact mode and a phase scan at a tip-sample distance $z_{\mathrm{tot}}$ of 80 nm. The phase signal of the cantilever oscillation as a function of lateral tip position is collected as the MFM raw data.

Figure 2

Characterization of MFM tip stray field by NV magnetometry. The tip is scanned at $z=80$ nm above the diamond surface. (a) Two-dimensional (2D) map of magnitude of the tip field $B_{}^{\mathrm{tip}}$. (b) 2D map of field angle β relative to the NV axis. The symmetry axis (dashed line) corresponds to the in-plane orientation of the NV axis. (c) map of $B_{\left|\right|}^{\mathrm{tip}}$. (d) 2D map of the derived $z$ component $B_z^{\mathrm{tip}}$ of the tip stray field. The 2D distribution of $B_z^{\mathrm{tip}}\left(x,y\right)$ is the key ingredient for quantum calibration of the MFM. (e) Sketch of the relative angle β of the tip stray field and the NV axis. (f) Tip-surface distance dependence of $B_{}^{\mathrm{tip}}$ with the tip positioned above the NV center. From interpolation (dashed line) a maximum tip stray field of 44 mT at the tip apex is derived.

Figure 3

Quantum calibrated tip stray-field distribution. (a), (b) Sections in $x$- and $y$ direction through the maximum of the $B_z^{\mathrm{tip}}$ data of Fig. 2 (blue solid symbols). The red lines show combined Gaussian-Voigt fits to the data. (c) Quantum calibrated tip stray-field distribution $B_z^{\mathrm{tip},\mathrm{NV}}\left(x,y\right)$ based on the extrapolated data fits of (a), (b). The fits are used to extrapolate the NV measured stray field to an area of 5.11 μm × 5.11 μm to consider longer-range magnetic interactions between tip and sample.

Figure 4

Quantum calibrated MFM. (a) MFM data of the Co/Pt multilayer sample with characteristic stripe domains. (b) Quantum calibrated $B_z^{\mathrm{QuMFM}}$ at a tip surface distance of 80 nm, obtained from a deconvolution of the MFM data with the instrument calibration function based on the quantum calibrated $B_z^{\mathrm{tip},\mathrm{NV}}$ of Fig. 3. (c) Calculated $B_z^{\mathrm{ref}}$ based on the discriminated domain pattern after deconvolution with $B_z^{\mathrm{tip},\mathrm{NV}}\left(x,y\right)$. (d) Classical qMFM image of the sample's stray field obtained from a deconvolution of the MFM data with a conventionally estimated instrument calibration function. The white ellipses in (b)–(d) mark characteristic differences of quantum calibrated QuMFM and classical qMFM (see text). (e), (f) Comparison of horizontal cross sections of $B_z$ of (b)–(d) in the center of the images. Uncertainty bands are plotted as shaded regions. (e) Comparison of $B_z^{\mathrm{QuMFM}}$ (b, red) and $B_z^{\mathrm{ref}}$ (c, blue). The data agree well within the uncertainty. (f) Comparison of $B_z^{\mathrm{QuMFM}}$ (b, red) and $B_z^{\mathrm{qMFM}}$ (d, black). The data again agree within the uncertainty bands. (g) Test of regularized deconvolution procedure applied in (b). The raw MFM data and the reconvolved data show very good agreement.