Quantitative interpretation of magnetic force microscopy images from soft patterned elements J. M. Garcı ´ a, a) A. Thiaville, and J. Miltat Laboratoire de Physique des Solides, University of Paris-Sud and CNRS, Orsay, France K. J. Kirk b) and J. N. Chapman Department of Physics and Astronomy, University of Glasgow, Scotland, United Kingdom F. Alouges Laboratoire de Mathe ´matique, University of Paris-Sud and CNRS, Orsay, France Received 6 March 2001; accepted for publication 4 June 2001 By combining a finite element tip model and numerical simulations of the tip–sample interaction, it is shown that magnetic force microscopy images of patterned soft elements may be quantitatively compared to experiments, even when performed at low lift heights, while preserving physically realistic tip characteristics. The analysis framework relies on variational principles. Assuming magnetically hard tips, the model is both exact and numerically more accurate than hitherto achieved. © 2001 American Institute of Physics. DOI: 10.1063/1.1389512 Magnetic force microscopy MFM 1,2 is now a widely disseminated technique aimed at imaging the micromagnetic structure of ferromagnetic materials. In the best cases, it al- lows for 20 nm lateral resolution with minimal sample preparation but with the drawback of a difficult quantitative analysis of the actual magnetization distribution. Early experiments 3 demonstrated a clear relation between charge distribution and MFM contrast. Such findings were subse- quently formalized by Hubert et al. 4 and, despite being ques- tionable, it is now generally assumed that MFM primarily is a charge mapping microscopy. Experiments performed on soft magnetic materials, especially, however clearly reveal the occurrence of tip induced perturbations. 5–9 Such difficul- ties may partly be alleviated when recording MFM images with low moment tips and large tip–sample distances, 10 at the potential expense, however, of sensitivity and/or lateral resolution. Even then, probe induced switching has been ob- served under field. 11 In this letter, we investigate, both ex- perimentally and theoretically, the MFM imaging process and introduce a simple framework allowing for its quantita- tive interpretation. The observed samples are thermally evaporated Ni 80 Fe 20 elements, 16 nm thick, prepared by electron-beam lithogra- phy and lift-off patterning on a Si 3 N 4 membrane. The MFM experiments have been performed using a NanoScope™ mi- croscope operated in the tapping/lift mode 12 and equipped with frequency detection. To first order, the frequency shift f is directly related to the force gradient according to: 2 k  f / f =- F /  z , f being the cantilever resonance fre- quency and k its spring constant. The probes were commer- cial Si cantilevers from NANOSENSORS™ having f80 kHz and k5 N/m mean characteristics. The tips were sput- ter coated with a Co 80 Cr 20 nominal composition alloy and magnetized along their pyramid axis prior to the experi- ments. We have found that only a tiny coating thickness range allows for successful imaging: if the Co 80 Cr 20 layer is thinner than 15 nm, no magnetic contrast appears, whereas a coating thicker than 30 nm gives rise to a sole strong uniform attractive contrast. Consequently, meaningful obser- vations requires finely selected tip moments. The images pre- sented in this letter have been acquired with a tip having a coating 20 nm thick. In the absence of tip–sample interaction, the 2 m-sized square elements should exhibit a perfect Landau–Lifshitz- type flux closure structure. However, if the tip–sample dis- tance z tip is kept small in order to ensure a good lateral reso- lution, the MFM images systematically display an apparent domain wall curvature producing a picture reminiscent of a four-bladed propeller Fig. 1. This apparent curvature proves insensitive to the scanning direction as demonstrated in Figs. 1a and 1b, precluding any nonequilibrium effect a Electronic mail: garcia@lps.u-psud.fr b Present address: Electronic Engineering and Physics Division, Univ. of Paisley, Scotland, UK. FIG. 1. MFM images of a 2 m2 m16 nm permalloy element. Lift height is z tip =20 nm except where mentioned. Conditions are: a Tip scanned from left-hand side to right-hand side trace, b tip scanned from right-hand side to left-hand side retrace, c reversed tip magnetization, and d lift height 45 nm and tip magnetization as in c. APPLIED PHYSICS LETTERS VOLUME 79, NUMBER 5 30 JULY 2001 656 0003-6951/2001/79(5)/656/3/$18.00 © 2001 American Institute of Physics Downloaded 16 Jun 2010 to 161.111.235.252. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp