Stray field measurement

The development of the scanning tunnelling microscope (STM) allowed scientists to determine the electronic topography of a sample with a resolution down to a few ångstroms (Binnig and Rohrer, 1985, Binnig et al., 1982, Binnig et al., 1983). The atomic force microscope (AFM) is a further development of the STM which exploits the deformation of a spring attached to a cantilever to allow the measurement of the force exerted on the sharp AFM tip (Binnig et al., 1986).

To characterise the magnetic properties of a sample, the standard diamond tip of the AFM is replaced by a ferromagnetic tip (Rugar et al., 1990, Sáenz et al., 1987) allowing the observation of magnetic domain structures by measuring the force gradient exerted on the tip by the stray field as a function of the position -- a technique known as magnetic force microscopy (MFM) . Rather than the tip following the surface contours of the sample as in STM, the MFM operates with the tip at a fixed $z$ point in space.

To assess the appropriateness of the two-dimensional approximation above, we compute the stray field and compare this to the experimental images provided through magnetic force microscopy.

The tip of an MFM is magnetic and we assume that it is a dipole $\mathbf{d}$. In the absence of knowledge of the precise magnetisation of the tip, this is the simplest approach possible (Barthelmeß et al., 2003). The signal which the MFM records is proportional to the gradient of the demagnetising field $\ensuremath{\mathbf{H}}_{\mathrm{de}}$.