Magnetic force microscopy

In MFM measurements we collect information about stray field distribution above sample, typically in one layer in fixed height above sample during one measurement. Together with this signal, we collect also topography, similarily to all the other SPM techniques.

The functions specific to MFM can be found in the Data ProcessingSPM ModesMFM. The individual modules and data processing algorithms discussed in the following paragraphs can be split into three categories:

The last category includes general functions such as data arithmetic which can add subtract images and generally combine them using arithmetic expressions or mutual crop which cuts images to the common sub-region (and is often a prerequsite for multi-data operations).

Addition and subtraction of measurements performed with two different probe magnetisation directions was proposed as the means of potential improvement of the process of splitting the magnetic and other contributions [Cambel11]. If measurements with different current direction are also available, we have four possible variants of how the electrostatic and magnetic forces are mutually oriented. All four can be comined to using data arithmetic to supress the noise in the data

Simulation of the stray field and MFM response

Several modules, differing by the algorithm used, simulate the stray field distribution in a plane above sample surface. None of them works with a general medium — e.g. using Poisson equation to solve the magnetic field distribution from the samle volume magnetisation — as this seems to be too computationally demanding for a general package like Gwyddion (at least for now). The modules are set up to simulate a few typical cases described in literature, where there are enough assumptions about sample magnetisation to make the stray field calculable by simple methods.

The module for handling perpendicular media is based on works of H. Hug [Hug98] and S. Vock [Vock11][Vock14]. The field distribution is calculated on the basis of known magnetisation distribution in a layer; in the z direction the magnetisation is constant for each position in the layer and changes only when we go in lateral directions. Initially, magnetisation is described as binary image (up/down magnetisation, defined by mask) and domain walls of given thickness can be added to it. The calculation is then done in Fourier space. Apart of the magnetic field distribution also results of using simple analytically known probe transfer functions can be provided (point charge, bar), and numerical derivatives of the resulting forces can be computed.

Assuming the surface charge distribution +σ(xy) at the top surface of the medium and opposite charge σ(xy) at the bottom surface we can use Fourier analysis to get the field distribution. First we convert the surface charges to the Fourier space:

Optionally we can even assume that in between of the domains there are Bloch walls of some width δw, convolving σ(xy) prior to the calculation with blurring operator

where the wall width can be expressed via exchange stiffness constant A and uniaxial anisotropy constant Ku as

Then we can calculate the Fourier components of the stray field in height z, generated from top surface and bottom surface of film of thickness d as

Via inverse Fourier transform we can then get the magnetic stray field above the sample:

While still staying in the Fourier space, we can also calculate the force acting on a simple bar probe of cross-section bx × by, length L and magnetisation Mt, by multiplying the stray field components length AHzz, d via the analytical force transfer function:

Screenshot of the perpendicular media simulation module.

Since Gwyddion is not just able to handle measured data, but also to synthetise many different artificial surfaces, there are many ways how to play with the perpendicular media module, creating various fields on more or less realistic samples. The following figure shows few examples of such simulated masks and resulting data.

Result of data synthetic modules used for MFM perpendicular media simulation: columnar film, fractal roughness, Ising model domains, grating, doughnuts.

For parallell media the range of potential calculations is much smaller — based on an analytical model the field above left and right direction oriented stripes (similar to a hard-disc) can be computed. This is based on [Rugar90], giving the following equations for individual transitions

where Mr, is the remanent magnetization of the magnetic layer, z the tip-sample separation, a is the transition area width, and d the layer thickness.

The stripes do not have to be of equal size. As a result, field intensities in x and z axes can be plotted, or the force in z based on known probe transfer functions (the same as for the perpendicular media) can be computed. Its derivatives can be computed as well (numerically).

Screenshot of the parallel media simulation module.

Similarily, analytical expressions can be used to simulate the current above a thin line with a electrical current passing through it. The current line module can be used for this, which is based on the equation given by [Saida03]:

for a stripe of width w in which the current I is flowing.

As all the simulation modules can be used also to add the result to existing data, by calling module multiple times we can also simulate a planar coil (or, more precisely, two lines with opposite directions of the current). This is demonstrated in the following figures.

Screenshot of the current line simulation module.

Stray field distribution above two parallel current lines with opposite current direction.

The algorithms for stray field calculation via Fourier transform can be also used to shift the field from one layer height to another. This works if we move further from the sample (blurring the data), but it does not work much when getting closer to the sample (the calculation diverges). Based on [Hug98] the procedure is as follows:

  1. The field Hz(r) is processed by Fourier transform to get its spectral representation AHz(k),
  2. Spectral components are multiplied by the factor exp(kd) where d is the distance between the two planes (positive or negative),
  3. Inverse Fourier transform is used to get back the field values in real space.

One can play with this algorithm in the field shift module as illustrated in the following figure.

Screenshot of the field shift module.

The same approach can be used for reverse operation — estmating the height difference between two data sets. Here a simple brute force search technique is used to find the shift in z direction that leads to best match between the two MFM responses. The user interface is shown in Fig. 8. Potential application is in searching for the real lift height difference instead of the reported one, however there are still potential caveats in real data if there are parasitic interactions.

Screenshot of the field shift estimation module.

Probe resolution and transfer function

Several ways have been proposed how to handle the issue of resolution of MFM probes. It is possible to analyse profiles across the perpendicular sample or utilise the power spectrum for probe resolution, e.g. by attempting to find where the power spectrum vanishes to noise level.

The method currently coarsely implemented in Gwyddion is based on probe transfer function estimation by Wiener filter deconvolution using FFT. This technique is not, in fact, specific to MFM and can be used universally for point spread function (PSF) estimation in any SPM regime, provided that the ideal sample response can be recorded or simulated. Data ProcessingStatisticsPSF Guess. The user interface of the module is and a typical performance on simulated noisy data is shown in the following figures.

Screenshot of the Wiener filter-based PSF estimation module.

Sample results of the PSF estimation module.

An alternative method of PSF estimation is the fitting of an explicit function form, for instance Gaussian. The free parameters of the function are sought to minimise the sum of squared differences between measured data and ideal response convolved with the PSF. Since the minimisation can be performed entirely in the frequency domain and does not require repeated convolution, this method is simple and fast, albeit somewhat crude. It is available as Data ProcessingStatisticsPSF Fit with two model functions currently implemented: fully symmetrical Gaussian and anisotropic Gaussian which can have different widths in the horizontal and vertical directions.

References

[Hug98] H. J. Hug, B. Stiefel, P. J. A. van Schendel, A. Moser, R. Hofer, S. Martin, H.-J. Gutherodt, S. Porthun, L. Abelmann, J. C. Lodder, G. Bochi and R. C. O’Handley: J. Appl. Phys. (1999) Vol. 83, No. 11

[Vock11] S. Vock, Z Sasvari, C. Bran, F. Rhein, U. Wolff, N. S. Kiselev, A. N. Bogdanov, L. Schiltz, O. Hellwig and V. Neu: IEEE Trans. on Magnetics, 47 (2011) 2352

[Vock14] S. Vock, C. hengst, M. Wolf, K. Tschulik, M. Uhlemann, Z. Sasvari, D. Makarov, O. G. Schmidt, L Schultz and V. Neu: Appl. Phys. Lett 105 (2014) 172409

[Rugar90] D. Rugar, H. J. Mamin, P. Guethner, S. E. Lambert, J. E. Stern et al.: J. Appl. Phys. 68 (1990) 1169

[Saida03] D. Saida, T. Takahashi: Jpn. J. Appl. Phys. 42 (2003) Pt. 1, No. 7B

[Cambel11] V. Cambel, D. Gregusova, P. Elias, J. Fedor, I. Kostic, J. Manka, P. Ballo: Journal of Electrical Engineering, 62 (2011) 37–43