Magnetic force microscope
Magnetic force microscopy (MFM) is a variety of atomic force microscopy, in which a sharp magnetized tip scans a magnetic sample; the tip-sample magnetic interactions are detected and used to reconstruct the magnetic structure of the sample surface. Many kinds of magnetic interactions are measured by MFM, including magnetic dipole–dipole interaction. MFM scanning often uses non-contact AFM (NC-AFM) mode. Overview. In MFM measurements, the magnetic force between the sample and the tip can be expressed as where formula_2 is the magnetic moment of the tip (approximated as a point dipole), formula_3 is the  magnetic stray field from the sample surface, and "µ" is the magnetic permeability of free space. Because the stray magnetic field from the sample can affect the magnetic state of the tip, and vice versa, interpretation of the MFM measurement is not straightforward. For instance, the geometry of the tip magnetization must be known for quantitative analysis. Typical resolution of 30 nm can be achieved, although resolutions as low as 10 to 20 nm are attainable. Important dates. A boost in the interest to MFM resulted from the following inventions: Scanning tunneling microscope (STM) 1982, Tunneling current between the tip  and sample is used as the signal. Both the tip and sample must be electrically conductive. Atomic force microscopy (AFM) 1986, forces (atomic/electrostatic) between the tip and sample are sensed from the deflections of a flexible lever (cantilever). The cantilever tip flies above the sample with a typical distance of tens of nanometers. Magnetic Force Microscopy (MFM), 1987 Derives from AFM. The magnetic forces between the tip and sample are sensed. Image of the magnetic stray field is obtained by scanning the magnetized tip over the sample surface in a raster scan. MFM components. The main components of an  MFM system are: Scanning procedure. Often, MFM is operated with the so-called "lift height" method. When the tip scans the surface of a sample at close distances (< 10 nm), not only magnetic forces are sensed, but also atomic and electrostatic forces. The lift height method helps to enhance the magnetic contrast through the following: Modes of operation. Static (DC) mode. The stray field from the sample exerts a force on the magnetic tip. The force is detected by measuring the displacement of the cantilever by reflecting a laser beam from it. The cantilever end is either deflected away  or towards the sample surface by a distance Δ"z" = "F"/"k" (perpendicular to the surface). "Static mode" corresponds to measurements of the cantilever deflection. Forces in the range of tens of piconewtons are normally measured. Dynamic (AC) mode. For small deflections, the tip-cantilever can be modeled as a damped harmonic oscillator with an effective mass ("m") in [kg], an ideal spring constant ("k") in [N/m], and a damper ("D") in [N·s/m]. If an external oscillating force "F" is applied to the cantilever, then the tip will be displaced by an amount "z". Moreover, the displacement will also  harmonically oscillate, but with a phase shift between applied force and displacement given by: where the amplitude and phase shifts are given by: Here the quality factor of resonance, resonance angular frequency, and damping factor are: Dynamic mode of operation refers to measurements of the shifts in the resonance frequency. The cantilever is driven to its resonance frequency and frequency shifts are detected. Assuming small vibration amplitudes (which is generally true in MFM measurements), to a first-order approximation, the resonance frequency can be related to the natural frequency and the force gradient. That is, the shift in the resonance  frequency is a result of changes in the spring constant due to the (repelling and attraction) forces acting on the tip. The change in the natural resonance frequency is given by For instance, the coordinate system is such that positive "z" is away from or perpendicular to the sample surface, so that an attractive force would be in the negative direction ("F"<0), and thus the gradient is positive. Consequently, for attractive forces, the resonance frequency of the cantilever decreases (as described by the equation). The image is encoded in such a way that attractive forces are generally depicted in black  color, while repelling forces are coded white. Image formation. Calculating forces acting on magnetic tips. Theoretically, the magneto-static energy ("U") of the tip-sample system can be calculated in one of two ways: One can either compute the magnetization ("M") of the tip in the presence of an applied magnetic field (formula_10) of the sample or compute the magnetization (formula_11) of the sample in the presence of the applied magnetic field of the tip (whichever is easier). Then, integrate the (dot) product of the magnetization and stray field over the interaction volume (formula_12) as and compute the gradient of  the energy over distance to obtain the force "F". Assuming that the cantilever deflects along the "z"-axis, and the tip is magnetized along a certain direction (e.g. the "z"-axis), then the equations can be simplified to Since the tip is magnetized along a specific direction, it will be sensitive to the component of the magnetic stray field of the sample which is aligned to the same direction. Imaging samples. The MFM can be used to image various magnetic structures including domain walls (Bloch and Neel), closure domains, recorded magnetic bits, etc. Furthermore, motion of domain wall can also be studied  in an external magnetic field. MFM images of various materials can be seen in the following books and journal publications: thin films, nanoparticles, nanowires, permalloy disks and recording media. Advantages. The popularity of MFM originates from several reasons, which include: Limitations. There are some shortcomings or difficulties when working with an MFM, such as: the recorded image depends on the type of the tip and magnetic coating, due to tip-sample interactions. Magnetic field of the tip and sample can change each other's magnetization, "M", which can result in nonlinear interactions. This hinders image interpretation. Relatively short lateral scanning  range (order of hundreds micrometers). Scanning (lift) height affects the image. Housing of the MFM system is important to shield electromagnetic noise (Faraday cage), acoustic noise (anti-vibration tables), air flow (air isolation), and static charge on the sample. Advances. There have been several attempts to overcome the limitations mentioned above and to improve the resolution limits of MFM. For example, the limitations from air flow has been overcome by MFMs that operate at vacuum. The tip-sample effects have been understood and solved by several approaches. Wu et al., have used a tip with antiferromagnetically coupled magnetic layers in  an attempt to produce a dipole only at the apex.