Magnetic force microscope

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 Δz = F_z/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.

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].[16]

If an external oscillating force F_z 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:[5][6][9]

${\displaystyle F_{z}=F_{o}\cos(\omega t),\;z=z_{o}\cos(\omega t+\theta )\,\!}$

where the amplitude and phase shifts are given by:

${\displaystyle z_{o}={\frac {\frac {F_{o}}{m}}{(\omega _{n}^{2}-\omega ^{2})+({\frac {\omega _{n}\omega }{Q}})^{2}}},\;\theta =\arctan \left[{\frac {\omega _{n}\omega }{Q(\omega _{n}^{2}-\omega ^{2})}}\right]\,\!}$

Here the quality factor of resonance, resonance angular frequency, and damping factor are:

${\displaystyle Q=2\pi {\frac {{\frac {1}{2}}kz_{o}^{2}}{\pi Dz_{o}^{2}\omega _{n}}}={\frac {1}{2\delta }},\;\omega _{n}={\sqrt {\frac {k}{m}}},\;\delta ={\frac {D}{2{\sqrt {mk}}}}\,\!}$

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.

${\displaystyle \omega _{r}=\omega _{n}{\sqrt {1-{\frac {1}{k}}{\frac {\partial F_{z}}{\partial z}}}}\approx \omega _{n}\left(1-{\frac {1}{2k}}{\frac {\partial F_{z}}{\partial z}}\right)\,\!}$

The change in the natural resonance frequency is given by

${\displaystyle \Delta f=f_{r}-f_{n}\approx -{\frac {f_{n}}{2k}}{\frac {\partial F_{z}}{\partial z}}\,\!}$, where ${\displaystyle f={\frac {\omega }{2\pi }}\,\!}$

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.

Theoretically, the magneto-static energy (U) of the tip-sample system can be calculated in one of two ways:[1][5][6][17] One can either compute the magnetization (M) of the tip in the presence of an applied magnetic field (${\displaystyle H}$) of the sample or compute the magnetization (${\displaystyle M}$) 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 (${\displaystyle V}$) as

${\displaystyle U=-\mu _{o}\int \limits _{V}{{\vec {M}}\cdot {\vec {H}}\,dV}\,\!}$

and compute the gradient of the energy over distance to obtain the force F.[18] 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

${\displaystyle F_{i}=\mu _{o}\int \limits _{V}{{\vec {M}}\cdot {\frac {\partial {\vec {H}}}{\partial x_{i}}}\,dV}\,\!}$

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.