Magnetic Resonance Force Microscopy:
A Detailed Summary

The physics of MRFM

In Magnetic Resonance Force Microscopy (MRFM), a small sample is suspended on a micromachined cantilever near a ferromagnetic tip whose shape creates an inhomogeneous magnetic field. The nuclear spins in the sample are polarized by the inhomogeneous magnetic field. A second, oscillating magnetic field is applied by an RF coil, which excites a spin resonance in the atoms in those regions of the sample where the magnetic resonance condition Wo = g * Bz , is met. Here, Wo is the frequency of the RF field, Bz is the strength of the inhomogeneous polarizing field, and g is the gyromagnetic ratio of the nucleus. By slow frequency or amplitude modulation of the RF field, a modulation in the nuclear magnetization of the resonant fraction of the sample occurs, leading to a modulation in the force between the sample and the magnetic tip. This force produces a measurable oscillation in the deflection of the cantilever, which is detected with an optical-fiber interferometer. A schematic drawing of this apparatus is shown below.

Since the magnetic field from the magnetic tip is non-uniform, the magnetic resonance condition is only satisfied in a small fraction of the sample. For an iron tip with a 100 nm radius of curvature, the field gradient at the surface is the partial derivative of Bz w/r/t z, which is approximately 1E10 Gauss/cm. For a typical resonance linewidth of 10 Gauss, the thickness of the region that satisfies the resonance condition is

which is sufficient to resolve individual atomic layers. By scanning the magnetic tip in three dimensions, and scanning the RF frequency, it will be possible to construct three-dimensional, elementally specific images.

The forces which arise from this arrangement of magnetic fields are small. The force exerted on the resonant spins by the inhomogeneous field of the tip is given by F = Mz*(dBz/dz) (using partials), where Mz is the component of the spin magnetic moment parallel to the field. During operation of the MRFM, this component is made to oscillate at the resonance frequency of the cantilever. For a single proton, the force is (1.4 x E(-23) emu)*(1E(10)Gauss/cm) = 1.4 E(-13) Dynes = 1.4 E(-18) N.

Detection of forces as small as 1E(-18) N requires a very careful displacement measurement. Optical interferometers can measure deflections smaller than an angstrom, and do not introduce significant interaction forces such as are found in STM or AFM. For the 1E(-18) N force to produce deflections comparable to an angstrom, the stiffness of the cantilever must be well below 1 N/m. Since the cantilever is to be driven with an oscillating force at its resonance, there is a quality-factor enhancement of the amplitude of the motion. As a result, the k/Q ratio must be nearly as small as 1E(-8) N/m.

An additional noise source must be considered. Micromechanical structures are subject to thermodynamic noise. In the case of micromechanical oscillators, thermal energy is present in the form of vibrations. This noise is often referred to as Brownian Noise because of the historical experiments in which the energy of a galvanometer was dissipated through random collisions with gas molecules. From a more general perspective, all micromechanical oscillators are in thermal contact with their surroundings. This thermal contact provides the loss mechanism for mechanical energy, as well as the transport mechanism which guarantees thermal equilibrium. By including the transport of thermal energy into the equation of motion of the mechanical oscillator, it is possible to calculate the power spectral density of the thermal noise induced motion of the oscillator. This contribution may be represented as a noise force, which then sets a limit to the minimum detectable signal force. This minimum detectable force is given by :

           (  4 kB T k B  )
Fmin = sqrt(--------------)
           (     Q Wo     )