K. J. Bruland

OPTIMAL CONTROL OF FORCE MICROSCOPE CANTILEVERS. II. MAGNETIC COUPLING IMPLEMENTATION

We describe the implementation of optimal controllers for damping the motion of cantilevers used in magnetic resonance force microscopy. We demonstrate that optimal control is achievable and that torsional magnetic coupling provides an effective actuation method. Cantilever Brownian vibrational amplitude was reduced from 2 to 0.16 A and resonant quality was reduced from 2000 to 5. Applied control fields were sufficiently small that they would not affect magnetic resonance phenomena.

Force-detected magnetic resonance in a field gradient of 250 000 Tesla per meter

We report the detection of slice-selective electron spin resonance with an external magnetic field gradient comparable to local interatomic gradients, using the techniques of magnetic resonance force microscopy. An applied microwave field modulated the spin-gradient force between a paramagnetic DPPH sample and a micrometer-scale ferromagnetic tip on a force microscope cantilever. A sensitivity equivalent to 184 polarized electron moments in a one-Hertz detection bandwidth was attained. We mapped the tip magnetic field with a resonant slice thickness of order one nanometer, thereby demonstrating magnetic resonance on length scales comparable to molecular dimensions.

Optimal control of force microscope cantilevers. I. Controller design

In magnetic resonance force microscopy (MRFM) experiments, magnetic forces couple to the motion of microscale cantilever beams. Extension of MRFM to the detection of single electrons will require both unprecedented force sensitivity and motional stability of the cantilever. We describe the principles and performance of optimal cantilever motion control. The method accounts for inherent noise processes and practical application of control forces. We show that active feedback control improves cantilever motional stability, enabling instrument designs of much higher sensitivity and faster imaging than passive designs. Experimental results of implemented cantilever control systems are presented in Part II.

Anharmonic modulation for noise reduction in magnetic resonance force microscopy

This article presents a new modulation technique for noise reduction in magnetic resonance force microscopy. Applied fields are modulated at frequencies that are not rational fractions of the cantilever resonant frequency, thus avoiding overtones that contribute to the noise level. An on‐resonance signal is obtained because the nonlinear sample magnetization acts as a frequency mixer of the two modulation frequencies, producing a net force modulation at the cantilever resonant frequency. These techniques are experimentally demonstrated using electron spin resonance in a <15 ng sample of diphenylpicrylhydrazil.

Optimal control of ultrasoft cantilevers for force microscopy

The goals of optimal control in force microscopy are: (1) to obtain favorable cantilever dynamic properties and (2) to control the cantilever to a desired amplitude, while (3) exerting as little control force as possible, and (4) preserving the force signal-to-noise ratio of the uncontrolled cantilever. This article describes the experimental implementation of an optimal controller that achieves these goals. The application of this controller to an ultrasoft cantilever with spring constant of 110 μN/m at 10 K reduced the resonant quality from 15 000 to 220, reduced the Brownian amplitude from 11.2 A to 1.4 A, used less than 7×10−17 N of control effort, left the force sensitivity unaltered at 9.8×10−18 N/ Hz, and demonstrated feedback control can force cantilever motion to track a reference input.

Thermal tuning of a fiber-optic interferometer for maximum sensitivity

We describe a fiber-optic interferometer that employs wavelength changes to achieve maximum sensitivity. Wavelength changes are induced by adjusting the operating temperature of the laser, eliminating the need for an actuator to vary the spacing between the sensing fiber and the object to be monitored. The instrument and techniques described are suitable for cryogenic, high vacuum applications such as magnetic resonance force microscopy, where space is limited and micromanipulation can be challenging. The noise floor of 1.6×10−3 nm/Hz is adequate for monitoring subangstrom displacement of force microscope cantilevers.

Detection of AC magnetic signals by parametric mode coupling in a mechanical oscillator

Parametric coupling has been demonstrated between mechanical vibration modes in a magnet-tipped microcantilever. An external magnetic field, coupled to the magnetic tip, pumps the effective spring constant at a frequency which is either the sum or the difference of the mode resonance frequencies. The presence of the pump field can be detected by driving one mode and observing the parametrically pumped excitation of the other mode, even though the pump frequency is off-resonance with respect to both mechanical modes. In a room temperature experimental realization, the magnetic flux coupling the pump field to the tip was approximately one flux quantum and the dominant noise source was the thermal vibration of the cantilever. Parametric mode coupling is a useful new design option in magnetic resonance force microscopy, whereby modulation is advantageously performed off-resonance to avoid parasitic excitations caused by stray couplings. Parametric coupling also provides a low-noise technique for amplifying mechanical oscillations. The reported experiment completes the set of all possible force microscope interaction Hamiltonians up to third order in time-dependent fields.