W. M. Dougherty

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.

Nanometer-scale magnetic resonance imaging

Magnetic resonance force microscopy (MRFM) images the three-dimensional spatial distribution of resonant spins by mechanical force detection. Image reconstruction in MRFM is challenging because the resonance occurs in a strongly curved shell that extends beyond the scan range. In contrast with conventional magnetic resonance imaging, where Fourier techniques work well, the curved-shell resonant geometry inherent to MRFM requires novel reconstruction methods. Here, we show the application of iterative reconstruction in an electron spin resonance imaging experiment with 80 nm voxels. The reconstructed image has a total scan volume of 0.5 cubic micrometers, and was generated by a magnetic resonant shell with a curvature radius of 2.3 μm. The imaged object was a paramagnetically doped solid with an obliquely tilted surface. The reconstructed image correctly identified the location and orientation of the surface, and mapped the spin distribution within the solid. Applications of MRFM include three-dimensional nanometer-scale mapping of dopant distributions in semiconductors, studies of magnetism of thin films, and spin diffusion physics. An ultimate goal of MRFM is the direct observation of molecular structure at the atomic scale.

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.

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.

The classical and quantum theory of thermal magnetic noise, with applications in spintronics and quantum microscopy

Thermal fluctuations generate magnetic noise in the vicinity of any conductive and/or magnetically permeable solid. This magnetic noise plays a fundamental role in the design of spintronic devices: namely, it sets the time scale during which electron spins retain their coherence. This paper presents a rigorous classical and quantum analysis of thermal magnetic noise, together with practical engineering examples. Starting with the fluctuation-dissipation theorem and Maxwells equations, a closed-form expression for the spectral density of thermal magnetic noise is derived. Quantum decoherence, as induced by thermal magnetic noise, is analyzed via the independent oscillator heat bath model of Ford et al. The resulting quantum Langevin equations yield closed-form expressions for the spin relaxation times. For realistic experiments in spintronics, magnetic resonance force microscopy, Bose-Einstein condensates, atomic physics, and solid-state quantum computing, the predicted relaxation rates are rapid enough that substantial experimental care must be taken to minimize them. At zero temperature, the quantum entanglement between a spin state and a thermal reservoir is computed. The same Hamiltonian matrix elements that govern fluctuation and dissipation are shown to also govern entanglement and renormalization, and a specific example of a fluctuation-dissipation-entanglement theorem is constructed. We postulate that this theorem is independent of the detailed structure of thermal reservoirs, and therefore expresses a general thermodynamic principle.

The design and control of a three-dimensional piezoceramic tube scanner with an inertial slider

Inertial sliders are often used to produce coarse relative positioning for scanning probe microscopes. This article describes the design, dynamic analysis, and control of a compact four-segment piezoceramic tube scanner employing an inertial slider. Velocity feedback control, implemented using two-segment piezoelectric sensing, was used to suppress undesired vibrations in the tube, and to improve scanner step uniformity. The control analysis was based on an empirical open-loop identification of the as-built tube behavior, which was also measured using the two-segment sensing technique. A reset integrator friction simulation predicted the overall system performance, and showed good agreement with experimental results.

Adaptive control of force microscope cantilever dynamics

Magnetic resonance force microscopy (MRFM) and other emerging scanning probe microscopies entail the detection of attonewton-scale forces. Requisite force sensitivities are achieved through the use of soft force microscope cantilevers as high resonant-Q micromechanical oscillators. In practice, the dynamics of these oscillators are greatly improved by the application of force feedback control computed in real time by a digital signal processor (DSP). Improvements include increased sensitive bandwidth, reduced oscillator ring up/down time, and reduced cantilever thermal vibration amplitude. However, when the cantilever tip and the sample are in close proximity, electrostatic and Casimir tip-sample force gradients can significantly alter the cantilever resonance frequency, foiling fixed-gain narrow-band control schemes. We report an improved, adaptive control algorithm that uses a Hilbert transform technique to continuously measure the vibration frequency of the thermally-excited cantilever and seamlessly a...