Antti Jaakkola

Piezoelectrically transduced single-crystal-silicon plate resonators

We report on the design, fabrication and characterization of piezoelectrically actuated single-crystal silicon plate resonators vibrating mainly in their bulk acoustic wave modes. Two resonator types are presented: one operates in the square extensional mode at 26 MHz with Q~18000 and motional resistance Rm~0.240 kOmega, while the other resonator features a resonance at 22 MHz with Q~51000 and Rm~1.5 kOmega. The resonators are characterized electrically and by scanning laser interferometry. Measured vibration fields are compared to simulated eigenmodes.

Determination of doping and temperature-dependent elastic constants of degenerately doped silicon from MEMS resonators

Elastic constants c₁₁, c₁₂, and c₄₄ of degenerately doped silicon are studied experimentally as a function of the doping level and temperature. First-and second-order temperature coefficients of the elastic constants are extracted from measured resonance frequencies of a set of MEMS resonators fabricated on seven different wafers doped with phosphorus (carrier concentrations 4.1, 4.7, and 7.5 × 10¹⁹ cm^-3), arsenic (1.7 and 2.5 × 10¹⁹ cm^-3), or boron (0.6 and 3 × 10¹⁹ cm^-3). Measurements cover a temperature range from -40°C to +85°C. It is found that the linear temperature coefficient of the shear elastic parameter c₁₁ - c₁₂ is zero at n-type doping level of n ~ 2 × 10¹⁹ cm^-3, and that it increases to more than 40 ppm/K with increasing doping. This observation implies that the frequency of many types of resonance modes, including extensional bulk modes and flexural modes, can be temperature compensated to first order. The second-order temperature coefficient of c₁₁ - c₁₂ is found to decrease by 40% in magnitude when n-type doping is increased from 4.1 to 7.5 × 10¹⁹ cm^-3. Results of this study enable calculation of the frequency drift of an arbitrary silicon resonator design with an accuracy of ±25 ppm between the calculated and real(ized) values over T = -40°C to +85°C at the doping levels covered in this work. Absolute frequency can be estimated with an accuracy of ±1000 ppm.

Temperature compensated resonance modes of degenerately n-doped silicon MEMS resonators

We model the temperature coefficients of resonance modes of degenerately n-type doped silicon resonators. By combining results from FEM-based sensitivity analysis and modelling of elastic constants of silicon with free carrier theory we are able to identify classes of resonance modes that can be temperature compensated via n-type doping. These include bulk modes such as the width/length extensional modes of a beam, Lamé/square extensional modes of a plate resonator, as well as flexural and torsional resonance modes. Our results show that virtually all resonance modes of practical importance can reach zero TCF when the resonator is aligned to a correct crystallographic orientation and when the n-dopant concentration is suitably selected.

Trapping atoms on a transparent permanent-magnet atom chip

We describe experiments on the trapping of atoms in microscopic magneto-optical traps on an optically transparent permanent-magnet atom chip. The chip is made of magnetically hard ferrite-garnet material deposited on a dielectric substrate. The confining magnetic fields are produced by miniature magnetized patterns recorded in the film by magneto-optical techniques. We trap Rb atoms on these structures by applying three crossed pairs of counterpropagating laser beams in the conventional magneto-optical trapping geometry. We demonstrate the flexibility of the concept in creation and in situ modification of the trapping geometries through several experiments.

Piezoelectrically actuated micromechanical BAW resonators

We report on piezoelectrically actuated 13 MHz silicon beam resonators operating in the fundamental length extensional BAW mode. The resonance is evoked using an aluminum nitride layer grown on top of the resonator. We demonstrate quality factors as high as Q~55000 in vacuum, and transduction factors of the order of eta~20 muN=V. The anchor loss is studied using designs with variations in the support beam width and in the resonator orientation (crystalline direction).

Experimental Determination of the Temperature Dependency of the Elastic Constants of Degenerately Doped Silicon

We study experimentally the temperature dependence of the elastic constants of degenerately doped silicon as a function of the doping level. First and second order thermal coefficients of the elastic constants are extracted from the temperature dependent resonance frequencies of a set of MEMS resonators fabricated on phosphorus, arsenic and boron doped wafers having maximum doping levels of 7.5 × 10¹⁹cm ^-3, 2.5 × 10¹⁹cm ^-3 and 3 × 10¹⁹cm ^-3, respectively. Trends in the behavior of the thermal coefficients as a function of doping are identified and discussed.

On the optimization of piezoelectrically actuated MEMS resonators

We address the numerical analysis of piezo MEMS square extensional resonators and show that interfacial losses are dominant. These are formulated in terms of the stress jump across the interface yielding a viscous type of dissipation. The model developed is then employed to perform an optimization of the pattern of the piezo-layer so as to excite selected mechanical modes.

Frequency splitting of the main mode in a microelectromechanical resonator due to coupling with an anchor resonance

We present an experimental study of the frequency scaling of the main, square-extensional mode in a piezoelectrically actuated plate resonator. The studied set consists of resonators of different plate sizes with identical anchors. The behavior of the square-extensional mode is analyzed using electrical impedance measurements and optical characterization of the mechanical vibration fields. The results reveal a detrimental anchor effect, where for certain plate sizes the square-extensional mode branch is split into two due to a coupled oscillation of the resonator plate and the anchors.

P2G-3 Piezotransduced Single-Crystal Silicon BAW Resonators

We report on the design, fabrication and measurement of 13-MHz piezoelectrically actuated single-crystal silicon beam resonators operating in the first length-extensional mode. The transduction mechanism is based on an aluminum nitride layer grown on top of the resonator beam. The resonators are measured to have a quality factor of Q ~ 20000 at p < 1 mbar and typical motional resistance of Rm ~ 3 kOmega. The electromechanical transduction factor is eta ~20 muN/V, representing a coupling of the same order as produced by 20 V over a 100-nm gap for capacitively coupled resonators. The quality factor is observed to be dependent on the crystal direction of the resonator beam. A qualitative explanation for this effect is given.

Design Rules for Temperature Compensated Degenerately n-Type-Doped Silicon MEMS Resonators

The first- and second-order temperature coefficients and the total temperature-induced frequency deviation of degenerately n-type-doped silicon resonators are modeled. Modeling is based on finite element modelling-based sensitivity analysis of various resonator geometries combined with the experimental results on doping-dependent elastic constants of n-type-doped silicon. The analysis covers a doping range from 2.4 × 1017 to 7.5 × 1019 cm-3. Families of resonance modes that can be temperature compensated via n-type doping are identified. These include bulk modes, such as the width/length extensional modes of a beam, Lamé/square extensional modes of a plate resonator, as well as flexural and torsional resonance modes. It is shown that virtually all resonance modes of practical importance can reach zero linear temperature coefficient of frequency when correctly designed. Optimal configurations are presented, where a total frequency deviation of ~150 ppm can be reached. The results suggest that full second-order temperature compensation familiar from AT cut quartz is not possible in silicon resonators with doping below 7.5 × 1019 cm-3. However, an analysis relying on extrapolated elastic constant data suggests the possibility of full second-order temperature compensation for a wide range of resonance modes when doping is extended beyond 1020 cm-3.