This paper presents a 3-axis Lorentz force magnetometer based on an encapsulated micromechanical silicon resonator having three orthogonal vibration modes, each measuring one vector component of the external magnetic field. One mode, with natural frequency (*f _{n}*) of 46.973 kHz and quality factor (

*Q*) of 14 918, is operated as a closed-loop electrostatically excited oscillator to provide a frequency reference for 3-axis sensing and Lorentz force generation. Current, modulated at the reference frequency, is injected into the resonator, producing Lorentz force that is centered at the reference frequency. Lorentz force in the first axis is nulled by the oscillator loop, resulting in force-rebalanced operation. The bandwidth and scale-factor of this force-rebalanced axis are independent of resonator

*Q*, improving the sensor's temperature coefficient from 20 841 ppm/ °C to 424 ppm/ °C. The frequencies of the other two modes are closely spaced to the first mode's reference frequency and are demonstrated to track this frequency over temperature within 1 ppm/K. Field measurements in these two axes are conducted open-loop and off-resonance, ensuring that the scale-factor is independent of

*Q*to first order and producing a measurement bandwidth of over 40 Hz.

Various types of magnetometers have appeared in recent years, including Hall-effect sensors,^{1} magnetoresistive (MR) sensors,^{2,3} and Lorentz force magnetometers.^{4–6} Among these, there is growing interest in 3-axis Lorentz force magnetometers for use in consumer electronic devices because they are low cost, low power, CMOS compatible, and can be fabricated with microelectromechanical systems (MEMS) accelerometers and gyroscopes in a single chip using a standard silicon MEMS fabrication process.^{5–11} Lorentz force magnetometers based on MEMS resonators can be generally classified into two groups based on the readout mechanism: amplitude modulation (AM) and frequency modulation (FM). In an AM sensor, the magnetic field strength is measured by monitoring the change in the amplitude of the resonator's motion. An ac excitation current at carrier frequency *f* applied to the resonator modulates the external low-frequency magnetic field, generating an AM Lorentz force centered at *f*. Generally, the carrier frequency is set to the resonator's natural frequency, thus the motion resulting from the Lorentz force is amplified by the *Q* of the sensor, which can be as high as few thousand in a vacuum ambient, maximizing the field sensitivity. To take advantage of *Q*-amplification, the carrier frequency must be precisely matched to the resonance frequency. However, it is difficult to keep the frequencies identical over the operational temperature range due to the resonator's temperature coefficient of frequency (*TCF*), resulting in unacceptably large variations in the scale-factor (*SF*) over temperature. To solve this problem, the resonator can be used as an oscillator whose frequency inherently tracks the resonator's natural frequency.^{11} However, the scale-factor is still temperature dependent due to the temperature dependence of *Q* (*TCQ*). In addition, the sensor's bandwidth (*BW*) is limited by the mechanical bandwidth of the resonator, which is inversely proportional to *Q* and equal to a few Hz for high-*Q* sensors. In an FM sensor, the magnetic field strength is measured by monitoring the change in the resonator's natural frequency.^{12–14} FM magnetometers have some advantages over AM magnetometers, including no bandwidth limitation due to *Q* and improved stability over temperature; however, they often have lower sensitivity than AM sensors.

Here, we demonstrate a 3-axis Lorentz force magnetometer relying on AM readout. In prior work, it was demonstrated that the magnetometer has three orthogonal vibration modes, each measuring one vector component of the external magnetic field.^{15} The in-plane mode is operated closed-loop as an oscillator, providing a frequency reference always at the in-plane mode's resonance frequency for Lorentz force generation. During operation, this mode is operated in closed loop using force rebalance and the other two modes are operated off-resonance, providing sensitivity and bandwidth that are independent of *Q*. The operation method demonstrated here provides a practical method for three-axis field measurement using a single MEMS structure while reducing the sensor's temperature sensitivity and achieving a wide bandwidth that is independent of the sensor's original mechanical bandwidth.

Fig. 1 shows (a) the 3-axis MEMS magnetometer and (b) its FEM simulation with a detailed schematic of the ac current, magnetic field, and Lorentz force directions. The magnetometer consists of a 1.8 × 1.8 mm^{2} resonator fabricated using a CMOS-compatible encapsulation process^{16} with a 40 -*μ*m thick highly doped single crystal silicon structural layer. The resistance between the contacts I_{B1}, I_{B2}, I_{B3}, and I_{B4} is measured to be about 60 Ω for each flexure. The resonator has three resonant modes, corresponding to in-plane, out-of-plane, and rocking modes excited by z-, y-, and x-axis field components, respectively. Magnetic field measurements are conducted using an ac bias current *I* injected into the structure through the contacts I_{B1}, I_{B2}, I_{B3}, and I_{B4}. Measurement in each axis is conducted sequentially by switching the direction of the current flow through the resonator. The applied *I* orthogonal to the magnetic field *B* generates a Lorentz force *F _{L}* which results in displacement of the structure that is measured through sense capacitors. In-plane motion (z-axis field) is sensed at a parallel-plate pair S

_{I}beside the structure with nominal gap of 1.2

*μ*m, while out-of-plane motion is sensed by taking the sum (y-axis field) or difference (x-axis field) of the signals read out through the capacitive pick-offs (S

_{z+}and S

_{z−}) located above the structure with nominal gap of 2

*μ*m. The magnetometer is wafer-level vacuum sealed at less than 1 Pa to reduce damping and to increase the quality factor (

*Q*).

The resonance frequencies of the three modes were designed to be close to each other so that one mode's resonance frequency can be used as a reference frequency for 3-axis sensing. The fabricated device has an in-plane mode natural frequency *f _{z}* = 47.861 kHz with a quality factor

*Q*of 15 200, while the out-of-plane and rocking modes' natural frequencies,

*f*and

_{y}*f*, are measured to be 46.826 kHz and 46.773 kHz with

_{x}*Q*of 4580 and 8230, respectively. While

*f*and

_{x}*f*are closely spaced, as desired,

_{y}*f*is almost 1 kHz higher than the expected value from FEM simulation, mainly due to overcompensation for over-etch in the device layout. Because

_{z}*f*is used as a frequency reference for the Lorentz current

_{z}*I*, it is desirable that the frequency offset (

*Δf*) between

*f*,

_{z}*f*, and

_{x}*f*be relatively small, since the magnetometer's sensitivity is inversely proportional to

_{y}*Δf*, as shown in Eq. (4) below. To reduce

*Δf*from the as-fabricated value, a dc tuning voltage

*V*is applied to the in-plane electrodes DR and S

_{T}_{I}, resulting in electrostatic tuning of

*f*via

_{z}where *m* is the effective mass, *k* is the stiffness, $\epsilon 0$ is the permittivity of the free space, *V _{b}* is the dc bias voltage applied to the resonator, and

*A*and

*g*are the overlapped area and air gap of the parallel-plate capacitors, respectively. During operation,

*f*is adjusted to be 46.973 kHz by setting

_{z}*V*of 14.5 V and

_{T}*V*of 2 V.

_{b}Fig. 2 shows the block diagram of the interface circuitry for the 3-axis magnetometer. In force-rebalanced mode, the feedback voltage *v* produces electrostatic force, *F _{E}*

*=*

*ηv*, where

*η*

*=*

*(*

*C/g*)

*×*

*(*

*V*

_{T}

*−*

*V*) is the resonator's electromechanical coupling with capacitance

_{b}*C*= 413 fF and air gap

*g*= 1.2

*μ*m. The force-rebalance loop ensures that this force is equal and opposite to the Lorentz force,

*F*

_{L}

*=*

*B*, where

_{z}LI*I*= 1.15 mA is the current,

*L*= 1400

*μ*m is the effective length of the current path through the resonator,

*B*is the z-axis magnetic field. Equating these forces yields the sensitivity,

_{z}From Eq. (2), the sensitivity is theoretically calculated to be 37.4 V/T with a gain of 100 used in the electronics. Unlike an open-loop AM Lorentz force sensor, the sensitivity is independent of the resonator's quality factor and instead depends only on parameters that are largely temperature invariant.

The resonator's out-of-plane and rocking modes are operated open-loop with an offset between their resonance frequencies and the carrier frequency *f _{z}* of the driven ac current. Using a second-order model of the resonator, the amplitude relation between the Lorentz force (

*F*) and resulting motion (

_{L}*z*) at

*f*is

_{z}where *m*, *Q,* and *f* are the effective mass, quality factor, and resonance frequency of the mode in question. Satisfying the following conditions: (i) *Q* ≫ 1, (ii) the frequency offset *Δf** **=** **f** **−** **f _{z}* is much lower than

*f*, and (iii)

*Δf*is much larger than the mechanical bandwidth

*f/*2

*Q*, the ratio between

*z*and

*F*can be simplified as

_{L}where *k* is the stiffness of the mode in question. For a dc magnetic field, the displacement sensitivity *S* for the out-of-plane and rocking modes can be derived from Eq. (4) as

The resulting displacement is converted into a voltage signal using capacitive sensing, and the final sensitivity (in V/T) is determined by multiplying Eq. (5) by the displacement-to-voltage scale factor.

Experiments were conducted at room temperature with 2 V dc bias on the moving structure, 14.5 V dc tuning voltage, and 1.15 mA bias current. The amplitude of the in-plane mode's vibration was closed-loop regulated at 11 nm, whereas the out-of-plane and rocking modes are operated off-resonance with *Δf* of 147 Hz and 200 Hz, respectively. A Helmholtz coil was used to characterize the sensitivity of the magnetometer, generating a dc magnetic field ranging from −400 *μ*T to 400 *μ*T. Fig. 3 shows the measured output response of each axis as a function of the input magnetic field. The magnetometer has a measured sensitivity of 37.5 V/T, 16.4 V/T, and 13.8 V/T for the in-plane, out-of-plane, and rocking modes, respectively.

Fig. 4 shows the measured temperature dependence of each mode's frequency, demonstrating a *TCF* of approximately −21 ppm/ °C, with each mode's *TCF* differing by less than 1 ppm/ °C. While each mode's frequency changes by 50 Hz when the temperature changes by 50 °C, the oscillator loop ensures the excitation frequency tracks the in-plane natural frequency, thereby minimizing the change in sensitivity in this axis due to the *TCF*. As can be seen in Eq. (5), the temperature dependence of the sensitivity for the out-of-plane and rocking modes can be mainly attributed to *Δf*. However, because the *TCF* difference among the resonant modes is less than 1 ppm/ °C, *Δf* is almost constant over temperature, ensuring good temperature stability in these axes.

The temperature variation also causes a change in each mode's *Q*. The temperature dependence of *Q* can be described using a parameter,^{17} γ,

where *T* is the absolute temperature. The value of γ for the in-plane mode is ∼3.5, and is ∼3.2 for both the rocking and out-of-plane modes. From Eq. (5), for the out-of-plane and rocking modes, off-resonance operation with *Δf* of 147 Hz and 200 Hz provides a sensitivity that is independent of *Q*. From (2), the closed-loop, force-rebalance operation also eliminates the dependence of the in-plane mode's sensitivity on *Q*. To observe the effect of *Q* variation on the sensitivity over temperature, the in-plane mode of the sensor was operated both with and without force rebalance. The sensitivity of the sensor is proportional to *Q*^{2} without force rebalance.^{11} Fig. 5 shows the measured *SF* over temperature, showing that the scale factor variation versus temperature improves from 20 841 ppm/ °C to 424 ppm/ °C as a result of force rebalance. The temperature dependence of force-rebalance operation can be mainly attributed to the temperature dependence of the electronic system and magnetometer structure parameters, such as the resistance and stiffness of the silicon flexures. Furthermore, the measured coefficient, γ, of ∼3.5 agrees with the temperature dependence of the scale-factor observed in operation without force rebalance.

For a sensor operated at resonance, the open-loop *BW* is determined by the mechanical bandwidth *BW _{mech}*

*=*

*f*2

_{n}/*Q*of the resonance.

^{8,11}However, using closed-loop operation for the in-plane mode and off-resonance open-loop operation for other two modes makes the resulting sensor bandwidth independent of the mechanical bandwidth of the sensor's resonant modes. Fig. 6 shows the frequency response of the magnetometer, obtained by sweeping the frequency of a constant amplitude ac magnetic field and measuring the corresponding output amplitude. The 3 dB bandwidths of the three axes are illustrated by the dotted lines. The measured

*BW*is increased by a factor of 25, reaching 40 Hz for the in-plane mode, independent of the

*Q*-limited

*BW*of 1.6 Hz. Although operating the out-of-plane and rocking modes with a frequency split of 147 Hz and 200 Hz provides a theoretical

_{mech}*BW*of 80 Hz and 109 Hz, a 3 dB bandwidth of 63 Hz is obtained for both these modes, which is determined by a low-pass filter used in the electronics.

In conclusion, we demonstrate a 3-axis Lorentz force magnetometer using a micromechanical resonator. The resonator has three resonant modes for x-, y-, and z-axis magnetic field sensing. During operation, only the in-plane mode used for z-axis field sensing is operated in closed loop to achieve and sustain oscillation of the resonator, providing a frequency reference for 3-axis sensing and Lorentz force generation. The other two resonant modes, used for x- and y-axis magnetic field sensing, are operated off-resonance in open-loop. The method demonstrated in this work significantly decreases the temperature dependence of the magnetometer's sensitivity both by using the resonator's natural frequency as a frequency reference and by providing operation that is independent from *Q*. The method also allows a larger bandwidth, independent from the *Q*-limited mechanical bandwidth of the sensor. Achieving operation with a larger bandwidth and lower temperature sensitivity makes the device compatible with consumer-grade magnetic sensor specifications.

This work was supported by the Defense Advanced Research Projects Agency (DARPA) Precision Navigation and Timing program (PNT) managed by Dr. Andrei Shkel and Dr. Robert Lutwak under Contract No. N66001-12-1-4260 and the National Science Foundation under Award No. CMMI-0846379. The work was performed in part at the Stanford Nanofabrication Facility (SNF), which was supported by National Science Foundation through the NNIN under Grant No. ECS-9731293. The authors would also like to thank the SNF staff, particularly M. M. Stevens for the timely assistance with the epitaxial reactor.