Device cross section A – A from Fig. 1 indicating four identical capacitors in dc representation. The electrostatic pull force on each capacitor is identical and is analyzed using the single-capacitor model in Fig. 3. 

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... double-ended tuning fork (DETF) resonators ( Fig. 1) were built to operate in flexural mode between 400 kHz and 2 MHz depending on the specific beam dimensions. The single-anchor design prevents substrate, package, and oxide stress from coupling to the beams and affecting the resonance frequency. Actuation of the symmetric mode is enforced by symmetrically placed drive electrodes on the sides of the two beams. The central electrode is used for sensing the motion of the resonator through capacitive coupling. The electrical configuration of the resonators is discussed later in Section IV-A. The resonators are fabricated in single crystal silicon using silicon-on-insulator wafers with a 20- μ m device layer. The resonators are encapsulated using the “epi-seal” encapsulation technology that has been discussed previously by Candler et al. [20]. In this process, the wafer-scale sealing is performed in an epitaxial deposition chamber at 1100 ◦ C in the presence of hydrogen and nitrogen gas and the silicon precursor dichlorosilane. This high-temperature step vaporizes native silicon dioxide, as well as any organics that are present in the resonator cavity. Such temperature- based removal of native oxide has been studied previously [21], [22]. Furthermore, the absence of oxygen and the presence of large amounts of hydrogen and nitrogen gas prevent any oxide from forming on the devices after sealing. As a result, this encapsulation technique gives us a test environment with zero native oxide. The estimated pressure inside the encapsulation is about 1 Pa [23] and, for our devices, yields quality factors of about 10 000–30 000. The devices have been shown to be very stable [23], [24] since they are protected against many of the common environmental issues such as dust, corrosive and oxidizing agents, and moisture. The primary stimulus that can physically affect the resonators at this point is the temperature, and that too can be controlled through the use of a temperature-controlled chamber. The fabrication of oxidized versions of these devices has been discussed by Melamud et al. [25]. Since thermal oxidation is a well-understood reaction-rate-limited process step, we can assume that oxidation, if carefully performed, creates a fairly uniform layer of oxide on all exposed silicon surfaces within the cavity. In addition, a variety of oxide growth conditions can be experimented with, and a variety of postoxidation treatments can be tested. For the purposes of this study, we restricted the oxidation to dry (oxygen ambient) and wet (steam grown) thermal oxides. The oxidation is performed in furnaces maintaining CMOS- grade cleanliness. Wet oxidation is performed at 1100 ◦ C in a steam ambient and takes 30 min–1 h for thick oxides. The furnace automatically performs a dry-oxidation step before and after the wet-oxidation step, resulting in a dry–wet–dry oxide stack. Dry oxidation is performed at 1100 ◦ C in a dry oxygen ambient for 5–9 h depending on the required oxide thickness. Since our thermal oxides are significantly thicker than a native oxide layer, they can survive the high-temperature sealing step. The as-designed presence of the thermal oxide in the finished devices has been confirmed through sectioning test wafers at the end of the process [25] and with the mapping of temperature–frequency characteristics [9]. In this paper, dry oxide of 0.35 μ m thickness and wet oxide of 0.42 μ m thickness are studied. III. M ODELING C HARGE E FFECTS Recent parallel work by Kalicinski et al. [26] models the effects of charge on resonator frequency. The presence of a thin native oxide is assumed on the silicon resonators, and the charge is modeled through an empirical correction to the dc bias voltage that they define as V shift . The effect of this parasitic voltage is mapped to the frequency through the equation f s ≈ f o 1 − A ( V dc /V PI ) 2 that demonstrates a quadratic dependence on the dc bias voltage ( V dc ) . This equation is derived from the work of Tilmans and Legtenberg [27] and is valid when the axial load on a resonant beam is zero, which is true for single-anchored devices. However, the pull-in voltage ( V PI ) description becomes ambiguous in cases with charged dielectrics since the observable pull-in voltage is different (and often variable) for positive and negative biasing [15], [28], [29]. The model presented in [26] assumes that the polarization of the dielectric is negligible. In general, for finite-thickness dielectrics, the voltage V shift described in the figures in [26] as the voltage across the dielectric will depend on the dc bias voltage applied to the device. The following sections present an analysis of the effects of charge and finite-thickness dielectrics based on the energy method without the need for a thin oxide simplification. We employ a different approach, where charge is associated to the change of frequency by affecting a softening modification to the effective spring constant of the resonator through f o ≈ f o (1 − k e / 2 k m ) . This method has been previously described by several studies [30]–[32] and is more general (for an arbitrary axial load, for instance) as long as the mechanical spring constant of the resonator k and its resonance frequency f can be modeled or determined. The electrostatic spring modification k e is dependent on the dc bias voltage and the geometry. Our model mathematically explains how dielectric charge changes the effective dc bias voltage applied to the device (and thus, k e ) by mapping its effects into an indirectly observable built-in voltage ( V bi in this paper, denoted V shift in [26]). This enables us, to some extent, to extract the amount of charge that is present in the system through frequency observations. The resonator beams form four capacitors with the three electrodes adjacent to them, as shown in Fig. 2. It is reasonable to treat these as parallel plate capacitors since the device layer thickness (20 μ m) and the beam length ( ∼ 200 μ m ) are much larger than the lithographically defined actuation gap of 1.5 μ m. Electrically, both stimulus and sense electrodes are held at a dc ground potential, although they do carry a small ac resonance signal. The mechanical structure is biased at a constant dc potential V bias . As a result, the dc electromechanical representation of all four of these capacitors is identical. Thus, the electrostatic pull force on each capacitor is also identical and can be modeled with the same common variables with a single- capacitor mass–spring lumped model, as shown in Fig. 3. This is a simplifying assumption for this analysis since the charge state of each individual capacitor is unknown. The following derivation goes through an analysis of the effects of charge and voltage on the electromechanical properties of a resonator that is operating in the linear regime. Although the following symbolic analysis is presented for the special case of a system where there are two symmetric dielectrics, the methodology used in this derivation is broadly applicable, and the end result can be rederived for other electrostatic actuators (such as with one dielectric, unequal dielectrics, or multiple dielectrics). It is assumed that the charge is only present at the oxide–air interface of the dielectrics. The bulk-charge case is considered in Section III-C as an extension of the result obtained for surface charge. Since the nature of the charge in the dielectrics may be unknown and the bias voltage can vary, we cannot a priori assume the polarity of the surface charges in any part of the system. Therefore, the charge is assumed to be positive in all cases. Dealing with a negative charge simply involves a sign reversal, and the solutions will yield appropriate signs when negative charge is present. The proposed solution method is to determine the total energy in the system U sys . This includes the energy stored in the field within the capacitors U fields and the contribution of the bias voltage source U V = U V 0 + ∆ U V , where U V 0 is the unknown but constant energy associated with the voltage source before the x displacement “probe” is applied. The negative gradient of this energy with respect to the displacement of the mobile structure is the attractive force acting on the capacitor plates. Then, linearizing the force equation and accounting for both transduction capacitors around the resonator beam, one can obtain an effective spring constant for the beam. The effects of charge appear as a modification to the spring-softening effect that is usually caused due to bias voltage alone. The model of one capacitor with variable definitions is shown in Fig. 3. Here, ρ s 1 and ρ s 2 are uniform surface charge sheets assumed to be present at the oxide–air interface of each dielectric layer. The oxide thickness t ox is taken to be identical on both electrodes. The portion of the transduction gap that is only air (or vacuum) has a spacing g . For this analysis, the electrodes are assumed to be highly doped and are consequently assumed metallic in behavior. Accumulation and depletion effects of silicon are not considered. The electric fields within the oxides and air gap are indicated as E ox1 , E ox2 , and E gap , respectively. In addition, ρ i 1 and ρ i 2 are the surface charge densities induced on the electrode surfaces due to the applied voltages and electric fields in the system. These charge sheets are provided by the voltage source. Using Gauss’s law and the assumed metallic nature of the electrodes, the following expression must hold true for any biasing ...