Patent Publication Number: US-7586164-B2

Title: Micro-electro-mechanical system (MEMS) variable capacitor apparatuses, systems and related methods

Description:
RELATED APPLICATION 
   This application is a divisional patent application from and claims the benefit of U.S. patent application Ser. No. 10/736,283 filed Dec. 15, 2003, now U.S. Pat. No. 7,180,145, entitled “Micro-Electro-Mechanical System (MEMS) Variable Capacitor Apparatuses and Related Methods”, which is incorporated herein by reference in its entirety; and U.S. Provisional Patent Application Ser. No. 60/433,454, filed Dec. 13, 2002, also incorporated herein by reference in its entirety. 

   TECHNICAL FIELD 
   The present subject matter relates generally to micro-electro-mechanical systems (MEMS) apparatuses and methods. More particularly, the present subject matter relates to variable capacitor apparatuses and related methods utilizing MEMS technology. 
   BACKGROUND ART 
   Micro-electro-mechanical systems (MEMS) apparatuses and methods are presently being developed for a wide variety of applications in view of the size, cost and power consumption advantages provided by these devices. Specifically, a variable capacitor, also known as a varactor, can be fabricated utilizing MEMS technology. Typically, a variable capacitor includes an interelectrode spacing (or an electrode overlap area) between a pair of electrodes that can be controllably varied in order to selectively vary the capacitance between the electrodes. In this regard, conventional MEMS variable capacitors include a pair of electrodes, one that is typically disposed upon and fixed to the substrate and the other that is typically carried on a movable actuator or driver. In accordance with MEMS technology, the movable actuator is typically formed by micromachining the substrate such that very small and very precisely defined actuators can be constructed. 
   As appreciated by persons skilled in the art, many types of MEMS variable capacitors and related devices can be fabricated by either bulk or surface micromachining techniques. Bulk micromachining generally involves sculpting one or more sides of a substrate to form desired three dimensional structures and devices in the same substrate material. The substrate is composed of a material that is readily available in bulk form, and thus ordinarily is silicon or glass. Wet and/or dry etching techniques are employed in association with etch masks and etch stops to form the microstructures. Etching is typically performed through the backside of the substrate. The etching technique can generally be either isotropic or anisotropic in nature. Isotropic etching is insensitive to the crystal orientation of the planes of the material being etched (e.g., the etching of silicon by using a nitric acid as the etchant). Anisotropic etchants, such as potassium hydroxide (KOH), tetramethyl ammonium hydroxide (TMAH), and ethylenediamine pyrochatechol (EDP), selectively attack different crystallographic orientations at different rates, and thus can be used to define relatively accurate sidewalls in the etch pits being created. Etch masks and etch stops are used to prevent predetermined regions of the substrate from being etched. 
   On the other hand, surface micromachining generally involves forming three-dimensional structures by depositing a number of different thin films on the top of a silicon wafer, but without sculpting the wafer itself. The films usually serve as either structural or sacrificial layers. Structural layers are frequently composed of polysilicon, silicon nitride, silicon dioxide, silicon carbide, or aluminum. Sacrificial layers are frequently composed of polysilicon, photoresist material, polimide, metals, or various types of oxides, such as PSG (phosphosilicate glass) and LTO (low-temperautre oxide). Successive deposition, etching, and patterning procedures are carried out to arrive at the desired microstructure. In a typical surface micromachining process, a silicon substrate is coated with an isolation layer, and a sacrificial layer is deposited on the coated substrate. Windows are opened in the sacrificial layer, and a structural layer is then deposited and etched. The sacrificial layer is then selectively etched to form a free-standing, movable microstructure such as a beam or a cantilever out of the structural layer. The microstructure is ordinarily anchored to the silicon substrate, and can be designed to be movable in response to an input from an appropriate actuating mechanism. 
   MEMS variable capacitors have been fabricated that include a movable, capacitive plate (or electrode) that is suspended above first and second coplanar electrodes. The variable capacitor operates by applying a voltage across the first electrode and the movable plate so that the plate is deflected towards the first electrode by electrostatic attraction. As the movable plate moves, the spacing between the second electrode and the movable plate changes, thus changing the capacitance value between the second electrode and the plate. A signal line is usually connected to the second electrode and the plate to sense the change in capacitance for use in various Radio Frequency functions. One problem with this configuration is that the voltage supply is electrically connected to the signal line through the plate that can result in undesirable noise/interference or degradation of the signal on the signal line. Thus, this configuration may require additional components to combine/separate the signal and actuation voltage, leading to a more complex and costly implementation. Another problem is that the RF voltage exerts an equivalent force on the movable plate to that exerted by the intended control voltage, leading to control complexity and increased intermodulation. 
   Other known MEMS variable capacitors provide parallel-plate electrodes that move linearly. The electrodes of these variable capacitors are subject to suddenly “snapping down” towards one another after moving close enough to one another. These types of variable capacitors are also subject to microphonics and stiction problems. 
   Some MEMS variable capacitors are based upon electro-thermally actuated parallel-plate design. These types of variable capacitors are subject to reduced power handling capability due to gap reduction and the likelihood for breakdown occurrence. These variable capacitors also consume excessive power, especially if the electro-thermal actuation must be applied continuously to maintain the capacitance value. 
   Other MEMS variable capacitors utilize a massively-parallel, interdigited-comb device for actuation. These variable capacitors are so sensitive to parasitic substrate capacitance that they require either a high-resistivity substrate such as glass or the removal of the substrate beneath the MEMS device. Thus, this type of variable capacitor is not readily integrated into a conventional integrated circuit (IC) process. Additionally, the MEMS device is physically large because the capacitance dependence on the overlap of comb fingers requires large aspect ratios. These devices require excessive space and cause a low resonant frequency resulting in shock and vibration problems. 
   Therefore, it is desirable to provide novel variable capacitor apparatuses and related methods for MEMS applications that improve upon aforementioned designs. 
   SUMMARY 
   It is an object to provide a MEMS variable capacitor for electrically isolating the capacitive and actuation plates. It is also an object to provide a MEMS variable capacitor for reducing electrostatic instability. Further, it is an object to at least partially mechanically decouple the movable capacitive and actuation plates of a MEMS variable capacitor. It is therefore an object to provide novel MEMS variable capacitor apparatuses and related methods. 
   Some of the objects of the present disclosure having been stated hereinabove, and which are addressed in whole or in part by the present disclosure, other objects will become evident as the description proceeds when taken in connection with the accompanying drawings as best described hereinbelow. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Exemplary embodiments will now be explained with reference to the accompanying drawings, of which: 
       FIG. 1  is a top view of an exemplary MEMS variable capacitor; 
       FIG. 2A  is a cross-section side view of one embodiment of the variable capacitor shown in  FIG. 1 ; 
       FIG. 2B  is a cross-section side view of an alternative embodiment of the variable capacitor shown in  FIG. 1 ; 
       FIG. 2C  is a cross-section side view of another alternative embodiment of the variable capacitor shown in  FIG. 1 ; 
       FIG. 3  is a cross-sectional side view of the variable capacitor shown in  FIG. 2A  with the voltage applied to a movable actuation electrode and a stationary actuation electrode set greater than 0 Volts; 
       FIG. 4  is a top perspective view of a variable capacitor including a movable component suspended above a substrate; 
       FIG. 5  is a cross-sectional side view of the variable capacitor shown in  FIG. 4 ; 
       FIG. 6  is another cross-sectional side view of the variable capacitor shown in  FIG. 4 ; 
       FIG. 7  is a top view of variable capacitor shown in  FIG. 4 ; 
       FIG. 8  is a top perspective view of the variable capacitor shown in  FIG. 4  with the voltage applied to the actuation electrodes set to a voltage greater than 0 Volts for overcoming the resistive force of tethers; 
       FIG. 9  is a computer simulation model of the z-displacement of a movable component at its first resonance mode; 
       FIG. 10  is a computer simulation model of the z-displacement of a movable component versus actuation voltage; 
       FIG. 11  is a computer simulation model of the z-displacement of movable component for an actuation voltage of about 25 Volts; 
       FIG. 12  is a graph showing capacitance (pF) between stationary capacitive electrode and movable capacitive electrodes versus voltage applied to electrodes shown in  FIG. 5 ; 
       FIG. 13A  is a computer simulation model of the deformation of movable component for a residual stress value of 120 MPa; 
       FIG. 13B  is a computer simulation model of deformation of a movable component under a stress gradient between +10 and −10 MPa; 
       FIG. 14  is a computer simulation model of an equivalent circuit of the variable capacitor shown in  FIG. 4 ; 
       FIG. 15A ,  15 B, and  15 C are computer simulation models of deformation of movable component under a stress gradient between +10 and −10 MPa; 
       FIG. 16  is a computer simulation model of an exemplary elliptically-shaped interior portion with the same area under the same stress gradients; 
       FIG. 17  is a computer simulation model of the deformation of an interior portion for an acceleration of 100 g; 
       FIG. 18  is a graph showing different tether lengths and peripheral portion widths versus actuation voltage for a variable capacitor; 
       FIG. 19  is a graph showing different tether lengths and peripheral portion widths versus resonance frequency for a variable capacitor; 
       FIG. 20  is a computer simulation model of the z-displacement of the first resonance mode of a variable capacitor having a tether length of 75 micrometers and peripheral portion width of 75 micrometers; 
       FIG. 21  is a computer simulation model of the z-displacement of a variable capacitor having a tether length of 75 micrometers and peripheral portion width of 75 micrometers at an actuation voltage set at 14 Volts; 
       FIG. 22  is a graph showing displacement of the center of a variable capacitor having a tether length of 75 micrometers and peripheral portion width of 75 micrometers versus voltage applied to the actuation electrodes; 
       FIG. 23  is a computer simulation model of the z-displacement of an interior portion of a variable capacitor exposed to a temperature difference; 
       FIG. 24  is another computer simulation model of the z-displacement of an interior portion of a variable capacitor exposed to a temperature difference; 
       FIG. 25  is a computer simulation model of the deformation of an interior component having a tether length of 75 micrometers and peripheral portion width of 75 micrometers for an acceleration of 100 g; 
       FIG. 26  is a top perspective view of another exemplary variable capacitor; 
       FIG. 27A  is a cross-sectional side view of one aperture; 
       FIG. 27B  is a cross-sectional side view of another aperture; 
       FIG. 28  is a graph showing the cut-off frequency of a variable capacitor versus the number of apertures in an interior portion of the variable capacitor; 
       FIG. 29  is a graph showing the damping and tether forces versus frequency for different number of apertures; 
       FIG. 30  is a graph showing the damping coefficient as a function of the frequency for different numbers of apertures; 
       FIG. 31  is a graph showing harmonic analysis of a variable capacitor having an interior portion with 37 apertures; 
       FIG. 32  is a graph showing the cut-off frequency for different aperture numbers; 
       FIG. 33  is a graph showing the effective area of the capacitive electrode as a function of the number of apertures for four different cases according to the minimum distance between the gold layer and the opening of an interior portion; 
       FIG. 34  is a top view of exemplary cascade arrangement of a plurality of variable capacitors; 
       FIG. 35  is a top view of exemplary cascade of plurality of variable capacitors in a fanned-shape arrangement; 
       FIG. 36A  is a computer simulation model for an equivalent circuit of four variable capacitors arranged in parallel; 
       FIG. 36B  is the RF results of the computer simulation shown in  FIG. 36B ; 
       FIG. 37A  is a top perspective view of another exemplary variable capacitor utilizing a rectangular geometry including a suspended, movable component; 
       FIG. 37B  is a top perspective view of another exemplary variable capacitor including a suspended, movable component; 
       FIG. 38  is a cross-sectional side view of a variable capacitor having isolation bumps; 
       FIG. 39  is a cross-sectional side view of the variable capacitor shown in  FIG. 38  when actuation voltage has been applied to the actuation electrodes; 
       FIG. 40  is a cross-sectional side view of another variable capacitor having an isolation bump; 
       FIG. 41  is a cross-sectional side view of another variable capacitor having an isolation bump; 
       FIG. 42  is cross-sectional side view of another variable capacitor having isolation bumps; 
       FIG. 43  is a top perspective view of variable capacitor; 
       FIG. 44A  is a cross-sectional side view of the variable capacitor shown in  FIG. 43 ; 
       FIG. 44B  is a cross-sectional side view of an alternative embodiment of the variable capacitor shown in  FIG. 43 ; 
       FIG. 44C  is a cross-sectional side view of another alternative embodiment of the variable capacitor shown in  FIG. 43 ; 
       FIG. 45  is a cross-sectional side view of variable capacitor in an actuated mode; 
       FIG. 46  is a graph showing the harmonic behavior for variable capacitor; 
       FIG. 47  is a graph showing the frequency response for different distances of the movable actuation electrodes and the movable capacitive electrodes shown in  FIG. 43 ; 
       FIG. 48  is a top view of a schematic diagram of another examplary torsional variable capacitor; 
       FIG. 49  is a computer simulation model of deformation of a torsional variable capacitor of an array of 16 variable capacitors; 
       FIG. 50  is a graph showing the capacitance of a torsional variable capacitor versus an applied actuation voltage; 
       FIG. 51  is a computer simulation model of deformation of a movable component of a torsional variable capacitor under a stress gradient between +1 and −1 MPa; 
       FIG. 52  is a computer simulation model of the deformation of a movable component in a torsional variable capacitor for an acceleration of 100 g; 
       FIG. 53A  is a computer simulation model for an equivalent circuit of a torsional variable capacitor; and 
       FIG. 53B  is the RF results of the computer simulation model shown in  FIG. 53A . 
   

   DETAILED DESCRIPTION 
   It is understood that when a component such as a layer, substrate, contact, interconnect, electrode, capacitive plate, or conductive line is referred to herein as being deposited or formed “on” another component, that component can be directly on the other component or, alternatively, intervening components (for example, one or more buffer or transition layers, interlayers, electrodes or contacts) can also be present. Furthermore, it is understood that the terms “disposed on”, “attached to” and “formed on” are used interchangeably to describe how a given component is positioned or situated in relation to another component. Therefore, it will be understood that the terms “disposed on”, “attached to” and “formed on” do not introduce any limitations relating to particular methods of material transport, deposition, or fabrication. 
   Contacts, interconnects, electrodes, capacitive plates, conductive lines, and other various conductive elements of various metals can be formed by sputtering, CVD, or evaporation. If gold, copper, nickel or Permalloy™ (Ni x Fe y ) is employed as the metal element, an electroplating process can be carried out to transport the material to a desired surface. The chemical solutions used in the electroplating of various metals are generally known. Some metals, such as gold, might require an appropriate intermediate adhesion layer to prevent peeling. Examples of adhesion material often used include chromium, titanium, or an alloy such as titanium-tungsten (TiW). Some metal combinations can require a diffusion barrier to prevent a chromium adhesion layer from diffusing through gold. Examples of diffusion barriers between gold and chromium would include platinum or nickel. 
   Conventional lithographic techniques can be employed in accordance with micromachining of the variable capacitors. Accordingly, basic lithographic process steps such as photoresist application, optical exposure, and the use of developers are not described in detail herein. 
   Similarly, generally known-etching processes can be employed to selectively remove material or regions of material. An imaged photoresist layer is ordinarily used as a masking template. A pattern can be etched directly into the bulk of a substrate, or into a thin film or layer that is then used as a mask for subsequent etching steps. 
   The type of etching process employed in a particular fabrication step (e.g., wet, dry, isotropic, anisotropic, anisotropic-orientation dependent), the etch rate, and the type of etchant used will depend on the composition of material to be removed, the composition of any masking or etch-stop layer to be used, and the profile of the etched region to be formed. As examples, poly-etch (HF:HNO 3 :CH 3 COOH) can generally be used for isotropic wet etching. Hydroxides of alkali metals (e.g., KOH), simple ammonium hydroxide (NH 4 OH), quaternary (tetramethl) ammonium hydroxide ((CH 3 ) 4 NOH, also known commercially as TMAH), and ethylenediamine mixed with pyrochatechol in water (EDP) can be used for anisotropic wet etching to fabricate V-shaped or tapered grooves, trenches or cavities. Silicon nitride is typically used as the masking material against ethcing by KOH, and thus can be used in conjunction with the selective etching of silicon. Silicon dioxide is slowly etched by KOH, and thus can be used as a masking layer if the etch time is short. While KOH will etch undoped silicon, heavily doped (p++) silicon can be used as an etch-stop against KOH as well as the alkaline etchants and EDP. The preferred metal used to form contacts and interconnects is gold, which is resistant to EDP. The adhesion layer applied in connection with forming a gold component (e.g., chromium) is also resistant to EDP. 
   It will be appreciated that electrochemical etching in hydroxide solution can be performed instead of timed wet etching. For example, if a p-type silicon wafer is used as a substrate, an etch-stop can be created by epitaxially growing an n-type silicon end layer to form a p-n junction diode. A voltage is applied between the n-type layer and an electrode disposed in the solution to reverse-bias the p-n junction. As a result, the bulk p-type silicon is etched through a mask down to the p-n junction, stopping at the n-type layer. Furthermore, photovoltaic and galvanic etch-stop techniques are also suitable. 
   Dry etching techniques such as plasma-phase etching and reactive ion etching (RIE) can also be used to remove silicon and its oxides and nitrides, as well as various metals. Deep reactive ion etching (DRIE) can be used to anisotropically etch deep, vertical trenches in bulk layers. Silicon dioxide is typically used as an etch-stop against DRIE, and thus structures containing a buried silicon dioxide layer, such as silicon-on-insulator (SOI) wafers, can be used as starting substrates for the fabrication of microstructures. 
   An alternative patterning process to etching is the lift-off process. In this case, the conventional photolithography techniques are used for the negative image of the desired pattern. This process is typically used to pattern metals, which are deposited as a continuous film or films when adhesion layers and diffusion barriers are needed. The metal is deposited on the regions where it is to be patterned and on top of the photoresist mask (negative image). The photoresist and metal on top are removed to leave behind the desired pattern of metal. 
   As used herein, the term “device” is interpreted to have a meaning interchangeable with the term “component”. 
   As used herein, the term “conductive” is generally taken to encompass both conducting and semi-conducting materials. 
   Examples of the methods of the present subject matter will now be described with reference to the accompanying drawings. 
   Referring to  FIGS. 1-3 , different views of an exemplary MEMS variable capacitor, generally designated  100 , are illustrated.  FIG. 1  illustrates a top view of variable capacitor  100  including a movable component MC suspended over a substrate (designated  200  in  FIG. 2A ). Movable component MC can include a movable actuation electrode MAE and a movable capacitive electrode MCE disposed on a top surface thereof. Alternatively, movable actuation electrode MAE and a movable capacitive electrode MCE can be connected to a bottom surface of movable component MC or the top and bottom surfaces can each include a movable actuation electrode and a movable capacitive electrode. Additionally, one of movable actuation electrode MAE and a movable capacitive electrode MCE can be a completely conducting section of movable component MC rather than a layer. Movable component MC can comprise one or more layers of silica, alumina, un-doped semiconductors, polymers, and other non-conductive materials known to those of skill in the art. The material of movable component MC can function to electrically isolate actuation electrode MAE from capacitive electrode MCE and provide flexibility for deflecting. 
   Movable component MC can include a plurality of tethers T 1 , T 2 , T 3 , and T 4  connected to movable component MC for attaching movable component MC to posts (shown in  FIG. 2A ) or other suitable support structures, which may be the structural layer of movable component MC with a step formed by the edge of the sacrificial layer during fabrication. If the process is planarized, the support can be the whole “field” where the top surface is nearly planar. Posts P 1  and P 2  can be rigidly attached to a surface S of substrate  200  (shown in  FIG. 2A ). In this embodiment, tethers T 1 , T 2 , T 3 , and T 4  extend along an at least substantially straight line that can be at least substantially perpendicular to a line extending from a center C of movable component MC to the connection of tether T 1  to movable component MC. For example, tether T 1  extends along broken line  102 . Broken line  104  extends from center C to the point of attachment for tether T 1  and movable component MC. Broken lines  102  and  104  are at least substantially perpendicular. Alternatively, broken lines  102  and  104  can be at other suitable angles with respect to one another. Tethers T 1 , T 2 , T 3 , and T 4  can function as stress decouplers, in order to reduce the effects of in-plane stresses such as residual stress, mounting stress and thermal expansion mismatch stress. Additionally, tethers T 1 , T 2 , T 3 , and T 4  can reduce the impact of the gradient of the out-of-plane distribution of these in-plane stresses. Tethers T 1 , T 2 , T 3 , and T 4  can also reduce the impact of the average of the out-of-plane distribution. This results in making variable capacitor  100  less sensitive to process tolerances related to stress control. A voltage supply and signal line can be used to connect movable actuation electrode MAE and movable capacitive electrode MCE as shown with reference to subsequent figures. 
     FIG. 2A  illustrates a cross-sectional side view of one embodiment of variable capacitor  100 . Variable capacitor  100  can include substrate  200  comprising one or more layers, composites, or other combinations of silicon, alumina, silica, polymers and other suitable substrate materials known to those of ordinary skill in the art. A stationary actuation electrode SAE can be formed on surface  104  of substrate and positioned directly beneath movable actuation electrode MAE. Electrodes SAE and MAE can be connected to a voltage supply VS via conductive lines  106  and  108 , respectively. Voltage supply VS can apply a voltage across electrodes SAE and MAE. An equal and opposite electrical charge develops on electrodes SAE and MAE upon the application of a voltage. The equal and opposite electrical charge causes an electrostatic force to pull movable actuation electrode MAE, and movable component MC, towards stationary actuation electrode SAE in a direction indicated by direction arrow  110 . Tethers T 1 , T 2 , T 3 , and T 4  can produce a biasing force to oppose movement of movable component MC in direction indicated by arrow  110 . Movable component MC can move towards substrate  200  only when the voltage applied across electrodes SAE and MAE is great enough to overcome the resistive force of tethers T 1 , T 2 , T 3 , and T 4 . The voltage applied across electrodes SAE and MAE can be increased to deflect electrode MAE closer to electrode SAE than another position. Thus, the gap distance between electrodes SAE and MAE can be adjusted by controlling the voltage output by voltage supply VS. The voltage applied by voltage supply VS can be varied directly by an operator or other suitable electrical circuitry known to those of skill in the art for controlling the voltage output by a voltage supply. Movable component MC is shown in position when the voltage applied by voltage supply VS is 0 volts. 
   Variable capacitor  100  can also include a stationary capacitive electrode SCE attached to a base portion  112  disposed on substrate  200 . Stationary capacitive electrode SCE can be positioned closer to movable component MC than stationary actuation electrode SAE, spaced apart vertically from stationary actuation electrode SAE, and immediately above base portion  112 . Electrode SCE can be positioned directly below electrode MCE. Electrodes SCE and MCE can be electrically connected to a signal line SL for supplying a signal, typically AC, to variable capacitor VC from other electrical circuitry (not shown). Signal line SL can comprise of a highly-conductive metal such as gold, aluminum, silver, copper, or the like. Signal line SL can be connected to a high-frequency distribution network with minimum fixed capacitance. Typically, the electrical circuitry connected to signal line SL is sensitive to capacitance of variable capacitor  100 . Capacitive electrodes MCE and SCE can be moved to different positions with respect to one another when voltage is applied to actuation electrodes MAE and SAE for moving movable component MC. Capacitive electrodes SCE and MCE and actuation electrodes SAE and MAE can comprise any suitable type of metal, semi-metal, or doped semiconductor. Capacitive electrodes SCE and MCE can comprise a highly conductive metal, such as copper, gold, silver, aluminum, or the like. 
     FIG. 2B  illustrates a cross-sectional side view of an alternative embodiment of variable capacitor  100 . In this embodiment, movable component MC comprises a first portion  200  and a second portion  202 , wherein second portion  202  is positioned closer to substrate  102  than first portion  200 . Therefore, movable actuation electrode MAE and stationary actuation electrode SAE can be positioned further apart than the distance between movable capacitance electrode MCE and stationary capacitance electrode SCE to its attachment to first portion  200  because movable actuation electrode MAE is positioned on raised first portion  200 . The dual gap can be formed by two different thicknesses of sacrificial layer. 
     FIG. 2C  illustrates a cross-sectional side view of another alternative embodiment of variable capacitor  100 . In this embodiment, stationary actuation electrode SAE is buried in substrate  102 . Therefore, movable actuation electrode MAE and stationary actuation electrode SAE can be positioned further apart than the distance between movable capacitance electrode MCE and stationary capacitance electrode SCE to its attachment to first portion  200  because stationary actuation electrode SCE is buried in substrate  102 . The dual gap can be formed by two different thicknesses of sacrificial layer. 
   Additionally, in another alternative of  FIG. 2B , stationary capacitive electrode SCE can be positioned parallel with stationary actuation electrode SAE on substrate  102  such that electrode SCE and SAE are not in electrical communication. In this embodiment, the distance between capacitive electrodes MCE and SCE can be about 0.5 micrometers. Additionally, the distance between actuation electrodes MAE and SAE can be about 2.0 micrometers. 
   Referring to  FIG. 2C  illustrates a cross-sectional side view of another alternative embodiment of variable capacitor  100 . In this embodiment, stationary actuation electrode SAE is attached directly onto the top surface of substrate  102 . Stationary actuation electrode SAE can be buried in substrate  102 . This positioning can increase the distance between stationary actuation electrode SAE and movable actuation electrode MAE without adding the complexity of additional sacrificial layers. Substrate  102  can comprise a dielectric or other suitable substrate material. 
     FIG. 3  illustrates a cross-sectional side view of the embodiment of variable capacitor  100  shown in  FIG. 2A  with the voltage applied to electrodes MAE and SAE set greater than 0 Volts. With the applied voltage set greater than 0 Volts, movable component MC can be positioned closer to substrate  200  than when the applied voltage is set to 0 (as shown in  FIG. 2 ). 
   Referring to  FIGS. 4-8 , different views of an exemplary hexagonal-shaped implementation of a variable capacitor, generally designated  400 , are illustrated.  FIG. 4  illustrates a top perspective view of variable capacitor  400  including a movable component MC suspended above a substrate  402 . Movable component MC can include movable actuation electrodes MAE 1  and MAE 2  and a movable capacitive electrode MCE 1  attached to a top surface  404  of movable component MC. 
   Referring to  FIG. 4 , movable component MC can include a peripheral portion  406  and an interior portion  408 . In this embodiment, peripheral portion  406  is hexagonal in shape with a hollow interior for enclosing interior portion  408 . Interior portion  408  can be attached to peripheral portion  406  with connectors  410  and  412 . There should be at least two connectors according to this embodiment. The exact number of connectors in alternative embodiments can depend on the geometry and design rules of a specific design and process. Peripheral portion  406  can be attached to substrate  402  via a plurality of tethers T 1 , T 2 , T 3 , T 4 , T 5 , and T 6 . Tethers T 1 , T 2 , T 3 , T 4 , T 5 , and T 6  can include ends  414 ,  416 ,  418 ,  420 ,  422 , and  424 , respectively, attached to posts (shown in  FIG. 5 ). The posts or other support structures can be rigidly attached to substrate  402 . 
     FIG. 5  illustrates a cross-sectional side view of variable capacitor  400 . Variable capacitor  400  can include posts P 1  and P 2  or other suitable support structures for attachment to tethers T 3  and T 6 , respectively. Tethers T 1 , T 2 , T 4 , and T 5  (shown in  FIG. 4 ) can also be attached to posts (not shown) such as posts P 1  and P 2  for attachment to substrate  402 . Movable component MC can also include movable actuation electrodes MAE 3  and MAE 4  attached to bottom surface  500  and opposing electrodes MAE 1  and MAE 2 , respectively. Movable component MC can also include a movable capacitive electrode MCE 2  attached to bottom surface  500  and opposing electrode MCE 1 . Additionally, a movable actuation electrode (such as movable actuation electrode MAE 3 ) can be positioned on movable component MC directly opposing movable capacitive electrode MCE 1 . Electrodes MAE 1  and MAE 3  can be in electrical communication via a conductive interconnect CI 1  extending through movable component MC. Electrodes MAE 2  and MAE 4  can be in electrical communication via a conductive interconnect CI 2  extending through movable component MC. Electrodes MCE 1  and MCE 2  can be in electrical communication via a conductive interconnect CI 3  extending through movable component MC. Electrodes MAE 1 , MAE 2 , MAE 3 , MAE 4 , MCE 1 , MCE 2  can comprise the same conductive material and be matched in shape and dimension to its opposing counterpart on movable component MC for mechanical stress matching of interior portion  408  ( FIG. 4 ) of movable component MC. Alternatively, electrodes MAE 1 , MAE 2 , MAE 3 , MAE 4 , MCE 1 , MCE 2  can have different suitable shapes and comprise different materials for providing desired stress matching. 
   Variable capacitor  400  can also include stationary actuation electrodes SAE 1  and SAE 2  positioned on the top surface of substrate  402  and beneath movable actuation electrodes MAE 1  and MAE 2 , respectively. Alternatively, movable actuation electrodes MAE 1  and MAE 2  can comprise a single actuation electrode as can be appreciated by one of skill in the art. Variable capacitor  400  can also include a stationary capacitive electrode SCE positioned on the top surface of substrate  402  and beneath movable capacitive electrode MCE 2 . Movable actuation electrodes MAE 1 , MAE 2 , MAE 3 , and MAE 4  can be connected to a voltage supply VS via conductive line CL 1 . Stationary actuation electrodes SAE 1  and SAE 2  can be connected to voltage supply VS via conductive line CL 2 . Voltage supply VS can apply one voltage potential at movable actuation electrodes MAE 1  and MAE 2  and a different voltage potential at stationary actuation electrodes SAE 1  and SAE 2 . The equal and opposite electrical charge causes an electrostatic force to pull movable actuation electrodes MAE 1 , MAE 2 , MAE 3 , and MAE 4 , and movable component MC, towards stationary actuation electrodes SAE 1  and SAE 2  in a direction indicated by direction arrow  502 . Tethers T 1 , T 2 , T 3 , T 4 , T 5 , and T 6  can produce a biasing force to oppose movement of movable component MC in direction indicated by arrow  502 . Movable component MC can move towards substrate  402  only when the voltage applied across the stationary actuation electrodes (SAE 1  and SAE 2 ) and the movable actuation electrodes (MAE 1 , MAE 2 , MAE 3 , and MAE 4 ) is great enough to overcome the resistive force of tethers T 1 , T 2 , T 3 , T 4 , T 5 , and T 6 . Movable component MC is shown in position when the voltage applied by voltage supply VS is 0 Volts. In this embodiment, when voltage supply VS is 0 Volts, movable capacitive electrode MCE 2  is separated from stationary capacitive electrode by about 0.5 micrometers. Additionally, in this embodiment, when voltage supply VS is 0 Volts, movable actuation electrodes MAE 3  and MAE 4  can be separated from SAE 2  and SAE 1 , respectively, by about between 1.5 and 2.0 micrometers. 
   Variable capacitor  400  can also include a stationary capacitive electrode SCE attached to the top surface of substrate  402  and beneath movable capacitive electrode MCE 1  and MCE 2 . Electrodes SCE, MCE 1 , and MCE 2  can be electrically connected to a signal line SL for supplying a signal, typically AC, to variable capacitor  400  from other electrical circuitry (not shown). Movable capacitive electrodes MCE 1  and MCE 2  can be moved to different positions with respect to stationary capacitive electrode SCE when voltage is applied to movable actuation electrodes (MAE 1 , MAE 2 , MAE 3 , and MAE 4 ) and stationary actuation electrodes (SAE 1  and SAE 2 ) for moving movable component MC such that capacitance is changed between movable capacitive electrodes MCE 1  and MCE 2  and stationary capacitive electrode SCE. 
   Referring to  FIG. 6 , another cross-sectional side view of variable capacitor  400  is illustrated. The voltage applied across movable actuation electrodes (MAE 1 , MAE 2 , MAE 3 , and MAE 4 ) and stationary actuation electrodes (SAE 1  and SAE 2 ) is greater than a 0 Volts for overcoming the resistive force of tethers T 1 , T 2 , T 3 , T 4 , T 5 , and T 6 . With the applied voltage set greater than 0 Volts, peripheral portion  406  can be positioned closer to substrate  402  than when the applied voltage is set to 0 (as shown in  FIG. 2 ). Interior portion  408  can also move closer to substrate  402  when peripheral portion  406  is moved towards substrate  402  due to the attachment of interior portion  408  to peripheral portion  406  with connectors  410  and  412 . 
   Interior portion  408  can be substantially, mechanically isolated from peripheral portion  406  because interior portion  408  is only attached to peripheral portion  406  via connectors  410  and  412 . Therefore, the deformation of interior portion  408  is substantially limited when its peripheral portion  406  moves towards substrate  402 . If only two connectors are used as in this exemplary embodiment, connectors  410  and  412  can include a cross-sectional area large enough to suppress torsional motion. According to one embodiment connectors  410  and  412  are substantially wider than the thickness of movable component MC and substantially shorter than they are wide. Connectors  410  and  412  can range in width between 0.5 micrometers and  100  micrometers. The thickness of movable component MC can be between about 0.5 and 20 microns. The width of connectors  410  and  412  can be greater than 5 times the thickness. The length of connectors  410  and  412  can be about 5 micrometers. This is advantageous because interior portion  408  and its attached movable capacitive electrode MCE can remain substantially planar when moved towards substrate  402 . 
   Referring to  FIG. 7 , a top view of variable capacitor  400  is illustrated. Movable capacitive electrode MCE 1  can be connected to signal line SL via conduits C 1  and C 2  disposed on top of movable component MC. Conduits C 1  and C 2  can extend from movable capacitive electrode MCE along tethers T 6  and T 3 , respectively, for connection to signal line SL. 
   Referring to  FIG. 7 , movable capacitive electrode MCE 1  can have a hexagonal shape with a diameter d 1  of between about 25 micrometers and 2 millimeters. In one embodiment, peripheral component  406  has a width of about 45 micrometers. Alternatively, peripheral component  406  can range between 25 micrometers and 1 millimeter. Tethers T 1 , T 2 , T 3 , T 4 , T 5 , and T 6  can have a length between about 100 and 250 micrometers. 
     FIG. 8  illustrates a top perspective view of variable capacitor  400  with the voltage applied to electrodes MAE 1 , MAE 2 , MAE 3 , and MAE 4  and SAE is set to a voltage greater than 0 Volts for overcoming the resistive force of tethers T 1 , T 2 , T 3 , T 4 , T 5 , and T 6 . With the applied voltage set to a voltage greater than 0 Volts, movable component MC can be positioned closer to substrate  200  than when the applied voltage is set to 0 (as shown in  FIG. 2 ). 
   Simulations have demonstrated that the embodiment shown in  FIGS. 4-7  can achieve a high impedance control input (with minimum leakage up to about 100 Volts), an operating frequency of between 0 and 10 GHz, a series resistance of less than 0.5 ohms and typically less than 0.2 ohms, a vibration sensitivity of less than 0.5% capacitance variation for 0.3 g@1 kHz, and a control input cut-off frequency of greater and 20 kHz. 
   One important consideration concerns the harmonic behavior of the variable capacitor. The variable capacitor is typically operated in normal air conditions with a very small air gap (between about 0.5 and 0.01 micrometers). When the movable component acts as a piston, the air in the air gap between the movable component and the substrate can act as a squeeze-film and its effects can be strongly dependent on the frequency of the motion. Apertures can be formed in a movable component to reduce the effects of the air in the air gap between the movable component and the substrate. 
   The quality of resonance (Q) can also be measured for the embodiment shown in  FIGS. 4-7 . Generally, Q refers to power dissipation/(energy stored*radian frequency). There are two resonance qualities of interest with regard to this embodiment. One resonance quality of interest is the mechanical quality of resonance of movable component MC. This can typically be low due to air damping. However, if it is too low, it will slow down the response of variable capacitor  400 . A mechanical quality Q on the order of unity is desirable. This can be designed through the gap selected between movable component MC and substrate  402  and spacing in movable component MC and size/quantities of apertures (described below). 
   Another resonance quality Q is the electrical resonance quality of variable capacitor  400 . To first order, this resonance quality Q is provided by the following equation: 
           (     radian   ⁢           ⁢   frequency   ⁢           ⁢     1       *     ⁢   capacitance   ⁢     *         ⁢           ⁢   series   ⁢           ⁢   resistance     )         
This quality of resonance Q should be as high as possible, such as greater than 100. This can be achieved with a low resistance conduit.
 
   Another key parameter is the tuning ratio which is the ratio between the maximum and minimum capacitances achievable by the variable capacitor. This should be as high as possible with a value greater than 4 being useful and a value greater than 8 considered very desirable. This is achieved by enabling the gap between movable capacitive electrode MCE and station capacitive electrode SCE to be varied stably over a wide range and by low parasitics such as fixed capacitances at the edges and at the conduits. 
   A mechanical resonance frequency calculation can be performed for a small-signal excitation at an “undeformed” state of variable capacitor with voltage set to 0. Damping effects can be considered. Additionally, experiments demonstrate that the variable capacitor embodiment shown in  FIG. 4-7  can have a resonance frequency above 20 kHz. The first resonance mode occurs at 21.6 kHz. The displacement of movable component MC with respect to substrate  402  (z-displacement) for this mode is a “flapping” mode. 
     FIG. 9  illustrates a computer simulation model of the z-displacement of movable component MC at the first resonance mode of 21.6 kHz. The edges of movable component MC exhibit the largest displacement. The edges are in phase, meaning that the two edges are moving in the same direction. A second resonance mode occurs at 23.4 kHz. The second resonance mode is a “torsional” mode, where the edges move out-of-phase (one edge goes up while the other edge goes down). 
     FIG. 10  illustrates a graph showing displacement of center C (μm) of movable component MC versus voltage applied to electrodes MAE and SAE.  FIG. 11  illustrates a computer simulation model of the z-displacement of movable component MC for an actuation voltage of about 25 Volts. Although a gap ratio of 3 is nominally stable for parallel plate actuation, the deformation of the plates during actuation creates non-planarity and thus introduces instability. This is solved by increasing the gap ratio to grater than 3 to provide margin. However, increasing the gap ratio also increases the control voltage for a given capacitor gap so it should not be increased more than necessary. Typical embodiments have gap ratios of about 4. Deformation is not due to the electrostatic force acting on interior portion  408  (shown in  FIG. 4 ), but due to the tilt of peripheral portion  406  (shown in  FIG. 4 ) at points where interior portion  408  is attached to peripheral portion  406  (i.e., where connectors  410  and  412  shown in  FIG. 4  contact interior portion  408 ). Bending of moving capacitive electrodes MCE 1  and MCE 2  can have an adverse effect on the capacitance value. 
   Referring to again  FIG. 5 , any radio frequency (RF) signals on signal line SL can generate an electrical force on movable component MC due to the electrical charge generated on stationary capacitive electrode SCE and movable capacitive electrodes MCE 1  and MCE 2 . Because the electrical force is related to the square of the voltage and the area of actuation, the AC voltage can introduce a net DC force between stationary capacitive electrode SCE and movable capacitive electrodes MCE 1  and MCE 2 . For example, when an RF-signal of 0.5 V pp  is applied, the equivalent of a 0.18 DC Volts is applied between stationary capacitive electrode SCE and movable capacitive electrodes MCE 1  and MCE 2 . For example, when 15 Volts is applied over an air-gap of 1.5 micrometers, an equivalent pressure of about 885 Pa is generated. In contrast, for example, when 0.18 Volts is applied over an air-gap of 0.5 micrometers, an equivalent pressure of about 1.15 Pa is generated. Even for a displacement as high as 0.4 micrometers, the equivalent pressure of actuation electrodes MAE 1 , MAE 2 , MAE 3 , MAE 4 , SAE 1 , and SAE 2  is about 1645 Pa. In contrast, the equivalent pressure from 0.18 Volts applied over the remaining 0.1 micrometers is 29 Pa. Therefore, movable component MC position is primarily determined by actuation voltage until the RF gap is very small as long as the areas of the actuation electrodes are on the order of or significantly larger than the area of the capacitance electrodes. 
     FIG. 12  illustrates a graph showing capacitance (pF) between stationary capacitive electrode SCE and movable capacitive electrodes MCE 1  and MCE 2  versus voltage applied to electrodes SAE 1 , SAE 2 , MAE 1 , MAE 2 , MAE 3 , and MAE 4  shown in  FIG. 5 . The minimum capacitance in this embodiment with actuation voltage set at 0 Volts is about 2.1 pF. The capacitance ratio is about 1:3.6. 
   The robustness of variable capacitor  400  (shown in  FIG. 4 ) against residual stress deformations can be a good indicator of the robustness of variable capacitor  400  against temperature changes. Allowing movable component MC to rotate to a certain degree generates most of the residual stress effects only in the XY plane.  FIGS. 13A and 13B  illustrate different computer simulation models of the deformation of movable component MC.  FIG. 13A  illustrates a computer simulation model of the deformation of movable component MC for a residual stress value of  120  MPa (uniform stress across movable component MC). The displacement in the x and y directions in this example are smaller than 0.5 micrometers while displacement in the z direction is as small as 0.001 micrometers. Thus, the capacitance in this example is not adversely affected by either the residual stress or the difference in thermal expansion between the movable component MC and substrate  402  (shown in  FIG. 4 ). 
   The robustness of movable component MC (shown in  FIG. 4 ) against stress gradients is also important. As referred to herein, the stress gradient means the varying of the residual and thermal stress levels across the thickness of movable component MC. Stress gradients can typically range between 1 and 10 MPa.  FIG. 13B  illustrates a computer simulation model of deformation of movable component MC under a stress gradient between +10 and −10 MPa. The warping of interior portion  408  can have a great impact on the capacitance and capacitance ratio of variable capacitor  400  (shown in  FIG. 4 ). 
     FIG. 14  illustrates a computer simulation model, generally designated  1400 , of an equivalent circuit of variable capacitor  400  shown in  FIG. 4 . In this example, the SABER™ simulator (available from Analogy, Inc. of Beaverton, Oreg.) can be used for modeling variable capacitor  400 . Simulation model  1400  can include six beams for the tethers  1402 ,  1404 ,  1406 ,  1408 ,  1410 , and  1412  and associated beams with electrodes  1414 ,  1416 ,  1418 ,  1420 ,  1422 , and  1424 , respectively. Simulation model  1400  can also include a connector models  1426  and  1428 , and a capacitive electrode model CEM. 
     FIGS. 15A ,  15 B, and  15 C illustrate computer simulation models of the deformation of different interior portions (such as interior portion  408  shown in  FIG. 4 ) under a stress gradient between + and −MPa.  FIG. 15A  illustrates a square-shaped interior portion  1500  under the stress gradient.  FIG. 15B  illustrates a hexagonal-shaped interior portion  1502  under the stress gradient.  FIG. 15C  illustrates a circular-shaped interior portion  1504  under the stress gradient. Table 1 below indicates the maximum and minimum z-displacements for each of the three interior portion shapes. 
                   TABLE 1                  Maximum and minimum z-displacements for       three interior portion shapes                                 Maximum Z   Minimum Z   Delta Z                                                 square   1.893 μm   −0.804 μm   2.697 μm           hexagon   1.420 μm   −0.933 μm   2.353 μm           circle   1.256 μm   −0.930 μm   2.186 μm                        
Based on the simulation results, square-shaped interior portion  1500  provides the largest maximum displacement, which is located at the corners. The center of square-shaped interior portion  1500  provides the smallest displacement of the three interior portion shapes. This result is explained by the fact that the axis of square-shaped interior portion  1500 , with the same total area, is shorter than for the other two shapes. The average displacement is an indication of the sensitivity of the capacitance to stress gradients. For the maximum displacement, the most robust shape against stress gradient is the circular plate with the hexagonal design of  FIG. 4  being nearly as good.
 
   An interesting observation is that the iso-displacement curves are elliptical.  FIG. 16  illustrates a computer simulation model of an exemplary elliptically-shaped interior portion  1600  with the same area under the same stress gradients. In this example, the edges of the ellipse do not move upwards or downwards: the bending of the axis and the bending perpendicular to the axis compensate each other along the elliptical contour. The center of elliptically-shaped interior portion  1700  is almost 1.4 μm below the zero displacement point. The capacitance change is higher in elliptically-shaped interior portion  1700  than in circular-shaped interior portion  1504  (shown in  FIG. 15C ). 
   A low sensitivity to acceleration is an important requirement for varactor capacitor. In particular, the change of the capacitance due to vibration or acceleration is expected to be an important source of noise for the variable capacitor. For computer simulations, a constant acceleration was applied to an undeformed interior portion (such as interior portion  408  shown in  FIG. 4 ). Several values can be considered, showing a linear behavior of the displacement even for relatively high values of acceleration.  FIG. 17  illustrates a computer simulation model of the deformation of an interior portion  1700  for an acceleration of 100 g. The center displacement of interior portion  1700  is about 0.12 μm. Therefore, the acceleration sensitivity is about 1.2 [nm/g]. For a constant acceleration of 0.3 g, such as the value expected for the vibration, the maximum displacement is only 3.6 Å. The capacitance change under these conditions is lower than 0.5%. From the mechanical perspective, the cut-off frequency for the mechanical Low-Pass-Filter can be targeted to be higher than 20 kHz. Therefore, the response of interior portion  1700  to the acceleration will be fairly independent of the frequency, up to 20 kHz. In other words, a vibration of 0.3 g at 1 kHz will provide a capacitance change up to 0.5% of the capacitance. 
   Table 2 below indicates a summary of specifications for one embodiment of a variable capacitor such as variable capacitor  400  shown in  FIG. 4 . 
   
     
       
         
             
           
             
               TABLE 2 
             
           
          
             
                 
             
             
               Summary of Specifications 
             
          
         
         
             
             
             
          
             
                 
               Parameter 
               Value 
             
             
                 
                 
             
             
                 
               V control   
               27 V 
             
             
                 
               Resonance frequency 
               21.6 kHz 
             
             
                 
               C min   
               2.2 pF (dc) 
             
             
                 
               Capacitance ratio 
               maximum 1:3.6 
             
             
                 
               Vibration sensitivity 
               &lt;0.5%/0.3 g 
             
             
                 
                 
             
          
         
       
     
   
   The actuation voltage and resonance frequency of a variable capacitor such as variable capacitor  400  (shown in  FIG. 4 ) can be dependent upon the width of a peripheral portion (such as peripheral portion  406  shown in  FIG. 4 ) and the length of the tethers (such as tethers T 1 , T 2 , T 3 , T 4 , T 5 , and T 6  shown in  FIG. 4 ).  FIG. 18  illustrates a graph showing different tether lengths and peripheral portion widths versus actuation voltage for a variable capacitor (such as variable capacitor  400  shown in  FIG. 4 ).  FIG. 19  illustrates a graph showing different tether lengths and peripheral portion widths versus resonance frequency for a variable capacitor (such as variable capacitor  400  shown in  FIG. 4 ). As shown, a variable capacitor having a tether length of 75 micrometers and peripheral portion width of 75 micrometers can achieve a resonance frequency of 35.9 kHz. 
     FIG. 20  illustrates a computer simulation model of the z-displacement of the first resonance mode of a variable capacitor  2000  having a tether length of 75 micrometers and peripheral portion width of 75 micrometers. The resonance frequency of the first resonance mode is about 33.9 kHz. The resonance frequency of the second resonance mode is about 59.9 kHz. As shown in  FIG. 20 , interior portion  2002  remains relatively rigid and most of the deformation occurs at tethers T 1 , T 2 , T 3 , T 4 , T 5 , and T 6 , along with a tilt in peripheral portion  2004 . 
     FIG. 21  illustrates a computer simulation model of the z-displacement of a variable capacitor  2000  having a tether length of 75 micrometers and peripheral portion width of 75 micrometers at an actuation voltage set at 14 Volts. 
     FIG. 22  illustrates a graph showing displacement of the center of a variable capacitor  2000  having a tether length of 75 micrometers and peripheral portion width of 75 micrometers versus voltage applied to the actuation electrodes. 
     FIG. 23  illustrates a computer simulation model of the z-displacement of an interior portion  2300  of a variable capacitor  2302  exposed to a temperature difference of 100° Celsius. In this example, the z-displacement of interior portion  2300  is about 0.002 micrometers. Therefore, temperature has little effect on the capacitance values in this embodiment. 
     FIG. 24  illustrates a computer simulation model of deformation of an interior component  2400  having a tether length of 75 micrometers and peripheral portion width of 75 micrometers under a stress gradient between +10 and −10 MPa. 
     FIG. 25  illustrates a computer simulation model of the deformation of an interior component  2500  having a tether length of 75 micrometers and peripheral portion width of 75 micrometers for an acceleration of 100 g. The z-displacement is less than about 0.03 micrometers for 100 g acceleration (i.e., 0.1 nm for an 0.3 g acceleration). Therefore, the capacitance can be modified by a factor of about 0.04% with an air gap of 0.26 micrometers. 
   Referring to  FIG. 26 , a top perspective view of another exemplary variable capacitor, generally designated  2600 , is illustrated. Variable capacitor  2600  can include an interior portion  2602  having a plurality of apertures  2604  extending from a top surface  2606  to an opposing bottom surface (not shown). Apertures  2604  can also extend through a movable capacitive electrode MCE 1  attached to top surface  2606  and a movable capacitive electrode (not shown), if any, attached to the opposing bottom surface (not shown). Apertures  2604  can function to ventilate variable capacitor  2600 . In this embodiment, interior portion  2602  includes thirty-seven apertures that are evenly distributed on surface  2606 . Alternatively, interior portion  2602  can include 7, 27, 169, 721, or any suitable number of apertures. 
   Referring to  FIG. 27A , a cross-sectional side view of one aperture, generally designated  2700 , of interior portion  2602  is illustrated. In this embodiment, interior portion  2602  includes movable capacitive electrodes MCE 1  and MCE 2  attached to top surface  2604  and a bottom surface  2702 , respectively. In this embodiment, aperture  2700  is cylindrically-shaped with a diameter of about 5 micrometers. Additionally, in this embodiment, distance d between the edges of interior portion  2602  and movable capacitive electrode (MCE 1  or MCE 2 ) is between about 0 and 8 micrometers. 
   Referring to  FIG. 27B , a cross-sectional side view of another aperture, generally designated  2704 , of interior portion  2706  is illustrated. In this embodiment, a movable capacitive electrode  2708  can extend inside aperture  2704 . Movable capacitive electrode  2708  can conform to and contact movable capacitive electrode  2710 . This embodiment can be advantageous because the area of the capacitive electrode is not reduced as much for a given aperture size. 
     FIG. 28  illustrates a graph showing the cut-off frequency of a variable capacitor versus the number of apertures in an interior portion of the variable capacitor. Extrapolating from the graph, the cut-off frequency is about 20 kHz for 721 apertures. 
   The number of holes can be selected in order to half the distance between the outer row of holes and the edges of the hexagonally-shaped interior portion at every increment. This leads to a series of the number of holes as follows: 0, 1, 7, 37, 169, 721, etc. At 169 holes, the pitch is 27 micrometers in one embodiment. 
   Referring to  FIG. 29 , a graph showing the damping and tether forces versus frequency for different number of apertures is illustrated. In the low-frequency regime, the air acts as a damper. In the high-frequency regime, the air acts as a spring. 
     FIG. 30  illustrates a graph showing the damping coefficient as a function of the frequency for different numbers of apertures. The force at high frequency is relatively independent of the number of apertures because the volume of air being squeezed remains relatively constant. 
     FIG. 31  illustrates a graph showing harmonic analysis of a variable capacitor having an interior portion with  37  apertures. In the low-frequency regime, the variable capacitor is overdamped exhibiting thus a low-pass filter characteristic. At 62 kHz all curves show resonance peaks. The resonance frequencies are determined by the mass of the structures and the combined stiffness of squeezed film and of the solid structure itself. At this pressure (1 bar) and this air gap (0.25 micron) the stiffness of the air is approx. twice the stiffness of the structure itself. 
     FIG. 32  illustrates a graph showing the cut-off frequency for different aperture numbers. Table 3 below indicates the cut-off frequency for different aperture numbers. 
                   TABLE 3                  Apertures numbers and Cut-off Frequency                                 Cut-off           Number of   frequency           Apertures   [Hz]                                         1   33           7   51           37   312           169   2420                        
From extrapolation, 17 kHz can be expected as a cut-off frequency for 721 apertures. In one embodiment, a distance of 6 micrometers is provided between a capacitive electrode and edge of the interior portion at the aperture. In the configuration of this embodiment, each 5 micrometer diameter hole in the interior portion can have a capacitive electrode opening of 17 micrometers in diameter. Thus, resulting in an effective loss for the capacitance area.
 
     FIG. 33  illustrates a graph showing the effective area of the capacitive electrode as a function of the number of apertures for four different cases according to the minimum distance between the gold layer and the opening of the interior portion (8 μm, 6 μm, 2 μm and 0 μm). Regarding 8 μm, the capacitance is reduced by 70% for 25 apertures. Regarding 2 μm, the capacitance is reduced by 40% for 169 apertures. In the embodiment shown in  FIG. 27B  where the aperture is defined by the metal rather than the hole in the structural layer of movable component MC, the capacitance can be reduced by less than 15%. 
   In order to achieve larger capacitance values, the variable capacitor can be made large or two or more variable capacitors can be connected in parallel. The maximum size of the capacitor is constrained by mechanical considerations (including release time, mechanical resonance frequency, damping and stress deformation), and thus the parallel connection of smaller capacitors can be advantageous. Referring to  FIGS. 34 and 35 , different top views of exemplary cascade arrangements of a plurality of variable capacitors are illustrated. Referring specifically to  FIG. 34 , variable capacitors  3400  are arranged in a rectangular shape. Referring to  FIG. 35 , variable capacitors  3500  are arranged in a fanned-shape. A signal line (not shown) having a total length of about 600 micrometers and a width of about 5 micrometers (impedance matched to 50 ohms). The inductance inserted by this long signal line can result in a self-resonance frequency in the order of 10 GHz, showing a degradation of the quality factor even at frequencies such as 4 GHz. These interconnects should be kept short as possible. Thus, a center feed to the array is desirable to minimize parasitics and maximize self-resonance frequency. 
   The variable capacitor arrangements shown in  FIGS. 34 and 35  can have the specifications shown in Table  4  below. 
   
     
       
         
             
           
             
               TABLE 4 
             
           
          
             
                 
             
             
               Summary of Specifications 
             
          
         
         
             
             
             
          
             
                 
               Parameter 
               Simulated 
             
             
                 
                 
             
             
                 
               V control   
               14.6 V 
             
             
                 
               Resonance frequency 
               33.9 kHz 
             
             
                 
               C min   
               2.6 pF (neglecting 
             
             
                 
                 
               area loss due to 
             
             
                 
                 
               apertures) 
             
             
                 
               Capacitance ratio 
               maximum 1:4 
             
             
                 
               Q 
               Greater than 35 @ 
             
             
                 
                 
               4.5 GHz 
             
             
                 
               Cut-off frequency of 
               2.4 kHz (with 169 
             
             
                 
               the LPF 
               apertures) 
             
             
                 
                 
             
          
         
       
     
   
     FIGS. 36A and 36B  illustrate a computer simulation model, generally designated  3600 , and RF results of computer simulation model  3600 , for an equivalent circuit of four variable capacitors (such as variable capacitor  400  shown in  FIG. 4 ) arranged in parallel. Referring to  FIG. 36A , the HFSS electromagnetic, full-wave simulator (available from Ansoft Corporation of Pittsburgh, Pa.) can be used for modeling four variable capacitors  3602 ,  3604 ,  3606 , and  3608 . Model  3600  can include a connection block  3610  representing the connection of variable capacitors  3602 ,  3604 ,  3606 , and  3608 . Additionally, model  3600  can include a block  3612  representing a line out of the measurement pads. 
   Referring to  FIG. 36B , line  3614  shows that the capacitance does vary some with frequency due to the interconnecting scheme. Line  3616  shows the electrical resonance quality Q falling with frequency. In this example, resonance quality Q includes the degrading effects of the interconnects. A Smith chart, generally designated  3618 , shows that the circuit behaves as a capacitor over the whole frequency range. 
     FIGS. 37A and 37B  illustrate top perspective views of other exemplary variable capacitors. Referring specifically to  FIG. 37A , a top perspective view of another exemplary variable capacitor, generally designated  3700 , utilizing a rectangular geometry including a suspended, movable component MC. Movable component MC can include movable actuation electrodes MAE 1  and MAE 2  and a movable capacitive electrode MCE. In this embodiment, electrodes MAE 1 , MAE 2 , and MCE are attached to a top surface of movable component MC. Alternatively, electrodes MAE 1 , MAE 2 , and MCE can be attached on the underside of movable component MC or on both the top and bottom surfaces. Actuation electrode MAE 1  and MAE 2  and capacitive electrode MCE can be electrically isolated via movable component MC. 
   Variable component  3700  can also include tethers T 1 , T 2 , and T 3  attached to movable component MC and posts (not shown) for suspending movable component MC above a substrate  3702 . Stationary actuation electrodes SAE 1  and SAE 2  can be disposed on the top surface of substrate  3702  and directly beneath movable actuation electrodes MAE 1  and MAE 2 , respectively. A stationary capacitive electrode SCE can be disposed on the top surface of substrate  3702  and directly beneath movable capacitive electrode MCE. A voltage supply VS can be connected at one terminal to movable actuation electrodes MAE 1  and MAE 2  and at another terminal to stationary actuation electrodes SAE 1  and SAE 2 . Voltage supply VS can apply a potential difference between the movable actuation electrodes (MAE 1  and MAE 2 ) and the stationary electrodes (SAE 1  and SAE 2 ) such that, at after a voltage threshold V T  is achieved, movable component MC deflects towards substrate  3602 . Electrodes SCE and MCE can be electrically connected to a signal line SL for supplying a signal, typically AC, to variable capacitor  3600  from other electrical circuitry (not shown). 
     FIG. 37B  illustrates a top perspective view of another exemplary variable capacitor, generally designated  3704 , including a suspended, movable component MC. Movable component MC can include movable actuation electrodes MAE 1  and MAE 2  and a movable capacitive electrode MCE. In this embodiment, electrodes MAE 1 , MAE 2 , and MCE are attached to a top surface of movable component MC. Alternatively, electrodes MAE 1 , MAE 2 , and MCE can be attached on the underside of movable component MC or on both the top and bottom surfaces. Actuation electrode MAE and capacitive electrode MCE can be electrically isolated via movable component MC. 
   Variable component  3704  can also include tethers T 1 , T 2 , T 3 , T 4 , and T 5  attached to movable component MC and posts (not shown) for suspending movable component MC above a substrate  3706 . Stationary actuation electrodes SAE 1  and SAE 2  can be disposed on the top surface of substrate  3706  and directly beneath movable actuation electrodes MAE 1  and MAE 2 , respectively. A stationary capacitive electrode (not shown) can be disposed on the top surface of substrate  3706  and directly beneath movable capacitive electrode MCE. A voltage supply VS can be connected at one terminal to movable actuation electrodes MAE 1  and MAE 2  and at another terminal to stationary actuation electrodes SAE 1  and SAE 2 . Voltage supply VS can apply a potential difference between the movable actuation electrodes (MAE 1  and MAE 2 ) and the stationary electrodes (SAE 1  and SAE 2 ) such that, at after a voltage threshold VT is achieved, movable component MC deflects towards substrate  3706 . The stationary capacitive electrode and movable capacitive electrode MCE can be electrically connected to a signal line SL for supplying a signal, typically AC, to variable capacitor  3704  from other electrical circuitry (not shown). 
   According to one embodiment, isolation bumps can be included with a variable capacitor (such as variable capacitor  400  shown in  FIG. 4 ) for preventing movable capacitive electrode (such as movable capacitive electrode MCE 2  shown in  FIG. 5 ) and/or movable actuation electrode (such as movable actuation electrodes MAE 3  and MAE 4  shown in  FIG. 5 ) from contacting a stationary capacitive electrode (such as stationary capacitive electrode SCE shown in  FIG. 5 ) and/or stationary actuation electrodes (such as stationary actuation electrodes SAE 1  and SAE 2  shown in  FIG. 5 ). The use of isolation bumps can enable variable capacitors with high capacitance ratio and electromechanical stability. 
     FIGS. 38 and 39  illustrate different cross-sectional side views of a variable capacitor having isolation bumps. Referring to  FIG. 38 , a cross-sectional side view of a variable capacitor, generally designated  3800 , having isolation bumps IP 1 , IP 2 , and IP 3  is illustrated. Variable capacitor  3800  can include a movable component MC having movable actuation electrodes  3802  and  3804  positioned on a top and bottom surface  3806  and  3808 , respectively. Movable component MC can include movable capacitive electrodes  3810  and  3812  positioned on a top and bottom surface  3806  and  3808 , respectively. Variable capacitor  3800  can also include a substrate  3814  including a top surface  3816  having a stationary capacitive electrode  3818  deposited thereon. 
   Substrate  3814  can include one or more substrate layers, generally designated SL, including a stationary actuation electrode  3820  positioned therein. Substrate layers SL can also include a capacitor interconnect C 1  for connecting stationary capacitive electrode  3818  to a signal line (not shown). In this embodiment, substrate layers SL include a base substrate layer BSL, a first metal layer M 1 , a first substrate layer S 1 , a second metal layer M 2 , and a second substrate layer S 2 . 
   In this embodiment, one or more sacrificial layers (not shown) can be used during a fabrication process for constructing movable component MC ( FIG. 38 ). The sacrificial layers can subsequently be removed by a suitable process to form the gap, generally designated G, between movable component MC and substrate  3814 . In this embodiment, gap G can extend different distances between movable component MC and substrate  3814 . For example, a gap distance D 1  between movable actuation electrode  3804  and surface  3816  of substrate  3814  can be about 2.5 micrometers with a range between about 0.5 and 10 micrometers. In this embodiment, gap distance D 1  is the total of the following thicknesses: thickness of stationary capacitor  3818 , the thickness of a first sacrificial layer for forming gap G, and the thickness of a second sacrificial layer for forming gap G. Additionally, for example, a gap distance D 2  between isolation bump IP 1  and surface  3816  of substrate  3814  can be about 2.0 micrometers in the embodiment and can be somewhat smaller than the overall actuation gap limited only be the fabrication precision. In this embodiment, gap distance D 2  is the thickness of the first sacrificial layer. Additionally, for example, a gap distance D 3  between movable capacitive electrode  3812  and surface  3816  of substrate  3814  can be between about 0.5 and 20 micrometers. In this embodiment, gap distance D 3  is the thickness of the first and second sacrificial layers. Additionally, for example, a gap distance D 4  between isolation bump IP 2  and stationary capacitive electrode  3818  can be 2.0 micrometers and range from between about 0.5 and 20 micrometers. In this embodiment, gap distance D 4  is the thickness of the first sacrificial layer. Additionally, for example, a gap distance D 5  between isolation bump IP 3  and stationary capacitive electrode  3818  can be 2.0 micrometers and range from between about 0.5 and 20 micrometers. In this embodiment, gap distance D 5  is the thickness of the first sacrificial layer. 
   Referring to  FIG. 39 , actuation voltage has been applied to actuation electrodes  3802 ,  3804 , and  3820  for moving movable component MC to a closed position such that isolation bumps IP 1 , IP 2 , and IP 3  contact substrate  3814 . Isolation bumps IP 1 , IP 2 , and IP 3  can prevent movable capacitive electrode  3812  from contacting stationary capacitive electrode  3818 . In this embodiment, capacitive electrodes  3812  and  3818  can be separated by a distance of about 0.5 micrometers when movable component MC is in the closed position. 
   The equivalent actuation gap of the embodiment shown in  FIGS. 38 and 39  is provided by the following equation (wherein S 1  represents the thickness of first substrate layer S 1 , S 2  represents the thickness of second substrate layer S 2 , M 2  represent the thickness of second metal layer M 2 , SAC 1  represents the thickness of the first sacrificial layer, SAC 2  represents the thickness of the second sacrificial layer, M 3  represents the thickness of stationary capacitive electrode  3818  and ks represents the relative dielectric constant of the substrate): 
             Equivalent   ⁢           ⁢   electrical   ⁢           ⁢   gap     =           S   ⁢           ⁢   1     +     S   ⁢           ⁢   2     +     M   ⁢           ⁢   2         k   ⁢           ⁢   s       +     SAC   ⁢           ⁢   1     +     SAC   ⁢           ⁢   2     +     M   ⁢           ⁢   3             
In this embodiment, the equivalent electrical gap is about 5 micrometers. In this embodiment, the mechanical displacement is limited to the thickness of the first sacrificial layer. The actuation voltage scales as V α airgap^(3/2). For an air gap of 1.5 micrometers, the actuation voltage is 15 Volts. With an equivalent air gap of 5 micrometers, the actuation voltage is 91 Volts. With the variable capacitor  400  ( FIG. 4 ) including isolation bumps such as isolation bumps IP 1 , IP 2 , and IP 3  shown in  FIGS. 38 and 39 , a minimum capacitance of 0.5 picoFarads can be achieved. Additionally, in such a configuration, the capacitance ratio is about 4.
 
   Referring to  FIG. 40 , a cross-sectional side view of another variable capacitor, generally designated  4000 , having an isolation bump IP is illustrated. Variable capacitor  4000  can include a movable component MC having movable actuation electrodes  4002  and  4004  positioned on a top and bottom surface  4006  and  4008 , respectively. Movable component MC can include movable capacitive electrodes  4010  and  4012  positioned on a top and bottom surface  4006  and  4008 , respectively. Variable capacitor  4000  can also include a substrate  4014  including a top surface  4016  having a stationary capacitive electrode  4018  deposited thereon. 
   Substrate  4014  can include one or more substrate layers, generally designated SL, including a stationary actuation electrode  4020  positioned therein. Substrate layers SL can also include a capacitor interconnect C 1  for connecting stationary capacitive electrode  4018  to a signal line (not shown). In this embodiment, substrate layers SL include a base substrate layer BSL, a first metal layer M 1 , a first substrate layer S 1 , a second metal layer M 2 , and a second substrate layer S 2 . 
   In this embodiment, one or more sacrificial layers (not shown) can be used during a fabrication process for constructing movable component MC ( FIG. 40 ). The sacrificial layers can subsequently be removed by a suitable process to form the gap, generally designated G, between movable component MC and substrate  4014 . In this embodiment, gap G can extend different distances between movable component MC and substrate  4014 . For example, a gap distance D 1  between movable actuation electrode  4004  and surface  4016  of substrate  4014  can be about 2.5 micrometers with a range between about 0.5 and 10 micrometers. In this embodiment, gap distance D 1  is the total of the following thickness: thickness of stationary capacitor  4018 , the thickness of a first sacrificial layer for forming gap G, and the thickness of a second sacrificial layer for forming gap G. Additionally, for example, a gap distance D 2  between isolation bump IP and surface  4016  of substrate  4014  can be between about 0.5 and 20 micrometers. In this embodiment, gap distance D 2  is the thickness of the first sacrificial layer. Additionally, for example, a gap distance D 3  between movable capacitive electrode  4012  and surface  4016  of substrate  4014  can be about 2.0 and range from between about 0.5 and 20 micrometers. In this embodiment, gap distance D 3  is the thickness of the first and second sacrificial layers. The gap ratio is about 0.55 in this embodiment. 
   The equivalent actuation gap of the embodiment shown in  FIG. 40  is provided by the following equation (wherein S 2  represents the thickness of second substrate layer S 2 , SAC 1  represents the thickness of the first sacrificial layer, SAC 2  represents the thickness of the second sacrificial layer, M 3  represents the thickness of stationary capacitive electrode  4018  and ks represents the relative dielectric constant of the substrate): 
   
     
       
         
           
             Equivalent 
             ⁢ 
             
                 
             
             ⁢ 
             electrical 
             ⁢ 
             
                 
             
             ⁢ 
             gap 
           
           = 
           
             
               
                 S 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 2 
               
               
                 k 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 s 
               
             
             + 
             
               SAC 
               ⁢ 
               
                   
               
               ⁢ 
               1 
             
             + 
             
               SAC 
               ⁢ 
               
                   
               
               ⁢ 
               2 
             
             + 
             
               M 
               ⁢ 
               
                   
               
               ⁢ 
               3 
             
           
         
       
     
   
   In this embodiment, the equivalent electrical gap is about 3.3 micrometers. In this embodiment, the mechanical displacement is limited by the thickness of the first sacrificial layer. For an air gap of 1.5 micrometers, the actuation voltage is 15 Volts. With an equivalent air gap of 5 micrometers, the actuation voltage is 49 Volts. With the variable capacitor  400  ( FIG. 4 ) including isolation bumps such as isolation bump IP shown in  FIG. 40 , a minimum capacitance of 0.5 picoFarads can be achieved. Additionally, in such a configuration, the capacitance ratio is about 4. 
   Referring to  FIG. 41 , a cross-sectional side view of another variable capacitor, generally designated  4100 , having an isolation bump IP is illustrated. Variable capacitor  4100  can include a movable component MC having movable actuation electrodes  4102  and  4104  positioned on a top and bottom surface  4106  and  4108 , respectively. Movable component MC can include movable capacitive electrodes  4110  and  4112  positioned on a top and bottom surface  4106  and  4108 , respectively. Variable capacitor  4100  can also include a substrate  4114  including a top surface  4116  having a stationary capacitive electrode  4118  deposited thereon. 
   Substrate  4114  can include one or more substrate layers, generally designated SL, including a stationary actuation electrode  4120  positioned therein. Substrate layers SL can also include a capacitor interconnect C 1  for connecting stationary capacitive electrode  4118  to a signal line (not shown). In this embodiment, substrate layers SL include a base substrate layer BSL, a first metal layer M 1 , a first substrate layer S 1 , a second metal layer M 2 , and a second substrate layer S 2 . 
   Movable component MC can also include a planarization dielectric that is compatible with the process attached to bottom surface  4108 . A exemplary dielectric choice is to use silicon dioxide for the planarization dielectric. This planarization oxide PO can be non-conductive for preventing movable capacitive electrode  4112  from electrically communicating with stationary capacitive electrode  4118 . 
   In this embodiment, one or more sacrificial layers (not shown) can be used during a fabrication process for constructing movable component MC ( FIG. 41 ). The sacrificial layers can subsequently be removed by a suitable process to form the gap, generally designated G, between movable component MC and substrate  4114 . In this embodiment, gap G can extend different distances between movable component MC and substrate  4114 . For example, a gap distance D 1  between movable actuation electrode  4104  and surface  4116  of substrate  4114  can be about 2.5 micrometers with a range between about 0.5 and 10 micrometers. In this embodiment, gap distance D 1  is the total of the following thickness: thickness of stationary capacitor  4118 , the thickness of a first sacrificial layer for forming gap G, and the thickness of a second sacrificial layer for forming gap G. Additionally, for example, a gap distance D 2  between isolation bump IP and surface  4116  of substrate  4114  can be between about 0.5 and 20 micrometers. In this embodiment, gap distance D 2  is the thickness of the first sacrificial layer. Additionally, for example, a gap distance D 3  between planarization oxide PO and surface  4116  of substrate  4114  can be between about 0.5 and 20 micrometers. In this embodiment, gap distance D 3  is the thickness of the first and second sacrificial layers. 
   Regarding the embodiment shown in  FIG. 41 , the unactuated capacitance value is about 0.6 picoFarads. The capacitance ratio in this embodiment is about 13. The higher ratio (greater than 3 times the above embodiments assuming silicon oxide as the planarization oxide) and higher maximum capacitance (greater than 4 times the above embodiments assuming silicon oxide as the planarization oxide) enabled by having a dielectric in the gap provide more control in the circuit and allow the use of smaller variable capacitors to provide the required function. Higher dielectric constant materials that are compatible with the process can also be utilized for the planarization oxide with greater gains in ratio and maximum capacitance. 
   Referring to  FIG. 42 , a cross-sectional side view of another variable capacitor, generally designated  4200 , having isolation bumps IP 1  and IP 2  is illustrated. Variable capacitor  4200  can include a movable component MC having movable actuation electrodes  4202  and  4204  positioned on a top and bottom surface  4206  and  4208 , respectively. Movable component MC can include movable capacitive electrodes  4210  and  4212  positioned on a top and bottom surface  4206  and  4208 , respectively. Variable capacitor  4200  can also include a substrate  4214  including a top surface  4216  having a stationary capacitive electrode  4218  deposited thereon. 
   Substrate  4214  can include one or more substrate layers, generally designated SL, including a stationary actuation electrode  4220  positioned therein. Substrate layers SL can also include a capacitor interconnect C 1  for connecting stationary capacitive electrode  4218  to a signal line (not shown). In this embodiment, substrate layers SL include a base substrate layer BSL, a first metal layer M 1 , a first substrate layer S 1 , a second metal layer M 2 , and a second substrate layer S 2 . 
   In this embodiment, one or more sacrificial layers (not shown) can be used during a fabrication process for constructing movable component MC ( FIG. 42 ). The sacrificial layers can subsequently be removed by a suitable process to form the gap, generally designated G, between movable component MC and substrate  4214 . In this embodiment, gap G can extend different distances between movable component MC and substrate  4214 . For example, a gap distance D 1  between movable actuation electrode  4204  and surface  4216  of substrate  4214  can be about 0.8 micrometers. Alternatively, distance D 1  can range between about 0.5 and 20 micrometers. In this embodiment, distance D 1  is the thickness of the first and second sacrificial layers. Additionally, for example, a gap distance D 2  between isolation bump IP 1  and surface  4216  of substrate  4214  can be about 0.3 micrometers. Alternatively, distance D 2  can range from between about 0.2 and 19 micrometers. In this embodiment, gap distance D 2  is the thickness of the first sacrificial layer. For the largest ratio, the second sacrificial layer should be as thin as is feasible with suitable thickness control. Additionally, for example, a gap distance D 3  between movable actuation electrode  4212  and stationary actuation electrode  4218  can be about 0.5 micrometers. Alternatively, distance D 3  can range from between about 0.2 and 20 micrometers. In this embodiment, gap distance D 3  is the thickness of the first and second sacrificial layers after planarization to level the top of the sacrificial material to the level of the sacrificial material in the area where there is no portion of electrode  4218 . Additionally, for example, a gap distance D 4  between isolation bump IP 2  and stationary actuation electrode  4218  can be about 0.3 micrometers. Alternatively, distance D 4  can range between about 0.2 and 19 micrometers. In this embodiment, gap distance D 4  is the thickness of the first sacrificial layer. 
   Regarding the embodiment shown in  FIG. 42 , the maximum capacitance value is about 5 picoFarads, and the minimum capacitance value is about 2 picoFarads. The capacitance ratio in this embodiment is about 2.5. In this embodiment, the actuation voltage at the maximum capacitance is about 15 Volts. 
   A variable capacitor according to one embodiment can include a rotatable movable component attached to one or more torsional beams for providing resistance to the rotational motion. The movable component can be attached to the torsional beam such that the movable component has two “free” ends for rotating about the torsional beam. One or more movable actuation electrodes can be disposed on one end of the movable component. Additionally, one or more movable capacitive electrodes can be disposed on an opposing end of the movable component such that the attachment of the torsional beam is between the movable capacitive electrodes and the movable actuation electrodes. When the movable actuation electrodes are actuated, the movable actuation electrode can cause its corresponding end of the movable component to move downward and rotate the movable component about the torsional beams. Additionally, when the movable actuation electrodes are actuated, the opposing end of the movable component can move upward to displace the movable capacitive electrode from an associated stationary capacitive electrode for changing the capacitance of the variable capacitor. 
     FIGS. 43-45  illustrate different views of a variable capacitor, generally designated  4300 , including torsional beams TB 1  and TB 2 . Referring specifically to  FIG. 43 , a top perspective view of variable capacitor  4300  is illustrated. Variable capacitor  4300  can include a substrate  4306  having a pair of spaced-apart pivot posts P 1  and P 2  supporting torsional beams TB 1  and TB 2 , respectively. Torsional beams TB 1  and TB 2  can support a movable component MC for rotational movement of opposing ends of movable component MC about a pivot axis (generally designated with a broken line  4308  extending from a side of movable component MC). Torsional beams TB 1  and TB 2  can also provide resistance to the rotational movement of movable component MC. The center support of these torsional beams enables robust fabrication and operation of torsional variable capacitors using movable component MC layers with compressive intrinsic stresses. 
   Torsional beams TB 1  and TB 2  can provide vertical stiffness to limit vertical motion of movable component MC with respect to substrate  4306 . Further, torsional beams TB 1  and TB 2  can provide torsional softness to ease rotational motion of movable component MC.  FIG. 44A  illustrates a cross-sectional side view of one embodiment of variable capacitor in the direction indicated by lines L 1  and L 2  (shown in  FIG. 43 ). Referring to  FIG. 44A , in this embodiment, torsional beams TB 1  and TB 2  (shown in  FIG. 43 ) can have a rectangular cross-section and a beam of sufficient length to provide flexibility. Alternatively, torsional beams TB 1  and TB 2  can have any suitable cross-section shape, dimension, or length. Additionally, torsional beams TB 1  and TB 2  can include folded springs. Torsional beams TB 1  and TB 2  can comprise one or more layers of silica, alumina, un-doped semiconductors, polymers, and other non-conductive materials known to those of ordinary skill in the art. 
     FIG. 44B  illustrates a cross-sectional side view of an alternative embodiment of variable capacitor  4300 . In this embodiment, movable component MC comprises a first portion  4400  and a second portion  4402 , wherein second portion  4402  is positioned closer to substrate  4306  than first portion  4400 . Therefore, movable actuation electrodes (MAE 1  and MAE 2 ) and stationary actuation electrode SAE can be positioned further apart than the distance between movable capacitance electrodes (MCE 1  and MCE 2 ) and stationary capacitance electrode SCE to its attachment to first portion  4400  because movable actuation electrode MAE is positioned on raised first portion  4400 . The dual gap can be formed by two different thicknesses of sacrificial layer. 
   Referring to  FIG. 44C  illustrates a cross-sectional side view of another alternative embodiment of variable capacitor  4300 . Stationary actuation electrode SAE can be buried in substrate  4306 . This positioning can increase the distance between stationary actuation electrode SAE and movable actuation electrodes MAE 1  and MAE 2  without adding the complexity of additional sacrifical layers. 
   Substrate  4306  can also include a stationary actuation electrode SAE and a stationary capacitive electrode SCE formed on a surface  4310  thereof. Movable component MC can include movable actuation electrodes MAE 1  and MAE 2  attached to a top surface  4312  and a bottom surface  4314  (shown in  FIG. 44 ), respectively, of movable component MC. Movable actuation electrodes MAE 1  and MAE 2  can be positioned above stationary actuation electrode SAE. Movable actuation electrodes MAE 1  and MAE 2  can be attached to one terminal of a voltage supply (such as voltage supply VS shown in  FIG. 4 ) and stationary actuation electrode SAE can be attached to another terminal of the voltage supply for applying a potential difference to actuate variable capacitor  4300 . When actuated, movable actuation electrodes MAE 1  and MAE 2  can move towards stationary actuation electrode SAE for operatively moving movable component MC along pivot axis  4308 . 
   Substrate  4306  can also include a stationary capacitive electrode SCE attached to surface  4310 . Movable component MC can also include movable capacitive electrodes MCE 1  and MCE 2  attached to surfaces  4312  and  4314 , respectively. Capacitive electrodes SCE, MCE 1 , and MCE 2  can be electrically connected to a signal line (such as signal line SL shown in  FIG. 4 ) for supplying a signal to variable capacitor  4300  from other electrical circuitry (not shown). When variable capacitor  4300  is actuated to move movable component MC along pivot axis  4308 , movable capacitive electrodes MCE 1  and MCE 2  can be moved away from stationary capacitive electrode SCE to change the capacitance between stationary capacitive electrode SCE and movable capacitive electrodes MCE 1  and MCE 2 . 
   Referring to  FIG. 45 , a cross-sectional side view of variable capacitor  4300  in an actuated mode is illustrated. Movable actuation electrodes MAE 1  and MAE 2  are positioned closer to stationary actuation electrode SAE than in an unactuated position as shown in  FIGS. 43 and 44 . Movable capacitive electrodes MCE 1  and MCE 2  are positioned further from stationary capacitive electrode SCE than in the unactuated position shown in  FIGS. 43 and 44 . 
   Variable capacitor  4300  can achieve the specifications shown in Table 5 below. 
                   TABLE 5                  Summary of Specifications                             Parameter   Value                       V control     4.5 V           Resonance frequency     2 kHz           C min     0.9 pF           Capacitance ratio   1:2                        
The specifications indicated in Table 5 can be varied by changing the length of torsional beams T 1  and T 2  ( FIG. 43 ). A capacitance value of about 0.26 pF for variable capacitor  4300  can be obtained. Torsional beams T 1  and T 2  can have a length between about 25 and 175 micrometers.  FIG. 46  illustrates a graph showing the harmonic behavior for variable capacitor  4300  ( FIGS. 43-45 ).
 
   An important parameter effecting resonance frequency is rotational inertia of movable component MC. The rotational inertia of movable component MC equals the mass of movable actuation electrodes MAE 1  and MAE 2  and movable capacitive electrodes SCE 1  and SCE 2 .  FIG. 47  illustrates a graph showing the frequency response for different distances of movable actuation electrodes MAE 1  and MAE 2  ( FIG. 43 ) and movable capacitive electrodes SCE 1  and SCE 2  ( FIG. 43 ) from pivot axis  4308  ( FIG. 43 ). 
     FIG. 48  illustrates a top view of a schematic diagram of another examplary torsional variable capacitor, generally designated  4800 . Variable capacitor  4800  can include a movable capacitor MC having a top surface  4802 . A movable capacitance electrode MCE and a movable actuation electrode MAE can be attached to top surface  4802 . Variable capacitor  4800  can also include pivot posts P 1  and P 2  and torsional beams TB 1  and TB 2 . The dimensions of the components of variable capacitor  4800  are indicated in micrometers. 
   An array of variable capacitor such as variable capacitor  4300  shown in  FIGS. 43-45  can be arranged in parallel to achieve different maximum and minimum capacitances. For example, sixteen variable capacitors (such as variable capacitor  4300 ) can be arranged in parallel to achieve a maximum capacitance of 4 pF, a minimum capacitance of 2 pF, and a first resonance mode of 22. kHz.  FIG. 49  illustrates a computer simulation model of deformation of a torsional variable capacitor  4900  of an array of 16 variable capacitors (such as variable capacitor  4800  shown in  FIG. 28 ). The maximum displacement is located near movable capacitance electrode MCE. 
     FIG. 50  illustrates a graph showing the capacitance of a torsional variable capacitor (such as variable capacitor  4300  shown in  FIG. 43 ) versus an applied actuation voltage. 
     FIG. 51  illustrates a computer simulation model of deformation of a movable component of a torsional variable capacitor (such as variable capacitor  4300  shown in  FIG. 43 ) under a stress gradient between +1 and −1 MPa. The corners of movable component have a displacement of nearly 1 micrometer. 
   A torsional variable capacitor (such as variable capacitor  4300  shown in  FIG. 43 ) can include apertures in the movable component for decreasing the effects of damping. According to one embodiment, the apertures in a torsional variable capacitor can be up to three times larger than 5 micrometers. 
     FIG. 52  illustrates a computer simulation model of the deformation of a movable component in a torsional variable capacitor (such as torsional variable capacitor  4300  shown in  FIG. 43 ) for an acceleration of 100 g. The displacement of the outer edge of movable capacitance electrode  5200  is about −0.09 micrometers. For a 0.3 g acceleration, the displacement of the outer edge of movable capacitance electrode  5200  is about 0.27 nanometers, resulting in a capacitance change of less than about 0.05%. 
   When a long conductive line is used to connect two or more torsional variable capacitors (such as torsional variable capacitor  4300  shown in  FIG. 43 ) in parallel, the overall RF performance of the configuration can be downgraded. In particular, the inductance added by the connection can lower the self-resonance frequency. 
   Table 6 below indicates a summary of specifications for 16 torsional variable capacitors (such as variable capacitor  4300  shown in  FIG. 44 ) connected in parallel. 
   
     
       
         
             
           
             
               TABLE 6 
             
           
          
             
                 
             
             
               Specification Summary 
             
          
         
         
             
             
             
          
             
                 
               Parameter 
               Value 
             
             
                 
                 
             
             
                 
               V control   
               27 V 
             
             
                 
               Resonance frequency 
               22.4 kHz 
             
             
                 
               C min   
               0.12 pF × 16 
             
             
                 
               Capacitance ratio 
               maximum 1:2 
             
             
                 
               R(dc) 
               ≈1.5 ohms 
             
             
                 
               Vibration sensitivity 
               0.05% for 0.3 g 
             
             
                 
               Stress sensitivity 
               negligible 
             
             
                 
               Stress gradient 
               −1 μm 
             
             
                 
               deformation (for +/− 1 MPa) 
             
             
                 
               Cut-off frequency 
               Un-Damped System 
             
             
                 
                 
             
          
         
       
     
   
     FIG. 53  illustrate a computer simulation model RF results of computer simulation model for an equivalent circuit of a torsional variable capacitor (such as variable  4300  shown in  FIG. 43 ). Referring to  FIG. 53 , the HFSS electro-magnetic, full-wave simulator (available from Ansoft Corporation of Pittsburgh, Pennsylvania) can be used for modeling a torsional capacitor. Referring to  FIG. 53 , the resonance quality Q and Smith chart, generally designated  5300 , of a torsional variable capacitor (such as variable  4300  shown in  FIG. 43 ) is shown. 
   It will be understood that various details of the subject matter disclosed herein may be changed without departing from the scope of the subject matter. Furthermore, the foregoing description is for the purpose of illustration only, and not for the purpose of limitation.