Chapter 2



Chapter 1

Introduction

1.1 Thesis Aims

The purpose of this thesis is to characterize a new resonator being developed at The Charles Stark Draper Laboratory. First, expected theoretical behavior of the resonator will be summarized, along with a model based on these calculations suitable for use in filter design. Next, radio frequency (RF) measurement and parameter extraction procedures will be described, applied to prototype resonator devices, and the results analyzed. Finally, the possibilities of implementing this resonator in high-performance bandpass filters with be discussed using several design examples.

1.2 Project Origins and Goals

The Draper resonator is the chief element of a project which seeks to develop an array of RF channel-select filters made from MEMS resonators and integrated with an RF low noise amplifier, funded by the Defense Advanced Research Projects Agency (DARPA). To date, the available integrated filters have performance limits in very narrowband, low insertion loss applications; the large size of current non-integrated RF resonators and filters makes a compact, portable filter bank infeasible. Integrated MEMS resonators offer a promising solution for this difficulty, because of their high Q, mechanical tuning techniques (e.g. laser trimming), compatibility with conventional silicon active device processes, and small size at GHz frequencies (on the order of 10 μm).

The goal of this project was to create resonators with center frequencies ranging from 200 MHz ~ 1.5 GHz, selectable by device geometry, with high enough Q to allow bandwidths on the order of 100 kHz to 10 MHz. Approximately twenty of these resonators would be fabricated to form a filter bank onto a 1-10 mm2 chip with a low noise amplifier.

1.3 Chapter Summaries

The remainder of this thesis is organized into five chapters. Chapter 2 expounds upon the problem of integration between RF filters and the other circuitry required for communications devices. Chapter 3 describes three major miniature resonator technologies and their capabilities, and then presents the theoretical characteristics of the Draper resonator. Chapter 4 summarizes in detail the measurement of prototype Draper devices and the characterization procedures, which were first validated on known TFR resonators. Chapter 5 begins with an overview of bandpass filter design, and discusses in detail three filter topologies: simple ladder, dual-resonator ladder, and full lattice, with numerical examples using the theoretical Draper resonator parameters given. Next the manufacturing limits and tolerances of the resonator are discussed, and their effect on filter characteristics. Given the achieveable limits of the Draper resonator, the filter design process is described, focusing on the suitability of using the Draper resonator for a given set of requirements. Finally, several topics concerning filter bank design are discussed. Chapter 6 provides final conclusions for the thesis work and suggestions for future efforts.

Chapter 2

Background

2.1 Current Applications of Miniature Resonators in RF Communication Technology

As the number of wireless applications have increased, allocations of the frequency spectrum have grown increasingly crowded. Newer technologies have therefore pushed their carrier frequencies higher and higher, which complements the demand for smaller and more portable devices. Greater portability also gives rise to a demand for lower power consumption. In general, reducing the total size of a piece of circuitry enables higher frequency operation while reducing power consumption. In current RF devices, all the major active components can be integrated monolithically, resulting in a small size and the best performance due to minimal parasitics. However, the passive resonators and filters necessary for frequency selection and duplex function have no viable integrated option. The filters used in commercial wireless products today are on the order of millimeters in dimension [1, 2], which is still by far the largest single component of modern RF circuitry.

2.2 Typical Communications Front-End

A typical RF receiving circuit begins with an antenna to receive wideband transmissions, producing an analog electronic signal. The next major step is to bandpass filter this signal to extract the appropriate frequency band, with the possibility of an intermediate gain stage depending on the expected signal strength of the antenna signal and the insertion loss of the filter. After filtering, the signal is amplified and sent to a downmixing element,

Figure 2.1: Block diagram of typical RF circuitry for a wireless communications device.

and from there it is commonly sampled and processed digitally. In a duplex communications device, there will also be a transmit circuit which takes a signal modulated to the carrier frequency, amplifies it, bandpass filters it to prevent interference from being generated in other frequency bands, then broadcasts it via the antenna (Figure 2.1). The various amplifiers can be integrated with standard silicon fabrication technologies, but the filters to date have not been integrated in commercial devices. They are frequently built with ceramic transmission-line resonators, or surface acoustic wave (SAW) resonators, neither of which is compatible with silicon IC technologies [1, 3]. An integrated filter solution is highly desirable for its reduced size and consequently smaller parasitics, and to eliminate the packaging parasitics and longer interconnects due to switching between on-chip and off-chip circuitry, particularly in the cases where an additional amplifier stage is required between the antenna and filter.

2.3 Requirements for an Integrated Resonator

In order for a resonator to be integrated with silicon active devices, its production must be compatible with silicon IC fabrication processes. However, once this requirement is satisfied, the resonator must also be able to compete with the best non-integrated technologies in performance, reliability, and cost before it can be considered a worthwhile alternative to them. In addition to process compatibility, the resonator fabrication must either maintain a certain degree of consistency or a suitable tuning method must be available for it. Since integrated devices will be much smaller than discrete elements, fabricating them to the same relative tolerances becomes that much more difficult.

The other major improvement required of integrated resonators is higher device Q. First, as frequencies grow higher, the transition from passband to stopband of the bandpass filters must grow sharper to allow the same channel spacing, and the sharpness of this transition is proportional to Q. For example, if a filter allows a 50 kHz channel spacing at 1 MHz center frequency, then with the same Q at 1 GHz the minimum channel spacing would be 50 MHz. Most protocols of communication do not require more than a couple MHz of bandwidth to contain all the information they need, so such large channels are a very inefficient use of an increasingly valuable commodity. Second, as resonator dimensions shrink and operational frequencies rise, in general the impedance level of the resonator increases. At GHz+ frequencies, impedance matching becomes very important when connecting different circuit elements. In order for high-frequency filters to effectively couple to other active elements without losing too much signal strength to parasitics, the resonators should have high Q, since the impedance level at resonance is inversely proportional to Q, which in turn affects the matching impedances required. Unfortunately as resonator dimensions decrease, relative loss mechanisms tend to worsen, making higher Q more difficult to achieve.

Chapter 3

Miniature Resonators

3.1 Current Miniature Resonator Technologies

Several different types of piezoelectric resonators are under investigation to produce a viable integratable solution. This section presents a brief overview of the various competing technologies.

3.1.1 SAW

Surface Acoustic Wave (SAW) resonators are currently one of those most common filter elements used in commercial RF telecommunications products today, along with ceramic resonators. They generally are produced by depositing a pair of transducers onto a smooth piezoelectric substrate. Each transducer consists of thin metal multi-fingered electrodes interdigitated in a pair; thus there is an input port and an output port. When an oscillation is applied across the input terminal, the input transducer converts the electrical signal into acoustic vibrations. A surface wave vibration is generated which propagates in a direction perpendicular to the electrode fingers, strongly favoring the frequency whose half-wavelength equals the interdigital spacing. The output transducer converts this vibration back to an electrical signal which is detected at the output.

The frequency response is determined exclusively by the structure of the electrodes. They are suitable for use in many applications, but they have one major drawback: high insertion loss. Each transducer emits energy in both directions, resulting in 6 dB of total loss. The electrodes are very thin and thus have high resistance. Finally, intentional RF mismatches must be introduced to avoid a rippling phenomenon known as triple-transit echo. Total insertion losses of SAW filters range from 7 to 30 dB [4].

Conventional SAW resonators are the inline type, where the wave propagates in a straight line from input to output. These have a typical packaged area of about 15 mm x 6.5 mm [4]. A smaller type, the Z-path SAW, uses reflectors to bounce the acoustic wave twice to make a Z-shaped pattern, allowing package dimensions on the order of 5 mm x 5 mm [5].

|Author |Type |Q |Size |Frequency |Year |

|King, Gopani4 |inline SAW |- |15.3 x 6.45 mm²* |210 MHz |1999 |

|Franz5 |Z-path SAW |- |5 x 5 mm²* |210 MHz |1999 |

Table 3.1: Example SAW resonators listed with important device characteristics.

3.1.2 TFBAR

Thin Film Bulk Acoustic Resonators (TFBARs) are under heavy development by a number of research groups due to their low-temperature processing methods, which make integration with a silicon IC process much more likely, though other obstacles have barred successful integration to date. TFBARs require two basic ingredients to function: a transduction method to convert between electrical oscillations and acoustic vibrations, and an acoustic cavity to trap these vibrations. The former is provided by a piezoelectric membrane, and the latter by a large acoustic impedance mismatch at each interface of the membrane, so that most acoustic energy hitting the interfaces is reflected. The method of generating this impedance mismatch defines the type of TFBAR. The more traditional method creates an air/crystal interface by either etching away the substrate from the bottom, or using a sacrificial substrate layer right beneath the resonator which is etched away after the membrane is deposited, leaving a small gap. Alternatively, the resonator membrane may be deposited onto a stack of quarter-wavelength thick layers of acoustic materials forming a Bragg reflector [6]. These solidly mounted resonators (SMRs) enjoy greater structural stability than the air/crystal type since the resonating membrane and electrodes are fully supported from below.

The acoustic cavity contains the piezoelectric membrane sandwiched between two electrodes. An electrical signal of the right frequency across the electrodes excites a standing longitudinal wave in the membrane. The major loss mechanism, and limiting factor for Q, is the coupling at the cavity interfaces with the adjacent air and supporting substrate. The film is usually about 1-5 μm thick, with lateral dimensions anywhere from 100 to 1000 times the thickness [7].

In general, TFBARs surpass SAW resonators in every way, except for process simplicity and cost to manufacture [8]. Thus, due to this fact and the desire to invent a process fully compatible with silicon active devices, experimentation on these devices centers on novel processing methods.

|Author |Type |Q |Size |Frequency |Year |

|Lakin9 |SM-TFBAR |717 |- |644 MHz |1999 |

|Lakin10 |TFBAR |1090 |- |1.6 GHz |2001 |

|Plessky6 |SM-TFBAR |641 |0.033 mm² |2 GHz |1998 |

|Ruby11 |TFBAR |500-1300 |~0.001-0.1 mm² |2-4GHz |1994 |

Table 3.2: Example TFBAR resonators listed with important device characteristics.

3.1.3 Mechanical Resonators

Mechanically resonant structures such as deflecting cantilevers and flexural beams show promise as filter elements due to their high Q and possible silicon IC process compatibility. Their principle of operation is very simple: a simple symmetric structure with mechanically resonant modes is built out of a piezoelectric material. If an oscillating force at some frequency is applied to the structure, actuated either mechanically or piezoelectrically, the structure will vibrate and produce a charge separation proportional to the amplitude of its vibration. Frequencies near resonance will produce larger amplitude responses, which can be viewed as electrical signals if electrodes are attached at appropriate points on the structure. When used as a filter element, these types of resonators are usually one-port devices acting as variable impedances: a voltage signal is applied to the port and a current waveform is drawn from the source with amplitude varying with the frequency of the input signal.

A major limiting factor to date with this type of resonator is the loss associated with the anchor between the resonating element and the substrate. In addition, interfacing electrically to the resonating element also introduces loss in general. Thus a mechanical element which could achieve a Q of perhaps 100,000 in isolation may only get one-tenth that when modified as an electrical resonator. Another drawback is poor linearity with amplitude of vibration in many cases.

|Author |Type |Q |Size |Frequency |Year |

|Cleland12 |Flexural beam |21000 @ 4.2 K |3.3 x 2.4 μm² |82 MHz |2001 |

|Nguyen13 |Flexural beam |7450 |~30 x 10 μm² |90 MHz |2000 |

Table 3.3: Example mechanical resonators listed with important device characteristics.

3.2 Draper Resonator Bar

The Draper resonator bar was designed to allow production of arrays of RF communications filters integrated with silicon active circuitry on a single die. It is a mechanical resonator designed to operate at the fundamental longitudinal mode. Its range of dimensions should allow center frequencies from 200 MHz to 1.5 GHz while being compact enough to fit an entire filter bank into an area on the order of 10 mm². Preliminary loss analyses predict a device Q of 104 should be attainable [27].

3.2.1 Device Overview

Figure 3.1 provides a schematic of the resonator bar with the major geometrical parameters labeled. The bar is suspended over a 1 μm-deep well in the substrate by two tethers attached to the midpoints of its sides. The bar and the tethers are a single continuous film of aluminum nitride (AlN). Electrodes cover the top and bottom of the bar, producing an off-resonance characteristic of a parallel-plate capacitor.

Figure 3.1: Resonator bar schematic. The bar is suspended over an empty well while the tethers rest on the lower electrode and the substrate.

Figure 3.2: Fundamental longitudinal vibration. One cycle is pictured. The amplitude of motion is exaggerated to illustrate the motion.

Longitudinal Mode Resonance

The primary engineered resonance is a longitudinal vibration where the bar expands and constricts lengthwise (Figure 3.2). The midpoint of the bar, where the tethers attach, is a node, while the two ends of the bar experience the greatest amplitude of displacement. The magnitude of vibration will be on the order of nanometers.

The bar has many different natural modes of resonance with several at frequencies below that of the intended longitudinal mode of operation. However, due to symmetry and the placement of the electrodes, the charge contribution of these lower order modes at the device’s port should cancel to nearly zero. This calculation was confirmed by a finite elements simulation of the device’s I-V transfer characteristic (Figure 3.3).

Figure 3.3: Finite elements model of the transfer characteristic of a resonator bar with 0.78 GHz longitudinal resonance [14].

3.2.2 Analytic Model of Longitudinal Resonance

By applying the piezoelectric equations of state and standard electrostatics relations to the resonator bar, an analytic solution for the primary resonance may be obtained. This solution only models a single resonance, but is immensely helpful during filter design.

A Brief Introduction to Piezoelectric Materials Properties

Piezoelectric properties arise in certain crystals with asymmetric structure. Mechanical stressing causes a polarization of charge in these materials, with the converse occurring as well: an applied electrical field induces physical deformation. A right-handed Cartesian set of axes is traditionally introduced to facilitate mathematical analysis of the various piezoelectric interactions, in accordance with the IEEE Standard on Piezoelectricity [15]. An electric field in one direction will elicit, in general, a mechanical response along each of the three axes, and a mechanical stress in one direction will induce, in general, an electric polarization in each axis as well.

Linear Theory of Piezoelectricity

The piezoelectric equations of state relate the major mechanical variables of interest (e.g. stress, strain, displacement) to the electric field and charge polarization. These equations may be linearized by assuming all displacements and vibrational amplitudes are very small. The various partial derivatives appearing in these equations may now be considered constants, and define the various material parameters such as stiffness, permittivity, and the piezoelectric stress and strain constants [16]. In addition, because the phase velocities of acoustic waves are several orders of magnitude less than the velocities of electromagnetic waves, quasi-electrostatic conditions are assumed [15]. A more detailed treatment of the formulation of these equations and the definition of the mechanical and electrical field variables may be found in [15],[16], and [17].

Longitudinal Mode Analytic Transfer Function

The following analysis was originally performed by Dr. Amy Duwel, and is summarized here with her permission. Applying the piezoelectric equations of state to AlN results in the following constitutive equations:

(3.1) [pic]

c= stiffness matrix: N/m2 eT= piezoelectric coupling: C/m2

(3.2) [pic]

ε= dielectric constants: F/m

where T is the stress tensor, S is the strain tensor, E is the electric field vector, and D is the polarization charge vector. The axis along the bar’s length is subscript 1 ([pic]= x), across its width is subscript 2 ([pic]= y), and vertically through its thickness is subscript 3 ([pic]= z). Due to symmetry, T and S only have 6 independent elements, with the reduced subscript mapping as follows:

(3.3) [pic]

(3.4) [pic]

The next step is to apply force balance to Eqn. (3.1), substituting in displacement u and electric potential φ:

(3.5) [pic]

In this notation, any subscripts after the comma refer to derivatives with respect to that variable. A number of approximations are now made. First, for a longitudinal vibration in the x-direction, no y-dependence will be assumed, so u2 and all its derivatives are set to zero. Second, for the lowest-order longitudinal mode, set stress in the z-direction and x-z shear to be zero for all z, meaning T3 = T5 = 0. Third, the inertia in the z direction is considered negligible. This gives the following acoustic wave equation for the fundamental longitudinal mode in the x direction, coupled to φ:

(3.6) [pic]

Solution of the electrostatics equations begins by assuming AlN is non-conducting and thus its free charge density is 0:

(3.7) [pic]

Making the same approximations as above and some algebra results in this equation:

(3.8) [pic]

which has the form of Laplace’s equation for an anisotropic material. The system is described by the coupled equations (3.6) and (3.8), which are difficult to solve due to the x-z coupling. Eqn. (3.8) may be split into two parts, since the system is linear. If we let [pic], [pic] is the solution to (3.8) with the RHS set to zero and the voltage boundary conditions applied, and [pic] is the solution to (3.8) with zero voltage boundary conditions. For [pic], the solution can be approximated as:

(3.9) [pic]

under the constraint that the bar is much longer than it is thick. The boundary conditions are set as follows: at z = 0 (bottom electrode), φ = 0, at z = 2a (top electrode), φ = f(x). As a first order approximation, Eqn. (3.6) is solved using only [pic] on the RHS [14]. Plugging this solution into Eqn. (3.6) allows calculation of the total equation of motion of the fundamental longitudinal resonance due to a driving voltage function V(s):

(3.10) [pic]

where the following intermediate variables have been defined:

(3.11) [pic]

(3.12) [pic]

(3.13) [pic]

(3.14) [pic]

(3.15) [pic]

Finally, with the complete solution we can integrate D3 over the electrode area, and then differentiate with respect to time to express the current as a function of voltage:

(3.16) [pic]

3.3 Butterworth Van-Dyke Model

Figure 3.4: The Butterworth Van-Dyke model for a crystal resonator.

The Butterworth Van-Dyke (BVD) model is a common lumped element circuit model used by crystal filter designers to simplify the transcendental functions that completely characterize the resonators used as filter elements [19]. It generally provides an accurate fit for a single resonance plus the other regions of a resonator’s transfer function not near another resonance, modeling those parts as a capacitance. Qualitatively, the R-L-C branch determines the “series” resonance, where the impedance drops sharply to a minimum value of R at the frequency where the series inductance and capacitance cancel each other out. At some higher frequency, the loop reactance hits zero and causes a “parallel” resonance where most current will travel around the loop instead of past it.

3.3.1 BVD Impedance

The exact transfer function representing the BVD impedance is:

(3.17) [pic]

This function can also be expressed directly in terms of the major resonator figures of merit, series resonance ws, parallel resonance wp, and Q:

(3.18) [pic]

(3.19) [pic]

(3.20) [pic]

(3.21) [pic]

The first term in this impedance is the impedance of the static capacitance C0. If C ................
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