Patent Publication Number: US-11641187-B2

Title: Torsional mode quartz crystal device

Description:
INCORPORATION BY REFERENCE TO ANY PRIORITY APPLICATIONS 
     Any and all applications for which a priority claim is identified in the Application Data Sheet as filed with the present application are hereby incorporated by reference under 37 CFR 1.57. 
     This application is a continuation of U.S. patent application Ser. No. 16/548,675, filed Aug. 22, 2019, and entitled “TORSIONAL MODE QUARTZ CRYSTAL DEVICE,” the content of which is hereby expressly incorporated by reference in its entirety. 
    
    
     BACKGROUND 
     Field 
     The disclosed technology generally relates to quartz crystal devices and more particularly to quartz crystal devices configured to vibrate in torsional mode. 
     Description of the Related Art 
     A crystal oscillator is an electronic oscillator circuit that uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a precise frequency. This frequency can provide a stable reference signal, e.g., a clock signal, for digital integrated circuits, and to stabilize frequencies for radio transmitters and receivers. One type of piezoelectric resonator used is quartz crystal, and oscillator circuits using the quartz crystal are referred to as quartz crystal oscillators. 
     A quartz crystal oscillator vibrates at a stable frequency by being distorted by an electric field when voltage is applied to an electrode near or on the crystal. This property is known as electrostriction or inverse piezoelectricity. When the field is removed, the quartz, which oscillates in a precise frequency, generates an electric field as it returns to its previous shape, which in turn can generate a voltage. This characteristic of a quartz crystal can be modeled as an RLC circuit. Advantages of quartz crystal oscillators include frequency accuracy, stability, and low power consumption. Because of these advantages, quartz crystal oscillators are used in many consumer devices such as wristwatches, clocks, radios, computers and cellphones, to name a few. Quartz crystals are also found inside test and measurement equipment, such as counters, signal generators, and oscilloscopes. However, high reliability is needed to fully benefit from the advantages of the quartz crystal oscillators. 
     SUMMARY 
     In one aspect, a quartz crystal device configured for temperature sensing comprises a fork-shaped quartz crystal comprising a pair of elongate tines laterally extending from a base region in a horizontal lengthwise direction of the fork-shaped quartz crystal. Each of the tines has formed on one or both of opposing sides thereof a vertically protruding line structure laterally elongated in the horizontal lengthwise direction. The quartz crystal device further comprises a first electrode and a second electrode formed on the one or both of the opposing sides of each of the tines and configured such that, when a bias is applied between the first and second electrodes, the fork-shaped quartz crystal vibrates in a torsional mode in which each of the tines twists about a respective axis extending in the horizontal lengthwise direction. 
     In another aspect, a quartz crystal device configured for temperature sensing comprises a fork-shaped quartz crystal comprising a pair of elongate tines laterally extending from a base region in a horizontal lengthwise direction of the fork-shaped quartz crystal. The quartz crystal device further comprises a first electrode and a second electrode formed on the one or both of the opposing sides of each of the tines. Each of the first and second electrodes have surfaces facing each other in a widthwise direction of the fork-shaped quartz crystal and configured such that, when a bias is applied between the first and second electrodes, the fork-shaped quartz crystal vibrates in a torsional mode in which each of the tines twists about a respective axis extending in the horizontal lengthwise direction. 
     In yet another aspect, a quartz crystal oscillator device configured for temperature sensing comprises a quartz crystal device bonded to a package substrate. The quartz crystal device comprises a fork-shaped quartz crystal device comprising a pair of elongate tines laterally extending from a base region in a horizontal lengthwise direction of the fork-shaped quartz crystal, wherein each of the tines has formed on one or both of opposing sides thereof a vertically protruding line structure laterally elongated in the horizontal lengthwise direction and electrodes configured such that the fork-shaped quartz crystal vibrates in a torsional mode in which each of the tines twists about an axis extending in the horizontal lengthwise direction. The quartz crystal oscillator device further comprises an integrated circuit (IC) die electrically connected to the quartz crystal device and bonded to the package substrate. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1 A  schematically illustrates a plan view of a main surface of a fork-shaped torsional mode quartz crystal device. 
         FIG.  1 B  schematically illustrates a cross-sectional view of the fork-shaped torsional mode quartz crystal device illustrated in  FIG.  1 A , including a schematic electric field path. 
         FIG.  2 A  schematically illustrates a perspective view of the fork-shaped torsional mode quartz crystal device illustrated in  FIGS.  1 A and  1 B . 
         FIG.  2 B  schematically illustrates a perspective view of the fork-shaped torsional mode quartz crystal device illustrated in  FIG.  2 A , in which the tines are twisted and displaced in operation under a bias. 
         FIG.  3    illustrates a circuit representation of a quartz crystal device. 
         FIG.  4    is a graph illustrating motional resistance and motional capacitance as a function of a crystal thickness for the fork-shaped torsional mode quartz crystal device illustrated in  FIGS.  1 A and  1 B . 
         FIG.  5 A  schematically illustrates a plan view of a first main (top) surface of a fork-shaped torsional mode quartz crystal device comprising a mesa structure, according to embodiments. 
         FIG.  5 B  schematically illustrates a plan view of a second main (bottom) surface of the fork-shaped torsional mode quartz crystal device illustrated in  FIG.  5 A . 
         FIG.  5 C  schematically illustrates a cross-sectional view of the fork-shaped torsional mode quartz crystal device illustrated in  FIGS.  5 A and  5 B  at tip regions of the tines. 
         FIG.  5 D  schematically illustrates a cross-sectional view of the fork-shaped torsional mode quartz crystal device illustrated in  FIGS.  5 A and  5 B  at midsections of the tines. 
         FIG.  5 E  schematically illustrates a cross-sectional view of the fork-shaped torsional mode quartz crystal device illustrated in  FIGS.  5 A and  5 B  at a base region. 
         FIG.  6 A  schematically illustrates a perspective view of the fork-shaped torsional mode quartz crystal device illustrated in  FIGS.  5 A- 5 E . 
         FIG.  6 B  schematically illustrates a cross-sectional view of the fork-shaped torsional mode quartz crystal device illustrated in  FIG.  6 A . 
         FIG.  6 C  schematically illustrates a perspective view of the fork-shaped torsional mode quartz crystal device illustrated in  FIGS.  6 A- 6 B , in which the tines are twisted in operation under a bias. 
         FIG.  7    schematically illustrates different crystal cuts and orientations for different types of quartz crystal devices including a fork-shaped torsional mode quartz crystal device, according to embodiments. 
         FIG.  8 A  illustrates a simulation of a linear temperature coefficient of a fork-shaped torsional mode quartz crystal device as a function of an angle of tilt θ relative to the X-axis of a quartz crystal, according to embodiments. 
         FIG.  8 B  illustrates a simulated frequency versus temperature curve of a fork-shaped torsional mode quartz crystal device at an angle of tilt of −30° relative to the X-axis of a quartz crystal, according to embodiments. 
         FIG.  8 C  illustrates a simulated frequency versus temperature curve of a fork-shaped torsional mode quartz crystal device at an angle of tilt of 30° relative to the X-axis of a quartz crystal, according to embodiments. 
         FIG.  9    illustrates simulated frequency versus thickness for fork-shaped torsional mode quartz crystal devices with and without a mesa structure. 
         FIG.  10 A  is a graph illustrating simulated motional resistance as a function of frequency for fork-shaped torsional mode quartz crystal devices with and without a mesa structure. 
         FIG.  10 B  is a graph illustrating simulated motional capacitance as a function of frequency for fork-shaped torsional mode quartz crystal devices with and without a mesa structure. 
         FIG.  10 C  is a graph illustrating a simulated quality factor as a function of frequency for fork-shaped torsional mode quartz crystal devices with and without a mesa structure. 
         FIG.  11    is a graph illustrating simulated frequency and motional capacitance as a function of mesa thickness for a fork-shaped torsional mode quartz crystal device, according to embodiments. 
         FIG.  12 A  is a graph illustrating simulated motional resistance and motional capacitance as a function of a ratio of thicknesses between a line region and a recessed region of a tine for a fork-shaped torsional mode quartz crystal device comprising a mesa structure, according to embodiments. 
         FIG.  12 B  is a graph illustrating simulated frequency and quality factor as a function of a ratio of thicknesses between a line region and a recessed region of a tine for a fork-shaped quartz crystal device comprising a mesa structure, according to embodiments. 
         FIG.  13 A  is a graph illustrating simulated motional resistance and motional capacitance as a function of a ratio between a width of a line region and a tine width for a fork-shaped quartz crystal device comprising a mesa structure, according to embodiments. 
         FIG.  13 B  is a graph illustrating simulated frequency and quality factor as a function of a ratio of between a width of a line region and a tine width for fork-shaped torsional mode quartz crystal device comprising a mesa structure, according to embodiments. 
         FIG.  14 A  is a graph illustrating simulated motional resistance and motional capacitance as a function of a ratio between a length of a tip region of a tine excluding the line region a length of the tine including the line region for a fork-shaped torsional mode quartz crystal device comprising a mesa structure, according to embodiments. 
         FIG.  14 B  is a graph illustrating simulated frequency and quality factor as a function of a ratio between a length of a tip region of a tine excluding the line region a length of the tine including the line region for a fork-shaped torsional mode quartz crystal device comprising a mesa structure, according to embodiments. 
         FIG.  15 A  illustrates experimentally measured frequency and motional capacitance versus mesa thickness for a fork-shaped torsional mode quartz crystal device comprising a mesa structure, according to embodiments. 
         FIG.  15 B  illustrates an experimentally measured frequency versus temperature curve for a fork-shaped torsional mode quartz crystal device comprising a mesa structure, according to embodiments. 
         FIG.  16 A  illustrates a plan-view of a fork-shaped quartz crystal device comprising a mesa structure packaged in a ceramic package substrate, according to embodiments. 
         FIG.  16 B  illustrates a cross-sectional view of the packaged fork-shaped quartz crystal device illustrated in  FIG.  16 A . 
         FIG.  17 A  illustrates a plan-view of a quartz crystal oscillator device comprising a fork-shaped quartz crystal device packaged in a ceramic package substrate, according to embodiments. 
         FIG.  17 B  illustrates a cross-sectional view of the packaged quartz crystal oscillator device illustrated in  FIG.  17 A . 
         FIG.  18 A  schematically illustrates a plan view of a main (top) surface of a fork-shaped torsional mode quartz crystal device comprising a mesa structure and vibration isolation arms, according to embodiments. 
         FIG.  18 B  schematically illustrates a cross-sectional view of the fork-shaped torsional mode quartz crystal device illustrated in  FIG.  18 A  at tip regions of the tines. 
         FIG.  18 C  schematically illustrates a cross-sectional view of the fork-shaped torsional mode quartz crystal device illustrated in  FIG.  18 A  at a midsection of the tines and tip regions of the vibration isolation arms. 
         FIG.  19 A  illustrates a plan-view of a quartz crystal oscillator device comprising a fork-shaped quartz crystal device packaged in a ceramic package substrate, according to embodiments. 
         FIG.  19 B  illustrates a cross-sectional view of the packaged quartz crystal oscillator device illustrated in  FIG.  19 A . 
     
    
    
     DETAILED DESCRIPTION 
     The following detailed description is now directed to certain specific embodiments of the disclosure. In this description, reference is made to the drawings wherein like parts are designated with like numerals throughout the description and the drawings. 
     In the following, various embodiments of quartz crystal devices and quartz crystal oscillator devices are disclosed. In particular, the disclosed quartz crystal devices comprise a fork-shaped or tuning fork-shaped quartz crystal that is configured to vibrate in a torsional mode and/or operate as a temperature sensor. Advantageously, the quartz crystal devices are designed such that its vibrating frequency is both extremely sensitive to temperature and highly linear. The high sensitivity offers the ability to detect fine changes in temperature, where the degree of sensitivity can be tailored depending on various implementations disclosed herein. Further, a frequency-based technique has the advantage of being immune to amplitude noise in the measurement system, which is a feature not present in other types of temperature sensors such as a thermocouple, a thermistor, or resistance temperature detectors (RTDs). Furthermore, the disclosed fork-shaped quartz crystal devices can enable remote temperature sensing by using an antenna to pick up the frequency of the electromagnetic waves emitted by the fork-shaped quartz crystal device. 
     In various examples disclosed herein, a quartz crystal device configured for temperature sensing comprises a fork-shaped quartz crystal comprising a base region and a pair of elongate tines laterally extending from the base region in a lengthwise direction. The quartz crystal additionally comprises a first electrode and a second electrode formed on the one or both of opposing sides of each of the tines and configured such that, when a bias is applied between the first and second electrodes, the fork-shaped quartz crystal vibrates in a torsional mode in which each of the tines twists about an axis extending in the lengthwise direction. 
       FIG.  1 A  schematically illustrates a plan view of an example fork-shaped torsional mode quartz crystal device  100 .  FIG.  1 B  schematically a cross-sectional view of the fork-shaped torsional mode quartz crystal device illustrated in  FIG.  1 A . The quartz crystal device  100  comprises a base  112 , a first or left tine  114  projecting from the base  112  and extending generally along a lengthwise direction, e.g., x-axis. The fork-shaped quartz crystal device  100  additionally comprises a second or right tine  116  or prong projecting from the base  112  and extending generally along the same lengthwise direction. The tines can also be referred as arms or prongs. The first tine  114  and second tine  116  can be generally parallel to one another. In various quartz crystal devices disclosed herein, electrodes are formed on various surfaces thereof such that the tines are configured to vibrate at least in a torsional mode, as described below. 
     The base  112  supporting the first and second tines  114 ,  116  can include multiple portions. For example, in the illustrated example, the base  112  comprises a first or upper base portion  112 A from which the first and second tines  114 ,  116  immediately extend and a second or lower base portion  112 B forming a terminal portion of the quartz crystal device  100 . The first and second base portions  112 A,  112 B can be defined by a first and second width, respectively. The second base portion  112 B can be configured such that the fork-shaped torsional mode quartz crystal device  100  can be mounted to a ceramic package or other support structures. In the illustrated example, the second base portion  112 B configured for mounting on a ceramic package is wider than the first base portion  112 A connected to the first and second tines  114 ,  116 . Such configuration can reduce the amount of vibration leakage from the first and second tines  114 ,  116  to the base  112 , such that the fork-shaped torsional mode quartz crystal device  100  vibrates with increased efficiency. However, embodiments are not so limited and the first portion  112 A can have the same or larger width than that of the second base portion  112 B. 
     The fork-shaped torsional mode quartz crystal device  100  has various surfaces having formed thereon electrode patterns that are collectively adapted for vibration of the quartz crystal device  100  in a torsional mode. Referring to  FIG.  1 A , quartz crystal device  100  shows on a first or front side  118  ( FIG.  1 B ) a first main surface, e.g., a front surface. The electrode patterns on the first main surface include a first electrode pattern  124  of a first polarity, e.g., one of a negative or positive electrode pattern, and a second electrode pattern  128  of a second polarity, e.g., the other of the negative or positive electrode pattern. While not illustrated, the fork-shaped torsional mode quartz crystal device  100  also has on a second or rear side  120  opposite the first or front side  118  a second main surface, e.g., a back surface. The second main surface back surface also includes various portions of the first electrode pattern  124 ′ and the second electrode pattern  128 ′ formed thereon. As described herein and throughout the specification, it will be understood that, while electrode patterns labeled with different reference numerals, e.g., first electrode patterns  124  and  124 ′, may be formed on different surfaces, they may be electrically connected to each other and therefore be at the same potential. 
     As described herein and throughout the specification, relative positions of various features formed on a side of a quartz crystal device, e.g., the first electrode patterns  124 ,  124 ′ and second electrode patterns  128 ,  128 ′, may be described as viewed from the respective sides. As such, it will be appreciated that positions that are described as having opposite lateral positions, when disposed on opposing sides, may be referring to the same physical position. For example, in  FIG.  1 A , when viewed from the first side  118 , the first and second tines  114 ,  116  are positioned on left and right relative positions, respectively, while when viewed from the second side  120  ( FIG.  1 B ), the first and second tines  114 ,  116  would be positioned on right and left relative positions, respectively. 
     Referring to the plan view illustrated in  FIG.  1 A , viewed from the first side  118 , the first (e.g., negative) electrode pattern  124  formed on the top surface includes first to fourth first electrode portions  124 A- 124 D, and the second (e.g., positive) electrode pattern  128  formed on the top surface includes first to fourth second electrode portions  128 A- 128 D. The first electrode pattern  124  includes a first portion  124 A covering a first (right) side of the base  112  and extends upward therefrom in the x-direction in the form of the second thin strip portion  124 B along a right or outer edge portion of the second tine  116 , and terminates adjacent to the third portion  128 C of the second electrode pattern  128  formed on the terminal end portion of the second tine  116 . The first electrode pattern  124  additionally includes the third portion  124 C formed on the terminal end portion of the first tine  114 , extends downward therefrom in the x-direction in the form of the fourth thin strip portion  124 D along a left or outer edge portion of the first tine  114 , and terminates adjacent to the first portion  128 A of the second electrode pattern  128  formed on the second (left) side of the base  112 . 
     Still referring to the plan view illustrated in  FIG.  1 A , the second electrode pattern  128  formed on the top surface includes the first portion  128 A covering a second (left side) of the base  112  and extends upward therefrom in the x-direction and forks or splits into the second thin strip portion  128 B formed and extending upward along a left or inner edge portion of the second tine  116 , and a fourth thin strip portion  128 D formed along and extending upward along a right or inner edge portion of the first tine  114 . The fourth thin strip portion  128 D terminates adjacent to the third portion  124 C of the first electrode pattern  124  formed on the terminal end of the first tine  114 . The second thin strip portion  128 B extends to connect to the third portion  128 C of the second electrode pattern  128  formed on the terminal end portion of the second tine  116 . 
     While not illustrated, the combination of the first and second electrode patterns  124 ′,  128 ′ formed on the rear side  120  may generally be shaped similarly as the combined pattern of the first and second electrode patterns  124 ,  128  formed on the front side  118  as described above, except that their polarities are interchanged. The first (e.g., negative) electrode pattern  124 ′ formed on the bottom surface includes first to fourth portions  124 A′- 124 D′, and the second (e.g., positive) electrode pattern  128 ′ formed on the bottom surface includes first to fourth portions  128 A′- 128 D′. The second electrode pattern  128 ′ includes a first portion  128 A′ covering a first (right side) of the base  112  and extends upward therefrom in the x-direction in the form of the second thin strip portion  128 B′ along a right or outer edge portion of the first tine  114 , and terminates adjacent to the third portion  124 C′ of the first electrode pattern  124 ′ formed on the terminal end portion of the first tine  114 . In addition, the second electrode pattern  128 ′ includes the third portion  128 C′ formed on the terminal end portion of the second tine  116 , extends downward therefrom in the x-direction in the form of the fourth thin strip portion  128 D′ along a left or outer edge portion of the second tine  116 , and terminates adjacent to the first portion  124 A′ of the first electrode pattern  124 ′ formed on the second (left) side of the base  112  when viewed from the rear side  120 . 
     Additionally, the first electrode pattern  124 ′ formed on the bottom surface (not shown) includes the first portion  124 A′ covering a second (left side) of the base  112  and extends upward therefrom in the x-direction and forks or splits into the second thin strip portion  124 B′ formed and extending upward along a left or inner edge portion of the first tine  114 , and a fourth thin strip portion  124 D′ formed along and extending upward along a right or inner edge portion of the second tine  116 . The fourth thin strip portion  124 D′ terminates adjacent to the third portion  128 C′ of the second electrode pattern  128 ′ formed on the terminal end of the second tine  116 . The second thin strip portion  124 B′ extends to connect to the third portion  124 C′ of the first electrode pattern  124 ′ formed on the terminal end portion of the first tine  114 . 
     As described above, various portions of the first electrode pattern  124  on the first side  118  and the first electrode pattern  124 ′ on the second side  120  are electrically connected to each other. Similarly, various portions of the second electrode pattern  128  on the first side  118  and the second electrode pattern  128 ′ on the second side  120  are electrically connected to each other. The resulting relative positions of different portions of the first electrode patterns  124 ,  124 ′ and the second electrode patterns  128 ,  128 ′ in a cross-sectional view are illustrated in  FIG.  1 B .  FIG.  1 B  schematically illustrates a cross-sectional view of the fork-shaped torsional mode quartz crystal device  100  through the section A-A illustrated in  FIG.  1 A . The cross-sectional view of the first tine  114  shows a first main surface on the first side  118  having the fourth strip portion  124 D of the first electrode pattern  124  formed on the outer edge portion thereof, and the fourth strip portion  128 D of the second electrode pattern  128  formed on the inner edge portion thereof. On the other hand, a second main surface on the second side  120  has the second strip portion  124 B′ of the first electrode pattern  124 ′ formed on the inner edge portion of the first tine  114 , and the second strip portion  128 B′ of the second electrode pattern  128 ′ formed on the outer edge portion of the first tine  114 . The section A-A additionally illustrates a cross-sectional view of the second tine  116  showing a first main surface of the first side  118  having the second strip portion  124 B of the first electrode pattern  124  formed on the outer edge portion thereof, and the second strip portion  128 D of the second electrode pattern  128  formed on the inner edge portion thereof. On the other hand, the second main surface of the second side  120  has the fourth strip portion  124 D′ of the first electrode pattern  124 ′ formed on the inner edge portion of the second tine  116 , and the fourth strip portion  128 D′ of the second electrode pattern  128 ′ formed on the outer edge portion of the second tine  116 . 
     In operation, electrical pulses having a first (e.g., one of negative or positive) polarity may be applied to the first electrode patterns  124 ,  124 ′ (including the second and fourth strip portions  124 B,  124 D on the front main surfaces of the second and first tines  116 ,  114 , respectively, and the second and fourth strip portions  124 B′,  124 D′ on the back main surfaces of the first and second tines  116 ,  114 , respectively, as illustrated in  FIG.  1 B ). On the other hand, electrical pulses having a second (e.g., the other of negative or positive) polarity may be applied to the second electrode patterns  128 ,  128 ′ (including the second and fourth strip portions  128 B,  128 D on the front main surfaces of the second and first tines  116 ,  114 , respectively, and to the second and fourth strip portions  128 B′,  128 D′ on the back main surfaces of the first and second tines  114 ,  116 , respectively, as illustrated in  FIG.  1 B ). It will be appreciated that, as disclosed herein and throughout the specification, positive and negative polarities may describe relative polarities, such that voltages that have the same sign but different magnitudes may be described as having positive and negative polarities. While embodiments are not so limited, pulses having the first and second polarities may be applied to the quartz crystal device  100  simultaneously. 
     Still referring to  FIG.  1 B . when the first and second electrode patterns  124 / 124 ′,  128 / 128 ′ are biased as described herein, the resulting electric field generally follows, e.g., on average, the curvilinear paths indicated by the arrows, from the second strip portion  128 B on the front surface of the second tine  116 , the fourth strip portion  128 D on the front surface of the first tine  114 , the second strip portion  128 B′ on the back surface of the first tine  114  and the fourth strip portion  128 D′ on the back surface of the second tine  116 , to the second strip portion  124 B on the front surface of the second tine  116 , the fourth strip portion  124 D on the front surface of the first tine  114 , the second strip portion  124 B′ on the back surface of the first tine  114  and the fourth strip portion  124 D′ on the back surface of the second tine  116 , respectively. 
       FIGS.  2 A and  2 B  schematically illustrate perspective views of the fork-shaped torsional mode quartz crystal device  100  described above with respect to  FIGS.  1 A and  1 B .  FIG.  2 A  illustrates the quartz crystal device  100  in a rest state, while  FIG.  2 B  illustrates a fork-shaped torsional mode quartz crystal device  100  under a bias. When the first and second electrode patterns  124 ,  128  are subjected to negative and positive relative voltages, respectively, as described above, such that the electric field lines are directed from various portions of the second electrode patterns  128 ,  128 ′ to corresponding portions of the first electrode patterns  124 ,  124 ′ in a manner as described above, the first and second tines  114 ,  116  twist about the x-axis ( FIG.  1 A ) extending in the lengthwise direction, as illustrated in  FIG.  2 B . When the first and second electrode patterns  124 ,  128  are subjected to pulsed negative and positive voltages, the quartz crystal device  100  can toggle between the rest state illustrated in  FIG.  2 A  and the deformed state illustrated in  FIG.  2 B . 
       FIG.  3    illustrates an equivalent circuit representation of quartz crystal oscillators. The equivalent circuit representation illustrated in  FIG.  3    is sometimes referred to as Butterworth Van Dyke (BVD) equivalent circuit. Without being bound to any theory, the electrical behavior of a piezoelectric bulk oscillator, such as a quartz crystal device near the vicinity of its frequencies of resonance can be represented by the equivalent circuit  300 . The equivalent circuit  300  includes two parallel paths between terminal nodes. The first path includes a static capacitance (C 0 ) which can represent a electrode capacitance of the quartz crystal device. The second path includes a motional inductor (L 1 ), a motional capacitor (C 1 ) and a motional resistor (R 1 ), and may sometimes be referred to as the motional arm. The L 1 , C 1  and R 1  and can be associated with the inertia, stiffness and internal losses of the mechanical vibrating system, respectively. 
     A quality factor (Q) can be used to benchmark quartz crystal resonance. The Q factor can represent a ratio of energy stored versus energy dissipated in a vibration cycle. Higher Q factors are generally associated with higher frequency stability and accuracy of a quartz crystal device. The magnitude of the Q factor can be associated with the qualitative behavior of a quartz crystal device. Without loss of generality, for example, a system with a very low Q factor can be an overdamped system. Below a certain value of the Q factor, an overdamped system may not oscillate at all. 
     Without being bound to any theory, the Q factor of a quartz crystal device can sometimes be expressed by the following expression: 
                   Q   =     1     2   ⁢       π   ⁢   f     s     ⁢     R   1     ⁢     C   1                 [   1   ]               
In equation [1], f s  is the series resonance frequency, and the product of R 1  and C 1  is sometimes referred to as a motional time constant τ, which can vary, among other things, with the angle of cut and the mode of vibration. According to Eq. [1], a larger R 1  results in a lower Q. Thus, a relatively large value of C 1  may be desired for the operating resonance of a piezoelectric device.
 
     Without being bound to any theory, a quartz crystal device such as the quartz crystal device  100  described above with respect to  FIGS.  1 A- 1 B and  2 A- 2 B  vibrating in a torsional vibrational mode can have a quartz crystal frequency f that can be approximated by the following equation: 
                   f   ≈     k   ⁢     t   lw               [   2   ]               
In equation [2], k is a constant for a given geometry and cut angle of the quartz crystal, l or L represents the tine length, w or W represents the tine width and t represents the tine thickness.
 
     Various applications increasingly demand quartz crystal devices to be miniaturized. To fit a quartz crystal device into a miniature package, the size of the quartz crystal device may be correspondingly scaled. As indicated by Eq. [ 2 ] above, for a constant width (W), to keep the frequency a constant, the tine length (L) and crystal thickness (t) may be scaled simultaneously. 
       FIG.  4    illustrates a graph plotting experimental crystal parameters (R 1  and C 1 ) obtained as function of crystal thickness (t), from a quartz crystal device such as the fork-shaped torsional mode quartz crystal device  100  described above with respect to  FIGS.  1 A- 1 B and  2 A- 2 B . The graph of  FIG.  4    was obtained from a quartz crystal device vibrating in a torsional mode crystal packaged in 5.0 mm×1.8 mm ceramic package. As illustrated, the motional capacitance (C 1 ) increases substantially linearly with increasing crystal thickness (t). On the other hand, the motional resistance (R 1 ) decreases super-linearly with increasing crystal thickness (t). For example, as illustrated, R 1  can increase from about 30 kΩ to about 100 kΩ when the thickness is reduced from about 75 μm to about 50 μm. When the R 1  is too high, the quartz crystal device can display unstable frequency and can even stop oscillation. According to various embodiments described herein, quartz crystal devices having structures with reduced R 1  are adapted for miniaturization of packaged quartz crystal devices. 
     To address these and other needs, according to various embodiments, a fork-shaped torsional mode quartz crystal device configured for temperature sensing comprises a fork-shaped quartz crystal comprising a base and a pair of elongate tines laterally extending from the base in a lengthwise direction, where each of the tines has formed on one or both of opposing sides thereof a mesa structure. The mesa structure includes a line structure laterally elongated in the lengthwise direction. The line structure vertically protrudes above adjacent recessed surfaces and has sidewalls abutting the recessed surfaces. The quartz crystal additionally comprises a first electrode and a second electrode formed on the one or both of the opposing sides of each of the tines and configured such that the fork-shaped quartz crystal vibrates in a torsional mode. In particular, each of the first and second electrodes have surfaces facing each other in a widthwise direction of the fork-shaped quartz crystal and configured such that, when a bias is applied between the first and second electrodes, the fork-shaped quartz crystal vibrates in a torsional mode in which each of the tines twists about an axis extending in the lengthwise direction. 
       FIGS.  5 A- 5 E  schematically illustrate various views of a fork-shaped torsional mode quartz crystal device  500  comprising a mesa structure, according to embodiments. The quartz crystal device  500  has on a first or front side  518  ( FIGS.  5 A,  5 C -E) having a first or front main surface, and a second or rear side  520  ( FIGS.  5 B,  5 C- 5 E ) opposite the first or front side having a second or rear main surface.  FIGS.  5 A and  5 B  illustrate a first or front side and a second side or bottom side of the fork-shaped quartz crystal device  500 , respectively.  FIGS.  5 C,  5 D and  5 E  illustrate cross-sectional views of the quartz crystal device  500  taken at cross-sections B-B, C-C, and D-D, respectively, as illustrated in  FIG.  5 A . Some features of the fork-shaped quartz crystal device  500  comprising a mesa structure are similar to corresponding features of the fork-shaped quartz crystal device  100  without a mesa structure described above with respect to  FIGS.  1 A- 1 B , and a detailed description of some of the similarities may not be repeated herein for brevity. 
     In a similar manner as described above with respect to  FIGS.  1 A and  1 B , the fork-shaped quartz crystal device  500  comprises a fork-shaped quartz crystal comprising a base  512 , a first tine  514  projecting from the base  512  and extending generally along a lengthwise direction, e.g., x-axis, and a second tine  516  projecting from the base  512  and extending generally along the same lengthwise direction. The tines can also be referred as arms or prongs. The first tine  514  and the second tine  516  can be generally parallel to one another. In the illustrated configuration, either or both of the first and second tines  514 ,  516  have a substantially uniform width along respective lengths. However, embodiments are not limited and in other configurations, one or both of the first and second tines  514 ,  516  can have uneven or non-uniform widths along the respective lengths. For example, without limitation, either or both of the first and second tines  514 ,  516  can be configured such that some portions of the first and/or second tines  514 ,  516  are wider than some other portions thereof, or such that some portions define a linear, curved, or other suitable shaped tapering width along the respective lengths. In various configurations, the centerlines of the first and second tines  514 ,  516  can be generally parallel to each another or generally follow parallel paths. While in the illustrated configuration, the quartz crystal device  500  has a pair of tines  514 ,  516 , embodiments are not so limited and in some other configurations, any of the quartz crystal devices disclosed herein can have a fewer or greater number of tines projecting from the base  512 . For example, in some configurations, without limitations, any of the quartz crystal devices disclosed herein can have four or more tines projecting from the base, where any number of the tines are configured to vibrate at least in a torsional mode. 
     Still referring to  FIGS.  5 A and  5 B , the base  512  supporting the first and second tines  514 ,  516  can include multiple portions or regions. For example, in the illustrated example, the base  512  comprises a first base portion  512 A from which the first and second tines  514 ,  516  directly extend, and a second base portion  512 B that forms an end portion of the fork-shaped quartz crystal device  500 . In the illustrated embodiment, the first and second base portions  512 A,  512 B have the same or similar widths. However, embodiments are not so limited, and in some other embodiments, the first and second base portions  512 A,  512 B can be defined by different widths. The second base portion  512 B can be configured such that the quartz crystal device  500  can be mounted to a ceramic package or other support structures. 
     In the illustrated embodiment, a pair of notches, slots or indentations  532  recessed inward from opposing side surfaces of the quartz crystal device  500 . According to embodiments, when present, the notches  532  can reduce the amount of vibration leakage from the first and second tines  514 ,  516  to the base  512 , thereby increasing the efficiency of the fork-shaped quartz crystal device  500 . In the illustrated embodiment, the notches  532  are formed at an adjoining region between the first and second base portions  512 A,  512 B. However, embodiments are not so limited and in other embodiments, the notches  532  can be positioned vertically anywhere between the connected ends of the first and second tines  514 ,  516  and the terminal edge of the base  512 . The notches  532  can have any suitable shape including any polygonal (e.g., rectangular, triangular) or curved (e.g., semicircular, elliptical) shapes. 
     Referring to  FIGS.  5 A / 5 B and  5 C, the B-B cross-section illustrates that each of the first and second tines  514 ,  516  comprises a tip portion that generally forms a rectangular shape, in a similar manner as the quartz crystal device  100  described above with respect to  FIGS.  1 A and  1 B . As illustrated in  FIG.  5 C , the first and second main surfaces on first and second sides  518 ,  520 , respectively, of the fork-shaped quartz crystal device  500  at the tip portions of the first and second tines  514 ,  516  can be generally parallel, opposing planar surfaces that are connected by side surfaces thereof. 
     Referring to  FIGS.  5 A / 5 B and  5 D, the C-C cross-section illustrates that, unlike the quartz crystal device  100  described above with respect to  FIGS.  1 A and  1 B , each of the first and second tines  514 ,  516  of the quartz crystal device  500  comprises a mid-section portion in which each of the first and second tines  514 ,  516  comprises a mesa structure formed on or as part of one or both of opposing main surfaces on the first and second sides  518 ,  520 . In the mid-section, the first tine  514  comprises a first side on which a vertically protruding first line structure  514 A is formed as part of a first mesa structure, and a second side opposite the first side on which a vertically protruding second line structure  514 B is formed as part of a second mesa structure. The first and second line structures  514 A,  514 B extend in the lengthwise direction of the first and second tines  514 ,  516 , e.g., the x-direction. On one end, the first and second line structures  514 A,  514 B terminate adjacent to respective terminal end portions of the first and second tines  514 ,  516 , On the other end, the first and second line structures  514 A,  514 B terminate adjacent to connected end portions of the first and second tines  514 ,  516 , or to the base  512 , The first and second line structures  514 A,  514 B laterally divide the first and second (left and right) sides of the first tine  514  into first and second recessed portions  514 C,  514 D separated in a widthwise direction, e.g., the y-direction. In a similar manner, the second tine  516  comprises a first side on which a vertically protruding first line structure  516 A is formed and a second side opposite the first side on which a vertically protruding second line structure  516 B is formed. The first and second line structures  516 A,  516 B extend in the lengthwise direction of the first and second tines  514 ,  516 , e.g., the x-direction, and laterally divide the first and second (left and right) sides of the second tine  516  into first and second recessed portions  516 C,  516 D separated in a widthwise direction, e.g., the y-direction. 
     In the illustrated embodiment, the top surfaces of the line structures  514 A,  514 B,  516 A,  516 B are coplanar with adjacent base  512  and the top portions of the respective tines. However, embodiments are not so limited and in other implementations, the top surfaces of the line structures  514 A,  514 B,  516 A,  516 B may be higher or lower relative to the adjacent base  512  and the top portions of the respective tines. 
     In the illustrated embodiment, each of the first and second line structures  514 A,  516 B of the first tine  514  and the first and second line structures  516 A,  516 B of the second tine  516  forms a mesa structure having sidewalls that are generally perpendicular with respective abutting recessed portions. However, embodiments are not so limited, and in some other embodiments, the one or both sidewalls may form angles greater or less than 90°. For example, the one or both sidewalls may form 40-50°, 50-60°, 60-70°, 70-80°, 80-90° or an angle in a range formed by any of these values. In addition, when the sidewalls are tapered, the line structures  514 A,  514 B,  516 A,  516 B may or may not have a top surface. Accordingly, in some embodiments, the line structures have a trapezoidal or triangular cross sectional shapes. 
     Referring to  FIGS.  5 A / 5 B and  5 E, the base  512  of the quartz crystal device  500  generally has a rectangular shape in a cross sectional view. As illustrated in  FIG.  5 E , the main surfaces of the base  512  on first and second sides  518 ,  520  of the quartz crystal device  500  can be generally parallel and opposing planar surfaces that are connected by side surfaces of the base  512 . 
     Various portions of the quartz crystal device  500  have surfaces having formed thereon electrode patterns that are collectively configured to vibrate the tines of the fork-shaped quartz crystal device  500  in a torsional mode. The electrode patterns include a first electrode pattern  524  of a first polarity, e.g., one of a negative or positive electrode pattern, and a second electrode pattern  528  of a second polarity, e.g., the other of the negative or positive electrode pattern. 
     Referring to the first main (top) surface of the fork-shaped quartz crystal device  500  illustrated in  FIG.  5 A , the first (e.g., negative) electrode pattern  524  formed on the top surface includes first to fourth portions  524 A- 524 D, and the second (e.g., positive) electrode pattern  528  formed on the top surface includes first to fourth portions  528 A- 528 D. The first electrode pattern  524  includes a first portion  524 A covering a first (right) side of the base  512  and extends upward therefrom in the x-direction in the form of the second thin strip portion  524 B. The second thin strip portion  524 B extends along a right or outer edge portion of the second tine  516 , and terminates adjacent to the third portion  528 C of the second electrode pattern  528  formed on the terminal end portion of the second tine  516 . The first electrode pattern  524  additionally includes the third portion  524 C formed on the terminal end portion of the first tine  514 , extends downward therefrom in the x-direction in the form of the fourth thin strip portion  524 D along a left or outer edge portion of the first tine  514 , and terminates adjacent to the first portion  528 A of the second electrode pattern  528  formed on the second (left) side of the base  512 . 
     Still referring to the plan view illustrated in  FIG.  5 A , the second electrode pattern  528  formed on the top surface includes the first portion  528 A covering a second (left side) of the base  512  and extending upward therefrom in the x-direction and forking or splitting into the second thin strip portion  528 B formed and extending upward along a left or inner edge portion of the second tine  516 , and a fourth thin strip portion  528 D formed along and extending upward along a right or inner edge portion of the first tine  514 . The fourth thin strip portion  528 D terminates adjacent to the third portion  524 C of the first electrode pattern  524  formed on the terminal end of the first tine  514 . The second thin strip portion  528 B extends to connect to the third portion  528 C of the second electrode pattern  528  formed on the terminal end portion of the second tine  516 . 
     Referring to the second main (bottom) surface of the quartz crystal device  500  illustrated in  FIG.  5 B , the quartz crystal device  500  has formed thereon various portions of a first electrode pattern  524 ′ of the first polarity and the second electrode pattern  528 ′ of the second polarity formed thereon. The combined pattern of the first and second electrode patterns  524 ′,  528 ′ formed on the bottom side  520  may generally be shaped similarly as the combined pattern of the first and second electrode patterns  524 ,  528  formed on the front side  518  as described above, except that their polarities are interchanged. The first (e.g., negative) electrode pattern  524 ′ formed on the bottom surface includes first to fourth portions  524 A′- 524 D′, and the second (e.g., positive) electrode pattern  528 ′ formed on the bottom surface includes first to fourth portions  528 A′- 528 D′. The second electrode pattern  528 ′ includes the first portion  528 A′ covering a first (right side) of the base  512  portion and extends upward therefrom in the x-direction in the form of the second thin strip portion  528 B′ along a right or outer edge portion of the first tine  514 , and terminates adjacent to the third portion  524 C′ of the first electrode pattern  524 ′ formed on the terminal end portion of the first tine  514 . The second electrode pattern  528 ′ additionally includes the third portion  528 C′ formed on the terminal end portion of the second tine  516 , extends downward therefrom in the x-direction in the form of the fourth thin strip portion  528 D′ along a left or outer edge portion of the second tine  516 , and terminates adjacent to the first portion  524 A′ of the first electrode pattern  524 ′ formed on the second (left) side of the base  512  portion when viewing the rear side  520 . 
     Additionally referring to the rear side  520 , the first electrode pattern  524 ′ formed on the bottom surface includes the first portion  524 A′ covering a second (left side) of the base  512  portion and extends upward therefrom in the x-direction and forks or splits into the second thin strip portion  524 B′ formed and extending upward along a left or inner edge portion of the first tine  514 , and a fourth thin strip portion  524 D′ formed along and extending upward along a right or inner edge portion of the second tine  516 . The fourth thin strip portion  524 D′ terminates adjacent to the third portion  528 C′ of the second electrode pattern  528 ′ formed on the terminal end of the second tine  516 . The second thin strip portion  524 B′ extends to connect to the third portion  524 C′ of the first electrode pattern  524 ′ formed on the terminal end portion of the first tine  514 . 
     As described above, the fork-shaped torsional mode quartz crystal device  500  includes various portions of the first electrode pattern  524  on the first side  518  and the first electrode pattern  524 ′ on the second side  520  that are connected to each other. Similarly, various portions of the second electrode pattern  528  on the first side  518  and the second electrode pattern  528 ′ on the second side  520  are connected to each other. The resulting relative positions of different portions of the first electrode patterns  524 ,  524 ′ and the second electrode patterns  528 ,  528 ′ are further described below with respect to  FIGS.  5 C- 5 E . 
       FIG.  5 C  illustrates a cross-sectional view of the fork-shaped quartz crystal device  500  including a mesa structure, though the section B-B or through terminal end portions of the first and second tines  514 ,  516 . As described above with respect to  FIGS.  5 A and  5 B , the terminal portion of the first tine  514  has formed on the first and second sides  518 ,  520  the third portion  524 C of the first electrode pattern  524  and the third portion  524 C′ of the first electrode pattern  524 ′, respectively. The third portions  524 C and  524 C′ are connected by side portions  524 C″ formed on left and right side surfaces of the first tine  514 . In addition, the terminal portion of the second tine  516  has formed on the first and second sides  518 ,  520  the third portion  528 C of the second electrode pattern  528  and the third portion  528 C′ of the second electrode pattern  528 ′, respectively. The third portions  528 C and  528 C′ are connected by side portions  528 C″ formed on left and right side surfaces of the second tine  514 . Thus configured, the terminal end of portion the first tine  514  is wrapped or surrounded by the first electrode pattern having the first polarity on top, bottom and side surfaces thereof, and the terminal end portion of the second tine  516  is surrounded by the second electrode pattern having the second polarity on top bottom and side surfaces thereof. 
       FIG.  5 D  illustrates a cross-sectional view of the fork-shaped quartz crystal device  500  comprising a mesa structure through the section C-C ( FIG.  5 A ) at midsections the first and second tines  514 ,  516 . As described above, each of the first and second tines  514 ,  516  comprises a mesa structure formed on one or both of the first and second sides  518 ,  520 . Accordingly, the quartz crystal device  500  comprises electrode patterns that are formed synergistically with respect to the mesa structures, including the first and second line structures  514 A,  514 B formed on the first and second sides  518 ,  520  of the first tine  514  and the first and second line structures  516 A,  516 B formed on the first and second sides  518 ,  520  of the second tine  516 . 
     Referring to  FIG.  5 D , the first tine  514  has a first main surface on the first side  518  having formed thereon the fourth strip portion  524 D of the first electrode pattern  524  on the outer edge portion thereof, and the fourth strip portion  528 D of the second electrode pattern  528  on the inner edge portion thereof. The first tine  514  additionally has a second main surface on the second side  520  having formed thereon the second strip portion  524 B′ of the first electrode pattern  524 ′ on the inner edge portion thereof, and the second strip portion  528 B′ of the second electrode pattern  528 ′ formed on the outer edge portion thereof. Thus configured, the first and second main surfaces of the first tine  514  comprises a mesa structure including the first and second line structures  514 A,  514 B, respectively, which defines the topography of the electrode patterns formed thereon. For example, the fourth strip portion  524 D of the first electrode pattern  524  comprises a portion, e.g., a vertical portion, that covers at least a portion of a first (left) sidewall of the first line structure  514 A and a portion, e.g., a horizontal portion, that extends to cover at least a portion of a first (upper) recessed surface of the first recessed portion  514 C abutting the first sidewall of the first line structure  514 A. Similarly, the fourth strip portion  528 D of the second electrode pattern  528  comprises a portion, e.g., a vertical portion, that covers at least a portion of a second (right) sidewall of the first line structure  514 A and a portion, e.g., a horizontal portion, that extends to cover at least a portion of a first (upper) recessed surface of the second recessed portion  514 D abutting the second sidewall of the first line structure  514 A. In a similar manner as the first side  518 , the second side of the first tine  514  has formed thereon the fourth strip portion  528 B′ of the second electrode pattern  528 ′ comprising a portion, e.g., a vertical portion, that covers at least a portion of a first (left) sidewall of the second line structure  514 B and a portion, e.g., a horizontal portion, that extends to cover at least a portion of a second (lower) recessed surface of the first recessed portion  514 C abutting the first sidewall of the second line structure  514 B. Similarly, the second strip portion  524 B′ of the first electrode pattern  524 ′ comprises a portion, e.g., a vertical portion, that covers at least a portion of a second (right) sidewall of the second line structure  514 B and a portion, e.g., a horizontal portion, that extends to cover at least a portion of a second (lower) recessed surface of the second recessed portion  514 D abutting the second sidewall of the second line structure  514 B. Thus formed, each of the first electrode portions  524 D,  523 B′ and second electrode portions  528 B′,  528 D have surfaces facing each other in a widthwise direction (y-direction) and configured such that, when a bias is applied between the first and second electrodes  524 / 524 ′,  528 / 528 ′, the fork-shaped quartz crystal device  500  vibrates in a torsional mode in which each of the tines  514 ,  516  twists about the x-axis. 
     Still referring to  FIG.  5 D , the cross-sectional view through the section C-C shows the second tine  516  having an analogous structure as the first tine  514 , where the first line structure  516 A, the second line structure  516 B, the first recessed structure  516 C and the second recessed structure  516 D are formed in an analogous manner as the corresponding topography including the first line structure  514 A, the second line structure  514 B, the first recessed structure  514 C and the second recessed structure  514 D of the first tine  514 , respectively. In addition, the second strip portion  528 B of the second electrode pattern  528 , the second strip portion  524 B of the first electrode pattern  524 , the fourth strip portion  524 D′ of the first electrode pattern  524 ′ and the fourth strip portion  528 D′ of the second electrode pattern  528 ′ are formed in an analogous manner relative the underlying topography as the fourth strip portion  524 D of the first electrode pattern  524 , the fourth strip portion  528 D of the second electrode pattern  528 , the second strip portion  528 B′ of the second electrode pattern  528 ′ and the second strip portion  524 B′ of the first electrode pattern  524 ′, respectively, that are formed on the first tine  514 . 
       FIG.  5 E  illustrates a cross-sectional view of the quartz crystal device  500  though the section D-D ( FIG.  5 A ) at the terminal end portion of the base  512 . As described above with respect to  FIGS.  5 A and  5 B , the terminal end portion of the base  512  has formed on the first side  518  the first portion  524 A of the first electrode pattern  524  on the right side and the first portion  528 A of the second electrode pattern  528  on the left side. The terminal portion of the base  512  additionally has formed on the second side  520  the first portion  524 A′ of the first electrode pattern  524 ′ on the right side and the first portion  528 A′ of the second electrode pattern  528 ′ on the left side. The first portions  524 A and  524 A′ of the first electrode patterns  524  and  524 ′, respectively, are connected by a side portion  524 A″ formed on right side of the base  512 . The first portions  528 A and  528 A′ of the second electrode patterns  528  and  528 ′, respectively, are connected by a side portion  528 A″ formed on left side of the base  512 . As configured, the first (e.g., negative) electrode pattern  524 / 524 ′ partly surrounds the base  512  at the right side, and the second (e.g., positive) electrode pattern  528 / 528 ′ partly surrounds the base  512  at the left side. 
     In operation, electrical pulses having a first (e.g., one of negative or positive) polarity may be applied to the first electrode patterns  524 ,  524 ′ (including the second and fourth strip portions  524 B,  524 D on the front side  518  of the second and first tines  516 ,  514 , respectively, and the second and fourth strip portions  524 B′,  524 D′ on the back side  520  of the first and second tines  516 ,  514 , respectively, as illustrated in  FIG.  5 D ). On the other hand, electrical pulses having a second (e.g., the other of negative or positive) polarity may be applied to the second electrode patterns  528 ,  528 ′ (including the second and fourth strip portions  528 B,  528 D on the front side  518  of the second and first tines  516 ,  514 , respectively, and to the second and fourth strip portions  528 B′,  528 D′ on the back side  520  of the first and second tines  514 ,  516 , respectively, as illustrated in  FIG.  5 D ). While embodiments are not so limited, the portions of the first electrode patterns  524 ,  524 ′ may be at the same potential, and the portions of the second electrode patterns  528 ,  528 ′ may be at the same potential. In addition, pulses having the first and second polarities may be applied to the quartz crystal device  500  simultaneously. 
     When the first and second electrode patterns  524 / 524 ′,  528 / 528 ′ are biased as described herein, the resulting electric field generally follows, e.g., on average, the paths indicated by the arrows in  FIG.  5 D . Advantageously, compared to the curvilinear electric field paths followed by the respective electrodes illustrated for the quartz crystal device  100  described above with respect to  FIGS.  1 A and  1 B , the electric field paths include direct horizontal paths between the vertical portions of adjacently disposed electrodes of opposite polarities. For example, the electric field paths include a horizontal path from the vertical portions of the second strip portion  528 B on the left sidewall of the first line structure  516 A of the second tine  516 , the fourth strip portion  528 D on the right sidewall of the first line structure  514 A of the first tine  514 , the second strip portion  528 B′ on the left sidewall of the second line structure  514 B of the first tine  514  and the fourth strip portion  528 D′ on right sidewall of the second line structure  516 B of the second tine  516 , to facing vertical portions of the second strip portion  524 B on the right sidewall of the first line structure  516 A of the second tine  516 , the fourth strip portion  524 D on the left sidewall of the first line structure  514 A of the first tine  514 , the second strip portion  524 B′ on the right sidewall of the second line structure  514 B of the first tine  514  and the fourth strip portion  524 D′ on the left sidewall of the second line structure  516 B of the second tine  516 , respectively. It will be appreciated that the electric field paths indicated by the arrows between the vertical electrode portions are generally horizontal and in directions that are orthogonal to the electrode surfaces, In addition, the distance traversed by the electric field paths, e.g., the thicknesses of the first and second line structures  514 A,  514 B of the first tine  514  and the first and second line structures  516 A,  516 B of the second tine  516 , are substantially smaller than the corresponding paths traversed by the corresponding electric field lines described above with respect to  FIG.  1 B . 
     Advantageously, the resulting electric field strength is relatively stronger for a given voltage and/or about the same for a lower voltage, thereby enabling more efficient vibration by twisting the first and second tines  514 ,  516 , or torsionally vibrating the fork-shaped quartz crystal device  500 . Thus, relatively lower power may be dissipated in torsionally vibrating the quartz crystal device  500  compared to quartz crystal devices that are configured differently, e.g., the quartz crystal devices that do not have the line structures protruding on one or both sides of the tines. In addition, because relatively smaller volume of the quartz substrate is twisted in operation, the inventors have recognized that the mesa structures can substantially reduce the motional resistance (R 1 ), thereby further reducing the energy consumed in operation and increasing the quality factor (Q). As configured, the configuration of the quartz crystal device  500  can be particularly adapted for lower power and higher precision applications. 
       FIGS.  6 A- 6 C  schematically illustrate perspective views of the fork-shaped torsional mode quartz crystal device  500  described above with respect to  FIGS.  5 A- 5 E .  FIG.  6 A  illustrates the fork-shaped quartz crystal device  500  in a rest state, while  FIG.  6 C  illustrates the quartz crystal device  500  under a simulated bias conditions in which the tines are torsionally twisted.  FIG.  6 B  schematically illustrates a cross-sectional view of the tines including the electrodes. When the first electrode patterns  524 ,  524 ′ and the second electrode patterns  528 ,  528 ′ are subjected to negative and positive voltages, respectively, as described above, such that the electric field limes comprise relatively strong horizontal vector components between vertical portions of the second electrode patterns  528 ,  528 ′ and the corresponding vertical portions of the first electrode patterns  524 ,  524 ′, the first and second tines  514 ,  516  twist about the x-axis extending in the lengthwise direction with increased efficiency, as illustrated in  FIG.  6 C . When the first and second electrode patterns  524 / 524 ′,  528 / 528 ′ are subjected to pulsed negative and positive voltages, the fork-shaped quartz crystal device  500  can toggle between the rest state illustrated in  FIG.  6 A  and the deformed or twisted state illustrated in  FIG.  6 C . 
     The piezo-electrical effect in quartz can be utilized in different vibrational modes based the way the crystal wafers or blanks are cut out of the raw bulk crystal in regards of their orientation to the atomic lattice. Some commonly used vibrational modes include, e.g., flexural, extensional, face shear and thickness shear modes, to name a few. In order to utilize the desired vibration mode, the quartz crystals are cut in a certain angle with respect to the quartz lattice.  FIG.  7    schematically illustrates different crystal orientations for different types of quartz crystal devices including a fork-shaped torsional mode quartz crystal device, according to embodiments. Commonly used cuts of quartz crystal include, e.g., X-cut, Z-cut, NT-cut, AT-cut, BT-cut and SC-cut, to name a few. Referring to  FIG.  7   , according to various embodiments disclosed herein, the fork-shaped quartz crystal devices configured for torsional vibration are fabricated from a Z-cut quartz. The quartz crystals for torsional vibration have surface normal directions of main surfaces that are rotated at a tilt angle θ of about −50 degrees to +50 degrees, about −40 degrees to +40 degrees, or about −30 degrees to +30 degrees relative to a X-axis. In addition, the lengths of the tines of the quartz crystal devices extend in the X-axis of the quartz crystal. This is notably in contrast with tuning fork quartz crystal devices configured for flexural vibrations, in which the lengths of the tines extend in the Y-axis of a quartz crystal from which the devices are fabricated. For example, referring back to  FIGS.  5 A- 5 E , the top surfaces of the first and second line structures  514 A,  514 B of the first tine  514  the top surfaces of the first and second line structures  516 A,  516 B of the second tine  516  are rotated at the θ relative to the X-axis. 
     In the following, various physical characteristics of the fork-shaped quartz crystal devices according to embodiments are described. 
     One application for quartz crystal devices described herein is a temperature sensor. The quartz crystal devices are particularly suitable for a temperature sensor application because the frequency of the quartz crystal devices described herein have a substantially linear relationship with temperature. The relationship between the frequency of oscillation of a quartz crystal device and temperature may be represented by the following approximate polynomial expression:
 
( f ( T )− f ( T   0 ))/ f ( T   0 )=α( T−T   0 )+β( T−T   0 ) 2 +γ( T−T   0 ) 3 ,  [3]
 
where T 0  is a reference temperature, e.g., 25° C., f(T 0 ) is the frequency at the T 0 , f(T) is the frequency of the crystal subjected to the temperature T, and α, β and γ are coefficients that are independent of temperature. For temperature sensor applications, it is desirable that the temperature dependence of the f(T) be as linear as possible because it allows for direct and accurate conversion between temperature and frequency in a numerical or quasi-numerical manner. Thus, by measuring the f(T), which can be performed with a high degree of resolution, a correspondingly accurate temperature can be calculated based on the linear relationship. For a high degree of linearity of the f(T), the non-linear coefficients β and γ should be as small as possible relative to the linear coefficient α.  FIG.  8 A  illustrates a linear temperature coefficient α as a function of an angle θ for a fork-shaped torsional mode quartz crystal device, according to embodiments. According to various embodiments, α can have a value of about 1×10 −6 -1×10 −5 ° C. −1 , 1×10 −5 -2×10 −5 ° C. −1 , 2×10 −5 -3×10 −5 ° C. −1 , 3×10 −5 -4×10 −5 ° C. −1 , 4×10 −5 -5×10 −5 ° C. −1 , or a value in a range defined by any of these values, for instance 1×10 −5 -3×10 −5 ° C. −1 , when the quartz crystal device is configured according to any of configurations descried herein. Typical experimental and finite element analysis (FEA)-simulated α, β and γ values for one example configuration of the fork-shaped quartz crystal device similar to that described in  FIGS.  5 A- 5 E  are shown in TABLE 1 below.
 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Experimental Data 
                 FEA Simulation 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 α,/° C. 
                 3.44E−05 
                 3.61E−05 
               
               
                   
                 β,/° C. 2   
                 1.93E−08 
                 2.29E−08 
               
               
                   
                 γ,/° C. 3   
                 2.82E−11 
                 5.14E−11 
               
               
                   
                   
               
            
           
         
       
     
     According to various embodiments described herein, the quartz crystal devices can be operated at a frequency in a range between about 100 kHz and 500 kHz. Example frequencies that can be used to operate the quartz crystal device include, e.g., 172.0 kHz, 190.5 kHz, 262.144 kHz, 300.0 kHz, 325.0 kHz, and 350.0 kHz, to name a few. 
       FIGS.  8 B and  8 C  illustrates the linearity of the f(T) for example fork-shaped quartz crystal devices according to embodiments.  FIG.  8 B  illustrates a frequency versus temperature curve at an angle of tilt of −30° relative to the X-axis of a quartz crystal for a fork-shaped torsional mode quartz crystal device having a mesa structure according to embodiments.  FIG.  8 C  illustrates a frequency versus temperature curve at an angle of tilt of 30° relative to the X-axis of a quartz crystal for a fork-shaped torsional mode quartz crystal device having a mesa structure according to embodiments. The linearity coefficient α for the examples illustrated in  FIGS.  8 B and  8 C  were 3.9×10 −5 ° C. −1  and 9×10 −6 ° C. −1 , respectively. 
       FIG.  9    illustrates a frequency versus thickness curves for fork-shaped torsional mode quartz crystal devices with and without a mesa structure. Both structures demonstrate a substantially linear relationship between the thickness t or t 1  and the resonance frequency. In addition, fork-shaped torsional mode quartz crystal devices with a mesa structure demonstrate a substantially linear relationship between the thickness t 1  of the recessed regions of the tines and the resonance frequency. 
     As described above with respect to Eq. [1], the motional resistance (R 1 ), the motional capacitance (C 1 ) and the quality factor (Q) are figures of merit for quartz crystal devices. The inventors have discovered that, by forming the mesa structures on the tines of the fork-shaped torsional quartz crystal devices as described herein, the R 1 , C 1  and Q can be advantageously engineered for improved torsional vibration compared to fork-shaped torsional quartz crystal devices that do not have the mesa structures formed on the tines.  FIGS.  10 A- 10 C  illustrate qualitatively different frequency or thickness dependencies of the R 1 , C 1  and Q between the quartz crystal devices configured for torsional vibration without (e.g.,  FIGS.  1 A and  1 B ) and with ( FIGS.  5 A- 5 E ) line or mesa structures formed on the tines of the fork-shaped quartz crystal devices. 
       FIG.  10 A  is a graph illustrating the R 1  as a function of frequency for fork-shaped torsional mode quartz crystal devices with and without a mesa structure. As illustrated, without the line or mesa structures formed on the tines, the R 1  of the fork-shaped torsional quartz crystal device generally decreases with increasing frequency at least within the illustrated frequency range. In contrast, the R 1  of the fork-shaped torsional quartz crystal device having the line or mesa structures formed on the tines generally increases with increasing frequency at least within the illustrated frequency range. 
       FIG.  10 B  is a graph illustrating the C 1  as a function of frequency for fork-shaped torsional mode quartz crystal devices with and without a mesa structure. As illustrated, without the line or mesa structures formed on the tines, the C 1  of the fork-shaped torsional quartz crystal device generally increases with increasing frequency at least within the illustrated frequency range. In contrast, the C 1  of the fork-shaped torsional quartz crystal device having the line or mesa structures formed on the tines generally decreases with increasing frequency at least within the illustrated frequency range. 
       FIG.  10 C  is a graph illustrating quality factor (Q) as a function of frequency for fork-shaped torsional mode quartz crystal devices with and without a mesa structure. From  FIGS.  10 A and  10 B , it will be appreciated that, while the R 1  is generally lower and the C 1  is generally higher for fork-shaped torsional quart crystal devices with line or mesa structures on the tines compared to those without the line or mesa structures on the tines, the decrease in magnitude of the R 1  is greater than the increase in magnitude of the C 1 . In sum, as illustrated, the fork-shaped torsional quartz crystal devices with line or mesa structures advantageously have significantly higher value of the Q at least within the illustrated frequency range. 
     According to various embodiments, fork-shaped torsional quartz crystal devices with line or mesa structures have an R 1  of 1-10 kΩ, 10-20 kΩ, 20-30 kΩ, 30-40 kΩ or a value in a range defined by any of these values, a C 1  of 0.1-0.2 fF, 0.2-0.3 fF, 0.3-0.4 fF, 0.4-0.5 fF, 0.5-0.6 fF, 0.6-0.7 fF, 0.7-0.8 fF, 0.8-0.9 fF, 0.9-1.0 fF, 1.0-1.1 fF, 1.1-1.2 fF, 1.2-1.3 fF or a value in a range defined by any of these values or greater, and a Q of 10,000-20,000, 20,000-50,000, 50,000-200,000, 200,000-400,000, 400,000-600,000, 600,000-800,000, 800,000-1,000,000, or a value in a range defined by any of these values, for a frequency range including at least 172.0 kHz to 350.0 kHz, according to embodiments. 
     Advantageously, the shape of the fork-shaped torsional mode quartz crystal devices having mesa or line structures formed on the tines offer a greater number of physical parameters that can be optimized for the desired performance. For example, the physical parameters that can be optimized include the overall thickness of the quartz crystal device, the height of the line or mesa structure, the width of the line or mesa structure and the length of the line or mesa structure, to name a few. The technical effects of optimizing these parameters are described below. 
       FIG.  11    is a graph illustrating the frequency and the motional capacitance (C 1 ) as a function of the height of the line structure or the mesa thickness for fork-shaped torsional mode quartz crystal devices comprising a mesa structure, according to embodiments. As illustrated, the frequency and the C 1  can vary inversely as a function of the height of the line structure or the mesa thickness 
       FIG.  12 A  is a graph illustrating the motional resistance (R 1 ) and the motional capacitance (C 1 ) as a function of a ratio (t 1 /t) between a thickness (t 1 ) of the recessed region of a tine and a thickness (t) of the line region of the tine of the fork-shaped torsional mode quartz crystal devices, according to embodiments.  FIG.  12 B  is a graph illustrating the frequency and the quality factor (Q) as a function of the t 1 /t for the fork-shaped torsional mode quartz crystal devices. As illustrated, the R 1  can be lowered at a faster rate compared to the rate of increase of the C 1  by decreasing the t 1 /t, such that the Q can advantageously be significantly increased. According to various embodiments, the t 1 /t can be about 0.01-0.2, 0.2-0.4, 0.4-0.6, 0.6-0.8, 0.8-0.99, or have a value in a range defined by any of these values. 
       FIG.  13 A  is a graph illustrating the motional resistance (R 1 ) and the motional capacitance (C 1 ) as a function of a ratio (W 1 /W) between a width (W) of a line region and an overall width (W) of the tines of fork-shaped torsional mode quartz crystal devices comprising a mesa structure, according to embodiments.  FIG.  13 B  is a graph illustrating the frequency and the quality factor (Q) as a function of the W 1 /W for the fork-shaped torsional mode quartz crystal devices comprising a mesa structure. As illustrated, the R 1  can be lowered at a faster rate compared to the rate of increase of the C 1  by decreasing the W 1 /W, such that the Q can advantageously be significantly increased. According to various embodiments, the W 1 /W can be about 0.01-0.2, 0.2-0.4, 0.4-0.6, 0.6-0.8, 0.8-0.99, or have a value in a range defined by any of these values. 
       FIG.  14 A  is a graph illustrating the motional resistance (R 1 ) and the motional capacitance (C 1 ) as a function of a ratio (L 1 /L) between a length (L 1 ) of the tip regions of the tines excluding the line regions and a length (L) of the tines including the tip regions and the line regions for fork-shaped torsional mode quartz crystal devices comprising a mesa structure, according to embodiments.  FIG.  14 B  is a graph illustrating the frequency and the quality factor (Q) as a function of the L 1 /L for the fork-shaped torsional mode quartz crystal devices. As illustrated, the R 1  can be lowered at a faster rate compared to the rate of increase of the C 1  by decreasing the L 1 /L down to about 0.3, such that the Q can advantageously be increased. According to various embodiments, the L 1 /L can be about 0.01-0.2, 0.2-0.4, 0.4-0.6, 0.6-0.8, or have a value in a range defined by any of these values. 
     According to some embodiments, a length (L) of the tines of the fork-shaped torsional mode quartz crystal devices can be about 500-1000 μm, 1000-1500 μm, 1500-2000 μm, 2000-2500 μm, 2500-3000 μm, or any value in a range defined by these values, for instance 1778 μm. 
     According to some embodiments, a length (L 1 ) of the tip regions of the tines of the fork-shaped torsional mode quartz crystal devices can be about 100-300 μm, 300-500 μm, 500-700 μm, 700-900 μm, or any value in a range defined by these values, for instance 508 μm. The L 1  can also be defined by any of the L 1 /L values described above. 
     According to some embodiments, a length (L m ) of the base (e.g.,  512 ,  FIG.  5 A ) of the fork-shaped torsional mode quartz crystal devices can be about 100-300 μm, 300-500 μm, 500-700 μm, 700-1000 μm, or any value in a range defined by these values, for instance 635 μm. 
     According to some embodiments, a width (W) of a tine of the fork-shaped torsional mode quartz crystal devices can be about 100-200 μm, 200-300 μm, 300-400 μm, 400-500 μm, or any value in a range defined by these values, for instance 236 μm. 
     According to some embodiments, a width (W) of a line or mesa structure formed on a tine of the fork-shaped torsional mode quartz crystal devices can be about 1-20 μm, 20-40 μm, 40-60 μm, 60-80 μm, 80-100 μm or any value in a range defined by these values, for instance 50 μm. 
     According to some embodiments, a thickness (t 1 ) of a recessed region of a tine of the fork-shaped torsional mode quartz crystal devices can be about 40-60 μm, 60-80 μm, 80-100 μm, 100-120 μm, or any value in a range defined by these values, for instance 85 μm. 
     According to some embodiments, a thickness (t) of the fork-shaped torsional mode quartz crystal devices can be about 60-90 μm, 90-120 μm, 120-150 μm, 150-180 μm, or any value in a range defined by these values, for instance 120 μm. 
       FIG.  15 A  illustrates experimentally measured frequency and motional capacitance (C 1 ) versus mesa thickness for a fork-shaped torsional mode quartz crystal device comprising a mesa structure, according to embodiments.  FIG.  15 B  illustrates experimentally measured linearity of the f(T) and the R 1  of fork-shaped torsional mode quartz crystal devices, according to embodiments. The simulated and measured linearity coefficients α are in reasonably good agreement, at 3.61×10 −5 ° C. −1  and 3.44×10 −5 ° C. −1 , respectively. 
       FIG.  16 A  illustrates a plan-view of a packaged fork-shaped torsional mode quartz crystal device  1600  packaged in a ceramic package, according to some embodiments.  FIG.  16 B  illustrates a cross-sectional view of the packaged fork-shaped torsional mode quartz crystal device  1600  illustrated in  FIG.  16 A , which includes the fork-shaped torsional mode quartz crystal device  500  enclosed, e.g., hermetically sealed, in the ceramic package illustrated in  FIG.  16 A . The fork-shaped torsional mode quartz crystal device  500  may be, e.g., any one of the fork-shaped torsional quartz crystal devices described herein. The ceramic package includes a ceramic package substrate  1612  and a lid  1608  for housing the fork-shaped torsional mode quartz crystal device  500  according to various embodiments described above. 
     The ceramic package substrate  1612  comprises a ceramic frame having a plurality walls that form a cavity  1616  configured to house the fork-shaped torsional mode quartz crystal device  500 . The ceramic package substrate  1612  is configured to provide thermal and electrical communication between the fork-shaped torsional mode quartz crystal device  500  and the outside world, e.g., a PCB (not shown), through one or more metallization structures (not shown). One or more mounting pedestals  1620  ( FIG.  16 B ) may be formed inside the cavity  1616  of the ceramic package substrate  1620  that are configured for mounting the fork-shaped torsional mode quartz crystal device  500 . While not shown, the mounting pedestals  1620  may have formed thereon an inner metallization pad that is electrically connected to an outer metallization pad for providing electrical and thermal communication between the fork-shaped torsional mode quartz crystal device  500  and the outside world. In the illustrated embodiment, the fork-shaped torsional mode quartz crystal device  500  is bonded at two points on each of the pedestals  1620 . However, embodiments are not so limited and in other embodiments, the fork-shaped torsional mode quartz crystal device  500  may be bonded at one or three or more points. 
     The fork-shaped torsional mode quartz crystal device  500  may be bonded to the ceramic package substrate  1612  using one or more bonding layers  1624  ( FIG.  16 B ), e.g., a conductive paste layer such as a silver epoxy layer or a conductive silicon-based bonding layer. As described above, the fork-shaped torsional mode quartz crystal device  500  has formed thereon one or more electrode patterns, e.g., first electrode patterns  524 ,  524 ′ ( FIGS.  5 A- 5 E ) and second electrode patterns  528 ,  528 ′ ( FIGS.  5 A- 5 E ). The bonding of the fork-shaped torsional mode quartz crystal device  500  on the ceramic package substrate  1612 , e.g., on the mounting pedestals  1620  may be performed by bonding the electrode pattern formed thereon, mechanically and/or electrically, to a metallization structure formed on the ceramic package substrate  1612 , e.g., on the mounting pedestals  1620 . For example, the metallization structure of the ceramic package substrate  1612  and the electrode pattern on the quartz crystal device  500  may comprise gold, such that a bonding layer  1624  formed therebetween contacts gold on both sides. Referring to  FIG.  16 B , the fork-shaped torsional mode quartz crystal device  500  may be hermetically sealed by bonding the lid  1608  on rims of the ceramic package substrate  1612  using a substrate bonding layer  1628 . 
       FIG.  17 A  illustrates a plan-view of a packaged quartz crystal oscillator device  1700  comprising a fork-shaped quartz crystal device  500  packaged in a ceramic package, according to some embodiments.  FIG.  17 B  illustrates a cross-sectional view of the packaged quartz crystal oscillator device  1700  illustrated in  FIG.  17 A , which includes the fork-shaped quartz crystal device  500  enclosed, e.g., hermetically sealed, in the ceramic package illustrated in  FIG.  17 A , in a similar manner as described above with respect to  FIGS.  16 A and  16 B . The ceramic package includes a ceramic package substrate  1712  and a lid  1608  for housing the quartz crystal device  500  according to various embodiments described above. The fork-shaped quartz crystal device  500  may be packaged using a two-point bonding method similar to the packaged quartz crystal device described above with respect to  FIGS.  16 A- 16 B , e.g., using a bonding layer  1624  formed of conductive paste, in a similar manner as described above with respect to  FIGS.  16 A and  16 B . In addition, in the illustrated embodiment, an integrated circuit (IC) die  1704  comprising control circuitry for controlling, e.g., providing oscillation signals to, the quartz crystal device  500 , is additionally bonded to the ceramic substrate  1712  using a bonding layer  1624 . The IC  1704  may be electrically connected to the metallization structures formed in the ceramic package substrate  1712  through one or more wires  1716  bonded therebetween. The metallization structures may in turn be electrically connected the fork-shaped quartz crystal device  500  through one or more metallization structures formed in the ceramic package substrate  1712 . The metallization structures formed in the ceramic package substrate  1712  may in turn provide thermal and electrical communication between the quartz crystal device  500  and the IC  1704  and the outside world, e.g., a PCB (not shown). 
     In some embodiments, the IC die  1704  is disposed at a different vertical level from the fork-shaped quartz crystal device  500 . For example, in the illustrated embodiment, the IC die  1704  is disposed below the fork-shaped quartz crystal device  500  and laterally overlaps the IC die  1704  for compact packaging. However, embodiments are not so limited. For example, in other implementations, the fork-shaped quartz crystal device  500  may be formed below the IC die  1704 . In yet other implementations, the fork-shaped quartz crystal device  500  and the IC die  1704  may be laterally adjacently disposed to each other at similar vertical levels. Advantageously, the fork-shaped quartz crystal device  500  and the integrated circuit die  1704  can be processed simultaneously, including co-sintering the bonding layers for bonding the fork-shaped quartz crystal device  500  ad the IC die  1704 . 
       FIGS.  18 A- 18 C  schematically illustrate various views of a fork-shaped torsional mode quartz crystal device  1800 , according to embodiments. The quartz crystal device  1800  has a front side  518  having a first main surface, e.g., a front surface, and a rear side  520  ( FIGS.  18 B,  18 C ) opposite the first side having a second main surface, e.g., a rear surface.  FIG.  18 A  illustrates a first side (e.g., a top side) and a second side (bottom side) of the fork-shaped quartz crystal device  1800 , respectively.  FIGS.  18 B,  18 C  illustrate cross-sectional views of the fork-shaped quartz crystal device  1800  taken at cross-sections H-H and I-I, respectively, as illustrated in  FIG.  18 A . The fork-shaped quartz crystal device  1800  is similar to corresponding features of the fork-shaped quartz crystal device  500  described above with respect to  FIGS.  5 A- 5 E , except for the presence of one or more vibration isolation arms  1804  and  1804 , and a detailed description of the similarities may not be repeated herein for brevity. Referring to  FIGS.  18 A- 18 C , unlike the fork-shaped quartz crystal device  500  described above with respect to  FIGS.  5 A- 5 E  in which the first and second base portions  512 A,  512 B of the base  512  are coextensive in the width (y) direction, in the illustrated embodiment, the second base portion  512 B below the notches  532  further extend in both horizontal (x) directions to form a pair of vibration isolation arms  1804 ,  1804 . When present, the vibration isolation arms  1804 ,  1808  can reduce the amount of vibration leakage from the first and second tines  514 ,  516  to the base  512 , such that the quartz crystal device  1800  vibrates with increased efficiency and/or with higher quality factor. Each of the vibration isolation arms  1804 ,  1808  comprises a horizontal portion extending in a first direction, e.g., the y-direction, from the second base portion  512 B and a vertical portion that forms a right angle with the horizontal portion and extends in a second direction different from the first direction, e.g., the x-direction. In the illustrated embodiment, the distal end of the vertical portion is connected to a tip region that may be wider than the vertical portion. Other suitable shapes and configurations of the vibration isolation arms  1804 ,  1808  are possible. For example, the vibration isolation arms  1804 ,  1808  may have horizontal portions but not the vertical portions. In addition, the angle between the horizontal and vertical portions can be any suitable angle. In addition, the tip region can have shapes other than the rectangular shape in plan-view as illustrated in  FIG.  18 A . For example, the tip region can have any polygonal and curved shapes. In addition, referring to  FIG.  18 C , while in the illustrated embodiment, the vibration isolation arms  1804 ,  1808  generally have similar thickness as the first and tines  514 ,  516 , embodiments are not so limited, and the vibration isolation arms  1804 ,  1808  can be thicker or thinner than the first and second tines. In addition, while in the illustrated embodiment, the vibration isolation arms  1804 ,  1808  generally have a rectangular similar thickness as the first and tines  514 ,  516 , embodiments are not so limited, and the vibration isolation arms  1804 ,  1808  can be thicker or thinner than the first and second tines. In addition, the tip region can have shapes other than the rectangular shape in cross-sectional view as illustrated in  FIG.  18 C . For example, the tip region can have any polygonal and curved shapes. 
       FIG.  19 A  illustrates a plan-view of a fork-shaped quartz crystal oscillator device  1900  packaged in a ceramic package substrate, according to some embodiments.  FIG.  19 B  illustrates a cross-sectional view of the packaged quartz crystal oscillator device  1900  illustrated in  FIG.  19 A , which includes the fork-shaped quartz crystal device  1900  enclosed, e.g., hermetically sealed, in the ceramic package illustrated in  FIG.  19 A , in a similar manner as described above with respect to  FIGS.  16 A and  16 B . The packaged quartz crystal oscillator device  1900  is similar to the packaged quartz crystal oscillator device described above with respect to  FIGS.  17 A- 17 B , and the similarities may not be repeated herein for brevity. Unlike the device described above with respect to  FIGS.  17 A- 17 B , the fork-shaped quartz crystal device packaged in the ceramic package is the fork-shaped torsional mode quartz crystal device  1800  described above with respect to  FIGS.  18 A- 18 C  having one or more vibration isolation arms. The fork-shaped quartz crystal device  1800  the integrated circuit (IC) die  1704  may be bonded to the ceramic substrate  1712  and interconnected to each other and connected to the outside world in a similar manner as described above with respect to  FIGS.  17 A- 17 B . 
     Aspects of this disclosure can be implemented in various electronic devices. In particular, applications of the electronic devices include, but are not limited to, high temperature, high reliability and/or high shock and vibration environments, such as oil/gas exploration, geophysical services, avionics, aerospace, military, process control and other harsh industrial applications. However, applications are not so limited and examples of the electronic devices can include, but are not limited to, consumer electronic products, parts of the consumer electronic products, electronic test equipment, cellular communications infrastructure such as a base station, etc. Examples of the electronic devices can include, but are not limited to, a mobile phone such as a smart phone, a wearable computing device such as a smart watch or an ear piece, a telephone, a television, a computer monitor, a computer, a modem, a hand-held computer, a laptop computer, a tablet computer, a personal digital assistant (PDA), a microwave, a refrigerator, a vehicular electronics system such as an automotive electronics system, a stereo system, a DVD player, a CD player, a digital music player such as an MP3 player, a radio, a camcorder, a camera such as a digital camera, a portable memory chip, a washer, a dryer, a washer/dryer, peripheral device, a clock, etc. Further, the electronic devices can include unfinished products. Aspects of this disclosure can be particularly implemented in various wireless telecommunication technologies in which high power, high frequency bands, improved linearity and/or improved efficiency are desired, including military and space applications such as radars, community antenna television (CATV), radar jammers and wireless telecommunication base-stations, to name a few. 
     Unless the context clearly requires otherwise, throughout the description and the claims, the words “comprise,” “comprising,” “include,” “including” and the like are to be construed in an inclusive sense, as opposed to an exclusive or exhaustive sense; that is to say, in the sense of “including, but not limited to.” The word “coupled”, as generally used herein, refers to two or more elements that may be either directly connected, or connected by way of one or more intermediate elements. Likewise, the word “connected”, as generally used herein, refers to two or more elements that may be either directly connected, or connected by way of one or more intermediate elements. Additionally, the words “herein,” “above,” “below,” and words of similar import, when used in this application, shall refer to this application as a whole and not to any particular portions of this application. Where the context permits, words in the above Detailed Description using the singular or plural number may also include the plural or singular number, respectively. The word “or” in reference to a list of two or more items, that word covers all of the following interpretations of the word: any of the items in the list, all of the items in the list, and any combination of the items in the list. 
     Moreover, conditional language used herein, such as, among others, “can,” “could,” “might,” “may,” “e.g.,” “for example,” “such as” and the like, unless specifically stated otherwise, or otherwise understood within the context as used, is generally intended to convey that certain embodiments include, while other embodiments do not include, certain features, elements and/or states. Thus, such conditional language is not generally intended to imply that features, elements and/or states are in any way required for one or more embodiments or whether these features, elements and/or states are included or are to be performed in any particular embodiment. 
     While certain embodiments have been described, these embodiments have been presented by way of example only, and are not intended to limit the scope of the disclosure. Indeed, the novel apparatus, methods, and systems described herein may be embodied in a variety of other forms; furthermore, various omissions, substitutions and changes in the form of the methods and systems described herein may be made without departing from the spirit of the disclosure. For example, while blocks are presented in a given arrangement, alternative embodiments may perform similar functionalities with different components and/or circuit topologies, and some blocks may be deleted, moved, added, subdivided, combined, and/or modified. Each of these blocks may be implemented in a variety of different ways. Any suitable combination of the elements and acts of the various embodiments described above can be combined to provide further embodiments. The various features and processes described above may be implemented independently of one another, or may be combined in various ways. All possible combinations and subcombinations of features of this disclosure are intended to fall within the scope of this disclosure.