### Ultra-High Q

July 23, 2018

As I wrote in an earlier article (Trilobite Sex, March 13, 2017), biochemist and prolific science fiction author, Isaac Asimov (1920-1992), had a simple test to determine whether or not a person was a scientist. It was how they interpreted the word, unionized. Non-scientists will invariably think of a trade union, while scientists will think of a neutral atom; that is, an atom that's not ionized.

I've developed a similar test for electrical engineers (and some physicists) based on the definition of Q. About the only Q that persons not knowledgeable in electrical engineering recollect is Q from the James Bond films. "Q" is Bond's quartermaster, the person responsible for the supply of his high-tech spy gear, such as the early version of vehicle tracking that appeared in the 1964 film, Goldfinger.

Electrical engineers think of a different Q, the quality factor that relates the bandwidth to the frequency of a tuned circuit, or the damping of an excited resonator. Combination of an inductor, capacitor, and resistor in either a series or parallel LCR circuit provides an electrical resonance that can be used to produce electronic oscillators and filters. A simple AM radio circuit consists of an oscillator based on a parallel LCR combination with a variable capacitor for tuning, and several other parallel LCR stages for intermediate frequency filtering.

Series (left) and parallel (right) inductance-capacitance-resistance (LCR) tuned circuits. In each case, the resistance produces energy loss that dampens the resonance amplitude. In the series circuit, the signal source is an ideal voltage source having zero resistance. In the patallel circuit, the signal source is an ideal current source that has infinite resistance. (Created using Inkscape.)

The resistance in these circuits determines the extent to which the stored energy is dissipated and the amplitude of the resonance is damped. The quality factor, Q, is a measure of how well the circuit keeps its stored energy. For a series circuit, the Q is given by
where the inductance L is given in henrys, the capacitance C is given in farads, and the resistance R is given in ohms. Not surprisingly, the Q of a parallel LCR circuit is the inverse of this; viz.,

A high Q-factor indicates a sharp resonance. As shown in the following figure, the bandwidth at the half-power points on the amplitude vs frequency curve is related to Q and the resonance frequency fc. The actual half-power value is 3.0103 dB, but few instruments can measure that accurately, and few engineers would even care.

Q related to the 3-dB bandwidth.

The bandwidth is arbitrarily taken at the half-power (3 dB) point, which is a convenient measure.

Q is reciprocally related to bandwith, with a high Q indicating a narrow bandwidth.

(Created using Inkscape.)

While it's difficult to get a Q much higher than 100 in an LCR circuit, quartz crystals routinely achieve Q-values of 10,000 or more. Since energy loss in quartz crystal wafers is strongly influenced by their mounting, the attached electrodes, and acoustic coupling to air, crystals with high-Q are convex in shape to keep their energy away from the edge mounts, have light attachment wires, and are enclosed in a vacuum. In this way, Q-values in excess of 100,000 can be achieved. Quartz crystals are used to make crystal filters with a sharply-defined passband for radio receivers.

All this is predicated on the fact that the quartz crystal itself has few lattice defects. Originally, quartz crystals for resonators were mined as minerals. Herkimer, New York is one source of large, clear crystals of quartz, where they are found in dolomite rock. Quartz crystals are impressive in appearance, since they are large and double-terminated, exhibiting facets all around.

Crystal facets of quartz.

Quartz grows as left- and right-handed crystals, as shown.

(Left image and right image by "strickja," via Via Wikimedia Commons, modified.)

Laboratory-grown quartz crystals are produced by the same hydrothermal crystallization that forms them in nature. Silica, dissolved in hot water, is transported to a seed crystal to allow the crystal to grow to a larger size. The mining of natural quartz for electronic resonators essentially ended in the mid-20th century when manufactured quartz became available. The industrial synthesis of quartz was developed at Bell Labs by A. C. Walker and Ernie Buehler.[1-2]

Quartz crystals have become commodity items, but modern integrated circuit technology has given us an alternative in a new class of resonators, microelectromechanical system (MEMS) oscillators. These are essentially miniaturized mechanical structures, some of which exhibit extremely high-Q. The Q-factor of LCR circuits is easily calculated from the component values, and measurement of the resonant frequency and bandwidth of quartz and MEMS resonators can give Q by a different route, but there's a more fundamental technique of measuring the damping itself. If we set a device into resonance by "pinging" it with a quick voltage pulse, the resonator will ring with a decaying sine wave according to the well-known function
A(t) = Ao e-λt cos(t)
where A(t) is the amplitude as a function of time, Ao is the initial amplitude, and lambda (λ), called the log decrement, sets the rate of decay. The degree of decay, or damping, is related to Q as 2π(Energy Stored/Energy Lost per cycle).

Amplitude of a damped harmonic oscillator. In this case, the log decrement is 0.01, giving the equation shown on the plot. (Graph created using Gnumeric.)

A Q-factor of 100,000 is impressive, representing a 10 Hz bandwidth at 1 MHz, but how high a Q can a resonator have? That's the problem tackled by a team of scientists from the École Polytechnique Fédérale de Lausanne (Lausanne, Switzerland) and IBM Research–Zurich (Rüschlikon, Switzerland).[3-5] By engineering stress in the resonant ribbon of a micromechanical oscillator, they achieved a Q value of 800,000,000.[5]

It's possible to enhance the performance of devices by the addition of stress or strain into their materials. For example, mechanical stress in silicon produces transistors operable at higher frequencies.[3] Stress in semiconductor material used in solar cells results in a higher electron mobility that leads to greater efficiency.[4] Engineering stress into a resonator would increase the Q-factor if it diminishes damping and reduces the energy lost per resonant cycle.[5]

How stress and its resultant strain affects resonator quality Q.

(Fig. 1b of ref. 4. Click for larger image.)[4]

The research team was able to enhance the Q-value of a silicon nitride (Si3N4) nanobeam by etching it with a spatially non-uniform photonic crystal pattern.[4] This allowed a co-localization of strain and flexural motion in the nanobeam.[3] The structure also produced dissipation dilution in which the stiffness of a material is effectively increased without adding loss.[4] This study builds on previous work with membrane resonators in which a periodic pattern of holes was punched into a thin membrane, leaving just a small central island intact.[6] The hole pattern isolated the membrane resonator from its mounting so it behaved as if it were in a vacuum, and this increased the Q-value of the membrane.[5-6]

These high Q-values were measured by the ringdown technique, which showed Q-factors as high as 800 million, and the product of Q and frequency exceeding 1015.[3-4] There are quite a few applications for such high-Q resonators, including measurement of atto-newton forces.[4]

Width and stress profile of a resonator ribbon etched with 60 unit cells. (Fig. 2b of ref. 4. Click for larger image.)[4]

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