Nanoscale Engineering for Reducing Phase Noise in Electronic Devices

Volume dependence in handel's model of quartz crystal resonator noise

“…In our case,  3.95 • 10 Ns/m 2 [24], which turns into an estimation of the maximum quality factor of about 2.7910 6 . This value corresponds well with the one given in [3] (2.710 6 ).…”

“…Hence, no entirely satisfactory quantitative predictive model can be given in order to optimize the industrial processes or simply understand the spread of characteristics in a single fabrication batch. Two models have indeed attempted to give quantitative predictions of the level of 1/f noise in quartz resonators, the 1/f quantum noise theory of P. Handel [6] and the degenerated thermal boson theory of M. Planat [7], but the prediction of M. Planat was orders of magnitude above the best measurements, maybe because of the use of the traditional phononic density of states despite the fact that all the harmonics of the fundamental modes had already been taken into account in the calculation of the partition function per mode [8].…”

“…It has been applied to several physically different systems by P. Handel (see e.g. [6], [13][14][15][16][17][18][19]), among which quartz crystal resonators. In the case of Bulk Acoustic Wave (BAW) quartz resonators at their turn-over temperature (~350 K), the fluctuations are mostly caused by the three-phonons process corresponding to the interaction of a phonon from the main resonator mode with a thermal phonon.…”

1∕f Performance Limits of Scanning Tunneling Microscopes

“…Therefore, the fractional fluctuations in phonon emission rate are equal to the fractional fluctuations in relaxation rate: 5 sx/x = 5 sr/r-According to the general quantum 1/f formula, F"^ 5 r (/) = laAJf, where a = e^/hc is the fine structure constant and A = 2(AJ/ec)^/37i is the quantum 1/f effect in any physical process rate F [2], [3]. The current discontinuity AJ in the Ampere-Maxwell equation equals the discontinuity AdP/dt in the rate of the crystal dipole moment change dP/dt.…”

“…The largest contribution will not come from the main resonator 1/n factor, but from the 1/n factor corresponding to the high-frequency thermal modes that have n close to 1. Furthermore, n is the number of thermal phonons at the frequency <C0eff > of the quartz, which we set equal to unity [2], [3]. Each coherence volume e^ of the quartz crystal has independent dissipation fluctuation, with e being the phonon coherence length.…”