Anja Zimmer Friedrich-Schiller-University Jena, Germany 3 rd ILIAS-GW Meeting, London October 27 th 2006

1. CRYO/Q Relaxation mechanisms in solids Anja Zimmer Friedrich-Schiller-University Jena, Germany 3rd ILIAS-GW Meeting, London October 27th 2006

2. Dissipation stress strain

3. Dissipation per cycle depending on frequency

4. Mechanical losses Single anelastic process (single relaxation time): For small loss angles: Maximum at relaxation strength frequency of acoustic wave relaxation time

5. Mechanical losses Especially for stress induced transitions between states of minimal energy: relaxation time relaxation constant activation energy Boltzmann constant

6. Dissipation due to stress induced hopping of alkali-ions in a-quartz Si O W. P. Mason in Physical Acoustics, edited by W. P. Mason (Academic Press Inc., New York, 1965), vol. 3B, p. 247.

7. Measurement + Fit 910-7 s 3.4 meV phonons 510-13 s 53 meV defects 110-14 s 163 meV 410-13 s 194 meV Q-Measurement on crystalline quartz  3“  12 mm Measurement Damping Q-1 Temperature [K] mode shape: measured frequency: 11565 Hz @ 300 K calculated frequency : 10793 Hz

8.  3“  12 mm Q-Measurement on crystalline quartz Damping Q-1 Temperature [K] measured frequency : 11565 Hz 17115 Hz 61720 Hz @ 300 K calculated frequency: 10793 Hz 16987 Hz 61121 Hz

9. Mechanical losses in solids • External losses • e.g. suspension losses, residual gas damping… • Internal losses • „ideal“ solid: • thermo-elastic damping • interaction of acoustic waves with thermal phonons of the solid • interaction of acoustic waves with electrons of the solid • „real“ solid: • additional damping caused by defect induced losses

10. Dissipation caused by interaction of acoustic wave with thermal phonons Damping Q-1 Temperature [K] Landau-Rumer damping Akhieser damping

11. Dissipation caused by interaction of acoustic wave with thermal phonons Landau-Rumer damping Akhieser damping • Perturbation of equilibrium of thermal phonon distribution. • Reestablishment affords increase in entropy and such leads to a partly absorbation and attenuation of the acoustic wave. • Interaction of acoustic waves with individual thermal phonons.

12. Crystalline silicon • less than 1014 doping atoms per cm3: mechanical losses are dominated by interactions of acoustical waves with thermal phonons of the crystal • higher doping concentrations: dissipation by interactions with additional electrons and holes respectively increases W. P. Mason in Physical Acoustics, edited by W. P. Mason (Academic Press Inc., New York, 1966), vol. 4A, pp. 299.

13. n-doped silicon • Origin of dissipation: Movement / transition of conduction electrons between minima of energy in k-space • Minima occur along the six <100> directions • Unstressed crystal: Minima possess same energies and numbers of occupation • Stress along crystal axis rises energy of parallel located minima and lowers that of perpendicular ones • Consequence: Reestablishing equilibrium by flow of electrons from minima of higher energy to lower ones • delay = intervalley relaxation time is origin of an effective energy transfer from the acoustic wave to the thermal bath

14. p-doped silicon • Energy surfaces of the valence band become reshaped by stresses induced by acoustic waves. • It is assumed that a flow of holes from regions of higher energy to that of lower energy of the same surface occurs and not between surfaces.

15. directionality (anisotropy!) of and How to determine the mechanical loss factor? background damping (suspension, residual gas damping…) sum of internal anelastic processes total damping