Phonons in a half space

We analyze the interaction at the surface of a three-dimensional (3D) topological insulator among 2D electron states belonging to the Dirac cone close to the point of the Brillouin zone and the Rayleigh surface phonon mode. The model deals with an elastic continuum in the long-wavelength limit, in Random Phase Approximation (RPA). Screening of the electronic polarization is quite effective at small wave vectors. On the other hand, the absence of backscattering for the Dirac electrons at the Fermi surface is partly responsible for the reduced influence of the electron-phonon interaction in renormalizing the phonon dispersion at finite wave vectors. We infer that softening of the Rayleigh mode appears as unlikely, at least for the case of a clean and defect-free surface to which our approximate treatment applies. The dielectric response to virtual excitation of the Rayleigh phonon could drive the electron-electron interaction attractive at low frequencies, but the average weak coupling pairing interaction is found to be too small to induce a surface superconducting instability.

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“…9,10 The Rayleigh mode has a linear dispersion relation ω (0) q = c R q with velocity c R = 0.89 c t .…”

Section: The Modelmentioning

confidence: 99%

Electron-phonon interaction on the surface of a three-dimensional topological insulator

et al. 2013

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“…Where material parameters are necessary we shall assume a Cu film; however, the qualitative behavior we obtain is generic. The evaluation of F (ω) requires the vibrational eigenfunctions for a semi-infinite substrate with a free surface, which have been obtained in the classic paper by Ezawa [13]. The modes are labeled by a branch index m, taking the five values SH, +, −, 0, and R, by a two-dimensional wavevector K in the plane defined by the surface, and by a parameter c with the dimensions of velocity that is continuous for all branches except the Rayleigh branch m = R. With the normalization convention of Ref.…”

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confidence: 99%

“…The modes are labeled by a branch index m, taking the five values SH, +, −, 0, and R, by a two-dimensional wavevector K in the plane defined by the surface, and by a parameter c with the dimensions of velocity that is continuous for all branches except the Rayleigh branch m = R. With the normalization convention of Ref. [13] we have The range of the parameter c depends on the branch m, and is summarized in Table I. The frequency of mode mKc is cK.…”

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Hot electrons in low-dimensional phonon systems

Cleland

Geller

2005

Phys. Rev. B

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A simple bulk model of electron-phonon coupling in metals has been surprisingly successful in explaining experiments on metal films that actually involve surface-or other low-dimensional phonons. However, by an exact application of this standard model to a semi-infinite substrate with a free surface, making use of the actual vibrational modes of the substrate, we show that such agreement is fortuitous, and that the model actually predicts a low-temperature crossover from the familiar T 5 temperature dependence to a stronger T 6 log T scaling. Comparison with existing experiments suggests a widespread breakdown of the standard model of electron-phonon thermalization in metals.PACS numbers: 63.22.+m, 85.85.+j The coupling between electrons and phonons plays a crucial role in determining the thermal properties of nanostructures. The widely used "standard" model of low temperature electron-phonon thermal coupling and hot-electron effects in bulk metals [1, 2] assumes (i) a clean three-dimensional free-electron gas with a spherical Fermi surface, rapidly equilibrated to a temperature T el ; (ii) a continuum description of the acoustic phonons, which have a temperature T ph ; (iii) a negligible Kapitzalike thermal boundary resistance [3] between the metal and any surrounding dielectric, an assumption that is often well justified experimentally; and (iv), a deformationpotential electron-phonon coupling, expected to be the dominant interaction at long-wavelengths. In a bulk metal, the net rate P of thermal energy transfer between the electron and phonon subsystems is where V el is the volume of the metal, and

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“…(2.2.19) is satisfied, and the Hooke tensor satisfies the symmetry constraints in eq. (2.2.10) (see [118], and appendix B.3). Therefore, the eigenvectorsũ n (x) defined by solutions of,…”

Section: Phonons: Quantised Linear Elastodynamicsmentioning

confidence: 99%

Quantum Limits on Measurement and Control of a Mechanical Oscillator

Sudhir

2018

Springer Theses

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PAR Vivishek SUDHIR JOSEPH CONRAD, Heart of DarknessExperimental physics demands a certain degree of manual dexterity, the patience to tolerate the mundane, and an inexhaustible supply of optimism. Tobias hired me to work on an immensely sophisticated experiment despite any proof of my possessing these qualities; I am indebted to him for the belief he has placed in me. I would also like to acknowledge his dedication to fostering an environment where the pursuit of science is unencumbered by other worldly concerns. Stefan, Ewold and Samuel bore the brunt of a theorist trying to perform delicate experiments; the stern, yet encouraging, stewardship of this trio ensured that I enjoyed the culture shock. Dal Wilson was more than a colleague and a post-doc; his pragmatic approach to physics provided the ideal counter-point in our attempts to forge a deeper understanding of things ranging from vibration isolation to the subtleties of quantum mechanics. Without the patient efforts of Amir, Ryan and Hendrik in designing, fabricating, and testing of devices, none of the results reported here would have been possible. I hope I have been able to bequeath a better vision of the future of our experiment to Sergey. Conversations with Alexey, Daniel and Nathan have enabled me to vicariously learn how to measure microwave "photons". John Jost once taught me how to decorate a Christmas tree; since then he has imparted some wisdom on frequency metrology, and some on atomic physics. Together with Dal, Victor and Caroline have probably spent the most time with me outside the lab; it was fun to exercise an air of social normalcy with this bunch. Christophe has been an amazing sparring partner on every subject capable of being brought under scientific scrutiny.The personal space required for the pursuit of science comes through the sacrifice of many people. My family back home in India -parents, sister, grand-parents -have had to patiently wait for the brief interludes when I go home. No words can do justice to this and the very many other pleasures they have surrendered over the years for my sake. Before physics attracted me, I was charmed by someone else; I am lucky to have married her. Long time friends, mentorsAjith and Subeesh -have played pivotal roles in my being able to engage in physics; I hope to repay this enormous debt, some day.It was a privilege to have learnt from a few great teachers during my formative years. Their vision of an indelible unity in nature continues to motivate my study of physics. vii AbstractThe precision measurement of position has a long-standing tradition in physics. Cavendish's verification of the universal law of gravitation using a torsion pendulum, Perrin's confirmation of the atomic hypothesis via the precise measurement of the Brownian motion, and, the verification of the mechanical effect of electromagnetic radiation, all belong to this classical heritage. Quantum mechanics posits that the measurement of position results in an uncertain momentum; an idea developed to full maturity within the ...

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Phonons in a half space

Cited by 125 publications

References 7 publications

Electron-phonon interaction on the surface of a three-dimensional topological insulator

Electron-phonon interaction on the surface of a three-dimensional topological insulator

Hot electrons in low-dimensional phonon systems

Quantum Limits on Measurement and Control of a Mechanical Oscillator

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