Scattering of phonons by quantum-dislocation segments in an elastic continuum

Abstract

A canonical quantization procedure is applied to elastic waves interacting with pinned dislocation segments ("strings") of length L via the Peach-Koehler force. The interaction Hamiltonian, derived from an action principle that classically generates the Peach-Koehler force, is a power series of creation and annihilation operators. The leading term is quadratic, and keeping only this term the observable quantities of scattering processes are computed to all orders in perturbation theory. The resulting theory is characterized by the magnitude of kL, with k the wavenumber of an incident phonon. The theory is solved for arbitrary kL, and different limits are explored. A significant result at this level is the scattering cross section for phonons by dislocation segments. As a function of frequency, this cross section has a much richer structure than the linear-in-frequency behavior that is inferred from scattering by an infinite, static, dislocation. The rate of spontaneous phonon emission by an excited dislocation is computed as well. When many dislocations are present, an effective mass operator is computed in the Weak and Independent Scattering Approximation. The contribution of the cubic terms is computed to leading order in perturbation theory. They allow for a comparison of the scattering of a phonon by a string and the three-phonon scattering, as well as studying the dependence of scattering amplitudes on the temperature of the solid. It is concluded that the effect of dislocations will dominate for relatively modest dislocation densities. Finally, the full power series of the interaction Hamiltonian is considered. The effects of quantum corrections, i.e., contributions proportional to Planck's constant, are estimated, and found to be controlled by another wavenumber-dependent parameter kdq, where dq is a length proportional to √ . The possibility of using the results of this paper in the study of the phononic thermal properties of two-and three-dimensional materials is noted and discussed.

The Scattering of Phonons by Infinitely Long Quantum Dislocations Segments and the Generation of Thermal Transport Anisotropy in a Solid Threaded by Many Parallel Dislocations

“…Note that this is a quadratic interaction. Furthermore, following previous work [17], we have defined…”

“…In previous work [ 17 ], considering only the completely isotropic case, we have assumed segments of edge dislocations only, in which case they may be written in terms of the Burgers vector and as where , are long- and short-distance cutoff lengths, and …”

“…Note that this is a quadratic interaction. Furthermore, following previous work [ 17 ], we have defined and we will denote its complex conjugate by . Here, we have denoted and .…”

“…Finally, it is important to note that this estimate relied on using the exact dissipation rate computed from dislons propagating on an infinite string. However, experiments testing this result may be sensitive to the length of the dislocation line L , which undoubtedly yields a different notion of dislon self-energy because momentum along the axis is no longer conserved [ 17 ] as the exact equation of motion becomes infinitely coupled between the different modes of the string, and the identification of B with the imaginary part of the inverse propagator needs to be revisited. In this case, one possibility for a direct identification would be to simply identify B with the decay rate of a given dislon mode, probably corresponding to the first normal mode of the string.…”

“…An additional feature of the experiment of Sun et al [ 15 ] is that most heat-carrying phonons appear to have wavelengths shorter than 5 nm, which is much smaller than the apparent typical length of a dislocation in their setup, m. Therefore, understanding the interaction between phonons and dislocations where the wavelength of the former is much smaller than the spatial extent of the latter should prove to be essential in describing and explaining thermal transport anisotropies mediated by dislocations. In a previous publication [ 17 ], we have developed a quantum theory of phonons in interaction with dislocation segments of finite length. Here, we extend that formalism to infinitely long dislocations, and explore the consequences for the thermal transport properties of a material threaded with such many parallel dislocations.…”