Phonon Focusing and Electron Transport in Potassium Single Crystals. Review 2

Abstract—The effect of anisotropy of elastic energy on electron–phonon relaxation in potassium crystals is studied. The spectrum and polarization vectors of phonons are calculated, and the contributions of all vibrational modes to the drag thermopower and electrical resistivity of potassium crystals are analyzed. It is shown that the contribution of the t₂ slow quasi-transverse mode to these effects, which was not previously taken into account, at temperatures much lower than the Debye temperature $T \ll {{\theta }_{{\text{D}}}}$, exceeds the contribution of longitudinal phonons by an order of magnitude. However, at high temperatures ($T \ll {{\theta }_{{\text{D}}}}$), the contribution of longitudinal phonons to the electrical resistivity turned out to be 4 times greater than the total contribution of electron relaxation in the fast and slow quasi-transverse modes. By comparing the calculated total thermopower of potassium crystals with different dislocation densities with experimental data, the constants of the coupling of electrons with the shear components of vibrational modes is determined. For samples with different dislocation densities, the contribution of shear waves to the drag thermopower is 4–6 times greater than the contribution of longitudinal phonons. It is established that the maximum values of drag thermopower in perfect potassium crystals at low temperatures $T \ll {{\theta }_{{\text{D}}}}$ do not depend on the electron–phonon relaxation constants, but are completely determined by the second-order elastic moduli, crystal density, and electron density. In this case, the contributions of various modes to the drag thermopower turn out to be proportional to the ratio of the heat capacities of the individual modes. It is shown that shear waves make a significant contribution to the electrical resistivity of potassium crystals at low temperatures $T \ll {{\theta }_{{\text{D}}}}$. It is 4 times greater than the contribution to the electrical resistivity from longitudinal phonons and should be taken into account when analyzing the electrical resistivity of potassium. The distribution of phonons most efficient for electrical resistivity is determined, and the effect of the inelasticity of electron–phonon scattering on the electrical resistivity and drag thermopower of potassium crystals at low temperatures is analyzed. It is shown that its maximum is at $\hbar \omega _{q}^{\lambda } \approx 5{{k}_{{\text{B}}}}T$ and the energy of the most efficient phonons is distributed in the interval ${{k}_{{\text{B}}}}T \leqslant \hbar \omega _{q}^{\lambda } \leqslant 12{{k}_{{\text{B}}}}T$. The effect of anisotropy of elastic energy on the electron–phonon mutual drag and the electrical resistivity of potassium crystals at low temperatures is considered. In the hydrodynamic approximation, the momentum exchange between the electron and three phonon fluxes corresponding to the three branches of the vibrational spectrum is analyzed. The actual mechanisms of phonon momentum relaxation are taken into account: scattering at sample boundaries, dislocations, and in phonon–phonon umklapp processes. It is shown that, in the limiting case of strong mutual drag, the drift velocities of phonons of all polarizations and electrons are close and determined by the total rate of phonon relaxation in resistive scattering processes.