Inelastic extended-electron–localized-vibrational-state scattering rate

摘要: We calculate the inelastic scattering rate of extended-electronic states off localized-vibrational states. perform calculation for electrons on a fractal lattice interacting with either fractons, or phonons. show that respective rates have quite different temperature dependences, allowing one to distinguish between two forms excitations. dependence electronic fractons both degenerate and nondegenerate electron statistics.The statistics is proportional ${T}^{(2d\mathrm{\ensuremath{-}}\mathrm{theta})/(2+\mathrm{theta})}$, where theta exponent involved in range force constant, d Euclidean embedding dimension. This d=2 varies as (approximately) ${T}^{8/7}$ percolation network scalar forces, ${T}^{5/19}$ central bending opposed ${T}^{2}$ phonon excitations (both extended localized).For statistics, fracton regime ${\mathrm{Tk}}^{d\mathrm{\ensuremath{-}}2\mathrm{\ensuremath{-}}\mathrm{theta}}$, k wave vector. For k\ensuremath{\propto}${T}^{1/2}$, this leads ${T}^{(d/2)\mathrm{\ensuremath{-}}\mathrm{theta}}$\ensuremath{\sim}${T}^{1/5}$ percolating ${T}^{\mathrm{\ensuremath{-}}7/4}$ linear obtained case excitations, ${T}^{d}$${\ensuremath{\omega}}_{c}^{(1\mathrm{\ensuremath{-}}d)\mathrm{theta}/(2+\mathrm{theta})}$. it ${\mathrm{Tk}}^{d\mathrm{\ensuremath{-}}2}$${\ensuremath{\xi}}^{\mathrm{theta}}$. Here, ${\ensuremath{\omega}}_{c}$ crossover frequency separating from regime, \ensuremath{\xi} corresponding characteristic length. The results are conveniently expressed form scaling relations.