Elastic, Quasielastic, and Superelastic Electron Scattering from Thermal Lattice Distortions in Perfect Crystals

TL;DR

The paper reframes electron transport in perfect crystals by treating the lattice center-of-mass as a dynamical degree of freedom and enforcing exact total-momentum conservation via pseudomomentum exchange between foreground electrons and the lattice background. It shows that elastic, quasielastic, and mixed scattering channels arise at first order in a density-density electron–lattice interaction, with the Debye-Waller factor quantifying elastic recoil to the lattice background. This framework explains why momentum relaxation can be efficient in clean crystals without significant energy transfer, aligning with observations such as weak localization, quantum oscillations, and Planckian diffusion as an emergent elastic diffusion phenomenon. The results offer a coherent interpretation of experiments and provide a route to diffusive transport in a time-dependent elastic background, challenging the conventional view that momentum relaxation in metals requires inelastic phonon processes.

Abstract

In conventional treatments of electron transport, momentum relaxation within a perfect, defect free crystal is commonly assumed to require phonon creation or annihilation. Here we treat the crystal as finite and isolated, retaining the lattice center of mass (recoil) degree of freedom and enforcing conservation of total mechanical momentum alongside discrete crystal pseudomomentum. Starting from the density density form of the electron lattice interaction, we find that an electron in the interior of a perfect crystal admits strong, and in some regimes dominant, elastic momentum relaxing scattering channels, in which momentum is conserved by recoil of the lattice background without phonon excitation. In addition, we identify mixed quasielastic and superelastic channels in which phonon occupations change but do not account fully for the electron's momentum transfer. These results provide a microscopic basis for momentum relaxation that does not rely on local energy dissipation. They naturally reconcile a wide range of experimental observations, including weak localization, quantum oscillations, ultrasonic attenuation, and the separation of momentum and energy relaxation times, with predominantly elastic scattering in clean crystals. The framework clarifies how diffusive transport, including Planckian scale diffusion, can emerge from elastic dynamics in a time dependent lattice background.

Figures (2)

• Figure 1: Sliding Blocks. Floating in space, two blocks attract one another and slide frictionlessly along their common long axis, $x$. At the top, both blocks carry conserved mechanical momenta, and clearly $P_{\rm tot}=p_A+p_B$. At the bottom, the same relation holds, but $p_A$ and $p_B$ are now pseudomomenta; their sum nevertheless equals the conserved mechanical momentum. The two blocks serve as surrogates for a foreground electron (A) in a perfect background crystal (B), illustrating that both the foreground and the background acquire pseudomomentum in each other’s presence.
• Figure 2: The total momentum (black) remains constant as the foreground (electron, blue) and background (lattice, red) momenta jump abruptly at the collision time $t^*$. There is no delay in the total lattice momentum change, even though only a single atom initially carries the momentum. While this is elementary, we emphasize it because there is a common tendency to associate momentum flow directly with energy flow in the medium. Indeed there is momentum flow among particles, but the total is fixed immediately. The total appears in the wavefunctions.