Chapter 13: Propagation in a Crystal Lattice

The model begins by considering an electron confined to a simple one-dimensional line of atoms, extending the principles observed in the two-state system. Analyzing the time-dependent changes in the probability amplitude for the electron to reside at a specific atomic site leads to the determination of the system's states of definite energy. This mathematical framework demonstrates that the allowed electron energies are restricted to a specific range, forming a continuous energy band, where the energy is dependent on the wave number through a cosine relationship. The superposition of these definite energy states forms a wave packet or "clump," which moves through the lattice with a specific group velocity. This motion is characterized by the concept of effective mass (m eff​ ), a parameter that describes the electron’s dynamic response within the crystal environment, often differing significantly from its free-space mass. The principles are then scaled up to describe an electron moving in a three-dimensional lattice, where energy is dependent on three directional wave numbers and, in less symmetrical crystals, the effective mass becomes a tensor quantity. Furthermore, the analysis covers other charge carriers, such as a hole (the absence of an electron in a normally filled state, behaving as a positive particle) and a neutral exciton (representing the transfer of excitation energy). Finally, the chapter addresses the crucial physical scenario where the perfect lattice is disrupted by an impurity atom, leading to the scattering of the incoming electron wave into both reflected and transmitted components. If the impurity is sufficient to create a strong binding effect, the electron can be captured in a trapped state (a bound state) with energy below the conduction band, where the probability of finding the electron falls off exponentially. Critically, there is a theoretical connection showing that these bound states correspond mathematically to energies where the scattering amplitude becomes infinite.