hoppingmenetelmää

The hopping method is a technique used in computational physics and chemistry to study the dynamics of particles, particularly electrons, in disordered systems. It is an approximation that simplifies the complex quantum mechanical problem of particle transport. Instead of continuously solving the Schrödinger equation for the particle's wave function, the hopping method assumes that the particle transitions between discrete localized states or sites. These transitions, or "hops," occur with a certain probability that depends on the distance between the sites and the energy difference between them.

The core idea is that in a disordered system, such as an amorphous semiconductor or a porous material, the wave function of a particle does not extend indefinitely. Instead, it becomes localized, meaning the particle is confined to a specific region. When a particle is in one localized state, it can tunnel or "hop" to another nearby localized state. The rate of these hops is often described by the Miller-Abrahams formalism, which takes into account the spatial separation and energy difference between the initial and final states.

The hopping method is particularly useful for calculating transport properties like electrical conductivity at low temperatures, where thermal activation plays a significant role. By simulating a large number of hops, researchers can statistically determine macroscopic transport coefficients. This approach reduces the computational burden compared to methods that require a full quantum mechanical treatment of the entire system. However, it is an approximation and its accuracy depends on the validity of the localized state assumption and the chosen hopping rate formula.