In this blog post, we will examine how the wave nature of electrons is localized in solids through various types and physical principles.

Wave nature of electrons in solids and localization phenomenon

Solids are three-dimensional structures formed by atoms arranged in relatively fixed positions, and are divided into regular crystals and irregular amorphous solids depending on the arrangement of atoms. Many physical properties of solids can be explained by the wave nature of electrons within solids. Electron waves are generally expressed as complex numbers called displacements, which are represented by the product of amplitude and phase. The probability of an electron existing at a given position is given by the square of the displacement, and the phase is a function of time and space that represents the wave nature of electrons.

Localization Phenomena and Types of Waves

During propagation, electron waves may be trapped in a certain area under specific conditions, preventing them from moving freely. This phenomenon is called localization, and there are three main types:

1. Anderson localization
2. Weak localization
3. Dynamical localization

Among these, Anderson localization and weak localization mainly occur in amorphous solids, while dynamical localization can also occur in chaotic systems.

Anderson localization: Complete spatial confinement of electrons

Anderson localization refers to a phenomenon in which electron waves cannot propagate further and become completely trapped in space. In amorphous solids, atoms are arranged irregularly, so electron waves originating from a single location encounter numerous atoms and irregular collision paths as they move. At this point, the waves on each path have different signs (+/-) and cancel each other out, making it impossible for the wave to actually move forward. As a result, the probability of an electron starting at a certain position and reaching another position becomes almost zero, and the electron becomes spatially localized. The size of the space in which the waves are trapped is called the localization length, and the shorter the localization length, the greater the localization strength. For Anderson localization to occur, the phase of the electron waves must be precisely defined as a function of time and space, and such waves are called coherent waves. The degree of coherence is expressed as the coherence length, and generally, localization occurs when the coherence length is longer than the localization length. As the temperature rises, the interaction between electrons and the thermal fluctuations of atoms increase, causing the coherence to break down, and eventually the coherence length converges to zero, and localization no longer occurs. In addition, Anderson localization exhibits different characteristics depending on the spatial dimension. In a one-dimensional structure, there are few paths for electron waves to bypass obstacles, so localization occurs strongly and the structure becomes insulating. However, in three dimensions, electrons can bypass obstacles, so depending on the conditions, localization may not occur completely, and the structure may exhibit conductive properties.

Weak localization: Propagation interference due to wave path interference

Weak localization refers to the phenomenon in which electron waves are confined to a closed-loop path and become weakly trapped, and is mainly related to changes in electrical resistance. In amorphous solids, electron waves start from random points and propagate along various paths. Some of these may form closed-loop paths that return to their starting points, in which case the electrons can rotate in both clockwise and counterclockwise directions. Since the two paths have the same distance and structure, the phases of the waves are the same, and when these two waves interfere with each other, the displacement increases. The square of the displacement is the probability that the electron will be present at that position, which means that the probability of the electron returning to its starting point increases, and as a result, the movement of electrons is impeded and electrical resistance increases. However, in the presence of a magnetic field, the waves in the two directions have different phases, which weakens the interference effect and reduces or eliminates the weak localization phenomenon.

Dynamical localization: wave confinement in chaotic systems

Dynamical localization is a phenomenon of electron wave localization that occurs in chaotic systems. Chaos refers to a phenomenon in which slight differences in initial conditions lead to very large differences in results over time. In chaotic systems, particles generally spread out along complex paths and exhibit disordered mechanical motion. However, even under these conditions, electron waves do not spread but are confined within a certain area. This phenomenon occurs because the process of waves being reflected and refracted in a chaotic system is similar to the structural conditions of Anderson localization, in which waves pass through numerous atoms arranged irregularly within an amorphous solid.

Conclusion

Electron waves do not simply move like particles within solids, but their movement is restricted or suppressed in various ways depending on properties such as wave interference, phase, and coherence. Such localization phenomena greatly affect the mobility and conduction properties of electrons and serve as a core theory in various fields such as solid state physics, materials science, and nanoelectronics. Anderson localization, weak localization, and dynamical localization are important concepts that reflect the structural characteristics of solids, the interference of electromagnetic waves, and the dynamical complexity of systems, respectively, and are essential topics for understanding modern condensed matter physics.