Under certain conditions, such as extremely low temperatures and high magnetic fields applied perpendicular to the electron system’s plane, two-dimensional electron systems exhibit the deep phenomena known as the Quantum Hall Effect (QHE). In these circumstances, the Hall resistance which is calculated by dividing the Hall voltage by the applied current displays distinct, consistent values throughout a broad range of charge carrier densities or magnetic field intensities.
The definition of the quantum hall effect, its fundamental principles (Landau levels and disorder), its topological origin, quantum hall effect applications, experimental realisations, and new discoveries are all covered in this blog.
Quantum hall effect definition
Klaus von Klitzing won the 1985 Nobel Prize in Physics for his initial observation of the QHE in 1980. He discovered that the Hall resistance was precisely quantized, which means that it multiplied a fundamental quantity by integers to get accurate, consistent values. The Integer Quantum Hall Effect (IQHE) is the name given to this first discovery. The Fractional Quantum Hall Effect (FQHE) was later identified in 1982 as a result of the discovery of plateau values for fractional numbers. The FQHE is still regarded as an open research subject despite being a more complicated phenomenon. The interactions between electrons are essential to its existence.
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Fundamental Principles: Landau Levels and Disorder
The idea of Landau levels serves as the foundation for comprehending the IQHE. When a strong magnetic field is applied to a two-dimensional electron system, the electrons’ traditional circular trajectories are quantized into distinct energy levels. Numerous electron states can coexist at the same energy due to the strong degeneracy of these Landau levels. The Zeeman effect, which occurs when an electron’s spin aligns with a high enough magnetic field, causes these energy levels to further separate. The existence of disorder or impurities in the material is a key factor in the observed plateaus in Hall resistance.
The distinct, crisp Landau levels become broadened into energy bands due to disorder. Electron states may be localized or extended within these bands. Extended states are free to travel, whereas localized states are restricted to specific areas and do not contribute to the current flow. The material effectively acts as an insulator in the longitudinal direction when the Fermi energy the highest energy level occupied by electrons falls within an area with only localized states (often referred to as a “mobility gap” between Landau levels), and the Hall conductivity stays constant, creating the plateaus. One important feature of the effect is that the disorder has no effect on the exact values of these plateaus.
Topological Origin
A topological genesis is strongly suggested by the IQHE’s exceptional robustness and insensitivity to microscopic disorder and macroscopic system deformations. Topological quantum numbers are the integer values seen in the Hall effect. Berry’s phase, a notion associated with the geometric phase that a quantum system acquires, is closely linked to these numbers, which are theoretically referred to as Chern numbers. Because of this relationship, the material’s observable electrical properties are determined by its fundamental qualities, which are determined by its underlying topological structure.
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Quantum hall effect applications
There are many real-world uses for the QHE, especially in metrology, or the study of measuring.
Electrical Resistance Standard: The quantized Hall conductance has been adopted as a new practical standard for electrical resistance due to its exceptional precision, which has been measured to better than one part in a billion. The von Klitzing constant named for its discoverer is the foundation of this standard. For global resistance calibrations, a set customary value for this constant was established in 1990. The von Klitzing constant is now a precisely defined quantity because to the 2019 revision of the International System of Units (SI), which also included precise definitions for fundamental constants like Planck’s constant and the elementary charge.
Determination of the Fine-Structure Constant: The IQHE provides a very accurate and independent way to find the fine-structure constant. The strength of the electromagnetic interaction between elementary particles is described by this fundamental constant.
Experimental Realizations and New Discoveries
When a two-dimensional electron system arises in a thin surface layer of silicon-based MOSFETs (Metal-Oxide-Semiconductor-Field-Effect-Transistors), the QHE was first noticed. It has now been extensively researched in various semiconductor materials, such as heterostructures of gallium arsenide.
The QHE has been seen in graphene and other materials more recently. A special half-integer quantum Hall effect is seen in graphene. The unique relativistic-like energy dispersion of graphene and the degeneracy between electrons and holes at its Dirac points are the causes of this half-integer behaviour. Interestingly, compared to traditional semiconductor devices, the QHE in graphene may be seen at significantly higher temperatures, occasionally even getting close to room temperature.
The demonstration of the QHE in Bi2O2Se thin films is a relatively new and interesting development. Even at extremely high magnetic fields, researchers found no evidence of odd-integer quantum Hall states in thicker Bi2O2Se films, just even-integer ones. A “hidden Rashba effect” is blamed for this peculiar behaviour. This action results in opposite spin polarizations that essentially cancel each other out due to local breaking of inversion symmetry within the [Bi2O2]$^{2+}$ layers of the material.
However, the intrinsic asymmetry brought about by the top surface and bottom interface produces a “net polar field” when the thickness of the Bi2O2Se film is decreased to a single unit cell and grown on a SrTiO3 substrate. Both odd- and even-integer quantum Hall states are observed as a result of the “global Rashba effect,” which lifts the band degeneracies found in thicker films. This study emphasizes the intricate relationship between spin-orbit coupling, a material’s structure, and the occurrence of quantum Hall phenomena.
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