This article explains the quantum Hall effect (QHE) phenomenon, including its discovery, explanation, types, and applications in metrology and quantum computing.
Quantum Hall Effect Phenomena
The quantum Hall effect (QHE) is a remarkable phenomenon that occurs in two-dimensional electron systems subjected to a strong magnetic field. It is a quantum-mechanical effect, which means that it can only be explained by the laws of quantum mechanics. The discovery of the QHE led to a Nobel Prize in Physics in 1985.
Discovery of the QHE
The QHE was first discovered by Klaus von Klitzing in 1980 while he was studying the electrical conductivity of a two-dimensional electron gas. He found that the electrical conductivity of the gas had very sharp plateaus at certain magnetic field strengths. These plateaus occurred at values of the magnetic field that were multiples of a fundamental constant known as the magnetic flux quantum. The magnetic flux quantum is given by:
Φ0 = h/2e
where h is Planck’s constant and e is the elementary charge. The discovery of the QHE was a significant achievement because it demonstrated that electrical conduction in a two-dimensional electron gas is quantized and that the quantization is related to fundamental constants of nature.
Explanation of the QHE
The QHE can be explained by considering the behavior of electrons in a two-dimensional electron gas in the presence of a strong magnetic field. When a magnetic field is applied perpendicular to the plane of the electron gas, the electrons move in circular orbits around the field lines. The size of the orbits is determined by the strength of the magnetic field.
At very low temperatures and high magnetic fields, the electrons in the two-dimensional electron gas occupy only the lowest energy level, known as the Landau level. The Landau level is quantized, which means that it can only take on certain discrete energy values. As the magnetic field is increased, the Landau level splits into smaller levels, each with a different energy. The energy spacing between these levels is proportional to the strength of the magnetic field.
The electrical conductivity of the electron gas is related to the number of electrons that can move through the sample. At certain magnetic field strengths, the energy gap between the Landau levels matches the energy of a photon of the same frequency as the radiation used to probe the sample. When this happens, the electrons absorb the photons and move to a higher energy level, creating a gap in the energy spectrum of the electrons. This gap prevents the electrons from moving through the sample, which leads to a plateau in the electrical conductivity.
The QHE has important applications in metrology because the plateaus in the electrical conductivity are very precise and can be used to accurately measure fundamental constants of nature.
Integer and Fractional Quantum Hall Effect
There are two types of quantum Hall effect: integer and fractional. The integer quantum Hall effect (IQHE) occurs when the plateaus in the electrical conductivity occur at integer multiples of the magnetic flux quantum. The fractional quantum Hall effect (FQHE) occurs when the plateaus occur at fractional multiples of the magnetic flux quantum. The FQHE is a more complex phenomenon than the IQHE and requires a stronger magnetic field and lower temperatures to observe.
The FQHE can be explained by the concept of quasiparticles. In the FQHE, the electrons in the sample form bound states with other electrons and form quasiparticles. These quasiparticles have a fractional charge and obey fractional statistics. The existence of these quasiparticles is a direct consequence of the strong correlations between the electrons in the sample.
Applications of the Quantum Hall Effect
The quantum Hall effect has important applications in metrology, particularly in the determination of the Planck constant, the fine-structure constant, and the von Klitzing constant. The von Klitzing constant is a fundamental constant of nature that is used to define the ohm, the unit of electrical resistance.
The QHE also has applications in the development of quantum computers. The FQHE can be used to create and manipulate qubits, which are the building blocks of quantum computers. The strong correlations between the electrons in the sample make the qubits more stable and less susceptible to decoherence.
Conclusion
The quantum Hall effect is a remarkable phenomenon that occurs in two-dimensional electron systems subjected to a strong magnetic field. The discovery of the QHE led to a Nobel Prize in Physics in 1985. The QHE can be explained by considering the behavior of electrons in a two-dimensional electron gas in the presence of a strong magnetic field. The IQHE and FQHE are two types of QHE, with the FQHE being a more complex phenomenon that requires a stronger magnetic field and lower temperatures to observe. The QHE has important applications in metrology and the development of quantum computers.
