What Do You Know About The Quantum Hall Effect?

The quantum Hall effect is observed in two-dimensional electron systems. This phenomenon occurs when a strong magnetic field is applied perpendicular to the plane of a two-dimensional sample, causing the electrons to move in circular orbits. As the magnetic field strength increases, the energy levels of the electrons become quantized, resulting in the Hall resistance exhibiting plateaus at specific values. This effect is a manifestation of the quantization of electron charge and is a key discovery in the field of condensed matter physics.

von Klitzing discovered the Quantum Hall Effect (QHE) and its relation to fundamental physical constants. The QHE refers to the quantization of the Hall resistance in a two-dimensional electron gas subjected to a magnetic field. Von Klitzing's discovery earned him the Nobel Prize in Physics in 1985.

The Chern number is a topological invariant that characterizes the topology of a system in condensed matter physics. It can be used to determine the topologically inequivalent Hamiltonians defined on the Brillouin zone. By calculating the Chern number, one can identify the presence of non-trivial topological properties in a system, such as the existence of topological insulators or superconductors. Therefore, the Chern number provides a reliable method for distinguishing between different topological phases and determining the topological equivalence or inequivalence of Hamiltonians.

Landau levels are the cause of Integer Quantum Hall Effect (IQHE). In a strong magnetic field, the electrons in a two-dimensional electron gas (2DEG) occupy discrete energy levels called Landau levels. These levels are quantized due to the magnetic field, and the energy gap between them is proportional to the strength of the magnetic field. The IQHE occurs when the Fermi level lies within one of these Landau levels, resulting in a quantized Hall resistance. The presence of Landau levels is a key factor in understanding the phenomenon of IQHE.

Scanning-force microscopy allows the measurement of the position-dependence of the Hall potential. This technique involves scanning a sharp probe over a sample surface and measuring the forces between the probe and the sample. By mapping the forces at different positions, the Hall potential can be determined, which is a measure of the voltage generated across a conductor when a magnetic field is applied perpendicular to the current. Therefore, scanning-force microscopy is the correct answer for this question.

IQHE (Integer Quantum Hall Effect) and FQHE (Fractional Quantum Hall Effect) both exhibit the same ultimate physical effect, which is the quantization of the Hall resistance. In both cases, the Hall resistance is observed to take on discrete values in units of the von Klitzing constant. This effect occurs in two-dimensional electron systems subjected to a strong magnetic field and low temperatures. The mechanism behind IQHE and FQHE may differ, but they both share this common physical phenomenon.

The correct answer is "Presence of impurities that shield from Coulomb force". The IQHE (Integer Quantum Hall Effect) is a phenomenon that occurs in a two-dimensional electron gas subjected to a strong magnetic field. It is characterized by the quantization of the Hall resistance in integer multiples of a fundamental constant. The presence of impurities in the system can create localized states that act as scatterers, effectively shielding the electrons from the Coulomb force and allowing the formation of well-defined Landau levels. This impurity scattering plays a crucial role in the IQHE.

In 1988, it was proposed that there was a quantum Hall effect without Landau levels. This suggests that the conventional understanding of the quantum Hall effect, which relies on the presence of Landau levels, may not be the only explanation for this phenomenon. This discovery opened up new possibilities for understanding the quantum Hall effect and its underlying mechanisms.

Formation of Anyons is responsible for FHQE (Fractional Hall Effect). Anyons are exotic quasiparticles that emerge in two-dimensional systems under the influence of a strong magnetic field. In the presence of a magnetic field, electrons in a two-dimensional electron gas can form bound states called anyons, which have fractional charges and obey fractional statistics. These anyons are responsible for the observed fractional quantum Hall effect, where the Hall conductance takes on quantized fractional values. Therefore, the formation of anyons is the correct answer for the cause of FHQE.

The correct answer is 1975 because in that year, the integer quantization of the Hall conductance was first predicted. This discovery, known as the Quantum Hall Effect, revolutionized the field of condensed matter physics and led to the Nobel Prize in Physics being awarded to Klaus von Klitzing in 1985.