The fractional quantum Hall effect (FQHE) is a remarkable phenomenon that occurs when a two-dimensional electron gas is subjected to a strong magnetic field. This effect, discovered in 1982 by Daniel Tsui, Horst Störmer, and Arthur Gossard, has revolutionized our understanding of quantum physics. In the FQHE, the electrons form collective, quantized states that exhibit fractional quantum numbers, such as 1/3 or 2/5 of the electron charge. These states have exceptional electrical and transport properties, including dissipationless current flow at certain specific fractions e/i of the Hall resistance, where i is an integer. The FQHE has profound implications for condensed matter physics and has led to significant advances in areas such as topological insulators and quantum computing.
The Fractional Quantum Hall Effect
The fractional quantum Hall effect (FQHE) is a quantum mechanical phenomenon that occurs in two-dimensional electron gases (2DEGs) at very low temperatures and high magnetic fields. It is characterized by the emergence of new quantum states with fractional values of the Hall conductance, which are not quantized in integer multiples of e2/h, the standard quantum of conductance in the integer quantum Hall effect (IQHE).
The FQHE was first observed in 1982 by Daniel Tsui, Horst Störmer, and Arthur Gossard at the Bell Laboratories. They found that the Hall conductance of a 2DEG in a strong magnetic field exhibited plateaus at fractional values of e2/h, such as 1/3 e2/h and 2/3 e2/h. This observation was unexpected and challenged the prevailing understanding of the IQHE, which predicted integer quantization of the Hall conductance.
The FQHE is a result of the interplay between quantum mechanics and the magnetic field. In a magnetic field, electrons in a 2DEG experience a Lorentz force that causes them to move in circular orbits. The magnetic field also quantizes the energy levels of the electrons, forming discrete Landau levels. At low temperatures, the electrons occupy the lowest Landau level, which is non-degenerate and has a spin degeneracy of two.
The FQHE occurs when the magnetic field is strong enough to cause the electrons to form a strongly correlated state known as a Laughlin state. In a Laughlin state, the electrons form a regular array of vortices, which are quantized magnetic flux tubes. The number of vortices is determined by the filling factor ν=nh/eB, where *n is the number of electrons per unit area and B is the magnetic field strength. For example, at a filling factor of ν=1/3, there is one vortex for every three electrons.
The Laughlin state is a very stable state and is characterized by a number of unusual properties. For example, the Hall conductance of a Laughlin state is quantized at νe2/h, where ν is the filling factor. This is in contrast to the IQHE, where the Hall conductance is quantized at integer multiples of e2/h.
The FQHE has a number of potential applications, including the development of new electronic devices and quantum computers. It is also a subject of active research in condensed matter physics, and there is still much that is not understood about this fascinating phenomenon.
Question 1:
What is the significance of the fractional quantum Hall effect?
Answer:
The fractional quantum Hall effect (FQHE) is a quantum mechanical phenomenon observed in two-dimensional electron systems at extremely low temperatures and high magnetic fields. It is characterized by the emergence of plateaus in the Hall resistance at specific fractional values of the filling factor, which is the ratio of the number of electrons to the number of flux quanta in the system.
Question 2:
How does the FQHE demonstrate the quantization of energy levels?
Answer:
The FQHE provides experimental evidence for the quantization of energy levels in two-dimensional electron systems. The observed plateaus in the Hall resistance occur at specific values of the filling factor because these values correspond to integer ratios of flux quanta to electrons, resulting in the formation of well-defined and quantized energy levels.
Question 3:
What are the applications of the FQHE?
Answer:
The FQHE has potential applications in various areas of condensed matter physics and engineering. It has been used as a metrological tool for precise measurements of fundamental constants, such as the fine structure constant. Additionally, the FQHE may contribute to the development of novel electronic devices, including ultra-high-mobility transistors and quantum computing systems.
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