Free Electrons in Uniform Magnetic Field: Landau Levels and Quantum Hall Effect
The Zeeman effect, which we discussed in some detail in Sect. 14.2 , originates from the interaction between the magnetic moment of an electron bound to an atom and a uniform magnetic field. Experimentally this effect is often observed in atomic gases bu
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Free Electrons in Uniform Magnetic Field: Landau Levels and Quantum Hall Effect
The Zeeman effect, which we discussed in some detail in Sect. 14.2, originates from the interaction between the magnetic moment of an electron bound to an atom and a uniform magnetic field. Experimentally this effect is often observed in atomic gases but can also manifest itself with bound electrons in semiconductors and dielectrics. What is important is that the quantum states of the electrons in all these cases belong to discrete energy eigenvalues. In metals and in the conduction band of semiconductors, on the other hand, the energy levels of electrons belong to the continuum spectrum, and in some instances, electrons can even be treated as free particles. The interaction between such unbound, almost free electrons and the uniform magnetic field results in some fascinating effects which had played and are still playing an important role in physics. Among older phenomena owing their existence to this interaction are the Hall effect (the emergence of an electric current in the direction perpendicular to the electric field), the de Haas–van Alphen effect (oscillations of magnetization of the metal with increasing magnetic field), and the Shubnikov–de Haas effect (oscillations of conductivity with the magnetic field). The Hall effect, which is a purely classical phenomenon, was discovered by American physicist Edwin Hall in 1879. It is interesting to note that Hall first worked as a high school principal before embarking on Ph.D. studies in physics, which he did at Johns Hopkins University. The discovery of the effect, which now bears his name, was part of his doctoral work. He became a professor of physics at Harvard University in 1895, but did not produce anything even remotely as significant as his student discovery. Both de Haas effects owe their existence to the quantum nature of electrons and were discovered in 1930 in the laboratory of Dutch physicist Wander Johannes de Haas. Pieter M. van Alphen was at that time de Haas’ student, while Lev Shubnikov was a Soviet physicist working with de Haas as a visiting scholar. After his return to the Soviet Union, Shubnikov was falsely accused of espionage and in 1937 executed by the People’s Commissariat for Internal Affairs (NKVD)—Soviet analog of Nazi Gestapo (Secret State Police). © Springer International Publishing AG, part of Springer Nature 2018 L.I. Deych, Advanced Undergraduate Quantum Mechanics, https://doi.org/10.1007/978-3-319-71550-6_16
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16 Free Electrons in Uniform Magnetic Field: Landau Levels and Quantum Hall Effect
One of the most fascinating and fairly recent magnetic field effects discovered in the system of unbound electrons is the quantum Hall effect and even more exotic fractional quantum Hall effects. The former was discovered by German physicist K. von Klitzing in 1980, who showed experimentally that the so-called Hall conductance (I will explain what it means later, have patience) in a twodimensional electron gas is independent of the geometry and detail
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