The Quantum Hall Effect

One of the basic rules of electricity says that in order to maintain an electrical current I in a wire, a voltage U is necessary that is proportional to the current. This is Ohm's law, U = RI, where the constant of proportionality is the resistance, R. If the wire with the current is exposed to a magnetic field, the Lorentz force acts on the wire, normal to both the direction of the current and the magnetic field. If the current flows through a sheet, as a result of the Lorentz force on the electrons in the current, a voltage can be measured that is oriented along the direction of the Lorentz force. This is the Hall voltage, named after the American physicist Edwin Herbert Hall, who was the first to observe this phenomenon. The Hall effect is used today, for example, to measure magnetic fields.

Ohm's law and the Hall effect can be understood in terms of classical physics: electrons travelling through the wire bump into the atoms of the material of the wire, which causes the resistance. However, as we know for example from the study of the electronic band structure of materials, electric current is carried by particles subject to the rules of quantum mechanics. And if the temperature of the conducting material is low enough, thus reducing the strength of thermal effects, quantum effects can become apparent, leading to such striking phenomena as superconductivity. In a superconductor, charges set in motion do not scatter, since quantum mechanics doesn't allow it, and hence, there is no resistance.

Another quite strange quantum phenomenon can be observed if the current is constrained to a very thin sheet, so that it's essentially two-dimensional, and if a strong magnetic field is applied normal to the sheet. In this case, there is a special class of states for the electrons called Landau levels, and depending of the occupancy of the Landau levels, a new macroscopic phenomenon similar to superconductivity can set in: the Quantum Hall Effect.

[Source: K. v. Klitzing, G. Dorda, M. Pepper: New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett. 45 (1980) 494-497, Figure 1.]

The experimental data shown in the plot stem from the experiment in which the Quantum Hall effect was discovered. In the experiment, shown schematically in the inset, a thin sheet structure of semiconducting material, carrying a constant current of 1 µA, was exposed to a strong magnetic field at the temperature of liquid helium, and the voltage drop along the current and the Hall voltage normal to the current and the magnetic field have been measured. These voltages are plotted on the vertical axis, and the curves are labelled as U_H for the Hall voltage, and U_pp. The horizontal axis shows another voltage, the so-called gate voltage V_g, which squeezes the electrons in a thin, two-dimensional sheet, and thus determines the occupancy of the Landau levels. Now, with varying gate voltage, a curious phenomenon can be observed in the Hall and standard voltage: the standard voltage drops to zero several times, meaning that the current flows without resistance, as in a superconductor, and at the same time, the Hall voltage takes on constant values. There is series of step-like plateaux in the Hall voltage, where it stays constant.

Now, what looks like quite an esoteric effect - take some contrived semiconductor structure, cool it down to a few Kelvin, put it in the strongest magnetic field your high-field lab can provide - has an extremely interesting twist to it:

The Hall resistance, the quotient of Hall voltage and the current, in the step-like structures comes in a regular series, R = R_K/n, where n is an integer, and R_K, the von Klitzing constant, is R_K = 25.812807557 kΩ. And the really fascinating thing is that as a consequence of the theory of Landau levels, the von Klitzing constant does not depend on the details of the material, but is universal: It is given by R = h/e² = μ₀ c/2α, where h is Planck's constant, e is the charge of the electron, and α = 1/137 is the fine structure constant, the coupling constant of electrodynamics.

Thus, the quantum Hall effect can be used, for example, to set a standard for electrical resistance, or as a means to measure the fine structure constant. Klaus von Klitzing, the German physicist who discovered the effect in February 1980 and immediately grasped these implications, was awarded the Nobel Prize in Physics 1985 for the discovery of the quantized Hall effect.

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Klaus von Klitzing's Nobel Lecture The Quantized Hall Effect gives a good introduction into theory and experiment of the Quantum Hall Effect (PDF file).

More details can be found in the proceedings of the Poincaré Seminar of November 13, 2004, dedicated to the Quantum Hall Effect. Among the talks:

Klaus von Klitzing: 25 Years of Quantum Hall Effect (QHE) A Personal View on the Discovery, Physics and Applications of this Quantum Effect (PDF file)

Benoît Douçot and Vincent Pasquier: Physics in a Strong Magnetic Field (PDF file), about Landau levels etc...

Beat Jeckelmann and Blaise Jeanneret: The Quantum Hall Effect as an Electrical Resistance Standard (PDF file), about the implications of the Quantum Hall Effect for metrology.

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This post is part of our 2007 advent calendar A Plottl A Day.