——————————————————————————————— A Different Kind of Quantization: The Quantum Hall Effect The quantum Hall effect is one of the most remarkable of all condensed-matter phenomena, quite unanticipated by the physics community at the time of its discovery in 1980 [2]. The basic experimental observation is the quantization of resistance, in two-dimensional systems, to an extreme precision, irrespective of the sample’s shape and of its degree of purity. This intriguing phenomenon is a manifestation of quantum mechanics on a macroscopic scale, and for that reason, it rivals superconductivity and Bose–Einstein condensation in its fundamental importance. 0 Magnetic Field FIG. 1: The basic setup of Edwin Hall’s 1878 experiment. The graph on the right shows that Hall resistance increases linearly with the magnetic field. I. WHAT IS THE HALL EFFECT? CLASSICAL AND QUANTUM ANSWERS The classical Hall effect was named after Edwin Hall, who, as a 24-year-old physics graduate student at Johns Hopkins University, discovered the effect in 1878. His measurement of this tiny 1 effect is regarded as an experimental tour de force, preceding by 18 years the discovery of the electron. The classical Hall effect offered the first real proof that electric currents in metals are carried by moving charged particles. As is shown in Figure 1, the basic setup of Hall’s experiment consists of a very thin sheet of conducting material, which is subjected to both an electric field E~ and a magnetic field B~ . The electric field, lying in the plane of the conductor, makes the charges in the conductor move, setting up an electric current I. The magnetic field, on the other hand, is perpendicular to the conductor, and according to the classical laws of electricity and magnetism, it exerts a so-called Lorentz force on these moving charges, pushing them sideways in the plane, perpendicular to the current I. This results in an induced voltage, perpendicular to both I and B, which is known as the Hall voltage. The Hall resistance, usually denoted by Rxy, is deduced from Ohm’s law, V = IR, so it equals the ratio of the transverse Hall voltage to the longitudinal current I. The Hall conductance, which is the reciprocal of the Hall resistance, is denoted by σxy. Just over a century later, in 1980, a quantum-mechanical generalization of Edwin Hall’s classical effect was discovered by German physicist Klaus von Klitzing. What today is called the “integer quantum Hall effect” took the physics world deeply by surprise. Von Klitzing was investigating the transport properties of a certain semiconductor device (namely, a silicon MOSFET, short for “metal-oxide semiconductor field-effect transistor”) at very low temperatures and in high magnetic fields. His experiment revealed two novel features closely related to, but quite distinct from, what Hall had found (which was a purely linear increase in Hall resistivity with the magnetic field). These features were: • Plateaus — that is, step-like structure — in Hall resistance (or conversely, plateaus in Hall conductivity, since conductance is the reciprocal of resistance), as can be seen in Figure 2. • Not just plateaus, but quantization of resistivity, meaning that the Hall resistivity assumes periodically spaced values — namely, integer multiples of a new fundamental constant e2 σH = h (where e is the charge on the electron and h, as always, is Planck’s constant). This very surprising discovery earned von Klitzing the Nobel Prize in physics in 1985. Appendix A contains some excerpts from the announcement of the Nobel Prize [1]. The quantum Hall effect arises in two-dimensional electronic systems, commonly known as two-dimensional electron gases, which are immersed in a strong magnetic field. There are a variety 2 FIG. 2: On the left, German physicist Klaus Klitzing and the discovery of the quantum Hall effect. On the right, the integer quantum Hall effect, characterized by quantized plateaus of resistance. More specifically, the resistivity of the material assumes only values that are integer multiples of the fundamental physical e2 constant h . of techniques to construct two-dimensional electron gases. Von Klitzing’s 1980 discovery relied on the existence of a two-dimensional electron gas in a semiconducting device. By the middle of the 1960s, such systems could be physically realized, thanks to the great technological progress that followed the invention of the transistor roughly twenty years earlier. Figure 3 schematically shows an example of a two-dimensional electron gas, in which the energy bands in a gallium-arsenide/aluminum-arsenide alloy are used to create a quantum well. Electrons from a silicon donor layer fall into the quantum well to create the electron gas. 3 FIG. 3: Schematic illustration of a rectangularly-shaped quantum well in a semiconductor, which becomes the home for a two-dimensional electron gas. The horizontal axis represents an ordinary coordinate of physical position, while the vertical axis represents energy. (To create the energy barriers that define the well, it is common to use an alloy of gallium arsenide and aluminum arsenide, as is shown in this figure, rather than pure aluminum arsenide.) The two dark circles represent positive silicon ions that have donated electrons to the quantum well. (Of course the gas consists of millions of electrons, not just two.) The wave function of the lowest energy level for an electron caught in the quantum well is indicated by the dashed line. II. A CHARGED PARTICLE IN A MAGNETIC FIELD: CYCLOTRON ORBITS AND THEIR QUANTIZATION A. Classical Picture We open our explanation of the quantum Hall effect by giving a purely classical description of the motion of an electron with charge −e in the presence of a uniform magnetic field B~ , which we will assume is directed along the z-axis. If the electron has velocity ~v, it will experience a Lorentz force −e~v × B~ due to the magnetic field, and its motion will be described by Newton’s equation F~ = m~a, broken down into x- and y-coordinates, as follows: 4 mx¨ = −eBy_ (1) my¨ = +eBx_ (2) The general solution to these coupled differential equations is periodic motion in a circle of arbitrary radius R: ~r = R(cos(!t + δ); sin(!t + δ)) (3) eB Here, ! = m is known as the classical cyclotron frequency, and δ is an arbitrary phase associated with the motion. Note that the period of the orbit is independent of the radius. In fact, the radius R and the tangential speed v = R! are proportional to each other, with constant of proportionality !. Thus a fast particle will travel in a large circle and will return to its starting point at exactly the same moment as a slow particle traveling in a small circle. Such motion is said to be isochronous, much like that of a harmonic oscillator (e.g., a pendulum with small amplitude), whose period is independent of the amplitude of the oscillation. B. Quantum Picture To treat the problem quantum-mechanically, we must solve the corresponding Schrodinger¨ equation, and, as in the case of the hydrogen atom, we find that this yields a quantized set of electron energies: 1 eB E = !(n + );! = (4) n ~ 2 m where n = 1; 2; 3; :::. These quantized energy levels are known as Landau levels, and the corresponding wave functions as Landau states, after the Russian physicist Lev Landau, who pioneered the quantum-mechanical study of electrons in magnetic fields. The Landau wave functions are products of Gaussian functions (bell-shaped curves) and certain polynomials called Hermite polynomials. Without going into detail, we will simply state 5 FIG. 4: The quantization of electron orbits in a magnetic field results in equally-spaced energy levels — eB Landau levels. The spacing of these levels is proportional to the classical cyclotron frequency ! = m . The colorful drawing on the right shows many electrons simultaneously executing cyclotron orbits, where each colored plane in the stack represents a different Landau level. that these wave functions describe waves that spread out over a characteristic distance known as the magnetic length lB: r l = ~ (5) B eB The existence of both a natural length scale lB and a natural energy scale ~! in this physical situation is a purely quantum phenomenon (as can be seen from the presence of ~ in the formulas for both of these natural units). In the classical situation, there was no such natural unit of length. 6 Note that the new length scale is nonetheless closely related to the classical cyclotron frequency eB of an electron in a magnetic field, m . Associated with the natural quantum-mechanical magnetic length lB is a natural quantum-mechanical unit of area: h=e 2πl2 = (6) B B This formula reveals a natural physical interpretation for the magnetic length, which is that the area 2 h that it determines — namely, 2πlB — intercepts exactly one quantum of magnetic flux, Φ0 = e . We can thus write: Φ 2πl2 = 0 (7) B B Furthermore, it can be shown that each Landau level is highly degenerate, with the degeneracy factor ν being given by the total number of flux quanta penetrating the sample of area A, BA h ν = ; Φ0 = (8) Φ0 e The model we have just discussed allows one to understand the quantization of Hall conductivity, as is shown in Appendix C. The crux of the matter is that if a Landau level (with ν quantum states) is completely filled, the Hall conductance is quantized. Furthermore, the quantum number associated with Hall conductivity is determined by the number N of filled Landau levels. Thus if the Fermi energy lies in the N th gap, then the transverse or Hall conductivity is: e2 σ = N (9) xy h Since the Fermi energy resides in a gap between bands (see Figure 5), quantum Hall systems are insulators, and yet, strangely enough, they exhibit Hall conductivity.