The Quantum Hall Effect - Landau Levels

Transcription

1 The Quantum Hall Effect - Landau Levels FIG. 1: Harmonic oscillator wave functions and energies. The quantization of electron orbits in a magnetic field results in equally-spaced energy levels Landau levels. The spacing of these levels is proportional to the classical cyclotron frequency ω = eb m. Quantum Mechanics of Electrons in a Magnetic Field The quantum treatment of a planar electron in a transverse magnetic field B comes from replacing the x and y components of the momentum vector p = (p x, p y ) as follows: 1

2 p x i x p y i y ebx (Here we are using the magnetic vector potential A = B(0, x, 0), called the Landau gauge.) The resulting Schrödinger equation is: 1 2m [( i x )2 + ( i y ebx)2 ]ψ(x, y) = Eψ(x, y) (1) Since there is no explicit dependence on y anywhere in the above equation, we look for solutions of the form ψ(x, y) = f(x)e ikyy. This leads to the following equation for f(x): 2 d 2 f 2m dx m (ebx k y) 2 = Ef(x) (2) This equation is identical to the Schrödinger equation for a one-dimensional harmonic oscillator, whose eigenfunctions and eigenvalues are: ψ(x, y) = φ n (x k y eb ) eikyy ; E n = ω c (n + 1/2) (3) where ω c equals the cyclotron frequency eb, and where the functions φ m n are harmonic-oscillator wave functions labeled by the same quantum number n that labels the discrete energy levels. Here 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. New Length & Energy Scale & Degeneracy of Landau Levels The Landau wave functions are products of Gaussian functions (bell-shaped curves) and certain polynomials called Hermite polynomials. These wave functions describe waves that spread out over a characteristic distance known as the magnetic length l B : 2

3 l B = eb (4) The existence of both a natural length scale l B and a natural energy scale ω 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. Note that the new length scale is nonetheless closely related to the classical cyclotron frequency of an electron in a magnetic field, eb. m Associated with the natural quantum-mechanical magnetic length l B quantum-mechanical unit of area: is a natural 2πl 2 B = h/e B (5) This formula reveals a natural physical interpretation for the magnetic length, which is that the area that it determines namely, 2πlB 2 intercepts exactly one quantum of magnetic flux, Φ 0 = h. We can thus write: e 2πl 2 B = Φ 0 B (6) Note that the eigenvalues of this system, unlike its eigenfunctions, are independent of k y. This results in degenerate levels whose degree of degeneracy is equal to ν = BA/φ 0 (A being the area of the sample) ( See Below).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, ν = BA Φ 0, Φ 0 = h e (7) 3

4 The model we have just discussed allows one to understand the quantization of Hall conductivity, as is shown below. 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: σ xy = N e2 h (8) See below for the derivation. Since the Fermi energy resides in a gap between bands, quantum Hall systems are insulators, and yet, strangely enough, they exhibit Hall conductivity. How is it possible for a system that, in the bulk, is an insulator, to conduct current? This mystery will be resolved by the intuitive semiclassical arguments namely the skipping orbits. Quantization of the Hall Conductivity We start with the expression for σ xy, the classical Hall conductance [? ]. Here, Q is the total electrical charge of the system, and A is its area. σ xy = Q BA (9) To obtain a quantum expression for Hall conductivity [? ], we consider a system with N filled Landau levels. (This means there is no longitudinal current, so the longitudinal conductance σ xx is equal to zero.) As each Landau level is ν-fold degenerate, Q = Nνe = NBAe 2 /h. (Here we have used equation (??), which gives ν = BA/Φ 0 = BAe/h.) This gives us the desired result: σ xy = N e2 h (10) The validity of this expression has been proven more rigorously, even in the presence of impurities. However, although this explains quantization, it does not give a topological basis for it. To establish topological basis for this quantization, we need to consider a crystal in a magnetic field. 4

5 I. TWO-DIMENSIONAL CRYSTAL IN A MAGNETIC FIELD Swiss physicist Felix Bloch, in 1928, discovered that electrons inside a crystal (in the absence of a magnetic field) have energy values that lie in Bloch bands that is, their energies range across a continuous swath of possible values, without any gaps. Bloch also found that such electrons have quantum-mechanical states (now called Bloch states, of course) that can be described as standing waves inside the lattice. Such waves of course oscillated periodically, with a period depending on their energy. FIG. 2: This schematized diagram shows a pure atomic energy level splitting into two when two identical atoms are brought together, and into three when three identical atoms are brought together. (The splitting is due to the Pauli exclusion principle, which forbids electrons to occupy the same state.) When huge numbers of atoms (10 24 of them, say) come together to form a periodic crystal lattice, then what was originally a single energy level splits into a cluster of enormously many energies, which are so close to each other that they essentially constitute a continuum. This continuum of energies is called a Bloch band. Thus inside a Bloch band there are no energy gaps, but between Landau levels there are gaps. A fascinating question was therefore this: What would happen when these two contrasting situations one with a continuous spectrum and an energy-dependent period, the other with a discrete spectrum and an energy-independent period were physically combined? More concretely, what would happen to electrons in a crystal when a magnetic field was turned on? 5

6 E Fermi Energy Single Atom FIG. 3: A schematic depiction of the difference between the energy levels belonging to a single atom, to a metal, and to an insulator. The discrete energy levels belonging to an isolated atom evolve into energy bands belonging to a crystal, as each atom s structure is modified by the close approach of other atoms. Inside the bands, the allowed energies take on a continuum of values. Two neighboring bands are separated by a band gap or simply a gap a region of forbidden electron energies. The shaded regions in the figure represent levels that are occupied. In a metal, the Fermi energy lies inside a partially filled band. In an insulator, the Fermi energy lies inside the energy gap. Would the band to which they belonged somehow split into a set of mini-bands perhaps narrow Bloch bands, perhaps thick Landau levels separated by mini-gaps? And how many such narrow bands or thick levels would there be? And how would they be spaced? And would the electron s behavior be periodic, and if so, with what period? How, in short, would nature manage to reconcile these profoundly opposing situations? None of this was at all obvious, and solving the Schrödinger equation for such a hybrid situation was, to say the least, mathematically very challenging. 6

7 II. THE MARVELOUS PURE NUMBER φ We also saw earlier that Russian physicist Lev Landau, just two years later, in 1930, discovered that electrons immersed in a magnetic field (in vacuum, not in a crystal) have very sharp energy values Landau levels that are evenly spaced, like the rungs of a ladder, with wide energy gaps separating them. Landau also found that such electrons have quantum-mechanical states that correspond to classical circular orbits of various radii, yet all these different states share the same period of oscillation (the reciprocal of the cyclotron frequency, which was the classical frequency of electrons making circular orbits in a magnetic field). In other words, this quantum-mechanical period is independent of the energy the Landau electron possesses. There is one extremely important new physical quantity that arises when the two situations are combined, and that is the pure number φ ( pure in the sense that it has no units attached to it), which measures the magnetic field s strength in an extremely natural fashion. How can a pure number, with no units at all, tell how strong a magnetic field is? The answer is not too subtle and very revelatory; in fact, this all-important quantity φ can be thought of in (at least) three conceptually different but mathematically equivalent ways, described below. It all hinges on the fact that there is a fundamental amount of magnetic flux the flux quantum, equal to hc/e that emerges intrinsically out of quantum mechanics. This minuscule quantity is an inherent fact about our universe, just as are the speed of light and the charge on the electron. Given that this tiny amount of flux is the natural chunk of flux, it is as if nature had handed us a measuring stick on a silver platter! This beautiful and generous favor on nature s part must not be ignored. A first way of thinking about the variable φ, then, is as how many flux quanta pass through a unit cell of the lattice. After all, the flux passing through a unit cell is proportional to the magnetic field B (it equals a 2 B in the case of a square lattice, which is what we are focusing on for the moment), and the flux quantum hc/e is the natural unit the unit par excellence with which to measure that flux. And indeed, counting the number of flux quanta passing through a unit cell gives a dimensionless answer a pure number, a flux divided by another flux namely, a2 Be hc. In truth, nothing could be more natural than this dimensionless way of measuring the strength of a magnetic field in a crystal. (By the way, this trick won t work to measure a magnetic field in a vacuum, because the trick involves flux, which involves area, and in a vacuum, unlike in a crystal, there is no scale defining a natural chunk of area.) 7

8 A second way to compute exactly the same number, yet conceptually quite different, vividly reflects the fact that φ is the fruit of the marriage of two unrelated parents the isolated crystal and the isolated magnetic field. How can one find a pure number that combines the unrelated genes coming from each of φ s parents? Well, the idea is to compute the ratio of two natural geometrical areas, one coming from each of the two parents. These geometrical areas can be thought of as the genes to exploit. In particular, the salient area having to do with the crystal alone is the area of a lattice cell (a 2 ). The salient area having to do with the magnetic field alone is the area of a circle (a Landau orbit) that intercepts exactly one flux quantum (this is hc eb ). Now take the ratio of these two natural areas, and voilà! You get a pure number, and a very simple calculation shows that it is equal to the previous number. A third way to compute the same number reflects, once again, φ s dual origins. This time we again take a dimensionless ratio, but this new ratio involves two different genes namely, the ratio of two natural time intervals, one coming from each of the two parents. The salient time interval having to do with the crystal alone is the time taken by an electron with momentum h/a to cross a unit cell (h/a is the canonical momentum for a crystal electron, and the associated velocity is h a, with m being the mass of the electron); this time is therefore am = a2 m (h/am) h. The salient time interval having to do with the magnetic field alone is the cyclotron period (the reciprocal of the cyclotron frequency eb, which, as was mentioned before, is independent of the size of the Landau mc orbit). As before, take the ratio of these two quantities, and voilà once again! The units cancel, you get a pure number, and another very simple calculation shows that it is equal to the previous two numbers. Seeing φ as a ratio of two independent entities of the same sort (whether they are areas or time intervals), one coming from each parent, is a fundamental and insight-giving way to look at what φ means. Also, seeing φ as the number of flux quanta threading a lattice cell is another deep way to look at what φ means, equally fundamental and insight-giving. Whichever may be your favorite way of conceiving of what φ means, in any case it is a pure number that naturally measures the magnetic-field strength in the hybrid situation. Indeed, φ is the key parameter at the very heart of the situation. [1] < press.html> 8

9 [2] Klaus von Klitzing, 25 Years of Quantum Hall Effect, A Personal View on the Discovery, Physics and Applications of this Quantum Effect, Vol. II, Séminaire Poincaré, (2004), page 1. [3] Topology and Quantum Hall Effect, Chapter 9 in Field Theories of Condensed Matter Systems by Eduardo Fradkin, Addison-Wesley. 9