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Week 13: Quantum Mechanics of a Crystal in a Magnetic Field Using Important Deadlines (A) Dec 6: Submit solutions to All Home Work Problems (B) Dec 8: Will be asked to come to the board to do these problems and explain them (C) Nov 29 & Dec 1 presentation of the project in class. List of Home Work Problems αZ 2 (1) (a)Using relativistic theory, show that upto leading power in α, E = En[1 + ( 2n ) ] αZ 2 n 3 (b) For the elliptic orbits, the Sommerfeld formula E = En[1 + ( ) ( − )] n = nr + nφ 2n nφ 4 reduces correctly to the formula for the circular orbits. (2) See Week3.pdf: two home work problems (3) See Week4.pdf: Three Home Work Problems. (4) See Week8.pdf: Home Work. I. QUANTIZATION OF HALL CONDUCTIVITY In Landau Level picture, quantization comes from filled Landau level. Note, each level is degenerate with degree of degeneracy ν . If there are N filled Landau levels, Hall conductance is 2 σxy = Ne =h. To explain topological root of this quantization, we need to consider the electrons moving in a crystal. A simple model as described below was used by David Thouless to show that the quantum number N of the Hall conductivity has topological origin. 1 II. 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. III. DESCRIBING ELECTRONS IN A CRYSTAL IN A MAGNETIC FIELD The story starts out with one of the simplest and most famous equations of classical physics, relating two ubiquitous quantities in mechanics: kinetic energy and momentum. For a classical particle of mass m, the kinetic energy E is a quadratic function of the momentum p: p2 E = 2m What is the corresponding equation in a crystal ??? Tight Binding Approximation Electrons can tunnel only to its nearest neighbor. H = E0(cos px + cos py) ≡ E0(cos pxa=~ + cos pyb=~) (1) H (x) = (x + a) + (x − a) + (y + b) + (y − b) (2) Note that eipxa=~ (x) = (x + a). The operator eipxa=~ acts as a translation operator that translates x to x + a. 2 To introduce magnetic field, ~p ! ~p − e=cA~. Using Landau gauge A = (0; xB; 0), we get the following equations for the two non-zero components of the operator: @ p = −i x ~@x @ e p = −i − Bx y ~@y c − ie Bxa ie Bxa E0[Ψ(x + a; y) + Ψ(x − a; y) + e ~c Ψ(x; y + a) + e ~c Ψ(x; y − a)] = EΨ(x; y) Like any self-respecting Schrodinger¨ equation, this is an eigenvalue equation that admits of solutions for the wave function Ψ only for certain values of the energy E on the right side. Finding those values of E as a function of the magnetic field φ is our “Holy Grail”, so to speak. At this point, it will be useful to introduce some notational conventions. Firstly, we will divide both sides by E0, giving us E=E0 on the right side, but we will replace this ratio by the letter E (this is just a rescaling of the energy, and turns E into a dimensionless quantity). Secondly, since it seems that only lattice points (ma; na) are involved, with m and n running over integer values only, we can replace the continuous variables x and y by the discrete variables ma and na, or even by just m and n. Lastly, we can introduce the pure number φ = Ba2=(h=e), into this equation. All of this will give us the following equation: Ψ(m + 1; n) + Ψ(m − 1; n) + e−2πimφΨ(m; n + 1) + e2πimφΨ(m; n − 1) = EΨ(m; n) And just think: this is no longer a differential equation, but a difference equation, since it only involves discrete positions in a two-dimensional lattice. That’s quite a surprise! The wave function Ψ(m; n) is a product of two independent functions m and χn, with χn being purely periodic along the x-axis (or the n-axis, if you prefer). (This is actually not just a ikyn guess but a necessary consequence of our prior assumptions.) If we set χn = e and plug it in, then the equation simplifies as follows: 3 2πimφ−ky −2πimφ+ky m+1 + m−1 + [e + e ] m = E m All that’s left of χn, which represented the wave function’s behavior along the y-axis, is the number ky in the two exponents. Other than that, there is no more trace of χn. Everything comes down to behavior along just the x-axis (i.e., the m-axis). The sum of the two exponentials in the square brackets actually amounts to twice a cosine, so we can finish up our simplification work as follows: m+1 + m−1 + 2 cos(2πφm − ky) m = E m (3) This equation was derived in 1955 by Philip Harper and is known as the Harper’s equation. Note, again, the two-dimensional problem is reduced to one-dimensional problem. IV. THE MARVELOUS PURE NUMBER φ 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 a2B 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 4 to measure that flux. And indeed, counting the number of flux quanta passing through a unit cell a2Be gives a dimensionless answer — a pure number, a flux divided by another flux — namely, 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.) 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 (a2). The salient area having to do with the magnetic field hc alone is the area of a circle (a Landau orbit) that intercepts exactly one flux quantum (this is eB ). Now take the ratio of these two natural areas, and voila!` 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 h a a2m is am , with m being the mass of the electron); this time is therefore (h=am) = h . The salient time interval having to do with the magnetic field alone is the cyclotron period (the reciprocal of the eB cyclotron frequency mc , which, as was mentioned before, is independent of the size of the Landau orbit). As before, take the ratio of these two quantities, and voila` 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.
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