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

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)

Hψ(x) = ψ(x + a) + ψ(x − a) + ψ(y + b) + ψ(y − b) (2)

Note that eipxa/~ψ(x) = ψ(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 ψm 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 ψm+1 + ψm−1 + [e + e ]ψm = Eψm

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:

ψm+1 + ψm−1 + 2 cos(2πφm − ky)ψm = Eψm (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. 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

5 measures the magnetic-ﬁeld strength in the hybrid situation. Indeed, φ is the key parameter at the very heart of the situation.

      −ikx 2C1 1 0 0 . . e ψ1 ψ1        1 2C 1 0 0 . 0   ψ   ψ   2   2   2         0 1 2C 1 0 . 0   ψ   ψ   3   3   3         ......   .  = E  .               ......   .   .               ......   .   .        ikx e . 0 0 0 1 2Cq ψq ψq

±ikx where Cn = cos(2πnφ − ky). The lower-left and upper-right corner terms e in this matrix

ikx reﬂect Bloch’s theorem, which assumes periodic boundary conditions — namely, ψn = e ψn+q. 1 1 1 In a handful of very simple cases, such as φ = 1, 2 , 3 , 4 , the eigenvalues E can be determined analytically. However, for q > 4, the above matrix-eigenvalue problem can only be solved numerically. Rational vs Irrational Flux

p Let φ = q .

For rational ﬂux, Harper equation describes a system that is periodic with period q. Such systems can be studied using “ Bloch Theorem”.

Bloch’s theorem Named after physicist Felix Bloch, Bloch’s theorem states that the energy eigenstates of an electron moving in a crystal (a periodic potential) can be written in the following form:

Ψ(r) = eik·ru(r) (4)

where u(r) is a periodic function with the same periodicity as that of the underlying potential — that is, u(r) = u(r + a). The exponential preceding the periodic function u is a kind of helical wave, or “corkscrew”, which multiplies the the wave function by a spatially-changing phase that twists cyclically as one moves through space in a straight line.

6 Some Analytic Results

(I) Analytic expressions for the energy dispersions E(kx, ky) for a few simple cases

(a) For φ = 1, the energy spectrum consists of a single band given by

E = 2(cos kx + cos ky).

(b) For φ = 1/2, the two bands having energies E+ and E− are given by p 2 2 E± = ±2 cos kx + cos ky.

(c) For φ = 1/3, the three bands having energies E0,E1, and E2 are given by √ √ 2 1 Ei = 2 2 cos(θ ± i 3 π). Here, θ = 3 arccos[(cos 3kx + cos 3ky)/2 2].

(d) For φ = 1/4, the four bands having energies E++,E+−,E−+, and E−− are given q 1 2 by the expression E±± = ± 4 ± 2[3 + 2 (cos 4kx + cos 4ky)] .

V. CENTRAL CONCEPTS OF CONDENSED-MATTER PHYSICS

• Periodic potentials

A periodic electric potential V (r) arises automatically in a crystal because the positively charged nuclei in the crystal are arranged in a perfect lattice — an endlessly repeating spatial pattern. The periodicity of the electric potential is expressed by the equation V (r) = V (r + a), where a is a lattice vector. The primitive unit cell of the crystal is the smallest building block that, when periodically placed next to itself in space, yields the full crystal lattice.

• Non-interacting electrons and electron gases

In crystals, electrons move under the inﬂuence of the just-described periodic potential, and their behavior and properties are described by the laws of (non-relativistic) quantum mechanics. It is rather surprising that many aspects of matter can be understood by assuming

7 that the electrons in such a system never interact with one another, but that is the case, and in this book only such phenomena are discussed. A system of non-interacting electrons is known as an “electron gas”.

• Pauli’s exclusion principle

In our description of non-interacting electrons, we need to take into account a very deep quantum effect that arises due to the fact that we are dealing with particles (electrons in particular) that are all perfectly identical. This feature has no classical analogue, since in classical physics one can always distinguish identical-seeming particles simply by following their distinct trajectories. In other words, in classical physics, identical particles simply do not exist; any two particles can always be distinguished. But in the quantum world, this deeply intuitive property fails to hold.

Electrons, which are the main actors in solid-state systems, are literally indistinguishable from each other, and they obey what is known as the “Pauli exclusion principle”, named after the Austrian-born physicist Wolfgang Pauli (who, over the course of his life, held Austrian, German, Swiss, and American citizenships). In 1945, Pauli was awarded the Nobel Prize in physics for the discovery of this central quantum-mechanical principle, which states that no two electrons can ever occupy the same quantum state simultaneously. Actually, Pauli’s principle is more general than this, as it applies not just to electrons but to every type of fermion. A fermion is a particle that inherently possesses half-integer spin — that is, an

~ angular momentum that is an odd multiple of 2 . Electrons are fermions, since they have 1 spin 2 . Pauli’s full exclusion principle states that two identical fermions can never occupy the same quantum state simultaneously.

On the other hand, there are identical particles that can occupy the same quantum state, and such particles are called bosons. A boson is a particle that inherently possesses an integral amount of spin — that is, an angular momentum that is an integer multiple of ~. Photons are bosons, since they have spin 1. As such, they are not subject to the exclusion principle; any number of photons can occupy the same quantum state, and indeed they tend to do exactly that.

• Bands and band gaps

The term “electronic band structure” (or just “band structure” for short) denotes the set of

8 energy values that electrons in a solid may take on. The allowed energy levels are limited to certain intervals of the energy axis called “Bloch bands”, “energy bands”, “allowed bands”, or simply “bands”. The ranges of energy values that an electron may not take on are called “band gaps” or “forbidden bands” . Band theory derives these bands and band gaps by solving the Schrodinger¨ equation to determine the quantum-mechanical eigenvalues for an electron in a periodic lattice of atoms or molecules.

The reader, having seen that in many circumstances quantum mechanics gives rise to a set of isolated, discrete energy levels, and nothing like the continuous range of values comprising an energy band, might well ask: Why do solids have bands, as opposed to discrete levels, of energy? This is an excellent question, and a sketch of the answer is as follows.

The electrons in a single isolated atom have discrete energy levels. When multiple atoms join together to form into a molecule, their wave functions overlap in space, and because of the Pauli exclusion principle, their electrons cannot occupy the same state (meaning they cannot have the same energy). Therefore, each discrete eigenvalue (energy level) splits into two or more new levels, all clustered close to the original level. As more and more atoms are brought together, the allowed energy levels have to split into more and more new levels (again because of the exclusion principle), and thus the cluster of energy eigenvalues becomes increasingly dense and widens. Eventually, when there are so many atoms periodically spaced together that they constitute a macroscopic crystal, the allowed energy levels are so astronomically numerous and are clustered so densely along certain portions of the energy axis that they can be considered to form continua, or bands. Band gaps are essentially the leftover ranges of energy not covered by any band.

If electrons completely fill one or more bands (i.e., occupy all the levels in them), leaving other bands completely empty, the crystal will be an insulator. Since a filled band is always separated from the next higher band by a gap, there is no continuous way to change the momentum of an electron if every accessible state is occupied. A crystal with one or more partly filled bands is a metal.

And ﬁnally, in semiconductors, there is a relatively small energy gap between the highest ﬁlled band and the empty band — a gap small enough that thermal or other kinds of excitation can make electrons “jump across” it (keep in mind that they are not jumping

9 in physical space but in energy space). At low temperatures, a semiconductor’s conduction band will have no electrons in it, and so there will be no conduction. However, as the temperature rises and electrons gain thermal energy, some of the levels in the conduction band may become filled by electrons. Current can then flow through the sample. It turns out that the introduction of even a very low percentage of a “doping material” (a deliberate impurity) into a semiconductor is sometimes sufficient to increase conductivity dramatically.

• Fermi energy

A key notion in band theory is the Fermi level or Fermi energy, denoted EF . This is the highest energy level occupied by a crystal electron when the crystal is at a temperature of absolute zero. The Fermi level can thus be thought of as the “surface” of the “sea” of electrons in a crystal — but it must be remembered that this “sea” lies in the abstract space of energy levels, not in a physical space.

The position of the Fermi level relative to a crystal’s band structure is a crucial factor in determining the crystal’s electrical properties (e.g., its propensity to conduct electric current). At absolute zero, the electrons in any solid systematically fill up all the lowest available energy states, one by one by one. In a metal, the highest band that has electrons in it is not completely full, and hence the Fermi level lies inside that band. By contrast, in an insulating material or a semiconductor, the highest band that has electrons in it is completely filled, and just above it there is an energy gap, and then above that, a band that is completely empty. In such a case, the Fermi level lies somewhere between the highest filled band and the empty band above it. In a semiconductor at zero degrees Kelvin, no electrons can be found above the Fermi level, because at absolute zero, they lack sufficient thermal energy to “jump out of the sea”. However, at higher temperatures, electrons can be found above the Fermi level — and the higher the temperature gets, the more of them there will be.

The Fermi velocity vF is the velocity of an electron that possesses the Fermi energy EF . 1 2 It is determined by the equation EF = 2 mvF , which, when solved for the unknown quantity p vF , gives the formula vF = 2EF /m.

• Bloch’s theorem

Named after physicist Felix Bloch, Bloch’s theorem states that the energy eigenstates of an electron moving in a crystal (a periodic potential) can be written in the following form:

10 Ψ(r) = eik·ru(r) (5)

Inside a crystal, the noninteracting electrons are not free, but are Bloch electrons moving in a periodic potential. When the potential is zero, the solutions reduce to that of a free

2 2 ik·r ~ k particle with Ψ(r) = e and with energy E = 2m . The existence of Bloch states is the key reason behind the electronic band structure of a solid.

• Crystal momentum

The vector k of a given eigenstate for a crystal electron is that state’s Bloch vector. When multiplied by ~, the Bloch vector gives the so-called crystal momentum ~k of that state. Although crystal momentum has the same units as momentum, and although in some ways an electron’s crystal momentum acts very much like a momentum, it should nonetheless not be conﬂated with the electron’s momentum, because unlike momentum, crystal momentum is not a conserved quantity in the presence of a potential.

• Brillouin zone

This notion was developed by the French physicist Leon´ Brillouin (1889–1969). For any crystal lattice in three-dimensional physical space, there is a “dual lattice” called the reciprocal lattice, which exists in an abstract space whose three dimensions are inverse lengths. This space lends itself extremely naturally to the analysis of phenomena involving wave vectors (because their dimensions are inverse lengths). If we limit ourselves to crystals whose lattices are perfectly rectangular (as has generally been done in this book), then given a lattice whose unit cell has dimensions a × b × c, the reciprocal lattice’s unit cell will have 1 1 1 dimensions a × b × c . This cell is called the ﬁrst Brillouin zone. The various locations in the Brillouin zone — wave vectors — act as indices labeling the different Bloch states (since there is a one-to-one correspondence between wave vectors and Bloch states). Thus each point in the Brillouin zone is the natural “name” of a quantum state.