———————————————————————————————

A Different Kind of Quantization: The Quantum

The 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 , 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 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(ωt + δ), sin(ωt + δ)) (3)

eB Here, ω = 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 = Rω are proportional to each other, with constant of proportionality ω. 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 = ω(n + ), ω = (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 ω = 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 ~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.

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. 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 given below. Then, for the next two chapters, what will occupy

7 FIG. 5: On the left side, the so-called “Landau fan” plots Landau levels, labeled by their quantum numbers n = 0, 1, 2..., as a function of magnetic field strength. The zigzagging red line shows how the Fermi energy varies with field strength for a fixed electron density. The staircase on the right side shows the quantum Hall states whose quantum numbers N = 1, 2, 3, ... correspond to the Fermi energy lying in the first, second, third (etc.) gap between Landau levels. us is the deep topological basis underlying the quantization of this strange type of conductivity in the presence of a periodic lattice.

C. Semiclassical Picture

The circularity of the Landau orbits followed by electrons in a two-dimensional electron gas immersed in a magnetic field shows why a sample with such a gas will be an insulator: there is no net flow in any direction, since a circle is a closed loop. However, as is explained below, this insulator, although it does not conduct current in its interior, does conduct current along its edges (although only in one direction along each edge). This effect takes place for purely topological reasons, independent of the details of the sample’s geometry. In any high-precision measurements, such as those involving the quantum Hall effect, a

8 natural question arises about the consequences of the finite size of the sample — specifically, effects that take place at the sample’s boundaries. To describe such effects, we focus on electrons confined inside a slab of finite width. Near the edges of such a sample, the confining potential well produces an upward bending of the Landau levels (see Figure 6). Wherever a bent Landau level intercepts the Fermi energy, a one-dimensional “edge channel”, or “skipping orbit”, is formed. Classically, this can be envisioned as shown in Figure 7 — namely, as an electron bouncing along the edge of the sample in a succession of semicircles. Soon after the discovery of the quantum Hall effect, the importance of these edge channels in the transport properties of the two-dimensional gas was recognized, and several edge-related theories were then developed, based on different approaches, and there is now ample experimental evidence that the current in a Hall bar is indeed flowing very close to the sample’s edges.

The edge of a quantum Hall system is a one-way street for electrons.

FIG. 6: Schematic energy-level diagram for a two-dimensional electron gas in a magnetic field with an infinite confining potential at the edges of the sample, as was shown in Figure 3. The normally flat Landau levels (see Figure 4) bend upwards as they approach the walls of the potential well. States below the Fermi energy EF are occupied (solid blue circles). The edge channels are located where the Fermi energy cuts across the Landau levels.

As is shown in Figure 7, although electrons in the bulk of the sample cannot move, the electrons along the sample’s edge can and do move. An electron near the edge of the sample

9 FIG. 7: In the middle of the lefthand panel is shown a circular orbit (brown) of an electron inside a two-dimensional electron gas in a thin sample, as well as two bouncing trajectories (red) of electrons at the sample’s two edges. Such classical trajectories, called “skipping orbits”, consist of a sequence of successive semicircles. The skipping orbits on both edges of the sample carry current; however, since the two edges are located on opposite sides of the sample, the currents they carry flow in opposite directions. By contrast, the circular orbits inside the sample carry no current. In the righthand panel we see that the edge states not only exhibit “one-way traffic flow” but are also unaffected by the presence of impurities — the electrons simply go around any impurity, never getting reflected backward. doesn’t have enough room to complete its classical circular orbit. Instead, it will hit the boundary of the sample, be reflected by it, will start another circle and will hit the boundary again, and so forth. Since the electron is continually moving in one direction, we can think of the sample’s edge as a one-dimensional “wire”. However, this wire is very different from an ordinary one-dimensional wire. In this wire, the electron can move only in one direction, whereas in a normal one-dimensional wire, electrons can move in either direction. This one-way current is very important when impurities are taken into consideration. In an ordinary one-dimensional wire, electrons will be reflected back by an impurity, and such reflections are the main reason that there is resistance at low temperatures. In the case of a one-way edge state, however, electrons cannot be reflected by an impurity, since they cannot move in the “wrong” direction. Therefore, whenever an electron encounters an impurity, it simply goes around it and keeps moving forward. The fact that no reflections take place implies that impurities do not disrupt the transport resulting from the edge states. This is why the quantization of the Hall

10 conductivity is so precise and so independent of the number of impurities. To summarize this semiclassical explanation of the quantum Hall effect, an energy gap results from the quantization of the closed circular orbits that electrons follow in a sample sitting in a magnetic field. The interior of a quantum Hall sample is thus inert, like an insulator. However, at the sample’s boundary, a different type of motion occurs, which allows charge to flow in one-dimensional edge states, though only in one direction at each edge. This unidirectionality makes those states insensitive to scattering from impurities, and this explains why the Hall resistance is so precisely quantized. Remarks

There are special conditions under which the Hall conductance exhibits plateaus at certain fractional values. This effect — the fractional quantum Hall effect — cannot be explained using the framework of a non-interacting two-dimensional electron gas, since electron–electron interactions are key in producing the fractional plateaus. The topic is therefore beyond the scope of this book.

Appendix A: Excerpts from the 1985 Nobel Prize Press Release [1]

Summary

When an electric current passes through a metal strip there is normally no difference in potential across the strip, if measured perpendicularly to the current. If, however, a magnetic field is applied perpendicularly to the plane of the strip, the electrons are deflected towards one edge and a potential difference is created across the strip. This phenomenon, termed the Hall effect, was discovered more than a hundred years ago by the American physicist E. H. Hall. In common metals and , the effect has now been thoroughly studied and is well understood. Entirely new phenomena appear when the Hall effect is studied in two-dimensional electron systems, in which the electrons are forced to move in an extremely thin surface layer between, for example, a metal and a semiconductor. Two-dimensional systems do not occur naturally, but, using advanced technology and production techniques developed within semiconductor electronics, it has become possible to produce them. For the last ten years there has been reason to suspect that, in two-dimensional systems,

11 what is called Hall conductivity does not vary evenly, but changes “step-wise” when the applied magnetic field is changed. The steps should appear at conductivity values representing an integral number multiplied by a natural constant of fundamental physical importance. The conductivity is then said to be quantized. It was not expected, however, that the quantization rule would apply with a high accuracy. It therefore came as a great surprise when in the spring of 1980 von Klitzing showed experimentally that the Hall conductivity exhibits step-like plateaus which follow this rule with exceptionally high accuracy, deviating from an integral number by less than 0.0000001. Von Klitzing has, through his experiment, shown that the quantized Hall effect has fundamental implications for physics. His discovery has opened up a new research field of great importance and relevance. Because of the extremely high precision in the quantized Hall effect, it may be used as a standard of electrical resistance. Secondly, it affords a new possibility of measuring the earlier-mentioned constant, which is of great importance in, for example, the fields of atomic and particle physics. These two possibilities in measurement technique are of the greatest importance, and have been studied in many laboratories all over the world during the five years since von Klitzing’s experiment. Of equally great interest is that we are dealing here with a new phenomenon in quantum physics, and one whose characteristics are still only partially understood.

Appendix B: Quantum Mechanics of Electrons in a Magnetic Field

As was stated in Chapter 6, 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 = (px, py) (these are “c-numbers”, in Dirac’s terminology) by a pair of “q-number” operators, as follows:

∂ p → −i x ~∂x ∂ p → −i − eBx y ~∂y

(Here we are using the magnetic vector potential A~ = B(0, x, 0), called the “Landau gauge”.) The resulting Schrodinger¨ equation is:

12 1 ∂ ∂ [(−i )2 + (−i − eBx)2]ψ(x, y) = Eψ(x, y) (B1) 2m ~∂x ~∂y

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)eikyy. This leads to the following equation for f(x):

2 d2f 1 − ~ + (eBx − k )2 = Ef(x) (B2) 2m dx2 2m ~ y

This equation is identical to the Schrodinger¨ equation for a one-dimensional harmonic oscillator, whose eigenfunctions and eigenvalues are:

k ψ(x, y) = φ (x − ~ y ) eikyy ; E = ω (n + 1/2) (B3) n eB n ~ c

eB where ωc equals the cyclotron frequency m , and where the functions φn are harmonic-oscillator wave functions labeled by the same quantum number n that labels the discrete energy levels — that is, the Landau levels. Note that the eigenvalues of this system, unlike its eigenfunctions,

are independent of ky. This results in degenerate levels whose degree of degeneracy is equal to

ν = BA/φ0 (A being the area of the sample), as was mentioned earlier in this chapter.

Appendix C: Quantization of the Hall Conductivity

We start with the expression for σxy, the classical Hall conductance [2]. Here, Q is the total electrical charge of the system, and A is its area.

Q σ = (C1) xy BA

To obtain a quantum expression for Hall conductivity [5], we consider a system with N filled

Landau levels. (This means there is no longitudinal current, so the longitudinal conductance σxx is equal to zero.) As each Landau level is ν-fold degenerate, Q = Nνe = NBAe2/h. (Here we

have used equation (8), which gives ν = BA/Φ0 = BAe/h.) This gives us the desired result:

13 e2 σ = N (C2) xy h

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. That will be done in the following chapters.

[1]

[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, Seminaire´ Poincare,´ (2004), page 1. [3] D. Langbein, Physical Review 180, 633 (1969). [4] Introductory papers: (a) Kane and Mele, Science, Vol. 314 (2006), page 1692; (b) Xiao-Liang and Shou-Cheng Zhang, Physics Today, page 34 (2010). [5] “Topology and Quantum Hall Effect”, Chapter 9 in Field Theories of Condensed Matter Systems by Eduardo Fradkin, Addison-Wesley.

14