The Josephson effect is a dramatic macroscopic manifestation of quantum mechanics: If you separate two superconductors by a thin insulating barrier, a current can flow across the junction with no voltage applied. Similar behavior was recently seen in a system that involves neither superconductors nor superfluids. Experimenters from Caltech and Bell Labs, Lucent Technologies, formed two parallel layers of electrons, separated them by a very small distance, and measured the tunneling conductance between them, that is, the differential change in interlayer current with voltage. 1 As the voltage dropped to zero, the researchers saw an enormous peak in the conductance—much more than could be explained by ordinary tunneling. Such strongly enhanced conductance, shown in figure 1a, suggests that electrons in the two systems form a collective system much like a superfluid.
Figure 1. Current-voltage characteristics of a quantum Hall bilayer. (a) The high peak in conductance (dI/dV) suggests that electrons behave collectively. (b) The current passes through zero very steeply, although it is not quite discontinuous, as it would be if it were a conventional Josephson junction.
Figure 1. Current-voltage characteristics of a quantum Hall bilayer. (a) The high peak in conductance (dI/dV) suggests that electrons behave collectively. (b) The current passes through zero very steeply, although it is not quite discontinuous, as it would be if it were a conventional Josephson junction.
The electron layers were actually quantum Hall states, that is, two-dimensional electron gases in a high transverse magnetic field. The charges in each layer move in quantized cyclotron orbits about the magnetic flux lines, performing what Xiao-Gang Wen of MIT describes 2 as a “quantum dance of electrons.” Over a decade ago, theorists began to realize that if two such quantum Hall states were brought close enough together, the electrons in the two layers would feel one another’s presence and join in a coordinated dance.
The tunneling of electrons is usually strongly suppressed because it’s very hard to squeeze an electron from one layer into the highly ordered pattern of the other. In the recent experiment, however, Ian Spielman and James Eisenstein of Caltech and Loren Pfeiffer and Ken West of Bell Labs found that when electrons were as close to those in the opposite layer as to those in their own layer, it would take only a tiny voltage to get a huge change in tunneling current. (The electron layers form in doped quantum wells in a gallium aluminum arsenide heterostructure. The distance between layers is fixed, but experimenters can control their effective separation with the magnetic field and gate voltages.)
Over the years, theorists have predicted that a quantum Hall bilayer would exhibit a number of exotic properties. Still, they were astonished to see the predicted collective behavior manifested in such a spectacular manner. Five groups of theorists published their take on the experiment in a single issue of Physical Review Letters. 3–7
Which layer?
Figure 2, suggested by Steven Girvin (Indiana University), depicts the bilayers at relatively large (top) and small (bottom) separations. (See Girvin’s article in Physics Today, June 2000, page 39.) In the Caltech-Bell Labs experiment, the ratio of electrons to magnetic flux quanta, that is, the filling factor v, is in each layer. At large separation, each electron is associated with two magnetic flux quanta, and the electrons in the two layers act independently of one another. When the layer separation is small, interlayer interactions become important, and the two layers now behave as if they were a single layer in a v = 1 state, having each electron paired with a single flux quantum. Every electron stays away from the flux quanta associated with the other electrons, no matter what layer those electrons are in.
Figure 2. Quantum Hall bilayers are depicted by electrons (red circles) associated with magnetic flux quanta (blue lines). (a) At large separation, each electron is associated with two flux quanta and is little influenced by electrons in the other layer. (b) At small separations, the electrons move in a collective state, each associated with a single flux quantum and each keeping its distance from the others.
Figure 2. Quantum Hall bilayers are depicted by electrons (red circles) associated with magnetic flux quanta (blue lines). (a) At large separation, each electron is associated with two flux quanta and is little influenced by electrons in the other layer. (b) At small separations, the electrons move in a collective state, each associated with a single flux quantum and each keeping its distance from the others.
This collective state lowers the energy penalty for an electron to move between layers: In effect, each electron has a space reserved for it in the other layer. Thus, it’s impossible to know in which layer an electron sits. You can describe such a system in terms of two complementary variables: the total number in the two layers (N 1 + N 2) and the imbalance between the two layers (N 1 − N 2). The first is precisely specified, and the latter becomes indeterminate.
In 1989, Herbert Fertig (University of Kentucky) described the microscopic quantum ground state for this system and predicted the existence of a zero-energy collective oscillation called a Goldstone mode. 8 In 1992, Wen and Anthony Zee (University of California, Santa Barbara) used different language to describe how the symmetry associated with conservation of N1 − N 2 (in the absence of tunneling) could be broken. 9 They predicted the giant Josephson-like peak in the tunneling conductance. (Other theorists made closely related predictions. 10 )
In the ground state of the broken N 1 − N 2 symmetry, each electron is in a linear superposition of the two layers, described by a phase angle φ. Every electron has the same value of this phase, and the symmetry is broken when the system spontaneously (but arbitrarily) picks a specific value of that phase. The smallest excitation of this ground state consists of a very slow change of the phase from one electron to the next, and that excitation is the Goldstone mode. It’s a very long wavelength excitation; the changes occur so gradually that its energy is virtually zero.
In a superfluid, the gradient of the phase is the superfluid velocity. Hence, the Goldstone mode is associated with an oscillation in the electron density of the bilayer: The electrons in the two layers move back and forth out of phase with one another.
To show that a Goldstone mode is present in the quantum Hall bilayers, one needs to measure the dispersion curve for such an excitation, that is, how the energy of the mode varies with momentum. In a recent experiment, 11 Eisenstein and his collaborators excited the Goldstone mode with a given momentum by applying a magnetic field parallel to the electron layers. The Lorentz force gives extra momentum to the electrons as they tunnel, and that momentum goes into creating the Goldstone excitation. Applying the field splits the zero-bias conductance peak, and the voltage at which the two side peaks appear gives the energy of the Goldstone mode. The dispersion curve thus determined by the Caltech-Bell Labs group was linear, as predicted, although its details are not in full agreement with calculations.
Various interpretations
Theorists have come up with different ways to picture the quantum Hall bilayer, although these pictures are all formally equivalent. Some theorists describe the combined quantum Hall state in the language of magnetism. They define a pseudospin vector that points up if the electron is in layer 1 and down if it is in layer 2. When the electrons might be in either layer, the system is described as a linear combination of the up and down vectors. The result is a vector lying in the xy plane in pseudospin space whose orientation is determined by the phase angle φ already introduced. In the pseudospin picture, the system has the symmetry of a two-dimensional ferromagnet, with all spins aligned in an arbitrary direction.
In another view, an electron in one layer is attracted to a hole in the other layer, and the pair forms an exciton that spans the junction. The polarization of each exciton is unknown because one can’t tell where the electron is. The collective state is then a condensate of these excitons, which behaves as a superfluid.
A third picture, which further develops the excitonic description, is that of the DC Josephson effect. As presented by one pair of theorists, the excitonic condensate of the quantum Hall bilayer is governed by the same equations as the Josephson junction, with φ playing the role of the phase difference between superconductors. 5 In the views of these theorists, the excitonic condensate embodies the essence of the Josephson effect: It’s a coherent system with an indeterminate number of excitations carrying charge in each layer.
Nagging questions
As tempting as it is to make a complete analogy with the Josephson effect, the Caltech-Bell Labs experiment does not yet fit in all regards. In particular, at zero voltage the experimenters observe zero current, not the finite supercurrent one expects from a Josephson junction (see figure 1(b)). According to Eisenstein, it is not yet clear whether the zero current is intrinsic to the way tunneling works at v = 1, or is caused by some unknown extrinsic experimental effect. The slope of the I−V curve near V = 0 keeps getting steeper (and the associated conductance peak taller and sharper) as he and his team improve the experiment. So, Eisenstein believes, the jury is still out on whether they have a Josephson junction.
Sankar Das Sarma of the University of Maryland, College Park, believes the evidence for a collective mode in the quantum Hall bilayer, but does not think we can claim a full understanding of it until we can reproduce its quantitative as well as qualitative features. To that end, two groups recently tried to understand in detail the observed height and width of the zero-conductance peak in the Caltech-Bell Labs experiment. 3,4 They examined the role of disorder in this system. At the same time, Das Sarma and two colleagues explored the possible ground states of the quantum Hall bilayer and found a large number of them, including some that exhibit only a quantum Hall effect or coherent electrons. 7 In yet other work, a group from Indiana University examined the evolution of the bilayer system between the extremes of large and small separation. 6
