Novel phenomena in dilute electron systems in two dimensions

Abstract

For the past two decades, all two-dimensional systems of electrons were believed to be insulating in the limit of zero temperature. Recent experiments provide evidence for an unexpected transition to a conducting phase at very low electron densities. The nature of this phase is not understood and is currently the focus of intense theoretical and experimental attention.

A two-dimensional system of electrons or “holes” (a hole, or missing electron, behaves like a positively charged electron) is one in which the positions of the electrons and their motion are restricted to a plane. Physical realizations of such systems can be found in very thin films, sometimes at the surface of bulk materials, in layered “quantum well” systems such as GaAs/AlGaAs that are specifically engineered for this purpose, and in silicon metal oxide-semiconductor field-effect transistors (MOSFETs). Two-dimensional electron systems have been studied for nearly 40 years (1) and have yielded a number of important discoveries of physical phenomena that directly reflect the quantum mechanical nature of our world. These include the integer Quantum Hall Effect, which reflects the quantization of electron states by a magnetic field, and the fractional Quantum Hall Effect, which is a manifestation of the quantum mechanics of many electrons acting together in a magnetic field to yield curious effects like fractional (rather than whole) electron charges (2).

For nearly two decades it was believed that in the absence of an external magnetic field (B = 0), all two-dimensional systems of electrons are insulators in the limit of zero temperature. The true nature of the conduction was expected to be revealed only at extremely low temperatures; in materials such as highly conducting thin films, this was thought to require temperatures that are unattainably low—in the microKelvin range. Based on a scaling theory for noninteracting electrons (3), these expectations were further supported by theoretical work for weakly interacting electrons (4).

Confirmation that two-dimensional systems of electrons are insulators in zero field was provided by a beautiful series of experiments in thin metallic films (5) and silicon MOSFETs (6, 7), where the conductivity was shown to display weak logarithmic corrections leading to infinite resistivity in the limit of zero temperature. It was therefore quite surprising when recent experiments in silicon MOSFETs suggested that a transition from insulating to conducting behavior occurs with increasing electron density at a very low critical density, n_c ≈ 10¹¹ cm⁻² (8–10). These experiments were performed on unusually high quality samples, allowing measurements at considerably lower electron densities than had been possible in the past. First viewed with considerable skepticism, the finding was soon confirmed for silicon MOSFETs fabricated in other laboratories (11), and then for other materials, including p-type SiGe structures (12), p-type AlAs/AlGaAs heterostructures (13–16), and n-type AlAs (17) and GaAs/AlGaAs heterostructures (18).

It was soon realized that the low electron (and hole) densities at which these observations were made correspond to a regime where the energy of the repulsive Coulomb interactions between the electrons exceeds the Fermi energy (roughly, their kinetic energy of motion) by an order of magnitude or more. For example, at an electron density n_s = 10¹¹ cm⁻² in silicon MOSFETs, the Coulomb repulsion energy, U_c ≈ e²(π n_s)^1/2/ɛ, is about 10 meV, while the Fermi energy E_F = π n_s ℏ² /2m* is only 0.55 meV. (Here e is the electronic charge, ɛ is the dielectric constant, and m* is the effective mass of the electron). Rather than being a small perturbation, as has been generally assumed in most theoretical papers to date, interactions instead provide the dominant energy in these very dilute systems.

Experiments.

The band structure of a silicon MOSFET consisting of a thin-film metallic gate deposited on an oxide layer adjacent to lightly p-doped silicon, which serves as a source of electrons, is shown schematically in the Inset to Fig. 1a. A voltage applied between the gate and the oxide–silicon interface causes the conduction and valence bands to bend, as shown in the diagram, creating a potential minimum that traps electrons in a two-dimensional layer perpendicular to the plane of the page. The magnitude of the applied voltage determines the degree of band bending and thus the depth of the potential well, allowing continuous control of the number of electrons trapped in the two-dimensional system at the interface.

Figure 1.

(a) Resistivity as a function of electron density for the two-dimensional system of electrons in a high-mobility silicon MOSFET. The different curves correspond to different temperatures. Note that at low densities, the resistivity increases with decreasing temperatures (insulating behavior), while the reverse is true for higher densities (conducting behavior). The Inset shows a schematic diagram of the electron bands to illustrate how a two-dimensional layer is obtained (see text). (b) Resistivity as a function of temperature for the two-dimensional system of electrons in a silicon MOSFET. Different curves are for different electron densities.

For a very high-mobility (low-disorder) silicon MOSFET, the resistivity is shown at several fixed temperatures as a function of electron density in Fig. 1a. There is a well defined crossing at a “critical” electron density, n_c, below which the resistivity increases as the temperature is decreased, and above which the reverse is true. This can be seen more clearly in Fig. 1b, where the resistivity is plotted as a function of temperature for various fixed electron densities. A resistivity that increases with decreasing temperature generally signals an approach to infinite resistance at T = 0, that is, to insulating behavior; a resistivity that decreases as the temperature is lowered is characteristic of a metal if the resistivity tends to a finite value, or a superconductor or perfect conductor if the resistivity tends to zero. The crossing point of Fig. 1a thus signals a transition from insulating behavior for n_s < n_c to conducting behavior at higher densities (n_s > n_c). Similar behavior obtains in other materials at critical densities determined by material parameters such as effective masses and dielectric constants. The value of the resistivity at the transition in all systems remains on the order of h/e² ≈ 26 kΩ, the quantum unit of resistivity.

The electrons’ spins play a crucial role in these low-density materials, as demonstrated by their dramatic response to a magnetic field applied parallel to the plane of the two-dimensional system. An in-plane magnetic field couples only to the electron spins and does not affect their orbital motion. The parallel-field magnetoresistance is shown for a silicon MOSFET in Fig. 2 for electron densities spanning the critical density n_c at a temperature of 0.3 K. The resistivity increases by more than an order of magnitude with increasing field, saturating to a new value in fields above 2 or 3 Tesla (19, 20) above which the spins are presumably fully aligned. The total change in resistance is larger at lower temperatures and for higher mobility samples, exceeding two or three order of magnitude in some cases. Although first thought to be associated only with the suppression of the conducting phase, the fact that very similar magnetoresistance is found for electron densities above and below the zero-field critical density indicates that this is a more general feature of dilute two-dimensional electron systems.

Figure 2.

For different electron densities, the resistivity at 0.3 K is plotted as a function of magnetic field applied parallel to the plane of the two-dimensional system of electrons in a high-mobility silicon MOSFET. The top three curves are insulating, while the lower curves are condcuting, in the absence of a magnetic field. The response to parallel field is qualitatively the same in the two phases, varying continuously across the transition.

Some Open Questions.

Strongly interacting systems of electrons in two dimensions are currently the focus of intense interest, eliciting a spate of theoretical attempts to account for the presence and nature of the unexpected conducting phase. Most postulate esoteric new states of matter, such as a strongly paramagnetic low-density conducting phase first considered by Finkelshtein (21–24), a perfect metallic state (25), non-Fermi liquid behavior (25–27), and several types of superconductivity (28–31).

A number of relatively more mundane suggestions have been advanced that attribute the unusual behavior shown in Figs. 1 and 2 to effects that are essentially classical in nature. These include a vapor/gas separation in the electron system (32), temperature- and field-dependent filling and emptying of charge traps unavoidably introduced during device fabrication at the oxide–silicon interface (33), and temperature-dependent screening associated with such charged traps (34). Although some may strongly advocate a particular view, all would agree that no consensus has been reached.

A great deal more experimental information will be required before the behavior of these systems is understood. Information will surely be obtained in the near future from NMR, tunneling studies, optical investigations, and other techniques. One crucially important question that needs to be resolved by experiment is the ultimate fate of the resistivity in the conducting phase in the limit of zero temperature. Data in all two-dimensional systems showing the unusual metal–insulator transition indicate that, following the rapid (roughly exponential) decrease with decreasing temperature shown in Fig. 1b, the resistivity levels off to a constant, or at most weakly temperature-dependent, value. The temperature at which this leveling off occurs decreases, however, as the transition is approached (similar leveling off was seen near the magnetically induced superconductor–insulator transition in thin films, see refs. 35 and 36). The question is whether the resistivity of dilute two-dimensional systems tends to a finite value or zero in the zero-temperature limit as the transition is approached. If the resistivity remains finite, this would rule out superconductivity (28–31) or perfect conductivity (25). The question may then revert to whether localization of the electrons reasserts itself at very low temperatures, yielding an insulator as originally expected. There are well known experimental difficulties associated with cooling the electron system to the same temperature as the lattice and bath (that is, the temperature measured by the thermometer), and these experiments will require great skill, care, and patience.

An equally important issue is the magnetic response of the electron system. Superconductors expel magnetic flux and are strongly diamagnetic, while Finkelshtein’s low-density phase would give a strongly paramagnetic signal. There are very few electrons in a low-density, millimeter-sized, 100 Å-thick layer, and measurements of the magnetization will be exceedingly difficult.

In closing, we address the crucial question regarding the nature of the apparent, unexpected zero-field metal–insulator transition: do these experiments signal the presence of unanticipated phases and new phenomena in strongly interacting two-dimensional electron systems, or can the observations be explained by invoking classical effects such as recharging of traps in the oxide or temperature-dependent screening? Some recent experiments suggest the former.

A Princeton University–Weizmann Institute collaboration (37) has demonstrated that the magnetic field-induced phase transition between integer Quantum Hall Liquid and insulator (the QHE–I transition) evolves smoothly and continuously to the metal–insulator transition in zero magnetic field discussed in this paper, raising the possibility that the two transitions are closely related. This conjecture is supported by the strong similarity between the temperature dependence of the resistivity in zero magnetic field and in the Quantum Hall Liquid phase (38).

Additional insight may be provided by a comparison of the “critical” resistivity, ρ_c, at the zero-field metal–insulator transition and the critical resistivity, ρ_{QHE–I}, at the QHE–I transition measured for the same sample. Fig. 3a shows values of the zero-field critical resistivity, ρ_c, for a number of samples of different two-dimensional electron systems: ρ_c varies by an order of magnitude, between ≈10⁴ and 10⁵ Ω and exhibits no apparent systematic behavior. In contrast, Fig. 3b shows that the ratio ρ_c/ρ_{QHE–I} is close to unity when measured on the same sample for three different materials. Since the QHE–I transition is clearly a quantum phase transition, this suggests that the zero-field transition is a quantum phase transition as well. The intriguing relationship between critical resistivities for these two transitions shown for only a very few samples in Fig. 3b clearly needs further confirmation.

Figure 3.

(a) The critical resistivity, ρ_c, which separates the conducting and insulating phases in zero magnetic field is shown for several two-dimensional systems for which the transition occurs at different electron (or hole ) densities, shown along the x axis. Although the critical resistivity is of the order the quantum unit of resistivity, h/e² ≈ 26 kΩ, it varies by about a factor of 10. (b) For several materials, measurements of ρ_c and ρ_QHL−I on the same sample yield ratios ρ_c/ρ_QHE−I that are near unity. Here, ρ_c is the critical resistivity separating the conducting and insulating phases in the absence of magnetic field, and ρ_QHE−I is the critical resistivity at the transition from the Quantum Hall Liquid to the insulator in finite magnetic field. Data were obtained for p-GaAs/AlGaAs heterostructures from refs. 13–16 and 36, for n-GaAs/AlGaAs from ref. 18, and for p-SiGe from refs. 12 and unpublished data of P. T. Coleridge.

Future work will surely resolve whether, and what, exciting and unanticipated physics is required to account for the puzzling and fascinating recent observations in two dimensions.