The emergence of the integer and fractional quantum Hall (QH) effects in two-dimensional electron systems (2DES) exposed to a perpendicular magnetic field can be viewed as a magnetic field induced phase transition from a two dimensional electron gas to an incompressible liquid. Hallmarks in transport of these incompressible liquids are a quantization of the Hall conductance σxy together with a vanishing of the longitudinal conductance σxx in the limit of zero temperature. At still higher values of the magnetic field, in the quantum limit when all electrons reside in the lowest Landau level, a transition to a solid phase, the 2D Wigner crystalline state, appears. Disorder pins the crystal and breaks it up into domains. As a result, the two-dimensional electron system becomes insulating. Another type of Wigner crystal can form even in the quantum Hall state. In this case, quasi-particles in the partially filled Landau levels crystallize. This Wigner crystal is bound to get pinned as well and hence localizes electrons in the bulk. This may cause an increase of the quantum Hall plateau width.
From a thermodynamic point of view, the compressibility usually goes down when a system undergoes a transition from a gas to a liquid or from a liquid to a solid. For the transition of a 2D electron gas (a Fermi liquid) into a quantum Hall liquid, a drop of the compressibility has been confirmed. For the WC which is supposed to be the ground state of a low density 2D electronic system, the compressibility not only drops but can turn negative. Here we report anomalies in the evolution of the chemical-potential of a 2DES within the quantum Hall regime. The chemical potential of one 2DES is measured in a bilayer configuration and is determined from the density variation induced in the second nearby 2D layer. The jump of the chemical potential due to condensation in a quantum Hall liquid is interrupted by two shoulders sitting symmetrically around integer filling. Our observations are consistent with the expected thermodynamic behavior when a Wigner crystal of quasi-particles appears as one moves away from exact integer filling.
Experiments were carried out on GaAs/AlGaAs bilayer systems consisting of two 19nm GaAs quantum wells separated by a 9.6 nm AlAs/GaAs superlattice barrier. Prepatterned backgates and thermally evaporated metallic top gates are used to achieve density tuning as well as separate contacts to the individual layers. Unless otherwise stated, measurements were performed at the base temperature of a dilution refrigerator (approximately 20mK).
Fig. 1: (a) Schematic of the measurement circuit. The two layers are separately contacted. The densities are controlled by the top gate (Vt) and the backgate (Vb). (b) Illustration of the feed-back measurement. The bottom layer is used as the sensing layer. (c) Solid: Dmt as a function of Vt measured at 0.2 and 0.3T. Dotted: conductance of the top layer (arbitrary units). Curves for different magnetic fields are vertically offset. (d) Dmt and its numerical derivative with respect to nt obtained at 3.5T. Dashed: expected slope when the top layer gets fully incompressible.[less]
Fig. 1: (a) Schematic of the measurement circuit. The two layers are separately contacted. The densities are controlled by the top gate (Vt) and the backgate (Vb). (b) Illustration of the feed-back measurement. The bottom layer is used as the sensing layer. (c) Solid: Dmt as a function of Vt measured at 0.2 and 0.3T. Dotted: conductance of the top layer (arbitrary units). Curves for different magnetic fields are vertically offset. (d) Dmt and its numerical derivative with respect to nt obtained at 3.5T. Dashed: expected slope when the top layer gets fully incompressible.
The basic principle of the measurement technique is illustrated schematically in Figs. 1(a),(b). It aims at determining the chemical potential of the top layer by monitoring the density change in the bottom layer. Both layers are grounded at the central contact. The conductivity of each layer is measured separately by applying a low voltage excitation to each with a different frequency. The densities of each layer can be tuned by adjusting DC voltages to the top and bottom gates: Vt and Vb. To record the behavior of the chemical potential of the top layer, mt, as a function of the carrier density or filling factor, Vt is swept. Since the electrochemical potential of this layer remains fixed, the change in the chemical potential induced by Vt is compensated for by a change of the electrostatic potential equal in size, but opposite in sign. The top layer will therefore act as a gate and its modified electrostatic potential affects the density in the bottom layer. If the backgate voltage is tuned such that the conductivity of the bottom layer exhibits a steep slope as shown in Fig. 1 (b), the small change in the electrostatic potential of the top layer, will cause a large variation in the conductivity of the bottom layer. The required readjustement of the backgate voltage to return to the original value of the bottom layer conductivity allows extracting the change in the chemical potential of the top layer.
Figure 1 (c),(d) displays typical traces we obtained. In brief, features consistent with previous experiments are: (1) The chemical potential of the low density 2DES decreases with increasing density. This negative slope becomes larger at higher B-field. It reflects the negative compressibility due to the exchange interaction in a low-density two-dimensional electron system. (2) An effective mass of 0.063me was obtained from a linear fit to the energy gaps of even filling factor QH states at different B-fields. (3) QH states with odd filling factors show large energy gaps that reflect the exchange-enhanced g-factor.
Fig. 2: Conductance of the bottom layer (a) and its derivative (b) plotted against nt with increasing temperature at B=1.4T. Curves are vertically offset. Panels in the bottom show the corresponding conductance of the top layer. Dashed lines are guides to the eye.[less]
Fig. 2: Conductance of the bottom layer (a) and its derivative (b) plotted against nt with increasing temperature at B=1.4T. Curves are vertically offset. Panels in the bottom show the corresponding conductance of the top layer. Dashed lines are guides to the eye.
There are however also features in the traces of Figs. 1(c),(d), which were not reported previously. They are of main interest here. For magnetic fields as small as B=0.3T, the chemical potential exhibits an unexpected additional bump at the center of each conductance minimum associated with the condensation into an integer quantum Hall state. This anomaly develops further into a double hump at higher fields (lower filling factor). For instance at n=1 and 3.5T in Fig. 1(d), these two humps are marked by grey shaded areas. The numerical derivative of the chemical potential with respect to the average density of the top layer reveals a central incompressible peak at n=1 surrounded by two satellite peaks. The same features were observed also on other samples. The anomalous features mark the range where a Wigner crystal with negative compressibility exists.
Further evidence for Wigner crystallization stems from temperature dependent measurements as shown in Fig. 2. By raising the temperature up to 300mK, the anomalies become less and less distinguishable although the ν=1 QH state still shows a wide plateau. This corroborates that the anomalous features are associated with a more fragile phase such as the Wigner crystal. As the conductance plateaus shrink at higher temperatures, the humps move closer to the center of the QH state (see panel b).