Letter


Nature Physics 4, 936 - 939 (2008)
Published online: 5 October 2008 | doi:10.1038/nphys1094

Subject Category: Condensed-matter physics

Wigner crystallization in a quasi-three-dimensional electronic system

B. A. Piot1, Z. Jiang2,3, C. R. Dean1, L. W. Engel2, G. Gervais1, L. N. Pfeiffer4 & K. W. West4


When a strong magnetic field is applied perpendicularly (along z) to a sheet confining electrons to two dimensions (xy), highly correlated states emerge as a result of the interplay between electron–electron interactions, confinement and disorder. These so-called fractional quantum Hall liquids1 form a series of states that ultimately give way to a periodic electron solid that crystallizes at high magnetic fields. This quantum phase of electrons has been identified previously as a disorder-pinned two-dimensional Wigner crystal with broken translational symmetry in the xy plane2, 3, 4, 5, 6, 7, 8. Here, we report our discovery of a new insulating quantum phase of electrons when, in addition to a perpendicular field, a very high magnetic field is applied in a geometry parallel (y direction) to the two-dimensional electron sheet. Our data point towards this new quantum phase being an electron solid in a 'quasi-three-dimensional' configuration induced by orbital coupling with the parallel field.


The formation of an electron solid has been observed previously at very high magnetic fields where less than 1/5 of the lowest Landau level describing the orbital dynamics is occupied by electrons, or at zero magnetic field in extremely dilute two-dimensional (2D) systems realized on helium surfaces 9. Recently, a 1D Wigner crystal was also reported for electrons in carbon nanotubes 10. Here, we present evidence for another possibility where the crystallization would occur in a 'quasi-3D' electronic system, evolving continuously from a disorder-pinned 2D state. In our work, a 2D electron gas (2DEG) is rotated inside a magnetic field by a tilting angle theta, so that an in-plane magnetic field Bparallel is added parallel to the 2DEG. Provided this field be large enough, it has been proposed theoretically11 that the electron solid energy would become lower than that of the fractional quantum Hall (FQH) liquid at Landau level filling factors Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com, where ns is the electron density, significantly higher than for the conventional 2D Wigner crystal in which nuless than or similar to1/5. A tilt-induced liquid-to-insulator transition has been observed in a 2D hole system12, 13, 14 in which the insulating phase, initially located at nuless than or similar to1/3, was shifted close to nu=2/3 with increasing theta. However, such a tilt-induced transition for electrons has so far been elusive. Here, combining a relatively 'thick' high-mobility 2DEG and very high parallel magnetic fields Bparallel to enhance orbital coupling, we report a transition from a FQH liquid to an insulating phase in an electron system.

In Fig. 1, we report in a 3D plot the longitudinal resistance, Rxx, of sample C as a function of the perpendicular magnetic field for different tilting angles. In the perpendicular configuration, when theta=0°, we observe the usual FQH series characterized by a vanishing longitudinal resistance due to the opening of a many-body gap in the density of states. At higher perpendicular magnetic fields, for filling factors lower than nu=1/5, a divergence of Rxx corresponding to the onset of a deep insulating state (Rxx>800 kOmega), previously identified as a disorder-pinned Wigner Crystal, is observed6, 7, 8. A re-entrance of the insulating state (Rxxapprox400 kOmega) is also observed in a narrow region between nu=1/5 and 2/9 (ref. 15). As the tilting angle is further increased, the onset of the divergence shifts to lower perpendicular fields, signalling the appearance of an insulating phase at higher filling factors, and approaching nu=2/3 at theta=68.35°. The temperature dependence of Rxx at theta=68.35° and Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com is shown in the inset. At this magnetic field, Rxx increases exponentially with 1/T, as observed in the zero-tilt insulating phase below nu=1/5 (refs 15, 16), thus confirming the similarity of this new quantum phase with the 2D Wigner crystal phase at zero tilt. At this angle, we also observed a critical field Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com separating a metallic (partRxx/partT>0) from an insulating region (partRxx/partT<0) for which Rxx does not depend on temperature, characteristic of a metal-to-insulator transition17. We also observe FQH states (such as at nu=1/3) that remain robust for intermediate angles even though the neighbouring (higher) filling factor region becomes insulating, showing up as resistance peaks in Fig. 1. This is reminiscent of the re-entrant insulating phase observed in the pure perpendicular field case.

Figure 1: Longitudinal resistance Rxx versus tilt and filling factor nu.

Figure 1 : Longitudinal resistance Rxx versus tilt and filling factor |[nu]|.

The resistance of sample C as a function of the perpendicular magnetic field Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com (or equivalently the filling factor nu) for different tilting angles theta and Tsime35 mK. White lines are actual data, and the trend in between is extrapolated as a guide to the eye. The resistance value is emphasized by a colour map, with the black lines corresponding to equi-resistance. Inset: Temperature dependence of Rxx at theta=68.35° and Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com.

Full size image (85 KB)

To further investigate the nature of this new insulating quantum phase, we have measured the FQH gap at nu=1/3,2/5 as a function of the tilting angle theta. In Fig. 2, the longitudinal resistance Rxx of sample A at nu=1/3 is plotted on a semi-log scale as a function of the inverse temperature 1/T, for various tilting angles. Rxx clearly follows a thermally activated behaviour where Rxxproportional toe(-Delta/2kBT) over a few decades, enabling us to extract the thermally activated gap, Delta. The resulting values are reported in Fig. 3a as a function of the total magnetic field, Btotal. At high tilting angles (total magnetic field), both the 1/3 and 2/5 gaps are reduced. For the 1/3 FQH state, the weakening of the gaps cannot be accounted for by spin effects because its ground state is fully spin-polarized. Similarly, at the large Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com involved here, the nu=2/5 FQH state is most likely spin-polarized18, 19, so no spin-related effects are expected to reduce the gap. In addition, pure disorder-related effects occurring at high theta (ref. 20) cannot account for the gap reduction observed here. The calculation of Rxx at 41.96° with disorder-induced gap suppression due to an increase in Landau level broadening21 (dashed line in Fig. 2) does not qualitatively agree with the observed linear trend. The Landau level broadening used to fit the onset of the Rxx decrease is seven times larger than measured from the Shubnikov de Haas oscillations at zero tilt. This is inconsistent with the observation of well-developed FQH states at this angle, theta=41.96°.

Figure 2: Thermal activation plots at nu=1/3 for sample A.

Figure 2 : Thermal activation plots at |[nu]|=1/3 for sample A.

The longitudinal resistance Rxx as a function of 1/T for different tilting angles theta (filled symbols). All data error bars (accuracy of the resistance measurement) are smaller than the symbols. The solid lines are fits enabling the extraction of the thermally activated FQH gaps. Calculation of Rxx for (pure) disorder-induced gap suppression at 41.96° is shown with a dashed line (see the main text). Inset: Schematic representation of the sample rotation in the magnetic field.

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Figure 3: Activation energy gaps and relevant energy scales.

Figure 3 : Activation energy gaps and relevant energy scales.

a, Thermally activated gap at nu=1/3 (triangles) and nu=2/5 (circles) as a function of the total magnetic field Btotal. Sample A (filled symbols) and B (open symbols). b, Thermally activated gap normalized with respect to their theta=0° value, at nu=1/3 (triangles) and nu=2/5 (circles) as a function of the parallel magnetic field Bparallel. The dotted line is a guide to the eye. Magnetic length lBparallel associated with Bparallel (dashed line) and average thickness of the 2DEG in the z direction left fencedzright fence (solid line) (right scale). All data error bars (estimated from the goodness of the linear fit in Fig. 2) are smaller than the symbols. c, Energy diagram (see the main text). All energies are calculated with respect to the bottom of the conduction band (dashed horizontal line) and are shown to scale.

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In Fig. 3b, we plot the same thermally activated gap normalized by their value at theta=0°, as a function of the parallel magnetic field Bparallel. Interestingly, a similar collapse of the FQH states is obtained as a function of Bparallel. More strikingly, the gap reduction begins at Bparallelapprox7 T, which is very close to the parallel magnetic field Bparallelapprox9 T where the magnetic length Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com associated with Bparallel approaches the average thickness of the 2DEG in the z direction, left fencedzright fence=8.3 nm (calculated using ref. 22). This is strong evidence for the collapse of the FQH gaps to be directly related to orbital coupling with the parallel field, becoming stronger when electrons are confined to dimensions smaller than the initial 2DEG thickness. For Bparallel>10T, the electron gas now defines a 'quasi-3D' system, with the quantum well now primarily providing a sample with a finite width in the z direction.

An estimate can be obtained for the angle thetac1/3 for which the nu=1/3 gap vanishes completely by extrapolating the Bparallel trend observed in Fig. 3b. The resulting angle, thetac1/3gapapprox58plusminus5°, correlates very well with the angle at which the insulating phase appears at nu=1/3, estimated from Fig. 1 to be thetac1/3Rxxapprox61plusminus2°. This reinforces our interpretation for the collapse of the FQH gap at nu=1/3, and the related emergence of an insulating phase to be driven by orbital coupling to the parallel field.

It is instructive to consider all relevant energies as tilt is increased, and these are shown schematically in Fig. 3c. In the xy plane, the conventional quantum Hall effects and associated Landau levels are separated by the cyclotron energy Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com, represented here for the perpendicular magnetic field where the nu=1/3 FQH state is observed in sample A, Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com. The spin gap, Deltas, is shown for the N=0 Landau level, and DeltaFQH gives the magnitude of the nu=1/3 FQH gap. Along the confinement axis z, we use calculations for the confinement energy of a 40-nm-wide quantum well as given in ref. 23, and report here the first three electric sub-bands, E0, E1 and E2. The Fermi energy EF is indicated in the absence of a magnetic field, showing that all electrons are confined in the first sub-band, E0. When a sufficient parallel magnetic field is applied, magnetic levels take over the confinement levels in determining the z energy; these 'Landau-like' levels are separated by the 'parallel cyclotron energy' planckomegacparallel and they determine the energy in the zx plane24. When the parallel magnetic field is further increased, the occupation of the lowest 'Landau-like' magnetic level with a degeneracy eBparallel/h is reduced. We note, however, that we described for simplicity two independent subsystems in the xy and zx planes, whereas in reality they are naturally coupled in the x direction.

From the data in Fig. 1, we can construct a phase diagram for the 'quasi-3D' insulator at high tilt (parallel field). For this, we (arbitrarily) define a critical filling factor nuc1 corresponding to the smallest nu value for which a FQH state is observed, as well as a critical filling factor nuc2 corresponding to the largest nu value for which the resistance value exceeds h/2e2=12.91 kOmega. nuc1 can therefore be viewed as the liquid phase termination, and nuc2 as the onset of the insulating phase. The criterion for nuc2 is justified by the inset of Fig. 1, which shows the sample to be already insulating for this resistance value. In this context, the re-entrance of the insulating phase, usually observed between nu=2/9 and 1/5, is characterized by nuc2>nuc1. These critical filling factors nuc1 (circles) and nuc2 (triangles) are plotted in Fig. 4 as a function of the tilting angle theta, and at base temperature Tsime35 mK, with filled and open symbols denoting data obtained during two separate cooldowns. As the tilt angle theta is increased, a higher total magnetic field is required to achieve the perpendicular field necessary to observe the nu=1/5 FQH state and the neighbouring insulating region. This restricts our tilting range for that state to angles thetaless than or similar to42°, within the 45 T of our magnet indicated as a dashed–dotted line. However, further tilt of the sample shifts the insulating phase to higher filling factors, so that it reappears within our field range at angles theta>60°. The persistence of the nu=2/3 FQH state at theta>70° results in having a re-entrant phase (nuc2>nuc1) in this region of the phase diagram.

Figure 4: Phase diagram for the 'quasi-3D' insulator at Tsime35 mK.

Figure 4 : Phase diagram for the |[lsquo]|quasi-3D|[rsquo]| insulator at T|[sime]|35|[thinsp]|mK.

Critical filling factors nuc1 (circles) and nuc2 (triangles) (defined in the main text) as a function of the tilting angle theta. The open symbols are data extracted from Fig. 1, and the filled symbols were obtained on a separate cooldown. The Btotal=45 T line (dashed–dotted) determines the experimental observable range. Right panel: 'Parallel (total) filling factor' Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com (Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com) (defined in the main text) associated with the critical filling factor nuc2 (triangles). The vertical dashed line is the average value of Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com (Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com).

Full size image (62 KB)

The smooth crossover between the low-theta and high-theta regions of the phase diagram is evidence for a continuous distortion of the perpendicular-field-dominated 2D Wigner crystal to an electron solid stabilized in a 'quasi-3D' geometry. The steep divergence of Rxx preceding the 'quasi-3D' insulator, as well as the existence of re-entrant states, is analogous to what is observed for the 2D Wigner crystal in the perpendicular configuration. Recent numerical calculations11 have predicted the ground-state energy of the solid phase to be very close to the liquid under sufficient tilting angles, for example, theta>45° at nu=1/3. For strong parallel fields, Landau-like magnetic levels in the zx plane should be forming as the 2DEG becomes a 'quasi-3D' system. The non-zero perpendicular field Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com is nevertheless absolutely required here to prevent the electronic system from being free along the y direction and the Landau quantization being smeared out by the free kinetic energy. At high tilt angles, Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com thus provides an effective magnetic confinement in the y direction so as to recreate a pseudo-2D system along zx. In this regime, theta>60°, we can define a parallel filling factor associated with the parallel field component, Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com, describing the occupation in the first zx magnetic level. Here, we have assumed ns to remain unmodified by the field axis. The right panel of Fig. 4 shows a scatter plot for the values obtained for Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com at the onset of the insulating phase nuc2 and theta>60°, where lBparallelless than or similar to6 nm. The transitions from a FQH liquid to an insulating phase in this regime occur at an average parallel filling factor Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com, shown by a dashed vertical line. We also define a total filling factor Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com, equal to Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com (nu) in the high Bparallel (Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com) limit. Using this definition, the transitions observed here occur close to 1/5 over the whole theta range, theta=[0,70°] (right panel in Fig. 4). The values of Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com at the transitions are similar to those associated with the onset of the conventional 2D xy Wigner crystal, at nu<1/5, therefore suggesting a possible stabilization of an electron solid by the total magnetic field due to a reduction of the occupation of the quantized orbital energy level. A continuous evolution of the liquid–insulator transitions can be seen in Fig. 4, where the phase boundary clearly mimics that of the total field. The solid phase would occur for lBparallel sufficiently small relative to the quantum well width, so that electrons may acquire the freedom to minimize their mutual repulsion by adjusting their positions in the z direction. We note, however, that the exact structure of this electron solid cannot at present be conjectured and remains an open question that will be better addressed by microwave experiments probing the pinning modes of the crystal.

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Methods

The 2DEGs studied here are 40-nm-wide modulation-doped GaAs quantum wells, all from the same wafer grown by molecular beam epitaxy. They are referred to as samples A, B and C, and have densities ns=1.05, 1.06 and 1.52times1011 cm-2 and corresponding mobilities mu=12(2), 8(2) and 14(2)times106 cm2 V-1 s-1, respectively. The samples were cooled in a dilution fridge with base temperature Tsime35 mK installed inside a hybrid superconducting/resistive magnet capable of reaching a total magnetic field of 45 T. Treatment with a red-light-emitting diode was used during the cooldown. Transport measurements were carried out using a standard low-frequency lock-in technique at low excitation current, Iexcapprox2–100 nA. The sample was tilted with respect to the total magnetic field Btotal (see Fig. 2, inset) using an in situ rotation stage. The tilting angle theta was determined from the shift of the resistance minimum of well-known integer quantum Hall states, according to Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com.



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Acknowledgements

This work has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada Fund for Innovation (CFI), the Canadian Institute for Advanced Research (CIFAR), FQRNT (Québec), the A. P. Sloan Foundation (G.G.) and the NSF under DMR-03-52738 (Z.J.). We thank H. L. Stormer and D. C. Tsui for helpful discussions, and J. Hedberg, G. Jones, T. Murphy and E. Palm for technical assistance. A portion of this work was carried out at the National High Magnetic Field Laboratory, which is supported by NSF Cooperative Agreement No. DMR-0084173, by the State of Florida and by the DOE.

Received 11 April 2008; Accepted 29 August 2008; Published online 5 October 2008.

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  1. Department of Physics, McGill University, Montreal, H3A 2T8, Canada
  2. National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA
  3. Department of Physics, Columbia University, New York, New York 10027, USA
  4. Bell Labs, Alcatel-Lucent Incorporation, Murray Hill, New Jersey 07974, USA

Correspondence to: G. Gervais1 e-mail: gervais@physics.mcgill.ca

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