Interaction-driven quantum Hall wedding cake–like structures in graphene quantum dots

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Science  24 Aug 2018:
Vol. 361, Issue 6404, pp. 789-794
DOI: 10.1126/science.aar2014

An electronic wedding cake

In nanostructures such as quantum dots, spatial confinement forces electrons to assume discrete energy levels. Quantization can also occur in an external magnetic field, where electrons' energies group into so-called Landau levels (LLs). Gutiérrez et al. explored the interplay between these two mechanisms and electronic interactions in a circulator resonator made of graphene. As an external magnetic field was increased, the electron quantum states transformed from atomic-like states to LL-like states. Electronic interactions caused a characteristic wedding cake–like shape of electronic density at high fields.

Science, this issue p. 789


Quantum-relativistic matter is ubiquitous in nature; however, it is notoriously difficult to probe. The ease with which external electric and magnetic fields can be introduced in graphene opens a door to creating a tabletop prototype of strongly confined relativistic matter. Here, through a detailed spectroscopic mapping, we directly visualize the interplay between spatial and magnetic confinement in a circular graphene resonator as atomic-like shell states condense into Landau levels. We directly observe the development of a “wedding cake”–like structure of concentric regions of compressible-incompressible quantum Hall states, a signature of electron interactions in the system. Solid-state experiments can, therefore, yield insights into the behavior of quantum-relativistic matter under extreme conditions.

Energy quantization owing to quantum confinement takes place when the particle’s de Broglie wavelength becomes comparable to the system’s length scale. Confinement can arise through spatial constraints imposed by electric fields or through cyclotron motion induced by magnetic fields. Together, confinement and quantization strengthen the effects of electron-electron interactions, providing a setting to probe a range of exotic phenomena in strongly correlated quantum systems. In the solid-state setting, different types of confined strongly correlated states and the transitions between them have been studied in quantum dots (QDs) in the presence of external magnetic fields (1). Evolution from atomic-like shell structure to magnetic quantization in QDs was first probed by using Coulomb blockade spectroscopy (24). For QDs at large magnetic fields—that is, in the quantum Hall regime—it is expected that Coulomb interactions and the redistribution of carriers between Landau levels (LLs) will lead to a characteristic “wedding cake”–like shape in the density of electronic states (Fig. 1E) (48). Although similar structures have been observed in ultracold atoms undergoing transition from the superfluid to Mott insulator (9, 10), the wedding cake–like structure has not been mapped spatially in a solid-state system.

Fig. 1 Schematic evolution of states with magnetic field in a graphene QD.

(A) The device geometry for the graphene QD resonator with a p-doped center inside an n-doped background used in the current experiment. Vb, sample bias voltage; Vg, gate voltage; VB=0(r), QD potential at B = 0. (B to E) Top panels show schematic of the potential profile (gray surface) and corresponding wave function density (orange surfaces), and bottom panels shows semiclassical orbits, as a function of applied magnetic field. Confined states start out as quasibound QD states [(B) and (C)] and condense into LLs (D) with increasing field. The corresponding screened potential (E) develops a wedding cake–like appearance through electron interactions. Semiclassical orbits start out as expected for a central force potential and then develop into cyclotron motion drifting along equipotential lines forming compressible (blue) and incompressible (yellow) density rings, as shown in (E). ψnm, electron wave functions with n and m radial and azimuthal quantum numbers, respectively; VB(r), effective potential, Eq. 3.

Graphene offers an ideal platform for this inquiry, as it hosts a fully exposed two-dimensional (2D) electron gas amenable to local probes (1117). Graphene circular p-n junction resonators (1822) with built-in local potentials (Fig. 1A) are particularly well suited to the present study; they circumvent the problems of edge roughness and edge impurities encountered in lithographically fabricated graphene QDs. Furthermore, they enable fine control of the confining potential as well as QD doping by means of local gate potentials, offering an opportunity to directly visualize the transition of electron states from spatial to magnetic confinement (Fig. 1, B to E).

In the absence of a magnetic field, confinement of graphene carriers in a p-n junction resonator gives rise to a series of quasibound single-particle states. These states result from oblique Klein scattering at the p-n interface (1822). At the same time, Klein tunneling, although present, is very weak for oblique scattering angles and thus has little impact on confinement (23). Analogous to atomic physics, the many-electron shell-like states are characterized by radial and azimuthal quantum numbers (n, m), forming a ladder of states within the spatially confined potential (Fig. 1B) (20, 22). In weak magnetic fields, energy levels exhibit abnormally large splittings, corresponding to time-reversed ±m states in the presence of the π Berry phase(22, 24). At higher fields, the system enters the quantum Hall regime, with confined states transitioning to highly degenerate LLs (Fig. 1, C and D). A signature of this transition is the formation in the electron density of wedding cake–like structures comprised of a series of compressible and incompressible electron liquid rings (Fig. 1E) (48). Extending the single-particle description to include Coulomb interactions is essential in this regime.

Our experiment involves spectroscopic mapping of a graphene QD by tunneling measurements. The QD is formed by ionized impurities in the hexagonal boron nitride (hBN) insulating layer acting as a fixed built-in confining potential [see Fig. 1A and (22, 25) for device fabrication]. The transition from spatial to magnetic confinement occurs when the magnetic length becomes smaller than the confining potential width. By following the bright envelope in the spectral map in Fig. 2A at zero field, we can model the screened confining potential, as seen by electrons, by Embedded Image, with the potential amplitude U0 ≈ 210 meV, radius R0 ≈ 104 nm, and the background value U = −55 meV, where r is the distance. This defines a characteristic length scale for the confining potential Embedded Image. Here is Plank’s constant h divided by 2π, e is the elementary charge, and vF ≈ 106 m/s is the graphene Fermi velocity. Such a potential gives rise to quasibound states with energy splitting ΔEℏvF/lV ≈ 20 meV. Application of a magnetic field B tends to confine electrons in a region of size Embedded Image and leads to the characteristic Landau quantization in graphene, Embedded Image, with the LL number N = 0, ±1, ±2, …. Each LL is highly degenerate and can host Embedded Image carriers per unit area, where g = 4 is the valley and spin degeneracy. Therefore, we expect a transition from atomic-like QD states to LL states occurring at lBlV, which gives a characteristic critical field of B ≈ 0.6 T.

Fig. 2 Visualization of the condensation of states from spatial to magnetic quantization.

(A to I) Experimental differential conductance [temperature T = 4.3 K], g(Vb, Vg, r, B), maps the local density of states as a function of applied magnetic field, from 0 to 3.5 T, showing manifolds of spatially confined QD states condensing into LLs at higher fields. The magnetic field and corresponding magnetic length lB is indicated at the bottom of each map. The 2D maps are radially averaged from a 2D grid of spectra. A smooth background was subtracted to remove the graphene dispersive background (25). The yellow arrows in (B) indicate the splitting of the m = ±1/2 degeneracy at r = 0 owing to the turn on of a π Berry phase. The blue arrows in (D) indicate the shell-like states merging into the N = 0 LL edge mode. (J) Energy positions of the n, m = 1/2 states (symbols) obtained from the maps in (A) to (I) at r = 0 are observed to evolve into separate LLs with increasing applied magnetic field. LL(0) splits into two peaks above 2.5 T, indicated by the open and solid square symbols. The experimental uncertainty, determined from fitting the peak positions in the spectra, represents one standard deviation and is smaller than the symbol size.

The transition results in an intricate evolution of QD states from spatial to magnetic confinement with increasing magnetic field, which is displayed in Fig. 2. The measured differential conductance signal, proportional to the local density of states (LDOS), shows the evolution of the QD states in the energy versus radial plane that cuts through the diameter of the QD. The zero-field shell-like QD states are well resolved in Fig. 2A under the bright concave band, which tracks the confining potential. States with m = ±1/2 have the largest weight in the center at r = 0, whereas states with a common radial quantum number n have a large weight in the form of arcs following the concave potential outline. The first critical field is reached by 0.25 T, where the ±m degeneracy is lifted owing to a π shift in the Berry phase of the QD wave function (22, 24), as seen by the doubling of the antinodes at r = 0 (arrows in Fig. 2B). The onset of the transition into the quantum Hall regime can be observed at B = 0.5 T (Fig. 2C), in agreement with the estimate above. Here states in the center of the resonator start to flatten out, have increased intensity, and shift lower in energy. Beginning at B = 1 T, various interior resonator states (arrows in Fig. 2D) merge into the N = 0 LL [LL(0)]. With progressively higher fields, the number of QD resonances decreases as they condense into the flattened central states, forming a series of highly degenerate LLs (Fig. 2, F to I). Beginning at about 2 T (Fig. 2F), LL(0) develops kinks near the QD boundary and an additional concave cusp near the center. Below, we argue that these are related to electron interaction effects. Additionally, LL(0) develops a splitting, which increases with field, whereas LL(−1) continually moves down in energy. In this field range, a decrease in conductance over a small energy range is also observed at the Fermi level, which we attribute to a Coulomb pseudogap (26, 27).

We now discuss the spatial pattern associated with the eigenstate evolution observed in Fig. 2. Experimental spatial maps of the differential conductance, corresponding to resonator LDOS wave function probability amplitudes, are obtained by taking a 2D slice in the x-y plane of the dataset in Fig. 2 at a specific energy (Fig. 3). Only a subset of the data is shown in Fig. 3, corresponding to specific energies of the prominent central states at r = 0 in Fig. 2, with increasing magnetic field for each column of maps (a complete view of the dataset can be seen in movie S1). The spatial extent of the m = ±1/2 states at the selected energies is observed at zero field in the first column of the maps in Fig. 3. As we increase field and progress from left to right in the figure, we observe the formation of rings that contract and, simultaneously, sharpen with increasing field. For higher-n states (progressing down in a column), more rings are seen. Some of these rings originate from the quasibound resonances that have not yet developed into LLs, and some reflect the presence of magnetic confinement. The former can have a relatively narrow spatial profile if they are dispersing up or down in energy. At a still-higher field (Fig. 3D), LL plateaus are formed, as seen in Fig. 2, and show up as bright rings in the spatial maps, as indicated by the arrows pointing to LL(+1) and the valley-split (28) LL(0+), LL(0−) state in Fig. 3D. When LL states cross, or are pinned at the Fermi level, they form compressible (metallic) rings and disks, which start to show Coulomb charging effects (13), as indicated by the fine quartet of rings in the center and outside edge of Fig. 3D (see also vertically dispersing lines in Fig. 4E and rings in Fig. 4G).

Fig. 3 Differential conductance spatial maps of QD states versus magnetic field.

(A to D) Differential conductance maps (T = 4.3 K) in the x-y plane, g(Vb, Vg, r, B), at a specific energy and at fixed magnetic fields from 0 to 4 T, as indicated at the top of each panel along with the magnetic length bar. The maps in each panel are at energies corresponding to prominent QD states observed in Fig. 2, at the sample bias voltages indicated at the top of each map. With increased magnetic field, various circular rings appear to get narrower, reflecting the drift states schematically indicated in Fig. 1, B to E. Charging of compressible rings develops at larger fields in (D), evidenced by the quartets of fine rings in the center and outside edge (see Fig. 4E for the corresponding radial map). A smooth background was subtracted from each differential conductance versus Vb spectra in the 2D grid to remove the graphene dispersive background (25).

Fig. 4 Electron interactions and the wedding cake–like structure.

(A) The effective potential, Eq. 3, for several magnetic field values (solid lines) and the VB=0(r) fit based on Fig. 2A (dashed line). (B) Carrier density and (C) LLs at B = 4 T as predicted by the model from Eq. 1. Screening produces compressible regions, where LLs are flat and pinned at the Fermi level, separated by incompressible regions (marked in gray). The size of incompressible regions is estimated in Eq. 4. The dashed line in (B) describes charge density in the compressible limit obtained by excluding the kinetic term K[n(r)] from the functional in Eq. 1. (D) LDOS map calculated with the screened potential from (A). A Fermi velocity of 1.2 × 106 m/s was used in the calculations to match the LL positions between theory and experiment at B = 4 T. (E) Experimental differential conductance map (T = 4.3 K), g(Vb, Vg, r, B), as a function of Vb and r at B = 4 T, showing the wedding cake–like structure in the LLs in the QD. (F) LDOS simulated by using the potential at B = 3 T and VB=0(r) from (A) (see text for discussion). (G) An x-y slice of the g(Vb, Vg, r, B) map in (E) at Vb = 6 mV (near the Fermi level), showing the inner compressible disk from LL(−1) and the outer compressible ring from LL(0), as schematically indicated in Fig. 1E. The solid yellow lines show where LL(0) and LL(−1) cut through the Fermi level, creating the compressible rings.

A notable and unexpected feature observed in Fig. 3 is the appearance of circular nodal patterns in the spatial maps of differential conductance, which are present even at zero field. The origin of these nodal patterns is not clear at present, but they can be attributed either to interactions, as they resemble the shell-like structure predicted for Wigner crystals (7, 2931), or to deviations from a rotationally symmetric confining potential. Deviations from perfect symmetry will partially lift the m-state degeneracy and give rise to nodal patterns. Moiré superlattice effects can be ruled out as an origin of the potential asymmetry because of the large angular mismatch (≈29°) between the graphene and hBN insulator for this device, which gives a superlattice period of ≈0.5 nm, much smaller than the nodal-separation length scales (22). A nonsymmetric potential can result from the shape of the probe tip, which gets imprinted in the QD potential shape from the electric field generated during the tip voltage pulse.

As a simple theoretical model, we use the edge-state picture of the quantum Hall effect. In a circular geometry, it yields a system of compressible and incompressible rings formed in the electron liquid (Fig. 1D) owing to the interplay between LLs and electron interactions (4). In our measurement, interaction effects are already observed at low fields, signaled by the shifting and flattening of the LL states in Fig. 2. A minimal model incorporating interactions at low fields is the energy functional (4)Embedded Image(1)where n(r) is charge density at position r. This functional describes the competition between carriers’ kinetic energy K[n(r)] and the effective potential VB(r) which includes the effects of screening. We approximate these quantities asEmbedded Image(2)Embedded Image(3)where ν = n/nLL is the filling fraction. These relations are valid in the limit Embedded Image. Here N = 0, ±1, ±2, … is the LL number, Vext is the electrostatic potential defining the dot, Embedded Image is the Coulomb interaction and Embedded Image is the screened electron charge [see (25) for details].

The calculated effective potential VB(r) is shown in Fig. 4A for a few magnetic field values. The joint effect of the magnetic field and interactions is the creation of a series of plateaus that form a wedding cake–like pattern of concentric rings within the dot. At the same time, the potential VB(r) is reduced compared to VB=0(r). Notably, this causes LL(0) to move toward the Fermi level in agreement with the energy dependence of LL(0) in Fig. 2. The extra concave features in the potential in the central region match those in the experimental maps of LL(0) in Fig. 2, F to I. The effect of interactions on the LLs is shown by comparing the LDOS with and without interactions in the left and right panels of Fig. 4F. Before interactions are turned on (right panel), the LLs seen through the LDOS essentially track the potential VB0(r). After including interactions (left panel), the evolution of the LDOS mimics that of potential VB(r): LLs shift to lower energy and flatten in the central region, in agreement with the evolution seen in the measurements (Fig. 2).

The incompressible and compressible rings become considerably clearer in higher magnetic field. The experimental spectral map in Fig. 4E shows the LLs becoming flat in the central region of the QD, even though the bare external electrostatic potential is concave (see QD outline in Fig. 2A), and then they progress sharply to a new energy level as new LLs become occupied, forming a wedding cake–like structure. Here LL(N), where N = −5 to 2, can be observed as plateaus in the center of the QD (Fig. 4E). Both LL(0) and LL(−1) cross the Fermi level at zero bias, as indicated by the yellow lines, forming a LL(−1) compressible disk in the center and an outer LL(0) compressible ring separated by an incompressible ring, as shown in the Fermi-level spatial map in the x-y plane (Fig. 4G). We observe Coulomb charging of these LLs as charging lines intersecting the LLs at the Fermi level and progressing upward at sharp angles in Fig. 4E. These lines correspond to a quartet of rings in Figs. 3D and 4G. The charging of the compressible regions occurs in groups of four, reflecting the fourfold (spin and valley) graphene degeneracy (13).

To understand these observations, we use a two-stage approach. We first use the mean field functional Eq. 1 to find LL occupancies and determine the screened potential VB(r) (25). We then use this potential to calculate the density of microscopic states, which can be directly compared to the measurements. The features seen in the measured LDOS can be understood by comparing them to a simple calculation of the LLs (25), shown in Fig. 4C. The highest LL that is partially filled can be obtained by counting the number of LLs that need to be populated to accommodate the carrier density equal to that in the fully compressible regime (dashed line in Fig. 4B). In the simulated LDOS map in Fig. 4D, we can identify the LL states, which track the screened potential VB(r) pictured in Fig. 4A. These states exhibit plateaus, as expected from theory (4, 5). This behavior is in good agreement with the experimental results shown in Fig. 4E.

The width of the observed incompressible ring can be estimated from the functional in Eq. 1 following the approach of (5) and yields the strip width (25),Embedded Image(4)The estimate in Eq. 4 is slightly greater than the width inferred from our measurement results shown in Fig. 4E. The discrepancy can partly be attributed to the result of (5), derived for LL spacing ΔεLL much smaller than the external potential, being used in the regime when ΔεLL is not small on the VB(r) scale.

Fingerprints of electron-electron interactions that are as clear and notable as the observed electronic wedding cake–like patterns are relatively rare in solid-state experiments. The measurements reported here suggest, as hinted by the charging lines and nodal patterns in the differential conductance maps, that even more exotic signatures of electronic interactions may be within experimental reach in future scanned-probe measurements at lower temperatures.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 and S2

References (3240)

Movie S1

Data S1

References and Notes

  1. Additional supplementary text and data are available in the supplementary materials.
  2. The uncertainty represents one standard deviation determined from a linear least-squares fit to the data.
Acknowledgments: We thank S. Blankenship and A. Band for their contributions to this project and M. Stiles for valuable discussions. Funding: C.G., D.W., and F.G. acknowledge support from the Cooperative Research Agreement between the University of Maryland and the National Institute of Standards and Technology Center for Nanoscale Science and Technology, grant no. 70NANB10H193, through the University of Maryland. C.L. and L.S.L. acknowledge support from the STC Center for Integrated Quantum Materials (CIQM) under NSF award 1231319. L.S.L. also acknowledges support from the U.S. Army Research Laboratory and the U.S. Army Research Office through the Institute for Soldier Nanotechnologies, under contract number W911NF-13-D-0001. J.F.R.-N. acknowledges support from the Harvard-MIT Center for Ultracold Atoms, NSF grant no. DMR-1308435. K.W. and T.T. acknowledge support for the growth of hexagonal boron nitride crystals from the Elemental Strategy Initiative conducted by the MEXT, Japan, and the CREST (JPMJCR15F3), JST. Author contributions: C.G., D.W., and F.G. performed the experiments. C.G. led the STM-STS data analysis. F.G. designed and fabricated the graphene device. J.A.S. designed and built the apparatus and conceived of the study. C.L., J.F.R.-N., and L.S.L. performed the simulations. K.W. and T.T. grew the hBN crystals used in the graphene device. All authors contributed to writing the manuscript. Competing interests: None declared. Data and materials availability: All data are available in the manuscript and supplementary materials.

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