Anomalous structure in the single particle spectrum of the fractional quantum Hall effect

Journal name:
Nature
Volume:
464,
Pages:
566–570
Date published:
DOI:
doi:10.1038/nature08941
Received
Accepted

The two-dimensional electron system is a powerful laboratory for investigating the physics of interacting particles. Application of a large magnetic field produces massively degenerate quantum levels known as Landau levels; within a Landau level the kinetic energy of the electrons is suppressed, and electron–electron interactions set the only energy scale1. Coulomb interactions break the degeneracy of the Landau levels and can cause the electrons to order into complex ground states. Here we observe, in the high energy single particle spectrum of this system, salient and unexpected structure that extends across a wide range of Landau level filling fractions. The structure appears only when the two-dimensional electron system is cooled to very low temperatures, indicating that it arises from delicate ground state correlations. We characterize this structure by its evolution with changing electron density and applied magnetic field, and present two possible models for understanding these observations. Some of the energies of the features agree qualitatively with what might be expected for composite fermions, which have proven effective for interpreting other experiments in this regime. At the same time, a simple model with electrons localized on ordered lattice sites also generates structure similar to that observed in the experiment. Neither of these models alone is sufficient to explain the observations across the entire range of densities measured. The discovery of this unexpected prominent structure in the single particle spectrum of an otherwise thoroughly studied system suggests that there exist core features of the two-dimensional electron system that have yet to be understood.

At a glance

Figures

  1. High field TDCS spectra show /`sash/' features: bright and dark diagonal lines across the spectrum.
    Figure 1: High field TDCS spectra show ‘sash’ features: bright and dark diagonal lines across the spectrum.

    The horizontal axis in each spectrum is the electron density in the quantum well, expressed as a filling factor ν with ν = 1 corresponding to completely filling the lowest spin-polarized Landau level. The vertical axis is energy E measured from the Fermi energy EF, with E>0 corresponding to injecting electrons into empty states in the quantum well and E<0 corresponding to ejecting electrons from filled states. Bright regions correspond to high SPDOS. a, Spectrum taken at a temperature of 100mK with a 4T perpendicular magnetic field. Features associated with the ν = 1 sash are highlighted with a blue arrow, while another sash about ν = 3 is indicated with a dashed blue arrow. b, 13.5T data, taken at 80mK. The ν = 1 sash is similarly indicated with blue arrows, while the ν = 1/2 sash about ν = 1/2 is indicated with green. Sharp downward steps in the spectrum corresponding to chemical potential jumps at filling fractions corresponding to fractional quantum Hall plateaus are indicated with yellow arrows. The contrast for positive and negative energies has been adjusted separately in this spectrum. In cf, selected regions of the spectrum in b are magnified and their contrast enhanced to ease identification of the features, indicated with arrows that match the colours in b.

  2. A  energy dependence indicates that the sashes originate with electron-electron interactions within the lowest Landau levels.
    Figure 2: A energy dependence indicates that the sashes originate with electron–electron interactions within the lowest Landau levels.

    a, Line cuts at several magnetic fields, all at 100mK, and at a variety of filling factors, showing a number of peaks in the SPDOS owing to the unexpected sashes. In b, the energy (horizontal) axis has been scaled by 1/ , collapsing the peaks associated with the ν = 1/2 and ν = 1 sashes. Residual mismatch in the peak locations is predominantly due to small mismatches in the filling fraction selected at each magnetic fields. c, Best-fit scaling factors averaged across all densities for each magnetic field, with a square-root dependence on B shown for reference. Error bars indicate the sample standard deviation for the best-fit scaling factor as a function of density. For fitting details and alternative functional dependences, see Supplementary Information.

  3. From the viewpoint of a composite quasiparticle, sweeping the density in a quantum well also sweeps the effective magnetic field.
    Figure 3: From the viewpoint of a composite quasiparticle, sweeping the density in a quantum well also sweeps the effective magnetic field.

    a, The effective field experienced by a 2CF as the electron density is swept through the first Landau level. As the effective field and particle density are swept, the filling factor of 2CF composite fermions ν* = |ν/(1-2ν)| changes (b). In c a Landau fan is shown in absolute energy, given by E = ħωc*(n+1/2) where ωc* is the 2CF cyclotron frequency and the composite fermion mass has been assumed to be constant as a function of ν. The Fermi energy shown in red: EF = ħωc*(nf+1/2), with nf the greatest integer less than ν*. The Fermi energy rises as the 2CF density increases in the quantum well. In d, the fan is shifted to place EF at E = 0, showing the fan as it is expected to appear in TDCS spectra. The 13.5T TDCS spectrum without (e) and with (f) the fan superimposed allows identification of the two ν = 1/2 sash features with a Landau level of the 2CF fan closest to the Fermi surface. Lines suggested by the fan but not observed have been included to show alignment with chemical potential jumps associated with known fractional quantum Hall states, as well as to demonstrate the origin of the asymmetry when high energy composite fermion Landau levels (lightest lines) are not observed.

  4. The [ngr] = 1 sash features are emphasized by taking an additional derivative of the data, easing comparison with simulated spectra from a model of electrons localized on a lattice.
    Figure 4: The ν = 1 sash features are emphasized by taking an additional derivative of the data, easing comparison with simulated spectra from a model of electrons localized on a lattice.

    a, The wide range of filling fractions over which the ν = 1 sash persists. Noise introduced in the image by taking the second derivative has been diminished by smoothing with a σ = 160μeV Gaussian. The onset energies of the ‘clouds’ of excitations are evenly spaced in energy, allowing a simple fan to describe all of the excitations. b, A sample spectrum from the lattice model. The axis is in units of e2/(4πεlb), roughly 16meV at 13.5T. The ‘sashes’ that move downward in energy as the density is raised originate with different types of sites that can be populated or depopulated. The origins of selected prominent features are indicated with cartoons of the relevant lattice situations in c. Each cartoon is accompanied by a replica of the spectrum for 0<ν<1 with the relevant portion highlighted. The hexagons represent lattice cells of a hexagonal lattice, with single particle states in the centre of each hexagon. Tunnelling an electron of either spin (yellow) onto a site adjacent to an occupied site (leftmost panel) may give rise to our ν = 1/2 sash, the lowest energy sash at low densities. A similar sash occurs at slightly higher densities (centre) corresponding to tunnelling an electron onto a site adjacent to two occupied sites. Tunnelling a minority spin (green) electron onto a site already occupied by a majority spin (red) electron (rightmost panel) appears to give rise to our ν = 1 sash, the highest energy sash at low densities. For reference, a breakdown of the spectrum by spin is provided in d, with the minority spins only (left), the majority spins only (centre), and the majority and minority spin states colour coded. Note that the minority carriers in the lowest Landau level are always completely spin polarized in this model if the on-site repulsion is larger than the peak-to-peak disorder amplitude (as is the case here).

Author information

Affiliations

  1. Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    • O. E. Dial &
    • R. C. Ashoori
  2. Alcatel-Lucent Bell Laboratories, Murray Hill, New Jersey 07974, USA

    • L. N. Pfeiffer &
    • K. W. West
  3. Present address: Princeton University, Princeton, New Jersey 08544, USA.

    • L. N. Pfeiffer &
    • K. W. West

Competing financial interests

The authors declare no competing financial interests.

Corresponding authors

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Author details

Supplementary information

PDF files

  1. Supplementary Information (806K)

    This file contains Supplementary Figures 1- 6 with legends, Supplementary Material 1-5 and Supplementary References.

Movies

  1. Supplementary Movie 1 (4.5M)

    This brief video shows the typical results from annealing the electron locations in the semi-classical model. Each frame shows the approximate ground state at a single density in a manner similar to that of supplemental figure 5d; the grey hexagons are empty lattice sites, shown using their Wigner-Seitz cells. Red sites are singly occupied, while yellow sites are doubly occupied. The solution is periodic with a 20 site period in each lattice direction. A single period is highlighted for clarity, and repeated in darker colours to demonstrate how the boundary conditions are met. For this calculation, the disorder was 0.2% of the Coulomb energy, and the setback was 2 magnetic lengths.

Additional data