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Full momentum- and energy-resolved spectral function of a 2D electronic system

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Science  17 Nov 2017:
Vol. 358, Issue 6365, pp. 901-906
DOI: 10.1126/science.aam7073
  • Fig. 1 Schematics of the tunneling device and principles of the energy-momentum selection process.

    (A) The vertical tunneling device used in the experiment. Two GaAs quantum wells (QWs; width of 20 nm each) are separated by an Al0.8Ga0.2As potential barrier (6 nm). Electrons in the top (probe) QW with nearly zero planar momentum probe electronic states in the bottom (target) QW. (B) The injection of an electron packet probes only empty available states in the target layer. (C) Diagram explaining the momentum selection mechanism. The in-plane field generates a momentum boost in the tunneling process to displace the zero planar momentum into Embedded Image in the bottom QW. (D) E-k dispersions in the presence of the in-plane field. In (B) to (D), the occupied states are shown in blue and red for probe and target layers, respectively. (E) Measured spectra with B|| along the crystallographic axis of [100] and [010] of GaAs. Multiple unoccupied QW subbands are visible.

  • Fig. 2 MERTS spectra at various densities n and extraction of self-energy.

    (A to C) Tunneling spectra measured at 1.5 K for various target QW densities (fig. S7 shows a full data set). In (A), three subbands due to the nonzero width of the target well are visible. In (C), data are shown for a gate voltage at which the target well first fully depletes. (D) A tunneling spectrum simulation with a self-energy term that accounts for impurity broadening and electron-phonon interactions. (E) A measured E-k spectrum. The blue curves show the bare band dispersion εkEF expected from detailed band structure calculations (17, 18). We extract the self-energy by measuring the peak locations kpeak and half-widths Δk of MDCs (curves cut along the momentum axis at energy E). The red bars are centered at kpeak and have widths of Δk. The yellow box marks an area of the spectrum considered in Fig. 4. In (A) to (E), positions of red and blue arrows mark expected positions of LO phonon features. (F) Extracted ImΣ values at various densities. The solid (dashed) curves are for data from the first (second) subband. The red and blue dashed (nearly vertical) lines indicate expected LO phonon features calculated the same way as the red and blue arrows in (A) to (C), respectively (17). Curves are offset vertically for clarity, and the thin horizontal dashed-dotted line indicates zero for each curve.

  • Fig. 3 Evolution of E-k spectra under perpendicular magnetic fields.

    (A) Measured spectra of the 2D electron system at a density of n ~ 2.5 × 1010 cm–2 for various perpendicularly applied magnetic fields. With increasing field, Landau quantizations with the cyclotron energy spacing Embedded Image become apparent. At EEF ≈ 40 meV, the spectrum displays a level splitting in the 5-, 8-, and 10-T data. Tunneling into Landau levels (LLs) produces features resembling line segments. The features resulting from tunneling into a Landau index greater than zero appear as line segments at nonzero k that are slanted away from horizontal in the 5-, 8-, and 10-T data. The slants arise from the nonzero width of our square QWs. (B) A simple model that explains the level splitting. When the LO phonon energy matches the energy difference between the first (blue lines) and second (orange line) subband, a resonance occurs (purple lines) to produce the strongly coupled limit of an electron and a phonon—i.e., a polaron. The splitting arises because a long-lived composite state of the zeroth LL in the lowest (first) subband together with a LO phonon Embedded Image forms and resonantly interacts with the zeroth LL of the second subband Embedded Image. (C) The splitting energies (blue circles) and error bars (vertical orange lines) are plotted as a function of magnetic field. The dashed line is a curve-fit of the four data points at nonzero B (from 2 to 10 T) to Embedded Image with fixed B0 = 0.52 T, giving the fit parameter Embedded Image. (D) The splitting diminishes greatly for a spectrum measured at higher density, n ~ 1.3 × 1011 cm–2, at 8 T. Color scales in (A) and (D) are calibrated so that the same currents give rise to the same colors and intensities (see also fig. S9).

  • Fig. 4 Electron-electron interaction and plasmon scattering.

    (A) A zoom-in of the MERTS spectrum of the yellow rectangle in Fig. 2E measured at 3 K. The yellow curve shows the expected bare band dispersion, and red dots are measured peaks from fits to the data. There is a subtle, but statistically significant, kink in the red data points, indicated by the yellow arrow. (B and C) ImΣ and ReΣ from the spectra at various densities, indicated in the center. In (C), the red curves represent a slowly varying background signal obtained by curve-fitting the data away from the kink structure to a third-order polynomial, and the blue curves are curve-fits to the data with an additional Lorentzian term. The arrows in (C) point to the center of this Lorentzian. The same arrow locations are reproduced in (B), denoting a corresponding broad step feature in ImΣ. Black curves in (B) are simple moving averages to guide the eye. The thin horizontal dotted line indicates zero for each curve. (D) Electron-plasmon interaction. The orange curves represent electron dispersion, and dashed-dotted curves visualize final electron states in the event of emitting a plasmon. There exists a threshold energy Ep above which an injected electron (blue circles) can lose energy by scattering a plasmon and decay to lower-energy states that exist above EF (red stars). (E) The locations of the features in (B) and (C) and how they change as a function of density. The red circles and error bars indicate peaks and widths of the Lorentzian function used to fit the data in (C). Also plotted are theoretically expected curves following (33). The blue dashed curve is from a semiclassical calculation, the red dashed-dotted curve is from a random phase approximation (RPA), and the black solid curve is for a RPA that accounts for the nonzero well width.

Supplementary Materials

  • Full momentum- and energy-resolved spectral function of a 2D electronic system

    Joonho Jang, Heun Mo Yoo, L. N. Pfeiffer, K. W. West, K. W. Baldwin, Raymond C. Ashoori

    Materials/Methods, Supplementary Text, Tables, Figures, and/or References

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    • Materials and Methods
    • Supplementary Text
    • Figs. S1 to S9
    • References

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