Coulomb-interaction induced coupling of Landau levels in intrinsic and modulation-doped quantum wells

We have performed two-dimensional Fourier transform spectroscopy on intrinsic and modulation doped quantum wells in external magnetic fields up to 10 T. In the undoped sample, the strong Coulomb interactions and the increasing separations of the electron and hole charge distributions with increasing magnetic fields lead to a nontrivial in-plane dispersion of the magneto-excitons. Thus, the discrete and degenerate Landau levels are coupled to a continuum. The signature of this continuum is the emergence of elongated spectral line shapes at the Landau level energies, which are exposed by the multidimensional nature of our technique. Surprisingly, the elongation of the peaks is completely absent in the lowest Landau level spectra obtained from the modulation doped quantum well at high fields.

Figure 1

Schematic of the experimental setup: The four phase-stabilized laser beams are provided by the MONSTR instrument [49, 50]. Three beams labeled ${\mathrm{A}}^{*}$, B, and C are used to generate the FWM signal, where ${\mathrm{A}}^{*}$ corresponds to the phase conjugate beam. The beams are aligned in the three corners of a square. The FWM signal generated at the sample propagates along the missing corner (direction $\overrightarrow{-k_a}+\overrightarrow{k_b}+\overrightarrow{k_c}$). A fourth beam labeled “Ref.” is used to trace the FWM and as the local oscillator for heterodyne detection. The samples are kept at 1.6 K inside the magneto-optical cryostat. The magnetic fields up to 10 T are applied perpendicular to the sample surface in Faraday geometry. The FWM signal is heterodyne detected and dispersed into a spectrometer. The Fourier transformed spectral interferograms lead to the 2DFT spectra. (Sample ID: VA0607.)

Figure 2

Experimental $S_I$ 2DFT spectra of the undoped GaAs quantum well at $(H,H,H,H)$ linear horizontal polarizations under high magnetic fields. The linear horizontal $(H,H,H,H)$ polarizations correspond to ${\mathrm{A}}^{*}$, B, C, and detection, respectively. The spectrally resolved FWM (blue line) and the absorbance (black line) are shown above the experimental spectra. Upper row: 2DFT spectra of the Landau levels from the undoped GaAs quantum well at magnetic fields of (a) 6 T and (b) 10 T. Bottom row: 2DFT spectra of the excitonic region from the undoped GaAs quantum well at magnetic fields of (c) 6 T and (d) 10 T.

Figure 3

(a) Absorbance of the modulation doped GaAs quantum well showing the two lowest Landau levels LL0 and LL1 at different magnetic field strengths from 0 to 10 T. Both Landau levels shift linearly with temperature, marked by the dashed lines. (b) Polarization dependent absorption spectra of the LL1 at 6.5 T. (c) Polarization dependent absorption spectra of the LL0 at 10 T. (d) Time-integrated FWM of the LL1 at 4.5 T (black squares) and LL0 at 10 T (blue squares). The red lines are the exponential fittings.

Figure 4

Left: Experimental $S_I$ 2DFT spectra using (${\sigma }^{+}{\sigma }^{+}{\sigma }^{+}{\sigma }^{+}$) polarizations of LL1 at three different magnetic fields (4, 4.5, and 5.5 T). Right: Experimental $S_I$ 2DFT spectra using (${\sigma }^{+}{\sigma }^{+}{\sigma }^{+}{\sigma }^{+}$) polarizations of LL0 at three different magnetic fields (8, 9, and 10 T).

Figure 5

Theoretical $S_I$ 2DFT spectra of the undoped GaAs quantum well under high magnetic fields, calculated using time-dependent DFT. 2DFT spectra of the Landau levels from the undoped GaAs quantum well at magnetic fields of (a) 6 T and (b) 10 T. 2DFT spectra of the excitonic spectral region from the undoped GaAs quantum well at magnetic fields of (c) 6 T and (d) 10 T. Theoretical $S_I$ 2DFT spectra from the modulation doped sample. $S_I$ 2DFT spectra of LL1 (e) at 5.5 T and LL0 (f) at 10 T, calculated using time dependent DFT.

Figure 6

The difference between the electron and hole charges at $B=10$ T. The blue and green circles estimate the effective sizes of the electron and hole clouds (full width at half maximum). The red arrow shows effective distance between the electron and hole clouds. The opposite sign of the wave functions leads to a charge density in the area between the circles that is much smaller than the average individual electron and hole charge densities.

Figure 7

(a) 2DFT spectrum of the undoped quantum well at field magnetic fields of 6 T. (b) The corresponding spectrum obtained by using the interaction functions obtained for a unit cell expanded four times in the in-plane directions.

Figure 8

Simplified schematic of the Landau levels and band structure in the undoped (left) and modulation doped (right) samples. The strong Coulomb interaction between the unscreened carriers in the undoped quantum well, depicted by the thick black arrows, leads to lifting the degeneracy of the Landau levels and coupling them to a continuum. This is depicted in a simplified manner by the broadened Landau levels for the undoped sample. In the modulation doped ($\delta$-doped) sample the screening generated by the electron doping significantly weakens the Coulomb interactions, leaving the Landau levels degenerate. The charge screening is schematically depicted by the pink cloud, whereas the Coulomb lines are weaker and the Landau levels are discrete.