Observation of exchange Coulomb interactions in the quantum Hall state at $\nu =3$

Coulomb exchange interactions of electrons in the $\nu =3$ quantum Hall state are determined from two inter-Landau-level spin-flip excitations measured by resonant inelastic light scattering. The two coupled collective excitations are linked to inter-Landau-level spin-flip transitions arising from the $N=0$ and $N=1$ Landau levels. The strong repulsion between the two spin-flip modes in the long-wavelength limit is clearly manifested in spectra displaying Coulomb exchange contributions that are comparable to the exchange energy for the quantum Hall state at $\nu =1$. Theoretical calculations within the Hartree-Fock approximation are in a good agreement with measured energies of spin-flip collective excitations.

Figure 1

(a) Schematic representation of the formation of collective modes at $\nu =3$ from single-electron transitions. The spin wave (SW) is described as a single spin-flip transition within half-filled LL${}_1$. MP and AP are formed as in-phase and antiphase combinations of two inter-LL transitions with $\delta S_z=0$ (green). Cyclotron spin-flip modes SF1 and SF2 arise from analogous combinations of inter-LL transitions with $\delta S_z=-1$ (red). (b) Dispersion curves of inter-LL excitations calculated for $B_{\perp }=5.3$ T within the first-order Hartree-Fock approximation are shown. Here the finite thickness of the 2D electron system (for sample A with a 24-nm quantum well) is taken into account via the geometric form factor. The dashed line represents the dispersion of the cyclotron spin-flip mode at $\nu =1$ under the same magnetic field. (c) Zoomed-in image of the long-wavelength region in (b) shaded light gray. The dashed vertical line indicates the experimental in-plane momentum $k^{*}=5.3\times {10}^4\hspace{0.5em}{\mathrm{cm}}^{-1}$. Open circles represent experimental data.

Figure 2

Inelastic light scattering spectra of the intra-LL SW mode at $\nu =3$ and $B=5.3$ T taken at different laser photon energies (shown at the left). Insets: SW dispersion curves calculated within the Hartree-Fock approximation[2] for a 24-nm-wide quantum well. The top inset shows the long-wave part and the bottom one displays the short-wave part of the SW dispersion. At the experimental in-plane momentum the energy of the SW is indistinguishable from $E_Z$.

Figure 3

Inelastic light scattering spectra of inter-LL excitations at $\nu =3$ and $B_{\perp }=5.3$ T taken at different incident photon energies (indicated on the right side of the spectra). The upper three spectra correspond to the resonant incident photon energies when electrons were promoted from the valence band to the second Landau level. Lower spectra were taken under resonant conditions when electrons were excited to the third Landau level. Gray vertical columns mark inelastic light scattering lines. The rest of the spectrum is composed of the luminescence bands, marked by slanted dashed lines. Note: The weak spectral feature between AP and SF1 in the lower spectra is a photoluminescence line since its spectral position changes upon variation of the laser photon energy.

Figure 4

Energy shift of SF2 from $\hslash {\omega }_c$ measured in three samples with different 2D-electron densities and quantum well widths. Filled symbols depict experimental values for samples A, B, and C. Open symbols are Hartree-Fock approximation calculation results obtained using the geometrical form factor for the quantum well size of each sample. The dashed curve corresponds to the Hartree-Fock approximation results calculated for the ideal 2D-electron system.

Figure 5

Magnetic field evolution of the inelastic light scattering spectrum of SF2 in the vicinity of $\nu =3$, taken at the fixed incident photon energy $\hslash {\omega }_{\mathrm{inc}}=1538.33$ meV. At $\left|\Delta B_{\perp }\right|~0.15$ T, the line nearly vanishes from the spectrum. The strong line in the left part of the spectrum originates from photoluminescence.