Low-Magnetic-Field Divergence of the Electronic $g$ Factor Obtained from the Cyclotron Spin-Flip Mode of the $\nu =1$ Quantum Hall Ferromagnet

We report an inelastic light scattering study of the cyclotron spin-flip mode in the two-dimensional electron system at filling $\nu =1$. The energy of this mode can serve as a probe of the many-body exchange interaction on short length scales. Its magnetic field dependence is compared with predictions based on Hartree-Fock theory. They agree well when including the nonzero width of the electron system. From the measured energies, the exchange enhanced $g$ factor is extracted. It diverges at small fields and differs largely from $g$ factors obtained via transport activation studies.

Figure 1

(a) Raman spectra at 8.5 T ($\nu =1$) for a 20 and 25 nm QW. The inset compares $\Delta (0,B=7.6\mathrm{T})$ extracted from experiment for three QWs (solid triangles) with a Hartree-Fock simulation (dashed line). The upper graphs display the dispersion of the Raman shift (RS) in the long wavelength limit for the CSFM (b) and the MP mode (c).

Figure 2

$\Delta (0,B)$ values extracted from Raman experiments for QWs with different widths (open symbols) in comparison with Hartree-Fock simulations, which take into account the nonzero width of the wave function via form factor $F(q)$ (solid lines). Dotted lines show the $\sqrt{B}$ behavior predicted by Hartree-Fock for the strict 2D case, i.e., when $F(q)=1$. Different symbols correspond to different heterostructures used for collecting the data. The form factor is calculated from the wave function, which itself is obtained from a self-consistent solution of the Schrödinger and Poisson equations. The inset pictorially explains the softening of the Coulomb interaction with $B$ when the FWHM of the wave function $l$ exceeds $l_B$. The middle panel shows on the right axis $-{\Sigma }_{00}/\Delta (0,B)$ (blue curves) and $-E_v^{10}/\Delta (0,B)$ (red curves) for a 25 nm QW (long-dashed curve) and the strict 2D case (short-dashed curve).

Figure 3

(a) Normalized values of $\Delta (0,B)$ (same symbols as in Fig. 2) for the 20, 25, and 30 nm QWs. The normalization coefficient is equal to the ratio of $\Delta (0,B)$ for a QW of width $W$ predicted by theory to the theoretical value of $\Delta (0,B)$ for a 25 nm QW. The solid line is the Hartree-Fock result for the 25 nm QW. The previously reported data point from Ref. 14 is also displayed as a blue diamond. (b) The exchange enhanced $g_{\mathrm{eff}}$ versus density (or equivalently $B$) calculated from the experimental data points as described in the text. Previously reported data from transport activation experiments are also shown: red squares [1], black diamonds [2], green (up-pointing) triangles [3], and blue (down-pointing) triangles [4].