Exchange energy enhanced $g$-factors obtained from Landau fan diagrams at low magnetic fields

We report on measurement of the electron-hole effective $g$-factors ($g_{\mathrm{eff}}^{*}$) depending on electron filling factors $\nu$ from magnetophotoluminescence spectroscopy at low magnetic fields $B<1$ T in low electron density regime in a GaAs/Al${}_{0.33}$Ga${}_{0.67}$As-gated quantum well. Enhancement of $g_{\mathrm{eff}}^{*}$ at odd $\nu$ is observed. The oscillatory behavior of $g_{\mathrm{eff}}^{*}$ is compared with results of a theory that takes into account the lowest-order exchange interaction of the screened Coulomb interaction. Good agreement of the observed $g_{\mathrm{eff}}^{*}$ with theoretical results is obtained except $\nu$ at around 3. The enhancement of $g_{\mathrm{eff}}^{*}$ at even $\nu$ with decrease in electron density has not been observed.

Figure 1

(a) Typical PL spectra in color-image plots for $n_s=5.0\times {10}^{10}$ cm${}^{-2}$ and (b) for $n_s=1.0\times {10}^{11}$ cm${}^{-2}$ at $T=20$ mK. (c) Magnetic field dependence of PL peak energies for $n_s=5.0\times {10}^{10}$ cm${}^{-2}$ and (b) for $n_s=1.0\times {10}^{11}$ cm${}^{-2}$ in (i) ${\sigma }^{-}$ (black) and (ii) ${\sigma }^{+}$ (red) circular polarizations. (e) Magnetic field dependence of PL peak intensities for $n_s=5.0\times {10}^{10}$ cm${}^{-2}$ and (f) for $n_s=1.0\times {10}^{11}$ cm${}^{-2}$ in (i) ${\sigma }^{-}$ (dots) and (ii) ${\sigma }^{+}$ (crosses) circular polarizations, for LL0 (black), LL1 (blue), LL2 (orange), LL3 (red), LL4 (green), and LL5 (dark green).

Figure 2

(a) Observed $\nu$-dependent $g_{\mathrm{eff}}^{*}$ for $n_s=$ (i) $2.9\times {10}^{10}$, (ii) $4.0\times {10}^{10}$, (iii) $5.0\times {10}^{10}$, (iv) $6.0\times {10}^{10}$, (v) $7.1\times {10}^{10}$, (vi) $8.6\times {10}^{10}$, and (vii) $1.0\times {10}^{11}$ cm${}^{-2}$. The origin of the vertical axis is shifted. Error bars in $g_{\mathrm{eff}}^{*}$ are shown for some representative cases. (b) Calculated $g$-factors using ${\Gamma }_0=0.16$ for $\nu B=$ (i) 4, (ii) 3, (iii) 2, and (iv) 1.

Figure 3

Magnetic field dependence of the observed $g$-factors $g_{\mathrm{eff}}^{*}$ (solid circles) and calculated $g_{\mathrm{eff}}^{*}$ using ${\Gamma }_0=0.16$ and $g_{\mathrm{eff}}^0=-2.0$ (solid curves) at (a) $\nu =3$, (b) 5, (c) 7, and (d) 10. $g_{\mathrm{eff}}^{*}$ $=$ $g_{\mathrm{eff}}^0$ at $\nu$ $=$ 10.