Spin susceptibility enhancement in a two-dimensional hole gas

The critical spin splitting needed to fully polarize a two-dimensional heavy-hole gas in a (Cd,Mn)Te quantum well thanks to the giant Zeeman effect is measured through photoluminescence and transmission spectroscopy. While the splitting between two circularly polarized peaks in photoluminescence remains equal to the bare spin splitting, the critical spin splitting which governs the polarization is enhanced, in good agreement with the prediction of a model of hole-hole interactions previously applied to conduction electrons. Consequences for carrier-induced ferromagnetism are discussed.

Figure 1

PL spectra of a ${\mathrm{Cd}}_{0.996}{\mathrm{Mn}}_{0.004}\mathrm{Te}$, $8\mathrm{nm}$ wide QW, in ${\sigma }^{+}$ polarization (solid line) and ${\sigma }^{-}$ polarization (dashed line), for two values of the carrier density, $p=1\times {10}^{11}{\mathrm{cm}}^{-2}$ (top) and $p=3\times {10}^{11}{\mathrm{cm}}^{-2}$ (bottom).

Figure 2

Transition energies in PL (down triangles, left scale) and transmission (up triangles, right scale) in ${\sigma }^{+}$ polarization (full symbols) and in ${\sigma }^{-}$ polarization (open symbols) for two values of the carrier density, (a) $p=3\times {10}^{11}{\mathrm{cm}}^{-2}$, and (b) $p=1\times {10}^{11}{\mathrm{cm}}^{-2}$; (c) Splitting between the ${\sigma }^{+}$ and ${\sigma }^{-}$ PL lines, as a function of the applied field, for different values of the carrier density from $p=0.7\times {10}^{11}{\mathrm{cm}}^{-2}$ (open triangles) to $3\times {10}^{11}{\mathrm{cm}}^{-2}$ (full circles). The solid lines show the Brillouin function describing the effect of the giant Zeeman effect in ${\mathrm{Cd}}_{0.996}{\mathrm{Mn}}_{0.004}\mathrm{Te}$.

Figure 3

Valence band spin splitting achieving complete polarization of the 2DHG (symbols: experimental data; lower set of lines: calculated), and Fermi energy (upper set of lines), as a function of the carrier density, assuming three values of the transverse mass of the heavy holes, $m^{*}=0.17m_0$ (diagonal approximation, dotted lines), $m^{*}=0.25m_0$ (axial approximation, dashed lines) and $m^{*}=0.22m_0$ (best fit, solid lines).

Figure 4

Enhancement factor of the spin susceptibility and spin splitting, as a function of the carrier density, using $m^{*}=0.22m_0$; the lines give the result of the calculation at complete polarization (solid line) and vanishing polarization (dashed line); symbols are experimental data.