Direct measurements of the spin and valley splittings in the magnetization of aSi∕SiGequantum well in tilted magnetic fields

Abstract

We present de Haas--van Alphen (dHvA) measurements on high-mobility two-dimensional electron systems formed in modulation-doped $\mathrm{Si}∕\mathrm{Si}\mathrm{Ge}$ (100) quantum wells and demonstrate directly the manifestation of the valley splitting in the magnetization. We resolve sawtoothlike magnetization oscillations at even filling factors which reflect the Landau quantization and the spin splitting of Landau levels in the electronic energy spectrum. At odd filling factors we observe the lifting of the valley degeneracy in Si at high magnetic field. The magnetization is a thermodynamic quantity that at low temperature reflects the ground-state energy of the interacting electron system. We can thus determine quantitatively the energetic splitting of the two occupied conduction-band valleys directly from the oscillation amplitude. Both valley and spin splitting are found to be enhanced by electron-electron interactions. The energy gap due to valley splitting is found to be $\ensuremath{\geqslant}0.8\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ at high perpendicular field ${B}_{\ensuremath{\perp}}$. From studies in tilted magnetic fields we find that the valley splitting is governed solely by ${B}_{\ensuremath{\perp}}$. From the spin splitting we recalculate an enhanced $g$ factor ${g}^{*}=2.9$ at $\ensuremath{\nu}=2$ including the influence of disorder. This is significantly larger than the band-structure $g$ factor of 2 in Si. We have successfully applied the coincidence technique for the dHvA effect and thus obtained a complementary means to determine the $g$ factor. It yields a constant value ${g}^{*}\ensuremath{\cong}3.2$ for filling factors $\ensuremath{\nu}\ensuremath{\geqslant}10$. A detailed analysis of the magnetization traces enabled us also to determine quantitatively the residual level broadening $\ensuremath{\Gamma}$ in this high-mobility $\mathrm{Si}∕\mathrm{Si}\mathrm{Ge}$ system. We obtain a small value of $\ensuremath{\Gamma}=0.15\phantom{\rule{0.3em}{0ex}}\mathrm{meV}\ifmmode\times\else\texttimes\fi{}{B}_{\ensuremath{\perp}}\phantom{\rule{0.3em}{0ex}}{[\mathrm{T}]}^{1∕2}$ for the $\mathrm{Si}∕\mathrm{Si}\mathrm{Ge}$ heterostructure of $200\phantom{\rule{0.2em}{0ex}}000\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{2}∕(\mathrm{V}\phantom{\rule{0.2em}{0ex}}\mathrm{s})$ mobility at $0.3\phantom{\rule{0.3em}{0ex}}\mathrm{K}$.