The two-dimensional electron system (2DES) in $\mathrm{Si}$ metal-oxide field-effect transistors consists of two distinct electron fluids interacting with each other. We calculate the total energy as a function of the density $n$ and the spin polarization $\zeta$ in the strongly correlated low-density regime, using a classical mapping to a hypernetted-chain (CHNC) equation inclusive of bridge terms. The ten distribution functions arising from spin and valley indices are calculated to obtain the total free energy, the chemical potential, the compressibility, and the spin susceptibility. The $T=0$ results are compared with the two-valley quantum Monte Carlo (QMC) data of Conti and Senatore [Europhys. Lett. 36, 695 (1996)] (at $T=0$, $\zeta =0$) and found to be in excellent agreement. Unlike in the one-valley 2DES, it is shown that the unpolarized phase is always the stable phase in the two-valley system, right up to Wigner crystallization at $r_s\sim 40$. Hence $g^{*}$ is insensitive to the spin polarization and to the density. The compressibility and the spin-susceptibility enhancement calculated from the free energy validate a simple approach to the two-valley response based on coupled-mode formation. The local-density approximation of density-functional theory is shown to fail, especially near $r_s=1$, even though the 2DES is uniform. The spin-susceptibility enhancement calculated from the coupled-valley response and directly from the two-valley energies is discussed. The three methods, QMC, CHNC, and coupled-mode theory, agree closely. Our results contain no ad hoc fit parameters and lead to general agreement with available experimental results.

• Received 25 February 2004

DOI:https://doi.org/10.1103/PhysRevB.70.035308