Magnetism in a Wigner solid and the Aharonov-Bohm effect: Experiment and theory

We have measured the magnetic field dependence of thermal activation energy $E_A$ in an insulating phase of a two-dimensional electron system formed in a Si inversion layer to investigate magnetic interactions among electrons. Experimental results are summarized as follows. (1) In a magnetic field parallel to the two-dimensional plane, $E_A$ increases linearly with the total strength $B_{\mathrm{tot}}$ of the magnetic field for $B_{\mathrm{tot}}\lesssim 3\mathrm{}\mathrm{T},$ while it takes a saturation value in the complementary high-$B_{\mathrm{tot}}$ region. (2) The normal component $B_{\perp }$ of the magnetic field causes a peculiar change in $E_A,$ where two minima appear at $B_{\perp }\approx {0.6N}_s{\phi }_0$ and $B_{\perp }\approx {1.1N}_s{\phi }_0$ for ${\phi }_0=h/e$. All the results are quantitatively explained by a model based on exchange interactions in a Wigner solid (WS) formed at $r_s\approx 8$. We assume that a three-particle ring exchange among localized electrons in the WS and a standard two-particle exchange between a thermally created mobile electron and a localized electron in the WS play dominant roles in determining the magnetic behavior of the system. At $B_{\perp }=0\mathrm{}$, the three-particle ring exchange interaction leads to the ferromagnetically ordered ground state of the WS. The antiferromagnetic two-particle exchange interaction favors magnetic states of a mobile electron different from the lowest spin-valley state of the localized electrons. Thermal activation, which involves a spin flip at low $B_{\mathrm{tot}}$ or a transition between the valley states which we call a “pseudo-spin-flip” at high $B_{\mathrm{tot}},$ explains the experimental result (1). Magnetic flux through the exchange path can change the nature of the ring exchange interaction due to the Aharonov-Bohm effect. A simple calculation using an exchange constant of $0.3\hspace{0.5em}\mathrm{K}$, which is consistent with the WKB calculation of Roger [Phys. Rev. B 30, 6432 (1984)], for the three-electron ring exchange interaction reproduces the experimental result (2). Theoretical values of $B_{\perp }{N}_s^{-1}/{\phi }_0$ at magnetic-flux-induced antiferromagnetic phase transitions, which occur in the pseudospin system for large values of $B_{\mathrm{tot}},$ agree with the observed minima in $E_A{(B}_{\perp })$.