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

We have measured the magnetic field dependence of thermal activation energy EA 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, EA increases linearly with the total strength Bₜₒₜ of the magnetic field for Bₜₒₜ3T, while it takes a saturation value in the complementary high-Bₜₒₜ region. (2) The normal component B_⊥ of the magnetic field causes a peculiar change in EA, where two minima appear at B_⊥≈0.6Nₛφ₀ and B_⊥≈1.1Nₛφ₀ for φ₀=h/e. All the results are quantitatively explained by a model based on exchange interactions in a Wigner solid (WS) formed at rₛ≈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_⊥=0, 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 Bₜₒₜ or a transition between the valley states which we call a ``pseudo-spin-flip'' at high Bₜₒₜ, 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 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_⊥Nₛ^-1/φ₀ at magnetic-flux-induced antiferromagnetic phase transitions, which occur in the pseudospin system for large values of Bₜₒₜ, agree with the observed minima in EA(B_⊥).