Theoretical studies on intervalley splittings in Si/SiO2 quantum dot structures

Using the coupled cluster method we investigatespin-s J ₁-J′ ₂ Heisenberg antiferromagnets (HAFs) on an infinite, anisotropic, two-dimensional triangular lattice for the two cases where the spin quantum number s = 1 and s = $frac{3} {2}$frac{3} {2}. With respect to an underlying square-lattice geometry the model has antiferromagnetic (J ₁ > 0) bonds between nearest neighbours and competing (J′ ₂ > 0) bonds between next-nearest neighbours across only one of the diagonals of each square plaquette, the same diagonal in each square. In a topologically equivalent triangular-lattice geometry, the model has two types of nearest-neighbour bonds: namely the J′ ₂ ≡ κJ ₁ bonds along parallel chains and the J ₁ bonds producing an interchain coupling. The model thus interpolates between an isotropic HAF on the square lattice at one limit (κ = 0) and a set of decoupled chains at the other limit (κ → ∞), with the isotropic HAF on the triangular lattice in between at κ = 1. For both the spin-1 model and the spin-$frac{3} {2}$frac{3} {2} model we find a second-order type of quantum phase transition at κ _c = 0.615 ± 0.010 and κ _c = 0.575 ± 0.005 respectively, between a Néel antiferromagnetic state and a helically ordered state. In both cases the ground-state energy E and its first derivative dE/dκ are continuous at κ = κ _c , while the order parameter for the transition (viz., the average ground-state on-site magnetization) does not go to zero there on either side of the transition. The phase transition at κ = κ _c between the Néel antiferromagnetic phase and the helical phase for both the s = 1 and s = $frac{3} {2}$frac{3} {2} cases is analogous to that also observed in our previous work for the s = $frac{1} {2}$frac{1} {2} case at a value κ _c = 0.80 ± 0.01. However, for the higher spin values the transition appears to be of continuous (second-order) type, exactly as in the classical case, whereas for the s = $frac{1} {2}$frac{1} {2} case it appears to be weakly first-order in nature (although a second-order transition could not be ruled out entirely).