Wigner-crystal phases in bilayer quantum Hall systems

Abstract

Electrons in double-layer quantum-well systems behave like pseudospin 1/2 particles where the up and down ``spin'' represent localized states in each of the layers. The magnetically induced Wigner crystals in these systems are therefore crystals of these pseudospin 1/2 particles. We have calculated the phase diagram of the bilayer Wigner crystals using a variational scheme which explores a continuum of lattice and spin structure. Five stable crystal phases are found. For given tunneling strength and layer separation, one typically encounters the following sequence of transitions as the filling factor is increased from zero (the same sequence also occurs if one increases the ``effective'' layer separation starting from zero, with tunneling strength and filling factor held fixed): (I) (One-component) hexagonal structure \ensuremath{\rightarrow} (II) centered rectangular structure \ensuremath{\rightarrow} (III) centered square structure \ensuremath{\rightarrow} (IV) centered rhombic structure \ensuremath{\rightarrow} (V) staggered hexagonal structure. Crystal I is a ferromagnet in pseudospin space. All other crystals (II--V) have mixed ferromagnetic and antiferromagnetic orders, which are generated by layer tunneling and interlayer repulsion, respectively. The relative strength of these two magnetic orders varies continuously with external parameters (i.e., the ratio of layer separation to magnetic length, the tunneling gap to Coulomb interaction, etc.). The lattice structures I, III, and V are ``rigid'' whereas II and V are ``soft,'' in the sense that the latter two vary with external parameters and the former three do not. Another important features of the phase diagram is the existence of a multicritical point and a critical end point, which allows all crystals (except V) to transform into one another continuously. While our findings are based on a variational calculation, one can conclude on physical grounds that the mixed ferromagnetic-antiferromagnetic order as well as the pseudospin-lattice coupling should be general features of most bilayer Wigner crystals.