Solid-Liquid Transition In A Two-Dimensional Electron System: A Monte Carlo Study

Seventy-one years ago,Wigner predicted that a system ofelectrons should have a crystalline form (WC) at sufficiently lowdensities and low temperatures because of strong Coulomb repulsion.The first experimental observation of a liquid to solid transition wasreported forty-five years later for a two-dimensional electron system(2DES) on a liquid helium surface. The 2DES was also realized inGaAs/AlGaAs heterojunction or in the interface of Si-metal-oxide-semiconductor-field-effect-transistor (Si-MOSFET). The2DES can exhibit rich physical phenomena and it has become afundamental model in condensed-matter physics and has beenactively investigated during recent decades.The density of the 2D electrons trapped on liquid helium is so lowthat the Fermi energy is much smaller than the experimentaltemperature. Therefore the system obeys classical statistics and itsthermodynamic properties are wholly determined by a dimension-less coupling constant Γ(=e2/akBT, whrer a is the averageinterelectron distance). Much theoretical work has been done on theWigner crystallization of the classical 2DES, however, about thenature of the transition from solid to liquid, i.e., the two-dimensional melting mechanism, there exists no decisiveconclusion.In this part, we use the canonical Metropolis Monte Carlo methodto probe the nature of the solid-liquid transition for a large classical2DES. The Ewald summation is employed to take care of thelong-range 1/r potential of the system under periodic boundaryconditions. In the simulation, we calculate various structural quan-tities, such as the pair distribution function, the snapshot of particleequilibrium configuration,six-coordinated particle ratio, the structurefactor and the positional and orientational correlation function. Weobtain the following results: (1) The system is a solid when Γ≥124 while a liquid when Γ≤122,the phase transition takes place in the region 122<Γ<124. This is inagreement with experimentally observed critical point Γ=137±15. (2) There exists no true long-range positional order in 2D electronsolid. The positional order is only quasi-long-ranged. This indicatesthat Mermin'theorem for system with hard potential can also beapplied to the softer 1/r potential electron system. (3) There remain some defects in the solid,which are tightly-bound pairs of dislocations. When the temperature is increased up to(or equivalently the density is decreased down to) the first criticalpoint, the dislocation pair dissociates and free dislocations appear. Adislocation is a bound pair of disclinations. Now the system has lostits positional order but still has some orientational order. When thetemperature increases to the second critical point, the disclinationpair dissociates, and one can see isolated disclinations. The systemonly has short-range positional and orientational orders, i.e., it is theisotropic fluid. (4) In the solid, intermediate phase and liquid, the positional andorientational correlation functions agree well with the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) predictions of the 2Dmelting mechanism. In the intermediate phase, we have observed,for the first time, the algebraic decaying form of the orientationalcorrelation function with the exponent less than the upper bound 1/4predicted by KTHNY theory. Hence the intermediate phase is astable one, with a short-range positional order and a quasi-long-range orientational order. (5) The transition from solid to liquid of the classical 2DESundergoes two stages. This transition is continuous, with an inter-mediate hexatic phase separating the solid and liquid phases. In the classical region, when the electron density is increased to acritical point, 2DES will change over to a crystal at low temperatures.If one increases the density further, the zero-vibrations enhance andthe quantum effects become important. Then at a higher density,Wigner crystal melts again into a highly degenerate quantum liquid.In this regime, the Fermi energy is much greater than theexperimental temperature, so the electrons obey quantum fermionstatistics. GaAs/AlGaAs heterojunctions provide a nearly idealbackground for such a two-dimensional quantum system. It is wellknown that the application of a strong perpendicular magnetic field(B), which quenches the kinetic energy and confines the electronswithin the magnetic length, facilitates the Wigner crystallization. Inthe strong magnetic field limit and at low temperature, the systemlies in solid; if the maghetic field is decreased, it transforms into thefascinating fractional quantum Hall liquid, i. e., the Hall resistancehas a quantized value and the magnetoresistance vanishes atfractional Landau level filling factor ν(=2π?cn/eB). Lots ofexperimental and theoretical researches have been devoted to thefractional quantum Hall effect (FQHE) and the solid-liquid transition,however, there still remain some controversies, e. g., the criticalfilling factor at which the transition occurs from series of FQHEstates to Wigner crystal, the ground state at low filling factors and themelting transition from solid to liquid as the temperature is raisedfromzero. In this part, we have generalized a path-integral formula for a 2Dquantum electron system at finite temperature and in the presence ofa strong perpendicular magnetic field, and used path-integral MonteCarlo approach to simulate such a system. This, for our knowledge,is the first time. We employ the newly developed multi-level block-ing method to cure the notorious fermion sign problem. Wedetermine the state of the system at low filling factor and extremelylow temperature, locate the phase boundary between the FQHEliquid and Wigner crystal, through computing the pair distributionfunction and Lindermann's ratio. The main results are given asfollows: (1) At ν=1/7, 1/9 and 1/11, and at very low temperature(T<100mK), the stable state of the system is Wigner crystal, while atν=1/3 and 1/5, the state lies in FQHE liquid. The transition occursnear ν=1/7. This is consistent with most theoretical and experimentalresults.The ground state is expected to be the same. (2) When the temperature is elevated, the solid will melt into theliquid like in the classical case. We obtain the finite tempera-ture-density phases at 1/7, 1/9 and 1/11. Most striking is thecoincidence of the quantum and the classical melting temperatures atν=1/11. At ν=1/7, the calculated melting temperature is T~220mK,which is in good agreement with T~200mK obtained by Goldman et...