Localization phase diagram for a disordered system in a magnetic field in two dimensions

A phase diagram for the localization-delocalization transition of a two-dimensional disordered semiconducting system in a perpendicular magnetic field B is investigated with a numerical method. Disorder originates from a random distribution of shallow impurities, measured in units of the impurity concentration c. Starting with a tight-binding Hamiltonian and an impurity state basis, the localization criterion is defined by means of the quantum connectivity of impurities. Finite-size scaling is employed to study the transition in the B-c parameter space. On this footing a phase diagram of the localization-delocalization transition in the B-c parameter space is calculated. At low concentrations c<c₁ approximately=0.246a^-2 where a is the impurity radius, all states are localized. Above c₁ two nose-shaped areas of a phase of delocalized states exist, the tips of which are found at (c₁,B₁)=(0.246+or-0.004a^-2, 0.013+or-0.001) and (c₃,B₃)=(0.67+or-0.03a^-2, 0.76+or-0.07) with the magnetic field given in terms of a²l ^-2, where l is the Lamor length. Both areas join at (c ₂,B₂)=(1.2+or-0.2a^-2, 0.233+or-0.009). States are well localized at B=0. An estimate of the localization length exponent is given. The transition is discussed in terms of orbital shrinking and interference effects, which are safely distinguished. The latter mechanism can account for a re-entrant behaviour with respect to the magnetic field. The metal-insulator transition is discussed as a function of the electron density in conjunction with the phase diagram. Results are compared with previous calculations within the zero differential overlap approximation.