Orbital Magnetism and Current Distribution of Two-Dimensional Electrons under Confining Potential

Abstract

The spatial distribution of electric current under magnetic field and the resultant orbital magnetism have been studied for two-dimensional electrons under a harmonic confining potential $V(\vecvar{r})=m ω_0^2 r^2/2$ in various regimes of temperature and magnetic field, and the microscopic conditions for the validity of Landau diamagnetism are clarified. Under a weak magnetic field $(ω_c\lsimω_0, ω_c$ being a cyclotron frequency) and at low temperature $(T\lsim\hbarω_0)$, where the orbital magnetic moment fluctuates as a function of the field, the currents are irregularly distributed paramagnetically or diamagnetically inside the bulk region. As the temperature is raised under such a weak field, however, the currents in the bulk region are immediately reduced and finally there only remains the diamagnetic current flowing along the edge. At the same time, the usual Landau diamagnetism results for the total magnetic moment. The origin of this dramatic temperature dependence is seen to be in the multiple reflection of electron waves by the boundary confining potential, which becomes important once the coherence length of electrons gets longer than the system length. Under a stronger field $(ω_c\gsimω_0)$, on the other hand, the currents in the bulk region cause de Haas-van Alphen effect at low temperature as $T\lsim\hbarω_c$. As the temperature gets higher $(T\gsim\hbarω_c)$ under such a strong field, the bulk currents are reduced and the Landau diamagnetism by the edge current is recovered.