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

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 $V(\vecvar{r})=m \omega_0^2 r^2/2$ 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 $(\omega_c\lsim\omega_0, \omega_c$ being a cyclotron frequency) and at low temperature $(T\lsim\hbar\omega_0)$, 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 $(\omega_c\gsim\omega_0)$, on the other hand, the currents in the bulk region cause de Haas-van Alphen effect at low temperature as $T\lsim\hbar\omega_c$. As the temperature gets higher $(T\gsim\hbar\omega_c)$ under such a strong field, the bulk currents are reduced and the Landau diamagnetism by the edge current is recovered.