Physics Letters A 167 (1992) 396—400 North-Holland PHYSICS LETTERS A Single-particle energy eigenstates for electrons on a sphere in an axial magnetic field Danhong Huang Department of Physics, University of Lethbridge, Let hbridge, Alberta TIK 3M4, Canada and Godfrey Gumbs Department of Physics and Astronomy, Iluifler College and the Graduate School, City University olNew York, 695 Park Avenue, New York, NY 10021. USA Received 24 April 1992; accepted for publication 5 June 1992 Communicated by L.J. Sham The single-particle energy eigenstates of a 2D EG, confined to the spherical surface in an axial magnetic field, are calculated. In strong-field limit, Landau quantization is complete, as it is for the planar 2D EG. Intermediate fields correspond to a crossover region between the zero-field free motion and the strong-field Landau quantization. The integral quantum Hall effect for a planar two- lished on a calculation of the single-particle energy dimensional (2D) electron gas (EG) is directly re- eigenstates for electrons in a linearly non-uniform lated to the Landau quantization of the single- magnetic field [3]. However, the calculations car- particle energy eigenvalues. There is a gap between ned out in that paper are only related to the weak the ground state of this planar 2D system and the field regime since there is an upper limit for both the excited states. Therefore, there is no energy dissi- gradient of the magnetic field and the size of the pation at low temperatures when the Landau-level sample containing the 2D EG. occupation number is an integer, i.e. the highest In this paper, we present calculations for the en- Landau level is completely full. The electrons in these ergy eigenstates of a 2D EG in a non-uniform mag- states are well localized within a spatial region of ra- netic field. The system we consider is a non- dius equal to the magnetic length LH. This system interacting EG constrained to the surface of a sphere was discussed by Laughlin [I] when interactions be- in an external magnetic field B along the polar :-axis. tween electrons are taken into account. The work of The normal component of this magnetic field varies Haldane [2] is concerned with electrons constrained with position on the sphere. There is complete to move on a sphere in the presence of the magnetic Landau quantization for our system univ in the strong field of a monopole located at the center of the sphere. magnetic field regime when the electrons are well lo- Since the dynamics of the electron motion in 2D is calized within a small region near the poles for which governed by the normal component of the magnetic their energy eigenstates become insensitive to the field, we define a uniform magnetic field as one whose global curvature of the sphere. In the absence of a normal component is constant. Although the ge- magnetic field, the electrons move freely on the sur- ometries for the systems discussed by Laughlin [1] face of the sphere. The intermediate magnetic field and Haldane [2] are different, the crucial feature regime corresponds to a crossover region between free they have in common is that the electrons move in electron motion and complete Landau quanttzatton. a uniform magnetic field. Recently, a paper was pub- The recovering of complete Landau quantization in 396 0375-9601 /92/$ 05.00 © 1992 Elsevicr Science Publishers B.V. All rights reserved.