Instability induced by exchange forces in a 2-D electron gas in a magnetic field with uniform gradient

The exchange interaction is investigated theoretically for electrons confined to a 2-D sample placed in a linearly varying magnetic field perpendicular to the plane. Unusual and interesting behavior is predicted: starting from zero, as one adds electrons to the system, its size will increase continuously but will then collapse. However this collapse will be reversed as the number crosses another critical value, which we estimate here. For electron parameters typical for 2DEG's, the instability could be observable at sufficiently low electron densities. A Hartree Fock equation is derived and investigated. We also show that in an appropriate asymptotic limit an approximate local potential can be derived. One key lesson is that the exchange interaction is large and cannot be reasonably excluded from any valid theoretical investigation.