The distribution of electrons in two parallel two‐dimensional electron sheets in close proximity at low temperatures is investigated. We find that due to the exchange interaction a symmetric structure may have a ground state in which all electrons are transferred to one of the sheets. This occurs if the separation between the two systems is sufficiently small and the two‐dimensional electron gas density is sufficiently low. The exchange interaction consequently leads to a bistability with respect to electron population in the two‐sheet system which may be useful for future electronic devices, such as memory elements.