Using both 0D and 1D fluid models, we revisit the formation of the breathing mode in Hall thrusters and show that it is an ionization instability associated with nonlinearity in the electron power absorption. As the plasma density increases, the axial electric field profile changes and the magnitude of the electric field is enhanced in the ionization zone. This causes a nonlinear increase in the power absorbed by electrons and an increase in the electron temperature and ionization rate factor that is able to partially compensate for the decreasing neutral density to keep the ionization rate high. This sets up a positive feedback mechanism where the electric field continues to be enhanced as the plasma density increases and, consequently, the neutral density needs to decrease even further before plasma growth can be halted. At this point, the neutral density is so low that the plasma can no longer be "sustained," and time is needed for neutrals to refill the thruster channel before "reignition" can occur and the process repeated. By treating the breathing mode as an AC excitation, a carefully designed external circuit can be used to counteract the change in the axial electric field by appropriately varying the anode voltage to stabilize the discharge.