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The tunneling resistance between two ferromagnetic metal layers that are separated by a thin insulator depends on the relative orientation of the magnetization M of each layer. In a memory device, independent switching of the magnetically soft layer is achieved by making the other layer either exchange-biased or magnetically hard. The repeated reversal of M of the soft layer by field cycling can demagnetize the other magnetically hard layer and thus erase the tunnel junction's memory. The M of exchange-biased structures was stable at least to 107cycles, whereas in hard structures, M generally decayed logarithmically with the number of field cycles. The decay was very sensitive to the thickness of the hard layer and the composition of the soft layer. However, no decay was observed when the moment reversal was accomplished by coherent rotation, establishing that demagnetization results from the formation and motion of domain walls in the soft layer during field cycling.