We have studied the Junction Magnetoresistance (JMR) and the Differential junction magnetoresistance (DJMR) for double tunnel junctions with magnetic metals in the Coulomb Blockade regime. Spikes are seen in both the JMR and the DJMR vs. voltage curves. They occur at those places where the current increases by a step. In all cases the large bias limit can be obtained by adding the resistances of each of the junctions in series. The JMR is positive in all the cases we studied, whereas the DJMR can be positive or negative as a function of the voltage. Moreover, the relative variation of the DJMR as a function of the voltage is larger than the variation of the JMR with the voltage.

The effects of quantum fluctuations due to directional anisotropy and frustration between nearest neighbors and next-nearest neighbors of the quantum spin-1 2 Heisenberg antiferromagnet on a square lattice are investigated using spin-wave expansion. We have calculated the spin-wave-energy dispersion in the entire Brillouin zone, renormalized spin-wave velocities, and the magnetization up to second order in 1 / S expansion for the antiferromagnetic Neél and collinear antiferromagnetic stripe phases. It is shown that the second-order corrections become significant with increase in frustration. With these corrections magnetizations and spin-wave velocities for both the phases become zero at the quantum critical points as expected from other numerical and analytical methods. We have shown that the transition between the two ordered phases are always separated by the disordered paramagnetic phase. the intermediate paramagnetic state at 1c is of second order and from the paramagnetic state to the collinear state at 2c is of first order. 52,53 A generalization of the frustrated J 1 -J 2 model is the J 1 -J 1 Ј-J 2 model where = J 1 Ј/ J 1 is the directional anisotropy parameter. 28,31 It is known that the spatial anisotropy reduces the width of the disordered phase. Extensive band structure calculations 46 for the vanadium phosphate compounds Pb 2 VO͑PO 4 ͒ 2 , SrZnVO͑PO 4 ͒ 2 , BaZnVO͑PO 4 ͒ 2 , and BaCdVO͑PO 4 ͒ 2 have shown four different exchange couplings: J 1 and J 1 Ј between the NN and J 2 and J 2 Ј between NNN. For example Ϸ 0.7 and J 2 Ј/ J 2 Ϸ 0.4 were obtained for SrZnVO͑PO 4 ͒ 2 . A possible realization of the J 1 -J 1 Ј-J 2 model may be the compound ͑NO͒Cu͑NO 3 ͒ 3 ͑Ref. 54͒ though recent band-structure calculations show a uniform spin chain model with different types of anisotropy and weak interchain couplings. 55 Within the spin-wave expansion the effect of directional anisotropy on the spin-wave-energy dispersion and the transverse dynamical structure factor has been studied before. 15 However, the effect of NNN frustration has not been incorporated in that study.For the J 1 -J 1 Ј-J 2 model using a higher order coupled cluster method Bishop et al. 45 reported existence of a quantum triple point ͑QTP͒ at Ϸ 0.60, Ϸ 0.33. Below this point they predicted a second-order phase transition between the quantum Neél and stripe phases whereas above it these two

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