Conditions for free magnetic monopoles in nanoscale square arrays of dipolar spin ice

We study a modified frustrated dipolar array recently proposed by M\"oller and Moessner [Phys. Rev. Lett. 96, 237202 (2006)], which is based on an array manufactured lithographically by Wang et al. [Nature (London) 439, 303 (2006)] and consists of introducing a height offset $h$ between islands (dipoles) pointing along the two different lattice directions. The ground states and excitations are studied as a function of $h$. We have found, in qualitative agreement with the results of M\"oller and Moessner, that the ground state changes for $h>{h}_{1}$, where ${h}_{1}=0.444a$ ($a$ is the lattice parameter or distance between islands). In addition, the excitations above the ground-state behave like magnetic poles but confined by a string, whose tension decreases as $h$ increases, in such a way that for $h\ensuremath{\approx}{h}_{1}$ its value is around 20 times smaller than that for $h=0$. The system exhibits an anisotropy in the sense that the string tension and magnetic charge depends significantly on the directions in which the monopoles are separated. In turn, the intensity of the magnetic charge abruptly changes when the monopoles are separated along the direction of the longest axis of the islands. Such a gap is attributed to the transition from the antiferromagnetic to the ferromagnetic ground state when $h={h}_{1}$.