  We investigate the influence of field-like torque and the direction of the
external magnetic field on a one-dimensional array of serially connected
spin-torque nano oscillators, having free layers with perpendicular anisotropy,
to achieve complete synchronization between them by analyzing the associated
Landau-Lifshitz-Gilbert-Slonczewski equation. The obtained results for
synchronization are discussed for the cases of 2, 10 and 100 oscillators
separately. The roles of the field-like torque and the direction of the
external field on the synchronization of the STNOs are explored through the
Kuramoto order parameter. While the field-like torque alone is sufficient to
bring out global synchronization in the system made up of a small number of
STNOs, the direction of the external field is also needed to be slightly tuned
to synchronize the one-dimensional array of a large number of STNOs. The
formation of complete synchronization through the construction of clusters
within the system is identified for the 100 oscillators. The large amplitude
synchronized oscillations are obtained for small to large numbers of
oscillators. Moreover, the tunability in frequency for a wide range of currents
is shown for the synchronized oscillations up to 100 spin-torque oscillators.
In addition to achieving synchronization, the field-like torque increases the
frequency of the synchronized oscillations. The transverse Lyapunov exponents
are deduced to confirm the stable synchronization in coupled STNOs due to the
field-like torque and to validate the results obtained in the numerical
simulations. The output power of the array is estimated to be enhanced
substantially due to complete synchronization by the combined effect of
field-like torque and tunability of the field angle.
