  Here, we present an analytical and numerical model describing the
magnetization dynamics in MgO-based spin-torque nano-oscillators with an
in-plane magnetized polarizer and an out-of-plane free layer. We introduce the
spin-transfer torque asymmetry by considering the cosine angular dependence of
the resistance between the two magnetic layers in the stack. For the analytical
solution, dynamics are determined by assuming a circular precession trajectory
around the direction perpendicular to the plane, as set by the effective field,
and calculating the energy integral over a single precession period. In a more
realistic approach, we include the bias dependence of the tunnel
magnetoresistance, which is assumed empirically to be a piecewise linear
function of the applied voltage. The dynamical states are found by solving the
stability condition for the Jacobian matrix for out-of-plane static states. We
find that the bias dependence of the tunnel magnetoresistance, which is an
inseparable effect in every tunnel junction, exhibits drastic impact on the
spin-torque nano-oscillator phase diagram, mainly by increasing the critical
current for dynamics and quenching the oscillations at high currents. The
results are in good agreement with our experimental data published elsewhere.
