  Spin-orbit torque (SOT) is a candidate technique in next generation magnetic
random-access memory (MRAM). Recently, experiments show that some material with
low-symmetric crystalline or magnetic structures can generate anomalous SOT
that has an out-of-plane component, which is crucial in switching perpendicular
magnetization of adjacent ferromagnetic (FM) layer in the field-free condition.
In this work, we analytically derive the threshold current of field-free
perpendicular magnetization switching using the anomalous SOT. And we
numerically calculate the track of the magnetic moment in a FM free layer when
an applied current is smaller and greater than the threshold current. After
that, we study the applied current dependence of the switching time and the
switching energy consumption, which shows the minimum energy consumption
decreases as out-of-plane torque proportion increases. Then we study the
dependences of the threshold current on anisotropy strength, out-of-plane
torque proportion, FM free layer thickness and Gilbert damping constant, and
the threshold current shows negative correlation with the out-of-plane torque
proportion and positive correlation with the other three parameters. Finally,
we demonstrate that when the applied current is smaller than the threshold
current, although it cannot switch the magnetization of FM free layer, it can
still equivalently add an effective exchange bias field H_{bias} on the FM free
layer. The H_{bias} is proportional to the applied current J_{SOT}, which
facilitates the determination of the anomalous SOT efficiency. This work helps
us to design new spintronic devices that favor field-free switching
perpendicular magnetization using the anomalous SOT, and provides a way to
adjust the exchange bias field, which is helpful in controlling FM layer
magnetization depinning.
