  The domain wall depinning from a notch in a Permalloy nanostrip on top of a
${\rm SiO_2/Si}$ substrate is studied theoretically under application of static
magnetic fields and the injection of short current pulses. The influence of
Joule heating on current-induced domain wall depinning is explored
self-consistently by coupling the magnetization dynamics in the ferromagnetic
strip to the heat transport throughout the system. Our results indicate that
Joule heating plays a remarkable role in these processes, resulting in a
reduction in the critical depinning field and/or in a temporary destruction of
the ferromagnetic order for typically injected current pulses. In agreement
with experimental observations, similar pinning-depinning phase diagrams can be
deduced for both current polarities when the Joule heating is taken into
account. These observations, which are incompatible with the sole contribution
of spin transfer torques, provide a deeper understanding of the physics
underlying these processes and establish the real scope of the spin transfer
torque. They are also relevant for technological applications based on
current-induced domain-wall motion along soft strips.
