  We investigate the nonlinear response of the domain wall velocity ($v$) to an
external electric field ($E_{ext}$) in ferroelectric Si-doped HfO$_{2}$ thin
film capacitors using piezoresponse force microscopy (PFM) and switching
current measurements. We verified the reliability of the PFM images of
ferroelectric domain switching by comparing the switched volume fraction in the
PFM images with the time-dependent normalized switched polarization from the
switching current data. Using consecutive time-dependent PFM images, we
measured the velocity of the pure lateral domain wall motion at various
$E_{ext}$. The $E_{ext}$-dependent $v$ values closely follow the nonlinear
dynamic response of elastic objects in a disordered medium. The thermally
activated creep and flow regimes were observed based on the magnitude of
$E_{ext}$. With a dynamic exponent of ${\mu}$ = 1, our thin film was found to
have random-field defects, which is consistent with the Lorentzian distribution
of characteristic switching time that was indicated in the switching current
data.
