  In this work, we investigate how the critical driving amplitude at the
Floquet MBL-to-ergodic phase transition differs between smooth and non-smooth
driving over a wide range of driving frequencies. To this end, we study
numerically a disordered spin-1/2 chain which is periodically driven by a sine
or a square-wave drive, respectively. In both cases, the critical driving
amplitude increases monotonically with the frequency, and at large frequencies,
it is identical for the two drives in the appropriate normalization. However,
at low and intermediate frequencies the critical amplitude of the square-wave
drive depends strongly on the frequency, while the one of the cosine drive is
almost constant in a wide frequency range. By analyzing the density of
drive-induced resonance in a Fourier space perspective, we conclude that this
difference is due to resonances induced by the higher harmonics which are
present (absent) in the Fourier spectrum of the square-wave (sine) drive.
Furthermore, we suggest a numerically efficient method to estimate the
frequency dependence of the critical driving amplitudes for different drives,
based on measuring the density of drive-induced resonances.
