We fit the $sim$0.1-500 MeV/nucleon H-Fe spectra in 46 large SEP events surveyed by Desai et al. (2016) with the double power-law Band function to obtain a normalization constant, low- and high-energy parameters $gamma_a$ and $gamma_b$; and break energy $E_B$. We also calculate the low-energy power-law spectral slope $gamma_1$. We find that: 1) $gamma_a$, $gamma_1$, and $gamma_b$ are species-independent within a given SEP event, and the spectra steepen with increasing energy; 2) $E_B$s are well ordered by Q/M ratio, and decrease systematically with decreasing Q/M, scaling as (Q/M)$^alpha$ with $alpha$ varying between $sim$0.2-3; 3) $alpha$ is well correlated with Fe/O at $sim$0.16-0.23 MeV/nucleon and CME speed; 4) In most events: $alpha<$1.4, the spectra steepen significantly at higher energy with $gamma_b$-$gamma_a >$3; and 5) Seven out of 9 extreme SEP events (associated with faster CMEs and GLEs) are Fe-rich, have $alpha >$1.4, have flatter spectra at low and high energies with $gamma_b$-$gamma_a <$3. The species-independence of $gamma_a$, $gamma_1$, and $gamma_b$ and the systematic Q/M dependence of $E_B$ within an event, as well as the range of values for $alpha$ suggest that the formation of double power-laws in SEP events occurs primarily due to diffusive acceleration at near-Sun CME shocks and not due to scattering in the interplanetary turbulence. In most events, the Q/M-dependence of $E_B$ is consistent with the equal diffusion coefficient condition while the event-to-event variations in $alpha$ are probably driven by differences in the near-shock wave intensity spectra, which are flatter than the Kolmogorov turbulence spectrum but still weaker compared to that inferred for the extreme events.
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