  We study a sample of 61 submillimetre galaxies (SMGs) selected from
ground-based surveys, with known spectroscopic redshifts and observed with
Herschel as part of the PACS Evolutionary Probe (PEP) and the Herschel
Multi-tiered Extragalactic Survey (HerMES) key programmes. We use the broad
far-infrared wavelength coverage (100-600um) provided by the combination of
PACS and SPIRE observations. Using a power-law temperature distribution model
to derive infrared luminosities and dust temperatures, we measure a dust
emissivity spectral index for SMGs of beta=2.0+/-0.2. Our results unveil the
diversity of the SMG population. Some SMGs exhibit extreme infrared
luminosities of ~10^13 Lsun and relatively warm dust components, while others
are fainter (~10^12 Lsun) and are biased towards cold dust temperatures. The
extreme infrared luminosities of some SMGs (LIR>10^12.7 Lsun, 26/61 systems)
imply SFRs of >500Msun yr^-1. Such high SFRs are difficult to reconcile with a
secular mode of star formation, and may instead correspond to a merger-driven
stage in the evolution of these galaxies. Another observational argument in
favour of this scenario is the presence of dust temperatures warmer than that
of SMGs of lower luminosities (~40K as opposed to ~25K), consistent with
observations of local ULIRGs triggered by major mergers and with results from
hydrodynamic simulations of major mergers combined with radiative transfer
calculations. Luminous SMGs are also offset from normal star-forming galaxies
in the stellar mass-SFR plane, suggesting that they are undergoing starburst
events with short duty cycles, compatible with the major merger scenario. On
the other hand, a significant fraction of the low infrared luminosity SMGs have
cold dust temperatures, are located close to the main sequence of star
formation, and thus might be evolving through a secular mode of star formation.
[abridged]
