  In the quasi-2D electron systems of the layered transition metal
dichalcogenides (TMD) there is still a controversy about the nature of the
transitions to charge-density wave (CDW) phases, i.e. whether they are
described by a Peierls-type mechanism or by a lattice-driven model. By
performing scanning tunneling microscopy (STM) experiments on the canonical
TMD-CDW systems, we have imaged the electronic modulation and the lattice
distortion separately in 2H-TaS$_2$, TaSe$_2$, and NbSe$_2$. Across the three
materials, we found dominant lattice contributions instead of the electronic
modulation expected from Peierls transitions, in contrast to the CDW states
that show the hallmark of contrast inversion between filled and empty states.
Our results imply that the periodic lattice distortion (PLD) plays a vital role
in the formation of CDW phases in the TMDs and illustrate the importance of
taking into account the more complicated lattice degree of freedom when
studying correlated electron systems.
