  In this paper we study Peierls instabilities for a half-filled
two-dimensional tight-binding model with nearest-neighbour hopping $t$ and next
nearest-neighbour hopping $t'$ at zero and finite temperatures. Two
dimerization patterns corresponding to the same phonon vector $(\pi, \pi)$ are
considered to be realizations of Peierls states. The effect of imperfect
nesting introduced by $t'$ on the Peierls instability, the properties of the
dimerized ground state, as well as the competition between two dimerized states
for each $t'$ and temperature $T$, are investigated. It is found: (i). The
Peierls instability will be frustrated by $t'$ for each of the dimerized
states. The Peierls transition itself, as well as its suppression by $t'$, may
be of second- or first-order. (ii). When the two dimerized states are
considered jointly, one of them will dominate the other depending on parameters
$t'$ and $T$. Two successive Peierls transitions, that is, the system passing
from the uniform state to one dimerized state and then to the other take place
with decrease of temperature for some $t'$ values. Implications of our results
to real materials are discussed.
