  We present detailed photoionization models of well aligned optically thin C
III absorption components at $2.1 < z < 3.4$. Using our models we estimate
density ($n_{\rm \tiny H}$), metallicity ($[C/H]$), total hydrogen column
density and line-of-sight thickness ($L$) in each C III components. We estimate
the systematic errors in these quantities contributed by the allowed range of
the quasar spectral index used in the ultraviolet background radiation
calculations. Our inferred $n_{\rm \tiny H}$ and overdensity ($\Delta$) are
much higher than the measurements available in the literature and favor the
absorption originating from gas associated with circumgalactic medium and
probably not in hydrostatic equilibrium. We also notice $n_{\rm \tiny H}$, $L$
and $[C/H]$ associated with C III components show statistically significant
redshift evolution. To some extent, these redshift evolutions are driven by the
appearance of compact, high $n_{\rm \tiny H}$ and high $[C/H]$ components only
in the low$-z$ end. We find more than 5$\sigma$ level correlation between
$[C/H]$ and $L$, $L$ and neutral hydrogen column density (N (HI)), N (HI) and
$[C/H]$. We show $L$ versus $[C/H]$ correlation can be well reproduced if $L$
is governed by the product of gas cooling time and sound speed as expected in
the case of cloud fragmentation under thermal instabilities. This allows us to
explain other observed correlations by simple photoionization considerations.
Studying the optically thin C III absorbers over a large $z$ range and probably
correlating their $z$ evolution with global star formation rate density
evolution can shed light into the physics of cold clump formation and their
evolution in the circumgalactic medium.
