  Antiferromagnetic spin fluctuations were investigated in the normal states of
the parent ($x = 0$), under-doped ($x = 0.04$) and optimally-doped ($x = 0.06$)
Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ single crystals using inelastic neutron
scattering technique. For all the doping levels, quasi-two-dimensional
antiferromagnetic fluctuations were observed as a broad peak localized at ${\it
Q} = (1/2, 1/2, l)$. At lower energies, the peak shows an apparent anisotropy
in the $hk0$ plane; longitudinal peak widths are considerably smaller than
transverse widths. The anisotropy is larger for the higher doping level. These
results are consistent with the random phase approximation (RPA) calculations
taking account of the orbital character of the electronic bands, confirming
that the anisotropic nature of the spin fluctuations in the normal states is
mostly dominated by the nesting of Fermi surfaces. On the other hand, the
quasi-two-dimensional spin correlations grow much rapidly for decreasing
temperature in the $x = 0$ parent compound, compared to that expected for
nearly antiferromagnetic metals. This may be another sign of the unconventional
nature of the antiferromagnetic transition in BaFe$_2$As$_2$.
