  The temperature and magnetic field dependence of the in-plane thermal
conductivity $\kappa$ of the iron-arsenide superconductor
Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ was measured down to $T \simeq 50$ mK and up to
$H = 15$ T as a function of Co concentration $x$ in the range 0.048 $ \leq x
\leq $ 0.114. In zero magnetic field, a negligible residual linear term in
$\kappa/T$ as $T \to 0$ at all $x$ shows that there are no zero-energy
quasiparticles and hence the superconducting gap has no nodes in the $ab$-plane
anywhere in the phase diagram. However, the field dependence of $\kappa$
reveals a systematic evolution of the superconducting gap with doping $x$, from
large everywhere on the Fermi surface in the underdoped regime, as evidenced by
a flat $\kappa (H)$ at $T \to 0$, to strongly $k$-dependent in the overdoped
regime, where a small magnetic field can induce a large residual linear term,
indicative of a deep minimum in the gap magnitude somewhere on the Fermi
surface. This shows that the superconducting gap structure has a strongly
$k$-dependent amplitude around the Fermi surface only outside the
antiferromagnetic/orthorhombic phase.
