  In this paper, we consider an electrodynamics of higher derivatives coupled
to a Lorentz-violating background tensor. Specifically, we are interested in a
dimension-five term of the CPT-odd sector of the nonminimal Standard-Model
Extension. By a particular choice of the operator $\hat{k}_{AF}$, we obtain a
higher-derivative version of the Carroll-Field-Jackiw (CFJ) term,
$\frac{1}{2}\epsilon^{\kappa\lambda\mu\nu}A_{\lambda}D_{\kappa}\square
F_{\mu\nu}$, with a Lorentz-violating background vector $D_{\kappa}$. This
modification is subject to being investigated. We calculate the propagator of
the theory and from its poles, we analyze the dispersion relations of the
isotropic and anisotropic sectors. We verify that classical causality is valid
for all parameter choices, but that unitarity of the theory is generally not
assured. The latter is found to break down for certain configurations of the
background field and momentum. In an analog way, we also study a dimension-five
anisotropic higher-derivative CFJ term, which is written as
$\epsilon^{\kappa\lambda\mu\nu}A_{\lambda}T_{\kappa}(T\cdot\partial)^{2}F_{\mu\nu}$
and is directly linked to the photon sector of Myers-Pospelov theory. Within
the second model, purely timelike and spacelike $T_{\kappa}$ are considered.
For the timelike choice, one mode is causal, whereas the other is noncausal.
Unitarity is conserved, in general, as long as the energy stays real - even for
the noncausal mode. For the spacelike scenario, causality is violated when the
propagation direction lies within certain regimes. However, there are
particular configurations preserving unitarity and strong numerical indications
exist that unitarity is guaranteed for all purely spacelike configurations. The
results improve our understanding of nonminimal CPT-odd extensions of the
electromagnetic sector.
