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, $frac{1}{2}epsilon^{kappalambdamu u}A_{lambda}D_{kappa}square F_{mu u}$, 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 $epsilon^{kappalambdamu u}A_{lambda}T_{kappa}(Tcdotpartial)^{2}F_{mu u}$ 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.
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