  When a stellar massive compact object, such as a black hole (BH), inspirals
into an intermediate massive black hole (IMBH), an
intermediate-mass-ratio-inspiral (IMRI) system forms. Such kind of systems are
important sources for space-based gravitational wave detectors including LISA,
Taiji and Tianqin. Dark matter (DM) minispikes may form around IMBHs. We study
the effect of dynamical friction against DM minispike on the evolution of
eccentric IMRI. For such investigation we construct the dynamical equations
which describes the evolution of eccentric IMRI under the effect of dynamical
friction. As dynamical friction is large for small velocity, the dissipation of
energy near apastron is more than that near periastron. This will greatly
enhance the eccentricity. For instance, with an initial semi-latus rectum of
$1\rm AU$, even a moderate DM minispike can make the eccentricity grow from
$0.3$ to $0.85$. In the extremal case the eccentricity could be enhanced to
near $1$. We also study a specific case which corresponds to an IMRI in the
center of a globular cluster (GC) and find the eccentricity can keep its value
above $0.95$ until the IMRI enters LISA band. These gravitational wave with
enhanced eccentricity by DM minispikes can be easly distinguished from that
without DM at $10^{-3}\rm Hz$ due to the eccentricity difference. These
anticipations can be tested by future space-based GW detectors such as LISA.
