  We investigate the thermal production of charm quark in the strongly
interacting quark-gluon plasma (sQGP) created in heavy-ion collisions at
relativistic energies. Our study is based on the off-shell parton-hadron-string
dynamics (PHSD) transport approach describing the full time evolution of
heavy-ion collisions on a microscopic bases with hadronic and partonic
degrees-of-freedom. The sQGP is realized within the effective dynamical
quasi-particle model (DQPM) which is adjusted to reproduce the lattice QCD
results for the thermodynamic observables of the sQGP. Relying that the DQPM
successfully describes the spatial diffusion coefficients $D_s$ from the lQCD,
which controls the interaction of charm quarks with thermal partons (expressed
in terms of strongly interacting off-shell quasiparticles), we evaluate the
production of pairs of anti-charm/charm quarks through the rotation of Feynman
diagrams using cross symmetry for elastic scattering of charm quark with
thermal partons ((anti-)quarks or gluons) in the sQGP by the exchange of the
incoming charm quark and outgoing light parton in elastic scattering diagrams.
The charm quark annihilation is realized by detailed balance. We find that the
number of produced thermal charm quark pairs strongly depends on the charm
quark mass in the QGP. While for the heavy charm quarks of mass $m_c=1.8$ GeV
it is subdominant compared to the primary charm production by binary
nucleon-nucleon collisions at RHIC and LHC energies, the numbers of primary and
thermal charm quarks become comparable for a smaller (bare) $m_c=1.2$ GeV.
Comparing with the experimental data on the $R_{\rm AA}$ of $D$-mesons in
heavy-ion collisions at RHIC and LHC energies, it is more favorable for charm
quarks in the QGP to gain additional mass due to thermal effects rather than to
have a low bare mass.
