  The surface tension of quark matter plays a crucial role for the possibility
of quark matter nucleation during the formation of compact stellar objects and
also for the existence of a mixed phase within hybrid stars. However, despite
its importance, this quantity does not have a well established numerical value.
Some early estimates have predicted that, at zero temperature, the value falls
within the wide range $\gamma_0\approx10-300{\rm\ MeV/fm^2}$ but, very
recently, different model applications have reduced these numerical values to
fall within the range $\gamma_0\approx5-30{\rm\ MeV/fm^2}$ which would favor
the phase conversion process as well as the appearance of a mixed phase in
hybrid stars. In magnetars one should also account for the presence of very
high magnetic fields which may reach up to about $ eB\approx 3-30\, m_\pi^2$
($B \approx 10^{19}-10^{20} \,G$) at the core of the star so that it may also
be important to analyze how the presence of a magnetic field affects the
surface tension. With this aim we consider magnetized two flavor quark matter,
described by the Nambu--Jona-Lasinio model. We show that although the surface
tension oscillates around its B=0 value, when $0 < eB \lesssim 10 \, m_\pi^2$,
it only reaches values which are still relatively small. For $eB \approx 5\,
m_\pi^2$ the B=0 surface tension value drops by about 30% while for $eB \gtrsim
10\, m_\pi^2$ it quickly raises with the field intensity so that the phase
conversion and the presence of a mixed phase should be suppressed if extremely
high fields are present. We also investigate how thermal effects influence the
surface tension for magnetized quark matter.
