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We consider a long topological Josephson junction formed on a conducting 2Dsurface of a 3D topological insulator (TI). The superconducting correlationsare proximity-induced by s-wave superconductors covering the surface. The 1Dspacing between the coverings is either unfilled or filled by a magneticinsulator. Generally, the Josephson current mediated by the TI surface isdetermined by scattering modes as well as by the states localized around thejunction (Andreev bound states or Andreev bands). We find out that it iscrucial to take into account both contributions to determine the current--phaserelation of the topological Josephson junction. We analyze the dependence ofthe Josephson current on the thickness of the junction as well as thedeviations from the sinusoidal shape of the current--phase relation.

Current--phase relation in a topological Josephson junction: Andreev bands vs. scattering states