  Recently, superconductors with higher-order topology have stimulated
extensive attention and research interest. Higher-order topological
superconductors exhibit unconventional bulk-boundary correspondence, thus allow
exotic lower-dimensional boundary modes, such as Majorana corner and hinge
modes. However, higher-order topological superconductivity has yet to be found
in naturally occurring materials. In this work, we investigate higher-order
topology in a two-dimensional Josephson junction comprised of two $s$-wave
superconductors separated by a topological insulator thin film. We found that
zero-energy Majorana corner modes, a boundary fingerprint of higher-order
topological superconductivity, can be achieved by applying magnetic field. When
an in-plane Zeeman field is applied to the system, two corner states appear in
the superconducting junction. Furthermore, we also discover a two dimensional
nodal superconducting phase which supports flat-band Majorana edge modes
connecting the bulk nodes. Importantly, we demonstrate that zero-energy
Majorana corner modes are stable when increasing the thickness of topological
insulator thin film.
