  Low Gain Avalanche Detectors (LGAD) for the High-Granularity Timing Detector
(HGTD) are crucial in reducing pileups in the High-Luminosity Large Hadron
Collider. Numerous studies have been conducted on the bulk irradiation damage
of LGADs. However, few studies have been carried out on the surface irradiation
damage of LGAD sensors with shallow carbon implantation. In this paper, the
IHEP-IME LGADs with shallow carbon implantation were irradiated up to 2 MGy
using gamma irradiation to investigate surface damage. Important characteristic
parameters, including leakage currents, breakdown voltage (BV), inter-pad
resistances, and capacitances, were tested before and after irradiation. The
results showed that the leakage current and BV increased after irradiation,
whereas overall inter-pad resistance exhibited minimal change and remained
above $10^9\ \Omega$ before and after irradiation. Capacitance was found to be
less than 4.5 pF with a slight decrease in the gain layer depletion voltage
(V$_{gl}$) after irradiation. No parameter affected by the inter-pad separation
was observed before and after irradiation. All characteristic parameters meet
the requirements of HGTD, and this design can be used to further optimization.
