We demonstrate an approach to create a high Q factor two-dimensional (2D) electromagnetic band gap (EBG) structure with composite defects in the center of the structures. By theoretical calculation, we show that a single-mode 2D EBG structure can be created with different ratios of diameter, d, to the lattice period, Λ, in this structure. This structure can achieve high Q factor and single mode operation with large d/Λ of the defect at the first circle around the removed rod in the center. With our structure, the single mode operation can be achieved within d/Λat the defect rods and d/Λat the others. In comparison, the condition of d/Λ should be strictly satisfied for the traditional design. These results provide very important guidelines for designing of the 2D EBG structure which can be used in traveling-wave tube device.