The dispersion relations of energy band structure and surface wave of left-handed and right-handed materials of photonic crystal with coating on the surface,as well as zero mean refractive index,are studied based on parameter matching by using the transmission matrix theory of plane wave and Bloch theorem.The results show that there are semi closed band gap,closed band gap and transmission band in the energy band structure of the left-handed and right-handed materials with zero mean refractive index,which is different from materials with positive refractive index.And the discrete energy level in the transmission band attenuates from high frequency to low frequency with oscillation and multiple degeneracy.the energy band structure attenuates and degenerates to zero with the increase of wave vector.The left-handed and right-handed materials of photonic crystal with coating on the surface support the propagation of both forward and reverse surface waves,and degenerate energy level splitting occurs in some of the semi closed or closed band gaps,forming a discrete dispersion curve with positive and negative values in the band gap.The frequency location and number of the dispersion curve can be modulated according to the thickness of coating on the surface.With the increase of coating thickness,the discrete dispersion curve moves to lower wave vector,and coupling splitting appears in the dispersion curve of the semi closed band gap in the high frequency region when the coating thickness reaches a certain value.However,the coating thickness has no effect on the multiple degeneracy of energy level and the degeneracy of transmission band and forbidden band.The characteristics of energy band structure and dispersion of surface waves of left-handed and right-handed materials of photonic crystal can provide guidance for the research and design of new optical waveguide devices.