Properties of the fundamental surface defect mode soliton have very important applications in all-optical control field. Based on nonlinear Schrdinger equation, properties of the fundamental surface defect mode, which can be supported by semi-infinite lattices with a local defect in Kerr-type nonlinear media, have been studied in details by employing numerical methods such as image-time method, nonlinear relaxation method and split-step Fourier algorithm. The research results show that the fundamental surface defect mode can only stay in the defect channel when the defect is attractive. However, when the defect is rejective, the fundamental surface defect mode can stay in the defect channel or not. In such a system, there are several critical values of the system parameters, and it is found that whether the fundamental defect mode can locate in the defect or not depends on whether the parameter value is higher or lower than its critical value when other parameters remain unchanged. In terms of physical mechanism, the competition between the optical lattices and nonlinear refractive index induced by optical beam leads to the occurrence of the critical behavior.