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The thermal conductance in graphene nanoribbon with a vacancy or siliconpoint defect (substitution of C by Si atom) is investigated by non-equilibriumGreen's function (NEGF) formalism combined with first-principle calculationsdensity-functional theory with local density approximation. An efficientcorrection to the force constant matrix is presented to solve the conflictbetween the long-range character of the {\it ab initio} approach and thefirst-nearest-neighboring character of the NEGF scheme. In nanoribbon with avacancy defect, the thermal conductance is very sensitive to the position ofthe vacancy defect. A vacancy defect situated at the center of the nanoribbongenerates a saddle-like surface, which greatly reduces the thermal conductanceby strong scattering to all phonon modes; while an edge vacancy defect onlyresults in a further reconstruction of the edge and slightly reduces thethermal conductance. For the Si defect, the position of the defect plays norole for the value of the thermal conductance, since the defective region islimited within a narrow area around the defect center.

First principle study of the thermal conductance in graphene nanoribbon with vacancy and substitutional silicon defect