The effect of DC bias on the growth of nanocrystalline diamond films on silicon substrate by microwave plasma chemical vapor deposition has been studied varying the substrate temperature (400, 500, 600, and $700^{circ}C$), deposition time (0.5, 1, and 2h), and bias voltage (-50, -100, -150, and -200 V) at the microwave power of 1.2 kW, working pressure of 110 torr, and gas ratio of Ar/1%$CH_4$. In the case of low negative bias voltages (-50 and -100 V), the diamond particles were observed to grow to thin film slower than the case without bias. Applying the moderate DC bias is believed to induce the bombardment of energetic carbon and argon ions on the substrate to result in etching the surfaces of growing diamond particles or film. In the case of higher negative voltages (-150 and -200 V), the growth rate of diamond film increased with the increasing DC bias. Applying the higher DC bias increased the number of nucleation sites, and, subsequently, enhanced the film growth rate. Under the -150 V bias, the height (h) of diamond films exhibited an $h=k{sqrt{t}}$ relationship with deposition time (t), where the growth rate constant (k) showed an Arrhenius relationship with the activation energy of 7.19 kcal/mol. The rate determining step is believed to be the surface diffusion of activated carbon species, but the more subtle theoretical treatment is required for the more precise interpretation.