Using an atomic force microscope (AFM) with a highly doped Si tip (0.01-0.02 Omega cm) we induced the oxidation of hydrogen-terminated Si(lll) surfaces by scanning the surface with an applied bias U-ox between sample tip (negative) and substrate (positive) in air of 60% relative humidity. The resulting oxide height was measured relative to the unoxidized substrate immediately afterwards using the same setup in the AFM mode. Typical oxide thicknesses of some nm were achieved. p-type as well as n-type samples were investigated and the oxidation time at each point was varied by varying the scan speed of the tip during oxidation. Some of the salient results are as follows : (i) There is a definite threshold voltage U-th such that no oxidation takes place below U-th; (ii) U-th depends on doping and is lowest (2.7 V) for n(+)- and highest (5.4 V) for p(+)-type material; (iii) U-th is independent of scan speed and thus oxidation time for speeds between 0.25 and 7 mu m/s. Assuming that the oxidation time is inversely proportional to the tip scan speed during oxidation we tried to fit the data to current models of oxide growth (parabolic growth law, Linear-quadratic growth law, and Mott-Cabrera mechanism). None of them fits the experimental data. Instead, we find that the results are well described by a power law of the form Z=(U-ox-U-th)alpha(0)(t(ox)/t(0))(gamma) where Z and t are the oxide thickness and the oxidation time, respectively, and alpha(0) is a factor that depends on experimental conditions such as doping, tip shape, and humidity. The exponent gamma has a value of 1/4 to within experimental uncertainty.