We have fabricated the Au/n-Si (D1), Au/Cu/n-Si (D2), Au/Cu(4nm)/n-Si (D3) and Au/Cu(2nm)/n-Si (D4) Schottky-barrier diodes (SBDs). The thickness of the Cu Schottky contact (SC) films for the D2, D3, and D4 diodes has been chosen as 100, 4 and 2 nm, respectively. We have investigated the thermal sensitivity from the voltage-temperature (V-T) characteristics of the SBDs at different current levels. The V-T measurements have been made in the temperature range from 10 K to 320 K with steps of 2 K, at the different current levels from 50 nA to 141.70 µA. The V-T curves have shown a good linearity degree for all SBDs. The slope dV/dT = α or thermal sensitivity coefficient α for each diode has decreased with increasing current level from 50 nA to 141.70 µA. But, it has been seen that the SBDs with the Cu SC have approximately the same α value as independent of metal thickness at the same current level. That is, the thermal sensitivity coefficient value has changed approximately from 2.48 mV/K at 50 nA to 1.82 at 141.70 µA for the SBDs with Cu Schottky contact as independent metal thickness. Furthermore, the α versus current level plots of the diodes have exhibited a linear behavior. The intercept α0 and slope dα/dI values of the α versus current level plots have been obtained as 2.80 mV/K and -0.0843 mV/(AK) for D2, and 2.85 mV/K and -0.092 mV/(AK) for D3 and 2.83 mV/K and -0.0876 mV/(AK) for D4. These values are very close to each other and the difference between the slope (dα/dI) values is small enough to be neglected.