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We demonstrate a 40 Gbit/s silicon Mach-Zehnder optical modulator driven by a differential voltage of 0.36 Vpp. The energy efficiency is as low as 32.4 fJ/bit which is near the power efficiency of the ring modulator. We analyze the relationship between the electrical bandwidth and the electro-optical (EO) bandwidth based on the electrical S parameter measurement. Because of the nonlinear response, the electro-optical bandwidths in the small-signal tests are is slightly different when the modulator is biased at different transmission points. But the EO response is much different when the optical phase change is large enough to cover the nonlinear and linear regions at the same time. The nonlinearity can greatly improve the EO response in large-signal test. In our experiment, the rise/fall (20%-80%) time decreases from 13 ps to 10 ps as the driving amplitude increases from 5 V to 6 V under the same reverse bias of 3 V.