Variance exploding (VE) based diffusion models, an important class of diffusion models, have shown state-of-the-art (SOTA) performance. However, only a few theoretical works analyze VE-based models, and those works suffer from a worse forward convergence rate 1/\text{poly}(T) than the \exp{(-T)} of variance preserving (VP) based models, where T is the forward diffusion time and the rate measures the distance between forward marginal distribution q_T and pure Gaussian noise. The slow rate is due to the Brownian Motion without a drift term. In this work, we design a new drifted VESDE forward process, which allows a faster \exp{(-T)} forward convergence rate. With this process, we achieve the first efficient polynomial sample complexity for a series of VE-based models with reverse SDE under the manifold hypothesis.