Electric current has been experimentally demonstrated to be able to drive the insulator-to-metal transition (IMT) in VO$_2$. The main mechanisms involved are believed to be the Joule heating effect and the strong electron-correlation effect. These effects are often entangled with each other in experiments, which complicates the understanding of the essential nature of the observations. We formulate a phase-field model to investigate theoretically in mesoscale the pure correlation effect brought by the current on the IMT in VO$_2$, i.e., the isothermal process under the current. We find that a current with a large density ($sim 10^1$ nA/nm$^2$) induces a few-nanosecond ultrafast switch in VO$_2$, in agreement with the experiment. The temperature-current phase diagram is further calculated, which reveals that the current may induce the M2 phase at low temperatures. The current is also shown capable of driving domain walls to move. Our work may assist related experiments and provide guidance to the engineering of VO$_2$-based electric switching devices.
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