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Extreme-mass-ratio inspiral (EMRI) is an important gravitational-wave (GW)source and it normally consists of one stellar-mass black hole (BH) whirlingclosely around a supermassive black hole (SMBH). In this Letter, we demonstratethat the small body, in fact, could be a BH binary (BHB). Previous numericalscatting experiments have shown that SMBHs can tidally capture BHBs to boundorbits. Here we investigate the subsequent long-term evolution. We find thatthose BHBs with a semi-major axis of $a\lesssim5\times10^{-3}$ AU can becaptured to tightly-bound orbits such that they will successfully inspiraltowards the central SMBHs without being scattered away by stellar relaxationprocesses. We estimate that these binary-EMRIs (b-EMRIs) could constitute atmost $10\%$ of the EMRI population. Moreover, we show that when theeccentricity of a b-EMRI drops to about $0.85$, the two stellar BHs willquickly merge due to the tidal perturbation by the SMBH. The high-frequency($\sim10^2$ Hz) GWs generated during the coalescence coincide with thelow-frequency ($\sim10^{-3}$ Hz) waves from the b-EMRI, making this system anideal target for future multi-band GW observations.

A New Type of Extreme-mass-ratio Inspirals Produced by Tidal Capture of Binary Black Holes