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Tunneling field-effect transistors (TFETs) have gained a great deal of interest recently due to their potential to reduce power dissipation in integrated circuits. One major challenge for TFETs so far has been to achieve high drive currents, which is a prerequisite for high-performance operation. In this paper, we explore the performance potential of a 1-D TFET with a broken-gap heterojunction source injector using dissipative quantum transport simulations based on the nonequilibrium Green's function formalism, as well as the carbon nanotube band structure as the model 1-D material system. We provide detailed insights into broken-gap TFET (BG-TFET) operation and show that it can, indeed, produce less than 60 mV/dec subthreshold swing at room temperature, even in the presence of electron-phonon scattering. The 1-D geometry is recognized to be uniquely favorable due to its superior electrostatic control, reduced carrier thermalization rate, and beneficial quantum confinement effects that reduce the off-state leakage below the thermionic limit. Because of higher source injection compared to staggered-gap and homojunction geometries, BG-TFET delivers superior performance that is comparable to MOSFET's. BG-TFET even exceeds the MOSFET performance at lower supply voltages (VDD), showing promise for low-power/high-performance applications.