Two-phase noble liquid detectors, with large target masses and effective background reduction, are currently leading the dark matter direct detection for WIMP masses above a few GeV. Due to their sensitivity to single ionized electron signals, these detectors were shown to also have strong constraints for sub-GeV dark matter via their scattering on electrons. In fact, the most stringent direct detection constraints for sub-GeV dark matter down to as low as 5 MeV come from noble liquid detectors, namely XENON10, DarkSide-50, XENON100 and XENON1T, although these experiments still suffer from high background at single or a few electron level. LBECA is a planned 100-kg scale liquid xenon detector with significant reduction of the single and a few electron background. The experiment will improve the sensitivity to sub-GeV dark matter by three orders of magnitude compared to the current best constraints.