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i don't think charge carrier density is zero ! either it connected or not to a battery.of course in "conductor".
it can increase when the conductor connected to a battery with ionized electron and left valance band to conductive band.
in conductor there are a lot of charge carrier in conductive band so when it connect to a battery it cues to generate an electricity field that force the carrier charge to move in specific direction , electron: opposite of electricity field direction and hole :straight.