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The effect of a DC bias field on the time-of-flight current waveforms of electron and hole drift in CdTe and CdZnTe is reported. Under a DC bias, the electron waveforms show more rapid decay than under a pulsed bias, whereas the hole waveforms evolve with time. Qualitatively similar behavior is observed in both CdTe and CdZnTe. Monte Carlo simulations of the current waveforms indicate that these results can be interpreted as quadratic band bending due to the emergence of a homogeneously distributed positive space charge with a concentration of 1010 to 1011 cm-3.