  We report on an experimental study of the effect of coherent surface-bulk
electron scattering on quantum transport in a three-dimensional topological
insulator Bi$_2$Te$_3$ nanoplate. The nanoplate is grown via van der Waals
epitaxy on a mica substrate and a top-gated Hall-bar device is fabricated from
the nanoplate directly on the growth substrate. Top-gate voltage dependent
measurements of the sheet resistance of the device reveal that the transport
carriers in the nanoplate are of n-type and that, with decreasing top gate
voltage, the carrier density in the nanoplate is decreased. However, the
mobility is increased with decreasing top-gate voltage. This mobility increase
with decreasing carrier density in the nanoplate is demonstrated to arise from
a decrease in bulk-to-surface electron scattering rate. Low-field
magnetotransport measurements are performed at low temperatures. The measured
magnetoconductivity of the nanoplate shows typical weak anti-localization (WAL)
characteristics. We analyze the measurements by taking surface-bulk
inter-channel electron scattering into account and extract dephasing times
${\tau}_{\phi}$, diffusion coefficients $D$ of electrons at the top surface and
in the bulk, and the surface-bulk scattering times ${\tau}_{SB}$ as a function
of top-gate voltage and temperature. It is found that the dephasing in the
nanoplate arises dominantly from electron-electron scattering with small energy
transfers. It is also found that the ratio of ${\tau}_{\phi}$/${\tau}_{SB}$ (a
measure of the surface-bulk electron coherent coupling) is decreased with
decreasing gate voltage or increasing temperature. We demonstrate that taking
the surface-bulk coherent electron scattering in our Bi$_2$Te$_3$ nanoplate
into account is essential to understand quantum transport measurements at low
temperatures.
