  Recent advances in monochromatic aberration corrected electron microscopy
make it possible to detect the lattice vibration with both high-energy
resolution and high spatial resolution. Here, we use sub-10 meV electron energy
loss spectroscopy to investigate the local vibrational properties at surface
and interface of an amorphous SiO$_2$ (a-SiO$_2$) thin film on Si substrate. We
find that each optical mode splits into three sub-modes, i.e., surface mode,
bulk mode and interface mode, which can be measured from different locations.
The pure surface modes can be measured in the vacuum near the surface, and the
pure interface modes are expected to be obtained either at the interface
location or in the Si, while inside the SiO$_2$ the measured signal is a
mixture of bulk, surface, and interface modes. The bulk mode has the largest
vibration energy and surface mode has the lowest. The energy of surface mode is
thickness dependent, showing a blue-shift as z-thickness (parallel to fast
electron beam) of SiO$_2$ film increases, while the bulk and interface modes
have constant vibration energy. The intensity of bulk mode linearly increases
with thickness being increased, and it drops steeply to zero near the surface
and interface (within a few nanometers). The surface modes decay slowly in the
vacuum following a Bessel function. The mechanism of the observed spatially
dependent vibration behavior is discussed and quantitatively compared with
dielectric response theory analysis. Our nanometer scale measurements of
vibrations properties provide useful information about the bonding conditions
at the surface and interface and thus may help to design better silicon-based
electronic devices via surface and interface treatments
