Biomedical Engineering Reference
In-Depth Information
electron-electron collisions, dynamic scatterings, or certain impurity scattering
processes in which an internal degree of freedom changes; the phase relaxation
length is often called the coherence length.
3. The electron Fermi wavelength, denoted as F .
When one or more dimensions of a device are smaller than the mean-free path and
the phase relaxation length, the number of scatterings reduces dramatically, and the
transport in the device is termed ballistic. In this case, the electrons behave no longer
as particles but as waves that follow all the reflection and refraction rules of common
light or acoustic waves. As will be seen later, the ballistic transport manifests over
distances of few hundreds of nanometers in carbon nanotubes (CNTs), graphene, or
high-mobility transistors at room temperature.
In Fig. 1.3 , we have schematically displayed a transistor with scaled down
dimensions. The transport is diffusive when the transistor has gate lengths of 1m
or greater and ballistic as soon as the gate channel shrinks to tens of nanometers. In
the ballistic transport regime, the carriers traverse the gate channel in a much shorter
time and with higher speeds.
The transport of ballistic charge carriers with electron effective mass m and
constant energy E can be modeled by the time-independent Schrodinger equation
2
2 f m ˛
r Œm ˇ
r .m ˛ ‰/ gC V‰ D E‰;
(1.2)
when the coupling phenomena between different electron bands can be neglected
( Dragoman and Dragoman 1999 ). In ( 1.2 ), ‰ denotes the envelope electron
wavefunction, which has a slow variation over the unit cell of the crystalline lattice
and V is the potential energy. The material-dependent parameters ˛ and ˇ are
gate
drain
source
diffusive transport
ballistic transport
Fig. 1.3 The scaling down of
a transistor
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