Biomedical Engineering Reference
In-Depth Information
Fig. 1.11 Geometry of
electron emission in vacuum
by tunneling through a
triangular potential barrier
V
f
T
A
B
V = 0
vacuum
x
x = 0
2 eF
1=3 m
2
j A j
T D
(1.18)
2 m 0
k
for incident electrons with energy E,massm, and wavefunction ‰.x/ D
A exp. ikx / C B exp. ikx /.In( 1.18 ), k D .2 mE =
2 / 1=2 and the coefficients A
and B are obtained from the continuity conditions imposed on the wavefunction
and its derivative at x D 0.
At low temperatures and high electric fields, and for m D m 0 , the current density
associated to the transmission coefficient in ( 1.18 )is
3 F p 2m 0 ' 3 ;
(1.19)
where the logarithmic term describes the effect of the transverse degrees of freedom
on the Fermi-Dirac distribution function. Formula ( 1.19 ) is the so-called Fowler-
Nordheim equation, which models the current-voltage characteristic of devices
based on field emission.
Let us consider now the case when electrons tunnel through a structure consisting
of two (or several) barriers that alternate with quantum well regions. Because
ballistic electrons propagate coherently, constructive or destructive interferences
appear between quantum waves that are only partially reflected and transmitted at
different interfaces. So we expect high and low transmission probability values in
a typical geometry such as that displayed in Fig. 1.12 , containing a quantum well
surrounded by two thin barriers. This phenomenon is analogous to the occurrence
of high- and low-intensity values associated with interference between coherent
light beams. A large transmission probability through a structure consisting of two
(or several) barriers, each of them with a low transmission probability, defines the
phenomenon of resonant tunneling.
This phenomenon can be described also with the transfer matrix method
developed above. For the case of a structure containing two barriers that surround
a quantum well with width L in which electrons propagate with a wavevector
k, the total transmission probability T is determined by the M 11 element of the
transmission matrix M through the whole structure, found by first multiplying the
transmission matrix M L of the left barrier with the diagonal matrix with elements
exp. ikL / and exp. ikL /, corresponding to free propagation across the quantum well,
Z T.E/ln 1 C exp E F E
k B T
dE / F 2 exp
em 0 k B T
2
4
J D
3
 
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