Biomedical Engineering Reference
In-Depth Information
Fig. 1.10
The geometry of a
tunneling structure consisting
of a single barrier
V
V
2
g
2
k
1
k
3
V
3
V
1
x
x =
0
x = L
In tunneling devices, the direction of electron propagation, denoted by x in the
expressions above, is determined by the direction of the applied electric field, and
the transfer matrix method remains applicable as long as ‰ denotes the x-dependent
part of the envelope wavefunction of electrons and the electron motion along the
longitudinal x direction can be separated from that along the transversey and
z
directions. The transmission probability through a single barrier, which is the
simplest tunneling structure displayed in Fig.
1.10
,isgivenby
4
v
1
v
3
T
D
C
Œ.
v
1
C
v
2
/.
v
2
C
v
3
/=
v
2
sinh
2
.
2
L/
;
(1.17)
.
v
1
C
v
3
/
2
where
v
2
D„
2
=m
2
and k
2
D
i
2
is imaginary in the barrier region labeled by 2
and surrounded by regions 1 and 3 with real wavenumbers.
From (
1.17
) it follows that, if
2
L
1, the transmission probability decreases
exponentially with the barrier width L: T
/
exp.
2
2
L/. At low temperatures,
when all electrons contributing to tra
nsport have ene
rgies around E
F
, this expression
can be rewritten as T
/
expŒ
2L
p
2m
2
.V
2
E
F
/=
„
(
Zhu 2001
). In this case, the
current density through the structure is proportional to T and therefore decreases
also exponentially with L, while the conductance is much smaller than G
0
.
The example of electron transmission through a single barrier is relevant for
many applications, such as scanning tunneling microscopy or vacuum microelec-
tronic devices, which contain flat panel field emission displays and electron sources
for microscopes. In these devices, electrons tunnel from a solid surface into vacuum
under the influence of a high electric field (
Zhu 2001
).
The potential barrier in the vacuum region, given by the electron affinity ,
takes a triangular shape in the presence of an applied electric field F , acquiring a
spatial dependence of
eFx
(see Fig.
1.11
). The wavefunction of the electron
in vacuum with mass m
0
is then a solution of the Schr odinger equation with a
triangular barrier, which, if ˛
D
0, is a superposition of Ai and Bi Airy functions:
‰
vac
./
D
A
vac
Ai./
C
B
vac
Bi./, with
D
Œ2m
0
=.
eF
„
/
2
1=3
.
eFx
E/.The
condition of current flow along the positive x direction imposes the following values
for the constant coefficients: A
vac
D
1, B
vac
D
i, so that the transmission coefficient
T across the barrier has the expression
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