Biomedical Engineering Reference
In-Depth Information
V
k
T
R
T
L
L
x
Fig. 1.12
Geometry of the double-barrier tunneling structure
followed by multiplication with the transmission matrix M
R
of the right barrier. The
final result is M
11
D
M
L;11
M
R;11
exp.
ikL
/
C
M
L;12
M
R;21
exp.
ikL
/,whereM
L;11
stands for the 11-element of the M
L
matrix, which gives
T
L
T
R
.1
p
R
L
R
R
/
2
T
D
C
4
p
R
L
R
R
cos
2
:
(1.20)
In (
1.12
), T
L
and T
R
are, respectively, the transmission probabilities through the left
and right barriers, R
L
D
1
T
L
and R
R
D
1
T
R
denote the related reflection
probabilities, and
D
kL
C
.argM
L;12
C
argM
R;21
argM
L;11
argM
R;11
/=2:
Equation (
1.20
) suggests that T can be close to unity if the energy resonance
requirement
D
.2n
C
1/
=2 ho
lds, with n integer, even if T
L
and T
R
have
small values, for which 1
p
R
L
R
R
Š
.T
L
C
T
R
/=2. The resonance transmission
probability, T
res
D
4T
L
T
R
=.T
L
C
T
R
/
2
, can become unity if T
L
D
T
R
, irrespective of
the values of T
L
and T
R
, or can be approximated with 4 min.T
L
;T
R
/= max.T
L
;T
R
/ if
T
L
and T
R
differ significantly. At resonance, when the energy of incident electrons
E equals a resonant value E
res
, besides being transmitted with high probability,
the electron travels through the structure faster than in off-resonance conditions. In
general, the devices based on resonant tunneling are ultrafast. Around resonance,
we have
L
R
.
L
C
R
/
2
=4
C
.E
E
res
/
2
;
T.E/
Š
(1.21)
where
L
D
.dE=d/T
L
=2 and
R
D
.dE=d/T
R
=2 (divided by
„
)are,
respectively, the rates at which an electron situated in the well leaks out of it through
the left and right barriers. Close to resonance, T is very sensitive to the values of
E and E
res
. The last parameter can be easily tuned by applying a bias across the
structure.
It is quite remarkable that, although the coherent nature of the electron transport
is explicitly used in deriving (
1.20
), which implies that electrons are transmitted in
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