w
min
w
h
2
ffiffiffiffiffiffiffiffiffiffiffi
2mE
b
p
(A1.7)
Note that (A1.7) is close to the Heisenberg distinguishability length a
H
(3.10) derived for tunneling.
Generally speaking, Eq. (A1.5) describes a situation of infinite height of the walls [11], and for
a finite barrier height, the solution is obtained numerically (see e.g. [11] for a detailed procedure).
However (A1.5) can still be used for o rder-of-magnitude estimates in the case of low barriers. For
example, let E
b
¼ k
B
T , then the effective barrier for a confined particle as a function of the well size is
plotted in Figure A1.2 calculated using approximation (A1.5) and by an exact numerical solution [11].
As can be seen, both approaches yield similar result for larger w, e.g. w ~ 10 nm and diverge for smaller
w, remaining however within a reasonable range for order-of-magnitude estimates. As an interesting
observation, the simple approximation (Eqs. A1.5 and A1.7) suggests w
min
~ 4 nm, where the effective
barrier for electron becomes very small. On the other hand, the exact solution says that at w ~ 4 nm the
effective barrier height is wk
B
T ln2, i.e. the Boltzmann’s limit for the minimum barrier height (3.18).
APPENDIX 2: DERIVATION OF ELECTRON TRAVEL TIME (EQ. 3.55)
The travel time of the electron along the distance L is determined by electron’s average velocity hvi:
s w
L
hvi
(A2.1)
In this treatment, ballistic transport is assumed as the best-case scenario, and thus constant acceleration
motion, for which
hvi¼
v
min
þ v
max
2
z
v
max
2
(A2.2)
(v
min
is assumed to be zero).
The maximum velocity v
max
can be found from the energy balance relation:
E ¼
mv
2
max
2
(A2.3)
From (A2.2) and (A2.3):
hvi¼
v
max
2
¼
ffiffiffiffiffiffi
E
2m
r
(A2.4)
and from (A2.1) and(A2.4) the electron’s travel time is:
s w L
ffiffiffiffiffiffi
2m
E
r
(A2.5)
Next, from Table 3.3:
s
H
a
H
¼
Z
2E
,
2
ffiffiffiffiffiffiffiffiffi
2mE
p
Z
¼
ffiffiffiffiffiffiffiffiffi
2mE
p
E
¼
ffiffiffiffiffiffi
2m
E
r
(A2.6)
Appendix 2: Derivation of electron travel time (EQ. 3.55) 87
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