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Chapter 5. Electron emission from solids
The total energy eigenvalue W is the same for both solutions,
and is given by
h
2 k
2
¯
W = 2m
= 2
1 mv x
2 .
(5.43)
For now we consider a single, specific energy W with associated
wave vector k.
The probability current is
ih ¯
� (x) − ¯
j(x) =
[ u(x) ¯
u
u(x) u
� (x) ] ,
(5.44)
2m
where a bar over a quantity indicates complex conjugation. Substituting u ± (x) above, we identify right- and left-propagating currents
¯
hk
j + (x ≤ 0) = + |a + |
2
m
¯
hk
j − (x ≤ 0) = − |a − |
2 ,
(5.45)
m
respectively, where j + is the current incident on the potential barrier from left to right inside the bulk at x ≤ 0, and j − is the
current reflected by the barrier from right to left. The algebraic
sum of these is the tunneling current transmitted through the barrier.
For x ≥ 0, the vacuum, Schr¨ odinger’s equation is
h
2 d
2
¯
−
+ U (x) u(x) = W u(x),
(5.46)
2m dx 2
where the potential energy U (x) is given approximately by
U (x) ≈ C − F x.
(5.47)
Chapter 5. Electron emission from solids
The total energy eigenvalue W is the same for both solutions,
and is given by
h
2 k
2
¯
W = 2m
= 2
1 mv x
2 .
(5.43)
For now we consider a single, specific energy W with associated
wave vector k.
The probability current is
ih ¯
� (x) − ¯
j(x) =
[ u(x) ¯
u
u(x) u
� (x) ] ,
(5.44)
2m
where a bar over a quantity indicates complex conjugation. Substituting u ± (x) above, we identify right- and left-propagating currents
¯
hk
j + (x ≤ 0) = + |a + |
2
m
¯
hk
j − (x ≤ 0) = − |a − |
2 ,
(5.45)
m
respectively, where j + is the current incident on the potential barrier from left to right inside the bulk at x ≤ 0, and j − is the
current reflected by the barrier from right to left. The algebraic
sum of these is the tunneling current transmitted through the barrier.
For x ≥ 0, the vacuum, Schr¨ odinger’s equation is
h
2 d
2
¯
−
+ U (x) u(x) = W u(x),
(5.46)
2m dx 2
where the potential energy U (x) is given approximately by
U (x) ≈ C − F x.
(5.47)