Page 120 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Free and Bound Electrons, Dimensionality Effects
⁄
dψ dx
must be fulfilled. This yields a system of four equations that
,
,
,
determine the four remaining coefficients a b b and c . The solu-
2 1 2 1
tions reads
(
2
2
–
a = ( exp ( ik a) exp – ( ik a)) k – k )D – 1 (3.48a)
II
2
II
II
I
(
(
b = 2exp ( ik – k )a)k k + k )D – 1 (3.48b)
1 I II I I II
(
(
b = 2exp ( ik + k )a)k k – k )D – 1 (3.48c)
I
II
2
I
II
I
c = 4k k D – 1 (3.48d)
1 I II
(
2
(
2
with D = exp ( ik a) exp[ ( ik a) k – k ) – exp ( i – k a) k + k ) ] .In
I II I II II I II
Figure 3.6 the real parts of the different superposing wavefunctions are
sketched for values E > U and E < U . There are several remarkable
0 0
features to discuss:
Reflection, • c is non-zero for E < U . In this case the particle behaves rather
1 0
Transmission wave-like. This is very similar to a light-wave going through materi-
and
als with different refractive index. There is always a reflected and a
Tunneling
transmitted part, i.e., a ≠ 0 and c ≠ . 0
2 1
• For E > U the situation changes. We find resonances for the trans-
0
⋅
mission at k a = n π with an integer. There is no reflection in
n
II
this case, whereas there is always a transmitted part, i.e., c ≠ . 0
1
Reflection and transmission are not uniquely defined by the amplitudes
R
of the partial waves alone. Therefore, we define the reflexivity as the
quotient of reflected and incoming current density; the quotient of trans-
mitted and incoming current density we call the transmitivity :
T
R = a 2 2 ⁄ a 1 2 = a 2 2 (3.49a)
T = c 2 ⁄ a 2 = c 2 (3.49b)
1 1 1
In the case where there is neither creation nor destruction of particles,
R + T = 1 must hold. In Figure 3.7 T is shown with increasing energy
E of the incoming wave. There are well defined resonances where the
Semiconductors for Micro and Nanosystem Technology 117