Page 112 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Free and Bound Electrons, Dimensionality Effects
ψ
I = ψ II (3.30a)
x = 0 x = 0
ψ ' = ψ ' (3.30b)
I II
x = 0 x = 0
ψ = ψ (3.30c)
II III
x = a x = a
ψ ' = ψ ' (3.30d)
II III
x = a x = a
Inserting (3.29a)-(3.29c) into (3.30a)-(3.30d), we obtain a linear system
of equations to determine A-D:

1 – 1 – 1 0 A
0 e ika e i – ka e – – κa B
= 0 (3.31)
κ – ik ik 0 C
0 ike ika – ike i – ka κe – κa D

A solution can only be found if the determinant of the coefficient matrix
in (3.31) is zero:

2
2
2κkcos ( ka) + ( κ – k )sin ( ka) = 0 (3.32)
2
2
2
With ak = ξ we have aκ = ( 2mU a ) —⁄ 2 – ξ = C – ξ 2 and we
0
may write (3.32) as
2
2ξ C – ξ 2
---------------------------- + tan ξ = 0 (3.33)
C – 2ξ 2
2
This equation must be solved either graphically, see Figure 3.4 and
Figure 3.5, or by numerical methods. A numerical solution of the dis-
cretized model for the discussed eigenvalue scenario may be counter-
2
2
checked in the special case with U « — ⁄ ( ma ) where there is exactly
0
(
one bound state found with E ≈ U 1 ma ⁄ ( 2— )) .
2
2
–
0 0
Note that only in special limiting cases can an analytical solution be
obtained easily. Usually, analytical methods end at a certain point and
Semiconductors for Micro and Nanosystem Technology 109