Page 288 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Inhomogeneities

1

(
u = --------c exp  jk + jk )ai – 1  ωt  (7.104a)
--- –

m m 1 2  4 
m  
1   1  
(
u = ---------c exp  jk + jk )ai + --- – ωt  (7.104b)
M M 1 2  4 
M  
so that the solutions become
2
ω =
E 2 (7.105)
−
------------- M +( [ m) + ( M + m) – 2Mm 1 – cos ( [ k + jk )a])]
(
2
1
aMm
With the identity
cos ( [ k + jk )a] = cos ( k a)cosh ( k a) – jsin ( k a)sinh ( k a) (7.106)
1 2 1 2 1 2
we find a condition for real ω
sin ( k a)sinh ( k a) = 0 (7.107)
2
1
Clearly, setting k = 0 gives the result for a bulk lattice. For k ≠ 0 , we
2
2
−
1 +
⁄
,
must have that k = iπ a , i = 0 + , − 2 …, . Thus the solution
1
becomes
2
ω =
E 2 (7.108)
−
------------- M +( [ m) + ( M + m) – 2Mm 1 – cos ( k a)cosh ( k a))]
(
1
2
aMm
ω
as long as is real. This condition is not fulfilled everywhere. In partic-
⁄
ular, at some value k for k = − π a the frequency becomes
+
2max 1
1
1
ω 2 = E  ----- + ----  (7.109)
---
k 2max a  M m 
beyond which no frequency is defined. This is clearly seen on the disper-
sion “surface” of Figure 7.13. In the range 0 ≤ k ≤ k we observe a
2 2max
completely filled band between the acoustic and optical branch of the dis-
Semiconductors for Micro and Nanosystem Technology 285