Page 73 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
The restriction to next-neighbor interactions and identical interatomic
force constants yields the following equations of motion when the
expressions (2.56) and (2.57) are inserted into the Lagrange equations
E
--- 2u
mu˙˙ im = – ( im – u iM – u ( i 1)M ) (2.58a)
–
a
E
Mu˙˙ iM = – ( iM – u ( i + 1)m – u ) (2.58b)
--- 2u
im
a
At rest the atoms occupy the cell positions (due to identical static forces)
⁄
( i 14)a and i +( 14)a . Hence we choose an harmonic atom dis-
⁄
–
placement ansatz for each atom of the form
1   1  
u im = --------c exp  jkai – --- – ωt  (2.59a)

4 
m
m  
1   1  
u iM = ---------c exp  jkai + --- – ωt  (2.59b)

4 
M
M  
The second ansatz can be written in terms of u
im
m  c M   ka 
u = ----- ------ exp j------ u (2.60)
iM M c  m   2  im
Equations (2.59b) and (2.60) are now inserted into (2.58a). Eliminating
common factors and simplifying, we obtain an equation for the ampli-
tudes c and c
m M

 2E ω 2  2E  ka 
2 
 ----------- – m  – ------------cos  ------
am aM c m
= 0 (2.61)
2E  ka   2E 2  c M
– -----------cos  ------ ------------ – ω M 
2  
am aM
This is an eigensystem equation for ω , and its non-trivial solutions are
obtained by requiring the determinant of the matrix to be zero. Perform-
ing this, we obtain

70 Semiconductors for Micro and Nanosystem Technology