Page 70 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Vibrating Uniform Lattice
where we have chosen the coordinates X
to describe the undeformed
configuration of the crystal, and the coordinates in the deformed crys-
x
tal, see Figure 2.19.
1D The complexity in applying the lagrangian formulation to a general 3D
Monatomic crystal can be avoided by considering a 1D model system that demon-
Dispersion
strates the salient features of the more involved 3D system. A large 1D
Relation
lattice of N identical bound atoms are arranged in the form of a ring by
employing the Born-von Karmann boundary condition, i.e.,
(
uN + 1) = uN() . From equation (2.31), the strain in the 1D lattice is
simply

1 T du
ε = --- ∇ +( u ∇ u ) = ------- (2.49)
11
2 dX
which gives a potential energy density of

1  du  2
U = ---E ------- (2.50)
2  dX

u
where is the displacement of the atom from its lattice equilibrium site
and is the linear Young modulus of the interatomic bond. Note that we
E
only consider nearest-neighbour interactions. The kinetic co-energy den-
sity is

* 1 2
T = ---ρu˙ (2.51)
2
⁄
where the mass density is ρ = ma for an atomic mass m and inter-
atomic spacing . The lagrangian density for the chain is the difference
a
between the kinetic co-energy density and the potential energy density,
*
–
L = T – U . Note that u = X – x and hence that u∇ = Id ∇ x and
u˙ = – x˙ . We insert the lagrangian density in equation (2.48) to obtain
d  du 
ρu˙˙ = – E------- ------- (2.52)
dX dX


Semiconductors for Micro and Nanosystem Technology 67