Chemistry Reference
In-Depth Information
where
F
A
and
F
B
describe the amplitude at sublattice points A and B,
is a band parameter,
k
=(
ˆ
ˆ
respectively,
γ
k
x
,
k
y
) is the wave-vector operator,
and
σ
x
,σ
y
) is the Pauli matrix. The equation of motion for the K'
point is obtained by replacing
σ
=(
σ
∗
in the above equation.
Electronic states in a carbon nanotube (CN) are obtained by imposing
generalized periodic boundary condition
F
(
r
+
L
)=exp(
σ
with
3)
F
(
r
) (up-
per sign for K and lower for K') in the circumference direction specified by
chiral vector
L
with
∓
2
πνi/
ν
=0 or
±
1 determined by the CN structure. We have
ν
1 for a semiconducting CN. The direction
of
L
is called chiral angle and denoted by
= 0 for a metallic CN and
ν
=
±
η
.
3. Acoustic Phonon
Acoustic phonons important in the electron scattering are described well
by a continuum model.
15
The potential-energy functional for displacement
u
=(
u
x
,u
y
,u
z
) is written as
[
u
]=
u
xy
S
(
1
2
u
yy
)
2
+
u
xx
−u
yy
)
2
+4
B
u
xx
+
,
U
dxdy
(
(2)
u
xx
=
∂u
x
∂x
+
u
z
R
,u
yy
=
∂u
y
u
xy
=
∂u
x
∂y
+
∂u
y
∂y
,
2
∂x
,
(3)
as in a homogeneous and isotropic two-dimensional (2D) system. The pa-
rameters
B
and
S
denote the bulk modulus and the shear modulus, respec-
tively (
B
=
λ
+
µ
and
S
=
µ
with
λ
and
µ
being Lame's constants). In carbon
nanotubes with finite radius
R
, we should add
u
z
/R
in the expression of
u
xx
as in the above equation, where the
axis is chosen along the circum-
ference direction. In order to discuss out-of-plane distortions, we should
consider the potential energy due to nonzero curvature. It is written as
U
c
[
u
]=
1
x
2
a
2
Ξ
∂
2
∂x
2
u
z
2
1
R
2
∂
2
∂y
2
dxdy
+
+
,
(4)
where Ξ is a force constant for curving of the plane. This curvature energy
is of the order of the fourth power of the wave vector and therefore is usually
much smaller than
[
u
].
In nanotubes, the phonon modes are specified by angular momentum
U
n
along the circumference direction. Figure 1 shows phonon dispersions
calculated in this continuum model.
15
The twisting mode with a linear
dispersion and the stretching and breathing modes coupled with each other
at their crossing are given by the solid lines. There exist modes
1
with frequency which vanishes in the long wavelength limit
q →
0. These
n
=
±
