Chemistry Reference
In-Depth Information
Now let us consider to inject electrons from left to right-side through
the sample.
When the chemical potential is changed from
E
=0,the
quantization rule of the dimensionless conductance (
g
K
+
) in the valley of
K
+
is given as
g
K
+
=
n,
(4)
where
n
=0
,
1
,
2
, ···
. The quantization rule in the
K
−
-valley is
g
K
+
=
n
+1
.
(5)
Thus, conductance quantization of the zigzag nanoribbon in the clean limit
near
E
= 0 has the following odd-number quantization, i.e.
g
=
g
K
+
+
g
K
−
=2
n
+1
.
(6)
Fig. 3. (a) Energy dispersion of zigzag ribbon with
N
= 10. The valleys in the energy
dispersion near
k
=2
π/
3
a
(
k
=
−
2
π/
3
a
) originate from the Dirac
K
+
(
K
−
)-point
of graphene. The red-filled (blue-unfilled) circles denote the right (left)-moving open
channel at the energy
E
0
(dashed horizontal line). In the left(right) valley, the degeneracy
between right and left moving channels is missing due to one excess right(left)-going
mode. The time-reversal symmetry under the intra-valley scattering is also broken. (b)
Schematic figure of scattering geometry at
K
+
and
K
−
points in zigzag nanoribbons,
where a single excess right-going mode exists for
K
−
point. But a single excess left-going
mode exists for
K
−
point. Here
n
c
=0
,
1
,
2
, ···
.
Since we have an excess mode in each valley, the scattering matrix has
some peculiar features which can be seen when we explicitly write the valley
dependence in the scattering matrix. By denoting the contribution of the
right valley (
K
+
)as+,andoftheleftvalley(
K
−
)as
−
, the scattering
matrix can be rewritten as
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
.
O
L
O
L
O
R
O
R
I
L
I
L
I
R
I
R
=
rt
tr
(7)
