Chemistry Reference
In-Depth Information
Here we should note that the dimension of each column vector is not identi-
cal. Let us denote the number of the right-going channel in the valley
K
+
or the left-going channel in the valley
K
−
as
n
c
. For example,
n
c
=1at
E
E
0
in Fig. 3(a). Figure 3(b) shows the schematic figure of scattering
geometry for
=
K
+
and
K
−
points. Thus, the reflection matrices have the
following matrix structures,
n
c
n
c
+1
n
c
+1
r
++
r
+
−
r
=
.
(8)
n
c
r
−
+
r
−−
The reflection matrices become non-square when the intervalley scattering
is suppressed, i.e. the off-diagonal submatrices (
r
+
−
,
r
−
+
and so on) are
zero.
When the electrons are injected from the left lead of the sample and
the intervalley scattering is suppressed, a system with an excess channel is
realised in the
K
−
-valley. Thus, for single valley transport, the
r
−−
and
r
−−
are
n
c
×
(
n
c
+1) and (
n
c
+1)
×n
c
matrices, respectively, and
t
−−
and
t
−−
are (
n
c
+1)
×
n
c
+1) and
n
c
× n
c
matrices, respectively. Noting the
(
r
†
−−
r
−−
r
−−
r
†
−−
r
−−
dimensions of
have
a single zero eigenvalue. Combining this property with the flux conserva-
tion relation (
r
−−
and
, we find that
and
t
−−
t
†
−−
has an eigenvalue equal to unity, which indicates the presence of a per-
fectly conducting channel (PCC) only in the right-moving channels. Note
that
S
†
S
SS
†
=
1
), we arrive at the conclusion that
=
t
−−
t
†
−−
does not have such an anomalous eigenvalue. If the set of
t
−−
t
†
−−
t
−−
t
†
−
−
eigenvalues for
is expressed as
{T
1
,T
2
, ··· ,T
n
c
}
,thatfor
is expressed as
, i.e. a PCC. Thus, the dimensionless
conductance g for the right-moving channels is given as
g
K
−
=
n
c
+1
{T
1
,T
2
, ··· ,T
n
c
,
1
}
T
i
=1+
n
c
T
i
,
(9)
i
=1
i
=1
while that for the left-moving channels is
n
c
g
K
−
=
T
i
.
(10)
i
=1
g
K
−
+ 1. Since the overall time reversal symmetry
(TRS) of the system guarantees the following relation:
We see that
g
K
−
=
g
K
+
=
g
K
−
,
(11)
g
K
−
=
g
K
+
,
